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Neglect hyperfine structure, spin-orbit coupling and the diamagnetic term, and set $b={\mu }_{B}B$, $\epsilon =\alpha a_0^2$. The Zeeman energy is $b(L_{\hat{B}}+2S_{\hat{B}})$. The electron spin is free, so every spatial result below is duplicated with an added $\pm b$.

For a magnetic field parallel to $z$, the old magnetic quantum number $m$ remains good and the first-order energy levels areThe two $s$ orbitals have zero orbital angular momentum, so only their electron spin contributes at this order.

For a magnetic field parallel to $x$, the expectation of $L_x$ in each individual $p_m$ orbital vanishes. One must still check the degenerate $m=\pm 1$ subspace: its projected $L_x$ is zero as well. Thus there is no first-order orbital splitting in a sufficiently weak transverse magnetic field. Spin still contributes $\pm b$.

The transverse orbital Zeeman mixing can be seen without assuming its size. Put $E_0=-\mathrm{Ry}/4+3\epsilon$, $E_t=-\mathrm{Ry}/4+21\epsilon$, and $\Delta =E_t-E_0=18\epsilon$. In the real $p_x,p_y,p_z$ orbital basis, $p_x$ is unaffected while $p_y,p_z$ have off-diagonal matrix elements of modulus $b$. Diagonalizing this two-dimensional Hamiltonian gives the spatial energy eigenvaluesWhen $b\ll \Delta$, their shifts relative to the earlier levels are $0,+b^2/\Delta ,-b^2/\Delta$, with $\pm b$ added for electron spin. The displayed square-root expression also handles $b$ comparable to $\Delta$, provided coupling to other principal shells stays negligible. The $2s$ orbital retains zero orbital correction.

These quadratic orbital estimates keep the enhanced contribution from mixing the split $2p$ levels. Minimal magnetic minimal coupling also produces $e^2{B}^2{r}_{\perp }^2/(8m_e)$; its expectation value is positive and must be added for a full order-$B^2$ calculation. For example it gives $e^2{B}^2{a}_0^2/(4m_e)$ for $1s$. It is parametrically smaller than the $2p$ mixing term when $\epsilon \ll \mathrm{Ry}$.

Let the dimensionful angular momentum operators obey $\mathbf{F}=\mathbf{I}+\mathbf{J}$ and take the interaction Hamiltonian as $H^{\prime }=A\mathbf{I}\cdot \mathbf{J}$. Sincethe energy eigenvalues at fixed $I,J$ arewhere $F=|I-J|,\dots ,I+J$. Subtraction gives the Landé interval ruleMany atomic conventions use dimensionless angular momentum operators and call the energy-valued constant itself $A$; then the right-hand side is $AF$. Each energy level has $2F+1$ magnetic sublevels with the same energy in zero field.

For fine structure, the effective interaction is $a_{LS}\mathbf{L}\cdot \mathbf{S}$ with dimensionless angular momentum operators. It originates in the spin-orbit interaction, including the magnetic interaction of electron spin with orbital motion and its relativistic correction. Within an isolated LS coupling atomic term, the labels are the electron configuration, $L,S,J,M_J$, and the interval between adjacent $J$ energy levels is $a_{LS}J$. This is smaller than the electrostatic separation of atomic terms.

For magnetic-dipole hyperfine structure, the nuclear magnetic dipole moment interacts with the magnetic field generated by the electrons. Contributions include the Fermi contact interaction of penetrating $s$ electrons, orbital magnetic fields, and the dipolar field of electron spin. Projected onto one electronic $J$ energy level, their rotationally invariant magnetic-dipole part has the form $a_{IJ}\mathbf{I}\cdot \mathbf{J}$. The labels are $I,J,F,M_F$, together with the approximate electronic labels such as $L,S$. This splitting is generally smaller than fine structure, reflecting the much smaller nuclear magnetic dipole moment. An electric nuclear quadrupole moment can add a different interaction and violate the simple Landé interval rule.

At the larger electrostatic energy scale, an effective two-spin exchange interaction can also be written $a_s{\mathbf{s}}_1\cdot {\mathbf{s}}_2$ within fixed orbitals. Its origin is the Coulomb interaction combined with the Pauli exclusion principle, rather than a large direct magnetic interaction. It separates singlet state and triplet state atomic terms; before spin-orbit coupling, useful labels include the electron configuration, $L,M_L,S,M_S$. General electrostatic atomic term separations need not obey a single Landé interval rule as $L$ varies.

In atomic units, $\hslash =m_e=e=4\pi {\epsilon }_0=1$, and the hydrogen atom hasThe allowed quantum numbers areHere the fixed electron spin quantum number is $s=1/2$. The energy eigenvalues are in hartrees; restoring units gives ${\hslash }^{2}l(l+1)$ and $\hslash m$ for the orbital operators and the corresponding factors for spin. The spin-independent hydrogen Hamiltonian does not split the two spin projections.

The $3d$ and $4s$ atomic orbitals of chromium are close enough in energy that electron-electron terms can overturn the simple filling order. The configuration $[\mathrm{Ar}]3d^{5}4s^1$ permits five singly occupied $3d$ orbitals and a parallel-spin $4s$ electron. Relative to $3d^{4}4s^2$, it removes the $4s$ pair and permits more parallel-spin exchange interactions, lowering the Coulomb-and-exchange contribution enough to offset the orbital promotion cost.

As qualitative bookkeeping, six parallel electron spins give $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ same-spin pairs, compared with $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ in high-spin $3d^{4}4s^2$. The individual exchange integrals are unequal, so this is an explanation of the tendency, rather than a quantitative atomic energy calculation. The observed ground state configuration is $\boxed{[\mathrm{Ar}]3d^{5}4s^1}$.^[ref]

Within a specified electron configuration in the LS coupling regime, Hund's rules select the lowest atomic term as follows. First maximize the total electron spin $S$ subject to the Pauli exclusion principle. Among the terms having that $S$, choose the largest total orbital angular momentum $L$. Finally, spin-orbit coupling places the smallest allowed $J=|L-S|$ lowest for a subshell less than half full, and the largest $J=L+S$ lowest for a subshell more than half full. At half filling the maximum-spin term has $L=0$, hence $J=S$. These rules rank terms within configurations; the exceptional neutral-atom configurations of chromium and copper must also be used.

Assume thermal equilibrium, nondegenerate carriers, and isotropic parabolic band dispersions, with the supplied effective masses interpreted as density-of-states masses. Take one effective valley with two electron spin states for each band; any additional valley multiplicity must instead be absorbed into the masses or explicit degeneracy factors.

Let the conduction band edge be $E_c$, the valence band edge $E_v$, and $E_g=E_c-E_v$. The spin-summed conduction density of states per volume isUsing the nondegenerate limit of the Fermi-Dirac distribution and the substitution $x=(E-E_c)/(k_{B}T)$ givesThe factor $\sqrt{\pi }/2$ in the supplied integral produces the normalization. Counting missing valence band electrons similarly givesMultiplication cancels the chemical potential, establishing the semiconductor mass-action law:The result does not require an intrinsic semiconductor: semiconductor doping changes $n$ and $p$ separately while preserving this product under the stated equilibrium and nondegeneracy assumptions.

For area $A$, allowed wavevectors have density $A/(2\pi {)}^2$. Include two electron spin states and fill a disk of radius $k_F$ at zero temperature. ThenThe free electron model has parabolic dispersion $E={\hslash }^2{k}^2/(2m_e)$, with its minimum chosen as zero, soThese are the Fermi wavevector and Fermi energy for a spin-degenerate two-dimensional free electron gas. An imposed spin polarization or additional valley degeneracy would change the state count.

The density of states $g(E)$ counts one-electron states per energy interval, including both electron spin orientations. It may be a total or a per-volume quantity; $U$ must have the matching normalization. The Fermi-Dirac distributionis the occupation probability of such a state. In the fixed-volume free electron model, the integration variable $E$ and the density of states do not depend on $T$. The occupation depends explicitly on $T$ and implicitly through the chemical potential $\mu (T)$ needed to preserve electron number. Thermal expansion or a changing electronic band structure would require additional effects outside this model.

At $T\ll T_F$, only states within order $k_{B}T$ of the Fermi energy change their occupation. There are order $g(E_F)k_{B}T$ such states, each carrying an excitation energy of order $k_{B}T$. Thus $U(T)-U(0)\propto g(E_F)(k_{B}T{)}^2$ and the electronic heat capacity is proportional to $g(E_F)k_B^{2}T$. Using the supplied coefficient,The shell argument fixes the scaling, not the numerical coefficient; the controlled Sommerfeld expansion fixes that coefficient at fixed electron number. Filled states far below $E_F$ cannot accept arbitrary thermal excitations because of the Pauli exclusion principle.

A material with paramagnetism develops a magnetization parallel to a weak applied magnetic field intensity, with positive magnetic susceptibility and no spontaneous magnetization in its paramagnetic phase. Molecular oxygen is an example; it has unpaired electron spins.

A material with diamagnetism develops an induced magnetization opposing the applied magnetic field, with negative magnetic susceptibility. A closed-shell atom such as helium is an example; its orbital response is diamagnetic.

A ferromagnet has a spontaneously ordered magnetization below its Curie temperature, even in the limit of vanishing applied magnetic field. Iron below its Curie temperature is an example. Different magnetic domains can cancel the net moment of an unmagnetized bulk sample, despite nonzero local spontaneous magnetization.

Several historical models represented the same broad magnetic observations in different ways:

James Clerk Maxwell's later treatment explicitly contrasts the induction model associated with Siméon Denis Poisson with Weber's permanently magnetic molecules and discusses the predicted saturation limit.^[ref] Its 1873 date is retained; the chapter is used to explain the earlier hypotheses, not attributed to an 1865 publication.

Experiments constrain these models. Cutting a magnet produces smaller magnets with both poles, rather than simply separating free north and south charges. A current loop and an ordinary magnet also have closely related external dipole fields. Such facts explain the utility of molecular dipoles and currents, but do not uniquely identify the microscopic constituents. Modern magnetic domains, electron spin and quantum magnetic ordering refine the account; classical loops and freely turning isolated molecular magnets are incomplete microscopic pictures of a real ferromagnet. An effective pole calculation must likewise be distinguished from claiming an observed isolated magnetic monopole.

The relevant sequence is electron capture followed by nuclear gamma decay:The nuclear spin angular momenta are $0\to 1\to 0$. The captured K-shell electron has total angular momentum $1/2$, so the initial atom-plus-electron system cannot be treated as having zero angular momentum. The atomic vacancy also produces subsequent X-rays or Auger electrons; these are not the selected nuclear transition.

Choose the neutrino momentum along $+z$ and the recoil momentum along $-z$. For a negative-helicity neutrino, $s_{\nu z}=-1/2$. With the two initial electron spin projections, angular momentum conservation allows the daughter to have $m_z=0$ or $+1$. A photon emitted exactly along the recoil direction is transverse, and the $1\to 0$ transition selects the $m_z=+1$ component. Its spin angular momentum points along $+z$, opposite to its momentum.

Thus in the selected collinear geometry the daughter and photon have the same negative helicity as the neutrino, while their spin angular momenta point opposite to the neutrino spin. Reversing all these spin projections gives the positive-helicity alternative. In normalized notation,The actual eigenvalues of $\mathbf{S}\cdot \hat{\mathbf{p}}$ are $\pm 1/2$ for the neutrino and $\pm 1$ for the photon; the equality concerns their normalized signs.

The nuclear gamma ray can be absorbed by ground-state samarium-152, excite the same level and be re-emitted: nuclear nuclear resonance fluorescence. Its energy in the absorber rest frame must overlap the nuclear absorption line within the relevant natural and thermal broadening. Atomic radiation from refilling the K shell does not have the required transition energy.

Let the excitation energy be $E_0$ and the nuclear mass be $M$. A free stationary emitter loses recoil energy while a free stationary absorber requires recoil energy:For $E_0\simeq 0.96\mathrm{MeV}$ the gap is about $6.5\mathrm{eV}$. A photon emitted along the daughter recoil receives a positive Doppler shift of approximately $E_0{p}_{\nu }/(Mc)$, whereas one emitted opposite the recoil is redshifted. Emission before the daughter slows down therefore selects photons roughly opposite to the neutrino. Comparable neutrino and photon energies nearly compensate the recoil mismatch; finite line widths and thermal motion permit overlap. Exact compensation need not hold for every atom, and this is not an assumption of recoil-free emission.

Magnetized iron supplies polarized electrons. Their Compton scattering cross section depends on the relative electron spin and photon helicity, so the iron attenuates the two circular polarizations differently. Reverse the magnetization and measure the resonance-selected count asymmetry $A=(N_{+}-N_{-})/(N_{+}+N_{-})$. With a calibrated analysing power $a$ and background correction, $A=a{P}_{\gamma }$ determines the sign and magnitude of the circular polarization. A scintillation counter such as NaI(Tl), coupled to a photomultiplier, determines the gamma ray energy from calibrated pulse height; a semiconductor detector could perform the same spectroscopic function.

electron spin is intrinsic angular momentum, independent of an orbital wavefunction's $\mathbf{r}\times \mathbf{p}$. It has $s=1/2$, soA spatial orbital therefore carries a two-component spinor degree of freedom. Noncommuting components do not describe an ordinary vector with simultaneously definite projections; measurements along one axis give only the two allowed values. Even a spherically symmetric $l=0$ orbital retains electron spin.

The associated magnetic moment is $\mathbf{\mu }=-g_{e}e\mathbf{S}/(2m_e)$ with $e>0$ and $g_e\simeq 2$. A spinor changes sign under a $2\pi$ rotation and returns under $4\pi$; that sign is observable only relative to another coherent amplitude, not as an isolated global phase. A literal classically rotating charged sphere does not account for this fixed half-integer representation, component algebra and spinor rotation law. Classical angular momentum provides useful analogies for torque and precession, but no ordinary classical orientation reproduces the full intrinsic quantum spin structure.

Count two electron spin states in each reciprocal-space cell of volume $(2\pi {)}^3/V$. Filling a Fermi sphere at zero temperature givesThe free electron model then givesThe energy zero is the bottom of the model's conduction band. The supplied carrier density already incorporates the one-electron-per-atom assumption; it must not be multiplied by another atomic count.

For weak fields in a linear isotropic material, magnetization is $\mathbf{M}={\chi }_m\mathbf{H}$, with magnetic susceptibility ${\chi }_m$. Paramagnetism has ${\chi }_m>0$, while diamagnetism has ${\chi }_m<0$. In either case, $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})={\mu }_0(1+{\chi }_m)\mathbf{H}$. These weak responses generally disappear when the applied field is removed, unlike hysteretic ferromagnetism.

A paramagnet can contain permanent magnetic moments from unpaired electrons or atomic angular momentum. Alignment lowers the energy $-\mathbf{\mu }\cdot \mathbf{B}$, while thermal disorder opposes it. For independent classical moments of magnitude $\mu$, their weak-field average projection is ${\mu }^{2}B/(3k_{B}T)$, soThis is Curie's law; it assumes weak magnetization and no important interactions between moments. With quantum angular momentum $J$, replace ${\mu }^2$ by $(g{\mu }_B{)}^{2}J(J+1)$. Strong-field magnetization saturates instead of growing linearly without limit. Conduction electrons also exhibit Pauli paramagnetism: only occupations near the Fermi surface readily redistribute between spin states, giving a nearly temperature-independent susceptibility when $T\ll T_F$.

Diamagnetism is an induced orbital response opposing the applied magnetic field. For bound electrons, a weak field changes orbital motion and induces an opposing magnetic moment. For an isotropic collection of atoms, the elementary orbital estimate iswhere the sum runs over the electrons of one atom. It is usually small and weakly temperature dependent. Delocalized electrons also have orbital Landau diamagnetism; for an ideal three-dimensional free electron gas with a quadratic isotropic dispersion and electron spin with $g=2$, ${\chi }_L=-{\chi }_P/3$. Both spin and orbital contributions must be included to determine the total response. Diamagnetism is not a consequence of permanent dipoles choosing to align against the field; that orientation would increase their energy.

For hyperfine coupling for a spin-one nucleus, let the electron spin be $s=1/2$ and the nuclear spin $I=1$. First couple $\mathbf{j}=\mathbf{L}+\mathbf{s}$, giving $j=L\pm 1/2$, then $\mathbf{F}=\mathbf{j}+\mathbf{I}$, giving $F=|j-I|,|j-I|+1,\dots ,j+I$.

For each $F$, the projection is $m_F=-F,-F+1,\dots ,F$; electron projections similarly satisfy $m_j=-j,\dots ,j$. Summing $2F+1$ gives 18 states for $L=1$ and 30 for $L=2$, agreeing with $(2L+1)(2s+1)(2I+1)$.

Notation qualification: if the printed leading “2” denotes the principal quantum number, a hydrogenic $2d$ bound state does not exist because $L\le n-1$; a physical $d$ state needs $n\ge 3$. If a doublet term symbol ${}^{2}P$ or ${}^{2}D$ was intended, its leading 2 instead means $2s+1$, and the angular-momentum table applies without fixing $n$. The answers above use the explicitly specified $L$ values, while retaining this ambiguity.^[ref]

Assume dilute, nondegenerate ideal electrons, protons and neutral atoms in thermal and chemical equilibrium, with charge neutrality $n_p=n_e$. Set ${\mu }_H={\mu }_p+{\mu }_e$, using each ideal species' translational factor $n{\lambda }_{\mathrm{th}}^3/g$ and hydrogen ground-state binding energy $-R$. Neglect excited states and take the heavy proton/atom thermal-wavelength ratio as one. With electron spin and neutral-atom degeneracies treated in the convention of the source, the remaining factor isDifferent partition-degeneracy conventions must be propagated consistently. Screening, dense-plasma level dissolution and departures from local thermodynamic equilibrium lie outside this derivation.

An experimental condensed matter physicist who coauthored an electrical readout method for electron spin states in a quantum dot. Different tunneling rates map the spin information onto distinguishable charge histories, followed by a fast charge measurement. Unlike simple energy-selective readout, this mechanism can work when the electron temperature exceeds the splitting between the states.^[ref]

A particle physicist who wrote a historical analysis of the Stern-Gerlach experiment. It distinguishes the experiment's original proposal and interpretation from its later connection with electron spin, and discusses the different problem of performing an analogous measurement on free electrons. This helps prevent later quantum concepts from being read back into the original experimental argument.^[ref]

An experimental physicist who coauthored a 2004 report of large room-temperature tunnelling magnetoresistance in iron electrodes separated by a crystalline magnesium-oxide barrier. Changing the relative magnetic alignment changes electrical electrical resistance, allowing a magnetic configuration to be read electrically. The work connects controlled crystalline interfaces with devices that use electron spin as well as charge.^[ref]