180
R. B. Firestone
Blatt and Weisskopf [2] derived the standard GSF analysis which consists
of single particle reduced matrix elements, B(σ L), for γ -ray transitions with
multipolarity σ L that is given by
B(EL) ↓=
γ (EL) · L[(2L + 1)!!] 2
8π(L + 1)e 2 b L
¯
hc
E γ
2L+1
B(ML) ↓=
γ (ML) · L[(2L + 1)!!] 2
8π(L + 1)μ 2
N b L
¯
hc
E γ
2L+1
(1)
where γ (σ L) is the transition width, ¯
hc = 1.9733 × 10 −11 MeV·cm,
e 2 = 1.43998 × 10 −13 MeV·cm, and μ 2
N = 1.59234 × 10 −41 MeV·cm 3 . The
photoexcitation,B(σ L) ↑, and deexcitation, B(σ L) ↓, matrix elements are related
by the spin dependent term
B(σ L) ↑=
2J f + 1
2J i + 1
B(σ L) ↓,
(2)
where J i is the spin of the initial state and J f is the spin of the final state for
photoexcitation.
Photonuclear experiments measure the photoexcitation cross section, σ γ (mb/MeV),
populating states above S n . The predominantly E1 average PSF, F
(γ ,n)
E1
↑, is related
to the cross section by detailed balance and was defined by Uhl and Kopecky [3] as
F
(γ ,n)
E1
↑ =
σ γ (E x , E1)
3π 2 ¯
h 2 c 2 E γ
=
(γ ,n)
E1
D · E
2L+1
γ
= ρ(E x , J
π ) ·
2J f + 1
2J i + 1
B(E1) ↓
C(E1)
= ρ(E x , J
π ) · f
(γ ,n)
E1
↑,
(3)
where D = 1/ρ(E x , J π ) is the average level spacing or inverse of the level density
and the dimensionless constant C(E1) = 9560 from Eq. 1. Notably the PSF is the
product of the GSF and the level density. The average photonuclear GSF, f
(γ ,n)
E1
↑, is
uniquely associated with the GDR and distinct from other possible E1 GSF modes.
The average photonuclear reduced matrix element is simply B(E1) ↑= 9560 ×
f
(n,γ )
E1
↑.
The photonuclear cross section for photon absorption is described by a
Lorentzian shape whose strength is determined by the dipole sum rule. The PSF
was elegantly described by the Brink-Axel (BA) formulation [4, 5] as
F
(γ ,n)
E1
↑= F BA
E1 ↑=
1
3(π ¯
hc) 2
i=2
i=1
σ G i E γ 2
G i
(E 2
γ − E 2
G i
) 2 + E 2
γ 2
G i
,
(4)
R. B. Firestone
Blatt and Weisskopf [2] derived the standard GSF analysis which consists
of single particle reduced matrix elements, B(σ L), for γ -ray transitions with
multipolarity σ L that is given by
B(EL) ↓=
γ (EL) · L[(2L + 1)!!] 2
8π(L + 1)e 2 b L
¯
hc
E γ
2L+1
B(ML) ↓=
γ (ML) · L[(2L + 1)!!] 2
8π(L + 1)μ 2
N b L
¯
hc
E γ
2L+1
(1)
where γ (σ L) is the transition width, ¯
hc = 1.9733 × 10 −11 MeV·cm,
e 2 = 1.43998 × 10 −13 MeV·cm, and μ 2
N = 1.59234 × 10 −41 MeV·cm 3 . The
photoexcitation,B(σ L) ↑, and deexcitation, B(σ L) ↓, matrix elements are related
by the spin dependent term
B(σ L) ↑=
2J f + 1
2J i + 1
B(σ L) ↓,
(2)
where J i is the spin of the initial state and J f is the spin of the final state for
photoexcitation.
Photonuclear experiments measure the photoexcitation cross section, σ γ (mb/MeV),
populating states above S n . The predominantly E1 average PSF, F
(γ ,n)
E1
↑, is related
to the cross section by detailed balance and was defined by Uhl and Kopecky [3] as
F
(γ ,n)
E1
↑ =
σ γ (E x , E1)
3π 2 ¯
h 2 c 2 E γ
=
(γ ,n)
E1
D · E
2L+1
γ
= ρ(E x , J
π ) ·
2J f + 1
2J i + 1
B(E1) ↓
C(E1)
= ρ(E x , J
π ) · f
(γ ,n)
E1
↑,
(3)
where D = 1/ρ(E x , J π ) is the average level spacing or inverse of the level density
and the dimensionless constant C(E1) = 9560 from Eq. 1. Notably the PSF is the
product of the GSF and the level density. The average photonuclear GSF, f
(γ ,n)
E1
↑, is
uniquely associated with the GDR and distinct from other possible E1 GSF modes.
The average photonuclear reduced matrix element is simply B(E1) ↑= 9560 ×
f
(n,γ )
E1
↑.
The photonuclear cross section for photon absorption is described by a
Lorentzian shape whose strength is determined by the dipole sum rule. The PSF
was elegantly described by the Brink-Axel (BA) formulation [4, 5] as
F
(γ ,n)
E1
↑= F BA
E1 ↑=
1
3(π ¯
hc) 2
i=2
i=1
σ G i E γ 2
G i
(E 2
γ − E 2
G i
) 2 + E 2
γ 2
G i
,
(4)