Pion absorption on polarized nuclear targets

PHYSICAL REVIE%' C VOLUME 46, NUMBER 6 DECEMBER 1992

Pion absorption on polarized nuclear targets

Mohammad G. Khayat, N. S. Chant, and P. G. RoosDepartment ofPhysics, University ofMaryland, College Park, Maryland 20742

T.-S. H. LeeArgonne National Laboratory, Argonne, Illinois 60439

(Received 21 July 1992)

The availability of polarized nuclear targets for nuclear reaction studies offers new research directions.In this paper we present the distorted wave impulse approximation formalism and calculations for two-nucleon pion absorption on polarized nuclear targets. %'e assume that the pion absorbs on a 'S& n-ppair and use phenomenological amplitudes to describe the mNN vertex. The effects of the distortions onthe incoming pion and outgoing protons, in contrast to the elementary ~NN vertex, are delineated forcertain special cases in which the formalism simplifies. Predictions of cross sections and vector and ten-sor analyzing powers are made for polarized target nuclei Li, Li, ' C, and ' N at pion incident energiesof 115, 165, and 255 MeV.

PACS number(s): 25.80.Ls, 24.10.Eq, 24.70.+s

I. INTRODUCTION

Recent advances in the technology of polarized targetshave made available several new nuclear targets withsufficient polarization to permit studies of nuclear reac-tions. This development has led in turn to a series of ex-periments, or proposed experiments, in which spin ob-servables are measured. Due to interference effects, spinobservables are generally more sensitive to small terms inthe transition amplitude than are unpolarized cross-section measurements. Thus one hopes to obtain a morecomplete understanding of either the reaction dynamicsor the underlying nuclear structure or response function.

A particular area of interest is reactions with the spinzero pions. Initial measurements have concentrated onpion elastic scattering, for example, He [1], Li [2], ' C[3], and ' N [4], although some inelastic scattering dataare available [2]. In addition, one measurement of thecharge exchange reaction ' C(m. +,n. )' N [5] has beenmade. Theoretical calculations of these processes havehad limited success. These calculations show sensitivityto both the reaction model and the nuclear structure.

Other possible pion-induced nuclear reaction studieswith polarized targets have not been studied experimen-tally, but theoretical studies have been made. Siegel andGibbs [6] have carried out calculations for the (n+, ri) re-action and pointed out the importance of the so-calledNewns [7) polarization which arises from the difFerentialattenuation of the incoming and outgoing particles.Chant and Roos [8] have examined spin observables forthe case of nucleon knockout by a pion from a polarizednuclear target. In these cases one obtains contributionsto the vector analyzing power from both the two-body~-nucleon interaction and the differential attenuation ofthe incident and emitted particles.

In the present work we will consider pion absorptionby a polarized nuclear target. We emphasize that thepion absorption mechanism is not well understood. In

particular, the previous distorted wave impulse approxi-mation (DWIA) calculations [9—11] of the A (rt+, 2p)Breaction have been rather successful in predicting thedependence on kinematic variables. However, there is amajor discrepancy in the magnitude of the cross section.The cause of this discrepancy is not understood, and mayrise in part from nuclear structure limitations and in partfrom the reaction model. The separation of these effectsis essentially impossible based on cross section data alone.The polarization observables will provide useful informa-tion for resolving the difficulties.

Our goal in this work is to provide guidance for thedesign of pion absorption measurements using polarizedtargets. We will carry out "baseline" calculations ofcross sections and analyzing powers for several exclusiveA(n+, 2p)B rea.ctions. To do so, we adapt the DWIAformulation of Chant and Roos [9]. It is assumed thatpion absorption occurs on a pair of nucleons in the targetnucleus, and the incident pion and outgoing protons arestrongly distorted by optical model potentials. We willelucidate the role of the contributions from these twodifferent dynamical effects. Note that possible three-bodyabsorption mechanisms, as suggested by recent measure-ments on He [12—14], do not directly induce nucleartransitions in the exclusive A (~+,2p)B reaction. Withinthe DWIA, such effects are assumed to be included in thepion optical potential.

To predict the polarization observables within thetwo-nucleon absorption model it would be desirable toexpand microscopic wave functions for the two partici-pating target nucleons in terms of a complete set of statesof relative motion, each to be associated with a corre-sponding two-particle absorption amplitude generatedmicroscopically (details are given in Sec. II). This ap-proach has been used by Ohta, Thies, and Lee [15].However, the existing m.XN models which have been re-viewed by Garcilazo and Mizutani [16) have not beenvery successful in describing the polarization observables

46 2415 1992 The American Physical Society

2416 KHAYAT, CHANT, ROOS, AND LEE 46

in the m. d~pp reaction and are, therefore, unlikely tolead to satisfactory results in more complicated nuclearsystems. We have noted previously that two-nucleonpion absorption is dominated by absorption on S& pnpairs. If we retain only this term, we can use the phe-nomenological amplitudes determined by Bugg, Hasan,and Shypit (BHS amplitudes) [17] which reproduce theobserved H(m. +,2p) analyzing powers quite well, andhence should be much more satisfactory for predictingpolarization observables for heavier targets. The use ofthese empirical amplitudes, instead of a single amplitudecorresponding to the dominant s-wave 5-S motion,

characterizes a major difference between the present cal-culations and our previous works on pion absorption.Clearly, this improvement is essential for the predictionof spin observables. In addition, the use of these ampli-tudes modifies somewhat the unpolarized cross section, aswill be presented for the ' O(m. +,2p)' N reaction.

In Sec. II we present the DWIA formalism and deriveexpressions for special cases of transitions in which thedistortion effects can be most clearly examined. In Sec.III we present a series of calculations for the polarizednuclei ' N, Li, Li, and ' C. A summary of the results isgiven in Sec. IV.

II. FORMALISM

Following Chant and Roos [9], we write the cross section for two-nucleon absorption on a nucleus with total angularmomentum J~, projection M~, leading to a final nucleus with total angular momentum Jz as

1L'os„(M„)= rose g g ', ' ', „' ' '

(—1) +'+ + E'EjgS„'s[(n, l,j, )(n2lzj2);JT)

M~

PcPd

xI2 L L' I L

S '

S J . (L'Aj'm'~JM)(JMJsMs~J„M„)T, ' „c d

PcPd

X(k', a', od', r, rd~t ~k;1Sjm;TN)

where ( k', o,'o d', r, rd ~

t~ k; ISj m; TN ) is the amplitude

for pion absorption on a pn pair in the state ~alSJ'm )leading to nucieons with spin projection 0.,', O.d',

TalL'A'

I tlPcPd

1 ( ) ( ) (+)G~1L'd3g(2L~+ 1)1/2 cr P u&Pd

(2)

and other quantities are defined in Ref. [9]. The details ofthe computation of GA' (R) are also outlined there. It is

significant that the sum over ulSjm is coherent.In the energy region near the 5 resonance, it was found

in Refs. [9] and [I5] that the (ir, 2p) react(on is dominat-ed by the absorption on a deuteronlike I =0, S=j =1nucleon pair. In the calculations which follow we will re-tain only these quantum numbers in evaluating Eq. (1).This will permit us to calculate the absorption matrix ele-ment (~t ~ ) using the BHS amplitudes for ~+d —+pp.Thus we obtain

I) l2 L

crs„(M„)= case g g ' ', '„' ' ' gS„'s [(n, l, j, )(nztzj~) JT) —,'

—,' S E(LASX~JM)

M~I It J& J2

PcPd

X(JMJqMs~J„M„)T, „(k',cr,'o d', r, rd ~t ~k;SX) TN )

,2

(3)

PcPd

and

TaLA 1 y', ', y „„y+G d g,(2L +1) ~ P ~dPd

PcPd

(4)

where the two-nucleon quantum numbers ISjm are sim-

I

ply replaced by the spin angular momentum SX andS =1. These are the expressions which are coded in thecalculations which follow.

For a given choice of absorption matrix element ( ~t~ )and nuclear spectroscopic factor S~~, one can see fromEqs. (3) and (4) that the absorption cross sections depend

46 PION ABSORPTION ON POLARIZED NUCLEAR TARGETS

visibly on the distortion of the incoming pion and outgo-ing protons. It is, therefore, useful to consider specialcases so that the distortion effects can be most clearly ex-amined. These special cases can be obtained by assuming

(i} that spin-orbit distortions for the emitted protons maybe ignored, and (ii) that only single values of L and J con-tribute to the transition. With these restrictions we canwrite

&sg(Mg)= top g g (LASXIJM)(JMJsMsIJ„M„)Tg~ (k', p', pd' ,r, r'dlt Ik;SX;TN& 2,MB AXM

where

TaL A 1 ' y'+'U(R)Y (P)d RBA(2L + 1)f/2 c d LA

and, for convenience, we have absorbed any structure factors into U(R), that is,

(6)

U(R ) Yt ~(P ) =c(~ ) ll J1 ~212J2

I, l2 L

gs„"~'[(tt,l,j, )(tt, l,j, ); JT] —' —' S V'2L+lG~ (R) .

Expanding the sum over quantum numbers in Eq. (5), and omitting reference to isospin quantum numbers, we obtain

os„(Mq )= cos g g (JMJttMttIJqMq ) (LASXIJM)(LA'SX'IJM)Tg q Tg „M ' " AX M

B1 c~d A~yt

x (k';p,'pd lt lk;SX & (k';p,'pd lt lk;SX' &

(M„)+o'""(M„),where the two terms correspond to the diagonal and cross terms in the summation, which are

(8)

and

os'g(M„)= „~, g g (JMJ,M, I J„M„)'( LAsxl JM)'I Tg„"I'I (k';p', p,"I tlk;sx&l'M A~MB~c~d

os„'"(M„)= tos g g (JMJsMs I J„Mq ) (LASXIJM)(LA'SX'I JM)Tg „Tg qM tt A+A MB~cI d y~y~

x (k';p,'pd lt Ik;sx &(k', p,'pd lt Ik;sx'&* . (10)

To display most clearly the physics content of ourstudies, we consider only coplanar reaction geometrieswith a target polarized normal to the scattering plane,chosen to be the axis of quantization (z). Furthermore,before considering calculations for specific nuclear transi-tions, we examine two cases in which Eq. (8) is particular-ly simplified. In these cases, of L =0, J=1 and L =1,J=1, the roles of the m.NlV vertex and the distortioneffects can be more clearly seen.

For L =0 transitions the cross terms in Eq. (10) vanish.By summing over p,

' and pd' and remembering that S = 1,the cross section becomes

where o d(X) is the free n+d~pp cross section for agiven deuteron spin projection X. We now note that thedistortion effects are isolated in I T~„I, which is indepen-dent of the magnetic quantum numbers Mz, X, and Mand hence can be taken outside of the summation. Thus,

Av

MB,™g™

os„(M~ )= cos g (JMJsMsI J~M~ } I T~„ I

MBrM

Xo d(X),

Using the above formula we can calculate polarizationobservables for specific initial and final nuclear spins. Weuse the following expressions for vector and tensoranalyzing powers:


Ao.(+—') —cr( ——')

2 2

+tot

A =[ [o ( —,

') —o ( —

—,'

) ] +—,' [o ( —,

')—o (

——,'

) ] j /o „,

A =[ [o.( —,

') +o (

——', ) ]—[o ( —,'

) +o (—

—,'

) ] j /o'„,

A», = I-,'[o(-', ) —o( ——', )]—[cr(-,')—o( —

—,')]]/cr...

A» = [cr(+1)—o (0}+o( —1) —cr(0)]/o „, (J„=1),

does not change this result significantly. We also notethat there can be no rank 3 analyzing power, even for aspin —,

' target.As a second case in which the cross section simplifies,

we consider L =1 transitions. In the general case ofLAO, the cross sections can no longer be separated into asimple product of a distortion term (dynamics) and thetwo-body m. +d —+2p cross sections. Coherent terms enterthe expressions in Eqs. (8)—(10). However, the equationscan be reduced somewhat in complexity by taking advan-tage of a symmetry property of the spherical harmonics,namely,

YLp(0, $)=(—1) +YL~(m —8,$} .

where cr, ,=g o (I ).Table I presents the results for specific transitions with

L =0.We note that for L =0 the vector (A~) and tensor

( A ) target analyzing powers are simply related to thetwo-body m+d~pp analyzing powers and are indepen-dent of distortion effects. Thus, although the cross sec-tion will be strongly affected by distortion effects and thenuclear structure, the analyzing powers reflect the m-NNvertex. Moreover, we will show in Sec. III that the in-clusion of spin-orbit terms for the emitted proton wave

Using this property, and restricting our considerations toa coplanar geometry, the terms in Tg„[see Eq. (6)] be-come mirror symmetric with respect to the scatteringplane; therefore, for our choice of the axis of quantiza-tion, these terms are identical for 0 and m —0. This inturn leads to the following selection rule: T~„=O ifL +A is odd.

If we now consider a pure L =1 transition startingwith a spin-one (J„=1) target, use of the above selectionrule leads to the following expressions for the differentialcross sections for each substate of the target spin:

TABLE I. Cross sections and vector and tensor analyzing powers for L =0 absorption only. Foreach case K=(2rrcos/AU)Ss„~ Tz„~'.

JB OBA MA=—JA

gBA BA~vvv

«d(+1)]co „d(0)«„d( —1)

gnd g 'Ird

vv

—,'«[o„d(+1)+cr d(0)]

z«[o d(+1)+o ~( —1)]z«[o d(0)+o d(

—1)]

1 g d2 v

1 g dvv

(K/10)[cr q(+ 1)+3cr d(0)+6cr d(—1)]

(«/10)[o d(+1)+4o d(0)+3o d( —1)]1«/10)[6o d(+ I )+3cr d(0)+o ~( —1)]

1 gmd2

—'A10

( «/3 ) [2cr „(+ 1 ) +o.„„(0)](«/3)[o d(0)+2o.„d( —1)]

«[6o pi+1)+ —,'cr g(0)+ —'o g( —1)]«[ ,'o. d(+1)+ —,'cr d(0)+ ,'cr „(——1)]—

1 grrd2

«d(+1)(K/3) [o ~d( + 1 ) +2o ~d(0)](«/3)[2o „10)+o d( —1)]«d( —1)

10 gmd12 V

1 md—Avv

46 PION ABSORPTION ON POLARIZED NUCLEAR TARGETS 2419

ops(Mg=0)= —Ii' iT' 'i o. d(+1)+iT'+'i o„d( —1)1

—T' 'T" g (k'&p', pd'itf ik, l, l)(k', p,'pd'itf ik, 1, —I)' —c.c. (14)

where c.c. denotes the complex conjugate of the termpreceding it, and v=2mcoz/iriv.

We immediately see that distortions have complicatedthe expressions considerably. However, we do obtain arelatively simple expression for A, namely,

A =—'v(iT" —iT' 'i )

o' d(0)2 BA

+tot

(15)

where o „,=QM o (M) as before.This formula clearly suggests that if distortions are ig-

nored then A =0 since iT"i

=iT' 'i in the plane-wave limit (PWIA). Thus, any observed vector analyzingpower for L =1 is largely due to the effects of distortionsin A (n+, 2p)8 reactions. Similar effects arising from dis-tortions are observed in (m., np) [8] and (p, 2p) [18] reac-tions.

III. DWIA CALCULATIONS

In performing these calculations we have modified theDWIA code THREEDEE [8,9] to calculate the cross sec-tions and analyzing powers of Eqs. (3) and (13). Thesecalculations require a number of input parameters anddata. Before considering specific calculations we specifythe input.

The functions G A describing the center-of-massmotion of the S& pair in the nucleus, as contained in Eq.(4}, were calculated microscopically as in Refs. [9—11].The nuclear structure input was taken from publishedshell-model calculations or, in some cases, calculatedfrom single configurations known to dominate the wavefunctions of the specific nuclear states. Using the resul-tant two-particle spectroscopic amplitudes for each tran-sition, single-particle wave functions were generated us-ing a Woods-Saxon potential with a geometry consistentwith electron scattering [19] and with appropriate bind-ing energies. These pn pair wave functions were thentransformed to relative and center-of-mass coordinatesand the overlap computed with the deuteron internalwave function. The resultant function describes thecenter-of-mass motion of that component of the target-residual nucleus overlap in which the relative motion ofthe pn pair is identical to the deuteron ground state.

The distorted waves for the outgoing protons were gen-erated using the optical-model potentials determined in

the global studies by Nadasen er al. [20]. These poten-tials have been shown to produce reasonable results forprotons in the 100—200 MeV energy range, even for lightnuclei [21].

In the present work we restrict ourselves to pion ener-gies 115 MeV& T &260 MeV; the choice of the lowerlimit is primarily dictated by the fact that the observedpolarization analyzing powers for H(m+, 2p} are verysmall for low energies; as discussed below, the upper limitis dictated by availability of detailed data for the two-body t matrix. For this energy range it is sufficient to cal-culate the incoming pion distorted wave using aKisslinger-type pion optical-model potential with param-eters from Cottingame and Holtkamp [22]. This poten-tial has been used in our previous calculations and gives arather good description of the kinematic dependences ofthe pion absorption data.

The n.+d ~2p amplitudes were taken from the work ofBugg and co-workers [17]. These BHS fitted amplitudesreproduce the experimental cross sections very well whencompared to the compilation of Ritchie [23]. A spline in-terpolation was used to obtain the amplitudes for energiesbetween those published. Because of the choice of datasets used in the fit, the available amplitudes are limited topion energies below about 256 MeV.

In Sec. III A below we reinvestigate the ' O(n. +,2p) re-action. We will examine the extent to which the previ-ously published DWIA results [9—11] can be improvedthrough the use of the BHS amplitudes. In Sec. III B wepresent detailed DWIA calculations for ' N(m +,2p),since ' N has been successfully polarized in the laborato-ry. The effects of distortions on the cross sections andpolarization observables, A„and A, for the specialcases discussed in Sec. II will be illustrated. Predictionsfor general cases with L &0 will also be presented, andthe effects of the spin-orbit potential for the outgoingprotons will be examined. In Sec. III C we calculate po-larization observables for other targets which have beenpolarized or present an opportunity to investigate theeffects of target spin on pion absorption reactions.

A. Results for ' O(m. +,2p)' N

In the previous DWIA calculations of Refs. [9—11] themNN matrix elements (iti) of Eq. (3) were calculatedfrom a single amplitude corresponding to the formationof a S2 5-N s-wave state which then decays into a 'D2p-p state. The nuclear transitions associated with other


(SINGLE AMPLITUDE) 50 j-107

(BHS) 50 /-107

possible pp partial waves were, therefore, neglected. Thissimplification is removed in the present work by using theempirical BHS amplitudes [17] to calculate the rrNX ma-trix element. It is, therefore, interesting and important toinvestigate whether the use of the BHS amplitudes willremove the discrepancies between the previous DWIApredictions for the ' O(tr+, 2p)' N reaction and the ex-perimental data.

It is instructive to first illustrate the dynamicaldifferences between the single amplitude and the BHSamplitudes by comparing the predictions of the J depen-dence of the cross sections. These results are presented inFig. 1 for ' O(m. +,2p)' N L =2 transition to J =1+, 2+,and 3+ states at 116 MeV. For each J we see that the useof BHS amplitudes changes significantly the calculatedcross sections both in their magnitudes and shapes. Moreimportantly, the J dependence calculated from BHS am-plitudes is less pronounced. This implies that in calcula-tions involving interference between amplitudes ofdifferent multipolarities, the predictions from using theBHS amplitudes can be very different from previousDWIA results.

In Fig. 2 we compare the data of Schumacher et al.[11] with the DWIA calculations. The shell model ofCohen and Kurath [24] is used to calculate the needednuclear structure amplitudes. The most notable improve-ment from using the BHS amplitudes is a better descrip-tion of the shapes of the cross sections for the 2+ (7.03MeV) and 3+(11.0 MeV) states. However, the difficultyin describing the shape of the cross section for the transi-tion to the 1+(0.0 MeV) state is not removed. This is nottoo surprising, since the Cohen-Kurath shell model wavefunctions are known to be inadequate for the 1+ groundstate of ' N from P-decay studies. This is because thelimited model space considered does not account for im-portant ground-state correlations involving more than 1A

excitation.In all cases, the calculated magnitudes of the cross sec-

tions are smaller than the data by a factor ranging fromabout 2 to 6. We note that the present DWIA calcula-tions involve three distorted waves and, therefore, thepredicted magnitudes of the cross sections are very sensi-tive to the accuracies of the employed optical potentials.The optical potentials used in this work are obtainedfrom fitting only the elastic cross sections. As is wellknown, the distorted wave functions generated from sucha phenomenological approach, which are used for calcu-lating nuclear transitions matrix elements, could be inac-curate in the nuclear interior. It is one of the objectivesof this paper to explore how the dynamics originatingfrom optical potentials can be delineated by performingnew experiments using polarized targets. The expressionsfor several special cases presented in Sec. II will be usedin such discussions later.

On the theoretical side, the next step to improve theDWIA predictions would be to use optical potentialsconstructed from not only fitting elastic cross sections,but also taking into account important dynamical effectsoccurring inside the nucleus. For example, the 6-nucleusinteraction, as described within the 4-hole model, shouldbe included in the pion optical potential. Proton opticalpotentials constructed from the NX G matrix may beessential for an accurate description of the outgoing pro-ton distorted waves. Our efforts in this direction will bepresented elsewhere. For present purposes of predictingpolarization observables which depend on the ratio ofcross sections, the DWIA model developed above shouldbe suScient.

50 F00 j.50

T,(Me V)

I

200 250

FIG. 1. DWIA L =2 energy sharing distributions for' O(m. +,2p)' N leading to 1+ (full), 2 (dashed), and 3+ (dotted)states using identical kinematics and structure. The proton an-

gles of 0, =SO' and I9~= —107 correspond to a quasifree setting.The top panel corresponds to the use of a single amplitude forthe m+d ~pp vertex, and the lower panel to the use of the Bugg,Hasan, and Shypit [17] amplitudes. The top panel correspondsto the calculations shown in Fig. 3 of Ref. [9], but with acorrected two-body transform.

B. Predictions for the ' N(m. +,2p)' C reaction

Since the ' N nucleus has been successfully polarizedin the laboratory, we will make predictions for guidingthe design of an experiment on the ' N(tr+, 2p)' C reac-tion. The calculations have been carried out for kinemat-ics such that one of the outgoing protons emerges at anangle of 50 with respect to the beam direction. This ischosen to maximize the analyzing power as suggested bythe data of ~ d~pp. The second proton is taken to beat the corresponding quasifree angle for which zero recoilmomentum of the residual nucleus is kinematically al-lowed. These angles are —107, —105, and —99' for the


considered incident pion energies of 115, 165, and 255MeV, respectively.

Within the DWIA developed in Sec. II, the strength ofa transition from the ' N(1+ ) ground state to a state in' C depends strongly on the orbital angular momentum Lof the deuteronlike S& pair in the ' N target nucleus.For the ' C ground state and first excited 2+(4.43 MeV)state, L can be only 0 and 2. The only states which canbe reached by an L =1 transition are 0 -2 states athigh excitation energy. Based on shell-model considera-tions, the excitation strengths for L & 2 transitions are ex-pected to be much weaker and are, therefore, not con-sidered in this work.

It is useful to examine first the L dependence of thecross sections and analyzing powers. The L =0 and 2nuclear structure amplitudes for the ' C ground-statetransition are taken from the shell model of Cohen andKurath [24]. The L =1 transition is assumed to be dueto the removal of a 1p, &21sl&2 pair of nucleons from ' N.As discussed in Sec. III A, we are particularly interestedin the distortion effects and will, therefore, compare theDWIA and plane-wave impulse approximation (PWIA)results.

The results for L =0 are displayed in Fig. 3. We ob-serve that the distortion effects are considerable for thecross sections, reducing the magnitude by about a factorof 6 at the maximum and washing out the minima whicharise in the PWIA calculation due to the nodes in thewave function. The analyzing powers are, however,unaffected by the distortion. This comes about becausethe L =0 cross sections depend on a single nuclear ma-trix element

~ Tz„~ as seen in Eq. (12). This nuclear reac-tion factor drops out in the calculation of the analyzingpowers. Hence, the predictions of A and Ayy in Fig. 3are simply related to the n.+d ~pp cross section o d(X).

Their variations with respect to the kinetic energy of thedetected proton is due to the Fermi motion of the deute-ronlike SI pair.

In Fig. 4 we present calculations for the L =1 transi-tion. The PWIA cross sections (dashed curves) have thecharacteristic minimum at the kinematics correspondingto zero recoil momentum for the residual nucleus. Theseminima are almost totally filled in by the distortioneffects. The overall effect of the distortion on the magni-tude of the cross section is similar to that of the L =0cases. As seen in Eq. (15), the vector analyzing power iszero in the PWIA since

~

T"~

=~

T' '~ in the absence ofdistortion effects. The predicted analyzing power isreasonably large. Its change in sign at the zero recoilmomentum point corresponds to the fact that, at thispoint, the recoil nucleus direction reverses. The predict-ed Ayy is large and negative and the distortion effects arealso significant.

The results for an L =2 transition to the ' C groundstate are shown in Fig. 5. The PWIA cross sections aresimilar to those in Fig. 4, with the minima at zero recoilmomentum being more pronounced due to the fact thatthe two maxima are better separated in energy for higherL values. In this L =2 case the expressions for theanalyzing power no longer simplify, and we expect con-tributions from both distortion effects and the two-bodya+d —+pp analyzing power. Nevertheless, distortionplays the dominant role in generating a very large vectoranalyzing power Ay, as can be seen by comparing theDWIA (solid) and PWIA (dashed) results in Fig. 5. Theeffect of distortion on A is much weaker, except nearthe minima.

We now turn to the presentation of predictions of real-istic calculations using the coherent admixture of L =0and 2 specified by the model of Cohen and Kurath

06-"02 04

N

0.2

C:0.0

C:

5.0

2.5

1+,0.00

1.5

1.0

0.5

0.0

0.6

,7.03

(d)

11.0

FIG. 2. Angular correlations for' O(m+, 2p)' N with one proton fixed at 50'and for various final states [11]. The DWIAcalculations correspond to the use of a singleamplitude for ~+d ~pp (dashed lines) and theBugg, Hasan, and Shypit [17] amplitudes (solidlines). The normalizations of the calculationsare 1+, 0.00 MeV (S, 3.0; B, 3.4); 1+, 3.95 MeV(S, 6.7; B, 6.4); 2+, 7.03 MeV (S, 3.7; B, 4.2);3+, 11.0 MeV (S, 1.9; B, 1.7), where S and Brefer to single amplitude and Bugg, Hasan, andShypit amplitudes, respectively.

0.0-150 —100 —50

0.0 —150I

-100 —50


e rer ~ r r r

0ElO

Al

I—

a ~~ ~

'~e

~ ~

s ~r

~ ~ a a ~Lr ~r ~ e ~ ~

~ a

r ~ ~

~ ~ ~~ ~ ~

, Ia

14

~e~ ~

a h ~ ~~ Fare e e ~

L

(P,r'K'

~ e ~ ~ ~ ~

I—

rr ~ \ ~F'

ssr~ erras ~

ar ~

La ~ s a

rrrrr r erare

~ r

~ ~ ~

L.... L....

L L....I

err e ~ rre ~ r ~

~a40 0 00 0~ ~ e ~ r ~ r ~

Ie r r r I r ~ r r

Ir r ~ ~

I~ r r r

Ir er ~ ~

~ ~ ~ rQ

~ I a ~ a a I ~ a ~ ~ I a ~

Ir ~ e r

Ie r r ~

. .I.~

Ie

~ a~ e

~ a~ ~

~ ~ ~ a~ re ~ r r ~r ~ ~

.I. .. .I. . . . I . .~

I~ r ~ ~

I~ e ~ ~

I~

~ I.~

I~

. .I.~ ~

I~

L.... 'r ~ r ~ r r ~ ~

Al0 00 Al0

~ml

~ ~

r ~

~ I

~ I~ I

~ r r \

~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~

I ~r

Ie

~ ~ a~ r

Ir

I~

~ ~~ ~

~ ~

~ ~'&ra ~

~ e

~ 4 /a

'~

a s ~e r r

+ OI n

~ 0-0' 0-n

T. w

-;8-'3

II' 0

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[24]. The results for the ground-state transition' N(m+, 2p)' C(0+, 0.0 MeV) are shown as the solidcurves in Fig. 6. The dotted curves are obtained by set-ting the spin-orbit potential for the outgoing protons tozero. It is seen that the spin-orbit effects are negligible,and hence, the expressions derived in Sec. II for the L =0and 1 transitions are valid in discussing the resultspresented in preceding figures. Similar small effects dueto the spin-orbit potentials for emitted protons are alsoobserved in DWIA calculations of A (m., ny) [8] andA (p, 2p) [25] reactions. We have omitted the spin-orbitpotentials in the remainder of the calculations.

The most interesting result in Fig. 6 is that the predict-ed analyzing power A is very large and is certainlywithin experimental reach. This is due to the fact thatthe transition to the ground state of ' C is predominantlyan L =2 transition which is influenced strongly by distor-tion, as seen in Fig. 5. This important feature is furtherillustrated in Fig. 7, in which the dotted curves are ob-tained by arbitrarily enhancing the L =0 component ofthe Cohen and Kurath nuclear structure amplitude by a

O

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T„- 165 MeV

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—-1.0~ r- 1.0

0.5 0.5

0.0 0.0

0 $0 100 150 200 250

—-0.5

T, (MeV)

FIG. 7. Energy sharing cross sections, vector analyzingpowers ( Ay ), and tensor analyzing powers ( Ayy ) fol' N(m. +,2p)' C(0+,g.s.) at 165 MeV incident pion energy. Thecurves correspond to DWIA calculations for the Cohen-Kurathwave functions (dashed lines) and with the L =0 amplitude dou-

bled (solid lines). The geometry is the same as Fig. 3.

factor of 2. %'e see that the predicted A is largelyunaffected by the increase in the contribution from L =0(see Fig. 3). Therefore, a precise measurement of A willprovide a good test of our treatment of distortion efFects.

Our predictions for ' N(m. +,2p)' C(2+, 4.43 MeV) aredisplayed in Fig. 8. Here the L =0 and 2 contributionsare comparable, and hence the distortion effects onanalyzing powers are less dramatic. Measurements forthis transition are, of course, also desirable for a morecomplete study of the role of distortion effects in the reac-tion dynamics.

C. Predictions for possible A {m+, 2p) experiments

To stimulate the planning of new experiments whichaddress the pion absorption mechanism, we also carryout calculations for Li, Li, and ' C targets. All havebeen successfully polarized for use in nuclear reactionstudies.

(1) Li(n. +,2p) He. For this transition to the Heground state we used a simple a-d cluster model to de-scribe the Li target. Such a description has had somesuccess in the past in describing (p,pd) [26] and (p,pa)[27] reactions on Li. The DWIA results are shown inFig. 9. Not surprisingly, given the choice of a clustermodel and the weak deuteron binding energy, the analyz-ing powers are just those of ~+0 ~pp smeared slightly bythe Fermi motion which is small. Again because of thestructure and the narrow momentum distribution of thedeuteron, the peak cross section is quite large. Thus Liis a good nucleus with which to begin a study of pion ab-sorption on polarized nuclei and to look for deviationsfrom simple absorption on a deuteron. These could arisefrom a variety of sources, such as the internal structure ofthe n-p pair or an L =2 a-d component where even asmall amplitude can have a large effect on the vectoranalyzin~ power.

(2) Li(n+, 2p) He. Calculations for the ground statetransition ( —,

' ~—', ) are presented in Fig. 10. Spectro-scopic amplitudes were calculated assuming that the Liground state is well described in LS coupling by a[3] P3/p wave function. This term has an amplitude of-0.98 in typical nuclear structure calculations [24,28].At low recoil momentum (near the peak in the cross sec-tion) the cross section is dominated by L =0. However,the L =2 component is large and dominates as the recoilmomentum increases (energies above and below thepeak). Because the L =2 vector analyzing power is somuch larger than that for L =0, A corresponds princi-pally to L =2, except very near the minimum recoilmomentum point. In the tensor analyzing power one seesclear contributions from the two L transfers in the re-gions where they dominate.

Transitions to the excited states of He are also expect-ed to be strong. We have carried out one such calcula-tion for the —,

'+ state at 16.8 MeV, by assuming the remo-

val of a ( is lp) pair with angular momentum l. =1 rela-tive to the core. The residual state was assumed to be de-scribed by coupling the s-shell hole to two p-shell nu-cleons in a [2] 5, state. With these structure assump-tions J =0, 1, and 2 terms all contribute with comparable


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amplitudes. These results are presented in Fig. 11, andare similar to those for ' N with L =1 (Fig. 4), althoughthe magnitude of the analyzing powers differ due to theinitial and final nuclear spins.

(3) ' C(n.+,2p) "B. For the ground state we have a —,'

to —, transition. This can proceed by a combination ofL =0 or 2 and with J=1, 2, or 3, the relative contribu-tions being determined by the p-shell wave functions.Calculations using the Cohen-Kurath wave functions areshown in Fig. 12. In this case, unlike that of the Liground-state transition, we see that the L =0 componentdominates leading to very small analyzing powers. Onlyin the cross section minima does one see contributionsfrom L =2 to the analyzing power.

IV. SUMMARY AND CONCLUSIONS

We have presented the DWIA formalism and calcula-tions of cross sections, and vector and tensor analyzingpowers for pion absorption on polarized nuclear targets.In this we have assumed that the absorption takes placeon S&, T =0 np pairs. The amplitudes for the

+(np)~pp proc. ess were taken from the work of Buggand co-workers [17], who have fitted experimental datafor the ~+d ~pp reaction. In calculations of' O(n+, 2p)' N we ha.ve shown that the use of these am-plitudes leads to a reduction in the J dependence ob-served in previous works.

The primary motivation for this work was to provideguidance in the design of new experiments which attemptto understand two-nucleon pion absorption. The ability

to measure analyzing powers using polarized targetsshould provide new sensitivities to the reaction dynamicsthrough the interference terms inherent in analyzingpowers.

For relatively simple cases we have shown that contri-butions to the vector analyzing power arise both from thetwo-body m+(np)~pp process and from the distortion ofthe incoming pion and outgoing protons by the residualnucleus. For the case of L =0 transitions, no distortioneffects contribute to the vector analyzing power. Thismakes such transitions excellent candidates for studies ofthe fundamental two-nucleon absorption process in nu-clei. Unfortunately, the vector analyzing power is small,so that a high quality experiment will be required.

For nonzero L transfers distortions dominate and leadto quite large analyzing powers. Thus such transitionsprovide the opportunity to study the overall treatment ofthe reaction mechanism, particularly the treatment of thedistortion of the incoming and outgoing particles. AnL = 1 transition is particularly attractive for such studies,since the vector analyzing powers are zero if there is nodistortion.

The tensor analyzing power appears to be rather in-sensitive to distortion effects and is, therefore, anothercandidate for studies of the sr+(pn)~pp process in thenuclear medium. However, at the present time such ex-periments are impractical.

Finally, if one can understand the reaction dynamicsbetter, there are nuclear structure aspects to such mea-surements. Due to the large differences between theL =0 and 2 vector analyzing powers, a measurement of

2428 KHAYAT, CHANT, ROOS, AND LEE

could serve to constrain nuclear structure calcula-tions which determine the relative amplitudes for L =0and 2.

We have also presented a series of calculations forspecific transitions for a variety of nuclei which have beenpolarized. These calculations will hopefully be helpful inthe planning of experiments.

ACKNOWLEDGMENTS

Support for this work has been provided in part by theNational Science Foundation and the U.S. Department ofEnergy, Nuclear Physics Division, under Contract W-31-109-ENG-38.

APPENDIX

The two-body H(n+, 2p) cross sections have been cal-culated using the parametrizations of Bugg and co-workers [17]. This parametrization provides H(n. +,2p)cross sections comparable to those of Ritchie [23]. Wealso find that the vector analyzing power for H(n. +,2p)calculated from Bugg and co-workers is in good agree-ment with the published data of Smith et al. [29].

Following the formalism of Mandl and Regge [30), wewrite the S matrix as

X PfPJI„I Pirr p

where the elements of this matrix can be written more ex-plicitly as

S(md, M )= g (l~(md —Mz)SM ~Jmd)I JI

Xaqi i (l 01mdl~md)

Yi '(8, $)Yi (0,0) .p

We have chosen the axis of symmetry (z) along the in-

coming beam; l and l are the relative orbital angular

These production amplitudes are taken from the phenom-enological work of Bugg and co-workers [17] who havetabulated the amplitudes for a range of energies, 400—800MeV in the pp laboratory system. We use a spline inter-polation for intermediate energies. Since these ampli-tudes are for pp~m. +d, we need to evaluate the ampli-tudes at a proton laboratory energy T, given a pion labo-ratory incident energy T„(the deuteron is assumed atrest); we obtain

m +md —2m +2md(T„+m )

2m'—m P

We can then write the differential cross section for7Td ~pp as

(md ~pp) =— g 10~S(md, M )~

mdM

where k is the c.m. pion incident momentum in (fm)Here we have used detailed balance to relate pion absorp-tion and production cross sections:

2do 4 kp dudQ

(nd ~pp) =— (pp ~nd),3 k dQ

where k is the proton momentum in the c.m. system.

momenta in the entrance channel (nd) and the exit chan-nel (pp), respectively. The quantum numbers m& and Mare, respectively, the spin states of the deuteron and thepp multiplet (triplet or singlet). The quantity J is the to-tal angular momentum of the m.d or pp system. Angularmomentum conservation and parity restrict l =l +1[3o].

The quantity aJI I are the pion production amplitudesrr p

and are related to the matrix elements pJI I by77 p

aJi i =V'(21&+1)I4npJi

[1]B. Larson, O. Hausser, E. J. Brash, C. Chan, A. Rahav, C.Bennhold, P. P. J. Belheij, R. S. Henderson, B. K. Jen-nings, A. Mellinger, D. Ottewell, A. Trudel, M. C. Vetter-li, D. M. Whittal, S. Ram, L. Tiator, and S. S. Kamalov,Phys. Rev. Lett. 67, 3356 (1991).

[2] S. Ritt, E. T. Boschitz, R. Meier, R. Tacik, M. Wessler, K.Junker, J. A. Konter, S. Mango, D. Renker, B. van denBrandt, V. Efimovyhk, A. Kovaljov, A. Prokofiev, R.Mark, P. Chaumette, P. Deregel, G. Durand, J. Fabve,and W. Thiel, Phys. Rev. C 43, 745 (1991).

[3] Yi-Fen Yen, B. Brinkmoller, D. Dehnhard, S. M. Ster-benz, Yi-Ju Yu, Brian Berman, G. R. Burleson, K. Cran-ston, A. Klein, G. S. Kyle, R. Alarcon, T. Averett, J. R.Comfort, J. J. Gorgen, B. G. Ritchie, J. R. Tinsley, M.Barlett, G. W. Hoffmann, K. Johnson, C. F. Moore, M.Purcell, H. Ward, A. Williams, J. A. Faucett, S. J. Greene,J. J. Jarmer, J. A. McGill, C. L. Morris, S. Penttila, N. Ta-naka, H. T. Fortune, E. Insko, R. Ivie, J. M. O'Donnell,D. Smith, M. A. Khandaker, and S. Chakravarti, Phys.Rev. Lett. 66, 1959 (1991).

[4] R. Tacik, E. T. Boschitz, R. Meier, S. Ritt, M. Wessler, K.Junker, J. A. Konter, S. Mango, D. Renker, B. van denBrandt, W. Meyer, W. Thiel, P. Chaumette, J. Deregel, G.Durand, J. Fabre, P. Amaudruz, R. R. Johnson, G. Smith,P. Weber, and R. Mach, Phys. Rev. Lett. 63, 1784 (1989);R. Meier, E. T. Boschitz, S. Ritt, R. Tacik, M. Wessler, J.A. Konter, S. Mango, D. Renker, B. van den Brandt, W.Meyer, W. Thiel, R. Mach, P. Amaudruz, R. R. Johnson,G. R. Smith, and P. Weber, Phys. Rev. C 42, 2222 (1990).

[5] J. J. Gorgen, J. R. Comfort, J. R. Tinsley, T. Averett, J.DeKorse, B. Franklin, B. G. Ritchie, G. Kyle, A. Klein,B. Berman, G. Burleson, K. Cranston, J. A. Faucett, J. J.Jarmer, J. N. Knudson, S. Penttila, N. Tanaka, B.Brinkmoller, D. Dehnhard, Yi-Fen Yen, S. Hgibrkten, H.Breuer, B. S. Flanders, M. A. Khandaker, D. L. Naples,D. Zhang, M. L. Barlett, G. W. Hoffmann, and M. Pur-cell, Phys. Rev. Lett. 66, 2193 (1991).

[6] P. B. Siegel and W. R. Gibbs, Phys. Rev. C 36, 2473(1987).

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Pion absorption on polarized nuclear targets