Physical origin of the reported azimuthal variation of the Verdet constant within the basal plane of tysonite at low temperatures

In a 1934 Letter, J. H. Van Vleck proved from the invariance of the quantum-mechanical partition function that the dielectric and permeability tensors of a crystal are linear and symmetric in weak applied fields, and deduced that in a hexagonal crystal the initial susceptibility and the Faraday rotation cannot depend on the direction chosen within the plane perpendicular to the optic axis. Yet Jean Becquerel's low-temperature magneto-optic measurements at Leiden on tysonite, a natural mixed rare-earth fluoride $(\mathrm{La},\mathrm{Ce},\mathrm{Nd},\mathrm{Pr})\mathrm{F}_3$ with uniaxial optics whose Faraday rotation is dominated by the Ce$^{3+}$ ion, reported exactly such an azimuthal dependence of the Verdet constant. Van Vleck could only attribute it to unspecified 'crystalline imperfections, or some other deviation from perfect hexagonality'; the companion analysis by Van Vleck and Hebb showed that the two azimuths studied (about 30 degrees apart) gave weak-field effective magneton numbers differing by roughly ten percent (0.523 versus 0.469, i.e. roughly twenty percent in the Verdet constants themselves) at liquid-helium temperatures, that any intrinsic azimuthal dependence of a perfect crystal appears only at fourth order in field in the saturation regime (negligibly small at the fields used), and that the discrepancy vanished at liquid-hydrogen temperatures and above. In the nine decades since, work on tysonite crystallography (the refined trigonal $P\bar{3}c1$ structure with staggered low-symmetry rare-earth sites), on the Ce$^{3+}$ crystalline electric field, and on modern room-temperature magneto-optics of synthetic $\mathrm{CeF}_3$ has neither reproduced nor refuted the azimuthal anomaly nor identified the responsible imperfection. The problem is to identify the specific physical origin of this symmetry-forbidden azimuthal dependence of the weak-field Verdet constant, or to demonstrate its absence on well-characterized tysonite-type crystals, establishing the historical report as an artifact of the original samples or method.

Background

The Faraday effect is the rotation of the plane of polarization of linearly polarized light that propagates through a medium parallel to an applied magnetic field; in the weak-field regime the rotation angle is $\theta = V H L$, where $L$ is the optical path length, $H$ the applied field, and $V$ the Verdet constant, a material property that in paramagnetic substances is dominated by the field-induced splitting and unequal thermal population of the Zeeman sublevels of the magnetic ion (the 'paramagnetic' rotation, quantitatively tied to the magnetic susceptibility $\chi$ by the Van Vleck-Hebb relation $V = A\chi + E$ with $A$ temperature-independent). In crystals, both the static response ($D_i = \sum_j \epsilon_{ij} E_j$ with $\epsilon$ the dielectric tensor) and its magnetic analogue are governed by second-rank tensors. For a crystal with a threefold or higher symmetry axis, symmetry requires the two principal values perpendicular to that axis, the 'optic axis' of such a uniaxial crystal, to be equal, so any property described by such a tensor is isotropic within the basal plane: it cannot depend on the azimuthal direction chosen in that plane. In a 1934 Letter, J. H. Van Vleck gave a quantum-mechanical proof that the dielectric and permeability tensors are indeed linear and symmetric in weak fields even in crystals of arbitrary symmetry. The nontrivial point is that the Stark and Zeeman energy levels of a magnetic ion in a crystalline electric field of low symmetry depend on the field components in complicated, generally irrational ways, so the simple linear form of the averaged response is not obvious; his proof uses the invariance of the partition function $Z = \mathrm{Spur}\, e^{-\mathcal{H}/kT}$ (a matrix trace, with $\mathcal{H} = \mathcal{H}_0 + \boldsymbol{\mu}\cdot\mathbf{E}$) under change of representation, which forces $Z$ into a simple power form in the field. As an immediate corollary he stated that in hexagonal crystals 'different initial susceptibilities or Faraday rotations in different directions perpendicular to the optic axis' are impossible, even including Heisenberg exchange and interatomic valence forces, provided saturation (nonlinear response in field) is neglected. Against this stands the experiment: tysonite is a natural mixed-crystal rare-earth fluoride $(\mathrm{La},\mathrm{Ce},\mathrm{Nd},\mathrm{Pr})\mathrm{F}_3$, optically uniaxial, whose Faraday rotation is dominated by the Ce$^{3+}$ ion (a $4f^1$ Kramers doublet ground state, where a Kramers doublet is a pair of levels degenerate by time-reversal symmetry in zero field, necessarily split linearly by a magnetic field). Jean Becquerel's Leiden measurements on this crystal (published as Leiden Comm. 211c and in the Becquerel-de Haas Leiden series of 1929-30; the later Physica 1, 383 (1934) paper reports the component along the optic axis, not these perpendicular ones) reported exactly what symmetry forbids: for light propagating within the plane perpendicular to the optic axis, the measured Verdet constant depended on the azimuth chosen even in weak fields, by roughly twenty percent between two azimuths, one parallel to a diagonal axis of the crystal and the other about thirty degrees from it (the corresponding weak-field effective magneton numbers, 0.523 and 0.469, differing by about ten percent), with the effect largest at liquid-helium temperatures (down to $1.5$-$1.7$ K) and absent at liquid-hydrogen temperatures and above. Van Vleck's Letter closed by attributing this to 'crystalline imperfections, or some other deviation from perfect hexagonality', without identifying them. The companion analysis by Van Vleck and Hebb quantified the conflict, showed that the low-symmetry local crystalline electric fields at the rare-earth sites (modeled as staggered rhombic fields, i.e. local fields with two nonequal transverse components, oriented at $120^\circ$ from site to site) generate azimuthal dependence only at fourth order in field within their saturation theory (far too small at the fields used), fitted the temperature dependence of both principal directions, and speculated that the discrepancy arose either from crystalline flaws or from Becquerel having used two different crystals. Two later developments are directly relevant and must be taken into account by any resolution. First, the tysonite structure was refined (Lowndes, Parrish and Perry, 1969) to the hexamolecular trigonal space group $P\bar{3}c1$ (point group $D_{3d}$, magnetic space group $P3'c'1$), with rare-earth ions at local sites of low ($C_2$) symmetry in staggered orientations, confirming the qualitative local-field picture used in 1934; crucially, a threefold axis is just as sufficient as a sixfold one to enforce basal-plane isotropy of second-rank response tensors, so the refined structure still forbids the reported anisotropy in a perfect crystal. Second, accurate measurement of Faraday rotation along birefringent (non-optic-axis) directions of uniaxial crystals became possible only recently with the generalized high-accuracy universal polarimeter (G-HAUP) of Nakagawa and Asahi, who measured the Verdet constant along the $a$ and $c$ axes of synthetic $\mathrm{CeF}_3$ at room temperature; their work documents why such measurements 'have been almost ignored': in birefringent directions the linear birefringence exceeds the magnetic circular birefringence by two to four orders of magnitude, and the naive rotation readout is an 'apparent rotation' entangled with the birefringence, a pitfall discussed in the older literature that Nakagawa and Asahi review (Chauvin, Ramaseshan and Tabor). This interplay between Faraday rotation and linear birefringence in imperfect crystals is precisely the arena in which the historical anomaly must be resolved.

Problem Statement

Identify the specific physical origin of the reported azimuthal dependence of the weak-field Verdet constant within the basal plane of tysonite (the natural mixed rare-earth fluoride $(\mathrm{La},\mathrm{Ce},\mathrm{Nd},\mathrm{Pr})\mathrm{F}_3$): for light propagating perpendicular to the optic axis, the two azimuths measured at Leiden (one parallel to a diagonal axis of the crystal, the other about thirty degrees from it) gave weak-field effective magneton numbers 0.523 and 0.469, about ten percent apart, i.e. roughly twenty percent in the Verdet constants themselves, with the discrepancy largest at liquid-helium temperatures (the data extending down to about $1.5$-$1.7$ K) and absent at liquid-hydrogen temperatures and above. Any intrinsic azimuthal anisotropy of the weak-field Verdet constant is forbidden by the crystal symmetry (threefold or higher axis), including in the refined trigonal $P\bar{3}c1$ structure, and local low-symmetry crystalline fields generate azimuthal effects only at fourth order in field in the saturation regime, negligibly small at the fields used. First, determine which imperfection, deviation from ideal symmetry, or measurement mechanism produced the historical observation. The identified mechanism must account, within a single consistent quantitative account, for both the observed magnitudes at the two azimuths (weak-field effective magneton numbers 0.523 and 0.469; saturation-method values 0.874 and 0.726) and the temperature dependence (largest at liquid-helium temperatures, absent at liquid-hydrogen temperatures and above). Alternatively, establish, at low temperatures and with a sensitivity sufficient to rule out an effect of the historical magnitude (roughly twenty percent in Verdet constant between azimuths about thirty degrees apart), that no such azimuthal dependence exists on well-characterized tysonite-type crystals, demonstrating the historical report to be an artifact of the original samples or method.

References

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10. D. Vojna, R. Yasuhara, O. Slezák, J. Mužík, A. Lucianetti, and T. Mocek, 'Verdet constant dispersion of CeF3 in the visible and near-infrared spectral range', Optical Engineering 56(6), 067105 (2017). DOI: 10.1117/1.OE.56.6.067105, https://doi.org/10.1117/1.OE.56.6.067105
11. K. Nakagawa and T. Asahi, 'Determination of the Faraday rotation perpendicular to the optical axis in uniaxial CeF3 crystal by using the Generalized-High Accuracy Universal Polarimeter', Scientific Reports 9, 18453 (2019). DOI: 10.1038/s41598-019-54174-2, https://doi.org/10.1038/s41598-019-54174-2
12. J. Becquerel, Communications from the Kamerlingh Onnes Laboratory, Leiden, No. 211c (low-temperature tysonite Faraday-rotation measurements in the direction perpendicular to the optic axis).
13. J. Becquerel and W. J. de Haas, Proc. Amsterdam Acad. 32, 523 (1929); ibid. 32, 536, 578, 590, 733, 874, 913, 926, 937 (1929-30); Leiden Communications Nos. 191, 193, 199, 211 (Faraday-rotation measurements on tysonite).