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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.

Take $(x_j,y_j,z_j)$ as fractional coordinates in the chosen unit cell. In the kinematic approximation the crystal basis structure factor isEquivalently, in Cartesian coordinates the phase is ${\mathbf{G}}_{hkl}\cdot {\mathbf{r}}_j$. Reversing every phase sign complex-conjugates the nonabsorbing amplitude and leaves its intensity unchanged. The atomic scattering factor is the Fourier transform of the atom's electron density, in electron scattering units:At zero scattering angle the normal form factor counts electrons: $f_j(0)=Z_j$ for a neutral atom, modified by the ionic charge for an ion. At increasing $Q$, different parts of the electron cloud scatter with different phases, reducing the coherent amplitude. Typical nonresonant atomic curves fall from their forward value toward small values at large angle. A localized cloud falls more slowly in reciprocal space. Anomalous scattering near an absorption edge adds energy-dependent complex terms and is excluded from this simple sketch.

For a fixed reflection and fixed instrumental and geometric corrections,The full amplitude also contains the lattice sum. At its reciprocal-lattice peak, $N_c$ coherently illuminated identical cells give a peak amplitude $N_{c}F$; peak height and integrated peak intensity have different size dependences because the peak narrows with crystal size. Powder multiplicity, polarization, absorption, thermal motion and measurement geometry also affect observed intensities. The basis factor determines relative interference, not every experimental prefactor.

The X-ray wavelength is $\lambda =hc/E=1.54980\times 10^{-10}\mathrm{m}$. Write the amplitude of one conventional unit cell as the face-centred cubic selection rule times the two-atom atomic basis amplitude:The first bracket is four when the indices have equal parity and zero otherwise. Among the allowed Bragg reflections, the zinc blende structure factor therefore givesHere the atomic scattering factors are taken as real. For the requested rough estimate set $f_{\mathrm{Ga}}\simeq 31$ and $f_{\mathrm{As}}\simeq 33$. The weak difference reflections remain allowed: unequal species do not have the extra exact extinction of the equal-species diamond limit.

The first six allowed values of $h^2+k^2+l^2$ are $3,4,8,11,12,16$. Bragg's law gives the measured scattering angle $2\theta =2{\sin }^{-1}(\lambda \sqrt{h^2+k^2+l^2}/2a)$. For powder diffraction, multiply the squared amplitude by the powder diffraction multiplicity $m_{hkl}$. The last two columns use $I_{111}=100$:The first intensity column is the geometry-independent multiplicity-and-structure estimate. The second additionally assumes an unpolarized beam and a common symmetric powder-diffractometer geometry, with Lorentz-polarization factorAn overall constant cancels in either normalization. A particular measured intensity set is not fixed by the atomic numbers alone: actual atomic scattering factors decrease with momentum transfer, anomalous scattering and absorption can matter, and the instrument geometry and polarization determine angular corrections. The two columns state the approximation rather than asserting identical peak heights for all instruments.^[ref]

In the one-band electron Hall effect model, $R_H=-1/(ne)$, with $e>0$. The measured Hall coefficient therefore givesA monovalent body-centered cubic lattice has two conduction electrons per conventional unit cell. Thus $n{a}^3=2$, andCounting two spin states within the Fermi sphere gives $k_F=(3{\pi }^{2}n{)}^{1/3}=9.0418\times 10^9{\mathrm{m}}^{-1}$. Using the specified effective mass in the dispersion,There is one mole of conduction electrons per mole of sodium. Its molar density of states at $E_F$ is $3N_A/(2E_F)$, givingUsing the volume density of states in c givesThe effective mass changes the dispersion and density of states; the electron Bohr magneton used for the spin coupling retains the bare electron mass. Replacing it by $e\hslash /(2m^{\ast })$ would be an additional, unjustified magnetic-moment assumption. A Hall factor or a multiband metal would also invalidate the simple density inference; the stated calculation uses the requested one-band approximation.

Within the stipulated localized, one-moment-per-atom spin model, the number density follows directly from mass density and molar mass:The face-centered cubic lattice is consistent with four atoms per conventional unit cell; no additional factor four is needed once $n$ has been obtained from the density. The Curie constant is $C=0.71549\mathrm{K}$. Identifying the model's $\theta =\lambda C$ with the specified Curie temperature givesAt very low temperature, a fully ordered single domain has $M_s=n{\mu }_B=8.4764\times 10^5\mathrm{A}{\mathrm{m}}^{-1}$. Its Weiss molecular field isThe cancellation of $n$ in the middle expression provides a useful check. The ordinary magnetostatic scale is only ${\mu }_0{M}_s\simeq 1.065\mathrm{T}$, smaller by the factor $\lambda$. Thus $B_{\mathrm{eff}}$ is not a plausible ordinary self-generated dipolar magnetic field of the nickel moments. It represents the exchange interaction energy in field units, not an additional Maxwell field of that magnitude. This does not assert that such large externally generated fields are forbidden. Nickel is an itinerant ferromagnet; the imposed one-${\mu }_B$-per-atom model is an estimate rather than its measured microscopic moment.

For the conventional body-centred cubic lattice, the crystal basis structure factor of identical atoms isIt vanishes for odd $h+k+l$ and equals $2f_{\mathrm{W}}$ for even $h+k+l$. Thus the first powder diffraction ring is the $\{110\}$ family, with interplanar spacing $d_{110}=a/\sqrt{2}$.

The measured deflection is the angle between incident and scattered rays, $2\theta =40.26^{\circ }$, so the Bragg angle is $\theta =20.13^{\circ }$. Using first-order Bragg diffraction, $2d\sin \theta =\lambda$, givesThe mass density follows from two atoms per conventional unit cell, with molar mass $M=0.1838\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ and Avogadro constant $N_A$:Using $40.26^{\circ }$ as the Bragg angle would confuse $\theta$ with $2\theta$ and give an incorrect lattice constant and mass density.

For the cubic crystal system, the Miller indices satisfyThe inverse squared interplanar spacings have ratios approximately $1:2.004:3.012:4.010$. With the lattice constant close to $0.4\mathrm{nm}$, the appropriate squared-index sums are $1,2,3,4$. Thus the four powder diffraction rings areThe braces collect all symmetry-related lattice planes with the same interplanar spacing. In particular, $(100)$ is forbidden for both a body-centered cubic lattice and a face-centered cubic lattice, but occurs for a primitive cubic lattice. These observations identify the Bravais lattice as primitive cubic lattice, with a multi-atom atomic basis, consistent with a perovskite structure.

Each ring independently gives $a_j=d_j\sqrt{j}$:Their average is $0.42413\mathrm{nm}$. Allowing for the rounded measured angles, a suitable final value isA larger centred unit cell could relabel reflections, but would conflict with the stated approximate lattice constant.

A consistent mechanism is a transition to G-type antiferromagnetism at the Néel temperature. On the manganese primitive cubic lattice, take the magnetic moments to beEvery nearest neighbour has the opposite magnetic moment. The nuclear crystal structure can retain its original lattice constant, while the magnetic order introduces the propagation wavevectorSumming the scattering amplitude from the magnetic moments over sites gives peaks at $\mathbf{q}=\mathbf{G}+\mathbf{Q}$, where $\mathbf{G}$ is a nuclear reciprocal lattice vector. Thus all three components of the new Miller indices, expressed relative to the original unit cell, are half-integers. The two smallest squared magnitudes in units of $(2\pi /a{)}^2$ areThe first predicts $2\theta \simeq 21.29^{\circ }$, agreeing with the new ring. The next squared magnitude is $19/4$, outside the measured angular range. Taking the observed first magnetic ring as the calibration, Bragg's law predictsUsing the independently fitted lattice constant gives $41.42^{\circ }$, within the input rounding. One ring alone does not uniquely determine a magnetic order; this proposed nearest-neighbour antiferromagnetic pattern explains both the observed ring and the count of two new rings in the specified interval.

The Bravais lattice is a face-centered cubic lattice. Four inequivalent representatives of its lattice points in the conventional cubic unit cell, in fractional crystal coordinates, areBoundary copies in the drawing do not constitute additional independent lattice points. Attach the two-atom atomic basisto each lattice point. The resulting aluminium phosphide crystal structure is the zinc blende structure. For example, the four phosphorus positions reduced into the conventional unit cell are $(1/4,1/4,1/4)$, $(1/4,3/4,3/4)$, $(3/4,1/4,3/4)$, and $(3/4,3/4,1/4)$, reproducing the heights in the source drawing. There are four aluminium and four phosphorus chemical atoms per conventional unit cell, but only two chemical atoms per primitive unit cell.

Subtracting the three acoustic phonon branches from the six displacement normal modes leavesAt small wavevector, these involve relative motion of the two atomic basis sites rather than a rigid translation of the primitive unit cell. Their angular frequencies remain nonzero at the zone centre. The count uses the primitive unit cell; using all eight chemical atoms in the conventional unit cell would spuriously multiply the number of distinct branches in the first Brillouin zone.

The ordered crystal structure consists of a primitive cubic lattice with the four-site atomic basisin fractional crystal coordinates. The drawing shows all boundary representatives of these sites. Eight corner gold chemical atoms, shared among eight unit cells each, contribute one gold chemical atom. Six face-centred copper chemical atoms, shared between two unit cells each, contribute three copper chemical atoms. Thuswhich gives the required stoichiometry. Although the unlabelled positions form a face-centered cubic lattice, a face-centering lattice translation exchanges gold and copper and therefore fails to preserve the ordered crystal structure. Its Bravais lattice is primitive cubic lattice.

Specify the normalization of the alloy sublattice order parameter rather than assuming that its endpoint values uniquely define it. Let $p_c$ be the probability of gold occupation at a corner site and defineThe complementary copper probability is $3(1-\rho )/4$, so the corner-site atomic scattering factor isThe stoichiometry provides one gold chemical atom per four-site unit cell. If all three face sublattices are equivalent, their gold occupation probability $p_f$ obeys $p_c+3p_f=1$. ThereforeAt $\rho =1$ the corner and face sites carry pure gold and pure copper; at $\rho =0$ they have identical statistical compositions. These formulas describe the branch with $0\le \rho \le 1$ and equivalent face-site averages.

For a long chain of length $L$ with periodic boundary conditions, each spin has $Ldk/(2\pi )$ quantum states in a wavenumber interval $dk$. Including two spin states, a full energy band contains $2L/a$ electrons. The state counting in a Bloch band therefore gives the energy densityThe integral of $1+y^2$ is $8/3$. For $ϵ>0$, the even square-root integral isFor example, this follows by setting $u=2y$, applying the supplied antiderivative, and evaluating both endpoints; the lower-end logarithm contributes $\ln ϵ$. HenceThe continuous zero-coupling limit is $\mathcal{U}(0)=2E_{\mathrm{ZB}}/(3a)$. This agrees with directly integrating the one-dimensional free electron model up to $k_F=\pi /a$ at two electrons per unit cell.

Define the dimensionless coupling $C=\pi |V_0|/E_{\mathrm{ZB}}$, so that $ϵ=C|\delta |$. Substitution into the energy density from part (b) gives, within the prescribed continuation of the two-wave nearly-free electron model,Its value at $\delta =0$ is understood by continuity. For $C|\delta |\ll 1$,Both electrons per doubled unit cell have been included. The fixed $2G$ harmonic affects the underlying energy bands beyond the two-wave approximation; the expression here is the energy density requested from part (b), rather than an exact solution for all harmonics of the full periodic potential.

Without the Peierls distortion, adjacent chemical atoms are separated by $a/2$. The primitive unit cell contains one chemical atom and contributes one electron. A energy band with two spin states accommodates two electrons per primitive unit cell, so the first energy band is half filled. Its true first Brillouin zone extends to $\pm 2\pi /a$, while the Fermi points are at $\pm \pi /a$.

Describing the same chain with a doubled unit cell of length $a$ merely folds the half-filled energy band into two branches that touch at those Fermi points. The unbroken lattice translation by $a/2$ prevents an artificial band gap there. A Peierls distortion breaks that smaller lattice translation and supplies the missing Fourier coefficient at $2k_F=G$; the lower folded energy band can then be completely filled and separated from the upper energy band. Thus the zero-band gap result at $\delta =0$ agrees with the original one-electron band filling.

Use the same normalization for the elastic energy and electronic energy density: if $\kappa {\delta }^2$ is an energy per unit cell, divide it by $a$ first; if it is already per length, use it directly. Write its coefficient per length as $K>0$ and put $A={\pi }^2{V}_0^2/(4a{E}_{\mathrm{ZB}})>0$. The small-distortion change in total energy density isNo minimization calculation is needed. For arbitrarily small nonzero $|\delta |$, the logarithm is negative with arbitrarily large magnitude. Hence the negative electronic Peierls energy gain, proportional to ${\delta }^2\ln (1/|\delta |)$, eventually exceeds any fixed positive quadratic elastic energy. There are distorted configurations with lower energy than the undistorted configuration for every finite $K$ and nonzero coupling. The ideal zero-temperature infinite-chain model therefore hasThe source's ${\delta }_{\mathrm{eq}}>0$ is correct when $\delta$ denotes a nonnegative distortion amplitude. If $\delta$ is signed, the energy density is even and the two signs describe equally stable patterns related by a lattice translation. For example, $V(x+a/2;\delta )=V(x;-\delta )$, so a negative equilibrium distortion has the same energy as the positive one. The zero-coupling case $V_0=0$ instead has its minimum at $\delta =0$. The argument assumes a coupled chain, complete lower-energy band filling and zero temperature; it does not by itself establish finite-temperature long-range order in an isolated one-dimensional system.

A Frenkel exciton is a tightly bound electron–hole excitation localized on approximately one molecular or ionic unit. Its radius is comparable to the lattice spacing, its exciton binding energy can be large, and transport is described through transfer of the local excitation between units. Molecular solids such as crystalline anthracene provide a natural example: the optical excitation is largely associated with an individual molecule, so a broad parabolic-band hydrogen model is not the most useful starting point.

A Wannier-Mott exciton extends over many unit cells. It is described approximately by a conduction electron and a valence hole with effective masses and a dielectric-screened Coulomb attraction. Its binding is usually smaller, and its levels resemble a hydrogenic series. gallium arsenide is a suitable example because its direct-gap band-edge carriers have light effective masses and substantial dielectric dielectric screening, supporting spatially extended excitons. Localization length and dielectric screening, rather than whether a material happens to be called a semiconductor, distinguish the two limiting pictures. Real materials can lie between the limits, so neither family is a universal sharp classification.

Use the scattering vector $\mathbf{q}={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}$; reversing its sign conjugates the following amplitudes without changing the intensities. In the kinematic approximation, the atomic scattering factor of atom $s$ isAtoms at $\mathbf{R}+{\mathbf{d}}_s$ therefore give the scattering amplitudeHere $A_e$ is the single-electron amplitude; $F$ is the unit cell structure factor, and $S_L$ is the geometrical lattice sum. If the crystal has $N_i$ repeats along ${\mathbf{a}}_i$, the latter factor is a product of geometric series, whose magnitudes containThe sharp constructive peaks occur when every $\mathbf{q}\cdot {\mathbf{a}}_i=2\pi h_i$. Thus a permitted reflection has wavevector change equal to a reciprocal lattice vector, with $F(\mathbf{G})\ne 0$ and the additional elastic condition $|{\mathbf{k}}_{\mathrm{out}}|=|{\mathbf{k}}_{\mathrm{in}}|$. A finite crystal broadens these peaks; it does not impose an exact delta-function selection on every scattered photon.

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.

The fluorite structure consists of a face-centered cubic lattice of calcium sites with all eight tetrahedral interstitial sites occupied by fluoride. In fractional coordinates of the conventional cubic unit cell, the calcium basis is $(0,0,0),(0,\frac{1}{2},\frac{1}{2}),(\frac{1}{2},0,\frac{1}{2}),(\frac{1}{2},\frac{1}{2},0)$, and fluoride occupies all combinations $(x,y,z)$ with $x,y,z\in \{\frac{1}{4},\frac{3}{4}\}$. The projection therefore superposes two fluoride sites at each quarter-coordinate position. Labels in the original diagram give fractional depth along $[001]$; periodic boundary sites are displayed on both edges.

Counting shared sites gives $8/8+6/2=4$ calcium sites and eight interior fluoride sites: four formula units per conventional cell.