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A galvanometer using oppositely directed, nearly equal magnetic dipole moments to reduce the net restoring torque of the approximately uniform terrestrial field. It still needs a weak restoring mechanism for a reproducible deflection; it does not shield the needles from that field.

The 2015 experiment used an air-driven nonmagnetic rotor containing polycrystalline gadolinium, with magnetic shielding and an external fluxgate magnetometer. Air drive reduced motor backgrounds. Measurements at 297–305 K were above the Curie temperature, at rotation frequencies up to 1.5 kHz. Reversing airflow reversed rotation and the magnetic signal; an empty capsule showed no speed-dependent signal. The readout measured stray magnetic field; geometry converted it to magnetization. Sample temperature was checked immediately after stopping. Heating and the conversion model limit inference.^[ref]

An additional analysis check follows from rotation symmetry. For signed sensor output $S$, separateAn ideal linear Barnett effect appears in the odd component; a fixed offset appears in the even component. This is a proposed diagnostic, not a claim about the paper's processing. A background correlated with direction can also be odd: reversal alone is insufficient. Empty-rotor checks, zero-speed readings, calibration of sensor polarity, and controlled temperature changes test different errors. Equatorial stray-field sign must also be converted to magnetization sign: for an axial magnetic dipole moment, the magnetic dipole field at its equator points opposite to that magnetic moment. The apparatus diagram makes this otherwise easy-to-miss sign change visible.

An original reduced model makes the calibration explicit. Let the torsion pendulum have restoring torque $-\kappa \theta$, with torsion constant $\kappa$, and viscous damping torque $-b\dot{\theta }$, with $b>0$. Allow an unwanted axial torque ${\tau }_p(t)$:Here ${\omega }_0$ is the undamped natural angular frequency and $Q$ the quality factor. The forced harmonic oscillator response must be included before a twist can be interpreted as a torque.

Using the complex amplitude convention $m_z(t)=\mathrm{Re}(\tilde{m}{e}^{\mathrm{i}\omega t})$ and similarly for $\theta$ and ${\tau }_p$, differentiation multiplies by $\mathrm{i}\omega$. Substitution givesThis is the torque-to-twist transfer function. With ${\tilde{\tau }}_p=0$ and real positive $\tilde{m}=m_1$, the twist amplitude isFor $\gamma <0$, the twist phase relative to $m_z$ is $\pi /2-atan2(b\omega ,\kappa -I{\omega }^2)$. At $\omega ={\omega }_0$ it is zero, although the reaction torque leads $m_z$ by $\pi /2$. A phase measured at the mirror is not automatically the phase of the applied torque.

A fixed-amplitude reference torque ${\tau }_c\cos \omega t$ measures $|H_{\tau }|$ directly. The Einstein-de Haas effect instead contains an extra factor $\omega$ when $m_1$ stays fixed. Consequently its twist amplitude peaks exactly at ${\omega }_0$ in this model, whereas the fixed-torque peak is at ${\omega }_0\sqrt{1-1/(2Q^2)}$ when $Q>1/\sqrt{2}$. To verify the first statement, divide the squared denominator of $A_{\mathrm{EdH}}$ by ${\omega }^2$: the quantity to minimize is $(\kappa /\omega -I\omega {)}^2+b^2$, minimized at ${\omega }_0$. Both response curves tend to zero at high frequency; only the magnetomechanical curve also tends to zero at zero frequency.

For a fully stipulated example, choose $I=2.00\times 10^{-10}\mathrm{kg}{\mathrm{m}}^2$, $\kappa =8.00\times 10^{-10}\mathrm{N}m/rad$, $b=8.00\times 10^{-11}\mathrm{N}\mathrm{m}s/rad$, $\gamma =-1.76\times 10^{11}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, and $m_1=1.76\times 10^{-4}\mathrm{A}{\mathrm{m}}^2$. These teaching inputs give ${\omega }_0=2.00rad/s$, $Q=5.00$, reaction torque amplitude $2.00\times 10^{-15}\mathrm{N}\mathrm{m}$ at ${\omega }_0$, andAn optical lever at distance $D=1.00\mathrm{m}$ gives small-displacement spot amplitude $x\simeq 2DA=25.0\mu \mathrm{m}$; its full peak-to-peak width is $4DA=50.0\mu \mathrm{m}$. For a fast reversal from $+m_1$ to $-m_1$, the same inputs give $\Delta L_{\mathrm{body}}=-2.00\times 10^{-15}\mathrm{kg}{\mathrm{m}}^2/s$ and initial $\Delta \dot{\theta }=-1.00\times 10^{-5}rad/s$, provided the external angular impulse is negligible.

One independent mechanical calibration adds known moment of inertia $\Delta I$, keeping $\kappa$ unchanged. If $T_a,T_b$ are the undamped oscillation periods before and after adding it, thenFor $\Delta I=I$ in the stipulated example, $T_a=\pi \mathrm{s}$ and $T_b=\sqrt{2}\pi \mathrm{s}$, recovering the chosen $I$ and $\kappa$. Real damped periods require a damping correction: a free decay has angular envelope $e^{-bt/(2I)}$ and oscillation angular frequency $\sqrt{\kappa /I-(b/(2I){)}^2}$. Measuring that decay determines $b/(2I)$. Calibrating the optical scale, mechanical response and change of magnetic dipole moment separately is what makes the inferred gyromagnetic ratio meaningful.

A current loop is a closed path carrying electric current. For a thin wire following a closed curve $C$, the volume integral defining its magnetic dipole moment becomes a line integral:The vector area $\mathbf{A}$ is $A\hat{\mathbf{n}}$ for a planar current loop. For a nonplanar current loop, use the vector area, rather than the scalar area of an arbitrarily chosen spanning surface. Reversing the electric current reverses $\mathbf{m}$ without changing the geometry. ^[ref].

For $N$ closely spaced identical turns carrying the same electric current, the magnetic dipole moments add: $\mathbf{m}=NIA\hat{\mathbf{n}}$. Each turn contributes its own vector area; unequal areas or differently oriented turns require a vector sum. As a calculated example, twelve circular turns of radius $R=3.0\mathrm{cm}$ carrying $I=0.40\mathrm{A}$ give $m=NI\pi R^2=1.36\times 10^{-2}\mathrm{A}{\mathrm{m}}^2$. This is an idealized example, not an experimental measurement.

The PHYWE guide Magnetic moment in the magnetic field makes the magnetic dipole moment measurable. A circular test current loop sits between a pair of Helmholtz coils, with separate supplies and meters for the test electric current $I$ and field-producing electric current $I_H$. The test current loop is mounted on a torsion dynamometer. The guide varies $I_H$, the number of turns, the test electric current, diameter, and the angle between the magnetic dipole moment and magnetic field. Its angular settings are spaced by $15°$. The leads hang loosely and are twisted together to suppress extra torque; the instrument zero is checked frequently because moving leads can shift it. Its reported results include approximately linear dependence on each electric current and on winding count, and approximately quadratic dependence on diameter.^[ref]

The readout is a balance of torques. According to the manufacturer's Torsion dynamometer 0.01 N operating instructions, a metallic band supplies the restoring torque. First set the indication to zero with the applied force absent and center the lever using the zero knob. Apply the load, then turn the indicating knob until the lever returns to its original zero position. The scale then reads a force equivalent, not a torque directly. The quoted scale division is $0.1\mathrm{mN}$. For a force applied perpendicular to the calibration lever arm at distance $\ell$, the conversion is $\tau =F\ell$; a current loop applies a couple, so its torque must be compared with the calibrated restoring torque.^[ref]

For a quantitative repeat, independently check that conversion with known forces at a measured perpendicular lever arm, and check the imposed magnetic field at the test current loop. These are proposed calibration checks, not additional measurements reported here. The ideal Helmholtz coil model predictswhere $N_H$ and $R_H$ are the winding count and radius of each field-producing Helmholtz coil, distinct from $N$ and $R$ of the test current loop. Derive $B_0$ by adding the on-axis magnetic field of a circular current loop at distances $R_H/2$ from both coils. Independently calibrating $B_0$ and the restoring torque lets a fitted torque amplitude determine $m$ as amplitude divided by $B_0$, to compare with $NI\pi R^2$.

Useful additional controls are to reverse only $I$, reverse only $I_H$, and reverse both: the signed torque should reverse in either single reversal and be unchanged in the double reversal. Record baselines with each electric current separately zero, and subtract a nearby zero reading to reduce torsion balance zero drift. Repeat angular settings in both directions to reveal lead-dependent offsets. These are suggested controls. Limits include finite magnetic field uniformity, uncertainty in effective winding area and angular zero, meter calibration, Joule heating, and residual lead torque. An apparently sinusoidal trace alone does not establish the absolute magnetic dipole moment. No raw laboratory data or new measured values are claimed here.

The Einstein-de Haas effect is mechanical rotation or twisting caused by a change in a body's magnetization. The microscopic carriers of magnetic dipole moment also carry angular momentum: changing their net orientation can transfer angular momentum to the body. A freely suspended ferromagnet can therefore turn when its magnetization reverses. A suspension instead stores part of the response as an elastic twist. This makes a microscopic property accessible through a macroscopic angle.

The central measurement compares a change of magnetic dipole moment with transferred angular momentum, giving a gyromagnetic ratio. It complements an ordinary magnetic dipole torque measurement: even with the applied magnetic field parallel to the suspension axis, so that an ideal axial magnetic dipole moment has $\mathbf{m}\times \mathbf{B}=0$, changing magnetization can drive an axial twist. The external drive supplies energy; conservation of angular momentum still requires accounting for the suspension and any external axial torque.

In 1915, Albert Einstein and Wander Johannes de Haas suspended a soft-iron cylinder by glass fibre between coaxial coils. Alternating magnetization drove torsional resonance; a mirror reflected lamplight onto a scale. They compensated Earth's magnetic field, tilted the coils, and rotated the suspension attachment to test unwanted magnetic dipole torques. Added known moment of inertia calibrated the mechanical response.^[ref]

Their paper acknowledges a failed phase determination and claims only ten-percent precision for the revised measurement. Its agreement with the orbital gyromagnetic ratio prediction was not a direct discovery of spin.^[ref] Comparing that orbital gyromagnetic ratio prediction with the electron spin g-factor explains why a successful qualitative demonstration of angular momentum transfer can coexist with an inadequate quantitative interpretation. The stated precision is the authors' estimate; it does not establish that all systematic errors were bounded by that amount.

The general control problem follows directly from the model. Unwanted magnetic dipole torque can drive the same mechanical resonance as the desired reaction torque. In the ideal geometry $m_z$ and $B_z$ alone produce no axial cross product, but transverse magnetic dipole moments, stray transverse magnetic fields and imperfect alignment can. Nulling stray magnetic fields and deliberately varying alignment test this contribution. Merely observing a large resonant twist cannot identify its cause.

For a nearly instantaneous magnetic response, a sinusoidal magnetization makes $m_z$ follow the drive while $-{\dot{m}}_z/\gamma$ is in phase quadrature. An ordinary magnetic contribution proportional to $m_z$ is then distinguishable by phase, after removing the mechanical frequency response. Hysteresis or delayed magnetic response can spoil this simple separation. The measured magnetization, rather than the coil electric current alone, must set the reference. Mechanical vibration, magnetostriction coupled through imperfect geometry, and changing suspension properties require separate checks; none is eliminated simply by reversing the coil electric current.

Retaining only the electric dipole induced in a small sphere gives $\mathbf{p}=4\pi {ϵ}_0{a}^3{\mathbf{E}}_{\mathrm{inc}}$. In the backward direction its differential scattering cross-section is $k^4{a}^6$, so the radar cross section is $4\pi k^4{a}^6$. This response model omits the sphere's induced magnetic dipole moment; it must be distinguished from perfect-conductor sphere dipole interference.

For external electrons on shell, factor out signed electric charge in the parity-preserving interaction vertex:The $u$ are Dirac spinors; ${\gamma }^{\mu }$ are gamma matrices. With on-shell renormalization scheme normalization $F_1(0)=1$, the Gordon identity gives $g_s/2=F_1(0)+F_2(0)$ and $a_e=F_2(0)$.^[ref]

This explains a subtlety: the Dirac equation already predicts a spin magnetic dipole moment with $F_2=0$. Rewriting its Dirac electromagnetic current with the Gordon identity produces a spin-dependent term even before any loop order correction. $F_2$ measures the additional response after that baseline is included. The limit $q\to 0$ is taken after extracting its coefficient; setting $q=0$ in the interaction vertex at the outset would hide it. This electromagnetic anomalous magnetic moment is not a failure of gauge invariance of the kind called a quantum anomaly.

The electron magnetic moment couples an electron's angular momentum to a magnetic field. Its intrinsic spin contribution isHere ${\mu }_B$ is the Bohr magneton, and $g_s$ is the positive electron spin g-factor under this sign convention. The minus sign makes magnetic dipole moment antiparallel to spin angular momentum. An electron in a chemical atom can also contribute an orbital magnetic dipole moment; the measured response of a whole atomic level combines these contributions.

Measuring a tiny departure of $g_s$ from $2$ tests how well a theory predicts an electron's interaction with its surroundings. The Dirac equation gives $g_s=2$ before radiative corrections; the anomalous magnetic moment makes a small correction experimentally accessible.^[ref] A useful strategy is to compare two atomic energy levels whose orbital and spin contributions have different weights, rather than trying to calibrate an absolute magnetic field more accurately.

The dimensionless coefficient in the spin part of the electron magnetic moment. It is distinct from a gyromagnetic ratio, which has units: in the angular convention, ${\gamma }_e=-g_{s}e/(2m_e)$. It is also distinct from the Landé g-factor $g_J$ of a combined atomic angular momentum and from a nuclear or hyperfine g-factor.

For spin one-half, the largest magnitude of a spin-component magnetic dipole moment is $g_s{\mu }_B/2$, not $g_s{\mu }_B$. Thus the ideal value $g_s=2$ gives one Bohr magneton. This follows directly from the spin angular momentum projections $S_z=\pm \hslash /2$.

Far outside a localized steady source of size $a$, at distance $r\gg a$, its multipole expansion starts with the magnetic dipole moment if that moment is nonzero. In vacuum, using SI units,where ${\mu }_0$ is the vacuum permeability, $\hat{\mathbf{r}}=\mathbf{r}/r$, and the magnetic field is the curl of the magnetic vector potential. The magnetic dipole field falls as $r^{-3}$, and the magnetic vector potential as $r^{-2}$. The localized steady-current moment identities remove the putative $r^{-1}$ term in the magnetic vector potential. No magnetic monopole is needed. If different current loops cancel the total magnetic dipole moment, a higher term of the multipole expansion supplies the leading distant magnetic field. ^[ref].

The approximation is not valid at the center of a finite current loop. For a circular current loop of radius $R$ and electric current $I$, the exact on-axis magnetic field of a circular current loop and the distant approximation areTheir ratio is $B_z/B_{z,\mathrm{dip}}=(1+R^2/z^2{)}^{-3/2}$: at $|z|=10R$, the approximate magnetic field exceeds the exact magnetic field by about $1.50\%$ relative to the exact value. Near the source, retain its geometry rather than replacing it with a point source specified only by its magnetic dipole moment.

The isotropic magnetic interaction of a nuclear magnetic dipole moment with electron spin density at the atomic nucleus. A nonrelativistic penetrating $s$ orbital can have $|{\psi }_s(0){|}^2\ne 0$ and a substantial contact contribution to hyperfine structure. A filled paired shell cancels its net contact spin density in the independent-particle approximation; unpaired occupation and core polarization can restore it. Relativistic and finite nuclear-size effects modify the point-contact description.

A dimensionless factor relating a system's magnetic dipole moment to its angular momentum, with a specified charge and mass convention. Electron spin, orbital motion, and a combined atomic level have different g-factors; the convention must be stated before comparing them.

The proportionality coefficient between a system's magnetic dipole moment and angular momentum when $\mathbf{\mu }=\gamma \mathbf{J}$ applies. In the angular convention, $\gamma$ is expressed in radians per second per tesla; $\gamma /(2\pi )$ is the corresponding coefficient in cycles per second per tesla. Its sign determines the relative orientations of the magnetic dipole moment and angular momentum.

A freely orientable classical magnetic dipole moment of fixed magnitude $m$ in a uniform magnetic field $B$ has angular canonical partition function $Z=4\pi \sinh x/x$, where $x=mB/(k_{B}T)$. Its internal energy is $U=-mB\coth x+k_{B}T$, and its fixed-field heat capacity is $C_B=k_B[1-x^2/{\sinh }^{2}x]$. At low temperature, two locally Cartesian tangent coordinates on the orientation sphere each contribute $k_{B}T/2$ by the equipartition theorem, giving $C_B\to k_B$. A discrete quantum spin in a nonzero magnetic field instead has a spectral gap and its heat capacity vanishes exponentially at sufficiently low temperature.

A small linearly magnetizable sample has a magnetic dipole moment induced by its applied field. In a fixed principal direction, write $m={\alpha }_B{B}_0$; this convention gives ${\alpha }_B$ units $\mathrm{A}{\mathrm{m}}^2{\mathrm{T}}^{-1}$. A uniformly magnetized ellipsoid of volume $V$ has ${\alpha }_B=V\chi /[{\mu }_0(1+N_d\chi )]$. The effective interaction potential energy at fixed source current is $-{\alpha }_B{B}_0^2/2$. The factor one half arises because the moment must be induced as the field is raised; differentiating $-m(B_0)B_0$ as though it described a permanent moment would count the response twice. The response model excludes remanence, nonlinear saturation and hysteresis.

An ion trap confines an ion using electromagnetic fields, allowing repeated interrogation of the same particle. The name also covers related traps for an electron. A Paul trap uses an oscillating electric quadrupole field; a Penning trap combines a static electric quadrupole field with a static magnetic field. Confinement creates time to measure an energy level, frequency or magnetic dipole moment; it does not itself cool the particle or identify its internal quantum state.

The obstacle is Earnshaw's theorem. In a charge-free neighbourhood the static electric potential $\Phi$ obeys Laplace's equation, ${\nabla }^2\Phi =0$. For a particle of fixed nonzero electric charge $Q$, the potential energy $Q\Phi$ is also a harmonic function. The maximum principle for harmonic functions excludes a strict interior minimum. For a quadratic example, the Hessian matrix has zero trace; it cannot have three positive eigenvalues. A static purely electric field therefore cannot supply three-dimensional stable confinement of a free point electric charge. Time variation in a Paul trap, and the velocity-dependent Lorentz force in a Penning trap, change the assumptions of this argument.

Alfred Kastler's 1966 lecture recalls his 1950 proposal and distinguishes the early ${}^{23}\mathrm{Na}$ atomic beam work from later sealed-vapour experiments. Its Fig. 5 shows subsequent sodium radio wave resonances, including multiple-photon transitions.^[ref]

The 1952 demonstration by Jean Brossel, Alfred Kastler and Jacques Winter used a sodium atomic beam, lamp light with circular polarization and fluorescence readout. Jean Brossel was a central experimental contributor; Alfred Kastler received the 1966 Nobel Prize in Physics alone.^[ref][ref]

The lecture also describes a ${}^{199}\mathrm{Hg}$ experiment: a pumping beam with circular polarization prepares the vapour, while a perpendicular probe converts transverse magnetic dipole moment Larmor precession into modulated absorption at a photodetector. Interrupting pump and RF together exposes decaying free Larmor precession. Its envelope measures transverse relaxation under that model; it is distinct from the sodium atomic beam fluorescence signal.^[ref]

MIT's later rubidium experiment pumps natural ${}^{85}\mathrm{Rb}$/${}^{87}\mathrm{Rb}$ with a discharge lamp filtered to D1 ($5S_{1/2}\to 5P_{1/2}$, near $795\mathrm{nm}$). A polarizer and quarter-wave plate precede a heated rubidium/neon cell. Helmholtz coils set a weak static magnetic field. Separate transverse RF mixing increases absorption, measured by photodiode transmission.^[ref]

Compare resonant and detuned RF with the optical beam fixed, then change the calibrated static magnetic field: a true magnetic sublevel resonance should move with its Zeeman splitting. Lamp drift, stray magnetic fields, imperfect circular polarization, optical attenuation and RF power broadening can change the observed signal. Consequently, a transmission dip alone does not determine every sublevel population. These are measurement-design considerations, not additional reported historical observations.

The Landé g-factor is the dimensionless electronic magnetic splitting factor relating a level's magnetic dipole moment to its total angular momentum. In the approximation where total orbital angular momentum $\mathbf{L}$ and total spin $\mathbf{S}$ couple to $\mathbf{J}=\mathbf{L}+\mathbf{S}$, take the orbital factor as one and the electron spin factor as $g_s\simeq 2$. For $J>0$,This follows by projecting $\mathbf{L}+g_s\mathbf{S}=\mathbf{J}+(g_s-1)\mathbf{S}$ onto $\mathbf{J}$ and using $2\mathbf{S}\cdot \mathbf{J}={\mathbf{J}}^2+{\mathbf{S}}^2-{\mathbf{L}}^2$. At $S=0$, it gives $g_J=1$; at $L=0$ and $J=S$, it gives $g_J=g_s$. A $J=0$ level instead has zero first-order electronic shift; the formula's zero denominator must not be used. This $g_J$ is distinct from the statistical weight of an atomic level $2J+1$, sometimes also written $g_J$.