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

2.1 General considerations
We want to deal with the single-particle motion in a time-dependent mean field U(x; q(t)), where
x denotes position, spin and isospin <strong>of</strong> the nucleon and q ≡{q n(t)} a set <strong>of</strong> time-dependent
collective parameters describing the shape <strong>of</strong> the nuclear system. The solution Ψ β(t) <strong>of</strong>the
time-dependent Schrödinger equation with the Hamiltonian
can be expressed as an expansion [57]
| Ψβ(t) >=
α
c αβ (t) exp
−i
H = T + U = −− ¯h2 ∇ 2
t
0
2m0
dt ′
ɛα(t ′
m0W (x; q, ) − ·
) q
+ U (x, q) (2.5)
/¯h
| ψ α(x, q) >, (2.6)
in terms <strong>of</strong> some orthonormal stationary basis functions | ψ α(x, q) > with the initial condition
c αβ(t =0)=δ αβ. In addition to the usual energy phase factor, a collective phase factor [58, 59]
has been introduced which is common to all states α. In atomic collisions, the velocity potential
W [55] is <strong>of</strong>ten allowed to depend on the atomic state α, so that the relative velocities can be
accounted for in specific transfer reactions. Such a refinement can also become important
for grazing nucleus-nucleus collisions. However, for central collisions which lead to compact
nuclear shapes, a common phase factor seems to be more appropriate [57]. This choice has the
advantage <strong>of</strong> leaving the stationary basis orthogonal and independent <strong>of</strong> the collective velocity.
Inserting the expansion (2.6) into the Schrödinger equation and projecting onto the state
ψ γ, we find the set <strong>of</strong> coupled first-order differential equations
i¯h ·
γβ= c
α
c αβ α
0
dt ′
{ɛα(t ′
− ɛγ(t ) ′
)}/¯h
with ɛα =< ψα | H | ψα > for the determination <strong>of</strong> the expansion coefficients cαβ. The coupling
terms are given by
20
(2.7)