Elastic and inelastic neutron scattering by silicon and sulphur

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Nuclear Physics 79 (1966) 231--240; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by pbotoprint or microfilm without written permission f r o m the publisher

ELASTIC AND INELASTIC NEUTRON SCATTERING BY S I L I C O N A N D S U L P H U R G. A. PETITT *, S. G. BUCCINOtt and C. E. HOLLANDSWORTH**t Duke University, Durham, North Carolina and P. R. BEVINGTON Duke University and Stanford University, Stanford, California Received 4 October 1965

Abstract: The energy spectra of neutrons scattered elastically and !nelastically to the first excited states of 2sSi and ~S have been measured using time-of-flight techniques. Angular distributions were taken for incident neutron energies of 2.45, 2.85, 4.00, 4.90, and 5.80 MeV and an excitation function was measured at 60°. The elastic scattering distributions are fairly well described by the Percy-Buck non-local potential optical model with slightly reduced well depths, corrected for the compound elastic component. The inelastic scattering cross sections show large fluctuations, but the angular distributions are fairly isotropic. For ~sSi, the inelastic scattering is fairly well described by the Hauser-Feshbach formula modified by Dresner's "width fluctuation" correction at low energies, but above 2.7 MeV the correction becomes too large. This may be an indication of the inapplicability of the theories in the region where ( / ' ) j ~ ~ Dj~. E i NUCLEAR REACTIONS ~sSi(n,n'), 3~S(n,n'), E ~ 2-6 MeV; measured a(E; En', 0).

i

1. Introduction Studies of the elastic a n d inelastic scattering of n e u t r o n s of several M e V incident energy b y time-of-flight ( T O F ) techniques o n target nuclei with A =__ 20 have been m a d e b y several authors 1- 5). The present study was p r o m p t e d by the desire to m a k e a similar study over a wider range of b o m b a r d i n g energies t h a n has been used in previous work. The very wide spacing of the first two levels in 2sSi a n d 3as facilitates r e s o l u t i o n of n e u t r o n s scattered to the first level in each of these nuclei i n the T O F spectra at incident energies u p to 6 MeV. It has also b e e n of interest to analyse these data in light of recent theoretical developments 6-1 o) which have led to corrections to the H a u s e r - F e s h b a c h formula. These corrections d e p e n d o n the assumed distribution of partial widths a n d the ratio of the average total width to the average level spacing in the c o m p o u n d system. * Presently at Georgia State College, Atlanta, Georgia. tt Presently at Tulane University, New Orleans, Louisiana. ~** Presently at U.S. Army Nuclear Defense Laboratory, Edgewood Arsenal, Maryland. This research was supported in part by the U.S. Atomic Energy Commission and the National Science Foundation. It is based in part on a dissertation submitted by G. A. P. in partial fulfillment of the requirements for the Ph.D. degree at Duke University. 231

232

G . A . PETITT et al.

2. Experimental Procedure The Duke University T O F system has been described in detail elsewhere ~z - a 3). For average incident neutron energies of 2.45, 2.85, 4.0, 4.9 and 5.8 MeV, differential cross sections for elastic and inelastic scattering of neutrons by 28Si, 3zS, and ~2C were obtained by recording the scattered neutron T O F spectra at angles from 30 ° to 150 °. The incident energy spreads at these energies were 130, 120, 275, 340 and 249 keV, respectively. The carbon data were used for normalization of the silicon and sulphur cross sections. The absolute values of the angle integrated cross sections obtained in this way are correct to within __+10 % for the elastic and _4-12 % for the inelastic cross sections. i

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The variation with energy of the cross sections was investigated in small energy intervals from 2.25 to 3.0 MeV and from 4 to 6 MeV with the detector at 60 °. Data were taken over the low-energy range in 12.5 keV steps with neutrons from the T(p, n)3He reaction using a T - Z r target 10 keV thick at threshold and in 25 keV steps with a 50 keV thick T - Z r target. In the high-energy range, data were taken in 100 keV steps with neutrons from the D(d, n)3He reaction. A gas target was used whose thickness varied from 200 keV to 100 keV over the energy range. The scattering samples were contained in identical stainless steel cans 5 cm high and 2.5 cm in diam. with 0.025 cm wall thickness and were positioned with their symmetry axes perpendicular to the scattering plane. To measure the background, the relative incident neutron flux was monitored with a Hansen-McKibben long counter placed at 60 ° and 240 cm from the source.

NEUTRON SCATTERING

233

The data were corrected for multiple scattering, flux attenuation and geometrical effects using a Monte-Carlo computer code M A N I A C 6 (ref. 14)). Only the ground and first excited states of 2sSi and 32S produced distinct peaks in the T O F spectra over the energy range studied, and the intensity of the inelastic peak for sulphur was too weak to be measured at energies below 3 MeV. A typical TOF spectrum in fig. 1 shows the elastic group and the inelastic group from the 1.772 MeV level in 28Si.

3. Theoretical Comparisons Several authors 6-s,~o,2o) have noted that the H-F formula is strictly correct only if (i) complete positive correlation exists between the reduced widths for scattering to different levels from the same entrance channels g, and with the same orbital angular momentum I and channel spin s, or (ii) if the average total width for given total angular momentum J and parity n is much greater than the corresponding average level spacing (F>j,~ >> Ds~. The second requirement is met generally for high excitation energies and where many exit channels are open. Experimental evidence indicates that the first condition is not generally met, but rather that the partial widths are uncorrelated. Therefore the average inelastic scattering cross sections at low energies can be calculated only if the distribution of partial widths is known. Lane and Lynn 6) and more recently Dresner 7) have calculated a "width fluctuation" correction to the H-F formula resulting from an assumed Porter-Thomas distribution of partial widths 2,, 22), and this correction is found to enhance the compound elastic scattering cross section at the expense of inelastic scattering, sometimes by as much as a factor of two. Elastic scattering angular distributions were calculated using the non-local potential optical model of Perey and Buck 15) with a modified version of the computer code ABACUS II 16). The resulting transmission coefficients were used to calculate the inelastic scattering angular distributions predicted by the Hauser-Feshbach (H-F) formula ~7) and the corrections of Dresner 7, ~8). Equivalent local potential optical model 1s, 29) calculations reproduced the elastic scattering distributions, but reduced the magnitude of the inelastic scattering cross sections. TABLE 1 Scattering a n d reaction cross sections in (b) 32S

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0.52 0.77 0.56 0.92 0.70

0.02 0.08

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4. Results T h e results are s u m m a r i z e d in table 1. The t o t a l cross sections a n d the (n, p ) a n d (n, cz) cross sections listed were o b t a i n e d b y other workers. The elastic a n d inelastic scattering cross sections were m e a s u r e d in this experiment. 4.1. DATA T h e elastic scattering a n g u l a r distributions are shown in fig. 2. Because o f large fluctuations with energy o f the cross sections for c o m p o u n d nucleus f o r m a t i o n agreement with the optical m o d e l p r e d i c t i o n s is expected to be only qualitative, b u t

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Hofstadter 2¢) has found that to within a few percent for nuclei with A > 12 the radius at which the nuclear density falls to half its maximum value varies as r = r o A ~ and the "skin thickness" remains constant. This suggests that the analogous nonlocal potential optical model parameters, the radius parameter ro and the diffuseness a, should not be varied. However, this implies that the average density of nucleons increases slowly with increasing A up to about A = 40. Since the strength of the STANDARD W E L L S - H F O"CE ------REDUCED WELLS-HFO-cE . . . . S T A N D A R D W E L L S - D R E S N E R O"CE . . . . R E D U C E D W E L L S - D R E S N E R £rc¢ ,~ ~..~'

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o f the same calculations with the real, imaginary and spin-orbit well depths reduced by 10%. The width fluctuation correction to the c o m p o u n d elastic cross section adds an approximately isotropic c o m p o n e n t to the elastic scattering angular distribution. SULFUR STANDARD W E L L S - H F

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237

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sections, Two of these represent calculations with the standard and reduced nonlocal potential well depths using the H-F formula to calculate the compound elastic and inelastic differential scattering cross sections. The other two result from making Dresner's correction to these curves. l 5.8 MeV 40~•

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F i g . 5. I n e l a s t i c s c a t t e r i n g a n g u l a r d i s t r i b u t i o n s . C u r v e s c a l c u l a t e d w i t h t h e P e r e y - B u c k o p t i c a l m o d e l p a r a m e t e r s a n d w i t h t h e w e l l d e p t h s r e d u c e d 10 ~ . A r r o w s i n d i c a t e t h e D r e s n e r c o r r e c t i o n s to the H-F formula.

The measured inelastic scattering angular distributions are shown in fig. 5. The theoretical curves shown for silicon are the results of the calculations using the H-F formula. The reductions resulting from applying Dresner's correction are shown by arrows whose tips are at the corrected values. 4.2.

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FROM

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At the minima, and to a lesser extent at back angles, the elastic scattering distributions for silicon are dominatedby compound elastic scattering, and fluctuationeffects preclude any precise agreementbetween theory and experiment. At forward angles the agreementis slightlybetter for the calculationswith reduced potential well depths. The rapid variation with angle of the elastic scattering angular distributions makes it difficultto compare theoretical and experimental differentialcross sections at the single angle at which the excitation curves were measured. At the lower energies the curves differ markedly from the data, but above 3 MeV the agreement is fairly good.

G . A . PETITT et aL

238

The near isotropy of the inelastic scattering angular distributions, although perhaps due in part to the energy averaging effect of the thick target used to measure the angular distributions, makes possible a valid comparison between theory and experiment at a single angle. Lind and Day 25) have measured the excitation of the first level in silicon up to 3 MeV. The fluctuations in their excitation curve are similar to those observed in our 60 ° excitation curve. At the lowest energies, the Dresner curve clearly matches the data much better than the H-F curve as shown in fig. 3. Above about 2.7 MeV, the energy average of the data fails midway between the two theoretical curves. There are two possible explanations for this behaviour: (i) The presence of a small direct reaction component may increase the average value of the cross section. However, the inelastic scattering angular distributions fail to show the asymmetry associated with direct reactions. The distributions are relatively symmetric about 90 °, which is consistent with the assumption that the scattering takes place by formation and decay of the compound nucleus. (ii) The H-F formula corrected by Dresner is expected to be valid in the region (F)j~ << Dj~ and in the region (F)j~ >> Dj~ where the Dresner correction approaches 1. In the intermediate region, the Dresner correction may be an overestimate. Moldauer ~o) has calculated a number of inelastic scattering cross sections with the assumption that (F)j~ ~ Dj~. These calculations involve parameters whose values are crude estimates but they represent a "first guess" at the kind of modification to be made to the Dresner formula. The assumed distribution of level spacings affects the results, and the magnitudes of the resulting cross sections generally fall between the magnitudes calculated using the H-F and Dresner formulae. This suggests that the behaviour of our excitation curve above 2.7 MeV may be explained by assuming that ( F ) s ~ _~ Dj~ in the compound system. The fluctuations in the excitation curves are presumably of the type described by Ericson 26). That is, they result from the interferences between the amplitudes of the many resonances which simultaneously dominate the cross section. Note that the prominent fluctuation in the inelastic scattering cross section at 2.85 MeV is not accompanied by a corresponding maximum in the compound elastic cross section, and the corresponding fluctuation in the total cross section can be ascribed to this one reaction channel alone. This lack of correlation between the fluctuations in various partial cross sections is evidence that the fluctuations are of a random nature and are not due to the "doorway states" postulated by Feshbach 27). 4.3. S C A T T E R I N G

FROM

82S

The elastic scattering curves for sulphur include a contribution from compound elastic scattering calculated assuming no competing reactions. As shown in table 1, 32S has large (n, p) and (n, ~) cross sections in this energy range, but appropriate transmission coefficients for the H-F formula cannot be calculated with any certaintly.

NEUTRON SCATTERING

239

The resultant overestimate of the compound elastic contribution is at least partly compensated for by the omission of the Dresner correction, which tends to increase the calculated cross section. As in the case of silicon, the curves predicted using the reduced well depths generally match the data better at the forward angles where shape elastic scattering dominates the cross section. The fluctuations in the sulphur excitation curves are smaller than those in the silicon curves by about a factor of two, and the predicted inelastic scattering cross sections are generally more than twice the measured cross sections. This is presumably due to the many decay channels that are not accounted for in the calculations. As in the case of the compound elastic cross section, competing reactions could decrease the calculated cross section by about a factor of two. The Dresner correction is expected to be negligible because of the many open decay channels. 5. Conclusions

The results of this experiment may be summarized as follows: the non-local potential optical model of Percy and Buck describes the elastic scattering of neutrons from ZSSi and 3zS fairly well, but there is an indication that the well depths should be reduced slightly from the optimum values for heavier nuclei. The inelastic scattering cross sections for silicon show large fluctuations and the average inelastic scattering cross section is generally smaller than the predictions of the H-F formula and larger than the predictions of the Dresner formula. The excess of the measured cross sections over the predicted values is probably not due to direct reactions since the angular distributions are, on the average, symmetric about 90 °. Instead the discrepancy may be due to the inapplicability of the theories in the region where (f')s~ ~- Dj~ for the compound system. The inelastic scattering cross sections for sulphur exhibit the predicted energy dependence but are smaller than the predicted values. This may reasonably be ascribed to the competition of (n, p) and (n, e) reaction, which cannot be treated properly in the calculations. The cooperation of the entire nuclear structure group of Duke University is gratefully acknowledged. One of us (G.A.P.) would like to thank Professor W. E. Meyerhof for the hospitality of the Stanford University Nuclear Structure Group. Computations were carried out in the Duke University Digital Computing Laboratory, supported in part by the National Science Foundation, and the Stanford University Computation Center. References

1) L. Cranberg and J. S. Levin, Phys. Rev. 103 (1956) 343 2) D. B. Thomson, L. Cranberg and J. S. Levin, Phys. Rev. 125 (1962) 2049 3) J. H. ToMe and W. B. Gilboy, Nuclear Physics 32 (1962) 610

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a . A . PETITT et al.

4) J. H. ToMe and W. B. Gilboy, Nuclear Physics 39 (1962) 300 5) K. Tsukada, S. Tanaka, M. Maruyama and T. Tomita, Physics of fast and intermediate reactors (IAEA; Vienna, 1962) p. 75 6) A. M. Lane and J. E. Lynn, Proc. Phys. Soc. AT0 (1957) 557 7) L. Dresner, Columbia University Report CU-175 (1957) p. 71 8) P. A. Moldauer, Phys. Rev. 123 (1961) 968 9) P. A. Moldauer, Revs. Mod. Phys. 36 (1964) 1079 10) P. A. Moldauer, Phys. Rev. 135 (1964) B642, 136 (1964) B947 11) H. W. Lewis et al., Rev. Sci. Instr. 30 (1959) 923 12) S. G. Buccino, C. E. Hollandsworth, H. W. Lewis, and P. R. Bevington, Nuclear Physics 60 (1964) 17 13) P. R. Bevington, C. J. Kapadia, C. E. Hollandsworth and G. A. Petitt, to he published 14) P. R. Bevington, S. G. Buccino, C. E. Hollandsworth and (3. A. Petitt, to be published 15) F. Perey and B. Buck, Nuclear Physics 32 (1962) 353 16) E. H. Auerbach, private communication 17) W. Hauser and H. Feshbach, Phys. Rev. 87 (1952) 366 18) A. B. Tucker, J. T. Wells and W. E. Meyerhof, Phys. Rev. 13/ (1965) Bl181 19) F. E. Bjorklund and S. Fernbach, University of California Radiation Lab. report U C R L 5028 (1958) 20) H. Feshbach, in Nuclear spectroscopy, part B, ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) Chapts. V. A., VI.D. 21) C. E. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 22) J. L. Rosen, I. S. Desjardins, J. Rainwater and W. W. Havens, Phys. Rev. 118 (1960) 687 23) J. R. Stehn, M. D. Goldberg, B. A. Magurno and Renate Wiener-Chasman, Brookhaven report BNL 325 (1964) 2nd. ed. Suppl. 2, Vol. I 24) R. Hofstadter, Revs. Mod. Phys. 28 (1956) 214 25) D. A. Lind and R. B. Day, Ann. of Phys. 12 (1961) 485 26) T. Ericson, Ann. of Phys. 23 (1956) 390 27) H. Feshbach, Ann. of Phys. 19 (1962) 287 28) K. Tsukada and Osamu Tanaka, J. Phys. Soc. Japan 18 (1963) 610