DYNAMICAL EFFECTS IN NEUTRON OPTICS AND SPIN INTERACTIONS

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DYNAMICAL EFFECTS IN NEUTRON OPTICS AND SPIN INTERACTIONS

M. Forte

To cite this version:

M. Forte. DYNAMICAL EFFECTS IN NEUTRON OPTICS AND SPIN INTERACTIONS. Journal de Physique Colloques, 1984, 45 (C3), pp.C3-249-C3-253. �10.1051/jphyscol:1984342�. �jpa-00224059�

DYNAMICAL EFFECTS IN NEUTRON OPTICS AND SPIN INTERACTIONS

M. Forte

Physics Division, J o i n t Research Centre - I s p r a Establishment, 21020 I s p r a (Val, I t a l y

Re'sume' - On e'tudie l a production d ' e f f e t s d'optique neutronique qui sont s e n s i b l e s 2 la diffusion non-en avant du spin d a n s l e s conditions de l a diffraction dynamique. On considgre l e s c a s d e s i n t e r a c t i o n s s p i n - o r b i t e e t d e s i n t e r a c t i o n s hypothe'tiques qui violent P e t T .

A b s t r a c t - The g e n e r a t i o n of neutron optical effects, which a r e s e n s i t i v e t o non-forward spin s c a t t e r i n g i n dynamical diffraction conditions, i s studied. C a s e s of s p i n - o r b i t and of supposed P and T violating i n t e r a c t i o n s a r e considered.

In the study of low-energy neutron s p i n i n t e r a c t i o n s , the m o s t s e n s i t i v e e x p e r i - m e n t s a r e based on neutron optical effects. We c o n s i d e r , f o r example, the P- violation effects, namely the s p i n rotation and the helicity-dependent t r a n s m i s s i o n due to weak c u r r e n t s i n the neutron n u c l e a r i n t e r a c t i o n , ,which have been observed with v a r i o u s n u c l e a r t a r g e t s and a r e reviewed i n different p a p e r s of t h i s Workshop.

Effects of the f i r s t and of the second kind a r e d e t e r m i n e d , r e s p e c t i v e l y , by the r e a l and by the i m a g i n a r y spin-dependent p a r t of the optical potential. In o r d i n a r y t a r g e t s , the optical potential i s proportional to the f o r w a r d coherent s c a t t e r i n g f a c t o r Fo of t a r g e t c e n t r e s , thus excluding the possibility of optical effects r e l a t e d to s p i n - o r b i t and t o o t h e r spin i n t e r a c t i o n s , violating both the P and T s y m m e t r y , which do not contribute to f o r w a r d s c a t t e r i n g .

In a p e r f e c t c r y s t a l , instead, the propagation of c o h e r e n t neutron waves, a p p r o a c h - ing diffraction conditions ko-lch =

=

Different a s p e c t s and c a s e s a r e considered i n a previous work /1/ (and r e f e r e n c e s t h e r e i n ) , looking a t spin effects i n the e m e r g e n t t r a n s m i t t e d and r e f l e c t e d neutron beams. We s h o r t l y r e v i e w a few r e l e v a n t points.

The s c a t t e r i n g amplitudes, f o r the considered spin interactions, a r e a n t i s y m m e - t r i c : f6(k, k t ) = - & ( k t , h). In t h i s c a s e , a non-monoatomic a c e n t r i c s t r u c t u r e i s n e c e s s a r y t o have spin t e r m s in F F . Assuming a complex f, = ( s t i s ' ) o3 and a r e a l n u c l e a r amplitude fn, and s e p a r a t i n g the contributions of e a c h amplitude t e r m , we have

The r e l e v a n t spin t e r m s , r e f e r r i n g the r e l a t i v e phase a n g l e s t o 6 , = 0, a r e given by

R ~ ( F F ) O / 1 ^F, I ^{= -2(F$} I s i n 6;, I ~ ( F F ) , / IFn I ^{= 2} s i n 6,

and produce, r e s p e c t i v e l y , spin rotation and s p i n t r a n s m i s s i o n effects. The r o l e of r e a l and i m a g i n a r y amplitudes i s exchanged, with r e s p e c t t o o r d i n a r y optics,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984342

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and t h e magnitude of s p i n effects i s e s s e n t i a l l y independent of IF,)

.

The e l e c t r o m a g n e t i c s p i n - o r b i t i n t e r a c t i o n g i v e s a p u r e i m a g i n a r y amplitude f S a i/.g.kxklFe, with Fe the e l e c t r i c a t o m f o r m f a c t o r . Related spin r o t a t i o n effects c a n be quite l a r g e , f o r example r o t a t i o n a n g l e s p e r c m a1

-

The n u c l e a r s p i n - o r b i t is usually negligible, but c a n be enhanced, with a complex amplitude, i n the p r e s e n c e of n e a r - t h r e s h o l d p - r e s o n a n c e s . R e f e r r i n g t o known n u c l e a r p a r a m e t e r s , but to hypothetical c r y s t a l s a m p l e s , we e s t i m a t e s p i n r o t a - tions a 1 2. ,., 1 0 - 3 and polarization t r a n s m i s s i o n e f f e c t s p e r c m = 1 0 - 3 "

10-4. A d i r e c t d e t e r m i n a t i o n of t h e s e p-wave effects a t threshold, with p a r t i c u l a r nuclei, m a y be i n t e r e s t i n g in connection with P-violation e x p e r i m e n t s .

A simultaneous violation of P and T i n the neutron-nucleon i n t e r a c t i o n would allow a t e r m f poc ~i G p ~ ~ ( h - k t ) i n n u c l e a r s c a t t e r i n g /2/. In the vicinity of a p - r e s o - nance, f pi s expected t o be s t r o n g l y enhanced and complex, in analogy with P- ~ odd T - e v e n i n t e r a c t i o n s / 3 / . We m a y a l s o r e f e r t o known enhanced P-violation ef- f e c t s and t o a s s u m p t i o n s on the r e l a t i v e coupling s t r e n g t h G P T / G ~ / 2 , 4 / , f o r a n indicative upper l i m i t e s t i m a t e : a1 (, f o r t h e r m a l neutrons and a 1 Z P1 d 10-4 a t r e s o n a n c e , in a g r e e m e n t with the e s t i m a t e f o r a n o t h e r kind of P- and T-viola- tion effects, a t the 0.75 eV p - r e s o n a n c e of 139 La /4/.

A m o s t advanced investigation of P and T violation i n fundamental interactions is the s e a r c h f o r a neutron e l e c t r i c dipole moment, i n f r e e - n e u t r o n r e s o n a n c e expe- r i m e n t s r e p o r t e d i n t h i s Workshop. The neutron a t o m i c s c a t t e r i n g includes a n EDM amplitude of the f o r m ~ E D M oc ipecr(h-&')re. An evaluation of the r e l a t e d s p i n r o t a t i o n in a q u a r t z c r y s t a l gives, f o r favourable r e f l e c t i o n s , a l ( r a d / c m ) % l o 1 9 !?

with R (cm) = p e / e . A conceivable EDM e x p e r i m e n t should be based on a back- r e f l e c t i o n configuration, i n the Bragg c a s e , to s u p p r e s s s p i n - o r b i t effects. P a r t i - c u l a r techniques and feasibility p r o b l e m s a r e d i s c u s s e d in / I / .

A l s o f o r the possibility of a n e x p e r i m e n t a l investigation of r e l a t i v e l y l a r g e effects, a c r u c i a l problem i s the existence of a suitable c r y s t a l t a r g e t . A c e n t r i c s t r u c t u r e s a r e not s o r a r e a n d , i n a few c a s e s , c r y s t a l s of good quality a r e available inclu- ding a v a r i e t y of nuclides, however, a question t o be f i r s t investigated c o n c e r n s the d e g r e e of perfection which i s r e a l l y c r i t i c a l t o o b s e r v e dynamical s p i n effects.

In the next p a r t of t h i s w o r k we a n a l z s e the g e n e r a t i o n of spin effects inside a p e r - fect c r y s t a l . C e r t a i n p a r a m e t e r s a r e defined i n the u s u a l way by boundary condi- tions which r e f e r t o a plane plate of thickness d , however we will put into evidence r a t h e r g e n e r a l r e l a t i o n s and r e s u l t s , r e l e v a n t f o r a f u r t h e r investigation of spin e f f e c t s i n non-perfect c r y s t a l s .

In the vicinity of a reflection, ko-kn = h, we c o n s i d e r the plane -wave solution in a p e r i o d i c a l potential

- ^A

where go = l / g o =Lo. fi and A ~ + / A , + - - = - z + / F . ^-

Continuity conditions a t the e n t r a n c e s u r f a c e , with a n incident wave 9E exp(i&. r),

d e t e r m i n e the d i r e c t i o n of fi, p a r a l l e l t o the ingoing n o r m a l , and

lco = k + (1/2) k Fogo% ( o r d i n a r y r e f r a c t i o n , taking Fo and F i n units of k2).

The phase velocities have the f o r m

H e r e X, with Zkg. h = h2, r e p r e s e n t s the deviation f r o m the c e n t r e of t h e d i f f r a c - tion line, o r r e s o n a n c e e r r o r , and _W i s the pendellosung frequency, while the r e - flection g e o m e t r y d e t e r m i n e s gh = kh. fi and g = l / g = go/gh. I c i s convenient t o s e - p a r a t e the n u c l e a r and the much s m a l l e r spin t e r m , writing FF = (FF),+FF)o, with (FF), % I Fn 1 (assuming a negligible a b s o r p t i o n ) and to introduce reduced units

.

w h e r e t i s the penetration depth in extinction length units, putting t = 0 a t the en- t r a n c e s u r f a c e . The spin effects a r e e s s e n t i a l l y due t o s p i n t e r m s i n the expozen- t i a l f a c t o r s of (Z), given by a reduced f o r m ( w h e r e got i s a coordinate along k o )

In o t h e r c a s e s we can safely a p p r o x i m a t e

In t h e Bragg c a s e , we implicitly exclude the r a n g e where w2 ,< 0 (no t r a n s m i s s i o n ) . The amplitude r a t i o i s d e t e r m i n e d by the boundary conditions %(O) = 0 i n the Laue c a s e and Ph(d) = 0 in the Bragg c a s e , obtaining r e s p e c t i v e l y

Ao+ = - + z ~ / ( z + - z ) and A = T Z F exp (Fiwgod) /N

- of.

with N the s u m of n u m e r a t o r s , t o have A o++Ao- = P (0) = P k(0) = 1. T h e r e f o r e , i n the Bragg c a s e , the wave a m p l i t u d e s include a pendglliisung f a c t o r , with a n effec- tive dependence on s p i n t e r m s .

The r e l a t i o n 1 ^of 1 2 / I ~ o , l ~ z - ' / z m e a n s that, f o r a sufficiently l a r g e r e s o - nance e r r o r ( s a y f o r w > , g1/2 in the B r a g g c a s e ) and corresponding t o i t s sign, t h e r e i s one dominant t r a n s m i t t e d wave, a s s o c i a t e d with the s m a l l e r of I z + I .

A qualitative validity of t h i s r e s u l t c a n be stated, i n g e n e r a l , by the physiczl a s s u m p t i o n I ^Ah+ / 2 + /(Ah- 1 < JAOt 1 ⁺ 1 ^AO- 1 ^2.

T o proceed, i t i s convenient t o c o n s i d e r the c u r r e n t d e n s i t i e s a s s o c i a t e d with '+'o(r) and qh(%, given by Lo, h ( ~ ) = ko, h(p*,, hyq, hh)- BY m e a n s of (2), d i s r e g a r d i n g t h e spin dependence, f o r the moment, and puttln k /A,+ l 2 ^{= I,+.,} - k /AhL1 = Ih+, .Zf - = R + , - we obtain the flux density, f o r any Io+)?o-.

With our boundary conditions, a s i m p l e a l g e b r a d e t e r m i n e s the splitting r a t i o of t h e incident flux bk = kgo into 4, and $ h a t the s u r f a c e s , reproducing well known r e s u l t s . In the Bragg c a s e , r e f e r r i n g t o pendellosung a v e r a g e s { I o t } = R ~ / J R + - R - J we obtain { @(t) } = { 9 o(d) 1 = kgo w/ x (outgoing t r a n s m i t t e d flux).

Extending t h i s f o r m a l i s m , we define s p i n c u r r e n t d e n s i t i e s , r e g a r d i n g Po, h a s s p a c e x s p i n functions:

w h e r e i m a y be omitted. We f i r s t d e s c r i b e a s m a l l spin t r a n s m i s s i o n effect, due t o wo = i b o 3 , ( ( b d ( < < l ) . To evaluate the flux 6 . 2 3 of 0 3 , we have to take the d i f - f e r e n c e of J. fi i n the s p i n s t a t e s 03 = t 1. The r e s u l t

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shows a l i n e a r growth of the s p i n flux v e r s u s t , while the i n i t i a l value fiS(0) =

,.

-

The equivalent f o r m C = -2bt ~ o + ( l + g ~ + ) - I ~ - ( ~ + ~ R - ) ] allows to v e r i f y t h a t , i n the Bragg c a s e , on e i t h e r side of the t r a n s m i s s i o n c u r v e , (that i s f o r -gRt < 1 and -gR- > l o r v i c e v e r s a ) the sign of Z i s uniquely defined and r e v e r s e s like x.

T h i s r e l e v a n t r e s u l t i s valid in g e n e r a l .

Our boundary conditions give Ih+ = Ih- and, t h e r e f o r e , 2 = - 2 b t ( I o t - ~ o - ) . In the Bragg c a s e , the pendelltjsung a v e r a g e i s ; C } = -2bt x/l X I . In the Laae c a s e

Z = g. S = gS0, with no s p i n effect i n J h (with no n u c l e a r absorption).

F i n a l l y , we c a n evaluate the neutron flux polarization, given by

In the Bragg c a s e , where P ( d ) i s the t r a n s m i t t e d beam polarization, we have

I n { P 3 ( t ) } , only the constant t e r m i s a n a v e r a g e . According t o (9) and (4), some enhancement of spin effects (not included i n o u r previous e s t i m a t e s ) i s expected n e a r the t o t a l reflection edge (w + 0) and it\ a s y m m e t r i c reflection w i t h l g \ > > l , i n the Laue c a s e too.

To d e s c r i b e the s p i n r o t a t i o n around f 3, due t o wu = a 3 , with I a d ] <<I, we m a y c o n s i d e r a n incident beam completely polarized along X I . The s a m e f o r m a l r e l a - tions a r e obtained but with -a, S 2 , P 2 , e t c . , in place of b, S3, P g , etc. The a p p r o x - imate s p i n r o t a t i o n i s then given by P2.

F o r a sufficiently l a r g e r e s o n a n c e e r r o r and a r e l a t i v e l y low reflection intensity, t h e r e i s a predominant Io+ o r I,-, with a s p i n phase-velocity t w o o r -wo, c o r - responding t o the e r r o r sign. In this a p p r o a c h , by dividing (7) by ( 5 ) , we obtain the s i m p l e r e l a t i o n P3x -2bgot ( x / l x l ) , w h e r e b cx g w - l , and the analogous f o r P2.

In g e n e r a l , i n the l i m i t ~ 2 > > ~ ~ F , t h a t i s W ^% X = g(8-$B), a solution (2) r e d u c e s t o one r e f r a c t e d y a v e propagating in a d i r e c t i o n n e a r ko, in a spin-dependent op- t i c a l potential ( F F ) u / 2 ( 8 - 8 B ) . T h i s f o r m , independent of boundary conditions, r e p r e s e n t s the effective dynamical o r "resonance" neutron s p i n i n t e r a c t i o n with the c r y s t a l .

A r e l e v a n t question c o n c e r n s the existence of c o h e r e n t s p i n effects in n o n - p e r f e c t c r y s t a l s , where the optical-potential periodicity i s a l t e r e d . In this c a s e , s i m p l e plane wave solutions like (2) do not apply, but m a y be r e p l a c e d following Takagi /5, 6/, by wave packets with a s p a c e r e p r e s e n t a t i o n

where D o ( r ) , Dh(x) a r e ( n a r r o w ) d i s t r i b u t i o n functions a r o u n d r e f r a c t e d and r e f l e c - t e d n e u t r o n t r a j e c t o r i e s , which s a t i s f y p a r t i a l d i f f e r e n t i a l equations of the kind

position, a p p e a r s i n X' = X-k-l(a/aqh)h. y ( r ) , w h e r e i s the d i s p l a c e m e n t of the a t o m s of a unit c e l l ( r e g a r d e d a s r i g i d ) with r e s p e c t t o a p e r f e c t r e f e r e n c e l a t t i c e , f o r which X i s defined. Usually, the second t e r m in XI i s a s s u m e d relatively s m a l l and slowly varying. In regions with no d i s t o r t i o n , the Takagi equations a r e s a t i s - fied by a packet of o r d i n a r y solutions (2). A s a f i r s t a p p r o a c h we c o n s i d e r the c a s e of a back r e f l e c t i o n , i n the Bragg c a s e , i n a s y s t e m composed of q u a s i - p e r f e c t blocks of a sufficient s i z e ( a t l e a s t s e v e r a l estinction lengths) with t i l t a n g l e s , r e - lative t o the r e f e r e n c e l a t t i c e , definitely l e s s than the i n t r i n s i c width of the r e f l e c - t i o n ( d m i n u t e s ) . T h e r e f o r e , i n a n y block, g = -1 and X t ( r ) X ( r e f e r e n c e value, d e t e r m i n e d by k). F o r neutrons t r a n s m i t t e d on a given side of the t r a n s m i s s i o n c u r v e , a s u b s t a n t i a l spin effect will r e s u l t by the accumulation of r e g u l a r i n c r e - m e n t s i n e a c h block while weak i r r e g u l a r spin changes through g r a i n boundaries m a y be d i s r e g a r d e d .

The influence of lattice d i s t o r t i o n c a n be studied by eq. (1 1) with (a/aqh) = -(a/aqo),

which m a y be r e w r i t t e n i n reduced units t=klFn1 qo, x1=x'/1Fn(, e t c . The equation a d m i t s a +ble solution iDo(x). I t is e a s i l y s e e n that, by introducing a s m a l l s p i n t e r m in FF = J F , ) ~ + ^{(FF), = IFn} 1' (l+ccr3), the solution i s v a r i e d , a p p r o x i m a t e - ly, by a f a c t o r exp[io3 b+(t)] - (for 6+ - <<l), such that

(L c/2w1)dt, with w' = (x12-1) 1 /2

T h e r e f o r e , a l s o in non-periodical solutions Do+, r e s p e c t i v e spin-dependent p h a s e - f a c t o r s c a n be s e p a r a t e d (with 6 + ( t ) depending En the neutron t r a j e c t o r y ) and s p i n c o h e r e n c e i s p r e s e r v e d . T h i s s i m p l e extension of the p e r f e c t - l a t t i c e c a s e (where w ' ( ~ ) = w and c/2w1 = a o r ib) s p e c i f i e s the qualitative d e s c r i p t i o n of s p i n effects i n the previous p i c t u r e , and allows to f o r e s e e qualitatively s i m i l a r effects i n d i s - t o r t e d blocks and in the whole c r y s t a l . Indeed, with reasonable physical a s s u m p - t i o n s , f u r t h e r considerations following the line of previous c a s e s , support the s i g n coherence of spin effects along neutron t r a j e c t o r i e s , when the t r a n s m i s s i o n o c c u r s on one side of the effective r e f l e c t i o n c u r v e (including d i s t o r t i o n and m o s a - i c broadening and a s s u m i n g no s a t e l l i t e s ) . In p a r t i c u l a r , a d e s c r i p t i o n with one wave-packet ( e i t h e r Do+, 6+ o r Do-, 6, ) i s meaningful in the l i m i t of a good t r a n s - m i s s i o n i n analogy with the p e r f e c t c r y s t a l c a s e .

In s u m m a r y , tke existence of c o h e r e n t s p i n effects d o e s not n e c e s s a r i l y r e q u i r e a long-range coherence of neutron waves but, a t l e a s t a sign coherence of the diffe- r e n t i a l spin-dependent phase velocity ( o r attenuation) along the neutron t r a j e c t o r i e s . The p r e s e n t d i s c u s s i o n i s p r e l i m i n a r y , and d o e s not c o n s i d e r , f o r instance, the r o l e of waves r e f l e c t e d and diffused by c r y s t a l i m p e r f e c t i o n s . B e s i d e s the e x - a c t back-reflection, o t h e r c a s e s should be considered. Finally, we suggest t h a t a n a l t e r n a t i v e m e c h a n i s m f o r the s e l e c t i o n of neutron waves with univocal spin e f f e c t s i s n a t u r a l l y o c c u r r i n g in ( m o d e r a t e l y ) absorbing c r y s t a l s and, p a r t i c u l a r l y , i n the anomalous t r a n s m i s s i o n ( a l s o i n the reflection) i n the Laue c a s e .

R e f e r e n c e s

/1/ FORTE M . , Conf. Neutron and i t s Application, Cambridge (1982) Contr.

4. B. 4 and J. Phys. G. 4 (1983) 745.

/2/ SZYMANSKI Z . , N u c l . P h y s . A E ( l 9 6 8 ) 385.

/3/ FORTE M. , Fundamental P h y s i c s with R e a c t o r Neutron and Neutrinos (1 977) Inst. Phys. Conf. S e r . 42 (1978).

/4/ BUNAKOV V. E. and GUDKOV V. P . , Leningrad, Nucl. Phys. Inst. P r e p r i n t 777 (1982).

/5/ TAKAGI S . , Acta C r y s t . 3 (1 962) 23.

/6/ TAKAGI S . , J. Phys. Japan, 26 (1969) 1239.