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.
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2. General Comments on Decays
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. PHYS684)
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3. Simplified Phase Space
Decay: A  a + b + c …..
Q = available kinetic energy
large Q  large phase space  higher rate
larger number of final state products possibly means more phase space and higher rate as more variation in momentums. Except if all the mass of A is in the mass of final state particles
3 body has little less Q but has 4 times the rate of the 2 body (with essentially identical matrix elements)
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4. 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….
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5. 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
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6. 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
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7. 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
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8. Alpha decay-Barrier penetration
One of the first applications of QM was by Gamow who modeled alpha decay by assuming the alpha was moving inside the nucleus and had a probability to tunnel through the Coulomb barrier
from 1D thin barrier (460) for particle with energy E hitting a barrier potential V and thickness gives Transmission = T
now go to a Coulomb barrier V= A/r from the edge of the nucleus to edge of barrier and integrate- each dr is a thin barrier
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9. Alpha decay-Barrier penetration
this integral isn’t easy, need approximations
see nuclear physics textbook (see square) Get
where K = kinetic energy of alpha. Plug in some numbers
see
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10. 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
if angular momentum changes, then a suppression of about for each change in L (increases potential barrier)
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11. Alpha decay-Decay chains
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12. Alpha decay-Energy levels
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
not for even-even nuclei (I=0). suppression of about for each change in L (increases potential barrier)
s 0
p 1
d 2
f 3
g 4
h 5
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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
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14. Energy vs A Alpha decay
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15. Lifetime vs Energy in Alpha Decays
Perlman, Ghiorso, Seaborg, Physics Review 75, 1096 (1949) 10
log10 half-life in years
-10 5 7
Alpha Energy MeV
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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
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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.....
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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
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19. Beta+ vs Electron Capture
from Particle Data Group
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20. 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
Pauli/Fermi understood this in 1930s electron neutrino discovered 1953 (Reines and Cowan) muon neutrino discovered 1962 (Schwartz +Lederman/Steinberger) tau neutrino discovered 2000 at Fermilab
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21. 3 Body Kinematics
While 3 body the nuclei are very heavy and easy approximation is that electron and neutrino split available Q (nuclei has similar momentum)
maximum electron energy when E(nu)=0
example
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22. Beta decay rate Start from Fermi Golden Rule
first approximation (Fermi). Beta=constant=strength of weak force
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
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23. Beta decay rate II
Orbital angular momentum suppression of for each value of L (in matrix element calculation)
look at density of states factor. Want # quantum states per energy interval
we know from quantum statistics that each particle (actually each spin state) has
3 body decay but recoil nucleus is so heavy it doesn’t contribute
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24. Beta decay rate III
Conservation of energy allows one to integrate over the neutrino (there is a delta function)
this gives a distribution in electron momentum/energy which one then integrates over. (end point depends on neutrino mass)
F is a function which depends on Q. It is almost loqrithmic
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25. actual. not “linear” due to electron mass
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26. Beta decay rate IV FT is “just kinematics”
measuring FT can study nuclear wavefunctions M’ and strength of the weak force at low energies
lower values of FT are when M’ approaches 1
beta decays also occur for particles
electron is now relativistic and E=pc. The integral is now easier to do. For massive particles (with decay masses small), Emax = M/2 and so rate goes as fifth power of mass
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27. Beta decay rate V
M=bM’  b is strength of weak interaction. Can measure from lifetimes of different decays
characteristic energy
strong energy levels ~ 1 MeV
for similar Q, lifetimes are about
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28. 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
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29. “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
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30. 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 (see Eq 14-4 and 14-5)
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
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31. Discovery of Parity Violation in Beta Decay by C.S. Wu et al.
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
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32. 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.
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33. 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
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34. 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
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35. Gamma Decays 5
26%
E MeV
11%
gamma
53%
gamma
conserve angular momentum and parity. lowest order is electric dipole moment. then quadrapole and magnetic dipole
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36. Mossbauer Effect
Gamma decays typically have lifetimes of around sec (large range). Gives width:
very precise
if free nuclei decays, need to conserve momentum. Shifts gamma energy to slightly lower value
example. Very small shift but greater than natural width
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37. Mossbauer Effect II
Energy shift means an emitted gamma won’t be reabsorbed
but if nucleus is in a crystal lattic, then entire lattice recoils against photon. Mass(lattice)infinity and Egamma=deltaM. Recoiless emission (or Mossbauer)
will have “wings” on photon energy due to lattice vibrations
Mossbauer effect can be used to study lattice energies. Very precise. Use as emitter or absorber. Vary energy by moving source/target (Doppler shift) (use Iron. developed by R. Preston, NIU)
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38. Nuclear Reactions, Fission and Fusion
2 Body reaction A+BC+D
elastic if C/D=A/B
inelastic if mass(C+D)>mass(A+B)
threshold energy for inelastic (B at rest)
for nuclei nonrelativistic usually OK
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39. Nuclear Reactions (SKIP)
A+BC+D
measurement of kinematic quantities allows masses of final states to be determined
(p,E) initial A,B known
8 unknowns in final state (E,px,py,pz for C+D)
but E,p conserved. 4 constraints4 unknowns measure E,p (or mass) of D OR C gives rest or measure pc and pd gives masses of both
often easiest to look at angular distribution in C.M. but can always convert
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40. Fission 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
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41. Spontaneous Fission
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42. Induced Fission
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43. Neutron absorption
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44. Fusion
“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)
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45. 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??)
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46. 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.
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47. 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
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