diff --git a/docs/_images/many_crystals_quartz.png b/docs/_images/many_crystals_quartz.png new file mode 100644 index 0000000..f68ae1a Binary files /dev/null and b/docs/_images/many_crystals_quartz.png differ diff --git a/docs/_images/single_crystal_quartz.png b/docs/_images/single_crystal_quartz.png new file mode 100644 index 0000000..23c3baf Binary files /dev/null and b/docs/_images/single_crystal_quartz.png differ diff --git a/docs/_images/testing_rocking_curves.png b/docs/_images/testing_rocking_curves.png new file mode 100644 index 0000000..5c5e53e Binary files /dev/null and b/docs/_images/testing_rocking_curves.png differ diff --git a/docs/_sources/gaussian_model_theory.rst.txt b/docs/_sources/gaussian_model_theory.rst.txt new file mode 100644 index 0000000..353fb63 --- /dev/null +++ b/docs/_sources/gaussian_model_theory.rst.txt @@ -0,0 +1,582 @@ +:orphan: + +.. _gaussian-model-theory: + +Gaussian model: theory and references +===================================== + +This page describes the scattering model and approximations underlying +:mod:`xrd_simulator.gaussian_crystal_model`. A worked example is provided in +the module documentation. + +.. contents:: On this page + :local: + :depth: 2 + +Gaussian subgrain representation +-------------------------------- + +Gaussian splatting represents a three-dimensional scene using Gaussian +primitives whose projected contributions can be rendered efficiently by +rasterization [Kerbl2023]_. Here, a polycrystal is represented by Gaussian +grains or subgrains, each with a Gaussian real-space density profile and a +narrow Gaussian distribution of lattice orientations. Under the approximations +below, each subgrain produces Gaussian-shaped diffraction spots on the +detector. Related Gaussian-basis methods for diffraction prediction and +partiality estimation have been demonstrated in serial crystallography +[Brehm2023]_. + +Several effects can contribute to diffraction-spot broadening: grain size, +detector point-spread, incident-beam bandwidth, incident-beam angular +divergence, lattice misorientation (mosaicity), and distributions of strain. +When these effects have Gaussian profiles and their deviations are small +enough to neglect nonlinear terms, they can be combined within a Gaussian +model. + +The model described here includes the real-space extent of the illuminated +subgrains and their mosaicity. The incident beam is monochromatic and +collimated, and each subgrain has a uniform strain tensor. A uniform strain +can shift a reflection but does not describe a distribution of lattice +spacings within that subgrain. Small sample rotations during an exposure are +approximated by additional Gaussian orientation broadening. + +.. note:: + + The Gaussian-splat rendering path does not apply a separate detector + point-spread convolution. The ``gaussian_sigma`` and + ``max_gaussian_kernel_radius`` detector settings retained in the worked + example therefore do not add this broadening to its Gaussian spots. + +Scattering theory +----------------- + +Scattered-beam distribution +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Let the incident beam have a phase-space density :math:`p(\mathbf{k})` +centred on :math:`\mathbf{k}_0`, with nominal wavenumber + +.. math:: + :label: gaussian-wavenumber + + k = \lvert\mathbf{k}_0\rvert = \frac{2\pi}{\lambda_0}. + +For a particular reflection, let :math:`f(\mathbf{q})` describe the subgrain's +reciprocal-space map (RSM) around a reciprocal-lattice vector +:math:`\mathbf{G}_0`. The scattered-beam distribution is written as + +.. math:: + :label: gaussian-scattering-integral + + I(\mathbf{p}) + = \int \mathrm{d}^{3}\mathbf{k} + \int \mathrm{d}^{3}\mathbf{q}\; + p(\mathbf{k})\,f(\mathbf{q})\, + \delta\!\left(\lvert\mathbf{k}\rvert-\lvert\mathbf{p}\rvert\right) + \delta^{(3)}\!\left(\mathbf{k}+\mathbf{q}-\mathbf{p}\right). + +The scalar Dirac delta enforces elastic scattering, while the +three-dimensional Dirac delta enforces momentum conservation. + +Incident-beam coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~ + +Following the coordinate construction used by Poulsen et al. +[Poulsen2017]_, write the incident wavevector as + +.. math:: + :label: gaussian-incident-wavevector + + \mathbf{k} + = \mathbf{k}_0 + + k\left( + \varepsilon\,\hat{\mathbf{k}}_0 + + \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + + \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp} + \right). + +A hat denotes a unit vector. The vector +:math:`\hat{\mathbf{k}}_{\parallel}` is perpendicular to +:math:`\mathbf{k}_0` and lies in the plane spanned by :math:`\mathbf{k}_0` +and :math:`\mathbf{G}_0`. Its sign is chosen so that +:math:`\mathbf{G}_0\cdot\hat{\mathbf{k}}_{\parallel}>0`. Complete the +right-handed basis with + +.. math:: + + \hat{\mathbf{k}}_{\perp} + = \hat{\mathbf{k}}_0\times\hat{\mathbf{k}}_{\parallel}. + +The coordinate :math:`\varepsilon` describes the relative wavenumber +deviation, and :math:`\zeta_{\parallel}` and :math:`\zeta_{\perp}` describe +small angular deviations of the incident beam. + +Reciprocal-space coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Define the nominal Bragg angle and the exactly aligned scattering vector by + +.. math:: + :label: gaussian-nominal-scattering-vector + + \begin{aligned} + \theta_0 + &= \arcsin\!\left(\frac{\lvert\mathbf{G}_0\rvert}{2k}\right), \\ + \mathbf{Q} + &= 2k\sin\theta_0\left( + \cos\theta_0\,\hat{\mathbf{k}}_{\parallel} + - \sin\theta_0\,\hat{\mathbf{k}}_0 + \right) \\ + &= 2k\sin\theta_0\,\hat{\mathbf{Q}}. + \end{aligned} + +The vectors :math:`\mathbf{G}_0` and :math:`\mathbf{Q}` coincide when the +reflection is exactly aligned. For a small misalignment, their difference is +parallel, to first order, to the rocking direction + +.. math:: + + \hat{\mathbf{q}}_{\mathrm{rock}} + = \cos\theta_0\,\hat{\mathbf{k}}_0 + + \sin\theta_0\,\hat{\mathbf{k}}_{\parallel}. + +Use dimensionless rocking, strain, and rolling coordinates to write + +.. math:: + :label: gaussian-reciprocal-coordinates + + \begin{aligned} + \mathbf{q} + &= \mathbf{G}_0 + + 2k\sin\theta_0\left( + q_{\mathrm{rock}}\,\hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right) \\ + &\approx \mathbf{Q} + + 2k\sin\theta_0\left( + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right), + \end{aligned} + +where + +.. math:: + :label: gaussian-rocking-offset + + \delta q + = \frac{ + (\mathbf{Q}-\mathbf{G}_0)\cdot\hat{\mathbf{q}}_{\mathrm{rock}} + }{2k\sin\theta_0}. + +This offset measures the reflection's misalignment and is equal to the +rocking angle to first order, with the sign convention above. The incident +coordinates :math:`(\varepsilon,\zeta_{\parallel},\zeta_{\perp})` and the +reciprocal-space coordinates +:math:`(q_{\mathrm{rock}},q_{\mathrm{strain}},q_{\mathrm{roll}})` separate +bandwidth, collimation, misorientation, and strain effects. + +Outgoing-beam coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~ + +The nominal scattered wavevector is + +.. math:: + :label: gaussian-nominal-outgoing-wavevector + + \mathbf{k}_h + = \mathbf{k}_0+\mathbf{Q} + = k\left( + \cos(2\theta_0)\,\hat{\mathbf{k}}_0 + + \sin(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + \right). + +Define the radial direction perpendicular to :math:`\mathbf{k}_h` by + +.. math:: + + \hat{\mathbf{k}}_{\mathrm{rad}} + = \cos(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + - \sin(2\theta_0)\,\hat{\mathbf{k}}_0. + +The outgoing wavevector can then be written as + +.. math:: + :label: gaussian-outgoing-wavevector + + \mathbf{p} + = \mathbf{k}_h + + k\left( + \varepsilon'\,\hat{\mathbf{k}}_h + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + \right). + +Here :math:`\psi_{\mathrm{rad}}` and :math:`\psi_{\mathrm{azim}}` are small +transverse angular deviations from the nominal outgoing direction. + +Linearized conservation equations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To first order, + +.. math:: + + \lvert\mathbf{k}\rvert\approx k(1+\varepsilon), + \qquad + \lvert\mathbf{p}\rvert\approx k(1+\varepsilon'). + +Energy conservation therefore sets :math:`\varepsilon'=\varepsilon`. +Momentum conservation gives :math:`\mathbf{q}=\mathbf{p}-\mathbf{k}` and, +after substitution, + +.. math:: + :label: gaussian-linearized-momentum + + \begin{aligned} + &2\sin\theta_0\Bigl[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \Bigr] \\ + &\qquad = 2\sin\theta_0\,\varepsilon\,\hat{\mathbf{Q}} + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} \\ + &\qquad\quad + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + - \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + - \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp}. + \end{aligned} + +.. note:: + + The incident-divergence terms have minus signs because the incident + wavevector is subtracted in :math:`\mathbf{q}=\mathbf{p}-\mathbf{k}`. + This corrects the corresponding signs in the original workflow. It does + not affect the monochromatic, collimated specialization below, where + both terms vanish. + +Monochromatic, collimated specialization +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The current model assumes + +.. math:: + + \varepsilon=\zeta_{\parallel}=\zeta_{\perp}=0, + \qquad q_{\mathrm{strain}}=0. + +The last condition excludes strain-distribution broadening; it does not +require the uniform subgrain strain to vanish. The conservation equation +reduces to + +.. math:: + + \begin{aligned} + &2\sin\theta_0\left[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right] \\ + &\qquad + = \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}. + \end{aligned} + +For a nondegenerate Bragg geometry, the three coordinate equations are + +.. math:: + :label: gaussian-scattering-constraints + + q_{\mathrm{rock}}=\delta q, + \qquad + \psi_{\mathrm{rad}}=0, + \qquad + \psi_{\mathrm{azim}}=2\sin\theta_0\,q_{\mathrm{roll}}. + +Thus, the scattered angular distribution samples the RSM along a line +parallel to :math:`\hat{\mathbf{k}}_{\perp}`, offset by :math:`\delta q` +from its centre. For a Gaussian RSM, this gives a one-dimensional Gaussian +angular distribution. The finite real-space extent of the illuminated +subgrain is subsequently projected onto the detector to produce a +two-dimensional spot. + +Anisotropic Gaussian orientation distributions +---------------------------------------------- + +Tangent-space approximation +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Consider a narrow orientation distribution centred on an orientation +:math:`g_0`. Approximate orientation space by its left tangent space at +:math:`g_0`, so that a small rotation vector :math:`\mathbf{r}` acts in +laboratory coordinates: + +.. math:: + :label: gaussian-tangent-rotation + + g\approx R_{\mathbf{r}}g_0 + \approx + \begin{bmatrix} + 1 & -r_z & r_y \\ + r_z & 1 & -r_x \\ + -r_y & r_x & 1 + \end{bmatrix}g_0. + +The Gaussian orientation density used in the workflow is + +.. math:: + :label: gaussian-orientation-density + + f(\mathbf{r}) + = \frac{2}{\sqrt{\pi}}\sqrt{\det\mathbf{T}}\, + \exp\!\left(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r}\right), + +where :math:`\mathbf{T}` is a symmetric positive-definite concentration +tensor. + +.. note:: + + The prefactor above retains the workflow's normalization convention. + With ordinary Euclidean tangent-space measure, + :math:`\int f(\mathbf{r})\,\mathrm{d}^{3}\mathbf{r}=2\pi`, not one. + It should therefore not be interpreted as a unit-normalized Euclidean + probability density without rescaling. + + For an exponential of the form + :math:`\exp(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r})`, the + corresponding normalized Gaussian has covariance + :math:`\tfrac{1}{2}\mathbf{T}^{-1}`. The implementation directly inverts + ``misorientation_tensor`` to obtain :math:`\mathbf{T}`. Consequently, + the square roots of the input tensor's eigenvalues are Gaussian width + parameters, not standard deviations. For an input eigenvalue + :math:`a^2`, the standard deviation is :math:`a/\sqrt{2}`. + The real-space Gaussian uses the same exponential convention for + ``shape_tensor``. This distinction matters because the class attribute + descriptions currently use the word "covariance" for these inputs. + +Pole density as a line integral +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For Miller indices :math:`(h,k,\ell)`, define the unit lattice direction in +crystal coordinates by + +.. math:: + + \mathbf{h} + = \frac{ + \mathbf{B}_0[h,k,\ell]^{\mathsf{T}} + }{ + \left\lvert\mathbf{B}_0[h,k,\ell]^{\mathsf{T}}\right\rvert + }, + \qquad + \mathbf{p}=g_0\mathbf{h}. + +Here :math:`\mathbf{B}_0` is the reciprocal-lattice basis matrix. In this +section, :math:`\mathbf{p}` denotes the mean lattice direction, rather than +the outgoing wavevector used in the scattering section. It is parallel to +:math:`\mathbf{G}_0`. Let :math:`\mathbf{y}` be a unit direction in +laboratory coordinates, parallel to a sampled reciprocal-space vector +:math:`\mathbf{q}`; it is not a Cartesian coordinate-axis label. + +The pole density, also called the pair-correlation function in texture +analysis, describes the density of lattice directions along +:math:`\mathbf{y}`. Its calculation normally involves an integral over a +circle in :math:`\mathrm{SO}(3)`. Within the tangent-space approximation, +replace this by an infinite line integral, parameterized as + +.. math:: + :label: gaussian-orientation-fibre + + \mathbf{r}(\lambda) + = \frac{\mathbf{p}\times\mathbf{y}}{\mathbf{p}\cdot\mathbf{y}} + + \lambda\mathbf{p} + = \mathbf{r}_0+\lambda\mathbf{p}. + +The parameter :math:`\lambda` here is a line-integration parameter, not the +X-ray wavelength. Substitution into the Gaussian density and completion of +the square give, with +:math:`d=\mathbf{p}^{\mathsf{T}}\mathbf{T}\mathbf{p}`, + +.. math:: + :label: gaussian-pole-density-integral + + \begin{aligned} + A(\mathbf{y},\mathbf{p};f) + &= \int_{-\infty}^{\infty} + f\!\left(\mathbf{r}_0+\lambda\mathbf{p}\right) + \,\mathrm{d}\lambda \\ + &= \frac{2\sqrt{\det\mathbf{T}}} + {\sqrt{d}} + \exp\!\left[ + -\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{r}_0 + + \frac{ + (\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{p})^2 + }{d} + \right]. + \end{aligned} + +Projected concentration tensor +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For directions close to the pole, make the further approximation +:math:`\mathbf{p}\cdot\mathbf{y}\approx1`. Define + +.. math:: + + \mathbf{M} + = \mathbf{T} + - \frac{ + \mathbf{T}\mathbf{p}\mathbf{p}^{\mathsf{T}}\mathbf{T} + }{d}. + +Let :math:`[\mathbf{p}]_{\times}` denote the cross-product matrix, +:math:`[\mathbf{p}]_{\times}\mathbf{y}=\mathbf{p}\times\mathbf{y}`. Then + +.. math:: + :label: gaussian-projected-concentration + + \mathbf{T}_{\mathbf{p}} + = [\mathbf{p}]_{\times}^{\mathsf{T}} + \mathbf{M} + [\mathbf{p}]_{\times}, + +and the pole density becomes + +.. math:: + :label: gaussian-pole-density + + A(\mathbf{y},\mathbf{p};f) + \approx \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left( + -\mathbf{y}^{\mathsf{T}}\mathbf{T}_{\mathbf{p}}\mathbf{y} + \right). + +Equivalently, using the Levi-Civita symbol :math:`\epsilon_{ijk}` and +summation over repeated indices, + +.. math:: + + [\mathbf{T}_{\mathbf{p}}]_{ij} + = p_k\,\epsilon_{lki} + \left( + T_{lm}-\frac{T_{la}p_a p_b T_{bm}}{d} + \right) + \epsilon_{mnj}\,p_n. + +This absorbs the cross products into the projected tensor and makes the +subsequent Gaussian expressions more compact. + +Reciprocal-space map +~~~~~~~~~~~~~~~~~~~~ + +The pole density is defined for unit-vector arguments. With no +strain-distribution broadening, extend it to a reciprocal-space map supported +on the local plane :math:`q_{\mathrm{strain}}=0`. + +Introduce the two-column basis matrix and coordinate vector + +.. math:: + + \mathbf{U} + = \begin{bmatrix} + \hat{\mathbf{q}}_{\mathrm{rock}} & \hat{\mathbf{k}}_{\perp} + \end{bmatrix}, + \qquad + \mathbf{u} + = \begin{bmatrix} + q_{\mathrm{rock}} \\ + q_{\mathrm{roll}} + \end{bmatrix}. + +In the dimensionless reciprocal-space coordinates defined above, the +workflow's local RSM is + +.. math:: + :label: gaussian-local-rsm + + f(\mathbf{q}) + \approx \delta(q_{\mathrm{strain}})\, + \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left[ + -\mathbf{u}^{\mathsf{T}} + \left(\mathbf{U}^{\mathsf{T}} + \mathbf{T}_{\mathbf{p}}\mathbf{U}\right) + \mathbf{u} + \right]. + +The Dirac delta expresses the absence of a distribution of lattice spacings +within the subgrain. Sampling this Gaussian at +:math:`q_{\mathrm{rock}}=\delta q`, as required by +:eq:`gaussian-scattering-constraints`, gives the scattered angular +distribution for the reflection. + +Illustrative tests +------------------ + +Comparison with a tetrahedral crystal +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The workflow compares a one-degree rotation of a single quartz crystal +shaped as a symmetric tetrahedron with a Gaussian approximation of the same +crystal. The crystal is much larger than the detector pixels, and the largest +scattering angles exceed 60 degrees, making radial spot elongation from the +projection geometry visible. + +Seven Gaussians approximate the tetrahedron: one isotropic Gaussian at its +centre and six prolate Gaussians along its edges. The Gaussian model uses a +small orientation spread. + +The comparison shows similar peak positions and shapes. The Gaussian model +also produces additional weak peaks because its rotational integration uses +a Gaussian time window, whereas the tetrahedral model uses a top-hat time +window. + +.. figure:: ../_static/single_crystal_quartz.png + :alt: Quartz diffraction patterns from tetrahedral and Gaussian models. + :align: center + :width: 100% + + Comparison of the tetrahedral crystal model with a seven-Gaussian + approximation during a one-degree sample rotation. + +High-resolution rocking curve +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A second test resolves a rocking curve and checks the shape of the resulting +three-dimensional reciprocal-space map. The workflow figure compares the +simulated profiles with the expected contributions from grain size and +misorientation. + +.. figure:: ../_static/testing_rocking_curves.png + :alt: Reciprocal-space map and profiles from a high-resolution rocking scan. + :align: center + :width: 100% + + High-resolution rocking-curve test showing the reciprocal-space map and + the contributions of grain size and misorientation to the simulated + profiles. + +References +---------- + +.. [Kerbl2023] Kerbl, B., Kopanas, G., Leimkühler, T. and Drettakis, G. + (2023). 3D Gaussian Splatting for Real-Time Radiance Field Rendering. + *ACM Transactions on Graphics*, **42**\ (4). + `doi:10.1145/3592433 `_. + +.. [Brehm2023] Brehm, W., White, T. and Chapman, H. N. (2023). + Crystal diffraction prediction and partiality estimation using Gaussian + basis functions. *Acta Crystallographica Section A: Foundations and + Advances*, **79**, 145–162. + `doi:10.1107/S2053273323000682 + `_. + +.. [Poulsen2017] Poulsen, H. F., Jakobsen, A. C., Simons, H., Ahl, S. R., + Cook, P. K. and Detlefs, C. (2017). X-ray diffraction microscopy based on + refractive optics. *Journal of Applied Crystallography*, **50**, 1441–1456. + `doi:10.1107/S1600576717011037 + `_. diff --git a/docs/_sources/index.rst.txt b/docs/_sources/index.rst.txt index 29d8fe8..61bc367 100644 --- a/docs/_sources/index.rst.txt +++ b/docs/_sources/index.rst.txt @@ -1,7 +1,3 @@ -.. toctree:: - :maxdepth: 2 - :caption: Contents: - .. include:: ../../README.rst ====================================== @@ -50,16 +46,16 @@ motion :members: :inherited-members: -templates +gaussian_crystal_model ====================================== -.. automodule:: xrd_simulator.templates - :members: - :inherited-members: +.. automodule:: xrd_simulator.gaussian_crystal_model + :members: + :inherited-members: -laue +templates ====================================== -.. automodule:: xrd_simulator.laue +.. automodule:: xrd_simulator.templates :members: :inherited-members: diff --git a/docs/gaussian_model_theory.html b/docs/gaussian_model_theory.html new file mode 100644 index 0000000..6e2c069 --- /dev/null +++ b/docs/gaussian_model_theory.html @@ -0,0 +1,626 @@ + + + + + + + + + Gaussian model: theory and references — xrd_simulator 0.0.0 documentation + + + + + + + + + + + + + + + + + + + +
+ + +
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+ +
+

Gaussian model: theory and references

+

This page describes the scattering model and approximations underlying +xrd_simulator.gaussian_crystal_model. A worked example is provided in +the module documentation.

+ +
+

Gaussian subgrain representation

+

Gaussian splatting represents a three-dimensional scene using Gaussian +primitives whose projected contributions can be rendered efficiently by +rasterization [Kerbl2023]. Here, a polycrystal is represented by Gaussian +grains or subgrains, each with a Gaussian real-space density profile and a +narrow Gaussian distribution of lattice orientations. Under the approximations +below, each subgrain produces Gaussian-shaped diffraction spots on the +detector. Related Gaussian-basis methods for diffraction prediction and +partiality estimation have been demonstrated in serial crystallography +[Brehm2023].

+

Several effects can contribute to diffraction-spot broadening: grain size, +detector point-spread, incident-beam bandwidth, incident-beam angular +divergence, lattice misorientation (mosaicity), and distributions of strain. +When these effects have Gaussian profiles and their deviations are small +enough to neglect nonlinear terms, they can be combined within a Gaussian +model.

+

The model described here includes the real-space extent of the illuminated +subgrains and their mosaicity. The incident beam is monochromatic and +collimated, and each subgrain has a uniform strain tensor. A uniform strain +can shift a reflection but does not describe a distribution of lattice +spacings within that subgrain. Small sample rotations during an exposure are +approximated by additional Gaussian orientation broadening.

+
+

Note

+

The Gaussian-splat rendering path does not apply a separate detector +point-spread convolution. The gaussian_sigma and +max_gaussian_kernel_radius detector settings retained in the worked +example therefore do not add this broadening to its Gaussian spots.

+
+
+
+

Scattering theory

+
+

Scattered-beam distribution

+

Let the incident beam have a phase-space density \(p(\mathbf{k})\) +centred on \(\mathbf{k}_0\), with nominal wavenumber

+
+(1)\[k = \lvert\mathbf{k}_0\rvert = \frac{2\pi}{\lambda_0}.\]
+

For a particular reflection, let \(f(\mathbf{q})\) describe the subgrain’s +reciprocal-space map (RSM) around a reciprocal-lattice vector +\(\mathbf{G}_0\). The scattered-beam distribution is written as

+
+(2)\[I(\mathbf{p}) += \int \mathrm{d}^{3}\mathbf{k} + \int \mathrm{d}^{3}\mathbf{q}\; + p(\mathbf{k})\,f(\mathbf{q})\, + \delta\!\left(\lvert\mathbf{k}\rvert-\lvert\mathbf{p}\rvert\right) + \delta^{(3)}\!\left(\mathbf{k}+\mathbf{q}-\mathbf{p}\right).\]
+

The scalar Dirac delta enforces elastic scattering, while the +three-dimensional Dirac delta enforces momentum conservation.

+
+
+

Incident-beam coordinates

+

Following the coordinate construction used by Poulsen et al. +[Poulsen2017], write the incident wavevector as

+
+(3)\[\mathbf{k} += \mathbf{k}_0 + + k\left( + \varepsilon\,\hat{\mathbf{k}}_0 + + \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + + \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp} + \right).\]
+

A hat denotes a unit vector. The vector +\(\hat{\mathbf{k}}_{\parallel}\) is perpendicular to +\(\mathbf{k}_0\) and lies in the plane spanned by \(\mathbf{k}_0\) +and \(\mathbf{G}_0\). Its sign is chosen so that +\(\mathbf{G}_0\cdot\hat{\mathbf{k}}_{\parallel}>0\). Complete the +right-handed basis with

+
+\[\hat{\mathbf{k}}_{\perp} += \hat{\mathbf{k}}_0\times\hat{\mathbf{k}}_{\parallel}.\]
+

The coordinate \(\varepsilon\) describes the relative wavenumber +deviation, and \(\zeta_{\parallel}\) and \(\zeta_{\perp}\) describe +small angular deviations of the incident beam.

+
+
+

Reciprocal-space coordinates

+

Define the nominal Bragg angle and the exactly aligned scattering vector by

+
+(4)\[\begin{split}\begin{aligned} +\theta_0 +&= \arcsin\!\left(\frac{\lvert\mathbf{G}_0\rvert}{2k}\right), \\ +\mathbf{Q} +&= 2k\sin\theta_0\left( + \cos\theta_0\,\hat{\mathbf{k}}_{\parallel} + - \sin\theta_0\,\hat{\mathbf{k}}_0 + \right) \\ +&= 2k\sin\theta_0\,\hat{\mathbf{Q}}. +\end{aligned}\end{split}\]
+

The vectors \(\mathbf{G}_0\) and \(\mathbf{Q}\) coincide when the +reflection is exactly aligned. For a small misalignment, their difference is +parallel, to first order, to the rocking direction

+
+\[\hat{\mathbf{q}}_{\mathrm{rock}} += \cos\theta_0\,\hat{\mathbf{k}}_0 + + \sin\theta_0\,\hat{\mathbf{k}}_{\parallel}.\]
+

Use dimensionless rocking, strain, and rolling coordinates to write

+
+(5)\[\begin{split}\begin{aligned} +\mathbf{q} +&= \mathbf{G}_0 + + 2k\sin\theta_0\left( + q_{\mathrm{rock}}\,\hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right) \\ +&\approx \mathbf{Q} + + 2k\sin\theta_0\left( + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right), +\end{aligned}\end{split}\]
+

where

+
+(6)\[\delta q += \frac{ + (\mathbf{Q}-\mathbf{G}_0)\cdot\hat{\mathbf{q}}_{\mathrm{rock}} + }{2k\sin\theta_0}.\]
+

This offset measures the reflection’s misalignment and is equal to the +rocking angle to first order, with the sign convention above. The incident +coordinates \((\varepsilon,\zeta_{\parallel},\zeta_{\perp})\) and the +reciprocal-space coordinates +\((q_{\mathrm{rock}},q_{\mathrm{strain}},q_{\mathrm{roll}})\) separate +bandwidth, collimation, misorientation, and strain effects.

+
+
+

Outgoing-beam coordinates

+

The nominal scattered wavevector is

+
+(7)\[\mathbf{k}_h += \mathbf{k}_0+\mathbf{Q} += k\left( + \cos(2\theta_0)\,\hat{\mathbf{k}}_0 + + \sin(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + \right).\]
+

Define the radial direction perpendicular to \(\mathbf{k}_h\) by

+
+\[\hat{\mathbf{k}}_{\mathrm{rad}} += \cos(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + - \sin(2\theta_0)\,\hat{\mathbf{k}}_0.\]
+

The outgoing wavevector can then be written as

+
+(8)\[\mathbf{p} += \mathbf{k}_h + + k\left( + \varepsilon'\,\hat{\mathbf{k}}_h + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + \right).\]
+

Here \(\psi_{\mathrm{rad}}\) and \(\psi_{\mathrm{azim}}\) are small +transverse angular deviations from the nominal outgoing direction.

+
+
+

Linearized conservation equations

+

To first order,

+
+\[\lvert\mathbf{k}\rvert\approx k(1+\varepsilon), +\qquad +\lvert\mathbf{p}\rvert\approx k(1+\varepsilon').\]
+

Energy conservation therefore sets \(\varepsilon'=\varepsilon\). +Momentum conservation gives \(\mathbf{q}=\mathbf{p}-\mathbf{k}\) and, +after substitution,

+
+(9)\[\begin{split}\begin{aligned} +&2\sin\theta_0\Bigl[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} +\Bigr] \\ +&\qquad = 2\sin\theta_0\,\varepsilon\,\hat{\mathbf{Q}} + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} \\ +&\qquad\quad + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + - \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + - \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp}. +\end{aligned}\end{split}\]
+
+

Note

+

The incident-divergence terms have minus signs because the incident +wavevector is subtracted in \(\mathbf{q}=\mathbf{p}-\mathbf{k}\). +This corrects the corresponding signs in the original workflow. It does +not affect the monochromatic, collimated specialization below, where +both terms vanish.

+
+
+
+

Monochromatic, collimated specialization

+

The current model assumes

+
+\[\varepsilon=\zeta_{\parallel}=\zeta_{\perp}=0, +\qquad q_{\mathrm{strain}}=0.\]
+

The last condition excludes strain-distribution broadening; it does not +require the uniform subgrain strain to vanish. The conservation equation +reduces to

+
+\[\begin{split}\begin{aligned} +&2\sin\theta_0\left[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} +\right] \\ +&\qquad += \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}. +\end{aligned}\end{split}\]
+

For a nondegenerate Bragg geometry, the three coordinate equations are

+
+(10)\[q_{\mathrm{rock}}=\delta q, +\qquad +\psi_{\mathrm{rad}}=0, +\qquad +\psi_{\mathrm{azim}}=2\sin\theta_0\,q_{\mathrm{roll}}.\]
+

Thus, the scattered angular distribution samples the RSM along a line +parallel to \(\hat{\mathbf{k}}_{\perp}\), offset by \(\delta q\) +from its centre. For a Gaussian RSM, this gives a one-dimensional Gaussian +angular distribution. The finite real-space extent of the illuminated +subgrain is subsequently projected onto the detector to produce a +two-dimensional spot.

+
+
+
+

Anisotropic Gaussian orientation distributions

+
+

Tangent-space approximation

+

Consider a narrow orientation distribution centred on an orientation +\(g_0\). Approximate orientation space by its left tangent space at +\(g_0\), so that a small rotation vector \(\mathbf{r}\) acts in +laboratory coordinates:

+
+(11)\[\begin{split}g\approx R_{\mathbf{r}}g_0 +\approx +\begin{bmatrix} + 1 & -r_z & r_y \\ + r_z & 1 & -r_x \\ + -r_y & r_x & 1 +\end{bmatrix}g_0.\end{split}\]
+

The Gaussian orientation density used in the workflow is

+
+(12)\[f(\mathbf{r}) += \frac{2}{\sqrt{\pi}}\sqrt{\det\mathbf{T}}\, + \exp\!\left(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r}\right),\]
+

where \(\mathbf{T}\) is a symmetric positive-definite concentration +tensor.

+
+

Note

+

The prefactor above retains the workflow’s normalization convention. +With ordinary Euclidean tangent-space measure, +\(\int f(\mathbf{r})\,\mathrm{d}^{3}\mathbf{r}=2\pi\), not one. +It should therefore not be interpreted as a unit-normalized Euclidean +probability density without rescaling.

+

For an exponential of the form +\(\exp(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r})\), the +corresponding normalized Gaussian has covariance +\(\tfrac{1}{2}\mathbf{T}^{-1}\). The implementation directly inverts +misorientation_tensor to obtain \(\mathbf{T}\). Consequently, +the square roots of the input tensor’s eigenvalues are Gaussian width +parameters, not standard deviations. For an input eigenvalue +\(a^2\), the standard deviation is \(a/\sqrt{2}\). +The real-space Gaussian uses the same exponential convention for +shape_tensor. This distinction matters because the class attribute +descriptions currently use the word “covariance” for these inputs.

+
+
+
+

Pole density as a line integral

+

For Miller indices \((h,k,\ell)\), define the unit lattice direction in +crystal coordinates by

+
+\[\mathbf{h} += \frac{ + \mathbf{B}_0[h,k,\ell]^{\mathsf{T}} + }{ + \left\lvert\mathbf{B}_0[h,k,\ell]^{\mathsf{T}}\right\rvert + }, +\qquad +\mathbf{p}=g_0\mathbf{h}.\]
+

Here \(\mathbf{B}_0\) is the reciprocal-lattice basis matrix. In this +section, \(\mathbf{p}\) denotes the mean lattice direction, rather than +the outgoing wavevector used in the scattering section. It is parallel to +\(\mathbf{G}_0\). Let \(\mathbf{y}\) be a unit direction in +laboratory coordinates, parallel to a sampled reciprocal-space vector +\(\mathbf{q}\); it is not a Cartesian coordinate-axis label.

+

The pole density, also called the pair-correlation function in texture +analysis, describes the density of lattice directions along +\(\mathbf{y}\). Its calculation normally involves an integral over a +circle in \(\mathrm{SO}(3)\). Within the tangent-space approximation, +replace this by an infinite line integral, parameterized as

+
+(13)\[\mathbf{r}(\lambda) += \frac{\mathbf{p}\times\mathbf{y}}{\mathbf{p}\cdot\mathbf{y}} + + \lambda\mathbf{p} += \mathbf{r}_0+\lambda\mathbf{p}.\]
+

The parameter \(\lambda\) here is a line-integration parameter, not the +X-ray wavelength. Substitution into the Gaussian density and completion of +the square give, with +\(d=\mathbf{p}^{\mathsf{T}}\mathbf{T}\mathbf{p}\),

+
+(14)\[\begin{split}\begin{aligned} +A(\mathbf{y},\mathbf{p};f) +&= \int_{-\infty}^{\infty} + f\!\left(\mathbf{r}_0+\lambda\mathbf{p}\right) + \,\mathrm{d}\lambda \\ +&= \frac{2\sqrt{\det\mathbf{T}}} + {\sqrt{d}} + \exp\!\left[ + -\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{r}_0 + + \frac{ + (\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{p})^2 + }{d} + \right]. +\end{aligned}\end{split}\]
+
+
+

Projected concentration tensor

+

For directions close to the pole, make the further approximation +\(\mathbf{p}\cdot\mathbf{y}\approx1\). Define

+
+\[\mathbf{M} += \mathbf{T} + - \frac{ + \mathbf{T}\mathbf{p}\mathbf{p}^{\mathsf{T}}\mathbf{T} + }{d}.\]
+

Let \([\mathbf{p}]_{\times}\) denote the cross-product matrix, +\([\mathbf{p}]_{\times}\mathbf{y}=\mathbf{p}\times\mathbf{y}\). Then

+
+(15)\[\mathbf{T}_{\mathbf{p}} += [\mathbf{p}]_{\times}^{\mathsf{T}} + \mathbf{M} + [\mathbf{p}]_{\times},\]
+

and the pole density becomes

+
+(16)\[A(\mathbf{y},\mathbf{p};f) +\approx \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left( + -\mathbf{y}^{\mathsf{T}}\mathbf{T}_{\mathbf{p}}\mathbf{y} + \right).\]
+

Equivalently, using the Levi-Civita symbol \(\epsilon_{ijk}\) and +summation over repeated indices,

+
+\[[\mathbf{T}_{\mathbf{p}}]_{ij} += p_k\,\epsilon_{lki} + \left( + T_{lm}-\frac{T_{la}p_a p_b T_{bm}}{d} + \right) + \epsilon_{mnj}\,p_n.\]
+

This absorbs the cross products into the projected tensor and makes the +subsequent Gaussian expressions more compact.

+
+
+

Reciprocal-space map

+

The pole density is defined for unit-vector arguments. With no +strain-distribution broadening, extend it to a reciprocal-space map supported +on the local plane \(q_{\mathrm{strain}}=0\).

+

Introduce the two-column basis matrix and coordinate vector

+
+\[\begin{split}\mathbf{U} += \begin{bmatrix} + \hat{\mathbf{q}}_{\mathrm{rock}} & \hat{\mathbf{k}}_{\perp} + \end{bmatrix}, +\qquad +\mathbf{u} += \begin{bmatrix} + q_{\mathrm{rock}} \\ + q_{\mathrm{roll}} + \end{bmatrix}.\end{split}\]
+

In the dimensionless reciprocal-space coordinates defined above, the +workflow’s local RSM is

+
+(17)\[f(\mathbf{q}) +\approx \delta(q_{\mathrm{strain}})\, + \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left[ + -\mathbf{u}^{\mathsf{T}} + \left(\mathbf{U}^{\mathsf{T}} + \mathbf{T}_{\mathbf{p}}\mathbf{U}\right) + \mathbf{u} + \right].\]
+

The Dirac delta expresses the absence of a distribution of lattice spacings +within the subgrain. Sampling this Gaussian at +\(q_{\mathrm{rock}}=\delta q\), as required by +(10), gives the scattered angular +distribution for the reflection.

+
+
+
+

Illustrative tests

+
+

Comparison with a tetrahedral crystal

+

The workflow compares a one-degree rotation of a single quartz crystal +shaped as a symmetric tetrahedron with a Gaussian approximation of the same +crystal. The crystal is much larger than the detector pixels, and the largest +scattering angles exceed 60 degrees, making radial spot elongation from the +projection geometry visible.

+

Seven Gaussians approximate the tetrahedron: one isotropic Gaussian at its +centre and six prolate Gaussians along its edges. The Gaussian model uses a +small orientation spread.

+

The comparison shows similar peak positions and shapes. The Gaussian model +also produces additional weak peaks because its rotational integration uses +a Gaussian time window, whereas the tetrahedral model uses a top-hat time +window.

+
+Quartz diffraction patterns from tetrahedral and Gaussian models. + +
+

Comparison of the tetrahedral crystal model with a seven-Gaussian +approximation during a one-degree sample rotation.

+
+
+
+
+

High-resolution rocking curve

+

A second test resolves a rocking curve and checks the shape of the resulting +three-dimensional reciprocal-space map. The workflow figure compares the +simulated profiles with the expected contributions from grain size and +misorientation.

+
+Reciprocal-space map and profiles from a high-resolution rocking scan. + +
+

High-resolution rocking-curve test showing the reciprocal-space map and +the contributions of grain size and misorientation to the simulated +profiles.

+
+
+
+
+
+

References

+
+
+[Kerbl2023] +

Kerbl, B., Kopanas, G., Leimkühler, T. and Drettakis, G. +(2023). 3D Gaussian Splatting for Real-Time Radiance Field Rendering. +ACM Transactions on Graphics, 42(4). +doi:10.1145/3592433.

+
+
+[Brehm2023] +

Brehm, W., White, T. and Chapman, H. N. (2023). +Crystal diffraction prediction and partiality estimation using Gaussian +basis functions. Acta Crystallographica Section A: Foundations and +Advances, 79, 145–162. +doi:10.1107/S2053273323000682.

+
+
+[Poulsen2017] +

Poulsen, H. F., Jakobsen, A. C., Simons, H., Ahl, S. R., +Cook, P. K. and Detlefs, C. (2017). X-ray diffraction microscopy based on +refractive optics. Journal of Applied Crystallography, 50, 1441–1456. +doi:10.1107/S1600576717011037.

+
+
+
+
+ + +
+
+
+ +
+ +
+

© Copyright 2021, Axel Henningsson.

+
+ + Built with Sphinx using a + theme + provided by Read the Docs. + + +
+
+
+
+
+ + + + \ No newline at end of file diff --git a/docs/genindex.html b/docs/genindex.html index 38ad151..8420211 100644 --- a/docs/genindex.html +++ b/docs/genindex.html @@ -145,8 +145,12 @@

D

@@ -190,9 +194,15 @@

G

  • gaussian_kernel (xrd_simulator.detector.Detector attribute)
  • gaussian_sigma (xrd_simulator.detector.Detector attribute) +
    • +
  • GaussianBeam (class in xrd_simulator.beam) +
    • +
  • GaussianPolycrystal (class in xrd_simulator.gaussian_crystal_model)
      • +
    • GaussianSubgrain (class in xrd_simulator.gaussian_crystal_model) +
    • generate_mesh_from_levelset() (xrd_simulator.mesh.TetraMesh class method)
    • generate_mesh_from_vertices() (xrd_simulator.mesh.TetraMesh class method) @@ -256,6 +266,8 @@

      M

    • mesh_sample (xrd_simulator.polycrystal.Polycrystal attribute)
    • miller_indices (xrd_simulator.phase.Phase attribute) +
      • +
    • misorientation_tensor (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)
    • module @@ -265,7 +277,7 @@

      M

    • xrd_simulator.detector
      • -
    • xrd_simulator.laue +
    • xrd_simulator.gaussian_crystal_model
    • xrd_simulator.mesh
      • @@ -296,6 +308,8 @@

      N

      O

      @@ -315,6 +329,8 @@

      P

    • peaks_detector_intersection() (xrd_simulator.detector.Detector method)
    • Phase (class in xrd_simulator.phase) +
      • +
    • phase (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)
    • phases (xrd_simulator.polycrystal.Polycrystal attribute)
      • @@ -333,6 +349,8 @@

      P

    • Polycrystal (class in xrd_simulator.polycrystal)
    • polycrystal_from_odf() (in module xrd_simulator.templates) +
      • +
    • position (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)
      • @@ -341,17 +359,19 @@

      R

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"from": 1, "gaussian": 0, "gaussian_crystal_model": 1, "high": 0, "illustr": 0, "incid": 0, "instal": 1, "integr": 0, "introduct": 1, "laue": [], "line": 0, "linear": 0, "linux": 1, "maco": 1, "map": 0, "mesh": 1, "model": 0, "monochromat": 0, "motion": 1, "orient": 0, "outgo": 0, "page": 0, "phase": 1, "pip": 1, "pole": 0, "polycryst": 1, "project": 0, "rai": 1, "reciproc": 0, "refer": 0, "represent": 0, "resolut": 0, "rock": 0, "scatter": 0, "simul": 1, "sourc": 1, "space": 0, "special": 0, "subgrain": 0, "tangent": 0, "templat": 1, "tensor": 0, "test": 0, "tetrahedr": 0, "theori": 0, "thi": 0, "window": 1, "x": 1}}) \ No newline at end of file diff --git a/docs/source/gaussian_model_theory.rst b/docs/source/gaussian_model_theory.rst new file mode 100644 index 0000000..353fb63 --- /dev/null +++ b/docs/source/gaussian_model_theory.rst @@ -0,0 +1,582 @@ +:orphan: + +.. _gaussian-model-theory: + +Gaussian model: theory and references +===================================== + +This page describes the scattering model and approximations underlying +:mod:`xrd_simulator.gaussian_crystal_model`. A worked example is provided in +the module documentation. + +.. contents:: On this page + :local: + :depth: 2 + +Gaussian subgrain representation +-------------------------------- + +Gaussian splatting represents a three-dimensional scene using Gaussian +primitives whose projected contributions can be rendered efficiently by +rasterization [Kerbl2023]_. Here, a polycrystal is represented by Gaussian +grains or subgrains, each with a Gaussian real-space density profile and a +narrow Gaussian distribution of lattice orientations. Under the approximations +below, each subgrain produces Gaussian-shaped diffraction spots on the +detector. Related Gaussian-basis methods for diffraction prediction and +partiality estimation have been demonstrated in serial crystallography +[Brehm2023]_. + +Several effects can contribute to diffraction-spot broadening: grain size, +detector point-spread, incident-beam bandwidth, incident-beam angular +divergence, lattice misorientation (mosaicity), and distributions of strain. +When these effects have Gaussian profiles and their deviations are small +enough to neglect nonlinear terms, they can be combined within a Gaussian +model. + +The model described here includes the real-space extent of the illuminated +subgrains and their mosaicity. The incident beam is monochromatic and +collimated, and each subgrain has a uniform strain tensor. A uniform strain +can shift a reflection but does not describe a distribution of lattice +spacings within that subgrain. Small sample rotations during an exposure are +approximated by additional Gaussian orientation broadening. + +.. note:: + + The Gaussian-splat rendering path does not apply a separate detector + point-spread convolution. The ``gaussian_sigma`` and + ``max_gaussian_kernel_radius`` detector settings retained in the worked + example therefore do not add this broadening to its Gaussian spots. + +Scattering theory +----------------- + +Scattered-beam distribution +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Let the incident beam have a phase-space density :math:`p(\mathbf{k})` +centred on :math:`\mathbf{k}_0`, with nominal wavenumber + +.. math:: + :label: gaussian-wavenumber + + k = \lvert\mathbf{k}_0\rvert = \frac{2\pi}{\lambda_0}. + +For a particular reflection, let :math:`f(\mathbf{q})` describe the subgrain's +reciprocal-space map (RSM) around a reciprocal-lattice vector +:math:`\mathbf{G}_0`. The scattered-beam distribution is written as + +.. math:: + :label: gaussian-scattering-integral + + I(\mathbf{p}) + = \int \mathrm{d}^{3}\mathbf{k} + \int \mathrm{d}^{3}\mathbf{q}\; + p(\mathbf{k})\,f(\mathbf{q})\, + \delta\!\left(\lvert\mathbf{k}\rvert-\lvert\mathbf{p}\rvert\right) + \delta^{(3)}\!\left(\mathbf{k}+\mathbf{q}-\mathbf{p}\right). + +The scalar Dirac delta enforces elastic scattering, while the +three-dimensional Dirac delta enforces momentum conservation. + +Incident-beam coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~ + +Following the coordinate construction used by Poulsen et al. +[Poulsen2017]_, write the incident wavevector as + +.. math:: + :label: gaussian-incident-wavevector + + \mathbf{k} + = \mathbf{k}_0 + + k\left( + \varepsilon\,\hat{\mathbf{k}}_0 + + \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + + \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp} + \right). + +A hat denotes a unit vector. The vector +:math:`\hat{\mathbf{k}}_{\parallel}` is perpendicular to +:math:`\mathbf{k}_0` and lies in the plane spanned by :math:`\mathbf{k}_0` +and :math:`\mathbf{G}_0`. Its sign is chosen so that +:math:`\mathbf{G}_0\cdot\hat{\mathbf{k}}_{\parallel}>0`. Complete the +right-handed basis with + +.. math:: + + \hat{\mathbf{k}}_{\perp} + = \hat{\mathbf{k}}_0\times\hat{\mathbf{k}}_{\parallel}. + +The coordinate :math:`\varepsilon` describes the relative wavenumber +deviation, and :math:`\zeta_{\parallel}` and :math:`\zeta_{\perp}` describe +small angular deviations of the incident beam. + +Reciprocal-space coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Define the nominal Bragg angle and the exactly aligned scattering vector by + +.. math:: + :label: gaussian-nominal-scattering-vector + + \begin{aligned} + \theta_0 + &= \arcsin\!\left(\frac{\lvert\mathbf{G}_0\rvert}{2k}\right), \\ + \mathbf{Q} + &= 2k\sin\theta_0\left( + \cos\theta_0\,\hat{\mathbf{k}}_{\parallel} + - \sin\theta_0\,\hat{\mathbf{k}}_0 + \right) \\ + &= 2k\sin\theta_0\,\hat{\mathbf{Q}}. + \end{aligned} + +The vectors :math:`\mathbf{G}_0` and :math:`\mathbf{Q}` coincide when the +reflection is exactly aligned. For a small misalignment, their difference is +parallel, to first order, to the rocking direction + +.. math:: + + \hat{\mathbf{q}}_{\mathrm{rock}} + = \cos\theta_0\,\hat{\mathbf{k}}_0 + + \sin\theta_0\,\hat{\mathbf{k}}_{\parallel}. + +Use dimensionless rocking, strain, and rolling coordinates to write + +.. math:: + :label: gaussian-reciprocal-coordinates + + \begin{aligned} + \mathbf{q} + &= \mathbf{G}_0 + + 2k\sin\theta_0\left( + q_{\mathrm{rock}}\,\hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right) \\ + &\approx \mathbf{Q} + + 2k\sin\theta_0\left( + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right), + \end{aligned} + +where + +.. math:: + :label: gaussian-rocking-offset + + \delta q + = \frac{ + (\mathbf{Q}-\mathbf{G}_0)\cdot\hat{\mathbf{q}}_{\mathrm{rock}} + }{2k\sin\theta_0}. + +This offset measures the reflection's misalignment and is equal to the +rocking angle to first order, with the sign convention above. The incident +coordinates :math:`(\varepsilon,\zeta_{\parallel},\zeta_{\perp})` and the +reciprocal-space coordinates +:math:`(q_{\mathrm{rock}},q_{\mathrm{strain}},q_{\mathrm{roll}})` separate +bandwidth, collimation, misorientation, and strain effects. + +Outgoing-beam coordinates +~~~~~~~~~~~~~~~~~~~~~~~~~ + +The nominal scattered wavevector is + +.. math:: + :label: gaussian-nominal-outgoing-wavevector + + \mathbf{k}_h + = \mathbf{k}_0+\mathbf{Q} + = k\left( + \cos(2\theta_0)\,\hat{\mathbf{k}}_0 + + \sin(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + \right). + +Define the radial direction perpendicular to :math:`\mathbf{k}_h` by + +.. math:: + + \hat{\mathbf{k}}_{\mathrm{rad}} + = \cos(2\theta_0)\,\hat{\mathbf{k}}_{\parallel} + - \sin(2\theta_0)\,\hat{\mathbf{k}}_0. + +The outgoing wavevector can then be written as + +.. math:: + :label: gaussian-outgoing-wavevector + + \mathbf{p} + = \mathbf{k}_h + + k\left( + \varepsilon'\,\hat{\mathbf{k}}_h + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + \right). + +Here :math:`\psi_{\mathrm{rad}}` and :math:`\psi_{\mathrm{azim}}` are small +transverse angular deviations from the nominal outgoing direction. + +Linearized conservation equations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To first order, + +.. math:: + + \lvert\mathbf{k}\rvert\approx k(1+\varepsilon), + \qquad + \lvert\mathbf{p}\rvert\approx k(1+\varepsilon'). + +Energy conservation therefore sets :math:`\varepsilon'=\varepsilon`. +Momentum conservation gives :math:`\mathbf{q}=\mathbf{p}-\mathbf{k}` and, +after substitution, + +.. math:: + :label: gaussian-linearized-momentum + + \begin{aligned} + &2\sin\theta_0\Bigl[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{strain}}\,\hat{\mathbf{Q}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \Bigr] \\ + &\qquad = 2\sin\theta_0\,\varepsilon\,\hat{\mathbf{Q}} + + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} \\ + &\qquad\quad + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp} + - \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel} + - \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp}. + \end{aligned} + +.. note:: + + The incident-divergence terms have minus signs because the incident + wavevector is subtracted in :math:`\mathbf{q}=\mathbf{p}-\mathbf{k}`. + This corrects the corresponding signs in the original workflow. It does + not affect the monochromatic, collimated specialization below, where + both terms vanish. + +Monochromatic, collimated specialization +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The current model assumes + +.. math:: + + \varepsilon=\zeta_{\parallel}=\zeta_{\perp}=0, + \qquad q_{\mathrm{strain}}=0. + +The last condition excludes strain-distribution broadening; it does not +require the uniform subgrain strain to vanish. The conservation equation +reduces to + +.. math:: + + \begin{aligned} + &2\sin\theta_0\left[ + (q_{\mathrm{rock}}-\delta q)\, + \hat{\mathbf{q}}_{\mathrm{rock}} + + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp} + \right] \\ + &\qquad + = \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} + + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}. + \end{aligned} + +For a nondegenerate Bragg geometry, the three coordinate equations are + +.. math:: + :label: gaussian-scattering-constraints + + q_{\mathrm{rock}}=\delta q, + \qquad + \psi_{\mathrm{rad}}=0, + \qquad + \psi_{\mathrm{azim}}=2\sin\theta_0\,q_{\mathrm{roll}}. + +Thus, the scattered angular distribution samples the RSM along a line +parallel to :math:`\hat{\mathbf{k}}_{\perp}`, offset by :math:`\delta q` +from its centre. For a Gaussian RSM, this gives a one-dimensional Gaussian +angular distribution. The finite real-space extent of the illuminated +subgrain is subsequently projected onto the detector to produce a +two-dimensional spot. + +Anisotropic Gaussian orientation distributions +---------------------------------------------- + +Tangent-space approximation +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Consider a narrow orientation distribution centred on an orientation +:math:`g_0`. Approximate orientation space by its left tangent space at +:math:`g_0`, so that a small rotation vector :math:`\mathbf{r}` acts in +laboratory coordinates: + +.. math:: + :label: gaussian-tangent-rotation + + g\approx R_{\mathbf{r}}g_0 + \approx + \begin{bmatrix} + 1 & -r_z & r_y \\ + r_z & 1 & -r_x \\ + -r_y & r_x & 1 + \end{bmatrix}g_0. + +The Gaussian orientation density used in the workflow is + +.. math:: + :label: gaussian-orientation-density + + f(\mathbf{r}) + = \frac{2}{\sqrt{\pi}}\sqrt{\det\mathbf{T}}\, + \exp\!\left(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r}\right), + +where :math:`\mathbf{T}` is a symmetric positive-definite concentration +tensor. + +.. note:: + + The prefactor above retains the workflow's normalization convention. + With ordinary Euclidean tangent-space measure, + :math:`\int f(\mathbf{r})\,\mathrm{d}^{3}\mathbf{r}=2\pi`, not one. + It should therefore not be interpreted as a unit-normalized Euclidean + probability density without rescaling. + + For an exponential of the form + :math:`\exp(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r})`, the + corresponding normalized Gaussian has covariance + :math:`\tfrac{1}{2}\mathbf{T}^{-1}`. The implementation directly inverts + ``misorientation_tensor`` to obtain :math:`\mathbf{T}`. Consequently, + the square roots of the input tensor's eigenvalues are Gaussian width + parameters, not standard deviations. For an input eigenvalue + :math:`a^2`, the standard deviation is :math:`a/\sqrt{2}`. + The real-space Gaussian uses the same exponential convention for + ``shape_tensor``. This distinction matters because the class attribute + descriptions currently use the word "covariance" for these inputs. + +Pole density as a line integral +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For Miller indices :math:`(h,k,\ell)`, define the unit lattice direction in +crystal coordinates by + +.. math:: + + \mathbf{h} + = \frac{ + \mathbf{B}_0[h,k,\ell]^{\mathsf{T}} + }{ + \left\lvert\mathbf{B}_0[h,k,\ell]^{\mathsf{T}}\right\rvert + }, + \qquad + \mathbf{p}=g_0\mathbf{h}. + +Here :math:`\mathbf{B}_0` is the reciprocal-lattice basis matrix. In this +section, :math:`\mathbf{p}` denotes the mean lattice direction, rather than +the outgoing wavevector used in the scattering section. It is parallel to +:math:`\mathbf{G}_0`. Let :math:`\mathbf{y}` be a unit direction in +laboratory coordinates, parallel to a sampled reciprocal-space vector +:math:`\mathbf{q}`; it is not a Cartesian coordinate-axis label. + +The pole density, also called the pair-correlation function in texture +analysis, describes the density of lattice directions along +:math:`\mathbf{y}`. Its calculation normally involves an integral over a +circle in :math:`\mathrm{SO}(3)`. Within the tangent-space approximation, +replace this by an infinite line integral, parameterized as + +.. math:: + :label: gaussian-orientation-fibre + + \mathbf{r}(\lambda) + = \frac{\mathbf{p}\times\mathbf{y}}{\mathbf{p}\cdot\mathbf{y}} + + \lambda\mathbf{p} + = \mathbf{r}_0+\lambda\mathbf{p}. + +The parameter :math:`\lambda` here is a line-integration parameter, not the +X-ray wavelength. Substitution into the Gaussian density and completion of +the square give, with +:math:`d=\mathbf{p}^{\mathsf{T}}\mathbf{T}\mathbf{p}`, + +.. math:: + :label: gaussian-pole-density-integral + + \begin{aligned} + A(\mathbf{y},\mathbf{p};f) + &= \int_{-\infty}^{\infty} + f\!\left(\mathbf{r}_0+\lambda\mathbf{p}\right) + \,\mathrm{d}\lambda \\ + &= \frac{2\sqrt{\det\mathbf{T}}} + {\sqrt{d}} + \exp\!\left[ + -\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{r}_0 + + \frac{ + (\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{p})^2 + }{d} + \right]. + \end{aligned} + +Projected concentration tensor +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For directions close to the pole, make the further approximation +:math:`\mathbf{p}\cdot\mathbf{y}\approx1`. Define + +.. math:: + + \mathbf{M} + = \mathbf{T} + - \frac{ + \mathbf{T}\mathbf{p}\mathbf{p}^{\mathsf{T}}\mathbf{T} + }{d}. + +Let :math:`[\mathbf{p}]_{\times}` denote the cross-product matrix, +:math:`[\mathbf{p}]_{\times}\mathbf{y}=\mathbf{p}\times\mathbf{y}`. Then + +.. math:: + :label: gaussian-projected-concentration + + \mathbf{T}_{\mathbf{p}} + = [\mathbf{p}]_{\times}^{\mathsf{T}} + \mathbf{M} + [\mathbf{p}]_{\times}, + +and the pole density becomes + +.. math:: + :label: gaussian-pole-density + + A(\mathbf{y},\mathbf{p};f) + \approx \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left( + -\mathbf{y}^{\mathsf{T}}\mathbf{T}_{\mathbf{p}}\mathbf{y} + \right). + +Equivalently, using the Levi-Civita symbol :math:`\epsilon_{ijk}` and +summation over repeated indices, + +.. math:: + + [\mathbf{T}_{\mathbf{p}}]_{ij} + = p_k\,\epsilon_{lki} + \left( + T_{lm}-\frac{T_{la}p_a p_b T_{bm}}{d} + \right) + \epsilon_{mnj}\,p_n. + +This absorbs the cross products into the projected tensor and makes the +subsequent Gaussian expressions more compact. + +Reciprocal-space map +~~~~~~~~~~~~~~~~~~~~ + +The pole density is defined for unit-vector arguments. With no +strain-distribution broadening, extend it to a reciprocal-space map supported +on the local plane :math:`q_{\mathrm{strain}}=0`. + +Introduce the two-column basis matrix and coordinate vector + +.. math:: + + \mathbf{U} + = \begin{bmatrix} + \hat{\mathbf{q}}_{\mathrm{rock}} & \hat{\mathbf{k}}_{\perp} + \end{bmatrix}, + \qquad + \mathbf{u} + = \begin{bmatrix} + q_{\mathrm{rock}} \\ + q_{\mathrm{roll}} + \end{bmatrix}. + +In the dimensionless reciprocal-space coordinates defined above, the +workflow's local RSM is + +.. math:: + :label: gaussian-local-rsm + + f(\mathbf{q}) + \approx \delta(q_{\mathrm{strain}})\, + \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\, + \exp\!\left[ + -\mathbf{u}^{\mathsf{T}} + \left(\mathbf{U}^{\mathsf{T}} + \mathbf{T}_{\mathbf{p}}\mathbf{U}\right) + \mathbf{u} + \right]. + +The Dirac delta expresses the absence of a distribution of lattice spacings +within the subgrain. Sampling this Gaussian at +:math:`q_{\mathrm{rock}}=\delta q`, as required by +:eq:`gaussian-scattering-constraints`, gives the scattered angular +distribution for the reflection. + +Illustrative tests +------------------ + +Comparison with a tetrahedral crystal +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The workflow compares a one-degree rotation of a single quartz crystal +shaped as a symmetric tetrahedron with a Gaussian approximation of the same +crystal. The crystal is much larger than the detector pixels, and the largest +scattering angles exceed 60 degrees, making radial spot elongation from the +projection geometry visible. + +Seven Gaussians approximate the tetrahedron: one isotropic Gaussian at its +centre and six prolate Gaussians along its edges. The Gaussian model uses a +small orientation spread. + +The comparison shows similar peak positions and shapes. The Gaussian model +also produces additional weak peaks because its rotational integration uses +a Gaussian time window, whereas the tetrahedral model uses a top-hat time +window. + +.. figure:: ../_static/single_crystal_quartz.png + :alt: Quartz diffraction patterns from tetrahedral and Gaussian models. + :align: center + :width: 100% + + Comparison of the tetrahedral crystal model with a seven-Gaussian + approximation during a one-degree sample rotation. + +High-resolution rocking curve +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A second test resolves a rocking curve and checks the shape of the resulting +three-dimensional reciprocal-space map. The workflow figure compares the +simulated profiles with the expected contributions from grain size and +misorientation. + +.. figure:: ../_static/testing_rocking_curves.png + :alt: Reciprocal-space map and profiles from a high-resolution rocking scan. + :align: center + :width: 100% + + High-resolution rocking-curve test showing the reciprocal-space map and + the contributions of grain size and misorientation to the simulated + profiles. + +References +---------- + +.. [Kerbl2023] Kerbl, B., Kopanas, G., Leimkühler, T. and Drettakis, G. + (2023). 3D Gaussian Splatting for Real-Time Radiance Field Rendering. + *ACM Transactions on Graphics*, **42**\ (4). + `doi:10.1145/3592433 `_. + +.. [Brehm2023] Brehm, W., White, T. and Chapman, H. N. (2023). + Crystal diffraction prediction and partiality estimation using Gaussian + basis functions. *Acta Crystallographica Section A: Foundations and + Advances*, **79**, 145–162. + `doi:10.1107/S2053273323000682 + `_. + +.. [Poulsen2017] Poulsen, H. F., Jakobsen, A. C., Simons, H., Ahl, S. R., + Cook, P. K. and Detlefs, C. (2017). X-ray diffraction microscopy based on + refractive optics. *Journal of Applied Crystallography*, **50**, 1441–1456. + `doi:10.1107/S1600576717011037 + `_. diff --git a/docs/source/index.rst b/docs/source/index.rst index 29d8fe8..61bc367 100644 --- a/docs/source/index.rst +++ b/docs/source/index.rst @@ -1,7 +1,3 @@ -.. toctree:: - :maxdepth: 2 - :caption: Contents: - .. include:: ../../README.rst ====================================== @@ -50,16 +46,16 @@ motion :members: :inherited-members: -templates +gaussian_crystal_model ====================================== -.. automodule:: xrd_simulator.templates - :members: - :inherited-members: +.. automodule:: xrd_simulator.gaussian_crystal_model + :members: + :inherited-members: -laue +templates ====================================== -.. automodule:: xrd_simulator.laue +.. automodule:: xrd_simulator.templates :members: :inherited-members: diff --git a/xrd_simulator/gaussian_crystal_model.py b/xrd_simulator/gaussian_crystal_model.py index b3b9e6c..63997b3 100644 --- a/xrd_simulator/gaussian_crystal_model.py +++ b/xrd_simulator/gaussian_crystal_model.py @@ -1,22 +1,206 @@ -from scipy.spatial.transform import Rotation as R +r"""Gaussian subgrain models for polycrystalline X-ray diffraction simulation. + +Represent a polycrystal as a collection of subgrains with Gaussian real-space +density profiles and narrow Gaussian orientation distributions. Each subgrain +is defined by its crystallographic phase, centroid, shape tensor, mean lattice +orientation, misorientation tensor, and uniform strain tensor. Both the shape +and orientation spread may be anisotropic. + +:class:`GaussianSubgrain` describes an individual subgrain, while +:class:`GaussianPolycrystal` assembles subgrains and renders diffraction +patterns using a :class:`~xrd_simulator.beam.GaussianBeam` and a +:class:`~xrd_simulator.detector.Detector`. + +The forward model computes beam-subgrain intersections, reflection +partialities, and scattered-beam angular distributions. Illuminated subgrain +shapes are projected onto the detector and combined with mosaicity-induced +angular broadening to form two-dimensional Gaussian spots. These contributions +are rasterized and summed into a detector intensity image. Small sample +rotations during an exposure are approximated by additional Gaussian +orientation broadening. + +The module also supports rigid-body transformations of the polycrystal and +pole-figure rendering. + +Notes +----- +The current implementation supports single-phase polycrystals and assumes a +monochromatic, collimated incident beam. Orientation distributions are +approximated in the tangent space about each subgrain's mean orientation, +requiring small misorientations. Uniform strain modifies reciprocal-lattice +vectors, but strain-distribution broadening is not included within an +individual subgrain. A distribution of strains can instead be approximated by +combining appropriately weighted diffraction patterns from subgrain +populations with different uniform strain tensors. + +.. seealso:: + + :doc:`Gaussian model: theory and references ` + Scattering geometry, Gaussian orientation distributions, reciprocal-space + maps, model assumptions, illustrative tests, and references. + +Examples +-------- +Simulate a static diffraction pattern from 2000 Gaussian quartz subgrains with +random positions, orientations, shapes, and anisotropic orientation spreads. + +Run the example from the repository root with ``xrd_simulator`` installed, so +that ``tests/data/quartz.cif`` is available. Real-space lengths are in +micrometres, the wavelength and unit-cell lengths are in angstroms, and +unit-cell angles are in degrees. Orientation-spread parameters are in radians. + +First, define the beam, detector, and crystallographic phase: + +.. code-block:: python + + import numpy as np + import matplotlib.pyplot as plt + from scipy.spatial.transform import Rotation as R + + from xrd_simulator.beam import GaussianBeam + from xrd_simulator.detector import Detector + from xrd_simulator.phase import Phase + from xrd_simulator.gaussian_crystal_model import ( + GaussianSubgrain, + GaussianPolycrystal, + ) + + rng = np.random.default_rng(0) + + gaussian_beam = GaussianBeam( + xray_propagation_direction=np.array([1.0, 0.0, 0.0]), + beam_centroid_position=np.array([0.0, 0.0, 0.0]), + wavelength=0.28523, + polarization_vector=np.array([0.0, 1.0, 0.0]), + long_axis_width=300.0, + long_axis_direction=np.array([0.0, 0.0, 1.0]), + short_axis_width=50.0, + short_axis_direction=np.array([0.0, 1.0, 0.0]), + ) + + detector = Detector( + det_corner_0=np.array([142938.3, -38400.0, -38400.0]), + det_corner_1=np.array([142938.3, 38400.0, -38400.0]), + det_corner_2=np.array([142938.3, -38400.0, 38400.0]), + pixel_size=(25.0, 25.0), + gaussian_sigma=1.0, + max_gaussian_kernel_radius=5, + ) + + quartz = Phase( + unit_cell=[4.926, 4.926, 5.4189, 90.0, 90.0, 120.0], + sgname="P3221", + path_to_cif_file="tests/data/quartz.cif", + ) + +Next, build the polycrystal. The helper generates a symmetric positive-definite +width tensor with a randomly oriented symmetry axis. Its eigenvalues are +``axis_1**2``, ``axis_1**2``, and ``axis_2**2``. Shape-width parameters range +from 50 to 200 micrometres, and misorientation-width parameters range from +0.01 to 0.03 radians. The distinction between these width parameters and +standard deviations is explained on the theory page. + +.. code-block:: python + + def make_random_tensor(axis_1, axis_2): + random_direction = rng.normal(size=3) + random_direction /= np.linalg.norm(random_direction) + return ( + axis_1**2 * np.eye(3) + + (axis_2**2 - axis_1**2) + * np.outer(random_direction, random_direction) + ) + + grain_list = [] + + for _ in range(2000): + position = rng.uniform(-500.0, 500.0, size=3) + shape_tensor = make_random_tensor( + rng.uniform(50.0, 200.0), + rng.uniform(50.0, 200.0), + ) + orientation = R.random(random_state=rng).as_matrix() + misorientation_tensor = make_random_tensor( + rng.uniform(0.01, 0.03), + rng.uniform(0.01, 0.03), + ) + + grain_list.append( + GaussianSubgrain( + phase=quartz, + position=position, + shape_tensor=shape_tensor, + orientation=orientation, + misorientation_tensor=misorientation_tensor, + strain_tensor=np.zeros((3, 3)), + ) + ) + + gaussian_polycrystal = GaussianPolycrystal( + grain_list, + max_misorientation=0.03, + max_grain_size=200.0, + ) + +Compute a diffraction pattern without sample rotation during the exposure: + +.. code-block:: python + + f = gaussian_polycrystal.diffract( + beam=gaussian_beam, + detector=detector, + verbose=True, + threshold=10.0, + ) + +Finally, plot the full detector image and a magnified region on the same +logarithmic intensity scale: + +.. code-block:: python + + log_intensity = np.log10(f.detach().cpu().numpy() + 1e-10) + + fig, axs = plt.subplots( + 1, 3, + figsize=(8, 4), + gridspec_kw={"width_ratios": (1, 1, 0.1)}, + constrained_layout=True, + ) + axs[0].imshow(log_intensity, cmap="jet", vmin=3, vmax=6) + img = axs[1].imshow(log_intensity, cmap="jet", vmin=3, vmax=6) + axs[0].set_title("Full detector") + axs[1].set_title("Magnified region") + axs[1].set_xlim(1500, 2200) + axs[1].set_ylim(2200, 1500) + fig.colorbar(img, cax=axs[2], label=r"$\log_{10}(I + 10^{-10})$") + plt.show() + +.. figure:: ../_static/many_crystals_quartz.png + :alt: Simulated quartz diffraction pattern and a magnified detector region. + :align: center + :width: 100% + + Representative output from the Gaussian workflow. The precise spot pattern + depends on the random subgrain realization. This image is retained from the + original workflow and need not match the seeded realization above exactly. +""" + +import time + import numpy as np import numpy.typing as npt import torch from torch import Tensor +from xfab.tools import form_b_mat -from xrd_simulator.phase import Phase -from xrd_simulator.utils import ensure_torch -from xrd_simulator.detector import Detector -from xrd_simulator.laue import _get_diffraction_arcsegment, _project_misorientation_tensor from xrd_simulator.beam import GaussianBeam +from xrd_simulator.detector import Detector +from xrd_simulator.laue import _get_diffraction_arcsegment from xrd_simulator.motion import RigidBodyMotion +from xrd_simulator.phase import Phase from xrd_simulator.scattering_factors import _polarization +from xrd_simulator.utils import ensure_torch -from xfab.tools import form_b_mat, genhkl_base -from xfab import sg - - -import time class GaussianSubgrain: """A `subgrain` has a 3D-gaussian density distribution and a narrow (<=0.1 rad) 3D @@ -33,21 +217,28 @@ class GaussianSubgrain: orientation : torch.Tensor | np.array Orientation of the centroid of the ODF as a rotation matrix, shape ``(3, 3,)`` misorientation_tensor : torch.Tensor | np.array - Covariance tensor of the grain orientation in the left-hand tangent space. + Covariance tensor of the grain orientation in the left-hand tangent space. aka. laboratory coordinates, shape ``(3, 3,)`` strain_tensor : torch.Tensor | np.array Strain tensor in laboratory coordinates, shape ``(3, 3,)`` """ - def __init__(self, + def __init__( + self, phase: Phase, - position: npt.NDArray | Tensor = np.array([0, 0, 0]), + position: npt.NDArray | Tensor = np.array([0, 0, 0]), shape_tensor: npt.NDArray | Tensor = np.eye(3), - orientation: npt.NDArray | Tensor = np.eye(3), # 3by3 Rotation matrix. - misorientation_tensor: npt.NDArray | Tensor = 0.0175**2 * np.eye(3), # Default one degree isotropic - strain_tensor: npt.NDArray | Tensor = np.zeros((3, 3,)), + orientation: npt.NDArray | Tensor = np.eye(3), # 3by3 Rotation matrix. + misorientation_tensor: npt.NDArray | Tensor = 0.0175**2 + * np.eye(3), # Default one degree isotropic + strain_tensor: npt.NDArray | Tensor = np.zeros( + ( + 3, + 3, + ) + ), ): - + self.phase = phase self.position = position self.shape_tensor = shape_tensor @@ -55,36 +246,53 @@ def __init__(self, self.misorientation_tensor = misorientation_tensor self.strain_tensor = strain_tensor + class GaussianPolycrystal: - """ Colection of ``GaussianSubgrain`` object representing a polycrystalline sample. + """Colection of ``GaussianSubgrain`` object representing a polycrystalline sample. Variables ``max_grain_size`` and ``max_misorientation`` can be supplied to speed up diffraction calculations. """ - def __init__(self, + def __init__( + self, grain_list: list[GaussianSubgrain], max_grain_size: float = 1000.0, - max_misorientation: float = 0.1, + max_misorientation: float = 0.1, ): - + phases_list = list(set([grain.phase for grain in grain_list])) n_phases = len(phases_list) if not n_phases == 1: - raise NotImplementedError('Multi-phase object has not been implemented yet.') + raise NotImplementedError( + "Multi-phase object has not been implemented yet." + ) self.n_grains = len(grain_list) self.max_grain_size = max_grain_size self.max_misorientation = max_misorientation self.phase = phases_list[0] - self.positions = torch.stack([ensure_torch(grain.position) for grain in grain_list]) - self.shape_concentration_tensors = torch.stack([ensure_torch(np.linalg.inv(grain.shape_tensor)) for grain in grain_list]) + self.positions = torch.stack( + [ensure_torch(grain.position) for grain in grain_list] + ) + self.shape_concentration_tensors = torch.stack( + [ensure_torch(np.linalg.inv(grain.shape_tensor)) for grain in grain_list] + ) - self.orientaions = torch.stack([ensure_torch(grain.orientation) for grain in grain_list]) - self.misori_concentration_tensors = torch.stack([ensure_torch(np.linalg.inv(grain.misorientation_tensor)) for grain in grain_list]) - - self.strains = torch.stack([ensure_torch(grain.strain_tensor) for grain in grain_list]) + self.orientaions = torch.stack( + [ensure_torch(grain.orientation) for grain in grain_list] + ) + self.misori_concentration_tensors = torch.stack( + [ + ensure_torch(np.linalg.inv(grain.misorientation_tensor)) + for grain in grain_list + ] + ) + + self.strains = torch.stack( + [ensure_torch(grain.strain_tensor) for grain in grain_list] + ) def diffract( self, @@ -94,10 +302,10 @@ def diffract( sample_translation: npt.NDArray | Tensor = np.zeros(3), sample_rotation_during_exposure: npt.NDArray | Tensor = np.zeros(3), verbose=False, - threshold = 6.0, - peaks_batch_size = 20000, + threshold=6.0, + peaks_batch_size=20000, ): - """ Render a single diffraction pattern. + """Render a single diffraction pattern. Parameters ---------- @@ -123,79 +331,121 @@ def diffract( wavelength = beam.wavelength sample_orientation = ensure_torch(sample_orientation) sample_rotation_during_exposure = ensure_torch(sample_rotation_during_exposure) - + # Rotate detector and incident beam by inverse of sample-rotation. - xray_propagation_direction = torch.einsum('ij,i->j', sample_orientation, ensure_torch(xray_propagation_direction) ) - detector_origin = detector.pixel_coordinates[0,0] + xray_propagation_direction = torch.einsum( + "ij,i->j", sample_orientation, ensure_torch(xray_propagation_direction) + ) + detector_origin = detector.pixel_coordinates[0, 0] W = np.stack([detector.zdhat, detector.ydhat]) - detector_norm = torch.einsum('ij,i->j', sample_orientation, ensure_torch(np.cross(W[0,:], W[1,:]))) - detector_origin = torch.einsum('ij,i->j', sample_orientation, ensure_torch(detector_origin)) - W = torch.einsum('ij,ui->uj', sample_orientation, ensure_torch(W)) + detector_norm = torch.einsum( + "ij,i->j", sample_orientation, ensure_torch(np.cross(W[0, :], W[1, :])) + ) + detector_origin = torch.einsum( + "ij,i->j", sample_orientation, ensure_torch(detector_origin) + ) + W = torch.einsum("ij,ui->uj", sample_orientation, ensure_torch(W)) pixellengths = torch.tensor([detector.pixel_size_y, detector.pixel_size_z]) - #Compute intersection of grains and beam - intersection_pos, intersection_shape_concentration_tensors, beam_intensity_factors, grains_hit\ - = beam._intersect( + # Compute intersection of grains and beam + ( + intersection_pos, + intersection_shape_concentration_tensors, + beam_intensity_factors, + grains_hit, + ) = beam._intersect( ensure_torch(self.positions), ensure_torch(self.shape_concentration_tensors), sample_orientation, sample_translation, self.max_grain_size, - threshold = threshold, + threshold=threshold, ) if verbose: - print(f'Beam-grain intersection took {time.time()-t0}.') + print(f"Beam-grain intersection took {time.time() - t0}.") t0 = time.time() # Simulate sample-rotation by adding a rotation to the grain misorientation sample_rotation_during_exposure = ensure_torch(sample_rotation_during_exposure) - rotation_vector = torch.einsum('ij,i->j', sample_orientation, sample_rotation_during_exposure) + rotation_vector = torch.einsum( + "ij,i->j", sample_orientation, sample_rotation_during_exposure + ) smeared_misorientation_tensors = torch.linalg.inv( - torch.linalg.inv(self.misori_concentration_tensors[grains_hit]) + torch.outer(rotation_vector, rotation_vector)) + torch.linalg.inv(self.misori_concentration_tensors[grains_hit]) + + torch.outer(rotation_vector, rotation_vector) + ) - #Construct some crystal and geometry information. + # Construct some crystal and geometry information. B = torch.Tensor(form_b_mat(self.phase.unit_cell)) - max_angle = detector._get_wrapping_cone(xray_propagation_direction, np.mean([0, 0, 0])) - self.phase._setup_diffracting_planes(wavelength=wavelength, min_bragg_angle=0.0, max_bragg_angle=max_angle+0.1) - + max_angle = detector._get_wrapping_cone( + xray_propagation_direction, np.mean([0, 0, 0]) + ) + self.phase._setup_diffracting_planes( + wavelength=wavelength, min_bragg_angle=0.0, max_bragg_angle=max_angle + 0.1 + ) + # Get miller indicies and structure factors miller_indices = torch.Tensor(self.phase.miller_indices) if self.phase.structure_factors is not None: - structure_factors = torch.sum(ensure_torch(self.phase.structure_factors) ** 2, axis=1) + structure_factors = torch.sum( + ensure_torch(self.phase.structure_factors) ** 2, axis=1 + ) miller_indices = miller_indices[structure_factors > 1e-6] structure_factors = structure_factors[structure_factors > 1e-6] else: # If no structure factors provided, use uniform intensity (all ones) structure_factors = torch.ones(miller_indices.shape[0]) - + # Get scattering vectors and scattering angle. - h = torch.einsum('ij,hj->hi', B, miller_indices) - p_vectors = torch.einsum('ghi,gij,kj->gkh', torch.eye(3)[None,:,:] - self.strains[grains_hit], self.orientaions[grains_hit], h) + h = torch.einsum("ij,hj->hi", B, miller_indices) + p_vectors = torch.einsum( + "ghi,gij,kj->gkh", + torch.eye(3)[None, :, :] - self.strains[grains_hit], + self.orientaions[grains_hit], + h, + ) p_vectors_norm = torch.linalg.norm(p_vectors, axis=-1) - theta_angle = torch.asin( p_vectors_norm * wavelength / 4 / np.pi ) + theta_angle = torch.asin(p_vectors_norm * wavelength / 4 / np.pi) # Filter out reflections far from the bragg-condition - dp = torch.einsum('i,ghi->gh', xray_propagation_direction, p_vectors) / p_vectors_norm - does_diffract = torch.abs( dp + torch.sin(theta_angle) ) \ - < threshold * (self.max_misorientation + torch.linalg.norm(sample_rotation_during_exposure)) #IDEA: Consider a per-gaussian max misorientation. + dp = ( + torch.einsum("i,ghi->gh", xray_propagation_direction, p_vectors) + / p_vectors_norm + ) + does_diffract = torch.abs(dp + torch.sin(theta_angle)) < threshold * ( + self.max_misorientation + torch.linalg.norm(sample_rotation_during_exposure) + ) # IDEA: Consider a per-gaussian max misorientation. # Exception if no grains diffract. if not torch.any(does_diffract): return torch.zeros(detector.shape) # Select the relevant reflections and flatten the grain- and hkl-indexes. - grain_does_diffract, hkl_does_diffract = torch.where(does_diffract) # Don't ask me what this line does - misori_concentration_tensors = smeared_misorientation_tensors[grain_does_diffract] + grain_does_diffract, hkl_does_diffract = torch.where( + does_diffract + ) # Don't ask me what this line does + misori_concentration_tensors = smeared_misorientation_tensors[ + grain_does_diffract + ] p_vectors = p_vectors[does_diffract] - shape_concentration_tensors = intersection_shape_concentration_tensors[grain_does_diffract] + shape_concentration_tensors = intersection_shape_concentration_tensors[ + grain_does_diffract + ] if verbose: - print(f'Bragg-condition filterin took {time.time()-t0}. ({does_diffract.shape[0]*does_diffract.shape[1]} -> {torch.sum(does_diffract)})') + print( + f"Bragg-condition filterin took {time.time() - t0}. ({does_diffract.shape[0] * does_diffract.shape[1]} -> {torch.sum(does_diffract)})" + ) t0 = time.time() # Find the parameters of the scattered rays. - mean_scattering_directions, log_partialities, normalization_factors, outgoing_beam_divergence_tensor = _get_diffraction_arcsegment( + ( + mean_scattering_directions, + log_partialities, + normalization_factors, + outgoing_beam_divergence_tensor, + ) = _get_diffraction_arcsegment( p_vectors, misori_concentration_tensors, xray_propagation_direction, @@ -203,60 +453,111 @@ def diffract( ) if verbose: - print(f'Reciprocal space part took {time.time()-t0}') + print(f"Reciprocal space part took {time.time() - t0}") t0 = time.time() # Splat grain realspace shapes (#NOTE Consider using the non-strained non-azimuthally shifted directions to simplify gradients) - detectorspace_grainshape_projections, projected_thicknes_scale_factors = self.splat_grainshapes( - mean_scattering_directions, shape_concentration_tensors, W, pixellengths,) + detectorspace_grainshape_projections, projected_thicknes_scale_factors = ( + self.splat_grainshapes( + mean_scattering_directions, + shape_concentration_tensors, + W, + pixellengths, + ) + ) if verbose: - print(f'Realspace proj took {time.time()-t0}') + print(f"Realspace proj took {time.time() - t0}") t0 = time.time() # Ray-trace onto detector plane pos = intersection_pos[grain_does_diffract] - ray_lengths = torch.einsum('xi,i->x', detector_origin[None, :] - pos, detector_norm) / torch.einsum('xi,i->x', mean_scattering_directions, detector_norm) - point_of_detector_intersection = pos + ray_lengths[:,None] * mean_scattering_directions - uv_coords = torch.einsum('xi,vi,v->xv',point_of_detector_intersection - detector_origin[None, :], W, 1/pixellengths) + ray_lengths = torch.einsum( + "xi,i->x", detector_origin[None, :] - pos, detector_norm + ) / torch.einsum("xi,i->x", mean_scattering_directions, detector_norm) + point_of_detector_intersection = ( + pos + ray_lengths[:, None] * mean_scattering_directions + ) + uv_coords = torch.einsum( + "xi,vi,v->xv", + point_of_detector_intersection - detector_origin[None, :], + W, + 1 / pixellengths, + ) # Smearing due to angular divergence of the scattered beam. - W_scaled = W * 1 / pixellengths[:, None] #TODO I think this assumes orthorgonal pixel directions. - divergence_smearing_tensor = torch.einsum('ui,xij,vj->xuv', W_scaled, outgoing_beam_divergence_tensor * ray_lengths[:, None, None]**2, W_scaled) - detspace_splat_concentration = torch.linalg.inv( torch.linalg.inv(detectorspace_grainshape_projections) + divergence_smearing_tensor) - intensity_spread_out_factor = torch.sqrt( torch.linalg.det(detspace_splat_concentration) / torch.linalg.det(detectorspace_grainshape_projections) ) + W_scaled = ( + W * 1 / pixellengths[:, None] + ) # TODO I think this assumes orthorgonal pixel directions. + divergence_smearing_tensor = torch.einsum( + "ui,xij,vj->xuv", + W_scaled, + outgoing_beam_divergence_tensor * ray_lengths[:, None, None] ** 2, + W_scaled, + ) + detspace_splat_concentration = torch.linalg.inv( + torch.linalg.inv(detectorspace_grainshape_projections) + + divergence_smearing_tensor + ) + intensity_spread_out_factor = torch.sqrt( + torch.linalg.det(detspace_splat_concentration) + / torch.linalg.det(detectorspace_grainshape_projections) + ) # Collect all intensity modifying factors - polarization_factors = _polarization(mean_scattering_directions, beam.polarization_vector) - solid_angle_factor = torch.abs(torch.einsum('xi,i->x',mean_scattering_directions, detector_norm)) - scalefactors = structure_factors[hkl_does_diffract] * projected_thicknes_scale_factors * np.exp(log_partialities) * normalization_factors * intensity_spread_out_factor\ - * beam_intensity_factors[grain_does_diffract]*polarization_factors*solid_angle_factor + polarization_factors = _polarization( + mean_scattering_directions, beam.polarization_vector + ) + solid_angle_factor = torch.abs( + torch.einsum("xi,i->x", mean_scattering_directions, detector_norm) + ) + scalefactors = ( + structure_factors[hkl_does_diffract] + * projected_thicknes_scale_factors + * np.exp(log_partialities) + * normalization_factors + * intensity_spread_out_factor + * beam_intensity_factors[grain_does_diffract] + * polarization_factors + * solid_angle_factor + ) # Discard reflections too far out on the rocking curve. - does_diffract = log_partialities > - 1 * threshold + does_diffract = log_partialities > -1 * threshold if verbose: - print(f'Raytracing took {time.time()-t0}') + print(f"Raytracing took {time.time() - t0}") t0 = time.time() n_batches = torch.sum(does_diffract) // peaks_batch_size + 1 f = torch.zeros(detector.shape) for peaks_batch in range(n_batches): - f += detector.render_gaussian_splats( - uv_coords[does_diffract][peaks_batch*peaks_batch_size:(peaks_batch+1)*peaks_batch_size], - scalefactors[does_diffract][peaks_batch*peaks_batch_size:(peaks_batch+1)*peaks_batch_size], - detspace_splat_concentration[does_diffract][peaks_batch*peaks_batch_size:(peaks_batch+1)*peaks_batch_size], - threshold + log_partialities[does_diffract][peaks_batch*peaks_batch_size:(peaks_batch+1)*peaks_batch_size], + uv_coords[does_diffract][ + peaks_batch * peaks_batch_size : (peaks_batch + 1) + * peaks_batch_size + ], + scalefactors[does_diffract][ + peaks_batch * peaks_batch_size : (peaks_batch + 1) + * peaks_batch_size + ], + detspace_splat_concentration[does_diffract][ + peaks_batch * peaks_batch_size : (peaks_batch + 1) + * peaks_batch_size + ], + threshold + + log_partialities[does_diffract][ + peaks_batch * peaks_batch_size : (peaks_batch + 1) + * peaks_batch_size + ], ) if verbose: - print(f'Rasterization took {time.time()-t0}') + print(f"Rasterization took {time.time() - t0}") t0 = time.time() return f - def splat_grainshapes( self, @@ -265,7 +566,7 @@ def splat_grainshapes( W: Tensor, pixellengths: Tensor, ): - """ Project the labratory space shape-concentration-tensors of a range of grains along a the scattering directions + """Project the labratory space shape-concentration-tensors of a range of grains along a the scattering directions into 2D detector pixels space. Parameters @@ -278,39 +579,50 @@ def splat_grainshapes( Pixel-direction unit vectors stacked, shape ``(2, 3)`` pixellengths : Tensor Pixel lengths, shape ``(2, 3)`` - + Returns ------- projected_shape_pixelunits : Tensor - Concentarion tensor of the projected grainshape in detector pixel units, shape ``(N, 2, 2)`` + Concentarion tensor of the projected grainshape in detector pixel units, shape ``(N, 2, 2)`` projected_thicknes_scale_factors : Tensor Intensity scaling factor due the projected thickness of the grain, shape ``(N,)`` """ # grain_volume = torch.sqrt(1/torch.linalg.det(shape_concentration_tensors)) - dSd = torch.einsum('xi,xij,xj->x', mean_scattering_directions, shape_concentration_tensors, mean_scattering_directions) - - inner_term = shape_concentration_tensors - torch.einsum( - 'xij,xj,xk,xkl->xil', - shape_concentration_tensors, - mean_scattering_directions, + dSd = torch.einsum( + "xi,xij,xj->x", mean_scattering_directions, shape_concentration_tensors, - ) / dSd[:, None, None] - + mean_scattering_directions, + ) + + inner_term = ( + shape_concentration_tensors + - torch.einsum( + "xij,xj,xk,xkl->xil", + shape_concentration_tensors, + mean_scattering_directions, + mean_scattering_directions, + shape_concentration_tensors, + ) + / dSd[:, None, None] + ) + W_scaled = W * pixellengths[:, None] projected_shape_pixelunits = torch.einsum( - 'ui,xij,vj->xuv', W_scaled, inner_term, W_scaled, + "ui,xij,vj->xuv", + W_scaled, + inner_term, + W_scaled, ) - return projected_shape_pixelunits, 1/torch.sqrt(dSd) - + return projected_shape_pixelunits, 1 / torch.sqrt(dSd) def transform( - self, - rigid_body_motion : RigidBodyMotion, - time : float = 1.0, - ): + self, + rigid_body_motion: RigidBodyMotion, + time: float = 1.0, + ): """Transform the polycrystal by performing a rigid body motion. This updates all the sample-information in-place. @@ -331,19 +643,27 @@ def transform( translation_vector = rigid_body_motion.translation * time # Rotate vectors: - self.positions = torch.einsum('ij,gj->gi', Rot_mat, self.positions-rigid_body_motion.origin[None,:])+rigid_body_motion.origin[None,:] + self.positions = ( + torch.einsum( + "ij,gj->gi", Rot_mat, self.positions - rigid_body_motion.origin[None, :] + ) + + rigid_body_motion.origin[None, :] + ) # Rotate lattices - self.orientaions = torch.einsum('ij,gjk->gik', Rot_mat, self.orientaions) + self.orientaions = torch.einsum("ij,gjk->gik", Rot_mat, self.orientaions) - #Rotate tensors - self.shape_concentration_tensors = torch.einsum('ij,gjk,lk ->gil', Rot_mat, self.shape_concentration_tensors, Rot_mat) - self.misori_concentration_tensors = torch.einsum('ij,gjk,lk ->gil', Rot_mat, self.misori_concentration_tensors, Rot_mat) - self.strains = torch.einsum('ij,gjk,lk ->gil', Rot_mat, self.strains, Rot_mat) - - #Translate - self.positions = self.positions + translation_vector[None, :] + # Rotate tensors + self.shape_concentration_tensors = torch.einsum( + "ij,gjk,lk ->gil", Rot_mat, self.shape_concentration_tensors, Rot_mat + ) + self.misori_concentration_tensors = torch.einsum( + "ij,gjk,lk ->gil", Rot_mat, self.misori_concentration_tensors, Rot_mat + ) + self.strains = torch.einsum("ij,gjk,lk ->gil", Rot_mat, self.strains, Rot_mat) + # Translate + self.positions = self.positions + translation_vector[None, :] # ------------------------------------------------------------------------------------------ # The methods below here are for computing polefigures, not needed for diffraction patterns. @@ -355,38 +675,48 @@ def render_polefigure( both_hemispheres: bool = False, max_misorientation: float = 0.1, ): - + # Make coordinate arrays if both_hemispheres: - polar, azim = np.meshgrid(np.linspace(0, np.pi, int(180//resolution_in_degrees)+1), - np.linspace(0, 2*np.pi, int(360//resolution_in_degrees)+1)) + polar, azim = np.meshgrid( + np.linspace(0, np.pi, int(180 // resolution_in_degrees) + 1), + np.linspace(0, 2 * np.pi, int(360 // resolution_in_degrees) + 1), + ) else: - polar, azim = np.meshgrid(np.linspace(0, np.pi/2, int(90//resolution_in_degrees)+1), - np.linspace(0, 2*np.pi, int(360//resolution_in_degrees)+1)) - - y_map = torch.tensor(np.stack([ - np.sin(polar) * np.cos(azim), - np.sin(polar) * np.sin(azim), - np.cos(polar) - ], axis=-1)) - + polar, azim = np.meshgrid( + np.linspace(0, np.pi / 2, int(90 // resolution_in_degrees) + 1), + np.linspace(0, 2 * np.pi, int(360 // resolution_in_degrees) + 1), + ) + + y_map = torch.tensor( + np.stack( + [ + np.sin(polar) * np.cos(azim), + np.sin(polar) * np.sin(azim), + np.cos(polar), + ], + axis=-1, + ) + ) + p, scale, T_proj = self.splat_onto_polefigure(hkl) patch_size = 16 - + f = self.rasterize_on_unitvector_map( y_map, p, scale, T_proj, - max_angle= 3*max_misorientation + (resolution_in_degrees*np.pi/180)*patch_size/2, + max_angle=3 * max_misorientation + + (resolution_in_degrees * np.pi / 180) * patch_size / 2, ) return f, polar, azim def splat_onto_polefigure( - self, - hkl: tuple[int], - ): + self, + hkl: tuple[int], + ): # A = form_a_mat(self.phase.unit_cell) # WARNING! xfab uses non-compatible A- and B-matrices. The B-matrix is considered the correct one. # B = 2 * np.pi * np.linalg.inv(A).T @@ -394,7 +724,7 @@ def splat_onto_polefigure( h = torch.tensor(B @ hkl) h = h / torch.linalg.norm(h) - levi_cita_symbol = np.zeros((3,3,3)) + levi_cita_symbol = np.zeros((3, 3, 3)) levi_cita_symbol[0, 1, 2] = 1 levi_cita_symbol[1, 2, 0] = 1 levi_cita_symbol[2, 0, 1] = 1 @@ -405,69 +735,91 @@ def splat_onto_polefigure( # TODO reduce the number of symmetries evaluated for low-multiplicity peaks n_symmetries = len(self.phase.rot) - + volumes = torch.sqrt(torch.linalg.det(self.shape_concentration_tensors)) - p_vectors = torch.einsum('gij,sjk,k->gsi', self.orientaions, ensure_torch(self.phase.rot), h) - + p_vectors = torch.einsum( + "gij,sjk,k->gsi", self.orientaions, ensure_torch(self.phase.rot), h + ) + # This is the trick: - pTp = torch.einsum('gsi,gij,gsj->gs', p_vectors, self.misori_concentration_tensors, p_vectors) - inner_part = self.misori_concentration_tensors[:, None, :, :] - torch.einsum( - 'gij,gsj,gsk,gkl->gsil', - self.misori_concentration_tensors, - p_vectors, - p_vectors, - self.misori_concentration_tensors, - ) / pTp[:, :, None, None] + pTp = torch.einsum( + "gsi,gij,gsj->gs", p_vectors, self.misori_concentration_tensors, p_vectors + ) + inner_part = ( + self.misori_concentration_tensors[:, None, :, :] + - torch.einsum( + "gij,gsj,gsk,gkl->gsil", + self.misori_concentration_tensors, + p_vectors, + p_vectors, + self.misori_concentration_tensors, + ) + / pTp[:, :, None, None] + ) projected_misorientation = torch.einsum( - 'gsj,ijk,gsil,lmn,gsm->gskn', + "gsj,ijk,gsil,lmn,gsm->gskn", p_vectors, levi_cita_symbol, inner_part, levi_cita_symbol, p_vectors, ) - - scale = 1 / n_symmetries / torch.sum(volumes) * volumes[:, None] * 2 * torch.sqrt(torch.linalg.det(self.misori_concentration_tensors))[:, None] / np.sqrt( pTp ) + + scale = ( + 1 + / n_symmetries + / torch.sum(volumes) + * volumes[:, None] + * 2 + * torch.sqrt(torch.linalg.det(self.misori_concentration_tensors))[:, None] + / np.sqrt(pTp) + ) return p_vectors, scale, projected_misorientation - + def rasterize_on_unitvector_map( self, - y : Tensor, - p : Tensor, - scale : Tensor, - T_proj : Tensor, + y: Tensor, + p: Tensor, + scale: Tensor, + T_proj: Tensor, max_angle: float, patch_size: int = 16, ): - + shape = y.shape[:2] min_dp = np.cos(max_angle) - n_patches_dim1 = (shape[0]-1)//patch_size+1 - n_patches_dim2 = (shape[1]-1)//patch_size+1 + n_patches_dim1 = (shape[0] - 1) // patch_size + 1 + n_patches_dim2 = (shape[1] - 1) // patch_size + 1 # Rasterization f = torch.zeros(shape) for patch_index_1 in range(n_patches_dim1): for patch_index_2 in range(n_patches_dim2): - # Figure out what splat lie in this pole figure patch - y_patch = y[patch_size*patch_index_1:patch_size*(patch_index_1+1), - patch_size*patch_index_2:patch_size*(patch_index_2+1)] + y_patch = y[ + patch_size * patch_index_1 : patch_size * (patch_index_1 + 1), + patch_size * patch_index_2 : patch_size * (patch_index_2 + 1), + ] patch_mean = torch.mean(y_patch, axis=(0, 1)) patch_mean_y = patch_mean / torch.linalg.norm(patch_mean) - include_index = torch.abs(torch.einsum('gsj,j->gs', p, patch_mean_y)) > min_dp - # If none, continue + include_index = ( + torch.abs(torch.einsum("gsj,j->gs", p, patch_mean_y)) > min_dp + ) + # If none, continue if not torch.any(include_index): continue # Evaluate gaussians - arg = -torch.einsum('pai,xij,paj->xpa', y_patch, T_proj[include_index], y_patch) + arg = -torch.einsum( + "pai,xij,paj->xpa", y_patch, T_proj[include_index], y_patch + ) vals = torch.exp(arg) * scale[include_index, np.newaxis, np.newaxis] - - f[patch_size*patch_index_1:patch_size*(patch_index_1+1), - patch_size*patch_index_2:patch_size*(patch_index_2+1)]\ - += torch.sum(vals, axis=0) + + f[ + patch_size * patch_index_1 : patch_size * (patch_index_1 + 1), + patch_size * patch_index_2 : patch_size * (patch_index_2 + 1), + ] += torch.sum(vals, axis=0) return f
      -
    • - xrd_simulator.laue + xrd_simulator.gaussian_crystal_model
    • diff --git a/docs/index.html b/docs/index.html index 2a7dac4..bf7f243 100644 --- a/docs/index.html +++ b/docs/index.html @@ -19,6 +19,7 @@ + @@ -124,6 +125,7 @@
    • Beam.load()
      • +
    • GaussianBeam
    • detector
      • @@ -171,13 +174,30 @@ +
    • gaussian_crystal_model +
    • templates
      • -
    • laue
      • @@ -206,9 +226,7 @@
      -
      -
      -https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true + https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true https://img.shields.io/pypi/pyversions/xrd-simulator.svg? https://github.com/FABLE-3DXRD/xrd_simulator/actions/workflows/python-package-run-tests-linux-py313.yml/badge.svg? @@ -582,7 +600,7 @@

      Documentation
      -diffract(beam: Beam, rigid_body_motion: RigidBodyMotion, min_bragg_angle: float = 0, max_bragg_angle: float = 1.5707963267948966, detector: Detector | None = None, verbose: bool = True) → Dict[str, Tensor | list][source]
      +diffract(beam: Beam, rigid_body_motion: RigidBodyMotion, min_bragg_angle: float = 0, max_bragg_angle: float = 1.5707963267948966, detector: Detector | None = None, verbose: bool = True) → dict[str, Tensor | list][source]

      Compute diffraction from the rotating and translating polycrystal.

      Simulates diffraction while the sample is illuminated by an X-ray beam. Returns peaks that can be rendered on a detector.

      @@ -1264,6 +1282,30 @@

      Documentation +
      +class xrd_simulator.beam.GaussianBeam(beam_centroid_position, xray_propagation_direction, wavelength, polarization_vector, long_axis_width=None, long_axis_direction=None, short_axis_width=None, short_axis_direction=None)[source]
      +

      Represents a monochromatic X-ray beam with a 2D gaussian cross-section.

      +
      +
      Parameters:
      +
        +

      • beam_centroid_position (torch.Tensor or numpy.ndarray) – A point on the center-line of the beam. (3,).

        • +

      • xray_propagation_direction (torch.Tensor or numpy.ndarray) – Propagation direction of X-rays, shape (3,).

        • +

      • wavelength (float) – X-ray wavelength in units of angstrom.

        • +

      • polarization_vector (torch.Tensor or numpy.ndarray) – Beam linear polarization unit vector, shape (3,). Must be +orthogonal to the X-ray propagation direction.

        • +

      • long_axis_width (float) – Width of the beam along the longest direction.

        • +

      • long_axis_direction (torch.Tensor or numpy.ndarray) – Direction of the major-axis of the beam, shape (3,). Must be +orthogonal to the X-ray propagation direction. If not given, the +beam is assumed circular with a width given by long_axis_width.

        • +

      • long_axis_width – Width of the beam along the shortest direction.

        • +

      • long_axis_direction – Direction of the minor-axis of the beam, shape (3,). Must be +orthogonal to the X-ray propagation direction.

        • +
      +
      +
      +
      +

      detector

      @@ -1660,6 +1702,32 @@

      Documentation +
      +render_gaussian_splats(uv_corrds: Tensor, scale_factors: Tensor, concentration_tensors: Tensor, threshold: Tensor, patch_size: int = 32)[source]
      +

      Basic 2D Gaussian rasterizer. Each gaussian has the expression:

      +
      +\[f = s \exp(-[u-u_0, v-v_0] S [u-u_0, v-v_0]^T).\]
      +
      +
      Parameters:
      +
        +

      • uv_corrds (Tensor) – Centroid pixel coordinates, shape (N, 2)

        • +

      • scale_factors (Tensor) – Intensity scale factors, shape (N,)

        • +

      • concentration_tensors (Tensor) – Shape concentration tensors, shape (N, 2, 2)

        • +

      • threshold (Tensor, shape (N,)) – Per gaussian threshold value used to truncate the Gaussians. Gaussians are set to zero +when the value falls bellow scale_factors*torch.exp(-theshold).

        • +

      • patch_size (int) – For evaluation the detector is split into patches of shize patch_size by patch_size.

        • +
      +
      +
      Returns:
      +

      detector_image – Detector image, shape self.shape.

      +
      +
      Return type:
      +

      Tensor

      +
      +
      +
      +
      peaks_detector_intersection(peaks_dict: Dict[str, Tensor | List[str]], frames_to_render: int = 1, verbose: bool = True) → Dict[str, Tensor | List[str]][source]
      @@ -1984,6 +2052,320 @@

      Documentation +

      gaussian_crystal_model

      +

      Gaussian subgrain models for polycrystalline X-ray diffraction simulation.

      +

      Represent a polycrystal as a collection of subgrains with Gaussian real-space +density profiles and narrow Gaussian orientation distributions. Each subgrain +is defined by its crystallographic phase, centroid, shape tensor, mean lattice +orientation, misorientation tensor, and uniform strain tensor. Both the shape +and orientation spread may be anisotropic.

      +

      GaussianSubgrain describes an individual subgrain, while +GaussianPolycrystal assembles subgrains and renders diffraction +patterns using a GaussianBeam and a +Detector.

      +

      The forward model computes beam-subgrain intersections, reflection +partialities, and scattered-beam angular distributions. Illuminated subgrain +shapes are projected onto the detector and combined with mosaicity-induced +angular broadening to form two-dimensional Gaussian spots. These contributions +are rasterized and summed into a detector intensity image. Small sample +rotations during an exposure are approximated by additional Gaussian +orientation broadening.

      +

      The module also supports rigid-body transformations of the polycrystal and +pole-figure rendering.

      +

      Notes

      +

      The current implementation supports single-phase polycrystals and assumes a +monochromatic, collimated incident beam. Orientation distributions are +approximated in the tangent space about each subgrain’s mean orientation, +requiring small misorientations. Uniform strain modifies reciprocal-lattice +vectors, but strain-distribution broadening is not included within an +individual subgrain. A distribution of strains can instead be approximated by +combining appropriately weighted diffraction patterns from subgrain +populations with different uniform strain tensors.

      +
      +

      See also

      +
      +
      Gaussian model: theory and references

      Scattering geometry, Gaussian orientation distributions, reciprocal-space +maps, model assumptions, illustrative tests, and references.

      +
      +
      +
      +

      Examples

      +

      Simulate a static diffraction pattern from 2000 Gaussian quartz subgrains with +random positions, orientations, shapes, and anisotropic orientation spreads.

      +

      Run the example from the repository root with xrd_simulator installed, so +that tests/data/quartz.cif is available. Real-space lengths are in +micrometres, the wavelength and unit-cell lengths are in angstroms, and +unit-cell angles are in degrees. Orientation-spread parameters are in radians.

      +

      First, define the beam, detector, and crystallographic phase:

      +
      import numpy as np
+import matplotlib.pyplot as plt
+from scipy.spatial.transform import Rotation as R
+
+from xrd_simulator.beam import GaussianBeam
+from xrd_simulator.detector import Detector
+from xrd_simulator.phase import Phase
+from xrd_simulator.gaussian_crystal_model import (
+    GaussianSubgrain,
+    GaussianPolycrystal,
+)
+
+rng = np.random.default_rng(0)
+
+gaussian_beam = GaussianBeam(
+    xray_propagation_direction=np.array([1.0, 0.0, 0.0]),
+    beam_centroid_position=np.array([0.0, 0.0, 0.0]),
+    wavelength=0.28523,
+    polarization_vector=np.array([0.0, 1.0, 0.0]),
+    long_axis_width=300.0,
+    long_axis_direction=np.array([0.0, 0.0, 1.0]),
+    short_axis_width=50.0,
+    short_axis_direction=np.array([0.0, 1.0, 0.0]),
+)
+
+detector = Detector(
+    det_corner_0=np.array([142938.3, -38400.0, -38400.0]),
+    det_corner_1=np.array([142938.3, 38400.0, -38400.0]),
+    det_corner_2=np.array([142938.3, -38400.0, 38400.0]),
+    pixel_size=(25.0, 25.0),
+    gaussian_sigma=1.0,
+    max_gaussian_kernel_radius=5,
+)
+
+quartz = Phase(
+    unit_cell=[4.926, 4.926, 5.4189, 90.0, 90.0, 120.0],
+    sgname="P3221",
+    path_to_cif_file="tests/data/quartz.cif",
+)
+
      +
      +

      Next, build the polycrystal. The helper generates a symmetric positive-definite +width tensor with a randomly oriented symmetry axis. Its eigenvalues are +axis_1**2, axis_1**2, and axis_2**2. Shape-width parameters range +from 50 to 200 micrometres, and misorientation-width parameters range from +0.01 to 0.03 radians. The distinction between these width parameters and +standard deviations is explained on the theory page.

      +
      def make_random_tensor(axis_1, axis_2):
+    random_direction = rng.normal(size=3)
+    random_direction /= np.linalg.norm(random_direction)
+    return (
+        axis_1**2 * np.eye(3)
+        + (axis_2**2 - axis_1**2)
+        * np.outer(random_direction, random_direction)
+    )
+
+grain_list = []
+
+for _ in range(2000):
+    position = rng.uniform(-500.0, 500.0, size=3)
+    shape_tensor = make_random_tensor(
+        rng.uniform(50.0, 200.0),
+        rng.uniform(50.0, 200.0),
+    )
+    orientation = R.random(random_state=rng).as_matrix()
+    misorientation_tensor = make_random_tensor(
+        rng.uniform(0.01, 0.03),
+        rng.uniform(0.01, 0.03),
+    )
+
+    grain_list.append(
+        GaussianSubgrain(
+            phase=quartz,
+            position=position,
+            shape_tensor=shape_tensor,
+            orientation=orientation,
+            misorientation_tensor=misorientation_tensor,
+            strain_tensor=np.zeros((3, 3)),
+        )
+    )
+
+gaussian_polycrystal = GaussianPolycrystal(
+    grain_list,
+    max_misorientation=0.03,
+    max_grain_size=200.0,
+)
+
      +
      +

      Compute a diffraction pattern without sample rotation during the exposure:

      +
      f = gaussian_polycrystal.diffract(
+    beam=gaussian_beam,
+    detector=detector,
+    verbose=True,
+    threshold=10.0,
+)
+
      +
      +

      Finally, plot the full detector image and a magnified region on the same +logarithmic intensity scale:

      +
      log_intensity = np.log10(f.detach().cpu().numpy() + 1e-10)
+
+fig, axs = plt.subplots(
+    1, 3,
+    figsize=(8, 4),
+    gridspec_kw={"width_ratios": (1, 1, 0.1)},
+    constrained_layout=True,
+)
+axs[0].imshow(log_intensity, cmap="jet", vmin=3, vmax=6)
+img = axs[1].imshow(log_intensity, cmap="jet", vmin=3, vmax=6)
+axs[0].set_title("Full detector")
+axs[1].set_title("Magnified region")
+axs[1].set_xlim(1500, 2200)
+axs[1].set_ylim(2200, 1500)
+fig.colorbar(img, cax=axs[2], label=r"$\log_{10}(I + 10^{-10})$")
+plt.show()
+
      +
      +
      +Simulated quartz diffraction pattern and a magnified detector region. + +
      +

      Representative output from the Gaussian workflow. The precise spot pattern +depends on the random subgrain realization. This image is retained from the +original workflow and need not match the seeded realization above exactly.

      +
      +
      +
      +
      +class xrd_simulator.gaussian_crystal_model.GaussianSubgrain(phase: Phase, position: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([0, 0, 0]), shape_tensor: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]), orientation: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]), misorientation_tensor: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([[0.00030625, 0., 0.], [0., 0.00030625, 0.], [0., 0., 0.00030625]]), strain_tensor: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([[0., 0., 0.], [0., 0., 0.], [0., 0., 0.]]))[source]
      +

      A subgrain has a 3D-gaussian density distribution and a narrow (<=0.1 rad) 3D +orientation distribution function.

      +
      +
      +phase[source]
      +

      Object representing the crystal structure of the grain.

      +
      +
      Type:
      +

      Phase

      +
      +
      +
      + +
      +
      +position[source]
      +

      Position of the grain centroid, shape (3,)

      +
      +
      Type:
      +

      torch.Tensor | np.array

      +
      +
      +
      + +
      +
      +shape_tensor[source]
      +

      Covariance tensor of the real-space grain shape. (3, 3,).

      +
      +
      Type:
      +

      torch.Tensor | np.array

      +
      +
      +
      + +
      +
      +orientation[source]
      +

      Orientation of the centroid of the ODF as a rotation matrix, shape (3, 3,)

      +
      +
      Type:
      +

      torch.Tensor | np.array

      +
      +
      +
      + +
      +
      +misorientation_tensor[source]
      +

      Covariance tensor of the grain orientation in the left-hand tangent space. +aka. laboratory coordinates, shape (3, 3,)

      +
      +
      Type:
      +

      torch.Tensor | np.array

      +
      +
      +
      + +
      +
      +strain_tensor[source]
      +

      Strain tensor in laboratory coordinates, shape (3, 3,)

      +
      +
      Type:
      +

      torch.Tensor | np.array

      +
      +
      +
      + +
      + +
      +
      +class xrd_simulator.gaussian_crystal_model.GaussianPolycrystal(grain_list: list[GaussianSubgrain], max_grain_size: float = 1000.0, max_misorientation: float = 0.1)[source]
      +

      Colection of GaussianSubgrain object representing a polycrystalline sample. +Variables max_grain_size and max_misorientation can be supplied to speed up +diffraction calculations.

      +
      +
      +diffract(beam: GaussianBeam, detector: Detector, sample_orientation: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]), sample_translation: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([0., 0., 0.]), sample_rotation_during_exposure: ndarray[Any, dtype[_ScalarType_co]] | Tensor = array([0., 0., 0.]), verbose=False, threshold=6.0, peaks_batch_size=20000)[source]
      +

      Render a single diffraction pattern.

      +
      +
      Parameters:
      +
        +

      • beam (GaussianBeam) – Object representing the x-ray beam hitting the crystal.

        • +

      • detector (Detector) – Object representing the x-ray detector.

        • +

      • sample_orientation (NDArray, optional) – 3-by-3 Rotation matrix representing the orientation of the sample

        • +

      • sample_translation (NDArray, optional) – 3-vector prepresenting the translation of the sample. (the sample is first +rotated about the origin, then translated)

        • +

      • sample_rotation_during_exposure (NDArray, optional) – 3-vector representing a small rotation carried out during the measurement

        • +

      • verbose (bool, optional) – If True print timing information. Default is False.

        • +
      +
      +
      +
      + +
      +
      +splat_grainshapes(mean_scattering_directions: Tensor, shape_concentration_tensors: Tensor, W: Tensor, pixellengths: Tensor)[source]
      +

      Project the labratory space shape-concentration-tensors of a range of grains along a the scattering directions +into 2D detector pixels space.

      +
      +
      Parameters:
      +
        +

      • mean_scattering_directions (Tensor) – Scattering direction unit vectors, shape (N, 3)

        • +

      • shape_concentration_tensors (Tensor) – Shape concentration tensors in laboratory coordinates, shape (N, 3, 3)

        • +

      • W (Tensor) – Pixel-direction unit vectors stacked, shape (2, 3)

        • +

      • pixellengths (Tensor) – Pixel lengths, shape (2, 3)

        • +
      +
      +
      Returns:
      +

        +

      • projected_shape_pixelunits (Tensor) – Concentarion tensor of the projected grainshape in detector pixel units, shape (N, 2, 2)

        • +

      • projected_thicknes_scale_factors (Tensor) – Intensity scaling factor due the projected thickness of the grain, shape (N,)

        • +
      +

      +
      +
      +
      + +
      +
      +transform(rigid_body_motion: RigidBodyMotion, time: float = 1.0)[source]
      +

      Transform the polycrystal by performing a rigid body motion.

      +

      This updates all the sample-information in-place.

      +
      +
      Parameters:
      +
        +

      • rigid_body_motion (RigidBodyMotion) – Rigid body motion object describing the polycrystal transformation +as a function of time on the domain time=[0, 1].

        • +

      • time (float) – Time between [0, 1] at which to call the rigid body motion.

        • +
      +
      +
      +
      + +
      +

      templates

      @@ -2222,13 +2604,6 @@

      Documentation -

      laue

      -

      Collection of functions for solving time-dependent Laue equations.

      -

      This module provides functions for arbitrary rigid body motions. It is mainly -used internally by the xrd_simulator.polycrystal.Polycrystal. However, -for the advanced user, access to these functions may be of interest.

      diff --git a/docs/objects.inv b/docs/objects.inv index cc264cf..30ad941 100644 Binary files a/docs/objects.inv and b/docs/objects.inv differ diff --git a/docs/py-modindex.html b/docs/py-modindex.html index cc701cb..d1310f5 100644 --- a/docs/py-modindex.html +++ b/docs/py-modindex.html @@ -101,7 +101,7 @@

      Python Module Index

          - xrd_simulator.laue + xrd_simulator.gaussian_crystal_model