Honeycomb – displacive

The honeycomb lattice is another example that may be geometrically frustrated. Using the forward Monte Carlo method, the Hamiltonian

\[E=K\sum_{\langle i,j\rangle}\big(|\pmb{u}_i-\pmb{u}_j|-d_{ij}\big)^2\]

is used to describe the nearest neighbor interactions where the spring constant is great than zero (\(K>0\)) and equilibrium distance parameter is \(d_{ij}=-2\|\pmb{u}\|\). Displacement of each atom is fixed with magnitude \(u\) and are free to be oriented in any direction.

Example of honeycomb lattice with displacive disorder

The displacement-pair correlations of an \(12\times12\times12\) 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.

Displacement-pair correlations

The diffuse scattering intensity is calculated over a range of 4 to 4 in each \(h\)-, \(k\)-, and \(l\)-direction with a bin size of 0.025 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 displacive disorder.

Crystal tab

Create a supercell for nonmagnetic refinement.

• Click on Load CIF file, navigate to the tutorials/honeycomb/ directory, and locate the honeycomb.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 honeycomb-displacive.nxs file.
• Click on Load NeXus file and locate the honeycomb-displacive.nxs file.
• Under the Rebin tab, change the step size to 0.125 in each \(h\)-, \(k\)-, and \(l\)-direction.
• Under the Crop tab, change the \(h\)-, \(k\)-, and \(l\)-range from 0 to 4.

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 displacement-pair correlations.

• Under the Three-dimensional tab, change Fraction to 0.16.
• 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 honeycomb.cif file.
• Click Calculate.

Recalculation tab

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