Neutron depolarization in matter with polarized nuclei

Physics Letters A 160 ( 1991 ) 197-200 North-Holland

PHYSIC S lETTERS A

Neutron depolarization in matter with polarized nuclei V.G. B a r y s h e v s k y

Institute of NuclearProblems,BelorussianState University,Minsk 220050, USSR Received 20 June 1991;accepted for publication I 1 September 199 l Communicated by A.A. Maradudin

The influence of fluctuations of a nuclear pseudomagnetic field on neutron spin rotation in matter with polarized nuclei is described. It is shown that nuclear pseudomagnetic field fluctuations lead to neutron spin depolarization. It is also shown that the attenuation of the polarization vector components of a neutron as a function of target thickness is described by a nonexponential law.

We have shown in ref. [ l ] that in matter with polarized nuclei neutron spin precession occurs which is attributed to the influence o f a nuclear pseudomagnetic field. The existence o f the nuclear pseudomagnetic field was proved experimentally by Abragam and Gl6ttle [2], Abragam and G o l d m a n [ 3 ] and Forte [ 4 ]. Recently, many successful experiments on the realization o f nuclear orientation at extremely low temperatures have been carried out. In particular, antiferromagnetic nuclear spin ordering in a LiH lattice has been discovered [3,5]. Nuclear orientation by light has been implemented [6]. Inquiry into the passage o f neutrons through such media allows us to discover not only the spin ordering effect itself, but also to study the dynamics o f polarized nuclear spin systems. Application of neutrons for studying matter with ordered nuclear spins is fully equivalent to the use o f neutrons in the investigation o f the magnetic structure of matter by means o f a magnetic neutron diffraction analysis. Here, the role o f fluctuations o f a magnetic field (magnetic m o m e n t ) in the case o f polarized nuclei is played by fluctuations o f a nuclear pseudomagnetic field [ 5,7 ] (pseudomagnetic moments [ 3 ] ). We should recall that the nuclear pseudomagnetic field is by two-three orders larger than the ordinary magnetic field generated by the magnetic moments o f the nuclei. The nuclear pseudomagnetic m o m e n t

is as many times larger than the nuclear magnetic moment. It is known (see, for example, ref. [ 8 ] ) that fluctuations o f a magnetic field on the nucleus lead to spin depolarization of polarized nuclei. It is shown in ref. [ 8 ] that as neutrons move in a nuclear pseudomagnetic field, depolarization of their spins occurs as well. This is due to the space and time fluctuations o f this field. In this case, the attenuation law o f the neutron polarization transverse components nonexponentially depends on the target thickness. Thus, let a neutron move in matter with polarized nuclei. The Schr6dinger equation, describing the propagation o f a coherent neutron wave in such matter has the form [ 1,7 ]

ih O~(r' t) - ( --~-~mAr+O(r,t) ) ~u(r,t),

(1)

where m is the neutron mass, O(r, t) is the effective potential energy operator o f interaction o f the neutron wave with the polarized target: 2/th 2

O(r, t ) = - - - p ( r , m

t)f(O)-lt~r.B(r, t) ,

(2)

p(r, t) is the number of nuclei in a volume unit o f matter, f(O) is the elastic coherent scattering amplitude through a zero angle, f ( 0 ) =a+flJ~.P(r, t), J+ 1 c~=~a

0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.

+

J +2---)-~a-,

fl=

a + -a2J+l

'

197

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PHYSICS LETTERSA

a ÷ is the scattering amplitude in a state with total moment of neutron and nucleus J + ½, a - is the same quantity for the state of J - ½, J is the magnitude of a nuclear spin, ix= 2S, S is the neutron spin operator, tr is the operator composed of the Pauli matrices ax, av, a:, P(r, t) = ( J) /J is the polarization vector of the nucleus at a point r at the moment t, ( J ) = Spp.d, Pn is the nuclear density spin matrix,/z is the neutron magnetic moment, B(r, t) is the magnetic field in the target. According to ref. [7], U(r, t) may be written in the form

O(r, t ) = V(r, t) -uo.G(r, t),

(3)

where V(r, t) is the part of the potential energy of the interaction of the wave with matter which does not depend on the spin, 27th 2

V(r, t) = - - - p ( r ,

t)a,

m

G(r, t ) = B ( r ,

t) + B~rr(r, t) .

t)P(r, t).

(5)

×exp[-(i/h)Et] ,

(7)

where k is the wave vector of a neutron incident on the target in vacuum, E=h2k2/2m. Substitution of (7) into (6) makes it possible to obtain the following set of equations for the amplitudes A. and phases 6.,

+ -1h l m V ( r , t ) A . ii

~Im( G)(nlazln)A.

2

- ~ I m ~ (nlo'SGlm)A., e x p [ i ( 6 m - 6 . ) ] , m=

-A"O6~

I

mh (k. V6.)A _ 2~ AAn+ 2~A.(V6.)2

1 lt + ~ R e V(r, t ) A . - ~ R e (G)(nla~ln)A. 2

- -~Re ~ (nlo'SGIm)A,.exp[i(6,.-6.)].

O~n h2 ih -~ = - 2~/xAu"+V(r' t)~u.

h

m= I

(8)

-u< G> ~u. 2

(6)

nl=l

In the absence of fluctuations of the effective potential energy owing to the diversification of the amplitudes a ÷ and a -, eq. (6), nevertheless, describes absorption and depolarization of neutrons. In this case the spin depolarization is caused by the processes of noncoherent scattering by the nucleus with spin flipping and without it. 198

~u.(r, t)=A.(r, t) exp[i6.(r, t)] exp(ik.r)

OA.ot - - h(k'V)A"rn - h (VA")'(V6")-2~

Assume that the field G(r, t) has the form G(r, t) = ( G ) +SG(r, t), where ( G ) is the average field in the target, 8G=G- (G). Let us choose the quantization axis along the average field ( G ) . The spinor ~,= ( ~ ) . In this case, the SchrSdinger equation ( 1 ) for the determination of a function ~,~ (n = 1, 2) may be represented in the following way,

- u ~' ~.,.

Consider now the influence of fluctuations of the effective field upon the behaviour of the neutron spin in the medium. Let us analyze the case with thermal neutrons. The typical space dimensions of fluctuations ~, attributed to the thermal motion of scatterers (or by inhomogeneities of the scatterer distribution in the target) are large in comparison with the wavelength of thermal neutrons 2 ~ 10-8 cm. This allows us to use the method of slow (smooth) amplitudes for solving (6). Represent ~u. in the form

(4)

The effective nuclear pseudomagnetic field is

2xh2flJ B~ff(r, t)= - - p ( r , rntz

11 November 1991

In virtue of 2<<~, as in the usual (without allowance for spin interaction) eikonal approximation, we can simplify the set (8) by rejection of the terms containing second-order derivatives with respect to coordinates and products of gradients. As a result, we have

Volume 160, number 2

OA. _ Ot

PHYSICS LETTERSA

11 November 1991

og.

I

(v.V)A. + ~- Im V(r, t)A.

1

0--t + (v.V)g. = - ~ Re V(r, t)

n

+ ~p R e + h/l Re 8G< nl az In)

- ~Im A.

(lOb)

2

--~Im h

~ A,,,exp[i(gm-g,)],

Let a particle move along the axis z. Eq. (lOa) implies

m=l

- A n 0g. Ot = ( v - V g . ) A . + ~1 Re V(r, t)A.

An(r, t) =Aon(Z-Vt) e x p ( - 2 n z ) ,

( 11 )

where

~ R e A.

2.=-

2

1

#

~ I m < V ) + ~ I m
- -PRe ~
v=hk/m.

(9)

It may seem at first sight that the neutron spin depolarization under the influence of the fluctuating pseudomagnetic field is due to processes flipping the neutron spin, i.e. to the transverse components of the fluctuating part 8G with respect to the direction < G ) . However, as we show below, neutron spin depolarization is observed in the absence of fluctuations of the direction G because of the density fluctuation too. The said phenomenon is analogous to that of nuclear spin depolarization observed in nuclear magnetic resonance owing to a magnetic field inhomogeneity in the target [8 ]. Thus, let the fluctuations 8G be attributed solely to density fluctuations. The direction 8G coincides with that of < G ) . We think also that fluctuations of the imaginary part of the potential are small compared to the imaginary part of the averaged potential. This makes it possible to ignore the influence of the fluctuations on a change in the wave amplitude A,. As a result, we obtain

OA. 1 Ot + (v'V)An = ~ Im < V)A. - ~Im
(lOa)

Ao. is the amplitude of the wave as it enters the target. It should be noted that 2 . > 0 , as always I m f ( 0 ) >0. Eq. (10b) is also easily solved: g.(r, t) =goAZ-Vt, to) - ( 1~vRe < V> - ~vvRe< G>) z +7.(r, t ) ,

,.(r, t ) = - i (h ReSV(z-vt+vr, t' ) to

- ~ReSG(z-vt+vt',t')
(12)

where L is the target thickness, go is the initial wave phase. Thus, in fact, we have determined the functions g.. Now we can find the neutron polarization vector P = <~ul~rlg/), Py=2Im~Tg'2,

Px = 2 Re g/T~2, ez=l~u~12-1~/212 .

(13)

We will assume further that the neutron spin during the entrance into the target is perpendicular to the nuclear polarization vector. This direction is chosen as axis x. As a result, we have P~ =2.4oz cos(~l - g 2 ) e x p [ - (21 +22)z] , Py = -2A~ sin (gl - g 2 ) e x p [ - (2,

"~-/~2)Z] ,

Pz =A~ [exp( - 2 ~ , z) - e x p ( - 2 2 z ) ] .

(14)

where 199

Volume 160, number 2

61

PHYSICS LETTERS A

__62 -- 2pRe ( G ) z fly I

tO

= ~ z + '~ w( t' ) dt' . l)

d to

According to (14), the neutron spin undergoes precession, as the depth of penetration into the target increases. Along with an uniform precession frequency g2= 2# Re (G) ~It, there is a fluctuating frequency (it is described by the second term in the difference 61 - 62). Let us average (14) over the distribution of these fluctuations. Inasmuch as the integral ( = i 02(t') d t ' , tO

is in fact the sum of many small terms, so the distribution function is given by a Gauss distribution dw(O- ~ e x p

d(,

(15)

where 2

l I November 1991

following way, G~(z)= (022)g~(r), (02 2) = ( 0 2 ( t ' ) 0 2 ( t ' ) ), g ( r = 0 ) = 1. Let us consider a typical law for the correlation attenuation [8], that is, we will assume that g ( z ) = e x p ( - Brl/re), where rc is the correlation time. By substituting g(r) into ( 16 ) and averaging the result over the time to taken for neutrons to enter into the target we will obtain the following expression for the transverse components of the polarization vector of neutrons, which have passed through a target having thickness L, /~x =cos [ (g2/v)z] e x p [ - (21 +22)L] ×exp { - (022)r~ [exp ( - v-L) - 1 + L-~I}, (17) /3~ is derived from/~x as a result of the replacement of cos by sin. According to ( 17), the fluctuations of a pseudomagnetic field lead to the nonexponential attenuation of the transverse neutron polarization. Analysis of the attenuation law permits us to determine the important cut-off characteristics ( 022 > and r~. It should be noted that the said applies equally to purely magnetic interactions of neutrons with a medium. In particular, the law (1 7) describes neutron depolarization in a medium with polarized electrons (atoms).

to

( ) stands for averaging over the parameter values, which define co(t' ). Now, if we average Ix, Py with the distribution (15), we will have

Px =2A~ j cos[ (f2/v)z+(] dw(() exp[ (21 +22)2] =2A 2 cos[ (f2/v)z] exp[ - (;tl +22)z1 X e x p ( - ½( ( 2 ) ) .

(16)

Let us introduce a correlation function of the frequency Go(t',t")=(02(t')02(t")) and take into account that for a stationary process GoAt', t" ) = G~(t'-t")=Go(r); G~,(r)=Go~(-r). It is convenient to introduce the function g~(z) in the

200

References [ 1 ] V.G. Baryshevsky and M.I. Podgorezkii, Zh. Eksp. Teor. Fiz. 47 (1964) 1050. [2] A. Abragam and H. Gl6ttle, C.R. Acad. Sci. 274 (1972) 423. [3] A. Abragam and M. Goldman, Nuclear magnetism. Order and disorder (Clarendon, Oxford, 1982). [4] M. Forte, Nuovo Cimento 18A (1973) 727. [5 ] V.G. Pokozanyev and G.V. Scrozkii, Sovo Usp. Fiz. Nauk 129 (1979) 615. [6] L.S. Vlasenko, N.V. Zavarizkii, S.V. Sorokin and V.G. Fleisher, Zh. Eksp. Teor. Fiz. 91 (1986) 1496. [7] V.G. Baryshevsky, Nuclear optics of polarized media (Belorussian University, Minsk, 1976). [8]A. Abragam, The principles of nuclear magnetism (Clarendon, Oxford, 1961 ).