Strong absorption model analysis of 15 MeV deuteron elastic scattering

Nuclear Physics 59 (1964) 641--650; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

S T R O N G ABSORPTION M O D E L ANALYSIS OF 15 MeV D E U T E R O N ELASTIC SCATTERING W. E. F R A H N

Physics Department, University of Cape Town and

L. P. C. JANSEN

Department of Physics, University of Stellenbosch, South Africa * Received 23 March 1964 Abstract: The angular distributions of 15 M e V deuterons elastically scattered by 23 target nuclei ranging from aluminium to lead are analysed by means of the strong absorption model. Generally consistent values of the three model parameters r0 (radius), d (diffuseness) and. t~/4A (real phase shift) are obtained. The systematic behaviour of the data is similar to that observed for other strongly absorbed composite projectiles such as He 3, alpha particles and heavy ions. The presence of all typical features arising from interference between diffraction-scattered and Coulomb-scattered. waves indicates that the elastic deuteron interaction is confined to the nuclear surface. Characteristic anomalies and deviations from the overall diffraction pattern for certain groups of target nuclei probably reflect nuclear structure effects.

1. Introduction Angular distributions of elastically scattered deuterons have been measured for a large number of target nuclei at various energies in the range from 3 to 30 MeV. (A bibliography of the experimental work is given in ref. i).) Basically the data follow the general diffraction pattern familiar from elastic scattering of alpha particles, He 3 and heavy ions. There are, however, certain systematic features which appear to be characteristic of deuteron scattering, and in some cases irregular differences between angular distributions for target nuclei with neighbouring mass numbers. Analyses of the data by means of models describing the average behaviour of the scattering interaction are useful for singling out those effects that are due to individual properties of target nuclei or to the particular structure of the deuteron. Most previous analyses of deuteron scattering were carried out by means of the complex potential model (CPM), and in many cases the calculations are in good agreement with the data. Recently, Perey and Perey 2) made an extensive investigation of 52 angular distributions for deuteron energies between 11 and 27 MeV, assuming a potential well (characterized by seven independent parameters) whose imaginary part describes a combination of volume and surface absorption. With t This work is based on a thesis submitted by L. J. to the University of Stellenbosch in partial fulfilment of the requirements for the M. Sc. degree. 641

64.2

W . la. FKAI.tN AND L. P. C. IANSEN

few exceptions, excellent fits to the data are obtained. However, the best-fit potentials are not uniquely determined; a given angular distribution can be fitted with comparable accuracy by considerably different parameter combinations. Although nonunique potentials are a general feature in CPM analyses of the scattering of composite particles, the multiplicity appears to be particularly great for deuterons. On the other hand, the various best-fit potentials yield practically the same values of the complex scattering function S(I) -- ~/l (see fig. 3 of ref. 2)). Another recent investigation by Halbert 3), who analysed 16 (d, d) angular distributions at 11-12 MeV by means of a six-parameter CPM with either volume or surface absorption, led to similar conclusions; widely different potential shapes are equivalent as to reproducing the experimental data, yet again the corresponding complex phases are essentially unique. The resulting distribution of I~1 shows the typical strong, absorption behaviour, i.e. a gradual transition, with increasing 1, from small values to unity over an intermediate range of partial wave numbers (see figs. I 1 and 12 of ref. 3)). Because of the multiplicity of potentials it is doubtful if the CPM description of deuteron scattering has a physical significance in the nuclear interior. One rather expects that the weakly bound deuteron would not be able to retain its identity except in the outermost fringe of the nuclear surface. For this reason, Percy and Perey 2) question whether the CPM can furnish more than a parametrization of the data in terms of a potential, and Halbert 3) considers the potentials simply as recipes for generating the correct asymptotic wave functions. However, the presence of redundant parameters in these potentials considerably complicates practical scattering analyses. An alternative approach, suggested by the uniqueness of the ~/t values, is the direct parametrization of the scattering matrix elements. This approach has been developed into a consistent analytical method for the phenomenological description of elastic nuclear scattering, called the generalized strong absorption model (SAM) 4' 5). By means of a simplified version of the SAM containing only three adjustable parameters, satisfactory agreement with the elastic scattering data for alpha particles, He 3 and heavy ions has been obtained 6). In the present paper this model is applied to deuteron scattering. Compared with CPM parametrization, the SAM analysis of experimental data is more straightforward and the resulting parameter values are essentially unique.

2. ~ a ~ s i s From the large number of deuteron scattering measurements available we choose the 15 MeV data of Jolly, Lin and Cohen 7). These data extend over a wide range of target nuclei so that systematic variations of the angular distributions with target atomic number can be studied. High energy resolution ensures complete elimination of inelastic scattering contributions. In the simplified SAM we assume that the nuclear (i.e. non-Coulombic) part of the scattering matrix elements can be approximated by Re r/l = g(t),

Im r/l = #dg(t)/dt,

(1)

DEUTERON SCATTERINO

643

g(t) = [1 + e x p ( T - t/A)]- 1

(2)

where t = 1+½ and

The resulting closed-form expressions for the ratio ¢r(0)/aR(0 ) of the differential scattering cross section to the Rutherford cross section are given by eqs. (4) and (8) of ref. 6). The three adjustable parameters T, A and # measure the "cut-off" orbital angular momentum, the width of the transition region in/-space and the real nuclear phase shift, respectively. The strength of the Coulomb interaction is determined by the value of mZ1 Z2 e 2 n , (3) h2k where k is the wave number, m is the reduced mass, and Z t and Z2 are the atomic numbers of projectile and target, respectively. The angular distribution features depend on the interference between Coulomb- and diffraction-scattered waves; the "critical angle" 0c = 2arctg (n/T)

(4)

separates the D region (0 > 0¢) where diffraction scattering is dominant, from the C region (0 ~ 0c) where Coulomb scattering predominates. In the simplified form described by eq. (1) the SAM applies to spin-zero projectiles and targets and to spherical nuclei. In the present analysis, spin-effects are assumed to be negligible. The spin interactions will affect the amplitude of the diffraction oscillations mainly at large angles, but only four of the 23 angular distributions of ref. 7) extend beyond 90 ° in the c.m.system. Certain target nuclei, such as Er, Yb, Ta and W, are known to have non-spherical equilibrium shapes. Appreciable deformation effects on the angular distributions will to some extent be reflected in the values of A and #. Procedures for analysing a given angular distribution in terms of the SAM were described in ref. 6). These methods make use of characteristic features in the experimental tr(0)/aR(0 ) distributions, such as the periods and amplitudes of the diffraction oscillations, the average slope of tr/trR at large angles, and the amplitude of the "rise". The determination of the parameters follows somewhat different lines according as the angular distribution is of the "oscillatory" type 1 or of the "smooth" type 2. The data of Jolly et al. can be roughly divided into type 1 distributions from A1 to Sn and type 2 distributions from Er to Pb. Because of irregularities observed for certain elements, and the fact that most of the data for medium-weight and heavy nuclei are confined to angles in the vicinity of 0o, where a/aR is not very sensitive to A and/~, the simple methods described in ref. 6) were in many cases not directly applicable. Nevertheless, aside from these exceptions, essentially unique "best-fit" values of T, A and # could be determined by searching a narrow region of three-dimensional parameter space, using an IBM 1620 computer for which eqs. (4) and (8) of ref. 6) were programmed.

644

W. E. FRAHN AND L. P. C. JANSEN 3. Results

F r o m T, A a n d # w e c a l c u l a t e t h e p a r a m e t e r s ro, d a n d # / 4 A , w h e r e r 0 a n d d a r e defined by

T = k R [ 1 - ( 2 n / k R ) l ÷,

(5)

R = ro(A~+A~2),

(6)

A = kd

1-(n/kR)

(7)

[1 - (2n/kR)] ~ '

and bt/4A = Im rh(T ) is the maximum value o f Im~h. F r o m semiclassical considerations we expect these parameters to be independent of energy and target atomic number *' 6) TABLE 1 Strong absorption model parameters for 15 MeV deuterons 0e

R

ro

d

Target

Z2

A,

n

(deg)

T

(fm)

(fm)

(fm)

/z/4d

A1 Ti Fe Ni Cu Zn Y Zr Nb

13 22 26 28 29 30 39 40 41

27.0 47.9 55.9 58.0 63.6 65.5 89.0 91.3 93.0

0.75 1.27 1.50 1.61 1.67 1.73 2.25 2.31 2.36

Mo

42

96.0

2.42

Rh Pd Ag Cd In

45 46 47 48 49

103.0 106.5 108.0 112.5 114.9

2.60 2.65 2.71 2.77 2.83

Sn Er Yb Ta W Pt Au Pb

50 68 70 73 74 78 79 82

120.0 167.3 173.1 181.0 183.9 195.1 197.0 207.2

2.88 3.92 4.04 4.21 4.27 4.50 4.56 4.73

12.8 19.1 22.6 25.7 24.9 25.1 31.4 45.0 45.2 35.7 45.7 36.8 42.6 47.3 40.3 39.7 50.3 42.5 47.1 55.2 56.4 53.5 60.7 59.8 58.4 63.9

6.69 7.54 7.49 7.08 7.57 7.76 8.00 5.57 5.68 7.34 5.75 7.29 6.66 6.06 7.39 7.67 6.01 7.27 6.61 7.49 7.53 8.35 7.29 7.82 8.15 7.58

6.69 7.73 7.88 7.64 8.09 8.31 8.99 7.09 7.24 8.57 7.36 8.58 8.27 7.86 8.97 9.25 8.02 9.00 8.55 10.43 10.59 11.41 10.70 11.37 11.68 11.48

1.57 1.58 1.55 1.49 1.54 1.57 1.57 1.23 1.25 1.48 1.26 1.47 1.39 1.31 1.49 1.52 1.31 1.47 1.38 1.54 1.55 1.65 1.54 1.61 1.65 1.60

0.54 0.41 0.46 0.43 0.47 0.50 0.38 0.30 0.30 0.64 0.34 0.63 0.33 0.37 0.46 0.51 0.33 0.75 0.35 0.45 0.52 0.57 0.49 0.44 0.55 0.51

--0.03 --0.04 0.00 0.01 --0.02 0.00 --0.05 0.12 0.13 --0.08 0.16 --0.02 --0.15 0.00 --0.04 --0.13 0.10 --0.10 --0.02 --0.09 --0.14 --0.28 --0.16 --0.24 --0.27 --0.25

0) (II) (I) (II)

(I) (II)

T h e p a r a m e t e r v a l u e s d e r i v e d f r o m a n a l y s i n g t h e d a t a o f J o l l y et aL a r e g i v e n i n t a b l e 1. T h e a n g u l a r d i s t r i b u t i o n s a ( 0 ) / a R ( 0 ) , c a l c u l a t e d w i t h t h e s e v a l u e s f r o m e q s .

DEUTERON

$CATTERINQ

645

[

I

I

t

(

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I.O

0.5

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1.0

oo o

oo

o

or(e) oR(e) 0.5

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I.O

I

0.5

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oO ~/

0.1 O.v,~

0

20

40

60 80 e(degreesl

I00

Fig. 1. Angular distribution o f 15 MeV deuterons elastically scattered by A1, Ti and Fe. The experimental data are from ref. 7) and the parameters are given in table 1.

I

I

I

20

40

60

I

I

80 I00 e(degrees)

120

140

Fig. 2. Angular distribution of 15 MeV deuterons elastically scattered by Ni, Cu, Zn and Y. See caption of fig. 1.

046

W.E.

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AND

L. P.

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I

I.O

[

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1

I

I

o o

-

°°

~c

°\

0.5

1.0

1.0 J

o-(e)

o" (e)

%(e) ~CNb

0.5

oo

0.~

,.,

I.O

o

e¢

'\ IMo

0.5

~

o

ol

~

o

ii

I.C

0.5

o 0 0

0.1

0.1 0

20

40

60

80

I00

e (degrees)

Fig. 3. Angular distribution of 15 MeV deuterons elastically scattered by Zr, Nb, Mo and Kh. S¢¢ caption of fig. 1.

120

140

I

0

20

40

60

80

I00

e(degrees)

Fig. 4. Angular distribution of 15 MeV deuterons elastically scattered by Pd, Ag, Cd and In. See caption of fig. 1.

DEUTERON

SCATTERING

647

I

[

,

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o

ec

I

I

I

1.0 I

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0.5

1.0

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Er

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0.5

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0.5

0.1

o.I 0

l

i

i

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20

40

60

80

]

i

I00

120

e(degrees)

'ig. 5. Angular distribution o f 15 M e V deuterons lastically scattered by Sn, Er, Y b a n d Ta. See caption o f fig. 1.

0.05 140

0

I

I

I

20

40

60

I

I

80 I00 (}(degrees)

T

120

140

Fig. 6. Angular distribution o f 15 MeV deuterons elastically scattered by W, Pt, A u and Pb. See caption o f fig. 1.

648

W.E.

FRAHN AND L. P. C. JANSEN

(4) and (8) of ref. 6), are shown in figs. 1-6. For Nb, Mo and In it was not possible to obtain unambiguous overall fits, and in table i two sets of parameters are given for each of these nuclei. The solid curves for Nb, Mo and In in figs. 3 and 4 correspond to sets (I) and fit the large-angle data, while the dashed curves pertain to sets (II) and fit the data at smaller angles. 4. Discussion

The calculated angular distributions reproduce the general trend of the data fairly satisfactorily. On the other hand, there are anomalies for certain groups of target nuclei. Before considering these, let us discuss the systematic behaviour. As described in detail in ref. 4), the SAM predicts the following variation of the main angular distribution features with increasing atomic number Z2: (i) damping of diffraction oscillations, (ii) steepening of the average slope of a/a R and (iii) decreasing amplitude of the "rise". It can be shown 4) that these features depend almost entirely on the Coulomb parameter n which increases linearly with Z2. In addition, for lighter nuclei where n is still small compared with T, we expect (iv) increasing 0c and (v) decreasing period of diffraction oscillations, with increasing mass number A 2. It is evident from figs. 1-6 that the data are generally in agreement with the predicted behaviour. A criterion for the consistency of the SAM analyses is the extent to which the parameters r o, d and ~/4A are constant. Aside from some exceptions to be discussed below, the values in table 1 are confined to relatively narrow ranges and meaningful averages can be formed. These are given in table 2. TABLE 2 Average SAM parameters for deuterons

(fm)

(fro)

/~14A

All values

1.48

0.46

--0.061

Excluding sets (I)

1.51

0.48

--0.086

Excluding sets (II)

1.48

0.44

--0.060

A feature characteristic of deuteron scattering is that the values of the real phase parameter ~t/4A are predominantly negative, especially for the heavy targets. This reflects the fact that for heavier nuclei the strong damping of the diffraction oscillations is not accompanied by a strong steepening of the average slope of a/aR at larger angles. In alpha particle and heavy-ion angular distributions the average a/oR at larger angles drops by several orders of magnitude, yet often the diffraction oscillations remain distinctly visible, and the rise in the Coulomb region has an appreciable

DEUTERON SCATTERING

649

amplitude. This behaviour is evidence of positive nuclear phase shifts*). For deuterons, on the other hand, the slopes of a/a R for heavy targets are so mild that non-negative values of # are incompatible with the observed quenching of the diffraction oscillations. This is corroborated by CPM analyses; for 12 MeV deuterons scattered by Cu, the values of Imr/l at intermediate l are negative, corresponding to 1~/4A ~ - 0 . 2 (see fig. 3 of ref. 2)). Anomalous deviations from the regular diffraction pattern are observed for the following two groups of nuclei: (a) Ti, Fe and Ni and (b) Zr, Nb, Mo, Rh and In. A common feature of the angular distributions for group (a) targets is the distortion of the second diffraction oscillation in the D region (0 > 0c). In group (b), the distributions for Nb, Mo and Ni can be fitted either at large or at small angles, corresponding to two different parameter sets (I) and (II), respectively. The parameters of sets (1) and those for Zr have exceptionally small values of ro and d, and relatively large positive values of #/4A, whereas the values of sets (II) are consistent with the parameters obtained for the other target nuclei. In some cases, the angular distributions are qualitatively different for elements with neighbouring mass or atomic numbers 7), such as Ni-Cu, M o - R h , Cd-In and In-Sn. These deviations from the smooth changes predicted by models describing average nuclear properties are probably due to differences in structure of the individual target nuclei. TABLE 3

Average SAM parameters obtained from angular distribution analyses for different projectiles Projectile

~0 (fm)

d (fm)

#/4.d

Deuterons

1.51

0.48

--0.09

He3

1.50

0.44

0.03

0t-particles

1.46

0.34

0.09

Heavy ions

1.44

0.28

0.26

In spite of the anomalies, the parameters obtained from our analysis are sufficiently consistent to allow a meaningful comparison of their average values with those derived from He 3, alpha particle and heavy-ion angular distributions 6). Table 3 indicates a systematic change of the average parameters with mass of the composite projectile; with increasing A~ the average diffuseness decreases, and the average phase shift increases from negative to positive values. The slight decrease in ~o is probably not significant. The average values for deuterons given in table 3 are those obtained by using sets (II) rather than the anomalous parameters (I) for Nb, Mo and In, yet the trend persists even if the latter are included.

650

W.E.

FRAHN AND L. P. C. JANSEN

For 15 MeV deuterons scattered by light target nuclei we have n < 1, and the "neutral" ~ersion of the SAM can be applied. In these cases the differential elastic scattering cross section is given by eq. (51) of ref. 4). This expression has recently been used in analysing angular distributions of 15.8 MeV deuterons scattered by Be 9, C 12, S32 and by Mg 24 in the range 9-22 MeV at the Pretoria cyclotron s). From our analysis we may ccnclude that deuteron scattering data in the intermediate energy region show the systematic diffraction behaviour which is typical of a strong absorption situation. The overall structure of the angular distributions is in accordance with the SAM assumption (1) on the form of the scattering matrix elements. This supports the semiclassical picture that deuteron interaction is confined to a surface zone of constant width with no appreciable penetration into the nuclear interior. References 1) 2) 3) 4) 5) 6) 7) 8)

E. W. Hamburger, Nuclear Physics 50 (1964) 66 C. M. Percy and F. G. Percy, Phys. Rev. 132 (1963) 755 E. C. Halbert, Nuclear Physics 50 (1964) 353 W. E. Frahn and R. H. Venter, Ann. of Phys. 24 (1963) 243 R. H. Venter, Ann. of Phys. 25 (1963) 405 R. H. Venter and W. E. Frahn, Ann. of Phys. 27 (1964) 401 R. K. Jolly, E. K. Lin and B. L. Cohen, Phys. Rev. 130 (1963) 2391 G. Heyman, M. J. Scott and R. L. Keizer, Bull. Am. Phys. Soc. 9 (1964) 55; M. J. Scott and R. L. Keizer, to be published