# Nuclear de-excitation

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```					Nuclear de-excitation
Outline of approach…
nucleus

ExB

• Electromagnetic radiation (quantized); v = c
• Power radiated P(,) depends on the nature of the
source (e.g., electric dipole, magnetic dipole, electric
simultaneously more than one; one will dominate.)
• Source uniquely describes P(,) and, conversely,
P(,) allows the determination of the source of the
• P(,) comes from E-M radiation theory (Maxwell)
• Multipole order of the radiation field: 2L
• L = 1 (dipole field); L = 2 (quadrupole); L = 3…
• “EL”  electric multipole of order L
• “ML”  magnetic multipole of order L

• Angular distribution of intensity of radiation field:
P2L cos    Legendre polynomials (from Maxwell’s Eq)
1
2  
P2  3cos2  1                      dipole field
1
8  
P4  35cos4   30cos2   3        quadrupole field
• The parity of the radiation field:
• (EL) = (-1)L
• (ML) = (-1)L+1
• Parity of E or M multipoles of same order is opposite.

• The total (integrated) radiated power in the classical
2L2
2(L  1)c                 2
P L                    
o L 2L  1!!c 
mL
(10.8)  (10.5)
if L=1
In quantum mechanics…
• The total (integrated) radiated power in the classical
2L2
2(L  1)c                 2
P L                    
o L 2L  1!!c 
mL
nuclear matrix
2L2                    element
2(L  1)c      
m fi L 
2
PL                     
o L 2L  1!!c 
m fi L     * mL  i dV
f

Transition operator
Transition between initial and
final nuclear states
In quantum mechanics…
These properties remain unchanged -
• Multipole order of the radiation field: 2L
• Angular distribution of intensity of radiation field
• The parity of the radiation field:
• (EL) = (-1)L
• (ML) = (-1)L+1
• Parity of E or M multipoles of same order is opposite.

In QM what is more meaningful is the decay rate - or
the probability per unit time of de-excitation =  …
In quantum mechanics…
These properties remain unchanged -
P L           radiated power: watts = joules/sec

h    E          Energy per quantum (photon) of
                                   frequency  ()

Therefore - the number of photons emitted per unit time is --

P L P L 
                      …from the nuclear source
     E

The form of the operators mfi is quite “classical” - it
represents the nature of the time-dependent charge
    distribution in the nucleus to produce this radiation field.
In quantum mechanics…
The transition matrix elements-
2L2
2(L  1)c      
m fi L 
2
PL                     
o L 2L  1!!c 
E-M matrix element
m fi L     * mLi dV
f

We need to know three quantities…
i           The initial state nuclear wave function
*
f

The final state nuclear wave function
m L         The transition operator
P L  P L 
Then, we can compute --               
     E
In quantum mechanics…
The transition matrix elements-
If we know these three quantities, we can compute --
1
8    1  2                             2
 
E
,m                         Q       Q   '
2  1!!c              m            m

Due to spatial coordinates                                   Due to intrinsic
magnetic moment

1
8    1  
2
2
 
M
,m                         M       M       '
2  1!!c              m                m
In quantum mechanics…
The transition matrix elements-
As an example, the electric and magnetic matrix
elements due to spatial coordinates can be written --
Z
Q m  e   rk Y m* ,   *  i d
f                    source term
k1
1 e Z
M   m                                            
  rk Y m* ,     * L k i d
 1 Mc k1
f

Note: each is a sum of Z integrals!                 ??  i ,  f ??
         '         '
The Q m and M m are the sum of A integrals and
involve the Pauli spin martices…

In quantum mechanics…
The transition matrix elements-
To get an estimate, try something “simple” --
Consider only a single proton transition:
 i  Rn ' r  j '   '
m'
,     Assume: R       '   r  ~constant
n
 f  Rn " r  j " "m" ,                  for 0  r  R

In general: average over initial states m’
                                         R  Ro A1/3
and sum over final states m”.

2       1
8   1 e 2 E                          2
 3  2
E    ,m                                          cR
2  1!! 4 o c  c               3

1
       E 
2
8    1 e 2          1
2                                           2
 3  2      2
M    ,m                       p 
             
                                cR
2  1!! 4 o c     1 m pc  c 
                            2 

In quantum mechanics…
The transition matrix elements-
Results for single proton transitions:

E   1  1.0 1014 A 2 / 3 E 3    M   1  5.6 1013 E 3
E   2  7.3 10 7 A 4 / 3 E 5    M   2  3.5 10 7 A 2 / 3 E 5
E   3  34 A 2 E 7               M   3  16A 4 / 3 E 7
E   4   1.1 105 A 8 / 3 E 9   M   4   4.5 106 A 2 E 9

   1                        E  
 105                           10 2
                             M  
Transition selection rules
Angular momentum & parity conservation
For E-M interactions:
Conserve E, p, L, 

Consider a transition: I P  I D  L


I P  I D  L  I P  I D           P   D  

Example:      I P  3 I D  2
                             
I P  0  I D  0  L


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