Nuclear Physics A 679 (2001) 549–562
www.elsevier.nl/locate/npe
Deeply bound pionic atoms on β-unstable nuclei
Y. Umemoto a , S. Hirenzaki a,∗ , K. Kume a , H. Toki b , I. Tahihata c
a Department of Physics, Nara Women’s University, Nara 630-8506, Japan
b Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan
c The Institute of Physical and Chemical Research (RIKEN), 2-1 Hirosawa, Wako, Saitama 351-0106, Japan
Received 27 April 2000; revised 4 July 2000; accepted 13 July 2000
Abstract
We study the structure and formation of deeply bound pionic atoms on β-unstable nuclei. We
calculate the eigenenergies of the pionic atoms using the model nuclear densities and the realistic
pion-nucleus optical potential. We also investigate the formation cross sections of the deeply bound
pionic atoms in the inverse reaction d(RI, 3 He)X using an effective number approach for wide range
of unstable nuclei with the closed neutron number so as to check the experimental feasibility.  2001
Elsevier Science B.V. All rights reserved.
PACS: 36.10.Gv; 25.70.-z
1. Introduction
Since the successful use of a new experimental tool, namely “beams of unstable
nuclei” [1,2], the properties of nuclei far from the stability line have been studied
extensively in many laboratories where secondary beams of unstable nuclei are available.
One of the exciting findings was the neutron halo structure around the 9 Li core in 11 Li [1].
Since then, the structure and reactions of β-unstable nuclei have been studied extensively
both theoretically and experimentally [3,4].
On the other hand the deeply bound pionic atoms, which were predicted to be quasistable [5–7], were observed using the (d, 3 He) reactions recently [8,9]. Since the repulsive
pion–nucleus interaction causes the formation of the pionic halo around the core nucleus [6,
7], the binding energies and widths of deeply bound pionic atoms are very sensitive
to the structure of the nuclear surface, in particular, to the neutron skin [10]. Hence,
we can expect to obtain new information on neutron/proton distribution of the unstable
nuclei by observing the deeply bound pionic atoms. The determination of the neutron
distributions for unstable nuclei is one of the most important subjects in the β-unstable
∗ Corresponding author. E-mail: [email protected]
0375-9474/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 3 5 4 - 7
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
nuclear physics [3]. In addition the (d, 3 He) reaction spectra with pionic atom formation
proceeds through the single-neutron pick up and are known to be very sensitive to the
single-particle properties of the neutron in nucleus [11]. Hence, we can also expect to
obtain the new information of the neutron single-particle states (nlj ) in unstable nuclei.
This is very interesting in the context of the possible change of the magic numbers far
from stability [12].
For the formation of the deeply bound pionic atoms on unstable nuclei, we use the
inverse kinematics technique, in which a light target such as deuteron is bombarded by
radioactive ion (RI) beam of intermediate energy and the recoiling light ejectile 3 He
is detected after forming pionic atoms in the heavy projectile [13,14]. We evaluate the
formation cross sections for inverse kinematics reactions d(RI, 3 He)X in order to examine
the experimental feasibility. Experiments in the inverse kinematics will be used for the
study of deeply bound pionic atoms in β-unstable nuclei.
2. Model for pionic atoms on unstable nuclei
To calculate the pionic bound state for unstable nuclei, we follow exactly the same
way as described in Ref. [11]. The Klein–Gordon equation is solved with the finite-size
Coulomb plus pion–nucleus optical potential [15]. In order to calculate the d(RI, 3 He)
spectra for wide range of the unstable nuclei, we adopt the theoretical density distributions
calculated by using the relativistic mean field (RMF) model [16]. We parameterized them
with the Woods–Saxon functional form by fitting the radius parameter to reproduce the
r.m.s. radii listed in Table A in Ref. [17] fixing the diffuseness parameter a = 0.5 [fm].
Radius parameters thus obtained are listed in Table 1 for all nuclei investigated in the
present work. For comparison, we also use the empirical density distributions in the same
Woods–Saxon form with radius parameter R = 1.18A1/3 − 0.48 [fm] and diffuseness
parameter a = 0.5 [fm] for both the proton and neutron densities.
Since the shapes of formation spectrum of the pionic atoms are sensitive to the singleparticle neutron configurations, we select the unstable nuclei with magic neutron numbers,
N = 20, 50, 82 and 126, which are expected to have simple closed shell structure for
neutron states. We assumed that there exists only one excited level in daughter nucleus
which couples to one neutron pick up process from a specific neutron orbit and also
assumed the normalization factors FO introduced in Ref. [11] to be 1 for all nuclei
considered here. This means that single neutron states in the target are fully occupied
without any configuration mixing.
The separation energies of neutron for the initial unstable nuclei are obtained by
the linear extrapolation from the data of stable nuclei. Since the experimental neutron
separation energies have certain systematics as a function of mass number [18], we use
this as a guide to determine the one neutron separation energies of β-unstable nuclei, which
have not been observed. As an example we show in Fig. 1 the mass-number dependence
of the neutron separation energies for N = 82 isotones. From the figure, we can see that
the mass-number dependence of the separation energies of the s1/2 , d3/2 and h11/2 levels
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
551
Table 1
Density parameters for N = 20, 50, 82 and 126 isotones for the Woods–
Saxon form obtained by fitting the r.m.s. radii calculated in Refs. [16,17]
using RMF model. Rn and Rp are the radius parameters of neutron and
proton density distributions. Rm is the nucleus matter density distribution.
The diffuseness parameter is assumed to be 0.5 [fm] for all cases
Neutron number
Nucleus
Rn
Rp
Rm
20
32 Mg
3.597
3.514
3.542
3.576
3.002
3.312
3.609
3.888
3.385
3.425
3.576
3.749
4.969
4.976
5.010
5.037
5.055
4.485
4.637
4.849
5.050
5.157
4.793
4.842
4.940
5.045
5.106
5.969
5.978
6.006
6.084
6.117
5.462
5.626
5.779
5.987
6.126
5.786
5.840
5.912
6.041
6.121
6.980
6.981
7.023
6.518
6.632
6.801
6.813
6.846
6.932
36 S
40 Ca
44 Cr
50
80 Zn
84 Se
90 Zr
96 Pd
100 Sn
82
130 Cd
136 Xe
142 Nd
148 Dy
154 Hf
126
200 W
208 Pb
216 Th
are well approximated by the linear function, while the d5/2 and g7/2 levels have slightly
deviate from the linear fit. However, we do not have enough information on the separation
energies for the unstable nuclei and we use the linear extrapolation throughout. For stable
region, the experimental data are used. In order to check the validity of our simple linear
extrapolation, we also plot in Fig. 1 the one-neutron separation energies obtained in
Ref. [19], which are based on the ground state masses of the nuclei and correspond to the
smallest one-neutron separation energies leading to the ground state of the daughter nuclei.
We find that our approach is reasonably good. We compile the separation energies obtained
in our linear extrapolation in Table 2 for all valence orbits for the nuclei considered here.
From the single-particle configuration shown in Table 2, we can expect that the isotones
with N = 20 and N = 82 are suited for observation of the pionic 1s state, while the isotones
with N = 50 and N = 126 are suited for observations of the pionic 2p state in the recoilless
nuclear reaction.
For the Lorentz transformation to inverse kinematics frame, the absolute value of nuclear
masses in the initial states are needed. We simply use MA = 12 (Mp + Mn )A with nuclear
mass number A when experimental data are not available. Since the nuclear mass in the
initial state is only used for the Lorentz transformation to the inverse kinematics frame,
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
Fig. 1. Neutron separation energies for the nuclei with N = 82. The experimental data of stable nuclei
are indicated by arrows in the figure. For unstable nuclei, the separation energies are obtained by the
linear extrapolation of the data for stable nuclei. Solid circles (s(1/2)n states) and squares (h(11/2)n
states) are results in Ref. [19], which correspond to the smallest one-neutron separation energy for
each nucleus leading to the ground state of the daughter nuclei.
the calculated spectra are insensitive to this mass. We have checked the insensitivity
numerically and the use of more accurate masses does not change present results.
To obtain the formation spectra of deeply bound pionic states, we use the effective
number approach described in Ref. [11]. Here we only give the formalism for calculating
the pionic formation cross sections in d(RI, 3 He)X reactions by inverse kinematics
technique, where the unstable nuclei are bombarded on the deuteron target and recoiled
3 He particle is observed. These measurements could be possible in the new facility at
RIKEN [20]. The formation cross sections of pionic states in heavy nuclei are expressed
generally as
dσ
dHe dEHe dA→3 He(A−1)π
X pHe 2Md MHe MA
1
Γπa
|t|2
,
(1)
=
√
2
2
2
2 /4
1/2
2π ( sπa − Mπa )2 + Γπa
(2π) λ (s, Md , MA )
−1
[lπ ⊗jn ]
where Mπa and Γπa are mass and width of pionic bound state consisting of pion and
√
daughter nucleus with (A − 1) nucleons, sπa the missing mass of the reactions, MA
the mass of target, and t the transition amplitude. λ1/2 (s, Md2 , MA2 ) is the Källen function
defined as,
λ(a, b, c) = a 2 + b2 + c2 − 2ab − 2bc − 2ac,
(2)
which is introduced to properly normalize the cross section by the initial flux.
For the normal kinematics, the above formula reduces to the same expression as that
used for A(d, 3 He) reaction in Ref. [11],
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
553
Table 2
Neutron separation energies in unit of MeV. For stable nuclei we use the experimental
values for the separation energies. The references are shown in the table. For unstable
nuclei, we evaluate the energies by linear extrapolation of data for the stable nuclei
(see text)
(N = 20)
32 Mg
36 S [26,27]
40 Ca [28]
44 Cr
d3/2
s1/2
d5/2
3.828
4.373
3.891
9.889
11.459
12.609
15.641
18.061
20.661
21.084
24.159
28.046
(N = 50)
80 Zn
84 Se
90 Zr [29]
96 Pd
100 Sn
g9/2
p1/2
p3/2
f5/2
7.214
7.133
8.347
8.723
9.076
9.245
10.192
10.566
11.972
12.560
13.066
13.422
14.662
15.580
15.726
16.094
16.524
17.692
17.570
17.937
(N = 82)
130 Cd
136 Xe [30]
142 Nd [31,32]
148 Dy
154 Hf
d3/2
s1/2
h11/2
d5/2
g7/2
6.176
6.628
6.600
8.113
6.888
8.060
8.350
8.590
9.640
8.890
9.828
10.021
10.585
11.054
11.173
11.698
11.741
12.600
12.559
14.511
13.538
13.447
14.600
14.042
17.052
(N = 126)
200 W
208 Pb [33]
216 Th
p1/2
f5/2
p3/2
i13/2
f7/2
6.202
6.860
7.255
7.303
9.930
7.367
7.937
8.264
9.000
9.710
8.532
9.014
9.273
10.697
9.490
dσ
dHe dEHe
=
dA→3 He(A−1)π
X dσ 1
Γπa
, (3)
Neff
2 + Γ 2 /4
d
2π
1E
3
πa
dn→ He π
−1
[lπ ⊗jn ]
with
1E = Q + mπ − BEπ + Sn − Md + Mn − MHe ,
(4)
where BEπ is the pionic atom binding energy and Sn is the neutron separation energy. Here
we neglect the recoil energy for the pionic atom, which is justified in recoilless kinematics
around Td = 500 MeV.
By comparing Eqs. (1) and (3), we can see the following correspondence in these
formulas,
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
dσ
d
dn→3 Heπ
Noting that the
Neff ←→
pHe 2Md MHe MA
|t|2 .
(2π)2 λ1/2 (s, Md2 , MA2 )
2Md MHe MA
1
|t|2
(2π)2 λ1/2 (s,Md2 ,MA2 )
(5)
is Lorentz invariant, we can evaluate the double
differential cross sections shown in Eq. (1) in the inverse kinematics from those in the
normal kinematics shown in Eq. (3) by multiplying the ratio of the momentum of emitted
3 He in each frame.
3. Numerical results and discussions
In this section, we show the numerical results for the structure and the formation of the
deeply bound pionic states in β-unstable nuclei with neutron number N = 82 in detail.
The complete results including the case of other nuclei are found in Ref. [21]. We consider
the nuclei which are expected to be produced in RI-beam factory in RIKEN with stronger
intensity than 105 [particle/second] [20].
We show the calculated binding energies and widths of the pionic bound states for these
unstable nuclei with N = 82 in Appendix A. In this mass region, we have experimental
data for 3d atomic state in Nd isotopes. Our result reproduce the data reasonably well. We
can see that the level spacings of the deeply bound pionic states in these nuclei are larger
than the level widths and, therefore, the pionic bound states in the β-unstable nuclei are
expected to be quasi-stable as in the case of stable ones. As shown in these tables, the
binding energies and widths are moderately influenced by the choice of nuclear density
distribution. The empirical density gives larger binding energies and widths for almost
all the states considered here. This is because the empirical density gives smaller r.m.s.
radius than that of RMF and the local part of the optical potential is repulsive reducing
the binding energy and the width of pionic states. The heavy isotones have larger binding
energies and widths due to the stronger Coulomb force. For the isotones with N = 82, the
pionic 1s state is expected to be observed clearly. The empirical and the RMF densities
give somewhat different pionic eigenenergies: the differences are around 135 and 71 keV
for binding energy and width for 1s state in 129 Cd, while 73 and 1 keV for 1s state in
153 Hf. Due to the strong Coulomb attraction, the binding energies are much larger for
153 Hf. For the case of N = 82 isotones, two nuclear densities give the largest differences
for 1s state binding energy and width in 129 Cd case. Obviously, the shallow states are
bounded by the pure Coulomb force and the binding energies are almost the same with
these two nuclear densities. The results in this table clearly shows the sensitivity of the
deeply bound pionic states to the nuclear optical potential. At present, we only examined
the dependence on the nuclear densities but we can expect to have similar sensitivity to the
isovector strength in the pion–nucleus optical potential. In order to obtain the feeling about
the isotone dependence of the pionic states, the binding energies and the widths for pionic
1s states are shown in Fig. 2. The upper (a) and lower (b) figures are the results with the
empirical and RMF densities, respectively.
By using the pionic wave functions described above, we have calculated the energy
spectra of d(RI, 3 He)X reactions. In the normal kinematics, the energy spectra are shown
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
555
Fig. 2. Binding energies (solid bars) and widths (hatched area) of pionic 1s state for the nuclei with
N = 82. For the nuclear density distribution, we use the Woods–Saxon form with (a) the empirical
value for the radius and the diffuseness parameters R = 1.18A1/3 − 0.48 [fm], a = 0.5 [fm] and
(b) the parameters obtained by fitting to the RMF results as listed in Table 1.
as a function of Q-value. In the case of inverse kinematics, the spectra are calculated as
a function of kinetic energy of the emitted 3 He THe . These two variables Q and THe are
related through the Lorentz transformation. As an example, we have plotted the relation Q
vs THe for the case of 136 Xe in Fig. 3. We can see that these two variables −Q and THe are
almost linearly proportional with each other. Then the spectrum in the inverse kinematics is
expected to have similar shape as in the case of the normal kinematics. In Fig. 3, we can see
He
that 1T
1Q ∼ 0.4. Thus, the experimental energy resolution 1Q ∼ 300 keV in the normal
kinematics corresponds to the 1THe ∼ 120 keV in the inverse kinematics. In what follows,
we assume this value for all the theoretical spectra in the inverse kinematics. Here, we
would like to mention the natural widths of the neutron–hole states. The widths for valence
shell states of N = 82 nuclei are shown in references cited in Table 2 and are much smaller
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
Fig. 3. The kinetic energy THe of recoiled 3 He in the inverse kinematics d(136 Xe, 3 He) is plotted as
a function of the Q-value in the normal kinematics 136 Xe(d, 3 He).
(∼ 10 keV) than the assumed experimental resolutions. Thus, in this paper we neglect the
natural widths of the neutron–hole states.
The results of the calculated spectra for d(142 Nd, 3 He) reaction are shown in Fig. 4.
In this figure, we compare results with the empirical WS density and those with the WS
density fitted to RMF result. We find the shapes of the spectra are almost same for both
cases and the absolute peak hight is slightly smaller for the result with the RMF fitted WS
nuclear density. These features of spectra are common for all N = 82 nuclei. In Fig. 5, we
show the results of the calculated spectra for other incident nuclei with neutron numbers
N = 82 for RMF fitted WS nuclear density case. The dominant peak consists of two
−1
subcomponents [(1s)π ⊗ (s1/2 )−1
n ] and [(1s)π ⊗ (d5/2 )n ]. These two components are
separately seen for light isotones but, for heavy isotones such as 148 Dy and 153 Hf, they
overlap with each other and it would be difficult to observe the individual pionic states in
detail.
For the cases of isotones with N = 20 and 50, the absolute value of the cross sections
increases with the mass number. Actually dominant peak height is around 5 [µb/(MeV sr)]
for 32 Mg and 12 [µb/(MeV sr)] for 44 Cr for [(1s)π ⊗ (s1/2 )−1
n ] configuration in N =
20 cases. For N = 50 case, the dominant configuration is [(2p)π ⊗ (p3/2 )−1
n ] and the
cross sections are 6 [µb/(MeV sr)] for 80 Zn and 21 [µb/(MeV sr)] for 100 Sn. But for the
cases of N = 82 and 126, the absolute value of the cross section saturates and does not
increase with the mass number. In the present calculations for the β-unstable nuclei, we
have assumed a simplified extreme single-particle picture for the incident and the daughter
nuclei. In a realistic situation, the strength in the spectrum will be fragmented and the
reaction spectrum involves more complex structure. The present calculation shows overall
structure of the spectrum and the typical aspect of the expected spectra in various mass
regions.
Finally, to see the dependence of our results on the pion–nucleus optical potential and
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
557
Fig. 4. Calculated energy spectra for d(142 Nd, 3 He) reaction with pionic atom formation at
TA = 250 MeV/nucleon. The WS density distributions are used with (a) the empirical parameters
and (b) the parameters obtained by fitting to the RMF results (see text for detail). In both
figures, dashed line and dotted line show the contributions from the [(1s)π ⊗ (s1/2 )−1
n ] and the
−1
[(1s)π ⊗ (d5/2 )n ] configurations, respectively. Experimental energy resolution is assumed to be
120 keV FWHM.
on the details of the nuclear density distribution, we calculate the expected d(142 Nd, 3 He)
spectra for bound pionic states formation using another optical potential obtained by
Konijn et al. and the full RMF nuclear density. We use the potential parameters “Present
fit, ξ =1” in Table 1 of Ref. [22]. The RMF density is calculated in Refs. [23,24] and is
provided numerically in Ref. [25].
The calculated d(142 Nd, 3 He) spectra for bound pionic states are shown in Fig. 6. Solid
line shows the result obtained with the empirical Woods–Saxon nuclear density and is the
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
Fig. 5. Calculated energy spectra for d(RI, 3 He) pionic atom formation reaction with (a) 130 Cd, (b)
136 Xe, (c) 148 Dy and (d) 154 Hf RI beam at T = 250 MeV/nucleon. The WS density distribuA
tion is used with the parameters obtained by fitting to the RMF results (see text for detail). In all
figures, dashed line and dotted line show the contributions from the [(1s)π ⊗ (s1/2 )−1
n ] and the
[(1s)π ⊗ (d5/2 )−1
n ] configurations, respectively. Experimental energy resolution is assumed to be
120 keV FWHM.
same as that of Fig. 4(a) except for quasi-free pion contributions. For dashed line, the pion–
nucleus optical potential determined by Konijn et al. [22] is used instead of Seki–Masutani
potential [15]. For dotted line, the RMF nuclear density in numerical form is used for the
calculation instead of the Woods–Saxon functional form. This result with the RMF density
resembles to that in Fig. 4(b), as expected, where the Woods–Saxon density fitted to the
RMF results is used. We find that all three results exhibit the similar spectrum shape and
absolute cross sections. Hence, we can expect that our theoretical predictions are stable
to model parameters. On the other hand, this fact also indicates that we need data with
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
559
Fig. 6. Calculated energy spectra for d(142 Nd, 3 He) reaction with pionic atom formation at
TA = 250 MeV/nucleon. Contributions from bound pionic states are only included. Solid line shows
the results with the Woods–Saxon nuclear density with the empirical parameters (ρemp ) and the
pion nucleus optical potential obtained by Seki–Masutani (SM) in Ref. [15]. Dashed line shows the
result with another pion–nucleus optical potential obtained by Konijn et al. in Ref. [22]. Dotted line
shows the result with the nuclear density obtained by RMF model (ρRMF ) [25]. Experimental energy
resolution is assumed to be 120 keV FWHM.
high energy resolution to deduce new information on nuclear density and/or pion–nucleus
potential.
We think that the studies of the deeply bound pionic states on β-unstable nuclei are very
interesting since we can expect to extract the information of the difference of the neutron
and proton density distributions from binding energies and widths. The 3 He energy spectra
also include the information of the single-particle states which itself is interesting in the
context of the structure of unstable nuclei.
4. Summary
We have investigated the deeply bound pionic atoms in β-unstable nuclei using the two
types of nuclear density distributions, empirical and the RMF densities, with the realistic
pion–nucleus optical potential. We have also calculated the reaction spectra d(RI, 3 He)
in the inverse kinematics. The incident β-unstable nuclei collide with the target deuteron
resulting the formation of final pionic states.
The purpose of the present work is to see the overall trends of the reaction spectra.
We, therefore, adopted the extreme single-particle picture for the target and daughter
nuclei. The reaction spectra are calculated for the cases with neutron magic numbers
N = 20, 50, 82 and 126. We show the results for N = 82 in detail in this paper. Similar
to the case of normal kinematics, the quasi-substitutional states are strongly populated in
these reactions. Among the target nuclei with specific neutron numbers, the cross section
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Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
becomes larger with the increase of the mass numbers for the case of light isotones N = 20
and 50 while absolute values of the cross sections stay almost the same for the case of
heavy isotones N = 82 and 126. Generally, the several subcomponents with pionic 1s or
2p states exhibit separate peaks for lighter isotones with specific neutron number N . For
heavier targets, these dominant peaks overlap with each other and the precise determination
of the binding energies and the widths seems to be difficult. In our present calculation, we
adopted simple nuclear models. In a more elaborate treatment we need to use the realistic
spectroscopic inputs. In this case we expect that the reaction strengths are fragmented
and the reaction spectra exhibit more complex structure. We can expect that the reaction
spectra reflects the detailed shell structure of the incident and the daughter nuclei. The
study of the shell structure is one of the interesting subjects for the β-unstable nuclei.
In this context, the d(RI, 3 He) reactions in the inverse kinematics are quite interesting to
pursue both experimentally and theoretically.
The smallness of the cross sections could cause problems for the experimental
observation. However, the difficulties may be resolved by detecting emitted 3 He for wide
angular range since the angular dependence of the cross sections for the case of inverse
kinematics d(RI, 3 He) are much milder than for the case of normal kinematics A(d, 3 He).
We hope that the present theoretical work motivates further experimental effort to study
the deeply bound pionic states and develop the physics of pionic atom spectroscopy.
5. Acknowledgement
We are grateful to K. Sumiyoshi for kindly providing results of the RMF model.
Y. Umemoto and S. Hirenzaki thank all members of RIBF Lab. of RIKEN for their
hospitalities during their stay. YU and SH also thank to H. Nagahiro and S. Yamamoto
for their helps to prepare the manuscript.
Appendix A
Calculated binding energies BEopt and widths Γ of pionic states in units of keV for N =
82 isotones. BEFC indicates the binding energies calculated with a finite-size Coulomb
potential only. The nuclear density distribution is assumed to be of the Woods–Saxon
form with two different density parameters, (1) The empirical values for the radius and
diffuseness parameters, R = 1.18A1/3 − 0.48 fm and a = 0.5 fm, and (2) The parameters
obtained by fitting the r.m.s. radii of RMF model, which are listed in Table 1.
Y. Umemoto et al. / Nuclear Physics A 679 (2001) 549–562
561
[N = 82]
(1)
(2)
Pionic atom
nl
BEopt (Γ )
BEFC
BEopt (Γ )
BEFC
π − −129 Cd
1s
2s
3s
2p
3p
3d
3464.(252.)
1289.(61.)
670.(23.)
2075.(105.)
931.(35.)
958.(2.)
5849.
1781.
844.
2142.
955.
956.
3329.(181.)
1258.(48.)
659.(19.)
2026.(69.)
915.(24.)
958.(1.)
5856.
1782.
845.
2142.
955.
956.
π − −135 Xe
1s
2s
3s
2p
3p
3d
4170.(340.)
1583.(87.)
830.(34.)
2601.(166.)
1171.(55.)
1216.(4.)
7012.
2194.
1050.
2702.
1207.
1212.
4039.(285.)
1552.(81.)
818.(33.)
2545.(124.)
1152.(43.)
1216.(4.)
6986.
2190.
1049.
2701.
1206.
1212.
π − −141 Nd
1s
2s
3s
2p
3p
3d
4925.(469.)
1902.(129.)
1005.(52.)
3181.(255.)
1436.(85.)
1507.(10.)
8212.
2640.
1275.
3321.
1486.
1499.
4803.(438.)
1873.(134.)
994.(56.)
3120.(211.)
1416.(74.)
1507.(9.)
8141.
2628.
1271.
3316.
1484.
1499.
π − −147 Dy
1s
2s
3s
2p
3p
3d
5728.(663.)
2246.(199.)
1195.(83.)
3813.(384.)
1726.(129.)
1832.(20.)
9441.
3115.
1518.
3993.
1791.
1817.
5619.(654.)
2216.(219.)
1182.(94.)
3750.(336.)
1706.(119.)
1832.(18.)
9267.
3084.
1507.
3978.
1786.
1817.
π − −153 Hf
1s
2s
3s
2p
3p
3d
6574.(956.)
2607.(312.)
1396.(134.)
4494.(568.)
2040.(193.)
2192.(39.)
10691.
3618.
1778.
4716.
2122.
2166.
6501.(955.)
2580.(338.)
1382.(147.)
4445.(522.)
2024.(187.)
2194.(37.)
10441.
3570.
1762.
4689.
2112.
2166.
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