Effects of diabaticity on fusion of heavy nuclei in the dinuclear model ...

the case <strong>of</strong> an independent particle model, the correlation function ψ In ′′
is expressed as [86]
(ω)
ψ ′′
=π¯h (ω)
|Fjk| j,k
2 j)[1 − n(ɛk)][δ(¯hω − ɛkj)+δ(¯hω + ɛkj)]. (5.6)
n(ɛ
ɛkj = ɛk − ɛj is the difference <strong>of</strong> single particle energies, n(ɛj) are the occupation numbers
Here,
Fjk =< j| and ˆ|k >the single particle matrix elements <strong>of</strong> the operator F ˆ The ψ F. ′′
singularity <strong>of</strong> δ—function type at ω =0
ψ with ′′
R
ψ ′′
=2πψ (ω) 0 δ(¯hω)+ψ ′′
(ω) hasa
R(ω), (5.7)
being regular at ω =0. ψ′′ R (ω) is given by a sum like the one in (5.6) but with the
(ω)
restriction j = k. At j = k, we find the contributions from the diagonal matrix elements
ψ 0 = k
|Fkk| 2 n(ɛk)[1 − n(ɛk)] = T0
k
∂n(ɛ)
∂ɛ
ɛ=ɛk
2 ∂ɛk
∂Q
. (5.8)
The last part in (5.8) was derived with a Fermi distribution for the occupation numbers, which
is characterized by the temperature T0. The value <strong>of</strong> T0 does not effectively go to zero with
decreasing excitation energy because each single particle level has a width due to the two-
body interaction. Indeed at zero excitation energy, the distribution <strong>of</strong> the occupation num-
deviates from a step function at least due to pairing correlations. In order to obtain a
bers
correlation function ψ smooth ′′
(ω), we substitute the δ—functions in Eq (5.6) by the Lorentzian
Γ/[π((¯hω ± ɛ kj) 2 +Γ 2 )]. The Lorentzian function with the double single particle width 2Γ is
applied because ¯hω is the transition energy between two single particle states [86]. Then using
Eqs. (5.4)-(5.8), we can write the friction coefficient in the following form
where
γ(0) = γ diag (0) + γ nondiag (0), (5.9)
γ diag (0) = ¯h
Γ
k
∂n(ɛ)
∂ɛ
ɛ=ɛk
2 ∂ɛk
∂Q
. (5.10)
For smaller temperatures T0 < 2 MeV, which are <strong>of</strong> interest here, γ diag (0) is much larger than
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