Target polarization effects in nuclear reactions

Nuclear Physics 39 (1962) 408---446; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

TARGET POLARIZATION EFFECTS IN NUCLEAR REACTIONS L. J. B. GOLDFARB t and D. A. BROMLEY Yale University, New Haven, Connecticut tt Received 14 June 1962 Abstract: Considerations of direct reactions, based on a distorted-wave formalism, show that the

effects of polarized targets represented by the tensors, pkx(a), are simply correlated to the polarization of the residual nucleus, represented by pkK(b) when the target is unpolarized, provided that the transferred total angular momentum j is unique. Two related processes are considered: (I) A beam of unpolarized particles is incident on an unpolarized target and the angular dependence of the reaction-product yield associated with each tensor pkK(a) is measured. (2) With no polarization in the entrance channel, the angular dependence o f the polarization o f the residual nucleus is measured by identifying the coefficient of the YkX(Ov,~ ) in the triple angular correlation involving the incident beam, the reaction product and the de-excitation gamma radiation. It is found that these angular dependences are identical; furthermore the relative magnitudes of the partial cross sections in (I) and (2), for a given reaction-product angle, lead to a determination ofj. Polarization tensors pkx(a) of rank k > 2jmax should not affect the angular distributions. No detailed assumptions are made concerning the nature of the projectile or reaction product, nor the spin-dependence of either the interaction or the channel distortions, except that such distortions be independent of the spins of the target and residual nuclei. Upper bounds may also be given, provided that j is unique, to both the magnitudes of the asymmetry in the angular distributions associated with a given polarization tensor and the magnitudes of the polarization of the residual nucleus. For a given rank of polarization these upper bounds are functions merely of a, b, andj. In several cases, because o f the accidental vanishing of Racah coefficients, selection rules may be given for polarization effects of a particular rank. The polarization correlation applies to a variety of direct processes, including knock-on phenomena. Experiments are proposed which help to disentangle direct-stripping amplitudes from those normally referred to as involving heavy-particle stripping where the polarization correlation is not simple unless additional assumptions are made concerning the spin dependence of optical-model potentials. It is demonstrated that provided j is unique, a series of measurements with polarized projectiles incident on polarized targets is sufficient to overdetermine the reaction amplitudes. Several processes are considered which involve spinless projectiles and reaction products and targets of integral spin. These need not be direct processes. So long as the target is appropriately polarized, observation of the reaction product, parallel or anti-parallel to the original motion, leads to a determination of the parity of the residual nucleus, and to the identification of non-zero spin states. CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409

NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

412

1. TARGET POLARIZATION

413

. . . . . . . . . . . . . . . . . . . . . . . . . . .

2. CORRELATIONS BETWEEN TARGET AND RESIDUAL NUCLEUS POLARIZATION 2.1. Polarized Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Polarization of the Residual Nucleus . . . . . . . . . . . . . . . . . . . . . t On leave from Department of Physics, The University, Manchester, England. tt This work is supported in part by the U. S. Atomic Energy Commission. 408

415 419

409

TARGET POLARIZATION EFFECTS

2.3. Limitations on Polarization Magnitudes . . . . . . . . . . . . . . . . . . . . 2.4. Pick-up Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Spin Independent Distortions . . . . . . . . . . . . . . . . . . . . . . . . .

424 428 429

3. APPLICATIONS TO SPECIFIC PROCESSES 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

Single Nucleon transfer processes . . . . . . . . . . . . . . . . . . . . . . Single Nucleon transfers with 1 = 1 . . . . . . . . . . . . . . . . . . . . . Single Nucleon transfers with 1 ~ 2 . . . . . . . . . . . . . . . . . . . . . Two Nucleon transfer processes . . . . . . . . . . . . . . . . . . . . . . Inelastic Scattering Processes . . . . . . . . . . . . . . . . . . . . . . . . Exchange Effects-Knock-on Reactions . . . . . . . . . . . . . . . . . . . . Exchange Effects-Heavy Particle Stripping . . . . . . . . . . . . . . . . . .

4. DETERMINATION OF REDUCED STRIPPING AMPLITUDES . . . . . . . . . .

. . . . . . .

431 432 434 435 436 437 437 441

5. USE OF POLARIZED TARGETS AS A SPECTROSCOPIC PROBE IN NUCLEAR REACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

Introduction T h e availability o f p o l a r i z e d - i o n sources for use in n u c l e a r - r e a c t i o n studies has a l r e a d y been r e p o r t e d 1) as have a n u m b e r o f theoretical investigations which b e a r o n the i n f o r m a t i o n derivable f r o m m e a s u r e m e n t s using such b e a m s 2 - 5). W i t h recent technological a d v a n c e s that have led to high static m a g n e t i c fields a n d i m p r o v e d cryogenic techniques 6 - a ) it n o w a p p e a r s entirely feasible to p r e p a r e p o l a r i z e d targets for studies o f nuclear processes. It is therefore a p p r o p r i a t e to consider w h a t a d d i t i o n a l i n f o r m a t i o n there is to be g a i n e d b y the use o f such targets. As is well k n o w n , nuclear p o l a r i z a t i o n is c h a r a c t e r i z e d b y several p a r a m e t e r s , the exact n u m b e r d e p e n d i n g u p o n the spin involved. It might be expected therefore that with this extra c o m p l i c a t i o n , e x p e r i m e n t a l studies, b a s e d on p o l a r i z e d targets, should p r o v i d e m o r e i n f o r m a t i o n b e a r i n g on the s p i n - d e p e n d e n c e o f nuclear p h e n o m m e n a t h a n is o b t a i n e d b y the m o r e usual p o l a r i z a t i o n m e a s u r e m e n t using u n p o l a r i z e d targets. The latter m e a s u r e m e n t usually involves either m e a s u r i n g the a s y m m e t r y o f the a n g u l a r d i s t r i b u t i o n associated with p o l a r i z e d projectiles or detecting the pol a r i z a t i o n o f the r e a c t i o n p r o d u c t s f r o m reactions i n d u c e d by u n p o l a r i z e d projectiles t C o m m e n t s c o n c e r n i n g the use o f p o l a r i z e d targets in studies o f direct reactions have a l r e a d y been presented by Satchler 9). O u r present c o n c e r n is to extend these considerations for direct r e a c t i o n s keeping to a m i n i m a l n u m b e r o f detailed a s s u m p t i o n s c o n c e r n i n g either: (a) the n a t u r e o f the interactions between reaction p a r t i c i p a n t s in each c h a n n e l o r (b) the direct i n t e r a c t i o n giving rise to the direct reaction. W e shall also c o m m e n t b r i e f l y on m o r e general b i n a r y nuclear interactions. t Our convention is to label the members of the entrance channel as projectile and target while the members of the exit channel are designated as reaction product and residual nucleus. The reaction product is to be interpreted as the observable in the exit channel.

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L. J. B. G O L D F A R B

AND

D. A. BROMLEY

We might consider the following example: The N14(d, p)N 15 reactions leading to residual states in N 15 are examined, both target and projectile having spin one. In one case we prepare a polarized deuteron beam and observe the angular distribution of the protons and in the other case we polarize the N 14 target, so that it is characterized by the same polarization parameters, and again observe the angular distribution of the protons. The question arises whether the polarization effects are the same in both cases. We shall demonstrate that they are not (see also S:~tchler 9)); moreover these polarization effects highlight the reaction in entirely different ways. It might be mentioned that even if a reaction were to proceed through definite intermediate states, the effects of the polarization of the two members of the entrance channel, if they are of the same spin, would not, in general, be simply related. The same is true of the polarizations of the two members of the exit channel having common spins. One case of interest is the reaction d + d ~ H 3 + p, where the polarizations of the proton and triton are shown to differ lo, 11). Likewise the photodisintegration of deuterons leads to unlike polarizations of the emitted nucleons, provided that the contributing radiative transitions are of multipolarity greater than one 12). The remarks which follow apply first to any direct process, typified by the transfer of energy and angular momentum between the projectile and target; in sect. 5 we shall consider polarization effects in non-direct processes. We shall compare, for an unpolarized incident beam, the angular asymmetries associated with a polarized target to the state of polarization of the residual nucleus, using an unpolarized target. The latter quantity may be found, in part, through the measurement of the angular correlation of the ~-ray, emitted in the de-excitation of this residual nucleus. The correlation in this case reflects the different populations of its spin states, and therefore the state of polarization of the residual nucleus - - albeit not in as much detail as is desirable. Such correlation measurements are already precluded if either the residual nuclear state is stable or its spin is 0 or ½. In both the latter cases the angular correlation is isotropic. Barring such situations, if one knows the nuclear spin involved in the decay and the multipolarity of the ~-ray, measurement of the angular correlations provides a polarimeter for the polarization of the residual nucleus. We shall find that as long as the direct reaction is attributed to a unique value of j, the total angular-momentum transfer in a reaction, identical angular distributions of the reaction product obtain in two different experimental situations: (1) The target is polarized, this being characterized by the spin tensors pk~(a), defined in terms of a target-polarization direction, bearing angles 0 and ~b relative to a set of coordinate-axes which are chosen to specify the reaction. This choice we leave open for later convenience. (2) The target is unpolarized, but measurements are made of the direction of the de-excitation ~-ray, labelled by requiring time coincidence with the reaction product. This provides a polarimeter whose efficiency is denoted by ek,,(b), which is now expressed in terms of the direction of motion of the ~-ray. We shall demonstrate that the angular dependence of the asymmetry as reflected

TARGET POLARIZATION EFFECTS

411

in the reaction-product angular distribution associated with a given tensor pk,c(a) in case (1) is identical to that of the polarization of the residual nucleus in case (2) where this is associated with the same values of k and ~. The latter angular dependence will be obtained similarly by selection from the general expression for the triple angular correlation the coefficient of the spherical harmonic Y~(Or, gPr) appropriate to the deexcitation gamma radiation. Furthermore, the relative magnitudes of the sets of coefficients (of pk,,(a) in case (1) and of ~k~(b) in case(2)) for a given angle of observation of the reaction product are expressed solely in terms of the nuclear spins and the values o f j and k - - thus allowing for a determination o f j . As an example we might consider the high cross-section stripping reaction, B 1° (d, p)B 11. (% 6.76 MeV)B 11. The dominant ground state de-excitation (83 %) proceeds with the emission of a pure E2 radiation and consequently allows for the measurement of tensors e2~(7) and 84~({) through study of the gamma radiation angular distribution relative to the direction of the projectile. These are detectable respectively through measurement of the Y~*(O r, c~r) and Y~*(Or, (or) terms in this angular correlation where 0r and ~br specify the direction of the y-ray. Polarization of the B 1° target on the other hand, leads to tensors pk~(3) with k ranging from 0 to 6, and to contributions in the angular distributions of the emitted proton involving coefficients Y~,*(O, ?p) where 0 and q5 now specify the incident polarization direction. Provided that the stripping occurs with a unique value of j, the ratio of the coefficients ofp2~(3) and e2~(½) as well as of p4~(3) and ~4~(½) are constants in each case (they are simple functions of the values of k and j). Most striking is the independence of these ratios on the direction of the emitted proton. The direct transition is normally interpreted as proceeding by an l = 1 transfer, so that j < ~. We shall show that on such an assumption, only tensors with k < 3 should appear in the correlation. Thus, the appearance of a contribution of e4~(-~) provides direct evidence for a competing reaction mechanism. The main assumption herein is the applicability of the distorted-wave formalism 13). This treatment involves the introduction of distortions in each channel with the tacit assumption that at the instant of occurrence of the direct reaction (e.g.), the instant of transfer of the neutron in the (d, p) process) the internal states of excitation o f the projectile and nucleus, on the one hand, and of the reaction product and residual nucleus on the other hand, are the same as those viewed asymptotically. We require only that the distortions be independent of the target spin or of the spin of the residual nucleus. This is justified on the basis of existing elastic-scattering information which shows no such dependence. We emphasize here that the equivalence of the two angular distributions noted above requires no assumptions concerning the nature o f the projectile or the reaction products, the dependence of the distortion on their spins nor of the spin-dependence of the interaction responsible for the entire process as long as the total angular-momentum transfer is unique. The situation is very different, on the other hand, for unpolarized targets, if we compare the polarization of the reaction products with the asymmetry in their angular distribution when the in-

412

L. J. B. GOLDFARB AND D. A. BROMLEY

coming projectiles are polarized. Such comparisons show great sensitivity to the spin dependence of the distortions 3, 4) and, in fact, provide useful probes as to the nature of this dependence. We shall proceed herein as follows: We define the tensors representing target polarization in sect. 1. This is followed in sect. 2 by the derivation of the above-mentioned result involving tile comparison of angular distributions, while in sect. 3 we consider applications to various specific processes. Deuteron-stripping and pick-up processes and other single-nucleon transfer processes are first dealt with and categorized according to the orbital angular momentum l of the transferred nucleon. This is followed by a brief discussion of the simultaneous stripping of two nucleons and heavy particle stripping, followed finally by inelastic nucleon scattering. We also include in this section a discussion of exchange effects as evidenced in "knock-out" processes for example. Sect. 4 contains a discussion of the relative advantages of using polarized beams, targets, or both, with a view to completely determining the scattering amplitudes specifying the direct process. Although we are concerned primarily with direct interactions, several remarks can be made concerning the use of a polarized target for spectroscopic studies independent of the type of reaction mechanism involved. This is done in sect. 5. Notation

The angular momentum quantum numbers a~ and bfl refer to the target and residual nucleus, respectivel.~ 14). If there is a decay to a final nucleus, either by y-emission z

•~ REACr,O, ~ f P~oo~cT S" ~, i

x

z

plane

'~\

T, RGET\PO',,,Z,~,ON ,,~ls\ /

I

Fig. 1. D e s c r i p t i o n o f the geometry p e r t i n e n t to studies with p o l a r i z e d targets a nd a n g u l a r - c o r r e l a t i o n measurements.

TARGET POLARIZATION EFFECTS

413

to a state of lower excitation in the same nucleus or by subsequent particle emission to a different nucleos we introduce the quantum numbers c~ to relate to this final nucleus. As usual, Latin and Greek symbols will designate angular momenta and their components, respectively. The direct reaction is typified by the transfer of angular momentum j with component ( iv the target-projectile system. This is further described through the coupling o f ! and s to f o r m j . We shall present the results for stripping processes and near the end of sect. 2 indicate the necessary modifications for pick-up processes. This presentation also refers to scattering phenomena. Incoming and outgoing wave~ are handled by means of partial-wave expansions, involving the coupling c f 11 and s~ to form j l for the entrance channel, while 12 and s 2 combine to give j2 for tbe exit channel. The relevant coupling schemes are expressed through the following relations: a + j = b , j x ~ - - j 2 - t - j -~ 11-[-S1 -~ (12"[-$2)'~j,

j = l--I-s.

We shall use a coordinate system with the z-axi~ alovg the direction of the incidem projectile and the y-axis, noimal to tbe plane of the direct-reaction, viz., along n = (k 1 x k2)/lk ~ x k2[ , where kx and k 2 are the wave vectors associated with the projectile and reaction product. This coordinate system and the vectors of interest are indicated schematicall) in fig. 1. Reference is made only to the centre of mass system o f the reaction.

1. Target Polarization The state of polarization of the target, wbicb is of spin a, may be described through the use of a spin density-matrix with elements (actlp[aa') where we have not yet specified the quantization axis. It is convenient to construct statistical tensors 15, 16) pk~(a) defined in terms of the density matrix as follows: pk,(a) = ~, (aa[p[a~')(-1)a-='(a~(, a--~'lkK),

(1.1a)

for values of k ranging in integral steps from 0 to 2a. Because of the orthogonal properties of the vector-addition coefficient (a~, a-c('[ k~), there exists a similar inverse relation: ( a a l p l a a ' ) = ~, p k , ( a ) ( - 1)a-"(a~, a-a']kl¢).

(1.1b)

These statistical or spin tensors completely characterize the polarization of the target. In particular, if there is an axis of polarization which defines the spin system, and the axis of quantization coincides with this direction (as provided by a magnetic field, say), then the density matrix is diagonal. It then follows from (1.1 a) that the only nonvanishing spin tensors correspond to n = 0 and we shall here represent them by fiko(a). One of these, fioo(a), corresponds to the choice of normalization of the system, while the other 2a terms represent the state of polarization. The diagonal elements of

414

L. J. B. GOLDFARBAND D. A. BROMI.,EY

p specify the population of each spin-substate, 0t, and these populations t are denoted by q, so that (a~l~la~t') = q,6ca,.

(1.2a)

It is convenient to normalize the system so that ~q~=

iz

1.

(1.2b)

As a consequence ~ko(a) is given by 17ko(a) = ~a-x E (aot, kOlaot)q,,,

~t

(1.2c)

where we denote the statistical factor x / ( 2 x + l ) by ~. An alternative possibility is to express Pk0(a) in terms of the expectation value of the component of the spin operator along the symmetry axis, ( ~ ) as well as of powers of this operator, e.g., ( ~ ) . Here, for example, the first three tensors can be shown to take on values 17)

(1.2d) /52o(a ) = a _ l {

5

}-t-['3(~ 2) - a(a + 1)].

(2a - 1)a(a + 1)(2a + 3) The tensor pk,~(a), SO defined, is an irreducible tensor is). (It transforms as Y~* under rotations of the coordinate system, where Yff is the normalized spherical harmonic.) It therefore follows that if the polarization axis of the target bears angles 0 and ~bwith respect to the coordinate system of the reaction (see fig. 1) the spin tensor, expressed in terms of this coordinate system, is given by

pk~(a) = ~kO(a)~ko(qb, O, O)

(1.3a)

= ~kO(a)[c- l(4rc)½Yk~*(O, ~b) =/Tko(a){(k--[xl)!['(k +

Ixl)!]- 1}-,1-(__1)r6r, ['l pk[X[(COS0)e-IK,.

(1.3b) (1.3c)

In the above expressions, ~o(~b, k 0, 0) is the rotation matrix la) corresponding to the rotation of the coordinate axes from that with z-axis along the incident beam direction to that with z-axis along the target polarization axis, i.e., the rotation through Euler angles (4, 0, 0) while P~(cos 0) is the associated Legendre function of the first kind. (See 3ahnke and Emde 19).) Particular values of pk,,(a), when referred to the reaction coordinate system are t The tensor ~,0(a) is identical to the function G~.(a) o r G v appearing in ref. e) and is proportional to the parameter f~ (see for example ref. v)).

415

TARGET POLARIZATION EFFECTS

listed below: poo(=) =

p,o(a) =

cos o,

Pt + l(a) = -T-Pro(a)2 -~r sin 0 e =~#,

P2o(a) = .52o(a){~ cos 2 0 - ½ } ,

(1.4)

3 p2+t(a) = -T-P2o(a) ~ sin 0 cos 0 e ~:'÷, P2±2(a) =/~2o(a) 3_~ sin2 0 e ;2'÷ 2~/6 We shall find in the next section that, given a value of k, only the real or imaginary parts of Pk,,(a) contribute to the angular distribution according as k is even or odd. It therefore follows from (1.3c) that tensors of even k contribute only cos K~b terms while tensors of odd rank lead only to sin ~¢~bterms. This clearly allows experimental test for effects attributable to a given value o f k through an examination of the angular distributions. One might search for contributions involving cos k¢ or sin k¢, (i.e., when k = r), according to the parity o f k . It is interesting to note that in this case the dependence on 0 is proportional to a power of sin 0 so that the most useful value of 0 would be ½~; moreover when 0 = ½~zall terms vanish i f k - r is odd. We shall illustrate this by considering the measurement suggested previously in the case of the Bl°(d, p)B 11 reactions. Here, because the target spin is 3, it follows that the specification of target polarization leads to spin tensors with k =< 6. The most instructive experimental measurements will be those of the azimuthal angular distribution of the outgoing protons at any fixed reaction angle 0'. Fourier analysis of the e-dependence of this angular distribution will exhibit the cos i¢¢ and sin K~bterms explicitly. From the relative magnitudes of the Fourier coefficients as we shalldemonstrate, it is possible to establish not only a lower bound on thej which is effective in producing the corresponding K but also the relative importance of the individualj values. It is clear however that for I¢ < 5, a single measurement will not suffice to identify the corresponding value of k; it will be necessary in such cases to repeat the azimuthal distribution measurements under different conditions of target polarization involving changes either in the polarization orientation or magnitude (e.g. through change in the magnetic field determining the polarization). In a particular case it may be possible to identify the responsible k value by noting whether elimination of particular Pk,, terms affects the Fourier component under examination, 2. Correlations between Target and Residual Nucleus Polarization

2.1. POLARIZED TARGETS Here we first examine the predicted angular distribution of the reaction-product, assuming that the target nucleus is polarized and that the incident beam is unpolariz-

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L . J . B . GOLDFARB AND D. A. BROMLEY

ed. The probability of detection of the reaction product with k 2 relative to the incident beam k 1 is expressed in general by the following:

W(k~, kz) = Tr (pie),

(2.1)

where pf is the final density matrix and e is the efficiency matrix (Coester and Jauch 16)) for detection of the final system. The detectors are assumed to be insensitive to polarization so that the elements of the efficiency matrix are simply given as ( bfls2 tr2lelbfl's2 a~) -- cSaa,6o2o,, •

(2.2)

The normalization differs from that adopted for the entrance channel in (1.2b) since here one averages over initial states and sums over final states. We note that the expression (2.1)is proportional to the angular distribution of the reaction product. The matrix, pf, is reJated to the initial density matrix, pi, through the direct-reaction operator R, so that pf = RpiR t. (2.3) The matrix element of R, i.e., the transition amplitude, is calculated using the distorted-wave formalism. For the present analysis, we merely assume that it may be reduced as follows:

(k2bflSEtr2lR[klao~sltrl) = ~(ao~,j(lbfl)(k2s2~2blXJ~t(O')lktsltrxa), J~

(2.4)

where 0' is the angle of observation of the reaction product. We refer to the coefficient of the vector-addition coefficient as the reduced amplitude t, introducing as a shorthand notation for this quantity, the symbol Zj°~2~r~ ~

(k2s2ff2

blXJgt(O')lkl sl at a).

The main point of this reduction is the independence of the reduced amplitude on the components of a and b. This is a valid expansion to the extent that the optical-model potentials used to describe the channel distortion are independent of the spins of the target or residual nucleus. To illustrate the validity of (2.4), we consider a general stripping process. The stripping amplitude is given by ( k2 b fls2 t r 2 [ R l k l a~sl trl ) - f d x d r l dr2 0t(x, rl)012o2(k2 r2)V(rl--r2, trl, ¢r2)~b,~(x)O.... (kl, r l , r2),

(2.5a)

where x, r 1 and r 2 refer to the coordinates of the target nucleus (treated as a core), the captured particle and the reaction product, respectively. Centre-of-mass effects are neglected for reasons of simplicity. We are here using the distorted-wave Born t We have in mind a stripping process or inelastic scattering process corresponding to a+j ~ b. The necessary modifications for the pick-up process where a = b-kj are given near the end o f the section.

TAKGBT POLAKIZATION EFFECTS

417

approximation so that the interaction in (2.5a) is given by V ( r t - r 2 , at, a2), which would be the neutron-proton interaction in the case of the (d, p) process, with rt trl referring to the neutron and r 2 a2 to the proton. The eigenfunctions ~,,~, and ~km~,~ are optical-model eigenfunctions which include the effects of distortion in entrance and exit channels and which have appropriate asymptotic behaviour. Upon integration over x, the amplitude reduces, as in (2.4) with the reduced amplitude given as

z2

=Z

Is

× f ~ , , 2 ( k 2 , r2)~tm~(rl)V(rx- r2, tr,, tr2)~k,,,,(kx, r,, r2)dr x dr 2 .

(2.5b)

The quantity Oj(~s)(ba) which is usually referred to as the reduced-width amplitude for stripping, is independent of ~ and ft. Another example occurs in the inelastic scattering of nucleons. We might consider scattering off a core with coordinates x and an outer nucleon with coordinates rt. We let r refer to the coordinates of the projectile. The amplitude, as a consequence of the distorted-wave Born approximation, involves a potential V(rt, r), acting between distorted states of the projectile. This potential acts on the outer nucleon and raises the nucleus to an excited state. The amplitude is written as (k2 b fls2 azlRIkl acts1 a l )

=

fdxdrxdr~,~B(x,

rl)d/~2,,(k2r)V(rxr)~bo,(x, rl)~s,,,(kxr ),

(2.6a)

where ~b,~,, and qJ,2-, are the distorted wave functions of the projectile and scattered nucleon. By expanding V(r x, r) in the form

V(r I , r) = ~ X~[,,)(r)x~t,,)(ra)Vj(~)(rt, r),

(2.68)

jls~

where Z~(z~)(r) is a tensor in coordinate and spin space, we again find using Racah algebra a) a reduction into the form given in (2.4). In this section we are concerned with the effect of polarized targets. The initial density matrix (2.3) for an unpolarized projectile beam incident on a polarized target, has elements given by

(aoq sxcrxlP'lao:', s x ~ > = (~x)-2~,1¢., Y~Pk,,(a)(--1)a-"'(a o~, a--~'lkx),

(2.7)

kx

if we use (1.1b) and are consistent with the normalization used in (1.2b) for our description of the spin states of the projectile. It therefore follows from equations (2.1), (2.4) and (2.7) that

W(kl, k2) = (~l) -2 E

jj'~"

E(a~,j¢lbfl)

ct~'fl

O'2~rlk/~

× (aog, j' ~'l bfl)(- 1)a-"(a~t, a-o~ ' [kx)[pk,,(a)Zj;o2-,Zj,;, ,,st* ]

(2.8a)

418

L. J. B. G O L D F A R B A N D D . A. B R O M L E Y

= 32(s,) -z Z (--1)"-b+~W(aajj '; k b ) ( j ( , j ' - ( ' l k - ~ : ) J~"~' 0"20"1 kl¢ ~r2~r1

×[pk,,(a)Zj~

(2.8a)

£r2~rI *

Z j, c ].

The angular distribution, which is proportional to W(k~, k2), may be shown to involve only the real part of [pk,,(a)Zj;~ , Zj.~, ~2~,*], as is expected. To see this, we merely interchange primed and unprimed symbols, replace r by - K and make use of some vector-addition identities. The unpolarized angular distribution, i.e., the angular distribution arising from an unpolarized projectile incident on an unpolarized target is denoted by (da/dI2) o. This obtains when k = x = 0, in which case p k x ( a ) = (a)-26kO 6K0. The expression (da]dI2)o differs from W(k~, k2) by a proportionality factor which depends on the normalization adopted for the description of incoming and outgoing waves, this factor being a function of the reaction kinematics, i.e. masses, linear momenta, etc. Whatever this factor may be, the angular distribution (da/dI2) PT to be associated with polarized targets, must involve the same proportionality factor and may be written as

•

=

(-0

o{l+[x(0')]

~2UIK

k~O

x W(aajj' ; kb)(j(,j'('lk-~c) Re {pk~(a; 0 )zj . . . . .(0)zr . . . . , *(0)}}, '

(2.8b)

where x(o')

= Z

A - , Zj; .2°, (0)r , 2, ](J)

(2.8c)

j~71a2

and

oc X(O').

L d f 2 _]o

(2.8d)

We shall now demonstrate that the measurement of the angular distribution of the reaction product (da/df2) Px provides a measure of the polarization state of the target nucleus. Just as we defined the polarization tensor in (1.1a), we may similarly de fine the efficiency tensor for detecting the target polarization e,,,(a) in terms of the efficiency matrix e:

ek,,(a) = ~
(2.9)

cruz'

The probability of detection of a system with polarization described by (a~xlp[a~') through a detector with an efficiency matrix,
W(a) = ~
(2.10b)

TARGET POLARIZATION EFFECTS

419

This reduces by (1.1b) and (2.9) to

W(a) = E Pk~(a)e*~(a) = Z pk~(a)(- 1)~ek-~(a) ks

(2.10b)

kx

where" we have used the Hermitian property of the tensor eke(a), viz., ~*~(a) = (-)~ek-~(a).

(2.10c)

The quantity W(a) must be proportional to the angular distribution of the reaction product (da/df2) PT, which is given by (2.8b). Hence a comparison of (2.10b) and (2.8b) allows for the determination of the efficiency term ~**(a) directly from the experimentally determined angular distribution. This is accomplished by reading off the coefficient of pk~(a) in the former expression. We find

~k~(a) = (X) -l~z Z (-1)"-b+;'W(aajj '; kb)(j~,J '-r'lt'~'~7~2~'w~2~l*

(2.11a)

j j ' ~" Crib2

where the normalization is such that Zoo(a) = a,

(2.11b)

and it is to be understood that (da/dI2) Pr is given in terms of these tensors as follows:

Earl

K

The measurement of the angular distribution of the reaction product (da/dQ) PT thus enables one, in principle, to evaluate the efficiency tensors eke(a), directly, assuming pk,~(a) is known. These tensors, eke(a), determine the efficiency of the angular distribution measurement as a polarimeter in obtaining information concerning the target polarization and clearly depend on 0', (the reaction-product observation angle) through the reduced amplitudes Z~d'~(O'). It is interesting that the efficiency tensors involve an incoherent summation over projectile and reaction-product spins with components a~ and az, but not over j and ~. This is in contrast to the situation where in one case the polarization of the reactionproduct is measured, and the incoherent summation is over j, ~, and al, and in the other case where measurement is made of the asymmetries in the angular distribution associated with polarized projectiles (da/df2) PP where the incoherent summation is over j, ~, and a 2. As an example, we refcr to the (d, p) process. In the expression for the outgoing proton polarization there is an incoherent summation over j, if, and the component of deuteron spin, whereas with a polarized target the asymmetry in the proton angular distribution involves an incoherent summation over the components of spin of the proton and deuteron, but not o v e r j and ~3). 2.2. T H E

POLARIZATION

OF THE

RESIDUAL

NUCLEUS

Let us consider an alternate possible measurement. Here both projectile and target are unpolarized, but we measure the polarization state of the residual nucleus. Thus,

420

L.J.B.

GOLDFARB AND D. A. BROMLEY

instead of (2.2) we have

= 6~,,~ ~ eke(b)(--1)b-#(bp ', b-fl'lkx).

(2.12)

kt¢

The manner of detection of the polarization of the residual nucleus is not our immediate concern. Suffice it to say that this information is contained in the expression for eke(b). We shall be concerned with evaluating the polarization tensor of the residual nucleus pk,,(b) and shall find that this is obtained by reading off the coefficient of e*~(b) in an angular distribution; however, it will be unnecessary to specify what e*,(b) is, for the present. The initial density matrix is simply given in this case for unpolarized target and projectile, as

= (a~x)-26o~,6,,~,,.

(2.13)

We now examine the angular dependence of the polarization of the residual nucleus as labelled by the angular distribution (da/dO) P~ of the reaction product. The polarization of the residual nucleus, for any given 0' will be determined by the triple correlation or other measurement which constitutes the polarimeter of efficiency %(b). Using (2.1), (2.3), (2.4), (2.12) and (2.13), we find, assuming a sharp state for the residual nucleus

W ( k l , k2) = 32(ast)-2 E (-1)°-b+~W(bbjj' ; k a ) ( - l Y +j''k JJ' W 0201kK

x(j(,j'-~'lk-x)Re

,,2,1Zj.~ ¢, 1 " }. (2.14a) {e,~(b)Zj¢

The corresponding angular distribution (da/dI2) rR is normalized, so that it reduces to (dtr/dt2)o, the basic unpolarized angular distribution, when k = x = 0 and ek,,(b ) = b3kotS~o.We then find da(0, ~b,; 0')] PR

I

= Vda(O')] [ 1 + [X(0')] -1 2 ( - 1)°-b +¢W( b bjj'; ka)( j(, j' - ~'l k - x) k d~ J o jj'~' 0"201K k#O

x(_l)J+J'-kRe{ek~(b; O~dp~)Zj; a2¢,(O)Zj, , *v,, , c °(0)}],

(2.14b)

where (da/df2)o is proportional to X(O') being given by the same expression as appears in (2.8b), and Or and ~b~ are angles specified through the polarimeter. The measurement of the reaction-product angular distribution (da/dI2) px provides a measure of the probability that the polarization of the residual nucleus, represented by pk,,(b) is measured by a detection process with efficiency characterized by eke(b). Then (da/dO) pR c~m be written as

TARGET POLARIZATION EFFECTS

4~1

7

[d_~lZ p.(b)ek.(b) [d~] ~,(_l).pk_.(b)ek.(b)" 0 k~¢

(2.15)

0 k~c

It therefore follows, on comparison with (2.14b) that the polarization of the residual nucleus is expressed as follows:

pk,(b) = S -t ~ (-1)°-b+¢'W(bbjj'; ka)(j~j'-~'lkx)(-1)J+J'-kZ~o.tZ~,2;°, '*,

(2.16)

O'2O"1

with the normalization such that Poo(b) --- (t;) -1 Comparison of (2.16) with (2.1 la) shows that ifj is unique, there exists a polarization correlation between pt~(b) and ek~(a), given by

W(bbjj; ka) (_ kb)

Pkjc(b) ~" ~ 2 W ( a a j j ;

1)k+2J~kx(a)"

(2.17)

We have thus established a direct correlation between the state of the polarization of the residual nucleus pkr(b), (as measured through a detection process with some known efficiency eke(b)) and the efficiency ek,(a) of an angular-distribution measurement in a direct-reaction process (the latter serving as a polarimeter for target polarization Pk,c(a)). We re-emphasize here that the quantities pkK(b) and ekR(a) are both to be determined as coefficients in angular-distribution measurements. We begin by polarizing a target in known fashion thus specifying explicitly pk.(a). We then measure (da/dO) PT and extract the e~,(a) as the coefficients of the individual pkr(a) terms in this angular distribution. We then carry out a second experiment in which we assume the efficiency e,kr(b ) is known and from the angular distribution of the reaction product (da/dQ) vR determine pk,~(b) as the coefficients of the individual known ek,~(b) terms. This relationship may be expressed simply by pk,(b, 0') ocek,(a, 0') with the proportionality factor as given by (2.17). This proportionality factor can be rewritten as a ratio of rational polynomials of the k th order in [b(b+ 1 ) + j ( j + 1 ) - a ( a + 1)] in one case and [a(a+ 1 ) + j ( j + 1 ) b(b+ 1)] in the other case 2o); however since these expressions become very complicated with increasing k, it is preferable to use the tabulations of the Racah coefficients themselves. Two classes of experiments suggest themselves. In the first, the target polarization is maintained (fixed 0 and ~b) and one examines the angular dependence of e*~(a, 0') with varying 0'., as given by the coefficient of pk~(a) in the expression (da/df2) PT. This is then compared to the angular dependence ofp~(b, 0') with varying 0' as reflected by the coefficient of eke(b) appropriate to the detection process used in the expression (dtr/df2) PR. In the second class of experiments 0' is kept fixed (for the maximum cross

422

L.J.B.

GOLDFARB AND D. A. BROMLEY

section (da(O')/df2)o, say) and either the target polarization is varied, or the polarimeter efficiency is varied (by measuring the angular correlation of?,rays, for example). The proportionality factor which is measured by the latter study, serves to determine the value of j. It is to be noted that both tensors in (2.17) refer to the same set of coordinate axes. A common method of detecting the polarization of the residual nucleus, i.e., determening pk~(b) is to measure the angular distribution of the y-rays arising in the de-excitation of the residual nucleus, provided that such a decay takes place, f f we assume that the emission of ?-rays of competing multipolarities L and L' is allowed, leading to a sharp nuclear state of spin, c, we find 24)

eke(b) =

(4n)-1• ~ (--

1)*+Z+VFk(LE; cb)~o(q~, 0~, O)(clZllb>(clEIIb>,

(2.18a)

LL'

where

Fk(LE; cb) = (-lf-b-xfi£f_/(L1E-1lkO)W(bbLE; kc),

(2.18b)

and (clLIIb> is the reduced matrix element (which is real) for emission of a y-ray of multipolarity L with a nuclear transition between spins b and c. I f the normalization is such that eoo(b) = $, we require that the reduced matrix elements satisfy t

Z ~ = L

4rL

(2.18c)

The angles 0~ and ~b~ in (2.18a) are the polar and azimuthal angles, respectively. that the emitted ~-ray makes with respect to the coordinate axes used for the description of the direct reaction (see fig. 1). Insertion of (2.18a) into (2.14b) leads to an expression appropriate to the triple angular correlation involving the directions of the incident beam, the reaction products, typified by (0', 0) and the ?-ray direction, typified by (Or, Cr) so that

W(O,,qS,;O')=

~

X-a3(4n) - t 0

E

(-1)'

jj'~'OlO2 kr>O

× ~/(k-I~cl)!W(bbjj; ka)(jl~, j'-- ~'Iktg)FR(LE; cb) × [(k+ Ixl)!]-~P~(cos O~)(2-O~o) x Re {zj~ . . . . .zj,~, ~01.} cos ~q~r,

(2.19)

where

t = j+j'+L+L'+a-b+?,. In deriving (2.19), we have used the fact that k is even, by virtue of the unique parity of the ?-radiation and also the relation a) Zi_ - , 2~- ~, = (--1) j+s'+s2-~-~l-°2 n a rto~i~7~2~1, (2.20) * This is precisely the condition that angular correlations be expressed in standard form x4), which obtains for a cascade decay involving the emission of two 7-rays when the angular correlation is of the form W(O)= ~ AkPk(cos0) with Ao = 1. In such a case, each of the emission matrix elements satisfies (2.18c).

TARGET POLARIZATION EFFECTS

423

where n°n~ denotes the product of the parities of the target and residual nuclei. This relation is also a consequence of parity conservation. It may be used, together with the fact that the spin tensors are Hermitian (2.10c) to demonstrate *) that as a result of (2.11) and (2.1b)

Pk"(b)

= (-- 1)kP**(b) = (-- 1)k+*Pk-*(b)' ~k~(a) = (-- 1)'~*~(a) = (-- 1)k+'ek_~(a).

(2.21)

It therefore follows 4) as noted in the introduction that according as k is even or odd, the real or imaginary parts of the spin tensors in (2.11a) and (2.16) contribute to what is observed. As noted in the introduction cos xq~ angular distributions are associated with even values of k, while odd values of k give rise to sin x~b distributions. Equation (2.19) is a case in point since k must be even. Furthermore, the only nonvanishing contributions of Pko are real and are associated with even k. The polarization correlation (2.17) can hold even i f j is not unique. It is merely necessary that k = 2imam,j=a~ being the largest value of j, since k arises from the vector summation o f j and j'. Unfortunately, this correlation is inapplicable for deuteronstripping processes if two values o f j contribute, since j is half integral and as a consequence the pertinent value of k would be odd. The polarization correlation does apply, in this restricted sense, to double stripping and to inelastic scattering generally (either of nucleons or heavier ions) since j is integral for these processes. It should be noted however that the pk,,(b) are, in principle, measurable for both even and odd values of k through examination of the scattering or reaction induced by the residual nucleus provided that the scattering or reaction is polarization-sensitive. Such measurements become feasible, particularly in the case of heavy ion projectiles where the linear momenta of the residual nuclei can be appreciable. Two types of experiments are suggested. In the first, heavy ions are incident on polarized targets; this situation has already been considered herein. In the second, the roles of projectile and target are interchanged, i.e., polarized heavy ions (e.g., Li 6, N 14, F 19 or Cl as) are incident on unpolarized targets. The case of a deuteron target is of particular interest. It is clear that the formalism given here is still valid since the centre of mass descriptions of the Lit(d, p)Li 7 and the d(Li 6, p)Li 7 reactions for example, are identical. The correlation (2.17) is seen to depend critically on the validity of the reduction, as expressed in (2.4). So long as the entrance and exit-channel distortions are insensitive to the target and residual nuclear spin, respectively, this reduction is valid. This precludes the dependence of the optical-model potentials on nuclear spins and strongly suggests the importance of optical model studies with polarized targets. It is clear that if there is this dependence on nuclear spin, the elastic-scattering experiments should show different results if the targets are polarized or unpolarized. Studies based either on the elastic scattering of polarized ions by unpolarized targets, or of unpolarized projectiles on polarized targets will be crucial in establishing the validity of most of the conclusions presented here.

424

L. J. B. GOLDFARB AND D. A. BROMLEY

At first sight it seems strange that the polarization correlation between pkx($2), the polarization tensor for the reaction product associated with no polarization in the incident channel, and ekK(sx), the asymmetry exhibited by the process arising from Pk,(Sl) is established 21) only if specific assumptions are made concerning the spindependence of the optical-model potentials. Should there be any distinguishing feature of the process as referred to the centre of mass system of the reaction? It is clear that the distinguishing element is the nonsymmetric treatment of the spin-dependence of the optical-model interaction. Even if there is a dependence on the spin of the target or residual nucleus, the polarization should differ for projectile and reaction product as compared to the target and residual nucleus. Only if the reaction participants are identical in either entrance or exit channel will there be a symmetical situation. 2.3. LIMITATIONS ON POLARIZATION MAGNITUDES We restrict ourselves to unique values o f j and define the statistical tensors for the transferred radiation, having angular momentum, j, as follows:

tk,(j ) = [ ~ [Z~'12] -1 ~ (j(Ij--(IkK)(--1)J-¢'~ ~o2~

~"

Z ~ ¢ ' Z ~ : 1.,

(2.22a)

~2~

so that in the particular case where k -- 0, where no information is available concerning the angular momentum sub-states of the transferred radiation,

too(J) = ( j ) - l .

(2.22b)

eke(a) = a2( - 1)~-b+ JW(aajj; kb)j2tk,(j),

(2.23a)

pk~(b) = (-- 1)°-b+k-JW(bbjj; ka))2tk,(j).

(2.23b)

We then have

It is seen then that for a unique value of j, both the tensors Pk,(a) and eke(b) show a simple factorization into a statistical tensor characterizing the state of polarization of the transferred radiation and a decoupling factor which arises from the coupling scheme: a + j = b. Indeed, if we consider quite generally various j-values, and we compare the polarization state of the residual nucleus to that of the target nucleus, we need refer to the projectile-reaction product system only in the sense that it provides us with a statistical tensor p,,(jj'), which we may define as follows:

p,~(jj') = a2~-2[X(O')] - 1 ~ ( - 1 ) J ' - C ( j ( , j . .-.~. . . .Ikx)Zj¢ .. Zj, c ~. .

(2.22c)

~'a2al

The polarization tensor pk~b(b) can then be reduced (14) as follows:

p,Jb) ---

j'

(ko o,

kaKakK

i J"

(i

ab

k~"/ b}

(2.22d)

TARGET P O L A R I Z A T I O N EFFECTS

49.5

If we assume Xo = ko = 0, we find (2.22d) reduces to (2.16). On the other hand, if we evaluate (2.22d) for x o = k~ = 0 and find the coefficient of pk...(a), this gives us 8k.,.(a) as in (2.11a). Aside from a normalization factor, tk,(j) is equivalent to Pk,(J,J). It thus follows that (2.22d) provides an extremely succinct characterization of measurements involving nuclear polarization in either or both the entrance and exit channels. Polarization correlations of the type covered by (2.22d) represent situations of somewhat greater complexity than are discussed in this paper; we shall defer detailed discussion of the implications o f (2.22d) to a later publication where we shall discuss polarization correlations involving not only the nuclear polarization but also the polarization of the projectile and reaction product. Let us now return to the case where the j-value is unique, and make applications of (2.23a) and (2.23b). We find for b = 0 that

ek~(a) = ( -

1)ka2tkr(a),

(2.24a)

but since our normalization is such that too(a ) = (d)-1 and Coo(a) = (a), it is clear that the factor 82 on the right-hand side of (2.24a) arises merely as a normalizing factor - - a matter of convention. Therefore, the asymmetry in the angular distribution associated with the target polarization pk~(a) provides a direct measure of the polarization state of the transferred radiation. Similarl), if a = 0, we find

pk~(b) = tk~(b),

(2.24b)

so that the polarization tensor of the residual nucleus, as revealed by a measurement of the angular correlation of de-excitation v-rays, say, is equivalent to the statistical tensor of the transferred radiation. The decoupling factor in both of the above cases is unity; in general, the decoupling factor is" less than unity and therefore it serves to attenuate the tensor tk~(j ). The tk,(j) terms are bounded; for example, if every amplitude Z ~ ~' vanishes except for j = ( this would correspond to the situation where the transferred radiation is fully pect to the quantization axis. Given the upper bounds for tk,(j). polarized wit r upper bounds for the magnitudes of both 8,,(a) and pk~(b) which we then ar~ and j, provided that the latter quantity is unique. Simple expresare funct" b r the upper bounds to tk~(j); however, since j is arbitrary, it is sions ca ,~eto show the explicit dependence of these tensors on the expecinstru~ agular momentum operators. We list the expressions for tk~(j) tatio~

~. 17)):

fl,%r ,-

, o ( j ) = ) -1,

%&

rio(j)-

~/3(Jz>

)x/j(j + 1)' t~±~(j) = Tx/3(Jx+ iJ,) ,

j~/2j(j + 1)

(2.23b)

426

L. J. B. GOLDFARB AND D. A. BROMLEY

t2o(J) =

~/513--j(j+ 1)] )x/i2j - l)j(j + l)(2j + 3)'

(2.23b)

t2+t(J) = ~ x/]5[<(Jx ~ ijr)j,+Jz(Jx-T-ij,)>], ]x/(2j- l)j(2j + 2)(2j+ 3) t2+2(j) = x / ~ < ( j x - T - ijr)(jx~ ij,)> j~/(2j-- 1)j(2j + 2)(2j + 3)' where we use the coordinate system appropriate for the description of the reaction with the axis of quantization along the z direction. These formulae enable one to find upper bounds for the tensors. Thus, ['tto(J)]ma x -- 4 3

j

[-t2o(J)']max

,/i31/ J

v;

J

(2.26a)

(2.i- 1)j

(2.26b)

+1'

[' (j+1)(2j+3) '

except for the case when j = 1 when the latter expression must be multiplied by 2. Expressions for the upper bounds of arbitrary tensors may be found in a similar fas hion. The decoupling factor does not show a simple algebraic form, although it is simple enough to evaluate for any particular example. For the purpose of our present discussion, we restrict ourselves to several simple situations. First, we deal with k = 1. Here, we know that it is only the component of polarization normal to the reaction plane, the y-component, which gives rise to any asymmetry. Likewise the polarization of the residual nucleus lies along the y-axis. We define the vector polarization for a particle of spin j by P(j) given by 1 J

P ( j ) - - .

(2.27a)

In the same way, we define % , ( j ) in terms of tile mean values of the angular momentum operators, ~ as measured by a polarimeter with efficiency denoted by e, so that, for example, a = Tr (ej).

We thus write

~to(J)= x/3)(j[J+ l])-½a, e,+,(j) = ~X/3j(2j[j+l])-½, etc., ~oo(J) = ),

and define the polarization-efficiencyvector, pa(j), such that /~(j) __ j - l < j > d .

(2.25b)

TARGET POLARIZATION EFFECTS

49~7

As has been previously discussed, only Im [811(a)] gives rise to an asymmetry in the angular distribution, and in terms of the quantities just defined, we have Im ['ell(a)] = a V . 3a .. P~y(a). r 2 ( a + 1)

(2.28)

Referring now to the imaginary part of ek~(a), as given in (2.23a), we find for the special case k = 1, P~

a)V j a+l W(aajj; kb)(-1)a-~+JPy(j). '

(2.29)

a

j-f-1

Noticing that the upper bounds for Py and pd are both unity, we have

P~y < a.~VjJ+ 1 a+--~l a [w(aajj; lb)l.

(2.30)

Now since

[W(aajj; bb)[= ( a ( 2 a +

1)(2a + 2)j(2j+

1)(2j+2))-½1b(b+

1 ) - a ( a + 1 ) - j ( j + 1)[, 42.31)

it follows that

l~y < [b(b+ 1 ) - a(a + 1 ) - j ( j + 1)]

(2.32)

2a(j+l) As an illustration we considerj = ½ and ½for the special cases b = a+½ and b = a -½. We find for j = ½ P~y < ~

or

ka+l, a

(2.33a)

according as b = a+½ or a - ½ , while i f j = ½ pd < y

:

3--a 5a -

or

a+4 - - , 5a

(2.33b)

according to the same possibilities for b. The latter relation presents the remarkable feature that i f a = 3, b = -~and k = ½, there would be a selection rule operating against an asymmetry (because W(3 3 { ½; 1 ~) vanishes.) If we neglect target polarization of rank greater than k = 1, we can write the angular distribution associated with target polarization Py(a) as follows: da_ dl2

F d q VI+ 3a Py(a)l~y(a)l Ld~2Jo k a+ 1

"

(2.34)

Satchler 2o) makes the point that the coefficient of Py(a) can be near 1 for stripping processes, thus pointing to the efficacy of target polarization as a probe in the study of such processes. We note that f o r j = ½, the coefficient of Py(a) -< 1 for b = a - ½ , while for b = a+½ it never exceeds a/(a+ 1). On the other hand i f j = ½, the coefficient can be greater than 1, e.g. when a = 1, b = ½, where it is equal to ~.

428

L.J.B.

GOLDFARB AND D. A. BROMLEY

We indicate the corresponding results for the polarization of the residual nucleus corresponding to k = 1. The analogous relation to (2.29) is b+l Py(b) = ~JV j J 1 -b W(bbjj; ka)(--1)°-b+k-Jpy(j).

(2.35a)

The upper bound to the I~olarization of the residual nucleus, is thus found to be _/_

P,(b) =< .~Vj

j 1 b+lb IW(bbjj;

la)[.

(2.35b)

The particular case o f j = ½ yields b+l

Py(b) < ~ -

or

b

],

(2.36a)

according as b = a+½ or a - ½ , while for j = ½

Py(b) =< --b+4

or

--[b-31, 5b

5b

(2.36b)

according to the same possibilities for b. Given the forms of tk~(j), the analysis is easily extended to higher values of k. We shall not go into more detail here, but just remark that i f j = a, both tensors ek~ and tkr , refer to the same angular momentum so that the decoupling factor (_),,-b+j W(aaaa; kb) gives a true measure of the efficiency of the process as a polarimeter for target polarization pk,,(a). This decoupling factor represents the fraction of the maximum value attainable by ek,,(a) m this process, as compared to that obtained with the most effective polarlmeter. 2.4. P I C K - U P P R O C E S S E S

The above results apply with little amendment to pick-up processes. For these, the reduction of the transition amplitude may be made as follows:

(k2 bfls2<721Rlkl a~tsxax) - ~

(actl bfl, j ( ) ( k

2 s 2 tr 2

bl YJ¢lkl sl ~rla)

X

- db -~ ~ ( - 1)°-b-:(bfllact, JO ~ 1 "

(2.8e)

X

This has the effect of replacing (2.8a) by

W(kl,

(--1)a-b+¢(--1)~W(aajj; kb)

k2) = t~2~-2 ~. j j "~'kx

(2.8f) Re {Pk.( a ) Y.,j¢ , , , VJ'g'*~ -~,.

x (j(,j'-~'lk-x) SO that d,

d~

_

[do] [1+Y-'a2X(-1)°-b+'+"W(aajj';kb) d-~ o

~o

x(j~j'-~lk-x)

Re

-J¢ Y~2,, -J'¢'* }], (Pk,,(a)Yd2,,

(2.8g)

TARGET POLAIUZATION EFFIECTS

429

with Y - - ~ I3-xYJ~,,,I2,

(2.8h)

j~¢1¢2

and (dcr/dl2)o is proportional to Y as in (2.8d). The analogue to (2.11) involves multiplying one of the sides of the equation by ( - ) ~ and changing Z to F. The same prescription applies to (2.14b), (2.16) and (2.19). It is therefore clear that the polarization correlation (2.17) is again valid for pick-up processes. 2.5. S P I N I N D E P E N D E N T

DISTORTIONS

For the sake of completeness we add remarks applicable where no spin-dependent distortion is involved. The reduced amplitude Z~ 2"1 is factorized 4) as follows: Zj~¢' ~- (k2s2.tr 2 blXJg*lkl

sl trl a) ---- E (12, salj~)(s 2 a2 strlsx al)Z(,,)jx.

(2.37)

13.s#

With a little Racah algebra, we find the efficiency tensor for measurement of target polarization, e~,,(a), as found by the asymmetries arising from a polarized target (ref. 2.11), is given by ek,(a) = [X'] -182 E s-2ffW(ll'jj '; ks)W(aajj'; kb) jj'il's

( - 1)°-b+j+j'-~+k+'' ~, ( - 1)"-a'(12, l'-;t'[kx)Z(,,)jxZ(*T)j,x,,

(2.38a)

2,1.'

where X' = ~ I(~l)" IZ(,,)ja[2,

(2.38b)

j~.ls

and d~

OC

X'.

The correlation (2.17) between pk,~(b) and eke(a) is still valid since it encompasses arbitrary values of Z ~ ~1 e.g., that given in (2.37). One of the main points to be brought out by (2.38a) is that k is bounded by 2/=a, instead of by 2j=, x, as in (2.8a) for example, where we allow for spin-dependent distortion; this implies that asymmetries involving k > 2lm.xmight then be attributed to such distortion effects. This again emphasizes the importance of having high-spin polarized targets such that k values > 2/=ax are permitted unless precluded by spin-dependent distortions. The example Bl°(d, p)Bt1*(7, 6.76 MeV)B 11, noted earlier, with lm~ = 1, is a case in point. It will clearly be desirable to examine a wide range of such situations to establish, if possible, the systematic behaviour of the spin-dependent distortions as functions of both energy and atomic number.

430

L. J. B. GOLDFARB AND D. A. BROMLEY

Also, as pointed out by Satchler 9), the effects of the target polarization tensor eke(a) are directly proportional to the polarization of the reaction product pk~(S2) for deuteron stripping processes since both tensors have the same angular distribution, as revealed by the terms involving 2 and 2'. This is true, provided that we make the assumption that a single value of l is operative and factorize Z(ts)ja in the form (2.39a)

Z(ls)j2 ~ O(ls)jZ.12.

Expressions similar to (2.38a) may be found for pk~(b), so that using the fact that in this approximation, Pk~(S2) is proportional to ek~(Sx) (see ref. 21)) it follows that all the tensors, pk~(b), eke(a), Pk~(S2) and ek~(Sl) are mutually proportional. This leads to few useful results in most cases since either sl or s 2 is limited to ½ which therefore restricts k < 1. Furthermore, for the case of deuteron stripping, polarized deuterons with k = 2, give rise to no angular asymmetries if the targets are unpolarized. On the other hand, the polarization correlation between pk,(b) and eke(a) ((2.17)) holds even ifjis not unique, provided that/is unique. In this case, one finds pkK(b) = ( a ) - 2

7' = ~ ( j ) - 2 ( _ jj's

~ ¢ ~ : 1 ekK(a) '

1)a-b-~j),W(bbjj,; ka)W(lljj'; ks)Ot,~)jOtt~)j,,

~b = Z (,~)--2(__ 1 ) a - b - s + j + j , + k W ( a a j y ;

(2.39b)

kb)j)'W(lljj'; ks)O(t~)jO(t~)j,,

jj's

which therefore in the case of deuteron stripping leads to an estimate of the relative magnitudes of O(ts)t+ t and O(ls)t_½. Similarly, we might compare in general the polarization of the reaction product corresponding to an unpolarized entrance channel with the reaction-product asymmetry associated with a polarized target. Assuming unique values of l and s, it follows, making use of the expression for pk~(S2) given in ref. 4), that s2 ( -- 1)b-"+k+!-s Z W ( s s , ks 2 ; s 2 s)W(ljks; PkK(S2) = ~/5

sl)O~ts)jekK(a)

(2.39C)

J

x {Z ))'W(lljj'; ks)W(aajj' ; kb)O(ts)j Oq~)j,}- 1. j j"

The case of deuteron stripping allows for comparisons with k = 1 only.

3. Application to Specific Processes Having the general results of the previous sections, it may now be of value to consider their specific application to selected reaction processes. We begin this section with a consideration of deuteron stripping and pick-up processes and shall continue thereafter to more complex phenomena. In the deuteron-stripping or pick-up cases, and indeed in single-nucleon transfer processes in general, the reaction is most conveniently characterized by specification

TARGET

POLARIZATION

EFFECTS

431

of the orbital angular momentum l which the transferred nucleon carries between the target and projectile. We shall restrict our consideration initially to situations involving polarized targets but unpolarized beams. 3.1. SINGLE NUCLEON-TRANSFER PROCESSES Clearly with l = 0, the total transferred angular momentum is uniquely equal to 3. The factor ( j (. j ' - ('lkx) occurring in (2.1 la) then restricts the spin tensors affecting the angular distribution of the reaction products (da/dfl) vr so that k = 1 only, the tensor corresponding to k = 0 being a normalization parameter. This tensor, in turn, by (2.21) is purely imaginary which leads to the well known consequence that only the polarization component normal to the reaction plane contributes to (dtr/ dfl) PT. The effect of the target polarization is here restricted to a simple left-right asymmetry as measured in the reaction plane. It should be noted that when a > 3, polarizationtensors, pk~(a) of rank > 1 m a y b e required. However, any departure from isotropy in (dtr(~)/dQ) PT beyond the sin t~ dependence, characteristic of rank 1 polarizations, is evidence for one or more of the fo!lowing phenomena: (a) The value o f j exceeds 3, e.g., an l = 2 admixture is present w i t h j = 3. (b) The target-projectile distortion is dependent upon nuclear spin, thus vitiating the development presented herein. (c) Significant heavy-particle stripping amplitudes are involved. (d) Significant competitive, non-direct reaction amplitudes are involved. It is obvious from the above that the higher the value of a and therefore of k, the more information, in the form of higher-order sin ~:~ or cos K~ terms one may hope to obtain from the (da(~b)/dfl) PT measurements. The relative magnitudes of these various orders can provide further evidence regarding which of the above possibilities may be operative. In particular, terms with ~c > 2j' cannot arise f r o m j ' admixture. The case o f j = ½ only provides an excellent opportunity to test the assertion that the magnitude of the asymmetry associated with the vector polarization of the target, viz., pyd, as reflected in the angular distribution (2.34) is bounded as in (2.33a). If for example a = 1 and b = 3, say, then P~ __<2. I f a larger value of P~ is found this might be attributed to point (d), or if the value o f j is highly uncertain, so that the transition may proceed entirely t h r o u g h j = 3, it follows from (2.33b) that P~ < 1. Some light on points (b) and (d) referred to above, is provided by examination in one case of the elastic scattering associated with corresponding entrance and exit channels, and in the other case, of the reaction excitation function. A search for (a) would serve as a sensitive probe for the determination of l = 2 admixtures with the dominant l = 0 amplitudes. As is well known, precise determination of such admixturres from (da/dO)o is extremely hazardous since kinematic factors alone reduce the l = 2 relative to the l = 0 intensities in (da/dO)o, by more than an order of magnitude. Moreover, as has been repeatedly demonstrated in distorted-wave Born approximation analyses, distortion effects further mask any weak l = 2 contributions. We shall defer discussion of heavy-particle stripping until later in this paper.

~

L. J. B. GOLDFARB AND D. A. BROMLEY

The same vector-addition coefficient appearing in (2.16) ensures that only pt,(b) tensors are non-trivial in characterizing the residual nuclear polarization. Since the unique photon parity restricts k to even values in describing the angular correlation (2.19), this restriction implies an isotropic gamma correlation. As above, however, the effects (a), (b), (c) or (d) may operate to remove this restriction, hence also the predicted isotropy. In our example it is then of interest to examine the correlation involving the dominant cascade gamma de-excitation of the 9.19 MeV state in B t i via the {-, 4.46 MeV state to search for anisotropies in the angular distribution of the primary E1 radiation. Any such anisotropy, in combination with the results given above, would further restrict the range of perturbations. The interest in l = 0 processes in this connection has been developed previously z,,). As noted therein, polarization phenomena arise only as a consequence of the spindependent distortion of either or both projectile and reaction product waves. In particular, effects of deuteron tensor polarization are manifestations of the spin dependence of the deuteron distortion. It follows from (2.38a) and (2.17) that under these conditions target polarization has no observable consequences. It should be emphasized here that our discussion is in no way restricted in applicability to deuteron-induced reactions. In particular, it encompasses single-nucleon transfer processes such as (He 3, d), (He 3, ~), (t, d), (Li 7, Li6), (B 11, Bl°), (N 1., N13), etc. and their inverses; all of which have been studied, in many cases involving l = 0 transitions. Thus far we have restricted discussion to examples drawn from the light nuclei. With recent advances in experimental technology, both in terms of high resolution particle detectors such as multi-gap magnets and the soon to be available 25 MeV tandem electrostatic accelerators, it becomes possible to consider extending precise measurements of this type throughout the periodic table. This has important and interesting consequences in that it will permit systematic study of the perturbations already listed; moreover, in heavy nuclei the variety of accessible orbital admixtures is considerably greater and a greater amount of detailed information regarding the residual nuclear eigenfunctions becomes available, albeit at the cost of greater experimental sophistication. It should be noted that the polarized target techniques presented in this paper may play a much more important role in heavy nuclear studies, e.g., Ta181(d, p)Ta ls2, where the larger numbers of accessible experimental parameters become essential. 3.2. S I N G L E

NUCLEON

TRANSFER

WITH

l =

1.

Much of the discussion appropriate to the l = 0 situation is relevant here. We now note t h a t t w o j values become possible. The case o f j = ½is very similar to (3.1) above except that the effects associated with either polarization of the reaction product or the projectile are not simply disentangled. W h e n j = ½, k < 3 is permitted and in order to search for perturbations of the sort considered above, it becomes necessary to have a > 2.

TARGET POLARIZATION EFFECTS

433

Several interesting examples of the selection rules arising from the vanishing of the decoupling coefficients discussed in subsect. 2.3 may be considered here. We begin this discussion by presenting a table listing selected values of nuclear spins a n d j values for which the Racah coefficients W(aajj'; kb) or W(bbjj'; ka) vanish. The entries in table 1 are labelled appropriately for the effects of polarized targets, i.e. W(aajj'; kb) selection rules involving residual nuclear polarization may be obtained by interchanging a and b in table I. In the case of gamma-ray correlations it must be noted that k may assume only even values. TABUS 1 Representative parameters for which W(aaff; kb) =_ 0 a

b

k

j

j'

| |

-]4

2 2

2 t

2 2

2 2 2 2 2

1 | t 3 3

3 2 3 1 3

2 | 4 2 4

2 | 3 2 3

t

2

4

~

t

3 3 3

2 t 4 3

4 1 2 3

2 | 2 t

3 | 3 |

If we consider the situation a = 2, b = ½ a n d j = ½ only, if follows from (2.1 la) that no effects of target polarization of rank two can arise from the direct interaction. Similarly the situation a --- 3, b = ~ and j -- ½ only eliminates rank one target polarization effects. This would apply to the reaction Bt°(d, p)B11*(9.19 MeV); unfortunately the relevant experimental data are not yet available for comparison with this selection rule. As an example of the operation of a selection rule in the exit channel we consider the case a = ½, b = 2, j = ½. In this case the residual nucleus should show no second rank polarization arising in the direct interaction. This would require, for example, that the angular correlation involving the de-excitation gamma radiation, which depends upon W(2 2 ½ ½; 2 ½) should be isotropic. A pertinent example for which experimental data are available is the Be9(d, p) Be1°*(3.37 MeV) reaction 2a) which has been studied in the deuteron energy range from 2.5 to 3.9 MeV as well as at 7.78 MeV. In all cases a reaction plane angular correlation of the form !

w(0,,

t

= 0) = 1 +A2P2(cos 0;)

was obtained. At energies below 4 MeV the value of A2 varied markedly with deuteron energy reflecting the importance of non-direct reaction amplitudes. At Ed = 7.78 MeV

4~4

L. J. B. GOLDFARB AND D. A. BROMLEY

one has A 2 = - 0 . 3 3 +0.04. Again at this energy, the azimuthal correlation was determined to be

W(O', = ½~, ~'~) = 1 + B cos 4~'~, t

t

where B = 0.02___0.04. In both case 0 r and ~br are referred to a coordinate system with z axis along the recoil vector. The latter correlation shows consistency with the crude predictions of the plane wave approximation. The ~'act that the former correlation is not isotropic requires interference between j = ½ and j = ~ angular momentum transfers since either pure j = ½ o r j = ~ leads to predicted isotropy. This is a direct demonstration of the failure o f j j coupling as implied by an extreme shell model in this region of the p-sheU and is of course not unexpected. It does however emphasize the importance of similar (d, py) studies involving heavier nuclei where the extreme shell model has been shown to have much greater validity. By selecting situations which involve a vanishing decoupling coefficient from the partial listing contained in table 1, it will be possible to probe directly for deviations from the shell model predictions. In studies involving polarized targets, utilization of these selection rules can greatly facilitate the disentangling of effects associated with particular polarization tensor ranks. Particularly in reactions involving light nuclei where the applicability of the independent particle shell model has been well established it is usually possible apriori to select a single j value for consideration. In the case of heavier nuclei, especially those involving static deformations, it is well known that the nuclear states involved in the direct reactions have non-zero reduced-width amplitudes for more than a single j value. It is of great interest to determine the relative importance of these admixtures. This may be accomplished by comparison of the experimental data with the results of calculations utilizing formulae such as (2.8b) and (2.14b) where one inserts appropriate explicit expressions for the amplitudes Z~'I(O'). (See for example ref. 4)). It should be noted that in principle, the effects of projectile or reaction-product polarization also can be analysed to provide this information; however the former studies permit determination of the relative phases of the admixtures whereas the latter do not. This has the further consequence that the former technique gives promise of detecting relatively much smaller admixtures than is possible with the latter. 3.3. S I N G L E - N U C L E O N

TRANSFER

WITH

1~ 2

Much of the discussion of the preceding two sections has relevance here. In practice. it might be noted that the difficulties inherent in the extraction of unambiguous nuclear information increase rapidly with increasing l. As above, target spins a > j are required to provide tests of the mechanism involved or of the nuclear eigenfunctions. Clearly, it will be of interest to search for effects

TARGET POLARIZATION EFFECTS

4~

of polarization tensors of rank k > 2j, which would reflect the operation of the perturbations considered previously. The considerations presented above, particularly with reference to (2.8b) and(2.14b) suggest a series of stripping and pick-up experiments 24) somewhat analogous to those suggested by Bethe and Butler and subsequently carried out by King and Parkinson, among others, in which a search was made for small admixtures of different orbital angular momentum transfers to test the purity of the shell-model wave functions. We wish to consider now an alternate possibility where we invoke the spin-independent approximation as shown in (2.39b), (2.39c) in order to provide a probe sensitive to small admixtures of different total angular momentum transfers and which may be used to determine explicitly the various j single-particle components in a deformed nuclear state where such admixtures become possible. As a particular example, examination of the reactions A127(d, p)A128*(7)A128 in this fashion will provide direct information on the relative contributions from d! and d t single-particle admixtures to the single-particle neutron configurations in A12s. Exploitation of correlation measurements in this fashion will thus provide vital information bearing on the correctness of the single particle wave functions in distorted nuclei 25) which is not otherwise directly accessible. These studies can also be extended to provide information concerning the effects of the pairing interaction in even deformed nuclei, as considered initially by Belyaev26), and more recently by Kisslinger and Sorenson 27) and by Yoshida 28). This follows particularly since the pairing scheme explicitly involves a j-representation, i.e., the quasi-particles are characterized by their j-values. Here, the measurements would permit determination of the reduced-width amplitudes for each j; these in turn may be related to the coefficients u~ and v2 appearing in the pairing formalism. These coefficients are proportional respectively to the number of holes or number of nucleons present in a given shell, characterized b y j in the even nucleus. 3.4. TWO-NUCLEON TRANSFER PROCESSES We are concerned here with processes such as (t, p), (He a, n), (t, n), (He 3, p), (d, ~), etc., together with the corresponding inverse pick-up reactions wherein two nucleons are transferred. Because two nucleons are involved, they may be associated with a total s = 0 or s = 1. Angular momentum and symmetry considerations however restrict these values in manycases. In particular, (t, p) and (He 3, n) reactions, involving transfer of identical nucleons are restricted to s = 0 as a consequence of the Pauli principle, provided that we assume that the captured nucleons occupy similar orbits. On the other hand, angular momentum considerations alone limit s = 1 in the case of the (d, ct) reaction, for example. Many of these processes have already been analysed 29) and, in particular, many have been associated with 1 = 0, where here 1 refers to the total orbital angular momentum of the two nucleons. Such processes are particularly amenable to study using polarized targets.

436

L. .I. B. GOLDFARB AND D. A. BROMLEY

If we consider the (He a, p) process, either or both s = 0 or 1 are permitted and an 1 = 0 process wherein j # 0 must involve only the contribution corresponding to the triplet (s = 1) two nucleon configuration; this may then be identified by determination o f asymmetries in the angular distributions associated with rank-one target polarization. Asymmetries associated'with rank-2 polarization are, in fact, correlated to the polarization of the residual nucleus, as reflected by coefficients of e2,(b) in the angular correlation of y-rays involved in the de-excitation o f the residual nucleus. If l = 1, contribution o f mixtures of j-values destroys the simple correlation between p2~(b) and e2K(a) as expressed by (2.17); however this is recovered when we compare p4~(b) and e4K(a), since only j = 2 contributions are effective here. In general, restricting our considerations to a definite orbital angular momentum l = lo, say, transitions with s = 1 may be studied by identification of asymmetries or other effects associated with target polarization tensors of ranks 2lo + 1 and 2(lo + 1). Furthermore the polarization correlation (2.17) should apply to the k = 2(l 0 + 1) situation since this would be attributable to transitions with j = l o + 1 only. Selection rules may be invoked against effects on the angular distribution associated with target polarization of rank 2 and against rank 2 polarization of the residual nucleus provided that the j-value equals 2 and both a and b equal ½. This is again a consequence of the fact that W(½ ½ 2 2; 2 ½)vanishes. 3.5. I N E L A S T I C

SCA'VI'ERING

PROCESSES

This general classification includes processes such as (p, p'), (n, n'), (I3, n), (n, p), (d, d') etc; for convenience in discussion, however, we shall restrict consideration here to the inelastic scattering of nucleons, deferring discussion of exchange effects to the following subsection. Inelastic scattering of nucleons is usually interpreted as proceeding through a spinindependent interaction such that s = 0 and l = j. Actually, the vector s is associated with the vector combination S1-]-$ ~

S2

or

$1 ~

$2 + s -

I t therefore follows that in addition to the s = 0 amplitudes, those corresponding to s = 1 are also permitted, these latter transitions are usually referred to as reflecting spin-flip interactions. Again, the presence of such amplitudes should be indicated by observation of asymmetries associated with target polarization tensors of rank k = 2l+ I or 2l+2; we assume here that the relevant I value will be known from other measurements and the polarization correlation (2.17) should apply for k = 2•+2. It should also be noted that the polarization of the outgoing nucleon is almost completely correlated with the scattered-nucleon asymmetry associated with the polarization of an incoming nucleon, if we confine ourselves to polarized targets. This holds a) provided that we assume first, that the interaction responsible for the process is spin-

TARGET POLARIZATION EFFECTS

437

independent and second, that the energy loss to excitation of the target nucleus is a small fraction of the incident energy. In contrast, as we have demonstrated previously, the correlation (2.17) between the target polarization and that of the residual excited nucleus merely depends on the uniqueness of the j contribution and on other factors mentioned above. Again, several selection rules arise because of the zeros of the Racah coefficients in eqs. (2.11a) and (2.16). Thus in the inelastic scattering of a nucleus of spin 2 which proceeds t h r o u g h j = 2 excitation to a spin 1 state, there should be no asymmetry in the angular distributions arising from target polarization of rank 2. Similarly, excitation to a spin 3 state should show no asymmetry associated with first rank polarization. As a final example, a ease involvingj = 2 inelastic excitation of a spin ½ nucleus with no spin change, has a selection rule operating against effects of rank 2 polarization of the target or against residual nuclear polarization of rank 2. 3.6. EXCHANGE EFFECTS: KNOCK-ON REACTIONS We shall consider two types of exchange effects. The first relates mainly to inelastic scattering processes where we consider the process as one involving an interaction between the projectile-nucleon and one of the nucleons in the target. Either the projectile emerges as the scattered nucleon or one of the target nucleons is emitted with capture of the projectile; in either case the interaction leading to these processes acts only between the two nucleons involved and specifically does not involve other target nucleons. Customarily such processes have been classed as knock-on reactions. It may be shown readily that by recoupling the angular momenta involved, such as in the coupling of the target core with angular momentum Jc to Jt or J2 and by the interchange of sl and s2, one can arrive at formulae which are of the same form for both direct and knock out inelastic-scattering processes (see refs. a,s)). The essential difference is effectively that arising from interactions involving coordinate exchanges and different spin-dependence. This same remark applies to (d, p) processes, for example, where the proton is assumed to emerge from the target nucleus which, in turn, captures the deuteron. We can account for this by means of some interaction operator acting between the deuteron and the proton; however the end result is that the transition from a target to residual nucleus is accompanied by a transfer of angular m o m m tumj. The correlation given by (2.17) applies equally well to these types of processes. This mainly reflects the fact that this polarization correlation is independent of the nature of the interaction linking the projectile and reaction product. 3.7. EXCHANGE EFFECTS: HEAVY PARTICLE STRIPPING In considering stripping or nucleon-cluster transfer processes a more complicated situation can present itself. We shall begin by discussing the stripping situation for definiteness. In addition to the normal stripping amplitude which is illustrated schematically by fig. 2(a) we have an additional possible amplitude arising from the interaction of the

4~8

L . J . B . GOLDFARB AND D. A. BROMLEY

projectile with one of two assumed clusters in the target nucleus; in this case the target nucleus itself is stripped (fig. 2(b)). Such transitions are customarily referred to as involving heavy-particle stripping, as originally proposed by Owen and Madansky 30) in their analysis of the Bl~(d, n)C 12 reaction data, and in the analysis 31) of the ClZ(He 3, ~)C 12 reaction data. Sz

b

b

Sz

S~

a

S~

a

(a)

(b)

Fig. 2. Angular-momentum diagrams representing (a) the stripping of the incident projectile and (b) the stripping of the target or pick up of a nuclear cluster by the projectile.

In general, in heavy-particle stripping, the target nucleus is viewed as composed of clusters so that a = J+s

2.

The residual nucleus of spin b arises from the capture of a cluster of spin J by the projectile of spin s, as illustrated in fig. 2(b). In this case it is instructive to consider a fractional-parentage expansion of the target-nucleus wave function assigning the orbital angular momentum to the projectile and to the residual nucleus of spin b. This amounts to the stripping of the target nucleus by the projectile beam. In some cases one might consider the parentage of the residual nucleus of spin b and attribute relative angular momentum to the target nucleus and reaction product. Here we might refer to the process as the pick-up by the projectile beam of one of the target clusters. When the transferred cluster is well-defined, having definite parity, and the residual and target nuclei are of opposite parities, it is clear that only one of these two descriptions is valid. A difficulty arises in carrying out the analysis for the case of a polarized target. An attempt to reduce the stripping amplitude to the form (2.4) involves replacing j by J and a by st; however, the reduced amplitude now retains a dependence upon ax, assuming spin-dependent distorted waves in the incident channel. If, however, one ignores this distortion and assumes that the stripping operator is also independent of or1, it becomes possible to carry through the summation over components and one finds effective contributions from pk,(a) for k < 2J; in fact, with spin dependent distortions it is quite possible that an even larger upper bound would be indicated, therefore, in studies where J > j, the presence of heavy-particle stripping amplitudes may be established providing only that the target spin is sufficiently large to permit k values in excess of 2j.

439

TARGET POLARIZATION EFFECTS

It is of interest to consider whether such arguments are applicable to those reactions for which large heavy-particle stripping amplitudes have already been established. In the case o f B l l ( d , n)C 12 we find t h a t j = J = ½ hence no simple discrimination is possible b e t w e e n the processes on the basis of k values involved. In the case of C13(He 3, ~)C ~2, on the other hand, the fact that a = ½ already imposes the limit k < 1 in the reaction. An example of a process permitting this type of discrimination, but one which has not yet been studied in detail, is Bl°(d, p)BXl*(9.19 MeV) which has been shown to proceed via a strong l = 0 stripping process. This is shown schematically in fig. 3(a) while fig. 3(b) correspondingly represents the possible competing heavy-particle stripping process involving capture of a Be 9 cluster by the deuteron.

BI°(3+)

d(l÷)

Ca)

~I°(3+)

d(l+)

(b)

Fig. 3. Angular-momentum diagrams representing (a) the stripping of deuterons by B 1° and (b) the stripping of Bx° by deuterons or the pickup of Be ° by deuterons.

It is clear that this latter process can be attributed to either the pick up of the Be 9 by the deuterons or the stripping of B 1o by deuterons. If, on the other hand, the residual

nucleus were in the ~- configuration, and the transfer were associated with Be 9 of negative parity, parity requirements would prohibit describing this transition as the pick up of Be 9 by deuterons. T h e diagrams show that in the direct stripping casesj = ½, while heavy-particle stripping involves J = ~. Thus, only target-polarization tensors of rank 1 are effective unless heavy-particle stripping or pick up is present, in which case values of k up to 5 should be effective, even if one ignores spin-dependent distortion. Since the B 10 spin is 3, the target polarization can have non-zero terms of order up to 6, consequently examination of the reaction suggested, using a polarized target, will make possible unambiguous identification of heavy-particle stripping amplitudes if terms in the angular distribution corresponding to k or x > 1 occur. This will be of particular intelest in that it will permit, in cases such as that suggested, a disentangling of the effects of distortions and of heavy-particle stripping; as has been repeatedly emphasized, the distorted-wave Born approximation calculations frequently include backward maxima and similar phenomena previously considered as signatures of heavy-particle stripping. Use of a polarized target offers direct promise of removing this ambiguity. Another example concerns (p, ~) and (~, p) processes. A factor operating against ordinary stripping here is the large binding energy of the ~-particle. Provided that the

440

L. J. B. G O L D F A R B A N D D . A. B R O M L E Y

transition results in a residual nucleus with high spin, it is clear that the heavy-particle stripping will require large values of J while low values o f j are most favourable for the direct-stripping process. Polarization studies with such target nuclei will therefore be rewarding in testing the mechanism involved. Thus far we have considered only cases in light nuclei: as an example in heavier nuclei we may consider the process Ta lsl (d, p)Ta is2, which, with the recent improvements in technology already alluded to, now becomes accessible to detailed study. We assume the Ta: s 2 ground state assignment of 3- suggested by the fl-decay evidence and select Ta lsl as a target because of the possibility of polarizing it with existing tech, niques. The amplitudes which can contribute are represented schematically in figs. 4(a), (b) and (c) corresponding respectively to stripping of the deuteron by Ta t s 1, stripping of Ta is 1 by the deuteron and pick up of H f is o by the deuteron. I f now we examine the values o f j and J which pertain to the different diagrams we find that the maximum k values permitted are 3, 6 and 8 respectively for amplitudes (a), (b) and (c). In consequence we have at least, in principle, the possibility of disentangling these amplitudes since Pk~ terms up to k = 7 are defined by the Ta lsl polarization. By identifying which of (b) or (c) is dominant we throw light on the cluster characteristics of Ta is I and of Ta ~s2 respectively. Clearly there are a large number of reactions to which arguments of this sort can be applied.

d,,,,

d,,+,

d,,+,

(a) (c) Fig. 4. Angular-momentum diagrams representing the process Ta:SX(d, p) Ta :s2, interpreted as (a) the stripping of the deuteron by Ta :sl, (b) the stripping ofTa TM by the deuteron or (c) the pickup of Hf xS° by the deuteron. In each case the vertex with 0 indicates a parentage expansion of the nuclear wave function, while the vertex without 0 is associated with the process participants to which relative orbital angular momenta are attached. I f we no longer restrict consideration to deutel"on-induced reactions but consider transfer reactions such as (N 14, N 13) in heavy nuclei, where again a single nucleon transfer is involved, it appears entirely probable that the process analogous to heavyparticle stripping should play a much less important role. This follows because the cluster decomposition of the target which would be involved in such an amplitude is in general one with low probability, usually because at least one of the cluster members will be anomalously neutron-deficient. While it may well be instructive to consider such cases in heavy nuclei, it appears probable that considerably more information will result from measurements on relatively light targets.

TARGET

POLARIZATION

EFFECTS

441

The ultimate value of the techniques mentioned in this section depends of course on the number of appropriate experimental situations which exist. The restrictions are rather stringent; the maintenance of adequate target polarization assumes a very high magnetic field. The brute-force polarization technique under beam bombardment conditions almost certainly demands a target conductivity attainable only with metallic targets. Moreover, as noted above, the power of the technique increases markedly with increasing target spin; K 4° (ground state configuration 4-) offers an attractive possibility for such measurements despite the difficulties which its natural radioactivity implies. Interest might be given to Sc41 and Co 59 (ground-state configurations -~-) Zr 91, Pd 1°s and Mo 95 (ground-state configuration ~+), Nb 9s (ground-state configuration 9+), Trial (ground-state configuration -~+) among others, and many of the rare earths depending upon their availability. With present technology it is entirely feasible to also consider the production of polarized beams of Li 6, B 1°, N 14, C13s, etc. although this has not been accomplished as yet. When such beams become available it may be possible to invert the role of target and projectile so that our discussion of the effects of polarized targets becomes applicable to a much wider range of experimental situations. 4. The Determination of the Reduced Stripping Amplitudes In view of the various types of possible polarization measurements, the question arises whether measurements of the four types of polarization tensors, pk~(al, sk~(a), pk~(b) and Sk,(b), completely determine the amplitudes and, if not, to what extent we require simultaneous measurement of polarization effects of pairs of members of the reaction. We confine our attention to deuteron-stripping processes and consider measurements corresponding to a given energy and angle. Because of the restraint (2.21) we find that 6(2j+ 1) real parameters are to be determined, assuming aunique value ofj. Any measurement, no matter how complicated it may be, involves bilinear combinations of the amplitudes. The unpolarized angular distributions depend on Z0-20-1 2 0"20"1" j; . The quantities Pk,,(b) and ek~(a) lead to linear combinations Z O'20"1 j; Z j,;, , as is apparent from (2.11a) and (2.16). Similarly, the tensors Pk~(S2) and Sk~(Sl) depend 0"t 20-1 * respectively on linear combinations of Zj;0"20"I Zj~ and Zj;0"2#1 Z ff20"' j; 1 * . As an example of simultaneous polarization measurements, we consider the effects of a polarized projectile beam incident on a polarized target. This is represented by the measurement of ek,~o(a)skl~l(sl) which depends on combinations ofZ~.~°1Z~,~;~,c*. The general rule is that indices are repeated if no measurement is made of the polarization states signified by these indices. It is clear that measurement of the unpolarized angular distribution (dr/dr2), at a given energy and angle, of the outgoing nucleon polarization and of the asymmetries associated with a polarized deuteron beam provide one with 1, 1 and 4 pieces of information, respectively.

442

L. 1. B. GOLDFARB AND D. A. BROMLEY

If km~ is the maximum rank of the polarization tensors pkK(a) or eke(b), there are [½kma~(kma~+ 2) - ¼{(1 - ( -

)/max)}]

pieces of information to be derived from either set of tensors (but not together, i f j is unique). We shall assume that a and b are both not less than j, so that kma~ = 2]. Simultaneous measurement of pk,~,(S2)ekl~,(St) is found to lead to 12 pieces of information while ~k,~,(a)e~l~,(sl) leads to 4km~(kma x + 2) pieces of information. These results are listed in the accompanying table 2, assuming a unique value ofj. TABLE 2 Possible experimental determination of reduced amplitudes for j = ½, ~ or t

Number of real theoretical parameters

Number of available real experimental parameters

ek.x.(a)

ements (do'/d.Q)o ektxt(s=)

j value

pk2x2(s=)

½ t

or

×

pkb~b(b)

~ 1

pk2x2(s= )

4

1

1

1

4

1

7

1

4

1

17

e~,~, (S,) 12

ek=K=(a) ×

[6(2j+1)1

,kilt(S1 ) 12

12

12

60

24

12

140

36

We have not yet considered the complex experiments relating the polarization of the residual nucleus to that of the target nucleus as noted previously. It is clear that such measurements constitute a prolific source of experimental parameters in consequence o f (2.22d) which demonstrates that in the reaction it is possible to increase the maximum rank of the residual nuclear polarization relative to that of the target by an amount 2jma,. It should be emphasized here, however, that the maximum rank of the residual nuclear polarization is still limited by the magnitude of the nuclear spin involved. The main conclusion to be borne out from this analysis is that measurements of polarization effects o f the most simple kind are not sufficient to completely determine the stripping amplitudes for a given angle and energy; but that for a unique value of j, measurement of asymmetries arising from a polarized projectile on a polarized target is by itself, sufficient to allow for an over-determination of the amplitudes. It should be noted here that while the experimental difficulties inherent in the suggested studies with polarized projectiles on polarized targets a priori may appear formidable, many of these difficulties are self-compensating. In particular, the low beam intensities characteristic of all currently available polarized ion sources and accelerators do not pose a limitation in these studies since the permissible beam currents are already subject to stringent limitations imposed by thermal transfer requirements in the target polarization system. In consequence of these latter limitations, the experimental system to be utilized with polarized targets will almost necessarily be a multi-

443

TARGET POLARIZATION EFFECTS

channel one such that substitution of a polarized for an unpolarized beam will not introduce significant additional complexity ignoring here any such complexity associated with the polarized ion sources themselves. 5. The Use of Polarized Targets as a Spectroscopic Probe in Nuclear Processes

Thus far, we have confined most of our remarks to direct reactions. We also mention here the comparative roles displayed by target and projectile polarizations in reactions proceeding through well-defined intermediate states (e.g., compound system reactions). A usual characterization involves the channel-spin formalism, and polarization of either the target or the projectile amounts to polarization of the channel which is described through the polarization tensor pk~(S, S') where S and S' indicate channel spins. This tensor reduces as follows:

^ " ~,~1 Pk*(SS') = E Pk.*.(a)Pk,~,(sx)(k.~C.kl rCllkx)SS

sl {i1 a

S'

t

(5.1)

i" }

and since one or other of the tensors in the right side corresponds to no polarization, the 9-j symbol reduces to a Racah coefficient. The latter quantity is seen to represent physically the attenuation of the initial polarization through the channel-spin coupling. Clearly by assigning the polarization to the target instead of the projectile, the effect is to introduce a different attenuation factor. If, in fact both members of the entrance channel have a common spin, the factor differs only by the phase ( - 1)s+s'. The polarization effects are different in the two cases because of the coherent summation over channel-spin. Similar arguments apply to the exit channel. Polarized targets have their use as a spectroscopic tool independent of what mechanism we attribute to the reaction considered. In particular, reactions such as the inelastic scattering of spin-zero particles (e.g. alpha particles) may be useful in ascertaining the parity of the residual nucleus if we assume the target spin a to be integral and in the particular substate ~ = 0 only. As long as observation is made of the reaction product in the forward or backward directions only, the following argument may be applied: We assume any intermediate state which may or may not be well-defined and designate its spin by c. The reaction amplitude is then proportional to

~'. (110, aOlcO)(120, bOlcO)Rhaz2c,b,

(5.2)

c

where Ii and 12 are the orbital angular momenta in the entrance and exit channels and Rh°~2c~ is a reduced amplitude. Because of the properties of the vector-addition coefficients, it is clear that ,~°,~ = (_)o-b, (5.3)

J~4-4

L. J. B. GOLDFARB AND "D. A. BROMLEY

where 7r,~z~represents the product of the intrinsic parities of the target and residual nuclei. Observation of the reaction product at 0 = 0 or n, corresponding to a final nuclear state of known spin, leads to direct information concerning the relative parities of the two nuclei. Exactly the same argument has, in fact, been put forward 32) b y Litherland in connection with studies on reactions a3) such a s C 1 2 ( C 12, ct)Ne 2° involving only zero-spin nuclei in the entrance channel and a zero-spin reaction product or in inelastic alpha particle scattering of zero-spin particles on a spin zero target. Use of a polarized "target, as noted above, permits extension of this powerful approach to the more general case of non-zero but integral spins. It has also been noted 34)that a polarized deuteron beam which is entirely in the spin state with tr = 0 and which is incident on zero-spin targets can provide similar information; the argument given here extends this procedure to targets with integral spin. It must be pointed out that the preparation of such polarization in the incident channel may not be an easily attainable feat with existing sources. On the other hand, one may rule out final states with b = 0 by polarizing the target so that tr takes on non-zero values, since in these cases, the second vector-addition coefficient in (5.1) would be (/2 0, btrlO0 ), which vanishes. This argument applies independently of the parity of the final state and m a y provide a very convenient technique for rapid identification of zero-spin residual states through comparison of the relevant cross sections at 0 = 0 or rc and at one or more other angles where in the latter case the cross sections would not be expected to vanish as they do a* 0 = 0 or ~z. A suggestion by Bohr 35) is in similar vein. The following relation is found: =

(5.3)

where S1 " n and S2 " n are the components of channel spins along the normal to the reaction plane for entrance and exit channels, respectively. In particular, the suggestion 35) is made that one consider the inelastic scattering from spin-zero targets and search for those y-rays propagating along n, which are associated with de-excitation o f the inelastically excited state back to the ground state. In this case, since the y-rays can have no longitudi,'nal component (a. = ___1) it follows that the parity of the excited state must be odd. This suggestion can be extended to targets of non-zero (but integral) spin by polarizing the target so that the only spin components along n are even. Again, observation of de-excitation y-rays in this direction leading to any spin-zero state would require that the parity of the state, which this y-ray de-excites be negative. Similarly, we might test for the parity of a state with b = 0 by preparing a target with integral spin so that it is polarized with particular spin components along the direction n. I f a = 1 only, observation of the reaction product is impossible (in any direction) if the product o f initial and final parities of the nuclei is even. Likewise, a change in nuclear parity would be ruled out if ct were even only.

TARGET POLARIZATION EFFECTS

445

6. Conclusions

Arguments have been put forward which demonstrate the particular role played by polarized targets both in the study of nuclear-reaction mechanisms and in providing new spectroscopic information. The most critical assumption required in these arguments is the independence of the channel distortions on nuclear spins. Up to now no evidence has been presented for such a spin-dependence; however, it is clear that with the availability of polarized targets, an intensive study of this question might be made. One would hope to see an extension of optical-model studies dealing with elastic scattering by polarized nuclei of different mass number. Perhaps the most significant result arising from the considerations presented herein concenrns the identity of the angular dependence of the tensors Pk,(b, 0') and ek~ (a, 0') which characterize respectively the residual nuclear polarization state and the efficiency of the reaction as a polarimeter for the target polarization. Detailed study with polarized targets will serve to test for small j-admixtures in direct reactions and to trace out the non-direct contributions to the reaction amplitude. One will be able to confirm nuclear clustering, as is involved in heavy-particle stripping, by observing the asymmetries associated with polarized targets of high spin for example. Application has been made to a variety of nuclear reactions and the specific consequences of target polarization explored. It seems evident that the detailed information derivable from such studies more than justifies the very considerable complexity introduced into the experimentation. References 1) Prec. Intern. Symposium on Polarization Phenomena of Nucleons, Birkhauser Verlag, Basel

(1961) 2) R. Huby, M. Y. Refai and G. R. Satchler, Nuclear Physics 9 (1958) 4; see also D. Robson, Nuclear Physics 22 (1961) 34 3) L. J. B. Goldfarh and R. C. Johnson, Nuclear Physics 18 (1960) 353 4) L. J. B. Goldfarb and R. C. Johnson, Nuclear Physics 21 (1960) 462 5) R. H. Bassel, R. M. Drisco and G. R. Satchler, ORNL 3240 (1962) 6) L. D. Roberts and J. W. T. Dabbs, Ann. Rev. Nuclear Sci. 11 (1961) 175 7) M . J . Steenland and H. A. Tolhoek, Prog. in Low Temp. Phys. 2 (1957) 1961 8) E. Ambler, Prog. in Cryogenics 2 0960) 1 9) G. R. Satchler, Prec. Rutherford Jubilee Conference, Manchester, 1961, ed. by J. B. Birks (Heywood, London, 1962) 10) J. M. Rook and L. J. B. Goldfarb, Nuclear Physics 27 (1961) 79 11) M. Cini, Nuovo Cim. 8 (1952) 1007 12) W. Czyz and J. Sawicki, Phys. Rev. 110 (1957) 900 13) N. Austern, Fast Neutron Physics, Part II, ed. by J. B. Marion and J. L. Fowler (Interscience, New York, 1960) Ch. V D 14) S. Devons and L. J. B. Goldfarb, in Handbuch der Physik Vol. 42 (Springer Verlag, Berlin, 1957) 15) U. Fano, National Bureau of Standards Report NBS 1214 (1952), unpublished 16) F. Coester and J. M. Jauch, Helv. Phys. Acta 26 (1953) 3 17) L. J. B. Goldfarb, Nuclear Physics 7 (1958) 622 18) M. E. Rose, Elementary theory of angular momenta (John Wiley, New York, 1957)

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L. J. B. GOLDFARB A N D D . A. BROMLEY

19) E. Jahnke and F. Erode, Tables of functions with formulae and curves. (Dover Publications, New York, 1945) 20) L. C. Biedenharn, J. M. Blatt and M. E. Rose, Revs. Mod. Phys. 24 (1952) 249 21) G. R. Satchler, Nuclear Physics 3 (1958) 67 22) A. Bilaniuk and J. C. Hensel, Bull. Amer. Phys. Soc. 3 (1958) 188 23) S. A. Cox and R. M. Williamson, Phys. Rev. 105 (1957) 1799; R. T. Taylor, Phys. Rev. 113 (1959) 1293 24) H. A. Bethe and S. T. Butler, Phys. Rev. 88 (1952) 1045; J. S. King and W. C. Parkinson, Phys. Rev. 88 (1952) 141 25) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 26) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959) 27) L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32 No. 9 (1960) 28) S. Yoshida, Phys. Rev. 123 (1961) 2122;see also S.Yoshida, Proc. Rutherford Jubilee Conference, Manchester, 1961, ed. by J. B. Birks (Heywood, London, 1962) 29) D. A. Bromley, Proc. Inter. Conf. on Nuclear Structure, ed. by D. A. Bromley and E. W. Vogt (University of Toronto Press, 1960) pp. 272-365 30) G. E. Owen and L. Madansky, Phys. Rev. 105 (1957) 1766 31) Owen, Madansky and Edwards, Phys. Rev. 113 (1959) 1575 32) A. E. Litherland and G. J. McCallum, Can. J. Physics 38 (1960) 927; A. E. Litherland and A. J. Ferguson, Can. J. Physics 39 (1961) 788 33) E. Almqvist, D. A. Bromley, J. A. Kuehner and B. Whalen, Proc. Intern. Conf. on Nuclear Structure, ed. by D. A. Bromley and E. W. Vogt (University of Toronto Press, 1960) pp. 922 34) A. E. Litherland and E. Almqvist, private communication (1962) 35) A. Bohr, Nuclear Physics 10 (1959) 486