A focused X-ray probe scans a thin specimen while a detector records transmission. Scanning photon energy across an absorption edge adds local spectroscopic contrast, with spatial resolution set by the probe and scanning performance. Thickness and composition both affect transmission, so an energy stack and suitable references help distinguish them.

Use of an approximate structural model to obtain initial phase information for an experimental crystallographic dataset; the model must still be assessed against the diffraction data.

Determination of a crystalline structure from the diffraction of X-rays by an oriented single crystal. It can distinguish cage connectivities and substitution patterns that share an elemental formula and the same number of NMR environments.

A stiff chain's transverse wandering over contour length $s$ scales as $s^3/{\ell }_p$. Equating this to the squared confining radius gives the displayed deflection length when $R\ll {\ell }_p$. A free-energy cost of order $k_{B}T$ per deflection yields confinement free energy per inserted contour length of order $k_{B}T/\lambda$. Numerical prefactors depend on geometry and boundary constraints.

For an elastic filament with reference arclength $s$, curvature $c$, axial strain $ϵ$ and excess twist rate $\Omega$, a harmonic energy density is $(K_b{c}^2+S{ϵ}^2+K_t{\Omega }^2)/2$. Bending and torsional rigidities have units of energy times length; the stretch modulus has units of force. Factoring out $k_{B}T$ therefore makes $A,C$ lengths but $B$ an inverse length. Axial strain under tension is approximately $F/S$. Twist is controlled by torque and rotational boundary conditions, not force alone. A three-dimensional thermally fluctuating worm-like chain has persistence length $K_b/(k_{B}T)$.

A circular loop of radius $R$ has constant curvature $1/R$ and contour length $2\pi R$. Integrating the quadratic bending energy density $K_b/(2R^2)$ gives the displayed energy. With $K_b=k_{B}TA$ this is $\pi k_{B}TA/R$. Applied-force work, axial strain, twist and contact energies are separate terms; the loop formula alone does not determine an actual loaded loop's radius.

Forest hardening is resistance to dislocation glide caused by dislocations that cut across a glide plane. An advancing dislocation must negotiate their intersections, form or break junctions, or bow between obstacles. The resulting obstacle network is called a forest because it obstructs motion across the plane, even though it is made of line defects rather than solid inclusions.

The mechanism connects dislocation density with work hardening: a shorter spacing between effective obstacles generally demands greater applied shear stress. It also explains why different slip systems need not evolve independently; slip on one system changes the obstacles encountered by another, contributing to latent hardening. Comparing a specimen before and after controlled plastic deformation tests the increase in flow resistance, while microstructural observation tests whether an assumed obstacle mechanism is plausible. Not every change of yield stress is forest hardening; other microstructural changes can also strengthen a material.

Latent hardening is increased resistance on one crystal slip system caused by plastic slip on another slip system. The word latent emphasizes that the affected system need not have been active during the earlier loading. It is therefore a statement about interactions between deformation mechanisms, rather than simply about the total amount of deformation.

A simple crystal-plasticity model writes ${\dot{g}}_{\alpha }={\sum }_{\beta }{h}_{\alpha \beta }|{\dot{\gamma }}_{\beta }|$, where $g_{\alpha }$ is the critical resolved shear stress for system $\alpha$ and ${\gamma }_{\beta }$ is slip on system $\beta$. Off-diagonal coefficients represent cross-system hardening, while diagonal coefficients describe self-hardening.^[ref] Changing loading direction after predeformation can activate previously less-favoured slip systems and reveal this history dependence. Predicting the next system to move therefore requires both resolved stress geometry and the evolving resistances.

Conditions favouring a Z alkene in a Wittig reaction, often using a nonstabilized ylide under kinetic, salt-controlled conditions. Isolation or verification is still required when complete stereoselectivity is not established.

A Wittig reaction with $\mathrm{P}{\mathrm{h}}_3\mathrm{P}=\mathrm{C}{\mathrm{H}}_2$ that replaces a carbonyl oxygen by $=\mathrm{C}{\mathrm{H}}_2$. On a cyclic ketone it installs an exocyclic methylene without inserting a new ring atom.

For $X_1=x$, $X_2=y$ with $[X_1,X_2]=i/2$, normalize the phase-space function by $\int W(x,y)dxdy=1$. A coherent state of amplitude $\alpha$ has $W=2e^{-2|x+iy-\alpha {|}^2}/\pi$. Quadrature marginals are physical probability distributions, but the full function can be negative. Cross terms of an optical cat state give interference fringes, even when low-order moments approach those of a mixture.

In three dimensions, let $a=(3/4\pi n{)}^{1/3}$ be the mean-spacing radius and $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$ the effective Bohr radius. The Coulomb scale is $E_C\sim e^2/(4\pi \epsilon a)$ and the Fermi energy scale is $E_F\sim {\hslash }^2/(m^{\ast }{a}^2)$, so $E_C/E_F\sim a/a_B^{\ast }=r_s$. Decreasing density favors repulsion relative to delocalization energy, motivating a Wigner crystal. This scaling does not determine the numerical quantum melting value of $r_s$. A classical nondegenerate gas instead compares Coulomb energy with $k_{B}T$.

The 1979 experiment of C. C. Grimes and colleagues studied a two-dimensional sheet of electrons above liquid helium. It reported crystallization at densities $3\times 10^8$ to $9\times 10^8{\mathrm{cm}}^{-2}$, with melting temperatures $0.35$ to $0.65\mathrm{K}$. The sheet was classical: Coulomb energy competed with thermal motion, unlike the kinetic-energy comparison used for a zero-temperature degenerate Wigner crystal. It supplies evidence for electron ordering while keeping these two physical regimes distinct.^[ref]

Analytic continuation from real time to $t=-i\tau$ turns unitary evolution into $e^{-\tau H/\hslash }$. A thermal trace has extent $\beta \hslash$, or $\beta$ when $\hslash =1$. This converts oscillatory action phases into an Euclidean path integral where convergence permits it; it is not the actual laboratory time of a dissipative process.

The hypothesis that two early large-scale duplication rounds contributed to the multiple paralogous genomic regions of jawed vertebrates. Hox clusters are one informative set of loci; a single family does not by itself establish genome-wide duplication.

A fully penetrated rough-surface wetting model with $\cos {\theta }^{\ast }=r\cos \theta$, where $r$ is actual wetted solid area divided by projected area. Roughness amplifies the planar surface tendency: it lowers contact angle when $\theta <90^{\circ }$.