P461 - nuclear decays 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. ν ν β α / : : 1 4 2 4 2 + + → + → ± ± − − e N N He N N A n Z A Z A n Z A Z
Transcript
P461 - nuclear decays 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.
ννβ
α
/:
:
1
4242
++→
+→±±
−−
eNN
HeNN
A
n
ZAZ
A
n
ZAZ
P461 - nuclear decays 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)
dVolumeVV
Vrate
fiif
fif
ψψ
ρπ
∫=
=
*
2||2
h
particleeachdEdpp iiif
2∝ρ
P461 - nuclear decays 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)
)( statefinalmMassQ iA ∑−=
MeVQ
bodyDB
MeVQ
bodyDB
250513977018655279
3
264477018655279
2
00
0
=−−−=
−→
=−−=
−→
++
++
πρ
ρ
P461 - nuclear decays 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….
mesonsKs
eduWWsc
→
→+→ τνµνν ,,,
P461 - nuclear decays 5
Lifetimes
• just one channel with N(t) = total number at time t
• 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
22 −=−=
+′→
′′ XXXX NNZZ
XX α
MeVQA
AKE
smallm
pTpp
mmmQ
X
XX
HeXX
944
2
2
−−
≈
===
−−=
′′′
′
α
α
P461 - nuclear decays 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
h
)(2
)1(16 2
EVmk
eV
E
V
ET ka
−=
−= −
απεπε K
ZerdrE
r
ZemT c
r
r
c
n0
2
0
2
2 4
2)
4
2(
22exp( =−−≈ ∫
h
P461 - nuclear decays 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. Plus in some
numbers
• see
www.haverford.edu/physics-astro/songs/alpha.htm
απεπε K
ZerdrE
r
ZemT c
r
r
c
n0
2
0
2
2 4
2)
4
2(
22exp( =−−≈ ∫
h
)2/2exp(
)/2exp(
2
2
KMzZe
vzZeT
h
h
π
π
−=
−≈
31104)70exp(
)62
9314
197
)4.1(9022exp(
−×=−≈
∗∗
×∗∗∗
−≈
T
MeV
MeV
MeVF
MeVFT
π
P461 - nuclear decays 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 0.002 for each change in L (increases
potential barrier)
Nr
velocityf
2
α≈
2
2
2
)1(
mr
ll +h
P461 - nuclear decays 11
Alpha decay-Decay chains
4n
4n+2
P461 - nuclear decays 12
Alpha decay-Energy levels
• may need to have orbital angular momentum if sub-
shell changes (for odd n/p nuclei)
• Z= 83-92 1h(9/2) N=127-136 2g(9/2)
Z=93-100 2f(7/2) N=137-142 3d(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 0.002 for each change in L (increases
potential barrier) s 0
p 1
d 2
f 3
g 4
h 5
P461 - nuclear decays 13
Parity + Angular Momentum
Conservation in Alpha decay
• X Y + α. 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
6,4
3)(
)2,(3141#
)6,(1143#
25
211
25
2/5
211
2/11
231235
=⇒
=−≥⇒
=⇒⇒=
=⇒⇒=
+→
+
+
orbital
orbital
L
L
lddn
liin
ThU α
l
orbitorbitYX PPPPPP )1(,1 −==⇒= αα
P461 - nuclear decays 14
Energy vs A Alpha decay
P461 - nuclear decays 15
Lifetime vs Energy in Alpha
Decays
log10 half-life
in years
10
0
-10
Alpha Energy MeV
Perlman, Ghiorso, Seaborg, Physics
Review 75, 1096 (1949)
75
P461 - nuclear decays 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
)(
)(
)(
,1,
,1,
,1,
νν
νν
νν
nepMMe
peneMM
nepeMM
eAZAZ
eAZAZ
eAZAZ
→+→+
→++→
→++→
−−
−+
+−
P461 - nuclear decays 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.....
YX
eYeYeXe
eAZAZ
eYX
eeYeYeX
eAZAZ
eYYX
eeYeYeX
eAZAZ
AMAMQ
KKKZmmZmmm
YXeEC
mAMAMQ
KKmKZmmZmm
eYX
KKKAMAtomicMassQ
KKmKZmmZmm
eYX
−=
++++=++
+→+
−−=
+++++=+
++→
++=−=
+++++=+
++→
−−
+−
+
−+
−
ν
ν
ν
ν
ν
νβ
νβ
)()(
:
2
)()(
:
)()(
:
,1,
,1,
,1,
P461 - nuclear decays 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
ν+→+
∗=∗<=∆
=
=
− 7374
74
73
00055.2200093.
01693.7
01600.7
LieBe
umuM
uMBe
uMLi
e
P461 - nuclear decays 19
Beta+ vs Electron Capture
• from Particle Data Group
ν++→+ +eHpp 2
ν+→+ LieBe 77
ν++→ +eBeB 88
P461 - nuclear decays 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-oddeven-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 1/2
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
P461 - nuclear decays 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
Qm
mmmmmmmEK
mm
mmmE
energyconserveEmEm
momentumconservepp
EleteYX
x
eyxeyx
eee
x
eyx
e
yex
ey
≈−+−−
=−=
−+−
=
=−−
=
=++→
2
))((
)(2
)(
0
222
max
22
ν
ν
νν
smallkeVm
pK
m
EMeVmEp
MeVQm
mm
eAlMg
Al
eeee
e
→==
===−=
=⇒=
==
++→ −
2.02
5.5,75.2
8.200055.
981.26,9843.26
2
22
13,2712,27
13271227
γ
ν
P461 - nuclear decays 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
τβψψ
ρπ
dM
MRates
F
Final
∫=
≈
*
2||2
h
τψψβ dMMM ZZ∫ +=′′= *
1
++−
+++
±
→++→
+→±=∆−
→++→
=−=∆
01
)010(1:
00
0:
16221532
20422142
1
ν
ν
eSP
notiTellerGamow
eCaSc
iiiFermi AZZA
P461 - nuclear decays 23
Beta decay rate II
• Orbital angular momentum suppression of 0.001
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
n
nnFinal
dE
dNMRates =≈ ρρ
π 2||2
h
11
0218361736
=→±=∆
→++→ ++−
Li
eCaSc ν
dph
pdN
3
2
4π=
cKQp
dph
pdp
h
pdN
e
ee
/)(
443
2
3
2
−≅
=
ν
ννππ
P461 - nuclear decays 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
eeee
ee
Final
e
mmpK
hc
KQ
h
pM
Mdp
dNRates
−+=
−
=≈=
2/122
3
2
3
22
2
)(
)(
)(44||
2
||2
πππ
ρπ
h
h
)(||2
1max
2
73
45
ee EFMcm
TRate ×≅=
hπ
maxloglog eKAF ≈
P461 - nuclear decays 25
4.4
max
3
5.log4.4loglog
KF
KKAF e
≅⇒
+≈≈
actual. not “linear” due to
electron mass
P461 - nuclear decays 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
e
e
eK
e
νπ
νππ
++→
++→−−
−−
0
0
30/)( 5
max
22
max
0
EdppKQ ee
p
e =−∫
P461 - nuclear decays 27
Beta decay rate V
• M=βM’ β 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
3362 10010 FeVmjoule ∗=∗≅ −β
eVF
FeV
vol1.0
)10(
*1003
3
=≈β
147 1010 −− ≈⇒≈ strengthrelativestrong
weak
αβ
s
s
s
weak
EM
strong
10
16
23
10
10
10
−
−
−
≈
≈
≈
τ
τ
τ
P461 - nuclear decays 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
•
),,(),,(
),,(),,(
πφθπφθ +−→
−−−→
rrP
zyxzyxP
RH
LHLprLP
ppPrrrr
rr
=×=
−=
)(
)(
P461 - nuclear decays 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
P461 - nuclear decays 30
Polarized Beta Decays
• 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)
21
21
6060
,45 ===
++→ −
sii
eNiCo ν
Co
Pnu
pe
Spin antinu-RH
Spin e - LH
P461 - nuclear decays 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.
neutron
Pe
Pe
θ
180ο−θ
mirror
Beta decay of a neutron in a
real and
mirror worlds:
If parity is conserved, then the
probability of electron
emission at θ is equal to that at
180o-θ.Selected orientation of neutron
spins - polarisation.
P461 - nuclear decays 32
Wu’s experiment
• Beta-decay of 60Co to 60Ni*. The
excited 60Ni* decays to the
ground state through two
successive γ 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.
P461 - nuclear decays 33
Wu’s results
• Graphs: top and middle - gamma anisotropy (difference in counting rate between two NaI crystals) -control of polarisation; bottom - βasymmetry - 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
P461 - nuclear decays 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
+=−+−=−=
=+→
=+→
+→
++
+−
)1)(1)(1()1(
...,102
,023
*
L
Nfinal PPP
momquadeL
edmL
NN
γ
γ
γ
γ
P461 - nuclear decays 35
Gamma Decays
1;202
)(122
32
*
=+=∆→
=+→
→
+→
+−
+−
−−
LGTi
changePLGT
GT
NN γ
E
MeV
5
0
3817Cl 3818Ar
26%
11%
53%
−2
+
+
−
0
2
3gamma
gamma
−
++
+−
=
=+→
=+→
1
;102
;023
PL
eqmL
edmL
γ
γ
γ
conserve angular momentum and parity. lowest order is electric dipole
moment. then quadrapole and magnetic dipole
P461 - nuclear decays 36
Mossbauer Effect
• Gamma decays typically have lifetimes of around
10-10 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
eVeVs
E 5
10
15
10sec10
10 −−
−
=≈≈∆=Γτh
)2
1(2 *
*
22
*
M
MM
M
MMEpp
AA
A
AAA
∆−∆=
−=⇒=
+→
γγ
γ
eVMeVE
MMeVM
005.13.
5.931*191,13.
−=⇒
==∆
γ
P461 - nuclear decays 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)
MeVEAA
MeVEAA
000000005.13.
000000005.13.
*
*
+=→+
−=+→
γ
γ
P461 - nuclear decays 38
Nuclear Reactions, Fission and Fusion
• 2 Body reaction A+BC+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
)(
2
)( 2222
icrelativistnonm
mmQK
MQm
mmmmQK
mmpEM
B
BAth
B
DCBAth
DCtottot
−+
−≈
∆=+++
−=⇒
+>−=
)(47.5
)(38.5)1(4
03.4)014102.22016049.3007825.1(
31
223
relMeVK
relnonMeVK
MeVuQ
HHHp
th
th
=
−=+=
−=∗−+=
+→+
P461 - nuclear decays 39
Nuclear Reactions (SKIP)
• A+BC+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 constraints4 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
Ωd
dσ
CMΘ
P461 - nuclear decays 40
Fission
• AB+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).
