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A surface reconstruction rearranges chemical atoms at a crystal surface relative to a simple termination of its bulk crystal structure.

An unoccupied site in an otherwise specified crystal structure. Equilibrium vacancies and an irradiation-driven vacancy supersaturation have different origins.

William Clegg coauthored research on crystal structure.^[ref]

A crystal structure specifies the chemical species and spatial arrangement of the chemical atoms or ions in a crystal. For an ideal periodic structure, separate two ingredients: a Bravais lattice specifies the translations that repeat the arrangement, and an atomic basis of a crystal specifies the chemical motif repeated at every lattice point. The outward shape, or crystal habit, is a different description: the internal arrangement is what this article describes.

William Giauque investigated chemical thermodynamics at very low temperatures. Calorimetry and magnetic cooling connect accessible microscopic states with entropy and thermal response.

A reduction in temperature makes previously negligible level spacings relevant to state populations. Low-temperature measurements therefore test how molecular rotation, spin, and crystal structure contribute to thermodynamic quantities.^[ref]

A chemist at Berkeley whose research and teaching focused on chemical thermodynamics.^[ref]

A Wyckoff letter is the conventional alphabetic label identifying a Wyckoff position within the tables for a space group. It is normally written beside the crystallographic site multiplicity, as in $1a$. Letters begin with $a$ for a position with maximal site-symmetry group order and proceed toward the general position. The letter is a label, not a coordinate or a chemical-species symbol.^[ref]

For example, the space group $P1$ has only translations. Every point has a trivial site-symmetry group, and the general position is labeled $1a$: one equivalent position per primitive unit cell, with free coordinates $(x,y,z)$.^[ref]

An original model makes the distinction from atom counting clear. Place one species at $(0.1,0.2,0.3)$ and another at $(0.4,0.1,0.7)$ in a generic $P1$ crystal structure. Both sites are described by $1a$, but they define different crystallographic orbits. The notation does not say that the entire structure contains only one atom per cell.

The space group, its setting and coordinate convention must accompany the letter. The same letter in another group need not denote the same symmetry or coordinates. Nor does matching multiplicity by itself identify the Wyckoff position, since distinct positions can have equal multiplicity.

A Wyckoff position of a space group consists of all points whose site-symmetry groups are conjugate subgroups of that space group. It classifies sites by their symmetry relationship within the whole group. A Wyckoff letter labels each such class in the conventional tables; the label is accompanied by its crystallographic site multiplicity.^[ref]

A Wyckoff position need not specify one particular atom or one fixed coordinate triplet. A general position usually has free fractional parameters. Choosing those parameters produces a crystallographic orbit, the collection of equivalent sites generated from that choice by the space group. Different parameter choices can give distinct orbits belonging to the same Wyckoff position.

An original comparison uses $P\bar{1}$, with operations $\mathbf{r}\mapsto \pm \mathbf{r}+\mathbf{n}$, where $\mathbf{n}$ is a lattice vector. Generic positions form pairs $\{\mathbf{r},-\mathbf{r}\}$ modulo lattice translations and have trivial site-symmetry groups. Many different pairs therefore belong to the same general Wyckoff position.

The origin and the half-cell point $(1/2,0,0)$ instead each have an inversion-containing site-symmetry group of order2. Their groups are isomorphic but not conjugate within this space group: a group operation sends the origin to an integer lattice point, never to that half-cell point. They occupy different special Wyckoff positions. Group order or multiplicity alone therefore does not uniquely identify a Wyckoff position.

This classification helps constrain independent atomic coordinates in a crystal structure; it describes allowed symmetry, not which chemical species occupies a site or its fractional occupation.

X-ray crystallographic structure inference is an inverse problem: infer a crystal structure from measured X-ray diffraction. A proposed unit cell and atomic basis of a crystal predict a pattern; structure determination asks which proposals survive comparison with that pattern. The useful separation is between directions and strengths. Peak positions constrain the reciprocal lattice and hence interplanar spacings; relative wave intensities constrain the motif through the crystal basis structure factor. Knowing a repeat distance does not locate all the chemical atoms within that repeat.

The lattice constant from two indexed reflections example isolates the geometric inference. Basis intensities and phase ambiguity then holds the translations fixed while changing the motif. Even complete, noiseless wave intensity data need not determine all wave phases: this is the phase problem. Chemical composition, symmetry, and additional measurements can supply constraints beyond the original pattern.^[ref]

X-ray crystallography infers a crystal structure from measured X-ray diffraction. Peak positions constrain the lattice geometry, while intensities constrain the atomic basis of a crystal; the inference also needs an experimental model and does not directly observe every atomic position.

X-ray diffraction is the angular redistribution of X-ray wave intensity through wave interference between scattered waves. In a crystal structure, repeated lattice translations select discrete outgoing directions by the Laue equations. A photographic plate can therefore turn invisible atomic periodicity into a visible constellation of spots. Their geometry constrains the Bravais lattice and its orientation; their wave intensities also depend on the crystal basis structure factor. Determining atomic positions belongs to X-ray crystallography and requires more information than recognizing a spot pattern.

The Laue X-ray diffraction experiment of 1912 established this phenomenon experimentally. The Laue method uses a fixed crystal structure with a range of incident wavelengths; the Bragg crystal reflection experiment of 1913 instead measured angularly resolved signals with a moving detector. These are different ways of finding the same coherent elastic scattering directions.

X-ray scattering is the redirection of X-rays through interaction with matter. The angular distribution and energy of the outgoing radiation carry information about the material. In elastic scattering the photon energy is unchanged; in inelastic scattering energy is exchanged. X-ray diffraction concerns organized interference of scattered waves, rather than a separate kind of radiation.

The scattering vector $\mathbf{q}={\mathbf{k}}_{\mathrm{out}}-{\mathbf{k}}_{\mathrm{in}}$ compares outgoing and incoming wave vectors. Consider two identical scatterers separated by $\mathbf{d}$, each with real elastic scattering amplitude $f$, and neglect multiple scattering. Their amplitude sum and intensity areTwo atoms can therefore scatter strongly in one direction and cancel in another. The measured pattern depends on arrangement, not merely on the number of atoms. For repeating lattice planes in a crystal, constructive interference gives Bragg's law $2d\sin \theta =n\lambda$.^[ref]

The1913 experiment by William Henry Bragg and William Lawrence Bragg made the angular pattern measurable with a rotating crystal and an ionization chamber in place of a spectrometer telescope. Slits selected a narrow incoming beam; the detector current was measured as its angle changed. Peaks and repeated orders constrained crystal structure and X-ray wavelengths, linking an electrical measurement to atomic-scale geometry.^[ref]