A regime in which particle-scale magnetic moments reverse thermally on the observation timescale, giving little quasistatic remanence despite internally ordered single-domain particles. Being single-domain is not sufficient: anisotropy barriers and measurement time determine whether reversal is rapid. A finite-frequency drive can still produce relaxation losses.

For motion perpendicular to a uniform magnetic field, the Lorentz force gives $r=p/(|q|B)$. An ultrarelativistic massive particle has $pc\simeq E$, hence $r\simeq E/(|q|Bc)$. Total energy and kinetic energy differ by the rest energy.

The Curie constant is the coefficient $C$ in Curie's law $\chi =C/T$ for an appropriate dilute or high-temperature paramagnet. It sets the strength of the weak-field magnetic susceptibility, rather than an ordering temperature. In an SI volume-susceptibility convention for number density $n$ of identical noninteracting moments,Here ${\mu }_{\mathrm{eff}}$ is the effective magnetic moment and ${\mu }_0$ the vacuum permeability. A mole-based susceptibility uses a correspondingly mole-based constant.

Plotting magnetic susceptibility against inverse temperature can estimate $C$ in the linear weak-field regime. Exchange interactions or thermal occupation of additional levels can change the dependence, so a fitted $C$ is meaningful only with its range and normalization stated. The factor $1/3$ reflects orientational averaging in the isotropic moment model; an anisotropic response requires more information than one scalar constant.

For a scalar quartic Landau free energy with $a=a_{0}t$ and positive quartic coefficient, the zero-field susceptibility is $1/(a_{0}t)$ above the transition and $1/(2a_0|t|)$ within a selected ordered phase below it. Their singular amplitudes have ratio two. Below the transition this is a one-sided pure-phase response: the globally minimizing magnetization jumps between opposite phases when the field crosses zero. The mean-field ratio is not asserted to be exact in every fluctuation-dominated system.

In a gauge with potentials independent of y, the canonical momentum $p_y$ commutes with the Hamiltonian. A simultaneous eigenbasis can be chosen as $e^{iky}{\Phi }_k(x)$, with $p_y=\hslash k$. Periodic length $L_y$ gives $k=2\pi n/L_y$. The kinetic y-momentum is instead $\hslash k-q{A}_y(x)$ and is not spatially constant. Confusing these two momenta loses the magnetic contribution to the velocity.

For $q\alpha k>0$, let $x_0=\sqrt{2\hslash k/(q\alpha )}$. Expansion of the magnetic-gradient quartic confinement at either minimum gives $V(x_0+\delta x)=(q\alpha {)}^2{x}_0^2(\delta x{)}^2/(2m)+\cdots$. The local oscillator frequency is therefore $|q\alpha |x_0/m$. Both lowest parity states have leading energy $\hslash {\omega }_d/2$ if $x_0$ greatly exceeds the oscillator width. Local anharmonicity changes their common energy, while quantum tunnelling gives their much smaller splitting.

In vacuum or an approximately nonmagnetizable plasma, the field energy per volume is $B^2/(2{\mu }_0)$. magnetic pressure has the same scale, but a magnetic field also exerts directional tension, so its stress is not universally an isotropic pressure. Material magnetization can require a different constitutive treatment.

In curvature coordinates the present sphere area is $4\pi (R_{0}r{)}^2$. Its transverse area distance is $R_{0}r$, whereas radial proper distance is $R_0{\int }_0^{r}dr/\sqrt{1-K{r}^2}$. Confusing them invalidates a flat-space distance identity in curved geometry.

At fixed positive redshift, intrinsic luminosity and present Hubble constant, a flat dust-vacuum model with $0<{\Omega }_{\Lambda 0}<1$ has luminosity distance between those of an Einstein-de Sitter universe and pure positive vacuum. The Hubble parameter of the dust-only model is largest at every positive integration redshift; that of pure vacuum is smallest. Therefore $c(1+z){\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$ is smallest for dust alone and largest for pure vacuum. Received radiative flux has the reverse ordering. This compares physical distances rather than arbitrarily normalized comoving coordinates.

A transient region where apical determinants and membrane traffic organize a new luminal surface during epithelial polarization.

An epithelial enclosure around a lumen, used experimentally to study coordinated polarity, division and surface traffic. Multiple lumina do not necessarily imply loss of polarity in every cell.