Semiclassical theory of polarization in direct nuclear reactions (II)

I 2B:2F: 2G

Nuclear Physics 74 (1965) 368--376; (~) North-Holland Publishing Co., Amsterdam Not to

be reproduced by photoprint or m i c r o f i l m without written p e r m i s s i o n from the publisher

SEMICLASSICAL THEORY

OF POLARIZATION

I N D I R E C T N U C L E A R R E A C T I O N S (II) A. Y. ABUL-MAGD and M. EL-NADI Atomic Energy Establishment, Cairo, Egypt, UAR

Received 30 March 1965

Abstract: The semiclassical theory of polarization in direct nuclear reactions is applied to stripping reactions from non-s internal states of the projectile, and the results obtained for the polarization of the reaction products are found to be similar to those obtained for stripping reactions from s-states. The circular polarization of gamma-rays emitted after stripping reactions is discussed and is related to the polarization of the outgoing particles. Finally, the cross-sections of direct reactions with polarized beams are evaluated and the derived expressions were found to agree with the quantal calculations. 1. Introduction In a previous w o r k 1), hereafter referred to as I, the polarization of outgoing particles and residual nuclei in direct nuclear reactions was considered in the semiclassical a p p r o a c h suggested by Biedenharn and Satchler 2). It was f o u n d that the semiclassical theory can give the correct expressions 3) for the polarization o f the reaction products when the captured or emitted particles have definite values o f orbital and total angular momenta. In the present w o r k we consider some cases which were not considered in I. Sect. 2 deals with the stripping reactions, in which the outgoing and the captured particles are in the projectile with values o f the orbital m o m e n t u m l b -~ 0. Introducing the orbital m o m e n t u m transfer, it is f o u n d possible to derive for these cases expressions similar to those derived in the usual deuteron stripping reactions with [b = 0. In sect. 3, the circular polarization of gamma-rays following stripping (or pickup) reactions is discussed. In sect. 4, direct nuclear reactions with polarized initial beams are considered. The cross-sections for such reactions are evaluated. The results for the stripping reactions are c o m p a r e d with the results of the exact quantal calculations 3, 4).

2. Stripping from Non-s Internal State of the Projectile Let us consider the following reaction: (a+b)+T

-~ a + ( b + T ) .

(A)

I f one neglects the spin-orbit coupling for the outgoing particle a, the following 368

369

NUCLEAR REACTIONS (II)

relations will govern the spin of the particle a:

(2.1) (2.2) (2.3) (2.4)

SaWJb = Sa+b, Sb + Ib =

Jb,

Sb+Lb = Jb, ST+J b :

Sb+T,

where the particles are denoted by the indices and where l u and L b a r e the orbital angular momenta and Jb and Jb are the total angular momenta of the particle b in the projectile and the residual nucleus, respectively. Now it is convenient to define the vector of the orbital angular momentum transfer: (2.5)

[ = L b- !b .

Using this definition, assuming that only one value of l contributes to the reaction and subtracting eq. (2.2) from eq. (2.3) one finds Jb + l = Jb.

(2.6)

Using eqs. (2.1), (2.4) and (2.6) and applying the semiclassical correlation techniques of Biedenharn z), one finds for the expectation value of the spin of the emitted particle a (2.7)

( S a ) ~--- (Sa ".Ib)()b " l ) ( l ) .

Similarly, one gets for the expectation value of the spin of the residual nucleus (2.8)

<~b+T> = ( $ b + T " 3 b ) ( 3 b " i ) < i > .

Here g is a unit vector in the direction of the vector s and it is assumed that the dominant contribution to the reaction arises fi'om one value of l. Now replacing the classical values of the angular momenta s by their quantal analogues h x / ~ 1), one obtains for the polarization of the outgoing particles (2.7a)

p z = Pm(l=>/l,

and for the polarization of the residual nucleus P~+T =

Pm, R/1,

(2.8a)

where Pm 4Sa Jb

Pm, R m

Sa+b(Sa+b+ 1)-- Sa(Sa+ 1)--jb(Jb + 1) Jb(Jb + a)--jb(Jb + 1)--l(l+ 1) jb+l I+1

_ _1

4Sb+xJb

ST(ST+ 1)--Sb+T(Sb+T+ 1)-- Jb(Jb + 1) jb(Jb + 1)-- Jb(Jb + 1)--l(l+ 1)

Jb + 1

l+ 1 (2.8b)

370

A. Y. ABUL-MAGDAND M. EL-NAD1

In the quantal theory of stripping reactions 3), the matrix element of the reaction may be written in the following way:

X Oeb(S a+blsajblbsb)O*b(sb+TIsTJuLbsb) M (SaO"a,jupblsa+b~.+b)(sbcrb, Ibmbljb,Ub) X (SbO'b, LbMblJb,.~/b)(STffT, Jb~/bJSb+TtTb+T) X ~eb(/b rob, L b mb) ,

(2.9)

where Oeb(Sa+blSaJblbSb)is the reduced width which represents the probability that the particle ( a + b ) to consist of the two particles a and b with relative angular momentum l b which is coupled with s b to produce Jb, and that this latter is coupled with s, to give the spin of the projectile Sa+b. The symbol % stands for the internal structure of the particle b. Similarly, Oeb(Sb+TlSxJbLbSb) is the reduced width which represents the configuration where the particle b is bound to the target nucleus T in its initial state and having the orbital angular momentum L b and the total angular momentum Jb. Now substituting the relation Z {O'b}(Sb~b, lb rob]Jb/~b)(Sbab, gb Mb[Jb ~ ' b )

= [Jb]~[Lb]-&(--) su+Lb+sb 2 {Ira} [l](Im, IbmblLbMb) X(jbPb' lm[Jb~/b) {jb

Ib slb)

Jb

into eq. (2.9), one finds I~T~,÷b,~ . . . . = Z

{eJbjbl"~/bflbm}OJeb,lJb(Satra'JbllblSa+btTa+b) x (jb/~b,

lrnlJb~gb)(STtrT, Jb~gblSb+T~rb+T)TRe(lm),

(2.10)

where e now refers to the totality of indices %, l b, s b and Lb,

ojb e, tsb = (__)Lb+sb+Sb[l][jb]½[Lb]--~: {Jb

lb S/b} O et.(sa+bls. Jb lb S6)

Lb Jb

X Oeb(Sb+TlSTJbLbSb) , (2.11)

92i~e(lm) = Z {rnt,, Mb}(Im, lbrnblLbMb)gJ~eb(Ibmb, LbMb).

(2.12)

It is seen that after introducing the orbital angular momentum transfer l, it is possible to write the transition amplitude (2.9) for stripping reactions from the non-s state in the form (2.10) which has the same structure as the transition amplitude of the stripping from the s-state. Thus it seems that the polarization in cluster stripping reactions from non-s states can be derived from the corresponding result in cluster stripping from the s-state, by replacing the spin or the captured particle by the valuejb and the orbital momentum in the final nucleus by the orbital momentum transfer l.

NUCLEAR REACTIONS (II)

371

However, if I is permitted to take more than one value, then, the quantal expression for this case may be written as 4): =

\

Sa

[s.]

-

Sa Jb

2 {tin}

X

Sa

Sa+b

2

"2 -- 1

lOe,,sbl ~]] 19)L(/m)l t }

. ib* g {ll ,m}Oe, vs b Oe,Jbts b

1' Jb Jlb}(I'm, l-mllO)~i~*~(l'm)~J~e(Im)

X (__)l+l'+m {Jb 1

1 Sa + b(Sa + b -~- 1) - Sa(S a -1- 1) --Jb(Jb ÷ 1) 2Sa

Jb + 1

]{~Jeb'IJb]2]~--e(;IT]O]21--1(--)Jb+JLV[Jb][Jb~b b[l] ÷ 1.33~ ~* .."",l+l'+m{~b l'Jib ) (it m~Oe, rSb Oe, tSbl,- ) Jb

X { E {ptH}

XE

...... Jb*

x (I'm, l-- m[lO)~*(l'm)~J~e(lm ). We note that in the case of stripping from the s-state in the projectile, the orbital angular momentum transfer in the reaction is simply the orbital momentum of the captured particle in the residual nucleus and eqs. (2.7) and (2.8) in this case coincide with the corresponding eqs. in I. Now let us consider the polarization of the reaction products in the following pick-up reaction: a+T ~ (a+b)+R,

T = R+b,

when the particles a and b are in a relative/b-state. The polarization in such reaction will be determined by the following relations:

s.+jb = S.+b,

(2.13)

Sb+ Ib = Jb,

(2.14)

Sb+Lb = Jb,

(2.15)

Jb + SR = ST"

(2.16)

Introducing the orbital angular momentum transfer (2.5) one may now relate the total angular momentum Jb of the picked up particle in the outgoing particle to its total angular momentum Jb in the target nucleus. Thus from eqs. (2.6), (2.13) and (2.16) one obtains (~a+b> = (~a+b ")b)(Jb " i)(i},

(2.17)

( $ R ) = (SR" 3 b ) ( 3 b " l b ) ( l ) "

(2.18)

372

A . Y . ABUL-MAGD AND M. EL-I~ADI

Hence, the polarization of the outgoing particles and residual nuclei will be P~+b = Pm(lz)/l,

(2,17a)

P~ = Pro,R(lz')/l,

(2.18a)

where Pm =

Sa(S a "{- 1) -- Sa + b(Sa + b "~ 1) - - J b ( J b + 1)

--

4Sa+ bJb

Jb+ t ×

Pm, R

Jb(Jb + 1)--jb(Jb + 1)-- l(l+ 1) /+1

(2.17b)

ST(ST+1)-- S.(SR+1)-- Jb(J + 1) 4SRJb

Jb-}- 1

X

Jb(Jb + 1)-- Jb(Jb + 1) -- l(l + 1) 1+1

(2.18b)

3. Circular Polarization of ~-Rays We now consider the case when the residual nucleus (b + T) in the stripping reaction represented by (A), is excited and decays to the ground state with spin s o emitting a y-quantum. It was shown by Satchler 5) that these y-rays are circularly polarized, and the degree of circular polarization C(0(9) of 7-rays emitted in the direction (0q~) is related to the spin tensor TRP((Sb+T)) of the residual nucleus, as follows:

W(O(a)C(Oc~) = Z {R = even, p)[4n/ZR+

1)]{aR(TRP(sb+T))YRp(O(9),

(3.1)

where

aR = Z (22'}FR(22'SoSb+ T)C~C~'' and c~ is the reduced matrix element of the 2~-pole 7-ray, FR(22'SoSR) is the y-ray correlation coefficient and W(0q~) is the angular correlation function 6). The eXpressions for the spin tensors (TRP(sb+T)) are given in ref. 4). The rank of this spintensor is R < 2sb+x, 2J b . If the spin of the residual nucleus or the total angular momentum Jh of the captured particle in the residual nucleus is less than 3 then only spin-tensors with rank 1 exist in eq. (3.1). Taking the quantization axis in the direction kb+a×ka, where ka +b and ka are the wave vectors of the incident and outgoing waves respectively, all the spin tensors but ( T 1°) vanish. Expressing this quantity through the polarization of the residual nucleus, one finds

W(O(9)C(Oc~) = aa[3Sb+ T/(Sb+T + 1)[Sb+T]] ~ COS OPTS.

(3.2)

N U C L E A R REACTIONS (II)

373

Substituting here the expression (2.8a) for the polarization of the residual nucleus, one obtains

W(04,)C(04,) = al[3Sb+ T/(Sb+T + 1)[Sb+d] ~ COS OPm,R(Iz)/t.

(3.3)

NOW, if the polarization of the outgoing particles P,~ is not zero, then one may express (lz) in terms o f P ,~ in the following way:

W(O(a)C(O$) = aI[3Sb+T/(Sb+T+ 1)[Sb+T]] ~ Pro, R_COS OP~a. Pm

(3.4)

Substituting eqs. (2.7b) and (2.8b) into eq. (3.4), one can find the following relation between the polarization of outgoing particles, the circular polarization and the angular correlation of the emitted 7-rays:

W(O(o)C(O~?) = Astripa 1 cos OPt,

(3.5)

where Astrip=Sa

Is

3 ?~ jb(Jb+ 1) b+T(Sb+T+i)(ZSb+T+I) Jb(Jb+l)

x ST(ST+l)--Sb+T(Sb+w+l)--Jb(Jb+l)jb(Jb+l)--Jb(Jb+l)+l(l+l) Sa+b(S, + b + 1)-- Sa(S, + 1) --Jb(Jb + 1) Jb(Jb + 1)--jb(jb + 1)-- l(l+ 1)"

(3.6)

The coefficient Astrip is calculated for some deuteron stripping reactions and the results are shown in table 1. Similarly, for pickup reactions one may write the following relations between the polarization of outgoing particles, the angular correlation and the circular polarization of emitted 7-rays:

W(OO)C(Od?) = Aplckupa'l PaZ+bCOS 0,

(3.7)

where I 3 ] ~ jb(jb+ 1) Apickup = Sa+b SR(SR+I)(2SR+I) )b(Jb+l)

×

ST(ST+I)--SR(SR+])--Jb(Jb+I) Jb(Jb+l)--Jb(Jb+l)--l(l+l) Sa(Sa -1-1) - - S a + b ( S a + b -[- 1)--jb(jb + 1) Jb(Ju + 1)--jb(jb + 1)-- I(1 + 1)'

(3.8)

al = 2 {22'}F1(22'So SR)CaCa'. We note that in the formalism under consideration the distorting potential is assumed to be independent of spin in both initial and final channels. Hence the expressions (3.5) and (3.7) may be useful in checking the validity of this assumption in different stripping and pick-up reactions, in addition to their importance in nuclear spectroscopic analysis and checking the direct character of the mechanism of these reactions.

374

A. Y. ABUL-MAGD AND M. EL-NADI TABLE 1

The coefficient Astrl p for some deuteron stripping reactions ST

Sb+ T

l

Jb

Astrlp

0

-~-

1

½

--3/~/2

~-

1

1

½

1

2

3

½

1

71-

--~/2

1 2

2~

5/V2 --3/~/2

1 1

½ ~

1/,/2 5/~/2

2

~

--3/~/2 1/,/2

1

½

1

,~.

2 2 3

23 ~ ~-

--3~/2/5 21/5~/2 --3/~/2

1

~

--3/~/2

2 2 3

,~ ~ ~-

9/5,/2 21/5~/2 --3/~/2

~/2

1

23

--3/~/2

2 2 3 3 4

:~ ~ ~ 7 7

9/5~,/2 3~/2/5 --3~/2/7 15/7V'2 --5/3~/2

2 3 3 4

~ ~ ,~ ~-

--3/~/2 15/7~2 27/7~/2 --3/~/2

4. R e a c t i o n s with Polarized Projectiles

It was shown by Satchler 7) that the differential cross-section for a nuclear reaction p r o d u c e d by polarized projectiles is related to the polarization o f particles emitted in the reverse reaction by the following relation: pol. =

~

unp. 1-- S+ 1 Pi "Pii

,

(4.1)

where Pi and P , are the polarization vectors o f a particle o f spin s with i referring to the particle initiating the reaction and ii to that emitted in the inverse reaction. In the case o f stripping reactions, for example, P~a will refer to the polarization o f the emitted particles in the corresponding pick-up reactions. N o t i n g that the sign o f the

NUCLEAR REACTIONS (II)

375

polarization is positive if the polarization vector is parallel to ka+b x k, in stripping reactions and to ka x k a + b in pickup reactions, and using the expressions (2.17) for Pii, one finds

•

S.+b+ 1 4Sa+bJb

unp.

X Sa(sa.4- 1 ) - S. + b(Sa+b+ 1)--Jb(Jb'4- 1) Jb(Jb + 1)--Jb(Jb-{- 1 ) -

jb+l

l(l + 1) Pi " (l)/1]

/

/+1

Here ( l ) is the expectation value of the orbital momentum in the stripping reaction. Using eqs. (2.7), one can express this quantity in terms of the polarization of emitted particles in the stripping reactions, and hence one finds ,

where 3Sa B, + b, b --

Sa(Sa+l)--Sa+b(Sa+b+l)--A(.]b+l)

S,+b+ 1 Sa+b(Sa+b+ 1)--S,(Sa+ a)--jb(Jb+ 1)

•

(4.3)

This result has been derived previously 3) on the basis of quantal calculations. In the case of pick-up reactions, following the same steps and substituting eqs. (2.18) and (2.8) into eqs. (4.1), one finds pol.

~

.

-- Ba, a+bPi. Pa+b],

(4.4)

where P, +b is the polarization of outgoing particles (a + b) in the same reaction with unpolarized incident beam, and Ba, a+b -- 3Sa+b Sa+b(Sa+b4- 1)--Sa(Sa+ 1)--jb(jb+ 1). Sa-~-1 Sa(Sa-4-1)--Sa+b(Sa+b-4- l)--jb(Jb-{- 1)

(4.5)

This coefficient is calculated for some pick-up reactions and the results are shown in table 2. It is interesting to note that, comparing the values for the coefficients Ba, a+b for pick-up reactions and the coefficients B, + b, a for stripping reactions 4), one finds TABLE 2

Cross-section of pickup reactions with polarized incident beams Reaction

(da/d.Q)pol '

(p, d), (n, d)a)

(da/d-Q)unp.(1 +½Pi" Pa+b)

(d, aHe), (d, t)

(da/d-Q)unp.(1 +3Pi" Pa+b)

(n, aHe), (p, t)

(da/d-Q)unp.(1 --Pi" Pa+b)

a) This result agrees with the quantal calculations of Vysotsky and Sitenko a).

376

A. Y. ABUL-MAGD

AND

M. EL-NADI

that the coefficients B are equal for b o t h these collisions when (a) the spin o f the incident projectile in the first process is equal to the spin o f the incident projectile in the second process and the spin o f the outgoing particle in the first process is equal to the spin o f the outgoing particle in the second and (b) when the total angular m o m e n t u m transfer Jb is equal in b o t h cases. Let us consider the direct exchange reaction a+T ~ b+R, where T = b+CandR=

a+C.

The cross-section with polarized beam can be related to the polarization Pa o f the residual nucleus R in the same reactions with unpolarized incident beam. Actually, substituting eqs. (20), and (22) o f I into eq. (4.1), one finds:

where

Ba b -- 6SR Ja(Ja-t-1) , Sa..}_1 Sc(Sc-t-a)--SR(SR"I-1)--Ja(Ja+I)

Ja(Ja-t-1)-Sa(Sa-t-1)--La(La-t-1)

(4.7)

Sa(SaWl)-Ja(Ja-Fl)-La(La-t-1 ) '

where Ja is the total angular m o m e n t u m o f the particle a in the residual nucleus. I n conclusion, the authors express their thanks to G. L. Vysotsky for helpful discussions and useful comments.

References 1) A. Y. Abul-Magd and M. E1-Nadi, Nuclear Physics 58 (1964) 439 2) L. C. Biedenharn, in Nuclear spectroscopy, ed. by F. Ajzenberg-Selove (Academic Press, New York, 1960) chapt. V.C; L. C. Biedenharn and G. R. Satchler, Suppl. Helv. Phys. Acta 6 (1961) 372 3) D. Robson, Nuclear Physics 22 (1961) 34 4) A. Y. Abul-Magd, M. E1-Nadi and G. L. Vysotsky, Nuclear Physics, to be published 5) G. R. Satchler, Nuclear Physics 16 (1960) 674 6) W. Tobocman and G. R. Satchler, Phys. Rev. 118 (1960) 1566 7) G. R. Satchler, Nuclear Physics 8 (1958) 65 8) G. L. Vysotsky and A. G. Sitenko, JETP (Soviet Physics) 4 (1959) 812