Transcript abg Decay Theory • Previously looked at kinematics now study dynamics (interesting bit). • QM tunnelling and a decays • Fermi theory of b.

abg Decay Theory
• Previously looked at kinematics now study
dynamics (interesting bit).
• QM tunnelling and a decays
• Fermi theory of b decay and e.c.
 g decays
Tony Weidberg
Nuclear Physics Lectures
1
a Decay Theory
• Consider 232Th Z=90 R=7.6 fm  E=34 MeV
E  Z1 Z 2
e2
4 0c
c
R
• Energy of a Ea=4.08 MeV
• Question: How does the a escape?
• Answer: QM tunnelling
Tony Weidberg
Nuclear Physics Lectures
2
radial wave function in alpha decay
iII
I
iI
Exponential decay of y
nucleus
Tony Weidberg
r
barrier (negative KE)
Nuclear Physics Lectures
small flux of real α
3
QM Tunnelling
y I  e xp(ikx)  A e xp( ikx)
y II  B e xp(Kx )  C e xp( Kx )
y III  D e xp(ikx)
V
k  2mE
K  2m (V0  E )
E
0
t
•
B.C. at x=0 and x=t for Kt>>1 and k~K gives for 1D rectangular barrier
thickness t gives T=|D|2=exp(-2Kt)
•
Integrate over Coulomb barrier from r=R to r=t
Tony Weidberg
Nuclear Physics Lectures
4
a-decay
DEsep≈6MeV per nucleon for heavy nuclei
DEbind(42a)=28.3 MeV > 4*6MeV
Neutrons
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Protons
Alphas
Nuclear Physics Lectures
5
 2t

 2

T  exp   K (r )dr   exp   2M( V(r )  Ea ) dr   exp( G )
 

 R

2
2Ze 2
V (r ) 
4 0r
2Ze 2
Ea 
4 0t
2  MZe2 

G  
  0 
1/ 2 t
1/ 2
 1 / r  1 / t 
dr ; r  t cos2 
dr  2t cos sind
R
1/ 2
1  MZe 2 
G 
   0 
t1 / 2  sin2  d
0
2
sin
  d  (1/ 2)  0  sin 0 cos 0 
0
1/ 2
  MZe t 
t  R;cos 0  R / t ; cos 0  0;0   / 2; G  

4   0 
2
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Nuclear Physics Lectures
6
Alpha Decay Rates
1/ 2
  MZe t 
G


4   0 
• Gamow factor
2
2 Ze 2
t
4 0 Ea
1/ 2
e  MZ 
G


4 0  2 Ea 
2
2
• Number of hits, on surface of nucleus
radius R ~ v/2R.Decay rate
( 2 Ea / m )

exp(G )
2R
Tony Weidberg
Nuclear Physics Lectures
7
Experimental Tests
• Predict log decay rate proportional to (Ea)1/2
• Agrees ~ with data for e-e nuclei.
• Angular momentum effects:
– Additional barrier
El 
l ( l  1)( c )2
2 Mc 2r 2
– Small compared to Coulomb but still generates large extra
exponential suppression. Eg l=1, R=15 fm El~0.05 MeV cf
for Z-90  Ec~17 MeV.
• Spin/parity
DJ=L
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parity change=(-)L
Nuclear Physics Lectures
8
Experimental Tests
Half-life (s)
1018
10-6
4
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Energy E (MeV)
Nuclear Physics Lectures
9
9
Fermi b DecayTheory
• Consider simplest
case: n decay.
• At quark level:
du+W followed by
decay of virtual W.
n  pe e ; d  ue e
d
u
u
n
d
u
d
p
eW-
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Nuclear Physics Lectures
( e)
10
Fermi Theory
• 4 point interaction (low energy approximation).
*
*
*
3
M if  G b y e (r )y (r )y p (r )y n (r )d r
 
 

 
y e (r )  exp(ik e .r ) ; y (r )  exp(ik .r ) ;q  k e  k


q ~ 1MeV / c R ~ 5fm  q.r ~ 1 / 40  exp(iq.r )  1
M if  G b  y p* (r )y n (r )d 3r
Tony Weidberg
Nuclear Physics Lectures
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Fermi Theory
• e distribution determined by phase space (neglect
nuclear recoil energy)
dN e  4pe2dp e / h3 ;dN  4p 2dp / h3


d2N  4pe2dp e / h3 4p 2dp / h3

p  (Ef  Ee ) / c ; p / Ef  1 / c
d N
2
16
2
6 3
h c
2
pe dp e (Ef  Ee ) dE f
2
• Use FGR : phase space & M.E. decay rate
Tony Weidberg
Nuclear Physics Lectures
12
Kurie Plot
I(p )  Ap 2 (Ef  Ee )
 A(Ef  Ee )
Coulomb correction 
Fermi function K(Z,p)
Continuous spectrum
neutrino
End point gives limit on
neutrino mass
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(I(p)/p2K(Z,p))1/2
p
2
Intensity
I (p )
Tritium b decay
18
Electron energy
(keV)
Nuclear Physics Lectures
Electron energy
(keV)
13
Selection Rules
• Fermi Transitions:
– e couple to give 0 spin: DS=0
– “Allowed transitions” DL=0  DJ=0.
• Gamow-Teller transitions:
– e couple to give 1 unit of spin: DS=0 or ± 1.
– “Allowed transitions” DL=0  DJ=0 or ± 1.
• “Forbidden” transitions:


 2
exp(iq.r )  1  (iq.r )  O(q.r )  ...
– Higher order terms correspond to non-zero DL.
Therefore suppressed depending on (q.r)2L
– Usual QM rules give: J=L+S
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Nuclear Physics Lectures
14
Electron Capture
• Can compete with b+ decay.
e  p  n   e
R
M if  G b y e (r )y p (r )y n* (r )y* (r )d 3r
0
• For “allowed” transitions.
R
Mif  G by e (0)y (0) y p (r )y n* (r ) d 3r
0
• Only l=0. n=1 largest.
y e ( 0)  
1 / 2
 Zme e 


2 
 40 
2
2 3
3/ 2
Zm ee 
2 Gb 
2
Mif  3 
M
F
L  4 0 2 
2
Tony Weidberg

e xp(ik .r )
;y  ( r ) 
L3 / 2
R
M F  y n* (r )y p (r )d 3r
Nuclear Physics Lectures
0
15
Electron Capture (2)
• Density of states:
dN 4 q 2 3 dN dN dq
 3 L ;

dq
h
dE dq dE
; E  q c
dN 4 q 2 3
 3 L
dE
hc
• Fermi’s Golden Rule:
w
2
2 dN
M

dE
2

16

E
Zm
e
 
e 
w  G 2b M F2
h4c3  4 0 2 
2 2
Tony Weidberg
3
Nuclear Physics Lectures
16
Anti-neutrino Discovery
• Inverse Beta Decay

n  pe  e ; ep  ne

• Same matrix elements.
2
2 6
M  G b MF L
2
• Fermi Golden Rule:
2
2 dN
w
M

dE
2 2 2 dN
w
G b MF

dE
Tony Weidberg
Nuclear Physics Lectures
17
Anti-neutrino Discovery (2)
• Phase space factor
dN 4p 2L3 dp

dE
h 3 dE
• Neglect nuclear recoil.
2
2 2
2 4
dp
E p c m c ;
 E / pc 2
• Combine with FGR dE
2 2 2 4pe Ee
w
Gb M F 3 2 3

hc L
; R  F
F  c/ L
3
3
16

peEe
2
  G 2b MF
Tony Weidberg
4 3
h c
Nuclear Physics Lectures
18
The Experiment
• For E~ 1MeV ~10-47 cm2
• Pauli prediction and Cowan and Reines.
 ep  ne 
e e  2g (prompt )
n  Cd  gs(9MeV, delayed)
Liquid Scint.
1 GW
Nuclear
Reactor
Tony Weidberg
H20+CdCl2
Shielding
Nuclear Physics Lectures
PMTs
19
Parity Definitions
r  r
; P[y (r )]  y (r )
P [y (r )]  y (r )
2
P (v )  v ; P (v1.v2 )  v1.v2
Lrxp
P( L)  L
• Eigenvalues of parity are +/- 1.
• If parity is conserved: [H,P]=0  eigenstates of H
are eigenstates of parity. If parity violated can have
states with mixed parity.
• If Parity is conserved result of an experiment should
be unchanged by parity operation.
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Nuclear Physics Lectures
20
Parity Conservation
• If parity is conserved for reaction a+b c+d.
ha hb ( 1)L
IN
 hc hd ( 1)L
FINAL
• Nb absolute parity of states that can be produced
from vacuum (e.g. photons) can be defined. For
other particles we can define relative parity. e.g.
define hp=+1, hn=+1 then can determine parity of
other nuclei.
• If parity is conserved <pseudo-scalar>=0 (see next
transparency).
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Nuclear Physics Lectures
21
 O p  y O py d r
*
y P O py d r
3
*
2
3
 O p    y PO p Py d r
*
3
 O p    (h p ) y Opy d r
2
*
3
 O p    y O py d r
*
3
<Op> = 0 QED
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Nuclear Physics Lectures
22
Is Parity Conserved In Nature?
• Feynman’s bet.
• Yes in electromagnetic and strong
interactions.
• Big surprise was that parity is violated
in weak interactions.
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Nuclear Physics Lectures
23
Mme. Wu’s Cool Experiment
60
Co(J  5)60Ni* (J  4) e e ;
60
Ni* 
Ni  g
60
• Align spins of 60Co with magnetic field.
• Adiabatic demagnetisation to get T ~ 10 mK
• Measure angular distribution of electrons and
photons relative to B field.
• Clear forward-backward asymmetry  Parity
violation.
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Nuclear Physics Lectures
24
The Experiment
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Nuclear Physics Lectures
25
Improved Experiment
q is angle wrt spin of
60Co.
Tony Weidberg
Nuclear Physics Lectures
26
g decays
• When do they occur?
– Nuclei have excited states cf atoms. Don’t worry
about details E,JP (need shell model to
understand).
– EM interaction << strong interaction
– Low energy states E < 6 MeV above ground state
can’t decay by strong interaction  EM.
• Important in cascade decays a and b.
• Practical consequences
– Fission. Significant energy released in g decays.
– Radiotherapy: g from Co60 decays.
– Medical imaging eg Tc.
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Nuclear Physics Lectures
27
Energy Levels for Mo and Tc
b decay leaves Tc
in excited state.
Useful for medical imaging
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Nuclear Physics Lectures
28
g Decay Theory (Beyond
Syllabus)
• Most common decay mode for nuclear excited
states (below threshold for break-up) is g decay.
• Lifetimes vary from years to 10-16s. nb long lifetimes
can easily be observed unlike in atomic. Why?
• Angular momentum conservation in g decays.
– intrinsic spin of g is1 and orbital angular momentum integer
 J is integer.
– Only integer changes in J of nucleus allowed.
– QM addition of J:
Ji  J f  J  Ji  J f
– Absolutely forbidden (why?): 00
Tony Weidberg
Nuclear Physics Lectures
29
g Decays
• Electric transitions


E  E0 exp[i ( k .r  t )]
  2
 3
E  E0 (1  ik .r  ( k .r )  O( k .r )
• Typically k~1 MeV/c r~ 1 fm k.r~1/200 
use multipole expansion. Lowest term is
electric dipole transitions, L=1.
2
2
*
3
H   y f er y i d r
• Parity change for electric dipole.
Tony Weidberg
Nuclear Physics Lectures
30
Forbidden Transitions
• If electric dipole transitions forbidden by angular
momentum or parity can have “forbidden”
transitions, eg electric quadropole.
• Rate suppressed cf dipole by ~ (k.r)2
• Magnetic transitions also possible:
• Classically: E=-m.B
• M1 transition rate smaller than E1 by ~ 10-3.
• Higher order magnetic transitions also possible.
• Parity selection rules:
– Electric: D=(-1)L
– Magnetic: D=(-1)L+1
Tony Weidberg
Nuclear Physics Lectures
31
Internal Conversion
• 00 absolutely forbidden:
• What happens to a 0+ excited state?
• Decays by either:
– Internal conversion: nucleus emits a virtual
photon which kicks out an atomic electron.
Requires overlap of the electron with the
nucleus only l=0. Probability of electron overlap
with nucleus increases as Z3. For high Z can
compete with other g decays.
– Internal pair conversion: nucleus emits a virtual
photon which converts to e+e- pair.
Tony Weidberg
Nuclear Physics Lectures
32