DRAFT - Universität zu Kölnschieck/SKRIPT/key4.pdf · for the study of the external and internal structure of the proton by elctron scattering). In nuclear physics, a ﬁeld of

DR

AFTScript/Key Nuclear-Reaction Experiments –

First Discoveries and Consequences

H. Paetz gen. SchieckInstitute of Nuclear Physics

Universität zu Köln

October 28, 2014

DRAFT

2

DR

AFT

Chapter 1

Preface

With the “age” of nuclear physics of just about 100 years it seems appropriateto consider in detail the historical aspects of how new knowledge appeared.On the one side, it is fascinating to look at theoretical developments in thefield of subatomic physics in step with the grand new ideas of the 20thcentury such as the the quantum theory and the theory of relativity openingentirely new loooks at the world. On the other, experimental progress gavehints as to “new physics” (e.g. the stepwise deciphering of the structure andcomposition of nuclei led to the idea of two new interactions, the weak andthe strong force and their theoretical dexcription) and also experiments hadand has to decide between alternative theoretical interpretations (e.g. “Areneutrinos Dirac or Majorana particles?”, “Do neutrinos have mass or not?”,or “Are electrons, neutrinos, muons, or protons point particles or extendedobjects?”.

In this respect it is worthwhile to study the often fascinating details ofearly (especially “first”) experiments. Quite often epoch-making results wereobtained with very simple means, see e.g. the discovery of nuclear fission.But all those experiments were based on earlier attempts that were carefullyrefined to yield unambiguous evidences, not pure serendipity (see e.g. theplanful outlay of the experiment to “find” the neutrino).

This text is designed to give an impression of the first experiments innuclear reactions. It is also motivated by an earlier book in German that Ifound often very useful myself: Erwin Bodenstedt;”Experimente der Kern-physik und ihre Deutung” (BI Wissenschaftsverlag, Mannheim/Wien/Zürich,1972). It had three volumes and covered also many nuclear-structure experi-ments whereas the present text is restricted to nuclear reactions. This seemsjustified, also because of increased specialization of the subfields of nuclearphysics.

In this book the “crucial” or “key” experiments that often, but not alwayshave been the first in a subfield, are described in some detail often includingoriginal drawings or setups because these may illustrate the igniting idea ofa new field better than later and more sophisticated setups. Neverthelessin many instances the later progress is often briefly indicated. Theoreticalbackground is given, but kept compact and, if necessary, the usual textbooksor original literature will have to be consulted. Therefore, at the end, a listof references for general reading or useful works is given.

DR

AFT

4 CONTENTS

Contents

1 Preface 3

2 Introduction 72.1 Rutherford and Evidence for the Nuclear Atom . . . . . . . . 72.2 The First True Nuclear Reaction . . . . . . . . . . . . . . . . 72.3 The Role of Accelerators . . . . . . . . . . . . . . . . . . . . . 92.4 The Neutron and the Correct Composition of Nuclei . . . . . . 92.5 Nuclear Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 10

3 Rutherford Scattering and the Atomic Nucleus 113.1 Rutherford Scattering Cross Section . . . . . . . . . . . . . . . 11

3.1.1 Minimal Scattering Distance d . . . . . . . . . . . . . . 143.1.2 Trajectories in the Point-Charge Coulomb Field . . . . 143.1.3 Quantum-Mechanical Derivation of Rutherford’s For-

mula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.4 Result of the Experiment . . . . . . . . . . . . . . . . . 183.1.5 Consequences of the Rutherford Experiments and their

Historic Significance . . . . . . . . . . . . . . . . . . . 18

4 The First True Nuclear Reaction and the Discovery of theProton 23

5 Extended Matter and Charge Distributions of Nuclei 295.0.6 Ansatz for Models . . . . . . . . . . . . . . . . . . . . 315.0.7 Expansion into Moments . . . . . . . . . . . . . . . . . 31

5.1 Hadron Scattering Experiments . . . . . . . . . . . . . . . . . 345.1.1 Nuclear Radii from Higher-Energy α-Particle Scattering 345.1.2 Heavy-Ion Scattering . . . . . . . . . . . . . . . . . . . 35

5.2 Elastic Electron Scattering – Hofstadter’s Experiments . . . . 365.3 Complementary Methods . . . . . . . . . . . . . . . . . . . . . 39

5.3.1 High Precision Laser Spectroscopy . . . . . . . . . . . 395.3.2 Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . 405.3.3 Matter-Density Distributions and Radii . . . . . . . . . 415.3.4 Hadronic Radii from Neutron Scattering . . . . . . . . 425.3.5 Special Cases – Neutron Skin . . . . . . . . . . . . . . 42

5.4 The Size and Shape Systematics of Nuclei . . . . . . . . . . . 42

6 Halo Nuclei and Farewell to Simple Radius Systematics 55

7 The Particle Zoo 61

DR

AFT

CONTENTS 5

7.1 The Pion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 First Production of the Antiproton in a Nuclear Reaction . . . 647.3 Discovery of the (Electron) Neutrino . . . . . . . . . . . . . . 657.4 Quasi-Elastic Electron Scattering – Excited Nucleons and the

Particle Zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.5 Deep-Inelastic Lepton Scattering – Partons inside Hadrons . . 73

8 Discovery of the Neutron (Nuclear Kinematics etc.) 79

9 First Precise Determination of the Neutron Mass and theBinding Energy of the Deuteron 839.1 The Photonuclear Disintegration of the Deuteron . . . . . . . 839.2 Neutron-Proton Capture . . . . . . . . . . . . . . . . . . . . . 85

10 First Nuclear Reaction with an Accelerated Beam; Cockroft-Walton Accelerator 91

11 Observation of Direct Interactions 9911.1 Elastic Scattering and the Optical Model . . . . . . . . . . . . 9911.2 Direct (Rearrangement) Reactions . . . . . . . . . . . . . . . . 10311.3 Stripping Reactions . . . . . . . . . . . . . . . . . . . . . . . . 10511.4 The Born Approximation . . . . . . . . . . . . . . . . . . . . . 108

12 Resonances and Compound Reactions 11312.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.2 Theoretical Shape of the Cross Sections . . . . . . . . . . . . . 11412.3 Derivation of the Partial-Width Amplitude for Nuclei (s Waves

only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11612.4 First Evidence of Resonant Nuclear Reactions . . . . . . . . . 117

13 Nuclear Reactions and Tests of Conservation Laws 12113.1 First Tests of Parity Violation in Hadronic Reactions . . . . . 12213.2 First Time-Reversal Tests . . . . . . . . . . . . . . . . . . . . 12513.3 NN Interaction and Isospin . . . . . . . . . . . . . . . . . . . . 129

13.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 12913.3.2 The Scattering Length . . . . . . . . . . . . . . . . . . 13013.3.3 Other Reaction Tests of Isospin Breaking . . . . . . . . 137

14 Scattering of Identical Nuclei, Exchange Symmetry andMolecular Resonances 14314.1 First observation of interference in the scattering of identical

nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14314.1.1 Identical Bosons with spin I = 0 . . . . . . . . . . . . . 14414.1.2 Identical Fermions with spin I = 1/2 . . . . . . . . . . 144

15 Nuclear Fission and Nuclear Energy 153

16 First Double Scattering and Polarization in p-4He and the(ℓ · s) Force 159

DR

AFT

6 CONTENTS

17 First Nuclear Reaction of an Accelerated Polarized Beamfrom a Polarized-Ion Source (Basel) 167

18 The Discovery of Giant Resonances 175

19 General Resources and Reading 187

Index 190

DR

AFT

Chapter 2

Introduction

Key experiments are those, which open up entirely new insights into unknown“territories” and start new fields of more detailed investigations in theseareas. One indicator of key experiments can be Nobel prize awards to theprincipal investigators (examples are Robert Hofstadter (NP 1961), JeromeIsaac Friedman, Henry Way Kendall, and Richard Edward Taylor (NP 1990),for the study of the external and internal structure of the proton by elctronscattering).

In nuclear physics, a field of science, which – by definition – has existedonly for about 100 years these key developments are: radioactivity and activeinvestigations of nuclei, either of their structure or their interactions in nu-clear reactions. Both are initimately connected with the continuing progressin the development of particle accelerators and in part nuclear reactors.

2.1 Rutherford and Evidence for the Nuclear

Atom

The famous Rutherford scattering experiment started the field of nuclear re-actions around 1911 in Manchester. The behavior of α particles elasticallyscattered from gold and other nuclei suggested a very small (from the pointof view of that time) and compact nucleus (i.e. containing most of the atom’smass and all of the positive charge Ze compensating that of the atomic elec-trons). Energetic particles from radioactive sources were used as projectiles(α’s from heavy elements such as “radium emanation” (222

86Rn) with suffi-ciently high energies and intensities). Even then scattering experiments weretedious: A MBq (in 4π solid angle) source corresponds to an incident “beam”current into a solid angle, small enough to define a reasonable scattering ge-ometry, of only ≈ 10−6 nA. Single scintillation events had to be counted byobserving them on a ZnS screen in the dark.

2.2 The First True Nuclear Reaction

Around 1917 Rutherford, using techniques similar to those of the famousscattering experiment, recognized that a different type of particle emerged

DR

AFT

8 CHAPTER 2. INTRODUCTION

from the interaction of α’s from radioactive sources with gas molecules. Ithad longer range in matter than the α’s and proved to be the nucleus ofthe hydrogen atom. Rutherford therefore had performed the first true nu-clear reaction with a rearrangement of the particles involved. Although ingas discharges the existence of negatively charged constituents (the atomicelectrons) and positive ions had been seen, only Rutherford identified theparticle which emerged from the

14N + α → 17O + p (2.1)

reaction as the very small nucleus of H and coined the term “proton”. Thus,he solved one part of the riddle of the structure and composition of nuclei;the other had to wait until the discovery of the neutron.

The nuclear charge number is identical to the element number of the pri-odic table and the Z dependence of the Rutherford cross section confirmedthe periodic system of the elements. The explanation of the existence ofisotopes and the correct placement of them in the chart of nuclides (Z vs.N) required the discovery of the neutron in 1932. Already Rutherford could– by comparing the measured scattering angular distribution of α particleson gold with his ansatz of a point-Coulomb interaction – conclude that thenucleus is an object smaller than the scattering distances (order of magni-tude: 1 fm = 1·10−15 m). The very fact that scattering at backward anglesoccured, showed that the scattering center had to be heavier than the α (thisis pure kinematics). The electron cloud relative to this is very large (order-of-magnitude radius: 1 Å = 1·10−10 m) and carries the charge -Ze such thatthe atom is exactly neutral.

After the invention of accelerators the use of α particles of much higherenergies with penetration into the target nucleus was possible and the ex-tension (the radius) of nuclei could be obtained by the onset of deviationsfrom the point-Coulomb scattering. A key rôle is played here by the chargeform factor and its Fourier transform, the charge-density distribution. Itexpresses how strongly the Coulomb potential of an extended (often simplyassumed to be homogeneous) charge distribution in the nuclear interior de-viates from that of a point charge or what the influence of the (hadronic)nuclear interaction on the observables is, see Fig. 5.3.

Using charged leptons as probes, which have no measurable extensionand do not feel the strong interaction, charge (and current) distributions innuclei and nucleons have been determined. At higher momentum transfer(i.e. at high energies and large scattering angles via inelastic or quasi-elasticscattering) excited states of the nucleon and later-on, (via deep-inelastic scat-tering), substructures of the nucleons (partons) were discovered that had allthe properties of quarks: 1/3 charges, spin 1/2h, color charge and confine-ment, characteristics of truly elementary particles (point shape, no internalstructure), and they proved to be sources of the strong, electromagnetic, andweak interactions, also by probing them with neutrinos.

DR

AFT

2.3. THE ROLE OF ACCELERATORS 9

2.3 The Role of Accelerators

It is evident that the use of radioactive sources imposed severe restrictions:fixed or very limited energy range and extremely low intensities. It is clearthat the field of nuclear ractions could only progress with the invention ofparticle accelerators. The first accelerator prototype important for nuclearphysics was the linear accelerator (“LINAC”) developed and published in1929 by Ralf Wideröe at the Aachen Institute of Technology, also laying theground for the betatron, which was realized by Kerst and Serber in 1940,and the cyclotron by Lawrence in 1931. Wideröe’s ideas also included thesynchrotron and storage-ring schemes. The first nuclear reaction initiatedwith accelerated beams was the reaction

p + 7Li → 2α (2.2)

by Cockroft and Walton in 1932 at the Cavendish Laboratory at Cambridgeusing a DC high voltage across several accelerating gaps and produced by theDelon/Greinacher voltage multiplication scheme. This and the ensuing de-velopments in nuclear and particle physics up to the present energies of up to14 TeV (at the Large Hadron collider LHC at CERN/Geneva) are intimatelyconnected with the achievements in accelerator physics and technology. Like-wise the development of detector technologies – from the first scintillators,later equipped with photomultipliers, to the cloud and the bubble cham-bers, the ionization chamber, Geiger-Müller counter, multiwire ionizationchambers, and the large field of solid-state detectors – was essential. Not un-justifiably accelerators have been called “tools of our culture” or “Engines ofDiscovery” (see e.g. the book with that title by Sessler and Wilson [SES07]).Their impact reaches now into social applications such as tumor diagnosisand therapy, materials identification and modification, age and provenienceanalyses in archaeology, geology, arts, environmental science etc.

2.4 The Neutron and the Correct Composi-

tion of Nuclei

With the detection of the neutron by Chadwick (1932) another branch ofnuclear physics and especially nuclear reactions opened up that only partlydepends on accelerators. Not only was the discovery of the neutron thekeystone to the fundamental structure of nuclei removing all kinds of incon-sistencies about e.g. nuclear isotopes, but immediately it incited Heisenbergto formulate the idea of charge independence of the nuclear interaction andthe fundamental symmetry of isospin.

The neutrality of the neutron facilitates the description of nuclear reac-tions. On the other hand, production of neutrons for nuclear reactions aswell as the detection methods are more complicated. Normally, except whenneutrons from nuclear reactions are used, the choice or selection of specificneutron energies requires additional methods such as moderation by elasticcollisions with light nuclei and/or chopper and time-of-flight facilities.

DR

AFT

10 CHAPTER 2. INTRODUCTION

Much of neutron work relies on neutrons from fission in reactors (anexample is the high-flux 660 MW research reactor with a thermal flux of> 1·1015 s−1cm−2, at the Institut Laue-Langevin (ILL Grenoble)) or on spal-lation neutron sources where intense proton beams in the GeV and mA rangeincident on (liquid) metal targets release many (up to 30) neutrons per pro-ton with high energies (a typical research center is the LANSCE facility witha proton LINAC, originally designed as meson factory at Los Alamos, NewMexico, another the spallation neutron source (SNS) at Oak Ridge, Ten-nessee, with 1.4 MW beam power and 4.8·1016 neutrons/s.)

The neutron has fundamental properties in its own right that have beenstudied:

• β decay

• The internal (quark + gluon) structure and charge and magnetic-momentdistributions. They have been studied e.g. by elastic and inelastic elec-tron scattering where deuterons and especially 3He served as neutrontargets. Polarized 3He is an almost pure polarized neutron target. Thecharge and magnetic-moment distributions inside the neutron are proofof its inner structure.

• The possible electric dipole moment and thus time-reversal and par-ity violations were studied where the absence of the Coulomb force isexperimentally advantageous.

• The wave nature of neutrons of low energies was studied in reflection,diffraction, and interference experiments.

• Especially ultracold neutrons offer many interesting properties and ap-plications, e.g. its interaction with the gravitational field or that of itsmagnetic moment with magnetic fields.

2.5 Nuclear Spectroscopy

We define nuclear spectroscopy as the science of learning all about the prop-erties of the thousands of nuclides, each with individual and also collectiveproperties. Aside from early studies of radioactive decays, nuclear reactionshave been the tool to investigate the action of nuclear forces (in the senseof an interplay of the strong interaction proper, the electromagnetic, andthe weak force). In high-density situations, e.g. in neutron stars, even thegravitational force enters the stage via the density dependence of the nuclearinteractions. The aim of modern nuclear spectroscopy is now moving awayfrom stable nuclei, from deformed highly excited nuclei with high angularmomenta on to the investigation of nuclei in the regions near the limits ofexisting nuclei with either high neutron excess, high neutron deficiency, orthe region of new elements, the superheavy nuclei. They can be characterizedby their isospin T = (N − Z)/A.

DR

AFT

Chapter 3

Rutherford Scattering and theAtomic Nucleus

We begin with a universal definition of the fundamental observable of nuclearreactions, the (differential) cross section that can be applied in classical aswell as quantum-mechanical descriptions.

Definition of Cross Section 3.1 The (differential) cross section is the num-ber of particles of a given type from a reaction, which, per target atom andunit time, are scattered into the solid-angle element dΩ (formed by the angu-lar interval θ...θ + dθ and φ...φ+ dφ), divided by the incident particle flux j(a current density!).

With no azimuthal dependence this definition yields the classical formula forthe cross section. With the number of particles jdσ = j · 2πbdb one obtains

(

dσ

dΩ

)

class

=2πbdb

2π sin θdθ=

b

sin θ·∣∣∣∣∣

db

dθ

∣∣∣∣∣. (3.1)

θ(b), which contains the dynamics of the interaction (the dynamics) is calleddeflection function. Its knowledge determines the scattering completely.

The measurements by Geiger and Marsden and the interpretation of theexperiment by E. Rutherford [RUT11, GEI13] constitute a milestone in ourunderstanding of the structure of nature and especially the true beginningof “nuclear physics”. Their apparatus carries all the features of modern scat-tering experiments, as can be seen in Fig. 3.1.

3.1 Rutherford Scattering Cross Section

For the derivation of the classical Rutherford scattering cross section weassume:

• The projectile and the scattering center (target) are point particles(with Gauss’s law it can be proved that this is also fulfilled for extendedparticles as long as the charge distribution is not touched upon)

• The target nucleus is infinitely heavy (i.e. the laboratory system coin-cides with the c.m. system)

DR

AFT

12 CHAPTER 3. RUTHERFORD SCATTERING AND THE ATOMIC NUCLEUS

RotatableVacuum Chamber

Ra Source ln a Lead Block

To Vacuum Pump

Target Foil

Microscope

Scintillator

Figure 3.1: The original setup of Rutherford’s, Geiger’s, and Marsden’s firstnuclear scattering experiment at Manchester 1908 to 1913.

• The interaction is the purely electrostatic point Coulomb force

FC = ± 14πǫ0

· Z1Z2e2

r2=C

r2(3.2)

with the Coulomb potential VC = ±C/r.

• “Classical” means that particles and trajectories are localized and nowave properties enter the description.

The classical scattering situation is shown in Fig. 3.2. The deflection functionis most simply determined by applying angular-momentum conservation andthe equation of motion in one coordinate (y):

L = mv∞b = mr2φ = mvmind (3.3)

and from this

dt = r2dφ/v∞b (3.4)

m∆vy =∫

Fydt

v∞ sin θ =C

mv∞b

∫ ∞

−∞φ sinφdt

=C

mv∞b

∫ π−θ

0sin φdφ =

C

mv∞b(1 + cos θ). (3.5)

After transformation to half the scattering angle the deflection function is

cot(θ/2) = mv2∞b/C = v∞L/C (3.6)

and

b =C

2E∞· cot

(

θ

2

)

. (3.7)

DR

AFT

3.1. RUTHERFORD SCATTERING CROSS SECTION 13

v

v

vmin

b+dbb

y

z

θ

θr

d0

φd

oo

oo

Figure 3.2: Classical Rutherford scattering.

DR

AFT


anddb

dθ=

C

2mv2∞

· 1sin2(θ/2)

=C

4E∞· 1

sin2(θ/2)(3.8)

and thus for the Rutherford cross section

dσ

dΩ=

1(4πǫ0)2

(

Z1Z2e2

4E∞

)2

· 1sin4(θ/2)

. (3.9)

Numerically:

dσ

dΩ= 1.296

(

Z1Z2

E∞(MeV )

)2

· 1sin4(θ/2)

[

mb

sr

]

. (3.10)

3.1.1 Minimal Scattering Distance d

For this quantity one needs additionally the energy-conservation law:

mv2∞

2=mv2

min

2+C

d. (3.11)

The absolutely smallest distance d0 is obtained in central collisions with:

E∞ =mv2∞

2=C

d0. (3.12)

From this and the angular-momentum conservation Eq. 3.3 the relation

b2 = d(d− d0) (3.13)

is obtained with the solution:

d =C

2E∞

1 +

√

1 + b24E2∞

C2

=d0

2

(

1 +1

sin θ/2

)

. (3.14)

The classical scattering distance in relation to the minimum distance d0 asfunction of the scattering angle is shown in Fig. 3.3.

3.1.2 Trajectories in the Point-Charge Coulomb Field

For the motion in a central-force field with a force ∝ r−2 classical mechan-ics shows that the trajectories for scattering, i.e. positive total energy, arehyperbolae, which can be derived using angular momentum and energy con-servation (with the Coulomb potential):

L = mr2φ = const (3.15)

E =mr2

2+

L2

2mr2+C

r. (3.16)

DR

AFT


0 50 100 150

1

10

100

Θ

d / d

0

Figure 3.3: Minimal scattering distance as function of the scattering angle.

In these equations dt can be eliminated. The integration of

dφ = − L

mr2

[

2m

(

E − C

r− L2

2mr2

)]−1/2

dr (3.17)

results in

r =L2

mC· 1

1 − ǫ cosφ. (3.18)

with b = L/√

2mE. With k = L2/mC and ǫ =√

1 + 4E2b2

C2 (the eccentricity)the standard form of conic sections is obtained

1r

=1k

(1 − ǫ cosφ). (3.19)

There is now a connection between impact parameter b, scattering angle θ,and (quantized) orbital angular momentum L=ℓh

b =12d0 cot

θ

2=

ℓh

p∞. (3.20)

3.1.3 Quantum-Mechanical Derivation of Rutherford’s

Formula

The point-Rutherford cross section can be derived quantum-mechanicallywith identical results. This can be done in two ways. One is to solve thecorresponding Schrödinger equation exactly, resulting in the regular and ir-regular Coulomb functions Fℓ and Gℓ as solutions. The other is to use theFirst Born approximation together with Fermi’s Golden Rule.

DR

AFT


Schrödinger Equation

The point-Rutherford cross section may be derived quantum-mechanicallyby solving the Schrödinger equation with the point (or extended) Coulombpotential as input. It has the form of a hypergeometric differential equation.

− h2

2µu

′′

+

(

C

r+h2

2µℓ(ℓ+ 1)r2

− h2k2

2µ

)

uℓ = 0. (3.21)

This equation may be written in its “normal” form with the Sommerfeldparameter ηS and ρ=kr:

d2uℓ(ρ)dρ2

+

(

1 − ℓ(ℓ+ 1)ρ2

− 2ηSρ

)

uℓ(ρ) = 0. (3.22)

It has the asymptotic solutions of the regular and irregular Coulomb Func-tions with the Coulomb phases σℓ = argΓ(ℓ+ 1 + iηS):

Fℓ −→ sin(kr − ℓπ/2 − ηS ln 2kr + σℓ), (3.23)

Gℓ −→ cos(kr − ℓπ/2 − ηS ln 2kr + σℓ). (3.24)

With the usual partial-wave expansion with incident plane waves theCoulomb scattering amplitude of the outgoing wave results:

ΨS −→ 1rei(kr−ηS ln 2kr)fc(θ), (3.25)

fc(θ) = −ηSe2iσ0 · eiηS ln sin2 θ/2

2k2 sin2 θ/2. (3.26)

The amplitude squared fC · f ∗C provides the Rutherford cross section, whichis identical to the classically derived equation.

1st Born Approximation

Starting points for appropriate descriptions are

• Fermi’s Golden Rule of perturbation theory

• The first Born approximation

For a “sufficiently weak” perturbation Fermi’s Golden Rule gives the transi-tion probability per unit time W:

W =2πh

|〈Ψout |Hint| Ψin〉|2 ρ(E)

=VmpdΩ4π2h4 · |Hif |2 . (3.27)

The density of final states ρ(E) = dn/dE, which enters the calculation canbe obtained from the ratio of the actual to the minimally allowed phase-spacevolumes:

dn

dE=V 4πp2dpdΩ

4π

(2πh)3dE, (3.28)

DR

AFT


E = p2/2m and dp/dE = m/p = E/c2p. Thus

ρ(E) =dn

dE= V

pmdΩ(2πh)3

= VpEdΩ

(2πh)3c2. (3.29)

W becomes the cross section according to the definition 3.1 on page 11 withthe incident particle-current density j = v/V = p/mV :

dσ =W

j=

W

( pmV

)=V 2m2dΩ

4π2h4 · |Hif |2 . (3.30)

The 1st Born approximation consists in using only the first term of the Bornseries with plane waves in the entrance and exit channels:

Φin =1√Vei~ki~r and Φout =

1√Vei~kf~r. (3.31)

If Hint = U(r) signifies a small time-independent perturbation then, with~K = ~kf − ~ki

|Hif | =∣∣∣∣

1V

∫

ei~K~rU(r)dτ

∣∣∣∣ (3.32)

anddσ

dΩ=(

m

2πh2

)2 ∣∣∣∣

∫

ei~K~rU(r)dτ

∣∣∣∣

2

= |f(θ)|2 . (3.33)

Inserting the Coulomb potential U(r)=C/r the classically calculated formulafor the Rutherford scattering cross section is obtained. The cross sectionis (with the constant Z1Z2e

2/16 and the substitution u = iKr cos θ anddu = − sin θ dθ(iKr))

dσ

dΩ= const ·

∣∣∣∣

∫

ei~K~r · 1

rdτ∣∣∣∣

2

= const ·∣∣∣∣

∫ ∫ 1reiKr cos θ2π sin θ dθr2dr

∣∣∣∣

2

= const · 2π∣∣∣∣

∫ ∫r

iKreu du dr

∣∣∣∣

2

= const ·( 2πiK

)2 ∣∣∣∣

∫

r

(

eiKr cos π − eiKr cos 0)

dr∣∣∣∣

2

= const ·( 2πiK

)2 ∣∣∣∣

∫

r

(

e−iKr − eiKr)

dr∣∣∣∣

2

= const ·(2π · 2i

iK

)2 ∣∣∣∣

∫ ∞

0sinKr dr

∣∣∣∣

2

. (3.34)

The integral is undefined. This is circumvented by a screening ansatz afterBohr, which corresponds to the real situation of the screening of the pointCoulomb potential by the electrons of the atomic shell, with the screeningconstant α. With

∫ ∞

0e−αr sinKrdr =

K

K2 + α2(3.35)

DR

AFT


one obtains(

dσ

dΩ

)

R,s

=

[

2µZ1Z2e2

h2(α2 + 4k2 sin2(θ/2)

]2

(3.36)

with the momentum transfer K = 2k sin(θ/2) for elastic scattering. Thiscross section is finite for θ → 0 . By letting the screening constant go tozero a cross section results, which is identical with that from the classicalderivation:

(

dσ

dΩ

)

R

= limα→0

(

dσ

dΩ

)

R,s

=

(

Z1Z2e2

4Ekin

)2

· 1sin4(θ/2)

. (3.37)

However, for all applications where there is interference the Rutherfordamplitude has to be used including its (logarithmic) phase. Typical casesare that of identical particles (see Chapter 14) or of interference with nu-clear (hadronic) amplitudes. Normally one has to assume a fundamentallyquantum-mechanical description that only in special cases, i.e. when the rel-evant de Broglie wavelengths are small, may be approximated by classicalmethods. For this decision the Sommerfeld Criterion has been formulated.

λdeBroglie = h/p ≪ d. (3.38)

When choosing for a typical object dimension half the distance of the tra-jectory turning point d0 for a central collision the Sommerfeld criterion forclassical scattering is obtained

ηS =Z1Z2e

2

hv= Z1Z2

e2

hc· cv

= Z1Z2 · αβ

≫ 1 (3.39)

or numerically (for a heavy target)

ηS ≈ 0.16 · Z1Z2

√

Aproj

Elab(MeV )≫ 1. (3.40)

3.1.4 Result of the Experiment

The results of the Rutherford-Geiger-Marsden experiment are shown in Fig. 3.4.The figure exhibits the strong angle dependence of this cross section togetherwith the original data of Ref. [GEI13], adjusted to the theoretical curveshown.

3.1.5 Consequences of the Rutherford Experiments andtheir Historic Significance

Rutherford and his collaborators Geiger and Marsden (later also Chadwick)used α particles from radiactive sources as projectiles. Their energies wereso small that for all scattering angles the minimum scattering distances d

DR

AFT


20 40 60 80 100 120 140 160 1801x100

1x101

1x102

1x103

1x104

1x105

1x106

1x107

1x108

1x109

1x1010

Counts adjusted to sin-4( /2)

1/si

n4 (c.

m./2

)

Scattering angle c.m.

(deg)

Angular Dependence of the Rutherford Cross Section

Figure 3.4: The curve shows the angular dependence of the theoreticalRutherford cross section ∝ sin−4(θ/2). The points are the original data(that consisted of tabulated numbers of counts with no error bars, and nottransformed into cross-section values) of Ref. [GEI13], adjusted to the the-oretical curve, giving a nearly perfect fit (Nowadays data with at least anerror estimate or, better, error bars are mandatory).

DR

AFT


were large compared with the sum of the two nuclear radii of projectiles andtargets. The complete agreement between the results of the measurementsand the (point-)Rutherford scattering cross section formula shows this inaccordance with Gauss’s law of electrostatics: a finite charge distribution inthe external space beyond the charges cannot be distinguished from a pointcharge with an r−1 potential. In addition, the mere occurence of backward-angle scattering events proves uniquely by simple kinematics that the targetnuclei were heavier than the projectiles. Thus the existence of the atomicnucleus as a compact (i.e. very small and heavy object) was established (andThomson’s idea of a “plum-pudding” of negative charges from distributedelectrons, in which the positive charges of ions were suspended, was refuted).

Later, the energy dependence as well as the dependence on charge numberscould be fully corroborated leading to a confirmation and a few correctionsto the periodic table of the elements.

DR

AFT

Bibliography

[GEI13] H. Geiger, E. Marsden, Philos. Magazine, Series 6, 25, 604 (1913)

[RUT08] E. Rutherford, H. Geiger, Proc. Roy. Soc. A81, 162 (1908)

[RUT11] E. Rutherford, Philos. Magazine, Series 6, 21, 669 (1911)

[SES07] A. Sessler, E. Wilson, Engines of Discovery – A Century ofParticle Accelerators, World Scientific (2007).

DR

AFT

22 BIBLIOGRAPHY

DR

AFT

Chapter 4

The First True NuclearReaction and the Discovery ofthe Proton

The first “true” nuclear reaction (i.e. one with transmutation into differentparticles) was discovered by Rutherford in 1919 (after earlier work togetherwith Ernest Marsden, in which particles with larger range, 1H nuclei, thanthat of the α’s in scattering from different targets were observed :

α+ 14N → 17O + p (4.1)

using 6 MeV α’s from a radioactive source. Fig. 4.2 shows such an event ina cloud chamber. It also shows an event of “Rutherford” scattering of therecoil 17O nucleus on an 14N nucleus.

The cloud chamber, which is still unsurpassed as an instrument for visu-alizing such events but also cosmic rays etc. was invented by Charles Wilsonfollowing 1911, but developed for practical use by Blackett only since 1921,was not yet used by Rutherford. Consequently the 1H nuclei were identified aspart of all nuclei and Rutherford coined the term “proton”. However, the stillremaining puzzles about the true structure of nuclei were only resolved afterthe neutron was discovered by James Chadwick in 1932 (after Rutherfordhad already speculated about neutrons in nuclei and others had mistakenlyinterpreted the neutron radiation from the reaction α+ 9Be → 12C + n (withα’s from a polonium source) to be an energetic γ radiation).

With the proton an essential component of nuclei had been found. Withonly α particles, electrons and protons as the known particles of the timeproperties of nuclei could not be explained satisfactorily. By comparison ofthe characteristic X-ray spectra of different elements with the charge num-ber Z from Rutherford scattering it became clear that Z characterizes thechemical elements. Soddy found that in some cases the radioactive decaysof chemically identical elements could be different and called these differ-ent substances isotopes[SOD13], see also Ref. [FAJ13]. Already in 1919 withthe development of the high-resolution mass spectrograph by Aston et al.[AST19, AST20] it became clear that almost all elements had a number ofisotopes that could be separated according to their masses in the mass spec-trograph. The atomic masses of all isotopes were nearly, but not exactly,

DR

AFT

24CHAPTER 4. THE FIRST TRUE NUCLEAR REACTION AND THE DISCOVERY OF THE PROTON

Figure 4.1: Apparatus used by Rutherford from 1917 to 1920 to bombard14N with α particles. The emitted particle radiation of longer range wasidentified as consisting of Z=1, A=1 particles, forming the nucleus of thehydrogen atom, and for which Rutherford coined the word “proton” in 1919.

integer multiples of the atomic mass of hydrogen, the small deviations be-ing due to the binding energy of nuclei. The possible structure of atomicnuclei was difficult to explain: the could not consist of protons only, andthe partial compensation of the nuclear charge by electrons in the nuclei isexcluded by quantum mechanics (which, however, was developed only after1924). Rutherford postulated already in 1920 that there must be an addi-tional particle with no charge and atomic mass number 1 which he christenedneutron [RUT20]. It took until 1932 before Chadwick discovered the neutron(see Chapter 8).

DR

AFT

25

Proton

Alphas

NO

14

17

Figure 4.2: Cloud chamber photograph by Blackett of the first nuclearreaction α+ 14N → 17O+p observed by Rutherford in 1919 [RUT19, BLA25].

DR

AFT

26CHAPTER 4. THE FIRST TRUE NUCLEAR REACTION AND THE DISCOVERY OF THE PROTON

DR

AFT

Bibliography

[AST19] F.W. Aston, Phil. Mag. 38, 709 (1919)

[AST20] F.W. Aston, Phil. Mag. 39, 449 (1920)

[BLA25] P.M.S. Blackett, Proc. Roy. Soc. A 107, 349 (1925)

[FAJ13] K. Fajans, Phys. Z. 14, 131 and 136 (1913)

[RUT19] E. Rutherford, Phil. Mag., Series 6, No. 222, 37, 537 (1919)

[RUT20] E. Rutherford, Proc. Roy. Soc. A 97, 374 (1920)

[SOD13] F. Soddy, Chem. News 107, 97 (1913)

DR

AFT

28 BIBLIOGRAPHY

DR

AFT

Chapter 5

Extended Matter and ChargeDistributions of Nuclei

Naturally the study of the finite size of nuclei requires higher-energy projec-tiles, as has been indicated by classical arguments in Chapter 3. These can behadronic particles such as α’s with medium energies (around 50 MeV wouldbe sufficient due to the small de Broglie wavelength) or e.g. electrons needingmore like hundreds of MeV. Because interference effects between strong andCoulomb interactions occur and electrons are relativistic a classical descrip-tion is impossible.

The extension of the derivation of the Rutherford cross section to an ex-tended (especially a homogeneous and spherically-symmetric) charge distri-bution is simple and leads to the fundamental concept of the form factor.

We start with the Coulomb potential of such an extended homogeneousspherical charge distribution (Fig. 5.1). It is calculated with Gauss’s theoremof electrostatics:

V (r) =

ze2

4πǫ0

1r

for r > Rze2

4πǫ0

12R

(

3 − r2

R2

)

for r ≤ R(5.1)

In the exterior space the potential is identical with that of a point charge,continues at r = R to a parabolic shape in the interior of the distribution. Itis therefore to be expected that in the scattering with sufficiently high energythe scattering cross section would strongly deviate from the Rutherford crosssection as soon as the nuclear surface is touched. In addition, the onsetof the short-range strong interaction will influence the scattering, especiallyby absorption. For the calculation of the cross section an integral over thecontributions from all charge elements dq = Zeρ(~r)dτ to the potential U(~r) =−Z1Z2e2

R· e−αRρ(~r)dτ has to be performed.

U(~r′) = −Z1Z2e2∫

ρ(~r′)e−αR

Rdτ. (5.2)

By inserting this into the Born approximation Eq. 3.33 (with d~R = d~r′

DR

AFT

30 CHAPTER 5. EXTENDED MATTER AND CHARGE DISTRIBUTIONS OF NUCLEI

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

VC(r

)/V

C(R

)

r / R

Figure 5.1: Coulomb potential of a spherical homogeneous charge distribu-tion.

and ~R = ~r′ − ~r) one obtains:

dσ

dΩ=

(

Z1Z2e2m

2πh2

)2

·[∫

ρ(~r)ei~K~rdτ ·

∫ e−αR

Rei~K ~Rd~R

]2

=[

F ( ~K2)K

K2 + α2

]2

. (5.3)

The cross section factorizes into two parts, one of which (after a transitionto the limit α → 0) results again in the point cross section, the other in theform factor:

dσ

dΩ=

(

dσ

dΩ

)

point nucleus

·∣∣∣F ( ~K2)

∣∣∣

2. (5.4)

This separation is characteristic for the interaction between extended ob-jects and signifies a separation between the interaction (e.g. the Coulombinteraction) and the structure of the interacting particles.

For rotationally-symmetric problems the form factor has a simplified in-terpretation:

F (K) =∫

ρ(r) exp (i ~K~r)2πr2dr sin θdθ. (5.5)

On substitution u = iKr cos θ and du = −iKr sin θdθ this becomes

F (K) = 2π∫

ρ(r)eur2drdu

−iKr

=∫

ρ(r)4πr2dr ·(

sin(Kr)Kr

)

︸︷︷︸

purely real

. (5.6)

Thus the form factor is a folding integral of the density with the samplingfunction (in parentheses). This function is oscillatory and its oscillation

DR

AFT

31

“wavelength” 1/K (which depends on the energy of the transferred radiation)has to be adjusted to the rate of change of the density. If the oscillation istoo frequent the integral results in ≈ 0 revealing no information on ρ. If it istoo slow the sampling function is ≈ constant, and the integral results in justthe total charge Ze. Fig. 5.2 illustrates this for different momentum transferson a given nuclear density distribution. Experimentally the form factor isobtained as the ratio

(

dσ

dΩ

)

experimental

/(

dσ

dΩ

)

point, theor.

. (5.7)

The charge distribution (or more generally: the density distribution e.g. ofthe hadronic matter) is obtained by Fourier inversion of the form factor F:

ρc(~r) =1

(2π)3

∫

0→∞Fc( ~K

2)e(−i ~K~r)d ~K. (5.8)

This means that (in principle) for a complete knowledge of ρ(~r) F must beknown for all values of the momentum transfer. Since ρ(~r) for small ~r isgoverned by the high-momentum transfer components of ~K this cannot beachieved in practice. For this reason the following approximations may beused:

• Model assumptions are made for the form of the distribution: e.g.homogeneously charged sphere, exponential, Yukawa, or Woods-Saxonbehavior.

• The model-independent method of the expansion of ei ~K~r into moments.

5.0.6 Ansatz for Models

It is useful to get an impression of the Fourier transformation of differentmodel density-distributions as shown in Fig. 5.3: It is a general observationthat “sharp-edged” distributions lead to oscillating form factors (and there-fore cross sections), and smooth distributions to smooth form factors. Inagreement with our ansatz a δ distribution (characteristic for a point chargeor mass) corresponds to a constant form factor (This is called “scale invari-ance”).

5.0.7 Expansion into Moments

With the power-series expansion of ei ~K~r the form factor becomes

F ( ~K2) ∝∫

ρ(~r)

1 + i ~K~r − ( ~K~r)2

2!+ −...

dτ (5.9)

By assuming a spherically symmetric distribution (with pure r dependenceonly) and with a normalization such that for a point object the constant formfactor is 1, we have:

F ( ~K2) = 1 − const ·K2∫

0→∞r2ρ(r)dτ ± ... (5.10)

DR

AFT


0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

K=2 fm-1

K=1 fm-1

K=0.5 fm-1

K=0.1 fm-1

Typical Nuclear Density Distribution

Radial Sampling Functionssin(

Kr)/K

r

r (fm)

Figure 5.2: Sampling functions for different momentum transfers show thatin order to sample details of a given structure (e.g. the shape around theradius of a nuclear density (charge or mass) distribution) the momentumtransfer (given by the incident energy and the scattering angle) has to havean appropriate intermediate value. In the example shown the value of K =0.5 fm−1 is suitable for sampling the region around the nuclear radius of5.0 fm. The vertical dotted lines indicate a 10 to 90% sampling region.

DRAFT

33

K

exp(-K2/2)

Gauß

K

Quadratic Dipole

1/(1+K2)2

K

FT of (r-R)

(sin(KR)/KR)2

r

exp(-r2)

Gauß

r

Exponential

exp(-r)

K

const(K)

r

(r)

Delta Distribution

K

FT of finitehomogeneous distribution

((sin(KR)-KRcos(kR))/(KR)3)2

r

r = R

HomogeneousDistribution

r

Yukawa

exp(-r)/r

K

Dipole

1/(1+K2)

r

(r - R)

Delta Distribution

Figure 5.3: Squares of the Fourier transforms – basically the form factors determining the shapes of the cross sections – of differentcharge-density distributions.

DR

AFT


The second term contains the average square radius 〈r2〉 = r2rms. For small

values of K2〈r2〉 one gets in a model-independent way (i.e. for arbitrary formfactors):

F ( ~K2) ≈ 1 − 16K2〈r2〉 (5.11)

Of course this approximation is becoming worse with smaller r (because oneneeds higher moments), i.e. if one wants to resolve finer structures.

5.1 Hadron Scattering Experiments

After accelerators were available charge and matter density distributions ofnuclei and their radii could be investigated by probing the distributions withhadronic projectiles. With light as well as with heavy ions, but also with neu-trons as projectiles it is evident that they are extended and possess structure.The consequence is that detailed statements about the density distributionsare difficult to make and may need the deconvolution of the contributionsfrom projectile and target nuclei. However, statements about nuclear radiiare possible, even with quite simple semi-classical assumptions such as ab-sorption between nuclei settung in sharply at a well-defined distance and pueCoulomb scattering beyon that distance. Systematic α scattering studies onmany nuclei (where we already have strong absorption at the nuclear sur-faces) revealed good A1/3 systematics for the nuclear radii. A dependence ofσα,α on R = R0(A1/3 + 41/3) was fitted to the data, assuming a sharp-cutoffmodel for the cross sections and taking into account the finite radii of bothnuclei. It yielded a radius constant of

R0 = 1.414 fm (5.12)

. However, when considering the range of the nuclear force for both nuclei ofabout 1.4 fm a radius constant of ≈ 1.2 fm resulted.

5.1.1 Nuclear Radii from Higher-Energy α-Particle Scat-tering

Already without detailed knowledge of the density distribution and of thepotential some quite precise statements about nuclear radii by scatteringof charged projectiles from nuclei were possible. One condition for this is,however, that the potential, which is responsible for the deviations fromthe point cross section is of short range, i.e. the charge distribution has arelatively sharp edge.

Most impressively these deviations from the point cross section appearwith diminishing distances between projectile and target in a suitable plot.Because the Rutherford cross section itself is strongly energy and angle de-pendent one may choose to plot the ratio

(

dσ

dΩ

)

exp

/(

dσ

dΩ

)

point, theor.

(5.13)

DR

AFT

5.1. HADRON SCATTERING EXPERIMENTS 35

as function of the minimum scattering (the apsidal) distance d. Thus dataat very different energies and angles can be directly compared (see Fig. 5.6in Section 5.1). If, in addition, one wants to check on the assumption ofthe systematics of nuclear radii to follow r = r0A

1/3 a universal plot for allpossible scattering partners by plotting the above ratio against d/(A1/3

1 +A

1/32 ) is useful. The experimental results show the extension of the charge

distribution and the rather sudden onset of (hadronic) absorption (providedthe interaction has a strong absorption term, which is typical for A ≥ 4.

Around 1954 α-particle beams from cyclotrons with energies much higherthan thos from radioactive sources became available, typically from about20 MeV to 40 MeV. In the classical Rutherford picture these energies werehigh enough that the colliding nuclei could be brought into contact in orderto “feel” the (hadronic) nuclear interaction in addition to the Coulomb field.According to this classical model the point of “grazing” depends on the energyand the scattering angle, see Eq. 3.14.

The key experiments were performed at the Brookhaven cyclotron with40 MeV beams on heavy nuclei. Using heavy targets has the significant ad-vantages

• The Coulomb interaction is quite strong.

• Quantum-mechanical interference or diffraction effects are small.

• The onset of strong absorption by the nuclear force is quite abrupt.

The following figures are from the key paper by Ref. [WEG55].The point of deviation from the Rutherford cross section is clearly visible

and corresponds todmin ≈ 1.7 fm (5.14)

. A best description of the data was achieved with the assumption thatnuclear radii (including that of the α particle) follow a

r = r0A1/3 (5.15)

law. With the additional assumption of a range of the nuclear force of ≈1.4 fm a “strong” radius constant of

r0 ≈ 1.45 fm (5.16)

resulted.

5.1.2 Heavy-Ion Scattering

As in α-particle scattering the strong absorption properties of the nuclearinteraction heavy-ion scattering experiments have been very useful to gaininsights into nuclear radii and other surface properties. A great number ofdifferent pairs of collision partners yielded very good systematics as shown inFig. 5.6. It becomes evident especially by plotting the relative cross sections

DR

AFT


against the distance parameter d, for which an assumed A1/3 dependence ofthe radii of both collision partners was applied

d = D0(A1/31 + A

1/32 )−1 (5.17)

with D0 the distance od closest approach, as calculated from energies andscattering angles. A well-defined sharp distance parameter of d0 = 1.49 fmfor the onset of absorption results. This corresponds to a universal radiusparameter of r0 = 1.1 fm if the range of the nuclear force is set to 1.5 fm.The simple model applied was to assume

• Pure point-Rutherford scattering outside the range of nuclear forces,

• Ratio of elastic to Rutherford cross section

dσ

dσR= 1 + Pabs(D) (5.18)

and

Pabs(D) =

0 for D ≥ D0,

1 − exp(D−D0

∆

)

for D < D0,(5.19)

with Pabs(D) the probability of absorption out of the elastic channel,D0 the interaction distance, and ∆ the “thickness” of the transitionregion.

The latter depends on the A of the nuclei involved and could be determinedwith good accuracy to be e.g. ∆ ≈ 0.33 fm for scattering of nuclei near 40Cafrom 208Pb.

5.2 Elastic Electron Scattering – Hofstadter’s

Experiments

Since all electrons (and all leptons) are considered to be point-particles theyare – as long as not the hadronic interaction region proper shall be probed– the ideal projectiles. They ”see“ the electromagnetic (and weak) structureof the nuclei. Of course, the treatment must be relativistic. Instead of theRutherford- (point-Coulomb) approach one has to use the proper theory.

Besides the relativistic treatment differences to the (classical) Rutherfordcross section come about by the lepton spin. The derivation of the correctscattering cross section relies on the methods of Quantum Electrodynamics(QED) and techniques such as the Feynman diagrams. Here only the resultswill be presented. The electromagnetic interaction between the electron and ahadron is mediated by the exchange of virtual photons, which is accompaniedby a transfer of energy and momentum. The wavelength of these photonsderives directly from the momentum transfer hK = 2(hν/c) sin(θ/2) to be

λde Broglie =h

hK= 1/K. (5.20)

The argument of diffraction limitation may also be formulated in the com-plementary time picture; it may be said that at long wavelengths, due to

DR

AFT

5.2. ELASTIC ELECTRON SCATTERING – HOFSTADTER’S EXPERIMENTS 37

the uncertainty relation, one needs long measurement times, in which theprojectile sees only a time-averaged picture of the object considered whilesmall wavelengths allow measurement times equivalent to snapshots of theobject or its substructures (partons).

Principally in lepton scattering at higher energies three distinct regions ofmomentum transfer can be distinguished:

• Elastic scattering at small momentum transfer is suitable to probe theshape of the hadrons. The resulting two form factors produce againthe charges and current (magnetic moment) distributions and the radiiof the hadrons by Fourier inversion.

• (Weakly) inelastic scattering at higher momentum transfer leads to ex-citations of the hadrons (e.g. Delta- or Roper excitations (resonances)of the nucleons). The form factors are quite similar to those from theelastic scattering, which means that we have some excited state of thesame nucleons.

• Deep-inelastic scattering is the suitable method to see partons insidethe hadrons. In this way in electron and muon scattering the quarksbound in nucleons and their properties (spin, momentum fraction) andalso the existence of sea quarks (s quark/anti-quark pairs were iden-tified). Especially the pointlike character of these constituents wasshown by the near constancy of the form factors (here called: structurefunctions with the momentum transfer (Bjorken scaling)).

Here only elastic scattering will be discussed in detail. In QED theory forthe differential cross section the Rosenbluth formula was deduced:

dσ

dΩ=

(

dσ

dΩ

)

point

·(

F 2E + bF 2

M

1 + b+ 2bF 2

M tan2 θ

2

)

. (5.21)

The point cross section (dσ/dΩ)point is a generalized Rutherford cross sectionand is calculable with the methods of QED (e.g. using Feynman diagrams).The most general form of this cross section (the Dirac scattering cross section)contains as main part the electrostatic scattering, a contribution from themagnetic (spin-dependent) interaction, which depends on the momentumtransfer, and a correction for the nuclear recoil:

(

dσ

dΩ

)

Dirac

=α2

4p20 sin4(θ/2)

[

1 +2p0

Msin2 θ

2

](

cos2 θ

2+

q2

2M2sin2 θ

2

)

..

(5.22)For small energies or momentum transfers the cross section simplifies to:

(

dσ

dΩ

)

Mott

=[2e2(E ′c2)]2

q4· E′

Ecos2 θ

2. (5.23)

The symbols used here mean: q = four-momentum transfer, b = −q2/(4m2c2),E ′, and E the energies of the outgoing and incoming electrons. FE and FMare the electric and magnetic form factors of the nucleons. Experimentally

DR

AFT


they are obtained from the measured data by least-squares fitting of theparameters of the theory, graphically through the Rosenbluth plot, i.e. byplotting (dσ/dΩ)exp/(dσ/dΩ)point against tan2(θ/2).

In analogy to the Rutherford cross section here the form factors (or struc-ture functions) are Fourier transforms of the charge and current-density dis-tributions (or: distributions of the (anomalous) magnetic moments). Likethere, these distributions result from Fourier inversion of the form factors,and at the same time quantitative values of the shape and size of the nucleonsare obtained.

The measured form factors as functions of q2 are normalized such thatfor q → 0 they become the static values of the electric charge and magneticmoments. Except for the electric form factor of the neutron all others arewell described by the dipole ansatz corresponding to a density distributionof an exponential function.

An early model for the charge-density distribution was – besides the ho-mogeneously charged sphere with only one parameter, its radius – a modifiedWoods-Saxon distribution with three parameters, because, besides the radiusparameter r0 and the surface thickness a, also the central density ρ0 must beadjustable because it varies especially in light nuclei:

ρc(r) =ρ0

1 + er−r

1/2

a

. (5.24)

The surface thickness t = 4 ln 3 · a signifies the 10 to 90% thickness rangecentered around r1/2. From this parametrization an electromagnetic radiusconstant of r1/2 = 1.07 fm, a surface-thickness parameter of a = 0.545 fm,and a central density of ρN = 0.17 nucleons/fm3 or 1.4 · 1014 g/cm3 for nu-clei with A > 30 have been derived. The description of “modern” densitydistributions is not so simple because the nuclei have individual and morecomplex structure even if the essential features such as the three parametersdo not vary too much. The detailed structure information is obtained frommodel-independent approaches such as Fourier-Bessel expansions. Radii aregiven as rms radii or converted into the equivalent radii R0. R0 is the radiusof a homogeneously charged sphere of equal charge using the relation

rrms =√

3/5R0. (5.25)

The definition of the (model-independent) Coulomb rms radius is

rrms = 〈r2〉1/2 =( 1Ze

∫ ∞

0r2ρC(r)4πr2dr

)1/2

. (5.26)

The results of elastic electron scattering on the proton show that theproton – different from heavier nuclei – has no sharp surface. Its densitydistribution has been described appropriately by an exponential with an rmsradius of

rrms ≈ 0.888 fm (5.27)

DR

AFT

5.3. COMPLEMENTARY METHODS 39

5.3 Complementary Methods

The scattering methods of determining density distributions and radii of nu-clei are well complemented by methods, which rely on the influence of theextended nuclear charge distribution on atomic levels. Laser-spectroscopymehods as well as exotic-atom methods are sensitive enough to compete withthem. As a model the interaction of an extended (homogeneous, spherical)charge distribution with atomic electrons shows the salient features of themodifications from a point charge. Fig. 5.9 depicts the charge-density dis-tribution together with the potential as functions of r. The Coulomb-energydifference ∆EC is determined by the integral over the potential difference forthe two cases

∆EC =∫ R

0eρe(r)∆ΦN(r)4πr2dr (5.28)

4πe ρe(r)︸︷︷︸

|ψe(0)|2

∫ R

0

C

2R

(

3 − r2

R2

)

︸︷︷︸

homogeneous

− C

R︸︷︷︸

point

r2dr (5.29)

≈ −|ψe(0)|2 · Ze2

4πǫ0· 4πR2

10(5.30)

For the 1S ground state of the hydrogen atom the electron’s radial wavefunction is

ψnℓm ∝(Z

a0

)3/2

(5.31)

leading to

∆EC ≈ −25

· Z4

4πǫ0· e2 · R

2

a30

, (5.32)

assuming a "contact interaction” at the nuclear center.

5.3.1 High Precision Laser Spectroscopy

Although the effects of nuclear size and shape on the energies and transi-tions of atomic electrons are very small the very high precisions reached inlaser-spectrospoy measurements allows to reach results comparable to othermethods (such as muonic atoms). The most intense efforts have been spenton the spectroscopy of the hydrogen atom, especially on the measurementof the famous Lamb shift, i.e. the energy separation between the 2S and2P states, which are predicted to be degenerate in Dirac theory but split bydifferent effects of QED (among them vacuum fluctuations and polarizationof the vacuum). This is why the measurement of the Lamb shift to very highprecision is essential.

The interpretation of the measurements requires the evaluation of the in-fluence of nuclear effects, especially the size (radius and shape) of the nucleus.The present state is that the precision of the measurements (in atmic spec-trosopy as well as in muonic atoms) is now so high that thee.g. the protonradius has become the final limitation to higher precision of the Rydberg con-stant and comparisons to QED. The results obtained for the proton radius

DR

AFT


agree within the errors with those from medium-energy electron scattering.However, a very recent, very puzzling disagreement (by 5 standard devia-tions!) with results from muonic atoms is unresolved, see next subsection.

5.3.2 Muonic Atoms

The 1S (n = 1) Bohr radius of a negative particle with mass m a0 is ∝ m−1

thusa0(µ−)a0(e)

=me

mµ−

= 4.75·10−3 (5.33)

and

∆EC(µ−)∆EC(e)

=|ψµ(0)|2|ψe(0)|2 =

[

a0(e)a0(µ)

]3

(5.34)

=(mµ

me

)3

= 210.53 = 9·106 (5.35)

testifying to a large increase in sensitivity of the position of energy levels andtransitions between them for muonic atoms as compared to electronic ones.However, because the level energies are larger by a factor mµ/me = 210.5 ,the transitions are in the X-ray region.

The first experiments with muons (the term “µ”-mesons is erroneous be-cause muons are leptons, mesons are hadrons underlying the strong interac-tion) could be performed in the 1950s after cyclotrons with sufficient energybecame available to produce pions (π mesons), which in turn decay weaklyvia

π− → µ− + νµ + 34 MeV (5.36)

(lifetime τ = 2.603·10−8 s, corresponding to a flight path of cτ = 7.8 m forhighly relativistic pions). By magnetic deflection and time-of-flight tech-niques the muons are separated from electrons and formed into a muonbeam in a muon channel. Their lifetime at rest is τ = 2.197·10−6 s andcτ = 658.65 m. Besides the properties of the muons their interactions withnuclei were studied. Negative muons, after slowing down by energy-loss pro-cesses in matter, may be captured by the nuclear Coulomb field and forcedinto outer Bohr orbits of the muonic atoms formed, from where they cascadedown into the ground state, thereby emitting characteristic X-rays. Refer-ences for the earliest such investigations are Refs. [FIT53, COO53, WHE53].In Ref. [FIT53] muonic X-rays on a number of nuclei were measured, of whichtwo examples are shown in Figs. 5.12 and 5.13. The Figs. 5.10 and 5.11show the apparatus used for these experiments. From the shift betweenthe 2p-1s transition energies of point and extended-charge distributions thenuclear-radius systematics of r = r0A

1/3 was established with a best value of

r0 = 1.17 . . . 1.22 fm (5.37)

and a muon mass of mµ = 210me. More modern methods used Ge(Li), Si(Li),HPGe, crystal spectrometers, and LAAPD photodiode detectors.

Nuclear radii from muonic atoms are often more precise than those fromlepton scattering but they are in a way complementary in relation to the

DR

AFT

5.3. COMPLEMENTARY METHODS 41

radius region probed they measure different moments). Thus, the resultsof both methods can be combined (Fig. 5.14). The distributions are quitewell reproduced by “mean-field” calculations, see e.g. [FRO87, DEC68]. Thesalient results of these investigations are:

• From the distributions a central density is derived, which for heaviernuclei is constant in first approximation. This and the systematics ofradii are characteristic for nuclear forces; their properties are: shortrange, saturation and incompressibility of nuclear matter, and suggestthe analogy to the behavior of liquids, which led to the development ofcollective nuclear models (liquid-drop models, models of nuclear rota-tion and vibration).

• The radii of spherical nuclei follow more or less a simple law r = R0A1

3 .For the radius parameter R0 = 1.24 fm is a good value. From Coulomb-energy differences of mirror nuclei a value of R0 = (1.22 ± 0.05) fm hasbeen derived.

• The surface thickness of all nuclei ist nearly constant with a 10-90%value of t = 2.31 fm corresponding to a = t/4 ln 3 = 0.53 fm. This isexplained by the range of the nuclear forces independent of the nuclearmass number A.

• The nucleons have no nuclear surface. The charge and current as well asthe matter densities of the proton follow essentially an exponential dis-tribution. For the neutron the charge distribution is more complicatedbecause volumes of negative and positive charges must compensateeach other to zero notwithstanding some complicated internal chargedistribution that originates from its internal quark-gluon structure.

• The rms radii for the current distributions of protons and neutrons andthe charge distribution of the protons are 0.88768(69) fm (accepted CO-DATA value) in agreement between atomic spectroscopy and electron-scattering results. Recently, with increased experimental precision anunresolved discrepancy between values from lepton scattering and muonic-atom work has been published [COD08, POH10], see Fig. 5.15. Therms charge radius of the neutron is −0.1161±0.0022 fm [EID04], which– with total charge zero – means that there must be positive and neg-ative charges distributed differently over the nuclear volume.

• Thus, nucleons are not “elementary”, but have complicated internalstructures.

5.3.3 Matter-Density Distributions and Radii

The matter density – apart from and independent of the charge or currentdistributions – can be investigated only by additional hadronic scatteringexperiments because neutrons and protons in principle need not have thesame distributions in nuclei.

DR

AFT


5.3.4 Hadronic Radii from Neutron Scattering

The total cross sections of 14 MeV neutron scattering under simple assump-tions have been shown to also follow a A1/3 law, see e.g. Ref. [SAT90],p. 32, cited from Ref. [ENG74]. The assumptions were that the sharp-edgedrange of the nuclear force was 1.2 fm and the total cross section σtot follows2π(R + λ)2 with R the nuclear (hadronic) radius, i.e. the nuclei are consid-ered to be black (totally absorbant) to these neutrons, which is not exactlyfulfilled, as the structures in this dependence show. These can be explainedwith the optical model, see below. The radius constant extracted from thissystematics is

Rhadr = 1.4 fm. (5.38)

In addition, there have been attempts to extract the neutron radius of208Pb from parity-violating electron scattering [ABR12].

5.3.5 Special Cases – Neutron Skin

Especially the question of a neutron skin in nuclei with neutron excess isinteresting and only recently such a thin skin was consistently shown toexist, see e.g. [TSA12] and references therein. Among the hadronic probesused have been protons, α’s, heavy ions, antiprotons, and, recently, also pionse.g. on 208Pb, 48Ca and others. The extraction of rms radii requires somemodel assumptions concerning the reaction mechanism and the interplay ofhadronic and Coulomb interactions. The pion results are derived from twosources: pionic atoms (in analogy to the derivation of the electromagneticradii from muonic atoms) and total reaction cross sections of π+ [FRI12].The neutron skin is related to the symmetry energy, which plays a rôle inthe in the mass formula of Bethe and Weizsäcker for the binding energiesof nuclei, especially for “asymmetric” nuclei with strong neutron excess, butalso for astrophysics and nuclear-matter calculations. The radius of neutronstars is closely related to the symmetry energy value in high-density nuclearmatter, see e.g. Ref. [TSA12].

Usually the quantity

δRnp = 〈r2〉1/2n − 〈r2〉1/2

p . (5.39)

The experimental values deduced from different experiments are on the orderof δRnp ≈ 0.2 fm.

5.4 The Size and Shape Systematics of Nuclei

Whereas the charge and magnetic distributions are best obtained with chargedprojectiles, for which the interaction is exactly known (e.g. electrons, whichdo not interact via the strong force), for neutrons one needs nuclear scat-tering models (e.g. the optical model, see Section 11.1). The assumptionthat neutron and proton radii of heavier nuclei are about equal has provedtoo simple with the evidence of neutron-halo and neutron-skin nuclei, seeChapter 6.

DR

AFT

5.4. THE SIZE AND SHAPE SYSTEMATICS OF NUCLEI 43

Figure 5.4: Cross-section angular distributions of 40 MeV α scattering fromheavy nuclei. The solid line is the pure point-Rutherford prediction.

DR

AFT


Figure 5.5: Same as Fig. 5.4, but plotted is the ratio of the measured crosssection and the calculated Rutherford cross section against the calculateddistance of closest approach, see Eq. 3.14, for α’s on Au. This plot showsthe relatively sudden onset of absorption by the nuclear interaction, whichallows, using the A1/3 law, the derivation of the range of the nuclear forceof about 1.4 fm. The data from Ref. [WEG55] has been augmented by datafrom Indiana University and University of Washington [FAR54, WAL55].

DR

AFT


Figure 5.6: Plot of the scattering cross sections (relative to the Ruther-ford cross section) as functions of the distance of closest approach ofmany different HI pairings. The figures are from Ref. [CHR73](right) andRef. [OGA78](left).

Figure 5.7: Connection between electron scattering cross sections and den-sity distributions on pointlike and extended nuclei.

DR

AFT


Figure 5.8: Differential cross section for 500 MeV electrons fitted by anexponential form factor.

DR

AFT


Homogeneous Charge Distribution

r

V(r) (r)ρ

Point Charge

Homogeneous Charge Distribution

Point Charge

r = R

Figure 5.9: Charge density distribution and Coulomb potential of a pointcharge compared to an extended homogeneous charge distribution and itspotential.

Figure 5.10: Experimental setup for the production of a beam of muons andslowing-down and capturing the nuons into Bohr orbits. After [FIT53].

DR

AFT


Figure 5.11: NaJ scintillator detector setup for the measurement of muonicX-rays, emitted in 2p-1s transitions to the ground states of different muonicatoms. After [FIT53].

Figure 5.12: Muonic X-ray spectrum of Pb obtained with a NaJ scintillationdetector and showing the large energy shift between a point charge and theactual extended-charge distributions. After [FIT53].

DR

AFT


Figure 5.13: Muonic X-ray spectrum of Ti obtained with a NaJ scintillationdetector and showing the large energy shift between a point charge and theactual extended-charge distributions. After [FIT53].

Figure 5.14: Charge density distributions of different doubly closed-shellnuclei with electron-scattering and muonic-atom data combined. After[FRO87].

DR

AFT


Figure 5.15: Unresolved discrepancies between determinations of the pro-ton’s rms radius by different methods. The accepted CODATA-06 value isrrms(p) = 0.88768(69) fm whereas the new muonic-atom value is rrms(p) =0.84184(67) fm. After [COD08, POH10].

Figure 5.16: Stanford facility.

DR

AFT


Figure 5.17: Stanford spectrometer.

DR

AFT


DR

AFT

Bibliography

[ABR12] S. Abrahamyan et al. (PREX Collaboration)Phys. Rev. Lett. 108, 112502 (2012)

[CHA56] E.E. Chambers, R. Hofstadter, Phys. Rev. 103, 1454 (1956)

[CHR73] P.R. Christensen„ V.I. Manko, F.D. Becchetti, R.J. Nickles,Nucl. Phys. A207, 33 (1973)

[COD08] CODATA-06, P.J. Mohr, B.N. Taylor, D.B. Newell,Rev. Mod. Phys. 80,633 (2008)

[COO53] L. Cooper, E. Henley, Phys. Rev. 92, 801 (1953)

[DEC68] J. Dechargé, D. Cogny, Phys. Rev. C 21, 1568 (1968)

[EID04] S. Eidelman at al. (Particle Data Group), Phys. Lett. B 592, 1(2004)

[ENG74] J.B.A. England, Techniques in Nuclear Structure Physics(Halstead, New York, 1974)

[FAR54] G.W. Farwell, H.E. Wegner, Phys. Rev. 93, 356 (1954)

[FAR54] G.W. Farwell, H.E. Wegner, Phys. Rev. 95, 1212 (1954)

[FIT53] Val. L. Fitch, J. Rainwater, Phys. Rev. 92, 789 (1953)

[FRI12] E. Friedman, Nucl. Phys. A 896, 46 (2012).

[FRO87] B. Frois, C.N. Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133(1987)

[HOF55] R. Hofstadter, R.W. McAllister, Phys. Rev. 98, 217 (1955)

[McA56] R.W. McAllister, R. Hofstadter, Phys. Rev. 102, 851 (1956)

[OGA78] Yu.Ts. Oganessian, Yu.E. Penionzhkevich, V.I. Man’ko,V.N. Poyanski, Nucl. Phys. A303, 259 (1978)

[POH10] R. Pohl, Nature 466, 213 (2010)

DR

AFT

54 BIBLIOGRAPHY

[TSA12] M.B. Tsang, J.R. Stone, F. Camera, P. Danielewicz, S. Gandolfi,K. Hebeler, C.J. Horowitz, Jenny Lee, W.G. Lynch, Z. Kohley,R. Lemmon, P. Möller, T. Murakami, S. Riordan, X. Roca-Maza,F. Sammarrucca, A,W. Steiner, I. Vidaña, S.J. Yennello,Phys. Rev. C 86, 015803 (2012).

[WAL55] N.S. Wall, J.R. Rees, K.W. Ford, Phys. Rev. 97, 726 (1955)

[WEG55] H.E. Wegner, R.M. Eisberg, G. Igo, Phys. Rev. 99, 825 (1955)

[WHE53] J.A. Wheeler, Phys. Rev. 92, 812 (1953)

[WAL55] N.S. Wall, J.R. Rees, K.W. Ford, Phys. Rev. 97, 726 (1955)

DR

AFT

Chapter 6

Halo Nuclei and Farewell toSimple Radius Systematics

At the “rims” of the valley of stability (the neutron or proton driplines) thereare a number of nuclei that have much larger radii than expected from thesystematics. 11B has about the same radius as 208Pb.

First experimental evidence of halos in 1985 were deviations of reactioncross sections σ = 4πr2 from the systematics expected as described in Chap-ter 5 in light nuclear isotopes far from the valley of stability, such as 11Li[TAN85], see also Refs. [OZA01, KRI12]. With increased interaction (ab-sorption) radii between nuclei in heavy-ion reactions also narrower momen-tum ditributions of breakup fragments in such reactions have been observed.[DOB06].

Also the deuteron has an extreme rms radius of about 3.4 fm. In all casesthe nuclei seem to have a halo of weakly bound neutrons (or protons), whichsurrounds a more strongly bound core. Different cores are possible, i.e. be-sides the strongly bound α making 5He and 6He one- and two-neutron halonuclei also 9Be forms some type of core. Generally indications of halo struc-tures are – among others – the exceptionally large cross sections in heavy-ionreactions, narrower momentum distributions of the nucleons in the nuclei,and larger radii, as compared to the A1/3 law. Fig. 6.1 shows the low-massportion of the chart of nuclides where halo nuclei have been found. The sci-entific interest in halo nuclei is manifold. They were among the first wherethe driplines have been reached. The results show that the shell structuresestablished for the valley of stability can be extended to “exotic” nuclei, butwith modifications of the closed shells, i.e. with new magic numbers emerg-ing. The low mass numbers invite application of microscopic theories such asFaddeev-(Yakubowsky), no-core shell models, Green’s function Monte Carlo(GFMC), and other approaches to test nuclear forces, e.g. three-body forces,or effective-field (EFT) approaches. Impressive results have been obtainedby such “ab initio” calculations, see e.g. Ref.[PIE01, DEA07]. A special roleis played by the so-called Borromean nuclei, i.e. those that consist of a coreplus two weakly (un)bound neutrons at large radii, and for which any of thetwo-particle subsystems are unbound (Example: 4He + n + n). They canbe treated by well-established three-body methods. Their name is derivedfrom the three intertwined Borromean rings that fall apart when one ring is

DR

AFT

56 CHAPTER 6. HALO NUCLEI AND FAREWELL TO SIMPLE RADIUS SYSTEMATICS

Stable Two−One−

One− Two− Four−

6

5

4

3

B

Be

Li

O8

7 N

Ne10

9 F

3 3 4 5 6

9 12 13 14

121187

8

11

86

1 20

1

22

87

9 10

11 12 13 1514 1614

2 He

H

18 2017 21 2322

15

9 10 11 14 15 16 17 18

21

20 22

N

19

17

CZ

21

16

13

10

1

13 1612

13 14 15 19 20

23 24191817

17 18 19 20 21

Proton Halo

Neutron Halo

7

7

A

6

19

20

10

9

3 4

21

9

A

A AA

A

5

16

13

11

10

12

14 15

16 17 18

21 22

8

Figure 6.1: Halo nuclei at the driplines of the chart of nuclides.

removed and hold together only when united, see Fig. 6.2. Many nucleosyn-thesis processes pass through nuclei that are neutron rich or neutron poor andare not well known. Thus, for nuclear astrophysics, a better understandingof all these reactions and their reaction rates is essential.

Since we deal with unstable (radioactive) nuclei the “radioactive-ion beams(RIB)” facilities, which are being developed are especially suited for theirinvestigation. These facilities collect, focus and accelerate nuclear reactionproducts in order to use them as projectiles in reactions. The Figs. 6.3 and6.4 show the properties typical for halo nuclei:

• They have radii, which are larger than predicted from the usual A1/3

Figure 6.2: Coat of arms and symbol of the Renaissance Borromean fam-ily (and other north Italian families like the Sforzas) at their castle on theBorromean island Isola Bella in the Lago Maggiore, Italy.

DR

AFT

57

systematics.

• Their density distributions reach further out than usual.

• In agreement with this they show narrower momentum distributions ofthe breakup fragments of the halo nuclei (one example: 19C → 18C+n,compared to 17C → 16C + n).

The latest discovery of a halo nucleus is that of 22C [TAN10] which showedan increased reaction cross section and an rms radius of rrms = 5.4 ± 0.9 fm,both larger than expected from the usual systematics.

DR

AFT

58 CHAPTER 6. HALO NUCLEI AND FAREWELL TO SIMPLE RADIUS SYSTEMATICS

A1/3

p (MeV/c)z

Rel

. cou

nt r

ate

−200 0 200100 0 5r(fm)

protons

all neutrons

1.25

last

two

neut

rons

−100 10 15 20

Den

sity

(nu

cleo

ns/c

m )3

C

Li

18

1116C

Figure 6.3: Fragment-momentum distribution and density distributions inhalo nuclei.

3.0

3.5

rms

radi

us (

fm)

6 8 10 12

Mass number A

7 9 11 14 13

BeLi

He

B

2.5

2.0

Figure 6.4: Radii of halo nuclei.

DR

AFT

Bibliography

[DEA07] D.J. Dean, Physics Today, November issue, p. 48 (2007)

[DOB06] A.V. Dobrovolsky et al., Nucl. Phys. A 766, 1 (2006)

[KRI12] A. Krieger et al., Phys. Rev. Lett. 108, 142501 (2012)

[OZA01] A. Ozawa, T. Suzuki, I. Tanihata, Nucl. Phys. A 693, 32 (2001)

[PIE01] S.C. Pieper, R.B. Wiringa, Ann. Rev. Nucl. Part. Sci. 51, 53(2001)

[TAN10] K. Tanaka et al., Phys. Rev. Lett. 104, 062701 (2010)

[TAN85] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida,N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi,N. Takahashi, Phys. Rev. Lett. 55, 2676 (1985)

DR

AFT

60 BIBLIOGRAPHY

DR

AFT

Chapter 7

The Particle Zoo

Although, after the discovery of the proton and the neutron, around 1935 thebasic structure of nuclei became clear, the neutrino as product of β decay wasstill a postulate as well as Yukawa’s exchange particle later called the pion.Then, new (elementary?) particles (such as the muon µ± and the positrone+) found in cosmic rays started to open the entirely new and complex worldof particle physics. The parallel development of accelerators immediatelysuggested the artificial production of these and the search for more suchparticles at higher and higher energies commensurate with increasing massesof these particles, a trend that continues up to these days (examples are theW± and Z0 exchange bosons, the top quark t, and the Higgs boson). A verylarge number of particles have been identified and classified according to theirproperties as leptons, hadrons, hyperons, baryons, fermions or bosons, see thetabulations of the particle data group with the most recent Ref. [RPP08]. Amore fundamental order was given to this zoo by the discovery of the buildingblocks of six leptons, six quarks, and their antiparticles, in the framework ofthe standard model

7.1 The Pion

Yukawa in 1935 postulated an exchange particle with a mass of ≈ 130 MeV/c2,commensurate with the range of the (hadronic) nuclear force of about 1.4 fm.For a time the muon, found in 1932 in cosmic radiation, was mistaken as thisparticle, but – being a heavy lepton – did not have the expected proper-ties. Only in 1947 the pion was detected as a component of cosmic rays[LAT47a, LAT47b, LAT47c] in photoplates, see Fig. 7.1, decaying as

π± → µ± + νµ. (7.1)

The sign of its charge could not be determined due to the lack of a strongmagnetic field. The pion was identified with a mass of ≈ 300 me by its shorterlifetime (range) and the ionization grain density being higher than that of themuon. All muon tracks observed were of about equal length, correspondingto a pion dacay at rest. Sometimes the pion decay would be accompanied bya number of strong hadronic tracks of short range, i.e α particles etc, form-ing a “star”, which can be taken as evidence of an hadronic interaction, i.e.

DR

AFT

62 CHAPTER 7. THE PARTICLE ZOO

track continued from A

µ

µ A

π

A

µ

Figure 7.1: Photoplate tracks of the decay of a cosmic pion into a muonand an invisible muonic (anti)neutrino, as seen through a microscope. Thetypical length of the muon track is ≈0.61 mm.

through the strong interaction between the pion and an emulsion nucleus, incontrast to the pion decay that is a weak-interaction process. Fig. 7.2 showssuch an event. Soon after this discovery – in 1948 – the artificial productionof pions in accelerators of sufficient energy, e.g. the 184” synchrocyclotronat Berkeley was successful [GAR48, BUR48, JON49]. A carbon target po-sitioned inside the magnetic field of the synchrocyclotron was bombardedwith protons of energies starting near the production threshold just below200 MeV. Positive and negative pions were deflected by the magnetic fieldin opposite directions and registered by two stacks of photoplates. Theseshowed not only microscopic tracks of pions but als their decay into muons(with typical track lengths of 0.6 mm) and the β deacy of the muons. Theproperties of pions could now be studied as function of their energies, e.gby scattering them from hydrogen. The interaction strength and the shapeof the differential cross sections pointed to the strong interaction as the onekeeping nuclei together (against the repulsive Coulomb force between theprotons). The spin and parity of the pions were determined to be J = 0−

(they are pseudoscalars), their isospin T = 1, meaning that they come as anisobaric triplet π±, π0.

Especially the study of the scattering of pions from protons proved tobe very fruitful. Resonances in excitation functions were indications of theformation of intermediate states, corresponding to excited states of the nu-cleons. The most prominent is the excitation of the ∆ resonance. Because ofthe high energies involved and thus the high number of decay channels theseexcited states are rather short-lived, i.e. the resonances are quite broad. The∆ has Jπ = 3/2+ and isospin 3/2 meaning that it comes as an iso-quartet.These quantum numbers require a spin flip whereas another such resonance,the Roper resonance has the same quantum numbers as the proton and can beinterpreted as an excited proton without change of internal structure. In thisway a large number of “particles” could be created, such as the baryons, whichencompass the nucleons and the hyperons ∆,Λ,Σ,Ξ, and Ω, the latter fourcarrying the new quantum number of strangeness. In the constituent-quarkmodel all baryons are made of three quarks u and d, quark-antiquark pairs,allowing for s and ¯mathrms, quarks, and gluons. In similar experimentsa number of mesons was discovered: The scalar mesons: ρ, ω, φ, and K∗

DR

AFT

7.1. THE PION 63

Figure 7.2: Photoplate tracks of the reaction of a cosmic (assumed negative)pion after slowing down and being captured in an atomic shell and cascadingdow into the K shell before being destroyed in an hadronic interaction witha nucleus forming a “star”. Outgoing from the star strong tracks from twoα’s from nuclear interactions are seen together with other (meson?) tracks.

DR

AFT


and the pseudoscalar mesons π, η, η′, and K. All mesons consist of quark-antiquark pairs. The multitude of particles is often called “The Particle Zoo”,into which only the assignment of quark combinations brought a perfect or-der.

7.2 First Production of the Antiproton in a

Nuclear Reaction

P.A.M. Dirac in 1928 created a relativistic form of the wave equation, theDirac equation with energy eigenvalues for a free electron

E = ±(p2c2 +mec2)1/2

[DIR28]. He associated the seemingly unphysical negative solution with ahole in an otherwise filled sea of negative-energy states and thus with theelectron’s antiparticle, the positron. After the discovery of the positron incosmic rays in a cloud chamber by Anderson in 1932 [AND32, AND33] andthe observation of positive and negative muons by Street and Stevenson[STR37] and Anderson and Neddermeyer in 1937/1938 [AND37, AND38]the question naturally arose of the existence of antiparticles for every particlewhich could be answered only by experiment. Simultaneously the conceptsof lepton-number and baryon-number conservation were developed.

An antiparticle has the same properties as the associated particle, exceptfor charge-related properties (charge, magnetic moment). Antiparticles anni-hilate when interacting with ordinary matter, i.e. their energy is radiated asneutral radiation such as photons or π0. Antiprotons had not been identifiedin cosmic rays before and the Bevalac synchrotron accelerator at Berkeleywas specifically designed to find the antiproton.

The conservation laws (energy and momentum conservation, and chargeconjugation) require that when bombarding a nuclear target with (high-energy) protons and with an antiproton in the exit channel the antiparticlemust be one of a particle-antiparticle pair (i.e. p + p). In addition the ejec-tile and recoil particles will appear, thus four particles are emitted, i.e. thereaction had to be

p+ p → p+ p+ p+ p,

in order to obey baryon number conservation for B = 2. Applying relativistickinematics for a fixed target particle with mass m2 we need a lab. energy ofincident beam particles with mass m1 for the energy available in the c.m.system of Ec.m.

Tlab. =E2

c.m.

2m2c2+ Ec.m.

(

1 +m1

m2

)

At threshold for the creation of the pair p +p Ec.m. = 2mpc2. The proton

lab. energy threshold for antiproton production on a fixed target of freeprotons is thus

Tthr = 6 ·mpc2 = 5.638 GeV. (7.2)

Such an energy could only be reached with synchrotron accelerators. In thiscase, the Bevatron at Berkeley, designed for a maximum beam momentum

DR

AFT

7.3. DISCOVERY OF THE (ELECTRON) NEUTRINO 65

of 6.3 GeV/c, corresponding to an energy of 5.4 GeV, was used to create thefirst antiprotons in 1955 [CHA55].

In the actual experiment a copper target was used. Because of the bindingof the protons in the heavier target the Fermi momentum of the nucleonscould be exploited to add some relative energy to the projectile-target systemwith the effect of lowering the threshold energy for p production. With theapproximate factor

1 − pF/Mpc (7.3)

the threshold momentum is lowered to a value of about 4.8 GeV/c. Fig. 7.3shows schematically the setup to produce antiprotons. How was the uniqueidentification of the antiprotons achieved? In view of an enormous back-ground of (negative) pions the following measures hed were taeken:

• . Use was made of the negative charge leading to a deflection andextraction opposite to the deflection of the protons in the magneticfield of the synchrotron.

• Near threshold the antiprotons move with half the beam velocity, i.e.with momentum 1.19 GeV/c to which the magnet system is tuned.

• The time of flight between two scintillation detectors about 10 m apartis measured by delayed coincidence techniques to obtain the velocityand thus get a handle on the mass.

• A combination of three Čerenkov detectors (which are velocity-sensitivedetectors), one a β=v/c threshold detector, the other with a narrow βwindow were used to register the antiprotons only and reject pions andvice versa, leading to a strong background suppression.

The system was tuned to transmit particles of mass mp, and a small variationof the beam momentum yielded a transmission curve for negative particlescentered about the proton mass, as shown in Fig. 7.4. Fig. 7.5 shows theproduction rate of antiprotons, relative to the rate of pions as function of theproton lab. energy starting from the expected threshold at about 4.2 GeV.

7.3 Discovery of the (Electron) Neutrino

W. Pauli formulated the Neutrino Hypothesis in 1930 [PAU30] as answer toopen questions concerning β decay. The continuous electron spectrum andthe spins of the particles called for a third decay particle. E. Fermi formulatedhis theory of the weak interaction in 1934. Nevertheless the first neutrino (inits electron flavor) was detected directly in experiments beginning in 1953by F. Reines and C.L. Cowan [REI53, COW56, REI59, REI60]. Only then βdecay was satisfacorily explained and conservation laws like lepton-numberconservation could be postulated.

The experiment was based on the reaction

ν + p → n + β+ (7.4)

DR

AFT


Figure 7.3: Experimental setup for the first detection of antiprotons at theBerkeley Bevatron.

DR

AFT


Figure 7.4: Transmission of detector arrangement for negative particles withmass of ≈ mp, i.e. of antiprotons p .

DR

AFT


Figure 7.5: Excitation curve of the antiproton production at the BerkeleyBevatron.

DR

AFT


(an inverse β decay). The two particles in the exit channel are a conditionfor sharp energies (line spectra) facilitating their detection. Fig. 7.6 depictsthe scheme of the detection setup. Antineutrinos come in large numbersfrom power reactors. Due to the neutron excess of primary fission prod-uct nuclei these undergo β decay with the emission of antineutrinos – ac-cording to baryon- and lepton-number conservation. Fluxes on the order of1013 cm−2s−1 are available, which – even with the very small weak-interactioncross section of typically 10−43 cm2 – provides for well-measurable event ratesof ≈ 5·10−3 s−1. The different parts of the experiment are:

• The neutrons from the reaction 7.4 are moderated (time scale: severalhundred µs) and captured in a liquid target consisting of CdCl2 in H2Othereby emitting γ rays that are registered in two large liquid scintilla-tor counters (in anticoicidence for background readuction). 113Cd (withan abundance of 12.26% in natural Cd) has a very large absorption crosssection for thermal neutrons (σ = 63, 600 b at En = 0.18 eV).

• The positrons lose energy, are finally stopped in the target, and an-nihilate into two 511 keV annihilation γ quanta emitted in oppositedirections in two scintillator tanks where they are registered in coin-cidence. The coincidence and anti-coincidence requirements led to anelaborate construction of large target and scintillator tanks

• The two (related) coincidence events are measured in an an additionaltime-delayed coincidence with variable delay time around the moder-ation time of the neutrons in the target. With the “true” events an

γ

γ

β +

νγ

Cd

γ

γ

γ

n p

511 keVannihilation

511 keVannihilation

Figure 7.6: Antineutrino reaction and detection scheme of the neutron-capture γ’s from Cd and positron-annihilation 511 keV γ’s in a suitable liquidscintillator containing a neutron moderator (water) and a Cd compound (e.g.CdCl2).

DR

AFT


average cross section of several measurements was

σ = 11 ± 2.6·1044 cm2. (7.5)

The authors announced the success of finding the neutrino on July 1959 ina telegram to W. Pauli at the ETH Zürich.

A later experiment that became famous for the measurement of solarneutrinos was based on the reaction

νe + 37Cl → 37Ar + e−, (7.6)

[DAV55, DAV64, CLE98]. In this experiment the small cross section wascompensated by a very large detector containing many tons of chlorine-compounds such as CCl4, see Fig. 7.7. The experiment ran for about 25years continuously. It turned out that the 37Ar produced could be extractedfrom the target liquid quantitatively and identified by its EC β decay

37Ar + e− → 37Cl + ν, (7.7)

which is accompanied by the emission of 3-5 atomic K-shell Auger electrons.Fig. 7.8 shows the miniature proportional counter used for measuring theactivity of the 37Ar β decay. The signals were selected by a pulse rise-timecondition imposed, thus reducing background. Great care had been observedin the selection of all materials to cut any background rate substantially belowthat of true solar neutrino events (≈ 1 event per week).

The setup was tested at the Brookhaven and a power-plant reactor andthen ran for many years 1.5 kilometers underground in the Homestake minein South-Dakota to avoid background events from terrestrial sources.

The results of these measurements were manifold:

• The experiment proved that antineutrinos are emitted from the fusionreactions in the sun’s interior.

Figure 7.7: View and schematic of the tank for the unique detection of solarneutrinos, from [CLE98]

DR

AFT


3 cm

Figure 7.8: Construction of the counter to measure the β activity of the 37Arproduced by the solar neutrinos and extracted from the chlorine compound,see [CLE98].

DR

AFT


• Neutrino and antineutrino are different particles: The cross sectionmeasured of σ < 0.9·10−45 cm2 per Cl atom is incompatible with thatexpected if neutrino and antineutrino were indistinguishable (“Majo-rana neutrino”). A consequence would that the neutrinoless double-βdecay would be allowed and that neutrinos must have mass. However,in contrast to the double β decay with emission of two antineutrinos,e.g. in the decay

10648Cd → 106

46Pd + 2e+ + 2νe, (7.8)

this process has not been found. Neutrino oscillations have been de-tected, however:

– At the Super-Kamiokande detector in Japan evidence of oscilla-tions of cosmic-ray neutrinos was found in 1998.

– In 2002 oscillations of solar electron antineutrinos to another fla-vor were seen by the Sudbury Neutrino Observatory in Ontariotwo kilometers underground, thus solving completely the “solar-neutrino puzzle”.

– Ocillations of antineutrinos from 22 different nuclear power plantsat different distances were detected by the KamLAND experimentin Japan in 2002 [EGU03, BAH03]. The detector was a massivekiloton liquid scintillator viewed by nearly 2000 photomultipliertubes in the Kamioka mine.

The intensity of the solar neutrino flux as measured by the Davis ex-periment was only about one third of that expected from model calcu-lations of solar models using all known parameters of the sun as input[CLE98, BAH95, SAC90]. The solution of this “solar neutrino puzzle”was that electron neutrinos have mass and thus can “oscillate”, i.e. pe-riodically transform into one or both of the other neutrino flavors νµand ντ (the evidence for cosmological as well as particle-physics rea-sons is very strong that there exist only three “families” of leptons –likewise as for the hadron families of quarks). With the existence ofthese oscillations the finite mass of the neutrinos is a fact – it remains,however, open what the masses of each of the three neutrino flavorsare.

DR

AFT

7.4. QUASI-ELASTIC ELECTRON SCATTERING – EXCITED NUCLEONS AND THE PARTICLE ZOO73

7.4 Quasi-Elastic Electron Scattering – Ex-

cited Nucleons and the Particle Zoo

Spectra of the inelastic electron scattering from the proton at relatively highenergies show besides the elastic peak a number of excited states (like innuclear scattering spectra), corresponding to excited nucleon states (nucleonresonances),

7.5 Deep-Inelastic Lepton Scattering – Par-

tons inside Hadrons

Once higher electron energies became available, finer structures of nucleonsand nuclei could be explored. The key experiment was performed by JeromeIsaac Friedman, Henry Way Kendall, and Richard Edward Taylor (NP 1990),see Ref. [TAY67] around 1966 and yielded clear evidence of “parton” struc-tures inside the nucleons that had all the properties of the quarks, postulatedtheoretically in 1964 by Murray Gell-Mann who coined the term “quark” af-ter James Joyce’s “Finnegan’s Wake”, and George Zweig [GEL64, ZWE64].They recognized that all known hadrons of the time could be fitted nicely intoa scheme (the “eightfold way” [GEL66]) of combinations of only three dif-ferent particles with certain properties. Most conspicuous was the necessityof their having fractional charges (±1/3e or ±2/3e), but also an additionaldegree of freedom called “color”. The existence of strange particles requiredthe introduction of a strange quark (the s quark) for which the discovery ofthe J/Ψ meson as an s – anti-s combination brought final evidence, as well astwo more types of quarks, the bottom (b) and finally the top quark (t) (andtheir antiparticles) closed the scheme. The evidence for the vector bosons(“gauge bosons’) mediating the strong interaction between quarks via virtualexchange – and thus responsible for the stability of nuclei – was first discov-ered around 1976 in electron-positron collisions at the PETRA acceleraatorat DESY/Hamburg and the particles were called “gluons”. The characteris-tics of these results were the occurence of “three-jet” events (i.e. the emissionof quark-antiquark pairs plus a sideways single jet from a neutral particle.The quarks as well as the gluons are “confined”, i.e. cannot be liberatedto appear as free particles, but show up as jets of hadrons instead. This isalso expressed by the idea of the quarks and gluons carrying the propertyof color and the postulate that free particles have to be color-neutral, i.e.do not carry color. Further information on the properties and hiostory of allparticles can be found in the regularly updated reviews of particle properties,see Ref. [RPP08].

The “scaling” behavior found in deep-inelastic electron scattering (as inthe classical Rutherford scattering) proves that these constituents apparentlyare pointlike and therefore truly “elementary” (to our present knowledge!).

The scattering of neutrinos from nuclei required a number of special prepa-rations. In order to form an “intense” beam of neutrinos relativistic kine-matics had to be applied, which causes reaction products to be emitted into

DR

AFT


Figure 7.9: Double differential cross section of deep-inelastic scattering of500 MeV electrons [BLO69].

DR

AFT

7.5. DEEP-INELASTIC LEPTON SCATTERING – PARTONS INSIDE HADRONS 75

narrow forward cones. The neutrinos were produced in the decay of pi-ons and muons that accompany the reactions of high-energy protons withsuitable solid targets. The invention of the Van der Meer neutrino horn[VDM61, GIE63] appreciably increased the neutrino beam density. In thismagnetic focussing device the beam of pions whose decay produces a beamof neutrinos is strongly focused in the forward direction with the same effecton the neutrinos. Only through this focusing the more recent long-distanceneutrino-oscillation experiments became possible.

The search for neutrino oscillations (transformation into other neutrinoflavors as function of the travelling distances or flight time) has led to verylarge detectors filled with neutrino-sensitive liquids and scintillators and sur-rounded by very many photomultipliers (SUPER-KAMIOKANDE, SUD-BURY OBSERVATORY, KAMLAND, GRAN SASSO etc.) looking for so-lar, cosmic, and reactor-produced neutrinos. Thus, neutrino oscillations havebeen discovered and the solar-neutrino puzzle has been solved.

Deep inelastic scattering of neutrinos from protons at CERN around 1979see e.g. Ref. [AGU92] as well as of muons [BEN90, AMA92] and electrons[WHI92] gave some evidence that inside the proton (and of course, otherhadrons) not only quarks but some other electrically neutral “material” ex-ists which was dubbed “gluons”. Only half of the momentum of the protonwas carried by the quarks, the other obviously by the gluons as shown by the“structure functions” (similar to the low-energy form factors) of the nucleons.More detailed experiments in which polarized particles were used as projec-tile and/or targets yielded “spin-structure functions” exhibiting the differentcontributions to the nucleon’s spin 1/2h. This global spin quantum numbercould not be explained by addition of the spins of the constituent quarksalone (“the spin crisis”) but also required contributions from the gluon spinand from orbital angular momenta.

DR

AFT


Figure 7.10: Big European Bubble Chamber at CERN.

Figure 7.11: Neutrino Scattering event in the Big European Bubble Chamberat CERN.

DR

AFT

Bibliography

[AGU92] M. Aguilar-Benitez et al., Particle Data Group, Phys. Rev. D45, 1 (1992)

[AMA92] P. Amaudruz et al., Phys. Lett. B295, 159 (1992)

[AND32] C.D. Anderson, Science 76, 238 (1932)

[AND33] C.D. Anderson, Phys. Rev. 43, 491 (1933)

[AND37] C.D. Anderson, S. Neddermeyer, Phys. Rev. 51, 884 (1937)

[AND38] C.D. Anderson, S. Neddermeyer, Phys. Rev. 54, 88 (1938)

[BAH95] J.N. Bahcall, M.H. Pinsonneault, Rev. Mod. Phys. 67, 781(1995)

[BAH03] J.N. Bahcall, M. Gonzalez-Garcia, C. Peña-Garay, J. HighEnergy Phys. 02, 009 (2003)

[BEN90] A.C. Benvenuti et al., Phys. Lett. B237, 592 (1990)

[BLO69] B.L. Bloom et al., Phys. Rev. Lett. 23, 930 and 935 (1969)

[BUR48] J. Burfening, E. Gardner, C.M.G. Lattes, Phys. Rev. 75, 382(1948)

[CHA55] O. Chamberlain, E. Segrè, C. Wiegand, T. Ypsilantis,Phys. Rev. 100, 947 (1955)

[CLE98] B.T. Cleveland, T. Daily, R. Davis jr., J.R. Distel, K. Lande,C.K. Lee, P. Wildenhain, J. Ullman, Astrophys. J. 496, 505(1998)

[COW56] C.L. Cowan, F. Reines, Science 124, 103 (1956)

[DAV55] R. Davis, Phys. Rev. 97, 766 (1955)

[DAV64] R. Davis, Phys. Rev. Lett. 12, 303 (1964)

[DIR28] P.A.M. Dirac, Proc. Roy. Soc. A117, 610 (1928)

[EGU03] K. Eguchi et al. (Kamland collaboration), Phys. Rev. Lett. 90,021802 (2003)

[GAR48] E. Gardner, C.M.G. Lattes, Science 107, 270 (1948)

DR

AFT

78 BIBLIOGRAPHY

[GEL64] M. Gell-Mann, Phys. Lett. 8, 214 (1964)

[GEL66] M. Gell-Mann, Y. Ne’eman, The Eightfold Way, Benjamin, NewYork (1966)

[GIE63] M. Giesch, B. Kuiper, B. Langeseth, S. Van der Meer, D. Neet,G. Plass, G. Pluym, B. de Raad, Nucl. Instr. Meth. 20, 58 (1963)

[JON49] S.B. Jones, R.S. White, Phys. Rev. 78, 12 (1949)

[LAT47a] C.M.G. Lattes, H. Muirhead, G.P.S. Occhialini, C.F. Powell,Nature 159, 694 (1947)

[LAT47b] C.M.G. Lattes, G.P.S. Occhialini, C.F. Powell, Nature 160, 453(1947)

[LAT47c] C.M.G. Lattes, G.P.S. Occhialini, C.F. Powell, Nature 160, 486(1947)

[PAU30] W. Pauli, Letter to the “Radioaktiven Damen und Herren”(”radioactive ladies and gentlemen”) assembled at a Tübingenmeeting, dated December 4th (1930)

[REI53] F. Reines, C.L. Cowan, Phys. Rev. 92, 830 (1953)

[REI59] F. Reines, C.L. Cowan, Phys. Rev. 113, 273 (1959)

[REI60] F. Reines, Ann. Rev. Nucl. Sci. 10, 1 (1960)

[RPP08] Particle Data Group, Review of Particle Properties,Rev. Mod. Phys. 80, 633 (2008)

[SAC90] J. Sackman, A.I. Boothroyd, W.A. Fowler, Astrophys. J. 360,727 (1990)

[STR37] J.C. Street, E.C. Stevenson, Phys. Rev. 52, 1003 (1937)

[TAY67] R.E. Taylor, Proc. Int. Symp. on Electron and PhotonInteractions at High Energies, Stanford (1967)

[VDM61] S. Van der Meer, CERN Report 61-7 (1961)

[WHI92] L. Whitlow et al., Phys. Lett. B282, 475 (1992)

[ZWE64] G. Zweig, CERN Report No. 8182/TH401 (unpublished)

DR

AFT

Chapter 8

Discovery of the Neutron(Nuclear Kinematics etc.)

Chadwick discovered the neutron in 1932 by correctly identifying the ener-getic radiation emitted from the reaction α+ 9

4Be → 126C+ 1

0n (induced by α′sfrom a Po source). The recoil energies transferred to the protons and 14N nu-clei of the filling gas of the ionization chamber were measured to be 5.7 MeVand 1.6 MeV, respectively (masses: 9

4Be: 9.011348 MeV; 14N: 14.002863 u)The value of the neutron mass (in u) obtained by Chadwick was 1.0067.

From range measurements of protons ejected as recoils from hydrogenousmaterial only a crude estimate of the neutron energies and its mass couldbe deduced. However, using the same apparatus but observing the reactionα + 11

5Be → 147N + 1

0n because of better-known mass values, he obtained themass of the neutron within the limits between 1.005 and 1.008. Today’s bestvalue is 1.008664904 ± 0.000000014 u.

Other nuclear physicists (among them the Curies) had erroneously inter-preted the energetic radiation as γ radiation with energies up to 50 MeV,transferring recoil energy by some Compton-like scattering process to theprotons or 14N nuclei. However, such energies of γ transitions cannot occurin nuclei. Only the assumption of a neutral particle with a mass near thatof the proton appeared consistent with all observations. Chadwick receivedthe Nobel prize in 1935.

Fig. 8.1 shows the extremely simple experimental setup used by Chadwick.Initially the measured mass of the neutron led to the assumption of theneutron being a (quasi-)bound proton-electron system, but very shortly thisidea was dismissed in favor of the neutron being an (elementary) particle ofits own.

Immediate consequences of the discovery:

• For the first time the model of nuclei as being composed of protonsand neutrons, the existence and “construction” of isotopes, the periodictable, and the chart of nuclides became unambiguous. It was proposedindependently by Heisenberg [HEI32a, HEI32b] and Iwanenko [IWA32].

• The neutron, due to its electric neutrality, proved to be an ideal projec-tile to penetrate nuclei and to perform all kinds of nuclear reactions atall energies, including the induced fission of heavy nuclei and creation

DR

AFT

80 CHAPTER 8. DISCOVERY OF THE NEUTRON (NUCLEAR KINEMATICS ETC.)

Figure 8.1: Apparatus used by Chadwick to discover the neutron, [CHA32b].

of heavier isotopes (in the laboratory and in nucleosynthesis). Neu-tron multiplication allowing a chain reaction after fission is the basis ofnuclear reactors for energy production as well as in the atomic bomb.

• The similarity of the properties of protons and neutrons led W. Heisen-berg [HEI32a] to postulate the symmetry of Charge Independence andthe conservation quantity for which E. Wigner later coined the termIsospin [WIG37], see also Section 13.3. Both have been very impor-tant in nuclear and particle physics, also because their symmetry laterproved to be slightly broken.

• The β decayn → p+ e− + νe (8.1)

is a prototype process caused by the weak interaction. The idea of theexistence of the particle class of Leptons, especially of neutrinos (an-tineutrinos) is intimately connected with this process. This decay wasmeasured already in 1914 by Chadwick displaying a continuous β spec-trum and thus could not be a two-particle decay, provided the kinemat-ics of energy and momentum conservation holds. Only, between 1930when Pauli formulated tentatively the Neutrino Hypothesis[PAU30],1934 when Fermi created his theory of β decay and weak interaction,and 1956 when the neutrino was detected directly by F. Reines andC.L. Cowan [COW56] β decays were satisfacorily explained.

• The neutron allows a large number of applications ranging from neutron-activation analysis, creation of medically-required isotopes by neutron-capture reactions in reactors, neutron radiography complementing X-ray studies in materials analysis, to the study of biological structures.

DR

AFT

Bibliography

[BEC69] F.D. Becchetti jr., G.W. Greenlees, Phys. Rev. 182, 1190 (1969)

[CHA32a] J. Chadwick, Nature 129, 3252 (1932)

[CHA32b] J. Chadwick, Proc. Roy. Soc. A 136, 692 (1932)

[COW56] C.L. Cowan, F. Reines, Science 124, 103 (1956)

[HEI32a] W. Heisenberg, Z. Physik 77, 1 (1932)

[HEI32b] W. Heisenberg, Z. Physik 78, 156 (1932)

[IWA32] D. Iwanenko, Nature 129, 798 (1932)

[PAU30] W. Pauli, Letter to the “Radioaktiven Damen und Herren”(”radioactive ladies and gentlemen”) assembled at a Tübingenmeeting, dated December 4th (1930)

[WIG37] E. Wigner, Phys. Rev. 51, 106 (1937)

DR

AFT

82 BIBLIOGRAPHY

DR

AFT

Chapter 9

First Precise Determination ofthe Neutron Mass and theBinding Energy of theDeuteron

After the discovery of the neutron by Chadwick and the implications for thestructure of nuclei it was essential to measure its properties more precisely,among them its spin and mass. The deuteron had just been discovered byUrey et al. [URE32] in 1932 by optical spectroscopy and its approximatemass was determined from the recoil factor (1 +me/mN )−1 in the energiesof spectral lines of optical spectra (me, mN are the masses of the electron andthe isotope investigated, resp.). Two principal methods could be applied tomeasure the bindiong energy of the deuteron.

9.1 The Photonuclear Disintegration of the

Deuteron

The photodisintegration of the deuteron

γ + d → p+ n

(analogous to the atomic photo effect) was used by Chadwick and Goldhaber[CHA34] to measure md. A strong γ source of Th C” (today 208Tl) with Eγ= 2.615 MeV was used to incite the reaction. D2 gas was used as filling gasin an ionization chamber and the energy of protons stopped in the chamberwas measured by the height of oscilloscope traces produced. A test with aRa C (today 214Bi) source with a γ emission line with Eγ = 1.764 MeV didnot produce any protons. Thus, the threshold for photodisintegration had tobe between these two values. The pulse heights of the proton traces allowedto deduce a kinetic energy of Ekin ≈ 250 keV. The neutrons must have hadthe same energy whence the binding energy BE of the deuteron

BE(d) ≈ 2.1 MeV.

DR

AFT

84CHAPTER 9. FIRST PRECISE DETERMINATION OF THE NEUTRON MASS AND THE BINDING

ENERGY OF THE DEUTERON

With Einstein’s mass-energy relation mdc2 = mpc

2 + mnc2 − BE(d) we

obtain the mass of the neutron

mn = md −mp +BE(d)/c2.

Bainbridge [BAI33] had measured the masses of the proton and deuteron andwith these values and error bars the mass of the neutron was

mn = 1.0080 ± 0.0005 u.

With later improved methods mn was determined more precisely by mea-suring the disintegration threshold using X-ray bremsstrahlung from an elec-tron accelerator and registration of the neutrons in a BF3 gas-filled propor-tional counter that makes use of the reaction 10B + n → 7Li +α+ 2.78MeV.One of these “precision” experiments was performed in 1959 by Mobley etal. [MOB50] at the pressurized ANL accelerator which had the specialtythat it could accelerate ions to ground potential and simultaneously elec-trons to a target inside the tank on high voltage where the experiment withbremsstrahlung γ’s could be performed. The ion beam served to determinethe energy of the electrons indirectly via a calibrated nuclear reaction.

It resulted in

BE(d) = 2.227(3) MeV and mn = 1.008982(3) u

The experimental arrangement with the electron accelerator as X-ray sourceis shown in Fig. 9.1, the target and detector setup in Fig. 9.2. The γ energy(i.e. the maximum energy of the bremsstrahlung spectrum) could be variedby varying the electron energy to pass the threshold for the reaction

2H(γ, n)1H.

It turned out that the intensity increase above threshold was quadratic suchthat a linear dependence of the square root of the yield would allow a preciseextrapolation to the threshold energy corresponding to the binding energy ofthe deuteron. The method and results are shown in Fig. 9.3.

Figure 9.1: View of the bremsstrahlung source, an electron accelerator usedin the experiment of Ref. [MOB50].

DR

AFT

9.2. NEUTRON-PROTON CAPTURE 85

Figure 9.2: The bremsstrahlung target consisted of a thin gold foil producinga gold bremsstrahlung spectrum, the reaction target was deuterium in theform of heavy water in a thin-walled container, and the neutron detector wasa 10BF3 proportional counter.

9.2 Neutron-Proton Capture

The inverse reaction (neutron-proton capture)

n + p → d+ γ

is another possibility to measure BE(d) by measuring the energy of the γ’sproduced with thermal neutrons. Before 1977 a number of such measure-ments had been performed, but gave conflicting results. Using new calibra-tion standards for γ energies [HEL79] Van der Leun et al. [VDL82] publishedthe results of a high-precision experiment using an n-type intrinsic Ge de-tector and another Ge(Li) detector for the γ spectrosopy. The setup of theexperiment is shown in Fig. 9.4. The relevant spectrum of the intrinsic Gedetector is shown in Fig. 9.5. The neutrons were produced by a 241Am +9Be source and thermalized in a paraffin cylinder. The result of

BE(d) = 2224575 ± 9 eV

in comparison to earlier data is shown in Fig. 9.6. The accepted best valuesat present ([NIS08] and [AUD03]) are

BE(d) = 2.224566 MeV and mn = 1.00866491600 u

DR

AFT



Figure 9.3: The method of extrapolation of Ref. [MOB50] to the thresholdenergy of the 2H(γ,n)1H reaction is explained.

DR

AFT

9.2. NEUTRON-PROTON CAPTURE 87

Figure 9.4: Schematic view of the sample and different neutron and γsources, the paraffin thermalizer, and lead shielding of the detector. Sourcesof 48V and 144Ce were used for calibration. From Ref. [VDL82].

Figure 9.5: The spectrum of p(n,γ)d γ’s of the n-type intrinsic Ge detectorwith an energy resolution of 1.75 keV at 1.33 MeV. Source of 48V and 144Cewers used for calibration. From Ref. [VDL82].

DR

AFT



Figure 9.6: The results of Ref. [VDL82] in comparison to earlier conflictingresults of Wapstra et al. [WAP77] (compilation of earlier data), Vylov etal. [VYL78], and Greenwood et al. [GRE80].

DR

AFT

Bibliography

[AUD03] G. Audi, A.H. Wapstra, C. Thibault, Nucl. Phys. A729, 337(2003)

[BAI33] K.T. Bainbridge, Phys. Rev. 44, 57 (1933)

[CHA34] J. Chadwick, M. Goldhaber, Nature 134, 237 (1934)

[GRE80] R.C. Greenwood, R.E. Chrien, Phys. Rev. C21, 498 (1980)

[HEL79] R.G. Helmer, P.H.M. Van Assche, C. van der Leun, Atomic Dataand Nucl. Data Tables 24, 39 (1979)

[MOB50] R.C. Mobley, R.A. Laubenstein, Phys. Rev. 80, 309 (1950)

[NIS08] http://physics.nist.gov/cuu/Constants/index.html

[URE32] H.C. Urey, F.G. Brickwedde, G.M. Murphy, Phys. Rev. 40, 1(1932)

[VDL82] C. Van der Leun, C. Alderliesten, Nucl. Phys. A380, 261 (1982)

[VYL78] Ta. Vylov, et al., Yad. Fiz. 28, 1137 (1978)

[WAP77] A.H. Wapstra, K. Bos, Atomic Data and Nucl. Data Tables 19,175 (1977)

DR

AFT

90 BIBLIOGRAPHY

DR

AFT

Chapter 10

First Nuclear Reaction with anAccelerated Beam;Cockroft-Walton Accelerator

In an early paper [COC30]) Cockroft and Walton very clearly outline thenecessity of using higher energy/higher intensity beams of projectiles forthe study of nuclear reactions. They refer to the use of α particles fromradioactive sources and the limitations connected with it: low intensities (aradium source with an activity equivalent to a decent charged-particle beamwould require hundreds of grams of radium!), very limited energy variabilityand limitation to α’s only etc. They knew that in order to initiate nuclearreactions between charged particles surmounting the Coulomb barrier wouldneed energies of many MeV. But recently G. Gamow et al. had shown that –via quantum-mechanical tunneling – nuclear reactions might be possible atmuch lower energies, with, however, reduced intensities [GAM29, GUE29].

With this knowledge they discuss in detail possibilities of creating highvoltages for particle acceleration, among them a Tesla-coil device, an ACdevice, and a pulse generator, and conclude that some kind of DC voltage-multiplication scheme using a high-voltage transformer, vacuum-tube recti-fiers, and capacitors could be suitable. In addition they also discuss methodsto produce low-energy high current ion beams in canal-ray tubes such as usedfor mass-spectrometry.

In 1932 Cockroft and Walton [COC32a, COC32b] published the resultsof their efforts to produce sufficiently high DC voltages to initiate a nuclearreactions. They developed a multiplier circuit using rectifier diodes and ca-pacitors that could withstand high voltages up to 200 kV. Such circuits hadbeen devised elsewhere [GRE14, SCH19, GRE21, SLE28], but Cockroft andWalton adapted them to their special needs. Fig. 10.1 explains the princi-ple of the circuit and the vacuum accelerating-tube design in several stagesconnected to different voltage levels from the rectifier. The use of severalaccelerating gaps has the advantage of smaller electric field strength per gapand better focussing properties for the ion beam. For a comprehensive surveysee Ref. [BAL59]. Figs. 10.2 and 10.3 show a schematic and a photographof the entire accelerator setup. The proton beam of up to 15µA could beaccelerated to about 720 keV and focussed only by the accelerating tube ar-

DR

AFT

92CHAPTER 10. FIRST NUCLEAR REACTION WITH AN ACCELERATED BEAM;

COCKROFT-WALTON ACCELERATOR

Figure 10.1: Schematic of the voltage multiplication circuit used in the first“Cockroft-Walton” accelerator and accelerator tube design.

Figure 10.2: Schematic of the accelerator setup.

DR

AFT

93

Figure 10.3: Photograph of the accelerator complex. In the box coveredby a black cloth the experimenter would sit and count scintillations on afluorescent screen e.g. as function of the scattering angle.

rangement with no extra focussing elements. In a first series of experimentsthe proton beam, described as a visible “pencil” exiting the thin mica win-dow into open air, was used to measure the proton range in different gases,see Fig. 10.4.

In a modified setup the protons impinged on a Li target under 45 in achamber shown in Fig. 10.4. The reaction particles were observed as scin-tillations on a ZnS screen with a microscope and also in a cloud chamberwhere by the thickness of the tracks they were identified as α particles. To-gether with a second such microscope setup on the opposite side frequentcoincident emissions of two α’s were registered such that the authors unam-biguously concluded that the

7Li + p → 2α+ 17.347 MeV (10.1)

had taken place. This was thus the first nuclear reaction initiated by artifi-cially accelerated projectiles (and also one of the first (the first?) coincidenceexperiments) [COC32b].

The authors varied the proton energy and saw a typical Coulomb thresholdbehavior. Then they subjected a remarkably large number of elements tothe proton beam and registered the relative number of counts per unit timeand beam current (i.e. they measured essentially the cross sections for (p,α)reactions).

The use of accelerated ions started the entire field of nuclear reactionsand its rapid development. The development of particle accelerators is aprerequisite not only for the study of nuclear reactions and the elucidation

DR

AFT



Figure 10.4: Reaction chamber setup with a vacuum pump system, a thinmica window allowing protons to exit or to impinge on a Li target. Inthis latter case (left) a microscope was used to count the reaction α’s asscintillations on a fluorescent screen. When using a second scintillator on theopposite side coincident events could be observed.

DR

AFT

95

of their mechanisms, but also for nuclear structure studies. The possibilityto excite many levels of thousands of nuclei and to study their formationand decay lead to the important shell model and different collective mod-els describing the rich landscape of nuclear shapes, their motion, and theirexcitations. The progress in our knowledge of these properties is thereforeintimately connected with the developments of accelerators. The quest forever higher energies is still with us, but also other properties such as higherenergy resolution, ease of changing energies and beam projectile type, andhigh beam currents for better statistics were driving the development up tothe highest energies such as that of the LHC with a relative energy (i.e. anenergy available in the c.m. system of E = 14 TeV. The Livingston plot inFig. 10.5 documents this. Fixed-target experiments – due to relativistic ef-fects – provide less and less relative energy available for particle interactionswith increasing lab. energy whereas colliders make the full relative energyof the head-on colliding beams available (with equal masses and lab. ener-gies of the two beams twice the lab. energy). For the study of nuclearreactions Van-de-Graaff machines and cyclotrons (in their modern form of(spiral-)sector focussing are best suited and have become the “workhorses”of low-energy nuclear physics. The simultaneous development of ion-sourcetechnologies, of solid-state detectors, and of magnetic spectrographs made thebraod spectrum of different investigations and discoveries of nuclear physicswith high-quality data possible. Examples are:

• Fine-structure studies on Isobaric Analog Resonances, possible onlythrough the high energy resolution and fine-tunability of the Tandem-Van-de-Graaff accelerator or

• Heavy-ion reaction studies as well as their use in γ spectroscopy throughthe ease of changing between many heavy-ion species in the beamsource, of energy, and charge states to obtain high beam energies.

Cockroft-Walton machines and cyclotrons are in use as first part in the chainof accelerators in intermediate and high-energy installations such as COSYand CERN. Accelerators in the low- and intermediate-energy domain havealso found many useful applications in medicine, art history, archaeology,and others.

DR

AFT



Figure 10.5: “Livingston plot”: Plot of the development of acceleratorsover the years with a doubling of the “available” energy approximately everyseven years. The approximate energy ranges of (low-energy) nuclear physicsproper, of intermediate-energy physics where the overlap and interactions ofquarks and nuclei are studied, and high-energy (or particle) physics whereall facets of the standard model are investigated, are indicated in the plot.Originally in Ref. [LIV54].

DR

AFT

Bibliography

[BAL59] E. Baldinger, Kaskadengeneratoren in: Encyclopedia ofPhysics/Handbuch der Physik, Nucl. Phys. Subseries, E. Creutz(ed.), Nuclear Instrumentation, Vol. 44 Springer, Heidelberg(1959)

[COC30] J.D. Cockroft, E.D.S. Walton, Proc. Roy. Soc. A129, 477 (1930)

[COC32a] J.D. Cockroft, E.D.S. Walton, Proc. Roy. Soc. A136, 619 (1932)

[COC32b] J.D. Cockroft, E.D.S. Walton, Proc. Roy. Soc. A137, 229 (1932)

[GAM29] G. Gamow, Z. Physik 52, 510 (1929)

[GRE14] H. Greinacher, Physikalische Zs. 15, 410 (1914)

[GRE21] H. Greinacher, Z. Physik 4, 195 (1921)

[GUE29] R.W. Guerney, E.U. Condon, Phys. Rev. 33, 127 (1929)

[LIV54] M.S. Livingston, High-Energy Accelerators, Interscience, NewYork (1954)

[SCH19] M. Schenkel, Elektrotechnische Zs. 40, 333 (1919)

[SLE28] J. Slepian, High-voltage direct-current system 1921, US patent,class 175-363, No. 1,666,473, disclosed/published (1928)

DR

AFT

98 BIBLIOGRAPHY

DR

AFT

Chapter 11

Observation of DirectInteractions

Together with compound nuclear CN reactions direct interactions DI are twoforms of reactions of composite nuclei that are best classified according totheir time behavior. They mark the extremes in this classification. The DIare processes, which occur at time scales of the traversal times of projectilespast target nuclei (typical times are 10−22 s). On the other end are the CNreactions where projectiles are more or less completely absorbed by the targetto form a compound nucleus living a long time during an equilibration processand decaying into open channels without memory of the formation process(typical times are 10−16 s).

11.1 Elastic Scattering and the Optical Model

Around 1950 it was noticed that the scattering data of intermediate-energynucleons (in the beginning mainly neutrons) could not be described well inthe framework of the compound-nucleus models. Angular distributions aswell as excitation functions showed marked diffraction patterns. Very earlyproton data were measured by Burkig et al.[BUR51] using 18.6 MeV protonson several nuclei as shown in Fig. 11.1. Early neutron elastic scatteringwas reported by Bratenahl et al. [BRA50]. For the description of the datatwo limiting cases of models inspired by phenomena in optics, the opaquemodel and the transparent model [FER49]. These data were then fit withtwo different optical model theories [LEL52]. For the one of them that gavea better fit to the diffraction pattern the authors coined for the first time theterm Optical Model. A picture of the first “Optical Model” fit is shown inFig. 11.2.

As standard literature on the optical model only a few references will begiven here: [HOD63, HOD67, MAR70] The radial Schrödinger equation forprotons (spin s = 1/2) reads:

[

d2

dr2+ k2 − ℓ(ℓ+ 1)

r2+ V f(r) + iWg(r) − VC(r) (11.1)

+ (Vs.o. + iWs.o.)h(r) ·

ℓ−(ℓ + 1)

]

u(±)ℓj (kr) = 0.

DR

AFT

100 CHAPTER 11. OBSERVATION OF DIRECT INTERACTIONS

Figure 11.1: Angular distributions of 84 MeV neutrons scattered from dif-ferent targets.

DR

AFT

11.1. ELASTIC SCATTERING AND THE OPTICAL MODEL 101

Figure 11.2: Fit with the first “Optical Model” .

DR

AFT


Here: V,W are real and imaginary parts of the central potential, VC theCoulomb potential, and Vs.o.,Ws.o. real and imaginary parts of the spin-orbitpotential, especially important for die description of polarization observables.For neutrons the Coulomb term vanishes. The solution of this equation ispart of the total scattering function

Ψ =1kr

∑

ℓjλ

iℓ [4π(2ℓ+ 1)]1/2 (ℓ0sµ|jm)(ℓλsν|jm)u(±)ℓj (kr)Y λ

ℓ (θ, φ)χµseiσℓ

(11.2)with

uℓj →r→∞12i

[

e−i(kr−η ln 2kr−ℓπ/2) − e2iδℓjei(kr−η ln 2kr−ℓπ/2+2σℓ)]

. (11.3)

δℓj are the complex nuclear scattering phases, σℓ = argΓ(1 + ℓ + iη) theCoulomb scattering phases, ηjℓ = e2iδℓj the “reflection coefficients”, η =Z1Z2e

2/hv the Coulomb (or Sommerfeld) parameter (see Eq. 3.39), and

k =√

2µEc.m.kin /h

2 the entrance-channel wavenumber.The potential form factor f(r) is defined in analogy to the shape of the

usual nuclear density or potential distributions that are used in the classicalshell model (Woods-Saxon form). The absorption occurs predominantly atthe nuclear surface. Thus, as form for g(r) at low energies one chooses thederivative of the Woods-Saxon-form factor, and for the spin-orbit term hthe Thomas form g(r)/r. At higher energies more absorption in the nuclearvolume is plausible, which is taken into account by a gradual transition fromthe surface absorption to volume absorption.

The best sets of parameters have been obtained by fits with χ2 mini-mization to a large number of data sets of cross sections as well as analyzing

r

r

r

Woods−Saxon

DerivativeWoods−Saxon

EmpiricalEnergy Dependence

τ=4.4σ

40 MeV

30 MeV

20 MeV

of W(r)

Figure 11.3: Form factors of the optical model. Upper: Woods-Saxon form ofthe real part f. Center: Derivative Woods-Saxon form g = f’ of the imaginarypart. Lower: Sliding-transition form of the surface-to-volume imaginary partas function of energy.

DR

AFT

11.2. DIRECT (REARRANGEMENT) REACTIONS 103

powers. The latter are important for fixing the (~L· ~S) potential and removingtypical ambiguities in the potential parameters. A few of these parametersets have become standards for the optical model. For nucleon scatteringthe parametrization most used is that of Greenlees and Becchetti [BEC69],especially because they provide a global set of parameters (i.e. valid overa large region of the periodic table). However, in special cases e.g. neardoubly-magic nuclei this set is not as good as a single fit. It is interestingthat the depth of the real potential corresponds closely to that of the shellmodel potential, similarly for the LS term. For light projectiles consisting ofA nucleons (deuterons, α particles etc.) one has potential depths that areA-fold the nucleon potential depths. For heavy-ion scattering there are quitedifferent approaches, partly with very shallow potentials. Fig. 11.3 shows theform factors and the behavior of imaginary potentials with energy. It canonly be mentioned in passing that steps to found the optical potentials onmore microscopic grounds have been undertaken by creating folding poten-tials. In these the potential of one nucleon of the projectile with the targetnucleus is folded with the nucleon density of the projectile nucleus and viceversa, or the same for both nuclei (double folding potentials).

11.2 Direct (Rearrangement) Reactions

The multitude of these reactions may be classified:

• Reactions without change of the mass number

– Elastic potential scattering (see above; description by the opticalmodel).

– Direct inelastic scattering [(p,p’γ), (α, α′), ...]. It leads preferen-tially to collective nuclear excitations (such as rotation, vibrationetc.).

– Quasi-elastic (charge-exchange) processes ((p,n), (n,p), (3He,t),(14N,14C),...). These lead e.g. to isobaric-analog states of thetarget nucleus.

• Reactions with change of the mass number

– Pickup reactions [One-nucleon transfer: (p,d), (d,3He), (d,t),...,few or multi-nucleon transfer: (p,α), (d,6Li),... ].

– Stripping reactions [One-nucleon transfer: (d,p), (d,n) ,(3He,d),...)few- or multi-nucleon transfer: (6Li,d), (α,p), (3He,p),...].

– Knockout reactions [(p,α), (p,p’),...].

– Direct breakup processes like knockout with few-particle exit chan-nels [(p,pp), (α, 2α),...].

– Induced fission is a special case of a rearrangement reaction re-sulting in larger debris.

– Processes of higher order (multi-step processes via excited inter-mediate states, coupled channels).

DR

AFT


Fe

FeFe

Fe

Fe

FeV

TiTi

Zr

Zr

Sn

Sn

4949

58

5757

54

51

Ni

Ni

120

60

5656

54

20 140 2080 80 140

(deg)Θ

58

yA (

)

120

90

Θ

90

60

c.m.

σ d /

d σ R

uth

Ni

Ni

Figure 11.4: Global fit of the optical model to elastic scattering data of14.5 MeV protons for a large nuclear mass range. The cross sections arenormalized to the Rutherford cross sections (i.e. to 1 at 0 ), the analyzingpowers are 0 at 0 . The arrows indicate the systematic variation of charac-teristic diffraction maxima with the target mass.

DR

AFT

11.3. STRIPPING REACTIONS 105

Figure 11.5: Angular and energy dependence of the cross section of elasticproton scattering from 90Zr calculated with standard Greenlees-Becchettiparameters of the optical model. The interference structure of the angulardistributions may be interpreted as “resonant” (single-particle) structures ofthe excitation function with widths typical for fast (i.e. direct) processes.They are also analogous to diffraction structures in classical optics.

Here only the simplest case of the stripping reaction will be discussed. Themany details of direct interaction processes are subjects of a large number ofbooks, see e.g. Refs. [AUS70, SAT83, GLE63, GLE83]. Standard computercodes such as DWUCKn (distorted wave code) and CHUCKn (coupled chan-nels distorted wave code) [KUNZ] are available. For induced nuclear fissionsee Section 15.

11.3 Stripping Reactions

Already a semi-classical ansatz provides a qualitative picture of the angulardistributions of stripping reactions. It explains the expected behavior withthe assumption of a rapid process that is localized at the nuclear surface andis non-equilibrated. The wave-number vector of the incoming deuterons is ~kd,those of the transferred nucleon and of the outgoing nucleons are ~kn and ~kp,respectively. They form a momentum diagram, from which the connectionbetween a preferred scattering angle θ and the transferred momentum andalso the angular momentum can be deduced:

pnR = hknR = hℓn (11.4)

For small θ

θ0 ≈ kn

kd=

ℓn

kdR(11.5)

Because of the quantization of ℓ there are discrete values of θ increasing withℓ. This qualitative picture is not changed when calculating the angular distri-

DR

AFT


butions quantum-mechanically. As an example for the reaction 52Cr(d,p)53Crthe angles of the stripping maximum in calculated in different ways are

θDWBA θPWBA θs.c.

ℓ = 0 00 00 00

ℓ = 1 180 130 130

ℓ = 2 340 190 260

ℓ = 3 490 300 390

ℓ = 4 640 400 520

(11.6)

The measured angular distributions of the cross sections show – in addi-tion to diffraction structures – marked stripping maxima, which often al-low the determination of the angular momentum of the transferred nucleon.Historically this feature was (and still is) important for the assignment of thefinal-nuclear states of a reaction to orbitals in the one-particle shell model.The energy relations in stripping reactions are such that the transferred nu-cleon near magic shells is preferentially inserted into low-lying shell-modelstates. Because of the spin-orbit splitting the complete assignment requiresthat also the total angular momentum j of the transferred nucleons is known.A good method is the measurement of the analyzing power of the strippingreaction, i.e. the use of polarized projectiles. In many cases the distinctionbetween the two possibilities for j can be made just from the sign of theanalyzing power alone. For an intuitive description of this fact there existsagain a simple semi-classical model (Newns). It is assumed that the inter-action happens at the nuclear surface and that we have a relatively strongabsorption in the nuclear matter. Thus the front side of the nucleaus di-rected towards the projectile contributes more strongly to the reaction thanthe backside of the target nucleus. In the front part the orbital angular mo-mentum vector points upward perpendicularly to the reaction plane whereasin the back half the orbital angular momentum points down. If the incidentdeuteron is polarized up or down perpendicularly to the scattering plane

l = 0 l = 1 l = 2 l = 3 l = 4

Θ (deg)

0.1

1.0

10

0.0160 120 60 120 60 120 60 120 60 120

σ d /d

(m

b/sr

)Ω

Figure 11.6: Characteristic and systematic features of the stripping maxi-mum as function of the transferred orbital angular momentum ℓ. The arrowsindicate the increase of the reaction angle of the stripping peak with ℓ.

DR

AFT

11.3. STRIPPING REACTIONS 107

j = 3/2

j = 1/2l = 1

E = 7.0 MeVd

4140Ca(d,p) Ca

20−0.6

0 806040

−0.4

−0.2

0

0.2

0.4

0.6

Ay

Θc.m.

p

p3/2

1/2

(deg)

Figure 11.7: Sensitivity (sign!) of the analyzing power of the strippingreaction to the total angular momentum j of the transferred nucleon.

DR

AFT


– under the assumption of the existence of a spin-orbit force – the trans-ferred nucleon in the scattering to the left ends preferentially in a state withj = ℓ+1/2 for the up case, in the down case with j = ℓ−1/2. The measuredanalyzing power

Ay =1Pd

Nup −Ndown

Nup +Ndown, (11.7)

will show opposite signs for the two cases. This behavior has been confirmedfor many examples not only for stripping reactions. If one wants to know thedegree, to which the transition considered is a single-particle transition thespectroscopic factor has to be determined. For that – at least approximately– quantitative theories are required (DWBA, CC-DWBA).

Because the single-particle strength is often strongly fractionated by theresidual interaction, i.e. spread out over many states in a range of energiesspectroscopic investigations on very many final nuclear states are necessary.Often these states are close together and high detector resolution to get acomplete picture is necessary. Especially useful tools for this purpose aremagnetic spectrographs with high resolution at tandem Van-de-Graaff accel-erators, also with polarized particle beams.

11.4 The Born Approximation

Here one uses the first Born approximation, i.e. the first term of the Bornseries. Starting from Fermi’s Golden Rule of perturbation theory, whichpredicts for the differential cross section

dσ

dΩ=

(2Ib + 1)(2IB + 1)2π2h4 µiµf

kf

ki|Tif |2 . (11.8)

one has to make assumptions about the transition matrix element.In the Plane Wave Born Approximation PWBA (also: Butler theory) for

the incoming and outgoing waves plane waves are used. Since the radial wavefunctions are Bessel functions one finds a simple diffraction pattern for thecross section

dσ

dΩ∝ [jℓ(kR)]2 (11.9)

For illustration Fig. 11.8 shows the few lowest-order spherical Bessel func-tions squared. The angle dependence of the stripping maximum is containedonly in the momentum relation k2 = k2

in + k2out − 2kinkout cos θ. Only in a

few simple cases the angular distributionen near the maximum are satisfac-torily described by PWBA. It also makes no statements about polarizationobservables and contains no information about nuclear structure.

Better results at least for forward angles are obtained with the DistortedWave Born Approximation DWBA. It was formulated with a number of ad-ditional and far-reaching assumptions:

• In the entrance and exit channels distorted waves are used, i.e. the wavefunctions are the solutions obtained from fits of the optical model (OM)to the elastic-scattering data in each pertaining channel at the proper

DR

AFT

11.4. THE BORN APPROXIMATION 109

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12x

Figure 11.8: The behavior of the squares of the lowest-order spherical Besselfunctions jℓ(kr) as functions of x = kr.

channel energy. E.g. for the description of the reaction A(d,p)B the OMwave functions from the fit to the data of the scattering A(d,d)A as wellas of B(p,p)B are needed. Thus – still in the first Born approximation– the diffraction and absorption of the incoming and outgoing waves inthe nuclear (and eventually the Coulomb) field, as well as the effect ofthe LS potential (see also Fig. ??) are taken into account.

• The nuclear initial and final states are shell-model states.

• The finite range of the nuclear forces is taken care of by a finite-rangeor even zero-range approximation.

• The T matrix is expanded into partial waves belonging to fixed angular-momentum transfer.

• The transfer matrix is factorized into a nuclear-structure dependentand into a kinematical part.

Thus the cross section reads:

dσ

dΩ=

µaµbπh4 (

mB

mA)4

2JB + 1(2JA + 1)(2sa + 1)

1kakb

∑

ℓsj

[

|Aℓsj|2∑

m

∣∣∣βℓmsj

∣∣∣

2]

(11.10)

The experimental cross section is a product of a fit parameter, the spectro-scopic factor Sℓj, and a theoretical cross section calculated in the frameworkof the DWBA with the assumption of single-particle states:

(

dσ

dΩ

)ℓj

exp

= Sℓj

(

dσ

dΩ

)ℓj

DWBA

(11.11)

In a stripping process the spectroscopic factor is the square of the am-plitude of a fragment of a single-particle state of the final nucleus. Because

DR

AFT


of this fractionization (which in reality is caused by the residual interactionof the many other nucleons) into many states with equal quantum numbersthe strengths of all these states have to be summed up. If a complete col-lection from all these states is possible one obtains the total strength, whichcan also be calculated because the number of nucleons N in a subshell isknown. Therefore sum rules can be applied, e.g. for single-particle stripping∑Sℓj = (2J + 1). Mathematically the spectroscopic factor is the overlap

integral between the anti-symmetrized k-particle final-nuclear state ΨA(i),into which the nucleon is inserted, and the single-particle configuration ofthe anti-symmetrized (k-1)-particles target, the nuclear ground state (core),and the single-particle wave function of the transferred k-th particle Ψ(j). Itthus gives the probability, with which a certain state is present in this con-figuration. When averaging over the strength distribution of all states thatare fractions of one single-particle state (e.g. while assuming a Breit-Wignerdistribution function) the position of the average energy provides the energyof the single-particle state, whereas the width of the distribution is a mea-sure of its lifetime, the (spreading width) Γ↓. It measures the decay of thesingle-particle state into the real nuclear states, split and spread out by theresidual interaction, and thus its strength.

DR

AFT

Bibliography

[AUS70] N. Austern, Direct Nuclear Reaction Theory, Wiley (1970)

[BEC69] F.D. Becchetti jr., G.W. Greenlees, Phys. Rev. 182, 1190 (1969)

[BRA50] A. Bratenahl, S. Fernbach, R.H. Hildebrand, C.E. Leith,B.J. Moyer, Phys. Rev. 77, 597 (1950)

[BUR51] J.W. Burkig, B.T. Wright, Phys. Rev. 82, 451 (1951)

[FER49] S. Fernbach, R. Serber, T.B. Taylor, Phys. Rev. 75, 1352 (1949)

[GLE63] N.K. Glendenning, Nuclear Stripping Reactions,Ann. Rev. Nucl. Sci. 13, 191 (1963)

[GLE83] N.K. Glendenning, Direct Nuclear Reactions, Academic Press,New York (1983)

[HOD63] P.E. Hodgson, The Optical Model of Elastic Scattering, Oxford,1963

[HOD67] P.E. Hodgson, The Optical Model of the Nucleon-NucleusInteraction, Ann. Rev. Nucl. Sci. 17, 1 (1967)

[KUNZ] P.D. Kunz, available from University of Colorado

[LEL52] R.E. Le Levier, D.D. Saxon, Phys. Rev. 87, 40 (1952)

[MAR70] P. Marmier, E. Sheldon, Physics of Nuclei and Particles, Vol. II,1087 ff., Academic Press, New York and London (1970)

[SAT83] G.R. Satchler, Direct Nuclear Reactions, Oxford, 1983

[YUL68] T.J. Yule, W. Haeberli, Nucl. Phys. A117, 1 (1968)

DR

AFT

112 BIBLIOGRAPHY

DR

AFT

Chapter 12

Resonances and CompoundReactions

12.1 Generalities

Resonances are a very general phenomenon in nature and therefore in all ofphysics. In classical physics they appear when a system capable of oscilla-tions is excited with one or more of its eigenfrequencies, which – dependingon the degree of damping – may lead to large oscillation amplitudes of thesystem. Nuclei are no exception. When tuning the system (changing the ex-citing frequency) these amplitudes pass through a resonance curve of Lorentzform. In particle physics most of the many known “particles” actually appearas resonances, i.e. as quantum states in the continuum, which decay withcharacteristic widths (or equivalently: lifetimes).

Resonances can be discussed in the energy picture where as function of en-ergy excursions of Lorentz form (Breit-Wigner form) with a width Γ appear,but also in the complementary time picture where they appear as quantumstates in the continuum, i.e. as states, which decay with finite lifetime τ .Between them there is the relation

Γ = h/τ (12.1)

.In nuclear physics resonances appear in the continuum (i.e., in scattering

situations, at positive total energy) when the projectile energy in the c.m.system plus the Q value of the reaction just equals the excitation energy ofa nuclear state.

The excitation functions of observables such as the cross section showcharacteristic excursions from the smooth background when varying the in-cident energy. The background may be due to a direct-reaction contributionfrom Coulomb or shape-elastic scattering or – in a region of high level den-sity – may be the energy-averaged cross section of unresolved overlappingcompound resonances; in this case resonant excursions would be due to door-way mechanisms. Likewise the scattering phases and scattering amplitudeschange in characteristic ways over comparatively small energy intervals. Nu-clei may be excited into collective modes such as rotations and/or vibrations

DR

AFT

114 CHAPTER 12. RESONANCES AND COMPOUND REACTIONS

of a part of the nucleons. At still higher energies new phenomena with highcross sections in charged-particle, neutron, γ, and π induced reactions appearinvolving up to all nucleons of a nucleus, the Giant Resonances. Fig. 12.1shows schematically the phenomena in different energy regions.

12.2 Theoretical Shape of the Cross Sections

A model assumption for resonances is – in contrast to direct processes – thatthe system goes via an intermediate state from entrance into the exit channel.For this case perturbation theory gives the following form of the transitionmatrix element

〈Ψout|Hint|Ψin〉 =constE − ER

. (12.2)

ER is the energy of the nuclear eigenstate. However, since it is a state in thecontinuum it is not a stationary but one, which decays in time. Such statesare best described by giving it a complex eigen-energy:

ER = ER + iΓ/2. (12.3)

The interpretation of the imaginary part is: the time development of a statehas the form eiEt/h, on the other hand the state decays with a lifetime τ ,whence

1/τ = Im(E) = Γ/2. (12.4)

The resonance amplitude thus has the form:

g(E) =F (E)

E − ER + iΓ/2. (12.5)

Excitation ofsingle nucleons

+collective motionof few nucleons

0 5 15 20 25 30E (MeV)x

Gro

und

Sta

teC

ross

−se

ctio

n

Excitation ofmany (all)nucleons

10

Sin

gle

Ove

rlapp

ing

Resonances Giant ResonanceBound States

Figure 12.1: The excitation of single resonances, overlapping resonances(with and without Ericson fluctuations) and giant resonances as functions ofthe energy in the continuum region above the bound-state energy.

DR

AFT

12.2. THEORETICAL SHAPE OF THE CROSS SECTIONS 115

The meaning of F(E) has to be determined. In the sense of Bohr’s indepen-dence hypothesis formation and decay of a resonance are independent (i.e.decoupled). Therefore, one writes the amplitude as the product of the prob-ability amplitude for its formation and its probability of decaying into theconsidered exit channel. In general, for one formation channel (the entrancechannel c) there will be several exit channels c’.

The width Γ of the Breit-Wigner function is inversely proportional to theformation probability P and is the integral over the cross section in the energyrange of the resonance:

P =∫σaAvaAV

· V p2aAdpaA

2π2h3 =∫σaAk

2indE

2π2h≈ k2

inF (ER)2π2h

∫dE

(E − ER)2 + Γ2/4

=k2

inF (ER)πhΓ

(12.6)

In equilibrium this is equal to the probability that the resonance re-decaysinto the entrance channel (purely elastic case). A measure for this is thepartial width ΓaA formed similarly as Γ, thus:

ΓaA/h =k2

inF (ER)πhΓ

, (12.7)

:F (ER) =

π

k2aA

ΓaAΓ (12.8)

By definition Γ is the sum of all partial widths over the open channels. Thus,the branching ratio for the decay into one definite channel bB ≡ c′ is equal toΓc′/Γ and the Breit-Wigner cross section for the formation of the resonancevia channel c and the decay via channel c’ is

σ(E) =π

k2in

· ΓcΓc′

(E − ER)2 + Γ2/4(12.9)

This derivation is simplified and must be carried out – when there isinterference with a direct background contribution and for the description ofa differential cross section via a partial-wave expansion near a resonance –with complex scattering amplitudes. For elastic s-wave scattering it resultsin a resonant scattering amplitude of the form:

Ares =iΓaA

(E − ER) + iΓ/2(12.10)

When a direct background is present, then, besides the pure resonanceterm and the pure direct (smooth) term, a typical interference term appears,which may be cosntructive or destructive. For σ we have then:

σtot = |Ares + Apot|2 = σres + σpot + 2Re(AresA∗pot) (12.11)

where Apot is the amplitude of the weakly energy-variable potential scatter-ing.

DR

AFT


12.3 Derivation of the Partial-Width Ampli-

tude for Nuclei (s Waves only)

The connection between the resonant scattering wave function and the wavefunction of the eigenstate of the nucleus is made by the R-matrix theory.Their basic features (for more details see [LAN58]) are approximately:

• The two wave functions and their first derivatives are matched contin-uously at nuclear radius (edge of the potential or similarly).

• The condition for a resonance is equivalent with the wave-function am-plitude in the nuclear interior taking on a maximum value. This hap-pens exactly if the matching at the nuclear radius occurs with a wavefunction with gradient zero (horizontal tangent).

Fig. 12.2 illustrates the conditions for resonance. The two conditions maybe summarized such that both logarithmic derivatives L (L0 for pure s waves)at the nuclear radius are exctly zero. With the form of the wave function inthe external region

u0(r) = e−ikr − η0eikr, r > a (12.12)

and the wave numbers in the external k in the nuclear interior κ we obtain

L0(E) =

(

a

u0

du0

dr

)

r=a

, (12.13)

ER

Nuclear Potential Well

Bound States

Scattering States

Exc

itatio

n F

unct

ion

Figure 12.2: Boundary conditions at the nuclear (potential) surface for theappearance of a resonance in the excitation function.

DR

AFT

12.4. FIRST EVIDENCE OF RESONANT NUCLEAR REACTIONS 117

which leads to the scattering function η0 as function of L0:

η0 =L0 + ika

L0 − ikae−2ika. (12.14)

Inserting this scattering function into the known expressions for elastic scat-tering and absorption (and with L0 = Re(L0) + iIm(L0)) the result is

σel =π

k2in

|1 − η0|2 =π

k2in

∣∣∣∣∣

[

e2ika − 1]

− 2ikaReL0 + i(ImL0 − ka)

∣∣∣∣∣

2

(12.15)

and

σabs =π

k2in

(

1 − |η0|2)

=π

k2in

[

−4kinαImL0

(ReL0)2 + (ImL0 − ka)2.

]

. (12.16)

By expanding ReL0 in a Taylor series and terminating it after the first term,by comparison – besides obtaining the resonance-scattering amplitude (withApot ∝ e2ika − 1) – one obtains the results:

σel =π

k2in

∣∣∣∣∣

(

e2ika − 1)

+iΓaA

(E −ER) + iΓ/2

∣∣∣∣∣

2

(12.17)

σabs =π

k2in

ΓaA(Γ − ΓaA)(E −ER)2 + Γ2/4

(12.18)

One sees that in agreement with our definition of absorption this encompassesall exit channels except the elastic channel.

12.4 First Evidence of Resonant Nuclear Re-

actions

In the years around 1934 Enrico Fermi and his collaborators (Amaldi, Pon-tecorvo et al. [FER34, AMA35a, AMA35b, FER36]) performed experimentsat Rome with neutrons, especially slow neutrons that were produced by slow-ing down fast neutrons in hydrogenous materials. The fast neutrons wereproduced in α, n reactions on nuclei such as 9Be with α’s from Radium orRadon sources. Their very systematic studies of the elastic scattering, butalso capture reactions revealed very different cross sections for different ele-ments. Very high absorption was observed for B and Cd, but only for veryslow neutrons, also on many other nuclei without any systematics betweenneighboring nuclei.

Leo Szilard [SZI35] concluded aleady that there must exist small energyregions with these high cross sections. Niels Bohr [BOH36] used these ob-servations to formulate his model of formation of a compound nucleus CNexplaining the properties of the high cross sections as resonances in the sys-tem of the target nuclei plus one neutron being captured to form a highlyexcited, rather long-lived nuclear state (lifetimes up to 106 times longer thanthe traversal time).

DR

AFT


E. Wigner [WIG55] gives a good account on these early developments. Hisstatement, cited from that text “Experimental work constituted, in my opin-ion, the most important step in the development” clearly points at Fermi’sstudies as key experiments leading to the CN model and theoretical devlop-ments hereafter.

DR

AFT

Bibliography

[AMA35a] E. Amaldi, E. Fermi, Ric. Sci. A6, 544 (1936)

[AMA35b] E. Amaldi, O. d’Agostino, E. Fermi, B. Pontecorvo, F. Rasetti,E. Segrè, Proc. Roy. Soc. A149, 522 (1935)

[BOH36] N. Bohr, Nature 137, 344 (1936)

[FER34] E. Fermi, E. Amaldi, O. d’Agostino, F. Rasetti, E. Segrè,Proc. Roy. Soc. A146, 483 (1934)

[FER36] E. Fermi et al., Ric. Sci. 1, 310 (1936)

[LAN58] A.M. Lane, R.G. Thomas, Rev. Mod. Phys. 30, 145 (1958)

[SZI35] L. Szilard, Nature 136, 950 (1935)

[WIG55] E. Wigner, Am. J. Phys. 23, no. 6 (1955)

DR

AFT

120 BIBLIOGRAPHY

DR

AFT

Chapter 13

Nuclear Reactions and Tests ofConservation Laws

Depending on the strength of the interaction a number of conservation lawsand their violation have been formulated. The following table lists all possi-bilities and relates them to various operators of the nucleon-nucleon interac-tion.

Table 13.1: Conservation Quantities and their Violation, and FundamentalInteractions: Conservation: +, Violation: -

Conservation Quantity STRONG EL-MAG WEAK

or Symmetry INTERACTION

Mass m/Energy E

Momentum p + + +

Angular Momentum L, S

Charge Q + + +

Isospin T + - -

Strangeness S

Charm C + + -

Beauty B, Topness T

Parity P + + -

Charge Conjugation C + + -

Baryon Number B + + +

Lepton Number(s) + +

Hypercharge Y + + -

Time Reversal T + + -

Charge Parity CP + + -

CPT + + +

DR

AFT

122 CHAPTER 13. NUCLEAR REACTIONS AND TESTS OF CONSERVATION LAWS

It is evident that the number of violations increases with increasing weak-ness of the interaction. It is also evident that each conservation law has tobe investigated separately for each interaction. For the weak interaction thecomplete violation of the parity symmetry corresponding to the invariance ofphysical processes against the mirror operation about the origin was the mostconspicuous and unexpected phenomenon (the “Wu experiment”) and trig-gered the host of further investigations of all possible conservation laws. Inaddition to static properties of particles, nuclei, or atoms, nuclear reactionsare a tool to study the effects of possible violations.

13.1 First Tests of Parity Violation in Hadronic

Reactions

Which observables are sensitive to parity violation? The parity operationentails

P~r = ~−rPr = −rPθ = π − θ

Pφ = π + φ

Pt = t

P~p = −~pPL = L

PS = S.

Under parity conservation the physics should not change under the mirroroperation, conversely under parity violation. Thus any physical quantitycontaining odd orders of ~r such as the electric dipole moment of a particle~µe = 〈q · ~r〉 is sensitive to parity violation.

The earliest attempt at measuring parity violation in a nuclear reactionwas performed by Tanner [TAN57] by comparing the 19F(p,α0)16O reactionat θ = 0 with 19F(p,α1γ)16O yield across a resonance in 20Ne at Ep =340 keV.Underparityconservationtheyieldofthefirstreactionispredictedtobezero.Infact, itshowednosigno indication of a parity-violating amplitude was detected. Note: The weakcontribution to the hadronic interaction is expected to be on the 10−7 level,except when some enhancement mechanism would amplify this ratio (e.g. ininterfering neutron resonances or when the P-invariant amplitude is small asin cross-section minima).

Pseudoscalar observables (expectation values of products of a polar andan axial vector) are also sensitive, such as the correlation of the momentumvector of the electrons emitted from a 60Co source and the spin of the Conuclei aligned by a strong magnetic field in the famous Wu experiment.

In nuclear reactions the appearance of a longitudinal spin polarization ,i.e. a polarization of the exit-channel particles along the direction of emission(usually designated as z’ axis) is only possible under parity violation. Polar-ization is defined as the expectation value of a spin operator. Similarly the

DR

AFT

13.1. FIRST TESTS OF PARITY VIOLATION IN HADRONIC REACTIONS 123

corresponding longitudinal analyzing power Az, the response of a nuclear re-action to the longitudinal polarization of an incident beam, has to disappearunder parity conservation.

At Los Alamos, using the 15 MeV proton beam of an FN tandem Van-de-Graaff accelerator with the LANL lambshift polarized-ion source [MCK68]the first such experiment was undertaken to look for parity violations inproton-proton elastic scattering (later also in ~p-4He elastic scattering). As-suming that in the strong and Coulomb interactions parity is conserved onlythe weak-interaction contribution could be responsible for a violation, andvery small effects, i.e. values of Az could be expected. This turned out tobe the case. Normally the transverse polarization of the incident beam in-troduces an azimuthal dependence into the cross section that normally is φindependent. In the case of a longitudinal polarization there cannot be a φdependence. Therefore, a detector design that integrates over all azimuthalevents could be used. The total cross sections of events with the polariza-tions of the incident protons alternating in the forward and the backwarddirections served as a measure of the longitudinal analyzing power. The twoimages of Fig. 13.2 show the (quite simple) scheme of the experiment anddetails of the detector apparatus. In order to obtain the necessary statisticsnot single scattered protons were counted but their current produced in a4π scintillator-photomultiplier arrangement was integrated and normalizedto the incident proton beam current. The final result was Az = (−1.7 ± 0.8),a clear signal of parity violation.

Similar, but more refined experiments of scattering of longitudinally po-larized protons from unpolarized protons followed at different laboratories,listed in Table 13.2.

Th smallness of the effect required not only good statistics, i.e. long run-ning times under stable experimental conditions, but also careful evaluationof systematic errors (such as from small transverse polarization components).

The DDH model is based on describing the NN interaction by mesonexchange where on vertex is strong, the other weak and parity-violating.

Table 13.2: Results of the ~pp scattering experiments

Ep,lab (MeV) Az×107 Laboratory Year Reference

13.6, -(1.5± 0.5) Bonn 990 [EVE91]

15.0 -(1.7±0.8) LANL 1978 [NAG79]

45 -(3.2± 1.1) SIN/PSI 1980 [BAL80]

45 -(2.21± 0.84) SIN/PSI 1984 [BAL84]

45 -(1.5± 0.22) SIN/PSI 1987 [KIS87]

221 +(0.84±0.29 ± 1.7) TRIUMF 2001 [BER01]

800 +(2.4±1.1) LANL 2001 [YUA86]

5300 +(26.5± 6 ± 3.6) ANL 1986 [LOC84]

DR

AFT


Figure 13.2: Setup of the first hadronic parity-violation experiment at LosAlamos [NAG79].

DR

AFT

13.2. FIRST TIME-REVERSAL TESTS 125

PS

I

TR

IUM

F

BO

NN

LAN

L

DDH

LAN

L

Figure 13.3: Measured values of the longitudinal analyzing power in ppscattering at 13.6 MeV (Bonn), 15 MeV (Los Alamos), 45 MeV (SIN/PSI),221 MeV (TRIUMF), and 800 MeV (LANL), in comparison to predictions ofthe standard model (DDH: Desplanques, Donoghue, and Holstein [DES80]).

The weak interaction hse been parametrized by six weak meson-nucleon cou-pling constants describing π, ρ, and ω exchanges. Different experiments,among them the one described here, would in principle allow the determi-nation of each of them separately. So far the two constants hppρ and hppωhave been constrained by the pp experiments. For details see the latestreview of Ref. [HAX13] and references therein. In Ref. [HOL09], besidesa history of hadronic parity violation, new approaches in the frameworkof model-independent effective-field theories replacing the “classical” DDHcomparisons are discussed.

13.2 First Time-Reversal Tests

The CPT theorem (stating the invariance of physical processes and systemsunder the combined operations of charge conjugation C, the parity operationP, and time reversal T) is one of the most fundamentally accepted theoremsin physics. Its validity must and can, however, be investigated directly as wellas by checking the three operations C, P, and T, separately. After a completeviolation of P and C symmetries had been shown for the weak interaction,the validity of invariance under the combined CP operation was assumed,but only until a weak violation of CP was discovered for the K0 − K0 andlater also for other systems. This immediately aroused interest in checkingindependently the time-reversal invariance T.

The time reversal is somewhat special because – unlike for the other sym-metries – there is no conserved quantity (quantum number) connected with

DR

AFT


it (for more detailed discussions cf. Refs. [FRA86, PGS14]). The reason is thespecial nature of the operator of time reversal: it is anti-linear and unitary= anti-unitary and acts on operators as

Tt = −tT~r = ~r

T~p = −~pTL = −LTS = −S.

which forbids certain static observables such as the electric dipole moment.In nuclear reactions time reversal is usually interpreted as reversal of motion,relating the input channel (aA) of a reaction with the exit channel (bB). Forelastic scattering both are identical. There are three main possibilities totest the time-reversal invariance in nuclear reactions:

• By performing the time-reversal operation directly on momentum andspin vectors of a nuclear reaction then – after an additional rotation –the physics of the reaction and its inverse should be the same. This isillustrated for the c.m. system of proton-proton scattering by Fig. 13.4.The quantities measured were the pp triple-scattering Wolfenstein pa-rameters) A and R for the two situations A and B in Fig. 13.4, usingwire spark chambers to register the protons. Under time-reversal in-variance the relation

tan θ =A+R′

A′ − R(13.1)

holds with θ the lab. scattering angle, A, A’ the transverse and lon-gitudinal final polarizations, respectively, for an 100% longitudinallypolarized incident beam, and R, R’ the same parameters for a 100%

Figure 13.4: Situation of incident and exit momenta and polarization vectorsbefore and after the time-reversal operation of reversing spins and momentain A leading to situation B in the center-of-mass system. The angles χi andχf are arbitrary. The situations must be Lorentz-transformed to describethem in the lab. system. From Ref. [HAN67].

DR

AFT

13.2. FIRST TIME-REVERSAL TESTS 127

transversely polarized initial beam [SPR61]. For an initial polarizationof unity in situations A and B we obtain final polarizations PA and PB

PA = (R sinχi + A cosχi) sin(χf + θ)

+ (R′ sinχi + A′ cosχi) cos(χf + θ),

PB = (−R sinχf + A cosχf) sin(θ − χi)

+ (−R sinχf + A′ cosχf ) cos(θ − χi).

The difference between PA and PB

PA − PB = [(A+R′) cos θ − (A′ −R) sin θ] sin(χi + χf )

vanishes if Eq. 13.1 holds. This comparison thus tests Eq. 13.1. Theexperiment was performed at 430 MeV at the cyclotron of the Universityof Chicago. The difference between the two polarizations that shouldbe equal if T is valid was

∆P = 0.0006 ± 0.0028.

Thus, no violation of time-reversal invariance was found and only upperlimits for its violation could be established.

• The (vector) analyzing power of a forward reaction is identical with the(vector) polarization of the inverse reaction, produced with unpolarizedincident projectiles

(Ay)→ = (Py)← (13.2)

with the caveat that the target spin must not be 0.

The earliest (key) experiments on possible time-reversal violations werethe more difficult polarization/analyzing-power difference measurements.Oxley at al. [OXL53] had measured P − A = 0.01 ± 0.06 in pp scat-tering but without explicit reference to time reversal. The first suchexperiments were undertaken in two labs simultaneously by Hillman etal. in 1957/58 [HIL58] and by Abashian et al. [ABA58]. One has tokeep in mind that at that time no polarized ion sources were availableand the polarization of proton beams and its measurement had to beprovided by double or triple scattering, causing strong restrictions asto intensity and choice of angles and energies. Proton scattering onseveral nuclear targets was performed. The combined result of bothexperiments was

P − A = −0.014 ± 0.014

• Violations of the principle of Detailed Balance which states that – iftime-reversal invariance holds – up to phase-space factors (spin factorsand momentum factors which cancel for elastic scattering) the crosssection of a reaction is equal to the cross section of its inverse (takenat the same cm. energy and scattering angle):

(dσdΩ

)

→(dσdΩ

)

←

=(2sa + 1)(2sA + 1)(2sb + 1)(2sB + 1)

· kin

kout

. (13.3)

DR

AFT


It is a question of fundamental importance which reaction mechanismshould be chosen to obtain maximum sensitivity of the (weak) time-reversal violating reaction amplitude in the presence of a strong non-violating amplitude. The conclusion was that a complicated reactionwith many channels such as compound processes would be better thansimpler direct processes [HEN59, MOL68]. A key experiment of thiskind was the measurement of Bodanki et al. in 1959 [BOD59], see alsoRefs. [BOD66, WEI68]. They chose the reaction

12C(α, d)14N − 13.574 MeV

with Eα,lab = 41.7 MeV from the Washington cyclotron and its inverse.The results of both cross sections is shown in Fig. 13.5. It is obviousthat the two sets of data agree qualitatively well. It is not quite easy toquantify the degree of (non-)agreement, and the authors only estimatedthe non-invariant contribution to <3%.

Later the same group investigated the reactions 24Mg + d 25Mg +p with <0.4% non-invariance [BOD66, WEI68].

Appreciably higher accuracy was obtained in experiments on the 24Mg+ α 27Al + p reactions [VWI68, BLA83]. In this experiment es-pecially the deep minima of Ericson fluctuations were used for com-parison. The authors give an upper limit on T-reversal violation of≈ 2·10−3. For a discussion of the determination of the “significancelimit” of such “null” experiments see Refs. [KLE74, HAR86, HAR90].

• It should be mentioned here that in the future storage-ring acceleratorexperiments are planned to measure the electric dipole moment of theproton and the deuteron. They would make use of spin-polarized beamsand the very weak interaction of the dipole moment with electric fields

Figure 13.5: Plot of the angular distributions of the reactions 12C(α,d)14Nand 14N(d,α)12C.

DR

AFT

13.3. NN INTERACTION AND ISOSPIN 129

summing up via the millions of revolutions of the beam in the ring.One such project (another at BNL, 2011) is proposed by the JEDIcollaboration at COSY-Jülich [ENG12] with a planned upper limit ofµe = 1·10−28 e·cm which for the neutron stands at 1·10−28 e·cm. Atthat level new physics might appear.

13.3 NN Interaction and Isospin

13.3.1 Generalities

Shortly after the discovery of the neutron 1932 and knowledge of its proper-ties such as mass and spin Heisenberg developed the concept of isospin (orisobaric spin) as a new symmetry. The most fundamental question connectedwith this concept is whether isospin is a conserved quantity and under whichinteractions it might be (eventually weakly) violated. Like in many otherinstances the study of this quantity in heavier nuclei (or their reactions) maybe obscured by nuclear structure. Therefore, naturally first the reactions be-tween pure nucleons were investigated, at very low energies. The scatteringlength is a useful concept to study isospin and its possible violation in thenucleon-nucleon or at most in nucleon-few-nucleon interactions.

Only a brief survey of the isospin operator (T or T ) properties will be givenhere (for a more detailed introduction into the field see e.g. the resourceslisted in Ref. [PGS14]).

• Formally it behaves just as the spin operator S: Commutation rules,isospin raising or lowering operators, Expectation values of T 2 and T3

(index 3 instead of z, because T has no spatial representation), 2T+1substates forming isospin multiplets etc.

•√

T (T + 1) is an invariant scalar under isospin conservation, i.e. in-variant under rotations in isospin space, i.e. [Hstrong,T] = 0 is the def-inition of isospin invariance, also c harge independence CI. The valueT is determined by the number of substates. T3 measures the electriccharge of the members of the multiplet with Q = T3 + Y/2 with Y thehypercharge that for nuclei is the mass number A. Thus, the Coulombinteraction breaks isospin naturally. Non-trivial is, however, a possi-ble breaking within the strong interaction. A number of examples innuclear physics in general exhibit approximately fulfilled featured re-quired from isospin conservation but in detail show small indication ofsome breaking too (at the 1% level):

– Energies of members of isospin multiplets, corrected for the simpleCoulomb force, differ slightly (“Nolen-Schiffer anomaly”)

– Isospin-forbidden nuclear reaction channels have weaker, but notzero transition strengths, compared to allowed channels (12C(p,p)12Cover a T=1/2 and a T=3/2 (forbidden) resonance in 13N, the lat-ter being also much narrower).

DR

AFT


– For the reaction d + d → 4He + π0 that was believed to be trulyisospin-forbidden, only recently a very small cross section has beenmeasured [STE03, MIL06].

– There are more examples.

Henley and Miller [HEN79] have classified isospin breaking in four classesdepending on the form of the isospin breaking potential in the two-nucleon(NN) systems. It is important to distinguish between charge independenceCI (breaking.) and charge symmetry CS (breaking). The connection be-tween CIB, CSB, and isospin invariance is: The charge-symmetry (CS) op-eration consists of a 180 rotation about the 2-axis PCS = exp(iπT2) with[Hstrong, PCS] = 0 when CS holds, and nn and pp observables should be equal.

13.3.2 The Scattering Length

The scattering length a was introduced in 1974 by Fermi and Marshall[FER47]. It is related to the integrated cross section extrapolated to E → 0which can be described by just one parameter. An extension of the theory,the effective range theory, improves the description with just one additionalparameter reff . Both are essentially S-wave quantities. It is obvious that withone parameter it is impossible to describe details of any nuclear potential,but only an integral quantity independent of the specific interaction. Thedefinition of a reads

a = limk→0

tan δ0(k)k

where hk = p the beam momentum and δ0 the S-wave scattering phase shiftdescribing the reaction. For S waves the extrapolated cross section is

limk→0

σ = 4πa2.

In order to take into account the energy variation of the cross sections nearE = 0 an expansion of σ as function of the scattering length a was definedwith a second parameter reff , the effective range where σ = 4π[a(k)]2 with

1a(k)

=1a

− k2reff .

The NN system is fundamental, being free of nuclear structure effects andnevertheless basis of all efforts to describe more complex nuclei, starting withthe three-nucleon system in which already three-body forces come into play.Much progress has been achieved in describing such and higher-A systemswith the precise NN interaction as input, via Faddeev or EFT techniques.

The NN system has one bound state, the deuteron (an np, T=0, 3S1 +3

D1 mixture), the np spin singlet, T = 1 state 1S0 as well as all other ppand nn states are unbound. However, from the point of view of isospinconservation, especially from charge independence CI observables such as npand nn scattering lengths should be equal. The early low-energy experimentsin the NN systems have been instrumental in determining the salient featuresof the NN interaction especially in exploring the isospin properties.

DR

AFT


The np Scattering Length

For anp np total cross sections had to be determined at th lowest possibleenergies. This was not hampered by a Coulomb barrier, but on the other hansspecial methods to work with neutrons of well defined energies of sufficientintensities had to be developed. One method is to produce neutrons witha broad energy spectrum and selecting them in a reaction by their time offlight, the other to produce them with charged particle reactions such as9Be(d,n)10B in a cyclotron. From the large number of early experiments toelucidate the properties of the NN interaction a few key experiments will bepresented here.

• In the early experiments the total cross section of np scattering hadbeen measured and yielded large values but no detailed clues as to thenature of the interaction. A more realistic picture of the np interactionemerged after a strong spin dependence of the hadronic interaction wasobserved. The interaction differed so strongly between the triplet andthe singlet states that the first is slightly bound (the deuteron) the otherexists only as unbound scattering states. Thus, from the definitionand geometrical interpretation of the scattering length both shouldhave different signs. Already in 1937 large cross-section differences hadbeen noticed in neutron scattering from ortho and para hydrogen, seeRefs. [HAL37, BRI38], a hint of the strong spin dependence of the npforce, expressed by a spin-spin term in the nuclear potential.

In order to measure this spin dependence, and also directly to obtainthe signs of a, the coherent n-H2 scattering was investigated on ortho(SH2

= 1; parallel proton spins) and para hydrogen (SH2= 0; anti-

parallel spins) by Sutton et al. [SUT47]. The relation between totalortho and para np cross sections and the scattering lengths is

σ(H2, ortho) = 4π

(as + 3at

2

)2

+12

(at − as)2

(13.4)

= σ(H2, para) + 2π(at − as)2 (13.5)

The experiment consisted of a neutron beam from the reaction 9Be(d, n)10B(with deuterons from the 42” Los Alamos cyclotron), moderated byparaffin and energy-analyzed by time-of-flight like shown in 13.8 above.At 20 K H2 is to 99.9% para hydrogen (in thermal equilibrium afterabout 48 hours), whereas pure ortho-H2 cannot be prepared. However,H2 normally is 75% ortho and 25% para-H2 and a cross section differ-ence must be evaluated. The essential special part of the setup is thus atarget vessel containing H2, cooled by liquid hydrogen and sufficientlyinsulated, as shown in Fig. 13.6. A BF3 counter was used as neutrondetector.

The neutron scattering cross sections on ortho and para H2 are shownin Fig. 13.7 The careful evaluation of the data yielded the four observ-ables, shown in Table 13.3. The authors estimate the errors of theirresults to about 5%, much improved over earlier results, thus making

DR

AFT


Figure 13.6: Experimental setup with liquidH2 target vessel of Ref. [SUT47].

their measurements the first with reasonable errors allowing the veryimportant conclusions about the nature of the hadronic NN force.

Table 13.3: np cross sections and scattering lengths from the early measure-ments of Ref. [SUT47]. as and at are the singlet and triplet np scatteringlengths, σs and σt the corresponding partial cross sections that have to beadded incoherently with their statistical weights 1/4 and 3/4 to the totalcross section σ0.

From Ref. [SUT47]

as -23.4 fm

σs 68.8 b

at +5.2 fm

σt 3.6 b

σ0 19.8 b

• A typical experiment allowing np measurements at neutron energiesbetween 0.8 and 15 eV is that by Melkonian et al. [MEL49] usingapparatus developed by Rainwater et al. [RAI46] and used for neu-tron cross-section measuremets on a large number of different targets.Fig. 13.8 shows the experimental setup around the Columbia Univer-sity cyclotron. Fig. 13.9 shows the integrated cross section as func-tion of 1/E fitted with the assumption that the cross section follows

DR

AFT


Figure 13.7: The cross sections of thermal neutron scattering on ortho andpara H2 from Ref. [SUT47]. The data clearly favor a total cross section,extrapolated to E = 0, of 19.7 b, together with a range of the nuclear forceof 1.54 fm.

σ = σ0 + β/Ekin. σ0 is the pure pn cross section extrapolated to zeroenergy. The extrapolation E → 0 yielded

σ0 = 20.36(10)·10−24 cm2.

from which the absolute value of a scattering length of anp = 0.13 fmis deduced, but not the sign. The sign is either convention or must beobtained from interference terms in some observable. In any case, for asimple central repulsive potential, leading to unbound np states, a hasa negative sign. The following conclusions have been drawn:

– The np interaction is strongly spin dependent. It is repulsive inthe triplet, attractive in the singlet S-wave state.

– The “range” of the nuclear force is about 1.5 fm.

– The neutron is definitively a spin-1/2, not a spin-3/2 particle.

The pp Scattering Length

Charged-particle reactions require techniques different from neutron reac-tions, escially at low energies. The energy loss and finite range of protonsrequire a scattering chamber with good vacuum, and in case of gas targetsgas cells with very thin windows or windowless setups like filling the chamberwith the target gas and replacing an entrance foil with an efficient differentialpumping system (more modern versions have a target gas jet at much higherdensities). Towards lower energies the Coulomb (Rutherford) cross sectionbecomes very high and tends to obscure the nuclear part of the cross sectionwhich in turn is difficult to be disentangled. pp scattering is no exception,

DR

AFT


Figure 13.8: Setup for the production of very slow neutrons from the reac-tion 9Be(d,n)10B, after slowing down the neutrons in paraffine. The pulseddeuteron beam allows the selection of narrow energy intervals of the neutronsby µs gating and time-of-flight measurement along the 5.4 m flight path tothe hydrogen target. The figure was redrawn from [RAI46] for clarity.

and the identity of the entrance-channel particles leads to more complicatedformulas (see also the next chapter 14), not only for the Mott cross sectionitself, but also in the interference terms with the pure nuclear part. However,we have to consider only S-wave nuclear scattering that is described by onlyone single phase shift δ0. The relevant fromula to be used when extractingthe “nuclear” pp scattering length, is

(

dσ

dΩ

)

Coul

=

(

Z2e2

4E∞

)2

1sin4 θ

2

+1

cos4 θ2

−cos

(

ηS ln tan2 θ2

)

sin2 θ2

cos2 θ2

− 2ηs

sin δ0

cos

(

δ0 + ηS ln sin2 θ2

)

cos2 θ2

+cos

(

δ0 + ηS ln cos2 θ2

)

sin2 θ2

+4η2S

sin2 δ0

.

pp scattering experiments were performed as early as accelerators such ascyclotrons or Van-de-Graaff machines became available. Here two high-precision experiments from the 1950s will be described from which the ppscattering length was obtained [WOR52, KNE58]. Earlier pp experimentshave been surveyed in Ref. [BLA50].

DR

AFT


σ = 2034 + 0.69/E

1/E (E = NEUTRON ENERGY IN EV)

CR

OS

S S

EC

TIO

N P

ER

PR

OT

ON

(10

c

m )

19

20

21

22

23

0 0.5 1.0 1.5 2.0 2.5

−24

2

Figure 13.9: Result of the np cross-section measurement of Ref. [MEL49].The figure was redrawn for clarity.

The nn Scattering Length

The problem with nn cross sections is the lack of a neutron target. One hasto rely on target such as deuterons or 3He in which the effect of the neutroncan be separated from the known reaction part without it. This means in areaction such as n + d → p + n + n the kinematics of the exit channel canbe chosen such that e.g. the nn final-state interaction with a relative energyof Erel(nn)→ 0 can be studied. However, the binding of n and p in thedeuteron causes the neutron to be only quasifree requiring model-dependentassumptions (e.g. three-body-force effects) to extrapolate to the free neutron.Experiments of this type have been used to extract ann. Mostly kinematicallyincomplete measurements were done in which only one outgoing particle wasregistered and the nn final-state interaction would show up as a peak in itsspectrum. Kinematically complete experiments (two of the three outgoingparticles are measured in coincidence) have the advantage that competingreactions and background can be clearly identified and separated. Earlythree-particle breakup reactions had to rely on relatively crude theoreticalapproaches (e.g. the Watson-Migdal theory) to obtain a best fit to the dataand to extract low-energy parameters (for simple potential assumptions thedepth and width and scattering lengths). Two early (key) experiments arethose of Zeitnitz et al. [ZEI69] using the D(n,p)2n reaction yielding ann = -16.4 ± 2.1 fm and that of Baumgartner et al. [BAU66] using the 3H(d,3He)2nreaction yielding ann = -16.1 ± 1.0 fm among a number of other reactionsfrom different groups with results varying over a large range of values.

DR

AFT


Modern theories are based on Faddeev or EFT approaches and – especiallyfor the final-state interaction – give excellent fits and thus quite reliable valuesfor ann. Nevertheless two most recent breakup experiments using somewhatdifferent techniques [GON99, HUH00] yielded conflicting results of ann =-16.1 ± 0.4 fm and -18.7 ± 0.6 fm, respectively.

In view of uncertainties about reactions with three interacting hadronsanother type of experiment where the nn interaction is a pure two-bodyinteraction is believed to be more trustworthy. Two early examples arethe D(π−, γ)nn reaction [RYA64, RYA67] which, with only the γ registeredyielded ann between -15.1 and 19.1 fm, depending on the energy bin selectionin the spectra, and the D(π−, γnn) reaction [HAD65] in which all three exitparticles were measured and which yielded ann = -16.4 ± 1.3 fm. Fig. 13.10shows the principle of the experiment.

Figure 13.10: Experimental setup and kinematics of the D(π−, γnn) reaction[HAD65]

It should be mentioned that – in view of the inherent difficulties andunsatisfactory situation with the extraction of the nn scattering length twoexotic schemes for direct nn measurements have neen proposed: one is to usethe strong neutron flux in a pulsed high-flux reactor (YAGUAR), the otherto use the neutron flux of a nuclear explosion, a one-shot experiment withdestruction of the equipment.

Since these early key experiments the situation has not changed much.A summary on isospin breaking in the NN scattering lengths is shown in

DR

AFT


Table 13.4 with the presently more or less accepted values. Already theearly experiments as well as the new results show clearly isospin breaking,i.e. CSB and CIB.

13.3.3 Other Reaction Tests of Isospin Breaking

Other key experiments at higher energies and not based on scattering lengthwere performed at the TRIUMF and IUCF labs. in which the analyzingpower from a polarized neutron beam on an unpolarized proton target wascompared with the analyzing power of unpolarized neutrons an a polar-ized proton target. Under isospin invariance they should be the same butwere found to be different: ∆A ≡ An − Ap = (34.8 ± 6.2 ± 4.1)·10−4 , seeRefs. [ABE86, ABE89, ZHA98], a clear manifestation of class IV CSB. Thebeauty of the experiments consists in the use of a zero crossing of the dif-ference of analyzing power and polarization to enhance the sensitivity of theexperiment. Fig. 13.11 shows the scheme of the zero-crossing experiment,Figs. 13.12 the experimental setup at TRIUMF. With relatively good theo-retical predictions it was possible to partly disentangle several different con-tributions to the CSB, among them predominantly the ρ0 − ω mixing. Withthis improved knowledge on the microscopic sources of isospin breaking, alsoa handle for explaining the Nolen-Schiffer anomaly was obtained.

Table 13.4: Presently accepted scattering lengths and effective ranges

of the NN system

NN state aNN (fm) reff (fm)

pp 1S0 -17.1 ± 0.2 2.85 ± 0.04

nn 1S0 -18.8 ± 0.3 2.75 ± 0.11

np 1S0 -23.715 ± 0.015 2.75 ± 0.05

np 3S1 +5.423± 0.005 2.75 ± 0.05

DR

AFT


Figure 13.11: Scheme of the zero-crossing method of the analyzing power[ZHA98].

Figure 13.12: General and detailed experimental setup of the CSB experi-ment at TRIUMF [ZHA98].

DR

AFT

Bibliography

[ABA58] A. Abashian, E.M. Hafner, Phys. Rev. Lett.1, 255 (1958)

[ABE86] R. Abegg et al., Phys. Rev. Lett. 56, 2571 (1986)

[ABE89] R. Abegg et al., Phys. Rev. D 39, 2464 (1989)

[BAL80] R. Balzer, R. Henneck, Ch. Jacquemart, J. Lang, M. Simonius,W. Haeberli, Ch. Weddigen, W. Reichart, S. Jaccard,Phys. Rev. Lett. 44, 699 (1980)

[BAL84] R. Balzer, R. Henneck, Ch. Jacquemart, J. Lang,F. Nessi-Tedaldi, T. Roser, M. Simonius, W. Haeberli,S. Jaccard, W. Reichart, Ch. Weddigen, Phys. Rev. C 30, 1409(1984)

[BAU66] E. Baumgartner, H.E. Conzett, E. Shield, R.J. Slobodrian,Phys. Rev. Lett. 16, 105 (1966)

[BER01] A.R. Berdoz et al., E497 TRIUMF Collaboration,Phys. Rev. Lett. 87, 272301 (2001)

[BLA50] J.M. Blatt, J.D. Jackson, Rev. Mod. Phys. 22, 77 (1950)

[BLA83] E. Blanke, H. Driller, W. Glöckle, G. Genz, A. Richter,G. Schrieder, Phys. Rev. Lett. 51, 355 (1983)

[BOD59] D. Bodanski, S.F. Eccles, G.W. Farwell, M.E. Rickey,P.C. Robison, Phys. Rev. Lett. 2, 101 (1959)

[BOD66] D. Bodanski, W.J. Braithwaite, D.C. Shreve, D.W. Storm,W.G. Weitkamp, Phys. Rev. Lett. 17, 589 (1966)

[BRI38] F.A. Brickwedde, J.R. Dunning, H.J. Hoge, J.H. Manley,Phys. Rev. 54, 266 (1938)

[DES80] B. Desplanques, J.F. Donoghue, B.R. Holstein,Ann. Rev. Phys. (N.Y.) 124, 449 (1980)

[ENG12] R. Engels et al. for the JEDI collaboration,http://www2.fz-juelich.de/ikp/jedi/documents/proposals.shmtl

[EVE91] D. Eversheim, W. Schmitt, S. Kuhn, F. Hinterberger, P. vonRossen, J. Chlebek, R. Gebel, U. Lahr, B. von Przeworski,M. Wiemer, V. Zell, Phys. Lett. 256B, 11 (1991)

DR

AFT

140 BIBLIOGRAPHY

[FER47] E. Fermi, L. Marshall, Phys. Rev. 71, 666 (1947)

[FRA86] H. Frauenfelder, E.M. Henley, Nuclear and Particle Physics A:Background and Symmetries, Lect. Notes and Suppl. in Phys.,2nd printing, Benjamin/Cummings, Reading (1986)

[GON99] D.E. Gonzáles-Trotter, F. Salinas, Q. Chen, A.S. Crowell,W. Glöckle, C.R. Howell, C.D. Roper, D. Schmidt, I. Šlaus,H. Tang, W. Tornow, R.L. Walter, H. Witała, Z. Zhou,Phys. Rev. Lett. 83, 3788 (1999)

[HAD65] R.P. Haddock, R.M. Salter, M. Zeller, J.B. Czirr, D.R. Nygren,Phys. Rev. Lett. 14, 318 (1965)

[HAL37] J. Halpern, E. Estermann, O.C. Simpson, O. Stern,Phys. Rev. 52, 142 (1937)

[HAN67] R. Handler, S.C. Wright, L. Pondrom, P. Limon, S. Olsen,P. Kloeppel, Phys. Rev. 19, 933 (1967)

[HAR86] H.L. Harney, A. Richter, H.A. Weidenmüller,Rev. Mod. Phys. 58, 607 (1986)

[HAR90] H.L. Harney, A. Hüpper, A. Richter, Nucl. Phys. A 518, 35(1990)

[HAX13] Haxton, B. Holstein, arxiv1303.4132v2

[HEN59] E.M. Henley, B.A. Jacobsohn, Phys. Rev. 113, 225 (1959)

[HEN79] E.M. Henley, G.A. Miller in Mesons in Nuclei, eds. M. Rho,H.D. Wilkinson, North-Holland, Amsterdam (1979), p. 415

[HIL58] P. Hillman, A. Johansson, G. Tibell, Phys. Rev. 110, 1218 (1958)

[HOL09] B.R. Holstein, Eur. Phys. J. A 41, 279 (2009)

[HUH00] V. Huhn, L. Wätzold, Ch. Weber, A. Siepe, W. von Witsch,H. Witała, W. Glöckle, Phys. Rev. Lett. 85, 1190 (2000)

[KIS87] S. Kistryn, J. Lang, J. Liechti, Th. Maier, R. Müller,F. Nessi-Tedaldi, M. Simonius, J. Smyrski, S. Jaccard,W. Haeberli, J. Sromicki, Phys. Rev. Lett. 58, 1616 (1987)

[KLE74] G. Klein, H. Paetz gen. Schieck, Nucl. Phys. A 219, 422 (1974)

[KNE58] D.J. Knecht, S. Messelt, E.D. Berners, L.C. Northcliffe,Phys. Rev. 114, 550 (1958)

[LOC84] N. Lockyer, T.A. Romanowski, J.D. Bowman, C.M. Hoffman,R.E. Mischke, D.E. Nagle, J.M. Potter, R.L. Talaga,E.C. Swallow, D.M. Alde, D.R. Moffett, J. Zyskind,Phys. Rev. D 30, 1409 (1984)

DR

AFT

BIBLIOGRAPHY 141

[MCK68] J.L. McKibben, G.P. Lawrence, G.G. Ohlsen,Phys. Rev. Lett. 20, 1180 (1968)

[MEL49] E. Melkonian, Phys. Rev. 78, 1744 (1949)

[MIL06] G.A. Miller, A.K. Opper, E.J. Stephenson,Ann. Rev. Nucl. Part. Sci. 56, 253 (2006)

[MOL68] P.A. Moldauer, Phys. Rev. 165, 1136 (1968)

[NAG79] E.D. Nagle, J.D. Bowman, C. Hoffman, J. McKibben,R. Mischke, J.M. Potter, H. Frauenfelder, L. Sorensen, inProc. Conf. on High-Energy Physics with Polarized Beams andTargets, Argonne 1978, (G.H. Thomas, ed.) AIPConf Proc. (New York) 51, 224 (1979)

[OXL53] C.L. Oxley, W.F. Cartwright, J. Rouvina, E. Baskir, D. Klein,J. Ring, W. Skillman, Phys. Rev. 91, 419 (1953)

[PGS14] H. Paetz gen. Schieck, Nuclear Reactions – An Introduction,Lecture Notes in Physics 882, Springer, Heidelberg (2014)

[RAI46] J. Rainwater, W.W. Havens, Jr., Phys. Rev. 70, 136 (1946)

[RYA64] J.W. Ryan, Phys. Rev. Lett. 12, 564 (1964)

[RYA67] J.W. Ryan, Phys. Rev. 130, 1554 (1967)

[STE03] E.J. Stephenson, A.D. Bacher, C.E. Allgower, A. Gårdestig,C.M. Lavelle, G.A. Miller, H. Nann, J. Olmsted, P.V. Pancella,M.A. Pickar, J. Rapaport, T. Rinckel, A. Smith, H.M. Spinka,U. van Kolck, Phys. Rev. Lett. 91, 142303 (2003)

[SPR61] D.W.L. Sprung, Phys. Rev. 121, 925 (1961)

[SUT47] R.B. Sutton, T. Hall, E.E. Anderson, H.S. Bridge, J.W. de Wire,L.S. Lavatelli, E.A. Long, T. Snyder, R.W. Williams,Phys. Rev. 72, 1147 (1947)

[TAN57] N. Tanner, Phys. Rev. 107, 1203 (1957)

[VWI68] W. von Witsch, A. Richter, P. von Brentano, Phys. Rev. 169,923 (1968)

[WEI68] W.G. Weitkamp, D.W. Storm, D.C. Shreve, W.C. Braithwaite,D. Bodanski, Phys. Rev. 165, 1233 (1968)

[WOR52] Worthington, McGruer, Findley, Phys. Rev. 90, 899 (1952)

[YUA86] V. Yuan, H. Frauenfelder, R.W. Harper, J.D. Bowman,R. Carlini, D.W. MacArthur, R.E. Mischke, D.E. Nagle,R.L. Talaga, A.B. McDonald, Phys. Rev. Lett. 57, 1680 (1986)

[ZEI69] B. Zeitnitz, R. Maschuw, P. Suhr, Phys. Lett. 28B, 420 (1969)

[ZHA98] J. Zhao, R. Abegg et al., Phys. Rev. C 57, 2126 (1998)

DR

AFT

142 BIBLIOGRAPHY

DR

AFT

Chapter 14

Scattering of Identical Nuclei,Exchange Symmetry andMolecular Resonances

In quantum mechanics identical particles are indistinguishable. In a reac-tion between identical nuclei this applies to forward-scattered nuclei andbackward-scattered recoil nuclei. Independent from reaction mechanisms andtype of interaction these particles must interfere, i.e. classical or semi-classicalapproaches to describe such reactions are principally unsuited.

14.1 First observation of interference in the

scattering of identical nuclei

In addition to the forward-scattering Rutherford cross section there is a cor-responding recoil Rutherford term plus an interference term between both.Fig. 14.2 shows this behavior (which is analogous to that of the light inYoung’s double-slit experiment, but additionally shows the influence of spinand statistics).

In the scattering of identical particles a detector at the c.m. angle θ isunable to distinguish whether it registers forward-scattered ejectiles underθ or, under the angle π − θ, backward-emitted recoils. This is shown inFig. 14.1. The formal scattering theory (see below) shows that the angulardistributions must be symmetric around π/2 and therefore must be describedby even-order Legendre polynomials. Quantum-mechanically, in addition, itis to be expected that the forward- and backward-scattered particle wavesinterfere. In this case no classical description of the scattering process ispossible. In addition, the details of the interference depend on the spinstructure of the interacting particles: identical bosons behave differently fromidentical fermions, and when the particles have spin , 0 (i.e. always forfermions) the spin states must be coupled and superimposed in the crosssection with their spin multiplicities as weighting factors. The followingexamples, which can be tested experimentally will explain this.

DR

AFT

144CHAPTER 14. SCATTERING OF IDENTICAL NUCLEI, EXCHANGE SYMMETRY AND MOLECULAR

RESONANCES

π−Θ

Θ

c.m.

c.m.

Figure 14.1: Trajectories of identical particles in the c.m. system.

14.1.1 Identical Bosons with spin I = 0

Here[dσ/dΩ(θ)]B = |f1(θ) + f2(π − θ)|2 . (14.1)

14.1.2 Identical Fermions with spin I = 1/2

For the fermions the spin singlet cross section

[dσ/dΩ(θ)]s = |f1(θ) − f2(π − θ)|2 (14.2)

and the triplet cross section

[dσ/dΩ(θ)]t = |f1(θ) + f2(π − θ)|2 (14.3)

in the total (integrated) cross section must be added incoherently, eachweighted with their spin multiplicities:

[dσ/dΩ(θ)]F =14

|f1(θ) + f2(π − θ)|2 +34

|f1(θ) − f2(π − θ)|2 . (14.4)

In these two cases the interference has opposite sign, which e.g. at θ = π/2has the consequence that in the case of two bosons there is an interferencemaximum, for fermions a minimum. Under the special assumption that thereis no spin-spin force acting (fs = ft =f), and with f(θ) = f(π− θ) one obtainsfor identical fermions a decrease, for identical bosons an increase each by thefactor 2 as compared to the classical cross section.

For pure (Sub-)Coulomb scattering (meaning: Coulomb scattering at en-ergies sufficienttly below the Coulomb barrier) of identical particles the scat-tering amplitudes can be calculated explicitly (i.e. also summed over partialwaves) since we deal with the Rutherford amplitude known from scatteringtheory, see Section 3.1.3:

(

dσ

dΩ

)

Coul

=

(

Z2e2

4E∞

)2

1

sin4 θ2

+1

cos4 θ2

+2(−1)2s cos

(

ηS ln tan2 θ2

)

(2s+ 1) sin2 θ2

cos2 θ2

.

(14.5)

DR

AFT

14.1. FIRST OBSERVATION OF INTERFERENCE IN THE SCATTERING OF IDENTICAL NUCLEI 145

In addition to the forward-scattering Rutherford cross section there is a cor-responding recoil Rutherford term plus an interference term between both.Fig. 14.2 shows this behavior (which is analogous to that of light in Young’sdouble-slit experiment, but additionally shows the influence of spin andstatistics).

Figure 14.2: Experimental c.m. angular distributions of Coulomb scatter-ing of two identical bosons (12C) and fermions (13C) as well as of two non-identical particles of nearly equal masses and theoretical cross sections atElab = 7 MeV. The angular distribution for the non-identical particles isobtained when the spectra of the forward and backward scattered particlescannot be separated by the detector, which is the case for (nearly) equalmasses. Otherwise one would obtain a typical Rutherford distribution for theforward-scatterd particle and a distribution reflected about 90 for the recoilparticle. The data were measured by students of an advanced lab. course atIKP Cologne in 2003.

Above the Coulomb barrier, additional terms including interference terms,arise from the hadronic interaction. A special example is low-energy proton-proton scattering in which, for S-waves, one nuclear phase shift δ0 must beconsidered for which e.g. a “nuclear” scattering length app may be obtained(see preceding Section 13.3.2).

The first experiments designed specifically to study the scattering of iden-tical bosons and at the same time details of the heavy-ion interactions andstructure were performed by Bromley et al. [BRO60, BRO61]. Tandem Van-de-Graaff accelerators became available as well as the compact solid-statedetectors. Both are especially suited to study heavy-ion reactions, the accel-

DR

AFT


RESONANCES

erators because of excellent energy definition and stability and ease of chang-ing energy and targets, the latter because of the possibility of using manysmall detectors providing good angular resolution. Thus, measurements ofthe highly structured cross sections, both in angle and energy, were facili-tated. Fig. 14.3 shows the scattering-chamber setup used in Ref. [BRO61].

Figure 14.3: The scattering chamber setup of Ref. [BRO61] is shown. It istypical for tandem-VdG experiments with charged particles. Important are awell focussed or collimated beam, thin foil targets, well-defined solid angles atthe detectors, precise angle definition of the detector slits, and a Faraday cupat the exit for beam charge calibration. For heavy-ion experiments, however,this cannot be used because the charge state equilibrium of the incidentbeam at the reaction is unknown. Therefore, a cross-section calibration suchas using a heavy target (e.g. gold) and calculable Rutherford cross sectionhas to be used for normalization

The following Fig. 14.4 shows a selection of the angular distributions andexcitation functions for 12C-12C and 16O-16O elastic scattering at energiesbelow the Coulomb barriers such that the Mott cross section applies.

DR

AFT


DR

AFT


RESONANCES

It is remarkable that the excitation functions for the two systems are quitedifferent, i.e. relatively smooth for the 16O case, but with (quasi-)periodicstructures for the 12C scattering (intermediate or gross structures with super-imposed fine structure). This would be typical signatures for some doorwayphenomenon. In fact, these oscillations have been interpreted as rotationalstates of nuclear molecules which partly decay into compound-nuclear fine-structure states. The excitation functions follow the Mott cross section butwith increasing energy there is a relatively sharp onset of strong absorption(leading to compound-nucleus formation). The r dependence of the potentialcombined of Coulomb potential, an absorptive (attractive) nuclear potential,and the orbital angular-momentum barrier at higher ℓ could form a shallowminimum with rotational molecular states as shown in Fig. 14.5.

Figure 14.5: Schematic view of the possible shape of a potential allowing formolecular states

In contrast to these early measurements more detailed studies, also athigher energies exhibit (quasi-)periodic structures, partly with marked fine-structure oscillations, also in other systems such as 16O-16O, 12C-16O, 14C-14C, 18O-16O, and 18O-18O, see Ref. [SIE67]. Two cases are shown in Fig. 14.6.

DR

AFT


Figure 14.6: Excitation functions of 12C-12C and 16O-16O elastic scattering

For a detailed discussion of nuclear molecular states and attempts to de-scribe them, e.g. in the framework of the optical model with shallow or deepoptical potentials see Ref. [BET97].

DR

AFT


RESONANCES

DR

AFT

Bibliography

[BET97] R.R. Betts, A.H. Wuosmaa, Rep. Prog. Phys. 60, 819 (1997)

[BRO60] D.A. Bromley, J.A. Kuehner, E. Almqvist, Phys. Rev. Lett. 4,365 (1960)

[BRO61] D.A. Bromley, J.A. Kuehner, E. Almqvist, Phys. Rev. 123, 868(1961)

[GOL64] M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, NewYork (1964)

[JOA83] C. Joachain, Quantum Collision Theory, 3rd ed., North-Holland(1983)

[MOT30] N.F. Mott, Proc. Roy. Soc. A 126, 259 (1930)

[MOT65] N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions,Clarendon Press, Oxford (1965)

[SIE67] R.H. Siemssen, J.V. Maher, A. Weidinger, D.A. Bromley,Phys. Rev. Lett. 19, 369 (1967)

DR

AFT

152 BIBLIOGRAPHY

DR

AFT

Chapter 15

Nuclear Fission and NuclearEnergy

Nuclear fission – i.e. the disintegration of a heavy nucleus into two (some-times three) lighter nuclei of roughly the mass and charge numbers of theoriginal – can occur spontaneously in some heavy nuclei, but there are manyinstances of fission induced by a nuclear reaction. All kinds of combinationsof projectile and heavy target nuclei may undergo fission. Nuclei, excited tovery high (rotational) spin states could also fission.

After the discovery of the neutron by Chadwick in 1932 many groupsstarted investigations of the interaction between neutrons and nuclei. Theproperty of neutrality of the neutron made it very attractive as a probe ofnuclei and nuclear reactions especially at low energies where no Coulombbarrier hindered the reactions. Even with the simple method of using ra-dioactive sources such as Ra-Be and Hydrogen containing moderators (suchas water or paraffin) many new results could be obtained. The practical useof accelerators after 1932 opened additional possibilites as did much later theuse of nuclear reactors with high neutron fluxes.

In the 1930s E. Fermi and his collaborators tried to produce transuranicnuclei by adding a neutron to a known nucleus. They succeeded in the case of23993Np via the neutron capture of 238

92U and subsequent β decay. At the sugges-tion of L. Meitner Hahn’s group started doing similar experiments with thegoal of producing new elements. Only in 1938 O. Hahn and F. Strassmann(after L. Meitner had been forced to emigrate to Sweden) took the additon-ally produced activities, which looked like e.g. Ba and other medium-weightnuclei, seriously and identified these medium-weight products as fission prod-ucts.

The nuclear chemists Otto Hahn and Fritz Strassmann published a se-ries of papers in 1938/39, in which they clearly proved – with chemicalmeans – that after the irradiation of uranium (and thorium) with neutronsnot only transuranium elements were created, as was generally assumed, seeRef.[HAH38], but that unambiguously also medium-heavy isotopes of barium(and lanthanum and cerium) appeared [HAH39b].

The experimental setup was remarkably simple: Slow neutrons were pro-duced by a radium-beryllium source, followed by a paraffin moderator andused to irradiate e.g. a uranium compound such as uranyl nitrate. The

DR

AFT

154 CHAPTER 15. NUCLEAR FISSION AND NUCLEAR ENERGY

captured neutrons left the reaction products in excited states and the decayactivity was measured by a Geiger counter. A setup, which collects the dif-ferent pieces of apparatus, is exhibited at the Deutsches Museum, Munich,see Fig. 15.1.

Radium is chemically homologous to barium and thus both could appeartogether in chemical separations. Therefore, in the beginning, Hahn et al.assumed to have produced new isotopes of radium from α decays of transura-nium nuclei. With more refined chemical methods (“fractionated crystalliza-tion”) barium and radium could be separated, and no trace of enrichmentof radium (with its known half-life) could be found. Thus, they could onlyconclude that they had produced barium isotopes. It is interesting to read inthe original paper, how hard this conclusion from the point of view of physicswas for them, but as chemists they could not evade it. In translation: “Aschemists in principle we should rename the above scheme and replace thesymbols or Ra, Ac, and Th by Ba, La, and Ce. As ’nuclear chemists’ whoare in a certain way close to physics we cannot yet make up our mind forthis jump. Perhaps a number of strange accidents could still have simulatedour results” [HAH39a]. But only little later [HAH39b] they wrote in the con-clusion: “The generation of barium isotopes from uranium has finally beenproven” and also isotopes of Sr and Y as fragments, and similarly for thorium.This hesitation shows how improbable the only possible conclusion of fissionof heavy nuclei into appproximately equal debris appeared. It was acceptedknowledge that nuclei couldn’t fission. It is, however, hard to understand

Counter

Ra−Be NeutronSource

Paraffin Block forNeutron Moderation

O. Hahn’sLogbook

with Lead ShieldingsGeiger−Mueller Counters

High−Voltage Batteries

AmplifiersVacuum−Tube High−Voltage

Figure 15.1: Museum exhibit at the Deutsches Museum, Munich: the fa-mous Hahn-Meitner-Strassmann desk with the collected pieces of experimen-tal equipment and Hahn’s laboratory logbook. Very probably the equipmenthas been used mostly by L. Meitner, she being the physicist in the group ofnuclear chemists. The parts were actually used in different rooms.

DR

AFT

155

that Fermi and his group at Rome didn’t conceive fission during their longseries of experiments with neutrons produced from different sources such asRa-Be, Po-Be etc. on many different elements, among them uranium andthorium. The results were published in ten short communications (letters)in italian in the journal Ricerca Scientifica of the National Research Counciland summarized in the Refs. [FER34a, FER34b, FER34c, FER34d]. Thegroup was so fixated on producing “new” (transuranic) nuclei that even theidea of possible fission put forward by Ida Noddack already in 1934 [NOD34]was ignored (for a good account of this see e.g. Ref. [SEG70]). Part of thiswere energetic considerations and the lack of a plausible model (however,around that time the liqid-drop model just came in time for an explanation).

It took almost five years before the fission process was established. Veryquickly at many places the experiments were confirmed and the possibilityof fission was theoretically explained with the analogy of nuclei as liquiddrops, first by Lise Meitner and Otto Frisch [MEI39], then by N. Bohr andA. Wheeler [BOH39].

The first physical, as compared to chemical, proof of the fission processwas published by Otto Frisch in Ref. [FRI39]. He (at Copenhagen) used anionization chamber lined with uranium irradiated with neutrons produced byan Ra-Be α source with and without paraffin moderator. In the first case thenumber of fission events was approximately twice that of the unmoderatedcase. When using thorium as “target” fission fragments, but no dependenceon moderation were observed. The medium-mass fission fragments with theirhigh energies and high charge states (estimated to about 20) could travel afew mm into the ionization volume and caused high pulses in the connectedcounter whereas “transuranes” would not be able to leave the uranium lining.

Also very quickly two important consequences of the phenomenon of nu-clear fission were discussed.

• The fission process liberates a large amount of energy (≈ 200 MeVper fission), immediatly inciting the idea of technical (and military?)useability.

• The fission products are nuclei with high neutron excess. So, besides theslower process of β decay the emission of fast neutrons brought up theidea of a chain reaction that was later implemented in nuclear reactorsand nuclear weapons. Already in 1933 L. Szilard had conceived of achain reaction but applied for patents instead of publishing the idea,probably of fear of German developments. Proof of fission neutronswas first found in 1939 by Dodé et al. [DOD39] and their number perfission was determined first by v. Halban et al. [HAL39] As an examplean interesting paper on these considerations was published in June of1939 by S. Flügge [FLU39] (with the title: “Kann der Energieinhaltder Atomkerne technisch nutzbar gemacht werden?”).

DR

AFT

156 CHAPTER 15. NUCLEAR FISSION AND NUCLEAR ENERGY

DR

AFT

Bibliography

[BOH39] N. Bohr, J.A. Wheeler, Phys. Rev. 56, 426 (1939)

[DOD39] M. Dodé, H. v. Halban jun., F. Joliot, L. Kowarski,Compt. Rend. Acad. Sci. Paris 208, 995 (1939)

[FER34a] E. Fermi, Nature 133, 757 (1934)

[FER34b] E. Fermi, Nature 133, 898 (1934)

[FER34c] E. Fermi, Nature 134, 668 (1934)

[FER34d] E. Fermi, E. Amaldi, O. D’Agostino, F. Rasetti, E. Segrè,Proc. Roy. Soc. (London) A 133, 483 (1934)

[FLU39] S. Flügge, Naturw. 27, 402 (1939)

[FRI39] O.R. Frisch, Nature 143, 276 (1939)

[HAH38] O. Hahn, F. Strassmann, Naturw. 26, 756 (1938)

[HAH39a] O. Hahn, F. Strassmann, Naturw. 27, 11 (1939)

[HAH39b] O. Hahn, F. Strassmann, Naturw. 27, 89 (1939)

[HAL39] H. v. Halban, F. Joliot, L. Kowarski, Nature 143, 470 and 680(1939)

[HMS] Exhibit at Deutsches Museum, Munich

[MEI39] L. Meitner, O.R. Frisch, Nature 143, 239 (1939)

[NOD34] I. Noddack, Angewandte Chemie 47, 653 (1934)

[SEG70] E. Segrè, Enrico Fermi, Physicist, The U. of Chicago Press,Chicago and London (1970)

DR

AFT

158 BIBLIOGRAPHY

DR

AFT

Chapter 16

First Double Scattering andPolarization in p-4He and the(ℓ · s) Force

In 1949 the single-particle shell model of nuclear states was formulated byHaxel, Jensen, and Suess [HAX49], and M. Goeppert-Mayer [GOE49]. Itbecame the basis of nuclear-structure models up to today. The essential in-gredient in this model is the non-central spin-orbit force term in the nucleon-nucleus potential. Orbitals with a given orbital angular momentum ℓ canbe split by the spin-orbit force into doublets corresponding to states withJ = ℓ ± 1/2. Depending on the sign of this force the sequence of the levelscould be “normal”, i.e. (like in atomic physics) the level with the higher couldbe raised in energy, the other lowered or vice versa, i.e. “inverted”. It turnedout that the level ordering in nuclei is different from that in atomic states,meaning that the origin of the (ℓ · s) force was probably not electromagnetic.The splitting increases with increasing ℓ such that increasingly the spin-orbitforce lowers the energy of the lower state so much that in higher orbits it isnow in the next-lower harmonic-oscillator shell with opposite parity explain-ing isomeric states, and creates a large energy gap that is characteristic forthe shell model.

Up to 1949 the level ordering in 5Li∗ and 5He∗ was unsettled. The unboundground and excited states states could be studied as resonances in 4He +proton or 4He + neutron scattering. Fig. 16.1 shows the level schemes for 5Liand 5He as of 1966. With the help of phase-shift analyses predictions couldbe made for the polarization of nucleons scattered from 4He that differedappreciably for he two cases 2P1/2 and 2P3/2.

For low-energy neutrons polarization effects in the interaction of the neu-tron’s magnetic moment and magnetized material (i.e. the polarized elec-trons) had been investigated earlier. Theoretical investigations of the effectsof an (ℓ · s) force (see Refs. [SCH46, SCH48, WOL49]) on nuclear reactions,e.g. production and measurements of spin polarization, for fast neutrons orprotons predicted that

• the scattered particles would acquire spin polarization, and

• by time-reversal invariance (TRI), the scattering if done with polarized

DR

AFT

160CHAPTER 16. FIRST DOUBLE SCATTERING AND POLARIZATION IN P-4HE AND THE (ℓ · S)

FORCE

Figure 16.1: Simplified level schemes of 5Li and 5He as of 1966 [AIZ66]showing the P-wave splitting

DR

AFT

161

projectiles would experience a left-right asymmetry (i.e. an azimuthaldependence of the cross sections) : The reaction would show a non-zero analyzing power. TRI states that the polarization produced in aforward reaction is equal to the analying power in the correspondingbackward reaction, provided the reactions are measured at the samec.m. energies and scattering angles. For elastic scattering both are thesame.

• Combining both – a first scattering producing a polarization which inturn may be determined by a second scattering of the same kind, a dou-ble scattering experiment – would be unique in showing the existenceof the (ℓ · s) force.

• At the same time the polarization/analyzing power of the above reac-tions are predicted to be strongly dependent on the j value of the splitshell-model orbital.

Heusinkveld and Freier [HEU50] undertook the first double-scattering ex-periment

4He + p → 5Li → 4He + p. (16.1)

They used photoplates for detecting the doubly-scattered protons. Theirsetup is depicted in Fig. 16.2. Fig. 16.3 shows the predictions for the po-larization for the two cases of a “regular” or an “inverted” sequence of thestates P1/2 and P3/2. The predictions for the angular distributions yieldedmarked forward/backward angle double-scattering cross section ratios thatdepended on the level sequence. The results are shown in Fig. 16.4, a re-production of the table of results (ratios from the original. The conclusionsfrom the comparison with the data of this experiment are:

• The P wave in 5Li (and by isospin arguments also in 5He) is stronglysplit between P1/2 and P3/2 which is proof of a strong (ℓ · s) force.

• The level sequence is inverted, i.e. the origin of the force is not elec-tromagnetic as in the atomic case but a property of the nuclear forceitself.

• A scattering experiment n + 4He or p + 4He produces highly polar-ized nucleons. The polarization is measurable in a double-scatteringexperiment as a left-right asymmetry of the cross section of the secondscattering.

DR

AFT


FORCE

Figure 16.2: Apparatus to measure protons doubly scattered from 4He in agas target chamber. The protons were registered in the emulsion of photo-plates.

Figure 16.3: Predictions for the proton polarization to be measured in adouble-scattering experiment 4He(p, p)4He.

DR

AFT

163

Figure 16.4: Reproduction of the table of results of Ref. [HEU50] of thedouble-scattering ratios of 4He(p, p)4He. The ratios clearly favor the “in-verted” sequence of levels, i.e. the 2P3/2 assignment for the ground state and2P1/2 for the broad first excited state of 5Li.

DR

AFT


FORCE

DR

AFT

Bibliography

[AIZ66] F. Aizenberg-Selove, Nucl. Phys. A 78, 1 (1966)

[GOE49] M. Goeppert-Mayer, Phys. Rev. 75, 1969 (1949)

[HAX49] O. Haxel, J.H.D. Jensen, H.E. Suess, Phys. Rev. 75, 1766 (1949)

[HEU50] M. Heusinkveld, G. Freier, Phys. Rev. 85, 80 (1952)

[SCH46] J. Schwinger, Phys. Rev. 69, 681 (1946)

[SCH48] J. Schwinger, Phys. Rev. 73, 407 (1948)

[WOL49] L. Wolfenstein, Phys. Rev. 75, 1664 (1949)

DR

AFT

166 BIBLIOGRAPHY

DR

AFT

Chapter 17

First Nuclear Reaction of anAccelerated Polarized Beamfrom a Polarized-Ion Source(Basel)

Usually the quantities most measured in a nuclear reaction are differentialor total cross sections. For particles with spin these quantities are – formally–averages over incident-particle spin states and sums over outgoing statesor (for the total crosss section) also sums over scattering angles (i.e. overorbital angular momenta). In this way much information about the detailsof the interaction may be lost that might be hidden in the transition am-plitudes between the spin-substates, especially about the spin dependence ofthe nuclear interaction, which has, besides a central spin-independent term,several spin-dependent contributions such as ~L · ~S, spin-spin, and tensor forcecontributions.

Up to 1960 the only way to study the spin dependence of nuclear reactionswas to produce spin-polarized particles in nuclear reactions with such an ~L· ~Sforce acting or to measure the polarization of the outgoing particles with sucha reaction. The term double scattering for these experiments implies thedifficulties involved such as the dependence on energy and angle propertiesof these reactions and very small event rates.

Starting in 1956 the idea to use atomic properties (in this case the factthat the magnetic moment of the electrons is on the order of about 2000times larger than that of the nuclei, see the values of the Bohr and the nu-clear magneton) for a Stern-Gerlach type separation of spin states was firstformulated and realized by G. Clausnitzer, R. Fleischmann, and H. Schopper[CLA56, CLA59] at Erlangen. The electronic polarization of atoms is trans-ferred to the nuclei by the hyperfine interaction and –later – enhanced bysuitable radiofrequency transitions between substates. The beam intensitywas increased by separation of the Zeeman components in multipole fieldswith cylindrical symmetry (quadrupole or sextupole fields). For acceleratoruse suitable ionizers (electron-collision or ECR type ionizers) were devel-oped. For more details see Ref. [PGS12]. The Erlangen source was designedfor protons.

DR

AFT

168CHAPTER 17. FIRST NUCLEAR REACTION OF AN ACCELERATED POLARIZED BEAM FROM A

POLARIZED-ION SOURCE (BASEL)

The first nuclear reaction initiated by a polarized and accelerated beamfrom such a source was the 3H(~d, n)4He reaction on resonance at Ed = 107 keVat Basel [RUD61] which was the occasion for the first polarization confer-ence [BAS61]. The setup of this experiment with the atomic-beam polarizeddeuteron source connected to a cascade generator with 100 kV with a tritiumtarget and plastic scintillator neutron detectors in the high-voltage dome.Figs. 17.1 shows the experimental setup and Fig.17.4 the construction de-tails of the atomic-beam source. Under the assumptions that

• the reaction 3H(d,n)4He goes entirely through the resonant S-wave Jπ =3/2+ matrix element of 5He · ,

• the ionization of the deuterium atoms takes place in a very weak mag-netic field B → 0,

• and no depolarization by unpolarized residual gas or non-adiabatictransitions occurs,

a vector polarization of PZ = 0 and a tensor polarization of PZZ = −1/3(in today’s nomenclature) are predicted. Later improvments of polarized-ionsources consisted in

• Replacement of the capillary collimators by a simple canal as nozzle(which is possible with much increased (differential) pumping).

• Cooling of the nozzle leads to higher beam density, higher acceptanceof the Stern-Gerlach magnets, and higher ionization efficiency.

• Use of sextupoles (which have focussing properties on magnetic mo-ments instead of quadrupoles that can only deflect them).

• RF transitions together with ionization in a strong magnetic field pro-vide maximum values of polarization and high flexibility in changingthe sign and also type of polarization.

Other types of sources have been developed, based on the Lamb shift, ion-ization by intense colliding particle beams, or on optical pumping. Typicalare now beams with polarizations of about 95% of the theoretical values andmany tens of µA beam currents. A principal limitation to the intensity of theatomic beam lies in the gas dynamics around the dissociator/nozzle region(“intra-beam scattering”).

DR

AFT

169

Figure 17.1: Schematic showing the first atomic-beam polarized ion sourceconnected to an accelerator and used for the nuclear reaction 3H(~d, n)4Hethereby testing thes sensitivity of this reaction as a polarization analyzerand checking the theoretical assumptions on the 107 keV resonance in 5Heas a pure s-wave resonance allowing no vector polarization sensitivity. After[RUD61].

DR

AFT



Figure 17.2: Construction of the atomic-beam polarized ion source using anrf-discharge dissociator, a glass-capillary collimator, long quadrupole mag-nets for spin-state separation, and an electron-collision ionizer to produce apositive beam of partly tensor-polarized deuterons. After [RUD61].

Figure 17.3: Construction details of the beam-forming parts of the atomic-beam polarized ion source using an rf-discharge dissociator, a glass-capillarycollimator, and long quadrupole magnets for spin-state separation. After[RUD61].

DR

AFT

171

Figure 17.4: Angular distribution of the tensor analyzing power (here calledP33) of the reaction 3H(~d,n)4He measured at the first Basel source, comparedto the prediction and with a best fit yielding PZZ = −0.245. After [RUD61].

DR

AFT



DR

AFT

Bibliography

[BAS61] Proc. Int. Symp. on Polarization Phenomena of Nucleons, Basel1960, (eds. P. Huber, K.P. Meyer) Helv. Phys. Acta Suppl. VI,Birkhäuser, Basel (1961)

[CLA56] G. Clausnitzer, R. Fleischmann, H. Schopper, Z. Physik 144,336 (1956)

[CLA59] G. Clausnitzer, Z. Physik 153, 600 (1959)

[PS12] H. Paetz gen. Schieck, Nuclear Physics with Polarized Particles,Lecture Notes in Physics 842, Springer, Heidelberg (2012)

[RUD61] H. Rudin, H.R. Striebel, E. Baumgartner, L. Brown, P. Huber,Helv. Phys. Acta 34, 58 (1961)

DR

AFT

174 BIBLIOGRAPHY

DR

AFT

Chapter 18

The Discovery of GiantResonances

1 Giant Resonances are broad, resonance-like structures in excitation func-tions with large cross sections, excited by incident γ’s as well as in inelasticparticle reactions such as (p, p′), (p, γ), and (α, α′). Their unusually largewidth is a consequence of the high excitation energy with many open de-cay channels and thus decay probabilities. The excitation energies of giantresonances as well as their widths follow a simple systematics, and thus aresimilar between neighboring nuclei. Together with the large values of thecross sections this suggests a collective behavior of many (or all) nucleons.

With the advent of high-energy/high intensity betatrons [WID28, KER40,KER41a, KER41b] the upper energy edge of the continous spectrum ofbremsstrahlung γ’s was moved to >100 MeV. A number of experiments wereperformed after 1945 mainly to investigate different exit channels, that havebeen summarized by Ref. [EYG52]. The first experiments showing clear indi-cations of such broad structures in different exit channels such as (γ,activation),(γ,fission, (γ,n) were performed by Baldwin et al. [BAL46, BAL47, BAL48]using the 100 MeV betatron of General Electric Lab. at Schenectady, N.Y.The problem with bremsstrahlung excitation functions is that the measuredintensities have to be deconvoluted with the known bremsstrahlungs’s spec-trum. In all cases, for a number of target nuclei resonance-like excitationfunctions around energies of ≈20 MeV were found without explicitly callingthe structures “resonances” but discussing them in the framework of (statisti-cal) compound-nucleus theories. One example is shown in Fig. 18.1. Similarresults obtained for other exit channels (activation, fission) are evidence forthe decays of a common intermediate state. With higher resolution fine struc-ture of the giant-resonance peak has been found testifying to its charcter asan intermediate structure/doorway phenomenon.

Based on the early data a theoretical description of the mechanism leadingto the observed excitation functions was first given in 1948 by Goldhaber andTeller [GOL48] assuming a resonant dipole vibration (an E1 excitation) i.e.

1It should be noted that in the older literature the term “Giant Resonance” was alsoused for the total cross section behavior of neutron scattering on many nuclei in an energyregion where single resonances are not resolved and which is described by the OpticalModel, see e.g. Ref. [SAT90].

DR

AFT

176 CHAPTER 18. THE DISCOVERY OF GIANT RESONANCES

Figure 18.1: Photonuclear (γ,n) excitation functions on 12C before and afterdeconvolution of the bremsstrahlung spectrum produced by a betatron withEe = 100 MeV. From Ref. [BAL47].

a collective movement of the protons of nuclei against the neutrons. A semi-classical (hydrodynamic) model of the “giant resonances” followed in 1950by Steinwedel and Jensen [STE50] which delivered a reasonable account ofthe resonant energies over the periodic table. Improved measurements us-ing bremsstrahlung were performed by Fuller et al. [FUL58a, FUL58b]. Theproblems of these measurements were still the deconvolution with the con-tinuous spectrum and the difficulty of separating the (γ,2n) from the (γ,n)contribution. The methods used, however, were good enough to obtain theshape of the giant-resonance peak for many nuclei, also the double-humpstructure of the peaks in the region of rare earths (reported independentlyby Spicer [SPI58]), which had been predicted by Danos [DAN56, DAN58] andsimultaneously by Okamoto [OKA56, OKA58]. Fig. 18.2 shows one exampleof the results of the experiments. The limitations of using bremsstrahlung γ’sare overcome by using photons with sharp energies. Already in 1937 Botheand Gentner had used 17.4 MeV photons from the reaction 7Li(p,γ) to inves-tigate the nuclear photoeffect on many nuclei. Another method for experi-ments with giant-resonance studies developed by Bramblett et al. [BRA64]was to use nearly-monochromatic annihilation γ’s from positrons producedand accelerated in a LINAC accelerator to up to 30 MeV. The LINAC hadtwo stages: in the first electrons are accelerated to 10 MeV, then hit a Wtarget to produce copious positrons from pair creation. These are acceler-ated to between 8 and 28 MeV and produce annihilation photon pairs on aLiH target with variable energies. The photons in forward direction get anenergy of

hν = 1/2(2Te + 3m0c2) = Te + 0.77 MeV

with Te the kinetic energy of the positrons, m0 the electron rest mass. Theγ energy was measured by a NaJ(Tl) scintillation detector. Fig. 18.3 showsthe experimental arrangement. Fig. 18.4 is an example of a (γ,n) excitationfunction on 159Tb showing the splitting of the giant resonance. Bramblett etal. could deduce the intrinsic quadrupole moment

Q0 = (7.0 ± 1.1·10−24) cm2

of the deformed nucleus 159Tb from the ratio of the two peak energies, a valueconfirmed by Coulomb excitation.

DR

AFT

177

Figure 18.2: Giant resonance on Tb. From Ref. [FUL58b].

Figure 18.3: Setup to produce γ’s via positron in-flight annihilation anddetection of the (γ,n) neutrons in BF3 detectors. From Ref. [BRA64].

DR

AFT


Figure 18.4: Measurement of the (γ,xn) excitation function for 159Tb. FromRef. [BRA64].

Many giant resonance types have been found later. They can be classifiedaccording to their electromagnetic modes (or multipolarities), their isospin,or their motion types. These latter are

• The Breathing mode: The entire nucleus “breathes” without changingshape. Electromagnetically this is an E0 mode, its isospin is 0. It isplausible that this mode is related to the nuclear compressibility.

• The E1 mode: This is the classical dipole mode, in which proton andneutron fluids move collectively against each other.

• The M1, T = 1 mode: This is the magnetic-dipole Scissors mode, inwhich the motions of the proton and neutron fluids have a rotationalcomponent acting like the arms of scissors.

• The E2 or quadrupole giant resonance where protons and neutrons os-cillate collectiveley against each other such that an oscillating quadrupolemoment is formed.

• Higher order resonances such as M3, E4 etc. have been found, e.g. theisoscalar octupole (3hω) resonance.

• Relatively new developments concern the dipole (E1) Pygmy resonancein nuclei with high neutron excess where – it is assumed – the neutronskin oscillates against the remaining N = Z (T = 0) core. Radioactiveion-beam facilities will be able to investigate this phenomenon near theborders of stability (driplines) in more detail.

The first-discovered giant resonances discussed above were of the “classical”electric E1 dipole type. Besides γ interactions inelastic particle scattering,

DR

AFT

179

especially with α’s, is an important tool for measuring giant resonances. Thecross sections peak at very small forward angles and the measurements nor-mally require the use of magnetic spectrographs. Fig. 18.5 shows an exampleof such spectra together with the indication of the disentanglement of the dif-ferent contributions. Fig. 18.6 shows schematically the classification of thedifferent types of giant resonances. According to their collective characterthe resonance energies and widths vary systematically and slowly with themass number A. The energies of the resonance peaks are higher than thoseof “normal” excitations, and the cross sections of the exciting reactions arehigh. This suggests the interpretation of the giant resonances as collectiveexcitations of many (all) nucleons and – microscopically – the collective andcoherent excitation of many-particle-many-hole (p-h) states across differentharmonic-oscillator shells. As an example a collective Jπ = 1− state may beconstructed from excitation of N = 5 such p-h states from the p shell intothe sd shell

|(1p3/2)−1(1d5/2)〉1− . . . |(1p1/2)−1(1d3/2)〉1− (18.1)

A comprehensive survey of giant resonances up to 1999 is given by Ref. [HAR01].Inelastic scattering up to 1976 has been discussed in Ref. [BER76]. The morerecent M1 scissors-mode giant resonance is discussed in detail in Ref. [HEY10].Some properties of important giant resonances are collected in Table 18.6.

DR

AFT


Figure 18.5: Giant resonances, e.g. monopole, quadrupole and octupole ex-citations, in inelastic α scattering (α, α′) with ∆L = 1, 2, 3 at Eα = 152 MeV.High background makes the disentangling of different resonances difficult andmay require model assumptions. Measurements are best done at small for-ward angles. From [?].

DRAFT

181

Table 18.1: Properties of Selected Giant Resonances

Type of Excitation

Resonance Character ∆ L ∆ S ∆ T Energy Ex (MeV) Width (MeV) Preferred Method Relevance

ISGMR E0 0 0 0 (IS) 80A−1/3 3 - 5 (α, α′

) Nuclear

Compressibility

IVGMR E0 0 1 (IV) 59A−1/6 10 - 15 Charge Exch. (π±, π0)

ISGDR M1 1 1 0 (IS)

IVGDR E1 1 1 1 (IV) 31.2A−1/3 4 - 8 Photoabsorption Historically

+20A−1/6 First (1937)

ISGQR E2 2 0 0 (IS) 64.7A−1/3 90A−2/3 (p, p′

), (e, e′

)

IVGQR E2 2 0 1 (IV) 130A−1/3 5 - 15

ISGOR (LEOR) E3 3 0 0 (IS) 41A−1/3 Low-Energy OR

ISGOR (HEOR) E3 3 0 0 (IS) 108A−1/3 140A−2/3 High-Energy OR

GT M1 0 1 1 (IV) 5 - 10 (p, p′

), (p, n), (n, p) β decay

Pygmy Res. E1 1 0 1 (IV) 9 - 12 2 - 4 (p, n), (3He, t) Neutron Skin,

Symmetry Energy

Scissors Mode M1 1 1 1 (IV) const ≈ 3 (e, e′

) Rotational

Component

DR

AFT


∆ = T

L∆ = 0

L∆ = 1

L 2∆ =

n

p

pn n

p

pn

npDipole

Monopole

Quadrupole

p

n

Isoscalar Isovector IsoscalarSpin−flip

T∆ = T∆ = T∆ =

p

00 0

110 1

1

p n

∆ = S ∆ = S∆ = S∆ = S

p n n

Isovector (GT)

pp n

n

npn

p

p

p

p n

n

npn

p

n

n

n Pygmy

Resonance

Scissors Mode

Figure 18.6: Giant resonances classified according to their multipolarity ∆L,their spin and isospin changes ∆S, and ∆T .

DR

AFT

183

Nd

Nd

Nd

Nd

Nd

148

146

145

144

143

142Nd

Nd150

Figure 18.7: GDR in the photoneutron (γ,n) reaction on isotopes of Nd withincreasing neutron number N, showing onset of deformation. Adapted from[BER75].

DR

AFT


DR

AFT

Bibliography

[BAL46] G.C. Baldwin, G.S. Klaiber, Phys. Rev. 70, 259 (1946)



[BER75] B.L. Berman, S.C. Fultz, Rev. Mod. Phys. 47, 713 (1975).

[BER76] F.E. Bertrand, Ann. Rev. Nucl. Part. Sci. 26, 457 (1976)

[BER80] F.E. Bertrand et al., Phys. Rev. C 22, 1832 (1980)

[BOT37] W. Bothe, W. Gentner, Naturw. 25, 90, 126, and 191 (1937)

[BRA64] R.L. Bramblett, J.T. Caldwell, R.R. Harvey, S.C. Fultz,Phys. Rev. 113, B896 (1964)

[DAN56] M. Danos, Bull. Am. Phys. Soc. 1 135 (1956)

[DAN58] M. Danos, Nucl. Phys. 5, 23 (1958)

[EYG52] L. Eyges, Phys. Rev. 86,325 (1952)

[FUL58a] E.G. Fuller, B. Petree, M.S. Weiss, Phys. Rev. 112, 554 (1958)

[FUL58b] E.G. Fuller, M.S. Weiss, Phys. Rev. 112, 560 (1958)

[GOL48] M. Goldhaber, E. Teller, Phys. Rev. 74, 1046 (1948)

[HAR01] M.N. Harakeh, A. Van der Woude, Giant Resonances –Fundamental High-Frequency Modes of Nuclear Excitations,Oxford Studies in Nuclear Physics 24 Oxford SciencePubl. (1999).

[HEY10] K. Heyde, A. Richter, P. von Neumann-Cosel,Rev. Mod. Phys. 82, 2365 (2004).

[KER40] D.W. Kerst, Phys. Rev. 58, 841 (1940)

[KER41a] D.W. Kerst, Phys. Rev. 60, 47 (1941)

[KER41b] D.W. Kerst, R. Serber, Phys. Rev. 60, 53 (1941)

[OKA56] K. Okamoto, Progr. Theor. Phys. 15, 75 (1956)

DR

AFT

186 BIBLIOGRAPHY

[OKA58] K. Okamoto, Phys. Rev. 110, 143 (1958)

[SPI58] B.M. Spicer, Australian J. Phys. 11, 298 and 490 (1958)

[STE50] H. Steinwedel, J.H.D. Jensen, P. Jensen, Phys. Rev. 79, 1019(1950)

[WID28] R. Wideröe, Arch. Elektrotech. 21, 387 (1928)

DR

AFT

Chapter 19

General Resources and Reading

DR

AFT

188 CHAPTER 19. GENERAL RESOURCES AND READING

DR

AFT

Bibliography

[BRI71] D.M. Brink, G.R. Satchler, Angular Momentum, Oxford (1971)

[COD08] CODATA-06, P.J. Mohr, B.N. Taylor, D.B. Newell,Rev. Mod. Phys. 80,633 (2008)

[EDM60] A.R. Edmonds, Angular Momentum in Quantum Mechanics,Princeton (1960)

[EID04] S. Eidelman at al. (Particle Data Group), Phys. Lett. B 592, 1(2004)

[GOL64] M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, NewYork (1964)

[JOA83] C. Joachain, Quantum Collision Theory, 3rd ed.,North-Holland (1983)

[LOR79] H. Lorenz-Wirzba, P. Schmalbrock, H.P. Trautvetter,M. Wiescher, C. Rolfs, Nucl. Phys. A 313, 346 (1979)

[MAR70] P. Marmier, E. Sheldon, Physics of Nuclei and Particles, Vol. I,Ch. 11.2 ff., Academic Press, New York and London (1970)

[MOT65] N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions,Clarendon Press, Oxford (1965)

[NEW66] R.G. Newton, Scattering Theory of Waves and Particles,McGraw-Hill, New York (1966)

[NNDC] Natl. Nucl. Data Center, EANDC, Nucl. Reactions,http://www.nndc.gov

[PGS12] H. Paetz gen. Schieck, Nuclear Physics with Polarized Particles,Lecture Notes in Physics 842, Springer, Heidelberg (2012)

[PGS14] H. Paetz gen. Schieck, Nuclear Reactions – An Introduction,Lecture Notes in Physics 882, Springer, Heidelberg (2014)

[RED82] A. Redder, H.W. Becker, H. Lorenz-Wirzba, C. Rolfs,P. Schmalbrock, H.P. Trautvetter, Z. Phys. A305, 325 (1982)

[ROD67] L.S. Rodberg, R.M. Thaler, Introd. to the Quantum Theory ofScattering, Academic Press, New York (1967)

DR

AFT

190 BIBLIOGRAPHY

[ROL88] C. Rolfs, W.S. Rodney, Cauldrons in the Cosmos, University ofChicago Press, Chicago (1988)

[RPP08] Particle Data Group, Review of Particle Properties,Rev. Mod. Phys. 80, 633 (2008)

[SAT90] G.R. Satchler, Introd. Nucl. Reactions, 2nd ed., Ch. 3.7 ff.,McMillan, London (1990)

DR

AFT

Index

AcceleratorCockroft-Walton, 91Cyclotron, 95Livingston Plot, 95Tandem Van-de-Graaff, 95Van-de-Graaff, 95

AcceleratorsRole, 9

Born Approximation, 108Distorted-Wave DWBA, 108Plane-Wave PWBA, 108

Charge Distribution, 29Charge Independence, 80Charge, Current, or Matter Distribu-

tionsSampling, 31

Compound ReactionsCross Section, 114

Breit-Wigner Form, 115Conservation Laws, 121

Interaction Strength, 121Coulomb

potential, 14Coulomb Scattering

Identical Particles, 143, 145

DetectorsHistory, 9

DeuteronBinding Energy, 83np γ Capture, 85Photodisintegration, 83

Direct Interactions, 99

Electron ScatteringDeep-Inelastic, 73Elastic, 36Quasi-Elastic, 73

Giant Resonances, 175

Inelastic Scattering, 179Splitting, 176Theory

Goldhaber-Teller, 175Steinwedel-Jensen, 176

Types, 178

Halo Nuclei, 55

Identical Bosons, 144Identical Fermions, 144Intermediate Structures

Giant Resonances, 175Isospin, 80, 129Isospin Breaking

Henley/Miller Classification, 130

LeptonElastic Scattering

Rosenbluth Formula, 37

MatterDensity Distributions, 41Multipole Fields, 167

NeutrinosScattering, 73

NeutronDiscovery, 79Multiplication, 80Nuclear Structure, 9Scattering, 42Skin, 42

NN InteractionRange, 133Spin Dependence, 133

NN System, 130Nuclear Force

(ℓ · s), 161Nuclear Radii

A1/3 Law, 41α Scattering, 34Hadronic Scattering, 42

DR

AFT

192 INDEX

Heavy-Ion Scattering, 35Laser Spectroscopy, 39Muonic Atoms, 40

Nuclear Reaction3H(~d, n)4He, 168First, 7

Nuclear ReactionsCompound-Nuclear, 113Direct

Rearrangement, 103Stripping, 105

Optical Model, 99Resonances

History, 117Nuclear Spectroscopy, 10Nuclear Surface

Thickness, 41Nucleon-Nucleon Interaction, 129Nucleons

Density Distributions, 41Nucleus

Discovery, 7

Optical Model, 42

ParityLongitudinal Polarization, 123Pseudoscalars, 122

Parity ViolationHadronic Reactions, 122

Partial WavesIdenticalParticles, 144

Particle Zoo, 61Resonances, 73The Pion, 61

Polarization, 159Double Scattering, 161

Polarized-Ion Source, 167Potential

Coulomb, 12Screened, 18

Screened Colulomb, 29Proton

Discovery, 23

Radioactive Ion Beams, 56Reaction

Neutron-Proton Capture, 85Photonuclear, 83

Rutherford, 11Classical

Scattering Distance, 14Trajectories, 14

Cross SectionClassical, 12

Historic Significance, 18Scattering, 11

Scattering Length, 130nn, 135np, 131pp, 133

Schrödinger Equation, 99Symmetries

Exchange, 143

Time Reversal Violation, 125Time-Reversal Violation

Compound Reactions, 128Transfer Reactions, 103Transformation

Fourier, 31

Uncertainty Relation, 36

WavelengthVirtual Photons, 36

DR

AFT

List of Figures

3.1 The original setup of Rutherford’s, Geiger’s, and Marsden’sfirst nuclear scattering experiment at Manchester 1908 to 1913. 12rutherfordoriginal3

3.2 Classical Rutherford scattering. . . . . . . . . . . . . . . . . . 13rutherf

3.3 Minimal scattering distance as function of the scattering angle. 15minabst

3.4 The curve shows the angular dependence of the theoreticalRutherford cross section ∝ sin−4(θ/2). The points are theoriginal data (that consisted of tabulated numbers of countswith no error bars, and not transformed into cross-section val-ues) of Ref. [GEI13], adjusted to the theoretical curve, givinga nearly perfect fit (Nowadays data with at least an error es-timate or, better, error bars are mandatory). . . . . . . . . . . 19rutherfordcs1

4.1 Apparatus used by Rutherford from 1917 to 1920 to bom-bard 14N with α particles. The emitted particle radiation oflonger range was identified as consisting of Z=1, A=1 parti-cles, forming the nucleus of the hydrogen atom, and for whichRutherford coined the word “proton” in 1919. . . . . . . . . . 24ruthnew

4.2 Cloud chamber photograph by Blackett of the first nuclearreaction α + 14N → 17O + p observed by Rutherford in 1919[RUT19, BLA25]. . . . . . . . . . . . . . . . . . . . . . . . . . 25reaction1

5.1 Coulomb potential of a spherical homogeneous charge distri-bution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30coulomb1

5.2 Sampling functions for different momentum transfers showthat in order to sample details of a given structure (e.g. theshape around the radius of a nuclear density (charge or mass)distribution) the momentum transfer (given by the incidentenergy and the scattering angle) has to have an appropri-ate intermediate value. In the example shown the value ofK = 0.5 fm−1 is suitable for sampling the region around thenuclear radius of 5.0 fm. The vertical dotted lines indicate a10 to 90% sampling region. . . . . . . . . . . . . . . . . . . . . 32sampling

DR

AFT

194 LIST OF FIGURES

5.3 Squares of the Fourier transforms – basically the form fac-tors determining the shapes of the cross sections – of differentcharge-density distributions. . . . . . . . . . . . . . . . . . . . 33fourierneu

5.4 Cross-section angular distributions of 40 MeV α scattering fromheavy nuclei. The solid line is the pure point-Rutherford pre-diction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43wegner551

5.5 Same as Fig. 5.4, but plotted is the ratio of the measured crosssection and the calculated Rutherford cross section against thecalculated distance of closest approach, see Eq. 3.14, for α’son Au. This plot shows the relatively sudden onset of absorp-tion by the nuclear interaction, which allows, using the A1/3

law, the derivation of the range of the nuclear force of about1.4 fm. The data from Ref. [WEG55] has been augmented bydata from Indiana University and University of Washington[FAR54, WAL55]. . . . . . . . . . . . . . . . . . . . . . . . . . 44wegner552

5.6 Plot of the scattering cross sections (relative to the Ruther-ford cross section) as functions of the distance of closest ap-proach of many different HI pairings. The figures are fromRef. [CHR73](right) and Ref. [OGA78](left). . . . . . . . . . . 45christensen

5.7 Connection between electron scattering cross sections and den-sity distributions on pointlike and extended nuclei. . . . . . . 45hofstadternobel1

5.8 Differential cross section for 500 MeV electrons fitted by anexponential form factor. . . . . . . . . . . . . . . . . . . . . . 46stanford500

5.9 Charge density distribution and Coulomb potential of a pointcharge compared to an extended homogeneous charge distri-bution and its potential. . . . . . . . . . . . . . . . . . . . . . 47extendednuc

5.10 Experimental setup for the production of a beam of muonsand slowing-down and capturing the nuons into Bohr orbits.After [FIT53]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47fitchrainwater3

5.11 NaJ scintillator detector setup for the measurement of muonicX-rays, emitted in 2p-1s transitions to the ground states ofdifferent muonic atoms. After [FIT53]. . . . . . . . . . . . . . 48fitchrainwater4

5.12 Muonic X-ray spectrum of Pb obtained with a NaJ scintil-lation detector and showing the large energy shift betweena point charge and the actual extended-charge distributions.After [FIT53]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

DR

AFT

LIST OF FIGURES 195

5.13 Muonic X-ray spectrum of Ti obtained with a NaJ scintillationdetector and showing the large energy shift between a pointcharge and the actual extended-charge distributions. After[FIT53]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49fitchrainwater1

5.14 Charge density distributions of different doubly closed-shellnuclei with electron-scattering and muonic-atom data com-bined. After [FRO87]. . . . . . . . . . . . . . . . . . . . . . . 49frois10

5.15 Unresolved discrepancies between determinations of the pro-ton’s rms radius by different methods. The accepted CODATA-06 value is rrms(p) = 0.88768(69) fm whereas the new muonic-atom value is rrms(p) = 0.84184(67) fm. After [COD08, POH10]. 50kottmannrppuzzle

5.16 Stanford facility. . . . . . . . . . . . . . . . . . . . . . . . . . 50stanfordapp

5.17 Stanford spectrometer. . . . . . . . . . . . . . . . . . . . . . . 51stanfordspec

6.1 Halo nuclei at the driplines of the chart of nuclides. . . . . . . 56halochart1

6.2 Coat of arms and symbol of the Renaissance Borromean family(and other north Italian families like the Sforzas) at their castleon the Borromean island Isola Bella in the Lago Maggiore, Italy. 56borromean1

6.3 Fragment-momentum distribution and density distributions inhalo nuclei. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58momdens

6.4 Radii of halo nuclei. . . . . . . . . . . . . . . . . . . . . . . . . 58haloradien

7.1 Photoplate tracks of the decay of a cosmic pion into a muonand an invisible muonic (anti)neutrino, as seen through a mi-croscope. The typical length of the muon track is ≈0.61 mm. . 62Lattes11

7.2 Photoplate tracks of the reaction of a cosmic (assumed nega-tive) pion after slowing down and being captured in an atomicshell and cascading dow into the K shell before being destroyedin an hadronic interaction with a nucleus forming a “star”.Outgoing from the star strong tracks from two α’s from nu-clear interactions are seen together with other (meson?) tracks. 63lattes6

7.3 Experimental setup for the first detection of antiprotons atthe Berkeley Bevatron. . . . . . . . . . . . . . . . . . . . . . . 66chamberlain1

7.4 Transmission of detector arrangement for negative particleswith mass of ≈ mp, i.e. of antiprotons p . . . . . . . . . . . . . 67chamberlain2

DR

AFT

196 LIST OF FIGURES

7.5 Excitation curve of the antiproton production at the BerkeleyBevatron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68chamberlain3

7.6 Antineutrino reaction and detection scheme of the neutron-capture γ’s from Cd and positron-annihilation 511 keV γ’s ina suitable liquid scintillator containing a neutron moderator(water) and a Cd compound (e.g. CdCl2). . . . . . . . . . . . 69antineutrinodet1

7.7 View and schematic of the tank for the unique detection ofsolar neutrinos, from [CLE98] . . . . . . . . . . . . . . . . . . 70

7.8 Construction of the counter to measure the β activity of the37Ar produced by the solar neutrinos and extracted from thechlorine compound, see [CLE98]. . . . . . . . . . . . . . . . . 71hspropcounter

7.9 Double differential cross section of deep-inelastic scattering of500 MeV electrons [BLO69]. . . . . . . . . . . . . . . . . . . . 74bloom69

7.10 Big European Bubble Chamber at CERN. . . . . . . . . . . . 76BEBCCERN

7.11 Neutrino Scattering event in the Big European Bubble Cham-ber at CERN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76neutrinoevent

8.1 Apparatus used by Chadwick to discover the neutron, [CHA32b]. 80chadwickapparatus

9.1 View of the bremsstrahlung source, an electron acceleratorused in the experiment of Ref. [MOB50]. . . . . . . . . . . . . 84mobley1

9.2 The bremsstrahlung target consisted of a thin gold foil pro-ducing a gold bremsstrahlung spectrum, the reaction targetwas deuterium in the form of heavy water in a thin-walledcontainer, and the neutron detector was a 10BF3 proportionalcounter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85mobley2

9.3 The method of extrapolation of Ref. [MOB50] to the thresholdenergy of the 2H(γ,n)1H reaction is explained. . . . . . . . . . 86mobley3

9.4 Schematic view of the sample and different neutron and γsources, the paraffin thermalizer, and lead shielding of thedetector. Sources of 48V and 144Ce were used for calibration.From Ref. [VDL82]. . . . . . . . . . . . . . . . . . . . . . . . . 87vanderleun1

9.5 The spectrum of p(n,γ)d γ’s of the n-type intrinsic Ge detectorwith an energy resolution of 1.75 keV at 1.33 MeV. Source of48V and 144Ce wers used for calibration. From Ref. [VDL82]. . 87vanderleun2

DR

AFT

LIST OF FIGURES 197

9.6 The results of Ref. [VDL82] in comparison to earlier conflictingresults of Wapstra et al. [WAP77] (compilation of earlier data),Vylov et al. [VYL78], and Greenwood et al. [GRE80]. . . . . 88vanderleun3

10.1 Schematic of the voltage multiplication circuit used in the first“Cockroft-Walton” accelerator and accelerator tube design. . . 92cockx

10.2 Schematic of the accelerator setup. . . . . . . . . . . . . . . . 92cockroftsetup

10.3 Photograph of the accelerator complex. In the box covered bya black cloth the experimenter would sit and count scintilla-tions on a fluorescent screen e.g. as function of the scatteringangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93cockroftphoto

10.4 Reaction chamber setup with a vacuum pump system, a thinmica window allowing protons to exit or to impinge on a Litarget. In this latter case (left) a microscope was used tocount the reaction α’s as scintillations on a fluorescent screen.When using a second scintillator on the opposite side coinci-dent events could be observed. . . . . . . . . . . . . . . . . . . 94cockrofttarget

10.5 “Livingston plot”: Plot of the development of accelerators overthe years with a doubling of the “available” energy approxi-mately every seven years. The approximate energy rangesof (low-energy) nuclear physics proper, of intermediate-energyphysics where the overlap and interactions of quarks and nu-clei are studied, and high-energy (or particle) physics whereall facets of the standard model are investigated, are indicatedin the plot. Originally in Ref. [LIV54]. . . . . . . . . . . . . . 96livingplot

11.1 Angular distributions of 84 MeV neutrons scattered from dif-ferent targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Bratenahl

11.2 Fit with the first “Optical Model” . . . . . . . . . . . . . . . . 101Lelevier

11.3 Form factors of the optical model. Upper: Woods-Saxon formof the real part f. Center: Derivative Woods-Saxon form g =f’ of the imaginary part. Lower: Sliding-transition form of thesurface-to-volume imaginary part as function of energy. . . . . 102omff

11.4 Global fit of the optical model to elastic scattering data of14.5 MeV protons for a large nuclear mass range. The crosssections are normalized to the Rutherford cross sections (i.e.to 1 at 0 ), the analyzing powers are 0 at 0 . The arrowsindicate the systematic variation of characteristic diffractionmaxima with the target mass. . . . . . . . . . . . . . . . . . . 104

DR

AFT

198 LIST OF FIGURES

sigayom1

11.5 Angular and energy dependence of the cross section of elasticproton scattering from 90Zr calculated with standard Greenlees-Becchetti parameters of the optical model. The interferencestructure of the angular distributions may be interpreted as“resonant” (single-particle) structures of the excitation func-tion with widths typical for fast (i.e. direct) processes. Theyare also analogous to diffraction structures in classical optics. . 105om2

11.6 Characteristic and systematic features of the stripping maxi-mum as function of the transferred orbital angular momentumℓ. The arrows indicate the increase of the reaction angle of thestripping peak with ℓ. . . . . . . . . . . . . . . . . . . . . . . . 106stripmax

11.7 Sensitivity (sign!) of the analyzing power of the stripping re-action to the total angular momentum j of the transferrednucleon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107aydp1

11.8 The behavior of the squares of the lowest-order spherical Besselfunctions jℓ(kr) as functions of x = kr. . . . . . . . . . . . . . 109Bessel

12.1 The excitation of single resonances, overlapping resonances(with and without Ericson fluctuations) and giant resonancesas functions of the energy in the continuum region above thebound-state energy. . . . . . . . . . . . . . . . . . . . . . . . . 114giantres

12.2 Boundary conditions at the nuclear (potential) surface for theappearance of a resonance in the excitation function. . . . . . 116res2

13.2 Setup of the first hadronic parity-violation experiment at LosAlamos [NAG79]. . . . . . . . . . . . . . . . . . . . . . . . . . 124

13.3 Measured values of the longitudinal analyzing power in ppscattering at 13.6 MeV (Bonn), 15 MeV (Los Alamos), 45 MeV(SIN/PSI), 221 MeV (TRIUMF), and 800 MeV (LANL), incomparison to predictions of the standard model (DDH: Des-planques, Donoghue, and Holstein [DES80]). . . . . . . . . . . 125

13.4 Situation of incident and exit momenta and polarization vec-tors before and after the time-reversal operation of reversingspins and momenta in A leading to situation B in the center-of-mass system. The angles χi and χf are arbitrary. Thesituations must be Lorentz-transformed to describe them inthe lab. system. From Ref. [HAN67]. . . . . . . . . . . . . . . 126handlersit

13.5 Plot of the angular distributions of the reactions 12C(α,d)14Nand 14N(d,α)12C. . . . . . . . . . . . . . . . . . . . . . . . . . 128bodanskiresult

DR

AFT

LIST OF FIGURES 199

13.6 Experimental setup with liquid H2 target vessel of Ref. [SUT47].132suttontarget

13.7 The cross sections of thermal neutron scattering on ortho andpara H2 from Ref. [SUT47]. The data clearly favor a totalcross section, extrapolated to E = 0, of 19.7 b, together witha range of the nuclear force of 1.54 fm. . . . . . . . . . . . . . 133suttonresults

13.8 Setup for the production of very slow neutrons from the reac-tion 9Be(d,n)10B, after slowing down the neutrons in paraffine.The pulsed deuteron beam allows the selection of narrow en-ergy intervals of the neutrons by µs gating and time-of-flightmeasurement along the 5.4 m flight path to the hydrogen tar-get. The figure was redrawn from [RAI46] for clarity. . . . . . 134rainwatersetup

13.9 Result of the np cross-section measurement of Ref. [MEL49].The figure was redrawn for clarity. . . . . . . . . . . . . . . . . 135melkoresults

13.10Experimental setup and kinematics of the D(π−, γnn) reaction[HAD65] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136haddocksetup

13.11Scheme of the zero-crossing method of the analyzing power[ZHA98]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138abeggscheme

13.12General and detailed experimental setup of the CSB experi-ment at TRIUMF [ZHA98]. . . . . . . . . . . . . . . . . . . . 138abeggsetup

14.1 Trajectories of identical particles in the c.m. system. . . . . . 144ident

14.2 Experimental c.m. angular distributions of Coulomb scatteringof two identical bosons (12C) and fermions (13C) as well as oftwo non-identical particles of nearly equal masses and theoret-ical cross sections at Elab = 7 MeV. The angular distributionfor the non-identical particles is obtained when the spectra ofthe forward and backward scattered particles cannot be sep-arated by the detector, which is the case for (nearly) equalmasses. Otherwise one would obtain a typical Rutherford dis-tribution for the forward-scatterd particle and a distributionreflected about 90 for the recoil particle. The data were mea-sured by students of an advanced lab. course at IKP Colognein 2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145c12c13test

DR

AFT

200 LIST OF FIGURES

14.3 The scattering chamber setup of Ref. [BRO61] is shown. It istypical for tandem-VdG experiments with charged particles.Important are a well focussed or collimated beam, thin foiltargets, well-defined solid angles at the detectors, precise angledefinition of the detector slits, and a Faraday cup at the exitfor beam charge calibration. For heavy-ion experiments, how-ever, this cannot be used because the charge state equilibriumof the incident beam at the reaction is unknown. Therefore,a cross-section calibration such as using a heavy target (e.g.gold) and calculable Rutherford cross section has to be usedfor normalization . . . . . . . . . . . . . . . . . . . . . . . . . 146bromleysetup

14.4 Cross-section angular distributions and excitation functions of12C-12C and 16O-16O elastic scattering . . . . . . . . . . . . . . 147bromleywinkel

14.5 Schematic view of the possible shape of a potential allowingfor molecular states . . . . . . . . . . . . . . . . . . . . . . . . 148bettspotential

14.6 Excitation functions of 12C-12C and 16O-16O elastic scattering 149bettsanrf

15.1 Museum exhibit at the Deutsches Museum, Munich: the fa-mous Hahn-Meitner-Strassmann desk with the collected piecesof experimental equipment and Hahn’s laboratory logbook.Very probably the equipment has been used mostly by L. Meit-ner, she being the physicist in the group of nuclear chemists.The parts were actually used in different rooms. . . . . . . . . 154hahntisch

16.1 Simplified level schemes of 5Li and 5He as of 1966 [AIZ66]showing the P-wave splitting . . . . . . . . . . . . . . . . . . . 160A5isobar

16.2 Apparatus to measure protons doubly scattered from 4He ina gas target chamber. The protons were registered in theemulsion of photoplates. . . . . . . . . . . . . . . . . . . . . . 162Heusinkveld

16.3 Predictions for the proton polarization to be measured in adouble-scattering experiment 4He(p, p)4He. . . . . . . . . . . . 162Heusinkveld6

16.4 Reproduction of the table of results of Ref. [HEU50] of thedouble-scattering ratios of 4He(p, p)4He. The ratios clearly fa-vor the “inverted” sequence of levels, i.e. the 2P3/2 assignmentfor the ground state and 2P1/2 for the broad first excited stateof 5Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Heusinkveldtab

DR

AFT

LIST OF FIGURES 201

17.1 Schematic showing the first atomic-beam polarized ion sourceconnected to an accelerator and used for the nuclear reaction3H(~d, n)4He thereby testing thes sensitivity of this reaction asa polarization analyzer and checking the theoretical assump-tions on the 107 keV resonance in 5He as a pure s-wave reso-nance allowing no vector polarization sensitivity. After [RUD61].169baselquelle

17.2 Construction of the atomic-beam polarized ion source usingan rf-discharge dissociator, a glass-capillary collimator, longquadrupole magnets for spin-state separation, and an electron-collision ionizer to produce a positive beam of partly tensor-polarized deuterons. After [RUD61]. . . . . . . . . . . . . . . 170baselabs

17.3 Construction details of the beam-forming parts of the atomic-beam polarized ion source using an rf-discharge dissociator,a glass-capillary collimator, and long quadrupole magnets forspin-state separation. After [RUD61]. . . . . . . . . . . . . . . 170baselps

17.4 Angular distribution of the tensor analyzing power (here calledP33) of the reaction 3H(~d,n)4He measured at the first Baselsource, compared to the prediction and with a best fit yieldingPZZ = −0.245. After [RUD61]. . . . . . . . . . . . . . . . . . 171baselres

18.1 Photonuclear (γ,n) excitation functions on 12C before and af-ter deconvolution of the bremsstrahlung spectrum producedby a betatron with Ee = 100 MeV. From Ref. [BAL47]. . . . . 176baldwin1

18.2 Giant resonance on Tb. From Ref. [FUL58b]. . . . . . . . . . 177fuller

18.3 Setup to produce γ’s via positron in-flight annihilation and de-tection of the (γ,n) neutrons in BF3 detectors. From Ref. [BRA64].177bramblett1

18.4 Measurement of the (γ,xn) excitation function for 159Tb. FromRef. [BRA64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 178bramblett2

18.5 Giant resonances, e.g. monopole, quadrupole and octupole ex-citations, in inelastic α scattering (α, α′) with ∆L = 1, 2, 3 atEα = 152 MeV. High background makes the disentangling ofdifferent resonances difficult and may require model assump-tions. Measurements are best done at small forward angles.From [?]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

18.6 Giant resonances classified according to their multipolarity∆L, their spin and isospin changes ∆S, and ∆T . . . . . . . . 182

18.7 GDR in the photoneutron (γ,n) reaction on isotopes of Ndwith increasing neutron number N, showing onset of deforma-tion. Adapted from [BER75]. . . . . . . . . . . . . . . . . . . 183

DRAFT - Universität zu Kölnschieck/SKRIPT/key4.pdf · for the study of the external and internal structure of the proton by elctron scattering). In nuclear physics, a ﬁeld of

Documents