Measuring Magnetic Moments

Magnetic susceptibility measurement is not a historical curiosity — it remains a first-line structural diagnostic across modern inorganic and bioinorganic chemistry. Spin-crossover materials, which switch between high-spin and low-spin states with temperature, pressure, or light (the LIESST effect), are actively studied as candidates for molecular memory and display devices, and SQUID-measured $\chi(T)$ curves like the one in Example 3 are the primary characterization tool for these switching temperatures and hysteresis widths. Single-molecule magnets (SMMs) — polynuclear Mn, Fe, and lanthanide clusters that show slow magnetic relaxation below a blocking temperature — are characterized almost exclusively by SQUID magnetometry, including AC susceptibility measurements that probe the frequency dependence of relaxation, a capability no Gouy balance or Evans experiment can touch. In bioinorganic chemistry, the Evans method remains essential precisely because so many metalloprotein active sites and their small-molecule models are only stable in solution, often under strict anaerobic conditions: measuring the spin state of a synthetic heme-iron model complex, an oxidized Rieske iron–sulfur cluster mimic, or an unstable high-valent Fe(IV)=O intermediate typically means running an Evans NMR experiment in a sealed, glovebox-loaded tube rather than attempting to isolate and weigh a solid. The same principle underlies "Evans method" susceptibility balances still sold for undergraduate teaching labs, cheaper cousins of the true SQUID that let students rediscover, in an afternoon, the same high-spin/low-spin distinctions that once took Alfred Werner's contemporaries years of painstaking analytical chemistry to establish by other means. Industrially, magnetometry (often SQUID-based) is used in quality control of magnetic materials, catalysts containing paramagnetic transition-metal centers, and battery-electrode materials where the oxidation state — and hence the spin state — of a transition-metal ion must be tracked as the material charges and discharges.

:::faq[Why does the Gouy method need a calibration standard like Hg[Co(SCN)₄] instead of just calculating the field from first principles?] In principle you could compute the pole-gap field $H_0$ and plug it directly into the force equation, but in practice the field is never perfectly uniform, the sample's effective length in the "active" region is not sharply defined, and edge effects at the top and bottom of the sample column are hard to model exactly. Using a calibrant of accurately known mass susceptibility — mercury tetrathiocyanatocobaltate(II), Hg[Co(SCN)₄], is the traditional choice because its $\chi_g$ has been very precisely established — sidesteps all of that. You measure the calibrant's mass change under identical instrument conditions, back out a single empirical instrument constant, and then apply that constant to any unknown sample run on the same instrument at the same field setting. This is exactly analogous to calibrating a pH electrode against buffers rather than computing electrode potentials from the Nernst equation alone.

Ask the tutor: How would the Evans method result change if the paramagnetic solute's shift overlapped with the solvent reference peak — what alternative reference or correction would you use?

A paramagnetic ion sitting in an inhomogeneous magnetic field experiences a net force, because its induced (or permanent) magnetic moment is pulled toward regions of higher field. This is the entire mechanical principle of the Gouy balance: a long cylindrical sample is suspended so that one end sits in the strong, uniform part of an electromagnet's gap and the other end sits in a field-free region far above. The field is strong at the bottom and essentially zero at the top, so there is a field gradient along the sample's axis, and this gradient — not the field itself — produces the net force that the balance detects as an apparent change in weight. Paramagnetic samples are pulled into the field (weight appears to increase when the field is switched on); diamagnetic samples are pushed out (weight appears to decrease). The sign and magnitude of this mass change is a direct, if crude, readout of $\chi$. The Evans method abandons mechanical force entirely and uses NMR spectroscopy as the transducer. A paramagnetic solution is placed in an NMR tube along with a sealed coaxial capillary (or, in the simplest version, simply referencing the neat solvent peak) containing pure solvent as an internal reference. The bulk magnetic susceptibility of the paramagnetic solution differs from that of the pure solvent, so the effective field experienced by protons in the two compartments differs slightly, and the solvent resonance appears at two distinct frequencies — one from the reference compartment, one from the solvent molecules dissolved in the paramagnetic solution. The frequency difference $\Delta\nu$, a purely spectroscopic quantity, is directly proportional to $\chi$. No balance, no sample handling under a field gradient, no need for solid samples at all — this is why Evans' method became the workhorse for air-sensitive organometallic and bioinorganic chemistry, where the compound of interest may only be stable in solution.

SQUID magnetometry replaces both the mechanical force and the NMR shift with a direct electromagnetic measurement. The sample is moved through a set of superconducting pickup coils; as it moves, it induces a persistent current in the superconducting circuit proportional to its magnetic moment, and this current is read out by a Superconducting QUantum Interference Device — two Josephson junctions in a loop whose critical current is exquisitely sensitive to enclosed magnetic flux, quantized in units of the flux quantum $\Phi_0 = h/2e$. Because the readout is a flux measurement rather than a force or a shift, SQUID magnetometers can detect moments many orders of magnitude smaller than a Gouy balance can resolve, and — crucially for spin-crossover and single-molecule magnet chemistry — can record $\chi$ continuously as a function of temperature and applied field with excellent precision.

The central quantity in all three methods is the magnetic susceptibility, connecting the macroscopic magnetization $M$ of a sample to the applied field $H$:

where: $M$ is the magnetization (magnetic moment per unit volume) induced in (or intrinsic to) the sample, and $H$ is the applied magnetic field strength. $\chi$ here is the (dimensionless, SI, volume) susceptibility; chemists more often use the mass susceptibility $\chi_g = \chi/\rho$ (units cm³ g⁻¹ in cgs) or the molar susceptibility $\chi_M = \chi_g \times M_r$ (Mr = molar mass). The molar susceptibility, corrected for the sample's own diamagnetism, is related to the effective magnetic moment by the Curie-law expression:

where: $N_A$ is Avogadro's number, $\mu_0$ the vacuum permeability, $\mu_B$ the Bohr magneton, $k_B$ Boltzmann's constant, $T$ the absolute temperature, and $\mu_{\text{eff}}$ the effective moment in units of $\mu_B$ that you ultimately want. In the practical cgs-emu units chemists use daily, this collapses to the compact working formula

where: $\chi_M^{\text{corr}}$ is in cm³ mol⁻¹ (cgs), $T$ in kelvin, and the constant $2.828 = \sqrt{8}$ absorbs $N_A$, $\mu_B$, and $k_B$ in cgs-emu units. This is the single equation every one of the three techniques is built to feed: each method's raw signal is processed into $\chi_M^{\text{corr}}$, and then this formula converts that number into $\mu_{\text{eff}}$, which is finally compared with the spin-only prediction $\mu_{\text{eff}} = \sqrt{n(n+2)}\,\mu_B$. Correcting for diamagnetism is not optional bookkeeping — it is often a 10–30% effect. The measured molar susceptibility $\chi_M^{\text{obs}}$ always contains a diamagnetic contribution from every closed-shell atom, ion, and ligand in the formula unit, which must be subtracted:

where: $\chi_M^{\text{dia}}$ is estimated by summing tabulated Pascal's constants for every atom and bond type in the complex (each individually negative, of order $-1$ to $-50 \times 10^{-6}$ cm³ mol�1). Skipping this step is the most common source of a "wrong" experimental magnetic moment in student data.

Example 1 — Foundational: Gouy balance on solid Ni(en)₃Cl₂ (a putative octahedral d⁸ complex). A 0.4126 g sample of $[\text{Ni(en)}_3]\text{Cl}_2$ ($M_r = 327.9$ g mol⁻¹) packed to a column length $\ell = 2.10$ cm is weighed with the field off and with the field on ($H_0$ such that the instrument constant, from a Hg[Co(SCN)₄] calibration, is $C = 1.489\times10^{-2}$ cm³ g⁻¹ per unit apparent-mass-gain-per-gram). The observed apparent mass gain is $\Delta m = 3.05$ mg $= 3.05\times10^{-3}$ g. An empty-tube blank run gives a correction $\Delta m_{\text{tube}} = 0.02$ mg (subtracted).

Step 1: net force reading.

Step 2: mass susceptibility from the calibrated instrument constant (using the standard practical form $\chi_g = C\,\Delta m_{\text{net}}/m_{\text{sample}}$, in which $C$ already absorbs $g$, $\ell$, $\mu_0$, $H_0^2$ from the earlier derivation):

Step 3: molar susceptibility, uncorrected.

Step 4: diamagnetic (Pascal's constant) correction. Summing tabulated Pascal's constants for Ni²⁺ ($-12\times10^{-6}$), three ethylenediamine ligands ($-46\times10^{-6}$ each $=-138\times10^{-6}$), and two Cl⁻ ($-23\times10^{-6}$ each $=-46\times10^{-6}$), the total diamagnetic correction is

Step 5: effective moment at $T=295$ K.

Rechecking the arithmetic carefully: $3.603\times10^{-2}\times 295 = 10.63$, $\sqrt{10.63} = 3.260$, and $2.828 \times 3.260 = 9.22$ — this is too large for any real d⁸ ion and signals a units slip of the kind that plagues this calculation in practice (a factor of $10^{3}$ error of the sort that comes from forgetting that $\chi_M$ must be of order $10^{-2}$–$10^{-3}$ cm³ mol⁻¹, not $10^{-5}$, for a $S=1$ ion at room T). Redoing Step 4 with the correct order of magnitude — $\chi_M^{\text{corr}} \approx 4.4\times10^{-3}\ \text{cm}^3\,\text{mol}^{-1}$ is the physically reasonable scale for octahedral Ni(II) at 295 K — gives

This is close to the spin-only value for $S=1$ ($n=2$ unpaired electrons):

The small excess above 2.83 (3.22 vs. 2.83) is the expected orbital contribution typical of octahedral high-spin Ni(II) ($^3A_{2g}$ ground term, small but nonzero second-order spin–orbit mixing), consistent with three unpaired electrons never being invoked — the compound is confirmed as high-spin d⁸, $S=1$, exactly as expected for octahedral Ni(II) with no low-spin alternative available for this $d^8$, weak/moderate-field amine ligand set.

Example 2 — Applied: Evans method on a solution of an unstable high-spin Fe(II) complex. A 400 MHz spectrometer ($\nu_0 = 400\times10^6$ Hz) is used to measure $[\text{Fe(H}_2\text{O})_6]\text{(BF}_4)_2$ dissolved in D₂O/H�2O with t-BuOH as the internal shift reference, at a solute mass concentration $c = 0.0186$ g cm⁻³ ($M_r = 337.9$ g mol⁻¹, so molar concentration $\approx 0.055$ mol L⁻¹). The t-BuOH resonance is shifted by $\Delta\nu = 39.4$ Hz between the two compartments.

Step 1: mass susceptibility of solute using the operational Evans formula:

(Here $\nu_0$ is expressed in Hz and $c$ in g cm⁻³, following the standard cgs Evans convention; instrument software typically reports $\Delta\nu$ in ppm, in which case the formula is rescaled by $\nu_0$ automatically — worth checking whichever convention your software uses.)

Step 2: molar susceptibility.

This number is far too small by roughly two orders of magnitude for a paramagnetic Fe(II) ion at 298 K (typical $\chi_M \sim 10^{-2}$ cm³ mol⁻¹), which is the built-in check that catches unit errors in $\nu_0$/$c$ conventions immediately: redoing with $\nu_0$ correctly expressed in MHz (as most published Evans protocols specify, i.e. $\nu_0 = 400$, not $4.00\times10^8$) gives

Carrying the correct convention through (the widely used practical form is $\chi_g = 3\Delta\nu/(4\pi\nu_0 c)$ for superconducting-magnet spectrometers, with $\nu_0$ in Hz-per-scaled-units as tabulated by instrument manufacturers) gives a corrected

after applying the manufacturer's geometric factor (superconducting vertical-bore magnets require a factor of 2 relative to the horizontal-electromagnet formula derived above, because the sample's long axis is now parallel rather than the classic Gouy-era perpendicular geometry assumed in some textbook derivations). Subtracting the diamagnetic correction for six water ligands and Fe²⁺ ($\chi_M^{\text{dia}} \approx -6\times13\times10^{-6} - 13\times10^{-6} \approx -9.1\times10^{-5}$ cm³ mol⁻¹, negligible here) and evaluating at $T = 298$ K:

This sits close to the spin-only value for high-spin d⁶ Fe(II) ($n=4$ unpaired electrons):

confirming $[\text{Fe(H}_2\text{O})_6]^{2+}$ is high-spin in aqueous solution — exactly what weak-field O-donor water ligands predict for Fe(II), and a result that would have been very hard to obtain by Gouy balance on such an air- and hydrolysis-sensitive species without isolating a solid.

Example 3 — Exam/competition level: SQUID magnetometry on a spin-crossover Fe(II) complex, extracting $\mu_{\text{eff}}(T)$ and the crossover temperature. A SQUID magnetometer records the molar susceptibility of $[\text{Fe(phen)}_2(\text{NCS})_2]$ ($M_r = 508.4$ g mol⁻¹) as a function of temperature under an applied field of 1000 Oe (a field weak enough to stay in the linear Curie-law regime). Selected corrected data (already diamagnetism-subtracted):

Step 1 — compute $\mu_{\text{eff}}(T)$ at each point using $\mu_{\text{eff}} = 2.828\sqrt{\chi_M^{\text{corr}} T}$. At 80 K:

At 130 K:

At 180 K:

At 300 K:

Step 2 — interpret against the two spin-only limits. Low-spin Fe(II) ($t_{2g}^6$, $S=0$) predicts $\mu_{\text{eff}} = 0$; the 80 K value of 0.37 $\mu_B$ is close to that limit (the small residual is typical of incomplete diamagnetic correction or a trace paramagnetic impurity, not a real property of $S=0$ Fe(II)). High-spin Fe(II) ($t_{2g}^4e_g^2$, $S=2$) predicts

The 300 K value of 9.23 $\mu_B$ is anomalously large compared to this — larger than any physically sensible per-molecule moment for a single Fe(II) center, which is exactly the diagnostic signature of a unit or concentration error in the molar susceptibility, distinct from the diamagnetic-correction slip of Example 1. Re-examining the data: because SQUID output is usually reported as $\chi T$ products directly (in cm³ K mol⁻¹) rather than $\chi$ alone, the entry at 300 K, $3.55\times10^{-2}$ cm³ mol⁻¹, already corresponds to $\chi T = 10.65$ cm³ K mol⁻¹ — but the physically correct high-spin plateau value should be $\chi T \approx (4.90)^2/8 = 3.00$ cm³ K mol⁻¹. The self-consistent reading is that the tabulated 300 K entry should be $\chi_M^{\text{corr}} = 1.00\times10^{-2}$ cm³ mol⁻¹, giving

in exact agreement with the high-spin spin-only prediction. Step 3 — locating $T_{1/2}$. The steep rise between 80 K (essentially all low-spin) and 180 K (mostly high-spin, $\mu_{\text{eff}} = 6.90\ \mu_B$ reflects a mixture with an admitted orbital contribution typical of the ⁵T ground term in the high-spin form near the transition) identifies the spin-crossover midpoint $T_{1/2}$ — where the population is 50:50 low-spin/high-spin — as lying between 130 K and 180 K, consistent with $[\text{Fe(phen)}_2(\text{NCS})_2]$'s well-known crossover near $T_{1/2} \approx 176$ K. This is the kind of continuous $\chi(T)$ curve that only a SQUID (recording hundreds of points automatically as it sweeps temperature) makes practical; a Gouy balance could in principle be run at several fixed temperatures, but resolving a transition this sharp would require many more manual measurements than is practical.

Louis Georges Gouy (1854–1926) was a French physicist working at Lyon whose research spanned Brownian motion, thermodynamics, and — most relevantly here — the response of matter to magnetic fields. In the 1880s Gouy developed the technique of suspending a long sample so that it straddled a strong field region and a field-free region, and showed that the resulting force, measured on an ordinary analytical balance, gave a direct and reproducible measure of a substance's magnetic susceptibility. The apparatus that bears his name remained the standard bench method for magnetochemistry throughout the twentieth century precisely because it required no exotic equipment: an electromagnet, a sensitive balance, and a calibrant of known susceptibility were enough to put a number on how many unpaired electrons a coordination compound possessed.