Microscopic optical model analyses of proton and neutron elastic scattering cross sections

2.E : 2.L

Nuclear Physics A295 (1978) 301-308 : © North-Holland Publishing Co., Atnfterdant Not to be reproduced by photoprint or microfilm wltltout written permisfio~n firom tha publiaber

MICROSCOPIC OPTICAL MODEL ANALYSES OF PROTON AND NEUTRON ELASTIC SCATTERING CROSS SECTIONS A. LEJEUNE Université de Liège

T

and P. E . HODGSON

University ojOxjord tr Received 7 October 1977 Abstract : In this paper we investigate the physical properties of a microscopic optical model potential (OMP) derived from a realistic nucleon-nucleon interaction through nuclear matter calculations, using the local density approximation for nuclei. We calculate nucleon elastic scattering cross sections for a wide range of energies and nuclei . Comparison with experimental data shows that the shape and depth of the real part of the OMP are quite satisfactory . However, the depth of the imaginary part turns out to be a little too large. We suggest some plausible physical arguments to explain this discrepancy . The overall argument is however quite encouraging and it suggests that the study of some microscopic effects, neglected previously, would be worthwhile .

1. Introduction There have been many analyses of the experimental differential cross sections and polarizations of protons and neutrons elastically scattered at many energies from many nuclei. These analyses are of two main types : the phenomenological one in which a complex optical potential of selected analytical forth is adjusted to optimise the fit to the experimental data, and the microscopic one in which the potential is calculated from the nucleon-nucleon interaction and from some features of the target nuclei . The phenomenological analyses are plagued by ambiguities, in that many different potentials can often be found to fit the same set of data, while the microscopic analyses encounter the difficulties of a many-body calculation. To some extent, the two types of analyses are complementary in that the microscopic analyses can give the overall form of the potential and the phenomenological analyses can make the fine adjustment to give precision fits to the data') . It is important to refine the microscopic analysis to the point where it can give a precise fit to the data without any phenomenological adjustment . This is not only f Physique Nucléaire Théorique, Institute de Physique su Sart Tilman, B9timent B .S, B-4000 Liège 1, Belgium . f~ Nuclear Physics Laboratory, Keble Road, Oxford OX1 3RH, United Kingdom. 301

302

A. LEJEUNE AND P. E . HODGSON

aesthetically more satisfying, but it would enable distorted waves to be calculated whenever they are required in reaction analyses without the necessity of measuring and analyzing elastic scattering data . Furthermore, since the microscopic analysis also require a knowledge of the structure of the target nuclei, in particular of their proton and neutron density distributions, comparison with elastic scattering data should enable these features to be determined . The development of the microscopic analyses towards this goal has been reviewed by Jeukenne, Lejeune and Mahaux and has now reached the stage when fairly good fits to the elastic scattering data can be obtained without parameter adjustment, although they are naturally improved, when this is allowed. In particular, Brieva and Rook s) calculate the t-matrix by an "extended" reference spectrum method applied to nuclear matter, using the Hamada-Johnston °) nucleon-nucleon interaction. Then they obtain the potentials for finite nuclei by folding with the matter distributions. Contemporary and parallel work by Jeukenne, Lejeune and Mahaux s 6), where Brueckner's g-matrix is calculated in the "r" space from the Reid hard core interaction'), using an "improved" local density interaction 6) (LDA) gives good overall fits to the volume integrals of the phenomenological potentials. This work extends their calculations to give differential cross sections for direct comparison with the experimental data . In sect . 2, we present the formalism used to calculate the differei.tial cross sections and the context of our investigations . The results are discussed in sect . 3 and conclusions summarized in sect. 4. Z)

~

2. Calculation of the differential crow sections We recall, first, the main formulae and then we describe the set of nuclei studied and the range of energies. 2.1 . BASIC MICROSCOPIC FORMALISM

The microscopic model used in the present calculations has been described in ref. 6). We recall that the target density introduces itself through the LDA approximation Y(kF(r), E~ -" V(P(r) . ~ --" VE(r) .

We use the "improved" LDA in which the real and imaginary effective potentials are given by e) i~x1r) _ (t~aJn)-3,(Va(~~P( - Ir - r~l Zltnk~~ . T~g (r) _ (t~mJn)-3f Wx(r')exp(-Ir-r'I Z/t~dr' .

(2.1) (2.2)

The ranges tK and t,m are the only phenomenological .parameters in these formulae

MICROSCOPIC OPTICAL MODEL

30 3

and are adjusted to fit the volume integrals and the RMS radii of the optical potentials. For the matter distributions of the target nuclei we use those of Negele e) and, for light nuclei, the point-like nucleon mass distribution of Malaguti and Hodgson 9). An other important quantity that we shall use in this work, is the difference between the neutron. and proton half density radii related to the RMS radii by the expression ' °) The microscopic model does not yet include a spin-orbit potential, so this is treated phenomenologically . This is sufficiently accurate since we do not attempt at this stage to fit the polarizations . The optical model calculations were made using the codes MAGALI ") and JIB3 of Perey. 2.2 . RANGES OF NUCLEI AND ENERGIES

Calculations were made for the nuclei ' zC, '60, z'Al, a° Ca, seFe, sa . eo, ez, eaNi, eaZn, 9°Zr, " 6" 'zoSn, zoePb and z ° 9Bi for both proton and neutron scattering. This choice was made partly because many data exist for those nuclei which, moreover, are mainly spherical and are therefore well described by the simple optical model potential with no strong coupling to low-lying collective states . It has however the disadvantage that the density of compound nucleus states is less than for the average nuclei, and the reaction thresholds higher, so that the strength of the imaginary part of the our theoretical optical potential will be too big for these magic nuclei . The energy range 10-70 MeV was studied for proton scattering and 1 to 15 MeV for neutron scattering ; these are well within the accepted limits, 1-160 MeV, of validity of the microscopic optical model. 6' .6s Cu,

2.3 . RESULTS

The work by Jeukenne et al. e) showed that the calculated value of J,,/A, the volume integral per nucleon of the real potential, agrees well with those of the phenomenological potentials if trc = t,~, = 1 .2 fm and its RMS radius also agrees for nuclei from ' zC to seNi, but is rather too small for the heavier nuclei ('z°Sn and z°BPb). The volume integral per nucleon JwlA of the imaginary potential is globally too high for the reason already mentioned. This discrepancy decreases with increasing energy, as would be expected. The RMS radius of the imaginary potential is rather too small except for light nuclei . From this we conclude that the real part of the potential Y has about the correct magnitude but the imaginary ~rt Wwill have to be reduced to agree with the data . Such a change does not affect the RMS radii. The work is based on a great number of trials of different types of density and

304

A . LEJELJNE AND P . E . HODGSON

10'

10~

10'

a~ ffg .5

a fig .6

MICROSCOPIC OPTICAL MODEL

30 5

T~a~.e 1 Characteristics of the calculations giving the optimum fit to the elastic scattering data Energy

Projedile

Nuclei

Density distribution

30.3 30 .3 30.3 61 .4 61 .4 61 .4 IS 9 5.3 9 11 14.5 Il

p p p p p p n n n n n n n

~zC ~60 soCa s°Ni izoSn zoepb 'zC, Z'AI, `°Ca °°Ni, s° Zn, 9°Zr "6Sn, zo°Pb 'zC z'AI ` °Ca °°Ni 'z °Sn z° °Pb z°°Bi

MH N N MH N N MH MH MH N N N N

-_ -

dP

0.129 0.129

0.129 0.129

In all calculations, tro = t, m = 1 .2 fm . MH and N stand for Malaguti and Hodgso~s distribution') and for Negele's distribution °), respectively .

of different values of dop, trc and t; m with the purpose of improving the agreement between the experimental and theoretical RMS radii. We present here some typical results of the calculations and compare with experimental differential cross sections in figs . 1-6. These results are typical in the semse that they give the lowest Xz in regard to the trials mentioned below. In each figure, the full curves show the cross sections before scale factors adjustment [the quantities ~E{r) and T~E(r) are numerically given] and the dashed curves, the results of varying the scale factors of the real and imaginary potentials . Table 1 summarizes the different features of our displayed results. Several values of t, m (from 1 .2 to 2.5 fm) combined with different values of d p (from 0.0 to 0.3 fm) were tried for the medium and heavy nuclei . Some equivalent fits can be obtained either with large value of t;m and d p = 0, either with t;,~ = 1 .2 fm and dop = 0 or 0.129 fm for these nuclei. But if V and W are optimized, Fig. 1 . Differential cross sections for the elastic scattering of 30 .3 MeV protons by 'zC, '60 and `,°Ca compared with microscopic optical model calculations . The full curves correspond to the optimum choice of density distribution and range parameters (see table 1) . The dashed curves give the results of optimizing the potential strengths V and W. The data arc taken from ref ") . Fig. 2. Same as fig. 1 but for s°Ni, 'z°Sn and z°°Pb [ref. ")]. Fig. 3. Same as fig. 1 but for61 .4 MeV protons elastically scattered by' zC, z'AI, ` °Ca and s°Ni [ref. z°)] . Fig. 4. Same as fig. 1 but for 61 .4 MeV protons elastically scattered by 6°Zn, 9°Zr, "6 Sn and z °°Pb [ref. z°)] . Fig. 5. Same as fig. 1 but for neutron scattering. The data arc obtained for 'zC from ref. z'), for `°Ca from ref. zz) and for z'AI and s°Ni from ref. z'). Fig. 6. Same as fig. 1 but for neutron scâttering. The data are obtained for 'z °Sn and z° 9Bi from ref. z`) and for z° °Pb from ref. zs).

306

A . LEJEUNE AND P . E . HODGSON

the best overall results are obtained with t,m = 1 .2 fm and dp x 0.129 fm. This latter quantity is in agreement with the last published result on the neutron skin thickness ' Z). In any case, large values of d p ( x 0.3 fm) gave bad fits. An other argument which support the result t,m = 1 .2 fm derives from the fact that the range of an effective interacKion should be independent of the nuclei . The commonly proposed value is 1 .4 fm [ref. ia)]. It is notable that the results for `60 are poorer than for the other nuclei ; this can be ascribed to strong channel coupling. The discrepancy in the backward direction, sometimes present in our results, has been recently accounted foc, phenomenologically, by an 1-dependent potential 14). Such a component is not yet well understood from the present microscopic approach . We observe a steady improvement in the quality of the fit towards the heavier nuclei and higher energies . 3. Disca~ion and conclusions On the whole, the fits shown in figs . 1~ are encouraging for a microscopic calculation with the ranges t« and t, m as the only adjustable parameters. The need to decrease the strength of the absorbing potential shows that this ~rt of the calculation is in need of further refinement . The improvement in the quality of the fits for heavier nuclei and increasing energies is what would be expected because the defects of the LDA are more serious in the nuclear surface region which has relatively less effect on the cross sections as the size of the nucleus and the incident energy increase . Since Negele's mass distribution is valid only for intermediate and heavy nuclei, the distributions calculated for each nucleus individually give better results for the light nuclei . In the case of proton scattering, it was found that no combination of large t,m or clop could give the experimental value of
MICROSCOPIC OPTICAL MODEL

307

Teste 2 Rcnormalization factors for p elastic scattering Target

E,,b (Mew

"C ' 2C ' 60 ~'Al

30 .3 61 .4 30.3 61 .4 30.3 61 .4 30.3 61 .4 61 .4 61 .4 61 .4 30.3 30.3 61 .4

4oCa

seNi se~ 9°Zr " 6 Sn '~°Sn ~oe~ :oePb

Factor;

v

w

0.932 1 .104 0.972 1 .002 0.992 1 .006 0.962 1 .021 1 .058 1 .063 1 .101 1 .007 1 .027 1 .123

0.598 0.704 0.818 0.727 0.722 1 .014 0.726 0.845 1 .025 0.907 1 .015 0.822 0.888 0.965

T.~s~ 3 Renormali7ation factors for n elastic scattering Target

E,,b (MeV)

"C

15 9 5 .3 9 1I 14.5 11

"AI

40~

seNi

' =°Sn :oe~ zo9gi

Factors

V

W

1 .027 0.964 0.991 0.968 0.973 0.976 0.960

0.625 0.971 0.379 0.878 0.866 0.681 0.722

closed shells and so have untypically low densities of compound states and high reaction thresholds, and also because no account is taken of the c.m . motion . Concerning the imaginary part, the LDA may include a spurious absorptive part that corresponds to the channels where the c.m . of the target is excited 's) . A further consideration is that we calculate only the first order term in the low density Brueckner expansion of the mass operator tb) ; it is possible that higher order terms decrease the potential strengths. Preliminary studies t') show that the second and third orders terms can decrease the real part by about 10 ~ and the imaginary part by about 6 ~. All these effects, taken together could well account for the observed discrepancies. The general decreasing trend of our overall normalization factor Wdoes not agree with that one described by Brieva and Rook a) even in the comparable situation

30 8

A. LEJEUNE AND P. E. HODGSON

of simple LDA. This is puzzling because the previous physical arguments give weight to a decrease of W. The origin of this discrepancy may be twofold . Firstly, the nucleon-nucleon interaction are not the same. Secondly, the way of solving the nuclear matter problem is different. Further investigations would be of interest . Finally, we observe very good agreement between our resulting cross sections, after renormalization, and those calculated from an average phenomenological potential like that of Becchetti and Greenlees ' e). Our microscopic calculations of the elastic scattering of protons and neutrons by nuclei give a gopd overall account of the experimental data . This microscopic approach can be improved in several respects : (i) Investigating the importance of higher order terms. This is a many-body problem. (ü) The features of particular nuclei, such as shell effects for example, must be included in a self-consistent way. (iii) The c.m . problem must be taken into account, at least roughly. (iv) A microscopic spin-orbit term is desirable. This last point is under study . One of us (A .L.) would like to thank the Nuclear Physics Laboratory, Oxford, for its kind hospitality and the Fonds National de la Recherche Scientifique of Belgium for a grant. Useful discussions with F. A. Brieva, J.-P. Jeukenne and C. Mahaux are gratefully acknowledged . References

1) P. E. Hodgson, in Congrès International de physique nucléaire, ed. P. Gugenberger (Editions du CNRS, Paris, 1964) p. 309 2) J.-P. Jeukenne, A. Lejeune and C. Mahaux, Phys . Reports 2SC (1976) 83 3) F. A. Brieva and J. R. Rook, Nucl . Phys. A291 (1977) 299; A291 (1977) 317 4) T. Hamada and D. Johnston, Nucl . Phys. 34 (1%2) 382 S) J.-P. Jeukenne, A. Lejéune and C. Mahauz, Phys . Rev. C1S (1977) 10 6) J:P. Jeukeme, A. Lejeune and C. Mahaux, Phys . Rev. C16 (1977) 80 7) R. V. Reid, Ann. of Phys . 30 (1968) 411 8) J. W. Negele, Phys . Rev. Cl (1970) 1260 9) F. Malaguti and P. E. Hodgson, Nucl . Phys. A215 (1973) 243 ; A257 (1976) 37 10) W. D. Myers, Nucl . Phys . A204 (1973) 465 11) J. Raynal, Preprint D.Ph.-T/69-42, CEA, Saclay 12) E. Friedman and C. J. Hatty, Phys . Rev., to be published 13)~ G. R. Satcher, private oommuaication 14) R. S. Mackintosh and L. A. Cordero, Phys . Lett . 68B (1977) 213 15) G. E. Brown, private communication 16) J. HQfner and C. Mahaux, Ann. of Phys. 73 (1972) S25 17) J.-P. Jeukenne, A. Lejeune and C. Mahaux, Nukleonika 20 (1975) 181 18) F. D. Beochetti and G. W. Greenlees, Phys . Rev. 182 (1969) 1190 19) B. W. Ridley and J. F. Turner, Nucl. Phys . 38 (1964) 497 20) C. B. Fulmer, J. B. Ball, A. Scott and M. L. Whiten,.Phys. Rev. 181 (1969) 1565 21) D. Spaargaren and C. C. Jonker, Nucl . Phys. A161 (1971) 354 22) J. D. Reber and J. D. Brandenberger, Phys. Rev. 163 (1%7) 1077 23) D. E. Velkey, D. W. Glasgow, J. D. Hrandenberger, M. T. McElliatrem, J. C. Manthuruthil and C. P. Poirier, Phys. Rev. C9 (1974) 2181 24) J. C. Ferrer, J. D. Carlson and l. Rapaport, Nucl . Phys . A275 (1977) 325 2S) F. Perey and B. Buck, Nucl . Phys . 32 (1%2) 353