---
abstract: 'The x-ray cascade from antiprotonic atoms was studied for $^{106}$Cd, $^{116}$Cd, $^{112}$Sn, $^{116}$Sn, $^{120}$Sn, and $^{124}$Sn.  Widths and shifts of the levels due to strong interaction were deduced.  Isotopic effects in the Cd and Sn isotopes are clearly seen.  The results are used to investigate the nucleon density in the nuclear periphery.  The deduced neutron distributions are compared with the results of the previously introduced radiochemical method and with HFB calculations.' 
author:
- 'R. Schmidt$^1$, A. Trzci[ń]{}ska$^2$, T. Czosnyka$^2$, T. von Egidy$^1$, K. Gulda$^3$, F. J. Hartmann$^1$, J. Jastrzȩbski$^2$, B. Ketzer$^1$, M. Kisieli[ń]{}ski$^2$, B. K[ł]{}os$^4$, W. Kurcewicz$^3$, P. Lubi[ń]{}ski$^2$[^1], P. Napiorkowski$^2$, L. Pie[ń]{}kowski$^2$, R. Smola[ń]{}czuk$^5$, E. Widmann$^6$, S. Wycech$^5$'
title: 'Nucleon density in the nuclear periphery determined with antiprotonic x-rays: cadmium and tin isotopes'
---

INTRODUCTION
============

Antiprotonic atoms are a specific tool to study the strong interaction and the nucleon density in the nuclear periphery.  The strong interaction potential leads to widths and energy shifts of antiproton-atomic levels compared to the purely electromagnetic interaction.  The measurement of these widths and shifts gives information on the strength of the interaction, which is often expressed by an effective scattering length in the optical potential model [@bat81b], and on the nucleon density in the region where the annihilation takes place. 

In contrast to other methods which are sensitive to the charge distribution and usually probe the whole nucleus with the nuclear periphery giving only a small contribution, antiprotons are sensitive to the matter density at the nuclear periphery (they probe the nucleon density at distances about 2 fm larger than the half-denstity charge radius).  By investigating different isotopes of one element, the effect of additional nucleons can be deduced.  The major part of the effect comes from the higher nucleon density in the nuclear periphery of isotopes with more neutrons [@koe86].  Possibly isospin effects on the effective antiproton-nucleon scattering-length may also exist. 

Before our studies data of antiprotonic atoms had been collected for several elements [@bat95].  However, with a few exceptions, these were mainly light isotopes ($Z < 40$) and a number of them was investigated using natural targets under difficult antiprotonic beam conditions.  The aim of the PS209 collaboration was to measure with antiprotons from LEAR at CERN a large variety of elements and isotopes in order to provide a set of data for a new combined analysis [@bat89] to determine the nucleon density in the nuclear periphery.  This analysis is expected to yield knowledge about the neutron density in the annihilation region and better knowledge of the antiproton-nucleus interaction, e.g. about a density or isospin dependence of the effective scattering length [@bat95; @wyc01]. 

The results of the PS209 experiment were already reported in a number of conferences, see e.g. [@jas97; @har01; @trz01l].  In particular the last reference presents a comprehensive table of level widths and shifts determined in 34 monoisotopic or isotopically separated targets ranging from [$ ^{16}\mbox{O}$]{} to [$ ^{238}\mbox{U}$]{}.  In Ref.  [@trz01] these data were analyzed under the assumption of a two parameter Fermi (2pF) distribution of peripheral protons and neutrons.  A linear relationship of the difference between the neutron and proton root mean square radii (rms) $\Delta r_{np}$ and the asymmetry parameter $\delta =
(N-Z)/A$ was established (where $N,Z$ and $A$ are neutron, atomic and mass numbers, respectively).  Besides conference communications some more detailed reports on the evaluations of PS209 results were already published [@sch98; @har02] or are in preparation. 

In this publication results for tin and cadmium isotopes are presented.  The isotopes [$ ^{106}\mbox{Cd}$]{}, [$ ^{116}\mbox{Cd}$]{}, [$ ^{112}\mbox{Sn}$]{}, [$ ^{116}\mbox{Sn}$]{}, [$ ^{120}\mbox{Sn}$]{} and [$ ^{124}\mbox{Sn}$]{} have been investigated.  For four of these nuclei also the neutron-to-proton density ratio in the nuclear periphery could be measured using the radiochemical method [@jas93; @lub94; @lub98; @sch99]. 

 EXPERIMENTAL METHOD AND SETUP 
===============================

The principle of the method employed is described in Ref.  .  The antiprotons are captured into a high antiprotonic-atom orbit.  They cascade down towards levels with lower principal quantum number $n_{\overline{p}}$ by the emission of Auger electrons and x-rays.  In states with low $n_{\overline{p}}$ the orbit of the antiproton comes close to the nucleus and the interaction with the nucleus becomes large.  The resulting shifts and widths of the levels were partly evaluated and interpreted as explained in the next sections. 

The strong interaction width can be measured directly (via analysis of the line shape) if it is of the order of magnitude of the instrumental resolution (about 1 keV).  For many isotopes this is the case for the lowest visible transition.  The energy of the transitions may be measured with an accuracy of about 10 eV. Thus strong-interaction energy shifts which are larger than this value may be determined.  For those levels, for which the strong interaction width is of the order of the electromagnetic width (due to x- and Auger transitions), the strong interaction width was deduced from the intensity balance of the x-ray transitions feeding and depopulating the respective level [@koc68].  In the case of non-circular transitions the feeding transitions cannot be observed experimentally, as they are hidden by the much stronger intensities of the circular transitions.  In these cases the feeding intensities can be taken from cascade calculations if the cascade is sufficiently well known [@sch98]. 

The experiment was performed with the antiproton beam provided by LEAR of CERN.  The setup (cf.  Fig.  \[setup\]) is similar to that described in Ref.  .  Due to the small initial momentum of the antiprotons of 106 MeV/$c$ the scintillation-counter telescope, consisting of an anticounter S1 and a counter S2, was placed inside a chamber (with aluminum windows of thickness 12 $\mu$m) filled with helium.  This was necessary to avoid large an energy straggling and degradation for the low energy (6.0 MeV) antiprotons in air.  After passing the chamber window the antiprotons were stopped inside the target.  The properties of the different targets are listed in Table \[targets\]. 

The x-rays emitted during the antiproton cascade were measured with three Ge detectors (two coaxial detectors with an active diameter of 49 mm and a length of 50 mm and one planar detector with diameter 36 mm and thickness 14 mm) with a resolution (FWHM) of about 1 keV at 200 keV gamma ray energy.  The detectors were placed at distances of about 50 cm from the target at angles of 13$^{\circ}$, 35$^{\circ}$ and 49$^{\circ}$ respectively towards the beam axis.  The detector-target distance was adjusted in the way to obtain a good signal-to-noise ratio and simultaneously decrease the background produced by pions from the annihilation processes, which would obscure the x-ray lines and would have damaged the detectors.  This also allowed to avoid summing effects.  The x-rays were measured in coincidence with the antiproton signal in a time window which was extended up to 500 ns after the antiproton signal from the telescope counter.  The stability and efficiency of the detectors and the data acquisition system was checked by on-line and off-line measurements with calibration sources. 

 EXPERIMENTAL RESULTS
=====================

The x-ray spectrum from the target [$ ^{124}\mbox{Sn}$]{}, as taken with detector 1, is shown in Fig.  \[sn124spec\].  The upper right part shows the part of the spectrum with the last visible transition $n=8\rightarrow7$.

Those lines in the spectra which are not significantly broadened by strong interaction were fitted with Gaussian functions.  Their relative intensities are given in Table \[cd\_int\] and Table \[sn\_int\] for the Cd and Sn istopes, respectively.  For the fit of the transition $n=8\rightarrow7$ two Lorentzians convoluted with Gaussians were used.  The strong-interaction energy shifts were deduced from the measured energies of these transitions.  The energy shift is the difference between the energy calculated with a purely electromagnetic potential [@bor83] and the measured transition energy. 

The widths of the levels $(n,l)=(8,7)$ were determined from the measured intensity balance.  Small corrections for parallel transitions and for unobserved transitions from higher levels were taken from the calculated cascade [@sch98].  For the determination of the width of the level (9,7) all intensities of the feeding transitions were taken from the results of the cascade calculations.  The radiative and Auger widths (obtainded according to Ref.  [@leo74]) which were used for these calculations are summarized in Table \[cd\_rad\_aug\] for cadmium and in Table \[sn\_rad\_aug\] for tin.  Tables \[cd\_gam\_sh\] and \[sn\_gam\_sh\] give the measured widths and shifts for the cadmium and tin isotopes, respectively.  The variation of these observables due to the different number of protons and neutrons from [$ ^{106}\mbox{Cd}$]{} to [$ ^{124}\mbox{Sn}$]{} is clearly visible.  The widths for [$ ^{124}\mbox{Sn}$]{} are roughly twice as large as those for [$ ^{106}\mbox{Cd}$]{}.  The shifts turn from attractive or compatible with zero for [$ ^{106}\mbox{Cd}$]{} to repulsive for [$ ^{124}\mbox{Sn}$]{}.  The only observable which does not follow rather smooth systematics is the upper level $(n=8, l=7)$ width of [$ ^{106}\mbox{Cd}$]{}.  For all other nuclei presented in these tables the lower to upper level widths ratio is $\Gamma_{low}/\Gamma_{up} = 85 \pm 7$, whereas the same ratio is only about 50 in case of [$ ^{106}\mbox{Cd}$]{}. 

This effect is due to E2 resonance [@leo76] which in Cd nuclei mixes the $n,l=6,5$ and the $n,l=8,7$ states.  The difference between the energies of nuclear $2^+$ state and the corresponding antiprotonic-atom transition is 65 keV and 184 keV in [$ ^{106}\mbox{Cd}$]{} and [$ ^{116}\mbox{Cd}$]{}, respectively.  As the electric quadrupole moment is not very different for both nuclei [@ram87], the increase of the upper level width due to the mixing is more significant in [$ ^{106}\mbox{Cd}$]{} than in [$ ^{116}\mbox{Cd}$]{}.  This qualitatively explains the observed effect. 

To be more quantitative, the width of the $n,l=6,5$ level in Cd nuclei should be known.  This width was estimated by an extrapolation to $Z=48$ of the systematics presented in [@trz01zak] for lower Z nuclei.  The extrapolated value is $7.7\pm 2.5$ keV. This leads to the E2 induced width of $1.9 \pm 0.5$ eV and $0.34\pm0.08$ eV in [$ ^{106}\mbox{Cd}$]{} and [$ ^{116}\mbox{Cd}$]{}, respectively.  The $j^+$ and $j^-$ components of the upper level widths, corrected for the E2 effect, are also given in Table \[cd\_gam\_sh\].  The summary of the results (measured values) for [$ ^{106}\mbox{Cd}$]{} is shown in Fig.  \[cd\_sum\]

 DISCUSSION 
============

The region of tin isotopes with a closed $Z = 50$ proton shell constitutes one of the favorable parts of the Nuclear Chart for experimental and theoretical nuclear-structure studies.  During our investigation on antiprotonic atoms in this region we measured, besides the results reported in this paper, the level widths and shifts in even Te isotopes ($Z=52$) [@klo02].  In addition, using the radiochemical method [@jas93], we have determined the neutron halo factor, a quantity reflecting the composition of the outer nuclear periphery in [$ ^{106,116}\mbox{Cd}$]{}, [$ ^{112,124}\mbox{Sn}$]{} [@lub98; @sch99] and in [$ ^{128,130}\mbox{Te}$]{} [@lub98]. 

In the present discussion we will concentrate on the first two elements.  In our recent publication [@trz01] we have presented in detail our approach to determine the peripheral neutron distribution and differences between the neutron and proton mean square radii $\Delta r_{np}$ using observables gathered from antiprotonic atoms under the assumption of a two-parameter Fermi (2pF) neutron and proton distribution: $\rho(r) = \rho_0 \cdot \{1+exp(\frac{r-c}{a})\}^{-1}$, where $c$ is the half density radius, $a$ the diffuseness parameter (related to the surface thickness $t$ by $t=4\,\mbox{ln}3 \cdot a$) and $\rho_0$ is a normalization factor.  This approach is summarized below. 

Assuming identical annihilation probabilities on neutrons and protons the radiochemical experiment determines the halo factor, which is close to the normalized neutron to proton density ratio ($Z/N \cdot
\rho_n/ \rho_p$) at a radial distance $2.5 \pm 0.5$ fm larger than the half density charge radius.  Comparing the halo factor with the neutron to proton density ratio deduced from $\Delta r_{np}$ determined in other experiments one can conclude that for neutron rich nuclei it is mostly the neutron diffuseness which increases and not the half density radius [@trz01].  Although this conclusion was based on the very simple 2pF model of the nuclear periphery it is corroborated by much more sophisticated Hartree–Fock–Bogoliubov (HFB) calculations.  This is illustrated in Figs.  \[hfb\_2pf\] and \[hfb\_2pf\_rorat\], where the proton and neutron density distributions for [$ ^{124}\mbox{Sn}$]{} are compared for both models.  The HFB calculations were performed using SkP force [@dob84], giving $\Delta r_{np}$ value equal to 0.16 fm.  As the calculated proton ($c_p$) and neutron ($c_n$) half-density radii are almost identical, this rms difference is mainly due to the difference in the proton and neutron surface diffuseness.  The fitted 2pF distributions with the HFB $c_n$, $c_p$ and $\Delta r_{np}$ values closely approximate the HFB distributions.  In the peripheral region from 6.5 fm to 8.5 fm, e.g, the 2pF neutron distribution differs by less than 20% from that derived from HFB calculations.  A similar result was obtained for other investigated nuclei. 

The antiprotonic x-rays are analyzed using an optical potential with the antiproton-nucleon scattering length of the form $\overline{a}=
(2.5 \pm 0.3) \, + i (3.4 \pm 0.3)$ fm, as proposed for point-like nucleons in Ref.  [@bat95].  The method allows to study the nuclear density at a radial distances about 1 fm closer to the nuclear center than those examined in the radiochemical experiment. 

The peripheral bare proton densities in form of 2pF distributions are obtained [@trz01] from the experiments sensitive to the nuclear charge: electron scattering [@vri87] or muonic x-rays [@fri95].  The differences between experimental level widths and shifts and those calculated with parameters of the proton distributions are attributed to the neutron contributions to these observables.  Based on the analysis and the comparison described above, the half density radii of the proton and neutron distributions are assumed to be equal, $c_n=c_p$. The neutron diffuseness is considered as a free parameter, adjusted to agree best with the experimental lower and upper level widths (the lower level shifts were not taken to the fits, see comments below). 

Table \[dtpar\] illustrates this procedure for the Cd and Sn nuclei.  For the Sn nuclei the 2pF charge distribution determined using data from muonic atoms or from electron scattering differ significantly.  Only the electron scattering data lead to $\Delta r_{np}$ values compatible with the systematics gathered for other nuclei [@trz01] and with previous experiments [@kra94; @kra99].  Therefore these data were retained for further analysis.  In Fig.  \[gam\_sh\_rys\] the widths and shifts, calculated with density distributions from this Table and the scattering lengths given above are compared with corresponding experimental values.  It is evident that the potential used is able to reproduce simultaneously the lower and upper level widths for Cd and Sn nuclei whereas one has some problems with the level shifts (only for [$ ^{116}\mbox{Sn}$]{}, [$ ^{120}\mbox{Sn}$]{} and [$ ^{124}\mbox{Sn}$]{} the measured shifts are reproduced within the experimental errors). 

The analysis of the x-ray data as presented in Table \[dtpar\] allows to determine the normalized neutron to proton density ratio $Z/N \cdot \rho_n/\rho_p$ as a function of the radial distance at the periphery of the investigated nuclei.  As indicated above, the radiochemical experiment can be considered as giving the same ratio at a radial distance in the far periphery.  Figure \[rorat\_prc\] compares the results of these two experiments together with the normalized neutron to proton density ratio obtained from the Hartree-Fock-Bogoliubov (HFB) calculations.  For the sake of illustration the comparison is extended to some other nuclei not discussed in detail in the present publication.  For heavy Cd and Sn nuclei two experimental approaches are consistent within the experimental errors.  They are also in fair agreement with HFB calculations (a similar result is obtained for 15 other investigated nuclei, partly shown in Fig.  \[rorat\_prc\]). 

As already mentioned in our previous paper [@trz01] the situation is quite different for the lightest members of the Cd and Sn chains.  For these nuclei the analysis of the x-ray data gives densities consistent with the HFB model with Skyrme interaction as well as with recent calculations with Gogny force [@pom00].  The radiochemical experiment, however, seems to indicate a proton-rich nuclear periphery.  We encountered a similar problem for the two lightest members of the Ru and Sm isotopic chains.  In Ref.  [@wyc01] the role of a quasi-bound $\overline{\mbox{p}}\mbox{p}$ ($^{13}\mbox{P}_0$) state in nuclei with weakly bound protons was indicated as an explanation of this puzzle. ( For [$ ^{106}\mbox{Cd}$]{} and [$ ^{112}\mbox{Sn}$]{} the corresponding proton separation energies are 7354 keV and 7559 keV, respectively).  The formation of such a state would favor the annihilation on protons in comparison with those on neutrons and lead to a much smaller halo factor than really expected from the peripheral neutron and proton densities.  This explanation, although opening new research areas, would indicate that our radiochemical method is not as universal as we believed previously. 

The x-ray data, combined with proton distributions deduced from electron scattering experiments (Sn nuclei) and muonic atoms (Cd nuclei) allowed to determine the differences $\Delta r_{np}$ between neutron and proton rms radii.  The results are presented in Table \[dtpar\] and Fig.  \[drdelta\].  They are compared with other experiments as well as with the HFB model in Ref.  [@trz01].  The $\Delta r_{np}$ values in Figure \[drdelta\] for the [$ ^{116}\mbox{Cd}$]{} result differ from these in Ref.  [@trz01] as the correction for E2 was done and the level shift was excluded from the fit.  The [$ ^{106}\mbox{Cd}$]{} results are presented for the first time. 

SUMMARY AND CONCLUSIONS
=======================

Antiprotonic x-rays were measured in two even Cd and four even Sn nuclei.  The strong interaction level widths and shifts were determined.  The observed isotopic effects are attributed, at least to a large extent, to the increase of the difference between the neutron and proton rms radii with increasing neutron number. 

The interpretation of the collected data was done using a simple two parameter Fermi (2pF) model to describe the peripheral proton and neutron distribution.  It was verified that these simple distributions approximate rather well (within 20%) the distributions obtained from the HFB model in the region where the antiproton annihilation probability is significant.  The parameters of the proton distributions were obtained from literature, where 2pF charge distributions were determined from muonic-atoms or electron-scattering experiments. 

For neutron rich nuclei the peripheral neutron distributions deduced from the antiprotonic x-ray data are in good agreement with ealier radiochemical experiments.  This is, however, not the case for the lightest members of the investigated Cd and Sn isotope chains.  In these nuclei the radiochemical data indicate enhanced peripheral proton density in comparison with the neutron density.  Such a result is in contradiction with the x-ray data as well as with HFB model calculations.  It may be explained by the formation of a quasi-bound [$\overline{\mbox{p}}$]{}p states in nuclei with weakly bound protons. 

We thank the LEAR team for providing the intense, high-quality antiproton beam and Dr. Anna Stolarz of the Heavy Ion Laboratory in Warsaw and Katharina Nacke and Dr. Peter Maier-Komor of the Technical University Munich for the target preparation.  Financial support by the Polish State Committee for Scientific Research as well as the Accelerator Laboratory of the University and the Technical University of Munich is acknowledged.  This work was also supported by Deutsche Forshungsgemeinschaft. 

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TABLES {#tables .unnumbered}
======

     Target     $d$ (mg/cm$^2$)   $a$ (%)   number of $\bar{p}$ ($10^8$)  calibration sources
  ------------ ----------------- --------- ------------------------------ ------------------------------------
   $^{106}$Cd        40.0          76.5                  9                $^{137}$Cs, $^{152}$Eu
   $^{116}$Cd        64.5          93.0                  10               $^{137}$Cs, $^{152}$Eu
   $^{112}$Sn        65.6          94.7                  17               $^{137}$Cs, $^{152}$Eu
   $^{116}$Sn        46.8          93.0                  9                $^{137}$Cs, $^{152}$Eu
   $^{120}$Sn        65.3          99.2                  11               $^{137}$Cs, $^{152}$Eu
   $^{124}$Sn        70.1          97.9                  23               $^{133}$Ba, $^{137}$Cs, $^{152}$Eu

  : Target properties: thickness $d$, enrichment $a$, number of antiprotons used, and on-line calibration sources. 

\[targets\]

\[cd\_int\]

\[sn\_int\]

  ----------------------- ------------------- ---------------------- ------------------- ----------------------
                                                                                         
   \[-1.5ex\][$(n,l)$]{}   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$
           (8,7)                 4.95                  0.04                 4.70                  0.04
           (9,7)                 3.49                  0.06                 3.31                  0.05
           (7,6)                 9.86                  0.03                              
  ----------------------- ------------------- ---------------------- ------------------- ----------------------

  :  Radiative width $\Gamma_{\rm em}$ and Auger width $\Gamma_{\rm Auger}$ for those levels of $\bar{\rm p}$Cd where the strong interaction width was determined via the intensity balance.  Values in eV.

\[cd\_rad\_aug\]

  ----------------------- ------------------- ---------------------- ------------------- ---------------------- ------------------- ---------------------- ------------------- ----------------------
                                                                                                                                                                               
   \[-1.5ex\][$(n,l)$]{}   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$   $\Gamma_{\rm em}$   $\Gamma_{\rm Auger}$
           (8,7)                 5.79                  0.04                 5.67                  0.04                 5.56                  0.04                 5.46                  0.04
           (9,7)                 4.08                  0.06                 3.99                  0.06                 3.92                  0.06                 3.85                  0.05
  ----------------------- ------------------- ---------------------- ------------------- ---------------------- ------------------- ---------------------- ------------------- ----------------------

  : Radiative width $\Gamma_{\rm em}$ and Auger width $\Gamma_{\rm Auger}$ for those levels of $\bar{\rm p}$Sn where the strong interaction width was determined via the intensity balance.  Values in eV.

\[sn\_rad\_aug\]

[c|cc|cc]{} & &\
& $j=l+1/2$ & $j=l-1/2$ & $j=l+1/2$ & $j=l-1/2$\
$\Gamma(7,6)$ (eV) & 173$\pm$83 & 229$\pm$86 & 307$\pm$63 & 186$\pm$69\
$\varepsilon(7,6)$ (eV)& -32$\pm$27 & -20$\pm$29 & -15$\pm$22 & $-24\pm24$\
$\Gamma(8,7)$ (eV) & 3.5$\pm$0.7 & 4.2$\pm$0.8& 2.7$\pm$0.6& 3.3$\pm$0.7\
$\Gamma(8,7)^*$ (eV) & 1.4$\pm$1.0 & 2.4$\pm$1.0& 2.4$\pm$0.6& 3.0$\pm0.7$\
$\Gamma(9,7)$ (eV) & &\

  ------------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
                                                                                                                                                                                      
                                  $j=l+1/2$             $j=l-1/2$             $j=l+1/2$             $j=l-1/2$             $j=l+1/2$             $j=l-1/2$             $j=l+1/2$             $j=l-1/2$
     $\Gamma(7,6)$ (eV)          411$\pm$22            358$\pm$25            386$\pm$27            377$\pm$31            448$\pm$27            505$\pm$32            493$\pm$25            534$\pm$29
   $\varepsilon(7,6)$ (eV)        -9$\pm$16             -1$\pm$13             $12\pm18$             $36\pm19$             $26\pm17$             $37\pm20$             $26\pm17$             $63\pm16$
     $\Gamma(8,7)$ (eV)      4.1$^{+0.8}_{-0.7}$   4.3$^{+0.8}_{-0.7}$   4.7$^{+1.4}_{-1.1}$   5.2$^{+1.5}_{-1.2}$   4.9$^{+0.8}_{-0.7}$   6.4$^{+1.0}_{-0.8}$   5.5$^{+1.0}_{-0.9}$   6.8$^{+1.1}_{-1.0}$
     $\Gamma(9,7)$ (eV)                                                                                                                                                               
  ------------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------

  :  Measured level widths and shifts for the tin isotopes.[]{data-label="sn_gam_sh"}

[c|l|l|l|l|r|r|r|l|l|l|l|r|r|r]{} & &\
Isotope &$c_{ch}$&$t_{ch}$&$c_{p}$&$t_{p}$&$\Delta t_{np}$&$\chi^2$ &$\Delta r_{np}$& $c_{ch}$&$t_{ch}$&$c_{p}$&$t_{p}$&$\Delta t_{np}$&$\chi^2$& $\Delta r_{np}$\
$^{106}$Cd & 5.2875 & 2.30 & 5.329 & 2.000 &$^{c)}$ 0.43$^{+0.18}_{-0.25}$ & 0.2 & 0.15$^{+0.06}_{-0.09}$ & & & & & & &\
$^{116}$Cd & 5.4164 & 2.30 & 5.457 & 2.000 &$^{c)}$ 0.45$^{+0.10}_{-0.13}$ & 0.2 & 0.15$\pm$0.4 & 5.42 & 2.34 & 5.461 & 2.043 & 0.39$^{+0.11}_{-0.13}$& 0.2 & 0.13$^{+0.05}_{-0.09}$\
$^{112}$Sn & 5.3714 & 2.30 & 5.412 & 1.995 & 0.53$\pm$0.06 & 0.8 & 0.19$\pm$0.02 & 5.375 & 2.416 & 5.416 & 2.184 & 0.26$\pm$0.07 & 0.8

& 0.09$\pm$0.04\
$^{116}$Sn & 5.417 & 2.30 & 5.458 & 1.995 & 0.48$\pm$0.07 & 0.3 & 0.17$\pm$0.02 & 5.358 & 2.420 & 5.399 & 2.135 & 0.33$\pm$0.06 & 0.3 & 0.12$\pm$0.02\
$^{120}$Sn & 5.459 & 2.30 & 5.499 & 1.995 & 0.58$\pm$0.05 & 0.7 & 0.20$\pm$0.02 & 5.315 & 2.530 & 5.356 & 2.263 & 0.33$\pm$0.06 & 0.9 & 0.12$\pm$0.02\
$^{124}$Sn & 5.491 & 2.30 & 5.531 & 1.995 & 0.60$\pm$0.06 & 0.9 & 0.21$\pm$0.02 & 5.490 & 2.347 & 5.530 & 2.052 & 0.53$\pm$0.06 & 0.9 & 0.19$\pm$0.02\
\
\
\
\
\
\

\[dtpar\]

FIGURES {#figures .unnumbered}
=======

![ Schematic view of the experimental setup: S$_1$ – anticounter, and S$_2$ – counter of the telescope.[]{data-label="setup"}](fig1_aufbau96.eps){width="70.00000%"}

![  Antiprotonic x-ray spectrum from $^{124}$Sn measured with detector 1.  The inset shows the spectrum around the the transition $n=8\rightarrow7$.[]{data-label="sn124spec"}](fig2_sn124_at2.eps){width="8.6cm"}

![ Summary of measured shifts and widths for $^{106}$Cd and the excitation energy of the nuclear 2$^+$ state in this nucleus. ( The upper level widths are the values before correction for the E2 effect).[]{data-label="cd_sum"}](fig3_cd_levels.eps){width="65.00000%"}

![ Comparison of the HFB model (dashed lines) and the two parameter Fermi (2pF) density distributions (solid lines) for nucleus [$ ^{124}\mbox{Sn}$]{}.  The 2pF distributions were fitted to HFB model curves (half density radii $c_n = c_p = 5.55$ fm and the difference between neutron and proton rms radii, $\Delta r_{np} = 0.16$ fm).  The obtained 2pF diffuseness parameters are $a_p =0.45$ fm and $a_n=0.57$ fm.[]{data-label="hfb_2pf"}](fig4_hfb_2pf.eps){width="90.00000%"}

![ The same as for Fig.  \[hfb\_2pf\] but for the density ratio ($Z/N \cdot \rho_n / \rho_p$).  The cross indicates the halo factor measured in the radiochemical experiment [@sch99].[]{data-label="hfb_2pf_rorat"}](fig5_hfb_2pf.eps){width="45.00000%"}

![ Average widths and shifts of the levels (7,6) and widths of the levels (8,7) plotted versus $A$.  Open circles and dotted lines: cadmium isotopes (upper level widths corrected for the E2 effect), full circles and solid lines: tin isotopes.  The lines are calculated using the optical potential for point-like nucleons [@bat95] with the surface parameters given in Table \[dtpar\] (see also text).  Positive level shift corresponds to repulsive interaction.[]{data-label="gam_sh_rys"}](fig6_sncd_isoef.eps){width="80.00000%"}

![ Normalized neutron to proton density ratio ($Z/N \cdot \rho_n / \rho_p$) deduced from strong-interaction level widths and shifts (solid lines with indicated statistical errors) and charge distributions given in Ref.  [@vri87] (Sn nuclei) and Ref.  [@fri95] (other nuclei).  They are compared with $f_{halo}$ measured in the radiochemical experiments (marked with crosses at a radial distance corresponding to the most probable annihilation site) and with HFB model calculations (dashed lines).[]{data-label="rorat_prc"}](fig7_ro_rat.eps){width="80.00000%"}

![ Difference $\Delta r_{np}$ between the rms radii of the neutron and proton distribution as deduced from the antiprotonic atom x-ray data, as a function of $\delta = (N - Z) / A$.  The full line is the same as in Fig.  4 of Ref.  [@trz01].[]{data-label="drdelta"}](fig8_sncd_dr_vs_delta.eps){width="70.00000%"}

[^1]: Present addres: N. Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
