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The contribution of the neutron-nucleus component to parity-non-conserving effects in neutron-nucleus scattering
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
THE CONTRIBUTION OF THE NEUTRON-NUCLEUS COMPONENT T O P A R I T Y - N O N - C O N S E R V I N G E F F E C T S IN N E U T R O N - N U C L E U S S C A T T E R I N G Bertrand D E S P L A N Q U E S t and Santiago N O G U E R A 2 lnstitut Laue Langevin, 156 X, 38042 Grenoble Cedex, France
Received 2 April 1984
The contribution of the neutron-nucleus component to parity-non-conserving effects in neutron nucleus scattering is reexamined in the case of 139La. Consequences for models of the nucleon-nucleonweak interaction are discussed. The study of p a r i t y - n o n - c o n s e r v i n g (pnc) neut r o n - n u c l e u s interactions at thermal energies has b e e n very active these last few years, experimentally a n d theoretically. Large effects have been observed for n e u t r o n s with both transverse a n d l o n g i t u d i n a l polarizations, giving rise to a spin-rotation in the first case [1-3], a difference in the cross section for n e u t r o n s with different helicities in the second case [1,4-6]. Other effects such as a circular polarization of p h o t o n s or an a s y m m e t r y in their emission with respect to the n e u t r o n polarization [7,8] have also been observed in " i n clusive" reactions. O n the theoretical side, two questions have been raised. The first one concerns the nuclear m e c h a n i s m which produces large effects a n d which is now believed to be essentially due to the presence of p-wave n e u t r o n resonances n e a r threshold [9]. The second one concerns the part of the total n e u t r o n - n u c l e u s scattering state which is involved in observed p n c effects. Two c o n t r i b u t i o n s , c o r r e s p o n d i n g to extreme cases, have been considered here. O n e of them is assumed to come entirely from the n e u t r o n - n u c l e u s compon e n t of the scattering state, the nucleus being in its g r o u n d state [9-15]. A large effect may result in this case from the m a x i m u m overlap between such
1 On leave from Division de Physique Throrique, IPN, Orsay, France. 2 On leave from Departamento Fisica Te6rica, Universidad de Valencia, Valencia, Spain.
c o m p o n e n t s with opposite parities in the i n c o m i n g a n d o u t c o m i n g scattering states. This c o n t r i b u t i o n directly depends o n the pnc n e u t r o n - n u c l e u s force, whose intensity can be calculated from models of the weak n u c l e o n - n u c l e o n interaction. It is det e r m i n e d once the wave f u n c t i o n describing the n e u t r o n - n u c l e u s c o m p o n e n t is given. Conversely, a n d provided that this c o n t r i b u t i o n is the domin a n t one, it m a y be possible to learn something a b o u t the n u c l e o n - n u c l e o n pnc interaction. U n t i l now, this approach has led to too small results c o m p a r e d to experiment. The second c o n t r i b u t i o n [16-22] is assumed to come from the parity admixture of states of the c o m p o u n d nucleus present in the scattering state. The overlap between such states is expected to be small, b u t this suppression is c o m p e n s a t e d b y the fact that, within the nuclear volume, the scattering state has a larger probability to be in some c o m p o n e n t of the c o m p o u n d nucleus than in the n e u t r o n - n u c l e u s c o m p o n e n t . D u e to the complexity of the c o m p o u n d nucleus, the relationship between observables a n d the n u c l e o n n u c l e o n weak interaction is rather loose in this case. Present estimates show that this contribution is certainly a good candidate to explain p n c effects presently observed in n e u t r o n - n u c l e u s scattering. The difficulty to make a precise prediction prevents however to make a definite statem e n t as to its adequacy for a particular case. It is only in the absence of other c a n d i d a t e that it should be considered definitively the correct explanation.
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where the i n c o m i n g a n d o u t c o m i n g scattering states, I~p-+), are o b t a i n e d from solving the Schri3dinger e q u a t i o n
n i a n describing a nucleon moving in the average field p r o d u c e d b y some nucleus, while the second term describes the influence of the coupling of the e n t r a n c e channel to states of the c o m p o u n d nucleus. T h e quantity, ~0R(r ), represents the transition m a t r i x element between s o m e state of the c o m p o u n d nucleus a n d the n e u t r o n - n u c l e u s ent r a n c e channel before being i n t e g r a t e d over the n e u t r o n coordinate. It is generally u n k n o w n , b u t it is expected to be p e a k e d at the nuclear surface, since it is there that nucleons which u n d e r g o some transition are likely to be localized at the energy u n d e r c o n s i d e r a t i o n here. The u n p e r t u r b e d energies of the c o m p o u n d nucleus states a n d their r a d i a t i v e widths are respectively d e n o t e d E ° a n d F R. In terms of the quantities defined above, the a m p l i t u d e s c ~ w o u l d be expressed as
H~t.i.t I~p-+>=E [~p+),
a~ =/dr~b~(r)cpR(r)/[E-(E ° -
In this paper, we r e p o r t on results which show that the d e b a t e is still open. Essentially, we show that m o r e realistic estimates of the c o n t r i b u t i o n of the n e u t r o n - n u c l e u s c o m p o n e n t can lead to the right m a g n i t u d e for p n c effects o b s e r v e d in 139La (spin r o t a t i o n a n d a s y m m e t r y in the total cross section). T h e i m p r o v e m e n t s consist in treating the r e s o n a n c e s o b s e r v e d n e a r threshold as arising from the c o m p o u n d nucleus. U n t i l now, they were considered as p o t e n t i a l resonances. p n c a m p l i t u d e s relevant to n e u t r o n - n u c l e u s scattering can be expressed as fpnc = -- ( M r / 2 ~ r )
(q~-IVp.¢lq,+ ) ,
(1)
(2)
whereas M r represents the r e d u c e d mass ( M r = M × A/(1 + A)). W e m a y write [~b) as a sum of two terms: [~b-+) =
fdr~+p.(r)a+(r)[O) +~_,a~af¢[ 0).
(3)
a
T h e first one describes the n e u t r o n - n u c l e u s comp o n e n t in which we are interested here. T h e seco n d one represents the states of the c o m p o u n d nucleus. T h r o u g h residual interactions, they are c o u p l e d to the n e u t r o n - n u c l e u s e n t r a n c e channel. Q u i t e generally, the SchrOdinger e q u a t i o n (2) gives rise to a system of c o u p l e d equations. A f t e r expressing the a m p l i t u d e s c o r r e s p o n d i n g to the c o m p o u n d nucleus states, a ~ , in terms of the wave f u n c t i o n d e s c r i b i n g the n e u t r o n - n u c l e u s (singleparticle) c o m p o n e n t , solving the Schr6dinger e q u a t i o n (2) a m o u n t s to solve the following equation:
fdrHsp(r, r , P
I
+
t
--
+
E ) + s p ( r ) - E~ksp(r),
(4)
where n s p ( r , r ' , E ) = ½(
p2/ZMr + V ( r ) , a ( r -
+E s E-( E°-
iFs/2)"
r')}
(5)
T h e first term in (5) represents the usual h a m i l t o 304
iFR/2)].
(6)
T h e present a p p r o a c h offers the a d v a n t a g e of a c o n t i n u o u s transition to optical potentials. In particular, the usual i m a g i n a r y p a r t of these p o t e n t i a l s w o u l d result from the average over several resonances of the i m a g i n a r y p a r t of the last term in (5). This p a r t of the i n t e r a c t i o n is very i m p o r t a n t to account for essential features of n u c l e o n nucleus scattering, as for instance inelasticity or also the absence of very large scattering lengths which should b e e x p e c t e d in the case of an inert nucleus. I n the present p a p e r , we are interested in processes which take place in an energy range which is m u c h smaller than the average energy spacing b e t w e e n states of the c o m p o u n d nucleus. W e will therefore retain the exact expression of the last term in H ( r , r', E ) . D u e to differences in the strengths or in the energy d e n o m i n a t o r s , this last t e r m is not the same for states of o p p o s i t e parities. T h e n e u t r o n will therefore feel a different interaction d e p e n d i n g w h e t h e r it is in an s- or a p-wave. N u m e r i c a l m e t h o d s allow to solve eq. (4) quite generally. F o r our purpose, we will c o n s i d e r a simplified case which offers the a d v a n t a g e to be solvable a n a l y t i c a l l y on one hand, to p r o v i d e resuits easily c o m p a r a b l e to previous ones on the o t h e r h a n d [9-14]. W e first assume that the nucleus has no spin. The n u c l e o n - n u c l e u s i n t e r a c t i o n is
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d e s c r i b e d b y a square well p o t e n t i a l with the d e p t h V0 a n d the radius R. Consistent with the fact that the coupling of the n e u t r o n - n u c l e u s channel to the c o m p o u n d nucleus is c o n c e n t r a t e d at the surface, we take
cpR( r ) = gYim( r ) 8 ( r - R )/v/aTr r 2.
(7)
Such an a p p r o x i m a t i o n is likely to b e a p p r o p r i a t e in cases where the wave function, ~k+(r), is well defined, b u t m a y be q u e s t i o n a b l e in cases where it w o u l d have a n o d e in the vicinity of the surface. P a r a m e t e r s g, E ° a n d F R are related to the n e u t r o n w i d t h of the resonance, F~, its energy E R and its radiative width, which are all m e a s u r a b l e q u a n t i ties. In fact, precise k n o w l e d g e of g a n d E ° is not necessary here, results d e p e n d i n g directly on F~ a n d E R. S o m e distinction between E ° a n d E R has b e e n i n t r o d u c e d due to the existence of an energy-shift arising from the coupling of the comp o u n d nucleus state to the e n t r a n c e channel. Finally, as far as the strong interaction is concerned, o n l y the closest r e s o n a n c e is r e t a i n e d for each p a r t i a l wave. T h e pnc interaction is d e s c r i b e d b y an average, square-well type, n e u t r o n - n u c l e u s potential:
Vp° c = ( 2 ~ r / M 3 ) X ~ × ½ ( o "p , p ( O ) O ( R - r ) } , (8) where p(0) represents the nuclear d e n s i t y at the origin. The q u a n t i t y X~ has a direct relationship to the n u c l e o n - n u c l e o n interaction [23] a n d can be c o m p a r e d to the strength of the p r o t o n - n u c l e u s force, X~, p r e s e n t l y d e t e r m i n e d from p n c effects o b s e r v e d in several o d d - p r o t o n nuclei [24]. A p p r o x i m a t i o n s m a d e on the d e s c r i p t i o n of the strong interaction lead to solving the following equation:
~(r)
= j , ( r ) + e ia sin 8 [ B t ( k r ) + i j l ( k r ) ] ,
with k 2 = 2 M r E ,
r
+~-(,)=A}jt(Kr),
with K 2 = 2 M r ( V 0 + E ) .
(10)
T h e c o m p l e x phase shift which a p p e a r s in the a b o v e expression is d e t e r m i n e d b y the relation e ln/z e i ~ s i n S = e i~° s i n 8 ° + E R - E - i ( F R+Fn)/2
'
(11) where 8 ° represents the p o t e n t i a l p h a s e shift, whereas F,, F R a n d E R refer to the neutron reson a n c e properties. T h e relation (11) results from o u r p a r t i c u l a r a p p r o a c h , b u t its validity is quite general. In the low energy range of interest here, 8 o can be derived from the m e a s u r e d scattering length in the case of an s-wave together with the p r e s o n a n c e properties, it is generally of the o r d e r of - k R . F o r a p-wave, the k n o w l e d g e of 8 o, which is generally expected to be of the o r d e r of - ( k R ) 3 / 3 , is u n i m p o r t a n t as far as the second t e r m in eq. (11) is largely d o m i n a n t for nuclei u n d e r consideration. Finally, the q u a n t i t y A [ in eq. (10) is o b t a i n e d b y the c o n d i t i o n that the wave function be c o n t i n u o u s at r = R. It is given b y
A[- = J l ( k R ) + el8 sin 8 [ ~ l , ( k R ) + i j , ( k R ) ] j,(KR).
. (12)
T h e solution, + s+p ( r ) can b e easily o b t a i n e d . F o r an a n g u l a r m o m e n t u m / , it can b e written:
It is i m p o r t a n t to notice that the derivative of the wave function, ~k+(r), at r = R is not continuous, due to the presence of the term 8 ( r - R ) in eq. (9). This feature, together with the fact that the strength of this term is different for s- a n d p-waves, will m a k e the p n c a m p l i t u d e s strongly d e p e n d e n t on the p r o p e r t i e s of the average single particle p o t e n tial. This contrasts with previous a p p r o a c h e s [9-14] where the equality of the a b o v e strengths was m a k i n g the results d e p e n d e n t o n l y on the a s y m p totic p r o p e r t i e s of the n e u t r o n - n u c l e u s wave function. T h e calculation of the p n c a m p l i t u d e is now straightforward. W r i t i n g the f o r w a r d scattering a m p l i t u d e as
r>R
fpnc (0) = G ' < o . - k , )
p2
( -fffr - VoO( R - , ) +
g2
E-(E°-iFR/2) = E~b ~-~( r ).
8 ( r -- R ) ] 4~rR 2
+
) ffsp(r) (9)
(13) 305
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we get for very low energy neutrons (kR << 1):
as;
=
--( Mr//M 3) X ~ A [1 -- sin2( KR ) / ( KR )2]
[1 + e i ~ s i n ( 8 s ) / k R ] [ 1 + 3 e i8~ s i n ( S p ) / ( k R ) 3] X
[ s i n ( K R ) / K R ] [sin( KR ) / K R - cos( KR )] (14)
C o n s i s t e n t l y with the low energy limit c o n s i d e r e d here, factors involving p h a s e shifts in (14) m a y be written: 1 + e i8~ sin(8~)/kR = (1 + 8 ° / k R ) + ( F 2 / Z ) / ( E ~ - E - i F ~ / 2 ) ,
(15) 1 "4- 3 e iSp
sin(Sp)/(kR) 3
= (1 + 3 ~ ° / ( k R ) 3) + ( F P / 2 ) / ( E
p - E - iFP/2),
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= 168 meV, E p = 0.734 eV, 2gF~p = 0.73 x 1 0 - 7 eV, 1 + 8 J k R = - 0 . 9 8 , 1 + 3~p/(kR) 3 = 67, [ s i n ( K R ) / K R ][sin( K R ) / K R - cos( KR )] = - 0.042), whereas the previous a p p r o a c h would have led to an e n h a n c e m e n t of a b o u t - 1 0 . This last n u m b e r is small due to some cancellations, but, even in their absence, it should not be much larger than 100 (absolute magnitude). O u r study shows therefore that the c o n t r i b u t i o n of the n e u t r o n - n u c l e u s c o m p o n e n t might be m u c h larger than what could be expected from the S t o d o l s k y - F o r t e formula. A careful c o m p a r i s o n indicates that this should be generally the case when there are n e u t r o n resonances near threshold in both s- a n d p-waves. In o r d e r to m a k e a c o m p a r i s o n with the contrib u t i o n arising from the p a r i t y a d m i x t u r e of states of the c o m p o u n d nucleus, it is a p p r o p r i a t e to give here its expression [18,19]. In o u r notations, it is written as
(16) O u r expression for the c o n t r i b u t i o n of the singleparticle c o m p o n e n t to the weak a m p l i t u d e is significantly different from the one o b t a i n e d b y S t o d o l s k y [10] or F o r t e [9]. Phase-shifts enter in an o t h e r c o m b i n a t i o n . M o r e i m p o r t a n t is the factor which a p p e a r s in the d e n o m i n a t o r of eq. (14) a n d represents, up to a factor KR, the p r o d u c t of sa n d p - w a v e functions c a l c u l a t e d at the nuclear surface. It evidences the sensitivity, a n n o u n c e d above, to the d e s c r i p t i o n of the strong neut r o n - n u c l e u s i n t e r a c t i o n w i t h i n the n u c l e a r volume. In cases where it is quite small, corres p o n d i n g to a m i n i m u m in one of the s- or p-wave strength functions, it m a y give rise to a further e n h a n c e m e n t . Such cases should be h a n d l e d with caution, however, as c o r r e s p o n d i n g s- or -wave n e u t r o n widths are likely to be small within the p r e s e n t model, thus cancelling possible enhancem e n t resulting from the presence of an s- or p-wave n e u t r o n resonance near threshold. T o illustrate the difference between the present result a n d the one o b t a i n e d b y using the S t o d o l s k y - F o r t e formula, we quote the e n h a n c e m e n t factor with respect to the Born a m p l i t u d e for the case of 1 3 9 L a , which is free from the a b o v e ambiguities. O u r expression (14) would lead to an e n h a n c e m e n t of a b o u t 1550 ( V0 = 46 MeV, R = 6.7 fm, E s = - 4 8 . 6 eV, 2gF, °s 306
c;o= [(SlVp.clP> .... I × [ E - (Eg - iF~/2)] [ E - ( E ~ - i F P / 2 ) ] ' (17) where.... represents the pnc m a t r i x elem e n t between states of the c o m p o u n d nucleus. P r o v i d e d that 1 + 8 ° / k R = 0 a n d 1 + [3/(kR)318 ° = 0, as generally expected, b o t h contributions, G'p. a n d G~'.,. are quite similar as far as the energy d e p e n d e n c e is concerned. It follows that b o t h pictures will roughly give the same relations between the various observables. These are the a s y m m e t r y in the cross sections for different n e u t r o n helicities: P=
[o(+) - o(--)]/[o(+)
+ o(-)1
= 47r I m ( G ' ) / o ,
(18)
a n d the spin r o t a t i o n whose expression is
q~= - 47r9l R e ( G ' ) ,
(19)
where p is the a t o m i c density of the s a m p l e a n d l its length. I n s t e a d of q~, it m a y be interesting to c o n s i d e r the spin r o t a t i o n c o r r e s p o n d i n g to one m e a n free p a t h which is given by 4~( m . f . p . ) = - 4z- R e ( G ' ) / o .
(20)
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The well-known relationships of interest are
Pth/q'(m.f.p)
= -- Im( G ' ) / R e ( G ' )
FP(E - ES) + F~(E - E p) 2[(E-
E~)(E-
E p ) - F~FP/4] '
(21)
and Pres./~h.
=
(°th/°p)(2EP/rP) 2,
(22)
where % is the contribution to the cross section due to the p-wave resonance at its maximum. That both pictures give the same relationships should not surprise. To some extent, the d o m i n a n t contribution to G'.p. might be considered a contribution of the c o m p o u n d nucleus. After comparing with other approaches, it is appropriate to see what experiment would tell us about the strength of the pnc n e u t r o n - n u c l e u s force assuming that the contribution of the neut r o n - n u c l e u s c o m p o n e n t is the d o m i n a n t one. In this aim, we have chosen to work with 139La where a calculation along the lines developed in this paper seems to be reliable. Furthermore, several observables have been studied and their measurements [3-5] satisfy eqs. (21), (22) rather well. Fitting the spin rotation measured at the Institute Laue Langevin for this nucleus, q5 = - ( 2 1 . 9 _ + 2 . 9 ) × 10 -5 r a d / c m , implies the following value for the strength of the pnc n e u t r o n - n u c l e u s force: X~ = - 7 × 10 -6 .
(23)
A more realistic value, taking into account the spin of the nucleus neglected throughout this paper would be X~ = - 1 2 x 10 -6, while fitting the results for the observable P (eq. (18)) would give X~ = - 9 x 10 6. The above values are about - 3 times the strength of the pnc p r o t o n - n u c l e u s force, X p = 3.5 × 10 -6, which can be derived from pnc effects observed in several o d d - p r o t o n nuclei (19F, 41K, 175Lu, 181Ta) [23,24]. Such strengths would imply that the isovector part of the p n c n u c l e o n - n u c l e u s force has a strength larger than for the isoscalar part and has a sign such that both contributions add in the n e u t r o n - n u c l e u s force and, at the contrary, tend to cancel in the p r o t o n - n u c l e u s force. The above conclusions would contradict present
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results in 21Ne [25], where the absence of effect indicates that the strength of the pnc n e u t r o n nucleus force is much smaller than for the p r o t o n - n u c l e u s force ( [ X ~ [ < < X yp) [24]. They would also be difficult to reconcile with results in 18F [25] which put an upper limit on the isovector part of the n u c l e o n - n u c l e u s interaction ( I X ~ XP[ << 5 M 10-6). On the contrary, they might be consistent with the absence of effect in p r o t o n - a scattering at Ep = 45 MeV [26], provided that the cancellation between isovector and isoscalar contributions in this process be stronger than in more complex nuclei (19F, 41K, 175Ln and 181Ta). Assuming that the above strengths arise only from the isovector 7r exchange and isoscalar p exchange would imply that the corresponding pnc m e s o n nucleon coupling constants have the following values: f ~ = l . 3 × 1 0 6 and h0o-1.9×10 _ 6. Such values are slightly outside the reasonable range given for them in ref. [27]. As it supposes that the present analysis of pnc effects in several processes is not so reliable as usually believed, the above picture is not quite satisfying. It cannot be excluded however and, consequently, explaining pnc effects observed in 139La by a contribution from the n e u t r o n - n u c l e u s c o m p o n e n t is an hypothesis which has to be considered seriously. The puzzling situation which results from our analysis clearly requires further studies to distinguish between the various parts of the scattering state which c o m p o n e n t is responsible for the observed effects. In the case where the first terms of eqs. (15), (16), 1 +8°/kR or 1 +[3/(kR)318°p would not be quite negligible c o m p a r e d to the resonance part, some deviations with respect to the relations (21) and (22) should be expected. They could be observed by studying pnc effects within an energy range containing both s- and p-wave resonances [22]. The study of pnc effects in radiative processes m a y also help to disentangle the different sources for parity nonconservation. Thus, within the present model, the circular polarization of 3' issued from direct capture would be larger than what is observed. The corresponding total capture rate is however too small to consider that it is a reasonable explanation for observations in "inclusive" reactions. The model with parity admixture between states of the c o m p o u n d nucleus is 307
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d o i n g b e t t e r here, a l t h o u g h its p r e d i c t i o n s are also t o o s m a l l c o m p a r e d to e x p e r i m e n t [8,28]. It is n o t clear w h e t h e r the f u r t h e r studies b r i e f l y d e s c r i b e d a b o v e c a n p r o v i d e an u n a m b i g u o u s s i g n a t u r e o f the c o m p o n e n t of the s c a t t e r i n g state i n v o l v e d in effects p r e s e n t l y o b s e r v e d in t h e r m a l n e u t r o n - n u c l e u s i n t e r a c t i o n s . A t least, t h e y h a v e to b e d o n e . S h o w i n g t h a t there is n o r o o m in p r e s e n t e x p e r i m e n t a l results for a s i g n i f i c a n t c o n t r i b u t i o n f r o m the n e u t r o n - n u c l e u s c o m p o n e n t w o u l d i m p l y t h a t the s t r e n g t h o f the p n c n e u t r o n - n u c l e u s f o r c e is n e g l i g i b l e c o m p a r e d to the p r o t o n s t r e n g t h a n d t h e r e f o r e w o u l d s u p p o r t a p a r t i c u l a r m o d e l o f the w e a k n u c l e o n - n u c l e o n i n t e r a c t i o n w h i c h has b e e n successful in e x p l a i n i n g several o b s e r v a t i o n s [24]. O n the c o n t r a r y , the n e c e s s i t y to i n v o k e a large c o n t r i b u t i o n f r o m the n e u t r o n - n u c l e u s c o m p o n e n t w o u l d raise serious d o u b t s a b o u t the a p t i t u d e o f this m o d e l o f p n c n u c l e o n - n u c l e o n forces to d e s c r i b e p r o c e s s e s involving neutrons. W h i l e w e w e r e w r i t i n g this p a p e r , w h o s e m a i n results h a v e b e e n b r i e f l y d e s c r i b e d e l s e w h e r e [15,24], we l e a r n t t h a t results s i m i l a r to o u r s h a v e b e e n o b t a i n e d b y O l k h o v s k y a n d Z a i c h e n k o [29]. I n the p r e s e n t p a p e r , h o w e v e r , w e h a v e s h o w n h o w t h e results are c o n n e c t e d to the d e s c r i p t i o n o f the strong interaction. Approximations implicitly made b y t h e s e a u t h o r s h a v e b e e n fully e x p l a i n e d here. D e t a i l s h a v e b e e n g i v e n to e m p h a s i z e i m p r o v e m e n t s w i t h r e s p e c t to s o m e p r e v i o u s a p p r o a c h e s . F u r t h e r m o r e , we m a d e s o m e c o m p a r i s o n w i t h w h a t is k n o w n f r o m o t h e r processes. W e are i n d e b t e d to D r . B. H e c k e l for discuss i o n s w h i c h p a r t l y m o t i v a t e d the p r e s e n t work.
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References [1] M. Forte et al., Phys. Rev. Lett. 45 (1980) 2088. [2] B. Heckel et al., Phys. Lett. I19B (1982) 298. [3] B. Heckel et al., Workshop on Fundamental physics at reactors (Grenoble, 1983), J. Phys. (Paris) Colloque C3 (1984). [4] E.A. Kolomensky et al., Phys. Lett. 107B (1981) 272. [5] V.P. Alfimenkov et al., Nucl. Phys. A398 (1983) 93. [6] V.P. Alfimenkov et al., Workshop on Fundamental physics at reactors (Grenoble, 1983), J. Phys. (Paris) Colloque C3 (1984). [7] V.A. Vesna et al., JETP Lett. 36 (1982) 209. [8] V.M. Lobashov and V.A. Nazarenko, Workshop on Fundamental physics at reactors (Grenoble, 1983), J. Phys. (Paris) Colloque C3 (1984). [9] M. Forte, Inst. Phys. Conf. Ser. No. 42 (1978) 86. [10] L. Stodolsky, Phys. Lett. 50B (1974) 352. [11] D. Tadid and A. Barroso, Nucl. Phys. A294 (1978) 376. [12] A. Barroso and F.M. Margaqa, J. Phys. G. 6 (1980) 657. [13] G. Karl and D. Tadid, Phys. Rev. C 20 (1979) 1959. [14] J. Bernabru and S. Noguera, Nuclear physics, ed. C.H. Dasso (North-Holland, Amsterdam, 1982) p. 755. [15] B. Desplanques and S. Noguera, Proc. Intern. Conf. on Nuclear physics (Florence, 1983), Vol. 1. [16] R. Haas, L.B. Leipuner and R.K. Adair, Phys. Rev. 116 (1959) 1221. [17] R.J. Blin-Stoyle, Phys. Rev. 120 (1960) 181. [18] O.P. Sushkov and V.V. Flambaum, Sov. Phys. Usp. 25 (1982) 1. [19] V.E. Bunakov and V.P. Gudkov, Nucl. Phys. A401 (1983) 93. [20] G.A. Lobov, Soy. J. Nucl. Phys. 35 (1982) 822. [21] L. Stodolsky, Nucl. Phys. B197 (1982) 213. [22] S. Noguera and J. Bernab6u, Phys. Lett. 129 (1983) 125. [23] B. Desplanques and J. Missimer, Nucl. Phys. A300 (1978) 286; Phys. Lett. 84 B (1979) 363. [24] B. Desplanques, Workshop on Fundamental physics at reactors (Grenoble, 1983), J. Phys. (Paris) Colloque C3 (1984). [25] E.G. Adelberger et al., Phys. Rev. C 27 (1983) 2833. [26] R. Henneck et al., Phys. Rev. Lett. 48 (1982) 725. [27] B. Desplanques, J. Donoghue and B. Holstein, Ann. Phys. (NY) 124 (1980) 449. [28] V.V. Flambaum and O.P. Sushkov, preprint 1983 (Novosibirsk). [29] V.S. Olkhovsky and A.K. Zaichenko, Phys. Lett. 130B (1983) 127.
PHYSICS LETTERS
16 August 1984
THE CONTRIBUTION OF THE NEUTRON-NUCLEUS COMPONENT T O P A R I T Y - N O N - C O N S E R V I N G E F F E C T S IN N E U T R O N - N U C L E U S S C A T T E R I N G Bertrand D E S P L A N Q U E S t and Santiago N O G U E R A 2 lnstitut Laue Langevin, 156 X, 38042 Grenoble Cedex, France
Received 2 April 1984
The contribution of the neutron-nucleus component to parity-non-conserving effects in neutron nucleus scattering is reexamined in the case of 139La. Consequences for models of the nucleon-nucleonweak interaction are discussed. The study of p a r i t y - n o n - c o n s e r v i n g (pnc) neut r o n - n u c l e u s interactions at thermal energies has b e e n very active these last few years, experimentally a n d theoretically. Large effects have been observed for n e u t r o n s with both transverse a n d l o n g i t u d i n a l polarizations, giving rise to a spin-rotation in the first case [1-3], a difference in the cross section for n e u t r o n s with different helicities in the second case [1,4-6]. Other effects such as a circular polarization of p h o t o n s or an a s y m m e t r y in their emission with respect to the n e u t r o n polarization [7,8] have also been observed in " i n clusive" reactions. O n the theoretical side, two questions have been raised. The first one concerns the nuclear m e c h a n i s m which produces large effects a n d which is now believed to be essentially due to the presence of p-wave n e u t r o n resonances n e a r threshold [9]. The second one concerns the part of the total n e u t r o n - n u c l e u s scattering state which is involved in observed p n c effects. Two c o n t r i b u t i o n s , c o r r e s p o n d i n g to extreme cases, have been considered here. O n e of them is assumed to come entirely from the n e u t r o n - n u c l e u s compon e n t of the scattering state, the nucleus being in its g r o u n d state [9-15]. A large effect may result in this case from the m a x i m u m overlap between such
1 On leave from Division de Physique Throrique, IPN, Orsay, France. 2 On leave from Departamento Fisica Te6rica, Universidad de Valencia, Valencia, Spain.
c o m p o n e n t s with opposite parities in the i n c o m i n g a n d o u t c o m i n g scattering states. This c o n t r i b u t i o n directly depends o n the pnc n e u t r o n - n u c l e u s force, whose intensity can be calculated from models of the weak n u c l e o n - n u c l e o n interaction. It is det e r m i n e d once the wave f u n c t i o n describing the n e u t r o n - n u c l e u s c o m p o n e n t is given. Conversely, a n d provided that this c o n t r i b u t i o n is the domin a n t one, it m a y be possible to learn something a b o u t the n u c l e o n - n u c l e o n pnc interaction. U n t i l now, this approach has led to too small results c o m p a r e d to experiment. The second c o n t r i b u t i o n [16-22] is assumed to come from the parity admixture of states of the c o m p o u n d nucleus present in the scattering state. The overlap between such states is expected to be small, b u t this suppression is c o m p e n s a t e d b y the fact that, within the nuclear volume, the scattering state has a larger probability to be in some c o m p o n e n t of the c o m p o u n d nucleus than in the n e u t r o n - n u c l e u s c o m p o n e n t . D u e to the complexity of the c o m p o u n d nucleus, the relationship between observables a n d the n u c l e o n n u c l e o n weak interaction is rather loose in this case. Present estimates show that this contribution is certainly a good candidate to explain p n c effects presently observed in n e u t r o n - n u c l e u s scattering. The difficulty to make a precise prediction prevents however to make a definite statem e n t as to its adequacy for a particular case. It is only in the absence of other c a n d i d a t e that it should be considered definitively the correct explanation.
0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
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where the i n c o m i n g a n d o u t c o m i n g scattering states, I~p-+), are o b t a i n e d from solving the Schri3dinger e q u a t i o n
n i a n describing a nucleon moving in the average field p r o d u c e d b y some nucleus, while the second term describes the influence of the coupling of the e n t r a n c e channel to states of the c o m p o u n d nucleus. T h e quantity, ~0R(r ), represents the transition m a t r i x element between s o m e state of the c o m p o u n d nucleus a n d the n e u t r o n - n u c l e u s ent r a n c e channel before being i n t e g r a t e d over the n e u t r o n coordinate. It is generally u n k n o w n , b u t it is expected to be p e a k e d at the nuclear surface, since it is there that nucleons which u n d e r g o some transition are likely to be localized at the energy u n d e r c o n s i d e r a t i o n here. The u n p e r t u r b e d energies of the c o m p o u n d nucleus states a n d their r a d i a t i v e widths are respectively d e n o t e d E ° a n d F R. In terms of the quantities defined above, the a m p l i t u d e s c ~ w o u l d be expressed as
H~t.i.t I~p-+>=E [~p+),
a~ =/dr~b~(r)cpR(r)/[E-(E ° -
In this paper, we r e p o r t on results which show that the d e b a t e is still open. Essentially, we show that m o r e realistic estimates of the c o n t r i b u t i o n of the n e u t r o n - n u c l e u s c o m p o n e n t can lead to the right m a g n i t u d e for p n c effects o b s e r v e d in 139La (spin r o t a t i o n a n d a s y m m e t r y in the total cross section). T h e i m p r o v e m e n t s consist in treating the r e s o n a n c e s o b s e r v e d n e a r threshold as arising from the c o m p o u n d nucleus. U n t i l now, they were considered as p o t e n t i a l resonances. p n c a m p l i t u d e s relevant to n e u t r o n - n u c l e u s scattering can be expressed as fpnc = -- ( M r / 2 ~ r )
(q~-IVp.¢lq,+ ) ,
(1)
(2)
whereas M r represents the r e d u c e d mass ( M r = M × A/(1 + A)). W e m a y write [~b) as a sum of two terms: [~b-+) =
fdr~+p.(r)a+(r)[O) +~_,a~af¢[ 0).
(3)
a
T h e first one describes the n e u t r o n - n u c l e u s comp o n e n t in which we are interested here. T h e seco n d one represents the states of the c o m p o u n d nucleus. T h r o u g h residual interactions, they are c o u p l e d to the n e u t r o n - n u c l e u s e n t r a n c e channel. Q u i t e generally, the SchrOdinger e q u a t i o n (2) gives rise to a system of c o u p l e d equations. A f t e r expressing the a m p l i t u d e s c o r r e s p o n d i n g to the c o m p o u n d nucleus states, a ~ , in terms of the wave f u n c t i o n d e s c r i b i n g the n e u t r o n - n u c l e u s (singleparticle) c o m p o n e n t , solving the Schr6dinger e q u a t i o n (2) a m o u n t s to solve the following equation:
fdrHsp(r, r , P
I
+
t
--
+
E ) + s p ( r ) - E~ksp(r),
(4)
where n s p ( r , r ' , E ) = ½(
p2/ZMr + V ( r ) , a ( r -
+E s E-( E°-
iFs/2)"
r')}
(5)
T h e first term in (5) represents the usual h a m i l t o 304
iFR/2)].
(6)
T h e present a p p r o a c h offers the a d v a n t a g e of a c o n t i n u o u s transition to optical potentials. In particular, the usual i m a g i n a r y p a r t of these p o t e n t i a l s w o u l d result from the average over several resonances of the i m a g i n a r y p a r t of the last term in (5). This p a r t of the i n t e r a c t i o n is very i m p o r t a n t to account for essential features of n u c l e o n nucleus scattering, as for instance inelasticity or also the absence of very large scattering lengths which should b e e x p e c t e d in the case of an inert nucleus. I n the present p a p e r , we are interested in processes which take place in an energy range which is m u c h smaller than the average energy spacing b e t w e e n states of the c o m p o u n d nucleus. W e will therefore retain the exact expression of the last term in H ( r , r', E ) . D u e to differences in the strengths or in the energy d e n o m i n a t o r s , this last t e r m is not the same for states of o p p o s i t e parities. T h e n e u t r o n will therefore feel a different interaction d e p e n d i n g w h e t h e r it is in an s- or a p-wave. N u m e r i c a l m e t h o d s allow to solve eq. (4) quite generally. F o r our purpose, we will c o n s i d e r a simplified case which offers the a d v a n t a g e to be solvable a n a l y t i c a l l y on one hand, to p r o v i d e resuits easily c o m p a r a b l e to previous ones on the o t h e r h a n d [9-14]. W e first assume that the nucleus has no spin. The n u c l e o n - n u c l e u s i n t e r a c t i o n is
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d e s c r i b e d b y a square well p o t e n t i a l with the d e p t h V0 a n d the radius R. Consistent with the fact that the coupling of the n e u t r o n - n u c l e u s channel to the c o m p o u n d nucleus is c o n c e n t r a t e d at the surface, we take
cpR( r ) = gYim( r ) 8 ( r - R )/v/aTr r 2.
(7)
Such an a p p r o x i m a t i o n is likely to b e a p p r o p r i a t e in cases where the wave function, ~k+(r), is well defined, b u t m a y be q u e s t i o n a b l e in cases where it w o u l d have a n o d e in the vicinity of the surface. P a r a m e t e r s g, E ° a n d F R are related to the n e u t r o n w i d t h of the resonance, F~, its energy E R and its radiative width, which are all m e a s u r a b l e q u a n t i ties. In fact, precise k n o w l e d g e of g a n d E ° is not necessary here, results d e p e n d i n g directly on F~ a n d E R. S o m e distinction between E ° a n d E R has b e e n i n t r o d u c e d due to the existence of an energy-shift arising from the coupling of the comp o u n d nucleus state to the e n t r a n c e channel. Finally, as far as the strong interaction is concerned, o n l y the closest r e s o n a n c e is r e t a i n e d for each p a r t i a l wave. T h e pnc interaction is d e s c r i b e d b y an average, square-well type, n e u t r o n - n u c l e u s potential:
Vp° c = ( 2 ~ r / M 3 ) X ~ × ½ ( o "p , p ( O ) O ( R - r ) } , (8) where p(0) represents the nuclear d e n s i t y at the origin. The q u a n t i t y X~ has a direct relationship to the n u c l e o n - n u c l e o n interaction [23] a n d can be c o m p a r e d to the strength of the p r o t o n - n u c l e u s force, X~, p r e s e n t l y d e t e r m i n e d from p n c effects o b s e r v e d in several o d d - p r o t o n nuclei [24]. A p p r o x i m a t i o n s m a d e on the d e s c r i p t i o n of the strong interaction lead to solving the following equation:
~(r)
= j , ( r ) + e ia sin 8 [ B t ( k r ) + i j l ( k r ) ] ,
with k 2 = 2 M r E ,
r
+~-(,)=A}jt(Kr),
with K 2 = 2 M r ( V 0 + E ) .
(10)
T h e c o m p l e x phase shift which a p p e a r s in the a b o v e expression is d e t e r m i n e d b y the relation e ln/z e i ~ s i n S = e i~° s i n 8 ° + E R - E - i ( F R+Fn)/2
'
(11) where 8 ° represents the p o t e n t i a l p h a s e shift, whereas F,, F R a n d E R refer to the neutron reson a n c e properties. T h e relation (11) results from o u r p a r t i c u l a r a p p r o a c h , b u t its validity is quite general. In the low energy range of interest here, 8 o can be derived from the m e a s u r e d scattering length in the case of an s-wave together with the p r e s o n a n c e properties, it is generally of the o r d e r of - k R . F o r a p-wave, the k n o w l e d g e of 8 o, which is generally expected to be of the o r d e r of - ( k R ) 3 / 3 , is u n i m p o r t a n t as far as the second t e r m in eq. (11) is largely d o m i n a n t for nuclei u n d e r consideration. Finally, the q u a n t i t y A [ in eq. (10) is o b t a i n e d b y the c o n d i t i o n that the wave function be c o n t i n u o u s at r = R. It is given b y
A[- = J l ( k R ) + el8 sin 8 [ ~ l , ( k R ) + i j , ( k R ) ] j,(KR).
. (12)
T h e solution, + s+p ( r ) can b e easily o b t a i n e d . F o r an a n g u l a r m o m e n t u m / , it can b e written:
It is i m p o r t a n t to notice that the derivative of the wave function, ~k+(r), at r = R is not continuous, due to the presence of the term 8 ( r - R ) in eq. (9). This feature, together with the fact that the strength of this term is different for s- a n d p-waves, will m a k e the p n c a m p l i t u d e s strongly d e p e n d e n t on the p r o p e r t i e s of the average single particle p o t e n tial. This contrasts with previous a p p r o a c h e s [9-14] where the equality of the a b o v e strengths was m a k i n g the results d e p e n d e n t o n l y on the a s y m p totic p r o p e r t i e s of the n e u t r o n - n u c l e u s wave function. T h e calculation of the p n c a m p l i t u d e is now straightforward. W r i t i n g the f o r w a r d scattering a m p l i t u d e as
r>R
fpnc (0) = G ' < o . - k , )
p2
( -fffr - VoO( R - , ) +
g2
E-(E°-iFR/2) = E~b ~-~( r ).
8 ( r -- R ) ] 4~rR 2
+
) ffsp(r) (9)
(13) 305
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we get for very low energy neutrons (kR << 1):
as;
=
--( Mr//M 3) X ~ A [1 -- sin2( KR ) / ( KR )2]
[1 + e i ~ s i n ( 8 s ) / k R ] [ 1 + 3 e i8~ s i n ( S p ) / ( k R ) 3] X
[ s i n ( K R ) / K R ] [sin( KR ) / K R - cos( KR )] (14)
C o n s i s t e n t l y with the low energy limit c o n s i d e r e d here, factors involving p h a s e shifts in (14) m a y be written: 1 + e i8~ sin(8~)/kR = (1 + 8 ° / k R ) + ( F 2 / Z ) / ( E ~ - E - i F ~ / 2 ) ,
(15) 1 "4- 3 e iSp
sin(Sp)/(kR) 3
= (1 + 3 ~ ° / ( k R ) 3) + ( F P / 2 ) / ( E
p - E - iFP/2),
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= 168 meV, E p = 0.734 eV, 2gF~p = 0.73 x 1 0 - 7 eV, 1 + 8 J k R = - 0 . 9 8 , 1 + 3~p/(kR) 3 = 67, [ s i n ( K R ) / K R ][sin( K R ) / K R - cos( KR )] = - 0.042), whereas the previous a p p r o a c h would have led to an e n h a n c e m e n t of a b o u t - 1 0 . This last n u m b e r is small due to some cancellations, but, even in their absence, it should not be much larger than 100 (absolute magnitude). O u r study shows therefore that the c o n t r i b u t i o n of the n e u t r o n - n u c l e u s c o m p o n e n t might be m u c h larger than what could be expected from the S t o d o l s k y - F o r t e formula. A careful c o m p a r i s o n indicates that this should be generally the case when there are n e u t r o n resonances near threshold in both s- a n d p-waves. In o r d e r to m a k e a c o m p a r i s o n with the contrib u t i o n arising from the p a r i t y a d m i x t u r e of states of the c o m p o u n d nucleus, it is a p p r o p r i a t e to give here its expression [18,19]. In o u r notations, it is written as
(16) O u r expression for the c o n t r i b u t i o n of the singleparticle c o m p o n e n t to the weak a m p l i t u d e is significantly different from the one o b t a i n e d b y S t o d o l s k y [10] or F o r t e [9]. Phase-shifts enter in an o t h e r c o m b i n a t i o n . M o r e i m p o r t a n t is the factor which a p p e a r s in the d e n o m i n a t o r of eq. (14) a n d represents, up to a factor KR, the p r o d u c t of sa n d p - w a v e functions c a l c u l a t e d at the nuclear surface. It evidences the sensitivity, a n n o u n c e d above, to the d e s c r i p t i o n of the strong neut r o n - n u c l e u s i n t e r a c t i o n w i t h i n the n u c l e a r volume. In cases where it is quite small, corres p o n d i n g to a m i n i m u m in one of the s- or p-wave strength functions, it m a y give rise to a further e n h a n c e m e n t . Such cases should be h a n d l e d with caution, however, as c o r r e s p o n d i n g s- or -wave n e u t r o n widths are likely to be small within the p r e s e n t model, thus cancelling possible enhancem e n t resulting from the presence of an s- or p-wave n e u t r o n resonance near threshold. T o illustrate the difference between the present result a n d the one o b t a i n e d b y using the S t o d o l s k y - F o r t e formula, we quote the e n h a n c e m e n t factor with respect to the Born a m p l i t u d e for the case of 1 3 9 L a , which is free from the a b o v e ambiguities. O u r expression (14) would lead to an e n h a n c e m e n t of a b o u t 1550 ( V0 = 46 MeV, R = 6.7 fm, E s = - 4 8 . 6 eV, 2gF, °s 306
c;o= [(SlVp.clP> .... I × [ E - (Eg - iF~/2)] [ E - ( E ~ - i F P / 2 ) ] ' (17) where
[o(+) - o(--)]/[o(+)
+ o(-)1
= 47r I m ( G ' ) / o ,
(18)
a n d the spin r o t a t i o n whose expression is
q~= - 47r9l R e ( G ' ) ,
(19)
where p is the a t o m i c density of the s a m p l e a n d l its length. I n s t e a d of q~, it m a y be interesting to c o n s i d e r the spin r o t a t i o n c o r r e s p o n d i n g to one m e a n free p a t h which is given by 4~( m . f . p . ) = - 4z- R e ( G ' ) / o .
(20)
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PHYSICS LETTERS
The well-known relationships of interest are
Pth/q'(m.f.p)
= -- Im( G ' ) / R e ( G ' )
FP(E - ES) + F~(E - E p) 2[(E-
E~)(E-
E p ) - F~FP/4] '
(21)
and Pres./~h.
=
(°th/°p)(2EP/rP) 2,
(22)
where % is the contribution to the cross section due to the p-wave resonance at its maximum. That both pictures give the same relationships should not surprise. To some extent, the d o m i n a n t contribution to G'.p. might be considered a contribution of the c o m p o u n d nucleus. After comparing with other approaches, it is appropriate to see what experiment would tell us about the strength of the pnc n e u t r o n - n u c l e u s force assuming that the contribution of the neut r o n - n u c l e u s c o m p o n e n t is the d o m i n a n t one. In this aim, we have chosen to work with 139La where a calculation along the lines developed in this paper seems to be reliable. Furthermore, several observables have been studied and their measurements [3-5] satisfy eqs. (21), (22) rather well. Fitting the spin rotation measured at the Institute Laue Langevin for this nucleus, q5 = - ( 2 1 . 9 _ + 2 . 9 ) × 10 -5 r a d / c m , implies the following value for the strength of the pnc n e u t r o n - n u c l e u s force: X~ = - 7 × 10 -6 .
(23)
A more realistic value, taking into account the spin of the nucleus neglected throughout this paper would be X~ = - 1 2 x 10 -6, while fitting the results for the observable P (eq. (18)) would give X~ = - 9 x 10 6. The above values are about - 3 times the strength of the pnc p r o t o n - n u c l e u s force, X p = 3.5 × 10 -6, which can be derived from pnc effects observed in several o d d - p r o t o n nuclei (19F, 41K, 175Lu, 181Ta) [23,24]. Such strengths would imply that the isovector part of the p n c n u c l e o n - n u c l e u s force has a strength larger than for the isoscalar part and has a sign such that both contributions add in the n e u t r o n - n u c l e u s force and, at the contrary, tend to cancel in the p r o t o n - n u c l e u s force. The above conclusions would contradict present
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results in 21Ne [25], where the absence of effect indicates that the strength of the pnc n e u t r o n nucleus force is much smaller than for the p r o t o n - n u c l e u s force ( [ X ~ [ < < X yp) [24]. They would also be difficult to reconcile with results in 18F [25] which put an upper limit on the isovector part of the n u c l e o n - n u c l e u s interaction ( I X ~ XP[ << 5 M 10-6). On the contrary, they might be consistent with the absence of effect in p r o t o n - a scattering at Ep = 45 MeV [26], provided that the cancellation between isovector and isoscalar contributions in this process be stronger than in more complex nuclei (19F, 41K, 175Ln and 181Ta). Assuming that the above strengths arise only from the isovector 7r exchange and isoscalar p exchange would imply that the corresponding pnc m e s o n nucleon coupling constants have the following values: f ~ = l . 3 × 1 0 6 and h0o-1.9×10 _ 6. Such values are slightly outside the reasonable range given for them in ref. [27]. As it supposes that the present analysis of pnc effects in several processes is not so reliable as usually believed, the above picture is not quite satisfying. It cannot be excluded however and, consequently, explaining pnc effects observed in 139La by a contribution from the n e u t r o n - n u c l e u s c o m p o n e n t is an hypothesis which has to be considered seriously. The puzzling situation which results from our analysis clearly requires further studies to distinguish between the various parts of the scattering state which c o m p o n e n t is responsible for the observed effects. In the case where the first terms of eqs. (15), (16), 1 +8°/kR or 1 +[3/(kR)318°p would not be quite negligible c o m p a r e d to the resonance part, some deviations with respect to the relations (21) and (22) should be expected. They could be observed by studying pnc effects within an energy range containing both s- and p-wave resonances [22]. The study of pnc effects in radiative processes m a y also help to disentangle the different sources for parity nonconservation. Thus, within the present model, the circular polarization of 3' issued from direct capture would be larger than what is observed. The corresponding total capture rate is however too small to consider that it is a reasonable explanation for observations in "inclusive" reactions. The model with parity admixture between states of the c o m p o u n d nucleus is 307
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
d o i n g b e t t e r here, a l t h o u g h its p r e d i c t i o n s are also t o o s m a l l c o m p a r e d to e x p e r i m e n t [8,28]. It is n o t clear w h e t h e r the f u r t h e r studies b r i e f l y d e s c r i b e d a b o v e c a n p r o v i d e an u n a m b i g u o u s s i g n a t u r e o f the c o m p o n e n t of the s c a t t e r i n g state i n v o l v e d in effects p r e s e n t l y o b s e r v e d in t h e r m a l n e u t r o n - n u c l e u s i n t e r a c t i o n s . A t least, t h e y h a v e to b e d o n e . S h o w i n g t h a t there is n o r o o m in p r e s e n t e x p e r i m e n t a l results for a s i g n i f i c a n t c o n t r i b u t i o n f r o m the n e u t r o n - n u c l e u s c o m p o n e n t w o u l d i m p l y t h a t the s t r e n g t h o f the p n c n e u t r o n - n u c l e u s f o r c e is n e g l i g i b l e c o m p a r e d to the p r o t o n s t r e n g t h a n d t h e r e f o r e w o u l d s u p p o r t a p a r t i c u l a r m o d e l o f the w e a k n u c l e o n - n u c l e o n i n t e r a c t i o n w h i c h has b e e n successful in e x p l a i n i n g several o b s e r v a t i o n s [24]. O n the c o n t r a r y , the n e c e s s i t y to i n v o k e a large c o n t r i b u t i o n f r o m the n e u t r o n - n u c l e u s c o m p o n e n t w o u l d raise serious d o u b t s a b o u t the a p t i t u d e o f this m o d e l o f p n c n u c l e o n - n u c l e o n forces to d e s c r i b e p r o c e s s e s involving neutrons. W h i l e w e w e r e w r i t i n g this p a p e r , w h o s e m a i n results h a v e b e e n b r i e f l y d e s c r i b e d e l s e w h e r e [15,24], we l e a r n t t h a t results s i m i l a r to o u r s h a v e b e e n o b t a i n e d b y O l k h o v s k y a n d Z a i c h e n k o [29]. I n the p r e s e n t p a p e r , h o w e v e r , w e h a v e s h o w n h o w t h e results are c o n n e c t e d to the d e s c r i p t i o n o f the strong interaction. Approximations implicitly made b y t h e s e a u t h o r s h a v e b e e n fully e x p l a i n e d here. D e t a i l s h a v e b e e n g i v e n to e m p h a s i z e i m p r o v e m e n t s w i t h r e s p e c t to s o m e p r e v i o u s a p p r o a c h e s . F u r t h e r m o r e , we m a d e s o m e c o m p a r i s o n w i t h w h a t is k n o w n f r o m o t h e r processes. W e are i n d e b t e d to D r . B. H e c k e l for discuss i o n s w h i c h p a r t l y m o t i v a t e d the p r e s e n t work.
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