Nuclear Physics A388 (1982)153-172
© North-Holland Publishing Company

GLOBAL OPTICAL-MODEL POTENTIALS
FOR ELASTIC SCATTERING OF 6,7Li PROJECTILES

J. COOK

Wheatstone Laboratory, King's College, Strand, London WC2R 2LS, UK
and
Department of Physics, Florida State University, Tallahassee, Florida 32306, USA

Received 13 October 1981
(Revised 23 February 1982)

Abstract: Simultaneous fits have been made to 44 6Li data sets covering the mass range 24-208 and the energy range 13-156 MeV in order to determine an average ("global") optical-model potential for 6Li scattering. A similar study has been made for 25 7Li data sets over the same mass range and an energy range of 28-88 MeV to find an average 7Li potential. With Saxon-Woods form factors, constant values may be used for all parameters except for the depth of the imaginary potential, which decreases in magnitude with increasing mass. The necessity of energy dependence, Coulomb correction and (for 7Li) a symmetry term is investigated. The variation of the integral properties of the potentials is discussed, and also a comparison is made for the two projectiles. Application of the global potentials is made to inelastic scattering and single-nucleon transfer reactions.

1. Introduction

The optical model is the most widely used model to describe the elastic scattering of nuclei. However, the potential for one nucleus at a particular energy may reflect the peculiarities of that nucleus and therefore may not be suitable for neighboring nuclei at different energies. Since the optical model is a very crude representation of the many-body scattering problem, fluctuations in the potential parameters as a function of mass number and energy are to be expected. But if the optical model is to be a useful description of elastic scattering then there should be smooth trends in the parameters. Efforts have therefore been made for light projectiles [eg. ref. 1) for protons, ref. 2) for neutrons, ref. 3) for deuterons, ref. 4) for 3He and ref. 5) for heavy ions] to determine average ("global") optical-model potentials. The existence of average potentials means that the scattering is insensitive to the nuclear structure details of the projectile and target, but facilitates the description of distorted waves when no elastic scattering data are available. In addition to the utilitarian value of global potentials in DWBA calculations etc., the inability of such potentials to fit certain angular distributions may suggest anomalies in the scattering process or a greater than average sensitivity to the nuclear structure. The average mass and energy dependence of such properties as the volume integrals and rms radii of the potentials may also be determined in a way that is less dependent on the peculiarities of any one angular distribution.

There now exists a large amount of data for the elastic scattering of 6,7Li projectiles over wide mass number (A = 6 –208) and energy (E = 5 – 156 MeV) ranges. A complete compilation of 6,7Li optical-model parameters using Saxon-Woods potentials is given in ref. 6). However, large variations in the parameters exist and it is very difficult to determine any trends from this compilation. By comparing parameters for different targets at one energy 7-10), or the same target at different energies 11), investigations have been carried out to determine their mass and energy dependence, but previous to this paper no attempts have been made to find a global potential for lithium scattering. The aim of this paper is to establish global potentials for the two lithium isotopes by simultaneously fitting many sets of data.

Sect. 2 describes the optical-model analysis of 6,7Li cross section distributions. The selection of data, the parameterization of the optical potential and the fitting procedure are discussed. The results are presented in sect. 3, and are applied to inelastic scattering and single-nucleon transfer reactions in sect. 4.

2. Optical-model analysis

2.1. SELECTION OF DATA

Ref. 6) lists all of the measured elastic scattering cross section data for unpolarized 6,7Li beams up to the writing of this paper. However, only a carefully selected subset of this data has been used in the present work. The first restric­tion placed on the selection of data was the requirement that all data should be available in numerical form so that resort would not have to be made to digitizing graphically presented data. This prevented the use of some of the older measure­ments for which the author could not obtain numerical tabulations. Secondly, data for Z < 12 have been neglected since for light nuclei fluctuations in the param­eters are likely to be larger due to differences in the structure of the target nuclei. In addition, inelastic scattering channels may be strongly coupled with the elastic channel. This may be reflected in the requirement of energy dependence for lithium scattering from light nuclei but not when Z 14 [ref. 11)]. It is debatable even whether data for the magnesium isotopes (Z = 12) should be used since they have a large deformation and are strongly excited, but they have been included in order to determine the extent of the validity of the average potential obtained. Data 22,23) for 6Li + 40Ca at 30 and 34 MeV have also been neglected due to the occurrence of anomalous large angle scattering (ALAS). Finally, only data for even mass number targets have been used since for odd targets with the elastic scattering distributions are affected by couplings to the target ground-state quadrupole moments 10,12).

The 6Li data we have used extends over the mass range 24-208 and the energy range 13-156 MeV, although most of it is concentrated towards A < 60 and E < 100 MeV. The data includes the fixed energy surveys at 50.6 MeV [ref.7)] and 73.9 MeV [ref. 13)] from Michigan State University, 88 MeV from Oak Ridge10), 99 MeV from Indiana University 14), and 156 MeV from Karlsruhe 9,27). In addition we have used data for various targets for 28 – 50 MeV from Florida State University [refs. 15,22,28)] and 42 – 48 MeV from Canberra 25,26) and data for individual targets from refs. 16–18, 20–24). A total of 44 angular distributions were used, containing just over 2000 data points.

The 7Li data are much scarcer. No data exists for energies in excess of 90 MeV, and most of the data is restricted to E 50 MeV, with again an emphasis on the lighter targets. A wide range of data is available at 34 MeV from Florida State University [refs. 22,28–30)]; and surveys also exist at 52 MeV from Canberra 25,26,31), and at 88 MeV from King's College 33). There is also data for 54Fe at three different energies 34). Just under 1000 data points were used in a total of 25 angular distributions.

Each data set has associated with it systematic errors (in absolute normalization and angle calibration) and relative or statistical errors on each individual data point. As far as could be determined the original errors assigned by the experimenters for most data sets were statistical errors. In all cases these original errors have been retained and the normalizations have been adjusted to produce the best fits. The rationale behind this is explained in ref. 11). Average normalizations of 0.97 ± 0.08 and 0.99 ± 0.11 resulted for 6Li and 7Li respectively, showing that on average the data sets initially had the correct normalizations but with random variations in the normalization from one data set to another of typically 10%. It should be noted that the parameters obtained are probably dependent in some complicated way upon the handling of normalizations and statistical errors.

2.2. OPTICAL-MODEL PARAMETERS

The optical potential for elastic scattering is written as the sum of real and imaginary nuclear potentials and a Coulomb term, thus
(1)                                      

The most widely used form for the form factors fR(r) and fI (r) is the Saxon-Woods function
(2)                                      
where AT is the target mass number. The Coulomb term is taken as that for a uniformly charged sphere of radius RC = 1.3 fm. Following the light-ion studies, the depth of the real potential may be parameterized as
(3)                                      
The coefficients V1 and V3 represent the isoscalar and isovector strengths of the optical-model potential. Since 6Li has zero isospin V3 = 0, but it still remains to be determined what value V3 has for 7Li. V2 describes the energy dependence of the isoscalar potential, with E being the incident energy of the projectile. It is assumed here that the isovector part is energy independent. Finally, V4 is a Coulomb correction term 35) and should be zero if there is no energy dependence. The depth of the imaginary potential W0 may be parameterized in a similar way.

2.3. SEARCH PROCEDURE

All the cross-section data for each projectile were fitted simultaneously using a heavy-ion version of the global search code GENOA35) extended to accommodate 50 data sets and 2500 data points.

In the determination of average optical potentials there arises the problem of which parameters to vary and which to keep fixed. If too many are varied at once then ambiguous results may be obtained, but if too few are varied then the fits may be unsatisfactory. In light-ion studies [e.g. refs 1–4)] the geometry parameters have usually been kept fixed and the potential depths alone varied. More complicated parameterizations may be sought later, once the data have been reasonably well described.

In general, the same searching procedure was used for each projectile, although, because of the more restricted nature of the 7Li data and its consequently reduced sensitivity to the details of the potential, the 7Li calculations had to somewhat follow those for 6Li using the 6Li values as starting parameters. The aim of this paper is to find the simplest functional form of the 6,7Li optical potentials (within the constraints of Saxon-Woods form factors), which is consistent with good fits to the data over the whole of the mass number and energy ranges. To this end the first fits were made with rR fixed at 1.3 fm and starting from values of V0 = 110 MeV, rI = 1.7 fm and aR = aI = 0.85 fm. This choice of real potential well depth was used since at high energies when rainbow scattering occurs 9,20,21) the depth of the real potential is restricted to a small range of values. For rR = 1.3 fm this range is 110-120 MeV [ref. 11)]. It should be noted that the volume integral of real Saxon-Woods potentials using these parameter values is similar to that found from folding model fits [e.g. refs. 9,37)]. Using a mass-independent value of rI meant that the imaginary strength W0 had to decrease with increasing mass number. By fitting the higher energy data it was determined that the relation between W0 and A was approximately linear, but a quadratic dependence was allowed for greater flexibility. Although it has previously been shown 11) that 6Li potentials are independent of the incident energy, fits were also made with energy dependence in the real and imaginary potentials and with the Coulomb correction term in the real potential. Following the similarity of 6Li and deuterons in terms of the structure of the projectile, a parameterization of the imaginary diffuseness as
(4)                                      
was made, where . Here Mi are the magic numbers 8, 20, 28, 50, 82, 126 and N is the target neutron number. This parameterization was necessary for a global deuteron potential 3) where the last term described the variation of ai with neutron shell closure in the target. For 7Li the occurrence of a symmetry term in the real potential was also investigated. In general, an extra parameter was only included if it produced a reduction in c2 of at least 10%.

3. Results

3.1. 6Li SCATTERING

The first calculations were made with rR fixed at 1.3 fm in order to determine the coefficients for the mass dependence of W0. Initial adjustments were then made to the normalizations of the data sets, and these normalizations were used throughout until the final calculations. It was noted on the first calculation that both the 40,44Ca data sets at 88 MeV each contributed 20-25 % of the total c2. The fits appeared particularly bad and since it was known 38) that these data sets suffered from experimental difficulties they were omitted from subsequent searches.

Energy dependence was now added to both the real and imaginary nuclear potentials, and also a Coulomb correction term. Thus the real potential depth was written as in eq. (3) with V3 = 0. No improvement in the quality of the fits resulted, either with all three terms present, or with any subset of them. With rR = 1.3 fm the depth of the real potential was given by
(5)                                       .

The first two coefficients were well determined (±2 MeV and ±0.005) but the third coefficient showed a larger variation (±0.1 MeV). With these figures there is a net energy correction of 0.235 (E – 6.17 Z/A1/3) MeV. The second term in this correction is larger than might be expected for a "true" Coulomb repulsion term and might represent some form of A-dependence in the real potential. It should be noted that the amount of energy dependence found here is similar to that found for lighter ions 11), although it is not very meaningful since equally good fits can be obtained without any energy dependence at all.

Adding mass dependence in the imaginary diffuseness or a neutron shell closure term in the form given by eq. (a href="#eq4")(4) resulted in no improvement to the fits. However, in studies at fixed energies [ref. 8) at 99 MeV, and ref. 9) at 156 MeV] aI was found to vary in the form aI = a1 + a2A1/3.

rR was now allowed to vary and increased from 1.3 fm to 1.326 fm. In this search only a small reduction in c2 resulted. However, good fits could be obtained with any value of rR in the range 1.2 – 1.4 fm, and thus it must be concluded that the scattering is not very sensitive to the value of rR used.

Final re-adjustments were now made to the normalizations and the data refitted to obtain the final potential. The explicit form of the average potential for 6Li scattering is

(6)                                      

This very simple average potential, with constant parameters except for W0, fits most of the data well, including the rainbow scattering in fig. 6, with an average c2/N = 14.9 (excluding 40,44Ca at 88 MeV) corresponding to an average rms deviation between the experiment and theory of 39 %.

The fits to the 6Li data using the average potential of eq. (6) are grouped according to energy and are shown in figs. 1 – 6. There are certain deficiencies in some of the fits, although in general the fits are very good. The oscillations for 26Mg at 36 MeV have the wrong phase and magnitude, although at 88 MeV both the 24,26Mg distribu­tions are fitted well up to about 50°. This may indicate that some energy dependence is required when fitting Mg data. For energies in the range of approximately 35 – 50 MeV the oscillations for A < 55 are not as pronounced as the data might suggest. However, the distributions for heavier mass targets at these energies are fitted well, as are the distributions for A < 55 outside this energy range. The 40Ca distribution at 88 MeV is particularly poorly described in terms of the phase of the oscillations, although the phasing is correct at 99 MeV. A similar problem also exists for 44Ca at 88 MeV, although less pronounced in this case.

3.2. 7Li SCATTERING

Initially the 6Li parameters with rR fixed at 1.3 fm were used as starting values for the searches. The 7Li data required only small variations in the parameters compared with those for 6Li. Adjustments were made to the normalizations of the data after the first search and then the effect of energy dependence was investigated. Again it was found that no reduction in c2 occurred when energy dependence or Coulomb-correction terms were included. Similarly no improvement was found by parameterizing aI as in eq. (4). rR was now allowed to vary and the following potential resulted:

(7)                                      

With final adjustments to the normalizations this potential fitted the data well with an average c2/N = 19.1, corresponding to an average rms deviation between experiment and theory of 44%.

Since 7Li has an isospin of , symmetry (or isovector) terms in the potential depths may exist. When such a term was included in the real potential the depth became
(8)                                      
with rR = 1.295 fm and similar values for the other parameters as in eq. (7). The potential fitted the data with an average c2/N = 20.5 and therefore shows that a real symmetry potential is unnecessary for 7Li scattering. It might be expected that the symmetry coefficient should be negative [as for neutrons 2)] but the searches definitely led to a positive coefficient with a fairly well defined (±1 MeV) magnitude as above. It is possible that this term represents some mass dependence in the real potential, rather than being derived from an isovector interaction.

The fits to the data are shown in figs. 7–9 and are satisfactory in almost all cases. The only exceptions are for 54Fe at 48 MeV, where the fit appears to have the wrong phase and not to be oscillatory enough, and for 40,48Ca at 88 MeV where the minima are far too deep. It is likely, however, that these problems could be resolved if coupled channels fits were made including a coupling to the ground-state quadrupole moment of 7Li.

…

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