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The central system of protons and neutrons in an atom. Its mass number $A$ is its baryon number in ordinary nuclear states; $Z$ is the proton number and $N$ the neutron number.

The nucleus contains almost all of an atom's mass and carries positive electric charge $Ze$, where $e$ is the elementary charge. The atomic number $Z$ identifies the chemical element; changing $N$ at fixed $Z$ gives another isotope. The mass number $A=Z+N$ counts nucleons, rather than specifying the exact nuclear mass: nuclear binding energy lowers the mass of a bound system relative to its separated constituents. The simplest nucleus is a single proton; other atomic nuclei can contain many protons and neutrons bound by the nuclear force.

A nuclear quantum state has properties such as spin and a spatial nuclear charge distribution. These are different questions from merely counting nucleons. A small-angle elastic scattering measurement can reveal a nuclear charge radius, while a wider range of momentum transfer tests the distribution's shape. The Hofstadter nuclear electron scattering experiment made this connection between scattering patterns and nuclear structure experimentally concrete.

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]

Stereoscopy enables three-dimensional track reconstruction.^[ref] Calibrated camera positions and optical geometry are needed before a photographed arc becomes a physical radius of curvature. Neutral particles do not directly leave the same continuous ionization trail as charged particles; an interaction or decay can instead reveal them through charged secondary tracks. A photon's inferred path between interactions is not a photographed luminous line.

In a uniform magnetic field, with negligible stopping power and elastic scattering over the fitted segment, the transverse projection is a circle. The Lorentz force turns transverse momentum at angular rate $v_{\perp }/r$, givingHere $r$ is measured in the plane perpendicular to the magnetic field. The relation remains valid with relativistic physical momentum; it does not require $p=mv$. Thus curvature determines magnetic rigidity, not mass or kinetic energy alone. For chosen $|q|=e$, $B=1.5\mathrm{T}$ and $r=0.40\mathrm{m}$, $p_{\perp }\simeq 0.180\mathrm{GeV}/c$. This is an invented example, not a fit to a historical photograph. Reconstructing a three-dimensional helix also determines the component parallel to the magnetic field; the transverse radius of curvature alone does not. Determining the sign of electric charge additionally requires the field direction and the direction of travel, as in the charge-sign ambiguity of a curved track.

Choose a primitive cubic lattice with $a=0.400\mathrm{nm}$, one identical point scatterer per primitive unit cell, incident direction ${\mathbf{s}}_{\mathrm{in}}=(0,0,1)$, and the continuous wavelength band $0.080\le \lambda \le 0.160\mathrm{nm}$. This is a hypothetical model, not the zinc blende specimen of the Laue X-ray diffraction experiment of 1912. Write a candidate reciprocal lattice vector as $\mathbf{G}=(2\pi /a)(h,k,-m)$ with $m>0$, and set $S=h^2+k^2+m^2$. The Laue equations and elastic scattering giveA transmission photographic plate at $z=L>0$ intercepts only $s_{\mathrm{out},z}>0$. Its coordinates relative to the direct beam areFor $(h,k,-m)=(2,1,-1)$ and $L=50\mathrm{mm}$, $S=6$, $\lambda =0.13333\dots \mathrm{nm}$ and ${\mathbf{s}}_{\mathrm{out}}=(2/3,1/3,2/3)$, so $(X,Y)=(50,25)\mathrm{mm}$. Its length is one, and each component difference satisfies the corresponding Laue equations. This verifies a direction and a selected wavelength, without predicting an exposure or measured wave intensity.

The first panel has fourfold symmetry because its chosen Bravais lattice, beam axis and square detector share that symmetry. For the tilted panel, apply a rotation matrix $R$ to each index vector, set $v=R(h,k,l)$, and use $\lambda =-2a{v}_z/|v{|}^2$ and ${\mathbf{s}}_{\mathrm{out}}=(0,0,1)+(\lambda /a)v$. All indices with $|(h,k,l)|\le 2a/{\lambda }_{\min }=10$ suffice: elastic scattering bounds $|\mathbf{G}|\le 2|{\mathbf{k}}_{\mathrm{in}}|$. Thus the plotted spectral and angular selection is complete within this ideal point model and detector window. Multiples of an index vector select inversely scaled wavelengths and can land on the same spot; a white-beam spot need not identify a unique diffraction order.

A search for interactions of passing dark matter with a terrestrial target, for example nuclear recoils from elastic scattering. Predicting a count requires the local density and velocity distribution, interaction cross section, target response and detector acceptance. A background-compatible count gives a conditional interaction bound, rather than a measurement of the cosmological relic density.

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.

Scattering that resolves the constituents of a hadron at large spacelike momentum transfer, with an inclusive sum over final hadron states.

In electron scattering from a proton, write the reaction as $e^{-}p\to e^{-}X$: the outgoing electron is measured, while $X$ denotes all allowed final hadron states. Large momentum transfer supplies a short-distance probe; a large final invariant mass separates the deep continuum from elastic scattering and individual particle resonances. The useful surprise is that an extended proton can have a response characteristic of much smaller constituents. Deep inelastic scattering turns measurements outside the proton into constraints on its internal parton distribution functions.

The SLAC-MIT deep inelastic electron scattering experiment connects this definition to an actual measurement. Its evidence came from distributions of outgoing electron energies and scattering angles, rather than a photograph of a quark. Bjorken scaling concerns deep-inelastic structure functions after known scattering factors have been accounted for; it does not say that the raw differential scattering cross-section is constant.

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.

Robert Hofstadter's Stanford installation selected electrons up to $190\mathrm{MeV}$. Magnetic spectrometer scans counted shielded Lucite Cherenkov radiation pulses per monitor charge. Carbon-12 at $187\mathrm{MeV}$ and $80^{\circ }$ resolved $185.1\mathrm{MeV}$ elastic scattering and $180.7\mathrm{MeV}$ inelastic scattering ($4.43\mathrm{MeV}$ excitation). Calibration used forward-beam energy, hydrogen nuclear recoil and carbon-12 energy levels. Closed entrance slits checked background.^[ref]

The key distinction is between an electron counter and an energy measurement. Here the counter registers particles admitted by the magnetic spectrometer; changing its magnetic field selects another outgoing momentum. A peak is therefore reconstructed from a scan, rather than read as one pulse whose height equals the scattered electron energy. Dividing counts by monitored incident electric charge compensates for changes in beam intensity. A peak's height alone is insufficient: its area and the instrument's accepted momentum interval determine the selected yield.

An inelastic scattering peak corresponds to a different final nuclear quantum state, so including it in the elastic scattering yield would answer a different structural question. Conversely, finite energy resolution can hide sufficiently nearby excitations beneath an apparent elastic scattering peak. Background subtraction, target-loss corrections, angular acceptance and counter efficiency all enter before counts become a differential scattering cross-section. A dip in uncorrected counts is therefore not automatically a zero of a nuclear charge form factor.

Kamiokande, associated with Masatoshi Koshiba, was an underground water Cherenkov detector viewed by photomultipliers. Improved water purity and reduced background enabled solar neutrino observations. A concentration of reconstructed recoil directions toward the Sun distinguished a source signal above an approximately direction-independent background.^[ref]

For solar neutrinos, electron elastic scattering supplies a directional recoil, with different cross sections for active neutrino flavors. Neither the Homestake experiment deficit nor the SN 1987A neutrino burst alone established neutrino oscillation; later evidence compared complementary channels and propagation effects.^[ref]

For incident and outgoing unit propagation vectors ${\mathbf{s}}_{\mathrm{in}}$ and ${\mathbf{s}}_{\mathrm{out}}$, wavelength $\lambda >0$, and primitive lattice translation vectors $a_1,a_2,a_3$, the Laue equations areAll three must hold for the same wavelength and outgoing direction. The integers label a reciprocal lattice vector; they need not be coprime. For wavevectors ${\mathbf{k}}_{\mathrm{in},\mathrm{out}}=(2\pi /\lambda ){\mathbf{s}}_{\mathrm{in},\mathrm{out}}$, equivalentlyThis uses angular wavenumbers; the crystallographic convention without $2\pi$ instead uses $\mathbf{s}/\lambda$.^[ref]

For a short proof, translate a scatterer through $a_i$. Its incident wave phase changes by ${\mathbf{k}}_{\mathrm{in}}\cdot a_i$ and propagation to the same distant observation direction contributes $-{\mathbf{k}}_{\mathrm{out}}\cdot a_i$. The relative wave phase factor is therefore $\exp (-i\mathbf{G}\cdot a_i)$. It equals one for all lattice translations exactly when $\mathbf{G}$ belongs to the reciprocal lattice. A finite crystal lattice structure factor broadens the ideal maxima; the crystal basis structure factor can cancel them. Geometric phase matching is necessary for a sharp reflection, but does not guarantee a visible spot.

For elastic scattering, the additional constraint isIt is an Ewald sphere centered at $-{\mathbf{k}}_{\mathrm{in}}$ in the space of transfer vectors $\mathbf{G}$. The parent elastic Bragg scattering condition uses $q={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}=-\mathbf{G}$, so its displayed condition has the opposite sign. Both give Bragg's law for the same lattice planes.

In the Laue method, a fixed single crystal structure receives a collimated beam containing many wavelengths. Each reciprocal lattice vector can select its own wavelength; waves with different wavelengths need not interfere with one another. For $\mathbf{G}\ne 0$, the elastic scattering constraint gives the particularly useful testA candidate reflection requires ${\lambda }_{\mathbf{G}}>0$ within the source spectrum and an outgoing ray that reaches the detector. Varying wavelength changes the radius and center of the Ewald sphere while leaving the reciprocal lattice fixed. A monochromatic wave instead fixes the radius: rotating the crystal structure or varying the incident wavelength in a scan finds reflections sequentially. A white-beam photograph integrates many spectral selections in one exposure; it is not an angular scan at one known wavelength.

The Laue method is useful for orienting a single crystal structure: tilting it changes which reciprocal lattice vectors meet the spectral and detector conditions. Absolute scale has an ambiguity if the wavelengths are unknown. Scaling all direct lattice translations by $c>0$ and all selected wavelengths by $c$ leaves every outgoing direction unchanged. Independently known spectral information or length calibration is needed to remove this ambiguity.

Consider an original illustrative model with exactly two outgoing scattering channels: a central beam $0$ and one diffracted beam $g$. Let $t\ge 0$ be specimen thickness and $\kappa >0$ a constant coupling with units of inverse length. Normalize the probability amplitudes $A_0,A_g$ to incident probability current, assume equal longitudinal speeds, and specifyThese ordinary differential equations define a lossless toy model, not a fit to the silicon example or a complete theory of crystal scattering. Differentiating once more gives $A_0^{\prime \prime }=-{\kappa }^2{A}_0$, soThe identity $p_0+p_g=1$ expresses conservation within these two scattering channels. Alternatively, substituting the ordinary differential equations into $d(|A_0{|}^2+|A_g{|}^2)/dt$ makes the two cross terms cancel. Here ideal bright-field transmission electron microscopy accepts fraction $p_0$ and rejects fraction $p_g$; ideal dark-field transmission electron microscopy accepts precisely the rejected channel. This complementarity requires the declared two-channel completeness and absence of loss.

Even with known $\kappa$, thickness is ambiguous: $\kappa t=\pi /4$ and $3\pi /4$ both give $(p_0,p_g)=(1/2,1/2)$. Near $t=0$, $p_g\simeq (\kappa t{)}^2$; extending that approximation through a full oscillation would erase the ambiguity incorrectly. The model isolates reversible transfer between probability amplitudes; it does not say that every thicker specimen must look darker.

Real dark-field transmission electron microscopy usually collects only a subset of the directions rejected in bright-field transmission electron microscopy. Other elastic scattering directions, inelastic scattering and detector losses prevent a general two-image conservation rule. To keep loss and background separate, extend the counting model explicitly. Let $\ell$ be the fraction absent from both recorded beam channels, $p_0+p_g=1$ describe the remaining fraction, $N$ be incident count per exposure, ${\eta }_j$ the detector efficiency and $b_j$ its mean additive background. Then the expected value of recorded count $n_j$ isFor separately acquired equal-illumination exposures with known $b_j,{\eta }_j$, the two corrected mean counts sum to $N(1-\ell )$, not necessarily $N$. Removal from these channels is not synonymous with physical absorption: an electron can leave the chosen acceptance or change kinetic energy without being stopped in the specimen. Background subtraction removes a mean offset, but its counting variance remains.

Neutron scattering measures how a specimen changes the directions and kinetic energies of incident neutrons. Neutron diffraction uses the elastic scattering pattern to investigate crystal structure and magnetic order; inelastic neutron scattering uses energy transfer to investigate excitations such as phonons and magnons. The two measurements ask complementary questions: what arrangement repeats in space, and what disturbances can propagate through it?

The 1994 Nobel Prize in Physics recognized Clifford Glenwood Shull's diffraction work and Bertram Neville Brockhouse's spectroscopy.^[ref]

Nuclear scattering couples to atomic nuclei; magnetic scattering couples the neutron's magnetic moment to electronic magnetic moments. The latter underlies both magnetic neutron diffraction and magnon spectroscopy. An atomic number is not a nuclear neutron scattering weight: the relevant coefficient in the elementary coherent model is the nuclear neutron scattering length.

Treat the incident and outgoing neutrons as free nonrelativistic particles outside the specimen. Their wave vectors, de Broglie wavelengths, momenta and kinetic energies obeyHere $k_i=|{\mathbf{k}}_i|$, $k_f=|{\mathbf{k}}_f|$, $m_n$ is the neutron mass, and $\hslash =h/(2\pi )$ uses the Planck constant. Subtracting the outgoing projectile quantities from the incoming ones gives the momentum transfer and energy delivered to the specimen:Thus $\mathbf{Q}$ has units of inverse length, rather than momentum. A positive $\omega$ means the specimen gains energy; negative $\omega$ means it supplies energy to the neutron. If the angle between incident and outgoing propagation directions is $2\theta$, the law of cosines givesKnowing the detector angle alone therefore leaves $Q$ undetermined unless the outgoing energy is known or an elastic scattering assumption is justified.

For a reproducible unit conversion choose $m_n=1.67492750056\times 10^{-27}\mathrm{kg}$, $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $1\mathrm{eV}=1.602176634\times 10^{-19}\mathrm{J}$. The first is a recommended central value; the latter two conversions are exact.^[ref] With $1$ ångström equal to $10^{-10}\mathrm{m}$ and $1\mathrm{meV}=10^{-3}$ electronvolts, substitution into $E=h^2/(2m_n{\lambda }^2)$ givesSquare brackets here mean the numerical value in the indicated unit. For an original chosen example, ${\lambda }_i=2$ ångströms and ${\lambda }_f=2.5$ ångströms give $E_i\simeq 20.451\mathrm{meV}$, $E_f\simeq 13.089\mathrm{meV}$ and $\hslash \omega \simeq 7.362\mathrm{meV}$. At $2\theta =60^{\circ }$, $Q\simeq 2.879{\mathring{\mathrm{A}}}^{-1}$. These values stipulate an allowed projectile geometry; they do not assert that any particular specimen has an excitation there.

An elastic scattering event leaves the nuclear internal quantum state unchanged while transferring energy to its motion. Let a target of mass $M$ initially be at rest. Neglect the electron rest mass, but retain relativistic target recoil. Conservation of momentum and conservation of relativistic energy giveEliminating $q$ yields $M{c}^2(E-E^{\prime })=E{E}^{\prime }(1-\cos \theta )$ and henceThe static relation $q=2E\sin (\theta /2)/(\hslash c)$ follows only when $E/(M{c}^2)$ is negligible. For small recoil, $T_R\simeq {\hslash }^2{q}^2/(2M)$; this last approximation is nonrelativistic for the recoiling nucleus, not for the incident electron. The invariant spacelike transfer, expressed in energy-squared units, isso $Q/(\hslash c)$ and the spatial inverse length $q$ coincide only in the static limit. Recoil kinematics alone does not correct the static differential scattering cross-section: a finite-mass target also changes its kinematic prefactor and the appropriate current matrix elements.

For nonrelativistic elastic scattering of a WIMP of mass $m_{\chi }$ from a stationary nucleus of mass $m_N$, the reduced mass is ${\mu }_{\chi N}=m_{\chi }{m}_N/(m_{\chi }+m_N)$. In the centre-of-momentum description, a scattering angle $\vartheta$ gives momentum transfer squared $q^2=2{\mu }_{\chi N}^2{v}^2(1-\cos \vartheta )$. The target kinetic energy is $E_R=q^2/(2m_N)$, giving the displayed maximum at backscattering. In SI units, using masses in kilograms and velocity in metres per second gives joules. Low recoil energy and a falling recoil spectrum make threshold and response calibration important for dark-matter direct detection.

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.

The incident X-ray drives the electrons of an atom. Away from an absorption edge, and when the photon energy greatly exceeds the relevant electronic binding energies but is small compared with $m_e{c}^2$, each electron has approximately the same Thomson scattering amplitude. The much heavier nucleus makes a negligible contribution because its acceleration is much smaller.

For elastic scattering, the amplitudes from different bound electrons add coherently. In units of the single-electron amplitude, the atomic scattering factor is the Fourier transform of the bound-electron number density:Consequently, when the electron cloud is not well resolved by the scattering vector, its internal phase differences are small andThis is an approximation to the coherent scattering amplitude, not an assertion that independently scattered intensities add as $N_e$. At larger momentum transfer, destructive interference within the cloud reduces $f$; near absorption edges, the atomic scattering factor acquires dispersive corrections. Inelastic scattering is a separate contribution. In an ionic crystal, use the ion's bound-electron count rather than automatically using the neutral atom's atomic number.

The neutron is nonrelativistic at this kinetic energy. Its de Broglie wavelength isIn elastic scattering, Bragg's law gives $2d\sin \theta =\lambda$. The measured total deflection is $2\theta$, so it must be halved before taking the sine. The reflection order is absorbed into the Miller indices of the diffracting lattice planes. In the order of the four measured rings, the interplanar spacings areThe last digits give the numerical calculation, rather than accuracy beyond the supplied angular precision.