Triangular – occupational

The primitive triangular lattice is perhaps the simplest example of a geometrically frustrated lattice. Using the forward Monte Carlo method, the Hamiltonian

\[E=-J\sum_{\langle i,j\rangle}\sigma_i\sigma_j\]

is used to describe the nearest neighbor interactions where the interaction strength is less than zero (\(J<0\)). The occupancy parameter \(\sigma\) indicates either the presence of an atom (\(\sigma=+1\)) or vacancy (\(\sigma=-1\)).

Example of triangular lattice with occupational disorder

The occupancy-pair correlations of an \(16\times16\times16\) supercell are obtained from a forward Monte Carlo simulation. The first nearest neighbors are clearly negatively correlated while the second are positively correlated. In the case of third nearest neighbors, the correlation is also negative.

Occupancy-pair correlations

The diffuse scattering intensity is calculated over a range of -2 to 2 in each \(h\)-, \(k\)-, and \(l\)-direction with a bin size of 0.02 in each dimension. Averaging is done over 20 independent forward Monte Carlo simulations to improve the statistics.

Diffuse scattering intensity

RMC refinement

Setup, run, and analyze a refinement with occupational disorder.

Crystal tab

Create a supercell for nonmagnetic refinement.

• Click on Load CIF file, navigate to the tutorials/triangular/ directory, and locate the triangular.cif file.
• Create a supercell with size \(N_1=8\), \(N_2=8\), and \(N_3=8\).

Crystal tab

Intensity tab

Preprocess intensity obtained from forward Monte Carlo.

• Download the triangular-occupational.nxs file.
• Click on Load NeXus file and locate the triangular-occupational.nxs file.
• Under the Rebin tab, uncheck the integer box and change the step size to 0.08 in each \(h\)-, \(k\)-, and \(l\)-direction.
• Under the Crop tab, change the \(h\)-, \(k\)-, and \(l\)-range from 0 to 2.

Intensity tab

Refinement tab

Perform the refinement.

• Change the number of RMC Cycles to 100.
• Change the filter size to 1.0 pixel in each \(h\)-, \(k\)-, and \(l\)-direction.
• Change the temperature prefactor to 1.00e+05 and decay constant to 1.00e-04.
• Click on run and save refinement file.

Refinement tab

Correlations tab

Calculate three-dimensional occupancy-pair correlations.

• Under the Three-dimensional tab, change Fraction to 0.22.
• Click Calculate.
• Change Linear scaling to Logarithmic.

Correlations tab

Recalculation tab

Recalculate the intensity over the initial reciprocal space volume.

• In the table, change $h$, $k$ and $l$ filter size to 4.0.
• Change the Laue symmetry to cif which is inferred from the loaded triangular.cif file.
• Click Calculate.

Recalculation tab

• Navigate to File->Save to save all results into the refinement file.