Nuclear Physics A371 (1981) 381-392© North-Holland Publishing Company

DEUTERON ELASTIC SCATTERING IN THELARGE ANGULAR MOMENTUM LIMIT

R. C. JOHNSONUniversity gjSurrey, Guildford, Surrey, England

and

E. J. STEPHENSON/ndiana University Cyclotron Facility, Bloomington, IN 47405, USA

Received 16 February 1981(Revised 15 July 19ßl)

Abstract : The elastic scattering of spin-1 projectiles (deuterons) from spin-0 targets is discussed for thecase of L ~ S forces and large grazing angular momentum . In this limit, the projection of the spinperpendicular to the scattering plane is conserved. Deuterons in each of the three spin projectionsareshown to scatter independently from different potentials . In this limit, TZ , andXz = TZZ -f TZoare small, while other analyzing powers may be large.

1. Introduction

Although experiments with vector and tensor polarized deuterons are numerousat tandem energies t - 3), only recently have similar elastic scattering data becomeavailable at higher energies a, S). At these energies large values of the orbital angularmomentum become important . It is therefore appropriate to consider the characterof the spin-1 scattering observables in the semi-classical limit of very large angularmomentum . We will discuss this within the formalism of Hooton and Johnson 6),where the scattering matrix is expanded in terms of irreducible spin tensoroperators'). The expansion coefficients in this representation provide a particularlysimple form for our results.We will primarily investigate two effects. The first is the conservation in the

scattering process of the projection of the deuteron spin along the normal to thescattering plane. This result is true from parity conservation for the case of spin-~on a spin-0 target, but there is an additional requirement of large angular mo-mentum for the spin-1 case . With the added dynamical information that the classicaldeflection function is bounded and of one sign, we find a second effect . The scat-tering ofdeuterons in each projection is dominated by the spin-orbit force associatedwith a single, corresponding projection of the deuteron spin along the angular mo-mentum axis . In the large angular momentum limit, this permits the scattering to beseparated into three incoherent pieces associated with the three spin projections,

381

382

R. C. Johnson, E. J . Stephenson / Deuteron elastic scattering

each with its own potential . In this treatment, we will ignore the effects of tensorforces .We expect that our results will be increasingly important as the deuteron bom-

barding energy increases, and larger angular momenta are included in the scat-tering. The recent observation of rainbow effects in deuteron scattering at inter-mediate energies indicates that large angular momentum phenomena are playingan important role 5 ) . In addition, it will be seen that the extension of the rainbowscattering picture to describe analyzing powers s ) rests on the large angular mo-mentum separation of the problem discussed here .

The commutator is given by

where Qr is the usual Pauli spin matrix .

2. m-conservstion

The conservation of the projection of the deuteron spin perpendicular to thereaction plane is a property of scattering in the semiclassical limit where the angularmomentum 1 becomes large. The extent to which this conservation holds in a parti-cular case can be determined from the size of the commutator of the transitionmatrix, M = M(0, E), with the operator for spin perpendicular to the scatteringplane, S~ . The conservation of spin projection requires that [M, Sy] = 0 .The cross section is given in terms of the scattering matrix by e)

~ - Tr (MMT) .

(1)2s+ 1

We will consider only the case of elastic scattering from a spin-0 target ; then Mbecomes a square matrix of (2s+ 1)2 complex elements . It is useful to expand thescattering matrix in terms of the irreducible spin tensor operators')

(sm'~Tkq~sm~ = k'(sm, kq~sm'),

(2)

where ~ _ (2k+ 1)~ and -s 5 m, m' S s. This expansion is 6)

<m~IM~mi = ~<~~IT~IsmiMka = ~k(sm~, kq~sm)Mkv ,tw

kvwith the Mkq being complex expansion coefficients .

It is useful to contrast spin-1 with spin-~ projectiles . For spin--~, k <_ 1, and

M - (Moo+Mio- Mit Moo - Mio

CM~~y7 = iCf2(M1_1 - Mti)

-2Mlo

-2M~o f(M~~-M~-~)

With the y-axis chosen along the normal to the scattering plane, parity conserva-tion requires the M~ to satisfy e)

From this it can be shown immediately that Mlo = 0, M� = M, _ 1 , and[M, vy] = 0 . Thus, the projection of the spin perpendicular to the scattering planeis conserved in proton scattering as a result of parity conservation alone.

Applying parity conservation from the beginning to the spin-1 case leaves only5 independent expansion coefficients . The scattering matrix in terms of these is

M=

Moo+ Mzo

- J~(Mi i +Mzo)

Mzz

and the commutator is

where

R. C . Johnson, E. J. Stephenson / Deuteron elastic scattering

383

J~(Mii - Mzo) ßMzz

Moo - Mzo ~(Mii+Mzi)

-~(Mii - Mzi) Moo+Mzo

X = Mzi~

Y = fMzo- Mzz

For large l, the

-fX -Y ~X

fX Y -fX,

Thus, for spin-1, parity conservation alone is insufficient to insure the conservationof the spin projection .

Nevertheless, both Xand Ycan be shown to be small in the case where the angularmomentum is large and there are no tensor forces present. In that case, Mzo, Mziand Mzz differ in their angular dependence as seen in the first term of eq . (56) ofref. e) :

provided that B is not too close to 0 or n (or that sin 8 > 1 ll) . It follows that theratio of the contribution to Mzz andMzo from a particular large ! is approximately

- IzP,(6) _-~1(l+ 1)Pt(B)

~'

(11)

Pi(B)

1 15(21+ 1)z~4n

~ ~2I(1+ lx2!- lx21+3)J

~Mii P;(B) (9)

ssociated Legendre polynomials follow the rule

Pi(B) - -iZP,(9) (10)

384

R. C. Johnson, E . J. Stephenson l Deuteron elastic scattering

Hence in this limit MZZ - fMZO and Y - 0. In addition, the function P;(B)oscillates out of phase and with a magnitude of order 1 greater than P,(0). Thus, weexpect M2 , to be smaller in magnitude than MZO by a factor of order l. When onlylarge !-values contribute, X - 0. From eq . (8), we see that the conservation of thespin projection can be recovered in the semiclassical limit for spin-1 when theangular moments contributing to the scattering process become large .Even though the 1-value corresponding to a grazing collision may become large,

all partial waves contribute significantly to the scattering cross section . However,in considering the projection of the deuteron spin, what matters is the contributionof each partial wave to the commutator, or equivalently to the expression for theMZ9 in eq . (9). The coefficient preceeding each Legendre function has a magnitude

A realistic calculation of the elastic scattering matrix is needed in order to estimatethe contribution from various partial waves to the commutator .To make this estimate, we have chosen the elastic scattering of 80 MeV deuterons

from Sa Ni as an example. An optical model calculation was made using potentialparameter set L from Daehnick et al. 9). This parameter set is part of a global de-scription ofdeuteron elastic scattering data . Fig. l shows that C, rises with l, reaching a

C~

xrô45r-

ô

i ô

iti

Fig . 1 . Variation of the expansion coefficient C~ with angular momentum . See text for more details .

115(21+ 1 )_ _ ~z~_ _ __ (12)

where the M;;~ are given by eq . (34) of ref. e) as

1 ~Mu

lzl (21+3K21-1) t+ t r r t- ___- _ _ -21+1 [ ] ~1Mt (21+1)M,+(1+1)M,- ], (13)

101(1+1)

with the M', related to the usual scattering matrix bylt 2(S;- 1)

Mr_

4i~k ~(14)

R. C. Johnson, E. J. Stephenson / Deuteron elastic scattering

385IOs

102

ec .m .Fig . 2 . Magnitudes ofthe scattering matrix expansion coefïicients for 80 MeV deuterons scattering from

seNi .

maximum at l = 11 . Only l-values corresponding to near grazing wllisions createlarge contributions to the MZq , and through them to the commutator. For thisexample, we expect the approximation given in eqs. (10) and (11) will fail at thelevel of 1 /!peer, or at about 9 ~. Dynamically, the contributions to C, are large onlyfor those values of l where the scattering matrix has a strong, non-linear dependenceon j(j+ 1) . This dependence originates in the spin-orbit potential, and is most in-fluential for those partial waves whose centrifugal turning point is near the nuclearsurface.

Fig. 2 shows the magnitude of the M,~ coefficients for the elastic scattering of80 MeV deuterons on SSNi . For most of the angular range, the cross section issmooth and the relationships

Mzz = ~Mzo~

are approximately satisfied. There is about a factor of 10 separating ~Mz t ~ from thenearest coefficient . Near 0° and 180°, the relationship between Mzz andMzo is notwell satisfied.

This result is in accord with the usual classical limit for the scattering ofa particleof spin s in the presence of an I ~ s potential. In general, s ~ n is not constant (wheren is the normal vector to the scattering plane) because the spin-orbit force causesand s to precess about j. However, in the special case where s ~ ri = ±s or 0, s ~ n

becomes a constant of the motion . For s ~ ri = ±s, s lies along ! and no precessioncan occur. When s ~ n = 0, s is perpendicular to ! and there is no spin-orbit inter-

386 R. C'. Johnrai, E. J. Stc phensnn i Deuteron elastic scattering

action, hence no precession occurs (which also allows ! to lie along dthroughout thescattering). Quantum mechanically, when s and ! are not very large, there is someprecession for all these cases. Our derivation shows that the classical limit is ap-proached when only ! becomes large, even for particles where s is as small as 1 .The relationships indicated in eq . (15) also have an impact on the observables.

If they are satisfied, it follows that

Tz t - 0,

Xz = Tzz -~Tzo = 0.(16)

The vanishing of Xz also implies that its Cartesian counterpart, 2Axx +A yy , mustvanish . Using the relation Axx +Aye,+A== = 0, it follows that Axx = A=~.

3. Relationship of cross section and potential

The approximate conservation of the deuteron spin projection during scatteringsuggests the possibility of decomposing the scattering process into three independentpieces corresponding to the three projections of the deuteron spin perpendicular tothe scattering plane. Initially, we separate the scattering cross section into threepartial cross sections corresponding to the spin projections in the initial beam .Denoted by Q� tr ° and tr_ 1 , they are simply related to the cross section and Cartesiananalyzing powers through

Q = Qi +Q°+v_ 1 ,

In the last section it was shown that in the large angular momentum limit, the tr~,refer to a process in which the spin projection, m, is conserved during scattering . Inthis section we will show that in the large angular momentum limit, the angulardistributions for the three partial cross sections arise almost completely from thepotential with the corresponding value of the spin projection along the angularmomentum axis, namely j = 1+ 1, j = 1, and j = 1-1, ,respectively .The operators corresponding to the partial cross sections are given [in terms of

the Madison Convention 1°)] by

Pi = â1 -fi -1

~t 2 -fi-1 ~i 1

1 0 10 0 0 ,1 0 1

R. C . Johnson, E. J. Stephenson / Deuteron elastic scattering

387

1 ~ -1-

2i

2

2i ~ .

(18)- 1

-

2i

1Using these operators to compute the polarized cross section [see eq . (22) of ref. 6)]yields

6ti = IMoo+Mzo/f-~Mzzlz+3 lMt,lz +3lMz,lz+IMoo-~Mzolz+ ~Im[Ml i(2Môo -Mzo/~-~Miz)+ Mz i(v -~Mzz -3MZO/~)],

Qo = IMoo+Mzo/~+~Mzzlz~

(19)

The correspondence between these partial cross sections and the three pieces ofthe potential can be made by expanding the M,w in terms of the M-matrix used ineq . (13), and simplifying the resulting expressions in the large l limit . Before makingthe expansion, we will simplify the expressions for the partial cross sections usingeq . (15) :

Qti = IMoo-~/°Mzo±1ßM1~IZ,

Qo = IMoo+~Mzolz .

Consider first the simpler case of ts o . The expansions of Moo and Mzo are [withno tensor force and using eqs. (14) and (34) of ref. s)]

(20)

where the M; are related to the S-matrix elements according to eq. (14) . The sumin eq . (22) is dominated to order 1 /l by the M; matrix elements, precisely the elementsrequired by the association of the m = 0 partial cross section with the j = 1 pieceof the potential .

For the case of tr t and 6_ 1, we also need the expansion ofM11 , which is given by

Mii-f ~ 21+3 Mi+1-

21+1

Mi- 21-1 Mi_1Pi (cos6) .

(23

The expression can be simplified by assuming 1 to be large for significant values ofM; and eliminating terms of order 1/l. Then

1Moo = 12n~[(21+3)Mi+ '+(2l+1)M;+(2l-1)Mi -1]P,(cos6),

Mzo = (cos 6),il~a~ [-1M;+t+(21+1)M;-(1+1)M;-1]P,

(21)

Moo+Z~Mzo = 4n~[Mi +1 +(21+1)M;-Mi-1]P,(cosB), (22)

388

R. C. Johnson, E. J. Stephenson / Deuteron elastic scattering

Moo -V°Mio = 4~c~[lMi+t+lM~-t]Pt (cos 0),t

Qt, _ (1/4~)s I~M;+t [!P, cos B)±iP,' (cos 6)]i

+M;- t [lP, (cos B) + iP~ (cos B)] IZ .

(25)

Using the large 1 expansion for the Legendre function yields

lP, (cos B)±iP t (cos 8) =

21/(n sin B) exp [±i(!+~)9-~4in] .

(26)

The S-matrix of eq . (14) can be replaced by the phase shift S,~ _ $,~+i~,~ ($,~ and

n, ~ both real) as

S;[!P, (cos B) ± iP; (cos B)]

=

2l/(n sin 9) exp [ -2ri,~] exp [2iS,~+i(l+~)B+4i~ ] .

(28)

Fig. 3 . The central potential for 35 MeV deuterons scattering from SsNi with various angular moments.The heavy line is the locus of classical turning points with a vanishing potential derivative. The critical

energy at which the classical turning point vanishes is shown.

R . C . Johnson, E. J. Stephenson / Deuteron elastic scattering

389

The major contribution to the sum comes from the stationary phase point where

The constant term in eq . (14) makes no contribution, so this conclusion applies aswell to the M-matrix of eq . (25) . In terms of the deflection function, this is just

To satisfy this equation for only one choice of the sign, the deflection function mustbe bounded and predominantly of one sign . This is not universally the case, and sowe digress to discuss the dynamical considerations which make this so . The totalreal potential is the sum of central, Coulomb and centripetal pieces (for a givenvalue ofj-l in the presence of a spin-orbit force) . For various values of l, a familyof potentials such as those in fig . 3 is generated . Below some critical value of theenergy, there always exists a value of ! (regarded for the moment as a continuousvariable) where the total real potential has a turning point with zero slope (foundalong the locus marked by the dark line in fig . 3) . For that value of l, the projectilespirals into the nucleus and the classical deflection function is unbounded . At thecritical energy, this turning point ceases to exist . Above this energy, no orbitingl-values are found and the deflection function is bounded. In the case of deuteronson SSNi, the critical energy is about 35 MeV, and our remarks apply only abovethat energy . For most of the angular range the attraction of the nuclear well domi-nates, generating negative-going deflection functions like those shown in fig. 4.

(30)

Fig . 4 . The classical (JWKB) deflection functions for the scattering of the three spin projections of80 MeV deuterons from SaNi . See ref. s) for more details.

390 R. C'. Johnson, E. J. Stephcnson l Deuteron elastic scattering

For positive scattering angles, only one choice of the sign in eq . (30) leads to asolution . With a negative deflection function, only

can give a large contribution . Hence for a given projection m, both terms in eq . (25)cannot be large, and them = 1 and -1 partial cross sections thus depend on separate,corresponding pieces of the potential as

trt ,

4n ~~ [!Pt (cos 0) + iP; (cos 0)]M; t ' ~ Z .

(32)

L0

0.5

2.0

L0

0.5

1.5

L0

0.5 0

S', [/Pi (cos 0)+ iPi (cos 0)]

(31)

RATIO OF PERTURBED TOUNPERTURBED CROSS SECTION

60 120

ec.m .

Fig. 5. Ratio of the perturbed to unperturbed partial cross sections. See text for details .

PERTURBED V~~ L+ii

. . . ..~o___ ~i

...' 1. y

PERTURBED V~~ L

~t1PERTURBEO V~=L_i I

R. C . Johnson, E . J. Stephenson / Deuteron elastic scattering

391

The separation for Q, and a_, thus rests on both the large l limit and the use of thestationary phase approximation applied above some critical bombarding energy .

Analogously, one can consider the case of a mainly repulsive force producing apredominantly positive deflection function of opposite sign to that shown in fig. 4.The m = 1 and -1 partial cross sections would then be dominated by M; - ' andM;+', respectively, in the stationary phase approximation . The sign change occursbecause, for a classical orbit without spiralling, l and n (to which m is referred forthe scattering matrix and partial cross section) are parallel for an attractive force,but antiparallel for a repulsive force .A test of this separation was made for 80 MeV deuterons scattering from 58Ni .

Each piece of the potential (forj = l+ 1, j = 1, andj = !-1) was perturbed in turn(though not equally) by changing the size of the spin-orbit matrix element. Thiswas accomplished by multiplying the strength of the spin-orbit potential by 0.9 forj = I± 1 and 0.1 for j = 1. The fractional change in each of the three partial crosssections was then computed. These ratios of perturbed to unperturbed partial crosssections are shown in fig. 5. Beyond 140° the difFractive nature of the partial crosssections makes these ratios unilluminating . In each case the largest deviations areseen in the partial cross section corresponding to the perturbed piece of the poten-tial . The largest amount of crosstalk is observed between the m = 0 projection andthe m = 1 or -1 projections . The m = 1 and -1 projections are well separated.These optical model results support the arguments presented in this section .

4. Conclusions

We have investigated the consequences of large angular momentum for thescattering of spin-1 projectiles from spin-0 targets, with an intended applicationto the case of polarized deuterons . An expansion of the scattering matrix in termsof irreducible spin tensor operators makes the consequences of large angular mo-mentum clear with simple relationships among the second-rank expansion coef-ficients . From this study we have concluded that the elastic scattering process canbe regarded as the incoherent superposition of three pieces given by the three projec-tions of the deuteron spin . In the process it was essential to assume that tensorforces could be ignored and that the classical deflection function for scattering wasbounded and predominantly of one sign . Both of these latter assumptions are validin the semiclassical, or large angular momentum, approximation. With only spin-orbit forces to consider, most analyzing powers remain significant, with the excep-tion that TZ 1 = XZ = Axx - AZZ = 0.Once these results are given, the recent observations of rainbow scattering

effects in the deuteron elastic analyzing powers follow . They add dynamical contentto the results of this analysis, which shows that the separation of deuteron elasticscattering into three independent processes is valid for sufficiently high energies .

392

R. C. Johnson, E. J. Stephenson l Deuteron elastic .scattering

This work was performed under the auspices of the US National Science Founda-tion and the UK Science Research Council while one of us (R.C.J .) was visiting theIndiana University Cyclotron Facility . E.J.S. would like to thank Dr. J. T. Londerganfor useful suggestions in the revision of this paper. R.C.J . would also like to thankDrs. H. Nishioka and J. A. Tostevin for their comments on an earlier version ofthis paper.

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Hardekopf, Nucl . Phys . A334 (1980) 413, and references therein3) R. P. Goddard andW. Haeberli, Nucl. Phys. A316 (1979) 1164) K. Hatanaka, K. Imai, S. Kobayashi, T. Matsusue, M. Nakamura, K. Nisimura,T. Noro, H. Saka-

moto, H. Shimizu and J. Shirai, Nucl . Phys . A340 (1980) 935) E. J. Stephenson, C. C. Foster, P. Schwandt and D. A. Goldberg, Nucl . Phys . A359 (1981) 3166) D. J. Hooton and R. C. Johnson, Nucl . Phys . A175 (1971) 5837) D. M . Brink and G. R. Satchler, Angular momentum (Clarendon, Oxford, 1968)8) R . C. Johnson, Nucl . Phys. A293 (1977) 929) W. W. Daehnick, J. D. Childs and Z. Vrcelj, Phys . Rev. C21 (1980) 225310) The Madison Convention, Proc . Third Int. Symp . on polarization phenomena in nuclear reactions

(Univ. ofWisconsin Press, Madison, 1981) p. xxv