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When a single-particle energy is independent of its spin projection and all projections are available, the state degeneracy due to spin is $g_s=2S+1$. A spin-half gas has $g_s=2$ if both projections are available, while a fully polarized gas can have one populated spin species. A magnetic field that resolves Zeeman splitting requires counting the separate energies instead of multiplying one common occupancy by $g_s$.

A spin-flop transition is a field-induced reorientation of an antiferromagnet. In the classical easy-axis case, an applied magnetic field along the preferred axis can drive the nearly antiparallel sublattice magnetic moments into a canted arrangement whose dominant opposing components lie approximately transverse to the field. The moments acquire a net field-aligned component while retaining strong antiferromagnetic opposition.

The competition is between magnetic anisotropy favouring the easy axis, the exchange interaction favouring opposing moments, and the energy gained by responding to the applied field. The classical reorientation is a first-order phase transition; its neighborhood can include coexistence and magnetic domains.^[ref]

A field sweep measuring magnetization or magnetic susceptibility detects the changed response. A spin flop is a reorientation into a canted antiferromagnetic state; full alignment at much stronger field is a separate saturation process.

Spin inversion symmetry means that simultaneous reversal of every spin leaves the probability distribution or energy unchanged. For a finite spin model with this symmetry, the weights of opposite configurations pair up, so every observable that changes sign under the reversal has zero expected value. An applied magnetic field generally breaks this symmetry.

An original reduced model makes the cancellation explicit. Assume LS coupling, neglect nuclear spin, and write the electronic magnetic dipole moment as $-{\mu }_B(g_L\mathbf{L}+g_s\mathbf{S})/\hslash$. For $L=1$, $S=1/2$, the projection used to derive the Landé g-factor givesThese coefficients follow by substituting $J=3/2$ or $1/2$ into the angular-momentum projection formula; neither level measures pure spin alone. In a weak uniform magnetic field, adjacent electronic projections have spacing $h{f}_J=g_J{\mu }_{B}B$ when $g_J>0$. ThereforeThe common magnetic field and Bohr magneton cancel. The ratio determines $g_s/g_L$, not $g_s$ without further knowledge of $g_L$. For $g_L=1$, it gives $a_e=(R-2)/(R+1)$. Small departures from $R=2$ are magnified differently from small departures of $g_s$ from $2$; the two fractional changes should not be identified.

With the stipulated inputs $g_L=1$, $g_s=2.0024$ and ${\mu }_{B}B/h=100\mathrm{MHz}$, the two electronic frequencies are $66.5867$ and $133.4133\mathrm{MHz}$. Their ratio is approximately $2.0036043$, and inversion recovers $a_e=0.0012$. These are ideal electronic lines, not historical gallium data. Nuclear spin, mixing between levels and other deviations from LS coupling require a more complete model in an actual atom.

The spin projection is the component of a particle's spin angular momentum along a specified spatial direction. For the $z$ direction, an eigenstate of the angular momentum operator $S_z$ has measured value $m_s\hslash$, where $m_s=-s,-s+1,\dots ,s$ and $\hslash$ is the reduced Planck constant. For spin one-half, the two possible results are $+\hslash /2$ and $-\hslash /2$.

The direction belongs to the measurement arrangement: a state with definite $S_z$ generally has uncertain $S_x$. A Stern-Gerlach experiment separates the corresponding components through a spatially varying magnetic field. This is a component measurement, rather than a claim that the full spin angular momentum points along one of two classical directions.

In the two-spin exchange spectrum with unequal signed magnetic moments, the axial Hamiltonian conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm \hslash$, while the two mixed states lie in the $M=0$ block. Levels in different $M$ sectors may cross as the axial magnetic field changes because the conserved projection forces their coupling matrix element to vanish.

This is a symmetry qualification of the non-crossing rule. A nonzero isotropic exchange interaction does not by itself turn every field-dependent crossing into an avoided crossing. A transverse perturbation that breaks conservation of $M$ can mix the sectors and open a gap, provided its relevant matrix element is nonzero. Plotting the energy eigenvalues by their conserved projection therefore distinguishes protected crossings from the avoided splitting already present within the $M=0$ block.

Spin relaxation is the approach of nonequilibrium spin populations and coherences toward their equilibrium values through coupling to other degrees of freedom or environmental fluctuations. A spin polarized away from equilibrium can lose its net polarization even when the total angular momentum of the larger closed system is conserved.

For a simple ensemble in a fixed magnetic field, longitudinal relaxation with time $T_1$ restores the population difference along the field, while transverse relaxation with time $T_2$ damps the rotating transverse spin polarization. Static field inhomogeneity can produce an additional reversible spread of precession phases, so an observed free-induction decay time need not equal the intrinsic $T_2$. Spin-flip scattering changes populations; phase-randomizing processes can also suppress transverse coherence without individual flips. A Hanle spin-precession measurement probes the competition of precession, transport and relaxation.

Primary standards measurements of transport critical current, with specified temperature, magnetic field and electric-field criterion. NBS Special Publication 260-91 (1984). doi.org/10.6028/NBS.SP.260-91

Star formation is the conversion of interstellar gas into self-gravitating protostars and eventually stars. A region can contract when gravity overcomes support from gas pressure and other motions or fields. Cooling matters because it removes compression energy: during approximately isothermal fragmentation, rising mass density lowers the Jeans mass, permitting smaller regions to collapse independently.

When the gas becomes opaque and cannot cool efficiently, adiabatic suppression of fragmentation changes that behaviour. A first hydrostatic core and second protostellar collapse describe successive physical stages rather than two arbitrary labels for the same contraction. Accretion, rotation and magnetic fields affect the resulting objects, while stellar feedback limits how much gas becomes stellar mass. The initial mass function summarizes the resulting birth-mass distribution.

An electric field shifts and can split the energy eigenvalues of an atom. This is the Stark effect; its hydrogen atom ground-state case is quadratic in a weak field.

The observable in Stark effect spectroscopy is a change in a spectral line, rather than a direct reading of an individual energy eigenvalue. If an upper energy eigenstate $u$ emits a photon into a lower energy eigenstate $l$, the Planck constant $h$ relates its frequency to the difference:Here $\mathcal{E}$ denotes the applied electric field component along a chosen axis. Both levels can move, and electric-dipole selection rules for hydrogen decide which pairs produce observable spectral lines. This makes Stark effect spectroscopy a probe of energy eigenstates and their electric dipole moments: the slope $-\partial E/\partial \mathcal{E}$ is the corresponding dipole component by the Hellmann–Feynman theorem. The comparison with Zeeman splitting is useful: there an applied magnetic field supplies the perturbation.

A starspot is a relatively cool, dark region on a star's surface associated with a magnetic field. Starspots change the light curve as the star rotates. A planet crossing a starspot blocks less light than it would from an unspotted region, producing a local upward anomaly during an exoplanet transit; unocculted starspots can instead deepen the normalized exoplanet transit depth. They can also distort spectral lines and imitate a Doppler spectroscopy signal.

Boosting a static spatially periodic magnetic field produces temporal oscillations together with an electric field. Its spatial-grating four-wavevector remains spacelike, rather than the null four-wavevector of a vacuum plane wave. For a boost along the grating axis, $|E^{\prime }|/|B^{\prime }|=v<c$. Ultrarelativistic motion can make the field approximately wave-like along a prescribed trajectory without making it an exact source-free radiation field.

Time-independent crossed electric and magnetic fields can carry a nonzero energy flux $\mathbf{S}=\mathbf{E}\times \mathbf{H}$. A steady coaxial circuit is an example: radial electric and azimuthal magnetic fields carry energy axially from source to load. Static energy density does not imply zero flux.

Statistical reflection symmetry means that the joint distributions of a random field are unchanged by a simultaneous spatial reflection, with each field component transformed according to its vector or pseudovector character. For an isotropic magnetic field, it excludes a parity-odd magnetic helicity contribution to the two-point spectral tensor.

For a stationary, $z$-independent ideal magnetohydrodynamics flow with $E_z=0$, nonzero horizontal magnetic field and nonzero magnetohydrodynamic mass loading $k$, the displayed decomposition has $\mathbf{B}\cdot \nabla k=\mathbf{B}\cdot \nabla U=0$. The continuity equation gives the first invariant, and the ideal magnetohydrodynamic induction equation gives the second. Entropy and $V=u_z-B_z/({\mu }_{0}k)$ are also constant on each magnetic field line. Stationarity alone permits constant nonzero $E_z$, so it does not imply this decomposition.

An axial uniform magnetic field and localized radial differential rotation generate a steady toroidal magnetic field. The angular factor $\sin \theta \cos \theta$ is an eigenfunction of the toroidal angular diffusion operator with eigenvalue $-6$. Its radial amplitude satisfies $\eta (f^{\prime \prime }+2f^{\prime }/r-6f/r^2)=-B_{0}r{\Omega }^{\prime }$. Differentiating the integral in the title proves the formula. It is regular at the origin and decays at infinity when $\Omega$ is finite at zero and decays sufficiently rapidly.

Stefano Zapperi and Gianfranco Durin studied the statistics of Barkhausen effect avalanches in soft ferromagnets. Slowly changing the applied magnetic field produces bursts of magnetization as pinned domain walls move. Their 2000 work measured avalanche-size distributions in polycrystalline and amorphous materials and found distinct scaling classes in the slow-drive limit. Interpreting the data through the release of pinned domain walls connects irregular magnetic noise with a collective phase transition, including a quantitative relation between the distribution cutoff and the demagnetizing factor.^[ref]

A physicist and professor of theoretical physics of matter at Milan.^[ref]

Rotation influences convection, its helicity and differential rotation, which feed dynamo action. The generated magnetic field produces surface activity and strengthens the angular-momentum lever arm of a stellar wind. Wind braking lowers the rotation rate and ordinarily reduces unsaturated activity. Lorentz force feedback also changes the differential rotation and convective motions, saturating the dynamo on shorter time scales. Secular evolution, cyclic polarity reversal and irregular modulation are distinct time scales of this coupled system.

Stephan Stellmach is a researcher in geophysics and fluid dynamics. His University of Münster profile records work on density stratification, convective turbulence and magnetic field generation, alongside publications on rotating convection.^[ref]

His 2013 work with collaborators on fingering convection connects small-scale fluid dynamics to chemical transport inside stars. A stabilizing temperature gradient and an opposing composition gradient can have a stable combined mass density profile while still permitting fingers, because thermal diffusion and molecular diffusion act at different rates. Their paper develops a transport model supported by three-dimensional direct numerical simulations, with a prescription usable in one-dimensional models of stellar evolution.^[ref]

The bridge between dimensions is the practical point: a one-dimensional calculation of a star's evolution cannot explicitly resolve every three-dimensional finger. Estimating their average transport from local numerical simulations supplies a model for that unresolved motion. It is a physically tested approximation within the modeled parameter regime, not a direct observation of all mixing inside a star.

In 1922, Otto Stern and Walther Gerlach used an electrically heated oven to emit neutral silver chemical atoms. Two platinum apertures, a pinhole and a slit $0.03$–$0.04\mathrm{mm}$ wide, supplied collimation. The beam passed along a $3.5\mathrm{cm}$ knife-edge pole in a nonuniform magnetic field. At its end, a glass plate collected the condensed silver. Continuous pumping maintained vacuum; the thin deposit required development even after eight hours.^[ref]

The field-off image established the undeflected slit width. A comparison put field-on and field-off deposits on the same plate. Microscopy revealed two deflected bands, no detectable central component, and broadening attributed to the velocity distribution. Alignment and distance from the knife-edge affected the pattern; precise magnetic field gradient measurements were still pending.^[ref]

The apertures act as a collimator: they constrain the initial transverse positions and velocities, so a magnetic displacement can be distinguished from an initially broad beam. Vacuum reduces collisions that would destroy this preparation. Deposition integrates arrivals over the exposure; a darkened band is not a time-resolved record of individual chemical atoms. The field-off comparison tests whether the magnet changes the spatial distribution, while alignment checks address a geometric source of apparent asymmetry. Finite sensitivity qualifies the absence of a central deposit: it is an experimental upper limit, not a proof that exactly zero atoms landed there.