492
9 The Unification of Structure and Reaction Frameworks
determinant expansion. Moreover, r int is given in Eq. (9.135) as a function of
r and R CM , so that the intrinsic electromagnetic operator becomes an A-body
operator in terms of nucleon coordinates. Therefore, A-dimensional integrals
would have to be calculated, which is too expensive numerically.
B. The projectiles are well bound, so that their intrinsic wave function can be
expanded in a basis of harmonic oscillator states. Consequently, one can
calculate cluster states using harmonic shell model with the Lawson method,
so that they have the form:
|Ψ P = |Ψ CM ⊗ |Ψ P,int ,
where |Ψ P is the cluster state, |Ψ CM is the harmonic oscillator center-of-mass
ground state, and |Ψ P,int is the intrinsic state of the cluster. As |Ψ CM has an
angular momentum equal to zero, the only electric and magnetic sub-operators
(see Exercises V–VIII) leading to non-vanishing matrix elements when applied
to |Ψ P are those whose center-of-mass part is a scalar. Hence, all relative matrix
elements can be retrieved from matrix elements calculated with laboratory
coordinates.
C. Relative matrix elements are recovered from shell model matrix elements
by rewriting associated sub-operators using the results of Exercises V–VIII
and Eqs. (9.135), (9.136). Thus, from the knowledge of sub-operators matrix
elements issued from shell model and of one-body integrals in R CM , whose calculation is straightforward, one can deduce the relative electromagnetic operator
matrix elements entering
(f)
P,int |
i∈P
ˆ
O
L
M L
(i)|Ψ
(i)
P,int in Eqs. (9.54) and (9.55).
All many-body matrix elements can be calculated using recoupling algebra
once relative and center-of-mass matrix elements are known. Consequently, the
calculation of electromagnetic matrix elements in the many-nucleon cluster case
is as conveniently done as in the one-nucleon case, albeit more tedious.
Exercise X.
A. The main features concerning the use of a 12 C have already been considered in
Exercise IV of Chap. 7. Using only proton partial waves with ≤ 2 is justified
for the calculation of the low-energy spectrum of 15 F because single-particle
states belong only to the p and sd shells. Additionally, the 0p 1/2 proton shell is
occupied in 14 O and 15 F at Hartree–Fock level. Consequently, one can assume
that the excitations from the 12 C core to the valence space are small. Hence, they
can be neglected in this model.
Clearly, all neutrons are spectators in the 14 O(p,p) reaction when using an
inert core of 12 C. Moreover, as proton projectile energies are smaller than 6 MeV,
the partial wave expansion of projectile wave functions has small components
with ≥ 3. Consequently, the used model is also appropriate to describe the
excitation function of the elastic scattering reaction 14 O(p,p).
9 The Unification of Structure and Reaction Frameworks
determinant expansion. Moreover, r int is given in Eq. (9.135) as a function of
r and R CM , so that the intrinsic electromagnetic operator becomes an A-body
operator in terms of nucleon coordinates. Therefore, A-dimensional integrals
would have to be calculated, which is too expensive numerically.
B. The projectiles are well bound, so that their intrinsic wave function can be
expanded in a basis of harmonic oscillator states. Consequently, one can
calculate cluster states using harmonic shell model with the Lawson method,
so that they have the form:
|Ψ P = |Ψ CM ⊗ |Ψ P,int ,
where |Ψ P is the cluster state, |Ψ CM is the harmonic oscillator center-of-mass
ground state, and |Ψ P,int is the intrinsic state of the cluster. As |Ψ CM has an
angular momentum equal to zero, the only electric and magnetic sub-operators
(see Exercises V–VIII) leading to non-vanishing matrix elements when applied
to |Ψ P are those whose center-of-mass part is a scalar. Hence, all relative matrix
elements can be retrieved from matrix elements calculated with laboratory
coordinates.
C. Relative matrix elements are recovered from shell model matrix elements
by rewriting associated sub-operators using the results of Exercises V–VIII
and Eqs. (9.135), (9.136). Thus, from the knowledge of sub-operators matrix
elements issued from shell model and of one-body integrals in R CM , whose calculation is straightforward, one can deduce the relative electromagnetic operator
matrix elements entering
(f)
P,int |
i∈P
ˆ
O
L
M L
(i)|Ψ
(i)
P,int in Eqs. (9.54) and (9.55).
All many-body matrix elements can be calculated using recoupling algebra
once relative and center-of-mass matrix elements are known. Consequently, the
calculation of electromagnetic matrix elements in the many-nucleon cluster case
is as conveniently done as in the one-nucleon case, albeit more tedious.
Exercise X.
A. The main features concerning the use of a 12 C have already been considered in
Exercise IV of Chap. 7. Using only proton partial waves with ≤ 2 is justified
for the calculation of the low-energy spectrum of 15 F because single-particle
states belong only to the p and sd shells. Additionally, the 0p 1/2 proton shell is
occupied in 14 O and 15 F at Hartree–Fock level. Consequently, one can assume
that the excitations from the 12 C core to the valence space are small. Hence, they
can be neglected in this model.
Clearly, all neutrons are spectators in the 14 O(p,p) reaction when using an
inert core of 12 C. Moreover, as proton projectile energies are smaller than 6 MeV,
the partial wave expansion of projectile wave functions has small components
with ≥ 3. Consequently, the used model is also appropriate to describe the
excitation function of the elastic scattering reaction 14 O(p,p).