420
9 The Unification of Structure and Reaction Frameworks
M
M
L,M L
=
¯
hc
2mc 2
(2L + 1)!!
(L + 1) q L
i
g
i ∇ i
ˆ
j L (qr i )
qr i
Y LM L (Ω i )
· l i
+g
s
i ∇ i
ˆ
j
L (qr i )Y LM L (Ω i )
· s i
+
¯
hc
2mc 2
(2L + 1)!!
(L + 1) q L
i
g
s
i q ˆ
j L (qr i ) (s i · e i ) Y LM L (Ω i ) .
(9.54)
Consequently,
M
M
L,M L
=
¯
hc
2mc 2
(2L + 1)!!
(L + 1) q L
i
−
g
i
r i
L + 1
2L + 1
×
ˆ
j
L (qr i ) − (L + 1)
ˆ
j L (qr i )
qr i
[Y L+1 (Ω i ) ⊗ l i ]
L
M L
+
g
i
r i
L
2L + 1
ˆ
j
L (qr i ) + L
ˆ
j L (qr i )
qr i
[Y L−1 (Ω i ) ⊗ l i ]
L
M L
−
g
s
i
r i
L + 1
2L + 1
L(L + 1)
qr i
− qr i
ˆ
j L (qr i ) − L ˆ
j
L (qr i )
[Y L+1 (Ω i ) ⊗ s i ]
L
M L
+
g a
i
r i
L
2L+1
L(L+1)
qr i
−qr i
ˆ
j L (qr i ) + (L + 1) ˆ
j
L (qr i )
[Y L−1 (Ω i ) ⊗ s i ]
L
M L
+ g
s
i q ˆ
j L (qr i ) (s i · e i ) Y LM L (Ω i )
,
(9.55)
where
• i runs over all considered nucleons, with l i and s i its orbital and spin angular
momenta, respectively;
• Y LM L (Ω) is a spherical harmonic of order L and projection M L ;
• ˆ
j L is the Ricati–Bessel function of order L: ˆ
j L (qr) = qrj L (qr);
• r i , Ω i are the radial and angular coordinates of the nucleon i, respectively, and
e i = r i /r i ;
• e i is the dimensionless effective charge of the nucleon i, so that bare charge is 1
for proton and 0 for neutron;
• g
s
i is the dimensionless magnetic spin moment of the nucleon i and is 5.5857 for
proton and −3.8263 for neutron;