Scattering amplitude

A scattering amplitude is the coefficient of an outgoing asymptotic wave relative to the specified incoming wave.

A pole reached on an unphysical sheet of the scattering amplitude, often on the negative imaginary momentum axis for a short-range one-dimensional potential. Unlike a positive-imaginary-momentum bound-state pole, it does not give a normalizable exponentially decaying state on both sides.

The scattering amplitude at zero deflection angle. Its real part changes the phase of a coherent beam in matter; its imaginary part is related by the optical theorem to removal from that beam. Different forward amplitudes for a kaon and a neutral antikaon allow neutral kaon regeneration without fundamental CP violation.

A scattering amplitude evaluated with definite external helicities. Rotational covariance relates its angular dependence to a Wigner D-matrix. Summing final helicities and averaging an unpolarised initial mixture yields a spin-averaged squared matrix element.

With relativistically normalized external states, $\mathcal{M}$ is the reduced S-matrix element after removing the four-momentum conservation distribution. A relativistic scattering cross-section combines $|\mathcal{M}{|}^2$ with the Lorentz-invariant phase-space measure and the invariant flux factor. Identical-leg factorials in Feynman vertices and the identical final-state symmetry factor in the event integral have different roles.

Write the S-matrix as $S=1+iT$. Unitarity gives $-i(T-T^{†})=T^{†}T$, so the diagonal forward matrix element obeys $2\mathrm{Im}{T}_{ii}={\sum }_f\int d{\Phi }_f|T_{fi}{|}^2$. Thus the imaginary part of a forward scattering amplitude measures the total inclusive transition rate. For an electromagnetic source, the hadronic electromagnetic-current spectral density supplies the inclusive spectral sum.

For a real elastic scattering potential the first Born forward amplitude is real, while the leading total scattering cross-section is quadratic in the potential. The second Born amplitude supplies an imaginary part from its outgoing Green function. Symmetrizing the two interaction positions makes its imaginary part equal to $k/(4\pi )$ times the integrated squared first Born amplitude. The identity therefore compares quantities at the same perturbative order; a real first Born amplitude alone cannot test it against a nonzero scattering cross-section.

For an isotropic elastic amplitude normalized by an incident unit plane wave and outgoing $f{e}^{ikr}/r$, total cross section is $4\pi |f{|}^2$. The optical theorem gives the displayed identity. At a resonance where the real part of $1/f$ vanishes, $f=i/k$ and the cross section is $4\pi /k^2$.

Near a one-particle pole, a scattering amplitude factors into amplitudes coupling each side to the intermediate state, summed over its physical polarizations. The pole position gives its squared mass, while its residue encodes its spin and couplings.

Crossing symmetry relates scattering amplitudes by analytic continuation exchanging incoming and outgoing particles. A symmetric four-point scalar amplitude has equivalent descriptions under permutations of the Mandelstam variables.

With relativistically normalized external states, a two-particle initial state has differential cross-section $d\sigma =|\mathcal{M}{|}^{2}d{\Phi }_n/\mathcal{F}$, where $d{\Phi }_n$ is the Lorentz-invariant phase-space measure and $\mathcal{F}$ is the invariant flux factor. For identical unobserved final particles, divide the full labeled phase-space integral by their permutation multiplicity.

The lowest initial centre-of-mass frame energy at which a specified final state can be made. For free massive final particles in natural units, the minimum is the sum of their masses, attained when their relative momenta vanish. The cross section at the exact threshold can vanish even though the channel becomes kinematically accessible there.

At leading electromagnetic order and with negligible electron mass, the photon-channel relativistic cross-section is ${\sigma }_h=4{\pi }^2\alpha {\rho }_h=16{\pi }^3{\alpha }^2{\rho }_J/s$, in the stated hadronic photon spectral density and hadronic electromagnetic-current spectral density conventions. Initial electron and positron spins are averaged; final hadronic states are summed inclusively. Electroweak channels beyond photon exchange are separate contributions.

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is $R=N_c{\sum }_f{Q}_f^2$. It depends on flavor charges as well as the number of active flavors; it is not simply $N_c{N}_f$.

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude $|\lambda |$. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is ${\lambda }^2/(128{\pi }^{2}s)$. On a hemisphere representing each event once, the density is ${\lambda }^2/(64{\pi }^{2}s)$. Both conventions give total event cross-section ${\lambda }^2/(32\pi s)$.

For two incident particles in relativistic normalization, this is $2\sqrt{\kappa (s,m_1^2,m_2^2)}$, where $\kappa$ is the Källén function. For equal masses it is $2\sqrt{s(s-4m^2)}$.

The on-shell measure $d^{3}p/(2E)$ is Lorentz invariant. The product measure together with the four-dimensional Dirac delta distribution enforces four-momentum conservation. For equal masses in two-body scattering, $d{\Phi }_2=\sqrt{1-4m^2/s}d\Omega /(32{\pi }^2)$.

For two outgoing particles with centre-of-mass momentum magnitude $k$ and total energy $\sqrt{s}$, integrating the four-momentum delta function gives $d{\Phi }_2/d\Omega =k/(16{\pi }^2\sqrt{s})$. A factor $1/2!$ for identical outgoing particles is inserted when counting events over the full phase space, not into this labelled kinematic measure itself.

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.

This total decay-width formula applies in the parent's rest frame when the squared scattering amplitude is independent of direction and the daughters are distinct. For identical daughters an additional symmetry factor is needed.

When the full Lorentz-invariant phase-space measure treats $n$ identical final particles as labeled, each physical final configuration is counted $n!$ times. The unlabelled relativistic scattering cross-section therefore includes $1/n!$. For two identical scalars, this divides the full-angle result by two; equivalently, integrate over a region containing only one representative of each exchanged pair.

The momentum transfer in a scattering process is the difference between the incoming and outgoing wavevectors, commonly $Q=k-k^{\prime }$ up to a sign convention and a factor of $\hslash$.

The energy component of a transferred four-momentum in a specified frame. In fixed-target deep inelastic scattering the target rest frame gives $\nu =E-E^{\prime }$, where $E,E^{\prime }$ are incoming and scattered electron energies. It is frame dependent even though $Q^2=-q^2$ is Lorentz invariant.

In quantum scattering, a spatial feature on scale $\ell$ contributes Fourier components with wavenumber of order $1/\ell$. A momentum transfer of magnitude $q$ supplies wavenumber $q/\hslash$, giving the characteristic estimate $\ell \sim \hslash /q$. With the energy-like quantity $Q=qc$, this becomes $\ell \sim \hslash c/Q$. The full de Broglie wavelength uses $h/p$ and differs from the corresponding reduced wavelength by $2\pi$. Beam energy only bounds accessible momentum transfer; it does not determine the resolution of every event.

Elastic scattering preserves the total kinetic energy. For one particle scattered by a fixed target, the incoming and outgoing wavevectors therefore have equal magnitudes.

An elastic weak interaction of a neutrino with an electron. For a stationary electron of mass $m$ and an effectively massless incident neutrino of energy $E$, the maximum recoil kinetic energy in natural units is $T_{\max }=2E^2/(m+2E)$. This kinematic limit does not specify the interaction probability.

In coherent elastic scattering, amplitudes from the scattering constituents combine while their internal state is unchanged. For X-rays, recoil taken by a whole chemical atom can produce an effectively unshifted component alongside the Compton scattering component.

Diffraction is interference produced when waves scatter from spatial structure. A periodic structure concentrates the scattered intensity at reciprocal-lattice conditions.

The spatial distribution of intensity produced by interference of a wave passing an aperture or obstacle. A circular aperture's far-field point-source pattern is an Airy disk; a Rayleigh criterion comparison superposes the intensities of two mutually incoherent sources.

Diffuse scattering generated by fluctuating lattice displacements. Resolved one-phonon creation and annihilation transfer both energy and crystal momentum, with Bose factors $n_B+1$ and $n_B$. An energy-integrating diffraction measurement can record their summed intensity without resolving the small energy transfer. Phonon polarization, soft-mode frequencies and atomic scattering weights determine the pattern; static disorder supplies a distinct possible diffuse contribution.

Ewald sphere curvature is the curvature of the reciprocal-space surface sampled by elastic scattering. Accounting for its departure from a plane can improve very high-resolution electron microscopy reconstructions.

The obliquity factor weights the directional contribution of an aperture element in a scalar diffraction integral. In a fixed observation direction in the Fraunhofer diffraction regime, its variation across a sufficiently small aperture can be neglected along with the variation of the geometric amplitude denominator. Its angular dependence can still matter when comparing widely separated observation directions.

Electron diffraction is interference between scattered electron probability amplitudes. It connects the de Broglie wavelength of the incident beam to the arrangement of the scattering material. In elastic scattering from a fixed crystal, $|{\mathbf{k}}_{\mathrm{out}}|=|{\mathbf{k}}_{\mathrm{in}}|=2\pi /\lambda$, and constructive crystal scattering selects wavevector transfers $\mathbf{Q}={\mathbf{k}}_{\mathrm{out}}-{\mathbf{k}}_{\mathrm{in}}$ in the reciprocal lattice. The physical momentum transfer to the electron is $\hslash \mathbf{Q}$ in this convention.

Bragg's law expresses the same condition as $2d\sin {\theta }_B=n\lambda$, where $d$ is the spacing of the chosen lattice planes, $n$ the diffraction order, and ${\theta }_B$ the angle to those planes. The angle between incoming and outgoing propagation directions is $\chi =2{\theta }_B$ for that reflection geometry. In normal-incidence backscattering, a detector's angle $\varphi$ from the outward surface normal instead gives $\chi =\pi -\varphi$; neither detector colatitude nor angle to the surface normal should silently replace ${\theta }_B$. A finite-row electron diffraction model uses a different spacing, the spacing between sites along a surface row, and makes that distinction explicit.

diffraction in which repeated scattering and coupling between transmitted and diffracted beams significantly affect their amplitudes. Electron-diffraction intensity can then vary strongly with thickness and excitation error; it is not generally just the squared single-scattering crystal basis structure factor.

Transmission-electron diffraction from a region selected by an aperture in an image plane. Downstream lenses image the objective back focal plane to the detector, producing reciprocal-space rather than real-space information.

The Davisson–Germer experiment turned elastic scattering from nickel into evidence for a matter wave. The search became a deliberate wave test in 1926, with diffraction beams found early in 1927. The earlier crystal investigation had begun in 1925; the de Broglie wavelength proposal was already present in the 1924 thesis.^[ref]

A heated tungsten ribbon supplied electrons; gun apertures selected a narrow beam. Two $1\mathrm{mm}$ apertures were $8\mathrm{mm}$ apart; the gun ended $7\mathrm{mm}$ from a nickel crystal cut parallel to its $\{111\}$ planes. An adjustable electric potential difference set the speed. The beam struck normally inside an evacuated, field-free enclosure with target and outer electrodes at one potential. A double-box collector, $11\mathrm{mm}$ from the target with a $1\mathrm{mm}$ entrance, selected directions. A retarding electric potential difference rejected slower electrons; the inner collector fed a sensitive galvanometer. Moving the collector scanned colatitude; rotating the crystal scanned azimuth.^[ref]

The readout was collected electric current, not a photograph of electron paths. Voltage scans held the gun-plate current constant because incident-beam current could not be held constant; the overall current trend therefore needed caution. A strong feature developed at $54\mathrm{V}$ near $50^{\circ }$ colatitude. Surface heating distinguished crystal beams from adsorbed-gas beams; finite apertures and background shifted or broadened apparent peaks.^[ref]

At the bright beam's colatitude, an azimuth scan revealed three companion directions, matching crystal symmetry. The test compared wavelengths inferred from diffraction angles and known crystal spacings with wavelengths inferred from accelerating voltage and momentum; the 1937 lecture displayed this comparison against $U^{-1/2}$. Agreement across beam directions supplied more evidence than one angular peak.^[ref]

The inference is about coherent quantum phases in scattering probabilities. It does not establish a wavy particle trajectory, and a collector-current experiment does not resolve individual detection events. Predicting the full intensity additionally requires the material's scattering amplitude, its surface condition, and instrumental angular acceptance; the de Broglie wavelength alone predicts none of those strengths. The following finite-row electron diffraction model isolates phase addition without attempting to reproduce the nickel measurements.

Take $N$ equally spaced scattering sites at $x_j=jd$ on a line, a normally incident plane wave, equal site scattering amplitudes, and observation far enough away that outgoing directions are effectively parallel. Retain only one scattering event per amplitude, as in the Born approximation. For angle $\varphi$ from the outward normal, write the adjacent-site quantum phase difference as $\delta =kd\sin \varphi$, with $k=2\pi /\lambda$; reversing its overall sign leaves the intensity unchanged. Summing a geometric series givesAt $\delta =2\pi m$ use the continuous limit $I/I(0)=1$, rather than treating $0/0$ as a missing peak. The integer $m$ is the diffraction order: $d\sin \varphi =m\lambda$. This is a surface-row spacing, not the plane spacing in Bragg's law. The plotted intensity is normalized to its central value; it is not an integrated detection probability or an absolute collector electric current.

At $100\mathrm{V}$, $\lambda \simeq 0.12264\mathrm{nm}$ and the positive first-order angle is $\arcsin (\lambda /d)\simeq 29.4^{\circ }$. At $400\mathrm{V}$, the wavelength halves and this angle becomes about $14.2^{\circ }$. Increasing $N$ narrows each peak through more selective interference, while increasing momentum brings a fixed-order peak nearer the normal. Unequal site amplitudes, multiple scattering, a three-dimensional crystal lattice structure factor, and finite detector acceptance can change this simple picture; it is an explanatory model with hypothetical inputs.

In the Born approximation, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

In the kinematic approximation, each incident wave scatters once, so the total scattering amplitude is the coherent sum of single-atom amplitudes. Multiple scattering and depletion of the incident beam are neglected. The structure factor then determines systematic reflection absences. Strong multiple scattering can violate an absence predicted in this approximation.

For coherent elastic diffraction, the intensity associated with a given scattering amplitude is proportional to its squared modulus. Integrated powder diffraction intensities additionally include the powder diffraction multiplicity, Lorentz-polarization factor, Debye–Waller factor, absorption and instrument corrections. A peak's height is not its integrated intensity unless its profile width and shape are also specified.

Static fluctuations in which chemical element occupies a site produce scattering beyond coherent Bragg diffraction from the mean crystal structure. For independent binary occupations with probabilities $p$ and $1-p$, the variance of the site's atomic scattering factor is $p(1-p)|f_A-f_B{|}^2$. Writing each site amplitude as its mean plus a zero-mean fluctuation shows that the mean squared total amplitude is the coherent contribution plus a term proportional to the number of sites and this variance. Consequently a forbidden reflection of the average crystal structure can still sit on a diffuse background. Correlated occupations add nontrivial scattering vector dependence.

Coherent scattering from lattice planes satisfying $2d\sin \theta =n\lambda$. The angle between incident and diffracted rays is $2\theta$, not $\theta$.

A diffracted beam or powder peak associated with a reciprocal-lattice vector satisfying the elastic scattering condition. Its position follows from lattice spacings and its amplitude from the unit-cell structure factor. A reciprocal point whose structure factor vanishes does not produce an allowed reflection in the kinematic model.

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

A tetragonal Bravais lattice represented by conventional-cell points $(0,0,0)$ and $(1/2,1/2,1/2)$, with $a=b\ne c$ and right angles. A motif attached to each point determines the atomic structure and can give more than two atoms per conventional cell.

The set $\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2:n_1,n_2\in \mathbb{Z}\}$ for two independent planar translation vectors. Its primitive-cell area is $|\det ({\mathbf{a}}_1,{\mathbf{a}}_2)|$.

A two-dimensional Bravais lattice with perpendicular primitive vectors of the same length $\ell$. An orthogonal matrix $R$ gives its orientation, and its primitive area is ${\ell }^2$. Its nearest-neighbor graph is a square lattice; the set of crystal translations and the graph with chosen adjacency are related but distinct objects.

A lattice represented by a rectangular conventional cell with corner points and one center point. For conventional sides $2a,2b$, primitive vectors can be $(a,b)$ and $(-a,b)$. The special case $a=b$ is a square lattice with a rotated primitive basis.

A direct lattice vector is an integer linear combination of primitive translation vectors ${\mathbf{a}}_i$ of a Bravais lattice. Translating by such a vector preserves the lattice. Its dot product with every reciprocal lattice vector is an integer multiple of $2\pi$ in the angular-wavevector convention.

For identical touching atoms represented by hard spheres of radius $r$, a face-centered cubic lattice has conventional side $a_F=2\sqrt{2}r$ and four atoms per conventional cell. A body-centered cubic lattice has side $a_B=4r/\sqrt{3}$ and two atoms per conventional cell. Their primitive unit cell volumes are respectivelyEach primitive cell contains one atom in the monatomic model, so its mass density is atomic mass divided by $\Omega$. These are contact-geometry estimates; a fixed hard-sphere radius does not encode thermal expansion or phase-dependent electronic bonding.

A region whose translates tile a lattice's ambient space, together with the lattice points represented in it. A primitive unit cell represents one lattice point; a conventional unit cell may represent several to display symmetry.

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.

A fundamental cell of a Bravais lattice, of volume equal to the absolute determinant of any primitive lattice basis. Direct and reciprocal lattice primitive cell volumes multiply to $(2\pi {)}^d$ in dimension $d$ with the angular-wavenumber convention.

A tetragonal conventional cell with lattice points at its vertices and face centers describes a Bravais lattice. It is equivalent, after a rotation and a change of conventional cell, to a body-centered tetragonal description. The reciprocal lattice of a body-centered tetragonal lattice with conventional sides $a,a,b$ consists of $2\pi (h/a,k/a,l/b)$ with $h+k+l$ even; these points form a face-centered tetragonal conventional cell with sides $4\pi /a,4\pi /a,4\pi /b$.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

For equivalent scatterers on the four conventional fcc sites, the centering factor is $1+e^{i\pi (h+k)}+e^{i\pi (h+l)}+e^{i\pi (k+l)}$. It is four when the integer indices are all even or all odd, and zero otherwise. An additional atomic basis of a crystal can impose further extinctions. This rule concerns conventional cubic indexing.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

For a simple polygon whose vertices are lattice points in ${\mathbb{Z}}^2$, its area is $A=I+B/2-1$, where $I$ counts interior lattice points and $B$ counts boundary lattice points.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

An elongated reciprocal-space reflection caused by finite extent in the corresponding real-space direction. A foil of thickness $t$ has reflection breadth of order $1/t$ along its normal, increasing the range over which a reflection can intersect an Ewald sphere. The exact profile depends on the specimen shape and scattering model.

A plane wave whose phase is unchanged by every direct lattice translation: $e^{i\mathbf{k}\cdot \mathbf{R}}=1$ for all $\mathbf{R}\in \Lambda$. Its wavevector is a reciprocal lattice vector. In one dimension $ka=2\pi m$.

A reciprocal lattice vector $\mathbf{G}$ has the displayed property for every direct lattice vector $\mathbf{R}$. Thus $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. Fourier components of a lattice-periodic potential carry these wavevectors and couple plane-wave electronic states whose momenta differ by $\hslash \mathbf{G}$. This is the origin of the Bragg-plane mixing in a nearly-free electron model.

For free-electron dispersion proportional to $k^2$, states at $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$ are degenerate on this plane, obtained by equating their squared norms. A nonzero periodic-potential Fourier component can mix them and open an avoided crossing in the nearly-free electron model. First-zone boundaries are selected planes of this kind.

The reciprocal lattice of a body-centered cubic lattice is face-centred cubic, and that of a face-centered cubic lattice is body-centred cubic. In both cases its conventional cubic side is $4\pi /a$ if the direct conventional side is $a$. Thus a direct BCC-to-FCC phase transition reverses the reciprocal lattice types. The reciprocal primitive cell volume is $(2\pi {)}^3/\Omega$, so thermal expansion contracts reciprocal spacings, while a discontinuous change in direct primitive unit cell volume gives an inverse change in reciprocal cell volume.

A first-order structural transition need not continuously deform one perfect reciprocal array into another. Phase coexistence, new orientation variants, domains and loss of single-crystal alignment can produce two sets of diffraction peaks or many differently oriented sets. The two endpoint lattice types alone do not specify the transformation path or orientation relationship.

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

The cubic crystal system has cubic point symmetry. Its three cubic Bravais lattices are primitive, body-centered, and face-centered. The body-centered cubic lattice is one member of that classification.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

In a body-centred cubic lattice of conventional edge $a$, a lattice site has eight nearest neighbours along body half-diagonals at distance $\sqrt{3}a/2$, then six next-nearest neighbours along the coordinate axes at distance $a$. There are two sites per conventional cell, so identical atoms of mass $m$ and mass density $\rho$ obey $\rho a^3=2m$. Specifying a neighbour shell matters: the nearest and next-nearest distances differ.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

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.

A lattice translation moves every lattice point of a Bravais lattice onto another lattice point. Its vector is an integer combination of primitive lattice translation vectors. The Laue equations require the wave phase factor of every such translation to be one.

The coherent scattering amplitude associated with a specified spatial arrangement, obtained by summing the site amplitudes with their relative wave phases. Distinguish the crystal lattice structure factor, which sums over lattice sites, from the crystal basis structure factor, which sums the chemical atoms inside a unit cell. In a periodic crystal the two factors multiply. A basis factor can vanish even at an allowed reciprocal lattice vector, causing an extinct reflection.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

Sites at 000 and one half of each cubic coordinate give $F=f_A+(-1{)}^{h+k+l}{f}_B$. Identical sites produce a body-centered cubic lattice with absent odd-parity reflections. Distinct chemical species generally preserve only the simple cubic lattice translations, so those same odd reflections need not vanish. A geometric body center is not by itself a symmetry translation of the decorated crystal.

A vanishing crystal basis structure factor can depend on approximate equality of atomic scattering factors, rather than on an exact space group condition. For two inequivalent site classes, an amplitude $F=C(f_A-f_B)$ vanishes when $f_A=f_B$ but becomes weakly observable if bonding electron densities or thermal displacements differ. Chemical identity alone does not imply identical local atomic scattering factors. The diffraction intensity is quadratic in the small difference.

The scattering amplitude assigned to one atom or ion as a function of momentum transfer, normalized relative to a stated scattering convention. X-ray and electron factors are different; multiplying by basis phase factors produces the total unit-cell amplitude.

A diffraction reflection enabled by a larger chemical or structural repeat than that of a reference parent structure. In ordered $L{1}_2$ $A_{3}B$, sites $(0,0,0)$ and the three face centres give $F=f_B+f_A[(-1{)}^{h+k}+(-1{)}^{h+l}+(-1{)}^{k+l}]$. Equal-parity indices give $f_B+3f_A$; mixed parity gives $f_B-f_A$. The latter vanish when the two scattering factors are equal.

For elastic scattering with $k-k^{\prime }=q\in {\Lambda }^{\ast }$ and $|k|=|k^{\prime }|=k$,The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

This reciprocal-lattice condition is equivalent to Bragg's law, expressed in terms of the lattice-plane spacing and the wavelength.

Bragg's law selects directions of constructive interference in coherent elastic scattering from equally spaced lattice planes:Here $d>0$ is the perpendicular interplanar spacing, $\lambda >0$ the incident wavelength, $n$ the positive integer diffraction order, and $\theta$ the Bragg angle, measured between the incoming beam and the lattice planes. The scattering deflection, between incoming and outgoing propagation directions, is $2\theta$. Thus a plotted detector angle $2\theta$ must be halved before insertion into Bragg's law. The $\theta$ in the parent elastic Bragg scattering condition denotes the full deflection; the present $\theta$ denotes half of it. ^[ref]

The lattice planes organize wave interference throughout a crystal structure; they are not literal metal mirrors or separate reflecting surfaces. The picture assumes elastic scattering, hence an unchanged wavelength, and sufficiently stable relative wave phases across the contributing lattice planes. X-ray wavelengths comparable with atomic interplanar spacings make these directions experimentally accessible. The sine cannot exceed one: an order exists geometrically only if $n\lambda /(2d)\le 1$. For $\lambda >2d$, even the first diffraction order is inaccessible. A geometrically possible direction need not have nonzero wave intensity; the crystal lattice structure factor and crystal basis structure factor also matter. ^[ref]

The Bragg angle is the glancing angle from the incident beam to the selected lattice planes, with $0<\theta \le 90^{\circ }$. If $\alpha$ is instead the angle from the beam to the plane normal, $\alpha =90^{\circ }-\theta$ and Bragg's law becomes $n\lambda =2d\cos \alpha$. Defining the angle explicitly prevents a sine-versus-cosine error as well as a factor-of-two error in the detector deflection.

Choose $\lambda =0.154\mathrm{nm}$ and $d=0.200\mathrm{nm}$ as an illustrative model, not historical measurements. For the first diffraction order,The second diffraction order has $\sin \theta =0.770$ and $2\theta \simeq 100.707778^{\circ }$; the third would require $\sin \theta =1.155$ and is impossible. These are different diffraction orders of the same interplanar spacing at the same wavelength.

Conversely, a first-order reflection at a measured deflection $2\theta$ gives the interplanar spacingUsing $2\theta \simeq 45.287481^{\circ }$ recovers $d\simeq 0.200\mathrm{nm}$. This is the useful reversal: a known wavelength turns an angular measurement into an atomic length. An unassigned diffraction order or an unknown wavelength prevents that single measurement from determining $d$ uniquely.

For an isotropically compressed cubic crystal, $d\ln a/dp=-\kappa /3$. Differentiating Bragg's law at fixed wavelength gives the displayed derivative. Constant isothermal compressibility gives $a(p)=a_0{e}^{-\kappa p/3}$, not an exactly linear lattice parameter. The integrated prediction presumes no intervening structural phase change.

Compare two rays scattered into parallel outgoing directions from neighboring lattice planes. In the diagram, $A$ lies on the upper plane and $B$ on the lower plane, directly beneath $A$ with $AB=d$. Draw through $A$ the incoming and outgoing equal-wave phase fronts, perpendicular to their respective rays. They meet the lower ray at $C$ and $D$. Each right triangle gives $CB=BD=d\sin \theta$, so the lower ray travels the extra distanceThe resulting phase difference is $\Delta \phi =2\pi \Delta \ell /\lambda$. Constructive interference requires $\Delta \phi =2\pi n$, which proves Bragg's law in this ray construction. Comparing equal-wave phase fronts, rather than arbitrary ray endpoints, is what makes the distance comparison meaningful.

The same geometry connects to the reciprocal lattice. With wavenumber $k=2\pi /\lambda$, the elastic Bragg scattering condition gives a transfer magnitude $|q|=2k\sin \theta$. A reciprocal vector normal to these lattice planes has $|q|=2\pi n/d$. Equating them again yields Bragg's law. The ray and reciprocal lattice descriptions therefore express the same wave phase matching.

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]

William Lawrence Bragg introduced the plane interpretation; his father William Henry Bragg built the X-ray spectrometer. Their joint 1913 paper, The Reflection of X-rays by Crystals, reports the experiment. ^{[ref] [ref]}

An X-ray tube supplied radiation through lead slits to a crystal mounted with its cleavage face through the rotation axis. A rotating ionization chamber replaced a spectrometer telescope; electric current measured the scattered signal, and a circular scale supplied angular readings. They first scanned the chamber at fixed crystal orientation, then swept crystal and chamber together to follow the reflection. ^[ref]

Changing chamber gas checked detector dependence; readings on opposite sides checked face-setting errors. Repeated peaks with sines of half-deflections in integer ratios, together with matching absorption coefficients, distinguished repeated diffraction orders of the same wavelength from different radiation components. Comparing crystal faces constrained interplanar spacings. Their limits included slit broadening, face alignment and incomplete interception at glancing incidence; assigning absolute wavelengths still required a structural choice for $d$. ^[ref]

Bragg scattering is coherent elastic scattering from a periodic lattice, enhanced when the wavevector transfer is a reciprocal lattice vector. Finite crystals have narrow rather than infinitely sharp reciprocal-space peaks.

The identical two-atom motif adds a factor $1+\exp [i\pi (h+k+l)/2]$ to the FCC lattice structure factor. Among even-index FCC diffraction reflections, only those with $h+k+l$ divisible by4 survive. The allowed FCC reflection200 therefore vanishes for equal motif scattering amplitudes.

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.

A powder-diffraction camera with photographic film wrapped around a cylindrical specimen axis. Diffraction cones intersect the film in paired curved lines; angular position along the cylindrical film differs from the $R=L\tan 2\theta$ relation for a forward flat plate. Back reflections can also be recorded.

Diffraction from an oriented single crystal. A reflection requires both an allowed nonzero structure factor and access to its reciprocal-lattice direction and Bragg angle.

A geometry in which the diffracted beam traverses the specimen and exits another surface. Absorption and thickness affect whether it is observable.

Reflection diffraction from planes inclined to the specimen surface. Incident and exit angles relative to the surface differ; both must correspond to rays on the accessible side of an opaque specimen.

A diffraction geometry in which the incident and diffracted rays make equal angles with the specimen surface. The selected reciprocal-lattice vector is normal to that surface.

An instrument measuring X-ray diffraction intensity against scattering angle, sample orientation or photon energy.

Powder X-ray diffraction measures X-ray diffraction from many crystal grains with different orientations. For each allowed set of lattice spacings, suitable grains satisfy Bragg's law and produce a cone of diffracted directions, seen as a ring on an area detector or a peak in an angular scan. Several peak positions and systematic absences can identify a crystal structure even when a sample no longer retains one single-crystal orientation. Preferred orientation, strain and phase coexistence can complicate interpretation.