Neutron scattering from elemental silver

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 620 (1997) 249-264

Neutron scattering from elemental silver* A.B. Smith 1 Argonne National Laboratory, Argonne, IL 60439, USA University of Arizona, Tucson, AZ 85721, USA

Received 24 December 1996; revised 19 March 1997

Abstract Differential neutron elastic-scattering cross sections of elemental silver were measured from 1.5 ~ 10 MeV at scattering angles distributed between ~17 ° and 160°. Cross sections for ten inelastically scattered neutron groups were determined, corresponding to observed excitations of 3284-13,419-t-50, 7484-25, 9084-26, 11504. 38, 12864-25, 1507-t-20, 1623-4-30, 1835 + 20, and 1944-4-26 keV, all of which were composites of contributions from the two isotopes 107Agand t°gAg. The experimental results were interpreted in terms of the spherical-optical and rotational and vibrational coupled-channels models. ~) 1997 Elsevier Science B.V. PACS: -24.10.-i; 24.50.+g Keywords: Measured silver neutron do'/d/-/el and dtr/dOinel 1.5 ---* 10 MeV; Optical-statistical and

coupled-channels models; Model systematics

1. Introduction Elemental silver consists of the transitional nuclei l°TAg (51.8%) and l°9Ag (48.2%). Both of these isotopes have essentially the same low-energy structure characterized by a I / 2 - ground state (g.s.), followed by a ( 7 / 2 +, 9 / 2 +) doublet at ~ 1 1 0 keV, and a ( 3 / 2 - , 5 / 2 - ) doublet at ~ 3 6 0 keV [ 1]. Heavy-ion studies have attribute this structure to a quasi-particle configuration coupled to a slightly deformed (e.g. r 2 ,~ 0.12) eveneven prolate rotational core, with a variable moment of inertia and including coriolis perturbations [2]. This model leads to a negative-parity rotational band based upon the * This work is supported by the United States Department of Energy under contract W-31-109-Eng-38,and by the Nuclear and Energy Engineering Program at The University of Arizona. l E-mail: [email protected]. 0375-9474/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved. PH S0375-9474(97)00149-8

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A.B. Smith~Nuclear Physics A 620 (1997) 249-264

1 / 2 - g.s., then a positive-parity zlI = 1 band based upon the 9/2 + level at about 125 keV, and accounts for the "anomalous" yrast 7/2 + state. Alternate interpretations based upon triaxial models give similar results but involve larger quadrupole deformations (e.g. f12 0.2) [3,4]. At higher excitation energies high-spin rotational bands in the silver isotopes have been suggested [3-6], with very large, or "super", deformations similar to those reported in the neighboring palladium isotopes [ 7-9]. Neutron scattering at ~<10 MeV is sensitive to the collective properties of low-energy excitations. Unfortunately, there is very little experimental knowledge of neutron scattering from silver above ~1.5 MeV. The isotopes of silver lie near the peak of the fission-product yield curve thus their neutronic properties are of interest from the point of view of the fission-reactor fuel cycle. Since there are low-lying excited states in both isotopes with small spin differences in the scattering process, the inelastic-scattering cross sections should be large at the low energies of fission-reactor interest. The above considerations motivated the present work directed toward the experimental definition of neutron scattering from silver in the low-MeV range and its interpretation. Section 2 very briefly outlines the experimental methods, Section 3 presents the experimental results, Section 4 summarizes extensive model interpretations using spherical-optical and coupled-channels (rotational and vibrational) models, and Section 5 discusses and summarizes the results. A detailed description of this work is given in the laboratory report of Ref. [ 10].

2. Experimental methods All of the measurements were made using the fast-neutron time-of-flight method [ 11 ] and the Argonne ten-angle detection system. This method and apparatus have been amply described elsewhere [ 12] and therefore only details specific to the present measurements are outlined here. The measurement sample was a cylinder of high-purity metallic elemental silver 2 cm in diameter and 2 cm long. Below 4 MeV the 7Li(p,n)7Be neutron-source reaction was employed with incident-neutron energy spreads of 50 to 100 keV, and above 4 MeV the D(d,n)3He source reaction was used with energy spreads of ,.~300 keV at 4 MeV, decreasing to ~100 keV at 10 MeV [ 13]. The mean neutron energy was determined to within ~10 keV by control of the incident ion beam. The neutron sources were pulsed at a 2 MHz repetition rate with a burst duration of ,.~1 ns. Then ~5 m long collimated scattered-neutron flight paths were arranged about the sample with liquid-scintillation detectors placed at their ends. Below 4 MeV the silver cross sections were determined relative to the carbon total cross section using the method described in Ref. [ 14], and at higher energies relative to the H(n,n) scattering standard 15. All of the silver cross sections were corrected for beam-attenuation, multiple-event and angular-resolution effects using Monte Carlo techniques [ 16].

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

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8 (deg.) Fig. I. Differential elastic-scattering cross sections of elemental silver. The present experimental values are indicated by symbols and the results of Legendre-polynomial fitting by curves. Approximate incident energies are numerically noted in MeV. All of the measured values include some inelastic-scattering component as defined in the text. Throughout this paper differential data are illustrated in the laboratory coordinate system.

3. Experimental results 3.1. Elastic scattering The elastic-scattering measurements were made at incident-energy intervals of ,~100 keV from ,~1.5 to 3 MeV, at ~200 keV intervals from 3 to 4 MeV, and at ~500 intervals from 4 to 10 MeV. The experimental angular range extended from ~17 ° to 160 ° with measurements made at <40 angular intervals. The incident-energy spreads and the scattered-neutron resolutions were arranged so that the "elastic" scattering included contributions due to the inelastic excitation of the yrast 7/2 + and 9/2 + levels (i.e. included inelastic contributions resulting from excitation energies of <135 keV) up to incident energies of 4 MeV, and included inelastic contributions from the yrast 7/2 +, 9/2 +, 3 / 2 - and 5 / 2 - levels (i.e., a scattered-neutron resolution of ~450 keV) from 4 to 10 MeV. The estimated uncertainties associated with the differential values ranged from ~ 5 % below 4 MeV and 3-4% from 4 to 10 MeV, to larger amounts at the minima of the distributions. The experimental results are illustrated in Fig. 1. There is very little elemental silver neutron elastic-scattering information reported in the literature comparable with the present measurements. The ~< 4 MeV l°TAg elasticscattering results of Ref. [ 17] are consistent with the present elemental values.

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3.2. Inelastic scattering

At incident energies of ~< 4 MeV inelastic-scattering measurements were made concurrently with the elastic-scattering studies. Given the incident energy spreads, the relatively broad scattered-neutron resolutions and the complexity of the level structures of the contributing l°7Ag and l°9Ag isotopes, it was not possible to resolve neutron scattering from any explicit level in either isotope. Rather, ten scattered-neutron groups corresponding to apparent excitations of 328 4- 13,419 + 50, 748 + 25, 908 + 26, 1150 + 38, 1286 + 25, 1507 + 20, 1623 d: 30, 1835 4- 20 and 1944 + 26 keV were observed. The cited energy uncertainties are rms deviations of a number of measurements from the means and do not necessarily imply the distribution of individual excitation energies. In addition, a few high-resolution measurements indicated quite small cross sections for the excitation of levels at ~,110 keV. Most of the observed neutron groups can be reasonably associated with the reported levels in l°7Ag and l°9Ag as illustrated in Table 1. The corresponding angle-integrated cross sections were determined by fitting Legendre-polynomial expansions to the experimental values. The angle-integrated results were corrected for perturbations due to the second neutron group from the 7Li(p,n)TBe source reaction where appropriate. Similar corrections to the differential angular distributions were not attempted as at some excitation and incident energies the forward-angle perturbation from the source reaction was the dominant contribution to the observed neutron scattering. The uncertainties associated with the angle-integrated results are quite large (e.g. 10% to 30%), primarily due to correction procedures and statistics. The angle-integrated inelastic-scattering results are summarized in Fig. 2. They are reasonably consistent with the l°7Ag inelastic-scattering results of Ref. [ 17].

4. Model interpretations The model interpretations emphasized elastic-scattering data, essentially all of which come from the Argonne program. From 0.3 to 1.5 MeV the results of Vonach and Smith [ 18] form the lower-energy portion of the data base. From 1.5 to 4 MeV the present work was used, supplemented with the 1°TAg results of Smith et al. [ 17]. Below 4 MeV the measured values were averaged over ~200 keV in order to smooth any residual fluctuations and to reduce the size of the data base to more manageable proportions. The elemental and isotopic data were treated independently. From 4 to 10 MeV reliance was placed entirely on the present results. The literature contains very little silver elastic-scattering data at energies above ~1.5 MeV and none above 10 MeV. The elemental elastic-scattering data base used in the modeling is illustrated in Fig. 3. Consideration was given to the elemental neutron total cross sections taken from Refs. [ 19-22], and averaged over ~100 keV to smooth fluctuations. Comparisons were made with the present inelastic-scattering results and those of Refs. [17,18]. Consideration was also given to the strength functions taken from Ref. [ 23 ]. Throughout the interpretations the scattering real potential was assumed to have

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

253

Table 1 Observed excitation energies in keV. Isotopic components taken from Ref. [ 1] Ex (observed)

Contributing components lOTAg

0

328 4- 13 419 4- 50 748 4- 25

908 + 26

11504-38

1286 4- 25

15074-20

1623 4- 30

18354-20 19444-26

0 93 126 325 423 773 787

(1/2-) (7/2 + ) (9/2 + ) (3/2-) (5/2-) (11/2 + ) (3/2-)

922 (5/2 +) 95O ( 5 / 2 - ) 973 ( 7 / 2 - ) 991 (13/2 + ) 1061 (?) 1142 (1/2 + ) 1143 ( 5 / 2 - ) 1147 ( 7 / 2 - ) 1223 (5/2 + ) 1259 (3/2 + ) 1326 (3/2 + ) 1449 (?) 1465 ( 3 / 2 - ) 1481 (7) 1508 (?) 1570 (?) 1577 (15/2 + ) 1613 ( 1 / 2 - ) 1651 ( 1 / 2 - ) 1656 (?) 1685 (?) many many

lO9Ag 0 (1/2-) 88 (7/2 + ) 133 (9/2 + ) 311 ( 3 / 2 - ) 415 ( 5 / 2 - ) 697 (?) 702(3/2-) 707 (?) 724 (3/2 + ) 735 (5/2 +) 789 (?) 811 (?) 863 (5/2-) 869 (5/2 + ) 911 (7/2 + ) 912 ( 7 / 2 - ) 1092 (?) 1098 (5/2 + )

1200(?) t260 ( 1 / 2 - ) 1324 ( 3 / 2 - ) 1430 ( I / 2 + ) 1500(3/2-) 1524 ( 3 / 2 - )

1570 1589 1613 1658 1675

(?) (?) (1/2-) (1/2 + ) (?)

many many

the Saxon-Woods (SW) form, the surface-absorption imaginary potential the SWderivative form, and the spin-orbit potential (assumed real and non-deformed) the Thomas form [ 24]. Throughout the calculations the parameters of the spin-orbit potential given by Walter and Guss [ 25 ] were used. Where volume-absorption was considered it was assumed to have the SW form with the geometric parameters of the real potential. In the context of the present neutron scattering, the spherical optical model (SOM) is not a particularly attractive assumption as it does not account for the direct inelasticscattering processes that are very significant in the observed "elastic" and inelastic scattering above several MeV. Despite shortcomings, a silver SOM is of some interest

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

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E,,(MeV) Fig. 2. Angle-integrated inelastic-scattering cross sections of elemental silver as defined in the text. Symbols indicate experimental values where; i) solid circles are from the present work, ii) open circles from Ref. [ 181, and iii) crosses from the l°TAg results of Ref. [17]. Curves indicate; "S" SOM CN results, "R" ROTM direct-reaction results, "T" the total inelastic cross sections combiningCN and direct results, and "(3" are simple eyeguides. Observed excitation energies are numerically noted in keV (in some cases sums of contributions are given). in the provision of data for applications, and it can be used as a basis for D W B A calculations in more basic studies. The SOM was primarily obtained by chi-square fitting the elastic-scattering data base. The calculational tool was a version of the spherical opticalmodel code A B A R E X [26] that explicitly treated both the 1°TAg and l°9Ag isotopes of the element. Compound-nucleus processes were calculated using the Hauser-Feshbach formula [27] as modified by Moldauer to account for resonance width fluctuation and correlation effects [ 28]. The calculations employed twelve discrete excitations for l°TAg and thirteen for l°9Ag with the energies, spins and parities given in the Nuclear Data Sheets [ 1]. These discrete excitations extended to ~1 MeV. Higher-energy excitations were calculated using the statistical representation of Gilbert and Cameron [29]. At incident energies > 1.1 MeV the calculations combined the elastic and inelastic contributions in a manner consistent with the above-cited resolutions of the experimental data used in the data base (i.e. combining the elastic contribution with the first two or four inelastic components before evaluating chi-square). The fitting procedures followed the five-step

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8 (deg.) Fig. 3. Comparison of the elemental elastic-scatteringdata base with the results of the SOM calculations. Experimentalvalues are indicatedby symbolsand curvesdenotethe results of SOM calculations.Approximate incident energies are numericallynoted in MeV. regime long used at this laboratory [ 12]; i) first fixing the real-potential diffuseness, a~,, using six-parameter fitting varying real and imaginary potential strengths, radii and diffusenesses, ii) then five-parameter fitting to determine the real-potential radius, rv, iii) four-parameter fitting to fix the imaginary-potential radius, rw, iv) three-parameter fitting to obtain the imaginary-potential diffuseness, aw, and finally v) two-parameter fitting to determine real, Jr, and imaginary, J~, potential strengths (herein potential strengths Ji are presented as volume-integrals-per-nucleon and radii ri in the reduced form where the full radius R i = ri A l l 3 , unless otherwise explicitly stated). Following the above fitting regime, the SOM parameters of Table 2 were obtained. Model-parameters are given to sufficient precisions to permit accurate reproduction of the calculated results. These precisions do not necessarily imply accuracies which are generally approximately three significant figures. The SOM parameters provide the description of the elemental elastic-scattering data base illustrated in Fig. 3. A similar description of the 1°TAg elastic-scattering below 4 MeV was obtained. At lower energies, the SOM represents the compound-nucleus (CN) inelastic scattering as illustrated in Fig. 2. The measured total cross sections are compared with the SOM values in Fig. 4, and comparisons of SOM strength functions with those deduced from measurements are given in Table 3. The experimental data of the present work do not extend beyond lO MeV and into a region where volume absorption could be a potential concern [30],

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En(MeV) Fig. 4. Elemental silver total cross sections. Averages of the experimental values are indicated by symbols, referenced as defined in the text, and curves the results of SOM ("S") and ROTM ("R") calculations. Table 2 SOM parameters. Strengths (Ji) are given in terms of volume-integrals-per-nucleon (MeVfm 3) (except for the spin-orbit potential where strength is given in MeV), energy (E) is in MeV and geometrical dimensions are in fermis Real potential

Jv = 439.8 - 2.869E rv = 1.2710 av = 0.6029 Jw = 112.1 - 11.95E + 0.5759E 2 rw = 1.3215 aw = 0.4570 Vso = 6.027 - 0.015 E rso = 1.103 aso = 0.56

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Spin-orbit potential

Table 3 Measured and calculated strength functions of 1°TAg in units of 10 -4 Source

So

S~

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0.38 4- 0.07 1.09 0.79 0.55

3.8 4- 0.6 4.7 2.7 2.5

t h u s it is n o t s u r p r i s i n g t h a t t h e i n t r o d u c t i o n o f v o l u m e a b s o r p t i o n i n t o t h e S O M fitting p r o c e d u r e d i d n o t s i g n i f i c a n t l y i m p r o v e t h e d e s c r i p t i o n o f t h e m e a s u r e d values. C o m p r e h e n s i v e m u l t i - i s o t o p i c fitting u s i n g the c o u p l e d - c h a n n e l s m o d e l ( C C M )

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the m a n n e r e m p l o y e d f o r t h e S O M , is f o r b i d d i n g l y t i m e c o n s u m i n g . T h e r e f o r e , s o m e s i m p l i f y i n g a s s u m p t i o n s w e r e m a d e . It w a s a s s u m e d t h a t t h e target w a s a n e v e n n u c l e u s w i t h m a s s A = 108 a n d a g.s. J ~ -- 0 +. T h e first t w o e x c i t e d states in e i t h e r i s o t o p e

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

257

are of positive parity, have relatively large spin, and are weakly excited by neutron scattering, thus they were ignored. The third and fourth excited states are 3 / 2 - and 5 / 2 - levels in each isotope, are quite strongly excited by inelastic neutron scattering, and the excitations persist to relatively high energies where compound-nucleus (CN) contributions must be small. Thus the excitation of these two states has a significant direction-reaction component. It was assumed that this could be represented by the excitation of a 2 + level at 369 keV in the pseudo A = 108 target. Higher-lying discrete levels were assumed to be those of 1°gAg up to excitations of ,~1.0 MeV. This is a rather crude approximation but the corresponding excited levels of l°7Ag are very similar, and discrete CN excitations in the relatively narrow energy band of ,~0.5 to 1.0 MeV have a minor influence on the model determinations. Higher-lying excitations were statistically represented using the formalism of Ref. [29] as in the SOM derivation. Thus the CCM deals with a simple even-even nucleus with a g.s. of 0 +, a yrast collective 2 + excited state at 369 keV, and CN excitations extending to higher energies. Implicit in this is the assumption that the collective nature of the neutron interaction with l°7Ag and l°9Ag is essentially the same. This is reasonable in view of the similarity of the spectroscopy of all the odd silver isotopes [ 1 ] and the coulomb-excitation results of Ref. [ 31 ]. The model is an example of "the fictitious even-nucleus approximation" of Lagrange [32]. Heavy-ion studies suggest that the collective motion of the g.s. band should be rotational. On the other hand, proton-scattering studies indicate a vibrational interaction, and a neutron study has used a vibrational assumption [32]. Thus attention was given to both the rotational model (ROTM) and an alternative vibrational model (VIBM). The CCM fitting followed the five-step procedure of the SOM, using the coupledchannels code ECIS95 [33]. CN processes were treated in the same manner as for the SOM. Again, elastic and inelastic components were combined in the calculations to be consistent with the experimental resolutions. The entire ROTM fitting procedure was repeated for [32 values of 0.10, 0.15, 0.20, and 0.25. From comparisons of the calculated and measured elastic-scattering, inelastic-scattering and total cross sections (as discussed in Section 5) it was evident that/32 is ~0.2 d: 10%. The resulting ROTM potential parameters are given in Table 4. The ROTM and measured elastic-scattering cross sections are compared in Fig. 5, and comparisons of measured and calculated total cross sections are given in Fig. 4. ROTM strength functions are given in Table 3, and the calculate direct inelastic scattering is shown in Fig. 2. An analogous approach was used in determining the parameters of the VIBM model. A simple one-phonon vibrator was assumed and f12 limited to the 0.20 value of the ROTM. The resulting potential parameters are given in Table 5. The elastic and inelastic scattering and total cross sections calculated with the VIBM were essentially identical to those obtained with the ROTM. Strength functions calculated with the VIBM are given in Table 3.

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8 {deg.) Fig. 5. C o m p a r i s o n o f the elemental elastic-scattering d a t a base with the results o f R O T M calculations. The n o m e n c l a t u r e is identical to that o f Fig. 3. Table 4 R O T M p a r a m e t e r s obtained with the rotational a s s u m p t i o n a n d a /32 = 0.20, as described in the text. The n o m e n c l a t u r e is the s a m e as in Table 2 Real potential

I m a g i n a r y potential

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•Iv = 4 7 7 . 0 - 4 . 6 1 4 E rv = 1.2945 av = 0 . 6 6 9 0 Jw = 82.45 - 6 . 6 4 1 8 E + 0 . 3 9 2 6 E 2 rw = i . 4 5 9 4 - 0 . 0 1 2 6 4 E aw = 0 . 0 9 2 7 7 + 0 . 0 4 8 2 4 E ( s a m e as in Table 2)

Table 5 V1BM p a r a m e t e r s obtained with the vibrational a s s u m p t i o n a n d a / 3 2 = 0.20, as described in the text. The n o m e n c l a t u r e is the s a m e as in Table 2 Real potential

I m a g i n a r y potential

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Jv = 4 5 8 . 8 - 2 . 9 6 6 E rv = 1.3059 av = 6 . 6 9 3 8 Jw = 101.8 - 1 4 . 2 2 7 E + 1 . 0 0 6 1 E 2 rw = 1.4868 -- 0 . 0 1 5 1 1 E aw = 0.10311 + 0 . 0 8 5 8 8 2 E - 0 . 0 0 4 2 0 1 8 E 2 ( s a m e as in Table 2)

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

259

5. Discussion and summary

The SOM only crudely approximates the neutron interaction with silver. In some energy regions (e.g. about ~5.5 MeV) the calculated elastic scattering differs from the measured values by more than an order of magnitude (Fig. 3). Such discrepancies can not be attributed to the inelastic excitation of the first several states in the two contributing isotopes as that contribution was explicitly treated in the calculations, as cited above. At other energies the description is reasonably good. Furthermore, the fitting procedure behaved erratically yielding different parameters from low- and highenergy data where the direct inelastic component in the observed elastic distributions is different. The ROTM and VIBM calculations with /32 = 0.2 generally quite nicely describe the elastic-scattering (e.g. Fig. 5). Fitting with values of/32 less than 0.2 led to results progressively approaching those of the less suitable SOM as the/32 magnitude was reduced. The CCM models ignored inelastic-scattering contributions due to the 7/2 + and 9/2 + levels. These contributions are not large, as illustrated in Fig. 2. However, they are significant in the minima of the distributions below ~4 MeV, and have the largest effect near ~ 1.5 MeV where they are approximately twice the experimental uncertainty. Thus it is not surprising that the observed "elastic" scattering at such energies and angles tends to be larger than calculated with the CCMs. The SOM inelastic-scattering cross sections (Fig. 2) are in reasonable agreement with the observed values where the latter are primarily due to CN processes. Essentially the same CN results were obtained with the ROTM and VIBM as differences due to the use of transmission coefficients calculated with spherical or deformed potentials are small. However, the ROTM also provides an estimate of the direct inelastic excitation of the 328 ( 3 / 2 - ) and 419 ( 5 / 2 - ) keV levels which is in reasonable agreement with the measured values (as illustrated in Fig. 2 where the calculated 2 + results of the model are apportioned by 2J + 1). Results obtained fitting with/32 = 0.15, and = 0.10 gave progressively less suitable descriptions of the measured inelastic scattering. Those obtained with/32 = 0.25 also provided a reasonable description of the observed elastic and inelastic scattering, but not of the total cross section, as noted below. The quadrupole deformation,/32, of l°TAg and l°9Ag has been reported from coulomb-excitations studies of the yrast 3 / 2 - and 5 / 2 - levels [ 31 ] with results that are consistent with the/32 = 0.20 of the present work. The average of/32 values for neighboring palladium and cadmium targets is ~0.208, very close to that obtained in this work. Proton-scattering studies, assuming vibrational collective processes, resulted in a/32 for l°TAg approximately 10% larger than (and thus reasonably consistent with) the present value of/32 = 0.2 [34]. Some of the interpretations of spectroscopic heavy-ion studies based upon the assumption of a soft prolate rotor coupled with a quasi particle employ smaller /32 values of ,,~0.12 [2,5,35], while triaxial models have larger /32 values of ~0.20 ~ 0.23 [3,4]. The present neutron-scattering interpretations support the larger/32 values. Total cross sections calculated with the SOM and ROTM are compared with energy averages of the measured values in Fig. 4. The results obtained with the two models are essentially identical (and the ROTM and VIBM results are very similar), thus

260

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decreasing f12 from ~0.2 has little effect on trt. Increasing f12 to ~0.25 results in a markedly "flatter" and less suitable energy distribution of the total cross section. Both the SOM and ROTM give results within several percent of the measured values from ~1.5 ---, 7 MeV. At 10 MeV both results are higher then the measured quantities by ,~8%, and both are slightly lower below ~1.5 MeV. The energy-averaged measured total cross sections are probably systematically uncertain by .~2 ~ 3%, and the normalization of the present scattering results at 10 MeV is likely uncertain by .~3 --~ 4%, so the discrepancy between measured and calculated total cross sections is slightly larger than that due to the combined experimental uncertainties. Generally, global SOM's tend to give results 4 ~ 5% smaller than the average of the measured total cross sections at 10 MeV (e.g. Refs. [25,30,36] ). The s- and p-wave strength functions calculated with the present models are compared with those deduced from experimental measurements in Table 3. All of the calculated s-wave values are larger than the experimentally deduced quantities, the most so for the SOM. This mass region is near the minimum of the s-wave strength function and global SOM's typically over-predict the experimental values by rather large amounts (e.g. the models of Refs. [25,36] ). The calculated p-wave strength functions are in better agreement with the experimental values, with differences that are only modestly beyond the experimental uncertainty alone. The present models include a spin-orbit term with the parameters taken from Ref. [ 25 ]. That reference emphasizes polarization phenomenon and thus the potentials should give appropriate polarization results. However, that can not be experimentally verified as scattered-neutron polarization data are very limited [37]. The SOM, ROTM and VIBM real-potential strengths of the present work are similar. Generally, the real strengths of the collective potentials are slightly larger than that of the SOM (by .~20 MeVfm 3, or ~5%) as has been generally observed elsewhere [38]. There are some spherical silver proton potentials reported in the literature (13 are given in Ref. [39] ). These proton potentials are of variable quality. Taken in their entirety, they suggest a systematic strength of J, = 496.4-4.4213E MeV fm 3, when corrected for coulomb effects using the conventional formulation V~ = 0 . 4 Z / A 1/3. If two apparently anomalous proton potentials are ignored the J, = 477.7 - 3.4656E MeV fm 3. The slopes of these proton strengths are larger than that of the present SOM but values of the latter proton formulation differ from those of the present SOM by only 24 MeV fm3+ ..~ 8% over the comparable energy range of ~12 ~ 28 MeV. Given the uncertainties in the proton potentials, that is relatively good agreement. The proton J, values are related to the those of the present SOM through the well known expression J, = J0( 1 -4-(r/) [40], where ~7 is the asymmetry rl = ( N - Z ) / A , " + " ( " - " ) is for protons (neutrons), and sc is a constant. Comparing the above two proton J, expressions and the SOM result, one obtains ( values of 0.36 and 0.29, respectively. These sc values should be compared with the predictions of the theory of nucleon-nucleon forces [41,42] and other work which indicates sc = 0.45 ~ 0.5. They are very much smaller than often-quoted values of approximately unity which include isovector and size effects. Global potentials are generally deduced from considerations at energies well above the present scattering studies. This implies that dispersive contributions to the global

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

261

potentials are small. These dispersive effects are given by the well known expression +oo

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Jws(g') ~--27 dE',

(1)

where JHF is the local-equivalent Hartree-Fock potential, Jws is the strength of the surface-imaginary potential, and P denotes the principal value of the integral [43]. The limited energy scope of the neutron-scattering information and the complexity of the experimental resolution thwarts an effective dispersive interpretation of the present measurements. However, the integral of Eq. (1) can be estimated using one of the global models based upon higher-energy studies. That approach was followed using the method of Ref. [38], the global models of Refs. [30,36] and a fermi energy of -8.23 MeV. The results indicate that the present potentials, based upon neutron scattering at ~<10 MeV, should have real strengths ~20 ---+ 40 MeV fm 3 larger than an extrapolation of the global potentials to the same energy range. That difference is consistent with the larger strengths of the present potentials, and with low-energy trends of neutron models implied by (p,p) studies. As pointed out in Ref. [44], 8 MeV is a favorable energy for comparing SOM's. That energy lies within the present measurement range, is high enough to mitigate the effects of fluctuations and the dispersion integral, and yet is low enough to avoid the effects of uncertain volume absorption. Eight neutron scattering studies in the present mass region, from which SOM's were derived, have been reported by this group using Y [45], Zr [46], Nb [47], Rh [48], Pd [49], Cd [50], In [44], and Sn [51] targets. The asymmetry of these targets ranges from ~0.118 ~ 0.158, and several of them have closed neutron or proton shells. The average strength of these eight potentials at 8 MeV is J,, = 425.3 ± 11.5 MeV fm 3, where the uncertainty is the rms deviation from the mean. This result is consistent within the uncertainty with the 8 MeV value Jv = 416.8 MeV fm 3 of the present Ag SOM. A similar average of the eight r,, = 1.2707 + 0.0326 fm, in excellent agreement with the 1.2710 fm value of the present SOM. Similar comparisons of real-potential diffusenesses lead to a,, = 0.6732+0.0265 fm for the previously reported potentials compared to 0.6029 fm for the present SOM. In Ref. [44] the expression J~ = 234"2 [1 - 0'53 ( - ~ ) ]

( 1"1476 + A---~-0"4416"~] 3

was proposed for the systematic behavior of Jr, at 8 MeV. For l°8Ag this expression gives J,, = 416.15 MeVfm 3 which is in remarkable agreement with the 416.8 MeV fm 3 of the present SOM. Comparisons of the imaginary portion of the spherical potential are more sensitive to the detailed structure of the target, shell closures, etc., thus the imaginary parameters can be expected to scatter more than the real parameters. However, the 8 MeV imaginary potentials of seven of the above-cited targets lead to an average Jw = 64.3 + 8.3 MeVfm 3 (Cd results were not used as they appeared anomalous), an average rw = 1.3143 -4- 0.0217, and an average aw = 0.4695 -4- 0.0530 fm. These values are comparable to the present SOM results of Jw = 53.4 MeVfm 3, rw = 1.3215 fm and

262

A.B. Smith~Nuclear Physics A 620 (1997) 249-264

aw = 0.4570 fm. Thus the present SOM potential appears very similar to other such

potentials in this mass region at an incident energy of 8 MeV. There are some differences at other incident energies but they can reflect variations in the Fermi energy through the dispersion integral and varying collective effects which can be quite large in some cases. Similar comparisons of collective models in this mass region are not as rewarding as the number of such potentials is much more limited, the character and strength of the collective reactions vary widely, and in some cases it is not even clear whether a rotational or vibrational interaction (or a complex combination of the two) is involved. The present ROTM and VIBM geometric potential parameters are quite similar and generally larger than those of SOM. An exception is the imaginary diffuseness of the CCMs which increases with energy while that of the SOM is constant. A trend for aw to become small as E ~ 0 has been noted in previous studies at this laboratory. Generally, the present potentials are applicable only to energies of ~<10 MeV. Extrapolation to much higher energies leads to peculiar behavior, particularly of the imaginary potential, and is not valid. The ROTM and VIBM give essentially the same quality description of the neutron processes. Thus, while the available information on the neutron interaction with silver clearly indicates the importance of direct collective proceses it does not differentiate between a vibrational and a rotational interaction mechanism. Indeed, a successful study of proton scattering from l°7Ag used a vibrational model in both DWBA and CCM interpretations [34], and a limited neutron study was reasonably successful assuming a vibrational interaction [32]. On the other hand, some spectroscopic studies strongly suggest a rotational mechanism [ 2,3 ]. The present work does not resolve this dichotomy, but it does strongly suggest that/32 ~ 0,2 for the l°7Ag and l°9Ag isotopes, a value that is not consistent with some of the heavy-ion rotational models. A major limitation in the present work was the unavailability of reliable differential neutron-scattering data and (to a lesser extent) total cross sections at energies well above 10 MeV, particularly on an isotopic basis.

Acknowledgements The author is very much indebted to Dr. J. Raynal for the provision of the coupledchannels code ECIS95, and for instructions as to its use. The author is also appreciative of the data services of the National Nuclear Data Center, Brookhaven National Laboratory.

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