9.2 Generalization to Many-Nucleon Projectiles
449
The first operator on the right-hand side of Eq. (9.133) poses no problem as it
involves target states only, similarly to the one-nucleon projectile. However, one
cannot directly calculate the matrix elements of its second operator with a basis of
cluster channels, as it is written in terms of nucleon coordinates, whereas cluster
channels are expressed in terms of the R CM , P CM center-of-mass coordinates of
the cluster, and r i,int , p i,int relative coordinates of the nucleon i in the cluster with
respect to the cluster center-of-mass
R CM =
1
M P
i∈P
m i r i
and
r i,int = r i − R CM
(9.135)
P CM =
i∈P
p i
and
p i,int = p i −
m i
M P
P CM .
(9.136)
Consequently, the ˆ
O L
M L
(i) operator must be written as a function of the R CM , P CM
and r i,int , p i,int coordinates. The determination of the electromagnetic Hamiltonian
in relative and center-of-mass coordinates and the practical calculation of its matrix
elements are the subject of the next exercises. Exercises V–IX are technical, so that
readers can admit their results in a first reading.
Exercise IX
In Exercises VI–VIII, one has concentrated on the derivation of electromagnetic
operators involving clusters. In this exercise, one will show how to calculate
electromagnetic matrix elements from these operators.
A. Explain why a direct calculation of the relative matrix elements is impossible in
practice when using a Slater determinant expansion of cluster wave functions.
B. Show that the use of harmonic oscillator shell model is sufficient for considered
projectiles. Find a method to retrieve relative matrix elements with that
approach using N ¯
hω spaces.
C. Explain how the matrix elements of the electromagnetic Hamiltonian can then
be calculated.
9.2.10.2 Radiative Capture Cross Section
The radiative capture differential cross section writes similarly to Eq. (9.52):
dσ
dΩ γ
=
q
8πk
e 2
¯
hc
μ u c 2
¯
hc
1
2J int + 1
1
2J T + 1
×
J i ,c e ,L
f ||M L ||(J i ) c e
J
i ,c
e ,L
J f ||M L ||(J
i ) c
e
449
The first operator on the right-hand side of Eq. (9.133) poses no problem as it
involves target states only, similarly to the one-nucleon projectile. However, one
cannot directly calculate the matrix elements of its second operator with a basis of
cluster channels, as it is written in terms of nucleon coordinates, whereas cluster
channels are expressed in terms of the R CM , P CM center-of-mass coordinates of
the cluster, and r i,int , p i,int relative coordinates of the nucleon i in the cluster with
respect to the cluster center-of-mass
R CM =
1
M P
i∈P
m i r i
and
r i,int = r i − R CM
(9.135)
P CM =
i∈P
p i
and
p i,int = p i −
m i
M P
P CM .
(9.136)
Consequently, the ˆ
O L
M L
(i) operator must be written as a function of the R CM , P CM
and r i,int , p i,int coordinates. The determination of the electromagnetic Hamiltonian
in relative and center-of-mass coordinates and the practical calculation of its matrix
elements are the subject of the next exercises. Exercises V–IX are technical, so that
readers can admit their results in a first reading.
Exercise IX
In Exercises VI–VIII, one has concentrated on the derivation of electromagnetic
operators involving clusters. In this exercise, one will show how to calculate
electromagnetic matrix elements from these operators.
A. Explain why a direct calculation of the relative matrix elements is impossible in
practice when using a Slater determinant expansion of cluster wave functions.
B. Show that the use of harmonic oscillator shell model is sufficient for considered
projectiles. Find a method to retrieve relative matrix elements with that
approach using N ¯
hω spaces.
C. Explain how the matrix elements of the electromagnetic Hamiltonian can then
be calculated.
9.2.10.2 Radiative Capture Cross Section
The radiative capture differential cross section writes similarly to Eq. (9.52):
dσ
dΩ γ
=
q
8πk
e 2
¯
hc
μ u c 2
¯
hc
1
2J int + 1
1
2J T + 1
×
J i ,c e ,L
f ||M L ||(J i ) c e
J
i ,c
e ,L
J f ||M L ||(J
i ) c
e