Jana2020 is a new program for solving and refining regular, modulated and magnetic structures. The lecture will present new possibilities of this program in the field of magnetic structures.

In the last three years, a series of developments within the FullProf Suite [1], concerned with magnetic structures (both commensurate and incommensurate), have been performed. From the the first publication describing shortly the program FullProf [2] in 1993 many changes and re-writing of the code were done. In particular, the phase convention in the expression of the magnetic structure factor were changed. In 1993, we introduced for the first time the Simulated Annealing (SAnn) procedure for solving magnetic structures in a program called MagSan [2] that was later developed for incommensurate structures and embedded within FullProf. The method to make a symmetry analysis during many years was based in the Bertaut’s and Izyumov proposals [3-5] and we developed the program BasIreps to help the a priori construction of magnetic models to be refined. The use of magnetic space groups (MSG) was possible but, in the absence of appropriate tables or computing tools, the user had to construct the symmetry operators by hand. The treatment of incommensurate magnetic structures either was only possible by using basis vectors of irreducible representations or by constructing a series of 3D operators accompanied by a phase factor that was done by looking at the output of BasIreps and completing the information if the user was able to understand group theory.

The availability of new tools on the Web [6-8] and the creation of the Commission on Magnetic Structures of the IUCr has allowed the development of precise and unambiguous ways of describing magnetic structures using MSG and magnetic superspace groups (MSSG) [8, 9] by mean of magnetic CIF files. The team working in the FullProf Suite has accompanied these developments and created new tools to import these CIF files and convert them to input control files for FullProf. We have now the possibility of treating MSSG within FullProf with up to three independent modulation wave vectors, both for powders and single crystals, with automatic symmetry constraints determination for the amplitudes of modulations [10]. The displacement and thermal amplitudes are implemented but, for the moment, the calculation of structure factors using integration in internal coordinates is not yet available.

One important feature of FullProf is the use of SAnn, using the full powder diffraction pattern in which the components of the magnetic amplitudes are free parameters, in either crystallographic or spherical settings. This is very important for the powder case in which the loss of information may give rise to ambiguous or degenerate solutions. Moreover, the SAnn method may be used as an alternative to refinement because in such cases the least-squares refinement procedure diverges or it is unable to arrive to convergence.

In this talk, after summarizing the full set of changes performed during the last years (interoperability with the Bilbao Crystallographic Server [6, 7] and ISODISTORT [8], the use of magnetic Hall symbols [11], magnetic symmetry modes, etc.) I will present few recent examples of the use of FullProf in the magnetic structure determination of magneto-electric and multiferroic materials.

[1] https://www.ill.eu/sites/fullprof/

[2] Rodriguez-Carvajal J. (1993). Physica B. 192, 55.

[3] Bertaut E.F. (1968). Acta Cryst. A24, 217.

[4] Izyumov Yu. A., Naish V.E. and Ozerov R.P. (1991), Neutron Diffraction of Magnetic Materials, New York: Consultants Bureau.

[5] Rodriguez-Carvajal J. and Bourée F. (2012). EPJ Web of Conferences 22, 10, https://doi.org/10.1051/epjconf/20122200010.

[6] Aroyo M.I., Perez-Mato J.M., Capillas C., Kroumova E., Ivantchev S., Madariaga G., Kirov A. & Wondratschek H. (2006), Z. Krist. 221(1), 15.

[7] Aroyo M.I., Kirov A., Capillas C., Perez-Mato J.M. & Wondratschek H. (2006), Acta Cryst. A62, 115. http://www.cryst.ehu.es.

[8] Campbell B.J., Stokes H.T., Tanner D.E. & Hatch D.M. (2006), J.Appl.Cryst 39, 607. http://stokes.byu.edu/iso/isotropy.php

[9] Perez-Mato J.M., Ribeiro J. L., Petricek V. and Aroyo M. I. (2012), J. Phys.: Condens. Matter 24, 16320.

[10] Rodriguez-Carvajal J. and Villain J. (2019). C.R. Physique 20, 770, https://doi.org/10.1016/j.crhy.2019.07.004.

[11] González-Platas J., Katcho N.A. & Rodriguez-Carvajal J. (2021). J.Appl.Cryst 54, 338

Keywords: magnetic structures, simulated annealing, superspace groups

I thank my colleagues of the Diffraction Group at ILL and all the users of the FullProf Suite for the help in improving the programs.

MagStREXS is a crystallographic software dedicated to the analysis of resonant elastic X-ray scattering (REXS) diffraction data for the determination of magnetic structures that is under development at beamline P09 at PETRA III (DESY).

REXS is a powerful and element specific technique to study charge, spin, and orbital ordering in solids and thin films. Different types of data can be collected in a REXS experiment, although the analysis of these data is complex. The aim of MagStREXS is to facilitate this type of analysis to the non-specialist in this technique, and also to provide tools for the preparation of these experiments.

In this talk an overview of MagStREXS will be presented, together with some magnetic structures that have already been determined with it.

Lanthanide ions play a crucial role in various research fields. Much theoretical effort, that aims understanding and enhancing magnetic anisotropy in multiferroics and molecular magnetic materials, shows that the variation of magnetisation anisotropy is accompanied by important changes of 4f-electron, spin and orbital distributions. However, the experimental determination of the shape of these distributions is a non-trivial task especially in the case of unquenched orbital moment. Here, the procedure of magnetisation density reconstruction in lanthanides with unquenched orbital moment is developed, based on the iterative entropy maximization and the site susceptibility approach. The calculation were performed by recently developed code written as part of a crystallographic CrysPy library [1].

We illustrate the possibilities of the method by the first joint magnetisation density reconstruction and susceptibility refinement of locally anisotropic lanthanide pyrochlores RTi2O7 (R=Tb, Ho, Er and Yb) [2]. An oblate asphericity of Tb3+ density and prolate these of Ho3+ and Yb3+ was revealed (fig.1). Reconstructed distributions and refined susceptibility parameters are compared with these predicted by the crystal field theory in frame of single ion anisotropy model using McPhase software [3].

1. GitHub page of CrysPy library: https://ikibalin.github.io/cryspy/
2. H. Cao et al. Phys. Rev. Lett. (2009) 103, 056402
3. M. Rotter et al J. Phys.: Conf. Ser. (2011) 325 012005.

Magnetic x-ray standing waves (MXSW) - a combination of x-ray standing waves (XSW) [1] and x-ray magnetic circular dichroism (XMCD) - is a new method for direct investigation of magnetic structure of crystals and thin films on the atomic level. In the regular XSW technique a standing wave emerging in the region where incoming and Bragg reflected waves interfere is employed to study atomic positions in element specific manner. The standing wave has a periodicity of the lattice and moves by half of its period as the sample is rocked through the reflection domain. This movement across the lattice causes modulations in the amount of emitted fluorescence - their character is characteristic for a distribution of given atomic kind. This gives - in contrary to diffraction methods - a direct, element specific structural information. In MXSW, additional magnetic sensitivity is achieved by using circularly polarised incoming wave and magnetising the sample. The normalised difference between fluorescence yields recorded for each helicity/magnetic field orientation (XMCD signal) is proportional to the distribution of magnetic atoms and their magnetic moments. This makes MXSW site, element and magnetic sensitive method.

The theoretical framework of MXSW method is based on dynamical theory of x-ray diffraction and time-dependent perturbation theory. The first is used to describe the phenomena of the scattering of x-rays by the crystal lattice and yields a form of the wavefield inside the crystal for circularly polarised incident wave. The latter provides a tool to evaluate the absorption cross-section for the considered wavefield. What is obtained finally is an angular dependence of XMCD signal, which similarly as a single fluorescence yield in XSW method, exhibits variations dependent on the distribution of magnetic atoms.

The Fig. 1 shows an exemplary, simulated MXSW signal for the magnetite crystal, (004) reflection. The insets present schematically the magnetic structure and the positions of the standing wave at the low (marked by green colour) and high (blue) angular side of the reflection domain. Iron ions in the magnetite structure are arranged in two sublattices – octahedrally (marked by blue colour) and tetrahedrally (green) coordinated ones. Since the magnetic moment on each of two sublattices is different and oriented opposite, the contribution of the sublattices to the overall XMCD signal changes depending on the position of the nodes and antinodes of the standing wave. The variations are characteristic for this arrangement of the iron atoms and their shape would be different for any other one. Therefore, the MXSW signal directly provides information about the magnetic structure.

The first experiment aiming at proving the feasibility of the method and confirming the established theory was performed at PETRA III synchrotron on the single crystal sample of Pt₃Co alloy. The measurements were conducted at the Pt L₃ absorption edge. A clear variation in XMCD signal of the magnetic origin was observed. As a next step, an experiment on magnetite is planned to show the power of the method to probe the arrangement of the magnetic ions.

[1] Zegenhagen, J., Kazimirov, A., The X-Ray standing Wave Technique: Principles and Applications (2013).

The local minima of optimized non-linear functions are a reason for ambiguity of the obtained solutions in crystal structure analysis and magnetic structure analysis (Figure 1). In this study [1], semidefinite programming relaxation (SDR) was applied for the first time to the determination of magnetic structure. Use of SDR allows us the following things (i) completing the global optimization procedure in a very short time (less than several seconds in many cases), (ii) judging whether the obtained solution is truly global one, there are multiple good candidates, or the irreducible representations considered are wrong with 100 % probability. The solid foundation is provided by the duality theorem for convex optimization problems (Figure 2).

In general, the global optimization of SDR is applicable for estimating not only magnetic moment vectors but also atomic occupancies at a fixed set of coordinates x₁, …, x_m from the absolute values of structure factors. Therefore, the method can be also used to judge if x_i is an atomic site or a void.

In many cases, this problem has a unique minimum solution. In some cases, there are a few distinct solutions, all of which can be constructed from the output of SDR. In a very few cases, the existence of multiple solutions is suggested by the SDR result. This often occurs when the atomic coordinates x₁, …, x_m in the unit cell are periodic or almost periodic as in Figure 3. If the symmetry of the atomic coordinates is considered, the existence of such multiple solutions can be largely eliminated.

As a result, SDR can provide a numerical answer to the classical problem of the uniqueness of solutions in crystal structure analysis [2]. Global optimization of the SDR method is now being implemented into Z-Rietveld software [3] distributed for users of J-PARC (Japan Proton Accelerator Research Complex).