Glossary

Key terms used throughout ad_hoc_diffractometer, in alphabetical order.

B matrix

The matrix that encodes the reciprocal lattice and maps Miller indices h = (h, k, l)ᵀ to the scattering vector in Cartesian crystal-frame coordinates: Q_c = B h. Constructed from unit-cell parameters (a, b, c, α, β, γ) following Busing & Levy (1967), eq. 3. See Lattice and Direct Lattice in Crystallography.

Bisecting condition

A diffraction mode constraint in which the sample stage angle equals half the detector angle (ω = 2θ/2, or equivalently η = δ/2 for psic). Places the sample symmetrically between the incident and diffracted beams. See BisectingMode.

d-spacing

The interplanar spacing d_{hkl} for the reflection (h, k, l), related to the scattering vector magnitude by |Q| = 2π / d_{hkl} and to the Bragg angle by 2d sin θ = λ.

Diffraction mode

A named set of constraints that describes how forward() computes motor angles: which stages are free, which are held fixed, and which are coupled to other stages. See DiffractionMode and Switch Diffraction Modes.

Forward problem

Given Miller indices (h, k, l) and a UB matrix, find the motor angles that satisfy the Bragg condition. Returns a list of 0 to ~12 solutions depending on geometry and active diffraction mode. See forward() and Forward and Inverse Computations.

Inverse problem

Given motor angles and a UB matrix, find the unique Miller indices (h, k, l) in the Bragg condition. Unlike the forward problem, the inverse always has a unique solution once UB is established. See inverse().

Miller indices

Integer (or rational) triplet (h, k, l) that indexes a set of crystal lattice planes. Used throughout as the reciprocal-space coordinate of a Bragg reflection.

Monochromatic radiation

Radiation of a single, fixed wavelength λ. All diffraction calculations in this package assume monochromatic radiation — a fundamental assumption of the single-wavelength four-circle formalism. Spallation (time-of-flight) sources are out of scope.

Orienting reflection

A Bragg reflection measured at a known (h, k, l) whose measured motor angles are used to determine the U matrix. Typically two reflections (or1 and or2) are used to compute UB via Busing & Levy (1967), eqs. 23–27. See ub_from_two_reflections_bl1967() and Orient a Crystal.

Phi frame

The Cartesian reference frame attached to the innermost sample stage (φ for Eulerian geometries, kφ for kappa geometries). The UB matrix maps Miller indices to scattering vectors expressed in the phi frame.

Q (scattering vector)

The momentum transfer vector Q = k_f − k_i, where k_i and kf are the incident and scattered wave vectors. For elastic scattering |Q| = (4π/λ) sin θ = 2π / d{hkl}.

Sphere of confusion

The small three-dimensional volume swept out by the nominal sample position as all diffractometer stages rotate. Caused by mechanical tolerances; ideally a point but in practice a sphere of finite radius.

Stage

A single rotary axis of the diffractometer. Characterised by its axis direction (signed unit vector), parent stage, role (sample or detector), and optional motor limits. See Stage.

Serialization

The process of converting the complete diffractometer state to a Python dict (via to_dict()) and restoring it (via from_dict()). The dict contains only JSON-compatible types and can be saved to JSON (stdlib) or YAML (pyyaml). All major classes support this round-trip. See Save and Restore a Diffractometer Configuration.

Stack

An ordered chain of stages in which each stage sits mechanically on the one below it (its parent). The combined rotation matrix is the ordered product of individual stage matrices, from the base-mounted (outermost) stage to the innermost.

Two-theta

The total scattering angle 2θ between the incident and diffracted beams, related to d-spacing and wavelength by Bragg’s law: 2d sin θ = λ.

U matrix

The orthonormal orientation matrix that relates the Cartesian crystal frame to the phi-axis frame. U encodes how the crystal is mounted on the diffractometer; it is determined from one or more orienting reflections. See Orient a Crystal and Case Study: Coordinate Convention and UB Matrix.

UB matrix

The product U × B. Maps Miller indices directly to the phi-axis frame scattering vector: Q_φ = UB h. Can be determined from measured reflections without prior knowledge of the unit-cell parameters. See Orient a Crystal.

Vertical / horizontal scattering plane

The plane containing the incident beam, the sample, and the diffracted beam. Vertical — the scattering plane is vertical, the two-theta arm rotates about the transverse (horizontal) axis; typical for synchrotrons. Horizontal — the scattering plane is horizontal, two-theta rotates about the vertical axis; typical for laboratory instruments.

ψ (psi) angle

Two definitions appear in the literature: (1) You (1999): azimuthal angle of a reference vector about Q — a crystal-orientation diagnostic, constant for a given (hkl, UB). (2) Busing & Levy (1967): angle of sample rotation about Q relative to a reference orientation — the quantity physically varied in a ψ scan. See psi() and psi_trajectory().