The effect of the D-state of the deuteron in (d, p) and (p, d) reactions

i

2.G

[ I

Nuclear Physics A90 (1967) 289--310; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

T H E EFFECT OF T H E D-STATE OF T H E D E U T E R O N IN (d, p) and (p, d) R E A C T I O N S R. C. JOHNSON University of Surrey, London, S.W. 11, En.qland Received II July 1966

Abstract: The effect of the D-state of the deuteron in (d, p) and (p, d) reactions is discussed within the framework of the distorted waves Born approximation. Numerical estimates with plane waves arc used to suggest that the D-state may play an important role in certain reactions. The necessary modifications of previous work on polarization effects in the distorted waves theory are discussed in detail.

I. Introduction In the distorted waves theory of direct reactions (DWBA) (see refs. ,.2) and references contained therein), the transition amplitude is written as the matrix element of an operator V between wave functions describing the elastic scattering of the incident and outgoing projectiles by the target and residual nuclei. The approximation used for V depends on the particular projectiles and target nucleus involved in the reaction. For (d, p) and (p, d) reactions V is usually taken to be the free neutron-proton potential V,p. It is a common feature of nearly all calculations performed in the DWBA that the interaction V is assumed to be independent of terms coupling the spin and space coordinates of the projectiles. (An exception to this occurs in highenergy calculations of nucleon scattering 3) where the impulse approximation is valid and V can be approximated by the free two-body t-matrix.) This assumption has recently been shown to be inadequate for the excitation of collective levels by 40 MeV protons, especially in the treatment of polarization effects4). In this paper we give a preliminary analysis of the effects of the non-central components of V,p in (d, p) and (p, d) reactions. The standard DWBA matrix element 1.2) used here has the feature (see sect. 2) that these effects enter the matrix element only through the admixture of D-state into the deuteron ground state. At tirst sight this result suggests that the non-central effects give a very small correction because of the rather small D-state probability (Po < 10 %) in the deuteron 5). However, PD is a measure of the relative magnitude of the integrals of the squares of the S- and Dcomponents, whereas the DWBA matrix element is a sum of terms each of which depends linearly on a component of the deuteron wave function. Thus, in expressions for observable quantities there appear interference terms which involve the D-state wave function linearly, as well as terms quadratic in the D-state wave function. We 289

290

R.C. JOHNSON

shall refer to the former types of term as coherent D-state contributions and to the latter as incoherent D-state contributions. It is one of the purpose of this paper to investigate how far the S- and D-states contribute coherently to various experimental quantities. We also note that the deuteron wave function appears in the DWBA matrix element in conjunction with scattering state solutions of the Schr6dinger equation generated by the deuteron and proton optical potentials. Thus the most appropriate measure of the relative magnitude of S- and D-state contributions is afforded by the momentum space representative of the deuteron wave function. An important consideration is, therefore, that the D-state momentum state wave function is negligible compared to the S-state component only at very small momentum. This is discussed in detail in subsect 4.1. Particular emphasis is placed here on (d, p) reactions in which the neutron is captured with zero orbital angular momentum (l = 0). Certain simplifying features of these reactions have been discussed previously 6-8). For example, it was shown in ref. 7) that l = 0 cross sections are insensitive to the spin-dependence of deuteron and proton optical potentials in contrast to processes with l > 0. The analysis of subsect. 5.6 suggests that this is still true when D-state effects are taken into account. In addition, we find that the S- and D-states contribute incoherently to l = 0 processes. In general we find that there are always coherent D-state contributions to polarization effects, but again I = 0 processes have some distinctive features. These are discussed in detail in sect. 5, where the necessary modifications of previous work 7.9) are given. Apart from some very crude plane wave estimates in sect. 4, this paper contains no numerical results. The main emphasis here is on those features of the D-state effects that arise because of the special coupling of angular momenta implied by the DWBA matrix element. Full distorted wave calculations of the D-state effects are clearly required for a fully effective discussion, and are in progress in collaboration with F. Santos.

2. Distorted-waves theory In the usual form of the DWBA the transition matrix for a (d, p) reaction is approximated by the expression 1,2)


S20"2 ,

k2[t(d, p)lact, S l a l , kt> (-) <~s2~2(k2, P)~bp(~, n)[V, vl~,+~),(k,, P, n)~Pa~(~)>. (2.1)

Here, the spins of the deuteron, target nucleus, proton and residual nucleus are denoted by sl, a, s2 and b, respectively. In this paper angular momentum quantum numbers are represented by Latin letters and their components by the Greek equivalent.

(d, p) AND (p, d) REACTIONS

291

The ~,(+) are distorted waves describing the elastic scattering of the proton and the deuteron 1. The superscript (_+) refers to the usual boundary condition, and k t and k 2 are, respectively, the momenta of the deuteron and proton in the centre-of-mass system. If the elastic scattering is generated by spin-dependent potentials, the wave functions ~,(±) are of the form 6. ,o) (-) s2 (-) .... (k2, p) = Z- Z~'~(P)~'~(k2 , r,),

(2.2)

~'2

0(+)tb s~a~k'l

, P,

n)

=

r = r p - g r n,

Z q~.,(r, p, n)~c,.:,(k I , R), s~ (+)

(2.3)

R = ½(rp+(2-T)r,),

(2.4)

"/ = A/(A + 1), where A is the mass of the target nucleus. The function Z~,2(P)"2is a proton intrinsic spin wave function and q~,', is the internal wave function of the deuteron. In the absence of spin-dependent distorting potentials, the matrices ff,2~'2 (-) and ,//+) ~.o:,~ are proportional to the unit matrix in spin space. We shall not give here a discussion of the basic weak coupling approximation which leads to the form (2.1). We are concerned in this paper with the evaluation of (2. I ) as it stands, and, in particular, with effects due to the non-central nature of the neutronproton interaction V.p which appears explicitly in (2.1). We note that V,p appears in (2.1) in the product V,p ~"o',. From the Schroedinger equation satisfied by the deuteron wave function, we have Sl V,p c),,(r) = ( - ga + (hZ/M)V2)(a~ ',(r)

= Vo(r):.¢~', ,(r, p, n)+ va(r){gi', ,(r, p, n),

(2.5a) (2.5b)

where ed is the binding energy of the deuteron, and

Y/i,,s(r, p, n) = 2 (LA, s, a,lJ~)iLY~(r)z;;(p, n).

(2.7)

Ao"

In (2.6) uo(r) and u2(r) are, respectively, the S- and D-wave radial components of the deuteron wave function. Their normalization and phase coincide with the conventions of ref. ' ~), i.e., 4b;',(r, p, n) = Uo(r)q¢;' t ,(r, p, n)-u2(r)Yi~z'~ ,(r, p, n),

(2.8)

i°drr2((uo(r)) ~ + (u2(r)) 2) = 1.

(2.9)

+ W e a s s u m e t h a t t h e o p t i c a l p o t e n t i a l s d o n o t c o u p l e the spins o f t h e p r o j e c t i l e s a n d t a r g e t n u c l e i .

292

R . C . JOHNSON

In the usual treatment 12) of the matrix element (2.1) the coupling between spin and orbital angular momentum implied by the form (2.5b) is not properly taken into account. The second term in (2.5b) is completely neglected, and Vo(r) is calculated from (2.6) with u o given, for example, by the Hulth6n function

u~(r) = N o ( e - = ' -

e-#')lr,

(2.lO)

h2~12/M = gd,

fl ~ 7:t,

(2.11)

fo

~drr2(u~(r))2= 1.

(2.12)

This procedure corresponds to using an effective central neutron-proton force which binds the deuteron and gives the correct triplet effective range. With the advent of codes for full finite-range calculations a2) and the development of reliable approximations to finite range effects 13, 14), it now becomes possible to consider the effects of a more realistic neutron-proton force. The non-central nature of this force is reflected in the presence of the L = 2 term in (2.5b). In this paper effects due to this term will be referred to as D-state effects. There are, of course, other effects due to the non-central, two body force implicit in the nuclear wave functions and in the optical potentials used to generate the distorted waves in (2.1). In particular, Satchler 15) has shown, in a simple model, that the D-state of the deuteron generates spin dependent terms of the second rank tensor form in the deuteron optical potential. Very little is known about the details of these forces, and it would therefore seem reasonable to discuss the explicit and calculable D-state effects in (2.5b) on a separate footing.

3. Formulae for cross-sections and polarization effects With unpolarized incoming deuterons and target nucleus, the proton differential cross section and polarization can be expressed in terms of the tensors Pk2q2(S2) (0 < k2 < 2s2). The latter are given by ~6)

o(O)p,,.As2)

= E



0"2@"2

x

~- (fl, 0"2[t(d , p)l a, a l ) ( f l , a2lt(d, p)lct, a l ) * ,

(3.1)

where (fl, ctz[t(d, p)la, a l ) = (bfl, s2a2,k2[l(d, p)laT, slal, k l ) ,

(3.2)

and where the operators Tk2q2(s2) are the quantities defined in ref. 16). For arbitrary spin s2

02 a;I rk2,,2(s2)ls2 a2> = (-)'2-"'2(s 2 a2, s 2 - ailkz q2).

(3.3)

(d, p) A N D

(p, d)

REACTIONS

293

The quantity a(0) in eq. (3.1) is related to the differential cross section by = M*M*k2tr(O)

(d~2)

o

(3.4)

4nZh'*~2~2k, '

a(O) = ~ I(//, a21T(d, P)tU, a,>] z,

(3.5)

02¢710~/J

where M~' and M~' are the reduced masses in the deuteron and proton channels. We use the notation 2 = (2x + 1)~. We shall often use the fact that ~- ' tr(0) may be obtained by putting k2 = q2 = 0 in the right-hand side of eq. (3.1). The tensor Plq2(½) can be expressed in terms of the Cartesian components of the proton polarization vector P(-~)(=2(s(p))). We have 16)

Pl +_l(~) = T½(P.,(})T iPr(~)), P,o(½) = 2-1P--(½),

Poo(}) = 2-2

(3.6) (3.7)

With polarized incoming deuterons, the proton differential cross section can be written 16) do" = (do'_io( 1d~2 ,2 ,/ ,il e" +~,,o,q,2 Pk,q,(Sl)'k,q,(S,)) '

d~"22

(3.8)

where the tensors p~i,~,(s t) describe the polarization state of the incoming deuterons. They are defined in terms of the density matrix of the incident beam pO) through (Tr (p(i) ))fiktq,(Sl) (i)

=

Tr (Tk,q,(st)p(°),

(3.9)

where Tk,q,(SI) is defined in eq. (3.3). For kl = 1, we have p~i,~ _, = +½(Px(~1(l)~ip(yi)(l)),

ptl'o) = 2-*P(:il(l),

(3.10)

where (Tr (p(°))P(~'(I) = Tr (s0)p(").

(3.11)

The efficiency tensors ek,q,(sl) in (3.8) are given by h2

t

~r(0M',q,(si) = ~i ~ o" i ~7"1

x ~ *.

(3.12)

a2~fl

For ]cI = 1, these tensors may be used to define an efficiency vector PC(l). We have el~,(l) -- ~ ( P ; ( 1 ) ~ i ~ ( 1 ) ) ,

c,0 = 3P;(1)/2"*.

(3.13)

Eq. (3.8) can be written in terms of P~(1) and P(1)(l) as do"

-

\oszz'(d--~a--)o(l (i, • + .}p,i, " P"+ 2q, p2q,(l)e2q,(1)).

(3.14)

294

P.. C. JOHNSON

We insert the DWBA expression, eq. (2.1), into eqs. (3.1) and (3.12) and obtain ~2

,'(O)p,,~ds~) = 7 Tr (r~(s~)A~(s~)),

t~(O)e.*,q,(s,)= where the matrices

- j T - Tr

AJ(s2) a n d BJ(sl)

(Tk*q,(s~)B~(s,)),

(3.15) (3.16)

are given by

. . . (k,)> \v" .... t 2,P)@}.,)(n)l ~,~1¢ " .(+)

=

x <¢~5!~(k~ , p)~(,,(n)l ~ V.,pl~ .... (+~ (k~)> * ,

sL,,,(s,) -- E """-~'k ',~ .... t 2, o'2,~

(3.17)

~ (4" P)¢.m,)(n)lV,,pl¢,,,,,,,(k,)> X <¢ .... ( - ) (k2, P)~'im,(n)l "~ Vo~1¢'(+) .... (k,)> * .

(3.18)

In these formulae the overlap integral ~7) ¢,~.,~)(n)(s = ½) is defined by the integration over the target nucleus co-ordinates in eq. (2.1) in the usual manner l) (3.19) jlp

¢~(,.,)(n) =

"~Lj(n)Rjt(rn).

(3.20)

In eqs. (3.15) and (3.16) we have assumed for simplicity that only one value o f j

contributes to the sum in eq. (3.19), and we have suppressed the dependence of O~.(ts)(n) on the quantum numbers of the target and residual nucleus. The general formulae are obtained by summing o v e r j on the right.hand side of eqs. (3.15) and (3.16).

4. Plane wave approximation 4.1. DIFFERENTIAL CROSS SECTION A rough estimate of the effects of the D-state can be obtained by replacing the distorted waves in eqs. (3.17) and (3.18) by plane waves. It was shown in ref. is) that in this approximation the internal structure of the deuteron enters the proton difl'erential cross section as an angle-dependent multiplicative factor G(K) which is given by G(K) = (K 2+ u2)2((uo(K))2 + (u2(K))2), (4. I ) where

K = ½kl-k2,

h2oz2/M =

2.23 MeV.

(4.2)

The functions uo(K) and u2(K) are the radial parts of the S- and D-components of the deuteron wave function in momentum space

r2drA(Kr)u,.(r),

(4.3)

fi°K2 dK((uo( K)) +(u2(K))2) = 1.

(4.4)

uL(~) =

(½,0 -~

(d, p) AND (p, d) REACTIONS

295

For a small D-state probability ( < 10 ~ ) , uo(K) differs very little from the function given by eq. (2.10) (ref. 19)). The ratio

A(K) = uz(K)/uo(K)

(4.5)

is, therefore, a measure of the effect of the dcuteron D-state in (4.1). The dependence of A(K) on K can be estimated using the approximations of ref. " ) (chapt. II.5). We find A(K) = 2~'[QolKZF(K) (4.6)

= 0.40 K 2 F(K) (K in units of fm- '),

(4.7)

where Qo is the quadrupole moment of the deuteron s). The function F(K) is slowly varying and close to unity for K - t greater than the range of the neutron-proton force. The expression (4.6) is convenient because the sensitivity of A(K) to detailed properties of the neutron-proton-potential enters largely through the properties of the function F ( K ) for large K. In estimates where the variation of F(K) is important we shall use Yamaguchi's wave function 2o), which corresponding to a D-state probability of 4 ~ . This gives ~'(K) = 1.30 (1+(K//So)2) 2 (! +(K/-~2)2) ~ ' /~o = 5.76~,

f12 = 6.77:~,

c~ = 0.2316 fm - l .

(4.8) (4.9)

For the typical medium-energy reaction 4°Ca(d, p)4tCa,

Q = 4.19 MeV,

Ed = 12 MeV,

(4.10)

we find A z = 0.007 at 0 = 20:' (the first peak in the observed cross section 2~)), A 2 = 0.2 at 0 = 90 °. Thus, except possibly for small 0, the influence of the D-state term in eq. (4.1) is not particularly small. There are, moreover, good reasons for believing that these very crude plane wave calculations underestimate the D-state effects. (i) When proper distorted waves are used the functions Uo (K) and ue(K ) appear in the (d, p) matrix element evaluated at values of K corresponding to the momentum components in the distorted waves rather than the asymptotic momenta kt and k2. The quantity A(K) increases rapidly with K for K < 1 f m - t . (According to the Yamaguchi wave function 2o), (dA(K)/dK 2) > 0 for all K). It is plausible, therefore, that the high momentum components of the distorted waves in the nuclear interior will enhance the D-state effects over the plane wave estimates. This discussion may be compared with the similar argument in ref. ~2) concerning the influence of distortion on S-state finite-range effects, i.e., effects due to the variation of (K2+72)uo(K) in eq. (4.1) with K. It is pointed out in ref. ~2) that distortion introduces into the wave functions momenta that are lower as well as higher than are present in the plane waves. For a finite-range nucleon-nucleon force, ( K 2 + ~ 2) × uo(K) falls to small values as K increases, and, therefore, the plane wave results

296

g.c.

JOHNSON

(4.1) and (4.2) often overestimate the corrections for finite range at large momentum transfers. Nevertheless, because (K2+~2)u2(K) increases as K increases (for K < 1 fm -1) we expect that the use of plane waves does indeed underestimate the Dstate effects. (it) In expression (4.1) for G(K) the S- and D-components contribute incoherently. This is a special feature of the use of plane waves. It will be shown in sect. 5 that the spin-dependent terms in the deuteron and proton optical potentials introduce contributions to the (d, p) cross section that are linear in the D-state wave function u2. Medium-energy stripping reactions (with l > 0) fi.re known 2,) to be sensitive to the spin-orbit parts of the optical potentials in the angular region 90 ° to 120°. The plane wave estimate of the incoherent D-state contribution for the reaction (4.10) is already quite large (20 ~ ) in this angular region, so that detailed fits to the cross sections clearly require a proper investigation of D-state effects. 4.2. P O L A R I Z A T I O N E F F E C T S

The deuteron D-state is excepted to play an important role in the quantitative description of polarization effects in stripping reactions. Unfortunately, the D-state influences the vector polarization quantities (for example, P ( ~ ) a n d P'(I)) through complicated interference with the distortion produced by the deuteron and proton optical potentials, and simple estimates similar to those of subsects 4.1 are difficult to obtain. This is illustrated by the result of ref. ,8) where it is shown that if plane waves are used P(-~) and P~(I) vanish even if D-state effects are included. (It is well known 22) that P(½) and P ' ( I ) vanish in plane wave theory if D-state effects are neglected). In the next sections it will be shown that these interference effects can be delineated further in special cases; however, a complete discussion requires detailed numerical computation. There is one set of polarization quantities which does not vanish when plane waves are used, and which is amenable to a simple, very approximatc, treatment. This is the set of second rank tensor components of the deuteron polarization in a (p, d) reaction and the components of the second rank efficiency tensor for incoming polarized deuterons in a (d, p) reaction. For definiteness we shall considcr a (p, d) reaction induced by unpolarized protons. Formulae for the reaction ( b + p ) - - * ( d + a ) are most simply obtained from eqs. (3.15)-(3.18) by interchanging the suffices I and 2 and making the substitution " ' (.... - ) l kt.

~ n )IV.pig' .... ( + ) ( k . , p. n)> 2, P)~i(t~,( (-)

~

(+)

-+ ,

(4.1l)

where now k 2 is the momentum of the incident proton and k, the momentum of the outgoing deuteron. Using the plane wave formulae of appendix 2, we obtain * t The polarization parameters used here are related to the (T~.q> defined by Satchler ~0) and used in ref. 2~), by < Tkq,~ = g] p*kq(sl).

(d, p) AND (p, d) REACTIONS

297

p2q(s, ) = - 2(¢s 7r)t Y;*(K) (_A(K) + (}~)½d2(K)) 1 +A2(K) '

(4.12)

K = ½kl-k2,

(4.13)

where A(K) is the quantity defined in eq. (4.5). We emphasise that the formula (4.12) neglects contributions to the P2q(sl) from the central and spin-dependent terms in the deuteron and proton optical potentials, and is the contribution to the deuteron polarization arising from D-state effects alone. For these reasons the predictions of eq. (4.12) should not be taken too seriously. It is nevertheless of interest to compare the qualitative features of eq. (4.12) with the recent measurement 23) of the p2~(sl) for the reaction 9Be(p, d)SBe(Q = 0.56 MeV) at proton energies of 2.5 and 3.7 MeV. Preliminary calculations of the p:q(Sl) using the DWBA with spin-orbit term in the distorting potentials are reported in ref. 23). The calculations, which ignore the Dstate effects we are considering here, agree generally in sign with the measured values, but are consistently smaller than these by at least an order of magnitude. The P2q(Sl) given by eq. (4.12) are readily calculated using eqs. (4.6)-(4.9). For this low-energy reaction, the values of K entering the calculation are less than I fln- 1, so that F(K) in eq. (4.6) is very close to 1.30, and the numerical results are insensitive to the D-state probability assumed. As is expected we obtain poor agreement with experiment. It is of interest, however, that the rO2q obtained are larger than the distorted wave results reported in ref'. 23). At the largest angle at which measurements are reported (60 °) the results of eq. (4.12) are smaller than the measured values by a factor of about 2 only. The remarks in subsect. 4.1 concerning the effects of distortion on the D-state contributions are pertinent here. If, for example, as a rough correction for some o f t h e effects of distortion we evaluate K in eqs. (4.12) and (4.13) at K' = ~Ka(k,/k I ) -

Kp(k2/k2),

(4.14)

where Kd and Kp are the magnitude of the momenta corresponding to a deuteron and proton in a real potential well of depth 20 MeV, the value of p2o(Sl) predicted by (4.12) increases by a factor 8 at 0 = 60 °. There are very probably corrections to the weak coupling DWBA theory for this reaction other than D-state effects; although the D W B A calculations reported in ref. 2a) do fit the deuteron differential cross section quite well. However, our plane wave results, which are already larger than the predictions of standard distorted wave calculations, do suggest that the D-state of the deuteron gives a sizable contribution to the observed polarization. We have already remarked that even rough estimates of the effect of the D-state on vector polarization quantities are difficult to obtain. In this article we shall confine our discussion of these quantities to the features that arise because of the special coupling of angular momenta implied by the distorted wave matrix element. In the following sections we give a detailed discussion of the way in which the results of previous work 6.7) are modified by the D-state effects.

298

R.C.

JOHNSON

5. E f f e c t s o f the D - s t a t e in the D W B A

5.1. PRELIMINARY ANALYSIS

In order to develop further the expressions for polarization tensors in eqs. (3.15) and (3.16), we make the dependence of the matrix elements in eqs. (3.17) and (3.18) on spin projections more explicit. This will enable us to perform many of the summations over magnetic quantum numbers, and it will then be possible to pick out the contributions from various effects to the polarization tensors in a straightforward manner. We first consider the proton distorted wave. From eq. (2.2) we have ~(-)~k ~2~2k 2, P) = ~ ~2 ,,~

(-~

rp).

(5.1)

The matrix ¢o'2o2 <-) can be expanded in terms of the complete set of matrices Tt~2(Sz) defined in eq. (3.3). We therefore have quite generally (--) ~b..... (k2, P) = s2 ~ ( - )

32

(--) ff,2_~(k2,

s2 %)(Tt~2(s2)zo2(P)),

(5.2)

t212

where the functions g , ~ are given by ~2~L~}(k2, rp) = Tr (Tt~,2ff(-'(k2, %)).

(5.3)

Eq. (5.3) follows from eq. (5.2) and the properties 16) of the T,~: Tr (T,2~2Tt.t,. ) = 6,~.~fi,:~,~,

(5.4)

=

(5.5)

T0o(S2) = s2'.

(5.6)

For a proton (s2 = ½) the summation in eq. (5.2) involves 12 ~ 1, where the terms with t2 = 1 are non-zero only if the proton optical potential is spin dependent. We have introduced the factor ~2 into the definition (5.3) so that ~oo)(k2, %) is normalized in the usual manner, i.e., in the absence of Coulomb effects ~(oo)(k2, rp) ~ eik~"~+(ingoing spherical waves).

(5.7)

In the case of the deuteron, the situation is more complicated because of the presence of the deuteron D-state. From eq. (2.3), we have ~(+)/k • ,~,~ l , P, n) = ~ ~b~',.(r,p, n j~"(+~ ~ , , , ~ , ~ (k 1, R).

(5.8)

o'1

As in the proton case we can write =

'

a,)(-)

~b,_,(k, g),

(5.9)

tr

where for s~ = 1 the values taken by t satisfy t _< 2. The terms with t > 0 arise only when the deuteron optical potential is spin dependent. Now, however, we also have

299

(d, p) AND (p, d) REACTIONS

to take into account the spin-orbit coupling inherent in the deuteron internal wave function. With the notation of eqs. (2.5)-(2.9), we find V.vtP..... ( k l , p,

=

~',,-,,, 2, r, R)(Tt:,(sl)Z;',(p, n)),

(5.10)

/ltl

where

~ * ) = Z 3, ~W(Lslts I • sl t,) Z (--)L-a( tz' L--Altl zl)iLYfl.(r)vL(r)~b~+)(kx tL

R),

At

(5.11) and vL(r)(L -= O, 2) is defined in eq. (2.6). Eq. (5.10) is simular in structure to eq. (5.2). However, in eq. (5.10) there are contributions from t~ > 0 even if the deuteron optical potential is spin independent. Thus, if all terms with t > 0 in eq. (5.11) vanish, i~z+) is proportional to the D-state function v2(r). Expressions for the polarization tensors pk2q2(Sz) and eklq,(Sl) can be obtained by substitution of the expansions (5.2) and (5.10) into the basic formulae (3.15) and (3.16), and performing the summation over spin projections. The resulting formulae involve bilinear combinations of the matrix elements ( ~ b ~ ( k 2 , rp)i t ),ia(r,)Rj,(r,)[ ,TA+)/t.

r,

g)>,

(5.12)

weighted with Racah coefficients, etc., which are functions of t I, t z, /, j, st, s2, s, and the rank of the particular polarization tensor under consideration. These formulae are in fact not in the most convenient form for detailed numerical calculation. The usefulness of this formulation for our purposes lies in the fact that contributions from the various effects of interest (D-state, spin dependence of the optical potentials) are associated with definite values of tl and t: in eq. (5.12). It is clear from eq. (2.1) and eq. (2.5) that the DWBA transition matrix can be written t(d, p) = t s + t D,

(5.13)

where t s and to involve, respectively, the S- and D-state components of the deuteron wave function. The DWBA expressions for quantities measured in a (d, p) reaction can therefore be written

tr(O) = a(SS)+a(SD)+a(DD), tr(O)P(½) = P(SS) + P(SD) + P(DD),

(5.14a) (5.14b)

a(o)e'(1) = V~(SS) + P'(SD) + P~(DD),

(5.14c)

tr(0)e2q,(l ) = e2q,(SS)+e2q,(SD)+e2qt(DD),

(5.14d)

where our notation implies that P(DD), for example, is the contribution to a(O)PQ) that is quadratic in the matrix element to of eq. (5.13).

300

R . C . JOttNSON

We recall that a(0) is related through eq. (3.4) to the proton differential cross section corresponding to unpolarized deuterons, and P ( I ) is the polarization of the protons in the same reaction. The quantities P*(1 ) and e2q`(1) parameterize the proton differential cross section when the incident deuterons are polarized. It is well known 24) that P(½), P*(1) and t32,tl(l ) c a n also be interpreted, respectively, as the efficiency vector for incoming polarized protons and the vector and tensor polarization parameters of the outgoing deuterons in the time-reversed reaction. Thus our results have an immediate application to (p, d) reactions. In the next four subsections we discuss in detail the modifications to various quantities introduced by the (SD) and (DD) terms. Special emphasis is placed on the contribution from the coherent (SD) contributions, which are likely to give rise to the largets D-state corrections. We by no means imply, however, that the (DD) corrections are completely negligible. Some of the main results of our analysis are summarized in table I. TABLE 1 C o h e r e n t D - s t a t e c o n t r i b u t i o n s in the n o t a t i o n o f eq. (5.14) l plane waves approximation .

.

.

.

a(SD)

~ 0 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0

optical potentials .

.

.

only deuteron optical potential spin-dependent only proton optical potential spin-dependent deuteron and proton optical potentials both spin-dependent

.

.

.

.

.

.

.

.

.

.

.

.

.

zero

> 0 .

zero, P(,~) -= 0

zero .

spin-independent

P(SD)

.

.

0

.

.

.

.

non-zero .

.

.

.

.

.

.

.

.

v,v~(SD)

zero, P*(I ) -- 0

non-zero

zero

non-zero

.

zero

zero .

.

Pt(SD)

.

.

.

non-zero .

.

.

.

.

.

.

.

non-zero .

.

.

.

.

.

.

.

n o n - z e r o a)

non-zero

non-zero

non-zero

non-zero

non-zero

non-zero

non-zero

zero

zero

non-zero

non-zero

> 0

non-zero

non-zero

non-zero

non-zero

0

non-zero

non-zero

non-zero

non-zero

> 0

non-zero

non-zero

non-zero

non-zero

> 0 0

a) T h i s c o n t r i b u t i o n o n l y i n v o l v e s (Vd~o) z a n d h i g h e r p o w e r s (see subset. 5.6). 5.2. S P I N - I N D E P E N D E N T

DEUTERON

AND

PROTON

OPTICAL

POTENTIALS

In subsect 4.1 we have already quoted Dalitz's result 18) that in the plane wave approximation the deuteron S- and D-states contribute incoherently to the proton differential cross section, i.e., o ( S D ) = 0 in the notation of eq. (5.14). Perhaps surprisingly, we find that this result is still true if the plane waves are replaced by distorted waves generated by central optical potentials. Dalitz 18) also showed that the vector polarization quantities P(½) and P~(1 ) vanish if plane waves are used. (The second rank parameters e,2q,(l) were considered in subsect. 4.2.) It is well known 25) that in the absence of D-state effects the DWBA

(d, p) AND (p, d) REACTIONS

301

with spin-independent optical potentials predicts non-vanishing polarization satisfying IP(~)I < ~. P(½) = 0,

if

P~(I) = l = 0,

2P(½) e2q,(l) = 0.

(s.15)

We find that there are D-state corrections (which enter incoherently in the vector polarization quantities), which invalidate all the results of eq. (5.15). However, these results arc not of very great physical interest because the predictions (5.15) are already violated by the spin-dependence of the optical potentials 6) as well as being in disagreement with experiment 26). Formulae from which the D-state effects mentioned in this subsection can be derived are given in appendix 2. 5.3. SPIN-DEPENDENT DEUTERON AND PROTON OPTICAL POTENTIALS In general we find that the spin dependence of the deuteron and proton optical potentials introduces coherent D-state corrections to all the experimental quantities considered here, i.e., a(SD), P(SD), P~(SD) and e~¢,(SD) are all non-zero. However, for processes characterized by neutron capture with zero orbital angular momentum ( / = 0) considerably more than this can be said. It was shown in ref. 6), within the framework of the zero range DWBA and neglecting D-state effects, that the following selection rules and correlations are valid. With spin dependencc in deuteron optical potential only ( / = 0) P(~) = PC(l),

(5.16)

and with spin dependence in proton optical potential only (l = 0) P(½) = ~P'(1).

(5.17)

When both proton and deuteron optical potentials are spin dependent no simple correlation between P~(1 ) and P(½) exists 6). However, to first order in the deuteron and proton spin-orbit potentials (we assume here that they are both of the l.s type), it was shown in ref. 7) that P(½) = P ( p ) + P ( d ) , PC(l) =

z3P(p)+P(d),

(5.18) (5.19)

where P(p) and P(d) are the contributions to P(½) from the proton and deuteron spin-orbit potentials, respectively. The results (5.18) and (5.19) imply, for example, that ( P ( t ) - P ~ ( 1 ) ) is dependent of the parameters of the deuteron spin-orbit to a good approximation. It is of interest, therefore, to enquire how these results are modified by the deuteron D-state. Formulae for 1 = 0 processes are obtained from eqs. (3.15)-(3.18) and eqs. (5.2), (5. l I) in terms of the matrix elements t2r2 ( ) B,,,, = <~k,2,_,(k2, rp)Roi(,',)Y°(r,)l~',+,l(k,, r, R)>. - -

(5.20)

302

R.C.

JOHNSON

We obtain

a(O)PLq~(s2) = .~2_ 3" 1"k~(1, 2, k, k')Gk~q,_(l 2, k, k'), ~2 ~

(5.21)

with summation over k, k', t~, t'1, t2, t'z, where

rk2(l, 2, k, k') = b2~, ~,. ~, F, ~17; f,f,'(-)" +k*k'W(s2sks, ; s, s2) s2

x W(t~ t', s~ s~; ks~) s2 $2 t t2 t2 Gk~q2(l, 2, k, k') = ~ (-)"'+"2+c:+~'B't2,;~Bt/~:)* × (kq, k'q'lk2q2)(t', ~;, t,-r,lkq)(t2z2, t'2-r'zlk'q'),

(5.22)

(5.23)

with summation over q, q', rl, ¢1, r2, r~. Similarly we find ^4 S1

a(O)8*,q,(s,) = ~ Z Fk,(2, 1, k', k)Gk,q,(l, 2, k, k'),

(5.24)

where Fk,(2, 1, k', k) is obtained from eq. (5.22) by interchanging suffices 1 and 2 everywhere, together with k and k'. In subsects. 5.4-5.6, we consider in turn the modification of the results in eqs. (5.16)-(5.19) implied by formulae (5.21) and (5.24). 5.4. S P I N

DEPENDENCE

IN PROTON

OPTICAL

POTENTIAL

ONLY

(1 = 0)

We recall (see subsect 5.1) that a particular value of tl in eq. (5.20) is associated with that part of the deuteron distorted wave which depends on spin through a spin tensor of rank t~. Thus, if the deuteron optical potential is spin independent the only possible values of tl are 0 and 2, corresponding to S- and D-state effects, respectively (see the discussion immediately following eq. (5.11)). Eq. (5.23) shows that the contributions to a(SD) (k2 = 0), and P(SD) (k2 = 1), arise from k = 2. But from eq. (5.22) we see that Fk2(1 , 2, k, k') vanishes for k = 2, if s 2 = "~, and therefore tr(SD) = P(SD) --- 0.

(5.25)

A similar analysis of eq. (5.24) shows that P'(SD) # 0,

(5.26)

and therefore the correlation (5.17) is certainly invalidated by D-state effects. It is possible to show that provided the optical potentials conserve parity, but without further assumptions, the correlation (5.17) is satisfied by P(SS) and P~(SS). it is found from eqs. (5.21) and (5.24) that P(SS) is proportional to P'(SS) if terms with t 2 = t~ = 1 do not contribute. The latter can be eliminated using the parity condition given in eqs. (A.11) and (A.12) (c.f., the similar argument following eq.

(d, p) AND (p, d) REACTIONS

303

(5.32) in the next subsection). We therefore have

where we P(p, SD) symbol p from the

P(½) = P(p, SS),

(5.27)

P~(1) = ~P(p, S S ) + P ( p , SD),

(5.28)

have neglected the incoherent D-state corrections a ( D D ) etc. The quantity is the coherent D-state correction to P ' ( I ) which is absent in P(½). The attached to the quantities in eqs. (5.27) and (5.28) indicates that they arise proton spin-orbit force alone.

5.5. SPIN D E P E N D E N C E

IN D E U T E R O N

O P T I C A L P O T E N T I A L O N L Y ( 1 = 0)

In this case we have t 2 = t~ = 0 in eqs. (5.20)-(5.24), and these expressions now simplify to

,,(O)p,,%(~) = ~2~ ( - ) ~ w(~ ~k~ ~, ; s, ~) ~232

x ~ (--)c'~ 1 ~i W(tl Sl tl Sl ; s1 k2)nk2q~(tl, tl),

(5.29)

/ll'l

a(O)e~,q,(sl) = [~z~3 ~ ( - ) " ~ Y , W ( t , s t t ; s ~ ; s ~ k ~ ) H ~ q ~ ( t , , t ; ) , ,~2~2 tlt't

(5.30)

where

n~q(tl , tl) = E (-)"-~l(t', zl , tt-z,lkq)Bt°°,

B°°'c:,,

•

(5.31)

'~117' 1

It is shown in appendix i that if the optical potentials conserve parity, and if the tensors (5.29) and (5.30) are referred to a co-ordinate system with the z-axis perpendicular to the reaction plane B,°°, = 0, rl an odd integer.

(5.32)

This condition leads to the well-known result that P(½) and P~(I) are perpendicular to the reaction plane, i.e. P~o(½) and elo(1) are the only non-vanishing components of P lq2 and elq ~. But eq. (5.32) also has the consequence that HI o(1, 1) is proportional to the Clebsch-Gorden coefficient (10, 10110), which vanishes. Thus the only pairs of values of tl and t'~ that contribute to the summations in q s . (5.29) and (5.30) are (0,1), (1,0), (1,2), (2,1), (2,2).

(5.33)

Inserting these pairs into eqs. (5.29) and (5.30) we obtain [P(½)I- IP*(I)I = (a(O))-1ell1 o(2,2),

(5.34)

where c is a non-zero statistical factor. The quantity HI 0(2,2) can be evaluated from eqs. (5.31) and (5.32). We obtain H10(2,2) =

2 ½(IB2~I oo 2 - I B 2 oo (~) - 2 1 2).

(5.35)

304

R . c . JOHNSON

Thus, in order for the correlation (5.16) to be valid the dynamics of the (d, p) reaction must be such that H~o(2,2) vanishes. In ref. 6), where the result (5.16) was first given, the following assumptions were made: (i) neglect of the D-state of the deuteron, (ii) zero-range approximation for V,,p and (iii) neglect of spin-dependent terms in the deuteron optical potential that do not conserve orbital angular momentum (such forces, which are possible for a spin-I particle, are discussed in ref. 15)). With these approximations, explicit calculation shows that the matrix elements Btoo satisfy (z-axis along k 2 A kl, x-axis along kl) l'¢ 1 oo

ntl¢l

=

e

m0noo ~tl-rt

(5.36)

where 0 is the angle between k~ and k2. From eqs. (5.36) and 5.35) we see that assumptions (i)-(iii) ensure the vanishing of H~ o(2,2), and hence the validity of the correlation (5.16). On the other hand, if any of the assumptions (i)-(iii) are lifted, we find eq. (5.36) is no longer true, and therefore the right-hand side of eq. (5.34) does not vanish. However, there remains the possiblity that the quantities u2~ 2/:/00 2 may be considered negligible. This possibility is explored in the next subsection, where the spin-dependent forces are treated in first-order perturbation theory. In this subsection we have not separated out the coherent D-state corrections a(SD) etc. In subsect. 5.4, we found there were coherent D-state corrections only in P':(I). Inspection of eqs. (5.29)-(5.31) shows that with a spin-dependent deuteron optical potential a(SD), P(SD), P'(SD) are all non-zero. These terms will be given further consideration in the next subsection. 5.6. T R E A T M E N T O F S P I N - D E P E N D E N T F O R C E S BY P E R T U R B A T I O N T H E O R Y ( / -

0)

In this subsection we treat the spin-dependent terms in the deuteron and proton optical potentials in first-order perturbation theory 7), i.e. for a spin-dependent force of the form VS'°'f(r)l • s, we neglect quantities depending on (VS'°') 2 or higher powers. It is useful to recall the results and notation of subsect. 5.1. For the deuteron distorted wave we have (see eqs. (5.8 and (5.9)) TSlal\

=

1 ~ P'

qS~, (r, p, n)O~.,,,(k, R),

(5.37)

O'l

' a , ) ( - ) ¢O,_,(k, "e,,',,O,'lA+'¢ k i , R) = ~ e,
R).

(5.38)

t~

We shall assume that the deuteron spin-orbit force is of the form U~ ° = V g ° f ( R ) l

• s = 2l/~'°f(R) ~ (-)~T, _¢(l)T, dsl),

(5.39)

t

where T,o(l ) = 2-÷1~,

T,±,(l)

= -~-~(Ix-T-il,),

and T~,(s~) is the spin operator with matrix elements defined in eq. (3.3).

(5.40)

(d, p) A N D

(p, d) R E A C T I O N S

305

To first order in V] "°', we have 7) Sl ",(÷~(k,) = ¢'g+~(k,)4'o,+ ~(+~'r~'°'~'~+~n'-,o ,,~ ~,o ~,,,)4'2,

(5.41)

where

Gg+~= ( G - n o + i ~ ) /40 = Ko + Vo,

-1,

( G -/-/o)~,g +'(t,, ) = 0.

(5.42) (5.43)

Here, K o is the kinetic energy operator for the centre of mass motion of the deuteron and Ed the corresponding kinetic energy. The potential Vc(R ) is the spin-independent part of the deuteron optical potential, so that O~o+~(kl) is the distorted wave corresponding to zero spin-orbit force. Comparing eqs. (5.41) and (5.37) we see that in the first-order approximation ra'

iO'1\

I ~

=

R)3,,,,~,+(RIGo l~,g+~(k,)>.

(5.44)

This can be written in the form (5.38), but because of eq. (5.39), terms with t = 2 do not appear. The t = 0 and t = 1 terms are associated, respectively, with the first and second terms in eq. (5.44)) and t = 2 terms arise only when contributions of order (V~'°') 2 or higher powers are considered. Using this result it is readily shown from eqs. (5.11) and (5.20) that the matrix elements Bt°°, have the following properties: (i) B°T° is proportional to V,~$ . o . ., (ii) B°,° only involves the D-state wave function, v2(r). If t = 2 terms were allowed in eq. (5.11), B°~° would also depend on terms involving the S-state wave function. (iii) B °O only involves the S-state wave function Vo(r ) and is independent of V~'°'. We are now in a position to reconsider the results of subsect. 5.5. We refer first of all to the differential cross section cr(0), which can be obtained from eq. (5.29) by putting k 2 = q2 = 0 in the right hand side. We obtain

a(O) = (~})½[~2~

~t,,,w°°2.

(5.45)

IlTI

It was stated in subsect. 5.5 that a(SD) ~ 0 in the expression (5.45). However we see from the properties (i)-(iii) given above that a(SD) is proportional to (V~'°') 2 and is therefore small. Property {ii) also shows that ~-'2~, no0 2 is proportional to the square of the deuteron D-state wave function. Hence the function HI o(2,2) of eq. 1"5.35) is probably small, and to a good approximation, the result P(½) = P(d, SD),

(5.46)

PC(l) = P(d, SD)

(5.47)

is valid. The notation in eqs. (5.46) and (5.47) is meant to indicate that these quantities arise from the deuteron spin-orbit force, and that they involve contributions linear in the deuteron D-state wave function.

306

g.C. JOHNSON

These results are to be contrasted with those of subsect. 5.4, where the contribution from the proton spin-orbit force was considered. There it was found that the coherent D-state corrections to P(-~) and W(I) are different (sec eqs. (5.27) and 5.28)). In eqs. (5.46) and (5.47), however, the D-state corrcctions are the same. So far we have considered separately the contributions from deuteron and proton spin-orbit forces. However, in first order their contributions to cross sections and polarizations are simply additivc 7.8). Eqs. (5.27), (5.28), (5.46) and (5.47) give P ( [ ) = e(p, S S ) + P ( d , SD),

(5.48)

W(I) = 3~P(p, S S ) + P ( p , S D ) + P ( d , SD).

(5.49)

We emphasize that the results (5.48) and (5.49) apply only to l = 0 processes and are accurate to first order in the strenghts of the proton and deuteron l" s potentials. We have also neglected contributions quadratic in the deuteron D-state wave function. Thus, these results are not exact consequences of the DWBA matrix element. Experience with the first-order treatment of spin-orbit forces in stripping 9) and elastic scattering 27) has shown that this is an accurate approximation except when the polarization is very large, i.e. close to + 1. The quantitative effects of the deuteron D-state are not known at present; however it is plausible that the major corrections involve the D-state in a linear fashion. In this section we have assumed that the deuteron spin-dependent potential is of the I. s type. More general types of force, which involve spin tensors of rank 2, are also possible for the deuteron 15). It can be shown that, for l = 0 processes, these types of force modify the results given in eqs. (5.48) and (5.49) only to the extent of introducing terms of higher order than the first in their strenghts, and terms in which the D-state and not the S-state is involved. The results of the analysis of deuteron elastic scattering and polarization given in ref. 28) suggest that second rank tensor forces are very small, i.e. smaller than predicted by the theory of ref. ~5), where they are generated by the deuteron D-state and the nucleon spin-orbit forces. Thus, our present knowledge of these forces, together with the above statement concerning the manner in which they modify our results, both suggest they play a minor role in the determination of vector polarization quantities in l = 0 processes.

6. Concluding remarks In this section we discuss some possible implications of the results of sect. 5. Eq. (5.48) shows that for l = 0 processes the D-state affects the proton polarization only through interference with the deuteron spin-orbit force. This result may have important implications for the analysis of proton polarization effects. Thus, Hooper 9) has shown that fo, a reaction in which the deuteron can be considered to be strongly absorbed, the contribution to proton polarization from the poorlyknown deuteron spin-orbit force tends to be suppressed relative to the contribution from the proton spin-orbit force. There is therefore the possibility that D-state effects

(d, p) AND (p, d) REACTIONS

307

on proton polarization in l = 0 processes are relatively small. We note that Hjorth et al. 29) in contrast to Hooper 9) found that the deuteron spin-orbit force had an important effect on the proton polarization. Howcver, in ref. 29), only proccsses with l > 0 were considered, for which Hoopcr's analysis does not apply. Another aspect ofeqs. (5.48) and (5.49) is the fact that the same function P(d, SD) appears in both formulae. Thus, measurcment of P(½) and PC(l) would determine a quantity (P(~z)-P'(I)) which is very insensitive to the parameters of the deuteron spin-orbit force. Unfortunately, proton polarization measurements 2~) and experiments with polarized deuterons 30) performed so far for I = 0 processes do not involve the same energies and target nuclei. For processes with l > 0 it is not possible to find a linear combination of P(½) and P~(1 ) which eliminates thc tirst-order depcndence on the deuteron spin-orbit force. Part of this work was done while the author was at the University of Pittsburgh, where he was supported by the National Science Foundation. The helpful comments of N. Austern, L. R. B. Elton and L. J. B. Goldfarb are gratefully acknowledged.

Appendix A.1. SYMMETRY PROPERTY OF THE DWBA MATRIX ELEMENTS The cross section for a (d, p) matrix elements between plane t(d, p) can be expressed in terms in the deuteron and in the target need here is

reaction can be expressed exactly in terms of the wave states of certain operators t(d, p), where of the two-body interactions between the nucleons nucleus 31). The only property of t(d, p) we shall

R - ' t ( d , p)R = t(d, p),

(A.l)

P-~t(d, p)P = t(d, p),

(A.2)

where P is the parity operator for the system and R the rotation operator corresponding to an arbitrary rotation. Eqs. (A.1) and (A.2) express, respectively, the conservation of angular momentum and the conservation of parity. We therefore have

(kz,szaz,bfllt(d, p)[kl ,sla I, q~) = (k2,SEa2,bfll(RP)- it(d, p)RPlkl ,saal, act). (A.3) We choose R to be the rotation R(r~, z) that rotates any vector by rt about the z-axis. If we choose the z-axis to be perpendicular to the reaction plane the vectors k~ and k 2 are unchanged under the operation R(n, z)P. We also have 32)

R(rr, z)PIs I a l , act) = e -/~''' +~)rr(a)lsl a, , act),

(A.4)

R(Tr, z)PIs2az, bff,> = e-'*(°2+P'n(b)ls2a2, bfl),

(A.5)

where rr(a) is the parity of nucleus a, and we have used the fact that the internal

308

R . C . JOHNSON

state of the deuteron has even parity. Eq. (A.3) therefore gives (1-n(a)n(b)ei~(~'2+a-"'-~))(k2, s2a2, bfllt [k,, s, a , , e,a) = 0,

(A.6)

and, hence, the t-matrix elements vanish if

ei~('2+#-"-')n(a)n(b) = - 1 ,

(z-axis along k, ^ k2).

(A.7)

In the DWBA, the t-matrix is approximated by expression (2.1). The deuteron and proton optical potentials conserve parity and angular momentum so that the DWBA matrix element must also satisfy (A.7). Using the expansions (5.2) and (5.10) in eq. (2.1) we find. (~b~-,)2(k2, P)~bb#(¢, n)lV, pl~'~+,,),(k,, p, n)~a~(¢)>

= E (aot, j¢]bfl)(1)., salj()(s20.' 2 , s0.1st tr;) × (_)s2-o 2(s20.2, s 2 - 0 . ~ l t ~ 2 ) ( - ) '

×(-)

~2+~

i

° '(s, 0.,, ~, -0.;It,

Sl-

"

z,)

( - ) k 2)t.i Y~ a(~)Ri,(r~)l~,,,,(k,)), r (+) (~b,~,~(

(A.8)

with summation over ct, fl, 2, 0., ~, 0.2, 0.'J, t2, z2, it, zl- The Clebsch-Gordan coefficients in eq. (A.8) ensure that all the non-vanishing terms in the summation satisfy Z 2 - - Z t q- ,'~ =

(A.9)

0"2 " b f l - - 0 . 1 --Ct.

We also have

7z(a)rc(b) = ( - ) t ,

(A.10)

so that eqs. (A.7)-(A.9) give the result (-) .l ,l (+) (fft~,,(k2)' Yt (r,)Ri,(G)l~,,~,(k,))

= 0,

(A.11)

if (-)~+z+'~-" = - 1. Eq. (5.32) follows as a special case of this result. Similarly, if the y-axis is chosen to lie along k I

^

(A.12)

k2, we obtain

( - ) k 2)1.ly~, (~) r R i,(.)l~,,,,(k,)) r (+) hT,(+) \ ( A . 1 3 ) (~,,,~( = ( - , V,+,2+,~+t,+~,/a,(-) ,.~,,~_,~z.~v-;,o ,, ,-3,,v,,-,,/.

The result (A.13) is obtained by considering the invariance of T(d, p) under the operation R(n, y)P. A.2. S P I N - I N D E P E N D E N T

OPTICAL

POTENTIALS

In this case we have t2 = 0 in eq. (5.2), and t = 0 in eq. (5.11). Eq. (A.8) simplifies to (-)

(+) k

= ~ (act, j¢lbfl)(12, s0.lj¢)(s20. 2 , so.Is, 0.'1) x ~(L)-~(--)L+s'-'"(st0.~, s,--0.'~ILA)B~a(k~, k~),

(A.14)

309

(d, p) AND (p, d) REACTIONS

with summation over, (, 2, a, a], L, A, where

BZLaa(k, , k2) = <~,(oo)(k2)i' Yl~(r,)Ri,(r,)l i L yLa(r)vL(r)~bto+)(k,)>.

(A.15)

Inserting eq. (A.14) into eqs. (3.15) and (3.16), we find

y, w(,jkt; ts)

= Z

×w(s,s,Zr;Ls,)l's

s,

s ~,s2 s2

,7(o)/2,,,(s,) = Y.

Gtk~.q2(k, k', L, C),

(A.16)

k21

k, } ('i s' k, Gtk,,,.(k,k',L,E),

b2:,(-)kw(,jkt; t,)

X W(ss 2 ksl; s I s)

sl 12

(A.17)

with summation over k, k', L, L', where

Gt~,q,(k, k', L, E) = ~ ~k'(kq, k'q'lk~ ql) x (--)t-a(/2', l--2lkq)(--)L'-a'(LA, L". .--,a . . . . .Ir. . q )oLatzDL,a.,,,az'* (A.18) with summation over q, q', 2, 2', A, A'. We are interested in contributions to eqs. (A.16) and (A.17) that are linear in the D-state wave function, i.e. L = 0, L' = 2 and L = 2, L' = 0. These terms can arise from the terms in the summation with k' = 2 only. The differential cross section is obtained from the right-hand side of eq. (A.16) by putting k 2 = q2 = 0. This forces k' to equal k. The latter is restricted by the 9-j symbol in eq. (16) to satisfy 0 =< k =< 2s,

(A.19)

and therefore if s = ½ the contribution from terms with k = k' = 2 vanishes. Thus, as stated in subsect. 5.2, the S- and D-states contribute incoherently to the differential cross section if spin-dependent distortion is neglected. A similar analysis of eqs. (A.16)-(A.18) shows that there are coherent contributions to all the polarization quantities if l ~ 0. Finally, we briefly consider the consequences of using plane waves. In this approximation the matrix element (A.15) reduces to the expression

BtL~(k,, k2) = y L( a K ) Yt~"(q)B,L(K,q),

(A.20)

where

B,L(K, q) = (8rr)~(- i)LuL(K)

r2dr. j,(qr.)Rj,(r.),

(A.21)

y = A/(A+I),

(A.22)

o

K

=

½k~-k2,

q = k,-ykz,

and R o is the Butler cut-off radius ~,2). The quantity uz(K) is defined in eq. (4.3).

310

n . c . JOHNSON

A c o n s e q u e n c e o f t h e s p e c i a l d e p e n d e n c e o n A a n d 2 i m p l i e d b y eq. ( A . 2 0 ) is t h a t t h e r i g h t - h a n d side o f eq. ( A . 1 8 ) v a n i s h e s u n l e s s b o t h k a n d k ' a r e even. E q . ( A . 1 9 ) n o w r e s t r i c t s k t o b e z e r o , a n d t h i s i m p l i e s k ' = k2 in eq. ( A . 1 6 ) a n d k ' = k 1 i n eq. ( A . 1 7 ) . T h u s t h e t e n s o r s plq2(~,) a n d e l q l ( l ) b o t h v a n i s h in t h e p l a n e w a v e approximation.

T h i s r e s u l t w a s first g i v e n b y D a l i t z 18).

N o t e a d d e d in proof." S o m e o f t h e r e s u l t s a r e i m p l i c i t in t h e w o r k o f B o t z i a n 33).

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)

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