The optical potential for vector-polarized deuterons of 52 MeV

Nuclear Physics A339 (1980) 61-73; © North-BoUand Petbltfhttt~ Co., Muterrlatn Nbt to be reproduced by photopriat or microfilm without written parmirsion ttom the publh~her

THE OPTICAL POTENTIAL FOR VECTOR-POLARIZED DEUTERONS OF 52 MeV G. MAIRLE, K. T. KNöPFLE }, H. RIEDESEL and G. J. WAGNER Max-Planck-Institut für Kernphysik, D-ó9g0 Heidelberg, W Germany and V. BECHTOLD and L. FRIEDRICH

Kernforschungrtentrren Karlsruhe, Institutfür .lngewandte Kernphysik, D-75QD Karlsruhe, W Gernrtory Receivod 7 December 1979 Abstract : The vector analysing power for elastic scattering of 52 MeV polarised deuterons was measured for '~C, '60, '°Ne, 'sSi, '=S, `°Ar, ssNi, 9°Zr and '9'Au. The data were analysed together with proviously measured differential cross sections in the framework of the optical model. Heat-fit and average optical-model parameters were obtained both for surface and volume absorption . Fits with surface absorption are superior to those with volume absorption, especially for óeavier nuclei. Typical parameters of the spin-orbit part of the best-fit optical potentials are found to be Y, .e. - 5.5 MeV, r, .o . . I .15 frn and a, .o . . 0.4 fm, and there is no evgdena for an imaginary component. The volume integrals and rms radii of real, imaginary and ! ~ a potentials show a smooth mass deprndrnce and differ insignificantly for differrnt sets of potrntials. E

NUCLEAR REACTIONS "C,' 60, =°Ne, 2"Si, s=S, 4° Ar, ssNi, 9oZr,'9'Av(~, d), E = 52 MeV ; measured iT(~ ; deduced optical-model parameters Enriched targets.

1. Inbrodaction From the first comprehensive investigation of elastic deuteron scattering at 52 MeV on 27 complex nuclei ') ~ several sets of optical-model parameters were obtained. Part of the ambiguity arose from the insensitivity of the dil%rential cross sections to the spin-orbit potential. This is also true for the cross sections of inelastic deuteron scattering and transfer reactions. Therefore the ambiguity did not represent a problem and the optical potentials of ref. t) were successfully used for many DWBA calculations . When polarized deuterons of 52 MeV became available, the uncertainty of the spin-orbit potentials was felt immediately in attempts to describe the analysing powers of (~, s) reactions ~) in the framework of the DWBA. Therefore we decided to measure vector analysing powers of elastic deuteron scattering on a series of nuclei and to perform a systematic optical-model analysis of both existing differential cross. sections t) and vector analysing powers . s Supported by Deutsche ForacJtungsgemeinschaft. 61

62

G. MAIRLE et al.

In the past few years the elastic scattering of polarized deuterons at lower energies has been the subject of many detaíled investigations . A variety of data have been published for deuteron energies between 9 and 16 MeV [refs. a-lo)] and for 30 MeV [ref. ' 1)] . It could be shown that optical-model potentials with up to 18 parameters can describe the measured differential cross sections and all four analysing powers °'). It is the aim of the present investigation to demonstrate that optical potentials with a restricted number of parameters can reproduce the differential cross section and the iTl , component of the analysing power for the elastic scattering of 52 MeV deuterons on several nuclei distributed over the periodic system . This optical model study is thought not to be an end in itself, but the resulting parameters should be used in DWBA calculations for transfer reactions such that there is a direct and useful application to nuclear spectroscopy . From this point of view the knowledge of average potentials is desirable with fixed radius and diffuseness parameters and potential depths smoothly depending on mass and charge of the target nuclei. 2. Experimental procedure The experiments were performed with vector-polarized deuterons produced in a Lamb-shift ion source and accelerated by the Karlsruhe isochronous cyclotron iz). A beam of up to 40 nA was obtained on target with the analysing magnet of the beam transport system set in the achromatic mode. The energy spread of the beam was then 250 keV. Before and after the runs the beam polarisation was determined to P = 0.45 t 0.05 by utilizing the vector analysing power i T, 1 = 0.314 f 0 .035 of the ' ZC(d, d) reaction t 3) at B,,b = 47°. The long-time stability was better than 5 %. We used self-supporting Si, SeNi (99.9 ~ enriched), 9°Zr (99.3 ~ enriched) and Au targets which were 1-3 mg/cm2 thick. The gases C3H8, O2, a°Ne (99.9 ~ enriched), HZS and Ar were kept in a gas cell of 7 cm in diameter and 6 ~m Havar windows at a pressure of about 500 mbar. The scattered deuterons were detected by three dE-E telescopes each consisting of (1 .5 mm+1 .5 mm) dE Si surface barrier counters and 5 mm Si(Li) E-counters . We obtained an overall energy resolution oftypically 300 keV, resulting mainly from the energy spread of the beam . Except for 19'Au, the resolution was sufficient for separating the elastic scattering from inelastic peaks and even for resolving most of the strongly excited collective states . The analysing power iT, 1 of the reaction products was determined by measuring spectra with different beam polarisation P. During the measurements P was flipped every few minutes following a preset charge accumulation in the Faraday cup. The spectra were measuredtypically between laboratory angles of 8° and 90° in steps of 1° at forward angles, in steps of 2° at larger angles and in steps of 0.5° in regions of very pronounced structures of the analysing power. The relative uncertainties in the analysing-power measurements are primarily due to counting statistics, except

OPTICAL POTENTIAL

63

for some forward angles ; the absolute uncertainty results primarily from the error in the beam polarisation. 3. Retaalts The measured analysing powers of elastic deuteron scattering on ' Z C, ' 60,

2°Ne,

,~ 0 -0 .Y -0 .4 -0.0

ar ~ ~ er

erg

Ya

~

er

.~ 0

~ ~r -

~ - er

sn~

Fig. I . Measurod vector analysing power for the eWtic scattering of 52 MeV deuterom on `YC, ' 60, =°Ne, Y°Si, 'YS, 4°Ar, °°Ni, 9 °Zr aad ' 9'Au compared to optical-model calculations (i,. ~ 1 .25 fm, surface absorption, see table 1) .

64

r

G. MAIRLE et al.

~ss~ s~s~ ao~.~ saNi~ 9oZr and t 9 'Au are displayed in fig. 1 . The error bars are . purely ofstatistical origin . The measured effects are pronounced (IiTi i 15 0.75), and we thus assume that the extracted optical potentials are well defined. The differential cross sections given in fig. 2 are taken from an earlier measurement') of unpolarized deuteron scattering at 52 MeV. For experimental details see ref. t).

Fig. 2. Measured angular distributions of the differential cxoss section divided by the Rutherford cross section for the elastic scattering of 52 MeV deuterons compared to optical-model calculations (r,, = 1.25 fm, surface absorption, sa table 1) .

OPTICAL POTENTIAL

65

4. Parametrisatioo of the optical potentials We use optical potentials of the form

U(r) = Yc(r) - vh(r)- iWfz(r)+4iW'a,r"ß(r)+(h/m~F)~V..o.l ' sl4(r)~

(1)

Here Yc (r) is the Coulomb potential of a uniformly charged sphere of radius Rc = 1 .3A} fm. The radial form factors are defined as follows : 1'ß(r) _ (1+exp(r-r~A})/ay)-1, .Î2(r) _ (1+exp(r-rWA#)law)-1+ Ia(r) = r-'d/dr(1+exp(r-r..a.A})/a..a.)-1~

Volume absorption ( W' = 0) as well as surface absorption ( W = 0) was considered f . The parameters defining the potential were adjusted by an automatic search routine to minimize the quantity XZ per degree of freedom XZ =

N

1

N

n `~

2

CZdzZj]

.

The numbers z, are the measured dit%rential cross sections a(B,) or the vector analysing power iTll(Bi) with their related experimental errors dz,. The calculated quantities are denoted by z,; N is the number of experimental points and n the number of free parameters. The optical-model code MAGALI [ref. '4)] was used for the calculations . 5. Resolts of tóe optla~l-model caladations 5 .1 . BEST-FIT POTENTIALS

Since it is well known that the radius parameter ry and the potential depth V of the real part of the potential (eq. (1)) are highly correlated according to Vr"~ = coast, we stertod by searching for the best value of ry which then wind be used for the real potential of all nuclei. We performed calculations with parameter r,, fixed ; all the other eleven parameters of potential (1) ware adjustable. We found : (í) BY varying r,, between 0.9 fm and 1 .3 frn no pronounced minimwn of XZ(rr) could be found. The larger values of ry are slightly preferred for heavier nuclei. We decided to use fixed values of r~ = 1.15 fm or ry = 1 .25 fm. (ü) The search routine automatically diminished the volume absorptive part of the potential ( W ~ 0) f We take the opporhmity to note that a factor (4a,r .) û miming in the definition of the imaginary part of the optical potential in ref. ').

Teams 1

66

Beat fit optical potentiale with surface absorption')

(a) rv = 1,15 fm Az "C 's 0 2°Ne 2ssi

"S 4oA r ssNi 9°Zr '9'Au

(Me~

(fm)

(Me~

(fm)

(fm)

(Me~

(fm)

(fm)

s X

75 .1 81 .5 77 .2 79 .9 80 .3 79 .5 81,7 85 .7 90 .8

0.765 0.825 0.832 0.820 0.808 0.824 0,791 0,829 0.836

9.73 11 .6 12.3 12.8 11 .6 11 .4 11 .5 13 .5 18 .7

1 .32 1 .36 1,30 1 .23 1 .23 1 .16 1 .22 1 .27 1.31

0.722 0.701 0 .750 0.755 0.871 0.925 0.874 0.747 0,728

5.96 5.50 4.78 5.77 5,41 6,25 5.45 6.62 6.18

1.07 1.22 1.22 1.18 1.15 1.14 1.f6 1.20 1 .26

0.569 0.403 0.330 0.400 0.401 0.365 0.405 0.419 0.311

12.7 13.0 7.71 12.7 4.45 6.50 10,2 25 .4 3,09

0.714 0.777 0.775 0.754 0.754 0.772 0.709 0.743 0.590

10 .0 12 .8 13 .4 13.7 13.1 13,7 12,4 13.8 13 .5

1.37 1.37 1.31 1 .25 1 .25 1 .25 1 .21 1 .28 1 .23

0.675 0 .645 0.693 0.718 0 .790 0.761 0.835 0.719 0.962

5.45 5.12 4.60 5.19 5.02 5.61 4.73 5.43 6.07

1 .13 1.30 1 .28 1.25 1.25 1,25 1.23 1.27 1.01

0.560 0.373 0.303 0.370 0.380 0.366 0.363 0.377 0.282

13 .8 11,3 6.36 12.5 4.74 4.82 5.81 19.1 5.57

(b) rv = 1.25 fm '=C `s0 ~°Ne ~°Si "S 4oAr ss Ni 9°Za ` 9'Au

67 .4 73.4 69 .1 71 .2 72.7 73 .1 72,8 75 .3 80 .3

') For definition of the parameters see eq . (1).

(a) rv = 1,15 fm

Teams 2 Beat fit optical potentials with volume absorption')

Az

V (Me~

ay (fm)

W (Me~

r,r (fm)

a~. (fm)

V..o . (Me~

r, .o . (fm)

a,,o . (fm)

xs

'=C 's0 2°Ne 2°Si ~=S 4oAr '°Ní 9°T.r "'Au

80 .2 82 .2 78.2 79.3 79.8 78.7 T9.8 83,4 90;5

0.819 0.817 0.856 0.842 0.814 0,815 0,802 0.819 0.755

10 .4 10 .2 12 .3 11 .1 11 .4 9.76 9.57 9.61 9.01

1 .93 1.95 1 .79 1 .75 1 .70 1.71 1 .68 1 .70 1 .63

0.476 0.567 0.633 0.617 0.810 0.757 0.785 0,685 0.626

5.13 5.66 3.90 5,17 4,34 7.19 6.68 6.63 6 .80

1 .17 1 .11 1 .17 1 .13 1 .13 1 .11 1.11 1.10 1.23

0.510 0.480 0.361 0.402 0.342 0.367 0 .452 0.470 0.488

12 .8 19 .9 11 .3 16 .2 12 .6 8.13 13,1 46.7 6.54

0.777 0.773 0.808 0.783 0.737 0.740 0.708 0.T26 0.725

10 .8 10 .9 13 .5 12,2 12 .2 10,6 10 .6 10 .5 9.80

1 .93 1 .93 1.76 1.74 1 .68 1.68 1.62 1.65 1.63

0.470 0.551 0 .623 0.593 0 .811 0.751 0 .839 0.780 0.606

4.74 5.30 3.73 4.76 3.72 7.44 6.53 6.27 5.35

1.22 1.18 1 .21 1 .20 1,20 1 .17 1 .19 1 .19 1 .29

0.494 0.447 0.344 0.351 0.359 0.365 0.432 0.440 0.405

13 .6 19 .5 11 .3 16 .4 16 .8 9.98 14 .7 50 .1 10 .9

(b) rv = 1.25 fm `=C 's0 =°Ne '~Si 3'S '°Ar ssNi '°Zr '9~Au

71 .6 73 .7 69 .4 70 .7 70 .1 70 .3 70 .8 72 .4 76 .0

~ For definition of the parameters see eq . (1).

OPTICAL POTENTIAL

67

ifthe search was started with W ~ 0. This reflects the fad that surface absorption is generally superior to volume absorption (see subsect. 5.2). We decided to fit the data with either pure volume or pure surface absorption . Calculations with either type of absorptive potential may be required in DWBA calculations where the use of potentials with equivalent form factors in the entrance and exit channels minimizes some approximations in the standard DWBA treatment' s). As a result we obtained two sets (ry = 1 .15 and 1 .25 fm) of optical model parameters both for surface absorption (table 1) and volume absorption (table 2). The relative quality ofthe fits can be judged from the Xs values included in the tables . The dül'erential cross sections and analysing powers calculated with potentials with r1, = 1 .25 fm and surface absorption on the average show best agreement with the experimental data . They are given for comparison in figs. 1 and 2. Potentials with ry = 1.15 fm give slightly worse fits except for Au. Potentials with volume

10

12C J=100 (104) MsV"ims

=aaa (3.Oif)1m

5' 0

40Ar 10

E

J=131 (123)M~V"Im 3 =à2f(426)fm

5 0

10 5 0~ 0 Fig . 3. The radial dependence of the imaginary part of the beat-fit optical potentiela witó r~ = 1 .25 fm (tabka 1 and 2) for "C, `°Ar and ' 9 'Au . The values for the corresponding volume integrals J and rms radii (R ) of the awfaoe absorbing potentials (full lines) are inserted, those of the volume potentials (broken lines) are given in brackets . Note that potentiela with very dit%renl radii (R  > R .) show the same leila and give similar volwee integrals and rms radii.

68

G . MAIRLE et al.

absorption produce Xi values of about a factór of two larger than those obtained with surface absorption except for' 2C. Using volume absorption, no acceptable fit could be obtained for 9°Zr. In all four sets of potentials a relatively small variation of parameters with target mass is observed . The only striking difference between the parameters of tables 1 and 2 occurs for the radius of the imaginary potential rw = 1 .79 i0.17 fm for volume absorption and rW, = 1.26 f 0.13 fm for surface absorption . At first sight this might be surprising. But inspecting the radial dependence of both types of potentials, depicted for' 2C, °°Ar and ' 9'Au in fig. 3, one realizes that the tails, . which are important for strongly absorbed particles, are almost identical. In addition, it becomes plausible why light nuclei (' ZC) can be described equally well by optical potentials with surface or volume absorption, but that this is not the case for heavier nuclei where the imaginary potentials are quite different for distances r - R,,. 5 .2 . AVERAGE POTENTIALS

It is an empirical fact that in DWBA calculations average potentials are generally superior to best-fit potentials . An easy interpolation between potentials of measured nuclei is also often of advantage. We therefore searched for optical potentials with fixed values of the six geometrical quantities (radius parameters and diffuseness) such that all individual properties of the nuclei reflect themselves in the potential depths only . We started with a fixed value of ry = 1 .25 fm. The parameter which exhibited the smallest variation for the different nuclei was then fixed to its mean value (slightly rounded off) . This procedure was continued until all radius and diffuseness parameters were kept fixed. Interestingly, for some of the best-fitted nuclei this was accompanied by only a small increase of X2 . The results are given in table 3. Again surface absorption resulted in better fits than volume absorption for most cases. With this procedure a very simple potential was obtained for surface absorption : All radius parameters turned out to be equal (ry = rw- = r,.o. = 1.25 fm). There are equal diffuseness parameters a,, = aw. = 0.75 fm for the real and the imaginary potential and the diffuseness of the spin-orbit potential is a,.,. = 0.75/2 frn . The depths V of the real potentials show a typical dependence on the mass number A and charge Z of the target nuclei, which was parametrised following Perey ' e) as From the values of table 3 we calculated by linear regression a = 0.713±0.048 MeV and 6 = 68.19 f 0.35 MeV. If the increase of the well depth W' of the imaginary potential is assumed to be linear in A}, i.e.

OPTICAL POTENTIAL

69

Teat.s 3

Average optical potentials Srajace aluorptiwt

rv = r,r. = r, A, = 1.25 fm, av = a,r. ~ 0.75 fm, a, ., . = 0.375 fm

At:

V (MeV)

'=C '60 =° Ne ~°Si '~S 4oA r °sNi 9°Za ' 9~Au

71 .0 72 .5 67.4 70 .7 72.2 71 .8 73 .9 75 .1 78 .9

Voltort aásorptiow

W' (MeV)

V..e . (MeV)

X:

3.58 4.78 4.42 5.17 5.00 5.98 4.82 5.20 5.79

19 .7 13 .3 8.21 12 .4 6.86 6.05 10.4 19 .2 16.6

11 .4 12.5 12.9 13.0 13 .7 13 .4 I3 .5 13 .7 16.4

rv = 1 .25 fm, rp = 1.75 fm, r, .o. = 1.20 fm, av s 0 .75 fm, a,r = 0.65 fm, a,A. = 0.40 fm .

A=

V(MeV)

W (MeV)

V, .a . (MeV)

X~

'=C

69 .1 73 .4 65 .7 69 .1 71 .6 70.5 72.4 73 .5 74.6

12 .9 13.8 12.4 11 .5 11 .6 9.82 9.08 9.04 7.54

3.55 4.56 3.35 4.94 4.45 8.30 6.68 6.02 6.07

19 .2 23 .2 14.3 18.4 18.1 11 .1 35.7 67.9 16.1

16O

'°Ne se Si 33S

40~

se Ni 90~ ' 9~Au

') For defmition of tóe parameters see eq . (1).

we obtain c = 1 .12 f 0.06 MeV and d = 9.48 f 0.20 MeV. The depths of the spinorbit potential fluctuate about a value of V,.o. = 4.8 MeV with a maximum deviation of 1 .2 MeV. Starting with the geometrical parameters of the average potentials of table 3, we tried to improve the fits by including an imaginary component of the 1 ~ a potential with Thomas form, as proposed and used by Goddard and Haeberli 4). With fixed radius and diffuseness parameters a search was performed for the potential depths V, W, Y, .o. and w,.o. . As a result we obtained almost the same values as given in table 3 and values of W;.,, compatible with zero. Therefore, for the description of the present data an imaginary spin-orbit poUential is not necessary.

70

G. MAIRLE et al.

6. Volnme integrals and manare radii Following the ideas of Greenless et al . "), we were interested in quantities which are common to the different potentials that fit the data for a given target nucleus. Candidates for such quantities are the volume integrals per pair of interacting particles defined as J = AlA J V(r)dr a (RZ~~ and the mean-square potential radii defined by the relation (RZi

= A A (' raV(r~r, ~

where Aa is the mass of the deuteron and A the mass of the respective target nucleus. 8 E

7 6 5 4 3 400

E 300 200 100 0 Fig. 4. Means square radii (R ) and volume integrals J for best-fit potentials (r~ Q 1 .15 fm, surface absorption, table 1) of the real part (dots), imaginary part (triangles) and spin-orbit potential (circles). The corresponding valace for the different potential sets of tabka 1 aad 2 differ only slightly.

As a typical example we present in fig. 4 the rms radii and volume integrals of the real, imaginary and spin-orbit part of the optical potentials calculated with the bestfit parameters of table 1 (r,, = 1 .15 fm) and plotted versus Af. All rms radii increase

OPTICAL POTENTIAL

71

linearly with A} :
The rms radii
72

G. MAIRLE et al.

7. Summary The vector-analysing power for the elastic scattering of 52 MeV deuterons was measured on a series of nuclei from 1ZC to 19'Au. The data were analysed together with previously measured differential cross sections in terms of the optical model. Several sets of optical potentials dif%ring in the radius parameter of the real part of the potential and in the foam factor of the imaginary part were deduced which describe the measured data. Potentials with surface absorption are found to produce better fits than those with volume absorption . The related volume integrals, however, are very similar, andtheir rms radii show nearly perfect agreement despite the quite different radial dependence of the potentials. The best-fit spin-orbitpotentials arecharacterised by a welldepth of V,.o. = 5.5f 1.9 MeV, a radius parameter r,.o. = 1 .15±0.15 fm and a difuseness as .a . = 0.43 t 0.17 fm. The potentialsshow amaximum in thevicinity of R, .o ,, which is more pronounced for heavier nuclei than for light ones. The spin-orbit potentials of the differentpotential sets have, as a common feature, equal and nearly mass (A) independent volume integrals, and rms radii with (R; o .i } A- } = coast. An imaginary part of the l ~ s potential, as required °) to fit all four components of the analysing power, was not necessary for the present data. Comparison with the results of an earlier comprehensive analysis of differential cross sections ofelastically scattered deuterons at 52 MeV shows that the parameters of table 1 (r~ = 1 .25 fm) do not differ remarkably from those of set 1 of ref. '). Naturally, differences occur for the spin-orbit potentials which have been kept fixed in ref. ') at arbitrary values . Average parameters could be found, especially for surface absorptive potentials. They are characterised by fixed radius and diffuseness parameters . The free well depths ofthe real potentials show a linear dependence on the Coulomb parameter ZA -~, those of the imaginary potentials increase linearly with A~ and those of the spin-orbit potentials are rather constant. This parametrisation allows an easy interpolation between nuclei investigated in the present work, which is important for the application of the optical model potentials in DWBA calculations . The potentials have already been used in the analysis of several (~, i), (~, t) and (~, a) reactions on lp and 2s, ld shell nuclei. Depending on the type of optical potentials used for the exit channels, surface or volume absorbing potentials, respectively, from the present investigation produced convincing fits to differential cross sections and analysing powers of the transfer reactions and allowed to deduce reasonable spectroscopic information. This w~71 be the subject of a forthcoming paper. We thank Dr. H . Schweickert and the staff of the Karlsruhe cyclotron for their cooperation.

OPTICAL POTENTIAL

73

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