From ae63ac0b25a3049e8f0191e8aadc83b02035ec55 Mon Sep 17 00:00:00 2001
From: Axel Henningsson <nilsaxelhenningsson@gmail.com>
Date: Thu, 24 Sep 2026 08:56:34 +0200
Subject: [PATCH] Add Gaussian model documentation, example and theory

---
 docs/_images/many_crystals_quartz.png       | Bin 0 -> 242973 bytes
 docs/_images/single_crystal_quartz.png      | Bin 0 -> 90777 bytes
 docs/_images/testing_rocking_curves.png     | Bin 0 -> 119516 bytes
 docs/_sources/gaussian_model_theory.rst.txt | 582 +++++++++++++++++
 docs/_sources/index.rst.txt                 |  16 +-
 docs/gaussian_model_theory.html             | 626 ++++++++++++++++++
 docs/genindex.html                          |  48 +-
 docs/index.html                             | 399 +++++++++++-
 docs/objects.inv                            | Bin 1256 -> 1895 bytes
 docs/py-modindex.html                       |   2 +-
 docs/searchindex.js                         |   2 +-
 docs/source/gaussian_model_theory.rst       | 582 +++++++++++++++++
 docs/source/index.rst                       |  16 +-
 xrd_simulator/gaussian_crystal_model.py     | 676 +++++++++++++++-----
 14 files changed, 2744 insertions(+), 205 deletions(-)
 create mode 100644 docs/_images/many_crystals_quartz.png
 create mode 100644 docs/_images/single_crystal_quartz.png
 create mode 100644 docs/_images/testing_rocking_curves.png
 create mode 100644 docs/_sources/gaussian_model_theory.rst.txt
 create mode 100644 docs/gaussian_model_theory.html
 create mode 100644 docs/source/gaussian_model_theory.rst

diff --git a/docs/_images/many_crystals_quartz.png b/docs/_images/many_crystals_quartz.png
new file mode 100644
index 0000000000000000000000000000000000000000..f68ae1a053528c5b11f45c7ea5a8e72de065fe9a
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diff --git a/docs/_sources/gaussian_model_theory.rst.txt b/docs/_sources/gaussian_model_theory.rst.txt
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@@ -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 <https://doi.org/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
+   <https://doi.org/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
+   <https://doi.org/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 @@
+
+
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+              <div class="local-toc"><ul>
+<li><a class="reference internal" href="#">Gaussian model: theory and references</a><ul>
+<li><a class="reference internal" href="#gaussian-subgrain-representation">Gaussian subgrain representation</a></li>
+<li><a class="reference internal" href="#scattering-theory">Scattering theory</a><ul>
+<li><a class="reference internal" href="#scattered-beam-distribution">Scattered-beam distribution</a></li>
+<li><a class="reference internal" href="#incident-beam-coordinates">Incident-beam coordinates</a></li>
+<li><a class="reference internal" href="#reciprocal-space-coordinates">Reciprocal-space coordinates</a></li>
+<li><a class="reference internal" href="#outgoing-beam-coordinates">Outgoing-beam coordinates</a></li>
+<li><a class="reference internal" href="#linearized-conservation-equations">Linearized conservation equations</a></li>
+<li><a class="reference internal" href="#monochromatic-collimated-specialization">Monochromatic, collimated specialization</a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#anisotropic-gaussian-orientation-distributions">Anisotropic Gaussian orientation distributions</a><ul>
+<li><a class="reference internal" href="#tangent-space-approximation">Tangent-space approximation</a></li>
+<li><a class="reference internal" href="#pole-density-as-a-line-integral">Pole density as a line integral</a></li>
+<li><a class="reference internal" href="#projected-concentration-tensor">Projected concentration tensor</a></li>
+<li><a class="reference internal" href="#reciprocal-space-map">Reciprocal-space map</a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#illustrative-tests">Illustrative tests</a><ul>
+<li><a class="reference internal" href="#comparison-with-a-tetrahedral-crystal">Comparison with a tetrahedral crystal</a></li>
+<li><a class="reference internal" href="#high-resolution-rocking-curve">High-resolution rocking curve</a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#references">References</a></li>
+</ul>
+</li>
+</ul>
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+             
+  <section id="gaussian-model-theory-and-references">
+<span id="gaussian-model-theory"></span><h1>Gaussian model: theory and references<a class="headerlink" href="#gaussian-model-theory-and-references" title="Link to this heading"></a></h1>
+<p>This page describes the scattering model and approximations underlying
+<a class="reference internal" href="index.html#module-xrd_simulator.gaussian_crystal_model" title="xrd_simulator.gaussian_crystal_model"><code class="xref py py-mod docutils literal notranslate"><span class="pre">xrd_simulator.gaussian_crystal_model</span></code></a>. A worked example is provided in
+the module documentation.</p>
+<nav class="contents local" id="on-this-page">
+<p class="topic-title">On this page</p>
+<ul class="simple">
+<li><p><a class="reference internal" href="#gaussian-subgrain-representation" id="id6">Gaussian subgrain representation</a></p></li>
+<li><p><a class="reference internal" href="#scattering-theory" id="id7">Scattering theory</a></p>
+<ul>
+<li><p><a class="reference internal" href="#scattered-beam-distribution" id="id8">Scattered-beam distribution</a></p></li>
+<li><p><a class="reference internal" href="#incident-beam-coordinates" id="id9">Incident-beam coordinates</a></p></li>
+<li><p><a class="reference internal" href="#reciprocal-space-coordinates" id="id10">Reciprocal-space coordinates</a></p></li>
+<li><p><a class="reference internal" href="#outgoing-beam-coordinates" id="id11">Outgoing-beam coordinates</a></p></li>
+<li><p><a class="reference internal" href="#linearized-conservation-equations" id="id12">Linearized conservation equations</a></p></li>
+<li><p><a class="reference internal" href="#monochromatic-collimated-specialization" id="id13">Monochromatic, collimated specialization</a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#anisotropic-gaussian-orientation-distributions" id="id14">Anisotropic Gaussian orientation distributions</a></p>
+<ul>
+<li><p><a class="reference internal" href="#tangent-space-approximation" id="id15">Tangent-space approximation</a></p></li>
+<li><p><a class="reference internal" href="#pole-density-as-a-line-integral" id="id16">Pole density as a line integral</a></p></li>
+<li><p><a class="reference internal" href="#projected-concentration-tensor" id="id17">Projected concentration tensor</a></p></li>
+<li><p><a class="reference internal" href="#reciprocal-space-map" id="id18">Reciprocal-space map</a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#illustrative-tests" id="id19">Illustrative tests</a></p>
+<ul>
+<li><p><a class="reference internal" href="#comparison-with-a-tetrahedral-crystal" id="id20">Comparison with a tetrahedral crystal</a></p></li>
+<li><p><a class="reference internal" href="#high-resolution-rocking-curve" id="id21">High-resolution rocking curve</a></p></li>
+</ul>
+</li>
+<li><p><a class="reference internal" href="#references" id="id22">References</a></p></li>
+</ul>
+</nav>
+<section id="gaussian-subgrain-representation">
+<h2><a class="toc-backref" href="#id6" role="doc-backlink">Gaussian subgrain representation</a><a class="headerlink" href="#gaussian-subgrain-representation" title="Link to this heading"></a></h2>
+<p>Gaussian splatting represents a three-dimensional scene using Gaussian
+primitives whose projected contributions can be rendered efficiently by
+rasterization <a class="reference internal" href="#kerbl2023" id="id1"><span>[Kerbl2023]</span></a>. 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
+<a class="reference internal" href="#brehm2023" id="id2"><span>[Brehm2023]</span></a>.</p>
+<p>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.</p>
+<p>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.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The Gaussian-splat rendering path does not apply a separate detector
+point-spread convolution. The <code class="docutils literal notranslate"><span class="pre">gaussian_sigma</span></code> and
+<code class="docutils literal notranslate"><span class="pre">max_gaussian_kernel_radius</span></code> detector settings retained in the worked
+example therefore do not add this broadening to its Gaussian spots.</p>
+</div>
+</section>
+<section id="scattering-theory">
+<h2><a class="toc-backref" href="#id7" role="doc-backlink">Scattering theory</a><a class="headerlink" href="#scattering-theory" title="Link to this heading"></a></h2>
+<section id="scattered-beam-distribution">
+<h3><a class="toc-backref" href="#id8" role="doc-backlink">Scattered-beam distribution</a><a class="headerlink" href="#scattered-beam-distribution" title="Link to this heading"></a></h3>
+<p>Let the incident beam have a phase-space density <span class="math notranslate nohighlight">\(p(\mathbf{k})\)</span>
+centred on <span class="math notranslate nohighlight">\(\mathbf{k}_0\)</span>, with nominal wavenumber</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-wavenumber">
+<span class="eqno">(1)<a class="headerlink" href="#equation-gaussian-wavenumber" title="Link to this equation"></a></span>\[k = \lvert\mathbf{k}_0\rvert = \frac{2\pi}{\lambda_0}.\]</div>
+<p>For a particular reflection, let <span class="math notranslate nohighlight">\(f(\mathbf{q})\)</span> describe the subgrain’s
+reciprocal-space map (RSM) around a reciprocal-lattice vector
+<span class="math notranslate nohighlight">\(\mathbf{G}_0\)</span>. The scattered-beam distribution is written as</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-scattering-integral">
+<span class="eqno">(2)<a class="headerlink" href="#equation-gaussian-scattering-integral" title="Link to this equation"></a></span>\[I(\mathbf{p})
+= \int \mathrm{d}^{3}\mathbf{k}
+  \int \mathrm{d}^{3}\mathbf{q}\;
+  p(\mathbf{k})\,f(\mathbf{q})\,
+  \delta\!\left(\lvert\mathbf{k}\rvert-\lvert\mathbf{p}\rvert\right)
+  \delta^{(3)}\!\left(\mathbf{k}+\mathbf{q}-\mathbf{p}\right).\]</div>
+<p>The scalar Dirac delta enforces elastic scattering, while the
+three-dimensional Dirac delta enforces momentum conservation.</p>
+</section>
+<section id="incident-beam-coordinates">
+<h3><a class="toc-backref" href="#id9" role="doc-backlink">Incident-beam coordinates</a><a class="headerlink" href="#incident-beam-coordinates" title="Link to this heading"></a></h3>
+<p>Following the coordinate construction used by Poulsen et al.
+<a class="reference internal" href="#poulsen2017" id="id3"><span>[Poulsen2017]</span></a>, write the incident wavevector as</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-incident-wavevector">
+<span class="eqno">(3)<a class="headerlink" href="#equation-gaussian-incident-wavevector" title="Link to this equation"></a></span>\[\mathbf{k}
+= \mathbf{k}_0
+  + k\left(
+      \varepsilon\,\hat{\mathbf{k}}_0
+      + \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel}
+      + \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp}
+    \right).\]</div>
+<p>A hat denotes a unit vector. The vector
+<span class="math notranslate nohighlight">\(\hat{\mathbf{k}}_{\parallel}\)</span> is perpendicular to
+<span class="math notranslate nohighlight">\(\mathbf{k}_0\)</span> and lies in the plane spanned by <span class="math notranslate nohighlight">\(\mathbf{k}_0\)</span>
+and <span class="math notranslate nohighlight">\(\mathbf{G}_0\)</span>. Its sign is chosen so that
+<span class="math notranslate nohighlight">\(\mathbf{G}_0\cdot\hat{\mathbf{k}}_{\parallel}&gt;0\)</span>. Complete the
+right-handed basis with</p>
+<div class="math notranslate nohighlight">
+\[\hat{\mathbf{k}}_{\perp}
+= \hat{\mathbf{k}}_0\times\hat{\mathbf{k}}_{\parallel}.\]</div>
+<p>The coordinate <span class="math notranslate nohighlight">\(\varepsilon\)</span> describes the relative wavenumber
+deviation, and <span class="math notranslate nohighlight">\(\zeta_{\parallel}\)</span> and <span class="math notranslate nohighlight">\(\zeta_{\perp}\)</span> describe
+small angular deviations of the incident beam.</p>
+</section>
+<section id="reciprocal-space-coordinates">
+<h3><a class="toc-backref" href="#id10" role="doc-backlink">Reciprocal-space coordinates</a><a class="headerlink" href="#reciprocal-space-coordinates" title="Link to this heading"></a></h3>
+<p>Define the nominal Bragg angle and the exactly aligned scattering vector by</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-nominal-scattering-vector">
+<span class="eqno">(4)<a class="headerlink" href="#equation-gaussian-nominal-scattering-vector" title="Link to this equation"></a></span>\[\begin{split}\begin{aligned}
+\theta_0
+&amp;= \arcsin\!\left(\frac{\lvert\mathbf{G}_0\rvert}{2k}\right), \\
+\mathbf{Q}
+&amp;= 2k\sin\theta_0\left(
+    \cos\theta_0\,\hat{\mathbf{k}}_{\parallel}
+    - \sin\theta_0\,\hat{\mathbf{k}}_0
+  \right) \\
+&amp;= 2k\sin\theta_0\,\hat{\mathbf{Q}}.
+\end{aligned}\end{split}\]</div>
+<p>The vectors <span class="math notranslate nohighlight">\(\mathbf{G}_0\)</span> and <span class="math notranslate nohighlight">\(\mathbf{Q}\)</span> coincide when the
+reflection is exactly aligned. For a small misalignment, their difference is
+parallel, to first order, to the rocking direction</p>
+<div class="math notranslate nohighlight">
+\[\hat{\mathbf{q}}_{\mathrm{rock}}
+= \cos\theta_0\,\hat{\mathbf{k}}_0
+  + \sin\theta_0\,\hat{\mathbf{k}}_{\parallel}.\]</div>
+<p>Use dimensionless rocking, strain, and rolling coordinates to write</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-reciprocal-coordinates">
+<span class="eqno">(5)<a class="headerlink" href="#equation-gaussian-reciprocal-coordinates" title="Link to this equation"></a></span>\[\begin{split}\begin{aligned}
+\mathbf{q}
+&amp;= \mathbf{G}_0
+   + 2k\sin\theta_0\left(
+       q_{\mathrm{rock}}\,\hat{\mathbf{q}}_{\mathrm{rock}}
+       + q_{\mathrm{strain}}\,\hat{\mathbf{Q}}
+       + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp}
+     \right) \\
+&amp;\approx \mathbf{Q}
+   + 2k\sin\theta_0\left(
+       (q_{\mathrm{rock}}-\delta q)\,
+           \hat{\mathbf{q}}_{\mathrm{rock}}
+       + q_{\mathrm{strain}}\,\hat{\mathbf{Q}}
+       + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp}
+     \right),
+\end{aligned}\end{split}\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-rocking-offset">
+<span class="eqno">(6)<a class="headerlink" href="#equation-gaussian-rocking-offset" title="Link to this equation"></a></span>\[\delta q
+= \frac{
+    (\mathbf{Q}-\mathbf{G}_0)\cdot\hat{\mathbf{q}}_{\mathrm{rock}}
+  }{2k\sin\theta_0}.\]</div>
+<p>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 <span class="math notranslate nohighlight">\((\varepsilon,\zeta_{\parallel},\zeta_{\perp})\)</span> and the
+reciprocal-space coordinates
+<span class="math notranslate nohighlight">\((q_{\mathrm{rock}},q_{\mathrm{strain}},q_{\mathrm{roll}})\)</span> separate
+bandwidth, collimation, misorientation, and strain effects.</p>
+</section>
+<section id="outgoing-beam-coordinates">
+<h3><a class="toc-backref" href="#id11" role="doc-backlink">Outgoing-beam coordinates</a><a class="headerlink" href="#outgoing-beam-coordinates" title="Link to this heading"></a></h3>
+<p>The nominal scattered wavevector is</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-nominal-outgoing-wavevector">
+<span class="eqno">(7)<a class="headerlink" href="#equation-gaussian-nominal-outgoing-wavevector" title="Link to this equation"></a></span>\[\mathbf{k}_h
+= \mathbf{k}_0+\mathbf{Q}
+= k\left(
+    \cos(2\theta_0)\,\hat{\mathbf{k}}_0
+    + \sin(2\theta_0)\,\hat{\mathbf{k}}_{\parallel}
+  \right).\]</div>
+<p>Define the radial direction perpendicular to <span class="math notranslate nohighlight">\(\mathbf{k}_h\)</span> by</p>
+<div class="math notranslate nohighlight">
+\[\hat{\mathbf{k}}_{\mathrm{rad}}
+= \cos(2\theta_0)\,\hat{\mathbf{k}}_{\parallel}
+  - \sin(2\theta_0)\,\hat{\mathbf{k}}_0.\]</div>
+<p>The outgoing wavevector can then be written as</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-outgoing-wavevector">
+<span class="eqno">(8)<a class="headerlink" href="#equation-gaussian-outgoing-wavevector" title="Link to this equation"></a></span>\[\mathbf{p}
+= \mathbf{k}_h
+  + k\left(
+      \varepsilon'\,\hat{\mathbf{k}}_h
+      + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}}
+      + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}
+    \right).\]</div>
+<p>Here <span class="math notranslate nohighlight">\(\psi_{\mathrm{rad}}\)</span> and <span class="math notranslate nohighlight">\(\psi_{\mathrm{azim}}\)</span> are small
+transverse angular deviations from the nominal outgoing direction.</p>
+</section>
+<section id="linearized-conservation-equations">
+<h3><a class="toc-backref" href="#id12" role="doc-backlink">Linearized conservation equations</a><a class="headerlink" href="#linearized-conservation-equations" title="Link to this heading"></a></h3>
+<p>To first order,</p>
+<div class="math notranslate nohighlight">
+\[\lvert\mathbf{k}\rvert\approx k(1+\varepsilon),
+\qquad
+\lvert\mathbf{p}\rvert\approx k(1+\varepsilon').\]</div>
+<p>Energy conservation therefore sets <span class="math notranslate nohighlight">\(\varepsilon'=\varepsilon\)</span>.
+Momentum conservation gives <span class="math notranslate nohighlight">\(\mathbf{q}=\mathbf{p}-\mathbf{k}\)</span> and,
+after substitution,</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-linearized-momentum">
+<span class="eqno">(9)<a class="headerlink" href="#equation-gaussian-linearized-momentum" title="Link to this equation"></a></span>\[\begin{split}\begin{aligned}
+&amp;2\sin\theta_0\Bigl[
+    (q_{\mathrm{rock}}-\delta q)\,
+        \hat{\mathbf{q}}_{\mathrm{rock}}
+    + q_{\mathrm{strain}}\,\hat{\mathbf{Q}}
+    + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp}
+\Bigr] \\
+&amp;\qquad = 2\sin\theta_0\,\varepsilon\,\hat{\mathbf{Q}}
+    + \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}} \\
+&amp;\qquad\quad
+    + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}
+    - \zeta_{\parallel}\,\hat{\mathbf{k}}_{\parallel}
+    - \zeta_{\perp}\,\hat{\mathbf{k}}_{\perp}.
+\end{aligned}\end{split}\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The incident-divergence terms have minus signs because the incident
+wavevector is subtracted in <span class="math notranslate nohighlight">\(\mathbf{q}=\mathbf{p}-\mathbf{k}\)</span>.
+This corrects the corresponding signs in the original workflow. It does
+not affect the monochromatic, collimated specialization below, where
+both terms vanish.</p>
+</div>
+</section>
+<section id="monochromatic-collimated-specialization">
+<h3><a class="toc-backref" href="#id13" role="doc-backlink">Monochromatic, collimated specialization</a><a class="headerlink" href="#monochromatic-collimated-specialization" title="Link to this heading"></a></h3>
+<p>The current model assumes</p>
+<div class="math notranslate nohighlight">
+\[\varepsilon=\zeta_{\parallel}=\zeta_{\perp}=0,
+\qquad q_{\mathrm{strain}}=0.\]</div>
+<p>The last condition excludes strain-distribution broadening; it does not
+require the uniform subgrain strain to vanish. The conservation equation
+reduces to</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{aligned}
+&amp;2\sin\theta_0\left[
+    (q_{\mathrm{rock}}-\delta q)\,
+        \hat{\mathbf{q}}_{\mathrm{rock}}
+    + q_{\mathrm{roll}}\,\hat{\mathbf{k}}_{\perp}
+\right] \\
+&amp;\qquad
+= \psi_{\mathrm{rad}}\,\hat{\mathbf{k}}_{\mathrm{rad}}
+  + \psi_{\mathrm{azim}}\,\hat{\mathbf{k}}_{\perp}.
+\end{aligned}\end{split}\]</div>
+<p>For a nondegenerate Bragg geometry, the three coordinate equations are</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-scattering-constraints">
+<span class="eqno">(10)<a class="headerlink" href="#equation-gaussian-scattering-constraints" title="Link to this equation"></a></span>\[q_{\mathrm{rock}}=\delta q,
+\qquad
+\psi_{\mathrm{rad}}=0,
+\qquad
+\psi_{\mathrm{azim}}=2\sin\theta_0\,q_{\mathrm{roll}}.\]</div>
+<p>Thus, the scattered angular distribution samples the RSM along a line
+parallel to <span class="math notranslate nohighlight">\(\hat{\mathbf{k}}_{\perp}\)</span>, offset by <span class="math notranslate nohighlight">\(\delta q\)</span>
+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.</p>
+</section>
+</section>
+<section id="anisotropic-gaussian-orientation-distributions">
+<h2><a class="toc-backref" href="#id14" role="doc-backlink">Anisotropic Gaussian orientation distributions</a><a class="headerlink" href="#anisotropic-gaussian-orientation-distributions" title="Link to this heading"></a></h2>
+<section id="tangent-space-approximation">
+<h3><a class="toc-backref" href="#id15" role="doc-backlink">Tangent-space approximation</a><a class="headerlink" href="#tangent-space-approximation" title="Link to this heading"></a></h3>
+<p>Consider a narrow orientation distribution centred on an orientation
+<span class="math notranslate nohighlight">\(g_0\)</span>. Approximate orientation space by its left tangent space at
+<span class="math notranslate nohighlight">\(g_0\)</span>, so that a small rotation vector <span class="math notranslate nohighlight">\(\mathbf{r}\)</span> acts in
+laboratory coordinates:</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-tangent-rotation">
+<span class="eqno">(11)<a class="headerlink" href="#equation-gaussian-tangent-rotation" title="Link to this equation"></a></span>\[\begin{split}g\approx R_{\mathbf{r}}g_0
+\approx
+\begin{bmatrix}
+    1    &amp; -r_z &amp; r_y \\
+    r_z  &amp; 1    &amp; -r_x \\
+    -r_y &amp; r_x  &amp; 1
+\end{bmatrix}g_0.\end{split}\]</div>
+<p>The Gaussian orientation density used in the workflow is</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-orientation-density">
+<span class="eqno">(12)<a class="headerlink" href="#equation-gaussian-orientation-density" title="Link to this equation"></a></span>\[f(\mathbf{r})
+= \frac{2}{\sqrt{\pi}}\sqrt{\det\mathbf{T}}\,
+  \exp\!\left(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r}\right),\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{T}\)</span> is a symmetric positive-definite concentration
+tensor.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The prefactor above retains the workflow’s normalization convention.
+With ordinary Euclidean tangent-space measure,
+<span class="math notranslate nohighlight">\(\int f(\mathbf{r})\,\mathrm{d}^{3}\mathbf{r}=2\pi\)</span>, not one.
+It should therefore not be interpreted as a unit-normalized Euclidean
+probability density without rescaling.</p>
+<p>For an exponential of the form
+<span class="math notranslate nohighlight">\(\exp(-\mathbf{r}^{\mathsf{T}}\mathbf{T}\mathbf{r})\)</span>, the
+corresponding normalized Gaussian has covariance
+<span class="math notranslate nohighlight">\(\tfrac{1}{2}\mathbf{T}^{-1}\)</span>. The implementation directly inverts
+<code class="docutils literal notranslate"><span class="pre">misorientation_tensor</span></code> to obtain <span class="math notranslate nohighlight">\(\mathbf{T}\)</span>. Consequently,
+the square roots of the input tensor’s eigenvalues are Gaussian width
+parameters, not standard deviations. For an input eigenvalue
+<span class="math notranslate nohighlight">\(a^2\)</span>, the standard deviation is <span class="math notranslate nohighlight">\(a/\sqrt{2}\)</span>.
+The real-space Gaussian uses the same exponential convention for
+<code class="docutils literal notranslate"><span class="pre">shape_tensor</span></code>. This distinction matters because the class attribute
+descriptions currently use the word “covariance” for these inputs.</p>
+</div>
+</section>
+<section id="pole-density-as-a-line-integral">
+<h3><a class="toc-backref" href="#id16" role="doc-backlink">Pole density as a line integral</a><a class="headerlink" href="#pole-density-as-a-line-integral" title="Link to this heading"></a></h3>
+<p>For Miller indices <span class="math notranslate nohighlight">\((h,k,\ell)\)</span>, define the unit lattice direction in
+crystal coordinates by</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{h}
+= \frac{
+    \mathbf{B}_0[h,k,\ell]^{\mathsf{T}}
+  }{
+    \left\lvert\mathbf{B}_0[h,k,\ell]^{\mathsf{T}}\right\rvert
+  },
+\qquad
+\mathbf{p}=g_0\mathbf{h}.\]</div>
+<p>Here <span class="math notranslate nohighlight">\(\mathbf{B}_0\)</span> is the reciprocal-lattice basis matrix. In this
+section, <span class="math notranslate nohighlight">\(\mathbf{p}\)</span> denotes the mean lattice direction, rather than
+the outgoing wavevector used in the scattering section. It is parallel to
+<span class="math notranslate nohighlight">\(\mathbf{G}_0\)</span>. Let <span class="math notranslate nohighlight">\(\mathbf{y}\)</span> be a unit direction in
+laboratory coordinates, parallel to a sampled reciprocal-space vector
+<span class="math notranslate nohighlight">\(\mathbf{q}\)</span>; it is not a Cartesian coordinate-axis label.</p>
+<p>The pole density, also called the pair-correlation function in texture
+analysis, describes the density of lattice directions along
+<span class="math notranslate nohighlight">\(\mathbf{y}\)</span>. Its calculation normally involves an integral over a
+circle in <span class="math notranslate nohighlight">\(\mathrm{SO}(3)\)</span>. Within the tangent-space approximation,
+replace this by an infinite line integral, parameterized as</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-orientation-fibre">
+<span class="eqno">(13)<a class="headerlink" href="#equation-gaussian-orientation-fibre" title="Link to this equation"></a></span>\[\mathbf{r}(\lambda)
+= \frac{\mathbf{p}\times\mathbf{y}}{\mathbf{p}\cdot\mathbf{y}}
+  + \lambda\mathbf{p}
+= \mathbf{r}_0+\lambda\mathbf{p}.\]</div>
+<p>The parameter <span class="math notranslate nohighlight">\(\lambda\)</span> here is a line-integration parameter, not the
+X-ray wavelength. Substitution into the Gaussian density and completion of
+the square give, with
+<span class="math notranslate nohighlight">\(d=\mathbf{p}^{\mathsf{T}}\mathbf{T}\mathbf{p}\)</span>,</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-pole-density-integral">
+<span class="eqno">(14)<a class="headerlink" href="#equation-gaussian-pole-density-integral" title="Link to this equation"></a></span>\[\begin{split}\begin{aligned}
+A(\mathbf{y},\mathbf{p};f)
+&amp;= \int_{-\infty}^{\infty}
+     f\!\left(\mathbf{r}_0+\lambda\mathbf{p}\right)
+     \,\mathrm{d}\lambda \\
+&amp;= \frac{2\sqrt{\det\mathbf{T}}}
+        {\sqrt{d}}
+   \exp\!\left[
+       -\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{r}_0
+       + \frac{
+           (\mathbf{r}_0^{\mathsf{T}}\mathbf{T}\mathbf{p})^2
+         }{d}
+   \right].
+\end{aligned}\end{split}\]</div>
+</section>
+<section id="projected-concentration-tensor">
+<h3><a class="toc-backref" href="#id17" role="doc-backlink">Projected concentration tensor</a><a class="headerlink" href="#projected-concentration-tensor" title="Link to this heading"></a></h3>
+<p>For directions close to the pole, make the further approximation
+<span class="math notranslate nohighlight">\(\mathbf{p}\cdot\mathbf{y}\approx1\)</span>. Define</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{M}
+= \mathbf{T}
+  - \frac{
+      \mathbf{T}\mathbf{p}\mathbf{p}^{\mathsf{T}}\mathbf{T}
+    }{d}.\]</div>
+<p>Let <span class="math notranslate nohighlight">\([\mathbf{p}]_{\times}\)</span> denote the cross-product matrix,
+<span class="math notranslate nohighlight">\([\mathbf{p}]_{\times}\mathbf{y}=\mathbf{p}\times\mathbf{y}\)</span>. Then</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-projected-concentration">
+<span class="eqno">(15)<a class="headerlink" href="#equation-gaussian-projected-concentration" title="Link to this equation"></a></span>\[\mathbf{T}_{\mathbf{p}}
+= [\mathbf{p}]_{\times}^{\mathsf{T}}
+    \mathbf{M}
+    [\mathbf{p}]_{\times},\]</div>
+<p>and the pole density becomes</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-pole-density">
+<span class="eqno">(16)<a class="headerlink" href="#equation-gaussian-pole-density" title="Link to this equation"></a></span>\[A(\mathbf{y},\mathbf{p};f)
+\approx \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\,
+  \exp\!\left(
+      -\mathbf{y}^{\mathsf{T}}\mathbf{T}_{\mathbf{p}}\mathbf{y}
+  \right).\]</div>
+<p>Equivalently, using the Levi-Civita symbol <span class="math notranslate nohighlight">\(\epsilon_{ijk}\)</span> and
+summation over repeated indices,</p>
+<div class="math notranslate nohighlight">
+\[[\mathbf{T}_{\mathbf{p}}]_{ij}
+= p_k\,\epsilon_{lki}
+  \left(
+    T_{lm}-\frac{T_{la}p_a p_b T_{bm}}{d}
+  \right)
+  \epsilon_{mnj}\,p_n.\]</div>
+<p>This absorbs the cross products into the projected tensor and makes the
+subsequent Gaussian expressions more compact.</p>
+</section>
+<section id="reciprocal-space-map">
+<h3><a class="toc-backref" href="#id18" role="doc-backlink">Reciprocal-space map</a><a class="headerlink" href="#reciprocal-space-map" title="Link to this heading"></a></h3>
+<p>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 <span class="math notranslate nohighlight">\(q_{\mathrm{strain}}=0\)</span>.</p>
+<p>Introduce the two-column basis matrix and coordinate vector</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{U}
+= \begin{bmatrix}
+    \hat{\mathbf{q}}_{\mathrm{rock}} &amp; \hat{\mathbf{k}}_{\perp}
+  \end{bmatrix},
+\qquad
+\mathbf{u}
+= \begin{bmatrix}
+    q_{\mathrm{rock}} \\
+    q_{\mathrm{roll}}
+  \end{bmatrix}.\end{split}\]</div>
+<p>In the dimensionless reciprocal-space coordinates defined above, the
+workflow’s local RSM is</p>
+<div class="math notranslate nohighlight" id="equation-gaussian-local-rsm">
+<span class="eqno">(17)<a class="headerlink" href="#equation-gaussian-local-rsm" title="Link to this equation"></a></span>\[f(\mathbf{q})
+\approx \delta(q_{\mathrm{strain}})\,
+   \frac{2\sqrt{\det\mathbf{T}}}{\sqrt{d}}\,
+   \exp\!\left[
+       -\mathbf{u}^{\mathsf{T}}
+        \left(\mathbf{U}^{\mathsf{T}}
+              \mathbf{T}_{\mathbf{p}}\mathbf{U}\right)
+        \mathbf{u}
+   \right].\]</div>
+<p>The Dirac delta expresses the absence of a distribution of lattice spacings
+within the subgrain. Sampling this Gaussian at
+<span class="math notranslate nohighlight">\(q_{\mathrm{rock}}=\delta q\)</span>, as required by
+<a class="reference internal" href="#equation-gaussian-scattering-constraints">(10)</a>, gives the scattered angular
+distribution for the reflection.</p>
+</section>
+</section>
+<section id="illustrative-tests">
+<h2><a class="toc-backref" href="#id19" role="doc-backlink">Illustrative tests</a><a class="headerlink" href="#illustrative-tests" title="Link to this heading"></a></h2>
+<section id="comparison-with-a-tetrahedral-crystal">
+<h3><a class="toc-backref" href="#id20" role="doc-backlink">Comparison with a tetrahedral crystal</a><a class="headerlink" href="#comparison-with-a-tetrahedral-crystal" title="Link to this heading"></a></h3>
+<p>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.</p>
+<p>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.</p>
+<p>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.</p>
+<figure class="align-center" id="id4">
+<a class="reference internal image-reference" href="_images/single_crystal_quartz.png"><img alt="Quartz diffraction patterns from tetrahedral and Gaussian models." src="_images/single_crystal_quartz.png" style="width: 100%;" />
+</a>
+<figcaption>
+<p><span class="caption-text">Comparison of the tetrahedral crystal model with a seven-Gaussian
+approximation during a one-degree sample rotation.</span><a class="headerlink" href="#id4" title="Link to this image"></a></p>
+</figcaption>
+</figure>
+</section>
+<section id="high-resolution-rocking-curve">
+<h3><a class="toc-backref" href="#id21" role="doc-backlink">High-resolution rocking curve</a><a class="headerlink" href="#high-resolution-rocking-curve" title="Link to this heading"></a></h3>
+<p>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.</p>
+<figure class="align-center" id="id5">
+<a class="reference internal image-reference" href="_images/testing_rocking_curves.png"><img alt="Reciprocal-space map and profiles from a high-resolution rocking scan." src="_images/testing_rocking_curves.png" style="width: 100%;" />
+</a>
+<figcaption>
+<p><span class="caption-text">High-resolution rocking-curve test showing the reciprocal-space map and
+the contributions of grain size and misorientation to the simulated
+profiles.</span><a class="headerlink" href="#id5" title="Link to this image"></a></p>
+</figcaption>
+</figure>
+</section>
+</section>
+<section id="references">
+<h2><a class="toc-backref" href="#id22" role="doc-backlink">References</a><a class="headerlink" href="#references" title="Link to this heading"></a></h2>
+<div role="list" class="citation-list">
+<div class="citation" id="kerbl2023" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Kerbl2023</a><span class="fn-bracket">]</span></span>
+<p>Kerbl, B., Kopanas, G., Leimkühler, T. and Drettakis, G.
+(2023). 3D Gaussian Splatting for Real-Time Radiance Field Rendering.
+<em>ACM Transactions on Graphics</em>, <strong>42</strong>(4).
+<a class="reference external" href="https://doi.org/10.1145/3592433">doi:10.1145/3592433</a>.</p>
+</div>
+<div class="citation" id="brehm2023" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">Brehm2023</a><span class="fn-bracket">]</span></span>
+<p>Brehm, W., White, T. and Chapman, H. N. (2023).
+Crystal diffraction prediction and partiality estimation using Gaussian
+basis functions. <em>Acta Crystallographica Section A: Foundations and
+Advances</em>, <strong>79</strong>, 145–162.
+<a class="reference external" href="https://doi.org/10.1107/S2053273323000682">doi:10.1107/S2053273323000682</a>.</p>
+</div>
+<div class="citation" id="poulsen2017" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">Poulsen2017</a><span class="fn-bracket">]</span></span>
+<p>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. <em>Journal of Applied Crystallography</em>, <strong>50</strong>, 1441–1456.
+<a class="reference external" href="https://doi.org/10.1107/S1600576717011037">doi:10.1107/S1600576717011037</a>.</p>
+</div>
+</div>
+</section>
+</section>
+
+
+           </div>
+          </div>
+          <footer>
+
+  <hr/>
+
+  <div role="contentinfo">
+    <p>&#169; Copyright 2021, Axel Henningsson.</p>
+  </div>
+
+  Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a
+    <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a>
+    provided by <a href="https://readthedocs.org">Read the Docs</a>.
+   
+
+</footer>
+        </div>
+      </div>
+    </section>
+  </div>
+  <script>
+      jQuery(function () {
+          SphinxRtdTheme.Navigation.enable(true);
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+
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+</html>
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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 @@ <h2 id="D">D</h2>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.detector.Detector">Detector (class in xrd_simulator.detector)</a>
 </li>
-      <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.diffract">diffract() (xrd_simulator.polycrystal.Polycrystal method)</a>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.diffract">diffract() (xrd_simulator.gaussian_crystal_model.GaussianPolycrystal method)</a>
+
+      <ul>
+        <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.diffract">(xrd_simulator.polycrystal.Polycrystal method)</a>
 </li>
+      </ul></li>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh.dof">dof (xrd_simulator.mesh.TetraMesh attribute)</a>
 </li>
   </ul></td>
@@ -190,9 +194,15 @@ <h2 id="G">G</h2>
       <li><a href="index.html#xrd_simulator.detector.Detector.gaussian_kernel">gaussian_kernel (xrd_simulator.detector.Detector attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.detector.Detector.gaussian_sigma">gaussian_sigma (xrd_simulator.detector.Detector attribute)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.beam.GaussianBeam">GaussianBeam (class in xrd_simulator.beam)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal">GaussianPolycrystal (class in xrd_simulator.gaussian_crystal_model)</a>
 </li>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain">GaussianSubgrain (class in xrd_simulator.gaussian_crystal_model)</a>
+</li>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh.generate_mesh_from_levelset">generate_mesh_from_levelset() (xrd_simulator.mesh.TetraMesh class method)</a>
 </li>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh.generate_mesh_from_vertices">generate_mesh_from_vertices() (xrd_simulator.mesh.TetraMesh class method)</a>
@@ -256,6 +266,8 @@ <h2 id="M">M</h2>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.mesh_sample">mesh_sample (xrd_simulator.polycrystal.Polycrystal attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.phase.Phase.miller_indices">miller_indices (xrd_simulator.phase.Phase attribute)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.misorientation_tensor">misorientation_tensor (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
 </li>
       <li>
     module
@@ -265,7 +277,7 @@ <h2 id="M">M</h2>
 </li>
         <li><a href="index.html#module-xrd_simulator.detector">xrd_simulator.detector</a>
 </li>
-        <li><a href="index.html#module-xrd_simulator.laue">xrd_simulator.laue</a>
+        <li><a href="index.html#module-xrd_simulator.gaussian_crystal_model">xrd_simulator.gaussian_crystal_model</a>
 </li>
         <li><a href="index.html#module-xrd_simulator.mesh">xrd_simulator.mesh</a>
 </li>
@@ -296,6 +308,8 @@ <h2 id="N">N</h2>
 <h2 id="O">O</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.orientation">orientation (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
+</li>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.orientation_lab">orientation_lab (xrd_simulator.polycrystal.Polycrystal attribute)</a>
 </li>
   </ul></td>
@@ -315,6 +329,8 @@ <h2 id="P">P</h2>
       <li><a href="index.html#xrd_simulator.detector.Detector.peaks_detector_intersection">peaks_detector_intersection() (xrd_simulator.detector.Detector method)</a>
 </li>
       <li><a href="index.html#xrd_simulator.phase.Phase">Phase (class in xrd_simulator.phase)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.phase">phase (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.phases">phases (xrd_simulator.polycrystal.Polycrystal attribute)</a>
 </li>
@@ -333,6 +349,8 @@ <h2 id="P">P</h2>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal">Polycrystal (class in xrd_simulator.polycrystal)</a>
 </li>
       <li><a href="index.html#xrd_simulator.templates.polycrystal_from_odf">polycrystal_from_odf() (in module xrd_simulator.templates)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.position">position (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
 </li>
   </ul></td>
 </tr></table>
@@ -341,17 +359,19 @@ <h2 id="R">R</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.detector.Detector.render">render() (xrd_simulator.detector.Detector method)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.detector.Detector.render_gaussian_splats">render_gaussian_splats() (xrd_simulator.detector.Detector method)</a>
 </li>
       <li><a href="index.html#xrd_simulator.motion.RigidBodyMotion">RigidBodyMotion (class in xrd_simulator.motion)</a>
 </li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh.rotate">rotate() (xrd_simulator.mesh.TetraMesh method)</a>
 
       <ul>
         <li><a href="index.html#xrd_simulator.motion.RigidBodyMotion.rotate">(xrd_simulator.motion.RigidBodyMotion method)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.motion.RigidBodyMotion.rotation_angle">rotation_angle (xrd_simulator.motion.RigidBodyMotion attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.motion.RigidBodyMotion.rotation_axis">rotation_axis (xrd_simulator.motion.RigidBodyMotion attribute)</a>
@@ -376,15 +396,21 @@ <h2 id="S">S</h2>
         <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.save">(xrd_simulator.polycrystal.Polycrystal method)</a>
 </li>
       </ul></li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.beam.Beam.set_beam_vertices">set_beam_vertices() (xrd_simulator.beam.Beam method)</a>
 </li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.phase.Phase.sgname">sgname (xrd_simulator.phase.Phase attribute)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.shape_tensor">shape_tensor (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.splat_grainshapes">splat_grainshapes() (xrd_simulator.gaussian_crystal_model.GaussianPolycrystal method)</a>
 </li>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.strain_lab">strain_lab (xrd_simulator.polycrystal.Polycrystal attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.strain_sample">strain_sample (xrd_simulator.polycrystal.Polycrystal attribute)</a>
+</li>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.strain_tensor">strain_tensor (xrd_simulator.gaussian_crystal_model.GaussianSubgrain attribute)</a>
 </li>
       <li><a href="index.html#xrd_simulator.detector.Detector.structure_factor">structure_factor (xrd_simulator.detector.Detector attribute)</a>
 </li>
@@ -398,8 +424,12 @@ <h2 id="T">T</h2>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh">TetraMesh (class in xrd_simulator.mesh)</a>
 </li>
-      <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.transform">transform() (xrd_simulator.polycrystal.Polycrystal method)</a>
+      <li><a href="index.html#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.transform">transform() (xrd_simulator.gaussian_crystal_model.GaussianPolycrystal method)</a>
+
+      <ul>
+        <li><a href="index.html#xrd_simulator.polycrystal.Polycrystal.transform">(xrd_simulator.polycrystal.Polycrystal method)</a>
 </li>
+      </ul></li>
   </ul></td>
   <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="index.html#xrd_simulator.mesh.TetraMesh.translate">translate() (xrd_simulator.mesh.TetraMesh method)</a>
@@ -463,10 +493,10 @@ <h2 id="X">X</h2>
 </li>
       </ul></li>
       <li>
-    xrd_simulator.laue
+    xrd_simulator.gaussian_crystal_model
 
       <ul>
-        <li><a href="index.html#module-xrd_simulator.laue">module</a>
+        <li><a href="index.html#module-xrd_simulator.gaussian_crystal_model">module</a>
 </li>
       </ul></li>
       <li>
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 @@
       <script src="_static/sphinx_highlight.js?v=dc90522c"></script>
       <script src="_static/clipboard.min.js?v=a7894cd8"></script>
       <script src="_static/copybutton.js?v=f281be69"></script>
+      <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
     <script src="_static/js/theme.js"></script>
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" /> 
@@ -124,6 +125,7 @@
 <li><a class="reference internal" href="#xrd_simulator.beam.Beam.load"><code class="docutils literal notranslate"><span class="pre">Beam.load()</span></code></a></li>
 </ul>
 </li>
+<li><a class="reference internal" href="#xrd_simulator.beam.GaussianBeam"><code class="docutils literal notranslate"><span class="pre">GaussianBeam</span></code></a></li>
 </ul>
 </li>
 <li><a class="reference internal" href="#module-xrd_simulator.detector">detector</a><ul>
@@ -150,6 +152,7 @@
 <li><a class="reference internal" href="#xrd_simulator.detector.Detector.load"><code class="docutils literal notranslate"><span class="pre">Detector.load()</span></code></a></li>
 <li><a class="reference internal" href="#xrd_simulator.detector.Detector.render"><code class="docutils literal notranslate"><span class="pre">Detector.render()</span></code></a></li>
 <li><a class="reference internal" href="#xrd_simulator.detector.Detector.contains"><code class="docutils literal notranslate"><span class="pre">Detector.contains()</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.detector.Detector.render_gaussian_splats"><code class="docutils literal notranslate"><span class="pre">Detector.render_gaussian_splats()</span></code></a></li>
 <li><a class="reference internal" href="#xrd_simulator.detector.Detector.peaks_detector_intersection"><code class="docutils literal notranslate"><span class="pre">Detector.peaks_detector_intersection()</span></code></a></li>
 </ul>
 </li>
@@ -171,13 +174,30 @@
 </li>
 </ul>
 </li>
+<li><a class="reference internal" href="#module-xrd_simulator.gaussian_crystal_model">gaussian_crystal_model</a><ul>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain</span></code></a><ul>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.phase"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.phase</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.position"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.position</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.shape_tensor"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.shape_tensor</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.orientation"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.orientation</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.misorientation_tensor"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.misorientation_tensor</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.strain_tensor"><code class="docutils literal notranslate"><span class="pre">GaussianSubgrain.strain_tensor</span></code></a></li>
+</ul>
+</li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal"><code class="docutils literal notranslate"><span class="pre">GaussianPolycrystal</span></code></a><ul>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.diffract"><code class="docutils literal notranslate"><span class="pre">GaussianPolycrystal.diffract()</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.splat_grainshapes"><code class="docutils literal notranslate"><span class="pre">GaussianPolycrystal.splat_grainshapes()</span></code></a></li>
+<li><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.transform"><code class="docutils literal notranslate"><span class="pre">GaussianPolycrystal.transform()</span></code></a></li>
+</ul>
+</li>
+</ul>
+</li>
 <li><a class="reference internal" href="#module-xrd_simulator.templates">templates</a><ul>
 <li><a class="reference internal" href="#xrd_simulator.templates.s3dxrd"><code class="docutils literal notranslate"><span class="pre">s3dxrd()</span></code></a></li>
 <li><a class="reference internal" href="#xrd_simulator.templates.polycrystal_from_odf"><code class="docutils literal notranslate"><span class="pre">polycrystal_from_odf()</span></code></a></li>
 <li><a class="reference internal" href="#xrd_simulator.templates.get_uniform_powder_sample"><code class="docutils literal notranslate"><span class="pre">get_uniform_powder_sample()</span></code></a></li>
 </ul>
 </li>
-<li><a class="reference internal" href="#module-xrd_simulator.laue">laue</a></li>
 </ul>
 </li>
 </ul>
@@ -206,9 +226,7 @@
           <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
            <div itemprop="articleBody">
              
-  <div class="toctree-wrapper compound">
-</div>
-<img alt="https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true" src="https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true" />
+  <img alt="https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true" src="https://github.com/FABLE-3DXRD/xrd_simulator/blob/main/docs/source/images/logo.png?raw=true" />
 <a class="reference external image-reference" href="https://pypi.org/project/xrd-simulator/"><img alt="https://img.shields.io/pypi/pyversions/xrd-simulator.svg?" src="https://img.shields.io/pypi/pyversions/xrd-simulator.svg?" />
 </a>
 <a class="reference external image-reference" href="https://github.com/FABLE-3DXRD/xrd_simulator/actions/workflows/python-package-run-tests-linux-py313.yml/badge.svg"><img alt="https://github.com/FABLE-3DXRD/xrd_simulator/actions/workflows/python-package-run-tests-linux-py313.yml/badge.svg?" src="https://github.com/FABLE-3DXRD/xrd_simulator/actions/workflows/python-package-run-tests-linux-py313.yml/badge.svg?" />
@@ -582,7 +600,7 @@ <h1>Documentation<a class="headerlink" href="#documentation" title="Link to this
 
 <dl class="py method">
 <dt class="sig sig-object py" id="xrd_simulator.polycrystal.Polycrystal.diffract">
-<span class="sig-name descname"><span class="pre">diffract</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beam</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.beam.Beam" title="xrd_simulator.beam.Beam"><span class="pre">Beam</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">rigid_body_motion</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.motion.RigidBodyMotion" title="xrd_simulator.motion.RigidBodyMotion"><span class="pre">RigidBodyMotion</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">min_bragg_angle</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_bragg_angle</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1.5707963267948966</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">detector</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.detector.Detector" title="xrd_simulator.detector.Detector"><span class="pre">Detector</span></a><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">None</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Dict</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Tensor</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">list</span><span class="p"><span class="pre">]</span></span></span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/polycrystal.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.polycrystal.Polycrystal.diffract" title="Link to this definition"></a></dt>
+<span class="sig-name descname"><span class="pre">diffract</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beam</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.beam.Beam" title="xrd_simulator.beam.Beam"><span class="pre">Beam</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">rigid_body_motion</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.motion.RigidBodyMotion" title="xrd_simulator.motion.RigidBodyMotion"><span class="pre">RigidBodyMotion</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">min_bragg_angle</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_bragg_angle</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1.5707963267948966</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">detector</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.detector.Detector" title="xrd_simulator.detector.Detector"><span class="pre">Detector</span></a><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">None</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">dict</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Tensor</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">list</span><span class="p"><span class="pre">]</span></span></span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/polycrystal.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.polycrystal.Polycrystal.diffract" title="Link to this definition"></a></dt>
 <dd><p>Compute diffraction from the rotating and translating polycrystal.</p>
 <p>Simulates diffraction while the sample is illuminated by an X-ray beam.
 Returns peaks that can be rendered on a detector.</p>
@@ -1264,6 +1282,30 @@ <h1>Documentation<a class="headerlink" href="#documentation" title="Link to this
 
 </dd></dl>
 
+<dl class="py class">
+<dt class="sig sig-object py" id="xrd_simulator.beam.GaussianBeam">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">xrd_simulator.beam.</span></span><span class="sig-name descname"><span class="pre">GaussianBeam</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beam_centroid_position</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">xray_propagation_direction</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">wavelength</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">polarization_vector</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">long_axis_width</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">long_axis_direction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">short_axis_width</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">short_axis_direction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/beam.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.beam.GaussianBeam" title="Link to this definition"></a></dt>
+<dd><p>Represents a monochromatic X-ray beam with a 2D gaussian cross-section.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>beam_centroid_position</strong> (<em>torch.Tensor</em><em> or </em><em>numpy.ndarray</em>) – A point on the center-line of the beam. <code class="docutils literal notranslate"><span class="pre">(3,)</span></code>.</p></li>
+<li><p><strong>xray_propagation_direction</strong> (<em>torch.Tensor</em><em> or </em><em>numpy.ndarray</em>) – Propagation direction of X-rays, shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code>.</p></li>
+<li><p><strong>wavelength</strong> (<em>float</em>) – X-ray wavelength in units of angstrom.</p></li>
+<li><p><strong>polarization_vector</strong> (<em>torch.Tensor</em><em> or </em><em>numpy.ndarray</em>) – Beam linear polarization unit vector, shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code>. Must be
+orthogonal to the X-ray propagation direction.</p></li>
+<li><p><strong>long_axis_width</strong> (<em>float</em>) – Width of the beam along the longest direction.</p></li>
+<li><p><strong>long_axis_direction</strong> (<em>torch.Tensor</em><em> or </em><em>numpy.ndarray</em>) – Direction of the major-axis of the beam, shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code>. Must be
+orthogonal to the X-ray propagation direction. If not given, the
+beam is assumed circular with a width given by <code class="docutils literal notranslate"><span class="pre">long_axis_width</span></code>.</p></li>
+<li><p><strong>long_axis_width</strong> – Width of the beam along the shortest direction.</p></li>
+<li><p><strong>long_axis_direction</strong> – Direction of the minor-axis of the beam, shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code>. Must be
+orthogonal to the X-ray propagation direction.</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
 </section>
 <section id="module-xrd_simulator.detector">
 <span id="detector"></span><h2>detector<a class="headerlink" href="#module-xrd_simulator.detector" title="Link to this heading"></a></h2>
@@ -1660,6 +1702,32 @@ <h1>Documentation<a class="headerlink" href="#documentation" title="Link to this
 </dl>
 </dd></dl>
 
+<dl class="py method">
+<dt class="sig sig-object py" id="xrd_simulator.detector.Detector.render_gaussian_splats">
+<span class="sig-name descname"><span class="pre">render_gaussian_splats</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">uv_corrds</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scale_factors</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">concentration_tensors</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">threshold</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">patch_size</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">32</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/detector.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.detector.Detector.render_gaussian_splats" title="Link to this definition"></a></dt>
+<dd><p>Basic 2D Gaussian rasterizer. Each gaussian has the expression:</p>
+<div class="math notranslate nohighlight">
+\[f = s \exp(-[u-u_0, v-v_0] S [u-u_0, v-v_0]^T).\]</div>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>uv_corrds</strong> (<em>Tensor</em>) – Centroid pixel coordinates, shape <code class="docutils literal notranslate"><span class="pre">(N,</span> <span class="pre">2)</span></code></p></li>
+<li><p><strong>scale_factors</strong> (<em>Tensor</em>) – Intensity scale factors, shape <code class="docutils literal notranslate"><span class="pre">(N,)</span></code></p></li>
+<li><p><strong>concentration_tensors</strong> (<em>Tensor</em>) – Shape concentration tensors, shape <code class="docutils literal notranslate"><span class="pre">(N,</span> <span class="pre">2,</span> <span class="pre">2)</span></code></p></li>
+<li><p><strong>threshold</strong> (Tensor, shape <code class="docutils literal notranslate"><span class="pre">(N,)</span></code>) – Per gaussian threshold value used to truncate the Gaussians. Gaussians are set to zero
+when the value falls bellow <code class="docutils literal notranslate"><span class="pre">scale_factors*torch.exp(-theshold)</span></code>.</p></li>
+<li><p><strong>patch_size</strong> (<em>int</em>) – For evaluation the detector is split into patches of shize <cite>patch_size</cite> by <cite>patch_size</cite>.</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><strong>detector_image</strong> – Detector image, shape <code class="docutils literal notranslate"><span class="pre">self.shape</span></code>.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Tensor</p>
+</dd>
+</dl>
+</dd></dl>
+
 <dl class="py method">
 <dt class="sig sig-object py" id="xrd_simulator.detector.Detector.peaks_detector_intersection">
 <span class="sig-name descname"><span class="pre">peaks_detector_intersection</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">peaks_dict</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Dict</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Tensor</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">List</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">frames_to_render</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Dict</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Tensor</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">List</span><span class="p"><span class="pre">[</span></span><span class="pre">str</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span></span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/detector.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.detector.Detector.peaks_detector_intersection" title="Link to this definition"></a></dt>
@@ -1984,6 +2052,320 @@ <h1>Documentation<a class="headerlink" href="#documentation" title="Link to this
 
 </dd></dl>
 
+</section>
+<section id="module-xrd_simulator.gaussian_crystal_model">
+<span id="gaussian-crystal-model"></span><h2>gaussian_crystal_model<a class="headerlink" href="#module-xrd_simulator.gaussian_crystal_model" title="Link to this heading"></a></h2>
+<p>Gaussian subgrain models for polycrystalline X-ray diffraction simulation.</p>
+<p>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.</p>
+<p><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain" title="xrd_simulator.gaussian_crystal_model.GaussianSubgrain"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianSubgrain</span></code></a> describes an individual subgrain, while
+<a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal" title="xrd_simulator.gaussian_crystal_model.GaussianPolycrystal"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianPolycrystal</span></code></a> assembles subgrains and renders diffraction
+patterns using a <a class="reference internal" href="#xrd_simulator.beam.GaussianBeam" title="xrd_simulator.beam.GaussianBeam"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianBeam</span></code></a> and a
+<a class="reference internal" href="#xrd_simulator.detector.Detector" title="xrd_simulator.detector.Detector"><code class="xref py py-class docutils literal notranslate"><span class="pre">Detector</span></code></a>.</p>
+<p>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.</p>
+<p>The module also supports rigid-body transformations of the polycrystal and
+pole-figure rendering.</p>
+<p class="rubric">Notes</p>
+<p>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.</p>
+<div class="admonition seealso">
+<p class="admonition-title">See also</p>
+<dl class="simple">
+<dt><a class="reference internal" href="gaussian_model_theory.html"><span class="doc">Gaussian model: theory and references</span></a></dt><dd><p>Scattering geometry, Gaussian orientation distributions, reciprocal-space
+maps, model assumptions, illustrative tests, and references.</p>
+</dd>
+</dl>
+</div>
+<p class="rubric">Examples</p>
+<p>Simulate a static diffraction pattern from 2000 Gaussian quartz subgrains with
+random positions, orientations, shapes, and anisotropic orientation spreads.</p>
+<p>Run the example from the repository root with <code class="docutils literal notranslate"><span class="pre">xrd_simulator</span></code> installed, so
+that <code class="docutils literal notranslate"><span class="pre">tests/data/quartz.cif</span></code> 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.</p>
+<p>First, define the beam, detector, and crystallographic phase:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">scipy.spatial.transform</span> <span class="kn">import</span> <span class="n">Rotation</span> <span class="k">as</span> <span class="n">R</span>
+
+<span class="kn">from</span> <span class="nn">xrd_simulator.beam</span> <span class="kn">import</span> <span class="n">GaussianBeam</span>
+<span class="kn">from</span> <span class="nn">xrd_simulator.detector</span> <span class="kn">import</span> <span class="n">Detector</span>
+<span class="kn">from</span> <span class="nn">xrd_simulator.phase</span> <span class="kn">import</span> <span class="n">Phase</span>
+<span class="kn">from</span> <span class="nn">xrd_simulator.gaussian_crystal_model</span> <span class="kn">import</span> <span class="p">(</span>
+    <span class="n">GaussianSubgrain</span><span class="p">,</span>
+    <span class="n">GaussianPolycrystal</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
+
+<span class="n">gaussian_beam</span> <span class="o">=</span> <span class="n">GaussianBeam</span><span class="p">(</span>
+    <span class="n">xray_propagation_direction</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]),</span>
+    <span class="n">beam_centroid_position</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]),</span>
+    <span class="n">wavelength</span><span class="o">=</span><span class="mf">0.28523</span><span class="p">,</span>
+    <span class="n">polarization_vector</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]),</span>
+    <span class="n">long_axis_width</span><span class="o">=</span><span class="mf">300.0</span><span class="p">,</span>
+    <span class="n">long_axis_direction</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]),</span>
+    <span class="n">short_axis_width</span><span class="o">=</span><span class="mf">50.0</span><span class="p">,</span>
+    <span class="n">short_axis_direction</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">]),</span>
+<span class="p">)</span>
+
+<span class="n">detector</span> <span class="o">=</span> <span class="n">Detector</span><span class="p">(</span>
+    <span class="n">det_corner_0</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">142938.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">38400.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">38400.0</span><span class="p">]),</span>
+    <span class="n">det_corner_1</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">142938.3</span><span class="p">,</span> <span class="mf">38400.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">38400.0</span><span class="p">]),</span>
+    <span class="n">det_corner_2</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">142938.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">38400.0</span><span class="p">,</span> <span class="mf">38400.0</span><span class="p">]),</span>
+    <span class="n">pixel_size</span><span class="o">=</span><span class="p">(</span><span class="mf">25.0</span><span class="p">,</span> <span class="mf">25.0</span><span class="p">),</span>
+    <span class="n">gaussian_sigma</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span>
+    <span class="n">max_gaussian_kernel_radius</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
+<span class="p">)</span>
+
+<span class="n">quartz</span> <span class="o">=</span> <span class="n">Phase</span><span class="p">(</span>
+    <span class="n">unit_cell</span><span class="o">=</span><span class="p">[</span><span class="mf">4.926</span><span class="p">,</span> <span class="mf">4.926</span><span class="p">,</span> <span class="mf">5.4189</span><span class="p">,</span> <span class="mf">90.0</span><span class="p">,</span> <span class="mf">90.0</span><span class="p">,</span> <span class="mf">120.0</span><span class="p">],</span>
+    <span class="n">sgname</span><span class="o">=</span><span class="s2">&quot;P3221&quot;</span><span class="p">,</span>
+    <span class="n">path_to_cif_file</span><span class="o">=</span><span class="s2">&quot;tests/data/quartz.cif&quot;</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+<p>Next, build the polycrystal. The helper generates a symmetric positive-definite
+width tensor with a randomly oriented symmetry axis. Its eigenvalues are
+<code class="docutils literal notranslate"><span class="pre">axis_1**2</span></code>, <code class="docutils literal notranslate"><span class="pre">axis_1**2</span></code>, and <code class="docutils literal notranslate"><span class="pre">axis_2**2</span></code>. 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.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">make_random_tensor</span><span class="p">(</span><span class="n">axis_1</span><span class="p">,</span> <span class="n">axis_2</span><span class="p">):</span>
+    <span class="n">random_direction</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+    <span class="n">random_direction</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">random_direction</span><span class="p">)</span>
+    <span class="k">return</span> <span class="p">(</span>
+        <span class="n">axis_1</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+        <span class="o">+</span> <span class="p">(</span><span class="n">axis_2</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">axis_1</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
+        <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">outer</span><span class="p">(</span><span class="n">random_direction</span><span class="p">,</span> <span class="n">random_direction</span><span class="p">)</span>
+    <span class="p">)</span>
+
+<span class="n">grain_list</span> <span class="o">=</span> <span class="p">[]</span>
+
+<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2000</span><span class="p">):</span>
+    <span class="n">position</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mf">500.0</span><span class="p">,</span> <span class="mf">500.0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+    <span class="n">shape_tensor</span> <span class="o">=</span> <span class="n">make_random_tensor</span><span class="p">(</span>
+        <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">50.0</span><span class="p">,</span> <span class="mf">200.0</span><span class="p">),</span>
+        <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">50.0</span><span class="p">,</span> <span class="mf">200.0</span><span class="p">),</span>
+    <span class="p">)</span>
+    <span class="n">orientation</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="n">rng</span><span class="p">)</span><span class="o">.</span><span class="n">as_matrix</span><span class="p">()</span>
+    <span class="n">misorientation_tensor</span> <span class="o">=</span> <span class="n">make_random_tensor</span><span class="p">(</span>
+        <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">),</span>
+        <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">),</span>
+    <span class="p">)</span>
+
+    <span class="n">grain_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
+        <span class="n">GaussianSubgrain</span><span class="p">(</span>
+            <span class="n">phase</span><span class="o">=</span><span class="n">quartz</span><span class="p">,</span>
+            <span class="n">position</span><span class="o">=</span><span class="n">position</span><span class="p">,</span>
+            <span class="n">shape_tensor</span><span class="o">=</span><span class="n">shape_tensor</span><span class="p">,</span>
+            <span class="n">orientation</span><span class="o">=</span><span class="n">orientation</span><span class="p">,</span>
+            <span class="n">misorientation_tensor</span><span class="o">=</span><span class="n">misorientation_tensor</span><span class="p">,</span>
+            <span class="n">strain_tensor</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)),</span>
+        <span class="p">)</span>
+    <span class="p">)</span>
+
+<span class="n">gaussian_polycrystal</span> <span class="o">=</span> <span class="n">GaussianPolycrystal</span><span class="p">(</span>
+    <span class="n">grain_list</span><span class="p">,</span>
+    <span class="n">max_misorientation</span><span class="o">=</span><span class="mf">0.03</span><span class="p">,</span>
+    <span class="n">max_grain_size</span><span class="o">=</span><span class="mf">200.0</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+<p>Compute a diffraction pattern without sample rotation during the exposure:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="n">gaussian_polycrystal</span><span class="o">.</span><span class="n">diffract</span><span class="p">(</span>
+    <span class="n">beam</span><span class="o">=</span><span class="n">gaussian_beam</span><span class="p">,</span>
+    <span class="n">detector</span><span class="o">=</span><span class="n">detector</span><span class="p">,</span>
+    <span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+    <span class="n">threshold</span><span class="o">=</span><span class="mf">10.0</span><span class="p">,</span>
+<span class="p">)</span>
+</pre></div>
+</div>
+<p>Finally, plot the full detector image and a magnified region on the same
+logarithmic intensity scale:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">log_intensity</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">cpu</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="o">+</span> <span class="mf">1e-10</span><span class="p">)</span>
+
+<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span>
+    <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span>
+    <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span>
+    <span class="n">gridspec_kw</span><span class="o">=</span><span class="p">{</span><span class="s2">&quot;width_ratios&quot;</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)},</span>
+    <span class="n">constrained_layout</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
+<span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">log_intensity</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;jet&quot;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
+<span class="n">img</span> <span class="o">=</span> <span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">log_intensity</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">&quot;jet&quot;</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Full detector&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Magnified region&quot;</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">1500</span><span class="p">,</span> <span class="mi">2200</span><span class="p">)</span>
+<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mi">2200</span><span class="p">,</span> <span class="mi">1500</span><span class="p">)</span>
+<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="n">cax</span><span class="o">=</span><span class="n">axs</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$\log_</span><span class="si">{10}</span><span class="s2">(I + 10^{-10})$&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<figure class="align-center" id="id4">
+<a class="reference internal image-reference" href="_images/many_crystals_quartz.png"><img alt="Simulated quartz diffraction pattern and a magnified detector region." src="_images/many_crystals_quartz.png" style="width: 100%;" />
+</a>
+<figcaption>
+<p><span class="caption-text">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.</span><a class="headerlink" href="#id4" title="Link to this image"></a></p>
+</figcaption>
+</figure>
+<dl class="py class">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">xrd_simulator.gaussian_crystal_model.</span></span><span class="sig-name descname"><span class="pre">GaussianSubgrain</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phase</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.phase.Phase" title="xrd_simulator.phase.Phase"><span class="pre">Phase</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">position</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([0,</span> <span class="pre">0,</span> <span class="pre">0])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">shape_tensor</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([[1.,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">1.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">1.]])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">orientation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([[1.,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">1.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">1.]])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">misorientation_tensor</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([[0.00030625,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.00030625,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">0.00030625]])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">strain_tensor</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([[0.,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">0.]])</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain" title="Link to this definition"></a></dt>
+<dd><p>A <cite>subgrain</cite> has a 3D-gaussian density distribution and a narrow (&lt;=0.1 rad) 3D
+orientation distribution function.</p>
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.phase">
+<span class="sig-name descname"><span class="pre">phase</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.phase" title="Link to this definition"></a></dt>
+<dd><p>Object representing the crystal structure of the grain.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p><a class="reference internal" href="#xrd_simulator.phase.Phase" title="xrd_simulator.phase.Phase">Phase</a></p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.position">
+<span class="sig-name descname"><span class="pre">position</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.position" title="Link to this definition"></a></dt>
+<dd><p>Position of the grain centroid, shape <code class="docutils literal notranslate"><span class="pre">(3,)</span></code></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Tensor | np.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.shape_tensor">
+<span class="sig-name descname"><span class="pre">shape_tensor</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.shape_tensor" title="Link to this definition"></a></dt>
+<dd><p>Covariance tensor of the real-space grain shape. <code class="docutils literal notranslate"><span class="pre">(3,</span> <span class="pre">3,)</span></code>.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Tensor | np.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.orientation">
+<span class="sig-name descname"><span class="pre">orientation</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.orientation" title="Link to this definition"></a></dt>
+<dd><p>Orientation of the centroid of the ODF as a rotation matrix, shape <code class="docutils literal notranslate"><span class="pre">(3,</span> <span class="pre">3,)</span></code></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Tensor | np.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.misorientation_tensor">
+<span class="sig-name descname"><span class="pre">misorientation_tensor</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.misorientation_tensor" title="Link to this definition"></a></dt>
+<dd><p>Covariance tensor of the grain orientation in the left-hand tangent space.
+aka. laboratory coordinates, shape <code class="docutils literal notranslate"><span class="pre">(3,</span> <span class="pre">3,)</span></code></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Tensor | np.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py attribute">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianSubgrain.strain_tensor">
+<span class="sig-name descname"><span class="pre">strain_tensor</span></span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain.strain_tensor" title="Link to this definition"></a></dt>
+<dd><p>Strain tensor in laboratory coordinates, shape <code class="docutils literal notranslate"><span class="pre">(3,</span> <span class="pre">3,)</span></code></p>
+<dl class="field-list simple">
+<dt class="field-odd">Type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>torch.Tensor | np.array</p>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
+<dl class="py class">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianPolycrystal">
+<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">xrd_simulator.gaussian_crystal_model.</span></span><span class="sig-name descname"><span class="pre">GaussianPolycrystal</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">grain_list</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">list</span><span class="p"><span class="pre">[</span></span><a class="reference internal" href="#xrd_simulator.gaussian_crystal_model.GaussianSubgrain" title="xrd_simulator.gaussian_crystal_model.GaussianSubgrain"><span class="pre">GaussianSubgrain</span></a><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_grain_size</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_misorientation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0.1</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal" title="Link to this definition"></a></dt>
+<dd><p>Colection of <code class="docutils literal notranslate"><span class="pre">GaussianSubgrain</span></code> object representing a polycrystalline sample.
+Variables <code class="docutils literal notranslate"><span class="pre">max_grain_size</span></code> and <code class="docutils literal notranslate"><span class="pre">max_misorientation</span></code> can be supplied to speed up
+diffraction calculations.</p>
+<dl class="py method">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.diffract">
+<span class="sig-name descname"><span class="pre">diffract</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">beam</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.beam.GaussianBeam" title="xrd_simulator.beam.GaussianBeam"><span class="pre">GaussianBeam</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">detector</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.detector.Detector" title="xrd_simulator.detector.Detector"><span class="pre">Detector</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">sample_orientation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([[1.,</span> <span class="pre">0.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">1.,</span> <span class="pre">0.],</span> <span class="pre">[0.,</span> <span class="pre">0.,</span> <span class="pre">1.]])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sample_translation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([0.,</span> <span class="pre">0.,</span> <span class="pre">0.])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sample_rotation_during_exposure</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span><span class="p"><span class="pre">[</span></span><span class="pre">Any</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">dtype</span><span class="p"><span class="pre">[</span></span><span class="pre">_ScalarType_co</span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">Tensor</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">array([0.,</span> <span class="pre">0.,</span> <span class="pre">0.])</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">threshold</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">6.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">peaks_batch_size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">20000</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.diffract" title="Link to this definition"></a></dt>
+<dd><p>Render a single diffraction pattern.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>beam</strong> (<a class="reference internal" href="#xrd_simulator.beam.GaussianBeam" title="xrd_simulator.beam.GaussianBeam"><em>GaussianBeam</em></a>) – Object representing the x-ray beam hitting the crystal.</p></li>
+<li><p><strong>detector</strong> (<a class="reference internal" href="#xrd_simulator.detector.Detector" title="xrd_simulator.detector.Detector"><em>Detector</em></a>) – Object representing the x-ray detector.</p></li>
+<li><p><strong>sample_orientation</strong> (<em>NDArray</em><em>, </em><em>optional</em>) – 3-by-3 Rotation matrix representing the orientation of the sample</p></li>
+<li><p><strong>sample_translation</strong> (<em>NDArray</em><em>, </em><em>optional</em>) – 3-vector prepresenting the translation of the sample. (the sample is first
+rotated about the origin, then translated)</p></li>
+<li><p><strong>sample_rotation_during_exposure</strong> (<em>NDArray</em><em>, </em><em>optional</em>) – 3-vector representing a small rotation carried out during the measurement</p></li>
+<li><p><strong>verbose</strong> (<em>bool</em><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code> print timing information. Default is <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.splat_grainshapes">
+<span class="sig-name descname"><span class="pre">splat_grainshapes</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mean_scattering_directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">shape_concentration_tensors</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">W</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pixellengths</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.splat_grainshapes" title="Link to this definition"></a></dt>
+<dd><p>Project the labratory space shape-concentration-tensors of a range of grains along a the scattering directions
+into 2D detector pixels space.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>mean_scattering_directions</strong> (<em>Tensor</em>) – Scattering direction unit vectors, shape <code class="docutils literal notranslate"><span class="pre">(N,</span> <span class="pre">3)</span></code></p></li>
+<li><p><strong>shape_concentration_tensors</strong> (<em>Tensor</em>) – Shape concentration tensors in laboratory coordinates, shape <code class="docutils literal notranslate"><span class="pre">(N,</span> <span class="pre">3,</span> <span class="pre">3)</span></code></p></li>
+<li><p><strong>W</strong> (<em>Tensor</em>) – Pixel-direction unit vectors stacked, shape <code class="docutils literal notranslate"><span class="pre">(2,</span> <span class="pre">3)</span></code></p></li>
+<li><p><strong>pixellengths</strong> (<em>Tensor</em>) – Pixel lengths, shape <code class="docutils literal notranslate"><span class="pre">(2,</span> <span class="pre">3)</span></code></p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p><ul class="simple">
+<li><p><strong>projected_shape_pixelunits</strong> (<em>Tensor</em>) – Concentarion tensor of the projected grainshape in detector pixel units, shape <code class="docutils literal notranslate"><span class="pre">(N,</span> <span class="pre">2,</span> <span class="pre">2)</span></code></p></li>
+<li><p><strong>projected_thicknes_scale_factors</strong> (<em>Tensor</em>) – Intensity scaling factor due the projected thickness of the grain, shape <code class="docutils literal notranslate"><span class="pre">(N,)</span></code></p></li>
+</ul>
+</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py method">
+<dt class="sig sig-object py" id="xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.transform">
+<span class="sig-name descname"><span class="pre">transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rigid_body_motion</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="#xrd_simulator.motion.RigidBodyMotion" title="xrd_simulator.motion.RigidBodyMotion"><span class="pre">RigidBodyMotion</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">time</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/fable-3dxrd/xrd_simulator/blob/main/xrd_simulator/gaussian_crystal_model.py/"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#xrd_simulator.gaussian_crystal_model.GaussianPolycrystal.transform" title="Link to this definition"></a></dt>
+<dd><p>Transform the polycrystal by performing a rigid body motion.</p>
+<p>This updates all the sample-information in-place.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>rigid_body_motion</strong> (<a class="reference internal" href="#xrd_simulator.motion.RigidBodyMotion" title="xrd_simulator.motion.RigidBodyMotion"><em>RigidBodyMotion</em></a>) – Rigid body motion object describing the polycrystal transformation
+as a function of time on the domain <code class="docutils literal notranslate"><span class="pre">time=[0,</span> <span class="pre">1]</span></code>.</p></li>
+<li><p><strong>time</strong> (<em>float</em>) – Time between <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">1]</span></code> at which to call the rigid body motion.</p></li>
+</ul>
+</dd>
+</dl>
+</dd></dl>
+
+</dd></dl>
+
 </section>
 <section id="module-xrd_simulator.templates">
 <span id="templates"></span><h2>templates<a class="headerlink" href="#module-xrd_simulator.templates" title="Link to this heading"></a></h2>
@@ -2222,13 +2604,6 @@ <h1>Documentation<a class="headerlink" href="#documentation" title="Link to this
 </div>
 </dd></dl>
 
-</section>
-<section id="module-xrd_simulator.laue">
-<span id="laue"></span><h2>laue<a class="headerlink" href="#module-xrd_simulator.laue" title="Link to this heading"></a></h2>
-<p>Collection of functions for solving time-dependent Laue equations.</p>
-<p>This module provides functions for arbitrary rigid body motions. It is mainly
-used internally by the <a class="reference internal" href="#xrd_simulator.polycrystal.Polycrystal" title="xrd_simulator.polycrystal.Polycrystal"><code class="xref py py-class docutils literal notranslate"><span class="pre">xrd_simulator.polycrystal.Polycrystal</span></code></a>. However,
-for the advanced user, access to these functions may be of interest.</p>
 </section>
 </section>
 
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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
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      <tr class="cg-1">
        <td></td>
        <td>&#160;&#160;&#160;
-       <a href="index.html#module-xrd_simulator.laue"><code class="xref">xrd_simulator.laue</code></a></td><td>
+       <a href="index.html#module-xrd_simulator.gaussian_crystal_model"><code class="xref">xrd_simulator.gaussian_crystal_model</code></a></td><td>
        <em></em></td></tr>
      <tr class="cg-1">
        <td></td>
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index 77a7d7e..c49dbbd 100644
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+++ b/docs/searchindex.js
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\ 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
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--- /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 <https://doi.org/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
+   <https://doi.org/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
+   <https://doi.org/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 </gaussian_model_theory>`
+       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