Download Gamma Decay Supplement - Inside Mines

D EPARTMENT OF P HYSICS , C OLORADO S CHOOL OF M INES
PHGN 422: Nuclear Physics
γ Decay Supplement
Kyle G. Leach
October 21, 2016
1 I NTRODUCTION
Nuclei that are in an excited state generally decay via the emission of a γ ray or through internal
electron conversion to decrease the energy of the nucleus. The transitions can occur between
two excited states or an excited state and the ground state. This decrease in energy does not
change the isotope, it merely reconfigures the nucleons within the nucleus. In the γ-decay process, electromagnetic radiation of a specific energy is released when the nucleus undergoes a
transition from an excited state to a lower energy state. With internal conversion, the energy
that would be emitted through electromagnetic radiation instead liberates an atomic electron,
causing it to move into an unbound state. These two processes generally compete with each
other.
1.1 γ R ADIATION E NERGETICS
In order to understand the energy transfer in γ decay, we must consider the initial and final
states of the nucleus as well as its recoil momentum. Using conservation of energy and momentum, we obtain:
Conservation of Momentum:
Conservation of Energy:
p~R + p~γ = 0
E i = E f + E γ + TR ,
1
where:
TR =
p R2
2M x
=
p γ2
2M x
=
E γ2
2M x c 2
.
Therefore;
∆E = E γ +
E γ2
2M x c 2
,
(1.1)
where ∆E = E i − E f is the difference in the energies of the nuclear states involved. This expression can be solved for the γ-ray energy,
µ
¶
∆E
E γ ≈ ∆E 1 −
.
(1.2)
2m x c 2
We see that the energy released in the electromagnetic transition is slightly less than the energy
difference between the initial and final nuclear states due to the recoil of the daughter nucleus.
1.2 γ-R AY T RANSITION M ULTIPOLARITY
The emitted γ-ray photon can be understood in a simple model using classical electrodynamics,
where a radiation field can be described in terms of a multipole expansion. This classical theory
is then supplemented using a quantum mechanical description for the electric and magnetic
multipoles in terms of the power radiated for each:
λe (L) =
P e (L)
~ω
and λm (L) =
P m (L)
,
~ω
(1.3)
where the subscripts m and e represent magnetic and electric multipoles, respectively, L is the
multipolarity of the respective transition (see Table 1.1), and E γ = ~ω, where ω is the angular
frequency of the oscillating field. We can then expand the decay constants in multipoles by
expanding the radiation powers. In order to obtain order of magnitude expressions for the expected transition rates we can make a number of approximations. These are referred to as the
Weisskopf estimates. They are i) that the initial and final states are given by the single particle
wave functions ψi = R i (r )Y`i mi (θ, φ) and ψ f = R f (r )Y` f m f (θ, φ), and ii) the radial terms in the
wavefunctions are constant over the entire nuclear volume, and zero elsewhere. Once these approximations are made, the electric and magnetic multipole decay constants can be expressed
as [1]:
·
¸ µ ¶
2e 2 (L + 1)
3 2 E γ 2L+1 2L
λe (L) =
R ,
4π²0 ~L[(2L + 1)!!]2 L + 3
~c
(1.4)
and
λm (L) =
20e 2 ~(L + 1)
4π²0 c 2 m p2 L[(2L + 1)!!]2
·
3
L +3
¸2 µ
Eγ
~c
¶2L+1
R 2L−2 ,
(1.5)
2
Table 1.1: Properties and nomenclature for electromagnetic multipole radiation. [1]
Radiation Nomenclature
Symbol
Multipolarity (L)
Parity
Electric Dipole
Magnetic Dipole
Electric Quadrupole
Magnetic Quadrupole
Electric Octopole
Magnetic Octopole
.
.
.
E1
M1
E2
M2
E3
M3
.
.
.
1
1
2
2
3
3
.
.
.
-1
+1
+1
-1
-1
+1
.
.
.
where R is the nuclear radius, and E γ is expressed in MeV.
The total angular momentum (L) of the photon is subject to selection rules, which are related to
the angular momentum of the initial and final nuclear states by
|~
Ji − ~
J f | ≤ L ≤ |~
Ji + ~
Jf |
where L = 1, 2, 3... .
(1.6)
It is important to note that the angular selection rules do not include 0 → 0 transitions since they
can only be satisfied with L = 0, and there are no L = 0 photons. There are also parity selection
rules that are dependent on the angular momentum of the photon,
For E L transitions,
For M L transitions,
πi = π f (−1)L
πi = π f (−1)
L+1
.
(1.7)
(1.8)
For various initial and final nuclear spin and parity states there are in general a number of allowed γ-ray transitions that can occur. In the case where the lowest multipole permitted by
the selection rules is electric, it will dominate the decay. If the lowest allowed multipole L is
magnetic, there will, in general, be a competition between M L and E (L +1) multipole radiation.
1.3 I NTERNAL C ONVERSION
The internal conversion decay constant is, in general, a sum of the decay constants for the conversion of electrons from the various atomic shells (K, L, M, etc.). As mentioned previously, this
process competes with photon emission, which implies that the total decay constant for a transition between the initial and final nuclear states is a sum of the γ and internal conversion decay
constants, λ = λe + λγ , where the γ-decay constant is given above for M L and E L transition.
The internal conversion coefficient, α, is defined as the ratio of the decay constant for electron
3
conversion to the decay constant for γ emission,
α=
λe
,
λγ
(1.9)
which is then expressed in terms of the total decay constant
λ = λγ (1 + α).
(1.10)
The internal conversion coefficients can be calculated theoretically for each atomic shell [2].
An internal conversion coefficient, combined with a measurement of the γ-decay constant, will
therefore yield the total electromagnetic decay constant.
R EFERENCES
[1] Richard A. Dunlap. The Physics of Nuclei and Particles. Thomson Learning. Brooks/Cole,
2004.
[2] T. Kibédi et. al. BRICC 2.0b. http://www.nndc.bnl.gov/bricc, 2003.
4