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The crystallographic quaternions and their product law

This paper introduces crystallographic quaternions by generalizing the quaternion product law to work directly within crystal bases using a "cross tensor" derived from the metric tensor, thereby eliminating the need for coordinate transformations when calculating rotation compositions in crystallography.

Original authors: Cyril Cayron

Published 2026-07-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cyril Cayron

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe how a toy top spins, or how a dancer pirouettes. In the world of physics and engineering, we have a special mathematical tool called a "quaternion" to do this. Think of a quaternion as a super-efficient instruction manual for rotation. Instead of using a bulky, complicated list of numbers (like a matrix) to tell a computer how to spin something, a quaternion uses a compact code that is faster to calculate and less likely to glitch out. Usually, these instructions are written assuming the world is built on a perfect, square grid—like graph paper where every line is at a perfect 90-degree angle to the others. This is called a "Cartesian basis," and it's the standard playground for most math.

However, the real world of crystals isn't always made of perfect squares. Many crystals, like the ones found in metals or minerals, have atoms arranged in weird, slanted shapes. Their internal grid might be a stretched rectangle, a slanted box, or a hexagon. If you try to use the standard "square-grid" rotation instructions on these slanted crystals, you have to constantly translate your math back and forth between the slanted crystal world and the imaginary square world just to get the answer. It's like trying to bake a cake in a square pan but having to constantly measure your ingredients in a round bowl, then convert them back, then measure again. It works, but it's messy, slow, and prone to errors. This is the puzzle that scientists working with Electron Back Scatter Diffraction (EBSD)—a technique that takes pictures of crystal surfaces to see how they are oriented—have been facing. They need to know exactly how these slanted crystals are spinning, but the standard math tools feel like they are speaking the wrong language.

Enter Cyril Cayron, a researcher from the École Polytechnique Fédérale de Lausanne, who decided to stop translating and start speaking the crystal's native tongue directly. In his paper, he proposes a clever upgrade to the quaternion system. He argues that we don't need to force crystals into a square box to do the math. Instead, we can generalize the rules of the quaternion game so they work perfectly on the slanted, weird-shaped grids that crystals actually live in.

The secret sauce of this new method is a pair of mathematical "magic wands" that Cayron introduces. The first is a tool called the metric tensor, which is already known to scientists; think of it as a ruler that knows how to measure distances and angles on a slanted surface. The second is a brand-new invention by Cayron called the cross tensor. If the metric tensor is the ruler, the cross tensor is the protractor that knows how to handle "cross products" (a way of finding a direction perpendicular to two others) on a slanted grid. Cayron shows that these two tools are actually cousins; the cross tensor is just a specific, calculated version of the metric tensor's inverse.

By combining these tools, Cayron derives a new formula for multiplying quaternions. This new formula allows scientists to calculate how two rotations combine (like spinning a top and then tilting it) directly inside the crystal's own slanted coordinate system. No more translating back and forth to a square grid! The paper demonstrates that for non-cubic crystals, this direct approach is not only mathematically sound but potentially more efficient for computer programs.

The paper also tackles a common misconception: that you must use a Cartesian (square) basis to measure angles or distances in crystals. Cayron argues that this belief is unnecessary. While the old way of using "Rodrigues vectors" (another type of rotation code) is compact, it has a nasty flaw: it can blow up into infinity if you try to rotate something exactly 180 degrees, crashing computer programs. Quaternions avoid this infinity problem, but they were previously stuck needing the square-grid translation.

Cayron crunches the numbers to see if his new "crystallographic quaternions" are actually faster. He finds that for a single rotation calculation, the new method requires about 46 mathematical steps (floating-point operations), while the old method of translating to a square grid and back requires about 43 steps. At first glance, the old way looks slightly faster. However, the paper points out that this comparison is unfair because it ignores the hidden costs. If you count the extra steps needed to translate the data into the square grid at the start and out of it at the end, the old method actually takes more work overall. Furthermore, the new method makes it much easier to spot symmetries (like when a crystal looks the same after a spin) because the numbers stay simple and clean in the crystal's own language.

In short, the paper suggests that we can stop forcing crystals into square boxes. By introducing the "cross tensor" and updating the quaternion multiplication rules, we can do our rotation math directly on the crystal's natural, slanted grid. This makes the process of analyzing crystal orientations in Electron Back Scatter Diffraction cleaner, more elegant, and potentially faster, especially when dealing with the complex, non-cubic crystals that make up much of our material world. The author concludes that this approach could replace the current standard in computer programs used for these analyses, offering a more direct and robust way to understand how the microscopic world spins.

Technical Summary: The Crystallographic Quaternions and Their Product Law

Problem Statement
Unit quaternions are a standard tool in science for encoding rotations due to their computational efficiency and stability compared to matrix or Rodrigues vector methods. However, standard quaternion formulations are inherently tied to Cartesian bases. In crystallography, particularly for non-cubic systems, the crystal basis is non-Cartesian. Current practices in fields like Electron Back Scatter Diffraction (EBSD) require converting between the crystal basis and an arbitrarily attached Cartesian basis using a "structure tensor" to perform rotation compositions. This back-and-forth coordinate transformation is viewed as inefficient and relies on conventions (the structure tensor) that are not fundamental to the crystal lattice itself. The paper addresses the need for a method to perform rotation compositions directly within the crystal basis without intermediate Cartesian conversions.

Methodology
The author proposes a generalization of the quaternion product law to accommodate non-Cartesian crystal bases. The methodology involves three main theoretical steps:

Reformulation of Quaternion Definitions: Rotational quaternions are redefined relative to the crystal basis ( $\mathcal{B}_c$ ). A vector $\mathbf{u}$ is normalized not by the standard Euclidean norm, but by the metric tensor $\mathcal{M}$ , such that $\|\tilde{\mathbf{u}}\| = \sqrt{\tilde{\mathbf{u}}^t \mathcal{M} \tilde{\mathbf{u}}} = 1$ .
Introduction of the Cross Tensor: While the metric tensor $\mathcal{M}$ is well-known for calculating scalar products in non-Cartesian bases ( $\mathbf{u} \cdot \mathbf{v} = \mathbf{u}^t \mathcal{M} \mathbf{v}$ ), no equivalent tensor existed in the literature for the cross product. The author introduces the "cross tensor" ( $\mathcal{X}$ ), defined as the matrix formed by the cross products of the crystal basis vectors ( $\mathbf{b} \times \mathbf{c}, \mathbf{c} \times \mathbf{a}, \mathbf{a} \times \mathbf{b}$ ).
Derivation of the Generalized Product Law: The author demonstrates that the cross tensor is proportional to the inverse of the metric tensor scaled by the unit cell volume ( $V$ ). Specifically, $\mathcal{X} = \sqrt{\det(\mathcal{M})} \mathcal{M}^{-1}$ . This relationship allows the standard quaternion composition formulas (which rely on dot and cross products) to be rewritten using $\mathcal{M}$ and $\mathcal{X}$ , thereby enabling quaternion multiplication entirely within the crystal basis.

Key Contributions

The Cross Tensor: The primary theoretical contribution is the definition and derivation of the cross tensor $\mathcal{X}$ . This tensor allows for the calculation of cross products in a crystal basis using the same coordinate representation as the vectors, analogous to how the metric tensor handles scalar products.
Crystallographic Quaternion Product Law: The paper provides the explicit formula for the product of two crystallographic quaternions ( $\mathbf{q}_3 = \mathbf{q}_1 \mathbf{q}_2$ ). The scalar component uses the metric tensor for the dot product, and the vector component uses the cross tensor for the cross product.
Generalization of Rodrigues Vectors: The paper extends the composition law for Rodrigues vectors to the crystallographic context, providing a formula that similarly depends on $\mathcal{M}$ and $\mathcal{X}$ .
Vector Rotation Formula: A generalized formula for rotating a vector $\mathbf{v}$ by an angle $\alpha$ around an axis $\tilde{\mathbf{u}}$ in the crystal basis is presented, replacing the standard cross product with the cross tensor operation.

Results and Computational Analysis
The paper includes a computational cost analysis (measured in Floating-Point Operations or FLOPs) comparing the new crystallographic quaternion method against traditional Cartesian quaternion methods:

Direct Product Cost: Calculating the product of two crystallographic quaternions requires 46 FLOPs (28 multiplications, 18 additions), which is higher than the 28 FLOPs required for Cartesian quaternions.
Systemic Efficiency: The author argues that a fair comparison must include the overhead of the traditional method. Converting rotation axes between the crystal and Cartesian bases requires the structure tensor (adding ~15 FLOPs). Furthermore, the initial acquisition of EBSD data (converting Kikuchi patterns to Euler angles) typically involves matrix multiplications with the structure tensor (adding ~45 FLOPs).
Net Advantage: When the full workflow is considered, the crystallographic quaternion approach avoids the "back-and-forth" transformations entirely. All calculations remain in the crystal basis, potentially reducing the total computational load for the full EBSD processing pipeline.
Symmetry Handling: The paper notes that symmetry operations are more elegant and computationally simpler with crystallographic quaternions, as symmetry axes in the crystal basis often have integer coordinates (0, ±1), whereas they involve irrational numbers in Cartesian representations.

Significance and Claims
The paper claims that the introduction of the cross tensor and the generalized product law removes the necessity of using an arbitrary Cartesian basis and structure tensor for rotation calculations in crystallography. The author suggests that crystallographic quaternions can advantageously replace Cartesian quaternions in EBSD software. By working directly in the crystal basis, the method aligns with the intrinsic nature of the diffraction data (which is often derived directly in the crystal basis via spherical indexing) and eliminates unnecessary coordinate transformations. The paper posits that this approach is not only theoretically more fundamental (relying on the metric tensor rather than the convention-dependent structure tensor) but also potentially more efficient for the complete processing of non-cubic crystal orientation maps.

Technical Summary: The Crystallographic Quaternions and Their Product Law

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