Transcript Interaction of Charged Particles with Matter

1
Nuclear Beta Decay
W. Udo Schröder, 2009
Super Kamiokande (Japan) neutrino detector
50,000 t H2O) Cerenkov counter, 11,200 PMTs
Electron/Beta Spectrometry
Chadwick (1914): Some nuclides emit ewith continuous energy spectra “b rays”
2
b
Active
sample
a
60 Helmholtz coils
every 60 arranged
in a circle.
Current: ~1000 A
Magnet
g
Iron-free “Orange” spectrometer with axially symmetric toroidal magnetic field inside current loops
Setup used in
nuclear reaction
studies (counters
for coincident
particles & g-rays)
Nuclear Beta Decay
B
Radioactive Ra sample in a
magnetic field b = e-.
Observed later in decay of
neutrons and excited nuclei
(internal conversion) or nuclear
transmutation (b decay).
W. Udo Schröder, 2009
Different energies
correspond to
different locations
on focal detector
Circular e  orbit radius  in B field
pe  e  B  
Ee  pe2 2me  dEe   pe me  dpe
 m  dN
dN
  e
dEe
 pe  dpe
Energy spectrum constructed
from momentum spectrum
Electron and Beta Spectroscopy
Nuclei can deexcite via photon, (e+, e-) , or atomic-electron emission (internal conversion)
Conversion electron line spectrum for decay
of 203Tl state E*=280-keV
I 
g
Electron binding
energies in 203Tl
e
e e 


 eject
3
E*

I


, I 1  1
gs
Ee=E*-EB < 280keV
Nuclear Beta Decay
Nuclei transmute in b decay
Z , I 
dN
dEe

Q0
e  eject
 Z ± 1,( I , I  1) gs
gs
b spectrum is
continuous up
to Ee ≈ Q
Eemax  Q
Ee
Fixed differences Q and |DI| carried by more than one decay product  additional “neutrinos”
W. Udo Schröder, 2009
, 
The Neutrino Hypothesis
Dilemma: continuous e- spectrum would violate
energy/momentum balance in 2-body process.
Nuclear Beta Decay
4
Wolfgang Pauli (1930) postulates unobserved,
neutral particle (“neutron” later =“neutrino” (Fermi))
W. Udo Schröder, 2009
Evidence for Neutrino
Z , I 

Q0
e  eject
• Electron capture produces recoil momentum
5
 Z ± 1,( I , I  1) gs
gs
•Fixed decay energy (Q value  Dmc2)
but continuous e- spectrum
• e- has spin Ie=1/2
but |Ifinal-Iin|= 0, 1 typically
Nuclear Beta Decay
• Direct evidence by neutrino-induced reaction
pe
e-
37
Recoil Experiment
Ar  e  37Cl  
pN
TOF
distance
com
pN

i  observed
W. Udo Schröder, 2009
pi  0

i  observed
pi  pN
Auger eDetector
37Ar
cell
gas
Recoil
Detector
Fermi’s Neutrino Hypothesis
Enrico Fermi (1934): Adapt Dirac’s elm field theory to weak interactions.
Weak (beta-decay-type) interaction is similar to elm interaction between
currents. Range of weak interaction is rWI ≈ zero (relm  )
Nuclear Beta Decay
6
Electromagnetic CurrentCurrent Interactions
Fermi’s theory accepted as working hypothesis for weak interactions.
Neutrino properties predicted: spin=1/2, zero charge, zero mass.
Directly observed: 1956(Science)/1959(PR) by Fred Reines & Clyde Cowan
W. Udo Schröder, 2009
Direct Evidence for Neutrino
Savannah River
reactor experiment
(fission fragments decay
900 hrs with reactor on
Reines
Inverse
betaCowan
decay 250 hrs reactor off
 e  p  e  n
Experiment: s = 7·10-19b
LSc
(Cd)
tanks
Target
tanks
H2 O
7
e  e  annihilation
e   e   2 g  511keV 
Delayed n  capture g  rays
109
Cd 
Nuclear Beta Decay
nth 
W. Udo Schröder, 2009
110
Cd* 110 Cd  xg
prompt e+-delayed capture g coincidences
Elementary Modes of b Decay
8
Fermi’s zero-range (point-like) weak interaction, coupling constant GF
Different lepton families : electron, muon, tau All neutrinos have small
masses and 
neutrinos :  e , e
  ,    ,  (only upper limits known)
Nuclear Beta Decay
Nuclear b decay and electron capture
In energetics of decay, account
for electrons. Mass tables
apply to neutral atoms.
Example: EC “recycles” eb + decay of p produces ion
b
Bethge, Kernphysik
W. Udo Schröder, 2009
b
EC
b
to
K-hole
Beta Decays of Odd-A and Even-A Nuclei
m  A, Z   a  A   b  A  Z  g  A  Z 2  D
4as   mn  mp  me  c 2  A
b

m  mmin : Z A 
 
2g
2 4as  aC A2 3
Expand around ZA:
Mass parabola bottom of valley
odd-A
isobars
D=0
Nuclear Beta Decay
mc2
9

b
b
b b
ZA
W. Udo Schröder, 2009
Z


 11.2
MeV o  o

A

D   0 MeV A  odd
 11.2

MeV e  e
A

m(Z)  a( A)  D   b  Z  Z A 
2
Energetics of b Decay
Beta decay and EC (K)-capture
 Z, A   Z  1, A  e   e | b 
m(Z , A)  m(Z  1, A)
 Z , A    Z  1, A 

 e   e | b 
m(Z , A)  m(Z  1, A)  2me
10
 Z, A  e   Z  1, A   e | EC
1 extra e+
1 extra e-
m(Z , A)  m(Z  1, A)
Nuclear Beta Decay


 11
Example :  11
C  6e  

Be

6
e

e
  e  Qb
6
5




b
5
Qb>0
exotherm
1
Mass balance:
m(11 Be)c2  mec2   mec 2  Qb
m(11 C )c2 



b


Qb  m(11 C )c2  m(11 Be)c 2  2mec 2
Decay Q-value smaller by 2mec2 for b+ decay than for b-
W. Udo Schröder, 2009
Fermi Theory of b Decay
p
EC
core
core
e-
e
Isospin operators ˆ2 , ˆ3 ,ˆ analog to spin operators
f
11
i
n
Simple example: single nucleon orbiting core of
paired nucleons captures atomic 1s electron.
Isospin wave functions  p ,  n
i   p   r  p
Nuclear Beta Decay
Pif 
2
ˆ i
f H
WI
ˆ3  n    1 2   n
ˆ3  p    1 2   p
ˆ  p   n
ˆ  n   p
f   n    r  n
2
initial, final s.p. nuclear states
   Ef  Fermi’s Golden Rule
Perturbation
theory for i  f
Density of final
ME of weak
interaction H states per unit
energy
Weak Interaction Hamiltonian (point-like)



ˆ  G  ˆ   r  r    r  r    r  r
H
WI
F

p
e
n

p
n
W. Udo Schröder, 2009

GF: coupling constant,
ˆ: Isospin raising operator
 : delta distribution
Weak Transition Matrix Elements
ˆ i  d 3 r  *  r   G ˆ    r 
Hfi : f H
WI
f
F 
i

 i  r    e  r  p  r  core  r 
 f  r      r  n  r  core  r 
12
Lepton wave
functions
vary weakly
over nuclear
volume 
 e r 
 p r 
 n r 
2
104 fm
5 fm
Nuclear Beta Decay
 r 
2
2
2
r
104 fm
5 fm
r
 e ,  r 
2
Hfi
2
 GF2

d 3 r  f*  r ˆ i  r  
Nucl
2
 GF2  e (0)   (0)
2
 core  core
2

d 3 r  n*  r ˆ p  r 
Nucl
=1
W. Udo Schröder, 2009
=1, per def
2
2
  e ,  0 
2
Fermi Transition ME
Hfi
2
2
 G  e (0)  (0)
2
F
2
Hydrogen-like e- wave function
13
2
 e (0) 1s  2 
3
Z
e
3
 aB

2Zr
aB
Bohr Radius aB  5  10 4 fm
Plane-wave e wave function
Nuclear Beta Decay
  (r ) 
  (0 )
Hfi
2
2
1
e i k r
V
1 i k r

e
V
Normalization volume, drops
out in final calculations
2

1
V
2  Z3 1
G

3
 aB V
W. Udo Schröder, 2009
2
F
Fermi transitions
(“super-allowed”):
No change in I, 
For Pif need to evaluate
density (Ef) of final states:
neutron-neutrino relative
phase space
Neutrino Phase Space
Pif 
2
2  Z3 1
G
    Ef 
3
 aB V
2
F
=# final (n, ) states at energy Ef EC:
Ef ≈ E neglect nuclear recoil energy
 Dpx  Dpy  Dpz    Dx  Dy  Dz   h3
p
4 p2dp
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d 2n  4 p2dp dV h3
Dp
Nuclear Beta Decay
dV
Uncertainty
Relation
p  E c
dn
E2

V    Ef 
2
3
3
dE
2
c

2  2 2  Z 3 1   E2
2  Z3
2
2
Pif 


V

G
E


GF

F
 : gs
3
2 3 3
2 4 3
3
 aB V   2 c 
 c aB

Use experimental data for 7Be EC decay to determine GF 
GF ≈ 100 eV fm3. More exact average over many data sets:
GF ≈ 88 eV fm3
W. Udo Schröder, 2009
Branching in EC b Decay
2  Z3
Pif  G 2 4 3 3 E2
 c aB
15
 phase space depends on
Q = Emax 
rate  increases with Emax
0.86 MeV
7Be
3
2
E  E max  Q
2
  Emax   Ema
x
ex 0.478 MeV  Q  0.478MeV 

gs
Q2

EC
12%
I
0.48 MeV
EC
88%
Nuclear Beta Decay
2
F
0.0 MeV
7Li
1
2
2
ex  0.382 

 0.20

gs  0.861 

3
2

Experimental value correct
magnitude but disagrees
 ex

 gs

 0.115

exp
Reason: n ≠ p because of nuclear spin change 3-/2  1-/2
“forbidden” transition
W. Udo Schröder, 2009
2
Shape of the b± Spectrum
Beta decay other than EC
 N, Z  
 3-body final state
Neglect nuclear recoil energy.
d  ne  n 
2
2
Pif 
Hfi    Ef 
  Ef  


 N  1, Z  1  e   e


N

1,
Z

1

e
 e




16
dEf
dn
4 p2
4 V 1
2

V

E

E


max
e
dp
h3
h3 c 2
plane waves for e,  Hfi
2
p  E c
Ef  Emax  Ee  E
dne 4 pe2

V
3
dpe
h
 1 V 2 (problematic for e  , Coulomb)
Nuclear Beta Decay
Fixed Ee  dp dEmax  1 c
dne  dn 
V2
4 4
1
6
c3
pe2dpe  p2dp
dn
V2
2
2
dne 

p
dp
E

E
   Ef  dpe


e
e
max
e
4
6
3
dEmax 4
c
2
2
dNe GF Hfi
2
2


p

E

E
 max e 
dpe 2 3 7c3 e
W. Udo Schröder, 2009
momentum
spectrum
Shape of b± Spectrum/Coulomb Correction
Relativistic momentum-energy relation
Ee  W 
dW

dpe

 pe c 
2
 me c 2

2
Emax  Wmax  Q (neglect nucl. recoil )
pe c 2

 pe c   mec2
2
pe c  W 2  me2c 4

2
Nuclear Beta Decay
17
2
2
GF2 Hfi
GF2 Hfi
dNe
2

 peW  Wmax  W  
W
3 7 5
3 7 5
dW
2
c
2
c
dNe
dpe
W 2  me2c 4 Wmax  W 
2
Should use Coulomb e (r) ≠ plane wave.
Electron cloud acts as barrier for e+. Nonrelativistic numerical correction factor
(Fermi function)
2
2
2
F  Z , pe  :  e  0   efree  0  
1  exp   2 
bZ=0
b
 : 
+
e2 Z
e
2
 for
2
Barrier effect
W. Udo Schröder, 2009
pe
b

2
dNe GF Hfi
2
2

F
(
Z
,
p
)
p
E

E


e
e
max
e
dpe 2 3 7c3
Kurie/Fermi Plot
Kurie plot gives extrapolation to Emax of electron spectrum
2
2
dNe GF Hfi
2
2

F
(
Z
,
p
)
p
E

E


e
e
max
e
dpe 2 3 7c3
18
64Cu
b+ and b- Decays
 Linear Kurie Plot
 dNe
2
F
(
Z
,
p
)
p

e
e    Emax  Ee 
dp
 e

factor 
GF2 Hfi
2
2 3 7c3
Nuclear Beta Decay
Validity of Kurie Plot
•|Hfi| ≠ f(Ee)
• DI = 0 (allowed transitions)
• mc2≈ 0 eV
Owen et al. PR 76, 1726 (1949)
W. Udo Schröder, 2009
For DI ≠ 0  additional
correction factors
Kurie plots for forbidden
transitions
Neutrino Mass Effect
Correct decay energy for mc2:
Emax  Wmax  Emax  m c 2 ,
p2c 2  W2  m2c 4
19
dp
W
1
1
 
dEmax c
c W 2  m2c 4


2
12
dNe
GF2 Hfi
2
2
2 4
2 4

 F  3 7 4 W  me c Wmax  W  Wmax  W   m c


dpe
2
c

dNe
Kurie Plot
3H b - Decay
Ee (keV)
W. Udo Schröder, 2009
m ≠ 0 deviations of Kurie plot
from linearity at end point.
No direct evidence for mc2≠ 0
Indirect evidence (neutrino
oscillations) mc2 > 0.1 eV
Nuclear Beta Decay
Fpe2dpe

Total b± Decay Rate
Seek method to systematize data: Unit conversion
pe
2 3 7
W
 0 : 5 4 2
 :

:

2
me c GF
Hfi
mec
2
20
dNe
2

  2  1   max   
d
0
 
 max

d
1
mec
dNe
n2

d
t1 2
for F  1, m  0
Parameterization (Machner , 2005 )
b( Z )
f ( Z ,Emax )  a( Z )  E max


a( Z )  exp 5.553  7.3418 exp  Z 213.86 
b( Z )  4.148 exp  Z 51.6
Z  0 for b  , Z  0 for b 
Nuclear Beta Decay
Coulomb Correction :
f ( Z ,  max ) 
 max

d F ( Z ,  )     2  1    max   
Universal numerical
function, independent
of spectrum  Tables
2
1
Nuclear structure information
 
Hfi
2
0
W. Udo Schröder, 2009
n2
 f  Z ,  max  
t1 2
Hfi
2
 GF2

d 3 r  f*  r ˆ i  r 
Nucl
Phase space : f  Z ,  max  ,  0
2
b± Decay ft-Values
Experimental task: Emax, and t1/2
combination  nuclear matrix element
ft : f  Z ,  max  t1 2 
 0  n2
Hfi
2
21
B   0  n2   2787  70  s
Hfi
2
ft :
Hfi
2
 B ft
t1 2
1s
6·1014 y
Super allowed b transitions:
Large matrix elements, small ft
observed only for light nuclei
(“mirror nuclei”) and DI=0,±1
1st forbidden
17
7
allowed
super allowed
Frequency
of ft Values
Nuclear Beta Decay
Large ft: slow transitions, small|Hfi|2
B
Meyerhof, 1967
W. Udo Schröder, 2009
b
F 
 17
8 O
p
16
8
O
log ft  3.38
n
16
8
O
22
Nuclear Beta Decay
W. Udo Schröder, 2009
2
2
dNe GF Hfi
2
2

F
(
Z
,
p
)
p
E

E


e
e
max
e
dpe 2 3 7c3
Kurie Plot
Solid line corresponds
to mc2=100 keV
Nuclear Beta Decay
23
3H
W. Udo Schröder, 2009
 Linear Kurie Plot
 dNe
2
F
(
Z
,
p
)
p

e
e    Emax  Ee 
dp
 e

factor 
GF2 Hfi
2
2 3 7c3
Allowed and Forbidden b Decays
Kurie Plot
allowed decay
24
36Cl
36Cl
Kurie Plot
forbidden decay
Nuclear Beta Decay
1st
Ee (keV)
W. Udo Schröder, 2009
Nuclear Beta Decay
25
Double b Decay
W. Udo Schröder, 2009
Light Guide
Pumping Inlet
26
Polar
NaI
Counter
Anthracite
Scintillator
Count Rate/Count Rate warm
Parity Violation in b Decay
g Anisotropy average of both
counters, both field polarities
Sample
Ce/Mg Nitrate
Container

Count Rate/Count Rate warm
g
Equatorial
NaI Counter
Nuclear Beta Decay
g Anisotropy
a equatorial counter
b polar counter
W   2  W 0
W   2
b Anisotropy
t (min)
W. Udo Schröder, 2009