An eigenvalue of a graph's Laplacian matrix. For a finite undirected graph, these eigenvalues are nonnegative, and the multiplicity of zero equals the number of connected components.

Europium is the chemical element with atomic number 63 and symbol $\mathrm{Eu}$. It is a lanthanide. Its $+2$ state is unusually accessible compared with many neighboring lanthanides. Europium ions are used in luminescent materials, where changes in electronic energy are emitted as characteristic colors.^[ref]

Ytterbium is the chemical element with atomic number 70 and symbol $\mathrm{Yb}$. It is a lanthanide. It has important $+2$ and $+3$ chemistry. Its narrow optical transitions are useful in precision atomic experiments, showing how a chemical element can also provide a highly reproducible spectroscopic reference.^[ref]

For independent spin-one-half adsorbates whose two spin states are split only at the surface, the one-site partition function contains $2\cosh ({\mu }_{\mathrm{B}}B/k_{\mathrm{B}}T)$. Relative to the unsplit bulk spin degeneracy, this increases the Langmuir adsorption model affinity by a hyperbolic-cosine factor. At fixed temperature and gas pressure, increasing the magnitude of the magnetic field increases surface coverage in this model.

The chemical potential of an ideal single-occupancy lattice gas with occupied-site internal partition function $z$ and surface coverage $\theta$. The ratio $\theta /(1-\theta )$ accounts for the entropy of distributing particles among occupied and vacant adsorption sites.

An independent site has one empty state and an occupied set with binding energy $-ϵ$ and vibrational partition function $z_s$. Its occupied probability is $\theta =z_s{e}^{\beta (\mu +ϵ)}/[1+z_s{e}^{\beta (\mu +ϵ)}]$. The denominator enforces the one-molecule-per-site constraint even though the vapour may obey classical statistics. Multiple occupancy or lateral interactions require a different model.

Equating the site chemical potential with that of an internally structureless classical vapour gives $\theta =b_{s}p/(1+b_{s}p)$. The coefficient $b_s$ includes the bound vibrational partition factor, the translational thermal de Broglie wavelength, and the positive binding Boltzmann factor. It has inverse-pressure units. Half coverage occurs at $p=1/b_s$; an unspecified $z_s(T)$ prevents a numerical pressure scale, but not the functional isotherm.

Stepwise conductance as transverse propagating modes open in a clean constriction. Each fully transmitted spin-degenerate subband contributes $2e^2/h$; finite temperature, imperfect transmission and unresolved modes round the steps.

A multi-reservoir quantum transport relation. Here $I_i$ is conventional current injected from reservoir $i$, $N_i$ counts outgoing spin-resolved modes and $T_{ij}$ counts transmission from reservoir $j$ into $i$. Ideal voltage probes draw zero net current and equilibrate incident carriers. On a chiral integer Hall edge with $N$ modes, $I_i=(N{e}^2/h)(V_i-V_{i-1})$ for the stated circulation order.

For one directed spin-resolved channel, the density of longitudinal states per unit length and wave number is $1/(2\pi )$. Since $v={\hslash }^{-1}dE/dk$, velocity times density per energy is $1/h$. A reservoir occupation imbalance therefore contributes particle flux $\int \delta f(E)dE/h$; in zero-temperature linear response it is $\delta \mu /h$. Multiplying by charge and transmission gives the Landauer current per mode.

A perfectly transmitted spin-resolved channel contributes $e^2/h$ to two-terminal linear conductance. Two degenerate spin channels give $2e^2/h$. Flux normalization cancels group velocity against the longitudinal state density.

An element $t_{nm}$ is the amplitude for an incident unit-flux channel $m$ to emerge in outgoing unit-flux channel $n$. Then $|t_{nm}{|}^2$ is a transmission probability and $\mathrm{tr}(t^{†}t)$ sums transmitted channels. Unnormalized wave amplitudes require velocity ratios.

For an isotropic vector order parameter, an analytic scalar Landau free energy near the origin contains powers of $\mathbf{m}\cdot \mathbf{m}$. Odd powers of its magnitude are rotationally invariant but nonanalytic as functions of the Cartesian components at zero; excluding them requires analyticity, not merely writing the magnitude as a nonnegative scalar. With $d>0$, changing $b$ from positive to negative continuously creates minima at $|\mathbf{m}|=\sqrt{-b/(2d)}$, with an arbitrary direction.^[ref]

A scalar order parameter with symmetry $q\mapsto -q$ has only even powers in its zero-field free energy. A conjugate field contributes $-hq$. Without this symmetry, odd terms can be allowed and change the transition structure.

For $f=f_0+a(T-T_c)q^2/2+b{q}^4/4$ with $a,b>0$, the minimum is $q=0$ above $T_c$ and $q=\pm \sqrt{a(T_c-T)/b}$ below it. The stationary solutions must be checked for stability; below $T_c$ the central solution is a maximum.

A long-wavelength order-parameter description extending a local Landau free energy by gradient terms. For a one-component complex field, a common isotropic form is $\mathcal{F}=\int [\alpha |\Psi {|}^2+\beta |\Psi {|}^4/2+K|\nabla \Psi {|}^2]d^{3}r$. A charged field instead uses a covariant gradient. Static minimization determines equilibrium profiles; a dynamical law requires additional input. Ginzburg-Landau theory of superconductivity and the Ginzburg-Landau functional of a dilute Bose gas realize this structure with different microscopic normalizations.

An Arrott plot presents squared magnetisation against field divided by magnetisation. For the uniform SI Landau equation ${\mu }_{0}H=aM+b{M}^3$, it is a straight lineThe critical mean-field isotherm has $a=0$ and passes through the origin. Horizontal intercept $a/{\mu }_0$ changes sign across the transition; a positive vertical intercept estimates spontaneous magnetisation squared. The original criterion concerns internal magnetic field and the molecular-field regime. Demagnetizing fields and non-mean-field critical exponents require additional care.^[ref]

In a regular three-dimensional Fermi liquid, the Pauli exclusion principle leaves only a small set of final states available to a low-energy Landau quasiparticle. Writing $\xi =ϵ-\mu$, the leading decay rate has the scaleThe coefficient depends on the interaction and band structure.

At zero temperature, an excitation of energy $\xi >0$ creates an additional particle–hole pair. Conservation of energy leaves two small independent energy intervals, producing the quadratic phase-space factor. Its spectral width $\Gamma \sim \hslash /\tau$ then satisfies $\Gamma /|\xi |\to 0$ as $\xi \to 0$. This zero-temperature limit is the sense in which excitations become sharp at the Fermi surface. ^[ref]

At fixed nonzero temperature, the rate instead approaches a finite value proportional to $T^2$ as $\xi \to 0$. One must not infer a vanishing ratio $\Gamma /|\xi |$ in that different limit. A joint low-temperature, low-energy limit compares the width with the relevant excitation scale.

A reduction strategy that contracts the leftmost outermost reducible expression first. If a lambda calculus expression has a beta-normal form, this strategy reaches it, although the intermediate expressions can be large.