7.2 Halo Nuclei
321
where β j is a normalization constant, chosen so that it remains finite and nonzero
when κ → 0. Indeed, as exp(−κr) → 1 when κ → 0, the part of the normalization
integral involving u(k, r) from 0 to R becomes negligible, so that one obtains
β ∼
√
2 .
(7.7)
The expectation value of r 2 when κ → 0 follows:
+∞
0
r
2 u(k, r)
2 dr ∼ 2κ
+∞
R
r
2 exp(−2κr) dr
∼
1
2κ 2 .
(7.8)
Similarly, an analogous relation is found for neutron = 1 partial wave when
κ → 0. From Eq. (2.128, u(k, r) for = 1, j = 1/2, 3/2 and r ≥ R reads
u(k, r) = β j exp(−κr)(κ + 1/r) ,
r ≥ R ,
(7.9)
where β j is a normalization constant, which remains finite and nonzero when κ →
0 as u(k = 0, r) is integrable. Hence:
β → β
(0)
,
(7.10)
where β
(0)
j is the value of β j when κ = 0. The expectation value of r 2 for κ → 0
is straightforward to calculate
+∞
0
r 2 u(k, r) 2 dr ∼ β 2
+∞
R
exp(−2κr) (κ 2 r 2 + 2κr + 1) dr ∼
5
4κ
β
(0)
2
,
(7.11)
where the finite part of the expectation value of r 2 from 0 to R, is neglected as
it becomes negligible compared to its diverging asymptotic part. One then readily
obtains asymptotic limits of root-mean-square radii of neutron whose orbital angular
momentum verify ≤ 1:
2
1/2 ∼
1
2
κ
−1 ,
= 0 , j = 1/2
(7.12)
2
1/2 ∼
5
4
β
(0)
κ
−1/2 ,
= 1 , j = 1/2, 3/2 .
(7.13)
Contrary to the case for which ≥ 2, the root-mean-square radius of neutron
level with ≤ 1 is diverging when κ → 0. The effect of the centrifugal barrier
for = 1 is nevertheless visible, as the divergence of 2
1/2 , when κ → 0, is
slower than that occurring for = 0. One can also note that the dependence on
parameters of the potential disappears asymptotically for = 0, whereas for = 1