Nuclear Physics - Colorado School of Minesinside.mines.edu/~fsarazin/phgn310/PDFs/Slides/13b.Nuclear Physics.pdfFred Sarazin ([email protected]) PHGN300/310: Atoms, molecules and

PHGN300/310: Atoms, molecules and solidsFred Sarazin ([email protected])Physics Department, Colorado School of Mines

Nuclear Physics


Part I: Structure of the nucleus


• All the nuclei can be made with:p proton positively charge (+q)n neutron neutral (James Chadwick, 1932)

Z number of protons in the nucleusN number of neutrons in the nucleusA atomic number = Z + N

• The atoms are neutral:Charge of the nucleus: + Z.qElectron cloud is made of Z electrons of charge (–q)The electrons determine the chemical behavior à Z defines the Element

• Two nuclei with a same Z but different N (or A) are isotopes (of the same element)

Elementary constituents of the nucleus


• Examples: Hydrogen

Deuterium

Isotopes of the same element

Carbon

Uranium

The Z number is redundant with the element symbol & the N number is obsolete

• The symbol of an atomic nucleus is .where Z = atomic number (number of protons)

N = neutron number (number of neutrons)A = mass number (Z + N)X = chemical element symbol

Notations

𝑋"#$

𝐻&''

𝐻''(

𝐶**'(

𝑈',-.((-/

𝐶'(

𝑈(-/


Stable isotopes (~270), up to 209Bi (Z=83)

Unstable (radioactive) isotopes (>2000 and counting)

N=Z

The chart of nuclei


Enrico Fermi

Atom Nucleus

Force Coulomb Strong

Binding Energy The hydrogen atom: 13.6 eV

2H: 2.2 x 106 eV à 2.2 MeV

• Size: the order of magnitude is 1fm = 10-15m (a femtometer or fermi)

• Radius (assuming nucleus = sphere): R = R0.A1/3 with R0 = 1.2 fm

[Nucleus made of A nucleons, each of them has a R0=1.2fm radius àVnucleon = (4/3)pR0

3 ; Vnucleus = A . Vnucleon]

• Nuclear Density: 1017 kg/m3 à 100 million tons per cm3 !!! Density found in the core of a neutron star. Nuclear matter is uncompressible (properties of the strong force)

• Binding energy = [mass of the constituents – mass of the product] c2

Units and dimensions


• Calculate the density of 238U (M=238.05 u, 1u = 1.6605x10-27 kg)

• Based on this density, what would be the weight of a 1cm radius sphere of nuclear matter?

Exercise


• Short Range, only a few fm• Repulsive at very short range (< 1 fm)• Attractive over a range of a few fm

à a given nucleon only interacts with its next neighbors in the nucleus

• Nuclear Matter is uncompressible

• Negligible at long distances

• Same force for protons and neutrons

The nuclear / strong force


• Protons and Neutrons are Fermions (half-integer spin particles) à Fermi-Dirac statistics

• Density ~ constant“Skin” e.g. the nucleus has a diffuse surface

(Valid for light nuclei)

Nuclear properties

𝜌 𝑟 =𝜌&

1 + 𝑒 678 9⁄


Reminder: The hydrogen atomwe gain energy by bringing together the proton and the electron (13.6 eV)

Equivalence Energy ßà Mass: 𝐸 = 𝑚𝑐(

Binding energy (B) = [mass of the constituents – mass of the product] c2

= (mpc2+mec2) – mHc2

= 13.6 eVwith: mpc2 = rest mass of the proton = 938.28 MeV

mec2 = rest mass of the electron = 0.511 MeV = 511 keVè the Hydrogen atom weighs less than its constituentsè One needs to provide 13.6 eV to take the hydrogen atom apart

Any atom (defined by Z): B = mNucleusc2 + Z. mec2 – mAtomc2

àHow do we calculate mNucleusc2 ?

Binding energy


• The atomic mass unit (amu or u) is defined as 1/12th of the mass of 12C with all its electrons, that is:

– 1𝑢𝑐( = ''(𝑀 𝐶'( 𝑐( = '

'(6𝑚B + 6𝑚C + 6𝑚D 𝑐(

– 1u = 1.6605x10-27 kg = 931.5 MeV/c2

• All masses given in amu are the ATOMIC mass, i.e. they include Z electrons as well.

• In order to take into account the electron mass in the calculations, we replace (mp+me-) by mH, the mass of the hydrogen atom including the electron.

– 𝐵 𝑋#$ = 𝑍𝑚B + 𝑍𝑚C + 𝑁𝑚D 𝑐( −𝑚I𝑐( becomes:

– 𝐵 𝑋#$ = 𝑍𝑚J + 𝑁𝑚D 𝑐( −𝑚I𝑐(

• Hence, instead of using: mp = 1.007276u, one use mH = 1.007825u

Atomic mass unit (amu)


The nucleus:1H nucleus = free proton à Binding energy = 0 (but still has a rest mass)2H (deuterium) = one proton and one neutron

à 𝐵 𝐻'( = 𝑚J +𝑚D 𝑐( −𝑚K𝑐( = 2.2𝑀𝑒𝑉

𝑋:#$ 𝐵 𝑋#$ = 𝑍𝑚J + 𝑁𝑚D 𝑐( −𝑚I𝑐(

Example:56Fe (Z=26)

m56Fe = 55.934969 u (or amu, atomic mass unit = 931.5 MeV/c2)with: mH = 1.007825 u & mn = 1.008665 u

B(56Fe) = 26 x 1.007825u.c2 + 30 x 1.008665u.c2 – 55.934969u.c2

= 0.528461u.c2 = 492.3 MeVVery large number compared to B’s in atomic physics

Manufacturing 1g of 56Fe from protons and neutrons per second would generate 8.5.1011 Watt à 850 GW !!!

Binding energy of the nucleus


• Average binding energy produced by the strong force can be expressed by dividing the total Binding Energy of the nucleus by its mass number (B/A): B/A ~ 7-8 MeV is a typical value

Average binding energy


Part II: Radioactive decays


a

Coulomb Barrier (for the charged particles)

protonsneutrons

U

Square well potential (approx.)

• Filled levels – Quantified energy, Pauli principle. If all nucleons are in U<0, no nucleon can tunnel to the outside

à Stable nucleus• If unbound levels are filled (U>0), tunneling is

possible through the Coulomb barrierà Radioactive nucleus

Distance from thecenter of the nucleus

Tunnel effect (a-decay)Unbound levels (U>0)

Boundlevels (U<0)

A (simplified) model of the nucleus


Stable isotopes (~270), up to 209Bi (Z=83)

Unstable (radioactive) isotopes (>2000 and counting)

N=Z

The chart of nuclei


• Radioactive (Unstable) nuclei decay if there is an energetically more favorable condition, which it is trying to reach

• Change into another element: X àY + decay particle(s)

à a-decay: Emission of He-nucleus from the unstable nucleus

à b-decay: Emission of electron or positron (antiparticle of the electron) from the unstable nucleus

à Spontaneous fission: The radioactive nucleus breaks into so-called “fission fragments”

• An other decay mode (within the nucleus): X* à X + g’s

à g-decay: De-excitation from one excited state to a bound state (…or a less excited one !)

Radioactive decays


• Radioactive decay represents changes of an individual nucleus.

• HOWEVER: quantum mechanics prevents us from describing the decay of a single nucleus !!!

• Because, usually one looks at the decay of a large number of nuclei (N>>1), one can describe radioactive decay statistically.

• Decay is proceeding at a certain rate à Activity (A) [decays/s]

• Units:1 Becquerel = 1 Bq = 1 decay per second1 Curie = 1 Ci = 3.7 x 1010 decays per second (old unit still widely used)

(activity of 1g of radium)

Houston, we have a problem.

How to describe radioactive decays


• The activity of a certain sample (e.g. source) depends on the number N of radioactive nuclei and on the probability l (decay constant) for each nucleus to decay:

• Evolution with time: 𝑑𝑁 = −𝐴 S 𝑑𝑡 = −𝜆𝑁𝑑𝑡 with 𝑑𝑁, the number of nuclei that decayed during time 𝑑𝑡.

• By solving the differential equation, we get the radioactive decay law.

[1/s] [1/s]

Formalism (I)

𝐴 = 𝜆 S 𝑁

𝑁 𝑡 = 𝑁&𝑒7VW

with 𝑁 𝑡 = 0 = 𝑁& (boundary condition)


• Similarly, we find that the activity of a source change with time:

• From the decay constant l, one can define 𝝉 = 𝟏 𝝀⁄ , the mean lifetime

• Widely used: the half-life T1/2, the time after which half of the initial nuclei have decayed:

The half-life is characteristic to the decay of a given nucleus. This number (when known) is tabulated.

Formalism (II)

A 𝑡 = 𝐴&𝑒7VW

𝑇'/( =ln 2𝜆

= 𝜏 ln 2


• A sample of 18F is used internally as a medical diagnostic tool by observing this isotope’s positron decay (t1/2=110min). How much time does it take for 99% of the 18F to decay?

Exercise


• The counting rate from a radioactive source is 4000 counts per second at time t=0. After 10s, the counting rate is 1000 counts per second.

1. What is the half-life of the radioactive nucleus?

2. What is the counting rate expected after 20s?

Exercise


• Conservation of Energy

• Conservation of linear momentum

• Conservation of angular momentum

• Conservation of electric charge

• Conservation of mass number Aà The total number of nucleons is conserved, but Z and N can change

Conservation laws


• Let the radioactive nucleus 𝑋#$ be called the parent and have the mass 𝑀 𝑋#$ :

• Two or more products can be produced in the decay.• Let the lighter one be 𝑀a and the mass of the heavier one (daughter) be 𝑀K.• The conservation of energy is:

where Q is the energy released (disintegration energy) and equal to the total kinetic energy of the reaction products.

• If 𝑄 > 0, a nuclide is unbound, unstable, and may decay.• If 𝑄 < 0 , decay emitting nucleons do not occur.

Alpha, beta and gamma decays

𝑀 𝑋#$ = 𝑀K +𝑀a +𝑄𝑐(

𝑄 = 𝑀 𝑋#$ −𝑀K −𝑀a 𝑐(


a-particle = 𝐻𝑒(, : cluster in the nucleus – a: very tightly bound system

U

Tunneling effect (decay probability)

r

• Q-value energy (Q>0) for spontaneous decay. However, actual probability of decay can be very small (lifetime of 𝑋#$ high)

a-decay

𝑋#$ → 𝐷#7($7, + 𝛼

𝑄 = 𝑀 𝑋#$ −𝑀 𝐷#7($7, − 𝑀 𝐻𝑒(

, 𝑐(


Tunneling: a-decay with energy E2more likely than E1

a-decay


Exercise

• What is the Q-value of the reaction 𝑈.((-& → 𝛼 + 𝑇ℎ.&

((* ?

• Data: 𝑀 𝑈.((-& = 230.033927𝑢;𝑀 𝐻𝑒(

, = 4.002603𝑢;𝑀 𝑇ℎ.&((* = 226.024891𝑢

𝑄 = 𝑀 𝑋#$ −𝑀K −𝑀a 𝑐(


• How to calculate the kinetic energy of the a particle (and the kinetic energy of the daughter nucleus) ?– Conservation of the energy: 𝑄 = 𝐾K + 𝐾p– Conservation of the (linear) momentum: �⃗�s = 0 = �⃗�K + �⃗�p

NOTE: the kinetic energy of the a particle is well defined

a-decay: two-body kinematics

• One can show*:

𝐾p =𝑀K

𝑀K +𝑀p𝑄 ≈

𝐴 − 4𝐴

𝑄

* See derivation


• Weak interaction can transform a proton into a neutron or a neutron into a proton (It’s actually happening at the quark level)

• INSIDE a nucleus:• b- decay: 𝑛 → 𝑝 + 𝑒7 + �̅�C 𝑒7: electron / �̅�C: electron anti-neutrino• b+ decay: p → 𝑛 + 𝑒y + 𝜈C 𝑒y: positron / 𝜈C: electron neutrino• Electronic Conversion: p + 𝑒7 → 𝑛 + 𝜈C

• Example: b+ decay𝑝 → 𝑛 leads to amore stable configuration

U

neutrons protons

b-decay


• There are fundamental reasons why a (anti-)neutrino is included in the decay. Experimentally, this is the measured electron energy spectrum:

• At least one more particle needed (and not detected): neutrino – suggested by Pauli

Not a 2-body process (like the a-decay)

Why is a (anti-)neutrino needed?


b-decays

b- decay [excess of neutrons]:• Elementary process: 𝑛 → 𝑝 + 𝑒7 + �̅�C• Nuclear decay: 𝑋#$ → 𝑌#y'

$ + 𝑒7 + �̅�C

b+ decay [excess of protons]:• Elementary process: p → 𝑛 + 𝑒y + 𝜈C• Nuclear decay: 𝑋#$ → 𝑌#7'

$ + 𝑒y + 𝜈C

Electron conversion (from atomic orbit):• Elementary process: p + 𝑒7 → 𝑛• Nuclear decay: 𝑋#$ + 𝑒7 → 𝑌#7'

$ + 𝜈C


• Q-values: calculation using atomic mass units, watch out for the electrons*!

– b- decay: 𝑄{| = 𝑀 𝑋#$ −𝑀 𝑌#y'$ 𝑐(

– b+ decay: 𝑄{| = 𝑀 𝑋#$ −𝑀 𝑌#7'$ − 2𝑚C| 𝑐(

– Electron conversion: 𝑄}~ = 𝑀 𝑋#$ −𝑀 𝑌#7'$ 𝑐( − 𝐵D

b-decay Q-values

* See derivation


Stable nucleus

RadioactiveNuclei

Minimizing energy


• When the nucleus has undergone a a- orb-decay, the daughter nucleus may be in an excited state, e.g. protons, neutrons (or a combination of them) are not on the lowest energy levels possible in the potential well.

• The energy difference DE is radiated away by one or more g-rays (electromagnetic radiation):

Eg: few keV à few MeV

Example

DEg-radiation

E=hn

g-decay


Example: the decay chain of 238U

Naturally occurring radioactive nuclides


Naturally occurring radioactive nuclides


Part III: Nuclear reactions & applications


• Nuclear Reactions:

• Example:

Q = (Initial Mass) c2 – (Final Mass) c2 = (Mac2+MXc2) – (Mbc

2+MYc2)

Q = (Mac2+M14Oc2) – (Mpc

2+M17Oc2)

Note: can be generalized to any (low energy) reactions: X1+X2 àY1+Y2+Y3+…

Nuclear reactions

𝑎 + 𝑋 → 𝑏 + 𝑌

𝛼 + 𝑁', → 𝑝 + 𝑂'�

𝑋 𝑎, 𝑏 𝑌

𝑁', 𝛼, 𝑝 𝑂'�

or

or


• Some heavy isotopes can fission spontaneously into two so-called “fission fragments” (+ possibly some neutrons…)

Ex: 256Fm (t1/2=2.6 h)254Cf (t1/2=60.5 days)

• The fission fragments are statistically distributed over a large range of medium-mass nuclei and are usually radioactive (and b-decay back to stability) à How fast depends of the half-lives of the isotopes formed on the way !

(Spontaneous) Fission


• Because the neutron has no charge, it can penetrate the nucleus (no Coulomb barrier)

à Neutron used as a Probe.

• Example: 𝑛 + 𝑈.((-/ → 𝑈∗.(

(-* → 𝑍𝑟,&.. + 𝑇𝑒/(

'-, + 3𝑛

• 1938-39: Induced fission of 238Uà large release of energy (~200 MeV)à new neutrons are emitted !!! CHAIN

REACTION POSSIBLE ?

Induced fission

Leo Szilard


• Fission fragments are highly unstable because they are so neutron rich.

• Prompt neutrons are emitted simultaneously with the fission process. Even after prompt neutrons are released, the fission fragments undergo beta decay, releasing more energy.

• Most of the ~200 MeV released in fission goes to the kinetic energy of the fission products, but the neutrons, beta particles, neutrinos, and gamma rays typically carry away 30–40 MeV of the kinetic energy.

Thermal neutron mechanism


235U fission cross section

Faster Neutrons


• More than one neutron (on average) is emitted after every fission: possibility of triggering a chain reaction.

• Neutrons emitted are “fast neutrons”, need to be slowed down (“slow/thermal neutrons”). [Fission cross sections increases by 1/v]

• Chain Reactions:– Slightly more than one neutron

emitted per reaction à chain reaction critical

– Less than one neutron àsubcritical

– More than one neutron àsupercritical

Chain reaction


Controlled:

Nuclear Power Plant

Waste: long-lived fission fragments(ex: 90Sr, 137Cs…)

Uncontrolled:

Atomic Bomb (A-bomb)Hiroshima:

Energy released 1014 Joules = 20 kilotons of TNT

Nasty stuff (after blast): fission fragments !

Application of induced fission


Core:

• Moderator [hydrogen in water, beryllium, carbon in graphite]: à slows down the neutrons to “thermal” energies

• Control rods [Boron for example]à absorbs the neutron excess, control the criticality of the reactor

• Reflector [hydrogen in water]à allows backscattering of the neutrons back in the cells.

Nuclear reactor


Power comparison


• Fusion releases more energy per nucleon than fission. However, this process doesn’t occur spontaneously. One needs to ignite the reaction, then the energy produced has to self sustain the fusion process.

• Fusion of Hydrogen powers the Sun:

Fusion

4 𝐻' → 𝑠𝑒𝑣𝑒𝑟𝑎𝑙𝑠𝑡𝑒𝑝𝑠 → 𝐻𝑒, 𝑄 = 26.7𝑀𝑒𝑉,

Example


Controlled:

Nuclear Fusion Power Plant(projects: tokamaks, laser…)

ITER

Problem: Ignition !

Uncontrolled:

Hydrogen Bomb (H-bomb)requires an A-bomb to ignite fusion !!!

Enhanced yield:Hundreds of kilotons à Tens of Megatons !

(Hiroshima, 20 kt)

Application of induced fusion


Device: 60 tons, Yield: 10.4 MT

(Ivy) Mike – Oct 31st, 1952


First lithium-deuteride “dry” bomb:40%: 6Li à d, t, n…60%: 7Li + n à 6Li + 2n (Oops!!!)

Expected Yield: 4-8 MTMeasured Yield: 15MT !!!

Castle Bravo


Inertial Confinement

A fusion reactor?


• Medicine: X-rays, radioactive tracers (ex: 99mTc,…) + imagery, radiotherapy

• Engineering: g-radiography of material, neutron activation, thickness control…

• Archeology: 14C (t1/2 ~ 5730 years)

• Smoke detector

• Food safety

• Sterilization of medical instruments

• Oil and mining

• …

Other applications


14C is a radioactive isotope of carbon with a half-life of t1/2=5730y. It is produced in our atmosphere by the bombardment of 14N by neutrons (produced by cosmic rays).

1. Write the reaction that produces 14C.

14C can be absorbed by a living organism the same way than the stable carbon isotope 12C. The equilibrium ratio R0 = N(14C)/N(12C) in the atmosphere has been measured to be 1.20x10-12. When an organism dies, it ceases absorbing carbon (both 12C and radioactive 14C) and therefore the ratio R(t) changes over time as 14C nuclei decay.

Application: a bone suspected to have originated during the period of the roman emperors was found in Great Britain. The N(14C)/N(12C) ratio was determined to be 1.10x10-12.

2. Write the radioactive decay law for the 14C nuclei. Express the decay law using R(t) and R0.

3. How old is the bone according to the 14C dating method? Is the bone oldenough to have Roman origins?

Exercise


Neutron sources: PuBe or AmBe sources

Oil and mining application


Metastasis

Medical applications: imaging

99mTc – T1/2=6.01h


Radiotherapy delivered internallyRadiotherapy delivered externally

Medical applications: radiotherapy

Nuclear Physics - Colorado School of Minesinside.mines.edu/~fsarazin/phgn310/PDFs/Slides/13b.Nuclear Physics.pdfFred Sarazin ([email protected]) PHGN300/310: Atoms, molecules and

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