336
7 Wave Functions in the Vicinity of Particle-Emission Threshold
of partial widths Γ :
C =
|C | 2 =
Γ
μ
¯
h 2 (k)
.
(7.41)
In this expression:
Γ =
¯
h 2
μ
|C |
2
(k)
(7.42)
is given by the same formula as that obtained in Ref. [95], even though approximations and boundary conditions are different in the real-energy R-matrix approach
and in the complex-energy Gamow-state formalism [96, 97].
Figure 7.7 shows the radial overlap integrals for a 3
+
1 broad resonance in 8 B and
a narrow mirror resonance in 8 Li. The tails of real and imaginary parts of radial
overlap integrals can be seen to be well fitted with the outgoing Coulomb wave
functions of a complex argument k.
Fig. 7.7 Radial overlap
integrals I
a
bc;j (r) calculated
in Gamow shell model for the
J π = 3
+
1 resonance:
7 Be 3/2 − + p → 8 B 3 + (upper)
and 7 Li 3/2 − + n → 8 Li 3 +
(lower). 8 B in the 3
+
1 state is a
broad one-proton resonance
which results in a large
imaginary part of radial
overlap integral. The tail of
the radial overlap integral is
fitted by the Coulomb wave
function H
+
( , η)(kr) to
extract asymptotic
normalization coefficient as
J π = 3
+
1 states are unbound
(from Ref. [55])
0 2 4 6 8 10 12
0
0.2
0.4
0
0.2
0.4
r (fm)
Overlap integral
7 Li+n→
8
Li*
7 Be+p→
8
B*
J
π
=3
+
(a)
(b)
p 3/2
I p (r)
I n (r)
Re
Im
Re
Im