Institut Laue-Langevin International Student Summer Programme at ILL/ESRF 2017 Project Report Investigation of magnetic phase transitions using the Laue technique 4 – 29 September 2017 Author Supervisor Guido Homann ∗ Dr. Bachir Ouladdiaf This report presents the analysis of Laue diffraction patterns measured by the multiple CCD neutron diffractometer CYCLOPS at ILL. The magnetic structure of the metal oxide CaBaFe 4 O 7 is studied between 158 and 301 K. Two magnetic phase transitions can be observed at the ordering tempera- tures T 1 ∼ 278 K and T 2 ∼ 205 K. They are related to the magnetic prop- � 1 � agation vectors k 1 = (0 , 0 , 0) and k 2 = 3 , 0 , 0 , respectively. The phase transitions are revealed by evaluating the temperature dependence of the intensities of selected reflections. ∗ Universität Hamburg, guido.homann@physik.uni-hamburg.de
1 Introduction The first quantitative neutron diffraction experiments were carried out by Shull et al. in the 1940s [Shu95]. Still today, neutron scattering is used to study the atomic and magnetic structure of materials. Due to its magnetic moment [Alv40], the neutron does not only interact with the atomic nuclei but also with the un- paired electrons inside a magnetic sample. Since this electromagnetic interaction is comparable in strength to the strong interaction with the nuclei, neutron diffrac- tion is a powerful tool for the investigation of magnetic structures in condensed matter. This report aims to demonstrate the analysis of Laue diffraction patterns in order to examine the magnetic behavior of a CaBaFe 4 O 7 single crystal. The experimen- tal data were gained by Qureshi [Qur17] using the CCD neutron diffractometer CYCLOPS [Oul11] at the Institut Laue-Langevin in Grenoble. As shown in [Rav08] the compound CaBaFe 4 O 7 belongs to a class of metal oxides with a kagome sublattice. The magnetic properties of these materials range from 3D long-range order of both antiferro- and ferrimagnetism to spin-liquid behav- ior [Sch07]. This variety of phenomena motivates to study the magnetic phase transitions of CaBaFe 4 O 7 in detail. 2 Theoretical background After providing a summary of the fundamental aspects of neutron scattering, this section introduces the concept of magnetic propagation vectors. A more elaborate formulation of the underlying theory can be found in [Mar71, Squ78, Res14]. 2.1 Coherent elastic neutron scattering Elastic neutron scattering is the dominant process in diffraction experiments. This means that the scattering event does not cause any energy transfer. Hence, the wave vectors of the incident neutron k i and the scattered neutron k f have the same modulus. There can be, however, a momentum transfer (1) q = k i − k f . The following considerations are limited to the case of coherent scattering which is most relevant for the determination of nuclear and magnetic structures. Besides, the neutron beam is assumed to be unpolarized. 1
2.2 Nuclear neutron scattering Because of the very short-range interaction between a neutron and a nucleus, this type of scattering is isotropic. Therefore, it can be characterized by only one parameter, the scattering length b , and the potential takes the form V 1N ( r ) = 2 π � 2 (2) m b δ ( r − R ) for a neutron at r and a nucleus at position R . Moreover, � denotes the reduced Planck constant, and m the neutron mass. The potential stated above leads to a simple relation for the differential cross section: � d σ � (3) = b . dΩ 1N Now, one can proceed to the case of a single crystal where the scattering poten- tial transforms into a sum over all the contained nuclei. Their positions can be expressed through the position of their unit cell R α and their position inside the unit cell r j : (4) R α,j = R α + r j . Taking the temperature dependent Debye-Waller factor exp( − W j ) into account, the potential reads V N ( r ) = 2 π � 2 � (5) exp( − W j ) b j δ ( r − R α,j ) . m α,j The resulting differential cross section can be written as a sum over all reciprocal lattice vectors: � d σ = (2 π ) 3 � | F N ( q ) | 2 δ ( q − h ) � (6) N dΩ V 0 N h with the volume of a unit cell V 0 and the number of unit cells inside the crystal N . Also the nuclear structure factor was introduced in this step: � (7) F N ( q ) = exp( − W j ) b j exp(i q · r j ) . j The structure factor is affected by all the atoms in one unit cell and can become zero for certain configurations. The differential cross section in Eq. 6 implies a condition for the occurrence of reflections. It vanishes unless the scattering vector q equals a reciprocal lattice vector. In fact, the condition (8) q = h is equivalent to Bragg’s law and merely a consequence of the symmetry of the crystal lattice. 2
2.3 Magnetic neutron scattering Beyond nuclear scattering, the neutron magnetic moment µ n gives rise to magnetic scattering from the unpaired electrons of an atom. In terms of its gyromagnetic ratio γ = − 1 . 91 [Ber12] and the nuclear magneton µ N , the neutron magnetic moment has a magnitude of (9) µ n = γ µ N . The scattering potential is given by a sum of the dipole-dipole interactions of the neutron magnetic moment µ n with all the unpaired electrons of one atom: � � µ i × ( r − R i ) � − 2 µ B p i × ( r − R i ) � � V m ( r ) = − µ n · B el = − µ n . ∇ × | r − R i | 3 � | r − R i | 3 i (10) µ B represents the Bohr magneton whereas µ i is the spin magnetic moment of one unpaired electron. The momentum and the position of such an electron are de- noted by p i and R i , respectively. The differential cross section caused by this interaction is governed by two prop- erties of the scattering atom: the so-called magnetic form factor f ( q ) and that component of its magnetic moment µ ⊥ which is perpendicular to the scattering vector. Defining the constant p = 2 m (11) � 2 γ µ B µ n one finds � d σ � = p 2 f 2 ( q ) µ 2 (12) ⊥ . dΩ 1M 2.4 Magnetic propagation vectors Before discussing the final expression for the magnetic neutron scattering from a crystal, it is useful to introduce the concept of magnetic propagation vectors. The magnetic moments of an atom at the position R α,j can be expanded in a Fourier series over all reciprocal lattice vectors: � (13) µ α,j = m j, k exp( − i k · R α ) . k Those reciprocal lattice vectors with a non-vanishing Fourier component m j, k are referred to as the magnetic propagation vectors of a material. In this sense, the magnetic propagation vectors characterize the periodicity of a magnetic structure. The determination of the complete magnetic structure requires knowledge of the corresponding Fourier components as well. 3
Furthermore, the magnetic propagation vectors allow for a compact formulation of the differential cross section for magnetic neutron scattering from a single crystal: � d σ = (2 π ) 3 � | F M , ⊥ ( q , k ) | 2 δ ( q − h − k ) � (14) N dΩ V 0 M h , k Comparing the relation above with the differential cross section for the nuclear scattering in Eq. 6, there is one apparent difference, namely the additional sum over the magnetic propagation vectors involving a modified Bragg condition. It states that magnetic reflections occur for (15) q = h + k . This modified Bragg condition implies the emergence of additional peaks for mag- netic propagation vectors k � = 0 . In the case of k = 0 , the magnetic periodicity equals the nuclear one so that the intensity of the lattice reflections will increase. Actually, both contributions simply superimpose each other: � d σ � d σ � d σ � � � (16) = + . dΩ dΩ dΩ N M Of course, the magnetic structure factor F M , ⊥ differs from the nuclear one. It contains the magnetic form factor of each atom in the unit cell and vanishes for distinguished scattering vectors. � (17) F M , ⊥ ( q , k ) = p f j ( q ) m j, k exp(i q · r j ) j The notation with a ⊥ symbol indicates that only the magnetic moments with a projection perpendicular to q affect the scattering process. 3 Experimental setup 3.1 Laue technique The Laue technique [Fri12] is a wide-spread method for structure resolution with both X-rays and neutrons. As already mentioned, only reflections with q = h can be observed for the diffraction from a single crystal. They are determined by Bragg’s law [Bra13] which reads (18) nλ = 2 d hkl sin θ 4
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