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Neutron Scattering in Magnetism: Fundamentals and Examples

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Magnetic Measurement Techniques for Materials Characterization

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Abstract

In this chapter, an up-to-date survey of theoretical concepts and experimental results in the field of the neutron scattering techniques applied to magnetism, and their relevance to experiments in real materials, is presented. Main emphasis of the chapter is to enlarge the use of these techniques among the researchers (physicist, chemists, engineers, etc.) in the area of magnetism and magnetic materials. For that reason we do not enter deeply in topics as neutron production, neutron optics, detectors, instrumentation, etc., but we focused the chapter on the neutron scattering applications in such a field.

Along this chapter, together with some basic concept on quantum mechanics, solid-state physics, magnetism, and symmetry, we introduce the main theoretical concepts, peculiarities, and language, of the neutron scattering techniques, and we particularize it to the crystalline matter. In a second part, we briefly introduce the most commonly employed techniques (powder diffraction, single crystal diffraction, polarization analysis, inelastic, SANS, reflectivity, etc.) and how these techniques are applied to understand different magnetic phenomena and magnetic materials, trying to cover a large variety of topics with interest in magnetism. Every time, the examples showed here show how neutrons can enlighten problems that are quasi-impossible to be solved with other techniques.

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Notes

  1. 1.

    Occasionally, the symbol h will be used to denote one of the Miller indices, but the context will resolve any ambiguity.

  2. 2.

    Note that since we are dealing with s-wave scattering, the total spin of the system is conserved.

  3. 3.

    The scattering function is also called response function, dynamic scattering function, dynamic structure factor, or scattering law.

  4. 4.

    These expressions have to be manipulated carefully, since the Heisenberg operators \(\vec {R}_n(0)\) and \(\vec {R}_{n^{\prime }}(t)\) do not commute for t ≠ 0. For instance,

    $$\displaystyle \begin{aligned} \exp\{-{\mathrm{i}2\pi}\vec{q}\cdot\vec{R}_n(0)\}\exp\{{\mathrm{i}2\pi}\vec{q}\cdot\vec{R}_{n^{\prime}}(t)\}\neq\exp\{{\mathrm{i}2\pi}\vec{q}\cdot[\vec{R}_{n^{\prime}}(t)-\vec{R}_n(0)]\}. \end{aligned}$$
  5. 5.

    Equivalently the elastic nuclear scattering can be obtained from the t →∞ limit of Eq. (47).

  6. 6.

    Notice that the electrostatic energy operator commutes with the spin of each electron separately (but not with the single electron orbital angular momentum). At first sight, it seems that the multi-electron states can be characterized by the single-electron spins, not just by the total spin. However, the individual spins are not observables since the electrons are indistinguishable, and the single-electron spin states are not good quantum numbers. Indeed, the single-electron spin states are mixed in the antisymmetrization process, as can be seen in Eq. (107).

  7. 7.

    Let us recall that the spherical components of a vector operator \(\vec {V}\) are denoted by V M, with M = −1,  0,  1, and defined as \(V_{\pm 1}=\mp (V_x\pm {\mathrm {i}} V_y)/\sqrt {2}\) and V 0 = V z.

  8. 8.

    Other authors define the magnetic structure factor for the crystal by keeping the sum on the unit cells, l and l ′, in Eq. (142) in its definition.

  9. 9.

    Notice that this correlation is independent of time. The 1∕J 2 terms have to be neglected in the linear approximation for consistency. Higher-order corrections guarantee that the second equality in (157) holds to all orders in the 1∕J expansion.

  10. 10.

    The FullProf Suite has the necessary software to accomplish these tasks.

  11. 11.

    Notice that 1 cm3 contains ∼109 crystallites with size ∼10 μm or ∼1012 crystallites with size ∼1 μm.

  12. 12.

    In helical structures, driven by magnetic frustration, the period is usually shorter.

  13. 13.

    If these experiments are done in a magnetically ordered crystal, then the nuclear and magnetic lattices need to be the same.

  14. 14.

    ΔM S = ±1.

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Acknowledgements

Much of the work reported here was carried out during funding by the Spanish Ministry of Science and Innovation grant PGC-2018-099024-B-I00 or its equivalents before. JC is grateful for the hospitality received at the different neutron sources and in particular to the Institut Laue-Langevin and the Laboratoire Leon Brillouin. We are indebted to many discussions with our colleagues and co-workers, but in particular with Fernando Palacio, Javier Luzón, Garry McIntyre, José Alberto Rodríguez-Velamazán, Ángel Millán, Oscar Fabelo, Laura Cañadillas, Juan Rodríguez-Carvajal, Katsuya Inoue, Yusuke Kousaka, Yuko Hosokoshi, Kazuki Ohishi, Jun Kishine, Yoshiiko Togawa, Cristina Sáenz de Pipaón, Clara González, Clara Rodríguez, Miguel Pardo-Sáenz, Milagros Tomás, and Larry Falvello.

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  1. Aragón Nanoscience and Materials Institute (CSIC – Zaragoza University), Spanish National Research Council (CSIC), Zaragoza, Spain

    Javier Campo & Víctor Laliena

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  1. Javier Campo
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  2. Víctor Laliena
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Correspondence to Javier Campo .

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  1. Department of Condensed Matter Physics, University of Seville, Seville, Spain

    Victorino Franco

  2. Lake Shore Cryotronics, Westerville, OH, USA

    Brad Dodrill

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Campo, J., Laliena, V. (2021). Neutron Scattering in Magnetism: Fundamentals and Examples. In: Franco, V., Dodrill, B. (eds) Magnetic Measurement Techniques for Materials Characterization. Springer, Cham. https://doi.org/10.1007/978-3-030-70443-8_14

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