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Alfred Landé organized the magnetic response of coupled atomic angular momenta. The Landé g-factor relates a magnetic energy shift to the projection of total angular momentum, accounting for both orbital angular momentum and spin. For LS coupling, $g_J=1+[J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)]$ when $J\ne 0$, neglecting the small correction to the electron spin g-factor. For $L=0$ and $J=S=1/2$, it gives $g_J=2$. Different atomic terms therefore split differently in a weak magnetic field.^[ref]

A physicist.^[ref]

The anomalous Zeeman effect is Zeeman splitting whose pattern is not the simple orbital triplet. Spin and spin-orbit interaction make the upper and lower Landé g-factors differ. The same weak-field formula still applies when its assumptions hold: “anomalous” does not mean nonlinear in the magnetic field.

For example, take an electronic transition with $L_u=1,S_u=1/2,J_u=1/2$ and $L_l=0,S_l=1/2,J_l=1/2$, neglecting hyperfine structure. With $g_s=2$, the formula above gives $g_u=2/3$ and $g_l=2$. The two allowed $q=0$ transitions have dimensionless shifts $g_u{m}_u-g_l{m}_l=\pm 2/3$, while the $q=\pm 1$ transitions give $\pm 4/3$. Thus there are four frequencies, even though each level contains only two magnetic quantum numbers. This is an independent worked deduction; it should not be substituted for the unresolved widths in the 1896 experiment.

Curie's law is the inverse-temperature magnetic response of independent magnetic moments in the weak-polarization regime. Let each moment have angular momentum quantum number $J$, Landé g-factor $g_J$, and magnetic projection $g_J{\mu }_{B}m$, where $m=-J,\dots ,J$. Its Zeeman energy is $-g_J{\mu }_{B}Bm$. At zero field the projections are equally likely, so $⟨m⟩=0$ and $⟨m^2⟩=J(J+1)/3$. Differentiating their canonical partition function gives the total-moment response of $N$ independent moments,For a dilute weakly magnetized sample with number density $n=N/V$ and $B\simeq {\mu }_{0}H$, the dimensionless SI volume magnetic susceptibility is insteadThese conventions differ by volume, field and $g_J$ factors. A fixed nonzero field eventually saturates the moments as temperature falls, so the linear Curie's law regime requires $g_J{\mu }_B|B|J\ll k_{B}T$.^[ref]

The magnetic moment inferred from the weak-field Curie constant of a localized angular-momentum multiplet. Uniform zero-field occupation of its $2J+1$ substates gives $⟨m_J^2⟩=J(J+1)/3$, yielding ${\mu }_{\mathrm{eff}}=|g_J|{\mu }_B\sqrt{J(J+1)}$. Here $g_J$ is the Landé g-factor and ${\mu }_B$ the Bohr magneton. The dimensionless number $m_{\mathrm{eff}}={\mu }_{\mathrm{eff}}/{\mu }_B$ differs from the saturated projection $|g_J|J$.

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 $S_z=\pm \hslash /2$.

The Zeeman effect need not be diagonal between different $J$ multiplets. The familiar Landé g-factor gives the interaction projected into one isolated multiplet. Inter-multiplet matrix elements cause higher-order weak-field shifts and become important in the Paschen-Back effect.

An atomic addition of angular momentum scheme in which the individual orbital angular momentum vectors first combine into $\mathbf{L}$ and the individual spins into $\mathbf{S}$, then these combine into total angular momentum $\mathbf{J}$. When spin-orbit coupling is weak compared with the interactions between electrons described by electrostatics, $L$ and $S$ remain useful approximate quantum numbers. The Landé g-factor describes the resulting weak-field magnetic splitting.

The open shell contains one $f$ electron, so $L=3$ and $S=1/2$. For a less-than-half-filled shell, Hund's rules select the smaller total angular momentum, $J=L-S=5/2$, giving the free-ion ground level ${}^2{F}_{5/2}$. The Landé g-factor isIn a magnetic field weak compared with the spin-orbit splitting, the electronic Zeeman shifts are $\delta E_{m_J}=g_J{\mu }_{B}B{m}_J$, for $m_J=-5/2,-3/2,\dots ,5/2$. Thus six equally spaced magnetic substates appear. At $B=1\mathrm{T}$, the adjacent-level spacing isThe complete lowest-to-highest multiplet spread is five spacings,Giving both quantities removes the ambiguity in the word splitting. This is the free-ion, weak-field result; no crystal-field doublet or hyperfine structure is specified.

Stimulated emission adds a photon to an already occupied radiation mode; spontaneous emission can occur when that mode initially contains no photons. In the electric-dipole approximation the atom-field interaction is $H_{\mathrm{int}}=-\mathbf{d}\cdot \mathbf{E}$. Quantizing the radiation field makes the emission matrix element into a mode with occupation $n$ proportional toThis photon occupation enhancement of emission makes the emission rate into that mode proportional to $n+1$: the part proportional to $n$ is stimulated and the remaining part is spontaneous. Absorption is proportional to $|⟨n-1|a|n⟩{|}^2=n$. The photon energy is the atomic energy difference, up to any specified recoil or line broadening.

The stimulated contribution preferentially populates the stimulating mode, including its propagation direction and polarization; it provides optical gain. Spontaneous photons populate the available modes according to the transition's dipole matrix elements and radiation environment. They are not universally isotropic. An excited bare atom with the field vacuum is not an eigenstate of the fully coupled Hamiltonian, which resolves the apparent paradox of a nominally stationary excited state radiating.^[ref]

For lower and upper levels 1 and 2, with radiation spectral energy density $u_{\nu }$ per unit ordinary frequency, the rates areThermal detailed balance with the Planck spectrum givesThe statistical weights $g_i$ here are level degeneracies, not Landé g-factors. A definition of $B$ multiplying intensity or angular-frequency density would change the normalization factors.

For a specified free-space electric-dipole transition, summing radiation modes givesAppropriate sums over final magnetic substates and averages over initial substates are needed for an unresolved degenerate level. The radiative lifetime is $1/A_{21}$ only when this is the sole decay channel. In a laser, net stimulated gain requires $N_2/g_2>N_1/g_1$, as well as sufficient gain to overcome cavity loss; population inversion alone does not specify the lasing threshold.

The Zeeman effect comes from $-\mathbf{\mu }\cdot \mathbf{B}$. For an electron, the orbital and spin magnetic moments are approximately $-{\mu }_B\mathbf{L}/\hslash$ and $-2{\mu }_B\mathbf{S}/\hslash$, respectively. This explains both the sign and the relative factor two in the term linear in $B$. The second term is the spin-orbit interaction: the electron's motion through the atomic electric field produces a magnetic interaction in its rest frame. The relativistic correction, including Thomas precession, and the radial expectation value are incorporated into $K$. This effective Hamiltonian retains the linear Zeeman splitting and spin-orbit coupling, while omitting diamagnetic and hyperfine structure corrections.

Define the angular momentum ladder operators by $L_{\pm }=L_x\pm i{L}_y$ and similarly for $S_{\pm }$. Direct expansion givesso adding $L_z{S}_z$ gives the required scalar product. The off-diagonal terms change $m_{\ell }$ and $m_s$ in opposite directions. Consequently $J_z=L_z+S_z$ commutes with $H$, and $m_j$ remains a good quantum number at every magnetic field.

In the given product eigenbasis, the diagonal matrix elements areFor $\ell =1$ and $s=1/2$, the only off-diagonal couplings are between $(1,-1/2)$ and $(0,1/2)$, and between $(0,-1/2)$ and $(-1,1/2)$. Each relevant orbital ladder factor is $\hslash \sqrt{2}$ and each spin ladder factor is $\hslash$. Multiplying by $K/{\hslash }^2$ gives $\sqrt{2}K$. ThereforeAll other off-diagonal entries vanish either by the ladder endpoints or by conservation of $m_j$. This derives every entry of the displayed matrix.

The stretched states have no mixing and yieldThe $m_j=1/2$ block is $\left(\begin{array}{cc}-K& \sqrt{2}K\\ \sqrt{2}K& a\end{array}\right)$, while the $m_j=-1/2$ block is $\left(\begin{array}{cc}-a& \sqrt{2}K\\ \sqrt{2}K& -K\end{array}\right)$. Taking the trace and determinant of each $2\times 2$ block gives the remaining four energy eigenvalues:The subscripts indicate $m_j=\pm 1/2$ and the superscripts indicate the upper or lower root. These formulas are the Zeeman spectrum for orbital angular momentum one and remain valid for either sign of $K$.

For the usual ordering $K>0$, including sodium, the zero-field states follow fromThe $j=3/2$ quartet lies at $K$ and the $j=1/2$ doublet at $-2K$, separated by $3K$. In a weak magnetic field, the Landé g-factors giveThese label the curves near $B=0$. In a strong magnetic field, the Paschen-Back effect makes $(m_{\ell },m_s)$ the appropriate approximate labels:for the mixed states, with the stretched-state formulas already exact. The branch connections for $B>0$ areWithin a fixed $m_j$, the nonzero coupling prevents a crossing; this is an avoided crossing. Curves with different $m_j$ may cross because the Hamiltonian does not couple them.

For the sodium estimate, the zero-field frequency separation fixesThe lower $3s$ doublet has shifts $\pm a$. The Sodium D-line Zeeman pattern gives photon-frequency shifts $a/h$ times $\{\pm 2/3,\pm 4/3\}$ for D${}_1$ and $\{\pm 1/3,\pm 1,\pm 5/3\}$ for D${}_2$. Thus the highest-frequency D${}_1$ component and lowest-frequency D${}_2$ component approach each other by $3a/h$. Extrapolating the weak-field Zeeman splitting givesso the crossover occurs at a field of order $10\mathrm{T}$, where $a\sim K$ and the weak-field approximation is becoming marginal.

The exact eigenvalues refine the estimate within this Hamiltonian. The relevant photon energies, with a common zero-field reference omitted, are $E_{+}^{(-)}+a$ and $E_{-}^{(+)}-a$. The corresponding electric dipole transitions have $\Delta m_j=+1$ and $-1$, respectively. Equating them givesIts positive solution is $a/K=\sqrt{10}/3$, hencein the retained model. The crossing is between optical transition frequencies; the participating upper states themselves need not be degenerate.

Neglect hyperfine structure and use the electronic weak-field Zeeman splitting regime, where ${\mu }_{B}B$ is small compared with the spin-orbit interaction splitting. The sodium levels are $3s{}^2{S}_{1/2}$, $3p{}^2{P}_{1/2}$ and $3p{}^2{P}_{3/2}$. The Landé g-factor formula givesFor example, inserting $L=1,S=1/2,J=1/2$ gives $3/2+(3/4-2)/(2\cdot 3/4)=2/3$. Each sublevel with magnetic quantum number $m_J$ shifts by $g_J{m}_J{\mu }_{B}B$. Thus the lower level has shifts $\pm {\mu }_{B}B$, the upper $J=1/2$ level has shifts $\pm {\mu }_{B}B/3$, and the upper $J=3/2$ level has shifts $\pm 2{\mu }_{B}B$ and $\pm 2{\mu }_{B}B/3$.

The electric-dipole selection rule allows $\Delta J=0,\pm 1$ except $0\leftrightarrow 0$, and $m_u-m_l=0,\pm 1$. Both fine structure branches are allowed. The emitted photon energy isAll the allowed components of the sodium D-line Zeeman pattern are listed below; the last column is the photon shift in units of ${\mu }_{B}B$:These are four components of D1 and six of D2. Their strengths need not be equal; the table and arrows specify allowed transitions, not intensities.

The $3p{}^2{P}_{3/2}$ level lies above $3p{}^2{P}_{1/2}$, so D2 has the larger zero-field photon energy. ^[ref] In the stated weak-field regime its most positive shift is $5{\mu }_{B}B/3$. Since $\lambda =hc/E_{\gamma }$, the shortest wavelength is therefore produced byThe tempting $m_u=3/2\to m_l=-1/2$ pair has a larger difference of level shifts but is forbidden because $|m_u-m_l|=2$.

Use the electronic weak-field Zeeman splitting model, neglecting hyperfine structure. The Landé g-factor is $g_g=2$ for $3S_{1/2}$ and $g_e=2/3$ for $3P_{1/2}$. The shift of an absorption frequency is thereforeHere the magnetic field is weak relative to spin-orbit coupling, so $J$ remains a useful quantum number, but the spectral line widths are small enough to resolve its magnetic substates.

For linear polarization parallel to $z$, the optical electric field has only the field-axis $\pi$ component. The electric-dipole selection rule is $\Delta m=0$. Its two allowed transitions are $m_g=m_e=\pm 1/2$, givingFor propagation perpendicular to $z$ with linear polarization also perpendicular to $z$, the optical electric field is an equal-amplitude combination of the two field-axis circular components. It drives ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$. These connect $-1/2\to +1/2$ and $+1/2\to -1/2$, givingThus each orientation selects two lines, with twice the separation in the perpendicular-polarization case. Neither pair has an unshifted central component. This is polarization-selective Zeeman splitting of an alkali D1 line, an anomalous Zeeman effect rather than a normal orbital triplet.

Use the uncoupled angular momentum eigenstates $|m_L,\uparrow ⟩$ and $|m_L,\downarrow ⟩$, with $m_L=-1,0,1$. For the Hamiltonian operator actually written in the paper, these are the energy eigenstates, withThe initial $J=3/2$ states are not all energy eigenstates of this operator. To find their energy populations, first perform the Clebsch-Gordan decomposition. The extremal states are $|3/2,3/2⟩=|1,\uparrow ⟩$ and $|3/2,-3/2⟩=|-1,\downarrow ⟩$. Applying the total angular momentum lowering operator to the first givesSimilarly,The initial preparation is an equal incoherent mixture of the four coupled states. Each extremal state contributes population $1/4$ to its one energy eigenvalue, while each nonextremal state contributes $2/3$ and $1/3$ of its own $1/4$ population to the two displayed components. The resulting Zeeman splitting is thereforefor nonzero $B_z$. The relative populations are $\boxed{3:2:2:2:3}$ in this order; the two zero-energy states each have population $1/12$. These Zeeman populations from an unpolarized angular-momentum multiplet remain constant under subsequent evolution by the stated static Hamiltonian operator, although off-diagonal quantum coherence can evolve.

There is a distinct weak-field model in which a retained spin-orbit interaction keeps $J=3/2$ isolated and one projects the magnetic perturbation into that quartet. The Landé g-factor is then $4/3$, giving four shifts ${\mu }_B{B}_z(\pm 2,\pm 2/3)$ with equal populations. That calculation requires the additional projection or dominant spin-orbit interaction; it is not diagonalization of the full magnetic Hamiltonian operator specified here, which also mixes the $J=1/2$ sector.

Within a fixed $L,S,J$ angular momentum multiplet, the component of $\mathbf{L}$ along $\mathbf{J}$ follows from $2\mathbf{L}\cdot \mathbf{J}={\mathbf{J}}^2+{\mathbf{L}}^2-{\mathbf{S}}^2$. The corresponding result for $\mathbf{S}$ follows by interchanging $L$ and $S$. For the quantum state $|LSJ{M}_J⟩$,These are dimensionless expectation values in the convention of part (a). Substituting into the first-order time-independent perturbation theory formula $\Delta E=⟨H_Z⟩$ givesFor $g_s=2$, the Landé g-factor becomesThe magnetic sublevels must first be diagonalized within a degenerate multiplet; choosing $M_J$ along the magnetic field already does this. For $J=0$ the displayed fraction is undefined, but all vector expectation values vanish and there is no first-order Zeeman splitting. A higher-order shift may remain.

Measure each energy level by the wavenumber $\tilde{E}=(E-E_g)/(hc)$ in ${\mathrm{m}}^{-1}$. An upward electronic transition of vacuum wavelength $\lambda$ adds $1/\lambda$. Treat the quoted wavelengths as vacuum values to their limited accuracy; air corrections would be smaller than the precision justified here.

The strong ground-state absorption must be $5s^2{}^1{S}_0\to 5s5p{}^1{P}_1$ at $461\mathrm{nm}$. The weak ground-state electronic transition at $689\mathrm{nm}$ must reach $5s5p{}^3{P}_1$: the electric-dipole selection rule on $J$ excludes the other two triplet state energy levels, while weak spin-orbit coupling admixes singlet state character and relaxes the approximate spin selection rule.

Assume the usual ordering $E({}^3{P}_0)<E({}^3{P}_1)<E({}^3{P}_2)$, as expected from a positive effective spin-orbit coupling constant. The three nearby strong spectral lines then join these energy levels to a common $5s6s{}^3{S}_1$ energy level. Larger lower-state energy means a smaller gap and longer wavelength, so $679,688,707\mathrm{nm}$ correspond to $J=0,1,2$ respectively. The remaining $1.12\mu \mathrm{m}$ spectral line joins $5s5p{}^1{P}_1$ and $5s6s{}^1{S}_0$. Assume the latter is the upper energy level. These assignments and orderings agree with independently tabulated strontium energy levels.^[ref]

Starting with the two ground-state gaps, the construction isThe resulting energy levels, rounded to avoid suggesting spectroscopic precision, are

Both $461\mathrm{nm}$ and $1.12\mu \mathrm{m}$ exhibit the normal Zeeman effect. Each connects two singlet state energy levels, with Landé g-factor $g_J=1$ for the $J=1$ energy level and no first-order shift for $J=0$. Their frequency shifts are $0,\pm {\mu }_{B}B/h$. The weak $689\mathrm{nm}$ spectral line also has three components, but $g({}^3{P}_1)=3/2$ gives shifts $0,\pm 3{\mu }_{B}B/(2h)$; a three-component pattern alone does not establish the normal Zeeman effect. The $679\mathrm{nm}$ spectral line likewise has three components with $g({}^3{S}_1)=2$.

For $5s5d$, addition of angular momentum gives $L=2$ and $S=0,1$. The even-parity atomic terms and atomic fine-structure levels areThe electric-dipole selection rules exclude all connections to the even-parity $5s^2$ and $5s6s$ electron configurations. To the odd-parity $5s5p$ electron configuration, all seven allowed connections in pure LS coupling areHere a connection is listed independently of whether the $5s5d$ energy level is above or below its partner.

In a weak magnetic field, use the Landé g-factorFor the three ${}^3{D}_J$ energy levels, this gives $1/2,7/6,4/3$; for ${}^3{P}_1,{}^3{P}_2$ it gives $3/2$. The six-component spectral line is thereforeA component has frequency displacement $({\mu }_{B}B/h)(m_D/2-3m_P/2)$. The electric dipole transition with $m_D=m_P=0$ has zero matrix element, since the relevant $J=1\leftrightarrow 1$ rank-one Clebsch-Gordan coefficients vanish there. The remaining allowed pairs giveThus all six frequency positions $\pm 1/2,\pm 1,\pm 3/2$ are distinct. Perpendicular to the magnetic field, both the $\pi$ and $\sigma$ components are visible, with mutually perpendicular linear polarizations. Along the magnetic field the $\pi$ components disappear. This is the weak-field Zeeman effect, before the Paschen-Back effect changes the coupling scheme.

Use the ordered basis $(|1⟩,|2⟩)$ and ${\sigma }_z=\mathrm{diag}(1,-1)$, withA pure quantum superposition $|+x⟩=(|1⟩+|2⟩)/\sqrt{2}$ has $\mathbf{R}=(1,0,0)$ on the Bloch sphere. In contrast, a mixed state $\rho =p|1⟩⟨1|+(1-p)|2⟩⟨2|$ has $\mathbf{R}=(0,0,2p-1)$, inside the sphere for $0<p<1$. At equal populations its vector is zero. Equal populations alone therefore do not distinguish a coherent superposition from a mixture; their off-diagonal density matrix elements differ.

Choose the drive phase so the rotating-frame Hamiltonian is $H_{\mathrm{rot}}=\hslash (\Omega {\sigma }_x+\delta {\sigma }_z)/2$, where $\delta =\omega -{\omega }_0$ as specified. The Rabi vector is the angular-velocity vectorHere $\Omega$ is the on-resonance population oscillation angular frequency, not the unhalved coefficient of a cosine coupling. From $\mathbf{R}(0)=\hat{z}$, rotation about $\mathbf{W}$ at $W=\sqrt{{\Omega }^2+{\delta }^2}$ givesFor $\Omega \gg |\delta |$, the rotation axis is nearly transverse, the trajectory nearly a great circle, and transfer is almost complete. For $\Omega =\delta >0$, the axis bisects $+x$ and $+z$, the orbit lies in $R_x+R_z=1$, and $p_2=\frac{1}{2}{\sin }^2(\sqrt{2}\Omega t/2)$ never exceeds $1/2$. Each member remains a pure state in this coherent evolution.

For the Ramsey free interval, let atom $j$ have detuning ${\delta }_j=\delta +{\eta }_j$. Instantaneous ideal pulses give the Ramsey ensemble coherence factor $\chi (T)=⟨e^{i{\eta }_{j}T}⟩$. Averaging the free-evolution vector obtained in step (ii) givesDifferent atoms fan out around the equator, shortening the ensemble Bloch vector even though every individual's vector remains unit length. A broad smooth detuning distribution usually makes the average spiral towards the centre and damps the Ramsey fringes towards $1/2$. The dephasing time is of order $1/\Delta {\omega }_0\sim 5/\delta$. inhomogeneous ensemble dephasing is not necessarily irreversible decoherence of individual atoms; an appropriate spin echo can rephase static frequency offsets.

The given spread does not specify a probability distribution or whether it denotes an RMS width. An exact envelope is therefore not determined. The figure illustrates, rather than infers, a Gaussian distribution with RMS ${\sigma }_{\delta }=0.2\delta$, for which $\chi =e^{-{\sigma }_{\delta }^2{T}^2/2}$. A uniform distribution would instead give a sinc envelope and side lobes. Both have the same qualitative initial loss of contrast on the quoted inverse-spread scale.

In the rubidium-87 ground-state hyperfine structure, $J=1/2$ and $I=3/2$, so $F=1,2$. Taking $H_{\mathrm{hf}}=A\mathbf{I}\cdot \mathbf{J}/{\hslash }^2$ givesThe zero-field splitting is ${\Delta }_{\mathrm{hf}}=2A\simeq h\times 6.8\mathrm{GHz}$. Ignoring the small nuclear magnetic term, the weak-field Landé g-factors are $g_{F=1}\simeq -1/2$ and $g_{F=2}\simeq +1/2$. The diagram labels all three $F=1$ and five $F=2$ substates, with linear shifts $g_F{\mu }_B{m}_{F}B$.

The pair $|1⟩=|F=1,m_F=0⟩$ and $|2⟩=|F=2,m_F=0⟩$ is connected by a magnetic dipole transition with $\Delta F=1$, $\Delta m_F=0$, driven with its oscillating magnetic component along the quantization axis. Both first-order shifts vanish, giving the rubidium-87 clock transition first-order insensitivity near zero field. Its insensitivity does not imply absence of quadratic shifts.

For these two $m_F=0$ levels the Breit-Rabi formula gives the quadratic Zeeman shift of a clock transition:where the last expression neglects the nuclear term and sets $g_J\simeq 2$. HenceThis uses ${\mu }_B/h\simeq 1.40MHz/G$ and the supplied $A/h\simeq 3.4\mathrm{GHz}$. With physical moments defined by ${\mu }_i=-\partial E_i/\partial B$, the differential magnetic moment is ${\mu }_2-{\mu }_1\simeq -8.2\times 10^{-3}{\mu }_B$. If effective magnetic moment denotes the positive energy slope instead, that slope difference is positive. The magnitude and physical frequency sensitivity are the same. Retaining the full atomic factors gives the familiar quadratic clock-frequency coefficient about $575Hz/{\mathrm{G}}^2$; the simplified supplied parameters give about $576Hz/{\mathrm{G}}^2$.^[ref]

Use a far-detuned optical dipole trap, such as a red-detuned focused-beam trap, to confine both clock states. Their first-order magnetic moments vanish; their residual quadratic magnetic slopes are opposite, so a simple static magnetic-field minimum does not confine both in the same way. An optical potential acts through the electronic polarizability and can trap both ground hyperfine states. It is not perfectly state independent: differential AC Stark shifts, intensity noise and residual magnetic gradients still need control during Ramsey evolution.

A $3d^9$ shell has one hole in the otherwise filled $d$ shell. Hund's rules give $S=1/2$, $L=2$, and, for a more-than-half-filled shell, $J=L+S=5/2$: the free-ion term is ${}^2{D}_{5/2}$. Its Landé factor isDistinguish the Curie effective moment from a saturated projection:The question then explicitly asks us to use orbital quenching in the crystal. Ligand fields split and mix orbital states, suppressing their freely orientable angular momentum. The spin-only approximation has $g\simeq 2$, $S=1/2$, so ${\mu }_{\mathrm{eff}}=\sqrt{3}{\mu }_B$ but the two field-axis projections are $\pm {\mu }_B$. The latter, not $\sqrt{3}{\mu }_B$, belong in the tanh argument and saturation magnetization.

The sulfate and water environment supplies mainly closed-shell, weakly magnetic surroundings, stabilizes localized Cu2+ sites and separates their moments so that exchange can be weak on the temperature scale considered. Paramagnetism here is an approximation of weakly interacting local spins, not a claim that dilution guarantees no magnetic ordering at any temperature. Elemental copper has a filled $3d$ band and itinerant conduction electrons rather than one localized Cu2+ spin per atom. Pauli spin response and orbital/core contributions then replace a local-spin Curie law; ordinary pure copper has a weak net diamagnetic response.^[ref]

There is one Cu2+ per formula unit, hence $n_s=N_A/V_m=5.5148\times 10^{27}$ m${}^{-3}$. The spin-only weak-field SI volume susceptibility isThese neglect exchange, anisotropic $g$ factors and background diamagnetism. At high field and low temperature, saturation requires the nonlinear response in a; a field-independent Curie value then describes only the zero-field slope.

An $f$ shell has orbital quantum number $l=3$ and seven orbital states with $m_l=-3,-2,\dots ,3$. The first of Hund's rules maximizes total spin subject to the Pauli exclusion principle. Seven electrons occupy the seven orbitals singly with parallel spins, givingTheir orbital projections sum to zero, so the half-filled shell also has total orbital angular momentum $L=0$, total $J=7/2$, and Landé g-factor $g_J=2$. These further values identify the magnetic moment used below.

The Curie constant measures the squared magnetic moment averaged over all orientations of the $J$ multiplet, givingFor the usual positive Landé g-factor of the ion here, the absolute-value sign can be omitted. This effective paramagnetic moment differs from the maximum projected or saturation magnetic moment, ${\mu }_{\mathrm{sat}}=|\tilde{g}|J{\mu }_B$.