V
ω
j j
ijkl =
ZZ
dr 1 dr 2 ψ
*
i ðr 1 Þψ
*
j ðr 2 Þ
sin ω
j jr 12
r 12
ð1 − α 1 α 2 Þψ
*
k ðr 2 Þψ
*
l ðr 1 Þ
ð22Þ
The detailed procedure for computing the matrix elements (22) is presented in
Refs. [76–92]. All calculations are performed with using the numeral code
Superatom (version 98).
3 Results and Conclusions
3.1 The Parameters of the Optical Potential
In Table 1 we list the values of parameters for the proton and neutron c p , c n
À
Á
Fermi
distribution and nuclear spin I number of some nuclei.
In Table 2 there are presented the concrete values of some optical potential
parameters, which have been used in different calculations [15–39]. Let us explain the
used classification and abbreviations of corresponding sets of the optical potential
parameters: Tauscher, ξ = 0—Tau1; Tauscher, ξ = 1—Tau2; Batty et al.—Bat.;
Seki et al.—Sek; Nagels—Nag; de Laat-Konijin et al.—Laat, Serga-Shakhman—
Serg-Sha, this theory—Odes [16–22, 38, 39, 76–79]. Note that the parameters of the
optical potential in Table 1 (the initial set of parameters) were initially obtained by
calibration of the experimental data on pion-nuclear scattering for the light and
medium nuclei. Further application of the model to the heavy atoms and relatively
low-lying states showed imperfections (in some cases) of these sets of the parameter
values under theoretical studying heavy nuclei. This situation is called pion
anomalies. For example, experimental (the low-energy scattering of pions; LAMPF)
results for relatively low-lying states of pion at levels 3d, 2p, 1 s (in such heavy atoms
Table 1 Parameters for the
proton and neutron c p , c n
À
Á
Fermi distribution and nuclear
spin I number of some nuclei
Nucleus
Z
A
c p, Ф
c n, Ф
I
π
20
Ne
10
19.9924
2.963
2.912
0
24
Mg
12
23.9850
3.080
3.026
0
93
Mb
41
92.906
4.986
5.127
9/2
133
Cs
55
132.905
5.599
5.774
7/2
165
Ho
67
164.93
6.125
6.329
7/2
173
Yb
70
172.938
6.274
6.570
5/2
+
175
Lu
71
174.941
6.274
6.570
7/2
181
Ta
73
180.948
6.347
6.650
7/2
+
197
Au
79
196.967
6.454
6.850
3/2
+
205
Tl
81
204.974
6.587
6.881
1/2
+
208
Pb
82
207.977
6.652
6.892
0
209
Bi
83
208.98
6.688
6.870
9/2
−
237
Np
93
237.095
7.001
7.300
5/2
+
80
O. Yu. Khetselius et al.
ω
j j
ijkl =
ZZ
dr 1 dr 2 ψ
*
i ðr 1 Þψ
*
j ðr 2 Þ
sin ω
j jr 12
r 12
ð1 − α 1 α 2 Þψ
*
k ðr 2 Þψ
*
l ðr 1 Þ
ð22Þ
The detailed procedure for computing the matrix elements (22) is presented in
Refs. [76–92]. All calculations are performed with using the numeral code
Superatom (version 98).
3 Results and Conclusions
3.1 The Parameters of the Optical Potential
In Table 1 we list the values of parameters for the proton and neutron c p , c n
À
Á
Fermi
distribution and nuclear spin I number of some nuclei.
In Table 2 there are presented the concrete values of some optical potential
parameters, which have been used in different calculations [15–39]. Let us explain the
used classification and abbreviations of corresponding sets of the optical potential
parameters: Tauscher, ξ = 0—Tau1; Tauscher, ξ = 1—Tau2; Batty et al.—Bat.;
Seki et al.—Sek; Nagels—Nag; de Laat-Konijin et al.—Laat, Serga-Shakhman—
Serg-Sha, this theory—Odes [16–22, 38, 39, 76–79]. Note that the parameters of the
optical potential in Table 1 (the initial set of parameters) were initially obtained by
calibration of the experimental data on pion-nuclear scattering for the light and
medium nuclei. Further application of the model to the heavy atoms and relatively
low-lying states showed imperfections (in some cases) of these sets of the parameter
values under theoretical studying heavy nuclei. This situation is called pion
anomalies. For example, experimental (the low-energy scattering of pions; LAMPF)
results for relatively low-lying states of pion at levels 3d, 2p, 1 s (in such heavy atoms
Table 1 Parameters for the
proton and neutron c p , c n
À
Á
Fermi distribution and nuclear
spin I number of some nuclei
Nucleus
Z
A
c p, Ф
c n, Ф
I
π
20
Ne
10
19.9924
2.963
2.912
0
24
Mg
12
23.9850
3.080
3.026
0
93
Mb
41
92.906
4.986
5.127
9/2
133
Cs
55
132.905
5.599
5.774
7/2
165
Ho
67
164.93
6.125
6.329
7/2
173
Yb
70
172.938
6.274
6.570
5/2
+
175
Lu
71
174.941
6.274
6.570
7/2
181
Ta
73
180.948
6.347
6.650
7/2
+
197
Au
79
196.967
6.454
6.850
3/2
+
205
Tl
81
204.974
6.587
6.881
1/2
+
208
Pb
82
207.977
6.652
6.892
0
209
Bi
83
208.98
6.688
6.870
9/2
−
237
Np
93
237.095
7.001
7.300
5/2
+
80
O. Yu. Khetselius et al.