IL NUOVO CIMENTO VOL. VII, N. 3 1 ~ Febbraio 1958

Spin-orbit Coupling and Tensor Forces (*).

B. JANCOV~C~

Laboratoire de Physique de l'~cole Normale Supdrieure - Universitd de Paris

(riccvuto il 2 Luglio 1957)

R6suna6. - - Nous avons cherch6 si on peut expliquer les forces spin-orbite dans les noyaux complexes comme un effet du 2 ~ ordre des forces tensorielles. Suivant les m4thodes de Brueckner, nous calculons d 'abord une ampli tude de r6aetiorl modifi6e, pour les collisions de deux nucl4ons au sein de la mati~re nucl6aire. Nous utilisons la deuxi~me approximat ion de Born, et nous tenons compte du principe d'exclusion (lans les 4tats interm6diaires. Nous obtenons ensuite le potentiel spin-orbite moyen auquel est soumis un nucl6on en sommant sur routes ses collisions l ' ampl i tude de r6action. Le potent ie l spin-orbite ainsi obtenu est d 'un ordre de grandeur t rop faible et peut m~me avoir 16 mauvais sight. Nous discutons quelques implicat ions de ces rdsultats.

1 . - I n t r o d u c t i o n .

The on ly n o n - c e n t r a l forces which are k n o w n wi th c e r t a i n t y to ex i s t in t h e

t w o - b o d y n u c l e a r p r o b l e m (e.g. f r om t h e q u a d r u p o l e m o m e n t of t h e d e u t e r o n )

~re t h e t e n s o r forces. F o r t h e sake of s imp l i c i t y , i t m a y be ~sked if t h e s e forces

a re suff ic ient to a c c o u n t for t h e s p i n - o r b i t coup l ing which e x p l a i n s t h e shel l

m o d e l a n d t h e h igh ene rgy nuc l eon -nuc l eus p o l a r i z a t i o n e x p e r i m e n t s .

I n t h e ease of l i g h t nucle i , some ev idence has b e e n g iven t h a t th i s is i n d e e d

the case (~). W e he re t r e a t t h e p r o b l e m for h e a v y nuclei , us ing an a p p r o a c h

(*) Supported in par t by the United States Air Force through the European Office, Air Research and Development Command.

(1) E. P. W I ~ R and A. M. FEI~GOLD: Phys. Rev., 79, 221 (1950); A. M. FEIN- ~OLD: Phys. Rev., 101, 258 (1956); 105, 944 (1957); D. H. LYONS: Phys. l~ev., 105, q36 (1957).

SPIN-ORBIT COUPLING AND TENSOR FORCES 291

in the spirit of Brueckner 's model (2). A preliminary account of this work has already been given (3). Independent results obtained by L. S. KISSLINGER are in disagreement with ours (4).

We derive the spin-orbit effective potent ial from the reaction mat r ix for (( modified two-body collisions (5,6). This reaction matr ix will actua!ly be a ~>

reaction matr ix, which means tha t it is defined for the two-body collisions

inside the nuclear ma t t e r and tha t it takes into account the presence of other nucleons, through the use of the exclusion principle in the intermediate states

(statistical correlations). The modified reaction matr ix has a spin-linear par t

(~)

for the collision of two particles i and j having initial and final momenta k~, kj and k'i, k'j, respectively. ~ and a are the isotopic spin and spin operators.

We here assume tha t the effective potent ia l in which the i-th nucleon moves

is obtained by summing (1) on all two-body collisions with other nucleons j.

This assumption will be discussed in Sect. 7. A spin-orbit potent ia l may emerge (Sect. 2) from such a calculation only in a nucleus of non-uniform density: the spin-orbit potent ia l is a surface effect (7). However, it is convenient to compute (1) from the two-body forces in a medium of uniform density (Sect. 3). Therefore, we use the Thomas-Fermi approximation: the density ~(r) is varying, but, in the neighbourhood of each point, the nuclear s t ructure may be ap- proximated by a Fermi-gas with a local Fermi momentum

(2) ](r = [32~ ~(r)]~ .

2. - From the modified reaction amplitude to the spin-orbit potential.

The average spin-dependent potent ia l ai'V~ in which the i-th nucleon moves is obtained by adding the reaction amplitudes (1) for the collisions with all the other nucleons j. Thus, in momentum space, a mat r ix element of V;

(~) H. A. BETItE: Phys. Rev., 103, 1353 (1956). (3) B. JANCOVICI: Phys. Rev., 107, 631 (1957); Th~se (Paris, 1957). (4) L. S. KISSLINGER: Phys. Rev., 104, 1077 (1956). See note (0) in ref. (3). (5) S. FERNBACH, W. HECKROTTE and J. V. LEeORE: Phys. Rev., 97, 1059 (1955). (8) j . S. BELL and T. H. R. SKYRME: Phil. ]lag., 1, 1055 (1956). (7) E. FERm: quoted by R. M. STERNREIMER.

292

reads:

B. J A N C O V I C I

(3) T*(k;) r . . . . . I A I k i k ~ ) ( � 8 9 3 t 3 ) ( k ' ~ k ' i l A I k i k j ) J ~ J j ( k j ) ,

where the Tr are the Fourier transforms of the one-particle orbital states T~(r~):

(4) T(k) = fd~r exp [-- ik . r J T ( r ) . J

Eq. (3) can be brought, with some approximations, to the form of a usual spin-orbit potential (5,~). In our model, we assume a varying spherically sym- metrical nuclear density.

(5) 4 ~ T * ( r j ) ~ ( r j ) = ~(rj) , %

while around each point we describe the correlations by those of a (, local ~) Fermi gas of (( local )~ Fermi momentum [:

(6) 4 ~ ~*(rj)Tj(rj) ' = ~(r~)[3/4~ ~(r'j)] ~/exp[-- ~k." (re' -- r~)] d3~.

In such a model, using (6) and the momentum conservation:

(7) (k'ik'j I-,4 lk~kj) : (2~) 3 O(k~ § k:. - - k, - - k~)(k~ I~ I k~k~),

we may bring (3) to the form:

(s) P s

IE J k~) = ag 3r'j exp [,g. r,j~(r~)6(k~ - - k~ -~-g)[3/4~r] (rj

�9 fd%[~,(k~ § k~, kj) - - (�89 § 3 f i ) ( k ~ - - g l A [ k~k~)]. kj<f

The last integral in the expression (8) is a pseudovector depending only on k~ and g (after this last integration has been performed). We m a y expand this expression in successive powers of g, which would make successive derivatives of the density ~ appear in (8). We keep only the linear term in g. This means tha t we assume the spin-orbit effects to depend most ly on the first derivative of the density. Or, from another point of view, this approximat ion means

S P I N - O R B I T C O U P L I N G A N D T E N S O R F O R C E S 293

t ha t we only consider small angle (direct ~md exchange) scatterings of the i- th nucleon in the nuclear mat te r . This first non-vanishing t e rm of the ex- pansion mus t be of the form:

(9) (3/4zp) id~k,(kj § g f A I k~, k~) = g x k i E o ( k i , ~) "7- . . . , .)

(lO) (3/47~]a) f dakj(ki-- g IAI ~i, k~) = g X kiEl(ki , /) ~- . . . . kj</

l Using (9) and (10) in (8), and changing the integrat ion var iable rj into

r~, we obtain the famil iar expression

(11)

where

(12)

= fd3r~ r, )~ ~ (ri• k~), (k~/]~]k~) e x p [ - - i k ~ .~-ik,.'r~]a(k~, (r~) 1 ~

a(k~,/(r~)) = a0 + a, = i [ ~ E 0 ( k ~ , / ) - (�89 + 3fl)E,(k~, i ) ] .

Thus the energy of the i - th nueleon is the same as if i t were submi t ted to an effective spin-orbit force (s)

(13) a(k,, ](r,)) E ~ (l~.a,) .

In addit ion to l,, there is a veloci ty dependence in a because of k~. How-

ever, if i is the <( l~st ~ bound nucleon, ki is peaked ~round ] and we will jus t

take k~ ---- ](re). I f i is a scat tered high energy nucleon, ~2k~/2M must be taken as its kinetic energy inside the potent ia l well.

~.='- The calculation of the modified reaction amp l ude.

Our task is now to obtain the modified react ion ampli tudes (9) and (10)

f rom the interact ion between particles i and j, assuming tha t these particles

ure pa r t of ~ locally uniform Fermi gas of Fermi m o m e n t u m f. The re,~ction

(s) The a coefficient depends on both ki and (through ]) r~. This could lead to some difficulties, since these two variables do not commute. Actually, we can con- sider them now as classical variables in the same approximation as the Thomas-Fermi approximation; in the latter, a momentum (e.g. the Fermi momentum) is defined at each point, which is also only a classical concept.

294 B. J A N C O V I C I

matr ix O~j at energy E is given by tile integral equat ion

Q (14) O. = vi~ + ~'iJ E - - H Oi~ ,

where Q is the projector outside the states which are already occupied by other nucleons and H is the (( model )> hamil tonian for the uverage potential .

Here, we just use for (14) the second Born approximation, which gives the first non-vanishing spin-linear result in the case where the spin-dependent par t of vii is a tensor force. In this second order Born approximation, the reaction ampli tude is

~ ' ' ~q~qJ) ~ 2 ~ ( q ~ q j ] v . i ~k,). (15) (2u)-'~(2M/]/2) q, q~(k ik j ir~j, k~ + ks - - q, - - q~

q~>] q j > f

The momenta q~q~ of the intermediate state must be outside the Fermi sphere. We assume tha t v . has a tensor par t

(16)

where

(17)

v,j = [1 - - Z + Z('r~ . % ) ] u ( r . ) S . ( r . ) 4- ... ,

3 ~ i j ( r i j ) = ~ - ( o i �9 l ' i j ) ( ~ j" r~r - - (oi 'oj) ,

and we are only interested in the par t of (15) which is linear in the spins. By inserting (1.6) into (15), it is readily found tha t this par t is (1) where

(18)

and

(19)

a -- 1 - - 2 Z + 4• 2, fl = 2 Z - 4Z2,

(k ' r = : i ( 9 M / 1 6 ~ 2 h 2) 3 q ( k ' - - q ) •

~2(K)

. ( k ' - - q ) . ( k - - q ) w ( I k ' - T q l ) w ( ] k - - q t ) "

k ~ - - q 2 K k ' - - q 1 2 1 k - - q ] t

where we have inserted the relative momenta :

(20)

and

(21)

k = k i - - k j , t t

k ' = k ~ - - k j , q = q i - - q , ,

w(k) --=- fj, ,(kr) u(r)r ~ dr

K = ki + kj ,

S P I N - O R B I T C O U P L I N { ; A N D T E N S O R F O R C E S 295

(J2 is a spherical Bessel function if)). The domain of integrat ion over q, .(2, is defined by the conditions

(22) q,, q~> 1, i.e. Jq • (~)KI>/ .

/ ! We only need (19) for k s = kj + g and k~ = k , - -g . t e rm in an expansion in powers of g :

(23/ ( k , + g l A [ k , , k , l = i ( 9 M / 1 6 z " h " ) k=--q= [ ! q - - k l $2

and

(2~)

We just keep the first

. . . ,

(k:--glArk:, k,)=-i(aMli6~:t~:) ]d3qW(q--k I" Iq--k! .(2

w([q+kl) 2 (kxq ) [g . (q -- k)] w(Iq + kl) �9 g x ( q - - k ) - j q + - k r 2 -- ]c2~-q~ - iq~-k ' t~ +

[ ' § + 2(#,• g.(q + #+)rq+ k, ~lq + ~i lq + #,1 J] § ....

We need only the integrated values (9) and (10), the tensorial nature of which we know in advance, so tha t one may readily show tha t :

(25)

and

(2(.:)

9M 3 ~ f ;d'q ki ' (q - k)['w(!q-- k[) l' Eo(k i , ]J=i16~h24~/3 '~ dakj. k2_q~ [ { q - - k ! /

k j < f .~2

E , ( k , , i ) i 7+6.'7q~ 4:+t ~ '. ., I q - - k "

- k : q -~ i-q -,- ++i ~ + + [(k< .q)(k.q) -- (k,. k)q" + k:(k,.q) -- (k~. k)(k. q)].

1 ~ w(]q + _k!)[

lq~-k ~ ~lq+kl iq§

In Appendix I, we give a method for computing by a one-variable numerical

intcgr,~tion the integral {25), for any tensor force, the Bessel t ransform (211

(~) 1 ). M. MORSE and H. FESHBACH: Methods of Theoretical Physics (New York. 1953).

2 9 6 B . J A N C O V I C I

of wh ich is known. (25) is g iven b y the in tegra l

(27) E o ( k i , ] ) - - i 3 6 ( M / l ~ 2 ) V ~ , f -1 ~bA.(x; w L u ,] d x ,

U

where ~Sk,(x ) is a un ive r sa l func t ion 1/# is the r ange of the t ensor force.

t he l as t b o u n d nucleon, k~ = ], and Fo r

(28)

2

1

3 x 4 - 1 r : ~ (1 - - x~)~ Loo, x--~-- +

:1 + 2 x ( x + 1)2(x - - 2) L o g 2 + ~ x ( - - 2 x 3 - - 3 x 2 + 1 2 x 4 - 1 3 ) , if x < 1,

q~/x) = 2 x ( x - - 1)2(x4- 2) Log (x - - 1) 4- 2 x ( x + 1)2(x - - 2) L e g (x § 1) 4-

4- 4xs( - x 2 4- 3) L o g x 4- 2x ~ 4- 3 , if x ~ 1.

~b/x) is p l o t t e d on Fig. 1. One sees t h a t i t

changes sign when x increases , so t h a t the ~(x)with exclusion m a g n i t u d e and sign of (1) d e p e n d in a

(x) without exclusion sens i t ive w a y on the region where (21) is p e a k e d , t h a t is, on the shape of the t ensor

force a n d on the ra t io ]/#.

I n A p p e n d i x I I , we give a m e t h o d for

c o m p u t i n g the in tegra l (26) for k~ = ] b y

one -va r i ab le in tegra t ion , in the special case

of a t enso r force

0 -'-

,t \ / Fig. 1. - The functions q~y(x) for

a bound nucleon.

(29) u(r) = Vo #2r2 exp [ - - �89 2 ].

I t is found t h a t

co

MVY~ f : v (30) El(f , ]1 = - - i54~ .~2a10 (y) d y ,

0

where

N(y ) ---- exp [ - - ( 2 H # ~ ) y ] { ( 1 - - y )~( - - (466/315)y ~ - (272/315)y 4- (176/105)) 4-

+ y ~ - - (118/105)y 3 - - (272/315)y ~ 4- (1 136/315)y - - (176/105) 4-

+ ((16/35)y 4 - (16 /15}y~) (1 /~ /y ) tgh -1 ~/y + ( - - (8/15)y + (8/35))-

Log (1 - - y) 4- (]2/#2)[(1 - - y)~ ( - - (944 /945)y 2 - - (1541/945)y - / (211/135) ) 4-

4- (4/45)y 6 - - (17/]5)y ~ + 3y 4 - - (14/9)y ~ - - (62/15)y ~ 4- (529/105)y - -

- ( 2 1 1 / 1 3 5 ) ] } , it y < 1.

~ ( y )

+

S P I N - O R B I T C O U P L I N G AND T E N S O R F O R C E S 297

exp [ - - (212// , t~)y]((512/735)K 7 + ( (S32/525)y - - ( 1 4 4 8 / 5 2 5 ) ) K 5 q-

((8/315)7/2 - - (230/63)? I + (334 /105 ) )K 3 q- (y - - 1 ) ( - - (26/35)y 2 q- (2/7)y - -

( 2 2 / 1 0 5 ) ) K + (y - - 1)2((13/35)y 2 + (4/21)y - - (8/35))(2/V~y 1) .

( t~-i ~/2/.~/y - 1 - 1 - t~-~ {1/V9 - ~ ) ) - (5~2/735){27 +

+ ((576/175)y - - (8/25))@ 5 - - Q ' + ( - (1352 /315)y 2 q- (4 /45)y )Q 3 +

q- ((2/3)y q- (8/15))Q ~ q- ( - (92/105)y a q- (4/15)y")Q - (44/245)k 7 q-

q- ( (124/175)y - - (64/75))k 8 . k ] q- ( - - (16/105)y ~ q- (16 /45 )y )k ~ q-

+ ( - - 2y q- (8/15))k 2 -~- ( - - (16/35)y 3 q- (16/15)y2)k q- (16/21)q 7 +

q- ( - - 4y q- (16/15))q ~ q- ( (52/9)y 2 - - (28/9)y)q 3 q- ((4/3)y 3 - - (4/3)y2)q - -

- - (16/245)y: q- (124/175)y G - (1508 /525)y ~ + (1123 /315)y 4 4- (8/7)y 3 - -

- - (100/63)y 2 q- (4 0 3 4 / 1 5 7 5 ) y - - (1628 /1225) - - (2/~/y) [ ( - - (8/35)y 4 q-

q~ (8/15)y 3) t g h -1 ( 1 / V y ) q - ( - - ( 4 6 / 1 0 5 ) y 4 q : ( 2 / 1 5 ) y a) l~gh -~ ( V y / ( 1 + , / y - - l ) ) g -

q- ((2/3)y 4 - - (2/3)y 3) t gh -1 ~ / y / ( 2 y - - 1) q- ( - - (8/35)y ~ q- (8/15)ya) �9

t~'h -1 ~/2 = y] + ( - - (8/15)y + (8/35)) L o g (y(y - - 1 )~/2) § (/2I#2).

[ - - (16/63)K" § ( - - (16/15)y q- (128 /105) )K 7 q: ( - - (16/15)y 2 ~- ( 5 2 / 1 5 ) y - -

- - ( 2 8 / 1 5 ) ) K 5 ~ (y - - 1)((4/9)y" q- (16/9)y - - (8 /9 ) )K 3 - (4/3)(y - - 1)~K q-

- - (16/63)Q 9 q~ ( - - (128/105)y - - (8/105))Q 7 q- (4/9)Q + + (8 /5 )y :p ~ - -

- - (r 4 4- ((4/3)y 2 - - (16/15)y if- (16/35))02 - - (8/189)k 9 q-

q- ( (32/105)y I (16/35))k7 q~ (4/9)kr - - ( _ (8/15)y~ q_ (16/15)y)k~ _

- - (4 /3)yk ~ § ((4/3)y 2 - - (16/15)y q- (16/35))k 2 - - (8/27)q9 q-

q- ( (32/31)y - - (8/21))q v -~ ( - - (32/15)y 2 § (16/15)y)q ~ § (8/189)y 9 - -

- - (32/105)y s + (104/105)y 7 - (68/45)y ~ 4- (4/3)y ~ - - (4/3)y ~ q-

q- (88/45)y a - - (496/105)y 2 q- (64/15)y - - (416/315)]}7

w h e r e

K V 1 - - y q- 2 V y i ,

y.

if 1 < y < 2 .

N ( y ) exp [ - - (212/#2)y]{P(y, ~ / 2 y - - 1) - - e ( y , ~ 2 y ) } ,

298

where

(3].)

B. JANCOVICI

p(y, q) = (16/21)q7 _u ( _ 4y + (15/16))q 5 + ((52/9)y ~ -- (28/9)y)q ~ +

+ ((4/3)y 2 - (4/3)y2)q + (-- (4/3)y 4 @ (4/3)y3)(l/~/y) tgh -1 (~/y/q) ~-

4- (]2/#2) [ _ (8/27)q9 + ((32/2J)y -- (8/21))q7 +

+ (-- (32/15)y~ + (16/15)y)qq, if y > 2.

4. - Experimental and calculated results.

For a bound nucleon, (13) is the well-known spin-orbit potent ia l which is responsible for the shell-model. The analysis of low-lying energy levels may

determine an average value ~ through the nuclear surface for a [](ri), ](ri)]. This determinat ion cannot be very precise, because of various uncertaint ies; however ~50 and ~7N are fairly accounted for by ~ - 70 MeV (10 12 cm)5 (% The nuclei around 2~ indicate d ~ 57 (1% ~ must be positive to account

for the (( inversed doublet )> structure. We have computed the direct t e rm a0 of (12), through (27), for some po-

tentiMs, the Bessel transforms (21) of which are easy to obtain. These po- tentiMs are, together with their Bessel t ransforms:

(32a)

[ u(r) = Vo exp [--#~] ,

3 k 5k 2 + 3# 2 [ W(k) = ]'o ~ tg -1 ; - - # ]~(k~/~2)2 �9

(32b)

/ u(r) = 1,~, exp [ - -# r ] J #r '

re(k) = Vo - - ~2 t~ 1_ +

(32c) [ u ( r ) = l o ( ~ r 3 ] § #2r2 +

_ __ k ~ I w(k) =Vo 3(k2+~2).

/3r3 ) exp [--/zr], (11)

(32d)

[ u(r) = Vo#2r ~ exp [-- �89

/ e x p - -

(lo) R. J. BLIN-STOYLE: Phil. Mag., 46, (11) Such a singular potential is fed into

indicative purposes.

973 (1955). the second Born approximation only for

S P I N - O R B I T C O U P L I N G A N D T E N S O R FORCES 299

The exchange dependence is as i nd i ca t ed in (16). The c o m p u t e d va lues of ao

for these d i f ferent po ten t i a l s are l i s ted in Table I. The p a r a m e t e r s of the

po ten t i a l s were first t a k e n f rom classical po ten t i a l s (1~,13,~4) for (32 a, b, c), a n d

for (32d) we t r ied a p p r o x i m a t e l y to reproduce G a r t e n h a u s ' p o t e n t i a l (~5).

The resul t s o b t a i n e d in such a way are of the wrong sign. I t is easy to see

f rom (27) t h a t ao can be pos i t ive on ly for no t too s ingu la r forces w i th a long

range , since on ly such forces will c o n t a i n mos t ly low m o m e n t a , a n d thus ap-

preciable c o n t r i b u t i o n to the in t eg ra l in (27) will come m o s t l y f rom the region

where ~b~ is posi t ive. Accordingly , we t r ied to increase the r ange 1/#, decreas-

ing a t the same t ime ~o b y a fuctor e s t ima ted f rom F e s h b a e h ' s a n d Schwinger ' s

ca lcu la t ions (1G). The resul ts are also l i s ted in Table I. I t is seen t h a t a0 m a y

become posi t ive, b u t r ema ins too smal l b y an order of m a g n i t u d e .

TABLE I.

1/~ Potentials (10 -13

cm)

V 0 (MeV) Z

(a) exponential

(b) 1.529 YUKAWA 2.362

2.756

(d) 1.40 Mesic + c~

'1

(~) i 0.557 I

GART:ENHAUS i 0 . 7 8 8

i

0.75 50.8 0.984 22.6

8.11 2.39 1.61

1.13

46.2 17

0.875

1 . 2 9 ))

i Bound nucleon ' 300 MeV nucleon

a o in units of a 1 i n units of i a 0 in units of MeV(10-13cnI) 5 MeV (10 -13 cm) 5 ] MeV (10 -13 cm) ~

with exclu- sion

--4 5

< 1

3 40

without with exclu- exclu- sion sion

I

without with e x c l u - e x c l u -

sion sion

- - 1 8 - - - - 19-- i27 19 - - - - 15-- i16

7 - - i9 6 - - i 5

- - 9

< 0

- - 2 6 - - 1 3

- - 1 3 29

380 - - !

- - 1 2 0 - -

- - 5 0 1 - - 5.7 - - 29 - - 92

ao is the calculated direct part of the spi~l-orbit interaction coefficient. al is the calculated exchange part. The experimental value is a =ao~al = +70 MeV (10 -la cm) 5.

7 i - - 4 - - i 7 > 0

20 17

6 - - i 1 4

without exclu- sion

28 - - i30 2 8 - i18

13- - i10 14 - - i6

L

- - 3 - - i 8

8 - - i 1 5

(12) R. JASTROW: Phys. Rev., 81, 165 (1951). (13) H. H. HALL and J. L. POWELL: Phys. Rev., 90, 912 (1953). (14) A. MARTIIV and L. VERLET: Nuovo Cimento, 12, 483 (1954). (15) S. GARTENHAUS: Phys. Rev., 100, 900 (1955). (1~) H. FESHBACH and J. SCHWINOER: Phys. Rev., 84, 194 (1951).

300 B. JANCOVICI

I n all calculat ions, the local Fermi m o m e n t u m was t aken as ] =1 .27 .10 -~3 cm

through the whole nuclear surface, a value which would ra ther correspond to

the inside of the nucleus. I f the Thomas -Fe rmi approximat ion is valid, [

should ac tual ly be smaller th rough the nuclear surface, and this would lead

to still algebraically smaller values of ao.

Other models for the nuclear surface do not appreciably change the results.

A ten ta t ive description of the surface b y the model of an infinite potent ia l barr ier was made (17). We also t r ied to t ake into account possible higher

m o m e n t a in nuclear m a t t e r (is). 2all these a t t emp t s do not change the above

resul t t ha t the computed a0 is algebraically too small. I t might be interest ing to es t imate the impor tance of the Paul i principle

inasmuch as it is possible to consider exclusion effects as being less impor t an t

on the nuclear surface, a0 was also computed without taking into account

the exclusion effects in the in te rmedia te states, i.e. extending the integrat ions

to all values of q~, q~, q (27) is still valid, bu t ~b~(x) is now given b y the

second expression of (28) for all values of x. The results are l isted in Table I

I t is seen t ha t a0 increases, bu t t ha t a0 cannot a t ta in proper values for a

reasonable range 1/#. The exchange t e rm a~ was computed only for the potent ia l (32d), with

and wi thout exclusion. The results are listed on Table I. Again, p roper values

are not obtained.

5. - Expl ic i t calculat ion of the non-modi f ied react ion ampli tude.

The fai lure of obtaining proper values for a m a y be unders tood f rom another

poin t of view, in the special case where exclusion effects are not t aken into

a c c o u n t .

I n this case, it is possible to have an explicit expression for the forward and backward react ion ampli tudes, in second Born approximat ion , a t least

for the potent ia ls (32c, d) and for

(32e)

u(r) = V o ( 1 - ~ e x p [ - - # r ] ,

2k 2 w(k) = Vo#(k2 ~- # 2 ) ~ �9

Without exclusion effects, the react ion ampl i tude depends only on the

W. J . SWIATECKt: P'~'oe,. Phys. Soo., A 64, 226 (1951). (Is) G. F. CHEW ~nd M. L. GOLDBERGER: Phys. Rev., 77, 470 (1950); K. A. BRUECI(-

~'EB. J. EDEN and N. C. FRANCIS: Phys. Rev., 98, 1445 (1955).

S P I N - O R B I T C O U P L I N G A N D T E N S O R F O R C E S 301

relative momenta , and is of the form

(33) (E, k'~ IAIk,, k~) = i (2~) O(k', + k'~ - - ~ , - - }~)(k' • k )D .

F r o m (23) and (24), it is found that , for forward scat ter ing

(34) D=Do(k ) 1 8 M l ~ / ' . a k~ - - k .q lw ,~_ l)l {~a k ]

while, for backward scat tering

(35) D = Dl(k) 18 M ~ ~ f ~ w ( l q - - kl)

].( k ' q ' - - ( k ' q ) ' ] w ( [ q + k ' ) (k 'q) ' - -k~q ' ~ w![q +_k[)[ "[I - -k2+ - k , ~ j l q + k t ~ -4- ] q + k I ~ ] q + k l [q+kl']"

For potentials (32c, d, e), (34) and (35) provide the respective forward and backward ampli tudes:

M V 2 o [ 3#2(3k2+# 2) 3 ~ 3 t g - ~ k ] (36c) Do(k) = lz:~ ~ r [-- ~ + ~ 4- 2k - - '

(37e) MV~ [ 3# ~ 3#a(4k2 + #2) ,2k]

DI(~) = ~2~ ~ - - ~ + Sk"(2k~ + ~ ) tg- ~ ] .

(36d) ._ MV~

Do(k) = o~V:r h ~ ~"

.[ (3# a -? # k~ V/~ -- 2k3 6 ~ + 1 2 [ , ~] 2i

M V~o . (37d) Dl(k) = 6ux/~ h2#7

[ _ (;)1 k~ 4k3 k~ ~/~ Erf i exp - - . �9 - - 1 - - 2 / x ~ + , u3 exp -- ~ -

6:rMV~o -- 1 - - 24(k2/# ~) + 48(k4/# 4) (36e) Do(k) -- h2#7 [1 + 4(k~/#2)] a '

67~MV~) [ 3# s (37e) D~(k) -- h2#7 2k~(k2 + #~)(2k ~ +,u~) 2

3#5(8k 4 + 4k~/z ~ + # ' ) 2k3(2k2 + #2)3

2 0 - I l N u o v o C i m e n l o . ca

302 ]~. JANCOVICI

These ampli tudes are p lo t ted on Fig. 2. b y summing (36) and (37) on all collision, wi th the results:

f

(38) ao(/, ]) ---- [(1 - - Z) ~ ~- 3Z 2] ~ / d k ka(f2--k 2) Do(k) ,

o

f

o

These expressions m a y be used

I ~ ~, po t~r ia l c

potentlal d

o

-1

k i p ~

�9 0 o porenhal e

Fig. 2. - The reaction amplitudes D o and D 1 as functions of k/~. e( D o and D 1 are in units of 12~MV~/h2# 7 ; d) D o and D 1 are in units of 6~v/-~MV~o/t~2lu 7 ; e) D O and D1 are in units of

6nMV~o/h2# 7.

ao and at are respect ively ob ta ined

to check the corresponding results, with out exclusion, in Table I .

F r o m a qual i ta t ive point of view, it is interest ing to note t ha t the cont r ibu t ions

of the r ight sign (i.e. posit ive) to ao come

only f rom the high values of k in (38), for

which Do m a y become posi t ive if the

tensor potent ia l u(r) is not too singular

(Fig. 2). The low values of k always m a k e

Do negat ive (this last result m a y be shown to be independent of the potential) . The small net result comes f rom cancellation

between the two regions.

The sign of al is de te rmined i n (39) b y the exchange character of the tensor

potential . However , its magni tude was shown to be always too small, for reason-

able potentials.

6. - H i g h e n e r g i e s (300 MeV).

value of the same order of magni tude as in shell-model problems.

of a in the l i te ra ture (20) are a round 40 MeV (10 -13 cm) 5.

The strong polarizat ion observed in

the high scat ter ing of nucleons b y nuclei

m a y be accounted for b y a spin-orbi t potent ia l (13) (19) where a has an average

Values

(19) ]~. FERMI: ~Vuovo Cimento, I I , 407 (1954). (2o) W. HECKI~0~TE and J. V. LEPORE: .Phys. Rev., 94, 500 (1954); 95, 1109 (1954);

G. A. SNOW, R. M. STnRN~I~IMER and C. N. YANG: .Phys. Rev., 94, 1073 (1954); B. J. MAL]~NKA: Phys. Rev., 95, 522 (1954); R. M. ST]~RNHI~TMER: Phys. Rev., 95, 597 (1954); 97, 1314 (1955); 100, 886 (1955).

S P I N - O R B I T COUPLING AND TENSOR FORCES 3 0 3

Our calculations of the direct term can be readily extended to higher energies. (The exchange term is not expected to be important). To describe a scattering problem, we use the scattering amplitude rather than the reaction amplitude, by inserting a small imaginary part into the energy denominator of (14). The actual calculations was performed for k~ ----- 31; this corresponds to about 300 MeV for the incident nucleon. (27) is still valid but r now reads:

(40)

q)af(x) = (--4 +s) Log

+ x4----9 xa-- ~ x - - Log (x + 2) +

( 1 X4 + X ~ + 2x)Log (x + 3 ) + ( l x 4 - - + -2~

+ ~ x~ + y x + i~ ~ -~ ~ x ,

2+ --~x4+-~xa--~x + L o g ( l - - x ) +

\ X a + 2x) Log (3- -x) +

if O < x < l .

2 x a ]0 8 )~ x + l �9 3f(x)= --~ + ~ x ~ + 2 x + L o g ~ f ~ +

( 5 x 4 7 x 2 5 L 7 2 ) ~ x - - X ) ( X + l ) + 216 - - ~ +2--~ , 27x~ 9~ ~ Log + 2 ) ( x - - 1 ) +

+2xL~ l xa+lxa ~808 5 1 2 4x 2 36 ~ + x + ~ + 5 ~ + 9-~ +

5 7 2 5 7 +i~z - - ~ x 4 + ~ x 2 - ~ x 24 27x 2 + , if 1 < x < 2 .

~b3f(x ) = -- x 4 + - x 2 + 2 x + L o g x ~ +

+ (-- 2 x4 + l---~ x'-- 2x + ;) Log ~____ I

2 ( x + l ) ( x + 2 ) 2 5 - - ~ x LOg ( x _ l ) ( x _ 2) + ~x~ + 5 ' if x > 2 .

Re Oaf(x) and (1/~) I m

We also compute ao states. In that case, Re

qSaf (x) are plotted on Fig. 3. neglecting the exclusion principle in the intermediate qi(x) is the third expression (40) for all values of x, and

(41)

Im~b3s(x ) = g --~-{x , if x < l .

Imq~1(x)----~ -- x 4 + ~ - - - ~ x + , if 1 < x > 2 .

Imq~31(x)=0 , if x > 2 .

304 B. 3ANCOVICI

The results for a0 are listed on Table I for the previously considered po-

tentials. I t is interesting to see tha t the Pauli principle is still so impor tan t at

300 MeV, at least for the real par t of g~Re r

','x /1'

-0 " \X /7 ---without exclusion

-0.2

Fig. 3. - The function ~b~1(x ) for an incident 300 MeV nucleon.

ao, except for the singular potent ial (32c). This looks quite general, and can be unders tood easily, because, for regular potentials, most ly small m o m en tu m trans- fers occur in (21), and the small transfers are still forbidden by the exclusion prin- ciple, whatever the incident nucleon

energy is.

The sign of a 0 is correct, for regular potentials, bu t its magni tude is too small. This is in agreement with the commonly accepted idea tha t regular tensor forces do not provide enough polarization at high energy. Insert ion of singular forces (the Born approximat ion being no longer used) could probably lead to higher

polarizations. ao is complex. I t may be noted tha t

the real and imaginary parts are of op-

posite signs and of the same order of magnitude. Such a complex spin-orbit potent ia l has been proposed (~1) to explain the scattering f rom carbon, in the small-angle region where there is interference with the Coulomb scattering.

7. - D i scuss ion and conc lus ions .

In principle, the failure of tensor forces to account for the shell-model spin-

orbi t coupling is a strong indication for the existence of e lementary mutual

spin-orbit forces in the two-body interaction. However, our computat ion

involves several assumptions, some of which will be now discussed.

7"1. The Born approximat ion. - I t was shown in Sect. 5 that , in the case

where the exclusion effects are neglected, the negative value or smallness of ao r f rom the behaviour of Do(k) for small k. This behaviour was est imated

(21) W. HECKROTT]~: Phys. Rev., 101, 1406 (1956).

S P I N - O R B I T COUPLING AND TENSOR FORCES 305

by computat ions f rom the second Born approximat ion while the actual Do(k)

could be different. Actually, Do(k) ought to have a pole at low energy, since there is a bound state for the deuteron (e), and our approximat ion does not

reproduce this pole. However, in the case where exclusion effects are t aken into account the

Born approximat ion was claimed to be much more justified (~), at least in the case of regular forces. I t is still possible tha t singular tensor forces would

alter the results. Of course, in the second Born approximation, we showed tha t for the singular tensor forces (32c), things are ra the r worse, bu t for such a force, the Born approximat ion is certainly bad. Thus, our results, at best, are restr icted to smooth tensor forces, and, if singularities, hard cores, etc., could play a par t even at low energy, this could change our results.

7"2. The many-particle terms. - In the spirit of Brueckner~s theory, we have

computed only these terms of a per turba t ion series, which are contained in

the two-body reaction ampli tude (14). Up to the second order in the inter- action, these terms are the only ones which occur, only if momen tum is con- served in two-body collisions. This is indeed the case in the approximat ion

of infinite nuclear mat ter , and the assumption of momentum-conserva t ion has some consistency with a Thomas-Fermi approximation.

However, in the nuclear surface, there could occur terms tha t do not con- serve momentum. In the second order per turbat ion theory, these terms are for instance 3-particle groups, like vl~vla. Such terms have been previously

considered in light nuclei by FEINGOLD Ct al. (1). FEINGOLD could show that~ in this case, the most impor tan t par t of the vector forces are indeed such 3-body effective forces. The question therefore remains open to know if~ in heavier nuclei, contrar i ly to the spirit of Brueckner ' s model, such 3-particle terms are

important .

7 3. Elementary spin-orbit ]orees. - With the above restrictions, our cal- culations indicate tha t tensor forces cannot account for the spin-orbit coupling. Therefore, these results r a the r s t rengthen the idea of e lementary mutua l spin- orbit forces in the two-body interaction. F rom such forces, an average l . s

effect can of course be easily derived (~). The point of not assuming ele-

men ta ry spin orbit forces f rom the beginning was for simplicity, since tensor forces are anyhow known to exist, and also because the field theory does not

account for a spin-orbit two-body interaction. On another hand, there is

(~) J. HUGUES and K. J. LECouTEUR: Proc. Phys. Sou., A 63, 1212 (1950); I. TALMI: Helv. Phys. Aura, 25, 185 (1952); J. P. ELLIOTT and A. M. LANE: Phys. Rev., 96, 1160 (1954).

306 n. JANCOVICI

growing evidence from two-body scattering and polarization experiments tha t two-body spin-orbit forces do exist (23).

I am indebted to Dr. L. VERLET for helpful discussions�9

APPENDIX I

The direct term.

To compute (25), we make the change of variable

(A.1) z = q - - k .

Then

(A.2) �9 16~r2h 29M 4ze] 1..~ ~z 3k~z" ( ~ k j ) . z Fo(k~, ]) -- - , ~.~

[z+k~]>1 k~<J

We first perform the integration with respect to kj. The cases z < 2 / and z > 21 must be distinguished. The integration on angles of z is then performed. The final integration on z is the numerical integral (27) where x = z/21.

APPENDIX I I

The exchange term.

The calculation is too long to be given here. We just indicate the method. To compute (25), in the special case (29), we make use of the same kinds of methods as Euler (2~). We choose as variables q, k. The domain of integration E2

(23) 19. S. SIGNELL and R. E. MARSHAK: Phys. Rev., 106, 832 (1957). (24) H. EULEI~: Zeits. ]. Phys., 105, 553 (1937).

SPIN-ORBIT COUPLING AND T~NSOR FORCES 307

in q is of r e v o l u t i o n a r o u n d K = k i - k. W e first i n t e g r a t e o v e r p o l a r angles of q, K b e i n g t h e p o l a r axis . Then , k is de f ined b y k a n d K , a n d w e p e r f o r m t h e i n t e g r a t i o n ove r t h e p o l a r angles of k, i. e. ove r K . Of course , m a n y reg ions a p p e a r b e c a u s e of t h e l imi t s . W e a re l e f t w i t h a n i n t e g r a l ove r q a n d k. The f u r t h e r change of v a r i a b l e

k 2-q~ : 2 x , k s §

is done . T h e i n t e g r a t i o n ove r x can s t i l l b e p e r f o r m e d a n a l y t i c a l l y , a n d we a r e l e f t w i t h t h e n u m e r i c a l i n t e g r a t i o n (~0) ove r y.

R I A S S U N T 0 (*)

Abbiamo indagato se si possano spiegare le forze spin-orbi ta nei nuclei complessi come un effetto del second'ordine delle forze tensoriali. Secondo i metodi di Brueckner calcoliamo dappr ima un 'ampiezza di reazione modificata per le collisioni di due nucleoni entro la mater ia nucleare, usando la seconda approssimazione di Born e tencndo conto del principio d'esclusione negli s ta t i intermedi. Otteniamo in seguito il potenziale spin- orbi ta medio al quale ~ soggetto un nucleone sommando su tu t t e le sue collisioni l 'am- piezza di reazione. I1 potenziale spin-orbi ta cosi o t tenuto r isul ta di ordine di grandezza t roppo piccolo e pub anche esserc dota to del segno sbagliato. Discutiamo qualche con- seguenza di tal l r isultat i .

(*) Traduz ione a cura del la Redaz ione .