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Extra info for Theory of Nuclear Fission: A Textbook
36) extended to such states. We will restrict the discussion in the following to time-reversal invariant systems with pairs of degenerate states (Kramers degeneracy). In particular rotating systems are not time-reversal invariant. 7 of Ref. . The BCS Hamiltonian is ˆ = H ˆ+ ˆi ia i a + i≶0 1 V iı,jj a ˆ+ ˆ+ ˆj a ˆj . ) The transformation to quasiparticle operators simpliﬁes in the BCS theory compared to the HFB deﬁnition, Eqs. 16). 38) with real ui and vi . With this deﬁnition we follow the convention of Ref.
70), the kinetic energy densities τq , Eq. 69), and spin-orbit densities Jq = −i ϕi (r, σ, q)[∇ϕi (r, σ , q) × σ|σ|σ ] . 105) where ρ = ρp + ρn , τ = τp + τn , and ECoul is the Coulomb energy-density. The latter can be written as sum of a direct and an exchange contribution. The exchange contribution is usually approximated by the Slater form [162, 172] (cf. Eq. 142) below) ECoul (r) = e2 ρp (r) 2 d3 r 3 ρp (r ) − e2 |r − r | 4 3 π 1/3 ρ4/3 p (r) . 106) For systems whose ground-state wave-function is not time-reversal invariant, for example, for odd systems, the energy density has additional terms [115, 173].
The ﬁrst sections of this chapter are therefore devoted to the calculation of nuclear ground-state properties. With parameters ﬁtted to these properties, the construction of energy surfaces as functions of shape parameters is presented in the last section with a particular emphasis on secondary minima and saddle points. 1 Mean ﬁeld theories Many attempts have been made to obtain ground-state binding energies and radii on an a priori basis from empirical nucleon-nucleon potentials by solving the many-body Schr¨ odinger equation in the self-consistent Brueckner-HartreeFock approximation.