## Eric Gibbs and Dr. Branton Campbell, Physics

### Introduction:

This project focuses on solving magnetic crystal structures using symmetry-mode analysis (SMA). A crystal is an arrangement of atoms in a repeating spatial pattern with the smallest repeating unit being known as the unit cell. A magnetic crystal contains atoms with a non-zero magnetic moment. A magnetic crystal’s unit cell repeats both the atomic and magnetic structure of the crystal. Neutron diffraction is a common method for studying magnetic crystals as the diffraction pattern can be used to determine the arrangement of atoms and magnetic moments within the unit cell.

Structure solution uses a diffraction pattern to determine how the unit cell is arranged. One major hurdle is that the unit cell is often complex and describing the positions of each unique atom requires more information than is often available in a diffraction pattern. To reduce the number of free parameters, symmetries within the crystal are used to tie otherwise independent parameters together. Along with the translational repetitions, crystals often contain other symmetries within the unit cell, such as mirror planes or rotation axes. Symmetry principles can be applied to magnetic structures as well, though one must be cautious as to how the magnetic moment, a pseudo-vector, transforms when a symmetry operator is applied.

### Methodology:

Symmetry-modes are a group-theory motivated basis that uses the symmetries of a crystal to describe a structure in terms of how it has distorted from a high-symmetry parent structure. All of the parameters available within the symmetry of the distorted structure can be expressed as symmetry-modes, including the completely general set of parameters that result from no symmetry other than lattice translations being assumed (P1 symmetry). The symmetry-mode description is generated using Harold Stokes’ ISODISTORT software. Though SMA does not reduce the number of parameters, it does reduce the number of non-zero, or active, parameters. Thus the problem of structure solution can be simplified into two parts: Finding the active modes in the structure, and then refining these modes.

SMA has been used with success to determine both nuclear crystal structures, and magnetic structures with distortions that affect a single point in k-space, or single-k distortions. Our proposal was to generate a general method to analyze all magnetic structures, including multi-k structures.

La0.5Ca0.5MnO3 (LCMO) was chosen to test the abilities of magnetic SMA due to its multi-k low-temperature magnetic structure. We collected neutron diffraction data for LCMO at the POWGEN beam line at the Spallation Neutron Source in Oak Ridge, Tennessee. TOPAS academic was used to refine our structural model against the collected data. Data was analyzed using two previously developed techniques. The histogram approach requires the refined values each symmetry-mode available to the structure from about a thousand convergences. A histogram is made for each mode reflecting the most probable value of the mode on convergence. Modes that are centered about zero tend to be inactive while modes that converge on non-zero values tend to be active in the final structure. This gives a rough starting point. Modes are then included one at a time and then excluded one at a time from the structure and their impact is measured. After several rounds of including and excluding modes, one reaches a handful of active modes. These modes are then refined to give the final structure.

### Results:

Using the modes available within the previously published nuclear symmetry we were able to determine a nuclear structure very similar to previous models with a few additional structural details previously unnoticed. Once we had a good handle on the nuclear structure we then removed all symmetry constraints in order to detect the magnetic structure of LCMO a priori. The experimentally determined distortion has 16 Mn atoms and thus 48 independent parameters. Histograms of each of the 48 parameters were used to get an original set of active symmetry modes. Inclusion and exclusion runs were then used to refine this set. Due to the pseudo-symmetries of LCMO, there were several possible structures that could not be distinguished by diffraction. Most structures led to significant drops in symmetry, and not wanting to add unjustified complexity the highest symmetry resulting in a “best-fit” was detected as the LCMO’s symmetry. Working in this symmetry there were once again multiple best-fit structures. However, most of these models could be eliminated as unphysical. One model seemed more physical than the rest and was determined to be the final structure of LCMO.

### Discussion and Conclusions:

Ideally symmetry is detected from data before being used to simplify the structure, however often for lack of alternative symmetry is assumed, biasing the structure solution process. Using SMA we were able to start with no symmetry assumptions and using histograms and inclusion/exclusion runs were able to detect the symmetry of the structure. Unfortunately this structure has many pseudo-symmetries, which results in several structures can give a near optimal fit to the diffraction data. SMA was however able to detect all of these potential structures. Physical reasoning led us to the final LCMO structure.

SMA greatly simplifies the process of structure solution. In the past, SMA has been used to determine nuclear structures and has been used in a limited fashion to determine magnetic structures, but this is the first body of work that has determined a multi-k magnetic structure a priori, opening a new frontier for structure solution by diffraction. We are in the process of publishing a precise description of our work for a major crystallographic journal.