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Elastic scattering of 52 MeV deuterons
I
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Nuclear fihysics A l l l (1968) 265--288; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ELASTIC
SCATTERING
O F 52 M e V D E U T E R O N S
F. HINTERBERGER
Institut fiir Strahlen- trod Kernphysik der Unicersit?it Bonn G. MAIRLE, U. SCHMIDT-ROHR and G. J. W A G N E R
Max-Planck-lnstitut fiir Kernphysik, Heidelberg and P. T U R E K
Kernforschunysanlaye Jiilich, Kernforschunyszentrum Karlsruhe Received 5 February 1968 Abstract: Angular distributions for the elastic scattering of 52 MeV deuterons were measured from 27 nuclei ranging from hydrogen to uranium. The experimental data have been analysed in terms of the optical model. The inclusion of a vector spin-orbit term in the potential is necessary for nuclei with A > 100. For light nuclei with 12 ~_ A < 50, good fits to the experimental angular distributions could also be obtained without a spin-orbit potential. The optical-model parameters for nuclei with A ~_ 12 are discussed with respect to the dependence on the energy E, the charge Z and the mass A. The A-dependence of the real potential depth V can be understood as a consequence of the spatial polarization of the deuteron during the scattering process. Discrepancies between optical-model calculations and experimental angular distributions observed in the elastic scattering processes from XH, SHe, and 4He indicate contributions from transfer reactions.
E
N U C L E A R REACTIONS 1H, 3,aHe, 1~C, 14N, lsO, °-9,Z2Ne, -°~Mg, ZrAl, 2aSi, 328, 40Ar' 4s, r~q-i, .~4Fe' ~8,64Ni' ~gy, 93Nb' 103Rh' 140Ce' 181Ta' 19:Au' 20apb' 209Bi' 23sU (d, d); E = 52 MeV; measured a(0); deduced E- and A-dependence of optical-model parameters. Enriched targets.
1. Introduction In an e n e r g y r a n g e u p to 34.4 M e V , t h e a n g u l a r d i s t r i b u t i o n s o f elastically s c a t t e r e d d e u t e r o n s s h o w d i f f r a c t i o n p a t t e r n s w h i c h c a n be d e s c r i b e d well by t h e o p t i c a l m o d e l (refs. 1-8)). T h e k n o w l e d g e o f t h e d e u t e r o n o p t i c a l p o t e n t i a l is a n e c e s s a r y c o n d i t i o n f o r the s p e c t r o s c o p i c analysis o f r e a c t i o n s w i t h d e u t e r o n s in t h e e n t r a n c e o r exit c h a n n e l by t h e m e t h o d o f t h e d i s t o r t e d w a v e B o r n a p p r o x i m a t i o n ( D W B A ) . I n o r d e r to i n v e s t i g a t e t h e d e p e n d e n c e o f the o p t i c a l p o t e n t i a l o n the m a s s n u m b e r A, t h e c h a r g e Z a n d t h e s t r u c t u r e o f the n u c l e u s , o n e s h o u l d m e a s u r e the a n g u l a r distrib u t i o n s o f elastically s c a t t e r e d d e u t e r o n s f r o m m a n y t a r g e t nuclei. P r e v i o u s l y s u c h i n v e s t i g a t i o n s 1 - 8 ) w e r e p e r f o r m e d o n l y at a few lab e n e r g i e s b e t w e e n 11.8 M e V a n d 34.4 M e V . W i t h the c o n s t r u c t i o n o f i s o c h r o n o u s c y c l o t r o n s , o n e o b t a i n s e n e r g i e s w h i c h m a k e an i n v e s t i g a t i o n o f t h e " o p t i c a l " p r o p e r t i e s o f the nuclei v e r y p r o m i s i n g w i t h r e g a r d to the s h o r t e r de B r o g l i e w a v e l e n g t h . I n t h e case 265
266
F. ItINTERBERGERet
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of nuclei with A < 4, a deuteron lab energy of more than 50 MeV corresponds to a c.m. energy which lies near or above the total binding energy of the compound systems. With respect to the "few-nucleon" problems, the elastic scattering data may provide information about the importance of transfer processes in the elastic channel. Previously 9,1 o), we reported the measurement of the angular distributions for the elastic scattering of 52 MeV deuterons from 16 nuclei ranging from carbon to uranium and the preliminary results of an optical-model analysis. Since these publications, 11 further angular distributions have been measured and analysed. This paper is the comprehensive report of our systematic studies concerning the elastic scattering of 52 MeV deuterons. 2. Experimental procedure
The experiments have been performed with 52 MeV deuterons of the Karlsruhe isochronous cyclotron. The deflected beam is focussed with quadrupole lenses on the target without any diaphragms near the scattering chamber. The beam spot on the target is about 2 mm broad and 3 mm high. The quality of the beam is better than 10 m m • mrad. The angular distributions of the scattered deuterons are taken with a scattering chamber described previously t 1). The detectors are mounted outside the vacuum chamber on ring tables movable by remote control. We use dE-E telescopes consisting of 200 ~Lm-400/2m Si surface-barrier counters ( O R T E C ) as dE-counters and NaI(TI) scintillation counters as E-counters. A few differential cross sections are taken with a Ge(Li) detector as E-counter. In the case of the 3,4He targets, the differential cross sections at backward angles have been obtained by measuring the recoil nuclei scattered in forward angles with a telescope containing a 2 mm Si surface-barrier detector as E-counter. In the case of 1H(d, d)lH, the differential cross sections at backward angles are measured by the detection of the protons with a Ge(Li) detector as E-counter in the telescope. The corresponding experimental results are given as crosses in fig. 2 in comparison with the deuteron experimental points. There is good agreement of the data points in the overlap region. The over-all energy resolution is about 600 keV with the NaI(TI) detectors and about 250 keV with Ge(Li) and Si detectors as E-counters. A typical spectrum taken with a NaI(TI) detector is given in fig. 1. With the exception of 93Nb, 103Rh ' 181Ta ' 197AHand 238U,the elastic peak is well separated from the inelastic spectrum. But the shape of the spectra and the results on inelastically scattered deuterons from even-mass nuclei suggest the error from inelastic contributions to be small especially at forward angles. However we cannot exclude the possibility that the very smooth structure of the angular distributions of ~°aRh, 18tTa and 238U is caused partly by this lack of resolution. The angular resolution is 0.5 ° in the regions of sharp diffraction structure and between 0.5 ° and 1.2 ° at other angles. The precision of the relative angle position is better than 0.I °, the precision of the absolute angle position lelative to the 0 ° symmetry is +0.2 °.
267
ELASTIC DEUTERON SCATTERING
The energy of the beam has been determined using the cross-over technique. The method is to determine in a precise left-right measurement the angles at which deuterons elastically scattered from hydrogen have the same energy as those inelastically scattered from 12C leading to the excitation of the 4.433 and 9.629 MeV states. As the target, we use a foil of polyethylene (CH2) ~ and as detector a dE-E telescope with a Ge(Li) E-counter. The dependence of the cross-over angles on the energy of the primary deuteron beam is exactly computed relativistically. The result is E a = 51.7___0.2 MeV. I
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3. Targets The measurements on the nuclei 27A1, 48'5°Ti, 5¢Fe, 8 9 y , 9 3 N b ' tO3Rh ' 140Ce ' l S ' T a , t 9 7 A u , 2°8pb, 2°9Bi and 238U were performed with rolled foils of 1 mg/cm 2 to 10 mg/cm z thickness. The 12C, 2¢Mg, SSNi and 64Ni targets were self-supporting vacuum-evaporated foils of 250/~g/cm 2 to 1000 #g/cm s thickness. The 160 data points result from measurement on SiO2 and Si targets. The Si target was a 20 #m thick defective surface-barrier counter. The measurements on 1H, 3He, 4He, 2°Ne, 22Ne, asS and ¢°Ar were performed using a gas target with a diameter of 10 cm and a volume of about 1 1, which is inserted into the scattering chamber instead of the solid target. In the reaction plane, the particles pass through a slit which is covered by a 4.6 mg/cm 2 thick havar foil. The maximum pressure is 500 m m Hg. The pressure is continuously monitored with a Hg manometer during the measurement, and the volume density of the irradiated gas is computed according to the ideal gas law. The geometry factor G is computed according to the formulae of Silverstein (ref. 12)). The geometry was chosen so that the correction terms were smaller than 1 ~ . They are computed up to the fourth-order term assuming a theoretical opticalmodel angular distribution.
268
r. HINrrERBERGER et al.
]n the 22Ne, 24Mg, 48,5°Ti, 54Ee, 58,64Ni and 2°8pb measurement, we used isotopically enriched targets. The statistical error has been kept below 3 9/o, and the data have been corrected for the deadtime which was always smaller than 10 ~o. The reproducibility of a measurement with different geometries and different telescopes was better than 2 ~ . In total the relative error in the angular distributions lies below 5 ~ .
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269
ELASTIC DEUTERON SCATTERING
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271
ELASTIC DEUTERON SCATTERING
within 2 %, which is the reproducibility range of measurements with completely different geometries and dE-E telescopes. The main error of the absolute values results from uncertainties in the target thickness determination. The thickness of solid targets was measured both by weighing and by measuring the energy loss of alpha particles. For 160, 24Mg ' 28Si ' 48,50Ti ' 58'64Ni and 89y, a measurement of the absolute cross sections was prevented by the impurities in the targets. In these cases, the absolute cross section has been determined by fitting the experimental data to several optical-model calculations at forward angles. This procedure is justified by the observation that the c.m. calculations are very insensitive to parameter changes at forward angles. The volume density in the gas target is determined with an error of about 3 %. Therefore the total error in the absolute cross sections is about 10 % for solid targets and about 6 % for gas targets 4 /
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F i g . 5. P o s i t i o n o f t h e m i n i m a ( o ) a n d t h e m a x i m a ( I I ) o f t h e e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s as a f u n c t i o n o f A { . T h e c o r r e s p o n d i n g e x t r e m a are c o n n e c t e d b y s m o o t h curves.
4. Experimental results Figs. 2-4 show t the measured angular distributions. As in the case of lower energies the extrema move to forward angles with increasing mass number A according to the Tables of cross sections are readily provided by the authors.
F. HINTERBERGER et al.
272
diffraction law k R sin 0 = const (k is the wave number, R the radius of the nucleus and 0 the scattering angle). In fig. 5, the position of the maxima and minima is plotted versus A ~. The corresponding cxtrema are connected by smooth curves. In the region of 14°Ce, one extremum is vanishing. This effect of vanishing maxima or minima has been observed at lower energies in other mass regions by several authors l a). Deviations from the smooth curve of the minima near 28 ° in the region of 5°Ti and 54Fe and near 35 ° in the region of 89y are obvious, i.e. in the region of the closed N = 28 and N = 50 neutron shells. Other deviations at backward angles are within the error of the position of the extrema. 5. The
optical-model analysis
5.1. THE OPTICAL POTENTIAL The form of the local optical potential used in this analysis of 52 MeV deuteron scattering is the same as has been used in analyses at lower energies 1a). As the main part of absorptive processes is known to take place in the surface region, we assume a pure surface absorption term. The potential may be explicitly written as
U(r) = V f ( r ) + i W g ( r ) +
V~.o.h(r ) ( l . s),
f ( r ) = - {1 + exp [ ( r - Akrv)/av]} - 1, g(r) = - d / d r { 1 + exp [ ( r - A ~ r w ) / a w ] ) - 1 ,
(l)
h(r) = - (h/m~c) 2 1/r d/dr f ( r ) .
The geometrical parameters of the spin-orbit potential are the same as those of the real part of the potential. We take the potential of a uniformly charged sphere of radius R c = r c A ~ fm for the Coulomb potential Ue(r ) = Z e 2 / 2 R c ( 3 - r 2 / R c )
= Z e2/r
r < Re r > Rc.
(2)
The geometrical parameter r c has been kept fixed at r c = 1.3 fm. 5.2. SEARCH PROCEDURE The above optical-model potential contains the seven free parameters V, 14I, V~.... rv, av, rw and aw. A search procedure was performed in order to minimize the value N
Z2 = N - 1 ~ (acxp(0,)- ath,o~(0~)) z/Aa~xp(O,) , ,
(3)
i=1
N is the number of measured angles, trexp(0~) and tr,h,or(0~) the experimental and theoretical cross sections and Aa~xp(01) the experimental uncertainty at angles 01. We have taken AtYexp(Oi) = 0.10"exp(0i) uniformly for all points.
ELASTIC DEUTERON SCATI"ERING
273
The search procedure was performed using the automatic search code ABACUS written by Auerbach t4). Unfortunately this search code can treat only spin ½ particles. But the main effect of a spin-orbit potential with spin ½ is the same as with spin I, namely a smoothing of the backward diffraction structure. Moreover the discrepancies of a spin ½ and a spin 1 angular distribution computed with identical parameters are found to be small. Therefore it is justified to perform the automatic parameter search with the code ABACUS. The fit was improved in the final spin I calculations by a 10 ~ reduction of the imaginary potential depth W. The curves in figs. 2-4 have been calculated with the correct spin s = 1 of the deuteron by means of the code J U L I E ts) using the parameters in tables 1, 2 and 5. 5.3. O P T I C A L - M O D E L S E A R C H O F T H E D A T A IN T H E MASS R A N G E 12 ~ A ~ 238
In considering starting points for an analysis, one is concerned with discrete V and continuous Vr~ = const ambiguities of the optical potential13'16). The most successful choice 17-20) so far of the potential V of the deuteron is the sum of the neutron and proton optical potentials at half the deuteron energy, i.e. V lies in the interval 70 MeV to 110 MeV. Because of the Vr~ = const ambiguity, we kept the real parameter rv fixed and performed searches with the following two values: set I
rv = 1.25 fm,
V~tart =
75 (MeV) V~.o. :~ 0,
set 2
rv = 1.05 fm,
V~tar, = 100 (MeV) V~.o. ~ 0.
The starting points of the remaining parameters were taken from the values at lower energies 13). The preliminary results for a series of target nuclei of the optical-model analysis with parameter set 1 is reported in ref. 9) and with parameter set 2 in ref. 1o). The variation of the spin-orbit potential was found to be very unspecific. Therefore we used V.~.o"= 6 MeV throughout the following analyses. In tables 1 and 2, the best fit optical-model parameters of set I (r V = 1.25 fm) and set 2 (r v = 1.05 fm) are given for target nuclei with 12 < A < 238. The theoretical and experimental angular distributions are compared in figs. 3 and 4. Many opticalmodel calculations la) at lower energies have been performed without a spin-orbit potential. With increasing energy, however, one finds it necessary to include a spinorbit potential to obtain reasonable fits 8, 2 i). To clarify this question, at the energy of 52 MeV we also performed searches without a spin-orbit potential, i.e. set 1"
with rv = 1.25 fro,
Vs.o. = 0,
set 2*
with rv = 1.05 fro,
V.~.o. = 0.
In the mass range 12 < A < 64, one gets fits comparable to those including a spinorbit potential (tables 3 and 4). In the mass range 64 < A < 100, the fits are reasonable, but for A > 100 the angular distributions calculated without a spin-orbit potential show too distinct minima as one notices in fig. 6 for 2°spb.
TABLE 1 Parameter set 1; rv = 1.25 fro, r C = 1.3 fm Target
V (MeV)
av (fln)
Vs,o, (MeV)
rw (fm)
a~ (fm)
W (MeV)
g"
~2C NN i60 Z°Ne 2ZNe ~Mg ZrAl ~Si azS ~°Ar ~Ti 5°Ti S4Fe ~Ni 6*Ni sgy aSNb l°3Rh N°Ce l~lTa 19~Au z°~Pb "2°gBi z~sU
67.4 66.1 68.2 67.5 67.3 68.0 67.1 69.3 70.2 71.1 70.0 71.3 71.5 70.4 71.5 75.1 75.3 75.5 75.8 78.9 80.0 79.8 79.5 81.0
0.75 0.75 0.693 0.75 0.75 0.75 0.737 0.78 0.75 0.75 0.744 0.733 0.75 0.742 0.783 0.75 0.718 0.68 0.819 0.735 0.668 0.66 0.65 0.703
6 7.5 6 6 6 6 7 6.6 6 7 7.9 6.64 6 7 6.74 6 6 6 6.43 6 6 6 6 6
1.25 1.28 1.25 1.25 1.25 1.25 1.22 1.24 1.25 1,25 1.25 1.25 1.25 1,24 1.25 1.25 1.25 1.25 1.22 1.25 1.25 1.25 1.25 1.25
0.744 0.66 0.75 0.825 0.782 0.787 0.754 0.75 0.75 0.75 0.759 0.714 0.73 0.714 0.774 0.756 0.95 0.87 0.791 0,95 0.97 1.0 1.0 0.98
10.9 l t. 1 12.2 12.7 13.4 13.9 12.5 13.1 14.2 14.5 13.2 14.2 13.0 14.0 14.1 14.7 11.7 13.0 15.2 12.6 13.2 12.0 12.2 13.0
3.1 3.5 5.0 4.3 3.7 5.4 1.4 3.0 3.4 3.5 3,0 3.0 3.7 4.1 5.2 9.0 5.8 9.5 9,4 11.7 4.0 5.2 6.3 7.7
TABLE 2 Parameter set 2; rv = 1.05 fm, r C = 1.3 fm Target
V (MeV)
a,, (fm)
lzC ~4N 160 ZONe ZZNe ~-~Mg ~AI 28Si ~zS 4OAr 4aTi ~OTi raFe 58Ni 64Ni ~gy 9~Nb l°'~Rh ~4oCe lStTa l~rAu zo~Pb ~-°~Bi 23sU
83.1 84.0 85.0 84.6 84.5 86.1 86.5 86.9 87.5 88.9 90.0 89.0 92.0 93.1 93.5 97.0 97.1 98.3 102.6 106.0 105.5 106.4 106.3 106.1
0.80 0.80 0.810 0.90 0.9 0.787 0.839 0.83 0.85 0.85 0.85 0.852 0.889 0.845 0.898 0.924 0.893 0.903 0.924 0.95 0.94 0.945 0.96 0.793
I(~.o. (MeV) 7 7 6 5 6 6 6 6 6 7 7.9 7.9 6 6.8 6.7 5.9 6 6.1 6.8 6 6 6 6 7.6
r~ (fro)
aw (fro)
W (MeV)
gz
1.22 1.28 1.28 1.2 1.25 1.28 1.28 1.28 1,28 1.3 1.29 1,29 1.28 1.33 1.33 1.34 1.28 1.35 1.27 1.3 1.3 1.29 t .3 1,2
0.75 0.779 0.75 0.85 0.799 0.75 0.75 0.76 0.8 0.779 0.76 0.76 0.75 0.735 0.864 0.696 0.75 0.677 0.801 0.80 0.82 0.80 0.81 0.88
9.1 8.4 9.3 11.3 12.3 12.6 11.0 10.9 t 1.4 1 t.7 12.5 12.4 12.1 12.2 13.1 15.5 15.5 16.8 15.5 17.2 16.1 17.1 17.0 16.4
2.0 2.3 5.9 3.4 4.2 6.3 5.4 6.1 3.5 2.0 2.2 1.6 7.8 4,2 3.2 4.1 4.8 5.6 2,0 5.8 2.3 4.5 5.2 4.0
275
ELASTIC DEUTERON SCATTERING
TABLE 3 Parameter set 1"; Vs.o. = 0 MeV, rv = 1.25 fro, r C = 1.3 fm Target
V (MeV)
W (MeV)
12C 14N 160 2°Ne 2ZNe 24Mg 27A1 26Si s2S 4°Ar 4~Ti 6°Ti 64Fe 6SNi ~4Ni
71.8 68.9 76.8 67.4 68.3 67.5 69.8 73.0 73.4 74.1 72.5 74.0 72.3 75.6 76.2
11.0 10.2 11.9 11.7 13.4 14.8 12.9 12.3 14.3 14.4 15.1 14.0 12.8 15.5 15.0
av (fro) 0.700 0.70 0.75 0.80 0.75 0.71 0.705 0.75 0.75 0.75 0.733 0.75 0.75 0.75 0.75
rw (fm)
aw (fm)
1.25 1.25 1.25 1.25 1.25 1.15 1.18 1.25 1.25 1.25 1.21 1.25 1.25 1.25 1.25
Z2
0.70 0.70 0.75 0.80 0.75 0.775 0.778 0.75 0.75 0.75 0.739 0.75 0.75 0.75 0.75
1.3 0.8 6.9 3.3 3.8 4.9 2.9 5.9 4.0 3.9 4.0 6.1 6.0 12 7.9
TABLE 4 Parameter set 2*; Vs.o. = 0 MeV, r v = 1.05 fro, r c = 1.3 fm Target I~C 14N ~60 ZONe Z'~Ne 24Mg 2~AI 2sSi a2S 4°Ar 4STi S°Ti 64Fe 6SNi 64Ni
V
W
av
r~
aw
(MeV)
(MeV)
(fro)
(fro)
(fm)
87.7 92.1 98.6 88.6 84.9 83.5 85.5 89.5 89.0 92.3 89.6 93.2 91.1 98.7 98.7
9.48 10.6 11.3 11.9 11.6 11.6 9.76 10.2 12.6 12.5 11.7 12.3 11.5 14.0 13.3
0.75 0.85 0.85 0.90 0.85 0.746 0.80 0.84 0.85 0.85 0.843 0.85 0.86 0.85 0.87
1.20 1.20 1.33 1.20 1.33 1.31 1.30 1.28 1.33 1.33 1.26 1.33 1.28 1.33 1.33
0.75 0.75 0.75 0.85 0.75 0.778 0.82 0.810 0.75 0.75 0.820 0.75 0.80 0.75 0.78
Zz
3.1 2.6 20 6.1 5.9 6.4 2.7 8.9 5.4 2.4 3.6 6.0 8.1 14.5 8.1
5.4. OPTICAL-MODEL DESCRIPTION OF THE REACTIONS lH(d, d)lH, aHe(d, d)SHe AND 4He(d, d)4He T h o u g h t h e b a s i c a s s u m p t i o n s o f t h e o p t i c a l m o d e l as, f o r i n s t a n c e , t h e a v e r a g i n g o v e r m a n y r e s o n a n c e s o f t h e c o m p o u n d s y s t e m a r e p o o r l y fulfilled in t h e c a s e s u n d e r d i s c u s s i o n , we p e r f o r m e d a n o p t i c a l - m o d e l s e a r c h o f t h e e l a s t i c d a t a l H ( d , d ) I H , a H e ( d , d ) a H e a n d 4 H e ( d , d ) g H e . Fig. 2 s h o w s t h a t in e a c h c a s e t h e f o r w a r d a n g l e s a r e n i c e l y r e p r o d u c e d b y t h e o p t i c a l m o d e l . H o w e v e r it is i m p o s s i b l e t o get t h e diffraction structures at b a c k w a r d angles. Even with the r i g o r o u s variation o f starting
276
F. HINTERBERGER e t a l.
parameters, the calculation with or without a spin-orbit potential and the use of a Gaussian form factor instead of a Woods-Saxon form for the real potential
f(r)
= exp { - [ ( r - R ) / a ]
2}
gave no essential improvement. The different optical parameters found in this analysis with the value 7.2 for the total angular distribution and separated for forward angles 0.... < 90 ° and backward angles 0.... > 90 ° are compiled in table 5. ~O
~
I
I
t
I
i
~OFI ~,~,x
I
208pb • -~ -
0.1
..~, -
I
i
L
i
i
;
(d,d)
E a = 5 2 MeV
0001
00001
i
i
i
40*
i
r:~
i
,
~g3*
i
loo*
1~
ecM
Fig. 6. Optic-d-model fit w i t h o u t a s p i n - o r b i t potential in the reaction 2°sPb(d, d)2°sPb. The parameters are V = 111.3 MeV, r v ~ 1.05 fm, a v -- 0.85 fm, W = 20.5 MeV, rw = 1.2 fm, a w = 0.8 fm, Vs.o. = 0 MeV, r c = 1.3 fm, Zs = 28. TABLr 5 O p t i c a l - m o d e l - p a r a m e t e r sets of elastic s c a t t e r i n g with 52 MeV dc ut e rons on ' H , aHe a n d 4He Tar get IH IH XH 3He SHe SHe SHe 4He ~He ~He 4He 4He
V (MeV) 65.8 92.1 37.5 a) 61.4 67.0 80.5 82.5 73.2 70.0 91.8 86.0 98.7 a)
rv (fm)
av (fm)
Vs.o. (MeV)
rw (fm)
aw (fm)
W (MeV)
Z2
Zt"
Z22
1.25 1.05 0.0 1.25 1.25 1.05 1.05 1.25 1.25 1.05 1.05 0.0
0.501 0.500 1.918 0.599 0.590 0.700 0.66 0.614 0.592 0.700 0.661 1.918
0.0 0.0 0.0 0.0 6.0 0.0 6.0 0.0 6.0 0.0 7.0 0.0
1.2 1.02 1.3 1.01 1.25 1.05 1.14 1.01 1.25 1.10 1.15 1.66
0.517 0.511 0.466 0.598 0.630 0.650 0.640 0.647 0.634 0.750 0.639 0,591
4.59 7.67 5.10 9.03 8.10 8.94 9.00 11,5 8.2 8.34 8.9 6.75
5.8 3.9 17 25 15 27 27 22 21 28 26 15
0.6 0.8 3.0 6.5 8.2 7.2 6.0 0.9 1.4 3.0 2.1 4.9
14 8.5 38 51 49 55 58 39 37 47 44 24
In each case, r c has been kept fixed to r c = 1.3 fm. The value Z ~ is calculated using all me a s ure d angles, whereas the value Zt ~ is defined for the range 0 ° ~ 0c.m. "~ 90 ° and the value Ze2 for the range 90 ~ ~ 0c.m. ~ 180 ° . a) The real potential has a G a u s s i a n form instead o f a W o o d s - S a x o n form.
ELASTIC DELrTERON SCA'I-I'ERING
277
The failure of the optical model at backward angles might be caused by the lack of antisymmetrization. The optical model cannot treat the nucleon-transfer processes especially the nucleon-transfer reaction between identical states. In the last case, we have the following reaction mechanism: d t q- ( d a + C ) - - ~ ( d l + C ) + d 2. Experimentally such transfer processes cannot be distinguished from elastic scattering. Evidence for the existence of these transfer processes comes from the energy dependence of these effects. Unfortunately data on 1H, 3He and ~He deuteron elastic scattering have been published for only a few energies. 1OOO
~e
I
I
% o ~
I
i
i
I
I
4He (d,d)4He
1000
100
"-
ooo
10 °
l
~
°
i~i 100
°
"
"• •
~
165 ° ° ~,
". • •
o 10
10
l
347
I
(3=
I
40*
I
I
80*
I
I
120"
1
I
160" Oc.l,~
Fig. 7. Comparison of the elastic scattering data of 4He(d, d)4He taken at the following c.m. energies: £c.m. -" 14 MeV [re£. 2,)], Ec.m" ~ 16.5 MeV [ref. 29)] and Ec.m. = 34.7 MeV (this work). The solid curve for Ee.m. = 34.7 MeV is calculated using parameter set 1 (table 5). The optical-model parameters of the fit to the Ee.m. = 14 MeV data are F .... 67 MeV, r v = 1.25 fro, av ~ 0.59 fro, W = 3.95 MeV, r w - - 1.25 fro, a w - - 0 . 6 3 fro, F s . o . = 6 M e V , r c - 1.3 fm, Z ~---6.9. For the crosses at Ec.m. ~ 34.7 MeV, see fig. 2.
In the energy range 8 MeV < Ec.m. • 27 MeV, all the known angular distributions of the 1H(d, d ) l H reaction 2z-2s) show a strong increase of the differential cross sections with increasing angle 0 ..... for 0.... > 120' (see fig. 2). As there is no marked difference in the backward angular distribution over the energy range considered here, one cannot deduce information about the reaction mechanism from the energy dependence in the case of IH(d, d ) ' H .
278
F. HINTERBERGERet al.
The elastic scattering * 3He(d, d)3He has been measured at the following energies (refs. 26.27)): Elab ~ 14.4 MeV and 24 MeV < El, b < 27 MeV. Unfortunately, the angular distributions of the latter energy have been measured only in forward angles up to 0.... < 104°. In this angular range, an optical-model fit is easy to obtain just as in the case of the 52 MeV data. We shall discuss in detail only the 4He(d, d)4He reaction with respect to the energy dependence of the backward angular structures. In fig. 7, one finds the elastic angular distributions measured 2~,29) at c.m. energies Ec.m. = 14 MeV and E¢.m. = 16.5 MeV in comparison with our Ec.m. = 34.7 MeV data. The pronounced diffraction structure at backward angles is smoothed out at lower energies. For the E~.m. = 14 MeV data 27), we performed an optical-model search which gives a relatively good fit even at backward angles. The o.m. parameters are given in the caption of fig. 7. This comparison with the data at lower deuteron energies is a good argument that in the case of 4He(d, d)4He at a c.m. energy E¢.m. = 34.7 MeV, which lies 6.5 MeV above the total binding energy of 4He, there is a nucleon-transfer reaction dominating the backward angular range. It would be interesting to investigate the energy dependence of the pronounced structures at backward angles for 1H, 3He and 4He in the region Ee.m. : 50 MeV. This would give further information about the reaction mechanism as well as the fewnucleon compound systems 3He, 5Li and 6Li.
6. Results and discussion of the optical-model analysis 6.1. TOTAL REACTION CROSS SECTIONS Several authors have found agreement between measured total reaction cross sections and those calculated from the optical model 13) to within 20 % for deuteron energies up to 26 MeV. The values of a .... tion c',dculated with parameter set 1 differ less than 10% from those of parameter set 2. In fig. 8 is a plot of the calculated total reaction cross sections 6reaction versus A ~. The points are the average of an opticalmodel calculation with parameter set 1 resp. 2. A smooth curve is drawn through the points. The corresponding curves at the deuteron energies Eo = 11.8 MeV [refs. 1- 3,30)] and 26 MeV [ref. 31)] are drawn for comparison. To demonstrate the small effect of the Coulomb repulsion, we calculated the total reaction cross section of a 52 MeV "deuteron" with charge Z = 0. The result is given by the dashed curve in fig. 8. 6.2. THE RADIUS PARAMETER OF THE REAL POTENTIAL We began our analysis with a radius parameter rv fixed to the value rv = 1.25 fm. However, it was impossible to fit the elastic angular distribution 14°Ce(d, d)14°Ce with this parameter set 1. As fig. 4 shows, we obtained one additional maximum in the calculated angular distribution. Only with a radius parameter near r v = 1.05 fm, For the '1He(d,aHe)aH reaction, we have measured cross sections which are similar to those observed in the "~He(d.dpHe reaction at corresponding angles.
ELASTIC DEUTERON SCATTERING
279
we could get a reasonable fit. We therefore calculated in the case o f 2°8Pb(d, d) an angular distribution with parameters which are the arithmetic mean of sets I and 2. The result is an excellent fit to the data points (fig. 9). For the nuclei with ,4 < 64,
3.5
// .
30
//
///~/
25
"
////~ // // •
T .~
/#
15
52MeV •
•
//" /
1.0 /
I~V
0,.5 4He 160 'l
''l
50Ti54Fe 89y
'""'
'
~ll
I I
i t
140Ce 208pbA2/3
i
I
1
'
I'
Fig. 8. T o t a l r e a c t i o n cross s e c t i o n O'reaction as a f u n c t i o n o f A L T h e p o i n t s are t h e a v e r a g e o f a n o p t i c a l - m o d e l c a l c u l a t i o n u s i n g p a r a m e t e r set 1 ( t a b l e 1) a n d p a r a m e t e r set 2 ( t a b l e 2). A s m o o t h c u r v e is d r a w n t h r o u g h these p o i n t s . F o r c o m p a r i s o n , the m e a n A~ d e p e n d e n c e o f O'reaetion is g i v e n for the lab e n e r g i e s Ea -- 11.8 M e V [refs. i-3,1z, 20)] a n d E d = 26 M e V [ref. al)]. T h e d a s h e d c u r v e is a n o p t i c a l - m o d e l c a l c u l a t i o n o f a " d e u t e r o n " w i t h Z = 0 a n d the l a b e n e r g y E d = 52 M e V u s i n g the p a r a m e t e r s o f t a b l e 1, 2 a n d 5.
o '[0
l
I
I
I
I
I
I
l
I
I
I
l
I
I
I
l
I
I
I
0.1
O01
20*
40*
60*
80*
I
100"
120*
OcM
Fig. 9. Best fit to the '~°sPb(d, d)2°sPb d a t a u s i n g t h e f o l l o w i n g p a r a m e t e r set: V = 93 M e V , rv = 1.15 fro, av = 0.80 fm, W = 14.5 M e V , rw = 1.25 fm, aw = 0 . 8 0 fm, Vs.o. = 6 M e V , r c = 1.3 fro, Z z ~ 1.5.
280
F. HINTERBERGER et al.
the parameter set 1 with r V = 1.25 fm gives the better fits. We conclude that in addition to the pure A ~ law (Rv = A+rvwith rv = const), the radius of the real optical potential has a further dependence. The general behaviour of this is shown in fig. 10. II
I
w(fm) 14
I
I
I
I
l
I
I
I
I
I
I
I
l lll I 11..~ I I lOO 2ool A 89y 140Ce 208pb
12 10 08 06 Q4 02 0
I
J
Fig. I0. T h e ,4-dependence o f the radius p a r a m e t e r rv o f the real potential. 6.3. T H E D E P E N D E N C E O F T H E R E A L P O T E N T I A L D E P T H V U P O N C H A R G E Z, MASS ,4 AND ENERGY E
In figs. 11 and 12, the real potential depths V of this analysis are compared with those found at lower deuteron energies. They have been obtained with comparable
v~ev3
'
'
'
'
'
,
~,
90
85
/
/
•
i
l
o
~ o///c
.
80-
70
, , ,
S
,
,
o
,
,
2"15 MeV
_/~,,...5 2 IvleV
~///~,
,,~
/
• "
65 , , , "I
4He ~0
, "4"'
'
,
I
5a~54Fe 8gy
140Ce
2OSpb
Fig. I I. Real potcntial depths V as a function of Z / A - ' s for thc parameters of set I (table I). Thc 27.5 McV data 7) are transformed according V : ~ : 5 ~ const to the rca] potential radius rv -- 1.25 fro. The solid curves are calculated according relation (5) of the tcxt with V0 = 79 MeV, Ki = 1.5~ K~ = 0.35. T h e dashed curve would have resulted from a fit procedure using the real potential f o r m f D ( r ) o f e q . (12).
ELASTIC DEUTERON S C A T T E R I N G
281
g e o m e t r i c a l p a r a m e t e r s r v a n d a, a n d c o r r e s p o n d to a b o u t twice the n u c l e o n p o t e n t i a l (refs. 6, 7, 8,13)). F o r the r e d u c t i o n o f V f r o m the earlier d a t a to the r a d i u s p a r a m e t e r s u s e d in sets 1 a n d 2 (r V = 1.25 f m a n d rv = 1.05 fro), we h a v e a p p l i e d the r e l a t i o n
VrJ 5 = const.
vEMeV3
I
I
I I I I I I I I I I I I l i
130
125 -
120
115
110
344 MeV/ "~ 52 Me',/
105
100
,i
90 85
o.12"I"4"'I'6 '#'He 160
~' 8
~'
50"~i 54Fe89y !4'°Cer 2°8pb
a6Z/A~
Fig. 12. Real potential depths Vas a function of Z/A4: for the parameter set 2 (table 2). The 11.8 MeV data are from refs. a3, 2~), the 25.9 MeV data from ref. 6) and the 34.4 MeV data from ref. 8). They are transformed according Vr~.5 = const to the real potential radius r v = 1.05 fln. The solid curves are calculated according to eq. (5) with Vo -- 100 IVleV, K1 -- 2.5 and K~ .... 0.5. The dashed curve would have resulted from a fit procedure using the real potential formfD(r ) ofeq. (12). T h e solid c u r v e s in figs. 11 and 12 are c a l c u l a t e d with the c o n s t a n t s in t a b l e 6 (lines 6 a n d 7) u s i n g the f o l l o w i n g e x p r e s s i o n o f P e r e y 16), w h i c h d e s c r i b e s the m e a n d e p e n d e n c e o f V on the e n e r g y E a n d the C o u l o m b p a r a m e t e r Z/A'~:
V = V o + K 1Z/A + - K 2E,
(5)
V o, K 1 a n d K2 are c o n s t a n t s , E t h e c.m. energy, Z the c h a r g e o f the t a r g e t n u c l e u s a n d A the m a s s o f the t a r g e t nucleus.
F. H,NTERBERGER el aL
282
TABLE 6 T h e coefficients V0, K~ a n d K s
E,~b
rv
(MeV)
(fro)
Vo
K1
Kz
KI/ K.~
Ref.
(MeV)
11-27
1.15
81
2
0.22
9.1
tn)
11-27
1.3
75
1.14
0.42
2.72
a6)
91.2
1.03
0.57
1.78
25.9
1.13
27.5
1.21-1.265
1.7
~)
34.4
0.968 ÷ 0 . 0 2 9 , 4 3
27.5, 52
1.25
79
1.5
0.35
4.3
this work
11.8, 25.9, 34.4, 52
1.05
100
2.5
0.5
5
this work
0.89
r) s)
For definition, see eq. (5).
In the following section, we shall discuss the information contained in the parameters K, and/<2. The idea of expression (5) is to give within the framework of the effective mass approximation a linear dependence of the equivalent local optical potential V upon the kinetic energy Tin the nuclear interior (6)
V = Voo-~T.
This energy dependence arises partly from the local approximation of the optical potential and partly from an inherent velocity dependence of nuclear interactions 31). Inserting the expression for the kinetic energy T = E + V - Vc
(7)
in expression (6), one gets v = VoolO + ~)+[~/(1
+ ~)] v c - [ ~ l ( l
+
~)]E.
(8)
As the mean Coulomb potential Vc of a homogeneous charged sphere ol radius R c = 1.3 A+ fm is given by Vc = 1.33 Z / A + (MeV), one expects the following relation between K 1 and K2 comparing (8) with (5): K t / K z "~ 1.33. However, all known analyses 6-8, ,6) of deuteron elastic scattering gave values of K, essentially greater than the expected one of about 1.33 K2 (table 6). In an analysis of proton elastic scattering in the energy range 9-26 MeV, Percy 33) obtained a value of K , / K 2 between 1 and 1.33, where the value 1.33 (with K z = 0.3) is more realistic for higher energies. He considered an additional dependence of the proton optical potential upon the symmetry term ( N - Z ) / A . In the case of neutron scattering in the energy range up to 25 MeV, the energy dependence 32,34-) of the neutron optical potential depth gives also the value K z = 0.3. The values of K 1 shown in table 6 are essentially greater than 1.33 K2. This suggests that there exists another effect for deuterons which is responsible for an additional A-dependence of the optical potential depth V. This assumption is supported by the fact that the
ELASrlC DEUTERON SCATTERING
283
potential depth of heavier nuclei has a weaker A-dependence (figs. 11 and 12). In the following, we shall show that this additional A-dependence of the real potential depth may be understood by the effect of the finite size of the deuteron. If one derives the deuteron potential from the nucleon potential, a zero-order approximation gives the relation
Vd(r ) = V.(r)+ Vp(r).
(9)
The next step accounts for the spatial distribution of proton and neutron within the deuteron 17-20) Vd(r ) = f~2(X){
V.(e+½X)+ Vp(r-½X)}dX.
(10)
If one uses the Hulth6n wave function of the free deuteron instead of the unknown real spatial distribution of proton and neutron during the interaction with the nucleus @n(X) = B[exp ( - ~X){ I - exp ( - fiX)} ]/X
(ll)
and a Woods-Saxon form for the nucleon optical potentials, the following more realistic form factor for the deuteron optical potential is obtained: fa (r) =
f ½el)2 (x){f~Ts(l,+½Xl)+f~".~(Ir--~-Xl)}dX.
(12)
The integral was carried out numerically. The resulting form-factor fd (r) was then fitted with a Woods-Saxon form Jws (r). The geometrical parameters R N and aN of the nucleon Woods-Saxon form fwNs(r) were chosen so that the fitted Woods-Saxon form fws(r) had about the same geometrical parameters rv and av as sets I and 2, respectively. Fig. 13 shows the result of this procedure for 32S. Because of the finite distance between the nucleons in the deuteron, the central depth o f f d ( r ) is smaller than given by a Woods-Saxon form factor in the case of ]
lo ~ 08
I
~
I
L
]
i
I
5
6
7
8
- _ _. < j r , ( r , Ro,%)
- ~ ~ fD(r)/ " ~ \\y~fNs(r" '\ , RN,aN)
0.6
~ -
R,
02
aD
0 F i g . 13. C o m p a r i s o n
1
2
3
of the real potential
4
form factors f~s(r, c a s e o f 3-'S.
RN, aN),fd(r) a n d
r(fm) f w s ( r , R d , a d ) in t h e
r. HINTERBERGERet al.
284
l i g h t n u c l e i w i t h c o r r e s p o n d i n g l y s m a l l r a d i i Rv = A~rv. I n fig. 14, t h e d i f f e r e n c e K (13)
K = [fws(0)--fd(0)]/fws(0)
h a s b e e n p l o t t e d as f u n c t i o n o f t h e m a s s n u m b e r A f o r sets I a n d 2 g e o m e t r i e s . T h e c o r r e s p o n d i n g g e o m e t r i c a l p a r a m e t e r s a r e c o m p i l e d in t a b l e 7. I
I
I
I
I
I
I
I
I
I
I
i
I
K~ 12
10
I 8 6 4
2
I
0
40
80
120
160
200
240
Fig. 14. Geometrical correction K(A) for set 1 t o ) and set 2 to). TABLE 7
The correction K = (fws(O)--fd(O))/fws(O) and the geometrical parameters o f J ~ s ( r ) and o f f w s ( r ) fitted to fd(r) Set 1
A 4 12 20 32 50 89 140 208
Set 2
J'~s(")
f w s (r)
K (%)
rN
aN
rd
ad
1.48 1.38 1.35 1.32 1.295 1.28
0.21 0.4 0.42 0.44 0.45 0.45
1.25 1.25 1.25 1.25 1.242 1.244
0.64 0.76 0.75 0.76 0.76 0.75
11.3 7.5 5.2 3.3 2 0.9
1.265
0.40
1.245
0.70
0.1
fwNs(r) rN
1.31 1.21 1.16 1.13 1.12 1.10 1.08 1.075
fws(r)
aN
rd
0.2 0.42 0.48 0.52 0.55 0.60 0.68 0.70
1.05 1.05 1.046 1.044 1.056 1.055 1.046 1.049
K (%) ad
0.67 0.80 0.83 0.84 0.86 0.88 0.95 0.96
12 10 8 6 4 2.2 1.3 0.7
The parameters of the Hulth6n function are ~ = 0.231 fm -1, !3 == 1.207 fin -~ The
deuteron
optical
potential
C o r r e c t i n g t h e m as ( I + K ) V ,
depths
contain
this additional
A-dependence.
one obtains the depths which the search program
would have found using the more realistic form factor fn(r).
The corresponding
285
ELASTIC DEUTERON SCATTERING
II11111111
I l l l l l l l l l l l l l l l l l
'l"ll'lllltlllll
I Iit I I 1 I 1 I I lI:~_
-
~
-
I
~ _:
°
---o ~- p~
•"f
?
<
i
I I L]
11
1 1
l
t i i i i i 11
(Lu;) "e
Ii
-
iitlliiii
L
1 i ii
i i i i
E^al,,t] M
Iltll
t11'1'1'i'i'ilit
l ii
i ii
.
11
i i i i I
._L]
( w i ) ~J
h
.
L
! I:
~(wl)Me
'llilllllltllll|
......i ~ i i i i i i i i i l ~ L
•"
i i i i 11
-~o e
II
L
" "1:
0-
-\'~ .
-
~.
",L %i P
:-
II 0
I II CO 0
(mt
I111
III
(~ ~ 0 a
) ^e
Itllllilllll
~lllllllllllll
C~ 0 .....
~,oN M
( ~ l ) ''J
I
:l"
-~o [
I_L.Llllti!!_L
<
286
r. HINTERBERGERet al.
curves are d r a w n d a s h e d in figs. 11 a n d 12. However this geometrical c o r r e c t i o n is still insufficient to reduce the coefficient K ~ to the expected o r d e r o f m a g n i t u d e K 1 / K 2 ~, 1.33. The d a s h e d curves in figs. 11 a n d 12 give K 1 = 0.85, K t / K 2 = 2.4 for set 1 a n d K 1 = 1.5, K I / K 2 = 3.0 for set 2. The inclusion o f the finite size o f the d e u t e r o n using the Hulthdn function 4~H (x) reduces the value o f K t / K 2 by a b o u t 50 ~ to 2.4 and 3.0, respectively. These values are still a b o u t a factor o f two greater than the expected value o f K I / K 2 = 1.33. This m a y be interpreted as an indication for a spatial p o l a r i z a t i o n o f the d e u t e r o n d u r i n g the interaction with the nucleus, which leads to an increased average n e u t r o n - p r o t o n distance c o m p a r e d to the free deuteron. 6.4. AVERAGED PARAMETER SETS Using the o p t i c a l - p o t e n t i a l waves in D W B A analyses, one finds that averaged p a r a m e t e r sets give g o o d fits. Often these fits are even better than those calculated with individually searched p a r a m e t e r sets. One o f the p u r p o s e s o f this analysis is to give o p t i c a l - p o t e n t i a l sets which are c o m p a r a b l e with those f o u n d at lower d e u t e r o n energies. The general d e p e n d e n c e o f the real potential d e p t h V u p o n the charge Z , mass A a n d energy E was discussed in subsect. 6.3. The r e m a i n i n g p a r a m e t e r s are plotted against A ~ in figs. 15 and 16. S m o o t h curves have been d r a w n t h r o u g h the points. In table 8 is the analytical form o f these averaged parameters. F o r c o m p a r i s o n we give TABLE 8 Averaged parameter sets Set 1
rv = 1.25 fm, Vs.o. = 6 MeV, r c ~ 1.3 fm
V (MeV) W (MeV) av (fm) rw(fm) a~(fm)
79-i- 1.5Z/A-Y --0.35E 13 0.81 -- 0.024A~ 1.25 0.51 -I-0.076A:~
Set 2
rv = 1.05 fm,
V (MeV) W (MeV) av (fro) rw(fm) aw(fm)
5-]-2A4s 0.71 +0.04A~ 1.28 0.71 -t-0.02A~
34.4 MeVa)
rv =0.968~ 0.029A-~, Vs.o. ~ 7 MeV, re = 1.3 fm
V (MeV) W (MeV) a,. (fm) rw(fm) aw(fm)
90.2-~ 0.89Z/A~ 4.33 +2.2A~ 0.814 1.09+0.8A "~ 0.554 +0.059A't-
a) Ref. s).
Vs.o.:= 6 MeV, r C = 1.3 fm
IO0+2.5Z/A~s-0.5E
ELASTIC DEUTERON SCATTERING
287
the averaged parameter sets found at a deuteron energy E d = 34.4 MeV [ref. a)], which are comparable with the averaged parameter set 2. 6.5. RESULTS IN DWBA ANALYSES We have used the optical-model parameters in a series of DWBA analyses of (d, d') and (d, 3He) reactions with 52 MeV deuterons to calculate the distorted waves of the deuteron channels. in agreement with the discussion of the A-dependence of r v, we found that the inelastic deuteron scattering a5) from target nuclei with A __< 64 is better described by use of parameter sets with r v = 1.25 fro. In the case of the (d, 3He) reaction 36,37) o n 12C, 14N, 4s's°Ti and 51V, we used in the DWBA analyses the parameters of set 1 as well as those of set 2. The differences in the D W B A cross sections were less than 5 700. The Vr~ = const ambiguity cannot be solved with these data. In 12C(d, 3He) 11B and 14N(d, 3He)l 3C, the observed weak j-dependence of l = 1 transitions appears in the DWBA predictions if deuteron parameters with inclusion of a spin-orbit potential are used.
7. Conclusion It is concluded that elastic scattering of 52 MeV deuterons on target nuclei with A _>_ 12 may be described very well by the optical model. For heavier target nuclei, it is necessary to use an optical potential with spin-orbit term. The spin-orbit term is unimportant for the elastic scattering angular distributions of light nuclei. The angular distributions of elastically scattered deuterons from 1H, 3He and 4He show a structure which may be understood as a superposition of shape elastic potential scattering dominating at forward angles and nucleon-transfer processes dominating at backward angles. The coefficients K 2 of the energy dependence of the real potential depth are 0.35 and 0.5, for the two radius parameters used. The coefficients K 1 for the Z/A ~ dependence are 1.5 and 2.5. This large value of KI/K 2 can be explained only partly by the influence of the Coulomb potential. It indicates a large spatial polarization of the deuteron during the interaction with the nucleus. The two averaged parameter sets allow a good estimate of optical potentials at neighbouring energies as well as for other target nuclei. The optical parameters of this analysis were successfully applied in DWBA calculations. We thank Professor T. Mayer-Kuckuk for his interest in this work. We are indebted to Dr. G. Schatz and the staff of the cyclotron laboratory at Karlsruhe for their cooperation. We wish to express our appreciation to Dr. E. H. Auerbach for making available the ABACUS code and to Drs. R. H. Bassel, R. M. Drisko and
288
F. HINTERBERGERel a].
G . R. S a t c h l e r f o r m a k i n g a v a i l a b l e t h e c o d e J U L I E . o p e r a t o r s o f t h e I n s t i t u t fiir l n s t r u m e n t e l l e M a t h e m a t i k of the Rechenzentrum
Thanks are also due to the der Universitat Bonn and
d e s I n s t i t u t e s fiir H o c h e n e r g i e p h y s i k
der Universi~t
Heidel-
b e r g f o r t h e i n n u m e r a b l e I B M 7090 c o m p u t e r r u n s . T h e h e l p o f H . L i n k a n d H . M a c k h i n d a t a t a k i n g a n d h a n d l i n g is g r a t e f u l l y a c k n o w l e d g e d .
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37)
G. lgo, W. Lorenz and U. Schmidt-Rohr, Phys. Rev. 124 (1961) 832 T. Becker, U. Schmidt-Rohr and E. Tielsch, Phys. Lett. 5 (1963) 331 W. Fitz, R. Jahr and R. Santo, Nucl. Phys. h l 0 1 (1967) 449 C. Cindro and N. S. Wall, Phys. Rev. 119 (1960) 1340 J. L. Yntema, Phys. Rev. 113 (1959) 261 H. R. E. Tjin A Djie and K. W. Brockman Jr., Nucl. Phys. 74 (1965) 417 J. Testoni, J. Rosenblatt and S. Mayo, Nucl. Phys. 74 (1965) 481 E. Newman, L. C. Becker and B. M. Preedom, Nucl. Phys. AI00 (1967) 225 B. Duelli, F. Hintcrbcrgcr, G. Mairle, U. Schmidt-Rohr, P. Turek and G. Wagner, Int. Conf. on nuclear physics, Gatlinburg (1966) B. Duelli, F. Hinterberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. Wagner, Phys. Lett. 23 (1966) 485 B. Duclli, G. Mairle, U. Schmidt-Rohr, P. Turek and G. Wagner, Z. Naturf. 21a (1966) 969 E. A. Silverstein, Nucl. Instr. 4 (1959) 53 P. E. Hodgson, Advan. Phys. 15 (1966) 329 E. H. Auerbach, BNL-6562 (1962) R. H. Bassel, R. M. Drisko and G. R. Satchler, ORNL-3240 (1962) C. M. Perey and F. G. Percy, Phys. Rev. 132 (1963) 755 S. Watanabe, Nucl. Phys. 8 (1958) 484 G. Baumgacrtner, thesis, Technische Hochschule Miinchen (1966) unpublished J. Testoni and L. C. Gomes, Nucl. Phys. 89 (1966) 288 F. G. Perey and G. R. Satchler, Nucl. Phys. A97 (1967) 515 C. M. Perey and F. G. Percy, Phys. Rev. 152 (1966) 923 K. W. Brockman Jr. and W. T. H. van Ocrs, Nucl. Phys. 21 (1960) 189 D. O. Caldwell and J. R. Richardson, Phys. Rev. 98 (1955) 28 E. G. Auld, D. G. Crabb, J. G. M. Ewen, L. Bird, C. Whitehead and E. Wood, Nucl. Phys. A101 (1967) 65 J. H. Williams and M. K. Brussel, Phys. Rev. 110 (1958) 136 J. E. Brolley Jr., T. M. Putnam, L. Rosen and L. Stewart, Phys. Rev. 117 (1960) 1307 O. M. Bilaniuk and R. J. Slobodrian, Nucl. Phys. 50 (1964) 585 H. W. Brock and J. L. Yntema, Phys. Rev. 135 (1964) B678 W. T. H. van Ocrs and K. W. Brockman Jr., Nucl. Phys. 44 (1963) 546 E. C. Halbert, Nucl. Phys. 50 (1964) 353 S. Mayo, W. Schimmerling and M. J. Sametband, Nucl. Phys. 62 (1965) 393 F. Percy and B. Buck, Nucl. Phys. 32 (1962) 353 F. G. Percy, Phys. Rev. 131 (1963) 745 F. Bjorklund and S. Fernbach, Phys. Rcv. 109 (1958) 1295 F. Hinterberger, G. Mairlc, U. Schmidt-Rohr, P. "lurek and G. J. Wagner, Nucl. Phys. 106 (1968) 161 F. Hinterberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. J. Wagner, Z. Phys. 202 (1967)236 F. Hinterberger, G. Mairlc, U. Schmidt-Rohr, P. Turck and G. J. Wagner, Nucl. Phys. AI06 (1968) 161
~
Nuclear fihysics A l l l (1968) 265--288; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ELASTIC
SCATTERING
O F 52 M e V D E U T E R O N S
F. HINTERBERGER
Institut fiir Strahlen- trod Kernphysik der Unicersit?it Bonn G. MAIRLE, U. SCHMIDT-ROHR and G. J. W A G N E R
Max-Planck-lnstitut fiir Kernphysik, Heidelberg and P. T U R E K
Kernforschunysanlaye Jiilich, Kernforschunyszentrum Karlsruhe Received 5 February 1968 Abstract: Angular distributions for the elastic scattering of 52 MeV deuterons were measured from 27 nuclei ranging from hydrogen to uranium. The experimental data have been analysed in terms of the optical model. The inclusion of a vector spin-orbit term in the potential is necessary for nuclei with A > 100. For light nuclei with 12 ~_ A < 50, good fits to the experimental angular distributions could also be obtained without a spin-orbit potential. The optical-model parameters for nuclei with A ~_ 12 are discussed with respect to the dependence on the energy E, the charge Z and the mass A. The A-dependence of the real potential depth V can be understood as a consequence of the spatial polarization of the deuteron during the scattering process. Discrepancies between optical-model calculations and experimental angular distributions observed in the elastic scattering processes from XH, SHe, and 4He indicate contributions from transfer reactions.
E
N U C L E A R REACTIONS 1H, 3,aHe, 1~C, 14N, lsO, °-9,Z2Ne, -°~Mg, ZrAl, 2aSi, 328, 40Ar' 4s, r~q-i, .~4Fe' ~8,64Ni' ~gy, 93Nb' 103Rh' 140Ce' 181Ta' 19:Au' 20apb' 209Bi' 23sU (d, d); E = 52 MeV; measured a(0); deduced E- and A-dependence of optical-model parameters. Enriched targets.
1. Introduction In an e n e r g y r a n g e u p to 34.4 M e V , t h e a n g u l a r d i s t r i b u t i o n s o f elastically s c a t t e r e d d e u t e r o n s s h o w d i f f r a c t i o n p a t t e r n s w h i c h c a n be d e s c r i b e d well by t h e o p t i c a l m o d e l (refs. 1-8)). T h e k n o w l e d g e o f t h e d e u t e r o n o p t i c a l p o t e n t i a l is a n e c e s s a r y c o n d i t i o n f o r the s p e c t r o s c o p i c analysis o f r e a c t i o n s w i t h d e u t e r o n s in t h e e n t r a n c e o r exit c h a n n e l by t h e m e t h o d o f t h e d i s t o r t e d w a v e B o r n a p p r o x i m a t i o n ( D W B A ) . I n o r d e r to i n v e s t i g a t e t h e d e p e n d e n c e o f the o p t i c a l p o t e n t i a l o n the m a s s n u m b e r A, t h e c h a r g e Z a n d t h e s t r u c t u r e o f the n u c l e u s , o n e s h o u l d m e a s u r e the a n g u l a r distrib u t i o n s o f elastically s c a t t e r e d d e u t e r o n s f r o m m a n y t a r g e t nuclei. P r e v i o u s l y s u c h i n v e s t i g a t i o n s 1 - 8 ) w e r e p e r f o r m e d o n l y at a few lab e n e r g i e s b e t w e e n 11.8 M e V a n d 34.4 M e V . W i t h the c o n s t r u c t i o n o f i s o c h r o n o u s c y c l o t r o n s , o n e o b t a i n s e n e r g i e s w h i c h m a k e an i n v e s t i g a t i o n o f t h e " o p t i c a l " p r o p e r t i e s o f the nuclei v e r y p r o m i s i n g w i t h r e g a r d to the s h o r t e r de B r o g l i e w a v e l e n g t h . I n t h e case 265
266
F. ItINTERBERGERet
aL
of nuclei with A < 4, a deuteron lab energy of more than 50 MeV corresponds to a c.m. energy which lies near or above the total binding energy of the compound systems. With respect to the "few-nucleon" problems, the elastic scattering data may provide information about the importance of transfer processes in the elastic channel. Previously 9,1 o), we reported the measurement of the angular distributions for the elastic scattering of 52 MeV deuterons from 16 nuclei ranging from carbon to uranium and the preliminary results of an optical-model analysis. Since these publications, 11 further angular distributions have been measured and analysed. This paper is the comprehensive report of our systematic studies concerning the elastic scattering of 52 MeV deuterons. 2. Experimental procedure
The experiments have been performed with 52 MeV deuterons of the Karlsruhe isochronous cyclotron. The deflected beam is focussed with quadrupole lenses on the target without any diaphragms near the scattering chamber. The beam spot on the target is about 2 mm broad and 3 mm high. The quality of the beam is better than 10 m m • mrad. The angular distributions of the scattered deuterons are taken with a scattering chamber described previously t 1). The detectors are mounted outside the vacuum chamber on ring tables movable by remote control. We use dE-E telescopes consisting of 200 ~Lm-400/2m Si surface-barrier counters ( O R T E C ) as dE-counters and NaI(TI) scintillation counters as E-counters. A few differential cross sections are taken with a Ge(Li) detector as E-counter. In the case of the 3,4He targets, the differential cross sections at backward angles have been obtained by measuring the recoil nuclei scattered in forward angles with a telescope containing a 2 mm Si surface-barrier detector as E-counter. In the case of 1H(d, d)lH, the differential cross sections at backward angles are measured by the detection of the protons with a Ge(Li) detector as E-counter in the telescope. The corresponding experimental results are given as crosses in fig. 2 in comparison with the deuteron experimental points. There is good agreement of the data points in the overlap region. The over-all energy resolution is about 600 keV with the NaI(TI) detectors and about 250 keV with Ge(Li) and Si detectors as E-counters. A typical spectrum taken with a NaI(TI) detector is given in fig. 1. With the exception of 93Nb, 103Rh ' 181Ta ' 197AHand 238U,the elastic peak is well separated from the inelastic spectrum. But the shape of the spectra and the results on inelastically scattered deuterons from even-mass nuclei suggest the error from inelastic contributions to be small especially at forward angles. However we cannot exclude the possibility that the very smooth structure of the angular distributions of ~°aRh, 18tTa and 238U is caused partly by this lack of resolution. The angular resolution is 0.5 ° in the regions of sharp diffraction structure and between 0.5 ° and 1.2 ° at other angles. The precision of the relative angle position is better than 0.I °, the precision of the absolute angle position lelative to the 0 ° symmetry is +0.2 °.
267
ELASTIC DEUTERON SCATTERING
The energy of the beam has been determined using the cross-over technique. The method is to determine in a precise left-right measurement the angles at which deuterons elastically scattered from hydrogen have the same energy as those inelastically scattered from 12C leading to the excitation of the 4.433 and 9.629 MeV states. As the target, we use a foil of polyethylene (CH2) ~ and as detector a dE-E telescope with a Ge(Li) E-counter. The dependence of the cross-over angles on the energy of the primary deuteron beam is exactly computed relativistically. The result is E a = 51.7___0.2 MeV. I
I
40C
~30
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1ooj 0
~13
70
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58 50 395
fi
"i 200: U
I
14N (d,d') E d = 52 MeV
100
/ 200
300
Channel number L Fig. 1. Typical spectrum of the 14N(d, d') reaction at 0ia~ = 35 °.
3. Targets The measurements on the nuclei 27A1, 48'5°Ti, 5¢Fe, 8 9 y , 9 3 N b ' tO3Rh ' 140Ce ' l S ' T a , t 9 7 A u , 2°8pb, 2°9Bi and 238U were performed with rolled foils of 1 mg/cm 2 to 10 mg/cm z thickness. The 12C, 2¢Mg, SSNi and 64Ni targets were self-supporting vacuum-evaporated foils of 250/~g/cm 2 to 1000 #g/cm s thickness. The 160 data points result from measurement on SiO2 and Si targets. The Si target was a 20 #m thick defective surface-barrier counter. The measurements on 1H, 3He, 4He, 2°Ne, 22Ne, asS and ¢°Ar were performed using a gas target with a diameter of 10 cm and a volume of about 1 1, which is inserted into the scattering chamber instead of the solid target. In the reaction plane, the particles pass through a slit which is covered by a 4.6 mg/cm 2 thick havar foil. The maximum pressure is 500 m m Hg. The pressure is continuously monitored with a Hg manometer during the measurement, and the volume density of the irradiated gas is computed according to the ideal gas law. The geometry factor G is computed according to the formulae of Silverstein (ref. 12)). The geometry was chosen so that the correction terms were smaller than 1 ~ . They are computed up to the fourth-order term assuming a theoretical opticalmodel angular distribution.
268
r. HINrrERBERGER et al.
]n the 22Ne, 24Mg, 48,5°Ti, 54Ee, 58,64Ni and 2°8pb measurement, we used isotopically enriched targets. The statistical error has been kept below 3 9/o, and the data have been corrected for the deadtime which was always smaller than 10 ~o. The reproducibility of a measurement with different geometries and different telescopes was better than 2 ~ . In total the relative error in the angular distributions lies below 5 ~ .
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269
ELASTIC DEUTERON SCATTERING
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270
r. HINTERBERGER et aL 1
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271
ELASTIC DEUTERON SCATTERING
within 2 %, which is the reproducibility range of measurements with completely different geometries and dE-E telescopes. The main error of the absolute values results from uncertainties in the target thickness determination. The thickness of solid targets was measured both by weighing and by measuring the energy loss of alpha particles. For 160, 24Mg ' 28Si ' 48,50Ti ' 58'64Ni and 89y, a measurement of the absolute cross sections was prevented by the impurities in the targets. In these cases, the absolute cross section has been determined by fitting the experimental data to several optical-model calculations at forward angles. This procedure is justified by the observation that the c.m. calculations are very insensitive to parameter changes at forward angles. The volume density in the gas target is determined with an error of about 3 %. Therefore the total error in the absolute cross sections is about 10 % for solid targets and about 6 % for gas targets 4 /
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F i g . 5. P o s i t i o n o f t h e m i n i m a ( o ) a n d t h e m a x i m a ( I I ) o f t h e e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s as a f u n c t i o n o f A { . T h e c o r r e s p o n d i n g e x t r e m a are c o n n e c t e d b y s m o o t h curves.
4. Experimental results Figs. 2-4 show t the measured angular distributions. As in the case of lower energies the extrema move to forward angles with increasing mass number A according to the Tables of cross sections are readily provided by the authors.
F. HINTERBERGER et al.
272
diffraction law k R sin 0 = const (k is the wave number, R the radius of the nucleus and 0 the scattering angle). In fig. 5, the position of the maxima and minima is plotted versus A ~. The corresponding cxtrema are connected by smooth curves. In the region of 14°Ce, one extremum is vanishing. This effect of vanishing maxima or minima has been observed at lower energies in other mass regions by several authors l a). Deviations from the smooth curve of the minima near 28 ° in the region of 5°Ti and 54Fe and near 35 ° in the region of 89y are obvious, i.e. in the region of the closed N = 28 and N = 50 neutron shells. Other deviations at backward angles are within the error of the position of the extrema. 5. The
optical-model analysis
5.1. THE OPTICAL POTENTIAL The form of the local optical potential used in this analysis of 52 MeV deuteron scattering is the same as has been used in analyses at lower energies 1a). As the main part of absorptive processes is known to take place in the surface region, we assume a pure surface absorption term. The potential may be explicitly written as
U(r) = V f ( r ) + i W g ( r ) +
V~.o.h(r ) ( l . s),
f ( r ) = - {1 + exp [ ( r - Akrv)/av]} - 1, g(r) = - d / d r { 1 + exp [ ( r - A ~ r w ) / a w ] ) - 1 ,
(l)
h(r) = - (h/m~c) 2 1/r d/dr f ( r ) .
The geometrical parameters of the spin-orbit potential are the same as those of the real part of the potential. We take the potential of a uniformly charged sphere of radius R c = r c A ~ fm for the Coulomb potential Ue(r ) = Z e 2 / 2 R c ( 3 - r 2 / R c )
= Z e2/r
r < Re r > Rc.
(2)
The geometrical parameter r c has been kept fixed at r c = 1.3 fm. 5.2. SEARCH PROCEDURE The above optical-model potential contains the seven free parameters V, 14I, V~.... rv, av, rw and aw. A search procedure was performed in order to minimize the value N
Z2 = N - 1 ~ (acxp(0,)- ath,o~(0~)) z/Aa~xp(O,) , ,
(3)
i=1
N is the number of measured angles, trexp(0~) and tr,h,or(0~) the experimental and theoretical cross sections and Aa~xp(01) the experimental uncertainty at angles 01. We have taken AtYexp(Oi) = 0.10"exp(0i) uniformly for all points.
ELASTIC DEUTERON SCATI"ERING
273
The search procedure was performed using the automatic search code ABACUS written by Auerbach t4). Unfortunately this search code can treat only spin ½ particles. But the main effect of a spin-orbit potential with spin ½ is the same as with spin I, namely a smoothing of the backward diffraction structure. Moreover the discrepancies of a spin ½ and a spin 1 angular distribution computed with identical parameters are found to be small. Therefore it is justified to perform the automatic parameter search with the code ABACUS. The fit was improved in the final spin I calculations by a 10 ~ reduction of the imaginary potential depth W. The curves in figs. 2-4 have been calculated with the correct spin s = 1 of the deuteron by means of the code J U L I E ts) using the parameters in tables 1, 2 and 5. 5.3. O P T I C A L - M O D E L S E A R C H O F T H E D A T A IN T H E MASS R A N G E 12 ~ A ~ 238
In considering starting points for an analysis, one is concerned with discrete V and continuous Vr~ = const ambiguities of the optical potential13'16). The most successful choice 17-20) so far of the potential V of the deuteron is the sum of the neutron and proton optical potentials at half the deuteron energy, i.e. V lies in the interval 70 MeV to 110 MeV. Because of the Vr~ = const ambiguity, we kept the real parameter rv fixed and performed searches with the following two values: set I
rv = 1.25 fm,
V~tart =
75 (MeV) V~.o. :~ 0,
set 2
rv = 1.05 fm,
V~tar, = 100 (MeV) V~.o. ~ 0.
The starting points of the remaining parameters were taken from the values at lower energies 13). The preliminary results for a series of target nuclei of the optical-model analysis with parameter set 1 is reported in ref. 9) and with parameter set 2 in ref. 1o). The variation of the spin-orbit potential was found to be very unspecific. Therefore we used V.~.o"= 6 MeV throughout the following analyses. In tables 1 and 2, the best fit optical-model parameters of set I (r V = 1.25 fm) and set 2 (r v = 1.05 fm) are given for target nuclei with 12 < A < 238. The theoretical and experimental angular distributions are compared in figs. 3 and 4. Many opticalmodel calculations la) at lower energies have been performed without a spin-orbit potential. With increasing energy, however, one finds it necessary to include a spinorbit potential to obtain reasonable fits 8, 2 i). To clarify this question, at the energy of 52 MeV we also performed searches without a spin-orbit potential, i.e. set 1"
with rv = 1.25 fro,
Vs.o. = 0,
set 2*
with rv = 1.05 fro,
V.~.o. = 0.
In the mass range 12 < A < 64, one gets fits comparable to those including a spinorbit potential (tables 3 and 4). In the mass range 64 < A < 100, the fits are reasonable, but for A > 100 the angular distributions calculated without a spin-orbit potential show too distinct minima as one notices in fig. 6 for 2°spb.
TABLE 1 Parameter set 1; rv = 1.25 fro, r C = 1.3 fm Target
V (MeV)
av (fln)
Vs,o, (MeV)
rw (fm)
a~ (fm)
W (MeV)
g"
~2C NN i60 Z°Ne 2ZNe ~Mg ZrAl ~Si azS ~°Ar ~Ti 5°Ti S4Fe ~Ni 6*Ni sgy aSNb l°3Rh N°Ce l~lTa 19~Au z°~Pb "2°gBi z~sU
67.4 66.1 68.2 67.5 67.3 68.0 67.1 69.3 70.2 71.1 70.0 71.3 71.5 70.4 71.5 75.1 75.3 75.5 75.8 78.9 80.0 79.8 79.5 81.0
0.75 0.75 0.693 0.75 0.75 0.75 0.737 0.78 0.75 0.75 0.744 0.733 0.75 0.742 0.783 0.75 0.718 0.68 0.819 0.735 0.668 0.66 0.65 0.703
6 7.5 6 6 6 6 7 6.6 6 7 7.9 6.64 6 7 6.74 6 6 6 6.43 6 6 6 6 6
1.25 1.28 1.25 1.25 1.25 1.25 1.22 1.24 1.25 1,25 1.25 1.25 1.25 1,24 1.25 1.25 1.25 1.25 1.22 1.25 1.25 1.25 1.25 1.25
0.744 0.66 0.75 0.825 0.782 0.787 0.754 0.75 0.75 0.75 0.759 0.714 0.73 0.714 0.774 0.756 0.95 0.87 0.791 0,95 0.97 1.0 1.0 0.98
10.9 l t. 1 12.2 12.7 13.4 13.9 12.5 13.1 14.2 14.5 13.2 14.2 13.0 14.0 14.1 14.7 11.7 13.0 15.2 12.6 13.2 12.0 12.2 13.0
3.1 3.5 5.0 4.3 3.7 5.4 1.4 3.0 3.4 3.5 3,0 3.0 3.7 4.1 5.2 9.0 5.8 9.5 9,4 11.7 4.0 5.2 6.3 7.7
TABLE 2 Parameter set 2; rv = 1.05 fm, r C = 1.3 fm Target
V (MeV)
a,, (fm)
lzC ~4N 160 ZONe ZZNe ~-~Mg ~AI 28Si ~zS 4OAr 4aTi ~OTi raFe 58Ni 64Ni ~gy 9~Nb l°'~Rh ~4oCe lStTa l~rAu zo~Pb ~-°~Bi 23sU
83.1 84.0 85.0 84.6 84.5 86.1 86.5 86.9 87.5 88.9 90.0 89.0 92.0 93.1 93.5 97.0 97.1 98.3 102.6 106.0 105.5 106.4 106.3 106.1
0.80 0.80 0.810 0.90 0.9 0.787 0.839 0.83 0.85 0.85 0.85 0.852 0.889 0.845 0.898 0.924 0.893 0.903 0.924 0.95 0.94 0.945 0.96 0.793
I(~.o. (MeV) 7 7 6 5 6 6 6 6 6 7 7.9 7.9 6 6.8 6.7 5.9 6 6.1 6.8 6 6 6 6 7.6
r~ (fro)
aw (fro)
W (MeV)
gz
1.22 1.28 1.28 1.2 1.25 1.28 1.28 1.28 1,28 1.3 1.29 1,29 1.28 1.33 1.33 1.34 1.28 1.35 1.27 1.3 1.3 1.29 t .3 1,2
0.75 0.779 0.75 0.85 0.799 0.75 0.75 0.76 0.8 0.779 0.76 0.76 0.75 0.735 0.864 0.696 0.75 0.677 0.801 0.80 0.82 0.80 0.81 0.88
9.1 8.4 9.3 11.3 12.3 12.6 11.0 10.9 t 1.4 1 t.7 12.5 12.4 12.1 12.2 13.1 15.5 15.5 16.8 15.5 17.2 16.1 17.1 17.0 16.4
2.0 2.3 5.9 3.4 4.2 6.3 5.4 6.1 3.5 2.0 2.2 1.6 7.8 4,2 3.2 4.1 4.8 5.6 2,0 5.8 2.3 4.5 5.2 4.0
275
ELASTIC DEUTERON SCATTERING
TABLE 3 Parameter set 1"; Vs.o. = 0 MeV, rv = 1.25 fro, r C = 1.3 fm Target
V (MeV)
W (MeV)
12C 14N 160 2°Ne 2ZNe 24Mg 27A1 26Si s2S 4°Ar 4~Ti 6°Ti 64Fe 6SNi ~4Ni
71.8 68.9 76.8 67.4 68.3 67.5 69.8 73.0 73.4 74.1 72.5 74.0 72.3 75.6 76.2
11.0 10.2 11.9 11.7 13.4 14.8 12.9 12.3 14.3 14.4 15.1 14.0 12.8 15.5 15.0
av (fro) 0.700 0.70 0.75 0.80 0.75 0.71 0.705 0.75 0.75 0.75 0.733 0.75 0.75 0.75 0.75
rw (fm)
aw (fm)
1.25 1.25 1.25 1.25 1.25 1.15 1.18 1.25 1.25 1.25 1.21 1.25 1.25 1.25 1.25
Z2
0.70 0.70 0.75 0.80 0.75 0.775 0.778 0.75 0.75 0.75 0.739 0.75 0.75 0.75 0.75
1.3 0.8 6.9 3.3 3.8 4.9 2.9 5.9 4.0 3.9 4.0 6.1 6.0 12 7.9
TABLE 4 Parameter set 2*; Vs.o. = 0 MeV, r v = 1.05 fro, r c = 1.3 fm Target I~C 14N ~60 ZONe Z'~Ne 24Mg 2~AI 2sSi a2S 4°Ar 4STi S°Ti 64Fe 6SNi 64Ni
V
W
av
r~
aw
(MeV)
(MeV)
(fro)
(fro)
(fm)
87.7 92.1 98.6 88.6 84.9 83.5 85.5 89.5 89.0 92.3 89.6 93.2 91.1 98.7 98.7
9.48 10.6 11.3 11.9 11.6 11.6 9.76 10.2 12.6 12.5 11.7 12.3 11.5 14.0 13.3
0.75 0.85 0.85 0.90 0.85 0.746 0.80 0.84 0.85 0.85 0.843 0.85 0.86 0.85 0.87
1.20 1.20 1.33 1.20 1.33 1.31 1.30 1.28 1.33 1.33 1.26 1.33 1.28 1.33 1.33
0.75 0.75 0.75 0.85 0.75 0.778 0.82 0.810 0.75 0.75 0.820 0.75 0.80 0.75 0.78
Zz
3.1 2.6 20 6.1 5.9 6.4 2.7 8.9 5.4 2.4 3.6 6.0 8.1 14.5 8.1
5.4. OPTICAL-MODEL DESCRIPTION OF THE REACTIONS lH(d, d)lH, aHe(d, d)SHe AND 4He(d, d)4He T h o u g h t h e b a s i c a s s u m p t i o n s o f t h e o p t i c a l m o d e l as, f o r i n s t a n c e , t h e a v e r a g i n g o v e r m a n y r e s o n a n c e s o f t h e c o m p o u n d s y s t e m a r e p o o r l y fulfilled in t h e c a s e s u n d e r d i s c u s s i o n , we p e r f o r m e d a n o p t i c a l - m o d e l s e a r c h o f t h e e l a s t i c d a t a l H ( d , d ) I H , a H e ( d , d ) a H e a n d 4 H e ( d , d ) g H e . Fig. 2 s h o w s t h a t in e a c h c a s e t h e f o r w a r d a n g l e s a r e n i c e l y r e p r o d u c e d b y t h e o p t i c a l m o d e l . H o w e v e r it is i m p o s s i b l e t o get t h e diffraction structures at b a c k w a r d angles. Even with the r i g o r o u s variation o f starting
276
F. HINTERBERGER e t a l.
parameters, the calculation with or without a spin-orbit potential and the use of a Gaussian form factor instead of a Woods-Saxon form for the real potential
f(r)
= exp { - [ ( r - R ) / a ]
2}
gave no essential improvement. The different optical parameters found in this analysis with the value 7.2 for the total angular distribution and separated for forward angles 0.... < 90 ° and backward angles 0.... > 90 ° are compiled in table 5. ~O
~
I
I
t
I
i
~OFI ~,~,x
I
208pb • -~ -
0.1
..~, -
I
i
L
i
i
;
(d,d)
E a = 5 2 MeV
0001
00001
i
i
i
40*
i
r:~
i
,
~g3*
i
loo*
1~
ecM
Fig. 6. Optic-d-model fit w i t h o u t a s p i n - o r b i t potential in the reaction 2°sPb(d, d)2°sPb. The parameters are V = 111.3 MeV, r v ~ 1.05 fm, a v -- 0.85 fm, W = 20.5 MeV, rw = 1.2 fm, a w = 0.8 fm, Vs.o. = 0 MeV, r c = 1.3 fm, Zs = 28. TABLr 5 O p t i c a l - m o d e l - p a r a m e t e r sets of elastic s c a t t e r i n g with 52 MeV dc ut e rons on ' H , aHe a n d 4He Tar get IH IH XH 3He SHe SHe SHe 4He ~He ~He 4He 4He
V (MeV) 65.8 92.1 37.5 a) 61.4 67.0 80.5 82.5 73.2 70.0 91.8 86.0 98.7 a)
rv (fm)
av (fm)
Vs.o. (MeV)
rw (fm)
aw (fm)
W (MeV)
Z2
Zt"
Z22
1.25 1.05 0.0 1.25 1.25 1.05 1.05 1.25 1.25 1.05 1.05 0.0
0.501 0.500 1.918 0.599 0.590 0.700 0.66 0.614 0.592 0.700 0.661 1.918
0.0 0.0 0.0 0.0 6.0 0.0 6.0 0.0 6.0 0.0 7.0 0.0
1.2 1.02 1.3 1.01 1.25 1.05 1.14 1.01 1.25 1.10 1.15 1.66
0.517 0.511 0.466 0.598 0.630 0.650 0.640 0.647 0.634 0.750 0.639 0,591
4.59 7.67 5.10 9.03 8.10 8.94 9.00 11,5 8.2 8.34 8.9 6.75
5.8 3.9 17 25 15 27 27 22 21 28 26 15
0.6 0.8 3.0 6.5 8.2 7.2 6.0 0.9 1.4 3.0 2.1 4.9
14 8.5 38 51 49 55 58 39 37 47 44 24
In each case, r c has been kept fixed to r c = 1.3 fm. The value Z ~ is calculated using all me a s ure d angles, whereas the value Zt ~ is defined for the range 0 ° ~ 0c.m. "~ 90 ° and the value Ze2 for the range 90 ~ ~ 0c.m. ~ 180 ° . a) The real potential has a G a u s s i a n form instead o f a W o o d s - S a x o n form.
ELASTIC DELrTERON SCA'I-I'ERING
277
The failure of the optical model at backward angles might be caused by the lack of antisymmetrization. The optical model cannot treat the nucleon-transfer processes especially the nucleon-transfer reaction between identical states. In the last case, we have the following reaction mechanism: d t q- ( d a + C ) - - ~ ( d l + C ) + d 2. Experimentally such transfer processes cannot be distinguished from elastic scattering. Evidence for the existence of these transfer processes comes from the energy dependence of these effects. Unfortunately data on 1H, 3He and ~He deuteron elastic scattering have been published for only a few energies. 1OOO
~e
I
I
% o ~
I
i
i
I
I
4He (d,d)4He
1000
100
"-
ooo
10 °
l
~
°
i~i 100
°
"
"• •
~
165 ° ° ~,
". • •
o 10
10
l
347
I
(3=
I
40*
I
I
80*
I
I
120"
1
I
160" Oc.l,~
Fig. 7. Comparison of the elastic scattering data of 4He(d, d)4He taken at the following c.m. energies: £c.m. -" 14 MeV [re£. 2,)], Ec.m" ~ 16.5 MeV [ref. 29)] and Ec.m. = 34.7 MeV (this work). The solid curve for Ee.m. = 34.7 MeV is calculated using parameter set 1 (table 5). The optical-model parameters of the fit to the Ee.m. = 14 MeV data are F .... 67 MeV, r v = 1.25 fro, av ~ 0.59 fro, W = 3.95 MeV, r w - - 1.25 fro, a w - - 0 . 6 3 fro, F s . o . = 6 M e V , r c - 1.3 fm, Z ~---6.9. For the crosses at Ec.m. ~ 34.7 MeV, see fig. 2.
In the energy range 8 MeV < Ec.m. • 27 MeV, all the known angular distributions of the 1H(d, d ) l H reaction 2z-2s) show a strong increase of the differential cross sections with increasing angle 0 ..... for 0.... > 120' (see fig. 2). As there is no marked difference in the backward angular distribution over the energy range considered here, one cannot deduce information about the reaction mechanism from the energy dependence in the case of IH(d, d ) ' H .
278
F. HINTERBERGERet al.
The elastic scattering * 3He(d, d)3He has been measured at the following energies (refs. 26.27)): Elab ~ 14.4 MeV and 24 MeV < El, b < 27 MeV. Unfortunately, the angular distributions of the latter energy have been measured only in forward angles up to 0.... < 104°. In this angular range, an optical-model fit is easy to obtain just as in the case of the 52 MeV data. We shall discuss in detail only the 4He(d, d)4He reaction with respect to the energy dependence of the backward angular structures. In fig. 7, one finds the elastic angular distributions measured 2~,29) at c.m. energies Ec.m. = 14 MeV and E¢.m. = 16.5 MeV in comparison with our Ec.m. = 34.7 MeV data. The pronounced diffraction structure at backward angles is smoothed out at lower energies. For the E~.m. = 14 MeV data 27), we performed an optical-model search which gives a relatively good fit even at backward angles. The o.m. parameters are given in the caption of fig. 7. This comparison with the data at lower deuteron energies is a good argument that in the case of 4He(d, d)4He at a c.m. energy E¢.m. = 34.7 MeV, which lies 6.5 MeV above the total binding energy of 4He, there is a nucleon-transfer reaction dominating the backward angular range. It would be interesting to investigate the energy dependence of the pronounced structures at backward angles for 1H, 3He and 4He in the region Ee.m. : 50 MeV. This would give further information about the reaction mechanism as well as the fewnucleon compound systems 3He, 5Li and 6Li.
6. Results and discussion of the optical-model analysis 6.1. TOTAL REACTION CROSS SECTIONS Several authors have found agreement between measured total reaction cross sections and those calculated from the optical model 13) to within 20 % for deuteron energies up to 26 MeV. The values of a .... tion c',dculated with parameter set 1 differ less than 10% from those of parameter set 2. In fig. 8 is a plot of the calculated total reaction cross sections 6reaction versus A ~. The points are the average of an opticalmodel calculation with parameter set 1 resp. 2. A smooth curve is drawn through the points. The corresponding curves at the deuteron energies Eo = 11.8 MeV [refs. 1- 3,30)] and 26 MeV [ref. 31)] are drawn for comparison. To demonstrate the small effect of the Coulomb repulsion, we calculated the total reaction cross section of a 52 MeV "deuteron" with charge Z = 0. The result is given by the dashed curve in fig. 8. 6.2. THE RADIUS PARAMETER OF THE REAL POTENTIAL We began our analysis with a radius parameter rv fixed to the value rv = 1.25 fm. However, it was impossible to fit the elastic angular distribution 14°Ce(d, d)14°Ce with this parameter set 1. As fig. 4 shows, we obtained one additional maximum in the calculated angular distribution. Only with a radius parameter near r v = 1.05 fm, For the '1He(d,aHe)aH reaction, we have measured cross sections which are similar to those observed in the "~He(d.dpHe reaction at corresponding angles.
ELASTIC DEUTERON SCATTERING
279
we could get a reasonable fit. We therefore calculated in the case o f 2°8Pb(d, d) an angular distribution with parameters which are the arithmetic mean of sets I and 2. The result is an excellent fit to the data points (fig. 9). For the nuclei with ,4 < 64,
3.5
// .
30
//
///~/
25
"
////~ // // •
T .~
/#
15
52MeV •
•
//" /
1.0 /
I~V
0,.5 4He 160 'l
''l
50Ti54Fe 89y
'""'
'
~ll
I I
i t
140Ce 208pbA2/3
i
I
1
'
I'
Fig. 8. T o t a l r e a c t i o n cross s e c t i o n O'reaction as a f u n c t i o n o f A L T h e p o i n t s are t h e a v e r a g e o f a n o p t i c a l - m o d e l c a l c u l a t i o n u s i n g p a r a m e t e r set 1 ( t a b l e 1) a n d p a r a m e t e r set 2 ( t a b l e 2). A s m o o t h c u r v e is d r a w n t h r o u g h these p o i n t s . F o r c o m p a r i s o n , the m e a n A~ d e p e n d e n c e o f O'reaetion is g i v e n for the lab e n e r g i e s Ea -- 11.8 M e V [refs. i-3,1z, 20)] a n d E d = 26 M e V [ref. al)]. T h e d a s h e d c u r v e is a n o p t i c a l - m o d e l c a l c u l a t i o n o f a " d e u t e r o n " w i t h Z = 0 a n d the l a b e n e r g y E d = 52 M e V u s i n g the p a r a m e t e r s o f t a b l e 1, 2 a n d 5.
o '[0
l
I
I
I
I
I
I
l
I
I
I
l
I
I
I
l
I
I
I
0.1
O01
20*
40*
60*
80*
I
100"
120*
OcM
Fig. 9. Best fit to the '~°sPb(d, d)2°sPb d a t a u s i n g t h e f o l l o w i n g p a r a m e t e r set: V = 93 M e V , rv = 1.15 fro, av = 0.80 fm, W = 14.5 M e V , rw = 1.25 fm, aw = 0 . 8 0 fm, Vs.o. = 6 M e V , r c = 1.3 fro, Z z ~ 1.5.
280
F. HINTERBERGER et al.
the parameter set 1 with r V = 1.25 fm gives the better fits. We conclude that in addition to the pure A ~ law (Rv = A+rvwith rv = const), the radius of the real optical potential has a further dependence. The general behaviour of this is shown in fig. 10. II
I
w(fm) 14
I
I
I
I
l
I
I
I
I
I
I
I
l lll I 11..~ I I lOO 2ool A 89y 140Ce 208pb
12 10 08 06 Q4 02 0
I
J
Fig. I0. T h e ,4-dependence o f the radius p a r a m e t e r rv o f the real potential. 6.3. T H E D E P E N D E N C E O F T H E R E A L P O T E N T I A L D E P T H V U P O N C H A R G E Z, MASS ,4 AND ENERGY E
In figs. 11 and 12, the real potential depths V of this analysis are compared with those found at lower deuteron energies. They have been obtained with comparable
v~ev3
'
'
'
'
'
,
~,
90
85
/
/
•
i
l
o
~ o///c
.
80-
70
, , ,
S
,
,
o
,
,
2"15 MeV
_/~,,...5 2 IvleV
~///~,
,,~
/
• "
65 , , , "I
4He ~0
, "4"'
'
,
I
5a~54Fe 8gy
140Ce
2OSpb
Fig. I I. Real potcntial depths V as a function of Z / A - ' s for thc parameters of set I (table I). Thc 27.5 McV data 7) are transformed according V : ~ : 5 ~ const to the rca] potential radius rv -- 1.25 fro. The solid curves are calculated according relation (5) of the tcxt with V0 = 79 MeV, Ki = 1.5~ K~ = 0.35. T h e dashed curve would have resulted from a fit procedure using the real potential f o r m f D ( r ) o f e q . (12).
ELASTIC DEUTERON S C A T T E R I N G
281
g e o m e t r i c a l p a r a m e t e r s r v a n d a, a n d c o r r e s p o n d to a b o u t twice the n u c l e o n p o t e n t i a l (refs. 6, 7, 8,13)). F o r the r e d u c t i o n o f V f r o m the earlier d a t a to the r a d i u s p a r a m e t e r s u s e d in sets 1 a n d 2 (r V = 1.25 f m a n d rv = 1.05 fro), we h a v e a p p l i e d the r e l a t i o n
VrJ 5 = const.
vEMeV3
I
I
I I I I I I I I I I I I l i
130
125 -
120
115
110
344 MeV/ "~ 52 Me',/
105
100
,i
90 85
o.12"I"4"'I'6 '#'He 160
~' 8
~'
50"~i 54Fe89y !4'°Cer 2°8pb
a6Z/A~
Fig. 12. Real potential depths Vas a function of Z/A4: for the parameter set 2 (table 2). The 11.8 MeV data are from refs. a3, 2~), the 25.9 MeV data from ref. 6) and the 34.4 MeV data from ref. 8). They are transformed according Vr~.5 = const to the real potential radius r v = 1.05 fln. The solid curves are calculated according to eq. (5) with Vo -- 100 IVleV, K1 -- 2.5 and K~ .... 0.5. The dashed curve would have resulted from a fit procedure using the real potential formfD(r ) ofeq. (12). T h e solid c u r v e s in figs. 11 and 12 are c a l c u l a t e d with the c o n s t a n t s in t a b l e 6 (lines 6 a n d 7) u s i n g the f o l l o w i n g e x p r e s s i o n o f P e r e y 16), w h i c h d e s c r i b e s the m e a n d e p e n d e n c e o f V on the e n e r g y E a n d the C o u l o m b p a r a m e t e r Z/A'~:
V = V o + K 1Z/A + - K 2E,
(5)
V o, K 1 a n d K2 are c o n s t a n t s , E t h e c.m. energy, Z the c h a r g e o f the t a r g e t n u c l e u s a n d A the m a s s o f the t a r g e t nucleus.
F. H,NTERBERGER el aL
282
TABLE 6 T h e coefficients V0, K~ a n d K s
E,~b
rv
(MeV)
(fro)
Vo
K1
Kz
KI/ K.~
Ref.
(MeV)
11-27
1.15
81
2
0.22
9.1
tn)
11-27
1.3
75
1.14
0.42
2.72
a6)
91.2
1.03
0.57
1.78
25.9
1.13
27.5
1.21-1.265
1.7
~)
34.4
0.968 ÷ 0 . 0 2 9 , 4 3
27.5, 52
1.25
79
1.5
0.35
4.3
this work
11.8, 25.9, 34.4, 52
1.05
100
2.5
0.5
5
this work
0.89
r) s)
For definition, see eq. (5).
In the following section, we shall discuss the information contained in the parameters K, and/<2. The idea of expression (5) is to give within the framework of the effective mass approximation a linear dependence of the equivalent local optical potential V upon the kinetic energy Tin the nuclear interior (6)
V = Voo-~T.
This energy dependence arises partly from the local approximation of the optical potential and partly from an inherent velocity dependence of nuclear interactions 31). Inserting the expression for the kinetic energy T = E + V - Vc
(7)
in expression (6), one gets v = VoolO + ~)+[~/(1
+ ~)] v c - [ ~ l ( l
+
~)]E.
(8)
As the mean Coulomb potential Vc of a homogeneous charged sphere ol radius R c = 1.3 A+ fm is given by Vc = 1.33 Z / A + (MeV), one expects the following relation between K 1 and K2 comparing (8) with (5): K t / K z "~ 1.33. However, all known analyses 6-8, ,6) of deuteron elastic scattering gave values of K, essentially greater than the expected one of about 1.33 K2 (table 6). In an analysis of proton elastic scattering in the energy range 9-26 MeV, Percy 33) obtained a value of K , / K 2 between 1 and 1.33, where the value 1.33 (with K z = 0.3) is more realistic for higher energies. He considered an additional dependence of the proton optical potential upon the symmetry term ( N - Z ) / A . In the case of neutron scattering in the energy range up to 25 MeV, the energy dependence 32,34-) of the neutron optical potential depth gives also the value K z = 0.3. The values of K 1 shown in table 6 are essentially greater than 1.33 K2. This suggests that there exists another effect for deuterons which is responsible for an additional A-dependence of the optical potential depth V. This assumption is supported by the fact that the
ELASrlC DEUTERON SCATTERING
283
potential depth of heavier nuclei has a weaker A-dependence (figs. 11 and 12). In the following, we shall show that this additional A-dependence of the real potential depth may be understood by the effect of the finite size of the deuteron. If one derives the deuteron potential from the nucleon potential, a zero-order approximation gives the relation
Vd(r ) = V.(r)+ Vp(r).
(9)
The next step accounts for the spatial distribution of proton and neutron within the deuteron 17-20) Vd(r ) = f~2(X){
V.(e+½X)+ Vp(r-½X)}dX.
(10)
If one uses the Hulth6n wave function of the free deuteron instead of the unknown real spatial distribution of proton and neutron during the interaction with the nucleus @n(X) = B[exp ( - ~X){ I - exp ( - fiX)} ]/X
(ll)
and a Woods-Saxon form for the nucleon optical potentials, the following more realistic form factor for the deuteron optical potential is obtained: fa (r) =
f ½el)2 (x){f~Ts(l,+½Xl)+f~".~(Ir--~-Xl)}dX.
(12)
The integral was carried out numerically. The resulting form-factor fd (r) was then fitted with a Woods-Saxon form Jws (r). The geometrical parameters R N and aN of the nucleon Woods-Saxon form fwNs(r) were chosen so that the fitted Woods-Saxon form fws(r) had about the same geometrical parameters rv and av as sets I and 2, respectively. Fig. 13 shows the result of this procedure for 32S. Because of the finite distance between the nucleons in the deuteron, the central depth o f f d ( r ) is smaller than given by a Woods-Saxon form factor in the case of ]
lo ~ 08
I
~
I
L
]
i
I
5
6
7
8
- _ _. < j r , ( r , Ro,%)
- ~ ~ fD(r)/ " ~ \\y~fNs(r" '\ , RN,aN)
0.6
~ -
R,
02
aD
0 F i g . 13. C o m p a r i s o n
1
2
3
of the real potential
4
form factors f~s(r, c a s e o f 3-'S.
RN, aN),fd(r) a n d
r(fm) f w s ( r , R d , a d ) in t h e
r. HINTERBERGERet al.
284
l i g h t n u c l e i w i t h c o r r e s p o n d i n g l y s m a l l r a d i i Rv = A~rv. I n fig. 14, t h e d i f f e r e n c e K (13)
K = [fws(0)--fd(0)]/fws(0)
h a s b e e n p l o t t e d as f u n c t i o n o f t h e m a s s n u m b e r A f o r sets I a n d 2 g e o m e t r i e s . T h e c o r r e s p o n d i n g g e o m e t r i c a l p a r a m e t e r s a r e c o m p i l e d in t a b l e 7. I
I
I
I
I
I
I
I
I
I
I
i
I
K~ 12
10
I 8 6 4
2
I
0
40
80
120
160
200
240
Fig. 14. Geometrical correction K(A) for set 1 t o ) and set 2 to). TABLE 7
The correction K = (fws(O)--fd(O))/fws(O) and the geometrical parameters o f J ~ s ( r ) and o f f w s ( r ) fitted to fd(r) Set 1
A 4 12 20 32 50 89 140 208
Set 2
J'~s(")
f w s (r)
K (%)
rN
aN
rd
ad
1.48 1.38 1.35 1.32 1.295 1.28
0.21 0.4 0.42 0.44 0.45 0.45
1.25 1.25 1.25 1.25 1.242 1.244
0.64 0.76 0.75 0.76 0.76 0.75
11.3 7.5 5.2 3.3 2 0.9
1.265
0.40
1.245
0.70
0.1
fwNs(r) rN
1.31 1.21 1.16 1.13 1.12 1.10 1.08 1.075
fws(r)
aN
rd
0.2 0.42 0.48 0.52 0.55 0.60 0.68 0.70
1.05 1.05 1.046 1.044 1.056 1.055 1.046 1.049
K (%) ad
0.67 0.80 0.83 0.84 0.86 0.88 0.95 0.96
12 10 8 6 4 2.2 1.3 0.7
The parameters of the Hulth6n function are ~ = 0.231 fm -1, !3 == 1.207 fin -~ The
deuteron
optical
potential
C o r r e c t i n g t h e m as ( I + K ) V ,
depths
contain
this additional
A-dependence.
one obtains the depths which the search program
would have found using the more realistic form factor fn(r).
The corresponding
285
ELASTIC DEUTERON SCATTERING
II11111111
I l l l l l l l l l l l l l l l l l
'l"ll'lllltlllll
I Iit I I 1 I 1 I I lI:~_
-
~
-
I
~ _:
°
---o ~- p~
•"f
?
<
i
I I L]
11
1 1
l
t i i i i i 11
(Lu;) "e
Ii
-
iitlliiii
L
1 i ii
i i i i
E^al,,t] M
Iltll
t11'1'1'i'i'ilit
l ii
i ii
.
11
i i i i I
._L]
( w i ) ~J
h
.
L
! I:
~(wl)Me
'llilllllltllll|
......i ~ i i i i i i i i i l ~ L
•"
i i i i 11
-~o e
II
L
" "1:
0-
-\'~ .
-
~.
",L %i P
:-
II 0
I II CO 0
(mt
I111
III
(~ ~ 0 a
) ^e
Itllllilllll
~lllllllllllll
C~ 0 .....
~,oN M
( ~ l ) ''J
I
:l"
-~o [
I_L.Llllti!!_L
<
286
r. HINTERBERGERet al.
curves are d r a w n d a s h e d in figs. 11 a n d 12. However this geometrical c o r r e c t i o n is still insufficient to reduce the coefficient K ~ to the expected o r d e r o f m a g n i t u d e K 1 / K 2 ~, 1.33. The d a s h e d curves in figs. 11 a n d 12 give K 1 = 0.85, K t / K 2 = 2.4 for set 1 a n d K 1 = 1.5, K I / K 2 = 3.0 for set 2. The inclusion o f the finite size o f the d e u t e r o n using the Hulthdn function 4~H (x) reduces the value o f K t / K 2 by a b o u t 50 ~ to 2.4 and 3.0, respectively. These values are still a b o u t a factor o f two greater than the expected value o f K I / K 2 = 1.33. This m a y be interpreted as an indication for a spatial p o l a r i z a t i o n o f the d e u t e r o n d u r i n g the interaction with the nucleus, which leads to an increased average n e u t r o n - p r o t o n distance c o m p a r e d to the free deuteron. 6.4. AVERAGED PARAMETER SETS Using the o p t i c a l - p o t e n t i a l waves in D W B A analyses, one finds that averaged p a r a m e t e r sets give g o o d fits. Often these fits are even better than those calculated with individually searched p a r a m e t e r sets. One o f the p u r p o s e s o f this analysis is to give o p t i c a l - p o t e n t i a l sets which are c o m p a r a b l e with those f o u n d at lower d e u t e r o n energies. The general d e p e n d e n c e o f the real potential d e p t h V u p o n the charge Z , mass A a n d energy E was discussed in subsect. 6.3. The r e m a i n i n g p a r a m e t e r s are plotted against A ~ in figs. 15 and 16. S m o o t h curves have been d r a w n t h r o u g h the points. In table 8 is the analytical form o f these averaged parameters. F o r c o m p a r i s o n we give TABLE 8 Averaged parameter sets Set 1
rv = 1.25 fm, Vs.o. = 6 MeV, r c ~ 1.3 fm
V (MeV) W (MeV) av (fm) rw(fm) a~(fm)
79-i- 1.5Z/A-Y --0.35E 13 0.81 -- 0.024A~ 1.25 0.51 -I-0.076A:~
Set 2
rv = 1.05 fm,
V (MeV) W (MeV) av (fro) rw(fm) aw(fm)
5-]-2A4s 0.71 +0.04A~ 1.28 0.71 -t-0.02A~
34.4 MeVa)
rv =0.968~ 0.029A-~, Vs.o. ~ 7 MeV, re = 1.3 fm
V (MeV) W (MeV) a,. (fm) rw(fm) aw(fm)
90.2-~ 0.89Z/A~ 4.33 +2.2A~ 0.814 1.09+0.8A "~ 0.554 +0.059A't-
a) Ref. s).
Vs.o.:= 6 MeV, r C = 1.3 fm
IO0+2.5Z/A~s-0.5E
ELASTIC DEUTERON SCATTERING
287
the averaged parameter sets found at a deuteron energy E d = 34.4 MeV [ref. a)], which are comparable with the averaged parameter set 2. 6.5. RESULTS IN DWBA ANALYSES We have used the optical-model parameters in a series of DWBA analyses of (d, d') and (d, 3He) reactions with 52 MeV deuterons to calculate the distorted waves of the deuteron channels. in agreement with the discussion of the A-dependence of r v, we found that the inelastic deuteron scattering a5) from target nuclei with A __< 64 is better described by use of parameter sets with r v = 1.25 fro. In the case of the (d, 3He) reaction 36,37) o n 12C, 14N, 4s's°Ti and 51V, we used in the DWBA analyses the parameters of set 1 as well as those of set 2. The differences in the D W B A cross sections were less than 5 700. The Vr~ = const ambiguity cannot be solved with these data. In 12C(d, 3He) 11B and 14N(d, 3He)l 3C, the observed weak j-dependence of l = 1 transitions appears in the DWBA predictions if deuteron parameters with inclusion of a spin-orbit potential are used.
7. Conclusion It is concluded that elastic scattering of 52 MeV deuterons on target nuclei with A _>_ 12 may be described very well by the optical model. For heavier target nuclei, it is necessary to use an optical potential with spin-orbit term. The spin-orbit term is unimportant for the elastic scattering angular distributions of light nuclei. The angular distributions of elastically scattered deuterons from 1H, 3He and 4He show a structure which may be understood as a superposition of shape elastic potential scattering dominating at forward angles and nucleon-transfer processes dominating at backward angles. The coefficients K 2 of the energy dependence of the real potential depth are 0.35 and 0.5, for the two radius parameters used. The coefficients K 1 for the Z/A ~ dependence are 1.5 and 2.5. This large value of KI/K 2 can be explained only partly by the influence of the Coulomb potential. It indicates a large spatial polarization of the deuteron during the interaction with the nucleus. The two averaged parameter sets allow a good estimate of optical potentials at neighbouring energies as well as for other target nuclei. The optical parameters of this analysis were successfully applied in DWBA calculations. We thank Professor T. Mayer-Kuckuk for his interest in this work. We are indebted to Dr. G. Schatz and the staff of the cyclotron laboratory at Karlsruhe for their cooperation. We wish to express our appreciation to Dr. E. H. Auerbach for making available the ABACUS code and to Drs. R. H. Bassel, R. M. Drisko and
288
F. HINTERBERGERel a].
G . R. S a t c h l e r f o r m a k i n g a v a i l a b l e t h e c o d e J U L I E . o p e r a t o r s o f t h e I n s t i t u t fiir l n s t r u m e n t e l l e M a t h e m a t i k of the Rechenzentrum
Thanks are also due to the der Universitat Bonn and
d e s I n s t i t u t e s fiir H o c h e n e r g i e p h y s i k
der Universi~t
Heidel-
b e r g f o r t h e i n n u m e r a b l e I B M 7090 c o m p u t e r r u n s . T h e h e l p o f H . L i n k a n d H . M a c k h i n d a t a t a k i n g a n d h a n d l i n g is g r a t e f u l l y a c k n o w l e d g e d .
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