Model Model Inti

Introduction to Nuclei Physics

1

Wednesday, Mach, 23, 2011

Arif Hidayat

1. Nature of the Nuclear Force • Shape of the Nuclear Potential

• Yukawa Potential

• Range of Yukawa Potential

2. Nuclear Models • Liquid Drop Model

• Fermi Gas Model

• Shell Model

• A square well nuclear potential provides the basis of quantum theory with discrete energy levels and corresponding bound state just like in atoms – Presence of nuclear quantum states have been confirmed

through • Scattering experiments • Studies of the energies emitted in nuclear radiation

• Studies of mirror nuclei and the scatterings of protons and neutrons demonstrate – Without the Coulomb effects, the forces between two

neutrons, two protons or a proton and a neutron are the same • Nuclear force has nothing to do with electrical charge • Protons and neutrons behave the same under the nuclear force

– Inferred as charge independence of nuclear force.

2

Nuclear Potential

• Strong nuclear force is independent of the electric charge carried by nucleons – Concept of strong isotopic-spin symmetry.

• proton and neutron are the two different iso-spin state of the same particle called nucleon

– In other words, • If Coulomb effect were turned off, protons and neutrons

would be indistinguishable in their nuclear interactions

• Can you give another case just like this???

– This is analogues to the indistinguishability of spin up and down states in the absence of a magnetic field!!

• This is called Iso-spin symmetry!!!

3

Nuclear Potential – Iso-spin symmetry

• EM force can be understood as a result of a photon exchange – Photon propagation is described by the Maxwell’s

equation – Photons propagate at the speed of light. – What does this tell you about the mass of the

photon? • Massless

• Coulomb potential is

• What does this tell you about the range of the Coulomb force? – Long range. Why?

4

Range of the Nuclear Force

V r 1

r

Massless

particle

exchange

• For massive particle exchanges, the potential takes the form

– What is the mass, m, in this expression?

• Mass of the particle exchanged in the interaction – The force mediator mass

• This form of potential is called Yukawa Potential – Formulated by Hideki Yukawa in 1934

• What does Yukawa potential turn to in the limit m 0? – Coulomb potential

5

Yukawa Potential

V r

mcr

e

r

• From the form of the Yukawa potential

• The range of the interaction is given by some characteristic value of r. What is this? – Compton wavelength of the mediator with

mass, m:

• What does this mean? – Once the mass of the mediator is known, range

can be predicted – Once the range is known, the mass can be

predicted 6

Ranges in Yukawa Potential

mcr

eV r

r

re

r

mc

• Let’s put Yukawa potential to work • What is the range of the nuclear force?

– About the same as the typical size of a nucleus • 1.2x10-13cm

– thus the mediator mass is

• This is close to the mass of a well known p meson (pion)

• Thus, it was thought that p are the mediators of the nuclear force

7

Ranges in Yukawa Potential

2mc

mp

c

197164

1.2

MeV fmMeV

fm

2139.6 / ;MeV c

2

0 135 /m MeV cp

mp

• Experiments showed very different characteristics of nuclear forces than other forces

• Quantification of nuclear forces and the structure of nucleus were not straightforward

– Fundamentals of nuclear force were not well understood

• Several phenomenological models (not theories) that describe only limited cases of experimental findings

• Most the models assume central potential, just like Coulomb potential

8

Nuclear Models

• An earliest phenomenological success in describing binding energy of a nucleus

• Nucleus is essentially spherical with radius proportional to A1/3. – Densities are independent of the number of nucleons

• Led to a model that envisions the nucleus as an incompressible liquid droplet – In this model, nucleons are equivalent to molecules

• Quantum properties of individual nucleons are ignored

9

Nuclear Models: Liquid Droplet Model

• Nucleus is imagined to consist of

– A stable central core of nucleons where nuclear force is completely saturated

– A surface layer of nucleons that are not bound tightly

• This weaker binding at the surface decreases the effective BE per nucleon (B/A)

• Provides an attraction of the surface nucleons towards the core just as the surface tension to the liquid

10

Nuclear Models: Liquid Droplet Model

• If a constant BE per nucleon is due to the saturation of the nuclear force, the nuclear BE can be written as:

• What do you think each term does?

– First term: volume energy for uniform saturated binding

– Second term corrects for weaker surface tension

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Liquid Droplet Model: Binding Energy

BE

• This can explain the low BE/nucleon

behavior of low A nuclei

– For low A nuclei, the proportion of the

second term is larger.

– Reflects relatively large number of

surface nucleons than the core.

1a A 2 3

2a A

• Small decrease of BE for heavy nuclei can be understood as due to Coulomb repulsion – The electrostatic energies of protons have destabilizing

effect

• Reflecting this effect, the empirical formula for BE takes the correction term

• All terms of this formula have classical origin. • This formula does not explain

– Lighter nuclei with the equal number of protons and neutrons are stable or have a stronger binding (larger –BE)

– Natural abundance of stable even-even nuclei or paucity of odd-odd nuclei

• These could mainly arise from quantum effect of spins.

12


2 31 2BE a A a A 2 1 3

3a Z A

• Additional corrections to compensate the deficiency, give corrections to the empirical formula (again…)

– All parameters are assumed to be positive

– The forth term reflects N=Z stability

– The last term • Positive sign is chosen for odd-odd nuclei, reflecting

instability

• Negative sign is chosen for even-even nuclei

• For odd-A nuclei, a5 is chosen to be 0.

13


2 3 2 1 31 2 3BE a A a A a Z A

2

4

N Za

A

3 45a A

• The parameters are determined by fitting experimentally observed BE for a wide range of nuclei:

• Now we can write an empirical formula for masses of nuclei

• This is Bethe-Weizsacker semi-empirical mass formula

– Used to predict stability and masses of unknown nuclei of arbitrary A and Z

14


1 15.6a MeV 2 16.8a MeV 3 0.72 a MeV

4 23.3 a MeV 5 34 ; a MeV

2

, n p

BEM A Z A Z m Zm

c n pA Z m Zm

12

aA

c 2 32

2

aA

c 2 1 33

2

aZ A

c

2

42

N Za

Ac

3 452

aA

c

• An early attempt to incorporate quantum effects

• Assumes nucleus as a gas of free protons and neutrons confined to the nuclear volume

– The nucleons occupy quantized (discrete) energy levels

– Nucleons are moving inside a spherically symmetric well with the range determined by the radius of the nucleus

– Depth of the well is adjusted to obtain correct binding energy

• Protons carry electric charge Senses slightly different potential than neutrons

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Nuclear Models: Fermi Gas Model

• Nucleons are Fermions (spin ½ particles) so – Obey Pauli exclusion principle – Any given energy level can be occupied by at most

two identical nucleons – opposite spin projections

• For a greater stability, the energy levels fill up from the bottom to the Fermi level – Fermi level: Highest, fully occupied energy level (EF)

• Binding energies are given as follows: – BE of the last nucleon= EF since no Fermions above

EF – In other words, the level occupied by Fermion

reflects the BE of the last nucleon

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• Experimental observations show BE is charge independent

• If the well depth is the same for p and n, BE for the last nucleon would be charge dependent for heavy nuclei (Why?)

– Since there are more neutrons than protons, neutrons sit higher EF

17


Same Depth Potential Wells

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Neutron Well Proton Well

Nuclear b-decay

nFE

pFE…

…

en e p

• Experimental observations show BE is charge independent • If the well depth is the same for p and n, BE for the last

nucleon would be charge dependent for heavy nuclei (Why?) – Since there are more neutrons than protons, neutrons sit higher

EF

– But experiments observed otherwise • EF must be the same for protons and neutrons. How do

we make this happen? – Make protons move to a shallower potential well

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• What happens if this weren’t the

case?

– Nucleus is unstable.

– All neutrons at higher energy levels

would undergo a b-decay and

transition to lower proton levels

• Fermi momentum: • Volume for momentum space up to Fermi level • Total volume for the states (kinematic phase space)

– Proportional to the total number of quantum states in the system

• Using Heisenberg’s uncertainty principle: • The minimum volume associated with a physical

system becomes • The nF that can fill up to EF is

20

3

22

TOTF

Vn

p

Fermi Gas Model: EF vs nF

2F Fp mE

FpV

FTOT pV V V

x p

3

2stateV p

30

4

3r A

p

23

0

4

3FA r p

p

23

03

2 4

32FA r p

p

p

3

04

9

Fr pA

p

2 2F FE p m

34

3Fp

p

34

3Fp

p

2

Why?

• Let’s consider a nucleus with N=Z=A/2 and assume that all states up to Fermi level are filled

• What do you see about pF above? – Fermi momentum is constant, independent of the number

of nucleons

• Using the average BE of -8MeV, the depth of potential well (V0) is ~40MeV – Consistent with other findings

• This model is a natural way of accounting for a4 term in Bethe-Weizsacker mass formula

21

Fermi Gas Model: EF vs nF

2

AN Z

1 3

0

9

8Fp

r

p

or

FE

3

04

9

Fr pA

p

2

2

Fp

m

2 2 3

0

1 9

2 8m r

p

2

20

2.32

2

c

rmc

2.32 19733

2 940 1.2

MeV fmMeV

fm

• Exploit the success of atomic model

– Uses orbital structure of nucleons

– Electron energy levels are quantized

– Limited number of electrons in each level based on available spin and angular momentum configurations • For nth energy level, l angular momentum (l<n), one

expects a total of 2(2l+1) possible degenerate states for electrons

22

Nuclear Models: Shell Model

• Orbits and energy levels an electron can occupy are labeled by – Principle quantum number: n

• n can only be integer

– For given n, energy degenerate orbital angular momentum: l • The values are given from 0 to n – 1 for each n

– For any given orbital angular momentum, there are (2l+1) sub-states: ml

• ml=-l, -l+1, …, 0, 1, …, l – l, l

• Due to rotational symmetry of the Coulomb potential, all these sub-states are degenerate in energy

– Since electrons are fermions w/ intrinsic spin angular momentum , • Each of the sub-states can be occupied by two electrons

– So the total number of state is 2(2l+1) 23

Atomic Shell Model Reminder

2

• Exploit the success of atomic model

– Uses orbital structure of nucleons

– Electron energy levels are quantized

– Limited number of electrons in each level based on available spin and angular momentum configurations • For nth energy level, l angular momentum (l<n), one expects a

total of 2(2l+1) possible degenerate states for electrons

• Quantum numbers of individual nucleons are taken into account to affect the fine structure of spectra

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• Nuclei have magic numbers just like inert atoms

– Atoms: Z=2, 10, 18, 36, 54

– Nuclei: N=2, 8, 20, 28, 50, 82, and 126 and Z=2, 8, 20, 28, 50, and 82

– Magic Nuclei: Nuclei with either N or Z a magic number Stable

– Doubly magic nuclei: Nuclei with both N and Z magic numbers Particularly stable

• Explains well the stability of nucleus

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• To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential

–

– Details of nuclear potential not well known

• A few shapes of potential energies tried out

– Infinite square well: Each shell can contain up to 2(2l+1) nucleons

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Shell Model: Various Potential Shapes

2

2

20

mE V r r

27

Nuclear Models: Shell Model – Square

well potential case

NM n l=n-1 Ns=2(2l+1) NT

2 1 0 2 2

8 2 0,1 2+6 8

20 3 0,1,2 2+6+10 18

28 4 0,1,2,3 2+6+10+14 32

50 5 0,1,2,3,4 2+6+10+14+18 50

82 6 0,1,2,3,4,5 2+6+10+14+18+22 72

• To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential

–

– Details of nuclear potential not well known

• A few models of potential tried out

– Infinite square well: Each shell can contain up to 2(2l+1) nucleons

• Can predict 2, 8 and 50 but no other magic numbers

– Three dimensional harmonic oscillator:

• Predicts 2, 8, 20, 40 and 70 Some magic numbers predicted

28

Shell Model: Various Potential Shapes

V r 2 21

2m r

2

2

20

mE V r r

• Central potential could not reproduce all magic numbers

• In 1940, Mayer and Jesen proposed a central potential + strong spin-orbit interaction w/

– f(r) is an arbitrary empirical

function of radial coordinates and chosen to fit the data

• The spin-orbit interaction with the properly chosen f(r), a finite square well can split

• Reproduces all the desired magic numbers

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Shell Model: Spin-Orbit Potential

TOTV

Spectroscopic notation: n L j

Orbit number Orbital angular

momentum Projection of

total momentum

V r f r L S

• Spin-Parity of large number of odd-A nuclei predicted well – Nucleons are Fermions so the obey Pauli exclusion

principle

– Fill up ground state energy levels in pairs

– Ground state of all even-even nuclei have zero total angular momentum

• The shell model cannot predict stable odd-odd nuclei spins – No prescription for how to combine the unpaired

proton and neutron spins

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Predictions of the Shell Model

• Magnetic Moment of neutron and proton are

• Intrinsic magnetic moment of unpaired nucleons contribute to total magnetic moment of nuclei

– What does a deuteron consist of?

• Measured value is

– For Boron (10B5) , the 5 neutrons and 5 protons have the same level structure: (1S1/2)2(1P3/2)3, leaving one of each unpaired proton and neutron in angular momentum l=1 state

• Measured value is

• Does not work well with heavy nuclei 31

Predictions of the Shell Model

D

2.79p N 1.91n N

D

B

1.80B N

2.79 1.91N N N

p 2.79 N 1.91 N 0.88 Nn

0.86 N

2 N

el

m c N1

2 N

e

m c

p n orbit 1.88 N

• For heavy nuclei, shell model predictions do not agree with experimental measurements – Especially in magnetic dipole moments

• Measured values of quadrupole moments for closed shells differ significantly with experiments – Some nuclei’s large quadrupole moments suggests

significant nonspherical shapes – The assumption of rotational symmetry in shell model

does not seem quite right

• These deficiencies are somewhat covered through the reconciliation of liquid drop model with Shell model – Bohr, Mottelson and Rainwater’s collective model,

1953 32

Collective Model

• Assumption – Nucleus consists of hard core of nucleons in filled shells – Outer valence nucleons behave like the surface molecules in a

liquid drop – Non-sphericity of the central core caused by the surface motion of

the valence nucleon

• Thus, in collective model, the potential is a shell model with a spherically asymmetric potential – Aspherical nuclei can produce additional energy levels upon

rotation while spherical ones cannot

• Important predictions of collective model: – Existence of rotational and vibrational energy levels in nuclei – Accommodate decrease of spacing between first excite state and

the ground level for even-even nuclei as A increases, since moment of inertia increases with A

– Spacing is largest for closed shell nuclei, since they tend to be spherical

33

Collective Model

• Nuclei tend to have relatively small intrinsic spins

• Particularly stable nuclei predicted for A between 150 and 190 with spheroidal character – Semi-major axis about a factor of 2 larger than semi-minor

• Heavy ion collisions in late 1980s produced super-deformed nuclei with angular momentum of

• The energy level spacings of these observed through photon radiation seem to be fixed

• Different nuclei seem to have identical emissions as they spin down

• Problem with collective model and understanding of strong pairing of nucleon binding energy

• Understanding nuclear structure still in progress 34

Super-deformed Nuclei

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Model Model Inti

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Model Model Inti