as Ta, Bi et al.) have shown that the appropriate values of width due to strong
interaction by a factor two and more are less than the values specified within the
optical potential model with using the earliest parameterization [14, 15].
Such relatively significant deviations are typical for strong levels shifts.
A similar anomaly was detected in a case of the strong quadrupole shift of the
pion energy in 3d and 4f states. It should be noted that the results of the hyperfine
structure studying for heavy pionic atoms with pion at low (deep) lying states are
extremely scarce.
Earlier it is indicated a possibility of improving the consistency between theory
and experiment by taking into account of the strong interaction potential isovector
(absorption) terms. In fact, it is a more adequate account of increasing S-wave
repulsion for absorption of pion at 3d-level. In this regard, under the parameterization of the optical potential authors [33] left without changing the parameters
settings that are the most reliably identified, namely: ReB 0 , ImB 0, c 0 , c 1 , ReC 0 ,
ImC 0 .
At the same time the parameters whose values differ most strongly in different
sets, in particular, b 1 (plus parameters, which are usually not included so far in the
basic optical potential parameterizations, i.e. ImB1, ImC1), should be optimized.
This is achieved by receiving preliminary calculated relationships (illustrations are
given below) for the energy shifts and widths (for a number of states fof the
following systems:
20 Ne,
24 Mg,
93 Nb,
133 Cs,
175 Lu,
181 Ta,
197 Au,
208 Pb) upon the
b 1 , ImB 1 , ImC 1 parameter values; further, there were chosen such values that satisfy
the smallest standard deviation of the experimental values.
In Tables 3 and 4 we list the dependences of shifts and widths for the 4f, 3d
levels due to the strong pion nuclear interaction upon the parameter ImB 1 ImC 1
values for the pionic atom of
208 Pb (our data).
Further in Tables 5 and 6 we list the analogous dependences of shifts and widths
for the 4f, 3d levels due to the strong pion nuclear interaction upon the parameter
ImB 1 ImC 1 values for the pionic system
181 Ta.
Table 2 The values of some optical potential parameters, which have been used in different
calculations (see text)
Tauscher
ξ = 0
[16]
Tauscher
ξ = 1
[16]
Batty
ξ = 1
[17]
Seki
ξ = 1
[18]
Nagels
ξ = 1
[22]
Row78
ξ = 1
[19]
Laat
ξ = 1
[38, 39]
Odes
ξ = 1 [76,
79]
b 0
−0.0296
−0.0293
−0.017
0.003
−0.013
−0.004
0.007
0.003
m
− 1
π
b 1
−0.077
−0.078
−0.13
−0.143
−0.092
−0.094
−0.075
−0.094
m
− 1
π
ReB 0 0
0
−0.0475 −0.15
–
–
−0.18
−0.15
m
− 4
π
ImB 0 0.0436
0.0428
0.0475
0.046
–
–
0.058
0.046
m
− 4
π
c 0
0.172
0.227
0.255
0.21
0.209
0.23
0.266
0.21
m
− 3
π
c 1
0.22
0.18
0.17
0.18
0.177
0.17
0.40
0.18
m
− 3
π
ReC 0 –
–
–
0.11
–
–
0.07
0.11
m
− 6
π
ImC 1 –
–
–
–
–
–
−0.34
−0.25
m
− 6
π
Relativistic Quantum Chemistry and Spectroscopy of Exotic …
81
interaction by a factor two and more are less than the values specified within the
optical potential model with using the earliest parameterization [14, 15].
Such relatively significant deviations are typical for strong levels shifts.
A similar anomaly was detected in a case of the strong quadrupole shift of the
pion energy in 3d and 4f states. It should be noted that the results of the hyperfine
structure studying for heavy pionic atoms with pion at low (deep) lying states are
extremely scarce.
Earlier it is indicated a possibility of improving the consistency between theory
and experiment by taking into account of the strong interaction potential isovector
(absorption) terms. In fact, it is a more adequate account of increasing S-wave
repulsion for absorption of pion at 3d-level. In this regard, under the parameterization of the optical potential authors [33] left without changing the parameters
settings that are the most reliably identified, namely: ReB 0 , ImB 0, c 0 , c 1 , ReC 0 ,
ImC 0 .
At the same time the parameters whose values differ most strongly in different
sets, in particular, b 1 (plus parameters, which are usually not included so far in the
basic optical potential parameterizations, i.e. ImB1, ImC1), should be optimized.
This is achieved by receiving preliminary calculated relationships (illustrations are
given below) for the energy shifts and widths (for a number of states fof the
following systems:
20 Ne,
24 Mg,
93 Nb,
133 Cs,
175 Lu,
181 Ta,
197 Au,
208 Pb) upon the
b 1 , ImB 1 , ImC 1 parameter values; further, there were chosen such values that satisfy
the smallest standard deviation of the experimental values.
In Tables 3 and 4 we list the dependences of shifts and widths for the 4f, 3d
levels due to the strong pion nuclear interaction upon the parameter ImB 1 ImC 1
values for the pionic atom of
208 Pb (our data).
Further in Tables 5 and 6 we list the analogous dependences of shifts and widths
for the 4f, 3d levels due to the strong pion nuclear interaction upon the parameter
ImB 1 ImC 1 values for the pionic system
181 Ta.
Table 2 The values of some optical potential parameters, which have been used in different
calculations (see text)
Tauscher
ξ = 0
[16]
Tauscher
ξ = 1
[16]
Batty
ξ = 1
[17]
Seki
ξ = 1
[18]
Nagels
ξ = 1
[22]
Row78
ξ = 1
[19]
Laat
ξ = 1
[38, 39]
Odes
ξ = 1 [76,
79]
b 0
−0.0296
−0.0293
−0.017
0.003
−0.013
−0.004
0.007
0.003
m
− 1
π
b 1
−0.077
−0.078
−0.13
−0.143
−0.092
−0.094
−0.075
−0.094
m
− 1
π
ReB 0 0
0
−0.0475 −0.15
–
–
−0.18
−0.15
m
− 4
π
ImB 0 0.0436
0.0428
0.0475
0.046
–
–
0.058
0.046
m
− 4
π
c 0
0.172
0.227
0.255
0.21
0.209
0.23
0.266
0.21
m
− 3
π
c 1
0.22
0.18
0.17
0.18
0.177
0.17
0.40
0.18
m
− 3
π
ReC 0 –
–
–
0.11
–
–
0.07
0.11
m
− 6
π
ImC 1 –
–
–
–
–
–
−0.34
−0.25
m
− 6
π
Relativistic Quantum Chemistry and Spectroscopy of Exotic …
81