In 1966, Schieber, Lin, and Van Vleck analyzed the magnetic behavior of thulium iron garnet (Tm3_3Fe5_5O12_{12}, TmIG) and the isostructural thulium gallium garnet using a cubic crystal-field approximation for the Tm3+^{3+} ion plus a molecular-field exchange interaction, and were left with three unresolved questions of their own making: whether the approximately 88 K (about 61 cm−1^{-1}) optical absorption of TmIG is an electronic crystal-field level of Tm3+^{3+} or a vibrational transition; whether their self-described 'crude approximate' cubic-field treatment could be refined into a quantitative description; and why the fit improves when the exchange field at the Tm3+^{3+} site is taken proportional only to the tetrahedral iron sublattice magnetization (exchange energy about 18.7 K versus the roughly 19 K of independent estimates). Later work established that the garnet rare-earth site has orthorhombic D2D_2, not cubic, symmetry and that low-lying electronic levels of Tm3+^{3+} in garnets do exist, removing the original motivation for doubting them, but no work in the literature assigns the 88 K TmIG line, and a 2021 Physical Review Letters study of THz spin dynamics in TmIG leaves the attribution of a separate low-temperature absorption near 725 GHz (about 24 cm−1^{-1}) to Tm3+^{3+} crystal-field transitions unclear. The problem asks for a definitive origin assignment of the approximately 88 K (about 61 cm−1^{-1}) absorption of Tm3_3Fe5_5O12_{12} and for a quantitative microscopic description of the Tm3+^{3+} ion in TmIG under the correct D2D_2 site symmetry with sublattice-resolved exchange couplings, tested against the 88 K line and the macroscopic magnetic data.

Background

The rare-earth iron garnets R3_3Fe5_5O12_{12} are cubic ferrimagnetic insulators: iron occupies octahedral (a) and tetrahedral (d) sites whose moments couple antiparallel, and the trivalent rare-earth ion R3+^{3+} occupies dodecahedral (c) sites. The magnetic ion of interest here is Tm3+^{3+}, a 4f12^{12} 'non-Kramers' ion, meaning an ion with an even number of electrons and integer angular momentum, for which time-reversal symmetry enforces no doublet degeneracy, so the crystalline electric field of the surrounding oxygens can produce a nonmagnetic singlet ground state. In the free ion the 4f12^{12} shell has a 3H6^3H_6 ground multiplet with total angular momentum J=6J=6 (thirteen states). When the crystal field (the electrostatic field of the surrounding lattice, which splits a free-ion J multiplet into levels) leaves a singlet ground state with low-lying excited levels, the susceptibility arises mainly from field-induced mixing of those excited states; this nearly temperature-independent 'Van Vleck paramagnetism' is the regime in which thulium garnets were understood to sit in the 1960s.

Schieber, Lin, and Van Vleck (1966) fitted the measured susceptibility and magnetization of TmIG and of thulium gallium garnet (Tm3_3Ga5_5O12_{12}, a Van Vleck paramagnet with the same rare-earth sublattice but nonmagnetic Ga) with a cubic crystal-field approximation for Tm3+^{3+} plus an exchange field standing for the Fe-Tm interaction. Their fit was consistent only if the exchange field scaled with the magnetization of the tetrahedral iron sublattice alone, with a zero-temperature exchange energy of about 18.7 K, close to the roughly 19 K inferred independently by Caspari, Koicki, Koicki, and Wood (1964). They were troubled by an approximately 88 K (about 61 cm−1^{-1}) absorption reported for TmIG in the far-infrared spectra of Sievers and Tinkham (1963): their cubic-level scheme did not comfortably accommodate such a low electronic level, so they suggested it may be a vibrational transition, while conceding the evidence was not definitive and that the cubic theory was 'crude approximate'. Van Vleck himself published a companion note the same year on crystal-field parameters of rare-earth garnets, underlining how unsettled the parameters were.

The premise of the cubic approximation was corrected within a few years. The garnet c-site is only approximately cubic: its true point symmetry is D2D_2 (orthorhombic), and Pearson, Herrmann, and Buchanan (1968) showed that cubic models for Tm3+^{3+} in gallium and aluminium garnets predict no excited crystal-field levels below roughly 104 cm−1^{-1}, whereas measured spectra show levels well below that; a low-lying cubic triplet (in Koster notation a Γ5\Gamma_5 triplet) split by the noncubic D2D_2 field drastically changes the susceptibility, so 'cubic models are insufficient' for ions with near-degenerate cubic levels. Buchanan, Pearson, and Herrmann (1969) derived the D2D_2 selection rules for optical transitions of non-Kramers rare-earth ions in garnets and showed that earlier assignments needed reinterpretation. Jones (1968) measured the anisotropy of the 169^{169}Tm nuclear magnetic resonance in thulium gallium garnet, directly proving an orthorhombic local susceptibility, and Hodges and Jehanno (1980) extracted D2D_2 crystal-field parameter sets for the aluminium and gallium thulium garnets from 169^{169}Tm Mössbauer quadrupole splittings (Mössbauer spectroscopy is resonant nuclear gamma-ray absorption; the quadrupole splitting of the nuclear line measures the electric-field gradient the crystal field produces at the Tm nucleus), finding three of the nine D2D_2 parameters undetermined and discrepancies of order ten percent at low temperature. The D2D_2 crystal-field Hamiltonian has nine real parameters (BkqB_k^q with k=2,4,6k=2,4,6 and symmetry-allowed q=0,2,4,6q=0,2,4,6), which is precisely why data of several kinds must be combined.

On the exchange side, Wickersheim and Buchanan (1967) used Ce3+^{3+} as a spectroscopic probe in substituted garnets and found significant exchange contributions from both nearest (axial) and next-nearest (nonaxial) tetrahedral iron neighbors and none from octahedral iron neighbors, corroborating the tetrahedral scaling suggested by the 1966 fit, though in a probe system rather than TmIG itself. The low-temperature specific heat of TmIG (Onn, Remeika, Meyer, and Henderson, 1967) is consistent with the singlet-ground-state picture, and Streever and Caplan (1970) observed the 169^{169}Tm nuclear magnetic resonance in TmIG powder at 4.2 K, finding a strongly anisotropic hyperfine tensor and calling for single-crystal, field-dependent measurements.

TmIG has returned to prominence through THz magnonics and spintronics of magnetic insulators, where its low-energy spin dynamics are exploited in films. In that modern context, Blank, Grishunin, Mashkovich, Logunov, Zvezdin, and Kimel (2021) studied field-induced THz spin dynamics in ferrimagnetic iron garnets, identifying a collective exchange mode of the two sublattices (the Kaplan-Kittel mode, the uniform ferrimagnetic resonance in which sublattice magnetizations precess against each other) near 375 GHz in TmIG, and also observed a low-temperature absorption near 725 GHz whose attribution to Tm3+^{3+} crystal-field transitions they explicitly left open, citing the 1963 far-infrared spectra. The single-ion level assignment question of the 1966 paper is thus still live wherever TmIG low-energy absorption is interpreted.

Problem Statement

For the Tm3+^{3+} ion in thulium iron garnet Tm3_3Fe5_5O12_{12}, resolve the questions left open by Schieber, Lin, and Van Vleck (1966), restated under the site symmetry now known to be correct; the three open items of the source map onto the two linked parts below, with the question of the tetrahedral-only exchange scaling carried inside (ii). (i) Assign definitively the origin of the approximately 61 cm−1^{-1} (88 K) absorption of Tm3_3Fe5_5O12_{12} first reported in the far-infrared spectra of Sievers and Tinkham (1963): is it an electronic crystal-field transition of Tm3+^{3+} (shifted and mixed by the noncubic D2D_2 field and by the Fe-Tm exchange) or a vibrational transition (garnet lattice mode or local mode)? A definitive origin assignment must rule out the competing interpretation. (ii) Construct a quantitative microscopic description of the Tm3+^{3+} ion in Tm3_3Fe5_5O12_{12}: a quantitative D2D_2-symmetry crystal-field level scheme of the 3H6^3H_6 ground multiplet constrained by the available data, together with sublattice-resolved exchange couplings of Tm3+^{3+} to the octahedral (a) and tetrahedral (d) iron sublattices, replacing the 1966 'crude approximate' cubic-field treatment and its single phenomenological exchange field. The scheme and couplings must be consistent with the macroscopic magnetic data of the kind the source fitted (magnetic susceptibility and its anisotropy, high-field magnetization), must explain why the fit improves when the exchange field at Tm3+^{3+} is taken proportional only to the tetrahedral (d) iron sublattice magnetization (about 18.7 K at zero temperature versus the roughly 19 K of Caspari et al., 1964), and must settle the assignment of (i): they show either that an allowed electronic transition falls at approximately 61 cm−1^{-1} with the observed strength (electronic branch) or that no such transition exists (vibrational branch). The two parts share one objective, a microscopic account of the low-energy states of Tm3+^{3+} in TmIG: the level scheme and exchange parameters of (ii) are exactly what determine whether an electronic origin in (i) is quantitatively consistent, so the approximately 61 cm−1^{-1} line is the joint test of both.

References

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