Differential cross sections and polarization in (d, p) reactions from the WBP model

ANNALS OF PHYSICS: 52, 33-58

Differential

Cross

(1969)

Sections and Polarization from the WBP Model

in (cl, p) Reactions

C. A. PEARSON Department of Physics, University of Arizona, Tucson, Arizona 8.5721 AND

J. M.

BANG

AND

L. Pots*

The Niels Bohr Institute, Copenhagen, Denmark

Differential cross sections and proton polarization in (d, p) reactions predicted from the weakly-bound projectile model with nonspherical capturing interaction are compared with experiment for reactions with I,, ranging from 0 to 3. The calculations use standard nucleon-nucleus optical model parameters. The agreement found confirms the main features of the WBP mechanism, and suggests that the model may be useful for extracting values for ln, j, and accurate spectroscopic factors.

I. INTRODUCTION

The study of (d, p) reactions has had two main purposes, to discover the details of the reaction mechanism, and to fashion a tool for extracting information concerning the total angular momentum j, , orbital angular momentum 6’n, and spectroscopic factors for the state of the residual nucleus into which the neutron is deposited. Of special interest in the reaction mechanism is the extent to which the neutron and proton from the deuteron interact separately with the target nucleus, (three-body description) or as one particle (conventional DWBA). In this paper we present calculations with an approximate three-body model, the Weakly-Bound-Projectile (WBP) model (Z-3), both to test its mechanism, and its usefulness as a tool. The calculations include differential cross sections, and polarization P, of the outgoing proton in various (d,p) reactions on spin-zero target nuclei. The magnitude of differential cross sections calculated with spectroscopic factor S = 1 are reported for cases where S is expected to be close to unity. All the calculations are made with the one standard set of nucleon-nucleus optical-model parameters (4) which were extracted from fitting polarization and differential cross section measurements in elastic scattering. * On leave from the Central Research Institute for Physics, Budapest, Hungary. 33 595 /52/I-3

34

PEARSON, BANG, AND POCS

The simultaneous fits which have been obtained to the measured values for P, and the shape of differential cross sections confirm the main features of the WBP mechanism. They support the notion that the neutron and proton feel separate forces when interacting with the target nucleus rather than a force on the deuteron center of mass. At the same time these fits show that WBP calculations can be used to extract both &mand j,, from such measurements. Accurate spectroscopic factors are in principle more difficult to obtain than values for the discrete variables /,, , j, which can be determined exactly. The magnitude of the differential cross section may depend on features of the reaction mechanism other than those which most strongly influence the polarization and angle dependence of differential cross sections. Fitting the qualitative behavior for these does not guarantee values found for S. In the absence of a complete understanding of the stripping mechanism one can try an interpolation procedure based on a model, and apply it to (4~) measurements made over a range of bombarding energy, target nuclei, angular momentum, and energy transfer. The parameters of a useful model must be predetermined so that unambiguous predictions are made at all points in the range. The model should reproduce magnitudes of cross sections at points where spectroscopic factors are known, independent of the bombarding energy. It should also reproduce measurements of angular distributions and polarization over the whole range as an additional check on the interpolation. Without systematic fitting, prescriptions for obtaining spectroscopic factors will almost certainly be unreliable. Attempts to extract such information with the conventional DWBA model have been unsatisfactory in this respect.’ With the DWBA different sets of data have been fitted with different sets of parameters. The parameters which fit one set of data usually do not fit another. Much of the data which is sensitive to the spin-dependent forces have not been fitted at all. Some evidence indicates that spectroscopic factors extracted with the DWBA are incorrect (5). On the other hand, the WBP model has reproduced the observed j, dependence of angular distributions (6) over a range of bombarding energy, target nuclei, angular momentum, and energy transfer. The calculations discussed here fit the measured polarization, the angle dependence of differential cross sections, and the magnitude of such cross sections for cases where S is expected to be close to unity. Some of the requirements for an interpolation procedure are thus fulfilled. 1 It has recently been shown (20) that the revised version (21) of the Tanifuji-Butler (TB) model reduces to the original TB model when an error in its derivation is removed. The polarization reported for thesemodels by May and Truelove (19) and the calculations of Ref. (21) contain an additional mistake in expressions for the proton scattering matrix. The results shown do not represent actual predictions of the TFI model. Corrected formulae are given in an appendix to Ref. (3).

WBP-MODEL

35

CALCULATIONS

Calculations with a crude form of the WBP model have been previously reported (2,3). In this previous report the interaction which leads to the neutron capture was taken to be spherically symmetric. A lower radial cutoff in the neutron-capture integral simulated the reduction in the magnitude of this interaction inside the nuclear surface. In the present paper a more realistic, nonspherical potential is used. This not only refines the previous calculations but introduces an additional source of polarization for reactions with tn f 0. This polarization is especially important for angles near the stripping peak, where its sign is + for j, = 8%f 4 , respectively. In certain cases it is predominant. Even when polarization from the proton scattering modifies this simple sign rule, both the calculated and measured polarization show a characteristicj, dependence so that polarization measurements can serve to determine the value of j, . In Section II we discuss some details of our calculation. In Section III we show the proton polarization which stems directly from the neutron capture. In Section IV we compare calculated values for P, and differential cross sections with experiment. In Section V we draw some conclusions.

II.

CALCULATION

We consider only the case in which the deuteron energy is above the Coulomb barrier. The target has spin zero, and the final state can be represented as a neutron with j, = e, * 4 bound to the unexcited target-nucleus core. As frame of reference we choose the barycentric ‘frame with the Z-axis along kd and the Y-axis in the direction of kd x k, . In the same notation as Ref. (2), the WBP stripping amplitude is

x

In terms of this amplitude, parallel to the Y-axis is

I

dk,,

-J-[ K2+y2

the polarization

j&s]

Sug.upk

of the outgoing

3 &J

proton which is

36

PEARSON,

BANG,

AND

POCS

and the differential cross section

where rnz , rn$ are the usual reduced masses for proton and deuteron respectively. The factor which expresses the neutron capture is (4m)l and the capturing interaction

*Yl(r, , b)l

%+(r, , k,,))

(3)

is

r(r, , k,,) = .f I xp+(k,, , rp)12K,,(r, - r,) dr, . We follow Ref. (2) in using the zero range approximation interaction V,, , and replace (4) by

(4)

for the neutron proton

‘Urn ,k,p) =- z(K2 +r3Ix,+(m , k,~)12, where AK = ti(+kd - k,) is the relative momentum of the neutron and proton. It should be emphasized that this zero range approximation is made only within the neutron capture factor (3) and not throughout the whole stripping amplitude, i.e., the range of V,, is assumed short compared to that of typical variations in the neutron wave functions u(r,), u,+(r, , k,,) and the proton density 1xD+(rz,, k,*)j2. This is well justified provided that 1k,, 1<< 1 fm-l, and the nuclear interior does not contribute substantially to the capture factor (3) (2,3). The magnitude of the interaction (5) depends strongly through K on the angle between k,, and kd . From our calculations we have verified that the main contribution to the integration over dk,, comes from a small region of 8,) near the angle of the stripping peak. In this region the relative momentum UC of the neutron-proton motion and the capturing interaction (4) are small in magnitude, typically Y 2 5 MeV. This is essential for the self-consistency of the WBP model where proton scattering from the neutron has been neglected compared to protontarget-nucleus scattering. The wavefunction u,+(k n’ , r,) for the incident neutron is the solution of [Es - T, -

Vopt - V(r,, , &,,)I u,,+(rn , k,,) = 0,

(6)

where Es is the energy of the neutron in its final bound state and T,, is its kinetic energy operator. The wavefunction z&J for the bound neutron satisfies [EB

-

T,

-

VBoundl

&)

=

0.

(7)

WBP-MODEL

37

CALCULATIONS

the interaction %“(rn , k,,) VoIIt = VBound P Eqs. (6) and (7) differ only through in (6) which allows the incident neutron to propagate outside the bound state in the presence of the proton. The interaction Y(rn , k,t) (5) contains a constant part which comes from the plane wave part of x 9+. When the incident deuteron beam is a realistic packet the remainder of Y’(r, , k,,) cc l/rn2 for r, -+ co. We group the constant part of Y’& , k,,) with Es so that (6) becomes

If

[ Ewe - Tn - Vopt - E (K~ + y2)(1 - I xD+ IY] u,+(r, , k)

= 0

(8)

where Eiree = f

(P + y2) + Es

is the “free” neutron energy. From (8) we see that the wavefunction u,,+ has the asymptotic form u,+(r, , k,t) - eiknrrn + (out. spher. waves), with (fi2/2m)(k,,)2 = Erree , which is appropriate to a neutron in the wave packet representing the incident deuteron. The last term in the brackets on the left of (8),

f w2+ Y2N- Ixp+12),

@a)

is both weak and of “short range,” its maximum depth being 53 MeV. Because of this term Us+ represents a solution for scattering from a nonspherical potential and is difficult to compute. In the calculations reported here we have neglected (Sa) when evaluating uO+. Our approximate wavefunction then satisfies the same boundary conditions as u,+ and is computed for a spherical potential. In the limit, where Vopt = VBound this is equivalent to computing the amplitude (1) to first order in the weak interaction (8a). A. CALCULATING

THE AMPLITUDE

For calculating the stripping amplitude proton scattering matrix in the form Sup,&,

(1) it is convenient to substitute the

9 k,) = 6.psu,Wv - k,) - 2~iW,~ - 4) Tup,,,(kD~, k,)

so that the amplitude

(9)

(1) becomes a sum of two terms which in keeping with

38

PEARSON,

BANG,

AND

POCS

previous nomenclature (Z,2), we call the “unscattered” and “scattered” amplitudes, respectively. The separation (9) is made so that the second term of (9) may later be expanded as a finite sum over partial waves. To evaluate the amplitude (1) we must integrate over the angles 8,, , q+ of the proton momentum k,, . The cylindrical symmetry about the Z-axis ensures that the vP’ integration is trivial. Our main task is then to evaluate the neutron capture factor (3) as a function of the integration angle 8,~ . The incident-neutron wavefunction u,+(k ,,’ , r,) depends implicitly on the angle 6J,, in a nontrivial way through the magnitude of k,, . It must be numerically computed for each value of 8,~ in the integration mesh. On the other hand, the angle dependence of the two leftmost factors in (3) can be expressed in terms of spherical harmonics. It is therefore convenient to compute them separately and store their radial parts. For simplicity we assume a spectroscopic factor of unity and write the neutron final state

with SC, the neutron spin function and r~~~~,(r,J the usual bound state radial wavefunction. Since we have made the zero range approximation for V,, in evaluating the neutron-capturing interaction (5) we also neglect the small effects of the spin-orbit force on the proton density in (5), and expand this density as

(11) where &(I,)

= 47r 11 .(92d rep

+ 1)(2d' + 1) I(eoel0 I Lay x&n) x;(m)

(12)

and xc(m,J are the usual distorted wavefunctions without spin-orbit force. The first few terms in the multipole expansion (11) reproduce the important nonspherical features of the interaction (5). In the region exterior to the nuclear surface which contributes the main part of the neutron-capture factor (3), the typical difference between densities calculated with L,, = 4 and Lm, = 50 was found to be of the order of 1% of the total density, which is smaller than differences arising from ambiguities in optical model parameters. A refraction-diffraction maximum in the density (11) does occur inside the nucleus in the “shadow” hemisphere. To accurately reproduce this maximum which has small angular size, one must include more than four terms in the

WBP-MODEL

39

CALCULATIONS

expansion (11). However, because it occupies a very small volume in the integral (3) this maximum should not contribute strongly to the neutron capture. For the present calculations we have truncated the expansion (11) at L,,, = 4, partly because of the small memory of our computer, partly because the non-local and finite range effects we have neglected should strongly affect contributions to the amplitude (1) from the density maximum. It is preferable to study these contributions later when our calculation has been suitably improved. The radial dependence of typical coefficients uL(r,J is shown in Ref. (6). Densities calculated with them are shown in Figs. 1 and 2 of this paper (see Section III). The wavefunction u,+(r, , k,, , plz,) is expanded in partial waves

and with this substitution (3) becomes (4~)

I ~(r,,

b)/

together with (11) and (12) the neutron-capture

uo+(m, b))

= f W2 + y2)KMyn,Mn(kg,)exp[---i(M,

factor

- /4&f)%,I

where

x OLM(COS e,,) O~~(cos I?,,) and Bc”(cos 0) is the associated Legendre polynomial

(14) defined by

Ycm(8, fp) = (27r)-lle Bcm(cos 8) e”w, @p(cos e) = 0; d
40

PEARSON,

The “unscattered” ties (14)

amplitude

BANG,

AND

POCS

can be expressed directly

in terms of the quanti-

From (15) one can calculate the polarization due to the neutron capture alone. An example of such polarization is discussed in Section III. To obtain the complete WBP amplitude we expand the proton scattering matrix s,ptup(k,~ , k,) = W,

- k,*) 4.,r,, + &

,c (h&m 9 lh

I .im + w>

x Gi~,~m + w - ~1~I jm + i-w) htj - 1) W, x Y~+“-(ri~) Ypyi&), where the quantities Q are related to angular momentum j = G f $ by qn (Q - 1) in (16) tend to zero so that restricted. By substituting from (14), (15), and WBP amplitude:

- b) (16)

the complex phase shifts & for the total = exp(2&). For G 2 10 the coefficients the summations over t, j, m are severely (16) into (I), we obtain for the complete

x (cos Q) OG”d-“+n ccose,) [i - $$$I

I

(17)

For each reaction we have first calculated the quantities vGj , u=(m) associated with the proton optical model wavefunctions. The products z)j,d,(rn) uL(r,J were then formed and stored. These products are independent of the integration angle 8,t in (17) so that they were computed only once. The free neutron wavefunctions &Jk,v-J were integrated for each angle or, in the integration mesh, for values of /,, from 0 to Lmax + e, . The overlap integrals d~tJkd

= &, s Mr,J

~~,,&-,J x~,t,&~r,J rn dr,

were formed and the neutron capture factors K,n,Mn(cose,,) were assembled from them.

WBP-MODEL

CALCULATIONS

41

Some repetition has been avoided by noting the symmetry relation

K-,+-M, = (- lfn+ ‘2--i*+M*-unrKaa,,, . When the factors (18) are finally folded into the amplitudes (15) and (17), one obtains the additional symmetry relation,

Using (19) the polarization can be expressedas

(20) and the differential cross section

thus reducing the sum over M, . Finally, it is worth noting that the amplitude (17) depends on the outgoing angle ~9~ in a simplemanner. Someadvantage is gained by calculating this amplitude for all outgoing angIes simultaneously. B. PARAMETERS To determine the bound state radial wavefunction ujllG,(y,J we have used the potential v, = v [f(r) - a (A)’

; g Q * e]

where f(y) has the Saxon-Woods form: f(r) = [l + exp(r - @/a]-’ and have adjusted the depth V so that the neutron is bound with the appropriate binding energy. In all caseswe have used: Q:= 32, R = 1.25A1J3,a = .65f. For the scattered wavefunctions the nucleon-nucleus optical potentials had the form V(r) = - Vf(r) - iWg(r) - V&(r) Q * e + V,(r),

42

PEARSON,

BANG,

AND

POCS

where R = roA1j3,

f(r) = [I + exp (qj]-l, g(r) = -4b

$

[l + exp (*)I-’

(1 - S) + [l + exp ;+I]

h(r) = -kz

’

i $.

The proton potential was truncated as in Refs. (2,3). Because each set of angular distributions and polarization curves took several hours to compute on the Gier computer at the Niels Bohr Institute, it was not possible to systematically vary parameters. For the neutron and proton scattering, we took the parameters of Rosen et al. (6), except for the neutron imaginary potential. We tried various depths for this potential, partly because the imaginary potential of the simple optical model is poorly determined by elastic scattering data (e.g., the depths used by different authors differ by a factor of 2 (4), partly because the off-diagonal elements of V,, should contribute an extra imaginary part to the neutron potential used in the WBP model (I). The Rosen parameters are as follows:

V(MeV) W (MeV) R (fm) a (fm) b (fm) Vs (MeV)

Neutron

Proton

49.3-0.33E 5.75 1 25A1/3 0:65 0.70 5.5

53.8-0.33E 7.5 1 25A1’3 0:65 0.70 5.5

The neutron potentials we have chosen are all more strongly absorbing than the simple Rosen potentials. The main effect is to exclude the stripping reaction from the nuclear interior. In this respect they simulate the effect of finite range of VW and nonlocal effects on the bound neutron wave function, which are included in refined DWBA calculations, and which we have so far neglected. We denote the following choices of imaginary potential by A: B: C:

D:

W,, W, W, W,,

= 11 MeV; = 22MeV; = 8 MeV; = 5.5MeV;

s= s= s= s=

0 0 .33 0

(twice Rosen surface) (four times Rosen surface) (Rosen surface)

WBP-MODEL CALCULATIONS

43

All the above sets give similar results for polarization, j, dependence and differential cross sections. There was no reason to prefer any one of them. When finite-range and nonlocal effects are included we expect our calculations to depend very weakly on W, for strengths ranging from (l-5) x W, (Rosen). C. PROGRAM CHECKS

The integration routines were checked against known solutions. Previous calculations (2,3) with a spherically symmetric neutron capturing interaction were reproduced. The results reported here agree with those from an independent FORTRAN program written by one of us (C.A.P.). This FORTRAN program differs in structure from the original Algol program. It is worth noting that this new program takes approximately 1 minute for calculating a typical angular distribution on a CDC 6400. A study of finite range effects, D-state effects and parameter dependence will thus be possible in the future.

lII.POLARIZATION

FROM NEUTRON

CAPTURE

In Refs. (2) and (3) it was shown that a large, and often the largest, part of the WBP amplitude comes from those neutron-proton pair components of the incident deuteron in which the neutron travels slowly towards the target nucleus, and the proton travels towards the stripping peak. Although the approximations used here differ slightly from those of Refs. (2) and (3), this conclusion is unchanged. The protons observed at large outgoing angles have usually been scattered from the stripping peak. However, in the peak itself many protons survive unscattered. The polarization of these protons comes from the accompanying neutron capture. The neutron capture polarization as originally suggested by Newns (7) occurs when the neutron is captured predominantly in one hemisphere of the capturing volume, strongly orienting its angular momentum. Figures 1 and 2 show typical examples of the proton density appearing in the capturing interaction (3). In the neutron capture region between R and R + 5 fm, the average proton density in the hemisphere “illuminated” by the proton wave is several times that in the “shadow” hemisphere. Through the interaction (4) most neutrons are captured in the “illuminated” hemisphere. For the usual (d, p) reaction with Ed w 10 MeV, the energy transfer is such that k p % kd . At small proton angles k, = kd -k, is almost perpendicular to k, . The neutron capture then orients the angular momentum

&a= (m x k,)

44

PEARSON,

BANG,

AND

POCS

FIG. 1. Distribution of density for the proton optical-model wavefunction appearing in the interaction (5). Target nucleus is IsO and E, = 12.4 MeV. Optical model parameters are from Rosen et al. (4). Proton is incident from right rear and nuclear radius is at the center of dark band.

FIG.

2. As for Fig. 1 for scattering from Y3r at 11 MeV. Proton is incident from left rear.

WBP-MODEL

CALCULATIONS

45

FIG. 3. Angular distribution and polarization for reaction Wr(d,p) Wr (Q = 5.15 MeV, G, = 1, jn = 4, Ed = 11 MeV) calculated from “unscattered” amplitude (15). Parameters for the calculation were set B.

in the positive direction. Neutrons with spin i. are captured, according to the angular momentum jn = e, f 4 of the final state. Because the neutron and proton spins are parallel in the incident deuteron, the proton partnering the captured neutron also has spin f for j, = C,, + +, respectively. The polarization is not as large as the simple classical argument implies. It is well known that the maximum magnitude for the polarization which results from full alignment of the neutron angular momentum is 5 . In Fig. 3 we show a typical example of such polarization calculated directly from the “unscattered” amplitude (15) for a case with j, = e,, - 4. Near to the stripping peak this polarization N - l/3 indicating that the neutron angular momentum is strongly aligned in the direction of positive polarization. In the following section we show that the neutron capture polarization sometimes predominates over the proton scattering polarization near the stripping peak for reactions with /, = 1,2. Then a single measurement of proton polarization at an appropriate angle can determine the transferred angular momentum valuej, .

46

PEARSON, BANG, AND POCS

Even when the proton scattering causes deviations from the above sign rule, the polarization shows a characteristic j, dependence near the peak, and polarization measurements can be used to find jm values. IV. COMPARlSON

WITH

EXPERIMENT

A. POLARIZATION In Figs. 4-11 we compare calculated and measured curves for the polarization of the outgoing proton in various (d,p) reactions with unpolarized incident deuterons. Figures 4 and 5 show typical examples for reactions with C, = 0. In each figure the stripping angular distribution and polarization curves calculated with the same set of parameters are included. For comparison we show the polarization produced when protons with the energy of the outgoing proton scatter elastically from the

z = 5 b 2 2 r! s :

20

0

-20

-40

20

40

60 e CM.

80

100

I20

140

FIG. 4. Calculated and measured angular distributions and polarization for reaction YGr(d,p) YSr (Q = 3.11, Ed = 11 MeV). Measurements are from Ludwig and Miller (8). Solid angular distribution and polarization curves were calculated with set-A parameters, broken

polarization curve with set B. Broken-dashed curve is elastic scattering polarization.

WBP-MODEL

-60

IO

20

47

CALCULATIONS

30

40

50

60

TO

80

FIG. 5. Calculated and measured angular distributions and polarization for reaction **Si(d,p) Wi (p.s., Ed = 15 MeV). Polarization measurements are from Isoya and Marrone (9) and measured angular distribution from Blair (10). Solid angular distribution and polarization curves were calculated with set-A parameters; broken curve is elastic scattering polarization.

same proton potential used for the stripping calculation (we refer to this hereafter as elastic scattering). When &’= 0 the polarization comes mainly from the proton scattering, suitably averaged in the amplitude (1). Usually the stripping polarization corresponds closely to polarization from elastic scattering. The reason for this is discussed in Refs. (2) and (3). Beyond the stripping peak in Figs. 4 and 5 the stripping polarization is similar to elastic scattering. Near to the first minimum in the angular distribution the averaging of the proton scattering in the amplitude (1) and additional contributions from the spin-orbit force on the scattered neutron cause the calculated stripping polarization to deviate substantially from the elastic scattering polarization. In Fig. 4 it differs in sign, in agreement with the observed stripping polarization.

48

PEARSON, BANG, AND POCS

In Figs. 6 and 7 we compare calculated and measured angular distribution and polarization curves for the reaction W(d, p) W ground state (g.s.) In Fig. 6 the calculated curve corresponds to Ed = 10 MeV, while the measurements have been made over a spread of neighboring energies. The measured polarization shown varies slowly with bombarding energy2 over the range shown in Figs. 6 and 7, and we have combined the various measurements to provide more detailed experimental curves. The calculated polarization also varies slowly with energy between 10 MeV and 15 MeV. At very small angles both the predicted and the measured polarization increase in absolute magnitude with increasing energy. For angles slightly less than the stripping peak angle, the polarization comes mainly from the neutron capture and is negative in agreement with the rule of Section III. For small angles the absolute magnitude of the polarization exceeds 3 ,

I

I

IO

I

20

I

JO

I

40 ecu

I

50

I

60

I

70

I

80

I

90

FIG. 6. Calculated and measured angular distributions and polarization for reaction W!(d,p) W g.s. The measured angular distribution, (broken curve, Ed = 10 MeV) is from Hamburger (II). Polarization measurements are: A Johnson and Miller (Z2), (Ed = 10.8 MeV); A Juveland and Jentsche (13), (Ed = 11.9 MeV); o Allas and Shull (I@, (Ed = 10 MeV); ,o Bokhari et al. (IS), (Ed = 8.9 MeV). Set-C parameters were used for angular distribution and polarization calculated at Ed = 10 MeV.

p In contrast to the rapid variations observedbelow 10 MeV in Ref. (26).

WBP-MODEL

CALCULATIONS

49

FIG. 7. Same as Fig. 6, for reaction W(d, p) 13C g.s. Measured angular distribution (broken curve) and solid circular polarization points are from Isoya et a/. (16), (Ed = 15 MeV). Triangular points are from Juveland and Jentsche (13), (Ed = 11.9 MeV). Solid curves are calculated with set-C parameters, and broken polarization curve with W, = 1.3 x W, (set C).

the maximum for the neutron capture alone, the additional polarization coming from proton scattering. In Figs. 8 and 9 we compare calculated and measured polarization curves for two caseswith 8, = 2. As for the caseswith 8, = 1, the polarization is negative for angles slightly smaller than the stripping peak angle in agreement with the rule of Section III. However, as the angle increasesover the stripping peak, the polarization changes sign, showing that proton scattering cannot be neglected. In Figs. 10 and 11 we compare calculated and measured polarization curves for the reaction 40Ca(d,p) 41Cag.s. with ln = 3. For the samereasonsas in Figs. 8 and 9 we have included experimental points corresponding to two different bombarding energies.Both the experimental and calculated curves vary slowly over the energy range considered. Again the proton scattering is important even near to the stripping peak and the simple rule of Section III can be misleading. An extensive study of this reaction has been made with the conventional DWBA (23). The good fits shown here for the WBP model with standard parameters contrast with the fits of Ref. (23). 595/52/r-4

50

PEARSON,

BANG,

AND

POCS

-3oc IO

20

30

40 9CM

,l 50

1, 1 60

, 70

I-J a0

90

FIG. 8. Calculated and measured polarization for reaction %i(d,p) *%i (& = 2, Ed = 15 MeV, Q = 4.97 MeV). Measurements are from Isoya et al. (16). Set-C parameters were used for calculation.

FIG. 9. Calculated and measured angular distributions and polarization for reaction Z4Mg(d,p) 25Mg (g.s., Ed = 15 MeV). Measured angular distribution (broken curve) and polarization points are from Reber and Saladm (17). Set-C parameters were used for calculated curves.

WBP-MODEL

51

CALCULATIONS

IO

4

-20

5 e -30 2.i 2 -40 20

40

60

80 e CM.

100

I20

140

160

FIG. 10. Calculated and measured polarization for reaction Wa(d,p) *lCa g.s. Measurements are from Takeda et al. (22), at Ed = 11.4 MeV (open circles) and from Bercaw and Shull (18), at Ed = 10.0 MeV (solid circles). Broken curve (Ed = 10.9 MeV) is calculated with set-C parameters, solid curve (Ed = 12 MeV), with set-C, V, = 45 MeV.

20

40

60

80

120

140

160

ec!.4

FIG. 11. Calculated and measured polarization for reaction Wa(d, p) Wa g.s. (Ed = 10 MeV). Measurements are from Bercaw and Shull (18). Solid curve is calculated with set-C parameters, broken curve with the same parameters, except that the imaginary proton potential is reduced by20%.

52

PEARSON,

BANG,

AND

POCS

Near to the stripping peak the observed polarization and the polarization calculated from the WBP model shows a characteristic j, dependence even when proton scattering is important and the sign rule of Section III is not obeyed. The angular dependence of the polarization shown in Figs. 6 and 7 is typical for (L&P) reactions on zero-spin target nuclei with (j, = $, L,, = 1). The angular dependence shown in Figs. 8 and 9 is typical for (j, = Q , e, = 2), and that of Figs. 10 and 11 for (,jn = 5 , L’n = 3). For each of these values of tn the calculated polarization for the alternative value of j, is qualitatively different. At certain angles the two polarizations for j, = t,, f & differ in sign. B.

DIFFERENTIAL

CROSS

SECTIONS

We consider nine (a, p) reactions on target nuclei from 160 to g6Zr with orbital angular momentum transfer f% ranging from O-3. For each case the spectroscopic factor S is taken to be unity. In Figs. 12-15, we compare calculated and measured differential cross sections for reactions with en = 0. Two different values of the neutron imaginary potential are used in each figure, and each of the sets A to D is used at least once. The calculated curves depend weakly on the choice of neutron imaginary potential and there is no reason to prefer any particular set. The fits to the 160 angular distributions are noticeably worse than for heavier nuclei. This may be connected with the poor fits which Rosen potentials give for elastic scattering from 160. The curves in Fig. 14 for the reaction Y?.r(d, p) 8gSr (Q = 3.11, l,, = 0, Ed = 11 MeV) have been obtained using S = 1. The DWBA analysis for this reaction reported in Ref. (12) gives S = 0.3. In Figs. 16-19 we compare calculated and measured differential cross sections for reactions with e, = 2. The fits for heavier nuclei are again better than for laO. In Figs. 17-19, the less absorptive set-D parameters give slightly better agreement. In Fig. 20 we compare calculated and measured differential cross sections for the reaction 40Ca(d, p) 41Ca (g.s. C;, = 3, Ed = 10 MeV).

V. CONCLUSION

In Figs. 4-11, we have shown that proton polarization predicted from the WBP model for (d, p) reactions agrees with the measured polarization over a range of energy, target size and angular momentum transfer. In Figs. 12 to 20 we have shown that WBP calculations can reproduce the shape and magnitude of differential cross sections. Similar calculations fit the j, dependence of differential cross sections (6). In obtaining the curves shown in Figs. I-20 no attempt was made to system-

WBP-MODEL

FIG.

12. Differential

CALCULATIONS

cross sections for the reaction

lBO(d,p) I70 (.87 MeV,

53

G, = 0,

Ed = 12 MeV). The solid curve is calculated with set-D parameters, the broken curve with set A. Experimental

points are from Refs. (II) and (27).

FIG. 13. Differential cross sections for reaction ‘?3i(d,p) Y3i (g.s. I;, = 0, Ed = 15 MeV). The solid curve is calculated with set-D parameters, the broken curve tith set A. Experimental points are from Refs. (9) and (10).

54

PEARSON, BANG, AND POCS

I

20

40

60

80

100

I20

140

FIG. 14. Differential cross sections for the reaction **Sr(d,p) B9Sr (Q = 3.11, l,, = 0, Ed = 11 MeV). The solid curve is calculated with set-A parameters and the broken curve with set B. Experimental points are from Ref. (8).

,

20

i

40

60

,

I

80

100

I20

e 0.4 FIG. 15. Differential cross sections for the reaction Wr(d,p) OIZr (Q = 3.8 MeV, G, = 0, Ed = 10.88 MeV). The iolid curve is calculated with set-A parameters, the broken curve with set C. Experimental points are from Ref. (28).

WBP-MODEL

0.1

4

20

40

55

CALCULATIONS

!

60

80 ect.4

1

100

I20

140

160

FIG. 16. Differential cross sections for the reaction rBO(d,p) I70 (p.s. /, = 2, Ed = 12 MeV). The solid curve is calculated with set-D parameters, the broken curve with set A. Experimental points are from Ref. (27).

0.1

20

I 40

I 60 e CM

I 80

I 100

I20

FIG. 17. Differential cross sections for the reaction *OZr(d, p) OlZr (gs. G, = 2, Ed = 15 MeV). The solid curve is calculated with set-D parameters, the broken curve with set A. Experimental points are from Ref. (29).

PEARSON, BANG, AND POCS

56

FIG. 18. Differential cross sections for the reaction *“Zr(d, p) B7Zr (1.11 MeV, 4, = 2, Ed = 15 MeV). The solid curve is calculated with set-D parameters, the broken curve with set C. Experimental points are from Ref. (29).

0.11

20

40

60

80 e CM.

100

PO

140

FIG. 19. Differential cross sections for the reaction YSr(d, p) YSr (p.s. t,, = 2, Ed = 11 Mel? The solid curve is calculated with set-C parameters, the broken curve with set D. Experimental points are from Ref. (30).

WBP-MODEL

CALCULATIONS

57

e cm FIG. 20. Differential cross sections for the reaction %a@, p) %a (g.s. /, = 3, Ed = 10MeV). The solid curve is calculated with set-C parameters, the broken curve with set A. Experimental points are from Ref. (31).

atically vary parameters. The standard parameters of Ref. (4) extracted from fitting differential cross sections and polarization from elastic scattering were used. The changes made in the neutron imaginary potential were found to be unimportant. In the WBP model, the j, dependenceand the polarization are sensitive to the spin-orbit forces which act separately on the neutron and proton near to the nuclear surface. The successof the model in reproducing the details of these measurementscontrasted with the failure of the conventional DWBA suggeststhe neutron and proton feel largely independent forces close to the nucleus rather than a force which acts on their center of mass. The good fits of the WBP model to polarization measurements indicate that WBP calculations can be used to determine the value of j, . Such measurements are simpler using polarized deuteron beams. The vector analyzing power PD obtained for (d, p) reactions with polarized deuteron beams has been recently reported (24,25). The angular and j, dependenceof PD is similar to that for the polarization above. Calculations for PD will be discussedelsewhere. A new computer code has been written which has reduced the time for the above calculations so that D-state, finite range, and nonlocal effects may be comfortably included. With this code it should be possible to improve our fits and to choose a set of “best” parameters for an interpolation procedure to determine spectroscopic factors. As yet calculations have only been made near to closed shells. Before making

58

PEARSON,

BANG,

AND

POCS

use of this procedure in other regions it is necessary to carefully investigate such effects as inelastic excitation of the target nucleus which have been omitted from the WBP description and which are likely to be more important further from the closed shells. RECEIVED:

July

12, 1968 REFERENCES

1. C. A. PEARSON AND M. Coz, Nucl. Phys. 82, 533 (1966); Nucl. Phys. 82, 545 (1966); Ann. Phys. (N.Y.) 39, 199 (1966). 2. J. M. BANG AND C. A. PEARSON, Nucl. Phys. AlOO, 1 (1967). 3. J. M. BANG, C. A. PEARSON, AND L. Pots, Nucl. Phys. AlOO, 23 (1967). 4. L. ROSEN, J. G. BEERY, A S GOLDHABER, AND E. H. AIJERBACH, Ann. Phys. (N.Y.) 34, 96 (1065). 5. B. L. COHEN, R. A. MAYER, J. B. MOORHEAD, L. H. GOLDMAN, AND R. C. DIEHL (preprint). 6. C. A. PEARSON, L. Pots, AND J. M. BANG (to be published). 7. H. C. NEWNS, Pm. Phys. Sot. (London) A66, 477 (1953). 8. E. J. LUDWIG AND D. W. MILLER, Phys. Rev. 138, B364 (1965). 9. A. ISOYA AND M. J. MARRONE, Phys. Rev. 128, 800 (1962). 10. A. BLAIR, Unpublished doctoral dissertation, University of Pittsburgh (1960). II. E. W. HAMBURGER, Phys. Rev. 123, 619 (1961). 12. W. P. JOHNSON AND D. W. MILLER, Phys. Rev. 124, 1190 (1961). 13. A. S. JUV~LAND AND W. JENT~CHE, Phys. Reo. 110, 456 (1958). 14. R. G. ALLAS AND F. B. SHULL, Phys. Rev. 116, 996 (1959). 15. M. S. BOKHARI, J. A. COOKSON, B. HERD, AND B. WESSAKUL, Proc. Phys. Sot. (London) 72, 88 (1958). 16. A. ISOYA, S. MICHELE~, AND L. REBER, Phys. Reu. 128, 806 (1962). 17. L. H. REBER AND J. X. SALADIN, Phys. Rev. 133B, 1155 (1964). 18. R. W. BERCAW AND F. B. SHULL, Phys. Reu. 133B, 632 (1964). 19, R. M. MAY AND J. S. TRUELOVE, Ann. Phys. (N.Y.) 43, 322 (1967). 20. J. D. GARCIA AND C. A. PEARSON, Phys. Rev. Letters 21, 301 (1968). 21. S. T. BUTLER, R. G. L. HE\NITT, B. H. J. MCKELLAR, AND R. M. MAY, Ann. Phys. (N.Y.) 43, 282 (1967); M. TANKFUJI, Nucl. Phys. 58, 81 (1964). 22. M. TAKEDA, S. KATO, C. Hu, AND N. TAKAHASHI, in “Proceedings of the International Conference on Nuclear Structure, Kingston.” University of Toronto Press, Toronto, 1960. 23. S. A. HJORTH, J. X. SALADIN, AND G. R. SATCHLER, Phys. Rev. 138B, 1425 (1965). 24. T. YULE AND W. HAEBERLI, Phys. Rev. Letters 19, 756 (1967). AND S. ROMAN, Phys. Reu. Letters 20, 1114 (1968). 25. A. M. BAXTER, J. A. R. GRIFFITH, 26. J. E. EVANS, J. A. KUEHNER, AND E. ALMQVIST, Phys. Rev. 131, 1632 (1963). 27. J. L. ALTY, L. L. GREEN, R. HUBY, G. 0. JONES, J. R. MINES, AND J. F. SHARPEY-SCHAPER, Phys. Letters 20, 664 (1966). 28. R. L. PRESTON, H. L. MARTIN, JR., ANLI M. B. SAMPSON, Phys. Reu 121, 1741 (1961). 29. B. L. COHEN AND 0. V. CHUBINSKY, Phys. Rev. 131, 2184 (1963). 30. R. L. PRESTON, M. B. SAMPSON, AND H. L. MARTIN, Canadian J. Phys. 42, 321 (1964). 31. L. L. LEE, JR., J. P. SCHIFFER, B. ZEIDMAN, G. R. SATCHLER, R. M. DRI.%o, AND R. H. BASSEL Phys. Rev. 136B, 971 (1964).