9.2 Generalization to Many-Nucleon Projectiles
447
and
∇
ˆ
J LM L (q, r)
qr
· l
as a function of r int , p int , l int , s and R CM , P CM , and L CM similarly to the
spin-dependent operators of A from the previous results. Explain the physical
content of these operators.
Exercise VIII
In this exercise, one will show how to formulate electromagnetic operators
involving clusters for the calculation of radiative capture cross sections (see
Exercises VI–VII).
A. Show that the expressions involving ∇ int in Eqs. (9.54) and (9.55) read
∇ int
ˆ
J int (q, r int )
· l
= −
int + 1
2 int + 1
ˆ
j
int
(qr int ) − ( int + 1)
ˆ
j int (qr int )
qr int
1
r int
× [Y int +1 (Ω r int ) ⊗ l
]
int
+
int
2 int + 1
ˆ
j
int
(qr int ) + int
ˆ
j int (qr int )
qr int
1
r int
× [Y int −1 (Ω r int ) ⊗ l
]
int
∇ int
ˆ
J
int
(q, r int )
· s
= −
int + 1
2 int + 1
int (( int + 1)
qr int
− qr int
ˆ
j int (qr int ) − int ˆ
j
int
(qr int )
×
1
r int
[Y int +1 (Ω r int ) ⊗ s] int
+
int
2 int + 1
int (( int + 1)
qr int
− qr int
ˆ
j int (qr int ) + (( int + 1) ˆ
j
int
(qr int )
×
1
r int
[Y int −1 (Ω r int ) ⊗ s] int ,
where l is L CM , l int , r int × P CM , or R CM × p int .