GAMMA DECAY
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ELECTROMAGNETIC SPECTRUM
Energy
(eV)
μeV
meV
eV
keV
MeV
TeV
10-6
10-3
1
103
106
109
3
Public domain image (source: NASA)
DIPOLE RADIATION
Rate from Fermi’s Golden Rule + Density of states:
2⇡
W =
|h
~
3
!
2
2
|hr̂i| sin ✓ d⌦
f |V̂ | i i| ⇢(Ef ) =
3
2⇡c ~
2
Integrating over angles:
4 e2 ! 3
2
'(E1) =
|hri|
3
3 ~c
4
DIPOLE APPROXIMATION
In deriving the dipole emission formula we only kept lowest order
expansion:
†
†
† i~
k·~
r
~
~
A / ak e ~✏k ⇡ ak (1 + ik · ~r + . . . )~✏k ! ak~✏k
This yields the typical dipole emission pattern:
2
W / sin ✓ d⌦
In QM the angular distribution is related to the photon angular
momentum
5
BEYOND THE DIPOLE
Higher order terms in the expansion give rise to gamma
emission
with different angular-dependence pattern
and higher angular momentum for the gamma photon emitted
`
X
~
k
·
~
r
)
(i
†
† i~
k·~
r
~
A / ak e ~✏k ⇡ ak
`
`!
~✏k
Each l term contributes to a different decay rate.
6
MULTIPOLE RADIATION
Electric multipole
2
8⇡(` + 1) e
A(E`) =
`[(2` + 1)!!]2 ~c
✓
E
~c
◆2`+1 ✓
D E
|r̂|
~
⇡ R0 A1/3
Rates:
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A(E1) = 1.0 ⇥ 10 A
7
A(E2) = 7.3 ⇥ 10 A
2
A(E3) = 34A E
4/3
E
E
3
5
7
A(E4) = 1.1 ⇥ 10
7
2/3
◆2 D E
2`
3
c |~r̂|
`+3
-5
A
8/3
E
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MULTIPOLE RADIATION
Magnetic multipole
2
8⇡(` + 1) e E
>(M `) =
`[(2` + 1)!!]2 ~c ~c
Rates:
2`+1
✓
◆2 D E

✓
2`-2
3
~
c |~r̂|
µp
`+3
mp c
.(M 1) = 5.6 ⇥ 1013 E 3
.(M 2) = 3.5 ⇥ 107 A2/3 E 5
.(M 3) = 16A
4/3
E
7
.(M 4) = 4.5 ⇥ 10-6 A2 E 9
8
1
`+1
◆
WHICH TRANSITION?
The lowest multipole dominates:
Lower multipoles decay faster (higher rates)
Electric multipoles are faster than magnetic multipoles
➡
Why don’t we always only observe electric
dipole (E1) radiation?
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SELECTION RULES
The multipole l is related to the gamma angular momentum
the angular momentum must be conserved in gamma decay
Possible l:
|If
I i |  `  If + I i
Parity: (-1)l for Electric and (-1)l-1 for Magnetic: parity must be
conserved, Πγ=Πi Πf
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WHICH TRANSITION?
The lowest permitted multipole dominates
Electric multipoles are more probable than the same magnetic
multipole by a factor 100
)(El)
⇡ 102
)(M l)
Emission from the multipole l+1 is 10-5 times less probable than
the l-multipole emission
)(E, l + 1)
⇡ 10-5 ,
)(El)
)(M, l + 1)
⇡ 10-5
)(M l)
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WHICH TRANSITION?
Combining the two rules:
(E, l + 1)
⇡ 10
(M l)
3
(M, l + 1)
⇡ 10
(El)
,
Thus E2 competes with M1
But M2 does not compete with E1
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7
INTERNAL CONVERSION
In some cases energy is not released in the form of gamma
photons, but carried away by an electron:
A ⇤
Z
X
!
A +
Z
X
+ e-
This process is called Internal Conversion
It is the only process possible, when selection rules do not allow
any of the multipole transitions:
e.g. even-even nuclides, decay from a 0+ level
the photon cannot have zero angular momentum.
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MIT OpenCourseWare
http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
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