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NOVEL APPROACH TO POLARIZED NEUTRON SCATTERING

F. MEZEI lnstitut Laue Langevin, 156 X Centre de Tri, 38042 Grenoble Cedex, France

and

Central Research Institute for Physics, Budapest, Hungary

(Invited paper)

The strong interaction between neutron and electron spins has long made neutron scattering, and especially polarized neutron scattering, a privileged method in the study of magnetic phenomena. The use of polarized neutrons, however, was conventionally limited to polarization directions parallel to the magnetic field, thus only keeping track of a scalar projection of a vector phenomenon. In recent years, however, at several laboratories, various techniques have been developed to fully exploit the vector character of neutron polarization. These new vector polarization techniques lend themselves especially favourably to the study of domain structure and spontaneous magnetization in ferromagnets and of complex (non-coUinear non-centro-symmetrical) magnetic structures.

1. Introduction

The interaction between the magnetic moment of neutrons and a magnetic system is described by the Hamiltonian

/-I = - # • B = -2ytzNS • B, (1)

where # is the magnetic moment of the neutron given in terms of the nuclear magneton /z N and the neutron spin operator S = (~x, ~y, ~z) via the

c o n s t a n t 3' = - 1 . 9 1 3 1 , and B is the magnetic field. This happens to be of the same order of magnitude for common values of magnetic fields and ferromagnetic magnetizations (e.g. 0 .60x 1 0 - 7 ¢ V for B = 10kOe) as the nuclear interaction between neutrons and materials of usual densities, and small, (but not always ne- gligible) as compared to the kinetic energy of the neutrons available in research beams (0.001- 0.1 eV). This is why both nuclear and magnetic interactions give rise to two kinds of phenomena: optical effects on beams transmit- ted through samples or fields, and scattering effects.

The scope of these two phenomena is very different. In optical type studies we are in- terested in the change of the neutron wave function as a whole, while scattering experi- ments are mostly concerned with the, usually very low probability, occurrence of particular components in the neutron states due to in- teraction. Of course the aspects are not strictly separable, but in most cases they appear to be distinct.

A typical example of optical phenomena is the classical Stern-Gerlach effect: the beam is split, in the inhomogeneous magnetic field, into two distinguishable components which display different spatial and spin states. Indeed, in general, one can deal with change of both the spatial momentum and the spin part of the neu- tron states in the beam.

Considering the ratio between neutron kinetic energy and magnetic interaction, it is obvious that sizeable changes in neutron spin states oc- cur much more easily than any noticeable change in the momentum states. This is why in practice we can distinguish in neutron tran- smission work between refract ion effects in- volving neutron momentum changes and pure spin state changes with negligible refract ion effects. Thus for practical purposes (in view of both experimental technique and the type of information sought after) we can divide mag- netic phenomena into three classes:

I Scattering effects II Refraction effects

III Spin precession effects. It should be kept in mind that this division

makes only practical sense, and in a few par- ticular cases, one has to deal with two or all of these effects simultaneously so that they cannot really be distinguished.

Recent developments in polarized neutron technique had a considerable impact on the possibilities of Group III and I studies. It became possible to extend experimentally available information so that one can deal with neutron spin wave functions instead of the usual

Physica 86-88B (1977) 1049-1052 (~) North-Holland

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" u p " and " d o w n " densi ty matrices. This breakthrough made Group I I I - type exper iments much more flexible and powerful . On the other hand, this means that we can account for the expecta t ion values of all three componen t s of the neutron spin, which, in view of eq. (1), is obviously necessary to extract the max imum information on magnetic interactions.

2. Neutron spin states

The spin wave funct ion of a neutron most generally is given as

x = ~1T )~-/31 ~ >, (2)

where a and /3 are complex numbers , 11~211+11/3~11= l, and 11') and 14) are the fun- damental " u p " and " d o w n " spin state vectors with respec t to an arbitrarily chosen coordinate system. Apar t f rom an undetermined phase fac- tor, Eq. (2) can be written as

X=COS ½0l 1' ) +ei~ sin ~0l ~ )=]0, ~b) (3)

with proper ly chosen angles 0 and ~b. It is easy to show that

(0, ~bl2Sxt0, ~b) = sin 0 cos 6 = P~,

(0, 612~,10, 6) = sin 0 sin 6 = Py, (4)

<0, 612LI0, 6> -: cos 0 = Pz,

Px, Py and Pz are just the classical components of a unit vector P given by the polar angles 0 and ~b. So measuring the expecta t ion values of the three spin componen t s on a polarized beam of neutrons with allegedly identical spin states, one can determine 0 and ~b, i.e. the spin wave function. (Of course, these measurements can- not be made simultaneously, but one af ter the other.) This is in contras t to the classical ap- proach which only deals with the difference in the " u p " and " d o w n " probabilit ies, i.e. I1~112-11/3112= cos 0 = Pz. Extending the scope of interest to all three componen t s of spin polarizat ion is equivalent to extending the quantum-mechanica l descript ion to explicit spin states f rom a simple density matrix. The polarization of a neutron beam has to be con- sidered as a vector equal to the vector average of the spin polarization vectors P of individual neutrons in the beam.

The most e lementary effect of the interaction

(1) on the neutron state is the spin precession (Larmor precession). If, over the period of time we are concerned with, B can be taken as a constant vector in the z direction, the time dependence of state (3) is given by

10(t), ~b(t)) = e -(iv~NB/h)t cos ½00l 1' )

+e (i'ÈNB/h)t e i~° sin ½00[ ~ ).

Consequent ly

qb( t ) = dpo + (2 ytXNB/h )t = qbo + 09L t,

O(t) = 0o.

This type of 0(t), ~b(t) behaviour is called Lar- mor precession.

In the general case of group III exper iments it is easier to determine 0 and ~b f rom the classical equation

d P - - = O~L[B(t) × P] (5) dt

which follows f rom eq. (1) under the assumption that there are no refract ion effects, and con- sequently, the vector B ( t ) has the time depen- dence seen by the point-like neutron travelling through the space and time dependent real field B(r, t):

B( t ) = B( r ( t ) , t).

3. The new experimental technique

Neutron polarizer and analyser devices, either the classical ones (special Bragg reflecting crystals or totally reflecting magnetized bulk, or recently, evapora ted [1,2] mirrors) or the newly developed multilayers [3] and supermirrors [4] can only produce a beam polarized parallel to the magnetizing field on the device (z direction), or can be used to measure I1 112-11/3112= Pz only. For either producing or analysing the other two components of spin polarization, methods had to be invented to turn the spin direction in such a way that Pz could be interchanged with ano- ther component . E.g. a spin turn which inter- changes Pz and Py permits us to create a polarized beam in the y direction starting with a beam polarized classically in the z direction, and similarly to analyse polarization in the y direction. The first such method has been de-

veloped by Rekveldt [5] in Delft in 1969, and by now a number of them have been established which can handle all kinds of experimental conditions: monochromat ic [5, 8] or white beams [7], transmission [5,6,7], small angle scattering [8] or large angle scattering [6] ex- periments. All of these methods are based on a solution of eq. (5) in a special magnetic field configuration. Thus, by now this new, three- dimensional vector approach to neutron polarization work has become fully established and more and more frequent ly used.

4. New types of experiments

The first three dimensional polarization stu- dies have been published by Rekveldt [5] on domain structures in ferromagnets. The well known effect of depolarization of neutron beams on transmission through a ferromagnetic material is due to the fact that neutrons having different trajectories pass through different domains, so they experience different field pat- terns, and end up with very different polariza- tion directions via eq. (5). In the classical ap- proach this depolarization of course meant only a single parameter, i.e. the change in "up" and " d o w n " spin state population fractions, while the description of a realistic domain structure contains a large number of factors, such as distribution domain sizes, domain shapes, dis- tribution of domain orientation, correlations between neighbouring domains. Rekveldt has shown [9] that by three-dimensional extension of depolarization experiments, one can measure a 3 x 3 depolarization matrix which contains a lot more information about all these com- plexities and changes of domain structure e.g. near critical point, under influence of field and stress etc.

Drabkin and his co-workers in Leningrad developed a method to determine the value and orientation of magnetization in a single, though large enough domain using a sufficiently narrow beam and three dimensional polarization analy- sis [7, 10], and they obtained spectacular results e.g. in point-by-point topographic studies of the effect of sample shape on domain structure, or in magnetization measurements at zero fields near the ferromagnetic Curie point. Their dis- covery of a macroscopic magnetization direc- tion pattern, determined exclusively by the

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sample shape near the ferromagnetic Curie point, is of particular interest. They have also established the existence of a kinematic aniso- t ropy of neutron depolarization in ferromagnets , predicted by Maleev and Ruban [11].

The three-dimensional polarization analysis offers particularly interesting possibilities in the neutron diffraction study of complicated mag- netic crystal structures [12]. This is due to the fact that the relation between the incoming and diffracted beam polarization, which is charac- teristic of the structure studied, contains ten- sorial terms too [13]. The first experiment of this type has been done by Alperin [8] on a non- centrosymmetr icai crystal structure.

For completeness let us briefly mention a few novel type polarized neutron experiments too, which are not directly related to the main point of this paper. Shilstein, Yel'utin and Somenkov [14] and, independently, Sch/irpf [15] et al. es- tablished methods to study domain walls by neutron refract ion (Group II) i.e. essentially by the Stern-Ger lach phenomenon at the in- homogenei ty represented by the domain wall. Hayter , Jenkin and White [16] have shown that polarized neutron diffraction on selectively polarized nuclei can be a powerful tool in or- ganic crystal structure work. Finally, the effect of neutron spin memory, based on the deter- mination of spin states by vector polarization analysis, has been shown to be instrumental even in investigating non-magnetic scattering effects, namely in very high resolution studies of atomic motions (neutron spin-echo inelastic spect rometry [6]).

5. Conclusion

The novel, vector approach to neutron polarization work, as opposed to the classical "u p " or " d o w n " myth, offers a number of valuable possibilities in magnetism, especially in the study of problems related to magnetic domains and to complex magnetic crystal structures.

References

[1] G.M. Drabkin, A.I. Okorokov, A.F. Shchebetov. N.V. Borovikova, A.G. Gukasov, A.I. Yegerov and V.V. Runov, Preprint No. 183, Leningrad Institute for Nuclear Physics, (1975).

12] J. Penfold, J.C. Sutherland, W.G. Williams and J.B.

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Hayter, Preprint, No. 75-112, Rutherford Laboratory (1975).

[3] J.W. Lynn, J.K. Kjems, L. Passell, A.M. Saxena and B.P. Schoenborn, Proceedings of the Conference on Neutron Scattering, Gatlinburg, 1976 (in print).

[4] F. Mezei, Comm. on Physics 1 (1976) 81. F. Mezei and P. Dagleish, Comm. on Physics (in print).

[5] M. Th. Rekveldt, J. de Physique 32 (1971) C579. [6] F. Mezei, Z. Physik 255 (1972) 146. [7] A.I. Okorokov, V.V. Runov, V.I. Volkov and A.G.

Gukasov, 2hETF 69 (1975) 590. [8] H. Alperin, Proceedings ICM-73, III (Moscow, 1973)

128. [9] M. Th. Rekveldt, Z. Physik 259 (1973) 391; J. of Mag-

netism and Magn. Mat., in print.

[10] G.M. Drabkin, A.I. Okorokov and V.V. Runov, ZhETF Pis. Red. 15 (1972) 458.

[11] S.V. Maleev and V.A. Ruban, ZhETF 62 (1972) 415. [12] F. Mezei, Proc. of Conf. on Magn. and Magn. Mat.

(Boston 1973) p. 406. [13] M. Biume, Phys. Rev. 130 (1963) 1670. [14] S. Sh. Shilstein, N.O. Yerutin and B.A. Somenkov, Fiz.

Tvor. Tel. 16 (1974) 2008; ZhETF Pis. Red. 18 (1973) 318.

[15] O. Schiirpf, Physica 808 (1975) 289. O. Schiirpf, R. Siefert and Ch. Schwink, J. of Mag- netism and Magn. Mat. 2 (1976) 93.

[16] J.B. Hayter, G,T. Jenkin and J.W. White, Phys. Rev. Lett. 33 (1974) 696.