1. Nuclear Decays
Unstable nuclei can change N,Z.A to a nuclei at a lower energy (mass)
If there is a mass difference such that energy is released, pretty much all decays occur but with very different lifetimes.
have band of stable particles and band of “natural” radioactive particles (mostly means long lifetimes). Nuclei outside these bands are produced in labs and in Supernovas
nuclei can be formed in excited states and emit a gamma while cascading down.

2. General Comments on Decays – mostly skip
Use Fermi Golden rule (from perturbation theory)
rate proportional to cross section or 1/lifetime
the matrix element connects initial and final states where V contains the “physics” (EM vs strong vs weak coupling and selection rules)
the density of states factor depends on the amount of energy available. Need to conserve momentum and energy “kinematics”. If large energy available then higher density factor and higher rate.
Nonrelativistic (relativistic has 1/E also
Decay: A  a + b + c …..
Q = available kinetic energy
large Q  large phase space higher rate

3. Phase Space:Channels
If there are multiple decay channels, each adds to “phase space”. That is one calculates the rate to each and then adds all of them up
single nuclei can have an alpha decay and both beta+ and beta- decay. A particle can have hundreds of possible channels
often one dominates
or an underlying virtual particle dominates and then just dealing with its “decays”
still need to do phase space for each….

4. Lifetimes just one channel with N(t) = total number at time t
multiple possible decays. Calculate each (the “partial” widths) and then add up
Measure lifetime long-lived (t>10-8sec). Have a certain number and count the decays

5. Lifetimes
Measure lifetime medium-lived (t>10-13sec). Decay point separated from production point. Measure path length. Slope gives lifetime
short-lived (10-23 < t <10-16 sec). Measure invariant mass of decay products. If have all  mass of initial. Width of mass distributions (its width) related to lifetime by Heisenberg uncertainty.
100 10 1 Dx

6. Alpha decay Alpha particle is the He nucleus (2p+2n)
~all nuclei Z > 82 alpha decay. Pb(82,208) is doubly magic with Z=82 and N=126
the kinematics are simple as non-relativistic and alpha so much lighter than heavy nuclei
really nuclear masses but can use atomic as number of electrons do not change. Be 8 “lightest” alpha decay with lifetime of .07fs

7. Alpha decay-Barrier penetration – did already
from 1D thin barrier ( for particle with energy E hitting a barrier potential V and thickness gives Transmission = T
super quick - assume square potential

8. Alpha decay-Barrier penetration
3D Coulomb barrier V= A/r from the edge of the nucleus to edge of barrier and integrate- each dr is a thin barrier this integral isn’t easy, need approximations
see nuclear physics textbook . Get
where K = kinetic energy of alpha. Plug in some numbers

9. Alpha decay-Barrier penetration
Then have the alpha bouncing around inside the nucleus. It “strikes” the barrier with frequency
the decay rate depends on barrier height and barrier thickness (both reduced for larger energy alpha) and the rate the alpha strikes the barrier
larger the Q larger kinetic energy and very strong (exponential) dependence on this
as alpha has A=4, one gets 4 different chains (4n, 4n+1, 4n+2, 4n+3). The nuclei in each chain are similar (odd/even, even/even, etc) but can have spin and parity changes at shell boundaries

10. Alpha Decay T is the transmission probability per “incident” alpha
f=no. of alphas “striking” the barrier (inside the nucleus) per second = v/2R, If v=0.1c f=1021 Hz
Depends strongly on alpha kinetic energy

11. Alpha decay-Decay chains

12. Alpha decay-Energy levels
if angular momentum changes, then a suppression of about for each change in L (increases potential barrier)
may need to have orbital angular momentum if sub-shell changes (for odd n/p nuclei)
Z= h(9/2) N= g(9/2) Z= f(7/2) N= d(5/2)
so if f(7/2)  h(9/2) need L>0 but parity change if L=1  L=2,4
or d(5/2)  g(9/2) need L>1. No parity change L=2,4
s 0
p 1
d 2
f 3
g 4
h 5

13. Parity + Angular Momentum Conservation in Alpha decay
X  Y + a. The spin of the alpha = 0 but it can have non-zero angular momentum. Look at Parity P
if parity X=Y then L=0,2…. If not equal L=1,3…
to conserve both Parity and angular momentum

14. Uranium – not on tests
Uranium has 92 protons and mostly comes in 2 isotopes: U(235) with 143 neutrons and U(238) with 146 neutrons % of Uranium is U(238). But it is U(235) which chain reacts and so used in nuclear reactors and atomic bombs. “enriching” uranium increases the amount of U(235). The US has a stockpile of about 550 tons of enriched uranium for reactors on navy ships and the arsenal of nuclear weapons
The relative fraction of U(235) to U(238) and their lifetimes of 1 billion and 6.5 billion years can be used to estimate when the Uranium was produced. This is similar to Carbon 14 dating. The calculation gives a production date of about 5.8 billion years ago, which is maybe when a neutron star-neutron star collision occurred near where the Sun formed about 1 billion years later.
Detailed studies of the abundance of other radioactive elements heavier than lead can be used to determine the age of the Earth

15. Uranium – arithmetic Fraction U(238) = 99.27% Fraction U(235) = 0.72%
Lifetime U(235) = 6.52 billion years
Lifetime U(238) = 1.02 billion years
ASSUME at production equal amounts of U(235) and U(238)
Amount(today) = exponential(-time/lifetime)
Ln(.0072) = -4.9 = -time(1/ /6.52)
Time = 5.76 billion years ago – approximately when uranium was produced, maybe in neutron star-neutron star collision.
Have about 1 neutron star-neutron star collision every 10,000 years in the Milky Way galaxy sometimes called kilonova and will be 1000 times brighter than a Type Ia supernova
Had about 10 NS-NS collisions within 1000 LY of us in the billion years before the Sun and the other stars in our local cluster were formed

16. Beta Decays
Beta decays are proton  neutrons or neutron  proton transitions
involve W exchange and are weak interaction
the last reaction is electron capture where one of the atomic electrons overlaps the nuclei. Same matrix element (essentially) bit different kinematics
the semi-empirical mass formula gives a minimum for any A. If mass difference between neighbors is large enough, decay will occur

17. Beta Decays - Q Values
Determine Q of reactions by looking at mass difference (careful about electron mass)
1 MeV more Q in EC than beta+ emission. More phase space BUT need electron wavefunction overlap with nucleus.....

18. Beta+ vs Electron Capture
Fewer beta+ emitters than beta- in “natural” nuclei (but many in “artificial” important in Positron Emission Tomography - PET)
sometimes both beta+ and EC for same nuclei. Different widths
sometimes only EC allowed
monoenergetic neutrino. E=.87 MeV. Important reaction in the Sun. Note EC rate different in Sun as it is a plasma and not atoms

19. Beta Decay - 3 Body
The neutrino is needed to conserve angular momentum
(Z,A)  (Z+1,A) for A=even have either Z,N even-even  odd-odd or odd-oddeven-even
p,n both spin 1/2 and so for even-even or odd-odd nuclei I=0,1,2,3…….
But electron has spin 1/ I(integer)  I(integer) + 1/2(electron) doesn’t conserve J
need spin 1/2 neutrino. Also observed that electron spectrum is continuous indicative of >2 body decay
Decay rate depends on strength of weak interactions and on phase space  larger Q gives larger rate (smaller lifetime). Can also have suppressions if changing angular momentum or parity

20. 3 Body Kinematics
While 3 body the nuclei are very heavy and easy approximation is that electron and neutrino split available Q
maximum electron energy when E(nu)=0
example

21. Beta decay rate examples
Rule 1: parity of nucleus can’t change (integral of odd*even=0)
Rule 2: as antineutrino and electron are spin 1/2 they add to either 0 or 1. Gives either
Orbital angular momentum suppression of for each value of L

22. Gamma Decays
If something (beta/alpha decay or a reaction) places a nucleus in an excited state, it drops to the lowest energy through gamma emission
excited states and decays similar to atoms
conserve angular momentum and parity
photon has spin =1 and parity = -1
for orbital P= (-1)L
first order is electric dipole moment (edm). Easier to have higher order terms in nuclei than atoms

23. Gamma Decays in beta decays5
26%
E MeV
1.6 Mev
11%
gamma
53%
2.1 Mev
gamma
90%?
conserve angular momentum and parity. lowest order is electric dipole moment. then quadrapole and magnetic dipole

24. Parity Violation in Beta Decays
The Parity operator is the mirror image and is NOT conserved in Weak decays (is conserved in EM and strong)
non-conservation is on the lepton side, not the nuclear wave function side
spin 1/2 electrons and neutrinos are (nominally) either right-handed (spin and momentum in same direction) or left-handed (opposite)
Parity changes LH to RH RH LH

25. “Handedness” of Neutrinos
“handedness” is call chirality. If the mass of a neutrino = 0 then:
all neutrinos are left-handed all antineutrinos are right-handed
Parity is maximally violated
As the mass of an electron is > 0 can have both LH and RH. But RH is suppressed for large energy (as electron speed approaches c)
fraction RH vs LH can be determined by solving the Dirac equation which naturally incorporates spin

26. Polarized Beta Decays Spin antinu-RH Pnu pe Spin e - LH Co
Some nuclei have non-zero spin and can be polarized by placing in a magnetic field
magnetic moments of nuclei are small (1/M factor) and so need low temperature to have a high polarization
Gamow-Teller transition with S(e-nu) = 1
if Co polarized, look at angular distribution of electrons. Find preferential hemisphere (down)
Spin antinu-RH
Pnu pe
Spin e - LH Co

27. Discovery of Parity Violation in Beta Decay by C.S. Wu et al. 1957
Test parity conservation by observing a dependence of a decay rate (or cross section) on a term that changes sign under the parity operation. If decay rate or cross section changes under parity operation, then the parity is not conserved.
Parity reverses momenta and positions but not angular momenta (or spins). Spin is an axial vector and does not change sign under parity operation.
Beta decay of a neutron in a real and
mirror worlds:
If parity is conserved, then the probability of electron emission at q is equal to that at 180o-q.
Selected orientation of neutron spins - polarisation. q
180o-q
mirror Pe
neutron Pe

28. Wu’s experiment
Beta-decay of 60Co to 60Ni*. The excited 60Ni* decays to the ground state through two successive g emissions.
Nuclei polarised through spin alignment in a large magnetic field at 0.01oK. At low temperature thermal motion does not destroy the alignment. Polarisation was transferred from 60Co to 60Ni nuclei. Degree of polarisation was measured through the anisotropy of gamma-rays.
Beta particles from 60Co decay were detected by a thin anthracene crystal (scintillator) placed above the 60Co source. Scintillations were transmitted to the photomultiplier tube (PMT) on top of the cryostat.

29. Wu’s results
Graphs: top and middle - gamma anisotropy (difference in counting rate between two NaI crystals) - control of polarisation; bottom - b asymmetry - counting rate in the anthracene crystal relative to the rate without polarisation (after the set up was warmed up) for two orientations of magnetic field.
Similar behaviour of gamma anisotropy and beta asymmetry.
Rate was different for the two magnetic field orientations.
Asymmetry disappeared when the crystal was warmed up (the magnetic field was still present): connection of beta asymmetry with spin orientation (not with magnetic field).
Beta asymmetry - Parity not conserved

30. Nuclear Reactions: Fusion and neutron absorption
2 Body reaction A+BC+D: elastic if C/D=A/B inelastic if mass(C+D)>mass(A+B)
Neutron absorption an important inelastic reaction. Three light: H, Li, Boron, have large neutron absorption cross sections. H, B “cheap”.

31. neutron absorption in Boron
Happens in B10 (20%) and not B11 (80%). Gives off gamma and so can be medically use to deposit energy
National Nuclear Data Center
Cross sections ENDF data

32. Fission – mostly skip
AB+C A heavy, B/C medium nuclei
releases energy as binding energy/nucleon = 8.5 MeV for Fe and 7.3 MeV for Uranium
spontaneous fission is like alpha decay but with different mass, radii and Coulomb (Z/2)2 vs 2(Z-2). Very low rate for U, higher for larger A
induced fission n+AB+C. The neutron adds its binding energy (~7 MeV) and can put nuclei in excited state leading to fission
even-even U(92,238). Adding n goes to even-odd and less binding energy (about 1 MeV)
even-odd U(92,235), U(92,233), Pu(94,239) adding n goes to even-even and so more binding energy (about 1 MeV)  2 MeV difference between U235 and U238
fission in U235 can occur even if slow neutron

33. Neutron absorption in Uranium

34. Fusion H  Helium either in Sun or early universe or reactor
“nature” would like to convert lighter elements into heavier. But:
no free neutrons
need to overcome electromagnetic repulsion  high temperatures
mass Be > twice mass He. Suppresses fusion into Carbon
Ideally use Deuterium and Tritium, s=1 barn, but little Tritium in Sun (ideal for fusion reactor)

35. Fusion in Sun
rate limited by first reaction which has to convert a p to a n and so is Weak
s(pp) ~ barn
partially determines lifetime of stars
can model interaction rate using tunneling – very similar to Alpha decay (also done by Gamow)
tunneling probability increases with Energy (Temperature) but particle probability decreases with E (Boltzman). Have most probable (Gamow Energy). About 15,000,000 K for Sun but Gamow energy higher (50,000,000??)

36. Fusion in Sun II
need He nuclei to have energy in order to make Be. (there is a resonance in the s if have invariant mass(He-He)=mass(Be))
if the fusion window peak (the Gamow energy weighted for different Z,mass) is near that resonance that will enhance the Be production
turns out they aren’t quite. But fusion to C start at about T=100,000,000 K with <kT> about 10 KeV each He. Gamow energy is higher then this.

37. Fusion in Sun III
Be+HeC also enhanced if there is a resonance. Turns out there is one at almost exactly the right energy MeV
7.65 MeV
4.44 MeV