Refs:

Stars:

Hui, Yawei, Yaohua Liu, and Byung-Hoon Park. "Discovering Features in Sr $ _ {14} $ Cu $ _ {24} $ O $ _ {41} $ Neutron Single Crystal Diffraction Data by Cluster Analysis." arXiv preprint arXiv:1809.05039 (2018).

Hexagons:

Liu, Yaohua, Stuart Calder, Andrew May, and Yawei Hui. "Single crystal neutron diffuse scattering of layered ferromagnet Fe3-xGeTe2." Foundations of Crystallography 75 (2019): a229.

Octopuses:

Hui, Yawei, and Yaohua Liu. "Volumetric data exploration with machine learning-aided visualization in neutron science." In Science and Information Conference, pp. 257-271. Springer, Cham, 2019.

Simulated HDS from dilute isotropic point defects in a cubic crystal with anisotropic elasticity

Huang diffuse scattering describes the scattering near Bragg peaks caused by static displacements due to defects in crystals [1]. It was first explained by Eckstein (1945) and Huang (1947), and then further developed by many others, including Krivoglza, Dederichs, Trinkaus and Larson (1960s'-1970s') et. al.

Here shows a simple case of dilute point defects in a cubic crystal. In the dilute and non-interaction limits, each defect locally distorts the lattice and gives rise displacement field that can simply superimpose on each other. The effect of the difference in the scattering lengths between the defects and the host atoms is ignored. The resulted HDS can be calculated from the long-range displacement field due to the local defect in the continuum approximation, where the long-range displacement field reflects the symmetry of the local defect structure as well as the elastic properties of the host crystal. Therefore, we can obtain information on the elastic properties of the host crystal by studying the HDS pattern ( inelastic neutron scattering can be used to determine the phonon dispersion and is more straightforward for this purpose [2]) and determine the distortion field caused by the defect structure.

The HDS are simulated around three Bragg peaks: [1 0 0], [1 1 0] and [1 1 1], respectively, from the left to the right. The x-axes are parallel to the wave vector transfer. From the top to bottom, they are the sliced 2D maps of the scattering intensity, the 2D contour maps (log scale), and the line cuts along the high symmetry directions, respectively. There are several notable features in the simulations:

1. The overall features of the HDS patterns are different around the three Bragg peaks. In particular, HDS is no-longer symmetric w.r.t. the q-direction for the [1 1 1] peak. It reflects the anisotropic elasticity of the host crystal.

2. There are zero intensity planes going through the Bragg points. This is a simple consequence of the symmetry of HDS.

3. The intensity also shows the inversion symmetry w.r.t the Bragg peaks [3]: I(H+q) = I(H-q), where the H is the Bragg peak position, the q is the reduced wavevector transfer, q = K - H, and K is the wavevector transfer.

4. The characteristic q dependence (bottom row, the slope is -2 in the log-log plot): I (q) ~ 1/q^2 [4].

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[1]. The scattering near the Bragg peaks are not directly from the local displacement of the defect but from the long-range displacement field in response to the the local displacement. It can be understood based on the "WONDERLAND" rule. To directly probe the local defect structure (smaller r) we need to move to higher q (reduced wave vector transfer w.r.t the Bragg peaks). However, the information on the local displacement can still be available via the static displacement waves by coupling them to the long-range displacement.

[2] X-ray thermal diffuse scattering can also be used to determine phonon dispersion in crystal. T-C Chiang Lab @ UIUC has made significant advancements in early 2000's.

[3] The asymmetry term has been ignored in the calculation. The long-range displacement field gives rise to the symmetric term that is calculated here. The interference of the scattering from the long-range displacement field and the scattering of the defect itself will give rise to an asymmetric term, which is ignored here.

[4] It is related to [3]. The asymmetric term would has a 1/q dependence in the small q limit.

Ref

Theory of X-Ray and Thermal Neutron Scattering by Real Crystals, Chapter V, pp172-184, Krivoglaz, M. A. (1969), Springer.