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are both diagonal with respect to
l
,whereas
r
has no diagonal elements
(cf. the electric-dipole selection rule ∆
l
=
±
1). In the second term of
(4.1.10) we can, to leading order, replace
by the kinetic-energy oper-
ator and, if we also make the assumption ∆
l
= 0, this term transforms
like a second-rank tensor and so is quadrupolar. Symmetrizing
Q
p
with
respect to the expansion in spherical Bessel functions, and taking (
κ
·
r
)
r
outside the commutator, which is allowed because ∆
l
=0,wecanwrite
the second term in (4.1.10) as
H
(
κ
×
r
)(
κ
·
r
)
Q
r
,
with
r
=
r
/r
and
.
i
8
Q
r
=
Q
r
=
−
2
,r
2
]+[
∇
2
,r
2
]
{j
0
(
ρ
)+
j
2
(
ρ
)
}
{j
0
(
ρ
)+
j
2
(
ρ
)
}
[
∇
Thus the second term is a product of an angular and a radial operator,
which are both Hermitian. Our next assumption is that the radial part
of the wavefunction, as specified by the principal quantum number
n
,
and by
l
, is the same in the initial and the final state, i.e. that both
n
and
l
are unchanged. In this case,
<i
nl >
vanishes
identically, because
Q
r
is an imaginary Hermitian operator;
Q
r
=
Q
r
=
−
|
Q
r
|
f>
=
< nl
|
Q
r
|
Q
r
. If the radial part of the wavefunction is changed in the scattering
process, or if
is not diagonal in
l
, then the quadrupole moment leads
to an imaginary contribution to
K
(
H
), and gives a contribution to the
cross-section proportional to
κ
2
. In most cases of interest, however, this
term is very small.
The assumption that
κ
|
i>
and
|
f>
are linear combinations of
the states
(
nls
)
m
l
m
s
>
,where(
nls
) is constant, implies that the two
lowest-order terms in the expansion of
Q
p
in (4
.
1
.
9
b
) or (4.1.10) can be
neglected. Furthermore, the radial and angular dependences are then
factorized, both in the expansion of the operators and in the wave-
functions, so that the radial part of the matrix elements may be cal-
culated separately. Hence the orbital contribution
K
p
|
to
K
is approxi-
mately
)=
2
{
K
p
(
κ
j
0
(
κ
)
+
j
2
(
κ
)
}
l
,
(4
.
1
.
13
a
)
with
=
∞
0
∞
r
2
R
2
(
r
)
j
n
(
κr
)
dr
r
2
R
2
(
r
)
dr
=1
,
j
n
(
κ
)
;
(4
.
1
.
13
b
)
0
where
R
(
r
) is the normalized radial wavefunction. The assumption that
the final and initial states have the same parity implies that only the
terms in the expansion (4.1.8) for which
n
is odd may contribute to
K
p
.
By the same argument, the spin part
K
s
of
K
only involves the terms
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