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A Bravais lattice is an infinite periodic set of equivalent points $\mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3$, with integer $n_i$ and independent primitive translations ${\mathbf{a}}_i$. It describes translation symmetry, without assigning a chemical species to a point. An atomic basis of a crystal is the motif of atoms, with their species and relative positions, repeated identically at every lattice point. The resulting crystal structure consists of the lattice decorated by that basis; the basis can contain several atoms even when the lattice is primitive.

A primitive unit cell tiles space under lattice translations and contains exactly one lattice point, counting shared boundary points fractionally. Its volume is $|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$. A conventional unit cell is chosen to display useful crystal symmetry and may contain several lattice points. Thus a centered conventional cell is not primitive, but the crystal still has a primitive cell. Neither cell size nor the number of dots drawn on its boundary should be confused with the number of atoms in the crystal basis.

The structure factor is the coherent scattering amplitude of the contents of a unit cell at a specified momentum transfer. Contributions from its atoms add with phases fixed by their positions. At a reciprocal lattice vector indexed by Miller indices, the crystal basis structure factor isChanging every phase sign conjugates the amplitude for real scattering factors and leaves the intensity unchanged. The positions here are fractions of the chosen cell edges. An atomic basis of a crystal should not be confused with a set of primitive translation vectors.

In kinematic elastic scattering the diffraction intensity of a Bragg reflection is proportional to $|S_{hkl}{|}^2$, after including geometrical, polarization, thermal-displacement and experimental factors. Powder integrated intensities also include multiplicity. The many-cell interference factor selects reciprocal-lattice vectors satisfying the diffraction geometry; the unit-cell factor determines their relative amplitudes and additional extinctions. Ring positions determine spacings through Bragg's law, while intensity patterns and systematic absences constrain lattice centering and atomic arrangements. Intensity loses the overall complex phase, so structure determination is not a direct measurement of every individual atomic phase.

X-rays scatter from electron density, giving an atomic factor that changes with momentum transfer and chemical charge distribution. Neutrons scatter coherently from nuclei with bound coherent scattering lengths, which need not track atomic number; magnetic neutron scattering is an additional channel in magnetic materials. Their different elemental/isotopic contrasts make the two probes complementary.

A Bravais lattice is a set of points generated by integer translations along three independent primitive lattice translation vectors:Every lattice point has the same environment under lattice translations. A crystal structure consists of this Bravais lattice with an atomic basis of a crystal repeated at every lattice point; the set of atomic positions need not itself be a Bravais lattice.

The two positions in the conventional cubic cell of alpha-tungsten give a body-centred cubic lattice, with one tungsten atom per primitive cell and two per conventional cell. One possible set of primitive lattice translation vectors isTheir primitive-cell volume is $a^3/2$.

The beta phase has a primitive cubic lattice with an eight-atom atomic basis of a crystal, rather than a body-centred cubic lattice. A lattice translation must map the entire atomic arrangement onto itself. For example, the putative body-centering vector $(1/2,1/2,1/2)$ maps $(1/4,1/2,0)$ to $(3/4,0,1/2)$ modulo integer coordinates, where the given atomic basis of a crystal has no atom. None of the seven nonzero basis positions generates a translation that preserves all eight sites. ThereforeThe presence of atoms at both the origin and the body centre alone is insufficient to establish body-centering of the full crystal structure.

In the conventional cubic unit cell, the four face-centered cubic lattice positions contributeThe two-atom atomic basis of a crystal gives the zinc blende structure factorMixed-parity Miller indices have $S_{\mathrm{FCC}}=0$. With all indices even or all odd, $S_{\mathrm{FCC}}=4$. For real atomic scattering factors,Hence all same-parity reflections are generally permitted in GaAs, although the last class is weak when the two atomic scattering factors nearly coincide. These are the structural selection rules; a particular experiment must also satisfy the elastic Bragg scattering condition.

A motif can have fivefold symmetry. For example, repeat an identically oriented regular pentagonal molecular motif at every point of a square translation lattice. Each isolated motif has fivefold rotational symmetry, while the crystal structure as a whole does not: rotating about one motif fails to map the other motif centres onto lattice points. The symmetry of an atomic basis of a crystal need not be a symmetry of its Bravais lattice.

Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated interplanar spacing and first-order Bragg angle satisfyThe reflection labels $(hkl)$ use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set $n=1$ in Bragg's law for each label. The $(200)$ reflection is equivalently the second diffraction order of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.

For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated interplanar spacings are $0.400000$, $0.282843$, $0.230940$ and $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]

The primitive cubic lattice consists of all integer combinations of three perpendicular, equal-length primitive translations:Its conventional cubic unit cell is already primitive, with volume $a^3$ and one lattice point after shared-boundary counting. Each point has six nearest neighbors at distance $a$.

The reciprocal lattice is also simple cubic, with primitive vectors of length $2\pi /a$ in the usual diffraction convention. A primitive cubic Bravais lattice specifies translations rather than a chemical motif: placing several atoms at each lattice point changes the atomic basis of a crystal without automatically changing the translation lattice. This distinction is essential when interpreting crystal basis structure factors.

A member of a set of linearly independent vectors whose integer combinations generate every lattice point of a Bravais lattice. Such basis vectors specify translations; an atomic basis of a crystal specifies the chemical motif repeated under them.

Richard B. Frankel 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 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.

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.