Codex @codex  0

Loss of spatial overlap between interacting finite beams because their energy flux directions differ. In an anisotropic crystal, the direction of the Poynting vector need not coincide with the wave vector. For beam radius $w$ and relative walk-off angle $\rho$, an overlap length is of order $w/|\tan \rho |$. A full conversion calculation also includes beam profiles, diffraction and phase mismatch.

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 structural biologist studies the three-dimensional organization of biological macromolecules and how it relates to their function. Structural biology combines measurement with a macromolecular structural model: a representation supported by experimental observations, rather than a direct photograph of every atom. Common approaches include X-ray crystallography, cryo-electron microscopy and NMR spectroscopy.

In X-ray crystallography, a crystal produces a diffraction pattern from which researchers infer an electron density map and refine a molecular model. A concrete dataset is calmodulin structure1CLL in the Protein Data Bank. Its primary record reports X-ray diffraction at $1.70\mathring{\mathrm{A}}$ resolution and a refined model containing four calcium ions.^[ref]

An original unit conversion gives the scale:That resolution value is not a universal error bar of $0.170\mathrm{nm}$ on every modeled atom. Some regions can be well constrained while flexible or disordered regions are poorly resolved. In1CLL, the primary paper's abstract explicitly identifies poorly defined terminal residues.^[ref]

A structural biologist can then test a mechanistic proposal with complementary evidence. A model may suggest a calcium ion binding environment; comparing binding or activity measurements can test its functional consequences. A static arrangement alone does not establish every reaction step or rate. The scientific work lies in connecting the model, its uncertainty and the behavior of the molecular system.

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.

A nonlinear optics process producing a field at the sum of two input angular frequencies. The driven electric polarization contains a term proportional to the product of the input complex amplitudes. With undepleted plane-wave inputs, integration along a crystal gives an output amplitude proportional to $L{e}^{i\Delta kL/2}\mathrm{sinc}(\Delta kL/2)$; the sinc function therefore determines the ideal length and phase mismatch dependence. A nonzero initial output field also interferes with this generated field.

A neutral two-dimensional linear electronic spectrum has heat capacity proportional to $A{T}^2$, while three-dimensional acoustic phonons have Debye heat capacity proportional to $N{T}^3/{\Theta }_D^3$. Thus their ratio scales as $A{k}_B^2{\Theta }_D^3/({\hslash }^2{v}^{2}NT)$. At fixed crystal composition and shape, $A/N\propto 1/L$ for linear size $L$, favoring the electronic surface contribution in smaller or colder specimens. The continuum laws require thermally populated wavelengths small compared with sample size; their formal low-temperature crossover need not lie inside that regime.

A surface reconstruction rearranges chemical atoms at a crystal surface relative to a simple termination of its bulk crystal structure.

Theodore Maiman demonstrated stimulated optical emission from ruby in 1960. Optical pumping transferred energy into the chromium ions in the crystal, producing the conditions for stimulated emission near 694 nm. This showed that a fluorescent solid could act as an optical amplifier and oscillator, carrying the principle of the maser into the visible spectrum. The result tied the material's energy levels to a practical laser source.^[ref]

A physicist.^[ref]

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.

The tensor relating a small unconstrained thermal strain to a temperature change: ${\mathbf{\epsilon }}^{\mathrm{th}}=\mathbf{\alpha }\Delta T$. A crystal orientation rotates this tensor into laboratory coordinates.

An oscillation of transmitted or diffracted intensity with crystal thickness caused by coupled-wave electron diffraction. A uniform chemical composition can still show thickness fringes.

Alongside the 1927 Davisson–Germer experiment, George Paget Thomson developed an independent transmission test. Alexander Reid first used celluloid; Thomson used thin metal films. A narrow electron beam crossed a film about $10^{-6}\mathrm{cm}$ thick, reaching a photographic plate perpendicular to the beam. Rings appeared. Accelerating voltages of $20$–$60\mathrm{kV}$ set the de Broglie wavelengths; metal structures known from X-ray diffraction allowed quantitative comparison. A downstream magnetic field deflected ring-forming electrons like those passing through holes, identifying the carriers as electrons rather than X-rays. Celluloid's uncertain structure limited its quantitative interpretation.^[ref]

The Davisson–Germer experiment uses crystal reflection and angular electric current scans; this transmission camera records a spatial pattern behind the specimen. Both probe interference in electron diffraction, with different specimen preparations and readouts. A developed photographic plate records accumulated detections, not a direct image of the wavefunction.

Why rings? In an ideal film containing many small crystals with uniformly distributed orientations, a family of lattice planes with spacing $d$ can satisfy Bragg's law at every azimuth about the incident beam. First-order elastic scattering then forms a cone of opening half-angle $2\theta$, where $2d\sin \theta =\lambda$. A flat detector a distance $L$ downstream cuts this cone in a circle:The last expression requires small scattering angles. It explains a useful control: hold the specimen and camera distance fixed, increase accelerating voltage, and the ring radii should shrink with the electron accelerating-voltage wavelength. It predicts positions, not intensities: preferred crystal orientations, film thickness and repeated quantum scattering can alter a real pattern.

For a chosen example, take $L=0.200\mathrm{m}$ and $d=0.200\mathrm{nm}$. The exact relativistic electron accelerating-voltage wavelength gives $\lambda =8.589\mathrm{pm}$ at $20\mathrm{kV}$ and $4.176\mathrm{pm}$ at $80\mathrm{kV}$. The corresponding exact ring radii are $8.594\mathrm{mm}$ and $4.176\mathrm{mm}$. These are calculated examples with chosen spacings and camera distance, not Thomson's measurements. The higher voltage lies outside the range quoted from his lecture, deliberately illustrating a prediction rather than reconstructing his data.

A triple-axis spectrometer uses Bragg's law at an initial crystal monochromator to select ${\lambda }_i$ and at a second analyzing crystal to select ${\lambda }_f$. The sample sits between these selectors. The three axes refer to the monochromator, sample and analyzer rotations. The final counter measures accepted counts; the analyzer's wavelength selection makes those counts sensitive to outgoing energy even when the counter does not measure each neutron's energy directly.

For a constant-$\mathbf{Q}$ scan, change incident or outgoing energy while coordinating sample and analyzer settings so that ${\mathbf{k}}_i-{\mathbf{k}}_f$ stays fixed. Counts versus $E_i-E_f$ then probe excitations at one momentum transfer. Changing only the analyzer angle generally changes both transferred quantities. Incident-beam monitoring, analyzer response and the instrumental resolution must enter quantitative interpretation; the diagram's arrows specify information flow, not an absolute intensity calibration.

Choose a one-dimensional crystal with cell translation $a$ and two atoms per cell at $0$ and $d=a/2$. Here $d$ is an intracell displacement, distinct from the interplanar spacing $d_{\mathrm{p}}$ above. In a single-scattering approximation, a row of $N$ cells has scattering amplitudeThe sum selects reciprocal lattice positions $Q_h=2\pi h/a$ as $N$ becomes large. At these positions, the two-site crystal basis structure factor and its squared magnitude areConsequently $b_1=b_2=b$ gives even reflections of strength $4b^2$ and extinguishes the odd reflections. Choosing $b_1=b$, $b_2=b/2$ instead gives $9b^2/4$ and $b^2/4$: the odd/even ratio is $1/9$. Equal atomic numbers would not establish equal $b_j$ if the isotopes differ. The figure below plots these chosen coefficients, without assigning them to actual elements. Missing reflections can encode destructive wave interference within a cell, not the absence of matter at a site.

Two identical phase-matched crystals generate coherent harmonic amplitudes. In a dispersive gap, the harmonic from the first crystal accumulates a different phase from twice the transmitted pump phase. Their interference gives the displayed undepleted-pump intensity in the stated equal-coupling amplitude normalization. Depletion changes the fringe shape; in the ideal lossless identical-crystal model, an opposite relative phase can coherently undo the first conversion.

In a common birefringent down-conversion arrangement, the two daughter photons have the same polarization class, ordinarily both ordinary or both extraordinary relative to the crystal. The pump has a suitable different index for phase matching. This classification alone does not establish polarization entanglement.

Ultrasound is sound above the conventional upper limit of human hearing, approximately $20\mathrm{kHz}$. In a crystal, ultrasonic travel-time measurements determine long-wavelength elastic wave speeds, which are the small-wavevector slopes of the corresponding acoustic phonon branches.

Examination of particles in urine with a microscope, including cells, casts and crystals.

Walther Friedrich coauthored the experimental observation of X-ray diffraction by a crystal with Paul Knipping and Max von Laue. A beam transmitted through the sample produced separated spots on a photographic plate. Their geometry connected the periodic structure of the crystal with constructive interference of the scattered radiation. The apparatus turned a proposed wave test into an observable pattern, providing a method for investigating matter on atomic length scales.^[ref]

A physicist.^[ref]

William Henry Bragg developed experimental methods that turned crystal diffraction into X-ray spectroscopy. A crystal selects reflected radiation at particular angles, while a movable detector measures the reflected intensity. Relating these peaks to lattice-plane spacing connects an instrument reading with the wavelength of the incident radiation. His joint work with Lawrence Bragg established a quantitative relation between crystal geometry and X-ray reflection, helping make diffraction a practical probe of both radiation and structure.^[ref]

A physicist who worked on crystal scattering and Bragg's law.

Institutional biography: www.rigb.org/explore-science/explore/person/william-henry-bragg-1862-1942