$$^3 \overrightarrow {He} $$ (γ, π°)$$^3 \overrightarrow {He} $$ as the nuclear probeas the nuclear probe

PramAna, Vol. 17, No. 4, October 1981, pp. 337-352. © Printed in India.

SHe (v, 9 °) SHe as the nuclear probe

G RAMACHANDRAN, R S KESHAVAMURTHY and K VENKATESH*

Department of Physics, University of Mysore, Mysore 570 006, India

*Present address: Department of Physics, MS Ramiah Institute of Technology, Bangalore 560 054

MS received 8 June 1981

Abstract. The state of polarization of the recoil nucleus in ~, + SHe --, ~° + 8He as

well as the asymmetry in the differential cross-section when the initial SHe is polarized

are studied together with the differential cross-section taking into consideration the ,.% S' and D-state admixtures in the nuclear wavcfunctions. In view of the consider- able spin dependence in the photoproduction amplitudes these observables arc found to be quite sensitive to the small admixtures of S' and D-states in the nuclear wave- functions.

Keywords. Differential cross section; target asymmetry; recoil nucleus polarization;

S, S', D-state admixtures in; SHe nuclear wavefunction; nuclear structure; multipolc amplitudes.

1, Introdnction

The study of the three-nucleon ground state has been pursued with renewed interest during the last decade. Following the extensive electron scattering studies made earlier as part of the saga of high energy electron scattering experiments, we have now several theoretical solutions of the dynamical equations for the three-nucleon ground state with realistic potentials (see for example Sick 1981 ; Payne 1981 ; Drechsel 1980 where extensive references to the literature can be found). However it has been found that (McCarthy et al 1977) there is no agreement between experiment and theory especially around Q~ = 11 fm -~ for the charge form factor. A recent study of the magnetic form factor (Riska 1980) with simple wavefunction models a~d taking into account the pion and p-meson exchange current effect shows that the single- nucleon current contribution depends strongly on the D-state probability. Since the new data (Arnold et al 1978; Riska 1980; Sick 1981) on the SHe form factors at large Q2 indicate that the existing microscopic calculations of the wavefunctions are missing an important ingredient, it appears quite reasonable to use the wavefunctions for the nuclear systems following the traditional enumeration of Sachs (1953); Schiff (1964); Gibson and Schiff (1965), where free parameters can be adjusted in order to reproduce the experimental data. In fact this approach was used by Lazard and Marie (1973) in their discussions of photoproduction of charged pions on SHe.

Here, the photoproduction of neutral pions on SHe is studied since the reaction offers several advantages as a tool to study the nuclear structure (i) corrections due to Coulomb interaction between the pions and the nucleus in the final state or between the target and the beam in the initial state are absent, (ii) the momentum transfers 337

338 G Ramachandran, R S Keshavaraurthy and K Venkatesh

involved here are high even for comparatively low beam energies as this is a produc- tion process, (iii) the spin-dependent amplitudes in the photoproduction reaction are comparable to the spin-independent amplitudes unlike in electron scattering where the spin-independent Coulomb scattering is very dominant and the ~pin- dependent magnetic scattering is almost zero relatively except at the backward angles, and (iv) in view of the fact that observables like asymmetry and polarization are more sensitive probes to the small admixtures in the amplitudes, it is considered desirable to investigate the effects of S' and D-state admixtures in the waveftmctions by measur- ing the differential cross-section on polarized SHe nuclei or by measuring the recoil SHe polarization. In § 2 is outlined an elegant theoretical method to calculate the differential cross-section, target asymmetry and recoil nuclear polarization taking into account all possible admixtures of wavefunctions in the nucleus. In § 3 the admixture of states following basically the classification of S~.chs (1953), Schiff (1964, 1965) and Gibson and Schiff 0965) is considered specifically; typical numeri- cal estimates for the various observables are presented and the sensitivity of the observables to S' and D-state admixtures is pointed out. For simplicity the Gaussian forms for the radial wave-functions and the well-known photoproduction multipole amplitudes due to Berends and Donnachie (1975) are used taking into account all the four (l = 0, 1, 2, 3) partial waves.

2. Method of calculation

The earliest calculations on photoproduction of neutral pions on 3He were made by Ramachandran and Ananthanarayanan (1964), using the Ct3LN amplitudes and taking into account the S-state in the nuclear wavefunction. We follow essentially the same formalism here and take into account all the admixtures of S', P and D states in addition to the dominant S-state. The amplitude for the reaction may there- fore be written using PWtA in the form

3

( m ' l T j m ) = ( m ' I ~ t~exp(iQ.rj) lm), (1) j = l

where m and m' denote the initial and final magnetic quantum numbers of the nuclear spin states and tj denotes the basic amplitude for ? ¥ --> ,t°N in the momentum space on a nucleon whose position coordinates are denoted by rj. Q denotes the momentum transfer to the nucleus

Q = k - q, (2)

where k is the momentum of the incoming photon and q denotes the momentum of the outgoing ,r °. The nuclear wavefunction ~b" may be written following Sachs (1953);

Schiff (1964) and Gibson and Schiff (1965) in the form 8

#'--- ~ a, ~//~, (3)

i=l

where [ at I ~ denote the relative probabilities of each of the admixture states if ~m as well as each of the ~ ' are normalized to unity. Connection with the form of the

3He

(y, ~r °)

SHe as the nuclear probe

339 nuclear wave-functions generally used in the context of Faddeev calculations (Brandenburg

et al

1975, Hajduk

et al

1980) can readily be established by expanding each of the ~b~' in the form

= ~ C~ I [(Lp L,) L (½ S n) S] ½ m; (½ T~s) ½ ½),

(4)

~7

in terms of

L-S

coupling states where a denotes collectively the quantum numbers

- {Lp, z,, z, &~, s, r,.~) and

c? = ([(zp z,) L (½ &3) s] ½ m; (½ r2~) ½ ½ [ ~7).

Altcmativel) one can also expand each of the ~b~' in the form

(5)

~b~ = ~ B~ I [(Lp ½)

Jp (L, S2a) J,] ½ m; (½ T2a) ½ ½ ),

(6) in terms of

thej-j

coupling states, where fl denotes collectively the quantum numbers fl ~ -(Lp, Jm L,, S,3, :,, r~3 ) and

B~ = (((Lp ½) J, (L, Sza) Y,) ½ m; (½ r2s) ½ ½1 ~bT).

(7) An explicit evaluation of the coefficients C~ has recently been carried out by Keshava- murthy and Ramachandran (1981) and B~ are related to these coefficients through

fLp ½ J P l

s,~ = ~ L, s,3 J, [Jp] [41 ILl is] c~, (8)

{2

L S ½

where [j] = (2j + 1) 1/2 and .[ } denotes the standard Wigner 9j symbol. The Jacobi coordinates'~ and r or the equivalent R 1 and R~ of Gibson and Schiff (1965) are related to the nucleon coordinates rj through

-~p= r s - - - - r ~ + r s _ x/3 R1, (9a)

2 2

r = r 3 - - r 2 : - - g 2 ,

(9b)

in terms of which the amplitude (1) may be rewritten as

(m' I T[ m) = 3 (~b m, ] t z exp (iQ" R1/31/2) [ ~bm),

(1o)

after eliminating the over all momentum conservation factors in the usual way so that the differential cross-section may be written as

d~ _ _ q l

an k 4 I ( m ' l r l m ) l ~, (11)

E , m t m t

where ¢ denotes photon polarization.

Table 1.

S-state

340 G Ramachandran, R S Keshavamurthy and K Venkatesh

2.1 Evaluation of the matrix element

The matrix element (~bm, l t 1 exp(iQ'R1/311")l~m) may be evaluated in either of the two alternative ways corresponding to the expansion of the states following (4) or (6) respectively. Using (4), for example, we have

( m ' [ T l m ) = 3 ~ a T a t f ( ~ ) a / 2 R ~ d R z f R ~ d R 2C~'*C~

i,A a,a'

× ([(L'p L'~)L' (½ S~a)S' ] ½ m';(½ T~a) ½ ½

I

t 1 exp (iQ. R1/3 l/z) J [(L? L,) L(½ S2a) S] ½ m; (½ Tan ) ½ ½). (12) Noting that the operator acts only on the nucleon labelled l, we see readily that

L', = z , , = s23, I ; 3 : (13)

Moreover, if we are interested in calculating only those terms that are connected to the dominant S-state, the P states ~b a, ~b 4, ~b 5 get eliminated from our considerations.

Further Lp --- L since L, ~ 0. In this approximation the states that contribute to the summations in (12) are listed in table 1. The summation over T~a and S~a are not independent since antisymmetry requires that T~3 be zero if $23 ---- 1 and T~3 = 1 if $23 :- 0. It is very important to note here that while the quartet spin states are not connected to the dominant S-states in the context of the charge form factor, they contribute considerably to (12), since t 1 has the general structure

t 1 = i¢ 1 • K(I : 0) + L(I = 0) + *1, [i¢1" K(I = 1) + L(I = 1)1, (14) and the spin dependent amplitudes K are comparable to the spin independent ampli- tudes L. The isosc~lar amplitudes are denoted by putting 1 = 0 within brackets while the isovcctor amplitudes are denoted by I = 1. We shall not consider here the possible existence of isotensor amplitudes (Dombey and Kabir 1966; Donnachie and

The quantum numbers of the dominant S-state and states connected to the

$1 Lp L, L S~3 S T3,

0 0 0 1 ½ 0

¢1 o o o o ½ 1

o o o 1 ½ o

0 0 0 0 ½ 1

~6 2 0 2 1 I 0

~7 2 0 2 1 i 0

~s 2 0 2 1 ,} 0

3He

(7, ~r °)

aHe as the nuclear probe

341 Shaw 1972), since the isotensor amplitudes can contribute to the process only through the negligible isospin T--- 3/2 admixture of states in the nuclear wavefunction.

Using spherical tensor notation we write the operator in the form

(QRI~ l)l+n -A

tzex p (iQ.Rz/3z/~)

= 4~ ~ ~ ~ (i)'+" j,\ V'3 / (--

l I-0, I A

n=0,1

x [(¥, (]~i)®e) ~. (¥, (Q.)®K" (/))A] ~.oI (15)

where ~o°=1, Toi----~-iz, ~r~=l, d = o i , , 4i-~-:~(ulx-bicrl')'g°(l)'~L(l)

'

K i (/)----K (I). Using standard Raoah tcdmiques the matrix clement is now evaluated to give

(m'JT Im)

=24V:~- ~ a~a~ ~ C(½1½; ½0½)[I]

i,],a,a" i,n,~,Lp

× ST,, T~, W(Tzs ½ ½ I; ½ ½) C (½ a ½; m v m') (i) l+n (--

1)/+n-a

t p

X (-- 1) u (--1) S-S' [L'] Ill [SJ [S'] [,~1 In] ILl [LpJ (--1)

Lp-Lp-L+L

X 8S2 8

S~s 3L,

L" W

(]-Jr Lp

L' 1; LL~ W (Ss3 ½ S' n; S ½)

x n A C (Lp 1L'p;

000)

(Y, (QS@K" (I))A_F,,

(16)

' S' ½J

where the radial integrals F~ are given by

F,-~- R~ dRz R~ aRa CJ'* j` \ - ~ i a"

^{0 7)}

If we restrict ourselves in (16) n o w to those transitions envisaged in table I the radial integrals that are required are only

FSo s = [3~ 3/~ a 2

FoSS'=aia,(~)anme~' f f, v, jo(QR_'~R:R:dRidR,, (19,

~a/3 /

342 G Ramachandran, R S Keshavamurthy and K Venkatesh

~ o

⁼

f :,[ s,o,:,- ~R~a8~- s 1 (aof, + a8fs) ]

R t R~ dR 1 dR. z, (20)

where S, = R~ + R~ and $1 -= R~ -- R~. The radial functions fo vl can be chosen to have tractable analytic forms containing some adjustable parameters as has been done by Gibson and Schiff (1965) and Lazard and Marie (1973).

Noting further that the ZHe nucleus has spin ½ and introducing the nuclear spin operator, J we can write (m' [ T [ m) in the form

(m' J r l m> = (m' ] 2i J. 3f + ~e I m>, (21)

where £# and 3((" denote the spin independent and spin dependent amplitudes with respect to the nuclear spin, and are given by

i,j, a,a' I,I

× W(Tza ½ ½ I; ½ 1) (i) ~' (--1) Lp-L'o-I+L (-- 1) 2S+S'+'/*

× w (LSUS'; ½ 1) ~,. 8s2 ' s;, 8z.,L; W(L, Lp L' l; LL'e)

× [Z'] It] IS] [S'] [C] [].p]

w

(s~ ½ s' n; s ½)

X C (Lp 1L'p; 000) (r, ( Q ) ~ K " (I)) ° It, (22) Ofx-v ---24V'; ~

aTa' ~ C(½1½;½0½)[IlST,,T~,

i,], a, a' 1, n, I

× W(T~3

½ ½ I; ½ ½) (i),+.-1 (__ 1),+.-1 (__

1) s-s"

[L'] [t I IS l [S' l

×

In] [L] [Lp] (--

1)Lp--L'p--L+L' W

(L, Lp L' l; L L~)

X 3S, ' S~, 3L, L; W (S~.~ ½ S' n; S ½) C (L 0 l L' O; 000)

× n (r, (9.) ® K" (0)~_~ F,. 03)

' S '

If we confine ourselves to the transitions listed in table 1, these amplitudes have the following elegant form

.~

= 2 L, (FSoS + FSS')o " + L. (FSoS -- 2 FSoS" ),

(24)

aHe (y, ~r °) SHe as the nuclear probe

343 3f'tv = -- 2 ,--~,v'rr xt .~SS'0

+ (K.)tv (FSo s- 2 F~o S')

-- (16v'~'/ [Y2 ( Q ) ~ KXlX j~.SD (25)

where L~, Kp and L,, K, denote the spin-inde~ndent and spin-dependent amplitudes for neutral pion photoproduction on protons and neutrons respectively and are related to the I = 0 and 1 = 1 amplitudes in the usual way.

In the case of j - j coupling expansion for the wavefunctions using (6) we can write

( m ' l T ] m ) = 3 ~

a~'a,(ip/z f

R2, dR~ f R : d R ~ B ~ ' * i,j,#',fl

< . . . I o j

× ((I,; ½) (L' S£a) J,) ½ m, (½ T~a ) ½ ½ t 1

exp (i -111~

x ((Lo ½) Jp (L, S~a) J,) ½ m; (~ T~a) ½ ½ > , (26) which may be simplified using (15) and standard Racah techniques to give

(m'lTlm)=24vqr ~ a'a, ~ C(½;½;½0½)[I]

i,j, [3, ff I, n, ~t, l

× 8T23T'~a

W(T2a ½ ½ I; ½ ½) C (½ a ½; m v m')

(--1) v X (i)t+n (__. 1),+n-h (___ 1)Jp-J~

8jr j," SLrL,, 8S23 S~s

x

w(J, Jp ½ ~; ½

× 1 n A

z; ½ J'~

J~) [Jp] [J,~] [;q [zA It] In] c (Lp zL~; ooo)

(r, (~) ® K. (,0'~ o,,

⁽²⁷⁾

where the radial integrals G~ are given by

o, : (~),,~ f R~ d& f R~ dR2 B.~'" i. [QRfi B,~. (28)

We may once again express (27) in the convenient form (21) with L~ and oY" now given by

.Z--12V/2-~ ~

a*a~ ~ C(½1½;½0½) [I](i) ~l

i,j, fl, fl' I , l

× ST,, ris 8,. 8Ss ~ S;~ ~J, J; ~L, L; (-- 1)Jp+LP-'+a/a [Jp] [Lp] [1] [n]

× w(r2~ ½ ½ ;; ½ ½)

W(Lp ½ L'p ½; Jp 0 C (Lp tZ'p; 000)

x (Y, (Q) ® K" (;))0 G,, (29)

344

G Ramachandran, R S Keshavamurthy and K Venkatesh

3¢F_tv=24V'w ~

a~a, ~ C(½1½;½0½)[I]Sn, T~ '

i,],V,B l,n,I

× 3j, j; 8L, L; 3S, 3 S~3 (i)'+n-1 (--

l)t+"-I (-- 1)JP -- 6 [4] [Jp] [Lp]

x

Ill In] rv(r~3 ½ ½ t; ½ ½) rv (J, Jp ½ 1; ½ J~) c (zp l L~; ooo)

x t n 1 (r, (~) ® K" (t))b, a,. (3o)

L; kS;

2.2

Differential cross-section

The elegant form (21) into which we have reduced the matrix element in either case of L - S o r j - j coupling expansions enables us immediately to write down the differential cross section as

d~ q l

k2~._ [LP £~* -k Y/"3f*]. (31)

df~ _ff

2.3

Target asymmetry and recoil nucleus polarization

If the target nucleus is polarized, the state of polarization is characterized by the density matrix 0 t which has the form

p' = ½ [1 + 2J. P'], (32)

where P~ denotes the nuclear polarization initially. The final spin state of the nucleus is then characterized by the density matrix pr given by

p ~ = [i 2J. oY" -1- .~1 p' [-- 2 J . 3F* -[- ~q~*]

_ Tr ps [1 + 2 J . PQ, (33)

2

where ps denotes the polarization of the recoil nucleus.

The cross-section with polarized targets is simply given by

d-~d~ (P9 = ]c q ½ ~ Tr p~. (34)

ff

Denoting by ~ the unit vector perpendicular to the reaction plane

_ k x q (35)

I k x q l '

SHe (~, ~r °) SHe as the nuclear probe 345 the target asymmetry A is defined through

& .~) &

A = A . ~ = ~ - ~ ( - - ~ - ~ ( - - ~ ) dcr (~) q_ ( _ ~ ) d ~

where A is given by

(36)

~ ~:[(3r~e*- ~ ' y c * ) - (3r

^x

yc*)]

A = ' ( 3 7 )

z ( i ~ e l s + l arl ~)

6

With initially unpoladzed targets the polarization ps of the recoil nucleus is also readily calculated to give

i l ( ~ * ~ -- ~ 3 f * ) q- (3f x 3f*)

P ~ ' = ~ (38)

z l ~ e I n + l ~ c I n

E

3. Numerical results and discussion

For purposes of illustration we have estimated here the differential cross section do/dl2, the target asymmetry A and the recoil nuclear polarization ps numerically at laboratory photon energies 280, 340 and 400 MeV with x - z plane as the reaction plane. The spin-dependent and spin-independent amplitudes Kp, K, and Lp, L, have been calculated in the standard way (Donnaehie 1970) using the multipole amplitudes of Berends and Donnachie (1975). Since the purpose of this paper is mainly to show the sensitivity of the observables dcr/dl2, A and p s to the S' and D-state admixtures and not to explain theoretically any experimental data (which is not available at present) we choose for simplicity the Gaussian form for the radial wavefunctions.

Moreover, since ~b7 is believed to provide the dominant D-state admixture of about 8 % (Gibson and Schiff 1965) we set

a ~ = 0 . 9 , a~=0"02, a~=0"08; a s - - a s = 0 . (39) The radial wavefunctions used in this computation are

fx = N, exp [-- ~ a 2 (R~ + R~)], (40)

for the dominant S-state, with N~ = (3 a/2 aaflra),

vl, ~ = N,, Sz, n exp [-- ~ a s (R i + R~)], (41) for the S' state, where Ss = 2 R x • 1~, N, ~, = (35/n a1°18 93) with the normalization

8 (v~ q- v]) d s R 1 d s Ra = 1, (42)

P.--4

346 G Ramachandran, R S Keshavamurthy and K Venkatesh

and f7 (Ss) = N D exp [-- } a = (R~ ~- R~)], (43)

for the D state where N~) = (39/'. aa4/25 ,ra 2~). The normalization factor N D in (43) has been obtained using the Gibson and Schiff normalization for ~b~', i. e.,

/31/2 5 ~3\ r

f +t (44)

where R 2 = R~ ÷ R~. The parameter a has been chosen to be a = 0.384 fm -1.

We have chosen the expansion (4) for the ~b 7' and used in these calculations the expansion coefficients C~ tabulated explicitly by Keshavamurthy and Ramachandran (1981). The radial integrals have been evaluated analytically using the tables of inte- grals (Dwight 1961 ; Gradshteyn and Ryzhik 1965) to give

e s s _- ~ exp - ~ , (45)

=3 exp (-- Q2/18 a 2)

~ s ' ___ a~ a2 us iv,, (46)

81 ~v/3 " a IO

r : Q 2 a7 [10 _ _~a~] ( )

r SD = N: N D a 1 Q2 Q2

a ^TM 3 e exp -- ~ V~. (47)

The differential cross-section da/dta is plotted as a function of the production angle 0 in figures la, lb, lc for different energies. The target asymmetry is given likewise

24t

o:

(a)

t/ -~

60 120 180

0 (deq) Figure la.

. - . b

SHe (y, ¢r °) 3He as the nuclear probe 347

, i i , ,,,,

40 ~

^{i ~ \}

I \

!/'\1

32 I .-" "l

I!il ~!1

" il,

ol

^{6 0}

^,

¹²⁰

0 (deg)

(b)

180

/ ' \ (cl

161-- /:'"',\

°

0 30 90 150

e ((:leg)

Figures la, b, c. The differential cross section da/di2 at laboratory photon energies (a) 280, (b) 340 and (c) 400 MeV as a function of the pion photoproduction angle (0) (--) including S, S and D states with 9070, 2% and 8 70 respectively

(...) $ and S ' states with 96 70 and 4 7o respectively ( - - - - - - ) pure S-state with 10070, in figures 2a, 2b, 2¢ and the y component of the recoil nuclear polarization P~ in figures 3a, 3b, 30. An examination of these figures shows that the differential cross- section d,7/dn, the target asymmetry and recoil nuclear polarization are quite sensitive to the small admixtures of S' and D states. We therefore advocate planning of expe- riments to measure these observables as a means to study the nuclear wavefunctions

348

G Ramachandran, R S Keshavamurthy and K Venkatesh

Figure 2a 0 - 5

0 . 2

A 0.1

0

- 0 . 1 0

0 . 5

0.2

0.1

A 0

( a )

I I

I !

I !

I I

I I

I I

l I

I I

i I

/ I

/ I

/ I

/ !

J' °°°°°°°.°°°° °°°.*°°o..°~""~.. |

°°*°

I I I

6O 120 180

e (decj)

I b l

z - ,,,J , , , u , ,

A

I I

! I / I

-0.1

-0.2

Figure 2b

-0'3 E

0 ^_L

^{6 0 120}

^I

¹⁸⁰

^I

e (deg)

3He (y, ~'°) SHe as the nuclear probe 349 particularly in view of the active interest in developing polarized targets and in detec- ting polarization by several groups (Haeberli 1974; Carillon 1974; de Boer 1974).

The theoretical formulae derived in § 2 are stdticiently simple to be used by the experimentalists. They are also sufficiently general so that one can use any given set of nuclear wavefunctions. If in particular one employs the analytical forms (Hajduk et al 1980) which are given in the j - - j coupling basis, or the numerical solutions given in r. -- S coupling form (Brandenburg et a11975) rather than the Sachs, Gibson

, , , , ,

0,1 • (c}

0

A

- 0 . I ".. i

- 0 . 2

-o.3 I" I I .. I

0 6 0 120 180

O ( deg )

Figures 7.a, b, e. The target asymmetry (A) at laboratory photon energies, (a) 280, (b) 340 and (c) 400 MeV as a function of the pion photoproouction angle (0) ( ) including S, S" and D states with 90%, 2 ~ and 8% respectively.

(...) S and S' states with 96% and 4% respectively.

(-- - - --) pure S-state with 100%.

0 . 2

0.1

0

- 0 . 1

'°' /,,,

/ I

-- / |

/ / I

/ I

/ I

/

I I,,

O' 6 0 120

e (deg)

I ^,, 180

Figure 3a

350 G Ramachandran, R S Keshavamurthy and K Venkatesh

0.4

' 0 (b)

! I

I I / I

/ I

/ /

-0.2

- 0 . 4

I " I I I

0 6O i 2 0 180

e (deg) Figure 3b

and Schiff forms employed in this paper, the calculation is simplified further in that there are now no independent summations over i and fl or i and a i.e. a I C~ are simply replaced by aa C a with only a summation over a. Likewise at B~ are replaced by a/3 B/~ with a summation over fl but with no s~_~mmation over i. Thus the differential cross-section, target asymmetry and recoil nuclear polarization could be estimated corresponding to any given experimental situation, using a whole set of theoretical models using the formulae derived here. This facilitates to check the validity of the various microscopic calculations of the nuclear wavefunctions (see for example refs.

3-11 listed in Arnold et al 1978; Sick 1981) much more incisively than is possible at present using only the electron scattering data which itself has already led to a dis- agreement between experiment and theory. It is hoped that the information from the polarization and asymmetry studies advocated here could pinpoint the cause for this disagreement, since they present a perspective on the nuclear structure which is complementary to that obtainable from cross section studies.

Acknowledgements

Part of this work was done while one of the authors ((;R) was visiting the Mathe- matical Philosophy Group at the Indian Institute of Science. He is grateful to Prof. G N Ramachandran for his kind hospitality. The numerical calculations were

3He

(•, ¢r °) aHe as the nuclear probe

Oil

°°.°.°"'°"..%°o

-,,,

t 'V// I I

^'~'

.9 \

-(3.2

0 60 120

e (deq)

(c)

180

I

351

Figures 3a, b, c. The Y component of the recoil nuclear polarization (Pf) at labora- tory photon energies, (a) 280, (b) 340 and (c) 400 MeV as a function of the pion photo production angle (0)

(--) including S, S ' and D states with 90~o, 2 ~o and 8 % respectively (...) S and S ' states with 96~o and 4% respectively

( - - - ) pure S-state with 100~o.

performed on the DEC 1090 Computer at the Indian Institute of Science, Bangalore.

The authors are indebted to Prof. G N Ramachandran for making available the compater time for this purpose. Two of the authors (gsg and gv) wish to thank Prof. B Sanjeevaiah, for providing research facilities. RSK and gv are thankful to the Department of Atomic Energy (India) and the Council of Scientific and Indus- trial Research (India) for financial assistance.

References

Arnold R G e t at 1978 Phys. Rev. Left. 40 1429 Berends F and Donnachie A 1975 Nucl. Phys. 1184 342

Brandenburg R A, Kim Y E and Tubis A 1975 Phys. Rev. C12 1368

Catillon P 1974 Nuclear spectroscopy and reactions (ed.) Joseph Cerny (New York:Academic Press) p. 193

de Boer W 1974 Dynamic orientation of nuclei at low temperatures P h . D . Thesis, University of Technology, Delft, The Netherlands (CERN 74-11)

Dombey N and Kabir P K 1966 Phys. Rev. Lett. 17 730

Donnachie A 1970Hadronic interactions of electrons and photons, Proc. of the eleventh session of the Scottish Universities Summer School in Physics (New York: Academic Press), p. 119 Donnachie A and Shaw G 1972 Phys. Rev. D5 1117

Drechsel D 1980 Nucl. Phys. A335 17

Dwight H B 1961 Tables o f integrals and other mathematical data(New York : MacMillan) Gibson B F and Schiff L I 1965 Phys. Rev. B138 26

Gradshteyn I S and Ryzhik I M 1965 Tables of integrals, series and products (New York: Academic Press)

352 G Ramachandran, R S Keshavamurthy and K Venkatesh

Haeberli W 1974 Nuclear spectroscopy and reactions (ed.) Joseph Cerny (New York: Academic Press, p. 152

Hajduk Ch, Green A M and Sainio M E 1980 Nucl. Phys. A337 13

Keshavamurthy R S and Ramaehandran G 1981 J. Phys. G: Nucl. Phys. 7 867 Lazard C and Marie Z 1973 Nuovo Cimento A16 605

McCarthy J S, SiekI and Whitney R R 1977 Phys. Rev. C15 1396 Payne G L 1981 Nucl. Phys. A353 61C

Ramachandraa G and Ananthanarayanan K 1964 Nuel. Phys. 59 633 Riska D O 1980 NucL Phys. A350 227

Sachs R G 1953 Nuclear theory (Cambridge: Addison Wesley) p. 180 Sehiff L I 1964 Phys. Rev. 133 B802

Sick I 1981 NucL Phys. A354 37C