Schluesseltech_39 (1)

Fig. 10.15: Polarized neutron reflectivity and offspecular scattering for two AF-coupled

[Fe / Cr]N multilayers with N = 7 (top) and N = 19 (bottom) in

remanence field of 5 mT. Indicated are the AF superstructure Bragg peaks of order ½ in the NSF channels of the [Fe / Cr]7 system (1) and in the SF channels of the [Fe / Cr]19 system (2).

References

[1]P. Grünberg, R. Schreiber, Y. Pang, M.B. Brodsky, and H. Sowers, Phys. Rev. Lett. 57 (1986), 2442

[2]P.H. Dederichs, “Interlayer Exchange Coupling“, chapter C 3 in “Magnetism goes Nano“, 36th IFF Spring School, Forschungszentrum Jülich, series Matter and Materials, Vol. 26 (2005)

[3]G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39 (1989), 4828

[4]M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friedrich, and J. Chazelas, Phys. Rev. Lett. 61 (1988), 2472

[5]D.E. Bürgler, “Spin-Transport in Layered Systems“ , chapter E 5 in “Magnetism goes Nano“, 36th IFF Spring School, Forschungszentrum Jülich, series Matter and Materials, Vol. 26 (2005)

[6]U. Rücker, E. Kentzinger, “Thin Film Systems: Scattering under Grazing Incidence”, chapter D4 in “Probing the Nanoworld”, 38th IFF Spring School, Forschungszentrum Jülich, series Matter and Materials, Vol. 34 (2007)

[7]A. Paul, E. Kentzinger, U. Rücker, D.E. Bürgler, and P. Grünberg, Phys. Rev. B 70 (2004), 224410

[8]E. Josten, U. Rücker, S. Mattauch, D. Korolkov, A. Glavic, and Th. Brückel, J. Phys: Conf. Ser. 211 (2010), 012023

11.1 Introduction

One of the most important benefits of neutron scattering is the possibility to do inelastic scattering and by this way gain insight into the dynamics of materials as well as the structure. Neutrons tell us where the atoms are and how they move [1]. Although this feature is shared with inelastic x-ray scattering and dynamic light scattering, there is still a considerable range of slow dynamics in molecular systems which can be studied exclusively by inelastic neutron scattering.

This lecture can only present a short glimpse on the theoretical foundations and instrumental possibilities of inelastic neutron scattering. For those who are interested in more details, several textbooks can be recommended [2–6]. Also supplementary information on correlation functions [7] and Fourier transforms [8] may be found in earlier editions of this school.

Q

k'

2θ

k

Fig. 11.1: Definition of the scattering vector Q in terms of the incident and final wave vectors k and k . The black (isosceles) triangle corresponds to elastic scattering. The blue and red ones correspond to inelastic scattering with energy loss or gain of the scattered neutron, respectively.

11.2 Theory

11.2.1 Kinematics of neutron scattering

Up to this lecture it has always been assumed that the wavelength (or wave vector, or energy) of the neutrons is the same before and after scattering. The defining quality of inelastic neutron scattering is that this is not anymore the case. The neutrons may lose or gain energy in the collision with the nuclei implying that k = k. This implies that Q now does not anymore result from the isosceles construction drafted in black in Fig. 11.1 but from scattering triangles as those in blue and red. Application of the cosine theorem leads to the following expression for

q

Fig. 11.3: Dispersion relation in a linear chain with N = 40 atoms (Born-von-Karm´an´ boundary condition).

This relation does not contain the number of atoms anymore. For large N the points constituting the curve in Fig. 11.3 will get closer and closer, finally leading to the continuous function (11.10). The individual positions of these points depend on the boundary condition. But because they are getting infinitely dense for N → ∞ the exact boundary conditions do not matter for a large system.

It can be seen that the dispersion relation (11.10) is periodic in q. On the other hand, there are only N normal coordinates necessary to solve the N original equations of motion. This is exactly the number of wave vectors found in a q interval of length 2π/a. The usual choice is −π/a . . . π/a as a representative zone for the dispersion relation.

There are two modifications necessary when considering a real three-dimensional crystal instead of this simplified model: (1) The crystal is periodic in three dimensions. (2) The vibrations are governed by quantum mechanics. The first requirement leads to the consequence that instead of a scalar, one has to use a real wave vector, q → q = (qx, qy, qz) in reciprocal space. The interval defined in Fig. 11.3 changes into a polyhedron called the first Brillouin zone (Fig. 11.4) [9, 10]. It is constructed in the same way as the Wigner-Seitz cell in real space: The Brillouin zone contains all points which are closer to the origin than to any other lattice point. Its surfaces are the bisecting planes between the origin and its neighbours (in reciprocal space).

For every amplitude Ak equation (11.8) gives a valid solution of the equations of motion. This means that in the classical picture the vibrations can have any energy. The quantum mechanical treatment (which is too complex to be treated here in detail) leads to the result that only certain energies with a distance of Ωk are allowed. This quantisation implies that the vibrations can be treated as quasiparticles with the energy Ωk called phonons. The increase of the vibrational amplitude corresponding to an energy change of + Ωk is then seen as a creation, the inverse process as an annihilation of a phonon. Then it makes sense to define q as the momentum of the phonon. In this way the dispersion relation Ω(q) is similar to the relations shown in Fig. 4.2 of lecture 4 for real particles.