Microscopic origin, sign, and magnitude of the small exchange interaction in the antiferromagnetic ordering of the chrome alums

The chrome alums — hydrated chromium(III) double sulfates such as methylammonium chromium alum, CH$_3$NH$_3$Cr(SO$_4$)$_2$·12H$_2$O — order antiferromagnetically at about 0.015–0.020 K, and the ordering arises predominantly from magnetic dipole–dipole interactions between Cr$^{3+}$ ions that occupy a face-centered-cubic lattice while their trigonal crystalline-field axes are staggered along the four body diagonals of the cube. The molecular-field calculation of O'Brien (1956), summarized briefly in Van Vleck's 1957 report, reproduces the Néel temperature from pure dipolar coupling to within experimental error, but the measured isentropic magnetization curves and the entropy dependence of the zero-field susceptibility are recovered only when a small exchange interaction is added to the dipolar model; O'Brien's fit gives $v=-0.13$, the nearest-neighbor exchange in units of the nearest-neighbor dipolar coupling. The microscopic origin, sign, and precise magnitude of this exchange have never been determined: no direct measurement (for example, of pair spectra in magnetically diluted crystals) and no microscopic superexchange calculation for the chrome alums exists, and as late as 1972 the exchange was still characterized in the primary literature as small but not measured directly, its precise value open to debate. The problem is to determine this exchange interaction from evidence independent of the low-temperature observables it is invoked to explain, rather than by fitting, and to check that a dipolar-plus-exchange model with that value is consistent with the low-temperature thermodynamics of the chrome alums — the Néel temperature, the entropy–susceptibility relation, and the isentropic moment–field curves.

Background

The chrome alums are hydrated double sulfates of formula $M\mathrm{Cr}(\mathrm{SO}_4)_2\cdot12\mathrm{H}_2\mathrm{O}$, with $M$ = K, Rb, Cs, NH$_4$, or CH$_3$NH$_3$; the only paramagnetic species is the Cr$^{3+}$ ion ($3d^3$, spin $S=3/2$), and the magnetic ions are widely separated by the water molecules and sulfate groups of the alum framework. This dilution made the alums the classical working substances of adiabatic demagnetization — the refrigeration method in which a paramagnetic salt is magnetized in a field, thermally isolated, and then demagnetized, so that the spin entropy is absorbed at the expense of temperature and the millikelvin range is reached — and their low-temperature thermodynamics was measured in detail by the Leiden, Oxford, and Washington (NBS) groups (Daniels and Kurti; Gardner and Kurti; Hudson and McLane; Ambler and Hudson; Beun, Steenland, de Klerk, and Gorter).

In the alum structure the Cr$^{3+}$ ions lie on a face-centered-cubic lattice (an ion at each cube corner and face center), but the site symmetry is trigonal, the trigonal axis — the local crystalline-field axis set by the electrostatic environment of the surrounding ligands — pointing along one of the four body diagonals of the cube. Viewing the fcc lattice as four interpenetrating simple-cubic sublattices, each sublattice carries one of the four possible axis directions. Bleaney's paramagnetic-resonance spectra of the chromic sulfate alums (1950) showed that the crystals distort below about 90 K so that the chromium sites fall into several inequivalent environments; this does not destroy the four-sublattice picture. The $3d^3$ ion has its orbital angular momentum essentially quenched — suppressed by the crystalline electric field, so that the magnetic moment comes almost entirely from the spin — and its spin quartet is split by the trigonal field into two Kramers doublets (pairs of states that remain degenerate for an odd number of electrons in zero magnetic field), separated by roughly 0.25 K in methylammonium chromium alum. At the temperatures of interest only the lower doublet is populated, so each ion behaves as an effective two-state, Ising-like moment oriented along its own local trigonal axis.

The interactions between the ions are of two kinds. First, magnetic dipole–dipole interactions, whose energy between two moments separated by $\mathbf{r}$ varies as $(1-3\cos^2\theta)/r^3$; their characteristic temperature — the scale on which they compete with $kT$ — is only of order a hundredth of a kelvin in these salts, which is why ordering occurs near 0.01 K. The statistical mechanics of dipolar coupling in paramagnetic salts was developed by Van Vleck (1937) and applied to magnetic cooling by Hebb and Purcell (1937). In a cubic lattice of magnetically equivalent ions, the directional average of the dipolar sum vanishes, and it was long taken for granted that dipolar forces alone cannot generate a Curie or Néel point in a cubic compound (the Néel point being the critical temperature below which the antiferromagnetically ordered state sets in). Second, exchange interactions — the quantum-mechanical coupling of two spin moments arising from the overlap of their electronic wavefunctions, of which superexchange is the indirect variant transmitted through the orbitals of intervening ligand atoms — which dominate most magnetic salts but are expected to be extremely weak here, because the Cr$^{3+}$ ions are separated by many intervening water molecules and sulfate groups.

Daniels (with Kurti) had computed the dipolar sums for potassium chromium alum, and O'Brien extended this work. Her key result, published in full in O'Brien (1956) and summarized in Van Vleck's 1957 report, is that the staggering of the local crystalline-field axes defeats the cubic cancellation: within the molecular-field approximation (each ion feels the average moment of each of the other sublattices as an effective internal field), an antiferromagnetic transition appears even with zero exchange, at $T_N = 1.7\tau(\sigma+v)$, where $\tau$ is the characteristic dipolar temperature, $\sigma=0.78$ is a dimensionless dipolar lattice sum over the sublattices, and $v$ is the isotropic nearest-neighbor exchange measured in units of the nearest-neighbor dipolar coupling. For methylammonium chromium alum the pure-dipolar prediction ($v=0$) is $T_N\approx0.024$ K against observed values 0.015–0.020 K — agreement, in Van Vleck's words, practically within the limits of experimental error.

The comparison with experiment, however, required a small exchange. O'Brien computed the entropy, the susceptibility, and the isentropic magnetization curves — the curves of magnetic moment versus applied field at constant spin entropy, which are the bookkeeping curves of adiabatic demagnetization — and found that the measured zero-field susceptibility as a function of entropy below $T_N$ falls between her theoretical curves for $v=0$ and $v=-0.19$, at about $v=-0.13$; the same small exchange brings the isentropic moment–field curves into quite good agreement with experiment, and $T_N=\tau$ would require $v=-0.19$. In this convention, negative $v$ lowers the transition temperature below the pure-dipolar value. O'Brien noted that her conclusion — the exchange is small compared with the dipolar interaction — agreed with the contemporaneous electron-paramagnetic-resonance experiments on nearest-neighbor interactions in diluted chromium methylamine alum that she cited from the 1955 international low-temperature conference (these established only that the interaction is small and did not measure its value), and Baker's 1956 paramagnetic-resonance study of chromic methylamine alum supplied the crystalline-field parameters of the four inequivalent sites used by her and later calculations. But the value $v\approx-0.13$ is a fitted phenomenological parameter: no direct measurement of the Cr–Cr coupling, and no microscopic evaluation of the superexchange paths through the alum framework, was or has been available.

The question has stayed open at exactly this point. Van Vleck's 1959 review still characterized the Néel point as agreeing with the pure dipolar calculation within the roughly 25% experimental error. Meijer (1972), in the most refined theoretical treatment — a high-temperature expansion of the susceptibility of methylammonium chromium alum with the crystal field treated without approximation, using Baker's parameters and the tetragonal distortion — stated that the exchange is small and has not been measured directly, that the precise value inferred indirectly is open to debate, and that it would be far more satisfactory to determine $v$ from the deviations from the Curie law or from the ideal heat capacity above $T_N$; he also found that the four crystalline-field azimuthal angles are not equal, which modifies O'Brien's first-order results, confirming that the predictions are sensitive to the assumed field orientations. On the applied side, chromium alums remain refrigerants in adiabatic demagnetization refrigerators, and current modeling of chrome-caesium alum from single-ion terms alone (Gunderson et al. 2022) still fails to account quantitatively for the reduction of cooling capacity below about 100 mK — the engineering descendant of the same unresolved interaction question.

Problem Statement

Determine, by evidence independent of the low-temperature observables the exchange is invoked to explain rather than by fitting, the exchange interaction between nearest-neighbor Cr$^{3+}$ ions in the chrome alums — its microscopic origin as well as its sign and magnitude. The independently determined value must then be shown to be consistent with the measured low-temperature thermodynamics of the salt that established the need for a small exchange in the first place — the Néel temperature (0.015–0.020 K for methylammonium chromium alum; the same framework applies to the potassium, rubidium, and caesium analogues), the zero-field susceptibility as a function of entropy below the transition, and the isentropic (constant-entropy) magnetization curves of moment versus applied field. The fitted $v\approx-0.13$ is the earlier phenomenological number that an independent determination is to supersede, and the exchange interaction may not remain a parameter fitted to the same observables it is invoked to explain. This is the open core left by Van Vleck's 1957 report of O'Brien's calculation, which identified the need for 'a small amount of exchange' without determining its source or precise value.

References

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