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William Lawrence Bragg's 1913 structure paper connects a slit-selected beam, an oriented crystal and an ionization chamber to strength-versus-angle curves. Repeated sine ratios and matching absorption coefficients identify common radiation. Narrower slits resolve blended lines. Absolute wavelengths remain undetermined there; intercrystal comparisons first give relative scales. The Bragg crystal reflection experiment of 1913 covers the alignment controls.^[ref]

William Henry Bragg's 1915 Bakerian lecture, X-rays and Crystal Structure, separates the geometric and intensity arguments. Its Figure 9 compares reflection positions and strengths from three sodium chloride orientations; odd and even diffraction orders differ in strength along one orientation. The interpretation uses alternating scattering layers. Later, the calcite example identifies an internal oxygen coordinate that symmetry alone cannot fix. Improved wave intensity measurements revise a previously rough estimate. Thus a lattice assignment can precede determination of positions inside the unit cell; intensity interpretation also has experimental and theoretical limits. This is the father's 1915 lecture, distinct from the joint 1913 reflection paper.^[ref]

A binary crystal structure with one species at the corners of a primitive cubic unit cell and the other at its body centre. Each species has coordination number eight. Distinct chemical labels mean that the body-centre translation is not a translation symmetry; its Bravais lattice is primitive cubic, not body-centred cubic.

A crystallographic unit cell chosen to display the symmetry or a standard setting of a crystal structure. It tiles the periodic arrangement but need not be primitive. A conventional FCC cube contains four lattice points and a BCC cube contains two; a primitive unit cell contains one. These are lattice-point counts, not a universal count of atoms in a chemical motif.

Crystallography studies crystal structures and their symmetries. A structure determination connects measurements such as crystal scattering patterns to a model of the unit cell and its atomic basis of a crystal.

X-ray crystallographic structure inference makes that connection explicit: compare proposed crystal structures through the positions and wave intensities of their predicted reflections. The main difficulty is that a diffraction pattern encodes wave interference, so it must be interpreted through a model rather than read as a direct picture of the chemical atoms.

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.

For a concrete two-dimensional model, choose primitive lattice translation vectors $a_1=(2,0)$ and $a_2=(0,1.5)$ in arbitrary length units. At every lattice point $R=m{a}_1+n{a}_2$, with integers $m,n$, place species A at $R$ and a different species B at $R+(0.65,0.55)$. Translating by $a_1$ or $a_2$ preserves both the positions and their chemical labels. Translating by $(0.65,0.55)$ sends one A position to a B position, so it is not a translation symmetry of the labeled structure. A picture of dots with the labels erased can therefore conceal essential chemical information.

A primitive unit cell contains one lattice point, but may contain several chemical atoms. A conventional unit cell can contain several lattice points to display the symmetry more clearly. Thus the conventional cube of a face-centered cubic lattice contains four lattice points, whereas a body-centered cubic lattice has two. Neither lattice name alone specifies which chemical species occupies the repeated motif. The NaCl conventional cell and coordination example makes the distinction explicit.

This separation is useful experimentally: interplanar spacings constrain the repeating geometry, while relative wave intensities also depend on the atomic basis of a crystal through the crystal basis structure factor. The Bragg crystal reflection experiment of 1913 describes the X-ray spectrometer, angular scans and detector checks in detail. Its conversion from angle to length is developed under Bragg's law; a geometrically possible reflection can still have zero wave intensity.

William Lawrence Bragg's 1913 structure determination supplied a complementary test. He directed X-rays through sodium chloride sections cut parallel to a cube face and recorded spots on photographic plates. Comparing section thicknesses tested how transmission altered the pattern: the relative spot strengths changed. Comparing related salts led him to two interpenetrating face-centered cubic lattices, one for each kind of constituent. He explicitly distinguished evidence for that arrangement from the still-needed identification of a single chemical atom with each scattering center. The photographs were evidence for a structural model, not direct pictures of labeled ions.^[ref]

An $A{B}_2$ crystal structure with A on a face-centered cubic lattice and B occupying all eight tetrahedral sites of its conventional cubic unit cell. The coordination numbers are eight for A and four for B. The cubic cell contains four $A{B}_2$ formula units.

Fractional crystal coordinates are the dimensionless coefficients in $r=u{a}_1+v{a}_2+w{a}_3$ for specified unit cell vectors. They are not Cartesian lengths unless the vectors themselves are included. With primitive vectors, replacing $u,v,w$ by values differing by integers selects an equivalent position in another unit cell.

In the two-dimensional example under crystal structure, B has position $(0.65,0.55)$ in ordinary length coordinates, but fractional crystal coordinates $(0.65/2,0.55/1.5)=(13/40,11/30)$. Multiplying these coefficients by the two translation vectors recovers the drawn position. Reporting the axes is therefore part of reporting a structure.

A Frenkel exciton is an exciton whose internal excitation is concentrated on the scale of one molecule or unit cell. The local excitation can transfer between sites; localized internal structure does not require an immobile center of mass. The large-radius Wannier-Mott exciton approximation cannot simply be extrapolated to this limit.^[ref]

An ionic crystal repeats an arrangement of ions in three dimensions. A unit cell describes that repetition, whereas a formula unit describes composition. The two accounting devices answer different questions. Coulomb's law is not restricted to an arbitrarily drawn line connecting one cation to one anion: every ion contributes to the electric field experienced by the others.

A length parameter specifying a periodic unit cell. A conventional cubic edge is not always the nearest-neighbour distance.

Choose an original simulated model: a primitive cubic lattice with one identical point scatterer per primitive unit cell, edge $a=0.400\mathrm{nm}$, and incident X-ray wavelength $\lambda =0.200\mathrm{nm}$. These are stipulated lengths, not measurements of sodium chloride. Suppose orientation information has already assigned the two observed reflections the Miller indices $(100)$ and $(110)$. The detector deflection is $2\theta$, where $\theta$ is the Bragg angle.

Use the existing interplanar spacing formula and Bragg's law, with higher diffraction orders absorbed into the reflection indices:The simulated $(100)$ reflection has $\sin \theta =1/4$, hence $2\theta =28.9550^{\circ }$. The $(110)$ reflection has $\sin \theta =\sqrt{2}/4$, hence $2\theta =41.4096^{\circ }$. Their squared sines are $0.0625$ and $0.125$, in ratio $1:2$. Substituting the unrounded angles independently givesAgreement tests a proposed indexing and scale together; disagreement would prompt checks of the indexing, wavelength, angular zero, or the assumed unit cell geometry.

Two peak positions alone do not establish the indexing. Assigning the same two angles to $(110)$ and $(200)$ would give $N=2,4$ and $a=0.400\sqrt{2}\simeq 0.565685\mathrm{nm}$ instead. More reflections, orientation information and systematic absences help discriminate such assignments. An unobserved reflection can be inaccessible, weak or extinguished by the atomic basis; absence is not automatically evidence for a larger interplanar spacing.

Absolute scale also requires measurement calibration of the wavelength. To first order, differentiation of the inferred length giveswith angular changes in radians. A stipulated detector-zero error $\delta (2\theta )=+0.100^{\circ }$, with correct wavelength, biases the first inferred edge downward by approximately $0.338\%$ and the second by $0.231\%$. A common angular offset therefore need not cancel when two inferred lengths are compared. These are model sensitivity estimates, not historical instrument uncertainties.

In 1949, Clifford Shull and J. Samuel Smart compared powdered manganese(II) oxide at room temperature and $80\mathrm{K}$. Powder preparation supplies many crystal orientations. Their Fig. 1 records scattered-neutron intensity versus counter angle. Nuclear reflections remained, while the cold pattern gained strong reflections forbidden by the chemical unit cell. They indexed these with a magnetic unit cell having twice the chemical cell edge. The warm pattern already contained diffuse magnetic scattering: loss of sharp magnetic peaks does not mean loss of individual magnetic moments.^[ref]

The source was Oak Ridge's graphite-moderated reactor. In Early Development of Neutron Scattering (1994), Clifford Shull describes how the early two-axis instruments used a monochromating crystal to select the incident-neutron wavelength; a shielded neutron counter rotated around the specimen to record angular intensity. This specifies the source–monochromator–sample–counter path, without identifying which instrument revision produced the MnO trace.^[ref]

The control was the same compound above and below magnetic ordering, with retained nuclear peaks identifying the underlying chemical periodicity. The original paper labels an aluminium impurity peak and defers a complete magnetic-structure account. It supplies neither a chemical synthesis recipe nor detailed cold-stage construction. The inference is periodic magnetic order, rather than a claim that every possible structural change was excluded at arbitrary resolution.^[ref]

The connection to the measured angle is the elastic Bragg scattering condition. If the incident and outgoing wave vectors have equal magnitude $2\pi /\lambda$ and make an angle $2\theta$, their difference has magnitudeA longer real-space period supplies additional, smaller reciprocal lattice spacings, which can put magnetic reflections at new counter angles. In the one-dimensional toy model above, doubling the period changes the allowed spacing from $2\pi /a$ to $\pi /a$. The actual MnO indexing is three-dimensional; the toy row does not assert its reflection positions or relative peak strengths.

Louis Néel supplied the ordering theory; the 1949 diffraction work was a later experimental test. His 1970 Nobel Prize in Physics recognized antiferromagnetism and ferrimagnetism. Clifford Shull's neutron-diffraction award followed in 1994.^[ref]

Three Miller indices $(hkl)$ specify a lattice plane orientation in chosen unit cell axes. Choose a parallel plane away from the origin, express its intercepts in axis units, take reciprocals, and clear fractions. An infinite intercept gives index zero; a negative intercept gives a negative index, written with an overbar.^[ref]

For example, intercepts $(2a,3b,\infty )$ have reciprocal ratios $(1/2,1/3,0)$, so the orientation is $(320)$ after multiplying by six. Relative to the same axes, intercepts $(a,-b,\infty )$ give $(1\bar{1}0)$. Choosing a different parallel member changes the common intercept scale, not the orientation.

Primitive axes require relatively prime Miller indices for complete families. Centered conventional unit cells need not, as the FCC lattice $(200)$ example shows. Reflection labels can also retain higher diffraction orders. Use the reciprocal lattice for the normal vector with oblique axes; the direct vector $h{a}_1+k{a}_2+l{a}_3$ need not be perpendicular.

An atom is monovalent in a specified bonding or conduction model when its relevant valence is one. In the simple metallic free electron model, one monovalent atom per primitive unit cell supplies one conduction electron per cell. Counting the two spin states then fills half of a single energy band at zero temperature.

Ordinary solid sodium chloride has the rock salt structure. Choose a conventional cube of side $a$ with chlorine anions at the corners and face centers, and sodium cations at the edge centers and cube center. Corner sites are shared by eight cubes, face sites by two and edge sites by four. The unit cell counts are thereforeThere are four formula units per conventional unit cell, rather than four separate NaCl molecules.^[ref]

The Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors and atomic basis isThe determinant gives a primitive unit cell volume of $a^3/4$; that cell represents one cation and one anion. Four such volumes make the conventional cube. The union of all ion positions resembles a primitive cubic lattice of step $a/2$ only if the chemical labels are discarded: translation by $a(1,0,0)/2$ exchanges unlike ions and fails to preserve the chemical structure.

Each ion has six nearest unlike neighbors at distance $a/2$, along the positive and negative directions of the three cube axes. Thus the coordination number is six. Within either same-species face-centered cubic lattice, the nearest same-ion neighbors instead number twelve: their displacements are permutations of $(\pm a/2,\pm a/2,0)$, all of length $a/\sqrt{2}$. Since $a/\sqrt{2}>a/2$, those twelve are a more distant shell in the complete sodium chloride structure. The number twelve describes the same-ion FCC lattice neighborhood; six describes the nearest unlike-ion neighborhood.

The sum needs a stated interpretation. The original equation leaves the second bond outside explicit parentheses, with its index then formally free. The natural translation-invariant, equal-bond interpretation used here is the energy $-J{\sum }_j{\tau }_j({\sigma }_j+{\sigma }_{j+1})$. The subsequently mentioned $h$ has no term in that energy, so no magnetic field or single-ion anisotropy is added. Such an extra term would define another model and change the answer.

Let $K=\beta J$. Sum over each decorating spin before summing over the remaining spins. The decimation of an alternating mixed-spin chain gives the effective transfer matrix:No on-site entropy has been omitted: the term $1$ is the weight of the zero decorating spin. The normalized eigenvectors are $v_{\pm }=(1,\pm 1{)}^{\mathsf{T}}/\sqrt{2}$, with eigenvaluesThere are $L$ transfer steps but $2L$ physical sites, soThe free energy per two-site unit cell is twice this value. At $J=0$, the formula gives $Z_L=6^L$, as required by two states on one sublattice and three on the other. As $T\to 0$, the free energy per physical site tends to $-J$, the energy of an aligned chain with two satisfied bonds per unit cell. These checks also fix the factor of two in the site count.

A unit cell is a repeat region whose lattice translations tile the plane. A primitive unit cell represents exactly one lattice point and has the minimum repeat area for a lattice basis.

Cell A is nonprimitive; cell B is primitive. A represents two points and has area $4a^2$. B has four shared corner points and no additional interior point, represents one point and has area $2a^2$. Fractional corner counting refers to the tiling, not to fractional physical atoms.

Use the total scattering angle $\psi =2\theta$. For a cubic unit cell, Bragg's law and the Miller indices giveThe four measured values of ${\sin }^2\theta$, normalized to the first, are $1,1.330,2.655,3.649$. Multiplying by three gives approximately $3,4,8,11$, the initial allowed sequence for a face-centered cubic lattice:The FCC lattice factor is $1+(-1{)}^{h+k}+(-1{)}^{h+l}+(-1{)}^{k+l}$; it vanishes unless all three indices have the same parity. Thus the initial ratios differ from those of a primitive or body-centered cubic lattice. This is the conventional minimal cubic indexing; arbitrarily hiding additional weak reflections could make a short list alone ambiguous.

The low-pressure rock salt structure has an FCC Bravais lattice with one NaCl formula unit per primitive unit cell, or four per conventional cube. A possible atomic basis of a crystal is Na at zero and Cl at $(1/2,0,0)$ in conventional fractional coordinates. The two species have unequal atomic scattering factors, so both the odd and even FCC families can occur.

The four independent lattice constant estimates areTheir small scatter reflects the rounded angular data. The appropriate reported result is

The high-pressure lattice constant follows from its first $(100)$ reflection:Conserve the number of NaCl formula units. The low-pressure conventional unit cell contains four, whereas the high-pressure cell contains one. The volume per formula unit changes from $a_0^3/4$ to $a_P^3$, soThe sample contracts by about $39.5\%$. Comparing $a_P^3$ directly with $a_0^3$ would compare different amounts of material.

The FCC conventional unit cell has four atoms, so the cube containsatoms, neglecting surface corrections. A phonon is one excitation quantum of a vibrational normal mode; there can be many quanta per mode, and zero-point motion does not add half a phonon to the thermal population.

For the low-temperature thermal phonon number, integrate the Bose-Einstein distribution against the Debye model phonon density of states:At $10\mathrm{K}$, ${\Theta }_D/T=21.5$, so extending the upper limit to infinity has negligible effect at the precision required. The supplied integral yieldsThis is the thermally excited phonon number, not the number $3N$ of available modes.

The continuum approximation fails when the thermally occupied wavelengths become comparable with the cube size $L$. From the Debye temperature,A lowest nontrivial acoustic frequency is of order $v/L$; a standing-wave estimate ${\omega }_{\min }\sim \pi v/L$ corresponds to $\hslash {\omega }_{\min }/k_B\sim 0.25\mathrm{K}$. Periodic boundary conditions instead give $2\pi v/L$ and $0.50\mathrm{K}$. The precise spectrum depends on mechanical boundary conditions and branch speeds, neither of which is specified.

This finite-crystal phonon freeze-out requires summing discrete normal modes. For a finite positive lowest frequency, $N_{\mathrm{ph}}\sim g_{\min }{e}^{-\hslash {\omega }_{\min }/(k_{B}T)}$, rather than extrapolating the bulk $T^3$ law indefinitely. Rigid translations and rotations of an isolated crystallite are not internal thermal phonons. Surface acoustic modes can modify the intermediate crossover, but do not justify inventing a unique boundary spectrum.