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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.

A trophic level describes feeding position relative to primary producers. Assign primary producers level 1; in the illustrated food chain, zooplankton occupy level 2, small fish level 3, and large fish level 4. Following a single feeding path adds one level at each step. The labels make dependence on the initial resource visible, but a branching food web can give an organism several feeding paths. See Clark, Douglas and Choi, Biology 2e, 46.1: feeding positions and branching food webs.

An omnivore can obtain food from more than one trophic level. As an explicitly assumed diet-weighted example, suppose an omnivore gets half its assimilated food energy from level-1 primary producers and half from level-2 consumers. Define its trophic level as one plus the weighted mean level of its food: $1+(1/2)1+(1/2)2=2.5$. This computed position depends on the specified diet and weighting convention; it is not an immutable property of the species. A different life stage or diet can change it.

The truncation error in effective field theory is the uncertainty from omitted orders of a stated energy, derivative or loop expansion. It is commonly estimated from the first omitted terms using an explicit assumption about Wilson coefficients. Small energy alone does not justify ignoring an anomalously large coupling or a nearby omitted pole.

A tunnel diode is a p-n junction with sufficiently heavy semiconductor doping on both sides to make a thin depletion region and permit substantial interband tunnelling. The neutral regions are degenerate semiconductors. Its distinctive forward current-voltage characteristic contains a peak followed by a valley: over part of the curve, increasing the applied electric potential difference decreases the electric current. The Esaki tunnel-diode experiment made this behavior experimentally concrete.

The useful quantity is the local slope, not the ratio of total voltage to total electric current. With conventional electric current $I$ entering the positive-voltage terminal, defineNegative differential resistance means $g_d<0$, hence $r_d<0$ where defined. A forward operating point can still have $V>0$, $I>0$ and absorbed power $VI>0$. A small voltage increase then gives a negative current increment; this does not make the total electric current negative or turn the device into an unlimited source of energy. The external supply sets the operating point and supplies the energy available for an electronic amplifier.

An original idealized construction explains the shape. Put the n-type semiconductor on the left and the p-type semiconductor on the right. Take $q>0$ as the elementary charge and $V=V_p-V_n$. Increasing forward bias lowers the right-hand electron energy levels relative to the left. Denote the two reservoir electrochemical potentials by ${\mu }_n,{\mu }_p$, so that ${\mu }_n-{\mu }_p=qV$. At zero temperature, occupied left-hand conduction band states extend up to ${\mu }_n$; empty right-hand valence band states start at ${\mu }_p$. For transfer conserving electron energy, the available interval isIt is nonempty only when its upper endpoint exceeds its lower endpoint. Both an occupied initial quantum state and an empty final quantum state must exist at the same energy. The Pauli exclusion principle prevents transfer into an already occupied final quantum state.

For the original diagram, choose arbitrary energy units with $E_{c,n}=-0.5$, ${\mu }_n=0$, ${\mu }_p=-u$ and $E_{v,p}=0.5-u$, where $u$ is $qV$ in those units. The available interval has widthThus $L(0.25)=0.25$, $L(0.50)=0.50$, $L(0.70)=0.30$, and $L(1.15)=0$. Initially forward bias opens a transfer window; further forward bias closes it by moving the right-hand valence band edge below the left-hand conduction band edge. This is a state-counting illustration, not a formula for a measured electric current.

For reverse bias, the transfer direction changes: occupied right-hand valence band states can feed empty left-hand conduction band states. In the same construction, $u=-0.70$ opens the interval $0<E<0.70$ for this reverse transfer. A narrow depletion region allows a large reverse quantum tunnelling electric current at small electric potential difference, whereas the ordinary Shockley diode equation predicts only a small reverse saturation electric current before diode reverse breakdown. At zero biasing, equal reservoir occupations give zero net electric current; quantum tunnelling does not require a net equilibrium flow.

A finite-temperature refinement replaces sharp occupation cutoffs with the Fermi-Dirac distribution. An illustrative weak-transfer expression isHere $\mathcal{W}\ge 0$ describes allowed initial/final quantum states and their coupling, and $f_n,f_p$ are Fermi-Dirac distributions with the respective electrochemical potentials. The occupation difference follows by subtracting the reverse transfer factor from the forward one: $f_n(1-f_p)-f_p(1-f_n)=f_n-f_p$. An overlap interval alone does not determine $\mathcal{W}$: transmission probability, density of states, crystal momentum selection and phonons can matter. A complete germanium calculation is more demanding than multiplying two constant density of states factors. At larger forward bias, ordinary carrier injection and diffusion supply a rising current branch even after direct band overlap is lost.

Turbulent backscatter is transfer of kinetic energy from unresolved or smaller spatial scales into larger resolved scales. In a large eddy simulation, its sign is defined relative to the chosen spatial averaging filter and subgrid-scale stress. It can occur locally even when the mean energy cascade transfers energy toward small scales.^[ref]

With ${\tau }_{ij}=\overline{u_i{u}_j}-{\bar{u}}_i{\bar{u}}_j$, the stress force in the filtered momentum equation is $-{\partial }_j{\tau }_{ij}$. Multiplying by ${\bar{u}}_i$ givesThe first term transports kinetic energy through space. Because $\tau$ is symmetric, the second is ${\tau }_{ij}{\bar{S}}_{ij}$, the local exchange with unresolved motion. Thus the forward-transfer conventionmakes $\Pi <0$ a gain of resolved kinetic energy.

For a trace-free resolved strain $\bar{S}=\mathrm{diag}(\gamma ,-\gamma ,0)$ and stress $\tau =\mathrm{diag}(a,b,0)$, with nonnegative $a,b$,Taking $\gamma >0$ and $a>b$ produces backscatter even though the stress matrix is positive semidefinite. Positivity of a weighted covariance does not fix the sign of its contraction with an indefinite strain tensor.

The Smagorinsky model with nonnegative eddy viscosity instead gives $\Pi =2{\nu }_t{\bar{S}}_{ij}{\bar{S}}_{ij}\ge 0$. A dynamic subgrid-scale model can allow the reverse sign. Backscatter describes exchange between scales, not negative molecular viscous dissipation or creation of total energy.

A fixed-mass spherical shell has acceleration $\ddot{R}=-GM/R^2+\Lambda R/3$, in units $c=1$. At a finite maximum radius $R_m$, inward acceleration requires $\eta =\Lambda R_m^3/(3GM)<1$. Equality is the unstable static radius, approached only in the limiting separatrix case rather than a finite expansion-to-collapse turnaround. The effective potential reaches its maximum there, so an initially expanding bound inner-branch shell also needs energy below that maximum.

In cylindrical orthonormal bases the deformation gradient is $F=\left(\begin{array}{ccc}\rho /r& 0& 0\\ 0& r/\rho & \alpha r\\ 0& 0& 1\end{array}\right)$ and has unit determinant. One principal stretch is $\rho /r$; the squares of the other two are the roots of $z^2-[(r/\rho {)}^2+{\alpha }^2{r}^2+1]z+(r/\rho {)}^2=0$. The reference-volume energy per height is $E=2\pi \int \rho Wd\rho$. Virtual work gives end torque $M={\partial }_{\alpha }E$ and inner pressure $p=(2\pi a{)}^{-1}{\partial }_{a}E$ at fixed reference radii and height.

For a nonrotating uniform-stratification fluid under the Boussinesq approximation driven at $0<\omega <N$ by a localized vertical-wall velocity, positive vertical wavenumbers radiate down and right and negative ones up and right. The real velocity separates into functions of $Z_{+}$ and $Z_{-}$. Its in-phase component is half the imposed boundary profile on each beam; the quadrature component is its Hilbert transform partner. Thus compact localization at maximum boundary speed does not imply compact localization of all components at every phase. A finite-duration source contains several frequencies and spreads energy over their different ray angles and group speeds.

Consider one chosen model at a fixed total energy, with two open scattering channels, no absorption and no omitted outgoing modes. Channel 1 leaves the target unchanged; channel 2 excites it. Let $v_1=v>0$ and $v_2=v/2$. For the same projectile mass, conservation of energy then requires a target excitation $\Delta =3m{v}^2/8$. These are hypothetical parameters, not a fit to a particular atom. Each incoming or outgoing mode is normalized to unit probability current; the outgoing modes will have the angular profiles specified below.

Choose the S-matrixThis phenomenological unitary matrix specifies the channel conversion; it is not derived here from an interaction potential energy. For incidence in channel 1, its first column gives the outgoing flux amplitudesTo recover ordinary wave coefficients relative to an incident coefficient one, undo the velocity normalization:Thus $|c_2{|}^2=32/25>1$ is allowed: channel 2 travels more slowly. Its outgoing probability current fraction is $16/25$, not $32/25$. Renormalizing the unweighted squares would give $9/41$ and $32/41$, the wrong probabilities. More generally the ordinary coefficient map is $A=V^{-1/2}S{V}^{1/2}$, with $V=\mathrm{diag}(v,v/2)$; it obeys $A^{†}VA=V$. The conserved quantity is the squared coefficient norm weighted by velocity, which represents probability current.

Now specify an original angular realization of these two outgoing modes, with scattering angle $\theta$ and no azimuth dependence:The first normalization follows from $2\pi {\int }_{-1}^1(1+u^2)du=16\pi /3$, where $u=\cos \theta$. These angular profiles define the modes of this model; they are not asserted to follow from Born's approximation or an atomic potential energy. The joint directional probability densities are

Suppose a detector accepts the cone $D:0\le \theta \le \pi /3$, with full azimuth and unit efficiency. Its angular fractions within the channels areMultiplying the angular fractions by the outgoing flux weights givesFor $1600$ independent incident events, the expected values are therefore $171$ and $256$ accepted detections, compared with $576$ and $1024$ over all directions. The remaining expected $1173$ events fall outside the cone. The chosen incident number makes the expectations integers; actual counts fluctuate. Conditional on a detection inside the cone, the channel probabilities are $171/427$ and $256/427$, rather than $9/25$ and $16/25$. Angular acceptance changes the observed channel proportions even with equal detector efficiencies.

The final atomic states are orthogonal, so ignoring the atomic readout adds the two channel intensities; it does not produce $|b_1{g}_1+b_2{g}_2{|}^2$. Coherent paths that reach the same final state can instead interfere, as the existing probability amplitude and quantum superposition examples demonstrate. The distinction is physical distinguishability of final states, not whether the detector operator chooses to display their labels. Flux normalization and angular integration make the scattering interpretation useful: they turn complex outgoing waves into conserved event weights and then into the statistics of a specified detector.

For independent frequencies $1$ and $2$,so the eigenspace at energy $k+3/2$ has one basis state $|n,m⟩$ for each nonnegative solution of $n+2m=k$.

Two separately conserved bosonic species have two condensate fields and independent chemical potentials. A contact interaction energy density $U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ gives ${\mu }_1=U_{11}{n}_1+U_{12}{n}_2$ and ${\mu }_2=U_{12}{n}_1+U_{22}{n}_2$ in a uniform mixture. Coherent interconversion would change the conservation constraints and cannot be included while retaining independent particle-number constraints unchanged.

For motion confined to a plane but retaining potential energy $-\kappa /r$, separation in polar coordinates gives angular wavefunctions $e^{im\varphi }$ with integer $m$. The regular radial factor behaves as $r^{|m|}$ at the origin. Removing this power and the decaying exponential function leaves an Generalized Laguerre polynomial differential equation. polynomial termination requires $N=n_r+|m|+1/2$, yielding the displayed energy. At fixed $q=n_r+|m|$, the spinless quantum degeneracy is $2q+1$. This is not the logarithmic potential of electrostatics formulated entirely in two spatial dimensions.

For nonrelativistic electrons in area $A$, with spin degeneracy $g_s=2$ and energy $E={\hslash }^2{k}^2/(2m)$, the density of one-particle states is independent of energy:

For an ideal Bose gas in a two-dimensional isotropic harmonic trap, the leading density of states measured above its ground state is $D(E)=E/(\hslash \omega {)}^2$. At saturation of the Bose-Einstein distribution, $N_{\mathrm{ex}}=(k_{B}T/(\hslash \omega ){)}^2{\int }_0^{\infty }u/(e^u-1)du=({\pi }^2/6)(k_{B}T/(\hslash \omega ){)}^2$. The finite excited-state capacity gives the displayed critical temperature and condensate fraction $N_0/N=1-(T/T_c{)}^2$ below it. The appropriate trapped-gas thermodynamic limit fixes $N{\omega }^2$ as $N\to \infty$ and $\omega \to 0$; a finite trap exhibits finite-size rounding. The number density is inhomogeneous, and the physical chemical potential approaches the oscillator ground state energy, not an unshifted zero by convention.

For unit mass and angular frequency,The level with total occupation $N=n_x+n_y$ has energy $N+1$ and degeneracy $N+1$ in units where $\hslash =1$.

For a two-dimensional anisotropic quantum harmonic oscillator in the indicated range, the six lowest number states, counted with multiplicity, have occupations $(0,0),(1,0),(0,1),(2,0),(1,1),(0,2)$. Their excitation energies, divided by $\hslash$, are $0,{\omega }_1,{\omega }_2,2{\omega }_1,{\omega }_1+{\omega }_2,2{\omega }_2$. These are ordered because ${\omega }_2<2{\omega }_1$ and $2{\omega }_2<3{\omega }_1$; every omitted state has at least three quanta and costs at least $3\hslash {\omega }_1$. At equal angular frequencies, the two-dimensional isotropic harmonic oscillator has energy eigenvalue $\hslash \omega (N+1)$ and quantum degeneracy $N+1$, since there are $N+1$ ordered nonnegative pairs with sum $N$.

For two polarizations, $n$ atoms in area $A$, and $\omega =C|k{|}^{\alpha }$, mode counting givesAt low temperature its energy scales as $T^{1+2/\alpha }$ and its constant-volume heat capacity as $T^{2/\alpha }$.

For $k_z=v_z=0$ and $k_y\ne 0$, define $Z=i(k_x{v}_y-k_y{v}_x)$. Uniform rotation does not enter its evolution: $\dot{Z}=-\nu k^{2}Z$, $v_x=i{k}_{y}Z/k^2$ and $v_y=-i{k}_{x}Z/k^2$. With $T=k_x/k_y$ and $\mathrm{Re}=A/(\nu k_y^2)$, the perturbation energy ratio is $E(T)/E(T_0)=[(1+T_0^2)/(1+T^2)]\exp [-(T+T^3/3-T_0-T_0^3/3)/\mathrm{Re}]$. Leading waves can gain energy as shear reduces their wavevector magnitude, before viscosity and subsequent winding damp them.

// Target: hydrodynamic-stability.bigb

Let ${\mathbf{a}}_1,{\mathbf{a}}_2$ be linearly independent primitive translations of a two-dimensional Bravais lattice. Take one orthonormal orbital per site, on-site energy $E_0$, and equal real hopping integrals $-\lambda$ to the four neighbours at $\pm {\mathbf{a}}_1,\pm {\mathbf{a}}_2$. A Bloch state givesThe two phases can vary independently across a reciprocal primitive cell. Their cosine sum reaches both $2$ and $-2$, so the energy band ranges from $E_0-4|\lambda |$ to $E_0+4|\lambda |$, with electronic bandwidth $8|\lambda |$.

The independence assumption matters. If instead ${\mathbf{a}}_2=2{\mathbf{a}}_1$, the phases are locked. Writing $c=\cos (\mathbf{k}\cdot {\mathbf{a}}_1)$ gives $\cos \theta +\cos 2\theta =2(c+1/4{)}^2-9/8$, whose range is $[-9/8,2]$. For $\lambda >0$ that one-dimensional model has electronic bandwidth $25\lambda /4$, rather than $8\lambda$.

For $Q,L\ge 2$, a periodic Potts chain has $Q$ uniform ground states with energy $-JL$. A single domain wall cannot close around the ring. Its first excited states have two domain walls and energy $-JL+2J$. Choose the wall bonds in $\left(\genfrac{}{}{0ex}{}{L}{2}\right)$ ways and assign distinct states to the two resulting arcs in $Q(Q-1)$ ways. This proves the displayed degeneracy factor without identifying globally relabelled configurations.