Theory of polarization phenomena in direct nuclear reactions

Nuclear Physics 57 (1964) 421 --434; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permission from the publisher

T H E O R Y OF P O L A R I Z A T I O N P H E N O M E N A IN DIRECT NUCLEAR REACTIONS R. A. ABDEL-LATIF and G. L. VYSOTSKY t U.A.R. Atomic Energy Establishment, Cairo, U.A.R.

Received 13 December 1963 Polarization phenomena in low-energy deuteron stripping reactions in the vicinity o f a resonance in deuteron elastic scattering were considered. It was assumed that the proton width of this resonance is small and therefore the contribution of compound processes in the reaction channel was neglected. In this approximation a number of relations connecting quantities observed in polarization experiments was obtained. In the case in = 0 (In being the orbital momentum of the captured neutron), the considered mechanism can be responsible for the major part of the contribution to the polarization effects in the reaction channel. This is due to the appearance of these effects in the elastic channel because of the interference between potential and resonance scattering. In the case when the resonance is due to s-deuterons only, the cross section was found to depend on deuteron alignment.

Abstract:

1. Introduction In the last few years an appreciable araount of attention has been paid to the theoretical 1-5) and experimental investigation of the polarization phenomena accompanying direct nuclear reactions. The interest in these questions is connected with the fact that the polarization phenomena can give additional information on the mechanism of direct nuclear reactions. From this point of view the most widely investigated energy range is that starting from several MeV. In this energy region the method of distorted waves is used where distortion is caused by the elastic optical potential. This method was used for the study of polarization phenomena in refs. 1-3). In the past few years it was found 12) that the stripping mechanism can take place at very low energies (about 1-2 MeV). At these energies the contribution of the resonance processes can play an important role; in particular, the resonances in deuteron elastic scattering can appear at least for some nuclei for which the excitation energy is low enough to allow the isolated resonances to be excited. Buck and Satchler 11) investigated the influence of the resonance behaviour of the scattering phase on proton polarization. They found that if the number of phases contributing to the reaction is great, the resonance behaviour of one phase does not change the polarization appreciably. However, in their work the coupling of angular momenta into a compound spin was not considered. This is essentially taken into account by considering the interference between potential and resonance scattering due to which the polarization in the deuteron elastic channel can appear and the latter contributes to the polarization in the reaction channel. t On leave from Academy of Sciences of U.S.S.R. 421

422

It. A. ABDEL-LATIF AND G. L. VYSOTSKY

The appearance of polarization in the vicinity of a resonance in the elastic channel can be of importance not only to the (d, p) reaction but also to all direct processes. The experimental investigation of this phenomenon is of interest. Under the assumption that the proton partial widths of the resonances in the reaction channel and in the elastic channel are small (the resonance is observed for deuterons and is not observed for protons), we neglect the compound formation in the reaction channel and in the proton elastic channel. We take into account the resonance distortion in the deuteron elastic wave function only. Moreover, we shall assume that the contribution of the polarization resulting from the interference in the deuteron elastic channel is greater than that of the polarization connected with the spin-orbit interaction which we neglect. This assumption is reasonable if the elastic deuteron resonance is sufficiently strong. 2. General Theory In the present work we shall be interested in the most general qualitative features of polarization phenomena connected with such a mechanism, namely, in different relations between quantities observed in polarization experiments. These relations will be independent of the parameters of the interaction, the phases of potential scattering, the cut-off radius, etc. Under these assumptions the amplitude 0f the' (d, p) reaction will be of the form (in obvious notation)

•

m ~ ~|nJnmnmJn~n 2 "O'~jn(½#p½#Jlgd)(la'mn½#.ljnmj~) ×(j. rnjni#i[Jlzj)Jtnm." ~ Ol,a. fl( ldJdJo'*Jdld*) |njnJdjd*Idld* nlnmjnmJdmld*mdrtld*Mc~d*[~|*

×(Idmdlltdlj

d

" m i d Igi]Jc . myd)(Jd Mc)(Jd. , m J,a. t/~i* [J¢ Mc)(/d* md* l#d* lid.* m j *~ )

×(½#p½#n[l#*)(Inrnn½#Jjnmj~)(jnmjjlfflJ#j)YZ*,~d(~) J,.m.,d.md.} ,

(1)

where J,~m. = J'.-.','-d"

=

E-~-½iF4nitd*

f

dr~b~;)'(r)

k'*(knr) z*,°m. (r-r)(+) k,n(knR) ~bk~(r),

l'd*(kd R)..I... f ~ "'-)*"" t,J

(2)

k,o(knl~)kt°(knr) Y~* ,..(r) ,o.o x

kar

Here O~p(r) is the proton elastic scattering wave-function; Okd(r) is the non-resonance part of the deuteron elastic scattering wave-function; I and 0 are the Coulomb functions; fl and F are potential deuteron and total widths, respectively., To describe the polarization phenomena, one has to evaluate the quantity a-to) ~4KJLFRy ~uQarJ, which is connected in the following way with spin-tensors (T+!M), (T+LQ),

423

POLARIZATION PHENOMENA

and ( T +Rr> describing the polarization of the incident deuterons, initial nuclei, residual nuclei and outgoing protons, respectively: T r p f = j M QLE , (T+LQ>A ( J = [j]-~r,

QLFG R) ,

= [½]-~r,

T r p ~ = [J]½[½]~su,~"rQ(T+SU>A ( JM L

(4)

Is] = 2s+1, (5)

0 00)"

(6)

The quantity A~(uJ LQF~ ~) is given by

a

(J

L F R) Q G =

X

/4d/~f/lJ/~p

T. Z :u ZLQT r° T.Rr T.* ~d~,~j~ ~d~'d U,~', ~ ' J ~.~'~ ~'j~'.u'~',,

(7)

kq where T~,,,, = (-)~- ='(sms-m' [kq) for particles with spin s. In the case of unpolarized particles = 8kO5qo['S]-½.

It is convenient to perform the summation over magnetic quantum numbers by means of the graphical method ~6). • The quantity A(~J L F~Ro T) can be represented by a sum of three terms: A = A(°)+A(~)+A(2), (8) where A(°) depends on the non-resonance distortion only, AO) is the sum of the two interference terms and is of the order of fl/F, and A(2) is the term depending on resonance distortion only and is of the order of (fl/F) 2. We shall retain only the first two terms and neglect A(2). The quantity A(°) was calculated previously a-~o), and in the j~ representation is has the form

AO)( J

L F R) = B 2 Q G

~ Inl'njnJ'natbf

0 a t Vo ~.~A,"rnj'.~,-/

x [J]~[L]~[F]*[R]'~[j.]~[j'j~[j][1][al][a2][a3][b,][b2][b3]

x

½ R

i J, ll., a3

x Q+M-G

L

-T

axJtS"

½ ½ a2 j'ttaa 1'1 a 2 tlbl L b21 l" ax J~ /nJ[ln at bsJ[at J F ) a2

T+G-M-Q

where

~

Pb3T+~-u-Q(I~'In)'

(9)

4m2V V2m t2~ th2R fo = R+T+L+Q+J+G+l~+j.+j'~+b~+ba+a2+a3+l, pkq(l, l') = ~,, (-)r-m'(lrnl'- m'lkq)JtmJ*_m,, mm"

(10)

424

R. A. ABDEL-LAT1FAND G. L. VYSOTSKY

and It kl

12 k2

13 k~

,

are the 9j and 6.] symbols 16). The two interference terms A(~1), A(21)that make up the quantity A (1) are related by

(_)M+Q+G+TA(1), (-- JM - QL

A(11) (JM L F R ) Q

G

F

-G

--

~)

'

(11,

and this relation allows one to evaluate only A(~1). After the summation over magnetic quantum numbers, A t~) takes the form

A(t) (JM L F R ) = B 2 ~ O,MnOrn.r.fl(idJdJcj*l~)(--) fa Q G lnl'njnj'nalb, x {[J][LJ[F][R][jn][j'][jd][j~][at][as]}½[j][1][Jc][a2][aa][a4J[bl][b2] J¢ 1 x i I* b2 ½ e as 1 ½ j" i

I j*

al a2 In j b2 Jcllb2

j.

a4

a2 ;

(

a"

x ot+Q-G-T T-o~ dr.__~a+L+F+R+]a+lasr~(1)*~(l,

J

~(--)~

a3

G+T-ot

._LQ)(

1

ka 3 L a4J

½

G+ T - Mas- Q - ~ a3

°2

G+T-o: ~ - T

a,

o:+Q- T - G

)

_FG)

{Pa"(l"ld)PasT+t;-Q-M-'(ldl") I*~(2)*

(1 I' ~).

I " a l - ~'~,% *d ]l'aa¢~ - T - G + M + Q~, "d *n .IS

(12)

where (1) * pa~t(lnld) = ~ (--)~d*-md*(lnmnl*--m*lalOtl)Jt.m.ld*md*,

(13)

mnrdd*

with

Jt,~,,,,~l*d,,,*.,given

by (3) and

p(2) [1 tn-m'n (Idmdl.--m. , , as ~5)Ytd*md , (k:~ Jr.m'n ,,,:.'dl'n)= E (--) * nldm" n \kd/ with J~.~,. given by (2). Further,

( Jl J2 Ja J4 J5 } II 12 13 14 15 kl k2 k3 k4 ks is the so-called 15j-syrnbol of the 4th kind 16) and

{Jl J2 J3 J4 ) 11 12 13 14 kl k2 k 3 k4

(14)

POLARIZATION PHENOMENA

425

the so-called 12j-symbol for the first kind 16). Finally,

f i = Q+G+i+J'n+J*+l*+a3+a4+b2. From the definitions (10), (13) and (14) for P,~, ,-o~ nt:) and vo~,, ^t2) respectively, it follows that if the axis of quantization is chosen perpendicular to the reaction plane, the quantities q, ~x +0q must be even. This implies that for A

I~, Q'~ G ~):~,°,t~,'

Q

~G ~) +ACt) (~,'~Q ~G ~) '

M + Q + G + T = even,

(15) (16)

where At°), A~1) are given by (9) and (12) respectively. 3. Special Cases 3.1. s-DEUTERON RESONANCE, SPINLESS T A R G E T

Let us now consider some particular cases following from the general expression (15). First of all we shall consider the case when the resonance is due to the s-deuteron wave (ld = 0). In low energy experiments this situation is quite probable. The theory taking only spin-orbit interaction between the captured neutron and the nucleus into account is contained in the quantities A~°)(~ ~ ~ ~.). In this approximation the polarization phenomena in (d, p) reaction were considered by many authors. The cross-section of the reaction (d, p) in the case of polarized incident deuterons depends on the polarization of the deuterons but not on their alignment. The relation between the cross-section with polarized deuterons and the proton polarization when the deuterons are not polarized, in terms of the quantities A~°), has the following form:

~,o,~1o 0 o ~) 0 0

=~/6,

(17)

~'o'(°0 °°00 '0) ~o~ 0000 °0) 0

(18)

This leads to the known formula 4) (dtr)

~

= (l + 3pOp. pd) ( da) ~o"

(19)

However, in the case of a resonance caused by s-deuterons, the cross-section of the reaction appears to depend also on the alignment of the deuterons, although no polarization in the elastic channel occurs. This is explained by the interference between the elastic channel and the reaction channel.

426

R. A. ABDEL-LATIF AND G. L. VYSOTSKY

It could be easily verified that the following relation holds:

o 0 00) 0 AO)(~

0

= (_)tn+so+,+~ [1] ~r

0 0 10) 0 O

[½]t-

EO ×

,.j.o,~,o(-)

1. " ½ " ½( j'n j In jn In } Do]D.]

i

~*J'=

EO

0

½

i

½ 1

1

1

J¢

½

1

(20)

• ½ ", *[ In 1" jn j j" ]

lajn lnj'n[Jn]

m'.

[Jn]

1 ½ i i ½ ½ ½ 1 Jo I

The fact that the above mentioned effect is connected with the possibility of the deuteron spin-flip in the elastic channel can be easily seen from the following consideration. If the spin of the target nucleus is zero, no change of the deuteron spin projection is allowed and

00 00 00) =

O;

(21)

the cross-section ceases to depend on the deuteron alignment and eq. (20) becomes

00 A"'(O00 A.,(00000

= 46,

(22)

which in combination with (17), (18) and (20) yields the usual formula (19). However, this formula is now valid on account of the resonance distortion of the deuteron waves in the elastic channel. 3.2. s-DEUTERON RESONANCE, N O N - Z E R O T A R G E T SPIN

In the case when the spin of the target nucleus is not zero the dependence of the cross-section on alignment can be expressed in terms of quantities observed in polarization experiments. For this purpose let us consider the following three polarization experiments performed in the vicinity of the s-deuteron resonance. (i) The measurement of the cross-section with aligned incident deuterons (for the sake of simplicity we assume that the deuterons are not polarized and that from the parameters determining the alignment only (Ta+2°> is non-zero). (ii) Angular correlation measurements. From the experiments on angular correlations of y-quanta, one can determine the parameters with even F. This can be easily seen from the following formula giving the expansion of the angular dis-

POLARIZATION

427

PHENOMENA

tributions in terms of spherical harmonics:

W(~, tp)oc ~., {JF' j L} (L1L-IIF0)~ Y~(~,tp), e even

L

j

(23)

o

where the transition of multipolarity L occ~s from the state with spin j to the state with spin j'. From the parameters with even F thus obtained, we shall be interested only in . (iii) The measurement of the cross-section with aligned initial nuclei 14) (for simplicity we assume also that the nuclei are not polarized and the ali~ment is such that only is non-zero). Experiment (i) is described since A(°)(o~o o oo) = 0, by the following formula: (24) where (d~/dEJ)] and (d~/dl2)o axe the cross-sections in the ease of aligned and unpolarized deuterons, respectively. The quantity
<~:2o> (~)o=

~1~, ~+~' {:' (Ooooo~ Oo)++.(OoO~oo°o)/<~"

Finally experiment (iii) is described by

where (d~/dEJ)L is the cross-section with aligned initial nuclei, the alignment of which is determined by . It can easily be verified that the following relations hold (for definite/,):

E1 =

0~ X) +(~oo +(°o ~°oo ~) _

~. 0 " [ ~so+J'.ri 1½r,,1+ Jn J'~ [J]+J,,.r,, laJavlaJ'n~'--] LJn..I LJn_l j j o,~o

Ad':

,,+.D3 D3

n

i

(if only one value ofjn contributes to the reaction, then

{,o J: ~}

E 1 -

r j] + j

J

),

['i]+ {~" J" 21 i

j

j

i

In 2 Jn

2 jn

ln)

l,,J

428

E2

R . A . ABDEL-LATIF AND G. L. VYSOTSKY

a'l'(~o

0 0 0 0

2 o~ 0 0] _ [j]* (_)'=-J~+'+~r 0 00) [1]½ 0

to½½

"' ~t [½ij."'In2 Jn 1 1 O,ojoO,orn(--)j,~+j'n [J.]• ~[I.3 J=J'~ j j i Jo ( j'. J l~ j. In X Ol~jOl~j%(--)Jn[jn]½[jrn]½1 ½ i 2 ½ JnJ'n ½ 1 1 J~ 1 ~,,

E3 ~

A'"(~ 0 0 a,,,(~ oo00 ~) E,~'

(28)

{"1 "i}

E OlnJ~Ol~J%(--)J°[Jn]*[J'n]* 1 j~

X

,

j~d'n

j

1

i

i

J¢

2 n

,

(29)

J'n J In Jn In } X Ol=jaOl*J'a(--)Jn[jn]½[jPn]½ i ½ i 2 ½ jnj'n ½ 1 1 Jc 1

where

A J2 J3 A J5 1 It

kl

12 ls 14 ls k2 k3 k, k5

is the so-called 15j-symbol of the 2nd kind 16). The ratios El, E2, E3 are angle-independent quantities. They are determined by the neutron reduced widths and the above mentioned combinations of angular momenta. With the help of relations (27)-(29) and formulae (24)-(26) one can easily express the dependence of the cross-section on alignment through angular correlations and the cross-section with aligned nuclei. This yeilds the following formula: dG

<~:,o>

dff

(~)o

~,~,<~+~o> (~)o

- [j]~[1]~,(30)

where El, E2 and E a are given by (27), (28) and (29) respectively. Another formula relating experimentally observed quantities follows as a result of considering the following experiments:

429

POLARIZATION PHENOMENA

(1) The measurements of proton polarization, which is given by
[j]~r {A(O)(00 0 0 10) +A (t)(00 0 0 10)}. (31, Ill*Ill * 0 0

(2) The measurement of the cross-section with polarized deuterons

(~)~-(~)o [~,[,~,~+,o>~ ... {~,o,(looooo Oo)+A'"(lo°°oo ~)}'~' where we have assumed for simplicity that the deuterons are not aligned; otherwise, it is necessary to perform several experiments of this kind to eliminate the dependence on alignment. (3) The measurement of circular polarization of y-quanta in the (d, p?) reaction. Such an experiment allows one to determine the parameters (T#+F°) with odd F. We shall be interested only in ( T f lO). If from all the parameters ( T f ~°) with odd F only ( T f to) is non-zero then the circular polarization C of ?-quanta irradiated in the direction 8 is given by is)

C = (-)2J Ft(LLj[i )
(33)

where

Fk(LLj~]) = ( - )I-J-t[j]t[L](L1L- I[kO)W(jTLL, kj'), if the transition of multipolarity L occurs between states with spins j and j'. The parameter (Tf lo) is given by (T+I°) (d~)o-

[½]½ {A(°)(00 0 1 ~)+A(t)(00 [i]*[q~ 0 o

o, ~)} 0 0

,~,,

(4) The measurement of the cross-section with polarized initial nuclei (on the same grounds as in experiment (2) we assume that only the polarization (T~+ to) is nonzero). Then

(~)~-(~)o ~'~<~;~°>{A'°'(~'° ~)+a., (°oo ° [13 +

o

o

The quantitiesa(°)zt o -a(o)roo, -- ,o o o oO),A(O)(oOot o o), ,oo o o) and A(°)(o°o° o° ~) are proportional to Plo(l.ln); hence the ratio of any two of them does not depend on the proton angle. A similar situation occurs for the corresponding quantities Ao) which are proportional to - - 0~

~ ' V l n ~ t P I n - - o t ~ Pln--eLIdln~t

j"

430

R. A. ABDEL-LATIF

AND

G. L. VYSOTSKY

Thus formulae (31)-(34) can be rewritten as follows: d(r

~,~, (~), (~)o =

A(°)+A (1),

(36)

HtA(°)+G1A(t)'

(37)

[i]½[1]½ [½]½ (da) ~ 0 = H2 A (°)+ G2 A(1),

(38)

[/]~[½]~



[i]~}[1]~} [1]½ o (da) ~ o=

do (~)o [1],[½]~-":

(39)

= H 3 A ( ° ) + G 3 A (1).

<~+I0>

Here A(°) - A(°) (10 0000 ~), A(°)'(t) (~ 0 00 ~) 0 Ht, GI = 0 A(°),(1)(100 00 ~)

A(t)---A(t)(1000

00

~)'

A(O),O)(00100 ,

H2,

G2

~)

=

a(O),O)(100000

a,o,.,(~ 10oo ~) Ha, Ga =

(4o)

A(°)'(a)(; 0000 ~) The quantities H i, Gi can be easily evaluated by means of (9), (12). The above set of equations (36), (37), (38) and (39) allows One to relate any three of the four experimental quantities found on the left-hand sides of these equations. 3.3. NEUTRON CAPTURED WITH ZERO ANGULAR MOMENTUM Let us now proceed to another particular case following from the general expression (15). This is the case when the neutron is captured with zero orbital momentum (/,, = 0).

In the case 1. = 0 the polarization phenomena due to the neutron spin-orbit interaction and which are described by the quantities A(°)(~t ~ ~rgr) disappear. For, if in r) only one index of the rank of spin-tensors is non-zero, the corresponding A (°) is equal to zero: a,O,

(~00 ~)=a,O,(~L0 ~)=A,o,(~0~ ~) 0 0 Q 0 0 G

= A ( ° ) ( 0 0 0 o 0 R) =0"

(41)

POLARIZATION PHENObfHN&

431

If tWO of the upper indices are not zero the quantity A (°~ may be non-zero, but in this case it describes the trivial process of transferring the polarization from the initial system to the final system. Thus in the case 1~ = 0 all the information on the polarization is contained in the quantities ~(t)d ~ f Qr.Ge R~ TJ" We assume that only one value of the deuteron orbital momentum contributes to the elastic deuteron resonance (ld = l*) and consider the following quantities observed in polarization experiments: (a) The cross-section with polarized incident deuterons (d,/dfJ)j. In the case l~ = 0 this quantity is given by

\dO/o

[i]~

00 + X M < T d + 2 " > A O ' ( 2 0 0 00 ~)}"

(42)

(b) The proton polarization pO = ~/~o: (43) (c) Angular correlations in the (d, pT) reaction:

°

d-O o = [ l ] ' } [ i ]

*t

0

G

"

(44)

Angular distributions of 7-quanta determine o (F even). (d) In angular correlations the measurement of the circular polarization of yquanta determines o (F odd). (e) The measurement of the cross-section With oriented initial nuclei (we consider this experiment in spite of its difficulty for the sake of generality): ( d ~ ) L = ( d a ] + [j]½[½]t{oACl)(~ 1 0 ~) \dl'2] o [1] ½ 0 0 Q 0 Here for sh,iplicity we limit our consideration to the case when the polarization and the alignment of higher orders (terms with , , etc.) are not present. This is valid when the spin of the target nucleus i = ½ or 1 or if the orientation is such that , , etc. are zero. Otherwise, it is necessary to perform several experiments of this kind to eliminate the dependence on these parameters. The experiments (a)-(c) are typical polarization experiments. Experiment (d) is also quite accessible (see, for instance ref. t3) where the circular polarization of 7quanta in the reaction (d, p~) on B 1° was measured).

432

R. A. ABDEL-LATIF AND G. L. VY$OTSKY

In the case of a resonance with one definite value of the orbital momentum of the deuteron 1a, all the quantities A(t) with 1 in the upper line are proportional to --0~

IYldolPld--al-- Yld--gtP|dOt J'

and quantities ACa) with 2 in the upper line and with the corresponding projection M in the lower line are proportional to

- M-a

I'FldaFld--a T l~ld--al"laot j "

Therefore, the ratio of any two A (1) with 1 in the upper line and the ratio of any two A°) with 2 in the upper line do not depend on the proton angle. This allows one to find relations connecting quantities observed in polarization experiments. Consider, for instance, the cross-section with polarized deuterons (da/dl2)s (experiment (a)), the proton polarization ( T + lo> (experiment (b)) and the coefficients SMo
M

o}, (46)

M

A(~)(1 0 0 0 0 A(I)(~

0 0 00

°0)= (_)j_~_,o [1]~ ~) [½l'

f l(did l " J oJd "*l~)(--) Jd*Dd] • ½Dd] .* ½ _i -½j j~ ½ i

JdJd*

½ J¢ 1 jd]/1 1tdJ ,ij~• ld 1 ld 1}

"J~ J~ Ja Id ld ] i i 1 1 Z fl(/aJdJc j*ld)[jd]½[J~] ½

½ lj

jdjd*

½ J

and 00

a(1)(2

(47)

0

0

K2 = (__)M

½

= (_yj+,+,. [1]* [j]trl

Z # ( l d " Ja J eJd/d)(-)

Jdjd*

[Ja] [Ja]

i

1

Jd la la

X

E fl(/a Jd Je j*ld)[Jd]~[j~] ½

JdJa*

(JJ i1 J~ 2 ½ Jd J* ld ld 1

(48)

433

POLARIZATION PHENOMENA

In the case of i = 0, K 1 = x/½ and the term depending on the alignment in (46) vanishes. It is of interest to note that the same dependence of the cross-section on the deuteron polarization and the polarization of protons in the case of unpolarized deuterons was obtained by Goldfarb and Johnson 6) in the case In = 0 and taking into account spin-orbit interaction between deuterons and initial nuclei (the proton spin-orbit interaction was switched off). Apparently this result is connected with the possibility of the deuteron spin-flip in the elastic channel and does not depend on the way the deuteron spin is flipped. The consideration of the quantities o, o (determined in experiments (b), (d)) leads to the following formula:

o

[J]* rao '

(49)

where 0

r~ =

0

0 0 01

= (_y+,+,o+,~

[½]*

l-j]*

00)

E fl(IdJdJ©j~ld)[Jd]*[j~] ½

.raid* X

JdJd*

fl(ld Jd J, J*ld)[Jd]*[J~] ~

i

i

1

1

½ j ½½ (i i J J ½ ½1)"(50) J,

1

Jd Jd "

°*

½

Id ld 1

In the case i = 0, K 3 = - 1 and formula (49) takes the form

o

=-(rj

+,o )o,

(50

i.e. the polarization of the residual nuclei (j = ½) and the polarization of protons are equal in magnitude and opposite in direction. If we now consider the cross-section with oriented nuclei, the polarization of protons and the angular correlations we obtain

= [ ½ ] * [ j i l t , E aeo< Tf 1°>o 0

+[i]~[j]~K5~_,(-)°(Ti+Z°>(TZ-e)o, Q

(52)

434

R. A. ABDEL-LATIF AND G. L. VYSOTSKY

where

10 ~) o o

K4

(53)

(54)

K 5 = (-)Q o o oo

The quantities/(4 and K s can be obtained from the general expression (12) and as was stated previously they d9 not depend on the proton angle. Analogously, one can obiain relations including the cross-section with polarized deuterons, the cross-section with polarized nuclei and the circular polarization of v-quanta, etc. We should like to emphasize once more that the relations obtained depend solely on the ratio of the widths of the elastic resonance. The authors would like to express theirgratitude to Professor M. E1-Nadi for useful discussions and to Miss N. Dobryntchenko for her help in preparing the manuscript• One of the authors (G.L.V.) wishes to express his thanks to the authorities of the U.A.R. Atomic Energy Establishment for their hospitality.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

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