Single-crystal polarizers

Anderson, I. S.; Schärpf, O.

doi:10.1107/97809553602060000594

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.4, pp. 438-439

Section 4.4.2.6.1. Single-crystal polarizers

I. S. Anderson^a and O. Schärpf^f

4.4.2.6.1. Single-crystal polarizers

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The principle by which ferromagnetic single crystals are used to polarize and monochromate a neutron beam simultaneously is shown in Fig. 4.4.2.9 . A field B, applied perpendicular to the scattering vector $[\boldkappa]$ , saturates the atomic moments $[{\bf M}_{\nu}]$ along the field direction. The cross section for Bragg reflection in this geometry is $[({\rm d}\sigma /{\rm d}\Omega)=F_{N}({\boldkappa})^{2}+2F_{N}({\boldkappa})F_{M}({\boldkappa})({\bf {}P}\cdot {\boldmu})+F_{M}({\boldkappa}){}^{2}, \eqno (4.4.2.9)]$ where $[F_{N}({\boldkappa})]$ is the nuclear structure factor and $[F_{M}({\boldkappa}) = [({\gamma }/{2})r_{0}]\sum _{\nu }M_{\nu }\,f(hkl)\exp [2\pi (hx+ky+lz)]]$ is the magnetic structure factor, with f(hkl) the magnetic form factor of the magnetic atom at the position (x, y, z) in the unit cell. The vector P describes the polarization of the incoming neutron with respect to B; P =1 for + spins and P = −1 for − spins and $[{\boldmu}]$ is a unit vector in the direction of the atomic magnetic moments. Hence, for neutrons polarized parallel to B $[({\bf P}\cdot {\boldmu}=1)]$ , the diffracted intensity is proportional to $[[F_{N}({\boldkappa})+F_{M}({\boldkappa})]{}^{2}]$ , while, for neutrons polarized antiparallel to B $[({\bf P}\cdot {\boldmu}=-1)]$ , the diffracted intensity is proportional to $[[F_{N}({\boldkappa})-F_{M}({\boldkappa})]{}^{2}]$ . The polarizing efficiency of the diffracted beam is then $[ P=\pm 2F_{N}({\boldkappa})F_{M}({\boldkappa})/[F_{N}({\boldkappa})^{2}+F_{M}({\boldkappa}){}^{2}], \eqno (4.4.2.10)]$ which can be either positive or negative and has a maximum value for $[|F_{N}({\boldkappa})|= |F_{M}({\boldkappa})|]$ . Thus, a good single-crystal polarizer, in addition to possessing a crystallographic structure in which $[F_{N}]$ and $[F_{M}]$ are matched, must be ferromagnetic at room temperature and should contain atoms with large magnetic moments. Furthermore, large single crystals with `controllable' mosaic should be available. Finally, the structure factor for the required reflection should be high, while those for higher-order reflections should be low.

Figure 4.4.2.9| top | pdf |

Geometry of a polarizing monochromator showing the lattice planes (hkl) with |F_N| = |F_M|, the direction of P and $[\boldmu]$ , the expected spin direction and intensity.

None of the three naturally occurring ferromagnetic elements (iron, cobalt, nickel) makes efficient single-crystal polarizers. Cobalt is strongly absorbing and the nuclear scattering lengths of iron and nickel are too large to be balanced by their weak magnetic moments. An exception is ⁵⁷Fe, which has a rather low nuclear scattering length, and structure-factor matching can be achieved by mixing ⁵⁷Fe with Fe and 3% Si (Reed, Bolling & Harmon, 1973).

In general, in order to facilitate structure-factor matching, alloys rather than elements are used. The characteristics of some alloys used as polarizing monochromators are presented in Table 4.4.2.4. At short wavelengths, the 200 reflection of Co_0.92Fe_0.08 is used to give a positively polarized beam [ $[F_{N}({\boldkappa})]$ and $[F_{M}({\boldkappa})]$ both positive], but the absorption due to cobalt is high. At longer wavelengths, the 111 reflection of the Heusler alloy Cu₂MnAl (Delapalme, Schweizer, Couderchon & Perrier de la Bathie, 1971; Freund, Pynn, Stirling & Zeyen, 1983) is commonly used, since it has a higher reflectivity and a larger d spacing than Co_0.92Fe_0.08. Since for the 111 reflection $[F_{N}\approx -F_{M}]$ , the diffracted beam is negatively polarized. Unfortunately, the structure factor of the 222 reflection is higher than that of the 111 reflection, leading to significant higher-order contamination of the beam.

Table 4.4.2.4| top | pdf |
Properties of polarizing crystal monochromators (Williams, 1988)

	Co_0.92Fe_0.08	Cu₂MnAl	Fe₃Si	⁵⁷Fe:Fe	HoFe₂
Matched reflection $[\|F_{{N}}\|\sim \|F_{\rm {M}}\|]$	200	111	111	110	620
d spacing (Å)	1.76	3.43	3.27	2.03	1.16
Take-off angle $[2\theta _{{B}}]$ at 1 Å (°)	33.1	16.7	17.6	28.6	50.9
Cut-off wavelength, λ_max (Å)	3.5	6.9	6.5	4.1	2.3

Other alloys that have been proposed as neutron polarizers are Fe_3−xMn_xSi, ⁷Li_0.5Fe_2.5O₄ (Bednarski, Dobrzynski & Steinsvoll, 1980), Fe₃Si (Hines et al., 1976), Fe₃Al (Pickart & Nathans, 1961), and HoFe₂ (Freund & Forsyth, 1979).

References

Bednarski, S., Dobrzynski, L. & Steinsvoll, O. (1980). Experimental test on Fe₃Si(Mn) and Li_.5Fe_2.5O₄ crystals as polarizers for slow neutrons. Phys. Scr. 21, 217–219.Google Scholar

Delapalme, A., Schweizer, J., Couderchon, G. & Perrier de la Bathie, R. (1971). Étude de l'alliage Heusler (Cu₂MnAl) comme monochromateur de neutrons polarizés. Nucl. Instrum. Methods, 95, 589–594.Google Scholar

Freund, A. K. & Forsyth, J. B. (1979). Materials problems in neutron devices. Neutron scattering, edited by G. Kostorz, pp. 462–507. New York: Academic Press.Google Scholar

Freund, A. K., Pynn, R., Stirling, W. G. & Zeyen, C. M. E. (1983). Vertically focusing Heusler alloy monochromators for polarized neutrons. Physica (Utrecht) B, 120, 86–90.Google Scholar

Hines, W. A., Menotti, A. H., Budnick, J. L., Burch, T. J., Litrenta, T., Niculescu, V. & Raj, K. (1976). Magnetization studies of binary and ternary alloys based on Fe₃Si. Phys. Rev. B, 13, 4060–4068.Google Scholar

Pickart, S. J. & Nathans, R. (1961). Unpaired spin density in ordered Fe₃Al. Phys. Rev. 123, 1163–1171.Google Scholar

Reed, R. E., Bolling, E. D. & Harmon, H. E. (1973). Solid State Division Report, pp. 129–131. Oak Ridge National Laboratory, TN, USA.Google Scholar

Williams, W. G. (1988). Polarized neutrons. Oxford Series on Neutron Scattering in Condensed Matter, Vol. 1. Oxford: Clarendon Press.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.4, pp. 438-439