Displacement ice

Analogous to spin ice, atomic displacements along the vertices of the tetrahedra of a pyrochlore lattice can also be frustrated. Using the forward Monte Carlo method, the Hamiltonian

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

is used to describe the nearest neighbor interactions where the spring constant is great than zero (\(K>0\)). The displacement vectors \(\pmb{u}\) of the \(\mathrm{Ho}\) atoms are restricted to allow atomic displacements either in or out of the vertex of each tetrahedra which corresponds to the \(\langle111\rangle\) family of directions.

Example of pyrochlore lattice with displacive disorder

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

Displacement-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 displacive 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 \(N_3=4\).

Crystal tab

Intensity tab

Preprocess intensity obtained from forward Monte Carlo.

• Download the pyrochlore-displacive.nxs file.
• Click on Load NeXus file and locate the pyrochlore-displacive.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+03 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 $h$, $k$ 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.