Dinh Dang.pdf

where E γ
≡ ω is used to denote the energy of γ -emission. The normalization factor c
1
is defined so that the
total integrated photoabsorption cross section σ = ∫ σ( E ) dE satisfies the GDR sum rule SR GDR , hence
γ
γ
Emax
1 SR GDR
GDR
0
c = ∫ S ( Eγ ) EγdEγ
.
(11)
In heavy nuclei with A ≥ 40, the GDR exhausts the Thomas-Reich-Kuhn sum rule (TRK)
SR GDR = TRK ≡ 60 NZ/ A (MeV⋅ mb) at the upper integration limit E max
30 MeV, and exceeds TRK
( SR GDR > TRK ) at E
max
> 30 MeV due to the contribution of exchange forces. In some light nuclei, such as
16 O, the observed photoabsorption cross section exhausts only around 60 % of TRK up to Emax
30 MeV.
B. Thermal pairing
The standard finite-temperature BCS (FT-BCS) theory ignores fluctuations of the quasiparticle number.
As a result, the BCS breaks down at a critical temperature T c
0567 . ∆ ( T = 0) , which corresponds to the
sharp transition from the superfluid phase to the normal-fluid one. It has been known that, in finite systems
such as nuclei, thermal fluctuations smooth out this phase transition [6].
The modified BCS (MBCS) theory [7] proposes a microscopic way to include quasiparticle-number
fluctuations via the secondary Bogolyubov’s transformation
α
= − n α + n α , = n α − n α . (12)
†
jm 1
† †
j jm j jm jm
α
j jm
j jm
Using Eqs. (12) in combination with the original Bogolyubov’s transformation, one obtains the
transformation from the particle operators directly to the modified quasiparticle operators in the following
form
where the coefficients
u j and
a
= + , a = − , (13)
† † †
jm u jα jm v jα jm j jm
jm
u α v jα
jm
v j are related to the conventional Bogolyubov’s coefficients
u
j
u
j
and v
j
as
= u 1− n + v n , v j = v 1−n − u n . (14)
j j j j j j j j
†
The transformation of the pairing Hamiltonian (1) into the modified quasiparticles α jm and α jm has the
form identical to that obtained within the conventional quasiparticle representation with ( u j , v j ) replacing
† †
( u
j
, v
j
) and ( α jm , α jm ) replacing ( α
jm
, α
jm
), respectively. The MBCS equations, therefore, has exactly
the same form as that of the standard BCS equations, where the coefficients u
j
and v
j
are replaced with u j
and v j , i.e.
∑ ∑ (15)
∆= G Ω = G Ω − n u v − n −n u − v ,
j
2 2
juv
j j
j[(1 2
j) j j j(1 j)( j j)]
j
∑ ∑ (16)
N = Ω = Ω − n v + n − n − n u v ,
2
2
2
jv
j 2
j[(1 2
j) j j
2
j(1 j) j j]
j
j
The last terms at the rhs of these MBCS equations contain the quasiparticle-number fluctuations nj
(1 − nj
)
on j − th orbitals, which are not included in the standard FT-BCS theory.
C. EM cross sections of GDR and DGDR
The EM cross section σ
EM
is calculated from the corresponding photoabsorption cross section σ ( E γ
)
and the photon spectral function N ( E γ
) as
∫ ∫ (17)
σ N E σ E dE N E π ∞ −
= , = N E , b bdb.
mb ( )
EM
(
γ) (
γ) γ
(
γ) 2 e (
γ
)
bmin
73