Optical-model analysis of elastic scattering and polarization of 49.5 MeV protons on 148Sm

Nuclear Physics A154 (1970) 5 13 - 525; @

North-HoIland

Publishing

Co., Amsterriam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

OPTICAL-MODEL

ANALYSIS

AND POLARIZATION P. B. WOOLLAM, Wheatstone

OF ELASTIC SCATTERING

OF 49.5 MeV PROTONS

R. J. GRIFFITHS,

Physics Laboratory,

JOAN F. GRACE and V. E. LEWIS t

King’s CoNege, Strand, London Received

ON “*Sm

WC2R 2LS,

U.K.

1 June 1970

The measurement and optical-model analysis of polarized and unpolarized proton elastic scattering data from 14aSm at 49.5 MeV are presented as a preliminary to coupled channels calculations for all the even Sm isotopes. The differential cross-section and polarization measurements extend over an angular range of 10” to 118” c.m. Analysis has been made in terms of the simple optical model and also the recent reformulation of Greenlees, Pyle and Tang using nuclear-matter density distributions and a realistic two-body force.

Abstract:

E

NUCLEAR REACTIONS 14sSm @, p), (polarized p, p), E = 49.5 MeV; measured u(e), polarization (0) deduced optical-model parameters. Enriched ‘arget.

1. Introduction

The formulation of the optical model has been developed into a sophisticated account of the scattering and absorption of nuclear particles by nuclei. It has been most widely applied to the scattering of protons but in recent years its extension to the description of the scattering of heavier particles has greatly increased our understanding of these basic nuclear interactions. It is to be hoped that development of nuclear models will lead to a comprehensive nuclear theory, and there is evidence that microscopic and macroscopic models are eventually meeting on common ground. In terms of the optical model, the reformulation of Greenlees, Pyle and Tang ‘) has helped to reduce the empirical nature of the parameters, and the physically significant concepts that are well determined by such analyses are becoming clearer “). The information required for optical-model analyses includes elastic scattering, polarization and total reaction cross sections; sets of these data over a range of isotopes “) and energies have been shown to be of particular value. The use of coupled channel analysis ‘) (strong coupling approximation) has also enabled inelastic scattering cross sections and asymmetry measurements to be included and it has been demonstrated ‘) that the model will effectively describe both the spherically symmetric and distorted nuclei. In this context the isotopes of samarium are of particular interest because they change from spherical nuclei l 44Sm, l 48Sm and 150Sm which are vibrational to “‘Srn and ‘54Sm which show characteristic rotational energy level schemes “). t

Now at National

Physical Laboratory,

Teddington. 513

514

P. B. WOOLLAM

et al.

Very little information is available on the isotopes of samarium. Jolly and Moore ‘) identified the isobaric analog character of some of the levels using 9.6 MeV protons, and Wildenthal et al. “) showed from the (3He, d) and (d, 3He) reactions that many configurations are involved in the description of the ground state wave functions of nuclei in this mass region. More recently Edwards et al. ‘) using the neutron pick-up reactions (p, d) and (p, t) have described the isotopes 142,143,145,146Sm in terms of a simple unified model in which the open proton shell of 62 protons is treated as a collective quadrupole vibrator. The active neutrons (or neutron holes) outside the closed N = 82 shell are considered to interact with the vibrator in the weak coupling approximation and with each other through an effective two-body interaction. This work in particular has shown the need for a more comprehensive study of the opticalmodel parameters for these isotopes. The only proton scattering measurements prior to those reported here were by Stoler et al. “) at 16 MeV where lack of structure in the angular distributions limited the analysis, and some work at 55 MeV [ref. ‘“)I which though limited by resolution and statistics showed the general features. Recent work in which 50 MeV a-particles have been scattered from the distorted samarium nuclei 5’“) has emphasised the nuclear structure information to be obtained from these studies, and effectively demonstrated that the optical parameters obtained for the spherical nucleus 14*Sm were valid for a wide range of nuclei into the deformed region when used with coupled channel analysis with a suitable distortion. To initiate a further study of the interaction of protons with samarium, the polarization and differential cross sections for protons elastically scattered from the spherical nucleus 14*Srn have been measured and are presented here together with predictions from the regular optical model and the reformulated optical model of Greenlees, Pyle and Tang. No previous work has been published in which this model has been tested at 50 MeV. The parameters obtained will be used to analyse further the elastic and inelastic scattering cross sections for all the stable even samarium isotopes which have already been measured at a similar energy by Fulmer et al. “). As part of this programme the total reaction cross sections have also been measured 13). In particular it is expected that the spin-orbit parameters of the optical model will be well determined by the polarization data so that they may be fixed for future analyses. 2. Apparatus and experimental procedure A beam of 49.5 MeV protons from the linear accelerator at the Rutherford High Energy Laboratory was momentum analysed 14) and transported to the 66 cm diameter scattering chamber 1“) of th e n = +, 180” double-focussing magnetic spectrometer 16). U n p o1arized beam intensities of 5 x 1011 protons per second were integrated in a Faraday cup subtending an angle of & 5” at the centre of the scattering chamber. The polarization measurement used a 5 x lo* protons per second, 56 % polarized beam “) which was monitored in position and intensity by a split ion chamber ‘*).

“*smh?, PI

515

The beam polarization was continuously monitored by a carbon polarimeter “) which intercepted 20 y0 of the beam bursts and the beam energy was determined periodically throughout the measurements using a range telescope ““). The angular acceptance of the magnet was determined by an aperture plate to be t-0.5”. This was increased by the beam spot size of 3 mm x 3 mm and the beam divergence of kO.2”. The targets + were 95 % pure self-supporting metallic foils of 5 and 15 mg * cm-’ for the unpolarized and polarized measurements, respectively. Their thicknesses were determined to 1-2 % by few-energy Coulomb scattering 12). The change in mean scattering energy produced by the different thicknesses was considered negligible in terms of variation of optical-model parameters. The scattered protons were detected in the focal plane of the magnet by a sonic spark chamber “‘). An overall resolution of 100 keV was obtained, which was adequate to separate elastic scattering from oxygen contamination over the whole angular range. 3. Results Measurements of the elastic cross section were made from 30” to 105” c.m. and of the polari~tion from 10” to 118” c.m. The values obtained, corrected for dead time effects (0.1 %) are given in table 1. Smearing of the angular distributions due to the finite beam spot size, beam divergence and angular acceptance of the spectrometer were included in the theoretical predictions rather than subtracted from experimental results. The errors in the elastic cross sections are statistical, to which should be added absolute errors of +2 % for the dete~i~ation of target thickness and kO.25 % for each of the errors due to determination of solid angle and calibration of the current integrator. The asymmetry in the elastic scattering was determined by the single counter technique at a fixed angle with the incident proton spin, up then down, for equal integrated beam. The asymmetry E to first order in LIPI is given by

where U and D are normalised counts for spin-up and spin-down beam conditions, P(8) the pola~zation in the scattering and P” and dP1 are given by 2P1 = IP:I + IP:l, Pf and Pj being the polarizations

2APr = IP:l- lP%

of the two spin conditions of the beam, The statis-

tical uncertainty is given by SE =

IIv+o1’

l--E2 7,

+ Isotope Division, Oak Ridge National Laboratory.

516

P. B. WOOLLAM

et al.

TABLE l(a) 14sSm (p, p) cross sections at 49.5 MeV

ec.m. 30.14 32.15 34.16 36.17 38.18 40.19 42.20 44.21 46.22 SO.24 54.26 56.26 58.27

4Q 292. 131. 37.3 11.5 25.7 57.1 75.6 76.2 63.2 21.4 5.34 6.28 9.62

fAa(t’)

e

m

*Au@)

60.28 62.28 64.29 68.30 70.31 74.31 78.32 82.32 86.33 90.29 94.29 98.29 102.28

11.8 12.7 10.9 5.38 3.20 1.53 2.07 2.14 1.48 0.74 0.51 0.55 0.55

0.18 0.7 0.16 0.09 0.05 0.03 0.03 0.04 0.03 0.02 0.01 0.01 0.01

0.m.

4. 1.9 0.67 0.29 0.48 0.8 1.0 1.0 0.8 0.32 0.10 0.11 0.15

TABLE l(b) ‘*sSm @, p) polarizations

e

0.0.

10.16 12.19 14.22 16.25 18.28 20.31 22.34 24.36 26.39 28.42 30.45 34.5 36.2 38.2 40.2 42.3 46.3 50.3 52.3 54.3

p(e) -0.032 -0.063 -0.107 -0.109 -0.044 0.213 0.262 0.176 0.052 -0.057 -0.177 -0.748 -0.322 0.87 0.640 0.407 0.067 -0.359 -0.592 -0.215

at 49.5 MeV

&AP@)

e c.m.

pm

fAP(B)

0.008 0.013 0.012 0.029 0.018 0.023 0.014 0.012 0.007 0.012 0.012 0.033 0.043 0.021 0.034 0.023 0.018 0.030 0.094 0.058

56.3 58.3 60.3 62.3 66.4 70.4 72.4 74.4 77.4 80.4 82.4 86.4 90.4 92.4 94.4 98.4 104.4 110.4 116.3 122.3

0.722 0.954 0.767 0.523 0.229 -0.208 -0.196 0.423 0.91 1.075 0.790 0.437 0.134 0.51 0.296 0.846 0.948 0.764 0.223 0.49

0.041 0.043 0.042 0.029 0.045 0.035 0.057 0.053 0.05 0.076 0.048 0.045 0.058 0.12 0.061 0.061 0.058 0.071 0.086 0.08

from which the values of D(8) given in table 1 were calculated. Errors of fO.O1 in the polarization due to the beam spot position variations and a further error due to the calibration of the beam polarimeter amounting to less than 4 % have been included.

517

‘4swP, PI 4. Regular optical-model analysis The form of the regular optical model potential used was

where Vc(r) is the Coulomb potential for a uniformly charged sphere of radius 1.25A* fm, and where f(x) = (ex+l)-’ with

(r-rRA*)/uR,

xR =

x, = (r - r, A*)/q , x, = (r - r,

&)/us .

The computer code RAROMP “‘) used for both analyses minimises the quantity

Proton

elastic

r4eSm

at

scattering

493

from

MeV

-

Regular

----

Reformulated

optical

model model

fit fit

1

0.01 I_

2c ,

I 40

I

I 60

Centre-of-mass

I

I

00

angle.

I

I

100

0

1

deg

Fig. 1. Fits to the cross-section data using the regular and reformulated too small to be visible.

models. The error bars are

P. B.

518

WOOLLAM

et cd.

Starting from the average geometry of Fricke et al. 23), the parameters of the regular model were varied in a sequence of P’s, WV, W,, V,; r, , r,, r,; a,, a,, as until convergence was achieved. A similar search was made starting from the global parameters of Becchetti and Greenlees 24). The results are shown in figs. 1 and 2 and summarised in table 2. It appears that the best fit is obtained to the data with geometry parameters similar to those of ref. 23) with the exception of the real and spin-orbit diffusenesses which are larger. The real central potential agrees well with that predicted by ref. 24) but the imaginary depths are rather different in that W, is much smaller whilst W, is larger than would be expected at this energy.

Polarization in the elastic scattering of protons from ‘?Sm at 49.5 MN Regular optical model fit Reformulated

I -l.o

I

1

I

IO

20

30

I

I

40

50

I

I

60

70

Centre-of-mass

,

1

00

so

model fit

I 100

I ii0

1

angle, deg

Fig. 2. Fits to the polarization data using the regular and reformulated models.

5. Reform~at~ In the reformulation

optics-model

analysis

of the optical model the potential is written as

I&(r) = I%(r) - K.&W(O)

- i

+vi(&)‘y$q where Y&r) is the usual Coulomb potential and with I(r) = ~~~(~)~~(I~- rl)dq.

b.L,

(5)

best fit

Parameter

47.88

VRS

46.74

5.87

WV

1.168

5.22

0.75

0.75

Becchetti z4)

1.16

47.76

Fricke 23)

5.76

Vs

8.19

5.72

5.73

0.816

1.168

WV

46.55

akt

best fit

rR --~--_ll_-~

v,

~~~_

Parameter

TABLE2

1.32

0.623

0.630

0.630

ai

TABLE3

1.37

1.360

rz

1.281

0.605

1.184

rM

r,

1.01

IO.64

1.075

0.701

aM

parameters

6.2

5.89

5.61

v,

parameters

Reformulated optical-model

1.37

3.84

3.45

WD

Regular optical-model

5.485

* M

0.75

0.738

0.816

0,

8.1

A2

-z n

528

26

6.1

A2 d n

13.6

-M

AP2

15

16

8.3

A,= m

11.4

-i;i

AT2

217

20

7.5

N

AT2

mb

1775

OR

1687

I841

1817

oR mb

c”

W

520

p. a. wooLLAMet al.

TABLB4 Values of the volume integral per nucleon of the real central potential obtained in grids over (a) the rms matter radius and (b) the matter diiuseness parameter

(4 J&A (MeV . fm3)

JRsIA

*

(MeV * fmJ)

(fm) 5.001 5.259 5.516 5.774 6.031

0.4 0.5 0.6 0.7 0.8

339.0 370.6 372.3 365.1 397.0

351.6 365.1 364.2 365.8 376.5

The functional form of the direct part of the two-body force is taken to be f*(r) = exp ( - ~r)lflr and the matter density distribution p,(r) L(r)

= &rN&O)

(71

is introduced through

= P +exp

(k--d/~M)l-‘~

(8)

Channels other than the elastic are accounted for by the usual adoption of volume and derivative Saxon-Woods imaginary terms. The variable parameters in the model are VRs, WV, W,, V,, , r,, a,, r, and a,. These were varied in groups of potentials plus diffusenesses followed by potentials plus radii I). Best fit parameters are listed on table 3 and the fits to the data are shown in figs. 1 and 2. The fits to the data using this model are slightly worse than those obtained with the regular model. Differences are discernible in the fits to the cross-section data beyond 70”. In particular the oscillations of the reformulated model are smaller in amplitude than those of the regular model. This feature is also apparent in the fits to the polarization data where the amplitude of the reformulated model prediction is generally lower than that of the regular model over the whole angular range, Similar effects have been observed by Thomas ““) in the analysis of 63*6sCu at 50 MeV. A grid was made over (r’)*, the mean square radius of the direct part of the two-body force, varying all the parameters to convergence. Fig. 3 shows the optimum value to be 2.1 f ::: fm2 where the errors represent a criterion of a 50 y0 increase in A’. This value is in agreement with that found in ref. ‘) using this technique, and with other theoretical and experimental evidence, cited in ref. I). Fig. 4 shows the variations of converged A2 per point with rms matter radius obtained from a grid perpendicular to the line of constant mean square radius passing through the best fit point obtained in the earlier search. A sharply defined minimum is produced giving a value of 5.48 f 0.09 fm for {r ‘>$, which is thus determined to +3 %. By contrast fig. 5 shows the variation of A2 with the matter diffuseness parameter where . the optimum value is found to be 0.700 Zz*“,z: fm. This parameter is thus determined only to within f7 %.

148sm(P, P)

521

---w--s

4‘> 1.188-

wll

L

>

7282 i3280 '1.278

1.182-

LE _ b?

1.276 z

0.62

0.64-

0.60

LE _

0.58

c

0.62-

I 0

I 1.0

I

2.0

I 3.0

I

I

4.0

50

t 5


of A2 per point for cross-section and polarization data, together with variations of the reformulated model parameters, in a grid over (rz>d.

522

P. B. WOOLLAM

et d.

6

51

> r”

41

. >” 3

2’

401

30’ S .0 a ki a “a -

20

IO’ 7 5 2 3

52

5.4

5.6 X2,

Fig. 4. Variation ofA

5.6

6.0

fm

per point for the data together with variations in the strength parameters in a grid over *M’

523

Fig. 5. Varhtion

of A2 per point far the

data together with variations of the model parameters

grid over aM.

in a

524

P. B. WOOLLAM

et d.

Table 4 gives the values obtained for the volume integral per nucleon of the real central potential, J,,/A, in the grids over uMand (r ’ )M. * From these results it is clear that the limits imposed on (r”)& by the model, using the criterion of 50 % increase in A’, allow only very small changes in JRs of about 2 ‘A. Similarly the variations of a, within this criterion produce only marginal fluctuations in JRs. From these results and from the variations of the individual parameters shown in figs. 3 to 5, it appears that the best defined quantities in the model are the rms matter radius and the volume integral of the real potential. The well defined minima produced in A2 in the grids together with this conclusion show that the reformulated model holds well at this energy and in this mass region, over the angular range observed, even though rapid oscillations in the data provide a more stringent test than at lower energies. It can be shown from the formulation of this model “) that the mean square radius of the simple optical-model real potential is related to (r2>& and M by R = d.

(9

Using the optimum values of (r2& and (r2)d obtained from the grids the reformulated model predicts 32.2 rt 2.1 fm2 for (r ‘jR. The best fit parameters to the simple optical model produce a value of 32.1 fm2 using the relation:

The agreement in the value of this quantity between these two models is good, and well within the errors to be expected. The values of the reaction cross section predicted by both the regular and reformulated models are considerably larger than the experimental value of 1550& 100 mb found by Fulmer and Kingston 13). However, the imaginary absorption potentials predicted by global analysis 24) give a considerably lower value than the best fit obtained here. We gratefully acknowledge the heIp of the s&&of the Rutherford Laboratory linear accelerator and the financial support of the Science Research Council. We are also indebted to Dr. C. B. Fulmer and Dr. F. G. Kingston for information on the total reaction cross section prior to publication, and to Dr. V. R. W. Edwards, Dr. G. L. Thomas and Dr. G. J. Pyle for useful discussion. Our thanks are also due to Dr. G. J. Pyle for the use of the computer code RAROMP, and especially to Professor E. J. Burge who vitiated this work.

1) G. W. Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115 2) R. H. Boyd and G. W. Greenlees, Phys. Rev. 176 (1968) 1394 3) G. L. Thomas and E. J. Burge, Nucl. Phys. Al28 (1969) 545 4) T. Tamura, Rev. Mod. Phys. 37 (1965) 679

14ssm(33, P)

525

5) D. L. Hendrie, N. K. Glendenning, 3. G. Harvey, 0. N. Jarvis, H. H. D&m, J. Sandimos and J. Mahoney, Phys. Lett. 26B (1968) 127 6) P. Stoler, M. Slagowitz, W. Makofske and T. Kruse, Phys. Rev. 155 (1967) 1334 7) R. K. Jolly and C. F. Moore, Phys. Rev. 155 (67) 1377 8) B. H. Wildenthal, E. Newman and R. L. Auble, Phys. Lett. 27B (1968) 628 9) V. R. W. Edwards, N. K. Ganguly, D. G. Montague, K. Ramavataram, A. Zucker and D. J. Plummer, Proc. Conf. on properties of nuclear states, Montreal (August, 1969) contrib. 8 10) H. Kamitsubo, H. Ohnuma, K. Ono, A. Uchida, M. Imaizumi, S. Kobayaski and M. Sokiguchi, J. Phys. Sot. Jap. 22 (1967) 19 11) B. G. Harvey, D. L. Hendrie, 0. N. Jarvis, J. Mahoney and J.Valentin, Phys. Lett. 24B (1967) 43 12) C. B. Fulmer, private communication 13) F. G. Kingston and C. B. Fulmer, private communication 14) V. R. W. Edwards, J. A. Fannon, E. J. Burge, D. A. Smith and F. J. Swales, Nucl. Phys. A93 (1967) 370 15) J. A. Fannon, E. J. Burge, D. A. Smith and N. K. Ganguly, Nucl. Phys. A97 (1967) 263 16) F. J. Swales, RHEL~MlOl (1966) 17) A. P. Banford, PLA Progress report RHELjR136 (1966) 17 18) V. E. Lewis, E. J. Burge, A. A. Rush, D. A. Smith and N. K. Ganguly, Nucl. Phys, Al01 (1967) 589 19) A. Ashmore, B. Hird and M. E. Shepherd, NIRL/R/24 (1962) 74 20) R. C. Hanna and T. A. Hodges, Nucl. Instr. 37 (1965) 346 21) A. G. Hardacre, Nucl. Instr. 52 (1967) 309 22) G. J. Pyle, University of Minnesota report 600-1265-64 23) M. P. Fricke, E. E. Gross, B. J. Morton and A. Zucker, Phys, Rev. 156 (1967) 1207 24) F. D. Becchetti and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 25) G. L. Thomas, Nucl. Phys., to be submitted