1. Historical Note
William O. J. Boo was a professor of chemistry at the University of Mississippi from 1967 to 1992. He died on 7 September, 2011, leaving unfinished this manuscript which reviews his two decades of research in the lower-valence fluorides of vanadium and chromium. The manuscript has been completed by two of his former colleagues, Daniell L. Mattern and Robert M. Metzger. Professor Boo also had a long-standing interest in symmetry, constructing hook-and-loop-edged polygons to teach geometry symmetry principles to elementary-school students, including tilings, solid shapes, and packings as shown in Figure 4 below.
This study reviews the structures, transitions, ionic and magnetic ordering phenomena, and other anomalous behaviors of six lower-valence fluorides of vanadium and chromium, AxMF3, designated as (AxMF3), where A = K, Rb, or Cs; M = V or Cr; and x = 0.0 - 1.0. Descent in symmetry in oxygen-containing perovskites was reviewed by Goodenough  decades before the ceramic superconductors La2−xCuO4  and YBaCuO7−x  were discovered within that family . A recent article describes how charge ordering in lanthanide cuprates is related to unusual collective behavior, such as antiferromagnetism and superconductivity .
The alkali metal ions K+, Rb+, and Cs+ (the A+ ions) demonstrate the effects of their sizes and concentrations on crystal structures. V2+, V3+, Cr2+, and Cr3+ are first-row transition-metal ions with simple 3d electronic configurations. The F− ion is small and has small polarizability. It is a weak field ligand (compared to the O2- ion), and the V and Cr ions, which are octahedrally coordinated by F− ions, have high-spin electronic configurations (Figure 1). M-F bonds have mostly ionic character, but their covalency is strong enough to form 3-D network structures.
On the other hand, due to the weak bonding in the network, the structures easily adapt to the sizes and concentrations of the A+ ions. Hence, the corner-sharing MF6 octahedra are flexible enough to give rise to a variety of closed-network structures . The weak covalency of the M-F bonds also facilitates super-exchange interactions between nearest-neighboring magnetic centers.
Figure 2 outlines the temperature regions in which the various transitions appear to be occurring. At first, perovskite-like solid phases apparently form from the melt . Upon cooling, new structures are formed, designated α, β, and δ [8 - 10]; when these are cooled further, ionic ordering sets in [7 , 11 - 15]. Such transitions are characteristic of mixed-valence compounds [1 , 16]. Three ordered structures are possible in the α-phases, three in the β-phases, and two in the δ-phases . At very low temperatures (below 50 K), magnetic ordering sets in [11 - 14]. These transformations correspond to a descent in symmetry during the phase changes, from averaged and mixed-valent arrangements to more ordered ones.
Some anomalies are associated with specific phases, such as the metamagnetic transition that occurs with K0.49VF3 . In addition, behaviors associated with specific ions occur, such as the orbital quenching of V3+ .
In the remainder of this paper, we will describe the melting temperatures of the compounds, and then detail descriptions of the reconstructive transitions and the ionic and magnetic ordering that sets in as the temperature is lowered.
3. Melting Temperatures
The melting temperatures of the AxMF3 compounds  are shown in Figure 3. Remarkably, the AxCrF3 compounds all melt 200˚ lower than their AxVF3 analogs. The other striking feature of Figure 3 is that the melting temperature vs. x plots of the AxMF3 systems are all linear. These plots include the binary parent compounds VF3 and CrF3, which have cubic perovskite-like structures at high temperatures [7 , 17]. This suggests that the AxMF3 compounds also initially solidify to cubic-perovskite-like structures.
Figure 1. Electronic configurations of V2+, V3+, Cr2+, and Cr3+ in a weak octahedral field.
Figure 2. Temperature regimes for various likely ordering phenomena of the AxMF3 compounds. The wavy lines and the interpretation of the ordering processes above room temperature have partial support from DTA transitions .
4. Reconstructive Transitions
The high-temperature DTA data  suggest (Figure 2) a succession of reconstructive transitions between 900˚C and 650˚C, that break up the perovskite structure and form new lattices, designated α, β, and δ . Such α, β, and δ phases have only been reported for first-row transition-metal fluorides. (Note that MF3 and AMF3  are not mixed-valence compounds, and do not undergo reconstructive transitions. The variety of crystal structures of metal fluorides has been recently reviewed. ) Since a perovskite phase does form upon solidification [7 , 17], we surmise that single crystals of the lower-symmetry α, β, and δ phases will grow at temperatures considerably below the solidification temperatures.
The structural analogs of the AxMF3 compounds  include: cubic perovskite (SrTiO3) and rhenium oxide (ReO3) (x = 1.0, γ phase, each belonging to space group = #221) ; hexagonal tungsten bronze (HTB) (x = 0.167 to x = 0.31, α phase, space group P63/mcm = #185) ; tetragonal tungsten bronze (TTB) (x = 0.45 to x = 0.60, β phase, space group P4/mbm = #127)  or hexagonal BaTa2O6 (x = 0.43 to x = 0.52, β phase, space group P6/mmm = #191 , which is a structure similar to TTB ; and finally modified pyrochlore (MP) (x = 0.45 to x = 0.58, δ phase, space group = #227) . The value of x alone cannot discriminate between the β phase and the δ phase .
The structures of the AxMF3 compounds have topologies which can be described using packings of simple geometrical shapes: cubic perovskite with a packing of cubes; HTB with a packing of triangular and hexagonal prisms in the ratio 2:1 (Figure 4(a)); TTB with triangular, square, and distorted pentagonal prisms 2:1:2 (Figure 4(b)); and MP with a packing of tetrahedra and truncated tetrahedra 1:1 (Figure 4(c)). Each of the packings is used in the same way. The transition metal ions are located on all vertices, the F- ions approximately in the centers of all edges, and the A+ ions in the centers of some or all of the 3-dimensional holes.
Figure 3. Melting temperatures (˚C) of the AxMF3 compounds. Data from ref .
Figure 4. Geometrical configurations of (a) the HTB structure, showing the hexagonal unit cell (upper) and the ortho-hexagonal unit cell (lower); (b) the TTB structure, showing the tetragonal unit cell; and (c) the MP structure, showing the cubic unit cell.
A packing of cubes is a regular packing; a (1:1)—(tetrahedron:truncated-tetrahedron) packing is quasi-regular, while a (2:1)—(triangular-prism:hexagonal-prism) packing is semi-regular. The significance is that in each of these geometrical packings, the vertices (where transition metal ions reside) are congruent. The packing of triangular prisms, square prisms, and distorted pentagonal prisms has two sets of vertices, but all of the edges are of equal length. These packings each reveal that the MF6 (M = V, Cr) octahedra share corners with six other octahedra, and all M-F-M bond angles are approximately 180˚.
VF3 and CrF3 have lattices that collapse from cubic to rhombohedral near 500˚C [7 , 25]. These are not reconstructive transitions, since their network structures are displaced, but not broken.
5. Ionic Ordering
When they are cooled further, the AxMF3 compounds likely experience a variety of ionic ordering phenomena . The space-group symmetries of the ordered structures are sub-groups of the α, β, or δ parent structures .
In the AxCrF3 compounds, ionic ordering includes (Cr2+/Cr3+) electronic ordering, ordering of partially-filled A+ sites, and (Cr2+) Jahn-Teller ordering. These ordering phenomena probably occur cooperatively between 500˚C and 300˚C. The AxVF3 compounds undergo (V2+/V3+) charge ordering, ordering of partially filled A+ sites, and (V3+) Jahn-Teller ordering between 300˚C and 100˚C .
Three modulated structures are possible in the α phases: these correspond to 1/2-filled A+ sites (x = 0.167, space group Pnnm # 58); 2/3-filled A+ sites (x = 0.222, space group Cmcm #63); and 3/4-filled A+ sites (x = 0.250, space group Pmma #51) [12 , 13]. The α (0.222) and α (0.250) structures only form if samples are cooled slowly; α (0.167), however, is more tenacious, and forms no matter how rapidly the samples are cooled. Table 1 lists which modulated structures form in each of the six systems. The Jahn-Teller ions
Table 1. Modulated structures and distortion ratios in the α-phases (hettotype of HTB). At least in the Cr salts, note the “Goldilocks” effect, where the K+ ion is too small , and Cs+ is too large , but Rb+ is just right , to form all three modulated structures α (0.167), α (0.222), and α (0.250). Data from ref. .
V3+ and Cr2+ distort their α phase compounds from hexagonal to orthorhombic, even when no modulated structures are formed. The distortion of the hexagonal unit cell is expressed as |a|/( |b|); this ratio can be greater than, or less than, 1.000.
In the β-phases, three ionically-ordered structures are possible, in addition to the hexagonal BaTa2O6 structure. The three ordered structures are: a distorted BaTa2O6 structure (space group Cmmm #47), an ordered TTB structure (space group P42bc #106), and a distorted TTB structure (space group Pba2 #32) [7 , 9]. In the β-phase of KxVF3 the ionically-ordered TTB structure forms over the range x = 0.45 - 0.56. Only a trace amount of the ordered BaTa2O6 structure forms near x = 0.40 . In the β phase of KxCrF3 (x = 0.43 - 0.59), the hexagonal BaTa2O6 structure forms if the sample is cooled slowly . If the sample is cooled rapidly, the distorted TTB structure forms, and if the sample is cooled at an intermediate rate, the hexagonal BaTa2O6 structure forms at low x, the distorted TTB structure forms at high x, and the ordered BaTa2O6 structure forms in the intermediate region of x .
In the δ-phases, the fcc MP structure plus two ionically-ordered MP structures are possible. In the ordered structures, M2+ ions form linear chains along the <110> direction and M3+ ions form linear chains along the <110> direction. One of the ordered structures is body-centered orthorhombic MP (space group Imma #74). The second ordered structure is distorted to primitive orthorhombic MP (space group Pmna #53). The δ-phase of RbxVF3 (x = 0.45 - 0.52) has the fcc structure at x = 0.45 and the primitive orthorhombic structure for x above 0.45. In the δ-phase of CsxVF3 (x = 0.45 - 0.52), the fcc structure exists below x = 0.50 but the body-centered orthorhombic structure occurs above x = 0.50. The phases of RbxCrF3 (x = 0.45 - 0.56) and CsxCrF3 (x = 0.46 - 0.54) have primitive orthorhombic structures over their entire range of x .
6. Orbital Quenching of the V3+ Ion
The Curie-Weiss law may be stated as:
where χM is the molar magnetic susceptibility, CM is the Curie constant, θ is the Weiss constant, and T is the absolute temperature. A plot of 1/χM vs T usually displays a linear region for paramagnetic compounds, with the slope of that region equal to . For VF3 however, the plot displays 2 linear regions , because the orbital magnetic moment of V3+ is only partially quenched at high temperatures, but is totally quenched below 122 K. This phenomenon occurs in some (but not all) of the AxVF3 compounds as well . A plot of 1/χM vs T for K0.250VF3, shown in Figure 5, is an example of a compound in which the orbital
Figure 5. versus T of K0.25VF3. Data from ref .
moment of V3+ becomes totally quenched near 100 K.
For the AxVF3 compounds, CM = x CM [+2] + (1 − x) CM [+3], where CM [+2] and CM [+3] are the V2+ and V3+ components of CM, respectively. When the V3+ ion is totally quenched, the value of CM falls on a straight line that connects CM(VF3) at x = 0.0 with CM(AVF3) at x = 1.0. Figure 6 shows plots of CM vs x for KxVF3, RbxVF3, and CsxVF3 in which CM values were obtained from the temperature region 50 to 150 K. Figure 6 clearly shows that the V3+ orbitals in the α phases are totally quenched, those in the β phases remain partially quenched, and those in the δ phases become totally quenched if they have the fcc structure, but remain only partially quenched if they have the body centered or primitive orthorhombic structures.
7. Excitation of a Second Orbital State of Cr2+
In CrF2 a second orbital state approximately 116 cm−1 above the ground orbital state was observed . This excitation appears as a shoulder in a C(mag) vs T plot near 80 K, which leads to an additional Rln2 contribution to S (mag), making the total S (mag) Rln10 rather than Rln5. The magnetic susceptibility of CrF2 emulates those of the β phase KxCrF3 compounds, see Figure 7 . Evidence of Cr2+ in these compounds behaving the same as in CrF2, however, is inconclusive.
Figure 6. CM below 150 K versus x of the AxVF3 compounds. Data from refs [13 , 14].
Figure 7. versus T for K0.50CrF3. Data from ref .
8. Magnetic ordering
Magnetic interactions in the AxMF3 compounds are primarily antiferromagnetic, and weak ferromagnetic interactions are often difficult to detect in these complex structures. There are four observable magnetic parameters: CM (the Curie constant), TC (the Curie temperature), θ (the Weiss constant), and σo (the spontaneous magnetic moment extrapolated to 0 K).
The CM values of the AxVF3 compounds below 150 K are shown in Figure 6. Observed CM values of the AxCrF3 compounds are in good agreement with calculated spin-only values, except for the α (0.167) structure of RbxCrF3  and the β-phase of KxCrF3  (both at temperatures below 150 K). Possible explanations were given in the previous section for KxCrF3 and have been discussed before for α (0.167) of RbxCrF3 .
The TC value, the temperature at which long range magnetic ordering sets in, is usually seen as a maximum in a χM vs T plot. Most of the AxMF3 compounds, however, display small spontaneous magnetic moments (σ) that mask these maxima. In these cases, TC is determined by the onset of σ. Figure 8 shows
Figure 8. TN (●) and −θ (❍) for the six AxMF3 systems. Data from ref [13 - 15].
TC and −θ values of the six systems. In general, the TC values of the AxVF3 compounds are lower than their AxCrF3 analogs. The substitution of different A+ ions appear to have little effect on the magnetic properties. The three structures of the β phase of KxCrF3 have unique TC values: the hexagonal BaTa2O6 structure shows no evidence of TC; for the ordered BaTa2O6 structure, TC = 50 K; and for the distorted TTB structure, TC = 10 K. The TC values of the other phases appear to be essentially constant over their entire range of x.
θ values provide a measure of the strength and sign of magnetic interactions at temperatures above TN. Except for the β phases of KxCrF3, values of the Cr compounds are greater than their V analogs. The values of the α-phases of Cr are remarkably large compared with the β and δ-phases of Cr. Rb0.167CrF3 displays two θ values, θ from the high temperature region and from the low temperature region. The Heisenberg model  states that for ferromagnetic materials, , and that for antiferromagnetic materials, . The latter appears to be the case for most of the AxVF3 compounds. However, magnetic frustrations present in the structures of α, β, and δ should suppress TN values, making them much smaller than values [32 , 33]. This is the case with some of the β-phase compounds of KxVF3 and all of the AxCrF3 compounds, except those of the β-phase of KxCrF3. For antiferromagnetic interactions θ values are negative; for ferromagnetic interactions θ is positive. Overall, the Cr compounds display antiferromagnetic behavior, but the V compounds may possess a mixture of ferromagnetic and antiferromagnetic coupling.
Spontaneous magnetic moments in zero field (σo), extrapolated to 0 K for the six AxMF3 systems, are illustrated in Figure 9. The figure shows two kinds of spontaneous magnetic moments: ferromagnetic moments, shown as black areas, and ferrimagnetic moments, shown as hatched areas. The compounds which display no spontaneous moments are antiferromagnetic, and are shown as blank areas.
The V and Cr compounds show important differences in their magnetically ordered states. In these compounds, ferromagnetic order occurs from the canting of ordered spins; ferrimagnetic order means M2+
Figure 9. Spontaneous magnetic moments in zero field, extrapolated to 0 K, of the AxMF3 systems. Hatched areas correspond to ferrimagnetic compositions/phases, black areas to ferromagnetic compositions/phases caused by canting of spins, and blank areas represent compositions which have zero spontaneous magnetization. Data from refs [11 - 15].
spins are up while M3+ spins are down; and antiferromagnetic order means half of M2+ spins are up, half down, while half of M3+ spins are up, half down. Ferromagnetic ordering occurs in the α phases of all of the compounds and in the δ-phases of RbxVF3, CsxVF3, and RbxCrF3. The V compounds demonstrate ferrimagnetic ordering in the α (0.222) structures of KxVF3 and RbxVF3, in the β-phase of KxVF3 below x = 0.49, and in the fcc structure of the δ-phases of RbxVF3 and CsxVF3, but no ferrimagnetic ordering was observed in any Cr compound. The β-phase of KxCrF3 and the δ-phase of CsxCrF3 are antiferromagnetic over their range of x; the KxVF3 β-phase is antiferromagnetic above x = 0.49.
The ferrimagnetic mechanism of the β-phase of KxVF3 was described in detail previously,  as was that of the α (0.222) structure in KxVF3 and RbxVF3 . In the δ-phases of RbxVF3 and CsxVF3 ferrimagnetism occurs only in the fcc structure.
MF6 octahedra are canted in the HTB structure and the MP structure, but in the HTB and BaTa2O6 structures, there is no canting of the octahedra. The canting of spins (shown in Figure 9) is consistent with the canting of octahedra, except for the δ-phase of CsxCrF3, which, strangely enough, has no spontaneous moment. In general, the canting of spins is greater in the Cr compounds. Most notably, canting within the Cr α-phases increases with x.
9. Concluding Remarks
This paper identifies the solid-state transitions of the mixed valence AxMF3 compounds from their respective melting temperatures down to 4.2 K. This summary is organized to show connections of composition and structure with transitions and other events. The transitions occur in the following order:
melting à cubic perovskite à α, β, δ à ionic order à orbital quenching of V3+ à
à magnetic order.
Other features and events include:
· the V compounds melting 200˚ higher than their Cr analogs;
· the V compounds undergoing ionic ordering 200˚ lower than their Cr analogs;
· single geometric structure;
· variations of orbital quenching of V3+;
· differences of β-phases of V and Cr;
· differences of −θ/TN of V and Cr;
· metamagnetic transition of K0.49VF3;
· array of different magnetic interactions;
· canting of magnetic spins.
The study of the AxMF3 compounds is a good introduction to an unchartered area of solid-state chemistry. Hopefully, similar studies of other systems will follow. The future breadth and depth of solid-state chemistry will be determined by the curiosity, creativity, and perseverance of solid-state chemists. High-quality powder diffraction data followed by a careful Rietveld analysis, or else crystal structure determinations (if and when crystals are obtained) should in the future buttress the arguments made here about the inferred onset of charge ordering as the symmetry is lowered. The lower-valence transition metal fluorides may ultimately yield exciting new phenomena complementary to the overwhelming success of the lower-valence transition metal oxides. Now that both Paul Hagenmuller and his competitor William Boo are gone, the next generation of chemists will hopefully pick up the challenge.
This work was supported by the National Science Foundation and the University of Mississippi. We thank Prof. Jared Allred for helpful discussions.