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For a two-component gapped BdG Hamiltonian constrained to a plane, the winding number counts how many times its coefficient vector encircles the origin as crystal momentum traverses the Brillouin zone.

Work hardening is an increase in resistance to further plastic deformation caused by earlier deformation. A useful macroscopic test loads a metal beyond yield, unloads it and reloads in the same direction: further plastic flow then requires a larger stress. In a constitutive equation, this history is represented by evolving internal variables rather than by the current elastic strain alone.^[ref]

In a crystal, multiplication and interaction of dislocations can make later dislocation glide more difficult. Forest hardening and latent hardening describe particular aspects of that obstacle formation. Increased resistance to plastic flow is not the same as a large change in Young's modulus: the elastic unloading slope and the load required to restart permanent deformation measure different properties. Work hardening can delay necking by compensating for the shrinking loaded area.

The Wu experiment was performed by Chien-Shiung Wu, Ernest Ambler, Raymond Webster Hayward, Dale D. Hoppes, and Ralph Percy Hudson; its 1957 report was Experimental Test of Parity Conservation in Beta Decay. Their experiment supplied the evidence that the Lee-Yang proposal for a parity test had called for.^[ref]

A cobalt-60 surface layer on a cerium magnesium nitrate crystal was cooled by adiabatic demagnetization; a solenoid's magnetic field polarized its atomic nuclei. An internal anthracene counter detected electrons by particle scintillation; a light pipe carried flashes to a photomultiplier. Polar and equatorial gamma ray counters monitored nuclear polarization and effective temperature; bulk thermometry could miss surface heating.^[ref]

Reversing nuclear polarization reversed the electron excess; warming erased it with gamma ray anisotropy. Warm rates were field-independent. Cooling-field reversal checked remanence; poorly polarized sources checked magnetic deflection. Electrons favored the direction opposite spin.^[ref]

The gamma ray anisotropy diagnoses the preparation; the electron asymmetry diagnoses parity violation. These are different jobs. A directionally uneven gamma ray signal alone need not violate parity conservation: a distribution even in the cosine of the angle can distinguish an axis from its perpendicular without distinguishing its two ends. The spin-momentum correlation instead distinguishes those ends.

The inference requires control of changes in detection efficiency, electron transport and nuclear polarization. A fixed detector under spin reversal avoids physically exchanging detector positions, but still requires checks for magnetic field effects. The result establishes parity violation in this weak interaction process. A precise microscopic coupling requires more information than the existence or sign of a counting asymmetry; the following model makes that distinction explicit.

The fixed-volume equilibrium crystal shape obtained by minimizing orientation-dependent surface free energy. Facet support-plane distances are proportional to their interfacial free energies under the construction assumptions.

X-ray scattering is the redirection of X-rays through interaction with matter. The angular distribution and energy of the outgoing radiation carry information about the material. In elastic scattering the photon energy is unchanged; in inelastic scattering energy is exchanged. X-ray diffraction concerns organized interference of scattered waves, rather than a separate kind of radiation.

The scattering vector $\mathbf{q}={\mathbf{k}}_{\mathrm{out}}-{\mathbf{k}}_{\mathrm{in}}$ compares outgoing and incoming wave vectors. Consider two identical scatterers separated by $\mathbf{d}$, each with real elastic scattering amplitude $f$, and neglect multiple scattering. Their amplitude sum and intensity areTwo atoms can therefore scatter strongly in one direction and cancel in another. The measured pattern depends on arrangement, not merely on the number of atoms. For repeating lattice planes in a crystal, constructive interference gives Bragg's law $2d\sin \theta =n\lambda$.^[ref]

The1913 experiment by William Henry Bragg and William Lawrence Bragg made the angular pattern measurable with a rotating crystal and an ionization chamber in place of a spectrometer telescope. Slits selected a narrow incoming beam; the detector current was measured as its angle changed. Peaks and repeated orders constrained crystal structure and X-ray wavelengths, linking an electrical measurement to atomic-scale geometry.^[ref]