Transcript Seminar: Statistical Decay of Complex Systems (Nuclei)

FermiGasy
Addition of Angular Momenta
Angular momenta L1, L2 , direction undetermined
Projections conserved :m1, m2 , m  m1  m2  L1  L2  L  L1  L2
 (t )
1  const.
2  const.
1(t ), 2(t )  L(t ),  (t ), (t ) 
2
i (t )  i Larmor frequency
2
L  L(t )  Classical Probability :
Angular Momentum Coupling
2

P(L)   L t 
1

2
 2L L t

1
L2  m2  L21 sin2 1  L22 sin2 2 
1
 L1L2 sin 1 sin 2 cos(1  2 )
If (L1)  (L2 ) dipole interaction : L1 couples with L2  L
Coherent motion : m  conserved,   const.   (t ) L  L(t )
W. Udo Schröder, 2005
At large r
1  2  r1  r2
3
 decoupling
Angular Momentum Coupling
Quantal angular momentum eigen states j1  j2  ??
j
j
ˆ
J12 , ˆ
J1z : 1 ; ˆ
J22 , ˆ
J2 z : 2
m1
m2
3
Max alignment :
j1
j2
j1
j2
 dim  (2 j1  1)(2 j2  1) dimensionality
:
j1
j2
j1
j2

ˆ
2 j1
J
j1
j2
j2
??
 ??
j1
j2
j1
j2
2
Angular Momentum Coupling
Use ˆ
J 2   ˆ
J1  ˆ
J2   ˆ
J12  ˆ
J22  2ˆ
J1  ˆ
J2  ˆ
J12  ˆ
J22  ˆ
J1 ˆ
J2   ˆ
J1 ˆ
J2   2ˆ
J1z ˆ
J2 z


ˆ
2 j1
J
j1
j2
j2


2
 j1( j1  1) 
2
( j1  j2 )( j1  j2  1)
J
ˆ
Jz
j1
j2
j1
j2
W. Udo Schröder, 2005
j2 ( j2  1)  0  0  2 j1 j2 
 ( j1  j2 )
J 1
j1
j2
j1
j2
j1
j2
j1
j2
 m

2
j1
j2
j1
j2
j1
j2
j1
j2
J( J  1)

j1
j2
j1
j2
 The first

 ˆ
J2, ˆ
Jz
eigen state

Constructing J Eigen States
J , for example
Construct m  J spectrum successively by applying ˆ
ˆ
J
j1
j2
j1
j2
J  j1  j2
4

J2 
J1  ˆ
 ˆ
m  J 1
 2 j1

j1
j2
j1
j2
j1
j2
j1  1
j2

2 j1
+ 2 j2
j1
j2
j1  1
j2

2 j2
j1
j2
j1
j2  1
j1
j2
 eigen state
j1
j2  1
J  j1  j2 , m  J  1
Can you show this??
This is one specific linear combination of 2 states j1 , m1 and
j2 , m2
Angular Momentum Coupling
What about the other ? There should be 2 orthogonal combinations.
And : Further application of ˆ
J yields again only one specific linear
combination of 3 independent components
J  j1  j2
m  J 2

j1
j2
j1  2
j2
+
j1
j2
j1  1
j2  1
+
j1
j2
j1
j2  2
What about others ? There should be 3 orthogonal combinations.
W. Udo Schröder, 2005
Constructing J-1 Eigen States
We have this state:
J  j1  j2
 2 j1
m  J 1
j1
j2
j1  1
j2
+ 2 j2
j1
j2
 eigen state
j1
j2  1
J  j1  j2 , m  J  1
Two basis states j1 , m1 , j2 , m2  2 new orthogonal states
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J  ??
m  J 1
 2 j1
j1
j2
j1  1
j2
- 2 j2
j1
j2
j1
j2  1
Angular Momentum Coupling
Apply ˆ
J2  ˆ
J12  ˆ
J22  ˆ
J1 ˆ
J2   ˆ
J1 ˆ
J2   2 ˆ
J1z ˆ
J2 z
Use
ˆ
J1
j1
j2
m1
m2
J  j1  j2  1
m J

 2 j1
 j1
j2
j1  1
j2
- 2 j2
ˆ
J
to
m1   j1  m1  1
j1
is orthogonal
j1
j2
m1
m2
j1
j2
j1
j2  1
j1
j2
j1
j2
etc.
eigen state
J  j1  j2  1 ...etc.
m J
Condon-Shortley
Normalization conditions leave open phase factors  choose
asymmetrically <|J1z|> ≥ 0 and <a|J2z|b> ≤ 0
W. Udo Schröder, 2005
Clebsch-Gordan Coefficients
General scheme : Unitary transformation (between bases)
 j1
j2 j  j1
  

m
m1 , m2  m1 m2 m  m1
j2
j
6
j
m

j1

m1 , m2
m1
j2  j1

m2  m1
m2
 j1
  
m1 , m2  m1
j

m2 m 
j2

m1 , m2
j  j1

m2 m  m1
j2
j1
j2
j1
m1
m2
m1
j2
m2
j2
j
m2 m
ˆ
Representations of identity operator 1
j1 j2 :
Angular Momentum Coupling

m1 , m2
j1
j2
j1
j2
m1
m2
m1
m2
ˆ
1
j1 j2

j,m
j
j
m
m
ˆ
1
j1 j2
Orthogonality relations of CG coefficients :
 j j1
 
m1 , m2  m m1
j2   j1

m2   m1
j 

m2 m 
j2

m1 ,m2
j2
j  j1
m m1 m2
j1
j2
j
m1 m2 m
  j j  mm
=1
 j1
 
j , m  m1
W. Udo Schröder, 2005
j2 j   j j1

m2 m   m m1
j2 
j1



m2  j,m m1
j2 j
m2 m
j1
j2
m m1
m2
j
=1
  m1 m1  m2 m2
Recursion Relations
j2  j1
j2 j 
j1
j2
j1
j2
j
 





m1 m2
m
m
m1 , m2 m1 m2  m1 m2 m 
m1 , m2 m1 m2
j1
j2
j1
j2
j
j
j
j
ˆ
ˆ
J 
J
m1 m2 m - 1 m - 1
m
m
m1 , m2 m1 m2
j1
j
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ˆ
P
j , m 1

j
m-1
ˆ
J
j
m

m1 , m2
j1
m1
j2
j1
j2
m2 m  1 m1
m2
j

Angular Momentum Coupling
 j  m j  m 1

ˆ
J1- + ˆ
J2-
m
j


m1 , m2
j1
m1
j2
j
m2 m
j1
m1 - 1
ˆ
J1
j1
m1
j1
j2
m1 - 1 m2
 j1  m1  j1  m1 1

C jm :
 j  m j  m 1
W. Udo Schröder, 2005

m1 , m2
j1
m1
j2
j
m2 m
j2
m2 - 1
ˆ
J2 
j2
j1
j2
m2
m1
m2 - 1
 j2  m2  j2  m2 1

Recursion Relations for CG Coefficients
C jm
j1
m1
8
j1
j2
m2 m  1 m1
m2
j
j1
C j1m1 1
j2

j
j1
j2
m1  1 m2 m
m1
m2
 j  m j  m 1
C jm :
j1
 C j2m2 1
m1
j2
j
j1
j2
m2  1 m
m1
m2
j2
j
Projecting on <j1,j2,m1,m2| yields
C jm
Angular Momentum Coupling
j2
j1
j2
m1
j
m2 m  1
 C j1m1 1
j1
j2
j
m1  1 m2 m
 C j2m2 1
j1
m1 m2  1 m
Using ˆ
J  ˆ
J1  ˆ
J2  
C jm 1
j1
m1
j2
j
m2 m  1
 C j1m1
j1
j2
j
m1  1 m2 m
 C j2m2
j1
j2
j
m1 m2  1 m
Special values :
 j 0 j

 1
 m 0 m
W. Udo Schröder, 2005
 j1 j1 0  (1) j1  m1


m

m
2 j1  1
1 0
 1

0
0
 ???
Symmetries of CG Coefficients
Coupling depends on sequence ˆ
J1 , ˆ
J2


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Phase convention : non  diag.  ˆ
J1z   0,  ˆ
J2 z   0
 j1 j2 j 
j j j

  (1) 1 2
 m1 m2 m 
 j2 j1 j 


m
m
 2 1 m
Angular Momentum Coupling
 j1 j2 j3 
j3 j1 
j2  m2 2 j3  1  j2



  (1)
m
m
m
m

m
m
2
j

1
3
2
1
1
 1 2 3

 j1 j2 j3 
j1 j2 
j1  m1 2 j3  1  j3



  (1)
m
m
m
m

m
m
2
j

1
1
2
2
 1 2 3
 3
 j1 j2 j3 
j j j

  (1) 1 2 3
 m1 m2 m3 
m(m3 )  m1  m2
Triangular relation
Condon-Shortley :
Matrix elements of
J1z and J2z have
different signs
 j1 j2
j3 



m

m

m
3
1
2

Calculate CGs starting from max alignment : m  j
Then use recursion relations to obtain all
W. Udo Schröder, 2005
j, m  j
Explicit Expressions
A. R. Edmonds, Angular Momentum in Quantum Mechanics
 j1 j2 j 

   m, m1  m2 
m
m
m
 1 2

10
12

2 j  1  j1  j2  j  !( j1  m1 )!( j2  m2 )!( j  m)!( j  m)!  


j

j

j

1
!
j

j

j
!

j

j

j
!(
j

m
)!(
j

m
)!






 1
2
1
2
1
2
1
1
2
2 

j m
( j1  m1  s)!( j2  j  m1  s)!
s  j m
   1 1 1
s !( j1  m1  s)!( j  m  s)!( j2  j  m1  s)!
s 0
Angular Momentum Coupling
12
 j1 j2 j1  j2   2 j1  ! 2 j2  !  j1  j2  m1  m2  !  j1  j2  m1  m2  ! 



m
m
m

m
2
j

2
j
!
j

m
!
j

m
!
j

m
!
j

m
!











2
1
2
1
1
1
1
2
2
2
2 
 1 2 1


 j1 j2
j


 j1 m  j1 m 
12


2 j  1 ! 2 j1  !   j1  j2  j  !  j1  j2  m  !( j  m)!


j

j

j
!
j

j

j
!
j

j

j

1
!

j

j

m
!
j

m
!










2
1
2
1
2
1
2
 1

W. Udo Schröder, 2005
2 Particles in j Shell (jj-Coupling)
Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0 
spatially symmetric  jj1(r) = jj2(r). Construct consistent spin wf.
J
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 JM   j   j 
M
j  j  0  J  j  j  2j
 j j J
 JM (r1, r2 )  N  
  jm1 (r1 ) jm2 (r2 )   jm2 (r1 ) jm1 (r2 )
m1, m2  m1 m2 M 
N = normalization factor
Angular Momentum Coupling
Which J = j1+j2 (and M) are allowed?  antisymmetric WF JM
 j1 j2 j 
j j j
Use Pauli Principle and 
  (1) 1 2
 m1 m2 m 
 j2 j1 j 

 for 12 : 1  2
m
m
m
 2 1 
 j j J   j j J  
 JM (r1 , r2 )  N  

   jm1 (r1 ) jm2 (r2 )
m
m
m
m
m1 , m2 
 1 2 M   2 1 M  
W. Udo Schröder, 2005
 j j J
 N 1  (1)2 j  J   

(r )
(r )

 m , m m m M  jm1 1 jm2 2 For j1  j2  5 2 
2
1 2 1

2 j1  j  5  j  odd
 N 12
 j  0, 2, 4
Symmetry of 2-Particle WFs in jj Coupling
Antisymmetric function of 2 equivalent nucleons (2 neutrons or 2
protons) in j shell in jj coupling.
12
 JM (r1 , r2 ) 
 j j J
 
  jm1 (r1 ) jm2 (r2 )
m1, m2  m1 m2 M 
J  even
1) j1 = j2 = j half-integer spins  J =even
wave functions with even 2-p. spin J are antisymmetric
Angular Momentum Coupling
wave functions with odd 2-p. spin J are symmetric
jj coupling  LS coupling  equivalent statements
2) l1=l2=l integer orbital angular momenta  L
wave functions with even 2-p. L are spatially symmetric
wave functions with odd 2-p. L are spatially antisymmetric
W. Udo Schröder, 2005
Tensor and Scalar Products
Tensor product of sets of tensors T k1 and T k2
k
T
k
 k1 k2 k  k1
k1
k2
k


 T (1)  T (2)   
T (1)T 2 (2)

2

 1,2 1 2  1


13
Scalar product of sets of tensors T k (1) and T k (2)  number
T00
k 
 k k 0 k
(

1)
k
 T (1)  T (2)   
T (1)T (2)  
Tk (1)Tk (2)


0    0

2k  1


k
k
0
Angular Momentum Coupling
Vectors u and v : Rank k  1  2k  1  3 spherical components
Spin  0 scalar product
1  ux  iuy
u  v 0 
 
3 
2
1

u v
3
0
W. Udo Schröder, 2005

  v x  iv y   ux  iuy   v x  iv y 



  uz v z 
2  
2 
2 


Transforms like a J=0 object = number
Example: HF Interaction
14
Addition Theorem of spherical harmonics :
P  cos   
4
*
 Ym (1 , 1 )Ym (2 , 2 ) 
2 1m
   1
4
4
Y m (2 , 2 )
Ym* (1 , 1 )
2 1
2 1
m
m
 
r1 , r2
Electron  nucleus (sum over p) hyperfine interactions
Angular Momentum Coupling
ˆ  
H
int
i , p,
ei ep
ri
   1
1
m
m,
rp P (
ri  rp
ri rp
)


4
4
*
ep rpYm ( p , p )  
ei ri

 p 2  1
  i 2  1
0
 Tp  Ti   scalar product of separated tensors
0
protons electrons
only
only
W. Udo Schröder, 2005

Y m ( i , i )

1
Wigner’s 3j Symbols
Coupling  j1  j2  j3 equivalent to symmetric  j1  j2  j3  0
j2
15
j1
j3
j2
j1
j3 
 j1 j2 j3   j3 j3 0  (1) j3  m3

 j3  m1 m2 m3   m3 m3 0 
2 j3  1
 j1 j2 j3 



m
m
m
3
 1 2
Angular Momentum Coupling
Choose additional arbitrary phase factor for 3 j symbol
 j1 j2 j3  (1) j1  j2  m3


2 j3  1
 m1 m2 m3 
 j1 j2 j3 



m
m
m
3
 1 2
 j1 j2 j3   j3 j1 j2   j2 j3 j1 



 all cyclic
 m1 m2 m3   m3 m1 m2   m2 m3 m1 
 j1 j2 j3 
j1  j2  j3

(

1)


m
m
m
 1 2 3
W. Udo Schröder, 2005
 j2 j1 j3 

 any 2 columns
 m2 m1 m3 
Explicit Formulas
16
 j1 j2 j3   j1 j2 j3 
1

 j3 j3  m3m3
 


  2 j3  1
m1 , m2  m1 m2 m3   m1 m2 m3
 j1 j2 j3   j1 j2 j3 
2
j

1
  3
  m m m   m m m    m1m1  m2m2
j3 , m3
 1 2 3 1 2 3
Explicit (Racah 1942):
Angular Momentum Coupling
 j1 j2 j3 
j1  j2  m3


1
  m1  m2  m3 , 0  

  
 m1 m2 m3 


 j1 
j2  j3  !  j1  j2  j3  !   j1  j2  j3  !

 j1  j2  j3  !
 j1  m1  !  j1  m1  !  j2  m2  !  j2  m2  !  j3  m3  !  j3  m3  ! 
  1  z !  j1  j2  j3  z  !  j1  m1  z  !  j2  m2  z  ! 
z
z
All factorials must be ≥ 0
W. Udo Schröder, 2005
  j3  j2  m1  z  !  j3  j1  m2  z  !
1
Spherical Tensors and Reduced Matrix Elements


Spherical tensor of rank j  2 j  1 operators Tmj (m   j,....,  j )
 j ' transforming like angular  momentum ops.
Tmj   Dmj ' m  ,  ,  T m
m'
17
Tmj transfers angular
momentum to I  0 state
,
I 0
M 0
Tmj
:
,
0
0
 ,
j
m
, ,  = Qu. #
characterizing
states
In general LC of basis states
Angular Momentum Coupling
Tmj
,
,
j2
m2
j1

m1
Tmj

 , j3 , m3
,
j1
m1
N


j j1 j3
m m1 m3
 , j3 , m3
N
,
j3
N 
m3
j j1 j3
m m1 m3
,
j2
m2
,
degeneracy
not due to m3
j3
m3
 N
W. Udo Schröder, 2005
j2
m2
Tmj  ,
j1
m1
 N
m3
j j1 j2
m m1 m2
Wigner-Eckart Theorem
,
,
j3
j j1 j2
 dyn  geometry
m m1 m2
Wigner-Eckart Theorem
,
j2
m2
Tmj
18
,
,
j2
m2
j1
m1
Tmj
j j1 j2

N 
m m1 m2
,
j1
m1

(1) j2  m2
 j j1 j2 
j

  j2 T  j1
 m m1 m2 
Angular Momentum Coupling
Reduced (double  bar ) Matrix Element
j2
j1
j
 j2 T j  j1 contains all physics.
Conditions for non  zero :
3 angular momenta j1 , j, j2
" couple to zero "  m2  m1  m
Take the simplest ME to calculate
W. Udo Schröder, 2005
 j2 T j  j1
Examples for Reduced ME
Example 1 : const. operator  1
19
,
j2
m2
1 ,
j1
m1
   j1 j2  m1m2 
 0 j1 j1 

  j2 1  j1
m

m
0
1
1

 j1 j1 0  (1) j1  m1
Remember 

2 j1  1
 m1 m1 0 

Angular Momentum Coupling
(1) j1  m1
 j2 1  j1  2 j1  1   j1 j2

Example 2 : Vector operator  ˆ
J  ˆ
J
,
j2
m2
Jz  ,
j1
m1
 m1  j1 j2  m1m2 
 j 1 j1 
Look up  1

 m1 0 m1 
W. Udo Schröder, 2005

ˆ
J

use
ˆ
Jz 
ˆ
Jz
simplest
(1) j1  m1
 j1 1 j1 

  j2 J  j1

m
m
10 1

(1) j1  m1 m1
j1  j1  1 2 j1  2 
 j2 J  j1 
j1  j1  1 2 j1  2    j1 j2
Reduced MEs of Spherical Harmonics
2
m2
YML
1
m1
 L 1 2 
 (1) 2  m2 

m
m

M
1
2

2
YL
1

*
L
 d  Ym2 ( ,  ) YM ( ,  )Ym11( ,  )
20
2
(2
YML ( ,  )Ym1 ( ,  )  
Angular Momentum Coupling
1
,
2
Y
L
1
1
 1)(2L  1)(2  1)  1 L    1 L    *
Y ( ,  )



4
 m1 M    0 0 0 
 ( ) Y ( , )
 (1)
2
(2
1
 1)(2
2
 2 L 1
 1) 

0
0
0


Important for the calculation of gamma and particle transition probabilities
W. Udo Schröder, 2005
Isospin
Charge independence of nuclear forces  neutron and proton states
of similar WF symmetry have same energy  n, p = nucleons
Choose a specific representation in abstract isospin space:
Angular Momentum Coupling
21
1
Proton : 1 2   
0
0
Neutron : 1 2   
1
0 1
 0 i 
1 0 
Isospin matrices SU(2) 1  
;


;


2
3





1
0
i
0
0

1






1
Nucleon charge q  1   3 
2
1
Isospin operators tˆi :  i (i  1,2,3) analog to spin
2
tˆi , tˆj   i tˆk (cyclic i, j, k )  spherical tensor (vector ) tˆ1 : tˆ  (tˆ1  itˆ2 ), tˆ3


Transforms in isospin space like angular momentum in coordinate space
 use angular momentum formalism for isospin coupling.
W. Udo Schröder, 2005
2-Particle Isospin Coupling
22
Use spin/angular momentum formalism: t  (2t+1) iso-projections
t1
t2
mt1
mt 2
:
t1
t2
mt1
mt 2
Total isospin states T , MT 
Iso  antisymmetric :
0
Angular Momentum Coupling
can couple to t1  t2  T  t1  t2
1
T 1, MT  0   
0
W. Udo Schröder, 2005
1
0
mt 2
1(singlet ) state
(MT  1, 0,1)
MT
1
t2
1 2 (1)1 2 (2)  1 2 (2)1 2 (1)

2
T 1
1
MT
 t1 t2 T  t1

 

mt 1 , mt 2  mt 1 mt 2 MT  mt1
1
0
Iso  symmetric :
1 
MT  0
0
T  0, MT  0   
0
T 1, MT 1   
1 
T 0
T
3(triplet ) states
1
1 2 (1)1 2 (2) T 1, MT  1   
 1 
1
1
1
1 2 (1) 1 2 (2)  1 2 (2) 1 2 (1)

2
1 2 (1)1 2 (2)
2-Particle Spin-Isospin Coupling
Both nucleons in j shell  lowest E states have even J  T=1 !
For odd J  total isospin T = 0
23
 JM,TMT 
T J
 1
 j1 j2 J 
J T
 
  j1m1 (1) j2m2 (2)  (1)  j1m1 (2) j2m2 (1)T ,MT
2 m1,m2  m1 m2 M 
1
J  0 T  1

Angular Momentum Coupling
j1  j2  j   1
j2 JM ,T 1MT
 j j 0
  
  jm (1) j  m (2)   jm (2) j  m (1)T 1,MT
m1 , m2  m m 0 
3 states (MT=-1,0,+1) are degenerate, if
what should be true (nn, np forces are same)
ˆ, Tˆ  0
H


Different MT states belong to different nuclei T3 = (N-Z)/2
W. Udo Schröder, 2005
MT  1, 0,1
Angular Momentum Coupling
24
2-Particle Isobaric Analog (Isospin Multiplet) States
Corresponding
T=1levels in A=14
nuclei
T3=+
1
2n
W. Udo Schröder, 2005
T3=-1
2n holes
T3=0, pn
W. Udo Schröder, 2005
Angular Momentum Coupling
25
Separation of Variables: HF Interaction
26
Addition Theorem of spherical harmonics :
P  cos   
4
*
 Ym (1 , 1 )Ym (2 , 2 ) 
2 1m
   1
4
4
Y m (2 , 2 )
Ym* (1 , 1 )
2 1
2 1
m
m
 
r1 , r2
Electron  nucleus (sum over p) hyperfine interactions
Angular Momentum Coupling
ˆ  
H
int
i , p,
ei ep
ri
   1
1
m
m,
rp P (
ri  rp
ri rp
)


4
4
*
ep rpYm ( p , p )  
ei ri

 p 2  1
  i 2  1
0
 Tp  Ti   scalar product of separated tensors
0
protons electrons
only
only
W. Udo Schröder, 2005

Y m ( i , i )

1
Electric Quadrupole Moment of Charge Distributions
z
  r   arbitrary nuclear charge distribution with norm  d3r   r   Z
1
Coulomb
V (r )  e2  d 3r  p*  r  
 p  r 
interaction
r  r
Point
Charge e
r
r  r
1
r  r
1
27
|e|Z
r  r
  r   r 
2

1 
 r 
 r  1 
2
 1    cos    
3 cos   1  ...


r
r
r
2







1   r 
el
2 P2 (cos  )  Q02

r
   P (cos  ) 
quadrupole
r 0  r 
Nuclear Deform
Quadrupole moment Q  T2= Q2 -ME in aligned state m=j
Qz :
j
j
Qz (m) 
Q02
j
m
j
 j j 2
2

 j Q j
j
  j j 0
Q02
Qz (m)  (1) j  m
W. Udo Schröder, 2004
 spectroscopic Qz (m  j )
j 2  j j 2
 j j 2
2
j m   j

j
Q
j

Q
(

1)


 

m

m
m
0

m
m
0

j
j
0


 


j
2
3m  j( j  1)
Q
j(2 j  1)
Look up/calculate
3 j  Symbol  Qz  0 for j  1
Angular-Momentum Decomposition: Plane Waves
k
 r k
z

in z direction : e ikz  e ikr cos    c mYm ( ,  )
m
symmetric about z axis c
m
  m0
28
r
Plane wave can be decomposed into
spherical elementary waves
Spherical Bessel function

Angular Momentum Coupling
c 0 (r )  2  d sin  Y0 ( )  eikr cos    i


0
eikz  eikr cos    c 0Y0 ( ,  )   i
4 (2  1) j (kr )
4 (2  1) j (kr ) Y0 ( ,  )
For arbitrary direction k use Addition Theorem for Ym :
e
ik  r
m 
  i
W. Udo Schröder, 2005
m 
4 (2  1) j (kr ) Ym ( r , r )Ym*( k , k )
j (kr )
j-Transfer Through Particle Emission/Absorption
P
29
T
Angular Momentum Coupling
p+T 
W. Udo Schröder, 2005
Average Transition Probabilities
Fermi ' s Golden Rule : Pi  f 
i
Tk
mf
k
T
ji
mi
f (Ef )
If more than 1 initial state may be populated (e.g. diff. m)
 average over initial states
30
Angular Momentum Coupling
jf
for one i state and f f states
f
1
P 

2 j1  1 mi , mf
jf
mf
1
P 
2 j1  1 2k  1
ji
k
T
2
mi
jf T
k
Sum over all components of
ji
Tk
2
 ji k jf 
1

 

2 j1  1 mi , mf  mi  mf 
jf T
k
ji
 const .
2

= total if Tk transition probability
W. Udo Schröder, 2005
2
2
1
P   P 

2 ji  1
jf T
k
ji
2
2
W. Udo Schröder, 2005
Angular Momentum Coupling
31
W. Udo Schröder, 2005
Angular Momentum Coupling
32
Translations
cmi  m   cni  m VR  n  cmi Ei

Vmn

33

n
m  1,2,3
Angular Momentum Coupling
V(r)
V(x)
W. Udo Schröder, 2005
r
x