Neutron Scattering from Magnetic Materials part 14 pot

Inelastic neutron polarization analysis 389 Fig. 16. Energy dependence of I (ω) =I + (ω) −I − (ω) and I(ω)=I + (ω) +I − (ω) in CsMnBr 3 close to the antiferromagnetic point (1/3, 1/3, 1), at a temperature of 9.2 K and a field of 4 T. The difference I (ω) = I + (ω) −I − (ω) represents the polarization dependent (chiral) part of the scattering (I (ω) ∝ M ch ). As previously discussed (see Section 2.2), the chiral contribution should be an odd function of ω at ω/k B T 1, corresponding to the relation M ch =A ω/Γ ch [(1 +(ω/Γ ch ) 2 )] 2 , (56) where Γ ch characterizes the fluctuation rate of chiral fluctuations. Examples of spectra I(ω) (closed symbols) and I (ω) (open symbols) measured at 9.2 K ( T N )aregivenatFig- ure 16. According to the theory, A ∝(T −T N −φ C )P 0 , where φ C is the chiral crossover ex- ponent. In CsMnBr 3 , the polarized INS measurements [67] gave φ C ≈1.28 and β C ≈0.44 implying γ C ≈0.84. These values are found in good quantitative agreement with the Monte Carlo calculations of Kawamura [62], thus confirming the chiral universality class. Chiral fluctuations in the itinerant helimagnet MnSi. MnSi is a very interesting itinerant- electron weak ferromagnet with a Curie temperature T C ≈ 29 K [68,69]. The mag- netic structure in zero field is a long-period spiral described by the propagation vector k =(τ,τ,τ)with τ ≈0.017 and equivalents deduced by symmetry [70]. Theoretical work by Bak and Jensen [71] have shown that the spiral order in MnSi was due to the existence of a sizable Dzyaloshinskii–Moriya vectorial interaction term, resulting from the noncen- trosymmetric arrangement of Mn atoms in the cubic unit cell. Consequently, the spiral arrangement in MnSi must be very different from that arising in more conventional heli- magnets with competing exchange interactions because in MnSi it should be single handed, right or left depending on the sign of the Dzyaloshinskii–Moriya vector D DM =D DM ˆ k.In MnSi, LPA measurements [70] have clearly demonstrated that the spiral is right handed, even in zero field. The chiral nature of magnetic fluctuations above T C has been recently investigated by Roessli et al. [72], by measuring the difference between inelastic spectra taken with P 0 parallel and antiparallel to the scattering vector Q (that is to say associated with xx longitudinal components). Typical results are given in Figure 17 for temperatures 390 L.P. Regnault Fig. 17. Difference neutron counts for incident polarization parallel and perpendicular to Q in MnSi at T =31 K and T =40 K, showing the strength of chiral fluctuations close to T C . T = 31 K and T = 40 K which, among others, show the growing of chiral critical fluc- tuations as T approaches T C . Contrary to CsMnBr 3 , in MnSi the chiral fluctuations can be directly observed without disturbing the system by the application of a magnetic field (intrinsic chirality versus field-induced chirality). In most cases, the chiral contribution is deduced from LPA measurements of longitudinal components of the polarization along the scattering vector, namely xx or ¯xx. However, according to Section 2.3, the chiral terms can as well be obtained directly from SNP measurements of the transverse components P yx or P zx , which both depend on M ch . Such a determination has been successfully under- taken on the elastic contributions in MnSi below T C [73]. 3.5. Spherical polarimetry on CuGeO 3 In this section we will show how the SNP may help to solve the full problem, namely the determination of the complete set of correlation functions involving the nuclear and magnetic degrees of freedom. The spin-Peierls compound CuGeO 3 appeared to be a very suitable candidate to perform such a determination owing to the strong spin–lattice inter- actions existing in this material. In Section 3, it was shown that the nuclear structure factor associated with the hypothetic hybrid mode was very small, amounting to at best 0.3% of the main magnetic structure factor at q =π. However, if the hybrid correlation function be- tween the magnetic and structural degrees of freedom was strong enough, it might give rise to a sizable NMI term and, thus, a small rotation of the polarization away from P 0 should be detected. Maleyev [9] and Cepas et al. [76] have considered theoretically an S = 1/2 spin-Peierls system described by dynamical Heisenberg (H) and Dzyaloshinskii–Moriya (DM) interactions, both given by the generic term  i,j  α,β V αβ (R j −R i )S α i S β j , where V αβ (R j −R i ) =J β (R j −R i )δ αβ for the former and  γ D γ (R j −R i )ε γαβ for the latter, where the D γ coefficients are the components of the Dzyaloshinskii–Moriya vector which depend on the position of all ions involved in the superexchange paths. Maleyev [9–11], by using the standard perturbation theory, has found that the INMI terms are related to the Inelastic neutron polarization analysis 391 three-spin susceptibilities (or, equivalently, to the three-spin correlation functions), follow- ing the expression  N + Q S Q  ω =N −1/2  n,i,j b n  α,β  e −iQ·R n ,V αβ ij  ω  S α i S β j , S Q  ω . (57) In (57), the magnitude of NMI terms appears controlled by the first angle bracket which may be nonzero if the lattice excitations modulate both R n and the V components. For a system presenting no long-range order, as it is the case for CuGeO 3 , the scalar Heisen- berg exchange terms give no contribution to the NMI and only the Dzyaloshinskii–Moriya term can contribute. According to the general theory [10,11,76], the expression giving the INMI terms contain factors 1/(ω 2 − ω 2 λ ), where ω λ is the phonon energy and λ labels the different phonon (acoustic or optic) branches. Thus, the INMI terms must be a priori searched near scattering vectors for which the magnetic and nuclear contributions are not too far in energy. Obviously, the amplitude of INMI terms should also depend drastically on the polarization of the phonon involved in the interference process (remember that in an inelastic-neutron-scattering experiment one cannot measure simultaneously the spin com- ponents and the atomic displacements along the same direction). In order to verify these predictions, a spherical polarization analysis of the 2-meV mode on CuGeO 3 must be per- formed by using the CRYOPAD device. As previously explained, the procedure consists of determining, for each couple (Q,ω), the 18 components P αβ and P ¯αβ (α, β = x,y,z) of the polarization matrix after the appropriate background corrections. Table 2 gives the values of P αβ and P ¯αβ (α, β = x,y,z) and the corresponding error bars for the spin-Peierls mode at Q = (0, 1, 1/2) [74,75]. From Table 2, one can deduce the longitudinal compo- nents  P xx =(−0.963 ±0.005)P 0 ,  P yy =(0.005 ±0.006)P 0 ,  P zz =(−0.031 ±0.006)P 0 with P 0 =0.89±0.005, and where  P αα =(P αα −P ¯αα )/2 is the antisymmetric part which Table 2 Spherical polarization analysis on the 2-meV mode at Q =(0, 1, 1/2) in CuGeO 3 . Longitudinal and transverse components of the polarization matrix αβ P αβ (±P ) ¯αβ P ¯αβ (±P ) xx −0.868(6) ¯xx 0.866(6) xy −0.034(8) ¯xy 0.044(8) xz −0.015(8) ¯xz 0.025(8) yx 0.004(8) ¯yx 0.021(8) yy 0.001(8) ¯yy −0.008(8) yz 0.006(8) ¯yz 0.001(8) zx 0.006(8) ¯zx 0.015(8) zy −0.004(8) ¯zy −0.004(8) zz −0.031(8) ¯zz 0.026(8) 392 L.P. Regnault cancels at first-order the P 0 -independent terms. Undoubtedly, CRYOPAD allows a very precise determination of the longitudinal components, the limitation being mainly due to the statistical error on the counting (the intrinsic error is smaller than 0.0005!). In agree- ment with the LPA measurements, the quantity (  P xx +  P yy +  P zz )/P 0 =(−0.989±0.010) is very close to 1, which again means that the nuclear contribution associated with the putative hybrid mode at q = π should be very small: NN +  ω /σ 0 = (0.003 ±0.003),as previously found from LPA. Applying the set of relations given in Section 2.2, it has been possible to get relatively reliable estimates of the chiral term and the real and imaginary parts of INMI terms: M y −M z σ 0 =0.015 ±0.010, M ch σ 0 =0 ±0.003, R y σ 0 =0 ±0.004, I y σ 0 =−0.008 ±0.005, R z σ 0 =0.004 ±0.004, I z σ 0 =−0.015 ±0.005. Thus, within the error bars, the chiral term and the real parts of INMI terms appear to be very small, while the imaginary parts (especially I z ) appear finite, although at the limit of the accuracy of the present CRYOPAD version (roughly ±0.01). To summarize the results of the inelastic SNP on the spin-Peierls mode in CuGeO 3 , the magnitude of INMI terms is found to be at best 1% of the main magnetic signal. At q ≈ π and ¯ hω ≈ ∆ SP , the magnetic and lattice excitations can be considered as almost completely decoupled. The main reason for this negative result comes from the fact that in CuGeO 3 , ∆ SP ω ph . Hybrid modes may exist at other q or energy values, in particular those corresponding to the continuum [77]. Unfortunately, despite several tentatives, they have not been detected so far. 4. Conclusion In this chapter, we have shown how powerful the neutron polarimetry is for the investiga- tion of inelastic spectra in condensed-matter physics. The most frequently used method, namely the longitudinal polarization analysis (LPA), is best suited for all studies requiring the accurate determination of dynamical spin–spin correlation functions involving only the longitudinal components of the generalized susceptibility tensor. Among others things, the Inelastic neutron polarization analysis 393 LPA allows us to obtain separately the structural and various longitudinal magnetic struc- ture factors (e.g., the P 0 -dependent parts, chiral terms, anisotropy of various spin–spin correlation functions). Basically, the LPA can be used to solve problems for which the nuclear–magnetic interference terms can be neglected. If strong enough, such terms might produce a rotation of the polarization vector around the interaction vector different from π which can be only determined by spherical neutron polarimetry (SNP). In principle, the measurement of the nine coefficients of the polarization matrix should allow us to unam- biguously solve the full problem and, finally, should allow us to determine the various nuclear–nuclear, magnetic–magnetic and nuclear–magnetic correlation functions. References [1] O. Halpern and M.H. Johnson, Phys. Rev. 55 898 (1939). [2] O. Halpern and T. Holstein, Phys. Rev. 59 960 (1941). [3] S.V. Maleyev, Sov. Phys. JETP 13 595 (1961). [4] Yu. Izyumov and S.V. Maleyev, Sov. Phys. JETP 14 1168 (1962). [5] Yu. Izyumov, Sov. Phys. JETP 15 1167 (1962). [6] S.V. Maleyev, V.G. Baryakhtar and A. Suris, Sov. Phys. Solid State 4 2533 (1963). [7] M. Blume, Phys. Rev. 130 1670 (1963), and references therein. [8] R.I. Schermer and M. Blume, Phys. Rev. 166 554 (1968). [9] S.V. Maleyev, Physica B 267–268 236 (1999). [10] S.V. Maleyev, Physica B 297 67 (2001). [11] S.V. Maleyev, Physics-Uspekhi 45 569 (2002), and references therein. [12] G.T. Trammell, Phys. Rev. 92 1387 (1953). [13] S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vols. 1 and 2, Clarendon, Oxford (1987). [14] E. Balcar and S.W. Lovesey, Theory of Magnetic Neutron and Photon Scattering,Clarendon, Oxford (1989). [15] L. Van Hove, Phys. Rev. 95 249 (1954). [16] J. Schwinger, Phys. Rev. 73 407 (1948). [17] M. Blume, Phys. Rev. A 133 1366 (1964). [18] G.L. Squires, Introduction to the Theory of Thermal Neutron Scattering, Dover, Mineola, NY (1986). [19] R.M. Moon, T. Riste and W.C. Koehler, Phys. Rev. 181 920 (1969). [20] G.M. Drabkin et al., JETP Lett. 2 353 (1965). [21] S. Brehmer et al., Phys. Rev. B 60 329 (1999). [22] M. Muller et al., Phys. Rev. B 66 34423 (2002). [23] M. Matsuda et al., Phys. Rev. B 62 8903 (2000). [24] S.V. Maleyev, Phys. Rev. Lett. 75 4682 (1995), and references therein. [25] D.N. Aristov and S.V. Maleyev, Phys. Rev. B 62 R751 (2000), and references therein. [26] F. Tasset, Physica B 156–157, 627 (1989). [27] P.J. Brown, J.B. Forsyth and F. Tasset, Proc. Roy. Soc. London Ser. A 442 147 (1993). [28] H. Mikeska, J. Phys. C 13 2913 (1980). [29] K. Maki, J. Low. Temp. Phys. 41 327 (1981). [30] L.P. Regnault, J.P. Boucher, J.Rossat-Mignod, J.P. Renard, J. Bouillot and W.G. Stirling, J. Phys. C 15 1261 (1982), and references therein. [31] J.P. Boucher, L.P. Regnault, R. Pynn, J. Bouillot and J.P. Renard, Europhys. Lett. 1 415 (1986), and refer- ences therein. [32] K. Sasaki and K. Maki, Phys. Rev. B 35 257 (1987). [33] D. Vaknin et al., Phys. Rev. Lett. 58 2802 (1987). [34] J. Rossat-Mignod et al., Physica C 185–189 86 (1991). 394 L.P. Regnault [35] H.F. Fong et al., Phys. Rev. Lett. 75 316 (1995); H.F. Fong et al., Nature 398 588 (1999). [36] L.P. Regnault, P. Bourges and P. Burlet, in: Neutron Scattering in Layered Copper-Oxyde Superconductors, ed. A. Furrer, pp. 85–134, Kluwer Academic, Dordrecht–Boston–London (1998), and references therein. [37] S. Hayden, in: Neutron Scattering in Layered Copper-Oxyde Superconductors, ed. A. Furrer, pp. 135–164, Kluwer Academic, Dordrecht–Boston–London (1998), and references therein. [38] H.A. Mook et al., Nature 395 580 (1998). [39] P. Bourges et al., Science 288 1234 (2000). [40] J.M. Tranquada et al., Nature 375 561 (1995). [41] L.P. Regnault et al., Physica B 335 19 (2003). [42] L.P. Regnault et al., Physica B 345 111 (2004). [43] M. Hase, I. Terasaki and K. Uchinokura, Phys. Rev. Lett. 70 3651 (1993). [44] M. Nishi, O. Fujita and J. Akimitsu, Phys. Rev. B 50 6508 (1994). [45] L.P. Regnault, M. An, B. Hennion, G. Dhalenne and A. Revcolevschi, Phys. Rev. B 53 5579 (1996), and references therein. [46] J.P. Pouget, L.P. Regnault, M. Aïn, B. Hennion, J.P. Renard, P. Veillet, G. Dhalenne and A. Revcolevschi, Phys.Rev.Lett.72 4037 (1994). [47] M. Aïn, J.E. Lorenzo, L.P. Regnault, G. Dhalenne, A. Revcolevschi, B. Hennion and T. Jolicoeur, Phys. Rev. Lett. 78 1560 (1997). [48] J.P. Boucher and L.P. Regnault, J. Phys. I France 6 1939 (1996), and references therein. [49] J.E. Lorenzo, L.P. Regnault, B. Hennion, M. Aïn, F. Bourdarot, J. Kulda, G. Dhalenne and A. Revcolevschi, J. Phys.: Condens. Matter 9 L211 (1997). [50] G. Bouzerar, O. Legeza and T. Ziman, Phys. Rev. B 60 15278 (1999). [51] H.M. Rønnow, L.P. Regnault and J.E. Lorenzo, Physica B 350 11 (2004). [52] E. Pytte, Phys. Rev. B 10 4637 (1974); M.C. Cross and D.S. Fischer, Phys. Rev. B 19 402 (1979); J.W. Bray, L.V. Interrante, I.S. Jacobs and J.C. Bonner, in: Extended Linear Chain Compounds, vol. 3, ed. J.S. Miller, pp. 353–415, Plenum, New York (1982). [53] E.M. McCaron, M.A. Subramaniam, J.C. Calabrese and R.L. Harlow, Mater. Res. Bull. 23 1355 (1988). [54] M. Kato, K. Shiota and Y. Koike, Physica C 258 284 (1996). [55] Y. Mizuno, T. Tohyama and S. Maekawa, J. Phys. Soc. Jpn. 66 937 (1997). [56] T. Osafune, N. Motoyama, H. Eisaki and S. Uchida, Phys. Rev. Lett. 78 1980 (1997). [57] N. Motoyama, T. Osafune, T. Kakeshita, H. Eisaki and S. Uchida, Phys. Rev. B 55 3386 (1997). [58] M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Môri and K. Kinoshita, J. Phys. Soc. Jpn. 65 2764 (1997). [59] M. Matsuda and K. Katsumata, Phys. Rev. B 53 12201 (1996). [60] L.P. Regnault, J.P. Boucher, H. Moudden, J.E. Lorenzo, A. Hiess, U. Ammerahl, G. Dhalenne and A. Rev- coloevschi, Phys. Rev. B 59 1055 (1999). [61] C. Boullier, L.P. Regnault, J.E. Lorenzo, G. Dhalenne and A. Revcoloevschi, Physica B 350 40 (2004). [62] H. Kawamura, J. Phys.: Condens. Matter 10 4707 (1998). [63] S.V. Maleyev, Physica B 297 67 (2001). [64] T.E. Mason, Y. Yang, M.F.Collins, B. Gaulin and A. Harrisson, J.Magn. Magn. Mater.104–107 197 (1992). [65] S.V. Maleyev, V.P. Plakhty, O.P. Smirnov, J. Wosnitza, D. Visser, R.K. Kremer and J. Kulda, J. Phys.: Condens. Matter 10 951 (1998). [66] V.P. Plakhty, S.V. Maleyev, J. Kulda, J. Wosnitza, D. Visser and E. Moskvin, Europhys. Lett. 48 215 (1999). [67] V.P. Plakhty, J. Kulda, D. Visser, E.V. Moskvin and J. Wosnitza, Phys. Rev. Lett. 85 3942 (2000). [68] K.R.A. Ziebeck, P.J. Brown, J.G. Booth and J.A.C. Bland, J. Phys. F 11 L127 (1981). [69] K.R.A. Ziebeck, H. Capellmann, P.J. Brown and J.G. Booth, Z. Phys. B 48 241 (1982). [70] G. Shirane, R. Cowley, C. Majkrzak, J.B. Sokoloff, B. Pagonis, C.H. Perry and Y. Ishikawa, Phys. Rev. B 28 6251 (1983), and references therein. [71] P. Bak and M.H. Jensen, J. Phys. C 13 L881 (1980). [72] B. Roessli, P. Böni, W.E. Fisher and Y. Endoh, Phys. Rev. Lett. 88 237204 (2002). [73] M. Janoschek, PhD thesis, University of Munich (2004). Inelastic neutron polarization analysis 395 [74] L.P. Regnault, H.M. Ronnow, J.E. Lorenzo, R. Bellissent and F. Tasset, Physica B 335 19 (2003). [75] L.P. Regnault, H.M. Ronnow, C. Boullier, J.E. Lorenzo and C. Marin, Physica B 345 111 (2004). [76] O. Cepas and T. Ziman, in: Quantum Properties of Low-Dimensional Antiferromagnets, eds. Y. Ajiro and J.P. Boucher, pp.175–182, Kyushu University Press (2002). [77] M. Braden, B. Hennion, P. Pfeuty, G. Dhalenne and A. Revcolevschi, Phys. Rev. Lett. 83 1858 (1999). CHAPTER 9 Polarized Neutron Reflectometry C.F. Majkrzak Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA E-mail: chuck@rrdjazz.nist.gov K.V. O’Donovan Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, University of Maryland, College Park, MD 20742, USA and University of California, Irvine, CA 92697, USA N.F. Berk Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Contents 1. Introduction . . . . . . . . . . . 399 2. Fundamental theory of neutron reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 2.1.Waveequationinthreedimensions 402 2.2.Refractiveindex 404 2.3. Specular reflection from a perfectly flat slab: The wave equation in one dimension . . . . . . . . . 404 2.4. Specular reflection from a film with a nonuniform SLD profile . . . . . . . . . . . . . . . . . . . . 408 2.5.Bornapproximation 410 2.6.Nonspecularreflection 411 3.Spin-dependentneutronwavefunction 412 3.1. Neutron magnetic moment and spin angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 412 3.2. Explicit form of the spin-dependent neutron wave function . . . . . . . . . . . . . . . . . . . . . . 413 3.3.Polarization 415 3.4.Selectinganeutronpolarizationstate 417 3.5.Changinganeutron’spolarization 417 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji Published by Elsevier B.V. 397 398 C.F. Majkrzak et al. 4.Spin-dependentneutronreflectivity 424 4.1. Spin-dependent reflection from a magnetic film in vacuum referred to reference frame of film . . . 424 4.2. Magnetic media surrounding film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 4.3.Coordinatesystemtransformation 439 4.4.Selectionrules“ofthumb” 442 4.5.Three-dimensionalpolarizationanalysis 444 4.6.Elementaryspin-dependentreflectivityexamples 445 5. Experimental methods . . . . . . 448 6. An illustrative application of PNR 450 6.1.Symmetriesofreflectancematrices 451 6.2.Basis-independentrepresentation 453 6.3.Front–backreflectivityofidealizedtwists 455 6.4.PNRofactualsystems 457 Appendix 462 References 470 Polarized neutron reflectometry 399 1. Introduction Advances in our understanding of the structure and properties of matter have so often depended upon finding the right probe for studying a given problem. This was appreciated long ago, at the beginning of the evolution of modern science. In his masterpiece Faust, Johann Wolfgang von Goethe wrote about the legend of Doctor Faust, who bargained his immortal soul to the Devil, Mephistopheles, in exchange for unlimited knowledge. Early in the story, Faust ponders the relationship between humankind and the universe [1, Part I, Scene I]: Mysterious even in open day, Nature retains her veil, despite our clamors: That which she doth not willingly display Cannot be wrenched from her with levers, screws, and hammers. Fortunately,foranyone in the 21st century interested in the structure of condensed matter on the atomic and nanometer length scales, the considerable efforts of our predecessors have led to the development of a remarkable collection of sophisticated and exquisitely sensitive probes (far surpassing the capabilities of levers, screws and hammers). These newfound tools are so powerful, in fact, that making a pact with Mephistopheles may no longer be necessary! Polarized neutron reflectometry (PNR) is one such probe that is particularly well suited for determining the nanostructures of magnetic thin films and multilayers. Different types of magnetometers ordinarily yield only average magnetization values, integrated over the entire volume of the specimen, whereas other probes which do possess a higher degree of spatial resolution, such as scanning electron microscopy with polarization analysis (SEMPA) [2], are specifically surface sensitive because their relatively strong interaction with matter limits penetration. Together with magnetic X-ray scattering, PNR provides a unique means of “seeing” the vector magnetization with extraordinary spatial detail well beneath the surface. For neutrons this sensitivity to atomic magnetic moments comes about because the neutron itself possesses a magnetic moment and neutrons can be obtained with a wavelength comparable to interatomic distances. More specifically, the specular re- flection of polarized neutrons, namely, coherent elastic scattering for which the angles of incidence and reflection of the neutron wave vector relative to a flat surface are of equal magnitude, can be analyzed to yield the in-plane average of the vector magnetization depth profile along the surface normal. By measuring the reflectivity (the ratio of reflected to incident intensities) over a sufficiently broad range of wave vector transfer  Q, subnanome- ter spatial resolution can be achieved. The specular geometry is depicted schematically in Figure 1. Furthermore, by tilting  Q away from the surface normal, the resulting projection of  Q parallel to the surface of the film allows in-plane fluctuations of the magnetization, which give rise to nonspecular scattering, to be sensed. Unlike optical or electron mi- croscopy, neutron and X-ray reflectometry do not directly provide real-space images of the objects of interest. Because neutron and X-ray wavelengths are of the order of the di- mensions of the objects being viewed, the information about shape and composition that is contained in the reflected radiation pattern, including the vector magnetization depth profile, must be extracted by mathematical analysis. [...]... of the neutron is remarkably accurate in practice Thus, unless necessary to do otherwise, we will treat the neutron so Now the neutron interacts with matter primarily through a nuclear potential and a magnetic potential which affect the magnitude of k The strengths of these potentials are effectively characterized by scalar “coherent” scattering lengths, although for the magnetic coupling a scattering- angle-dependent... examine the interaction between the magnetic moment of the neutron and the atomic moments which are present in magnetic films 3.1 Neutron magnetic moment and spin angular momentum The magnetic moment of the neutron is associated with an intrinsic “spin” angular momentum which is quantized such that the neutron can occupy only one of two discrete energy states within a given magnetic field The energies E corresponding... complete set of equations of motion which govern spin-dependent neutron reflection from magnetic materials, we can use what we have learned thus far about reflection from nonmagnetic media and the spin-dependent neutron wave function to examine several important consequences of the spin on the observable behavior of the neutron 3.4 Selecting a neutron polarization state Recall our discussion at the end of... different from one another in the presence of a magnetic field Because of the magnetic energy (equation (31)), the refractive index for neutrons in a magnetic field is two-valued, i.e., the refractive index given by (10) for a purely nuclear medium, must now be generalized to include a magnetic contribution, n± = 2 1 − 4π(ρN ± ρM )/k0 ⇒ nz± = 2 1 − 4π(ρN ± ρM )/k0z , (33) where the magnetic scattering. .. rise to the magnetic potential) We will ignore absorptive as well as incoherent interactions for the time being and also make the assumption throughout that any nuclear spins in the materials considered are completely disordered (see, e.g., the book by Bacon [9] for a discussion) To properly account for the neutron magnetic moment and its interaction with a magnetic potential requires that the neutron. .. found to be E± ,magnetic = ∓µ B , (31) Polarized neutron reflectometry 413 where µ is the magnitude of the neutron magnetic moment (µ = −1.913 nuclear magnetons or −1.913 × 5.051 × 10−27 J/T) and B is the magnetic induction (which can be treated classically for our purposes: see, for example, the discussion by Mezei [17] of the implications of a magnetic interaction which is proportional to the magnetic induction... saturated ferromagnetic film, prepare it to be in an essentially pure spin + or “up” state, along the direction coincident with that of the applied magnetic 418 C.F Majkrzak et al Fig 8 Neutron reflectivity for a saturated ferromagnetic Fe mirror Two critical scattering vectors are observed, one corresponding to the sum of nuclear and magnetic SLDs, the other to their difference (the lower value) The magnetic. .. that is “adiabatic”, i.e., one for which the neutron spin and moment follow and stay aligned with the changing magnetic field direction.) A change of neutron polarization with respect to the magnetic field direction can be effected via a sudden transition in the laboratory, for example, by passage of a neutron beam through a thin current sheet [20] From the neutron s perspective, this is equivalent to... solution for reflection from the uniform slab of finite thickness Look at Figure 4 where the slab of Figure 3 is shown schematically in cross-section, with a thickness L and with a constant SLD ρ We can partition space into three distinct regions: Region I extending from z = −∞ to the boundary of the potential at z = 0; Region II from z = 0 to z = L in which the SLD is ρ; and Region III from z = L to z =... µB, (34) 414 C.F Majkrzak et al and where the “+” and “−” subscripts denote the neutron spin state and “N” and “M” refer to the nuclear and magnetic components of the SLD, respectively (see also (5)) Therefore, in a magnetic medium, k± = n± k0 so that Ψ± = e+ik± ·r = e+in± k0 ·r (35) As we shall soon see, however, the vectorial nature of the magnetic field and the spinor character of the neutron spin . Lovesey, Theory of Neutron Scattering from Condensed Matter, vols. 1 and 2, Clarendon, Oxford (1987). [14] E. Balcar and S.W. Lovesey, Theory of Magnetic Neutron and Photon Scattering, Clarendon,. . . . 413 3.3.Polarization 415 3.4.Selectinganeutronpolarizationstate 417 3.5.Changinganeutron’spolarization 417 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji Published. to whether the spin state of the neutron changes (“flips”) upon reflection from a magnetic film. In this particular example, the relative orientations of incident neutron polarization  P I and layer