P H Y S I C A L R E V I E W V O L U M E 1 3 9 , N U M B E R 3 B 9 A U G U S T 1 9 6 5

Polarization Effects in Elastic Electron-Deuteron Scattering* Wu-Ki TUNG

Department of Physics, Yale University, New Haven, Connecticut (Received 26 March 1965)

A relativistic calculation is made of the effects of polarization in elastic electron-deuteron scattering in order to provide a theoretical basis for the complete measurement of the deuteron electromagnetic form factors. The S matrix is calculated in the first Born approximation using the helicity representation for the incoming and outgoing particle states. The density matrix for the final state is obtained for an arbitrary initial state in the deuteron spin space. The two polarization experiments most likely to be feasible in the near future are examined in detail. These are (i) measurement of the spin alignment of the recoil deuteron for scattering of unpolarized electrons on unpolarized deuterons; and (ii) measurement of the differential cross section for scattering of unpolarized electrons on polarized deuterons. In particular, it is found that the interference term between the deuteron charge and quadrupole-moment form factors can lead to a meas­urable effect of 10-15% for a "reasonable" deuteron tensor polarization analyzer, or for a polarized deuteron target with percentage tensor polarization greater than about 0.1. These experiments provide a method for the separation of the deuteron charge and electric quadrupole form factors. Measurement of the latter provides a sensitive test of the Z)-state structure of the deuteron.

1. INTRODUCTION

FROM general invariance principles it can be shown1

that the deuteron electromagnetic vertex function requires three form factors, conventionally designated by Fc, FQ, and FM (the precise definitions are given in Sec. 2). These are real functions which depend only on the square of the 4-momentum transfer q2, and can be interpreted in the nonrelativistic limit as the Fourier transforms of the spatial distributions of the electric charge, electric quadrupole, and magnetic dipole moments of the deuteron. For q2=0, in particular, Fc, FQ, and FM are related to the corresponding static-electric and magnetic moments; with our normali­zation, FC(0) = 1, FM(0) = 2 M ^ ~ 1 . 7 1 , and FQ(0) = Af2<2~25.6. Experiments relevant to the determina­tion of these form factors have so far been confined to measurements of the differential cross section for the elastic scattering of unpolarized electrons and deuterons for q2< 18 F~2. The maximum information which one can obtain from these experiments are the values of FM2 and a certain combination of Fc2 and FQ2, the latter, at small momentum transfers, being essentially Fc2.

The charge and quadrupole-moment form factors can, in principle, be separated by performing experi­ments with polarized deuterons, or by measuring the (tensor) polarization of the recoil deuteron in the scat­tering from an unpolarized target. In this paper, we calculate these polarization effects and discuss the possibilities of extracting the desired information2 from some typical polarization experiments.

*This work was supported in part by the National Science Foundation.

1 V. Glaser and B. Jaksic, Nuovo Cimento 5, 1197 (1957). 2 M . Gourdin and C. A. Piketty [Nuovo Cimento 32, 1137

(1964)] calculated polarization effects for the special case of pure deuteron states polarized in the longitudinal and transverse directions. Their results can be easily obtained by specializing ours to the cases considered. The spin alignment of the recoil deuteron in the scattering of unpolarized electrons from un­polarized deuterons was calculated using the nonrelativistic theory of the deuteron by D. Schildknecht, Phys. Letters 10, 254

The motivation for obtaining the deuteron form factors from elastic e-d scattering need hardly be stressed. The form factors as functions of q2 provide the natural testing ground for any complete or semi-dynamical calculations on the structure of the deuteron. From a more modest point of view, the following points are of immediate interest. First, information on Fc can be used to supplement that obtained from inelastic e-d scattering in the determination of the charge form factor of the neutron. Secondly, and perhaps more usefully, the use of the deuteron form factors in conjunc­tion with the nucleon form factors determined from elastic e-p and inelastic e-d scattering should permit a closer examination of the theory of the deuteron itself. In this respect, the separation of FQ from Fc is especially interesting because the former depends sensitively on the poorly known Z)-state wave function of the deu­teron.3 Lastly, one hopes one can learn something about the magnitude of the meson exchange effects from these form factors.4

On the theoretical side, the elastic e-d scattering was first calculated by Jankus5 nonrelativistically in the impulse approximation treating the nucleons as point particles carrying charge and magnetic moment. The results were subsequently modified to allow for nucleons with electromagnetic structure.6 Extensive numerical calculations based on the Jankus theory were carried out by Glendenning and Kramer,7 using deuteron wave

(1964). The polarization of the electrons scattered from polarized deuterons has been considered by G. Ramachandran and R. K. Umerjee, Phys. Rev. 137, B978 (1965). A discussion of the e-d polarization problem using methods somewhat different from those of the present paper has recently been given by J. H. Scofield, Phys. Rev. 139, B47 (1965). Both treatments of the polarization are completely general and fully relativistic.

3 This point with respect to the polarization experiments was emphasized by L. Durand, III, Nucleon Structure (Stanford University Press, Stanford, California, 1964), pp. 28-41.

4 See, for example, Refs. 8, 9, and 10. 5 V. Z. Jankus, Phys. Rev. 102, 1586 (1956). 6 D. R. Yennie, M. M. Levy, and D. G. Ravenhall, Rev. Mod.

Phys. 29, 144 (1957). 7 N. K. Glendenning and G. Kramer, Phys. Rev. 126, 2159

(1962).

B547

B548 W U - K I TUNG

e*,X

e,A

d'.u'

d,u

FIG. 1. The single-photon-exchange diagram for elastic electron-deuteron scattering.

functions calculated from potentials which fit the low energy n-p scattering data. Gourdin8 made a calcu­lation using a semirelativistic approximation and ob­tained explicitly the connection between the three deuteron form factors mentioned above and the iso-scalar nucleon form factors. Relativistic corrections to Jankus theory were calculated by Jones9 using dis­persion relations in the "tail approximation." Meson exchange contributions to the deuteron form factors have been considered by Jones9 and Gourdin,8 and more recently by Adler and Drell.10 Although the results indicate that these are small, the details are still poorly understood, and some detailed experimental checks would be desirable.

In the following sections, we will first calculate the S matrix for elastic e-d scattering in the first Born approximation using the helicity representation for the incoming and outgoing particle states (Sec. 2). A brief summary of general formulas involving density matrices in the deuteron spin space and various polarization effects is then given (Sec. 3). These formulas are applied to the problem on hand and useful results applicable to all deuteron polarization effects are obtained in Sec. 4. These results are then discussed with particular refer­ence to two of the simpler (and, hopefully, more prac­ticable) polarization experiments, namely, (i) measure­ment of the spin alignment of the recoil deuteron from the scattering of unpolarized electrons on unpolarized deuterons, and (ii) measurement of the variation with the deuteron polarization of the differential cross section for scattering of unpolarized electrons on polarized deuterons.

2. THE S MATRIX

We will calculate the complete 5-matrix elements for elastic e-d scattering in the first Born approximation using helicity representation for the incoming and out­going particle states (Fig. 1):

= i - 1 (27 r ) 4 6(e , +^-^ - J )

X (e',\f | *<•> | *,A>rW I j , w I d#), (1) 8 M. Gourdin, Nuovo Cimento 28, 533 (1963). 9 H. F. Jones, Nuovo Cimento 26, 790 (1962). 10 R. J. Adler and S. D. Drell, Phys. Rev. Letters 13, 349 (1964).

where e, d are the four-momenta and X, M are the helicity quantum numbers (X, X'=d=^; /x, / / = ± 1 , 0) for the electron and deuteron plane-wave states, respectively. Our units are such that h=c=l7 and we will use the covariant normalization for the plane-wave states,

<P'X I M > = (2^)32^053(p-p')5xv.

The metric for the scalar product of two vectors is such tnar xv%v— x -w.

The electron vertex function is given by

(eV | y„(e) I e,\)=ieu\> (e')yvu\(e), (2)

where u\(e) and uy(ef) are spinors with helicity X and X', respectively.

If the deuteron is taken to be an elementary spin-one field, the requirements of Lorentz invariance, time-reversal invariancd, and gauge invariance limit the choice of the deuteron vertex function to the following most general form1-11 :

r - W I h™ I ,M>= CW) (f'*• f)] (d+d')v +W)[(r*'* • q) G"* • q)/M*] {d+df)v

+^(g8)[r/(f'*,*-g)-r/*(r'1-?)], (3) where f (£"') is the polarization vector for the deuteron state with momentum d (df) and helicity ii ( / /) ; q=d'—d is the four-momentum transfer and M is the mass of the deuteron. The deuteron form factors Fh

JF2, and Fz introduced in (3) are not directly related to the nonrelativistic multipole moments of the deuteron. For this reason, a more convenient set of form factors is given by Fc, FQ, and FM. 8 , 1 1 These functions cor­respond in the nonrelativistic limit to the Fourier transforms of the spatial distributions of the charge, electric quadrupole, and magnetic dipole moments in the deuteron. However, they may be defined more generally in the relativistic theory using the rotational properties of the deuteron current operator considered in the Breit frame for the deuteron.12 The two sets of form factors are related by the following linear trans­formation :

(4)

where T] = q2/^M2 is a convenient parameter which we shall use throughout this paper.

Since the explicit forms of the helicity spinors and

'Fa] FQ

FM^ —

[\+h tu(l+i?) 1 2(l+u)

I 0 0

-h " - 1

(l+v)-U2i

PFl ^ 3

1^2.

11L. Durand, III, Phys. Rev. 123, 1393 (1961). The identifi­cation of the charge, magnetic moment, and quadrupole moment of the deuteron given in this paper are incorrect because of an inadvertent replacement of M by the nucleon mass m. The quadrupole moment defined in this paper also refers in the non­relativistic limit to the expectation value of r2P2 (cosfl). See also the comments of Jones, Ref. 9.

12 A completely general treatment of the expansion of the electromagnetic vertex function for particles of arbitrary spin into electric and magnetic multipole moments was given by L. Durand, 111, P. C. DeCelles, and R. B. Marr, Phys. Rev. 126, 1882 (1962).

ELASTIC ELECTRON-DEUTERON SCATTERING B549

vectors are required we have to choose a definite coordinate system to carry out the calculations. This we take to be the center-of-mass system. The kinematics are as follows: k is the magnitude of the three-momen­tum of either of the particles, W is the total energy, and 6 is the electron-scattering angle. The rest mass of the electron is taken to be negligible as compared to its kinetic energy; therefore e0=k. Following the phase conventions of Jacob and Wick13 for the helicity states, we have

/ *±(0) \

\=bX±(0) /

x+ro)=Q, x_(o)=Q, (5)

/ x±(e)\ uje') = &m ) ;

\ ± x ± ( 0 ) /

/ c o s ^ \ / - s i n § 0 \ x+(d) = [ , xje)=l ,

\ s i n | 0 / \ cosjfl/

where ± stand for X= =b | ; and

?(d)= (OA/^, ~i/V2,0),

?>(d)=(-k/M,0,09do/M),

rw=(o,-i/v2,-*/vz,o), fi (df) = (0, cos0/v2, - i/yfl, - sin0/V2), (6)

j;0(d')=(-k/M, (d0/M)smd,0, (d0/M)cosd),

£-i(d')= (0, -cos0/v2, -i/y/2, sin0/V2), where d0= (M2+k2)112.

By substituting relations (5) and (6) into (1), (2), and (3), one can calculate all the ^-matrix elements in a straightforward, though rather tedious, manner. Since the result is necessarily complicated and not very illustrative we prefer to give the details in an Appendix and to present only the general features here. The result for S can be written in the following form:

Sxv; ^=i~x{^)W+df-e-d)

X(WkW/q2)cosidMx>»>;^, (7)

where for convenience we have extracted, from M, the factors which will lead in our final result for the polarized cross sections to the appearance of the familiar cross section for the scattering of an electron from a spinless particle. In the following, the electron spin indices X, X' will be written out explicitly while the dependence on H and \x will be put into matrix form. Our equations will therefore be 3X3 matrix equations in the deuteron spin space with JJL and yf as column and row indices, respectively. In this form

M±,±=A1F1(q2)+A,F,(q2)+A2^F2(q

2), (8)

M±* = 0. (9) 13 M. Jacob and G. C. Wick, Ann. Phys. (N. Y.) 7, 404 (1959).

The explicit forms for the matrices Ah A^, and A 3 are given in Appendix I. Here we only note two important facts, namely, (i) that the electron helicity flip terms all vanish in the approximation me=0,1 4 and (ii), that the matrices A1 and A 3 are the same for both transitions H—>+ and >—, while the matrices A2

± differ. However, A1 and A 3 are invariant under the reflection about the skew diagonal, A^^ —> ^4_M,_M/, while A2

+ and A2~ are simply interchanged by this operation. As a consequence, the use of polarized electrons in an e-d scattering experiment yields no information not already obtainable in the unpolarized case unless the deuteron target is polarized, or the spin alignment of the recoil deuteron is observed.15 The polarization of electrons scattered from polarized deuterons has been calculated recently by Ramachandran and Umerjee.2 However, the new information about the deuteron form factors which the measurement of the electron polarization would yield can be obtained more simply from other experiments in which the incident electron is un­polarized, and the spin of the scattered electron is not observed. Although our results are applicable to the general situation, we will henceforth confine our atten­tion to these simpler, and more nearly practicable, experiments.

We note finally that one can express the matrices M±~M±,± in terms of the charge and moment form factors Fc, FQ and FM by using the transformation given in Eq. (4). The result may be written as

M^AcFc^+AQFQiq^+AM^Mtf). (10)

The matrices Ac, AQ and AM± are given in Appendix I. A c and A Q are invariant under the reflection about the skew diagonal, while AM+ and AM~ are interchangedjyy this operation.

3. SUMMARY OF POLARIZATION FORMULAS

The spin-density matrix p/ for the final state of the electron-deuteron system may be expressed in terms of the initial density matrix pi and the matrices M as

Pf=MPiMf, (11)

where M is given by Eq. (8) or Eq. (10). Since we shall deal only with unpolarized electrons, and will assume that the spin of the scattered electron is not observed, it will be sufficient for our purposes to deal with the reduced density matrix for the deuteron obtained by

14 This is a consequence of the 75 invariance of the electron vertex function for me — 0. Alternatively, one can argue as follows: In the Breit frame for the electron, the two component reduction of the vertex functions is proportional to ^f(o,Xq)w; since q is along the z axis, the current only couples initial and final states with opposite spins, or equivalently, the same helicity. Since the helicity of a massless particle is not changed by a Lorentz trans­formation, the result remains true in an arbitrary frame. This is a special case of the general situation for the electromagnetic interactions of massless particles discussed by L. Durand, III , Phys. Rev. 128, 434 (1962).

15 This fact has been well known to several authors (private communication from L. Durand).

B550 W U - K I T U N G

taking the trace of Eq. (11) over the electron helicities. This operation will henceforth be implied.

Following Lakin16 we shall expand p in terms of a set of matrices TJM (J— 0 ,1 , 2; M = — / , • • • J) which have the same transformation properties under rotations as the spherical harmonics YJM. Hence for deuteron spin states

(\,m'\TJM\l,m)=N(J)(l,J,m,M\ \,m') = N'(J)(-l)m(l,l,-m,ni/\JM), (12)

where N and Nr are normalization factors which we shall choose such that

TY(TJMTJ'M^) = 3<)JJ'?>MM'- (13)

One can show easily that for a normalized 3X3 density matrix p,

P=ii: {TJM)TJM\ (14) J,M

where (TJM)=TI(PTJM), and T r p = l . I t may be

shown, in addition, that Y1JM\{TJM)\2<2 where the prime of the summation indicates that 7 = 0 is omitted. We define the percentage polarization P as

P = H Z ' \(TJM)\*J'\ J,M

where the contributions to the above sum from / = 1 and 7 = 2 are, respectively, the percentage vector and tensor polarization. After these preliminaries we turn to the details of two simple experiments involving polarization of the deuteron.

(i) Spin Alignment of Recoil Deuteron for an Unpolarized Target

For unpolarized target deuterons, pi is the unit 3X3 matrix, whence

Pf=MM*.

The spin configuration of the final deuteron state is determined by the expectation values of the TJM in that state,

(TjMh= [Tr(MWTJM)y[Ti(MMi)l, (15)

while the differential cross section for the scattering without observation of the deuteron spin state is given by

(Ar/<«2)o.m.= (2aV?2)2 l>s2(§0)]i TTMW .

I t is perhaps more convenient to evaluate this last quantity in the laboratory system. Then

(&r/<ffl)i.b= (da/dU)NS{Fc2(q2)+ ( V / 9 ) 2 W ) + f (l+v)vFM

2(q2)L2(l+v)t^(m+^} , (16)

where © is the laboratory scattering angle of the deu­teron, and (da/dQ)Ns is the cross section for the scatter-

i6 W. Lakin, Phys. Rev. 98, 139 (1955).

ing of an ultrarelativistic electron from a spinless particle,

(d<r/dQhs= ( a 2 / W ) c o s 2 ! 0 s in~^0 X [ l + ( 2 £ e / i r ) s i n 2 | 0 ] ~ 1 . (17)

The final polarization is usually measured by scattering the recoil deuteron on a second nuclear target, the analyzer. The differential cross section for the second scattering can be shown16 to be

(dcr/dQ)2= (dao/dtt)2{l+ (T20)f(T2oh

+ 2l{iTn)f(iT11)2— (r2i)/(r2i)2]cos02 + 2(r 2 2 ) / ( r 2 2 ) 2 cos20 2 }, (18)

where (dao/dQ)2 is the differential cross section for the nuclear scattering of an unpolarized incoming deuteron beam, the {TJM}^ are the equivalents of Eq. (15) for

scattering of unpolarized deuteron on the second target, and (02,02) are the polar angles for the twice-scattered deuteron with respect to its direction of motion after the first scattering. I t should be recalled that the deuteron spin parameters are given by Eq. (15) in the helicity representation for the deuteron spin. Thus, the deuteron spin is quantized along the direction of motion of that particle in the center-of-mass system (the phase conventions are those of Ref. 13 with the deuteron as "particle 2" or "particle 4"). The relativistic rotation of the deuteron spin connected with the Lorentz transformation from the center-of-mass system to the laboratory system can be neglected for ?7<<Cl. In this limit, the spin parameters in Eq. (15) refer to a recoil deuteron beam quantized along its direction of motion in the laboratory system.

(ii) Differential Cross Section for Scattering on Polarized Deuterons

If the deuteron target is polarized, the initial density matrix may be expanded as

Pi=i E (TjMhTjM* (19) J ,M

with at least some of the (TJM)I for / = 1 or 2 nonzero. The final density matrix is now given by

P / = * E {TjM)iMTjMW. (20) J,M

The differential cross section in this case is given by

(da/dQ^^ida/dQ)^ E (TJM)ZTVMWTJM\ (21) J,M

The M matrices may be expressed in terms of the laboratory scattering angle © using the expression

c o s ® - £ / ( £ + M ) COS0e.m. = .

1 - E c o s 0 / . ( E + M )

We shall discuss Eqs. (18) and (21) in the next section.

E L A S T I C E L E C T R O N - D E U T E R O N S C A T T E R I N G BS51

4. CALCULATIONS AND DISCUSSIONS OF RESULTS

From the formulas given in the last section one can see that in order to calculate the polarization or the cross section for elastic e-d scattering, the quantities one has to know are the matrices G=AOf t and Q' = MfM. These can be obtained from Eq. (10). Thus,

Tr(QT1M) = 0, M= 1 , 0 , - 1 (26)

a = § [ ( M + M + t ) + ( j t f _ M _ t ) ] = AcAcfFci(q2)+AQA(;F(?(qZ)

+ilAM+AM+i+AM-AM-^M2(qi)

+lAcAQ^+AQAc^Fc(q'i)FQ(qi)

+ lAQAM'^Au'AQ^FQ{f)Fu{qi)

+lAM'A^+AcAM'^FM(qi)Fc(q"),

where we have introduced the abbreviation

AM'=K(AM+)+(AM-)1.

(22)

The explicit form for 12, although easily obtainable from the A matrices, is again very complicated and will not be presented here. We point out only that it has four independent matrix elements, and is of the form

(23) 12= 1 12i 122 123

122 1*4 —122

123 —122 12i •

A similar calculation of the matrix

$l'=K(M++M+)+ (M-+M-)l

yields the result

12' = 12i —122 1*3

—122 124 122

1 O3 O2 12i 7 (24)

where the 12/s are the same as in (23). Therefore 12 and 12' differ only in the sign of 122 as is expected from time reversal arguments.

Because we are primarily interested in the medium-range structure of deuterons, corresponding to values of q2 much less than M2, it will be sufficient in the succeeding calculations to retain only the leading terms in the small parameter ri1/2= (g2/4Jkf2)1/2. For higher momentum transfers, on the one hand, there is at present no satisfactory theory of the multiparticle exchange-current contributions to the deuteron form factors; on the other hand, the cross section decreases so rapidly that experiments of this kind become almost impossible. Using this approximation and the symmetry relations

Tr(12rJ i l f) = £ Qafi(TjM)fia= (- 1)M Tr(127V,_M), «,P

one obtains

Trl2 = 212!+124= SFc2(q2)+ (S/3)rj2FQ

2(q2)

+ 2 [ l + 2 t a n 2 ( | 0 ) > F M 2 ( ^ 2 ) , (25)

Tr (12r20)=V2 (124- 12i) = |VZ (1 - 3 cosd)r]2FQ2 {(f)

+ (3 cos2*-2 cos(9+3)[2v2(l+cos6>)]-1

X ^ M 2 ( ? 2 ) + V 2 ( 1 - 3 cosd)VFc(q2)FQ(q2)

~6^l2($mhe)rfl2FQ{q2)FM{q2), (27)

Tr(Or 8 i )= V6122= (2/V3)(smd)rj2FQ2(q2)

+v3(tan§0) (l-co$d)7iFM2(q2)

+ 2^(smd)7jFc(q2)FQ(q2)

- 2 V 3 cosd(sec±dW2FQ(q2)FM(q2), (28)

Tr(12r22) = v3l23= - (l/^3)(l+cosd)v2FQ2(q2)

- iv3(3-cos*) r ? F^ 2 (^ )

- v S ( l + c o s 0 ) ^ c ( g 2 ) / W ) - 2^(smid)^2FQ(q2)FM(q2). (29)

One can also verify readily that

Tr(12T J Mf) = T r ( 1 2 r ^ ) . (30)

Several points should be immediately clear concerning these results:

(i) Equation (26) states that the vector part of the polarization always vanishes. This well known result is a consequence of the time-reversal and spatial-reflection invariances of the electromagnetic interaction.

(ii) The form factors always appear multiplied by certain powers of 77; in fact, the only combinations which enter Eqs. (25)-(29) are Fc, rjFQ and rjli2FM. We can obtain a rough estimate of the expected relative magni­tudes of various terms either by extrapolating the values of the form factors from those at #2=0, or by using the predictions of Jankus' theory as calculated by Glen-denning and Kramer.7 The result, for g2^~(4—9)mv

2, is

Fc:vFQ:rjll2FM^l:i:h

(iii) The absence in Eqs. (27)-(29) of the term in Fc2 which gives the largest contribution to the spin-independent part of the cross section in the present range of q2 indicates that polarization effects will be rather small. Fortunately, those effects are dominated by the charge-electric quadrupole interference term TJFCFQ (there are no charge-magnetic moment inter­ference terms). A measurement of this quantity, when combined with the results obtained from the spin-independent cross section, would provide detailed information not heretofore available on the £)-state structure of the deuteron. With these facts in mind we can consider the two polarization experiments which seem to be most nearly practicable at the present time.

(i) Spin Alignment of the Recoil Deuteron for Unpolarized Targets

Combining the results of Eqs. (18) and (26), we obtain the differential cross section for the second

B552 WU-KI TUNG

scattering of the recoil deuteron on a nuclear target as

(da/dty2= ( ^ 0 / ^ ) 2 [T rO] - 1 {Tr l2+( r 2 0 ) 2 Tr(ttT20) — 2{T2i)2 Tr(tiT21)coscl>2

+2(r22)2Tr(12r22)cos202}. (31)

The last two terms can be isolated by measuring the azimuthal variation of the second scattering cross section. The best choice of scattering angles depends on the signs and magnitudes of the nuclear polarization parameters (T231)2- To get some feeling for the general features, let us take cos0=J (0^70°). Then the azi­muthal asymmetries which result from the {T21)f and ( r 2 2 ) / contributions to the deuteron tensor polarization are about 1.1 (JT2I)2 and 0.8 (T22)2 times the isotropic term in the second scattering cross section. There is unfortunately rather little known17 concerning possible analyzer targets and the values of the corresponding parameters (T2M)2 in the relevant energy region, 35-50 MeV for the recoil deuteron in the laboratory system for q2~(4:— 9)mr

2. I t is of course desirable that the values of the \{TjM)2\ for the analyzer be as large as possible. However, even values as small as 0.1 may suffice. Thus, for | ( T W ) 2 | ^ 0 . 1 , the azimuthal asym­metry in the second scattering would be a 10-15% effect, and a measurement of this effect to an accuracy of 20% would yield significant information about FQ, hence, the deuteron ZJ-state wave function.

(ii) Differential Cross-Section for Scattering with a Polarized Deuteron Target

The results of Eqs. (21) and (30) in this case yield a differential cross section for the scattered electron of the

form

(dcr/dtt) iab = (d<j/d$) N s

Xt{TrG+<r20>< Tr(12r20)+2Re(r21), Tr(QT21) +2Re(r2 2),Tr(Or2 2)}. (32)

Here the unknown quantities are the initial polarization parameters (7W)». The explicit density matrices for pure deuteron states with longitudinal and transverse spin projections are given in Appendix II. For longi­tudinal polarization of the target, the various terms can be separated using the variations in the scattered intensity with spin projection, while for transverse polarizations, the separation can be effected using the azimuthal variations in the cross section. Several groups have been working on the preparation of a polarized deuteron target. Although no satisfactory polarized target has been produced to date, the preliminary results leave little doubt that such will be produced in the near future. As an order-of-magnitude estimate, using our previous numbers for (T2\)f and (T22)fy a measurable effect of 10-15% would be obtained with a 10% tensor polarization of the target.

ACKNOWLEDGMENT

It is a great pleasure for the author to thank Professor Loyal Durand, III, for suggesting this problem, for many very helpful discussions, and for a careful reading of the manuscript.

APPENDIX I

Here we present the explicit forms of the A matrices. In the following // is the row index and \x is the column index.

Ax= f( l + COS0) (do/^/2M)sind J(l-cos0)

- (d0/^2M)sind - (k2/M2)+ (d0

2/M2)cosd (do/^2M)sm6

J(l-cos0) - (d0/^2M)smd J ( 1 + COS0)

-43=277

J( l+cos0) - (<yV2M)sin0 _ i ( i + C O S 0 ) (d0/y/2M)sin0 - (d0

2/M2) (1 - cosfl) - (d0/^M)sin0 [-H1+COS0) (d0/^M)smd |(l+cos0)

A2+~- • (k/2W)

-2(l-cos0)cos|0

(l/y/7)Z(k/M)(l+cosd) — (d0/M) (1-3 cos0)]tan|0

1— COS0

-(1/V2)[(V*O(1+COS0) - (d0/M)(l-3 cos0)]tan§0

-(2doW/M2)(l-co$0)

(WfsflM)smd

1— COS0

• (W/^lM)smd

0

(LI)

(1.2)

(1.3)

The matrix A2~ may be obtained from A2+ by reflection about the skew diagonal. In the above equations, M is the deuteron mass, rj = q2/4cM2, 6 is the electron scattering angle in the center-of-mass system, while k is the mag­nitude of the three-momenta of the particles, d0 is the deuteron energy d0= (M2+k2)112, and W=k+d0 is the total

17 It has been suggested that various pickup reactions [e.g., He3(<2,£)He4; T>(d,p)Hz2 could be useful as deuteron polarization analyzers for specific energy ranges near isolated resonances. At high energies, J. Button and R. Mermod [Phys. Rev. 118, 1333 (I960)] measured the (TJMYS for the elastic scattering of deuterons on carbon and beryllium at 410 and 420 MeV, and found (T2M)~0.3. Earlier work by J. Baldwin, O. Chamberlain, E. Segre, R. Tripp, C. Weigand, and T. Ypsilantis [Phys. Rev. 103, 1502 (1956)] at 94-157 MeV failed to show significant values of the (T2MYS.

ELASTIC ELECTRON-DEUTERON SCATTERING B5S3

energy in that system. The matrices Ac, AQ and AM are defined in terms of the Aj given above by

AC=A1-[_2(1+V)y-1A,,

^0=-Mi+( l+§ i? ) [2 ( l+ i7) ] - 1 i4s ,

In the approximation )?<<Cl, these matrices simplify to

Ac= J(l+cos0) -(1/V5)sin0 | ( l -cos0) (1/V2)sin0 cos0 - (l/v2)siitf

[ |(l-cos0) (l/v2)sin0 J(l+cos<9) +0(v),

- 4 Q = V 3

|(l+cos<9) -(1/V2)sin0 - | (5+cos0) (1/V2)sin0 -3+cos0 - (1/V2)sin0 - | (5+cos0) (1/V2)sin0 i(l+cos0)

+OW,

^ M + = 7 ? l /2[ s i n^]-l-cos0 - ( l /2v2)( l -3 cos0)tan§0 - | ( l - c o s 0 ) (l/2V2)(l-3 cos0)tani0 l-cos0 (l/2v2)sin0

[ - 4 ( 1 - cos0) - (l/2v2)sin0 0

(1.4)

(1.5)

(L6)

+0(n) . (L7)

AM~ is again obtained from AM+ by reflection about the skew diagonal. The matrix AM

f introduced in Eq. (22),

AMf=(vii2Lsmm-i-v)

Kl-cos<9) (1/V2)cos0 tan|0 - | ( l - c o s 0 ) - (l/v2)cos0 tan|0 1 - cos0 (1/V2)cos0 tan|0 - | ( l - -cos0) - (1/V2)cos0 tan§0 | ( l -cos0)

APPENDIX II

+0(?f2). (1.8)

Here we give the density matrices for some pure deuteron spin states. Let the pure state be | m), with (m \ m) = 1. Then

p(m)= \m)(m\ , and

••(m) = (//|w)(m|/x), PMV

where |ju) and |/z') are helicity states used in the text.

(i) Longitudinal polarization:

P,(±D=i [r00=F (f )1/2r10- (i/v2) r 2 0 ] , PI(°) =*Croo+^r2 0] . (ii.2)

(ii) Transverse polarization:

Let the direction of quantization %' be in the (XY) plane and made an angle — <f> with & (Fig. 2). Then

(n . i )

FIG. 2. Kinematics for scattering with a transversely polar­ized deuteron target. (XZ) plane is the scattering plane; 0 is the scattering angle. The direction of polarization is along X'.

P*(±1)=i[^oo+ (i/2v2)r20=Fv5^r11+|v5e2^r22]+H.c. P / 0 ) = i [ r o o - (i/v2)r20-vJe2^r22]+H.c.

The reason for using — <j> rather than <f> to specify the direction of £' is the following. We have heretofore used a coordinate system (XYZ) such that the (XZ) plane is the reaction plane. However, in an experiment with a transversely polarized target, the natural coordinate system to use is (X'Y'Z) with the Xf axis as the direction of quantization of the target state. In this system the azimuthal angle of the scattered particle will be <j> (Fig. 2).