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5 Formulation and Implementation of the Gamow Shell Model
where a kk is a constant. Deduce that recoil terms in Eqs. (5.52) and (5.53)
cancel out.
E. Draw the conclusion that Eqs. (5.57–5.59) lead to the same results whether
laboratory or cluster orbital shell model coordinates are used, i.e., that recoil
corrections vanish.
Electromagnetic transitions of order L in the long-wavelength approximation are
functions of the following operators:
r
L
lab Y L (Ω lab )
(5.57)
r
L−1
lab [Y L−1 (Ω lab ) ⊗ l lab ]
L
(5.58)
r
L−1
lab [Y L−1 (Ω lab ) ⊗ s]
L .
(5.59)
One will show in Exercise III that in this approximation, all recoil terms in
electromagnetic transitions vanish in cluster orbital shell model. This exercise is
difficult and can be omitted without creating problems of comprehension in the
following.
Without the long wavelength approximation, the recoil corrections lead in
general to the many-body operators which are cumbersome to calculate. It is,
however, possible to calculate the recoil corrections exactly if the wave function
of the core is built from harmonic oscillator states. In this case, core wave functions
separate in relative and center-of-mass coordinates, and the recoil operator becomes
a one-body operator when written in center-of-mass coordinates (see Sect. 9.2.10 for
the exact calculation of electromagnetic operator matrix elements with cluster wave
functions). As occupied core states are always well bound, the use of harmonic
oscillator states for their description covers in fact all practical situations.
Recoil corrections for beta transitions are also vanishing or negligible. This is
obvious for allowed beta transitions, whose operators are spin and isospin tensors
and hence are independent of space coordinates. The recoil corrections of the firstforbidden beta decay operators without Coulomb corrections cancel out as their
spatial part is of rank one. It can be shown numerically that the Coulomb corrections
along with recoil correction in the first-forbidden beta decays are very small and
hence can be neglected.
Operators which might pose problem are the density operators. Indeed, they are
scalar operators so that the center-of-mass and valence parts lead to coupling matrix
between the intrinsic and center of mass parts of many-body wave functions. A
simple solution of this problem is to directly define density operators in cluster
orbital shell model coordinates, which is sound in practice as density is important
only in the valence space for both weakly bound and resonance nuclei.
5 Formulation and Implementation of the Gamow Shell Model
where a kk is a constant. Deduce that recoil terms in Eqs. (5.52) and (5.53)
cancel out.
E. Draw the conclusion that Eqs. (5.57–5.59) lead to the same results whether
laboratory or cluster orbital shell model coordinates are used, i.e., that recoil
corrections vanish.
Electromagnetic transitions of order L in the long-wavelength approximation are
functions of the following operators:
r
L
lab Y L (Ω lab )
(5.57)
r
L−1
lab [Y L−1 (Ω lab ) ⊗ l lab ]
L
(5.58)
r
L−1
lab [Y L−1 (Ω lab ) ⊗ s]
L .
(5.59)
One will show in Exercise III that in this approximation, all recoil terms in
electromagnetic transitions vanish in cluster orbital shell model. This exercise is
difficult and can be omitted without creating problems of comprehension in the
following.
Without the long wavelength approximation, the recoil corrections lead in
general to the many-body operators which are cumbersome to calculate. It is,
however, possible to calculate the recoil corrections exactly if the wave function
of the core is built from harmonic oscillator states. In this case, core wave functions
separate in relative and center-of-mass coordinates, and the recoil operator becomes
a one-body operator when written in center-of-mass coordinates (see Sect. 9.2.10 for
the exact calculation of electromagnetic operator matrix elements with cluster wave
functions). As occupied core states are always well bound, the use of harmonic
oscillator states for their description covers in fact all practical situations.
Recoil corrections for beta transitions are also vanishing or negligible. This is
obvious for allowed beta transitions, whose operators are spin and isospin tensors
and hence are independent of space coordinates. The recoil corrections of the firstforbidden beta decay operators without Coulomb corrections cancel out as their
spatial part is of rank one. It can be shown numerically that the Coulomb corrections
along with recoil correction in the first-forbidden beta decays are very small and
hence can be neglected.
Operators which might pose problem are the density operators. Indeed, they are
scalar operators so that the center-of-mass and valence parts lead to coupling matrix
between the intrinsic and center of mass parts of many-body wave functions. A
simple solution of this problem is to directly define density operators in cluster
orbital shell model coordinates, which is sound in practice as density is important
only in the valence space for both weakly bound and resonance nuclei.