The ideal activation criterion compares resolved shear stress with a critical resolved shear stress. Non-glide stresses, notably in some body-centred cubic crystals, can modify this ideal criterion.

The critical resolved shear stress is the shear stress on a specified slip system at which dislocation motion produces slip under stated conditions. For simple uniaxial loading, the resolved component is $\tau =\sigma \cos \varphi \cos \lambda$, with angles to the slip-plane normal and slip direction.

The threshold depends on material state, temperature, obstacles, and deformation history. A crystal can therefore begin to slip at different applied tensile stresses in different orientations even when the slip system has the same critical resolved threshold.

2016 paper: Public original PDF, physical page 800. Student-answer availability for this paper remains unverified.

For free-electron dispersion proportional to $k^2$, states at $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$ are degenerate on this plane, obtained by equating their squared norms. A nonzero periodic-potential Fourier component can mix them and open an avoided crossing in the nearly-free electron model. First-zone boundaries are selected planes of this kind.

A reciprocal lattice vector $\mathbf{G}$ has the displayed property for every direct lattice vector $\mathbf{R}$. Thus $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. Fourier components of a lattice-periodic potential carry these wavevectors and couple plane-wave electronic states whose momenta differ by $\hslash \mathbf{G}$. This is the origin of the Bragg-plane mixing in a nearly-free electron model.

A plane wave whose phase is unchanged by every direct lattice translation: $e^{i\mathbf{k}\cdot \mathbf{R}}=1$ for all $\mathbf{R}\in \Lambda$. Its wavevector is a reciprocal lattice vector. In one dimension $ka=2\pi m$.

An elongated reciprocal-space reflection caused by finite extent in the corresponding real-space direction. A foil of thickness $t$ has reflection breadth of order $1/t$ along its normal, increasing the range over which a reflection can intersect an Ewald sphere. The exact profile depends on the specimen shape and scattering model.

Let $Kx=a\phi (x)$ on a vector space, where $\phi$ is a linear functional. Its image of a linear map lies in the span of $a$. Applying $K$ twice gives $K^{2}x=a\phi (a)\phi (x)=\phi (a)Kx$. Mathematical induction then proves the displayed power formula. If $\phi (a)=0$, $K^2=0$, even when $K\ne 0$; this is a nilpotent operator. If $\phi (a)\ne 0$, then $K/\phi (a)$ is an idempotent linear map. In a real inner product space, the special case $\phi (x)=b\cdot x$ gives $\phi (a)=a\cdot b$.

The distance a downstream fluid element travels while losing a substantial fraction of its thermal energy. Multiplying the immediate downstream velocity by a radiative gas cooling time gives an order-of-magnitude length. A resolved layer instead uses $\ell =\int u(t)dt$, because mass density, temperature and speed change during cooling. It is distinct from the microscopic dissipative thickness of the shock wave.

Rotating a trap anisotropy can nucleate vortices in an atomic condensate. Expansion followed by absorption imaging resolves depleted cores, while varying rotation and waiting after removing the stirrer tests their preparation and persistence. The interpretation is quantized rotational response, rather than only the appearance of a condensed density profile.

Two surface electrodes whose depletion regions define a narrow conducting opening in an underlying two-dimensional electron gas. Gate voltage changes the transverse confinement and channel transmission without physically etching the conducting channel to its electrical width.

An injector and collector aligned across an open two-dimensional electron gas. A direction-sensitive signal between them probes ballistic electron transport. Contact resistance and diffuse current backgrounds must be separated from the direct beam contribution.

Slow widening of a constriction approximately conserves the transverse mode action, reducing transverse momentum as the width $W$ increases. A lowering band bottom also increases longitudinal kinetic energy. Together they can narrow the angular distribution of an emitted electron beam. An abrupt narrow aperture alone can instead diffract strongly.

A locally separable potential with an unstable longitudinal maximum and stable transverse confinement. Mode $n$ sees barrier top $E_n=U_0+\hslash {\omega }_y(n+1/2)$ and transmission $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$. It is a single-particle smooth-barrier model; interactions and bias-induced reshaping require additions.

At zero temperature, a rigid spin-degenerate transmission spectrum with symmetric source/drain bias has ${\mu }_{S,D}=E_F\pm eV/2$ and the displayed differential conductance. Half-integer plateaus in units $2e^2/h$ occur when the two bias-window edges sample different channel counts. The bias convention and electrostatic response matter.

A parallel field lifts spin degeneracy and separates threshold energies. The step size may change from $2e^2/h$ to $e^2/h$ when the two spin branches are resolved. Finite layer thickness can additionally alter orbital confinement.

The discrete Fourier transform of real displacement operators produces complex coordinates obeying $Q_k^{†}=Q_{-k}$. They are not independent real oscillators. The canonical expansion $Q_k=\sqrt{\hslash /(2m{\omega }_k)}(a_k+a_{-k}^{†})$ and $P_k=-i\sqrt{m\hslash {\omega }_k/2}(a_k-a_{-k}^{†})$ has independent occupation operators for all nonzero wavevectors, with the uniform free coordinate handled separately. At self-inverse zone-boundary wavevectors the same formulas reduce to one ordinary real oscillator.

2016 paper: Public original PDF, physical page 797. Student-answer availability for this paper remains unverified.

For a constant-coefficient braking law $\dot{\Omega }=-K{\Omega }^3$, integration gives $t={\tau }_c[1-(P_{\mathrm{birth}}/P{)}^2]$. The characteristic age approximates the actual elapsed time only when the birth period was short and torque evolution and kinematic timing contributions are negligible.