International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 5.3, pp. 655658
Section 5.3.3. Neutron spin, and diffraction by perfect magnetic crystals ^{a}Laboratoire Louis Néel du CNRS, BP 166, F38042 Grenoble Cedex 9, France, and ^{b}European Synchrotron Radiation Facility, BP 220, F38043 Grenoble, France 
A polarized neutron beam is represented by a twocomponent spinor, which is the coherent superposition of two states, of different amplitudes c and d, polarized in opposite directions along the spinquantization axis. The spinor components c and d are generally space and timedependent. We suppose that . The polarization vector P is defined as where the vector represents the set of Pauli matrices , and . The components of P are from which it is clearly seen that, unlike , the polarization components and depend on the phase difference between the spinor components c and d.
In a region of a vacuum in which a uniform magnetic field B is present, a neutron beam experiences a magnetic potential energy represented by the matrix being the neutron magnetic moment, if the directions of B and of the spinquantization axis coincide. Consequently, different indices of refraction , where E is the neutron energy, should be associated with the spinor components c and d; this induces between these spinor components a phase difference which is a linear function of the time (or, equivalently, of the distance travelled by the neutrons), hence, according to (5.3.3.1), a rotation around the magnetic field of the component of the neutron polarization perpendicular to this magnetic field. The time frequency of this socalled Larmor precession is , where h is Planck's constant.
A neutron beam may be partially polarized; such a beam is conveniently represented by a spindensity matrix ρ, which is the statistical average of the spindensity matrices associated to the polarized beams which are mixed incoherently, the density matrix associated to the spinor being The polarization vector P is then obtained as
In the common case of a nonpolarized beam, the spindensity matrix is It is easily seen that all components of P are then equal to 0.
Equation (5.3.3.2) is therefore applicable to the general case (polarized, partially polarized or nonpolarized beam). The inverse relation giving the density matrix ρ as function of P is
The spin and orbital motion of unpaired electrons in an atom or ion give rise to a surrounding magnetic field which acts on the neutron via the magnetic potential energy , where is the neutron magnetic moment. Since this is a longrange interaction, in contrast to the nuclear interaction, the magnetic scattering length p, which is proportional to the Fourier transform of the magnetic potential energy distribution , depends on the angle of scattering.
The classical relation shows clearly that the vector , which is the Fourier transform of , is perpendicular to the reciprocalspace vector s. If we consider the magnetic field as resulting from a pointlike magnetic moment at position , we get where is the permittivity of a vacuum and × denotes the cross product. can be Fouriertransformed into where is the projection of on the plane perpendicular to s (reflecting plane).
This result can be applied by volume integration to the more general case of a spatially extended magnetization distribution, which for a single magnetic ion corresponds to the atomic shell of the unpaired electrons. It is thus shown that the magnetic scattering length is proportional to , where is the projection of the magnetic moment of the ion on the reflecting plane.
For a complete description of magnetic scattering, which involves the spinpolarization properties of the scattered beam, it is necessary to represent the neutron wavefunction in the form of a twocomponent spinor and the ion's magnetic moment as a spin operator which is a matrix expressed in terms of the Pauli matrices . The magnetic scattering length is therefore itself a matrix: where is the dimensionless magnetic form factor of the ion considered and tends towards a maximum value of 1 when the scattering angle θ tends towards 0 (forward scattering). The value of is for Bohr magneton.
According to (5.3.3.4) or (5.3.3.5), there is no magnetic scattering in directions such that the scattering vector s is in the same direction as the ion magnetic moment . Magnetic scattering effects are maximum when s and are perpendicular.
The matrix (p) is diagonal if the direction of is chosen as the spinquantization axis. Therefore, there is no spinflip scattering if the incident beam is polarized parallel or antiparallel to the direction of .
It is more usual to choose the spinquantization axis (Oz) along . Let β be the angle between the vectors and s; the components of are then (, 0, ) if the y axis is chosen along . The total scattering length, which is the sum of the nuclear and the magnetic scattering lengths, is then represented by the matrix where b is the nuclear scattering length and with expressed in Bohr magnetons. The relations show clearly that the diagonal and the nondiagonal elements of the matrix (q) are, respectively, the spinflip and the nonspinflip scattering lengths. It is usual to consider the scattering cross sections, which are the measurable quantities. The cross sections for neutrons polarized parallel or antiparallel to the ion magnetic moment are These expressions are the sum of the spinflip and nonspinflip cross sections, which are equal to and , respectively. In the case of nonpolarized neutrons, the interference term between the nuclear and the magnetic scattering disappears; the cross section is then
In the general case of a partially polarized beam we can use the densitymatrix representation. Let be the density matrix of the incident beam; it can be shown that the density matrix of the diffracted beam is equal to the following product of matrices: . Using the relations between the density matrix and polarization vector presented in the preceding section, we can obtain a general description of the diffracted beam as a function of the polarization properties of the incident beam. Such a formalism is of interest for dealing with new experimental arrangements, in which a threedimensional polarization analysis of the diffracted beam is possible, as shown by Tasset (1989).
The most direct way to develop this dynamical theory in the twobeam case, which involves a single Braggdiffracted beam of diffraction vector h, is to consider spinor wavefunctions of the following form: as approximate solutions of the wave equation inside the crystal, where and are, respectively, equal to the nuclear and the magnetic potential energies multiplied by . In the calculation of in the twobeam case, we need only three terms in the expansions of the functions and into Fourier series: We suppose that the crystal is magnetically saturated by an externally applied magnetic field . is then proportional to the macroscopic mean magnetic field , where M is the magnetization vector and is the demagnetizing field. The results of Section 5.3.3.2 show that and are proportional to the projection of M on the reflecting plane.
The four coefficients , , and of (5.3.3.9) are found to satisfy a system of four homogeneous linear equations. The condition that the associated determinant has to be equal to 0 defines the dispersion surface, which is of order 4 and has four branches. An incident plane wave thus excites a system of four wavefields of the form of (5.3.3.9), generally polarized in various directions. A particular example of a dispersion surface, having an unusual shape, is shown in Fig. 5.3.3.1.

Schematic plot of the twobeam dispersion surface in the case of a purely magnetic reflection such that and that the angle between and is equal to . 
This is a much more complicated situation than in the case of nonmagnetic crystals, in which one only needs to consider scalar wavefunctions which depend on two coefficients, such as and , and which are related to hyperbolic dispersion surfaces of order 2, as fully described in Chapter 5.1 on Xray diffraction.
In fact, all neutron experiments related to dynamical effects in diffraction by magnetic crystals have been performed under such conditions that the magnetization vector in the crystal is perpendicular to the diffraction vector h. In this case, the vectors and are parallel or antiparallel to the vector which is chosen as the spinquantization axis. The matrices , and are then all diagonal matrices, and we obtain for the two spin states (±) separate dynamical equations which are similar to the dynamical equations for the scalar case, but with different structure factors, which are either the sum or the difference of the nuclear structure factor and of the magnetic structure factor : and are related to the scattering lengths of the ions in the unit cell of volume : The dispersion surface of order 4 degenerates into two hyperbolic dispersion surfaces, each of them corresponding to one of the polarization states (±). The asymptotes are different; this is related to different values of the refractive indices for neutron polarization parallel or antiparallel to .
In some special cases the magnitudes of and happen to be equal. Only one polarization state is then reflected. Magnetic crystals with such a property (reflections 111 of the Heusler alloy Cu_{2}MnAl, or 200 of the alloy Co–8% Fe) are very useful as polarizing monochromators and as analysers of polarization.
If the scattering vector h is in the same direction as the magnetization, this reflection is a purely nuclear one (with no magnetic contribution), since is then equal to 0. Purely magnetic reflections (without nuclear contribution) also exist if the magnetic structure involves several sublattices.
If h is neither perpendicular to the average magnetization nor in the same direction, the presence of nondiagonal matrices in the dynamical equations cannot be avoided. The dynamical theory of diffraction by perfect magnetic crystals then takes the complicated form already mentioned.
Theoretical discussions of this complicated case of dynamical diffraction have been given by Stassis & Oberteuffer (1974), Mendiratta & Blume (1976), Sivardière (1975), Belyakov & Bokun (1975, 1976), Schmidt et al. (1975), Bokun (1979), Guigay & Schlenker (1979a,b) and Schmidt (1983). However, to our knowledge, only limited experimental work has been carried out on this subject. Successful experiments could only be performed for the simpler cases mentioned above.
In this case, there is no average magnetization . It is then convenient to choose the quantization axis in the direction of and . The dispersion surface degenerates into two hyperbolic surfaces corresponding to each polarization state along this direction for any orientation of the diffraction vector relative to the direction of the magnetic moments of the sublattices. These two hyperbolic dispersion surfaces have the same asymptotes. Furthermore, in the case of a purely magnetic reflection, they are identical.
The possibility of observing a precession of the neutron polarization in the presence of diffraction, in spite of the fact that there is no average magnetization, has been pointed out by Baryshevskii (1976).
In polarized neutron diffraction by a magnetically saturated magnetic sample, it is usual to measure the ratio of the reflected intensities and measured when the incident beam is polarized parallel or antiparallel to the magnetization in the sample. This ratio is called the flipping ratio, because its measurement involves flipping the incidentbeam polarization to the opposite direction. This is an experimentally well defined quantity, because it is independent of a number of parameters such as the intensity of the incident beam, the temperature factor or the coefficient of absorption. In the case of an ideally imperfect crystal, we obtain from the kinematical expressions of the integrated reflectivities In the case of an ideally perfect thick crystal, we obtain from the dynamical expressions of the integrated reflectivities In general, depends on the wavelength and on the crystal thickness; these dependences disappear, as seen from (5.3.3.14), if the path length in the crystal is much larger than the extinction distances for the two polarization states. It is clear that the determination of or allows the determination of the ratio , hence of if is known. In fact, because real crystals are neither ideally imperfect nor ideally perfect, one usually introduces an extinction factor y (extinction is discussed below, in Section 5.3.4) in order to distinguish the real crystal reflectivity from the reflectivity of the ideally imperfect crystal. Different extinction coefficients and are actually expected for the two polarization states. This obviously complicates the task of the determination of .
In the kinematical approximation, the flipping ratio does not depend on the wavelength, in contrast to dynamical calculations for hypothetically perfect crystals (especially for the Laue case of diffraction). Therefore, an experimental investigation of the wavelength dependence of the flipping ratio is a convenient test for the presence of extinction. Measurements of the flipping ratio have been used by Bonnet et al. (1976) and by Kulda et al. (1991) in order to test extinction models. Baruchel et al. (1986) have compared nuclear and magnetic extinction in a crystal of MnP.
Instead of considering only the ratio of the integrated reflectivities, it is also possible to record the flipping ratio as a function of the angular position of the crystal as it is rotated across the Bragg position. Extinction is expected to be maximum at the peak and the ratio measured on the tails of the rocking curve may approach the kinematical value. It has been found experimentally that this expectation is not of general validity, as discussed by Chakravarthy & Madhav Rao (1980). It would be valid in the case of a perfect crystal, hence in the case of pure primary extinction. It would also be valid in the case of secondary extinction of type I, but not in the case of secondary extinction of type II [following Zachariasen (1967), type II corresponds to mosaic crystals such that the diffraction pattern from each block is wider than the mosaic statistical distribution].
References
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