Scattering of pions from
90Zr and
118Sn around the D resonance
M. AHMED(1), MD. A. RAHMAN(2) and H. M. SENGUPTA(2)
(1) Department of Physics, Shahjalal University of Science and Technology - Sylhet, Bangladesh
(2) Department of Physics, University of Dhaka - Dhaka, Bangladesh
(ricevuto il 17 Aprile 1997; revisionato il 19 Settembre 1997; approvato il 4 Febbraio 1998)
Summary. — The generalized diffraction model of Frahn and Venter is used to study the elastic and inelastic scattering of pions from 90Zr and 118Sn around the D
resonance. The interaction radius R, surface diffuseness d and the deformation parameter, leading to the lowest 21and 32states, are obtained.
PACS 25.80.Dj – Pion elastic scattering. PACS 25.80.Ek – Pion inelastic scattering. PACS 25.80.Gn – Pion charge-excharge reactions.
1. – Introduction
The pion-nucleus scattering around 200 MeV is well known to be dominated by a strong P-wave J 43O2, T43O2, D(1232) pion-nucleon resonance. This is termed as the D( 3 , 3 ) resonance of simply the ( 3 , 3 ) resonance. At the resonance, p1interacts much more strongly with protons than with neutrons. Likewise, the fundamental p2n interaction is much stronger than the p2p interaction. The p1 and p2 at energies around the D resonance can therefore be used to selectively probe, respectively, the density distributions of protons and neutrons in a nucleus.
The scattering of pions was studied by some of the present authors [1] around the D resonance from a number of nuclei within the framework of the generalized diffraction model of Frahn and Venter [2], also called the strong absorption model (abbreviated as SAM). The success of the model prompted us to undertake the present work on the SAM studies of the elastic and inelastic scattering of 163 MeV p6from 90
Zr and 118Sn. The two nuclei are so chosen that one has a closed neutron shell and the other has a closed proton shell in an attempt to see how far these effects are manifested in the elastic scattering, as well as inelastic scattering to collective states in view of the differential behavious of p6towards nucleons around the resonance. The experimental data are taken from Ullmann et al. [3].
2. – SAM analysis
The SAM is a simple geometrical model; the basic assumptions of the model are that only a narrow range of incident partial waves that are grazing the nuclear surface
contributions to the scattering process and that the size of the incident particle is much smaller than the nuclear size, i.e. the interaction is highly localized at the nuclear surface. The model starts with a direct parametrization of the scattering function hl as
follows:
hlexp [22isl] 4g(l)1im dg(l)Odl ,
(1)
where slis the Coulomb phase shift for the l-th partial wave and g(l) is a continuous
monotonic function of angular momentum l (4l11O2). A convenient form of g(l) is the Woods-Saxon form as given by
g(l) 4 ]11exp [ (L2l)OD](21. (2)
The function g(l) is thus characterized by a cut-off or critical angular momentum L , a diffuseness parameter D , with the requirement that its first derivative has a simple Fourier transform. The parameter m , more accurately mO4D, is a measure of the real nuclear phase shift. The interaction radius R and the surface diffuseness d are related to L and D , respectively, through the well-known expressions given below:
L 4kR(122hOkR)1 O2 (3)
and
D 4kd(12hOkR)(122 hOkR)21 O2, (4)
where h and k are, respectively, the Coulomb parameter and wave number.
The cross-section for elastic scattering under suitable approximations has been obtained in closed form (ref. [2]) in terms of the adjustable parameters L , D and m .
The formalism developed for the elastic scattering can be easily extended to include inelastic scattering as well. The strong absorption form of hl has been introduced by
Potgieter and Frahn [4] in the generalized scattering amplitude of Austern and Blair [5]. This leads again to a closed expression for inelastic cross-section for various multipole modes of excitation [4].
The amplitude for single excitation to a collective level for inelastic scattering of multipole L is given by fLM(u) 4 i 2( 2 L 11) 1 O2C L
!
l , l 8 il 2l 8[ ( 2 l 811)1 O2exp [s l1 sl 8] Q (5) Q ¯hlO¯lal 8 L00 Nl0 b al 8 L , 2MMNl0 b Yl 8M(u , 0 ) .The hl is as given in eq. (1) and other symbols are defined in ref. [4]. Clearly,
because of surface interaction, ¯hlO ¯l is confined to a narrow range around the cut-off
value l0 (L 4l01 1 O2 ). Using SAM conditions, closed expression are obtained for the differential cross-section, as given by
s (u) 4N fLM(u) N24 d2L(LO16p)(uOsin u)3 ](H21 H2 12)
!
M [a2LM(u) 1b2LM(u) ] Q Q [J2 NMN(Lu)1J 2 NMN 2 1(Lu) ] 12H2H1
!
M [a2LM(u) 2b2IM) u][JNMN2 (Lu) 2J 2
where
H64 [ 1 1 m
(
(uC6 u) ] F[D(uC6 u) ] , uC4 Coulomb scattering angle ,F(Du) 4
2Q Q
¯g(l) O¯l exp [2i(l2L) ] 4
(
pDu) Osinh (pDu))
,aLM(u) 1ibLM(u) 4S
!
k 42LL
ll 2Lexp [iluO2] al0L00 N(l01 l), 0 b al0L , 2MM(l01 l), ub
and dLis the deformation length (bLR).
Clearly the deformation length is the only free parameter and is given from the normalization. The ingradients are L , D and m and these are all fixed from the elastic scattering.
Further explicit forms of s(u) are given in refs. [4] and [6] for, respectively, quadrupole and octupole excitations.
3. – Results and discussion
Results of the SAM analyses are summarized in table I and fits to the angular distributions for elastic scattering of p6on90Zr and118Sn are shown in fig. 1.
The parameters are uniquely given from the analyses and no combination of the parameters L , D and m other than the ones shown in the table could be obtained that gave another minimum in the x2value. Uncertainties in the values of L and D are less than 5% and those in m are about 20%. The positions of maxima and minima in the angular distributions are governed by the grazing angular momentum L , while diffuseness parameter determines the overall slope of the angular distribution. The effect of the variation of m is felt only on the magnitudes of the cross-sections at the minima, little affecting the cross-sections elsewhere.
The interaction radius R and surface diffuseness d are obtained from relations (3) and (4), respectively. These are shown in table I. The radii of both the nuclei given from the elastic scattering of p2 are consistently larger than those given from the p1 scattering. This feature can be related to the selective scanning of neutron and proton distributions in a nucleus, respectively, by p2and p1around the D resonance, as also previously observed from a comparison of the elastic scattering of pions from two Ca isotopes, namely40Ca and48Ca (ref. [7]).
Angular distributions of pions inelastically scattered from 90Zr and 118Sn leading to the lowest 21 and 32 states in each nucleus [3] were analysed using the parameters derived from the corresponding elastic scattering (table I). Fits to the measured angular distributions are shown in figs. 2 and 3, respectively, from90Zr and118Sn. The quadrupole and octupole deformation lengths thus extracted are summarized in table II. The deformation lengths derived from p1 and p2 inelastic scattering at energies around the D resonance should correspond, respectively, to (bLR)p and (bLR)n. Also included in the table are those values given from the DWIA analyses at 163 MeV and 130 MeV [ 3 , 8 ]. The former data (ref. [3]) were in fact used in the present investigation within the framework of the generalized diffraction model [ 2 , 4 ]. There is a reasonably good agreement in the bLR values amongst different studies (table II).
Fig. 1. – The SAM analyses of the elastic scattering of 163 MeV pions for90Zr and118Sn.
The hydrodynamic model assumes equal neutron and proton collective vibrations in nuclei and accordingly one would expect (bLR)p4 (bLR)n. On the other hand, according to the schematic model [9], in closed proton shell nuclei the neutron vibration should dominate over the proton vibrations as the motion of protons will be locked and in closed neutron shell nuclei the vibrations are primarily due to protons. We would accordingly expect (bLR)nO(bLR)pE 1 for collective state in 90Zr and
TABLEI. – The SAM parameters giving the best fit to elastic scattering.
Nucleus L D m mO4D x2 ON R (fm) d (fm) p1-90Zr p1-118Sn p2-90Zr p2-118Sn 8.0 8.5 8.5 9.4 0.70 0.70 0.60 0.70 0.60 0.50 0.20 0.20 0.21 0.18 0.08 0.07 7.6 15.1 3.1 4.2 6.03 6.42 6.18 6.81 0.52 0.52 0.44 0.52
Fig. 2. – The SAM analyses of the inelastic scattering of 163 MeV pions leading to the 211and 312
states in90Zr.
(bLR)nO(bLR)pD 1 for118Sn. The extreme shell closure effect will of course be to some extent washed out by the core polarisation.
In agreement with the schematic model we observe (bLR)nE (bLR)p for both the collective states in90Zr, while for118Sn the neutron and proton deformation lengths are about the same, being in agreement to the hydrodynamic model. The deviation from the schematic model in the latter case should not be interpreted as the inadequacy of pions as probes for selective scanning of proton and neutron density distributions in nuclei, nor as shortfalls of the SAM or DWIA, but is a consequence of almost “in phase”, vibration of protons and neutrons in 118Sn, as also observed recently by Satchler [10]. This is understood as arising from the fact that the Sn isotopes are some of the best examples of spherical nuclei in the periodic table of elements. A deviation from the spherical symmetry shows up in different values (even if slightly) in the neutron and proton density distributions, hence in deformation parameters.
The experimental data analysed in the present work are taken from the measurements made at the LAMPF under exactly identical conditions and this would provide a consistent means of studying the isoscalar and isovector amplitudes, (bLR)0 and (bLR)1, respectively, for the transitions. These are related to the deformation lengths (bLR)p and (bLR)n through the well-known expressions given below (ref. [3],
Fig. 3. – As in fig. 2, but for118Sn.
TABLEII. – Summary of the inelastic scattering of pions. Nu-cleus Ex (MeV) Jp ( b LR)p(fm) ( bLR)n(fm) ( bLR)nO( bLR)p (a) (b) (c) (a) (b) (c) (a) (b) (c) 90Zr 2.18 21 1 0 .73 6 0.07 0 .66 6 0.05 0 .62 6 0.06 0 .50 6 0.05 0 .85 6 0.12 0 .75 6 0.10 118Sn 1.23 21 1 0 .81 6 0.08 0 .83 6 0.10 0 .82 6 0.08 0 .78 6 0.08 0 .79 6 0.08 0 .88 6 0.08 0 .96 6 0.14 0 .95 6 0.15 1 .07 6 0.14 90Zr 2.75 32 1 1 .06 6 0.11 1 .21 6 0.12 0 .96 6 0.10 0 .84 6 0.06 0 .91 60.14 0 .69 60.08 118Sn 2.33 32 1 1 .03 6 0.10 1 .14 6 0.14 0 .98 6 0.10 0 .98 6 0.10 0 .98 6 0.10 1 .02 6 0.09 0 .95 6 0.13 0 .86 6 0.14 1 .04 6 0.13 (a) From SAM, Ep4 163 MeV (present work).
(b) From DWIA, Ep4 163 MeV (Ullmann et al. [3]). (c) From DWIA, Ep4 130 MeV (Ullmann et al. [9]).
TABLEIII. – Isoscalar and isovector deformation lengths. Nu-cleus Ex (MeV) Jp ( b LR)0values (fm) ( bLR)1values (fm) (a) (b) (c) (a , a 8) (a) (b) (c) 90Zr 2.18 21 1 0 .67 6 0.04 0 .57 6 0.04 0 .42 6 0.06 (d) 0.38 (e) 0 .18 6 0.11 0 .14 6 0.08 118Sn 1.23 21 1 0 .79 6 0.06 0 .81 6 0.04 0 .86 6 0.08 0.65 (f) 0 .69 6 0.02 (g) 0 .70 6 0.20 0 .68 6 0.22 1 .05 6 0.61 90Zr 2.75 32 1 1 .00 6 0.07 1 .01 6 0.06 0 .77 6 12 (d) 0.64 (e) 0 .56 6 0.24 0 .61 6 0.33 118Sn 2.33 32 1 1 .00 6 0.07 1 .05 6 0.05 1 .00 6 0.66 0.73 (f) 0 .84 6 0.26 0 .53 6 0.16 1 .13 6 0.66 (a) From SAM, Ep4 163 MeV (present work).
(b) From DWIA, Ep4 163 MeV (Ullmann et al. [3]).
(c) From DWIA, Ep4 130 MeV (Ullmann et al. [9]). (d) Rozsa et al. [11].
(e) Martens and Bernstein [12]. ( f ) Bartand et al. [13]. (g) Finlay et al. [14]. for example): A( bR)04 [N( bR)n1 Z(bR)p] (6) and (N 2Z)( bR)14 [N( bR)n2 Z(bR)p] . (7)
These quantities are shown in table III, as given from the present work (SAM), as well from previous works (DWIA). Also included in the table are the (bLR)0values as obtained from the inelastic scattering of alpha-particles [10-13]. The alpha-nucleus interaction, like the pion-nucleus interaction at the D resonance, is also strongly localized at the nuclear surface and furthermore the inelastic scattering of alpha-particles excites only the isoscalar mode of vibration.
The deformation parameters as obtained from the inelastic scattering of pions are usually found to be somewhat larger than those given from other probes (Rahman et
al. [1] and references therein); it is not clear why. This feature is also noticed in the
isoscalar deformation length, which is larger than that from alpha-particles (table III). The (bLR)0values for the 32states are larger than those of the 21states in both the nuclei. This is consistent with the larger collectivity of the 32states than the 21states. It is of significance that (bLR)1values for the 211states in90Zr are small, being slightly positive or slightly negative in the two different models (SAM and DWIA), the difference between values is certainly within the limits of error.
4. – Conclusion
The usefulness of pion as a probe is again established from the studies of interaction of p6 on 90
Zr and 118Sn. The interaction radius given from p2 is consistently larger than that from the elastic scattering of p1 on the same nucleus. The dominant “proton-like behaviour” is observed in90Zr while almost “in phase” vibrational is noted in118Sn. Finally, the SAM with only three parameters is indeed a useful model for pions around the D resonance.
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