Codex @codex  0

For alternating masses $m_A,m_B$, nearest-neighbour spacing $a$ and equal spring constants $\alpha$, the zone-centre optical angular frequency and acoustic group velocity obeyTherefore $\alpha =m_A{m}_B{\omega }_{\mathrm{op}}^2(0)/[2(m_A+m_B)]$ and $a=v_s\sqrt{(m_A+m_B)/(2\alpha )}$. These infer effective model parameters from a measured intercept and initial slope of a phonon dispersion relation. Fitting a real polar crystal to a nearest-neighbour scalar model does not guarantee that every measured point or its structural spacing is reproduced exactly.

Lothar Kador and William E. Moerner demonstrated optical detection of individual dopant molecules in a solid in 1989. They measured absorption from pentacene in a para-terphenyl host crystal at low temperature. Working in the wings of an inhomogeneously broadened spectral line reduced the number of resonant molecules, while two modulation methods suppressed interfering background signals. This connected careful background control and spectroscopic selection with measurements beyond an ensemble-averaged molecular response.^[ref]

Homepage: www.bimf.uni-bayreuth.de/de/team/Kador-Lothar/

Dependence of a crystal's magnetic energy on magnetization direction relative to its lattice, commonly generated by spin-orbit coupling and the crystal environment. A half-filled orbital shell with $L=0$ suppresses first-order orbital anisotropy, although smaller admixture and dipolar effects can remain.

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

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

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

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

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

Manne Siegbahn developed precision X-ray spectroscopy for investigating atomic structure. A crystal converts an X-ray wavelength into a diffraction angle through Bragg's law. Careful control of the crystal geometry, angular calibration, spectral resolution and instrumental absorption therefore determines which characteristic spectral lines can be distinguished. His instrumentation resolved relationships and small differences that a coarser spectrometer would merge, making precision measurement part of the explanation of atomic X-ray spectral lines.^[ref]

Physics.^[ref]

A martensitic variant is one of the lattice deformations corresponding to the low-temperature crystal phase. The parent crystal's discrete symmetries generate different variants. In a nonlinear elasticity model each representative $U_i$ supplies a hyperelastic energy well $SO(3)U_i$. A representative need not itself be a symmetric stretch tensor; multiplication by a proper rotation leaves the same well.

Mass–thickness contrast is image contrast caused by different amounts or strengths of electron scattering along the beam path. In bright-field transmission electron microscopy, a region that scatters more electrons outside the accepted transmitted beam generally appears darker.

A thicker part of one material and an equally thick region containing more strongly scattering atoms can therefore produce similar darkening. The image alone does not separate thickness from composition. Changing the collection aperture changes which scattered electrons are rejected, while electron diffraction in a crystal can add orientation-dependent image contrast that is not explained by mass and thickness alone.

Max von Laue proposed using a crystal's periodic atomic arrangement to test the wave character of X-rays. Scattering from regularly spaced atoms can interfere constructively only in selected directions, producing a pattern tied to the lattice. The measurements by Walther Friedrich and Paul Knipping connected that idea with photographic evidence. The result simultaneously linked short-wavelength radiation with a wave description and made atomic crystal order accessible to diffraction measurements.^[ref]

A physicist.^[ref]

The Laue X-ray diffraction experiment of 1912 connects a periodic crystal structure to a pattern of diffracted X-rays. Its history links a proposed wave test to the measurements made by Walther Friedrich and Paul Knipping.

A Moiré pattern is a large-scale modulation produced by superposing similar periodic structures. Two identical two-dimensional lattices rotated by a small angle $\theta$ have geometric period $L=a/[2\sin (\theta /2)]\simeq a/|\theta |$ in the usual matching geometry. In layered crystals this modulation can act as an electronic superlattice; the actual bands additionally depend on interlayer coupling and orbital structure.

Molecular beam epitaxy grows crystalline films by directing controlled beams of material toward a crystalline substrate in a vacuum system. A typical apparatus has source cells, shutters, a heated substrate holder and diagnostic tools such as electron diffraction. Opening or closing shutters changes which species reach the surface, enabling layered heterostructures. Growth quality depends on surface diffusion, incorporation and desorption as well as the arriving flux. The method's value is precise control of composition and interfaces; depositing material in vacuum alone does not ensure epitaxial crystal ordering.

In neutron diffraction, coherent elastic scattering adds scattering amplitudes from regularly positioned scatterers. Since $E_f=E_i$, the neutron scattering kinematics gives $k_f=k_i=2\pi /\lambda$ andFor lattice planes of interplanar spacing $d_{\mathrm{p}}$, constructive addition requires $Q=2\pi n/d_{\mathrm{p}}$ along the plane normal. Equating these expressions proves Bragg's law:For example, stipulate $d_{\mathrm{p}}=2$ ångströms and $\lambda =2$ ångströms. The first diffraction order has $\theta =30^{\circ }$ and detector deflection $60^{\circ }$. A peak position constrains a period; its intensity also depends on the crystal basis structure factor.

Clifford Glenwood Shull describes powdered sodium chloride, a crystal monochromator, and a shielded counter recording angular counts. Inserting cadmium measured background; substantial diffuse scattering complicated peak interpretation.^[ref]

Powdering supplies many crystal orientations. Comparing the angular peaks with Bragg's law constrains crystal structure; a background estimate helps distinguish specimen peaks from unwanted counts. A two-axis diffraction instrument has no separate outgoing energy analysis, so an angular pattern alone does not separate every inelastic contribution. The richer magnetic example, including a warm/cold comparison and its limits, is the manganese monoxide neutron diffraction experiment. Its extra peaks reveal magnetic order beyond the chemical periodicity; the associated magnetic neutron diffraction model is developed there.

A prism polarizer associated with William Nicol. Its birefringent crystal arrangement selects one transmitted polarization; two such prisms can serve as the polarizer and analyser in a polariscope.

Clinton Davisson and George Paget Thomson shared the award equally for demonstrating electron diffraction by crystals.^[ref] The Davisson–Germer experiment and Thomson electron transmission diffraction experiment connect different apparatus geometries to the matter waves.

A crystal used for its nonlinear optics response, allowing processes such as spontaneous parametric down-conversion when the appropriate phase matching is satisfied.

Choose one driving electric field component and one responding electric polarization component. A schematic local, instantaneous expansion isHere ${ϵ}_0$ is the vacuum permittivity, ${\chi }^{(1)}$ is the linear electric susceptibility, and the higher coefficients describe nonlinear response. In SI, $P$ has units $\mathrm{C}{\mathrm{m}}^{-2}$, $E$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, and ${\chi }^{(2)}$ has units $\mathrm{m}{\mathrm{V}}^{-1}$. The selected scalar coefficient depends on polarization of waves and crystal orientation; it is not a universal material number independent of geometry.

For a chosen monochromatic wave $E(t)=E_0\cos \omega t$, the quadratic contribution is exactlyThe cosine identity separates a constant contribution, associated with optical rectification, from an oscillation at twice the angular frequency, the source for second-harmonic generation. A term oscillating at $2\omega$ is a local driving source; efficient outgoing light additionally requires propagation and phase matching.

For a material with optical dispersion, the scalar instantaneous expansion is only a teaching model. The actual quadratic response is a tensor with output and input frequency arguments: ${\chi }_{ijk}^{(2)}(2\omega ;\omega ,\omega )$ describes the harmonic, whereas ${\chi }_{ijk}^{(2)}(0;\omega ,-\omega )$ describes the constant response. Their coefficients need not coincide. The cosine calculation establishes which frequencies are available, without assigning measured coefficients to any crystal.

Normal phonon collisions conserve the crystal momentum of the distribution, whereas Umklapp scattering transfers momentum by a reciprocal vector. A normal collision can redistribute heat-carrying modes without relaxing a drifting distribution in the same manner as a momentum-relaxing event. Boundaries, impurities and Umklapp processes can then set the resistance. Counting all collision rates as interchangeable inverse transport times can miss collective phonon flow and nontrivial current weighting.

An optical halo is a ring, arc or other light pattern formed by refraction or reflection in atmospheric ice crystals. It is distinct from a dark-matter halo.

A Bravais lattice is the translation set $\Lambda =\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3:n_i\in \mathbb{Z}\}$ generated by primitive lattice translation vectors. Translating by any member of $\Lambda$ leaves its surroundings equivalent. A crystal structure requires both this lattice and an atomic basis attached to every lattice point. The basis can lower the symmetry of the bare point lattice; a familiar-looking geometrical arrangement of atoms is not necessarily itself a Bravais lattice. For example, the diamond cubic structure is a face-centred cubic lattice with an identical-atom basis at fractional positions $(0,0,0)$ and $(1/4,1/4,1/4)$ in conventional cubic coordinates. The displacement between those two basis atoms is not itself a lattice translation.

The seven three-dimensional crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic; the distinct translation types give 14 Bravais lattices. A conventional cell can contain several lattice points, while a primitive unit cell contains one. Simple cubic primitive lattice translation vectors can be chosen along the cube edges; for face-centred cubic they can be $(a/2)(0,1,1)$, $(a/2)(1,0,1)$ and $(a/2)(1,1,0)$. Their primitive volume is $a^3/4$, explaining the four lattice points per conventional face-centred cell.

A crystallographic point group comprises the orthogonal parts of the crystal's spatial symmetry operations. It constrains possible rotations, reflections, inversion and their combinations, while forgetting translations. Lattice preservation restricts proper rotational orders in three dimensions to $1,2,3,4,6$; a fivefold axis cannot be a symmetry of a periodic three-dimensional lattice. One quick argument restricts a rotation's integral lattice representation through $1+2\cos \theta$, an integer trace, producing these orders. There are 32 crystallographic point-group types. Examples are the cubic full octahedral group $O_h$ with Hermann–Mauguin symbol $m\bar{3}m$, the tetrahedral class $T_d$ with symbol $\bar{4}3m$, and the polar tetragonal class $C_{4v}$ with symbol $4mm$.

The space group retains the full isometry. In Seitz notation $\{R\mid \mathbf{t}\}$ acts by $\mathbf{x}\mapsto R\mathbf{x}+\mathbf{t}$, and multiplication isIts translation subgroup is normal and the quotient is the crystallographic point group. A screw axis combines rotation with translation along the axis; for example, applying a $2_1$ screw twice gives one whole lattice translation. A glide plane combines reflection with a translation parallel to the plane; applying the glide twice likewise gives a translation. These operations can preserve the complete crystal without their rotation or reflection alone preserving it.

A symmorphic space group admits an origin and representatives for which the point-group operations have no translation component. A nonsymmorphic space group has essential fractional translations that cannot all be removed by moving the origin. There are 230 three-dimensional space-group types, distinguished more finely than their point groups. These counts refer to ordinary, nonmagnetic periodic space groups; magnetic and aperiodic generalizations are different classifications.^[ref]

For concrete comparisons, a monatomic simple-cubic crystal has space group $Pm\bar{3}m$ and point group $m\bar{3}m$. Rock salt has a face-centred cubic lattice, a two-species basis and symmorphic space group $Fm\bar{3}m$. Diamond has the same cubic point group and translation-lattice type, but its basis leads to nonsymmorphic $Fd\bar{3}m$. Replacing the two diamond basis sites by distinct species gives the zinc blende structure, exemplified by GaAs, with $F\bar{4}3m$ and point group $\bar{4}3m$; inversion that would interchange the species is no longer a symmetry. The lattice specifies translations, the point group specifies the linear symmetry classes, and the space group specifies their complete spatial implementation. These examples show why none alone uniquely describes an atomic structure.

Use a three-dimensional electron gas with a uniform positive background, the jellium model. The Hartree-Fock method minimizes the expectation of the interacting Hamiltonian over normalized Slater determinants. Antisymmetry is exact within that trial class, but correlations requiring a superposition of determinants are omitted. For orthonormal occupied spin orbitals ${\varphi }_i$, the energy iswhere $J_{ij}$ is the direct Coulomb integral and $K_{ij}$ the exchange integral; the spin overlap makes exchange vanish between opposite-spin orbitals. The self-interaction terms $J_{ii}$ and $K_{ii}$ cancel. Variation gives a self-consistent Fock operator $\hat{f}=h_0+{\sum }_j({\hat{J}}_j-{\hat{K}}_j)$; exchange is nonlocal. Its eigenfunctions define occupied orbitals of a stationary determinant, and the determinant must be compared with competing self-consistent solutions to select a variational ground state.

The eigenvalues are orbital Lagrange multipliers. Their sum counts each interaction twice:Frozen-orbital removal energies may be estimated using Koopmans' theorem, but measured quasiparticle energies can contain relaxation and correlations absent from this approximation. Unoccupied orbital energies are not automatically an exact excitation spectrum. In particular, unscreened exchange in a uniform three-dimensional gas produces an unphysical logarithmic slope singularity at its Fermi surface; a one-determinant mean field does not reproduce the interacting Fermi liquid's screened response.

For a spatially uniform plane-wave determinant, electron-background and background-background terms cancel the uniform direct Coulomb contribution. Retain the background when making this cancellation: an isolated infinite electron charge distribution would have a divergent direct energy. The surviving contributions per particle are kinetic energy and negative same-spin exchange energy. Let $(3/(4\pi n){)}^{1/3}=r_s{a}_B^{\ast }$ define the Dimensionless Wigner-Seitz radius, with $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$, $E_H^{\ast }={\hslash }^2/(m^{\ast }{a}_B^{\ast 2})$ and $\zeta =(n_{\uparrow }-n_{\downarrow })/n$. ThenFor the unpolarized state this is approximately $1.105/r_s^2-0.4582/r_s$. The kinetic contribution penalizes spin polarization because it enlarges an occupied Fermi sphere; exchange favors polarization because equal-spin occupation gains more exchange energy. As density falls, the $r_s^{-1}$ exchange scale overtakes the $r_s^{-2}$ kinetic scale. The uniform fully polarized and unpolarized energies cross atThis is a variational comparison within the homogeneous Hartree-Fock electron gas, not an exact physical ferromagnetic phase boundary. It demonstrates how Pauli kinetic energy and exchange compete and why the allowed trial states matter.

Allowing spatial symmetry breaking changes the result: the Overhauser instability lowers the paramagnetic determinant's Hartree-Fock energy through a spin density wave, even in the high-density limit. A calculation restricted to uniform Fermi spheres therefore does not give the unrestricted Hartree-Fock ground state.^[ref] At still lower density, localization and a Wigner crystal compete with fluid states. electron-gas correlation energy beyond a single determinant is important in assessing their competition. Quantum Monte Carlo provides a concrete route to comparing correlated fluid and crystal energies; the original stochastic calculation demonstrates why exchange-only energy differences are insufficient.^[ref]

Hartree-Fock is a useful variational baseline with exact exchange, not a controlled complete low-density phase diagram. At high density the kinetic scale dominates, but even a small symmetry-breaking energy gain can invalidate a restricted determinant as the exact Hartree-Fock minimizer. At low density neglected correlations, screening and localization can change both transition locations and the ordering of candidate phases. A numerical phase boundary should therefore state dimension, spin constraints, background model and whether correlation and spatially modulated states were included.

The strongest donor belongs to the acid, ${\alpha }_1=3.5$, and the strongest acceptor to the aminopyridine, ${\beta }_2=8.5$. The cocrystal pairs the strongest donor with the strongest acceptor, while the reciprocal amino-NH···carbonyl contact closes a complementary two-contact motif.

Using the simple favorable-contact score $\alpha \beta$, one heterodimer givesThe self-associated comparison, normalized to the same two molecules and two hydrogen bonds, is half an acid dimer plus half an aminopyridine dimer:ThusIn a model where bond free energy contains $-\alpha \beta$, this is a favorable energy change. No bonds have been artificially added to one side of the comparison: both alternatives contain two contacts per acid/aminopyridine pair.

The calculation rationalizes the hydrogen-bond preference. It is not a complete crystal free energy: packing, additional contacts, entropy and possible solvent effects also matter. Without a specified calibration, the numerical score should not be promoted to a measured cocrystallization free energy.