Origin of the Ground Kramers Doublets for Co^{2+}(3d^{7}) Ions with the Effective Spin 3/2 Versus the Fictitious ‘Spin’ ½
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Abstract
Experimental spectroscopic and magnetic data for Co^{2+}(3d^{7}) ions in various systems are reviewed and critically examined. The focus is on Co^{2+} ions with the electronic spin S = 3/2, properties of which may be interpreted using the spin Hamiltonian with the effective S̃ = 3/2 or the fictitious ‘spin’ S (S′) = ½. Possible distinct ground states of Co^{2+}(3d^{7}) ions arising from crystal field energy levels are discussed. Distinctions between the concepts of the effective spin S̃ and the fictitious ‘spin’ S′ are outlined to clarify the terminological confusion encountered in literature. Sample cases of the ground state assignments and options for the ‘spin’ S′ = ½ origin are considered for better understanding of the Co^{2+} ions local environment in various systems, including low symmetry cases. Present study is motivated by potential applications of Co^{2+}(S̃ = 3/2) complexes exhibiting very large or moderate zero-field splitting as molecular nanomagnets.
1 Introduction
The electron magnetic resonance (EMR) spectra [1, 2] of various single crystals doped with Co^{2+}(3d^{7}) ions with the electronic spin S = 3/2, which yields nominally the effective S̃ = 3/2, should be described by the spin Hamiltonian (SH) including the zero-field splitting (ZFS) terms [3]. However, often Co^{2+} EMR spectra are interpreted using SH with the ‘spin’ S (S′) = ½, which includes only the Zeeman and hyperfine structure tensors g and A. In a number of papers, the origin of the observed ground Kramers doublet state, to which the so called ‘spin’ S′ = ½ is ascribed, is not specified.
The preliminary considerations were previewed in a poster presentation at the V Forum EMR-PL [4]. Here, we elucidate key aspects and present sample data. The possible mechanisms leading to such distinct ground Co^{2+}(3d^{7}) states are overviewed in terms of the underlying crystal field (CF) energy level schemes. The crucial concepts of the effective spin S̃ and the fictitious ‘spin’ S′ (that in fact does not represent the true spin) are clarified. The distinction between these concepts helps in clarifying the terminological confusion encountered in literature. Specific cases of assignments of the ground states and options for the origin of the ‘spin’ S′ = ½ are considered to obtain better understanding of the local environment around the Co^{2+} ions in various host crystals, including low-symmetry cases.
Additional motivation for the wider study of Co^{2+} (3d^{7}) ions in crystals arises from realization of potential applications of Co^{2+} complexes, which exhibit a variety of behavior associated with either the effective spin S̃ = 3/2 or the fictitious ‘spin’ S′ = ½. This offers potential applications ranging from the Co^{2+} compounds, which may be suitable as high-pressure probes for high-magnetic field and high-frequency EMR (HFM–EMR) measurements [5, 6, 7], to Co^{2+}-based molecular nanomagnets exhibiting very large or moderate ZFS, including single-molecule magnets (SMM) and single-ion magnets (SIM) [8]. As a case study, we investigate: (1) the polarized optical absorption and EMR spectra of Co-doped beryls and chrysoberyl [9], (2) EMR spectra of Co^{2+} (S′ = ½) ions in PbMoO_{4} and YAlO_{3} [10, 11]. The calculations are carried out using the crystal field analysis (CFA) package [12], which enables the complete diagonalization within the whole 3d^{N} configuration for arbitrary symmetry. The lowest energy levels and corresponding wave functions enable identifications of the particular observed ground Kramers doublet states. This mini review provides a primer for experimentalists to properly understand the crucial concepts of the effective spin S̃ and the fictitious ‘spin’ S′ for analysis and interpretation of EMR data. Full results and detailed considerations will be provided in subsequent publication.
2 Pertinent Aspects of Crystal Field (CF) and Spin Hamiltonian (SH) Theory
2.1 Distinction Between the High-Spin (S = 3/2) and the Low-Spin (S = ½) Configurations of Co^{2+}(3d^{7}) Ions
For the pure octahedral (OH), i.e., sixfold, coordination in the ‘weak’ CF the configuration (t_{2g})^{5}(e)^{2} with three unpaired electrons arises yielding the d-electron spins arranged as (↑↓, ↑↓, ↑) (↑, ↑) and the ground state is ^{4}T_{1(g})(^{4}F) with the high-spin (S = 3/2) and threefold orbital degeneracy. In the ‘strong’ OH CF, the configuration (t_{2})^{6}(e)^{1} arises with one unpaired electron (↑↓,↑↓, ↑↓) (↑) yielding the ground term ^{2}E_{(g)}(^{2}G) with the low-spin (S = ½) and twofold orbital degeneracy; see, e.g., the Table 6.1 of [22]. Note an inconsistency for the ‘strong’ OH CF case in the Table 3.1 of [23], where the spin value ‘1’ should be replaced by ‘½’ [22, 23]. As an example, the three- and one unpaired electron form the compounds [(NH_{4})_{2}Co(SO_{4})_{2}.6H_{2}O] and [K_{2}BaCo(NO_{2})_{6}], respectively, are named in [22, p. 128]. The correlation diagrams, see Figs. 6.5 and 6.6 in [22], between the energy levels for various 3d^{N} ions in the ‘weak’ and ‘strong’ CF (both for the octahedral and tetrahedral coordination discussed below) illustrate nicely also the two Co^{2+} configurations.
For the tetrahedral coordination (fourfold) and ‘cubic’ coordination (eightfold) [24], in the ‘weak’ CF the configuration (e)^{4}(t_{2})^{3} with three unpaired electrons arises yielding the d-electron spins arranged as (↑↓, ↑↓) (↑, ↑, ↑), and thus the ground state is ^{4}A_{2(g)}(F) with the high spin (S = 3/2); as an example the compound Cs_{2}CoCl_{4} is named in [22, p. 127]. For the TH coordination in the ‘strong’ CF the arrangement of the d-electron spins is controversial in view of several inconsistencies in the textbooks [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Nominally, by analogy, the configurations (e)^{4}(t_{2})^{3} with one unpaired electron could be expected yielding (↑↓, ↑↓) (↑↓, ↑) and thus, the ground state would be ^{2}E_{(g)} with the low spin (S = ½). However, doubts arise if this configuration is feasible. It appears that in the Table 2.3 of [23] the last arrow (↑) should be omitted and the number of unpaired electrons should be one not three. In the Table 3.2 of [2], the entries for the ‘weak’ and ‘strong’ CF case are identical and correspond to the high-spin case only. In the Table 6.1 of [20], the entry for the ‘strong’ CF case reads as that listed above for the ‘weak’ CF, but in the Table 9.3 of [22], where the ‘strong’ or ‘weak’ CF is not specified, only one case is listed: [(e)^{4}(t_{2})^{3}, ^{4}A_{2}]. In the Table 3 of [19], no low-spin case is listed for 3d^{7} ion in the TH coordination. Note that the Table 2.3 of [23] refers only to the ‘tetrahedral coordination’, whereas the corresponding Table 3.2 of [2] to the ‘4-, 8- and 12-coordination’. The controversy may be solved based on the correlation diagrams between the energy levels for Co^{2+}(3d^{7}) ion in the ‘strong’ CF and TH coordination, see Fig. 6.5 in [22]. In this case, the configuration (e)^{4}(t_{2})^{3} arises yielding the ground state as ^{4}A_{2(g)}(F), i.e., also the high-spin (S = 3/2) configuration, however, with different excited states than for the TH ‘weak’ CF case (see Fig. 1 below), since they arise from the higher term ^{2}G as ^{2}E_{(g)}, ^{2}T_{1(g)} and ^{2}T_{2(g)} [22]. Hence, based on the survey of textbooks and reviews, the existence of the low-spin configuration (↑↓, ↑↓) (↑↓, ↑) for Co^{2+}(3d^{7}) ions in the TH coordination in the ‘strong’ CF may be rather excluded.
The inversion of the CF energy levels for the sixfold coordination as compared with those for the four- and eightfold coordination is implied by the negative sign in the relationships, predicted based on the point charge model, between the octahedral cubic CF parameter, Dq(6), and that for the four- and eightfold coordination: Dq(8) = − (8/9)Dq(6) and Dq(4) = − (4/9)Dq(6), see, e.g., [13, 14, 15, p.468; 16, p. 62; 22, p. 41].
2.2 Distinction Between the Electronic Spin, the Effective Spin, and the Fictitious ‘Spins’
The crux of these problems appears to be the general lack of proper distinction between the various meanings of the ‘spin’, which were clearly categorized in the reviews [25, 26] and most recently in [3]. Importantly, the electronic spin S = ½ ascribed to the low-spin Co^{2+}(3d^{7}) configuration has to be distinguished from other ‘spins’ X, i.e., the effective spin \(\tilde{S}\) = ½ and the fictitious ‘spin’ S′ = ½, which may be ascribed to the specific subsets of Co^{2+} ground states with degeneracy, i.e., multiplicity (2X + 1) equal to two. As exemplified and discussed in details in the reviews [3, 25, 26], the meanings of the terms: physical versus effective Hamiltonian as well as electronic versus effective versus fictitious spin, are not well defined and are often confused with each other in the EMR-related literature.
Due to the action of the physical Hamiltonian [1, 2, 3], consisting of the free ion (FI) and CF terms: H_{phys} = H_{FI}+ H_{CF}, the ground state of a transition-metal ion in crystal may be either an orbital singlet (A) or an orbitally degenerate state (E, T), e.g., ^{4}A_{2}(F) and ^{4}T_{1g}(F), respectively, in Fig. 1. The orbital singlet ground state may be shortly denoted \(\left| {\left. \gamma \right\rangle } \right.\, \equiv \,\left\{ {\left| {\left. {\varGamma_{\gamma } } \right\rangle } \right.\,\left| {S, \, \left. {M_{S} \,} \right\rangle } \right.} \right\}\), where \(|\,\left. {\varGamma_{\gamma } } \right\rangle \,\) represents the orbital part, whereas \(\left| {\left. {S, \, M_{S} } \right\rangle } \right.\,\,\)—the spin part. Note that H_{phys} acts within the basis of the physical states of the whole configuration nl^{N}, including the singlet \(\left| {\left. \gamma \right\rangle } \right.\). The transitions between the spin levels within the \(\left| {\left. \gamma \right\rangle } \right.\) states are observed using EMR [1, 2, 3, 13, 14, 15, 16, 17, 18], whereas those between \(\left| {\left. \gamma \right\rangle } \right.\) and the higher lying multiplets (see, Fig. 1) using optical spectroscopy [19, 20, 21, 22, 23]. To describe only the former transitions, without being bothered with the higher lying levels, the concept of the effective spin Hamiltonian H_{eff} has been introduced [1, 2, 3, 13, 14, 15, 16, 17, 18, 25, 26, 27]. Important point is that the effective SH, unlike H_{phys}, acts only within its own basis of states, i.e., \(\left| {\left. {\tilde{S},\,\tilde{M}_{s} } \right\rangle } \right.\) of the effective spin operator S̃. Major advantage of H_{eff} is that it mimics the energy levels of the electronic spin states \(\left| {\left. {S, \, M_{S} } \right\rangle } \right.\,\,\) [3, 25, 26, 27]. To emphasize the different nature of the physical and effective quantities, the tilde (~) is used to distinguish the effective spin operator S̃ and its states from \(\left| {\left. {\tilde{S},\,\tilde{M}_{s} } \right\rangle } \right.\) the (total) electronic spin operator S and its states \(\left| {\left. {S, \, M_{S} } \right\rangle } \right.\,\,\). Since the values of the quantum numbers S and \(\tilde{S}\) are equal for the transition ions with an orbitally non-degenerate ground state, often the two types of related quantities are inappropriately mixed up in the EMR literature. The respective reviews may be consulted for a concise description of the microscopic spin Hamiltonian (MSH) theory, which underlies the original derivation of the effective SH H_{eff}; and the interrelationships between H_{CF} and H_{ZFS} [3, 25, 26, 27] as well as the low symmetry aspects in EMR [28].
2.3 Forms of Spin Hamiltonian Suitable for Spin 3/2 and ½
A note of caution is pertinent concerning terminological confusion between key notions occurring in the literature. The generic A = B confusion denotes the cases pertaining to incorrect referral to the quantities associated with the notion B (e.g., ZFS) by the names of the quantities associated with the notion A (e.g., CF). The quantities may comprise Hamiltonians, eigenfunctions, energy-level splitting, or associated parameters. Such terminological confusion arises due to disregarding the well-accepted meaning of the given notions: A and B. The most common form is the CF = ZFS confusion, whereas the inverse ZFS = CF confusion has crept into literature in recent years [3, 25, 26, 27, 29, 30]. Other types of terminological confusion have been identified [3], including the MA = ZFS confusion between the notion ‘magnetic anisotropy (MA)’ (including related notions: magnetic anisotropy energy (MAE) and single-ion anisotropy (SIA)) and ZFS. This confusion occurs mainly in magnetism literature as outlined in the reviews in [31, 32] and more recently in studies of magnetic adatoms on surfaces [33].
Concerning the relations between the SH in Eq. (1) and (2) and that in Eq. (3), we note that for the effective spin Hamiltonian, the ZFS magnitude does not affect the g-tensor at all. However, for the fictitious spin S′ = 1/2 Hamiltonian, the g’-tensor is indeed related to the effective spin ZFS parameters, see, e.g., [34, 35]. On the other hand, the symmetry of the g- and g’-tensors as well as the ZFS tensor is determined by symmetry of the system and their explicit forms depend on the selected axes in which these tensors are expressed.
3 The High-Spin (HS) Co^{2+} (S = 3/2) Cases
Structural and magnetic properties of transition-metal complexes of pyridine N-oxide, including cobalt(II) ions in octahedral surroundings have been reviewed by Carlin and De Jongh [36]. This review provides an excellent background for the crystal field effects and paramagnetic behavior of Co^{2+} ions. Although cobalt(II) was described as ‘(3d^{7}; L = 3; S = 3/2)’, i.e., an ion with the electronic spin S = 3/2, all systems have been described by the ‘effective’ spin S = ½ (so as discussed above, more properly the name fictitious ‘spin’ S′ = ½ should be used [3, 25, 26, 27, 28]).
Sample listing of experimental SH parameters for Co^{2+} (S̃ =3/2) ions in the four- and eightfold coordination
Co^{2+} system [Coord. no.], site sym. | Spin Hamiltonian parameters | Experimental method | Source | |||
---|---|---|---|---|---|---|
g _{ x} (g _{⊥} ) | g _{y} (g _{av} ) | g _{z} ( \(g_{||}\) ) | D [cm^{−1}] | |||
Small |D|: up to 19 [cm^{−1}] | ||||||
Co^{2+}: Cs_{2}ZnCl_{4} [4], pseudo T_{d}, C_{s} | 2.3 | 3.9 | 4.7 | 4.5 | EMR | [38] |
HgCo(NCS)_{4} [4], distorted T_{d}, AX | 2.251 | 2.220 | 2.168 | 10.8, 10.2 10.6 | MS | [39] |
YAG [4], T_{d}, O_{h} | 2.176 | 2.446 | 18 | EMR | [40] | |
Co(PPh_{3})_{2}Cl_{2} | 2.166(4) | 2.170(4) | 2.240(5) | − 14(3) | HFM-EMR | [41] |
[(3G)CoCl]^{+} | 2.36 | 2.15 | 12.7 | EMR | [42] | |
[4], pseudo T_{d}, C_{3v} | 2.30 | 2.17 | HFM-EMR | |||
Large |D|: 50–99 [cm^{−1}] | ||||||
2.32 | 2.32 | 2.35 | 50 | EMR | [43] | |
(PhP)_{2} [Co^{II} (SPh)_{4}] [4], OR, D_{2d} | 2.2 | 2.6 | − 70^{a} | EMR & MS | [44] | |
Very large |D|: over 100 [cm^{−1}] | ||||||
(PhP)_{2} [Co^{II} (SPh)_{4}] [4], OR, D_{2d} | 2.25 | 2.25 | 3.13 | − 100^{b} | EMR & MS | [44] |
Analysis of data in Table 1 indicates that even for small ZFS cases, HFM–EMR techniques [41, 42, 45, 46] may be very useful. However, for very large ZFS cases, these advanced techniques alone may not provide direct detection of the EMR transitions. Then, the ZFS parameters may be obtained from combined EMR and magnetic susceptibility measurements [44].
3.1 The Co^{2+} Ions with the Spin 3/2 in the Four- and Eightfold Coordination
Sample listing of experimental parameters for Co^{2+} (S′ = ½) ions
Co^{2+} site [Coord. no.], site sym | Spin Hamiltonian parameters | Source | |||||
---|---|---|---|---|---|---|---|
g_{x} (g_{⊥}) | g_{y} (g_{av}) | g_{z} (\(g_{||}\)) | A_{x}10^{−4} [cm^{−1}] | ||||
A_{x} (A_{⊥}) | A _{ y} | A_{z} (\({\text{A}}_{||}\)) | |||||
PbWO_{4} [4], T_{d}, C_{3v} | 6.20 | 4.50 | 2.07 | 323.3 | 201.7 | 82.7 | [10] |
PbMoO_{4} [4], T_{d}, C_{3v} | 6.26(1) | 4.47(1) | 1.97(1) | 359.6(1) | 264.4(1) | 76(1) | [34] |
CsCdCl_{3} (b) C_{3v} | 3.782(5) | 5.534(5) | 91.5(2) | 214.6(2) | [48] | ||
Y_{3}Ga_{5}O_{12} O_{h} | 2.665(2) | 7.027(2) | 15(3) | 307.5(2) | [49] |
The above survey of the ground-state assignments and illustrative examples of the Co^{2+}(S̃ = 3/2) and Co^{2+}(S′ = ½) cases provided in this Section may help in better understanding of the Co^{2+} ions local environment in various systems, including low-symmetry cases. Detailed survey and analysis of both cases will be provided in a full review, with focus on most recent reports and HFM–EMR studies, as, e.g., on (Ph_{4}P)_{2}[Co(SPh)_{4}] by Suturina et al. [50].
4 Summary and Conclusions
The Co^{2+}(3d^{7}) ions in crystals may occur in one of the two ‘configurations’ (or ‘states’): HS = the high-spin (S = 3/2) or LS = the low-spin (S = ½) configuration. For the tetrahedral (fourfold) and ‘cubic’ (eightfold) coordination, in the intermediate crystal field (CF), the ground state is ^{4}A_{2g}(F) with the high spin (S̃ = 3/2). For the TH coordination in the strong CF, the arrangement of the d-electron spins is somewhat controversial and several inconsistencies must be clarified. Survey of EMR studies of Co^{2+} systems has enabled systematic categorization of the origin of the ground states observed for Co^{2+} ions. Distinctions between the cases described by the effective spin S̃ = 3/2 (and S̃ = ½) and those by the fictitious ‘spin’ S′ = ½ have been elucidated. Implications of these distinctions for HMF–EMR measurements have been considered.
Notes
Acknowledgements
This work was partially supported by the research grant no. 2016/21/B/ST4/02,064 from the Polish National Science Center. Thanks to the anonymous reviewers for helpful comments and bringing a pertinent reference to our attention.
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