Charge ice

Metastable cubic water ice phase \(\mathrm{I_c}\) exhibits disorder of hydrogen protons such that only two sites are occupied and two sites are unoccupied on each tetrahedra. An analogous system with the pyrochlore lattice is constructed. 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 a hydrogen atom (\(\sigma=+1\)) or vacancy (\(\sigma=-1\)).

Example of pyrochlore lattice with occupational disorder

The occupancy-pair correlations of an \(8\times8\times8\) supercell are obtained from a forward Monte Carlo simulation. The first and second nearest neighbors are clearly negative.

Occupancy-pair correlations

The diffuse scattering intensity is calculated over a range of -6 to 6 in each \(h\)-, \(k\)-, and \(l\)-direction with a bin size of 0.04 in each dimension. Averaging is done over 20 independent forward Monte Carlo simulations to improve the statistics. A slice in the \((h,h,l)\)-plane is also calculated from -4 to 4 along \((h,h,0)\) and -6 to 6 along \((0,0,l)\).

Diffuse scattering intensity

RMC refinement

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

Crystal tab

Create a supercell for magnetic refinement.

• Click on Load CIF file, navigate to the tutorials/pyrochlore/ directory, and locate the pyrochlore.cif file.
• Create a supercell with size \(N_1=4\), \(N_2=4\), and \(N_3=4\).
• Change the atom to a \(\mathrm{Ho}\) atom.
• change occupancy to 0.5.

Crystal tab

Intensity tab

Preprocess intensity obtained from forward Monte Carlo.

• Download the pyrochlore-occupational.nxs file
• Click on Load NeXus file and locate the pyrochlore-occupational.nxs file.
• Under the Rebin tab, change the step size to 0.2 in each \(h\)-, \(k\)-, and \(l\)-direction.
• Under the Crop tab, change the \(h\)-, \(k\)-, and \(l\)-range from 0 to 6.

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 $h$, $k$ and $l$ to 1.
• Change Fraction to 0.2.
• Click Calculate.
• Change Linear scaling to Logarithmic.

Correlations tab

Recalculation tab

Recalculate the intensity over new axes.

• In the table, change $h$ and $k$ size to 201 and $l$ size to 301.
• Change $h$ and $k$ min to -4 and $l$ min to -6.
• Change $h$ and $k$ max to 4 and $l$ max to 6.
• Change $h$, $k$ and $l$ filter size to 3.0.
• Change the Axes to (hh0), (-kk0), (00l).
• Change the Laue symmetry to cif which is inferred from the loaded pyrochlore.cif file.
• Click Calculate.

Recalculation tab

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