180
4 Two-Particle Systems in the Berggren Basis
In order to calculate the B(E1) reduced transition probability, one has to use
an effective charge for the neutron. The latter has two origins: one arising from
the nuclear recoil and the other from the truncation of the configuration space.
The recoil effective charge for the E1 transition is equal to −Z/A. Taking into
account only the nuclear recoil, one obtains B(E1) = 0.199 e 2 fm 2 , almost twice
smaller than the experimental value B(E1) exp = 0.105 e 2 fm 2 . This indicates
that the considered model space is too small and hence the configuration mixing,
which reduces the transition probability, is insufficient. Indeed, the particle-hole
excitations from 10 Be are replaced in this model by a rotating core interacting with
a neutron through a deformed Woods-Saxon potential, which can simulate only a
fraction of the total configuration space. The experimental value of B(E1) reduced
transition probability is recovered by taking an effective charge which is smaller by
about 25% than the recoil effective charge.
Solutions to Exercises 1
Exercise I. The commutation relations: [P CM , R CM ] = −i ¯
h and [p rel , r rel ] = −i ¯
h
are directly obtained using Eqs. (4.1)–(4.4), the standard commutators [p 1 , r 1 ] =
[p 2 , r 2 ] = −i ¯
h, and the fact that operators p 1 , r 1 involving particle 1 commute with
those involving particle 2: p 2 , r 2 .
Exercise II. One only has to consider the kinetic operators in H CM + h rel . Using
Eqs. (4.1)–(4.4), (4.7), and (4.8), one directly obtains:
P 2
CM
2M CM
+
p 2
rel
2m rel
=
p 2
1
2m 1
+
p 2
2
2m 2
.
(4.35)
Exercise III.
A. As the Berggren basis is complete, energies and observables are independent
of the potential depth of the basis and of the form of the contour, as long as it
encompasses all resonances of interest (see Sect. (3.5.1)). Consequently, a small
change of parameters related to basis contours and potentials does not change
results significantly in practice.
B. The use of a Fermi function changes the Hamiltonian as it removes unstable
components at large distances. Consequently, results have to depend on the
parameters of the Fermi function. Conversely, once the Fermi function is fixed, it
generates moderate size matrix elements of the Hamiltonian. Such a Hamiltonian
can then be efficiently diagonalized with the Berggren basis. As the latter is
complete theoretically and in practice, results are independent of both basis
potentials and contours.
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.
4 Two-Particle Systems in the Berggren Basis
In order to calculate the B(E1) reduced transition probability, one has to use
an effective charge for the neutron. The latter has two origins: one arising from
the nuclear recoil and the other from the truncation of the configuration space.
The recoil effective charge for the E1 transition is equal to −Z/A. Taking into
account only the nuclear recoil, one obtains B(E1) = 0.199 e 2 fm 2 , almost twice
smaller than the experimental value B(E1) exp = 0.105 e 2 fm 2 . This indicates
that the considered model space is too small and hence the configuration mixing,
which reduces the transition probability, is insufficient. Indeed, the particle-hole
excitations from 10 Be are replaced in this model by a rotating core interacting with
a neutron through a deformed Woods-Saxon potential, which can simulate only a
fraction of the total configuration space. The experimental value of B(E1) reduced
transition probability is recovered by taking an effective charge which is smaller by
about 25% than the recoil effective charge.
Solutions to Exercises 1
Exercise I. The commutation relations: [P CM , R CM ] = −i ¯
h and [p rel , r rel ] = −i ¯
h
are directly obtained using Eqs. (4.1)–(4.4), the standard commutators [p 1 , r 1 ] =
[p 2 , r 2 ] = −i ¯
h, and the fact that operators p 1 , r 1 involving particle 1 commute with
those involving particle 2: p 2 , r 2 .
Exercise II. One only has to consider the kinetic operators in H CM + h rel . Using
Eqs. (4.1)–(4.4), (4.7), and (4.8), one directly obtains:
P 2
CM
2M CM
+
p 2
rel
2m rel
=
p 2
1
2m 1
+
p 2
2
2m 2
.
(4.35)
Exercise III.
A. As the Berggren basis is complete, energies and observables are independent
of the potential depth of the basis and of the form of the contour, as long as it
encompasses all resonances of interest (see Sect. (3.5.1)). Consequently, a small
change of parameters related to basis contours and potentials does not change
results significantly in practice.
B. The use of a Fermi function changes the Hamiltonian as it removes unstable
components at large distances. Consequently, results have to depend on the
parameters of the Fermi function. Conversely, once the Fermi function is fixed, it
generates moderate size matrix elements of the Hamiltonian. Such a Hamiltonian
can then be efficiently diagonalized with the Berggren basis. As the latter is
complete theoretically and in practice, results are independent of both basis
potentials and contours.
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.