Microscopic Calculation of Fission Fragment Mass Distributions at Increasing. . .
277
depends on ρ (g) (the particle-hole channel) and another one that also depends on
κ (g) (the particle-particle channel). Standard examples of functionals in the p.h.
channel are the Skyrme and Gogny functionals; in the pairing channel, energy
functionals are often derived from simple density-dependent, zero-range two-body
potentials.
The nuclear EDF approach can be extended to describe systems at finite
temperature. At T > 0, a quantum-mechanical system is not described by some
ket |Φ but by a density operator ˆ
D = e β ˆ
H /Z where ˆ
H is the exact Hamiltonian
of the system, Z = Tre −β ˆ
H the partition function and β = 1/kT . Determining the
density operator is a formidable task; in practice, the HFB approximation consists
in replacing it with a specific, quadratic form of creation and annihilation operators,
ˆ
H → ˆ
K. Using the statistical Wick theorem, it is then possible to show that there
is a one-to-one correspondence between the generalized density of the HFB theory
and the operator ˆ
K; we refer to [15, 16] for details about the formalism. In practice,
solving the finite-temperature HFB equation only requires modifying the expression
for the one-body density matrix and anomalous density according to
ρ kl =
V
∗ (1 − f )V
T
kl
+
Uf U
†
kl
(4a)
κ kl =
V
∗ (1 − f )U
T
kl
+
Uf V
†
kl
,
(4b)
where f μ = 1/(1 + e βE μ ) is the Fermi-Dirac statistical occupation of the
quasiparticle with energy E μ .
3 Large-Amplitude Collective Motion
The description of nuclear fission within the nuclear EDF approach often begins
with the introduction of collective variables q μ that are supposed to drive the
fission process. 1 Following the intuition of Meitner, Bohr, etc., we view fission as an
extreme deformation process: the collective variables are thus the parameters
that characterize the nuclear shape. In the EDF picture, these are typically the
expectation value on the reference state (1) of suitable operators such as, e.g., the
mass multipole moments. By solving the HFB equation under constraints on the
expectation value of such operators, one can construct a potential energy surface
(PES) which shows how the energy of the nucleus changes as a function of the
collective variables q. An example of such a PES for 236 U is shown in Fig. 1.
1 This is in fact not necessary in the time-dependent DFT approach to fission, where all degrees
of freedom encapsulated in the (now time-dependent) generalized density are treated on the same
footing [17–19].
277
depends on ρ (g) (the particle-hole channel) and another one that also depends on
κ (g) (the particle-particle channel). Standard examples of functionals in the p.h.
channel are the Skyrme and Gogny functionals; in the pairing channel, energy
functionals are often derived from simple density-dependent, zero-range two-body
potentials.
The nuclear EDF approach can be extended to describe systems at finite
temperature. At T > 0, a quantum-mechanical system is not described by some
ket |Φ but by a density operator ˆ
D = e β ˆ
H /Z where ˆ
H is the exact Hamiltonian
of the system, Z = Tre −β ˆ
H the partition function and β = 1/kT . Determining the
density operator is a formidable task; in practice, the HFB approximation consists
in replacing it with a specific, quadratic form of creation and annihilation operators,
ˆ
H → ˆ
K. Using the statistical Wick theorem, it is then possible to show that there
is a one-to-one correspondence between the generalized density of the HFB theory
and the operator ˆ
K; we refer to [15, 16] for details about the formalism. In practice,
solving the finite-temperature HFB equation only requires modifying the expression
for the one-body density matrix and anomalous density according to
ρ kl =
V
∗ (1 − f )V
T
kl
+
Uf U
†
kl
(4a)
κ kl =
V
∗ (1 − f )U
T
kl
+
Uf V
†
kl
,
(4b)
where f μ = 1/(1 + e βE μ ) is the Fermi-Dirac statistical occupation of the
quasiparticle with energy E μ .
3 Large-Amplitude Collective Motion
The description of nuclear fission within the nuclear EDF approach often begins
with the introduction of collective variables q μ that are supposed to drive the
fission process. 1 Following the intuition of Meitner, Bohr, etc., we view fission as an
extreme deformation process: the collective variables are thus the parameters
that characterize the nuclear shape. In the EDF picture, these are typically the
expectation value on the reference state (1) of suitable operators such as, e.g., the
mass multipole moments. By solving the HFB equation under constraints on the
expectation value of such operators, one can construct a potential energy surface
(PES) which shows how the energy of the nucleus changes as a function of the
collective variables q. An example of such a PES for 236 U is shown in Fig. 1.
1 This is in fact not necessary in the time-dependent DFT approach to fission, where all degrees
of freedom encapsulated in the (now time-dependent) generalized density are treated on the same
footing [17–19].