5.2 Translationally Invariant Shell Model Scheme: The Cluster Orbital Shell. . .
199
spin-orbit part of U i reads:
ˆ
U
(L S)
i
(r i,lab )(l i,lab · s i )
= ˆ
U
(L S)
i
(r i,lab )
r i + R CM,core
× p i
· s i
= ˆ
U
(L S)
i
(r i )(l i · s i )
+ (R CM,core · ∇ ˆ
U
(L S)
i
(r i ))(l i · s i ) + ˆ
U
(L S)
i
(r i )
R CM,core × p i
· s i
+
1
2
(R CM,core · Δ ˆ
U
(L S)
i
(r i ) · R CM,core )(l i · s i )
+ (R CM,core · ∇ ˆ
U
(L S)
i
(r i )) ·
R CM,core × p i
· s i
+ h.o. ,
(5.45)
where the first-order terms of Eq. (5.45) vanishes for the same reason as for the
central part and higher order terms are neglected as well.
The error made in the considered approximation is of the order of ΔU/M core .
Indeed, the core matrix elements involving R 2
CM,core are of the order of 1/M core ,
which can be checked analytically using harmonic oscillator wave functions.
Moreover, the Laplacian of a Woods-Saxon potential is about five times smaller
than the potential itself, and the derivative of the spin-orbit part of the one-body
potential is even smaller. The relative error on binding energy is of the order of 5%
when the α-particle core is used, which is the lightest core in the cluster orbital
shell model applications [13]. Hence in practical calculations, the error resulting
from neglecting the recoil term induced by the one-body term in Eq. (5.42) will not
exceed few percents.
However, there remains an important point related to the use of cluster orbital
shell model coordinates. As core and valence nucleons are not treated symmetrically, the antisymmetrization of many-body wave functions becomes difficult
to impose. This problem can be solved by noting that couplings between core
and valence nucleons always vanish using cluster orbital shell model. As core
and valence nucleons do not interact, the core and valence parts of many-body
wave functions do not have to be antisymmetrized. For this approximation to be
consistent, it is sufficient to demand that core states are orthogonal to valence states.
This procedure is called the orthogonality condition model [14]. In the calculation
of an observable, the operator associated to this observable must be calculated with
its recoil term included before being put to cluster orbital shell model form. Below,
we will show that the recoil terms of the operators providing mean-square radius,
electromagnetic, and beta transitions are either exactly treated or negligible.
Root-mean-square radius with the recoil correction included reads:
R ν;rms =
1
N ν
ν
(r ν,lab − R CM )
2
1/2
,
(5.46)
199
spin-orbit part of U i reads:
ˆ
U
(L S)
i
(r i,lab )(l i,lab · s i )
= ˆ
U
(L S)
i
(r i,lab )
r i + R CM,core
× p i
· s i
= ˆ
U
(L S)
i
(r i )(l i · s i )
+ (R CM,core · ∇ ˆ
U
(L S)
i
(r i ))(l i · s i ) + ˆ
U
(L S)
i
(r i )
R CM,core × p i
· s i
+
1
2
(R CM,core · Δ ˆ
U
(L S)
i
(r i ) · R CM,core )(l i · s i )
+ (R CM,core · ∇ ˆ
U
(L S)
i
(r i )) ·
R CM,core × p i
· s i
+ h.o. ,
(5.45)
where the first-order terms of Eq. (5.45) vanishes for the same reason as for the
central part and higher order terms are neglected as well.
The error made in the considered approximation is of the order of ΔU/M core .
Indeed, the core matrix elements involving R 2
CM,core are of the order of 1/M core ,
which can be checked analytically using harmonic oscillator wave functions.
Moreover, the Laplacian of a Woods-Saxon potential is about five times smaller
than the potential itself, and the derivative of the spin-orbit part of the one-body
potential is even smaller. The relative error on binding energy is of the order of 5%
when the α-particle core is used, which is the lightest core in the cluster orbital
shell model applications [13]. Hence in practical calculations, the error resulting
from neglecting the recoil term induced by the one-body term in Eq. (5.42) will not
exceed few percents.
However, there remains an important point related to the use of cluster orbital
shell model coordinates. As core and valence nucleons are not treated symmetrically, the antisymmetrization of many-body wave functions becomes difficult
to impose. This problem can be solved by noting that couplings between core
and valence nucleons always vanish using cluster orbital shell model. As core
and valence nucleons do not interact, the core and valence parts of many-body
wave functions do not have to be antisymmetrized. For this approximation to be
consistent, it is sufficient to demand that core states are orthogonal to valence states.
This procedure is called the orthogonality condition model [14]. In the calculation
of an observable, the operator associated to this observable must be calculated with
its recoil term included before being put to cluster orbital shell model form. Below,
we will show that the recoil terms of the operators providing mean-square radius,
electromagnetic, and beta transitions are either exactly treated or negligible.
Root-mean-square radius with the recoil correction included reads:
R ν;rms =
1
N ν
ν
(r ν,lab − R CM )
2
1/2
,
(5.46)