Abstract

The low-temperature magnetic properties and Néel temperature, TN, behavior of four silicate substitutional solid solutions containing paramagnetic ions are analyzed. The four systems are: fayaliteforsterite olivine [Fe22+SiO4-Mg2SiO4], and the garnet series, grossular-andradite [Ca3(Alx,Fe1x3+)2Si3O12], grossular-spessartine [(Cax,Mn1x2+)3Al2Si3O12], and almandine-spessartine [(Fex2+,Mn1x2+)3Al2Si3O12]. Local magnetic behavior of the transition-metal-bearing end-members is taken from published neutron diffraction results and computational studies. TN values are from calorimetric heat capacity, CP, and magnetic susceptibility measurements. These end-members, along with more transition-metal-rich solid solutions, show a paramagnetic to antiferromagnetic phase transition. It is marked by a CP λ-anomaly that decreases in temperature and magnitude with increasing substitution of the diamagnetic component. For olivines, TN varies between 65 and 18 K and TN for the various garnets is less than 12 K. Local magnetic behavior can involve one or more superexchange interactions mediated through oxygen atoms. TN behavior shows a quasi-plateau-like effect for the systems fayalite-forsterite, grossular-andradite, and grossular-spessartine. More transition-metal-rich crystals show a stronger TN dependence compared to transition-metal-poor ones. The latter may possibly show superparamagnetic behavior. (Fex2+,Mn1x2+)3Al2Si3O12 garnets show fundamentally different magnetic behavior. End-member almandine and spessartine have different and complex interacting local superexchange mechanisms and intermediate compositions show a double-exchange magnetic mechanism. For the latter, TN values show negative deviations from linear interpolated TN values between the end-members. Double exchange seldom occurs in oxides, and this may be the first documentation of this magnetic mechanism in a silicate. TN behavior may possibly be used to better understand the nature of macroscopic thermodynamic functions, CP and S°, of both end-member and substitutional solid-solution phases.

Introduction

The majority of rock-forming minerals contains transition metals. Iron, Fe2+ and/or Fe3+, is the most abundant element in terms of concentration, but Ni2+, Mn2+/3+, Cr3+, and Ti4+ can also be considered major elements in some cases. Transition metals, even in small concentration, can play a key role in determining optical, magnetic, and various transport properties in crystals. Thermodynamic behavior can also be affected by them. Their presence affects large-scale Earth processes as in redox reactions and deep mantle melting, for example. The property of paleomagnetism is based on the ability of a mineral to retain a memory of Earth’s paleogeomagnetic field during crystallization.

At the simplest level, magnetism in minerals results from partially occupied d-shells of transition-metal ions (minerals with f electrons can also be magnetic, but for rock-forming minerals these electrons are less important in terms of magnetic behavior). The resulting physical property is a magnetic dipole moment generated by the spin of the electrons. In terms of classical physics, the spin can be described by an electron spinning in either a clockwise or counterclockwise (or spin up and spin down) manner. In quantum terms, this is given by the spin quantum number, where MS = +½ or MS = –½. Magnetic behavior in crystals is determined by the type and strength of the various interactions between the electron spins. These interactions can be of the simple dipole type or more complex ones involving additional intervening atoms (Goodenough 1963; Blundell 2001). All spin interactions are a function of temperature.

Detailed study of the magnetic behavior of crystals in the mineralogical sciences is relatively young (see Parks and Akhtar 1968, for an early work and references therein) and not extensive. In contrast, in physics and material sciences the amount of research made on the magnetic behavior of crystals is enormous. In the late 1940s, important theoretical concepts were developed, synthesis experiments on various composition spinel(ferrite)- and garnet-structure crystals were started and investigations on their magnetic properties were made (e.g., Néel 1948; Geller 1967; Winkler 1981). Many of these phases contain rare earth elements with partially occupied f-orbitals, but Fe2+,3+ with d-electrons is important in many cases.

In contrast, little study has focused on the magnetic properties of rock-forming silicates and especially for substitutional solid solutions. The level of scientific understanding is minimal to nonexistent. In these systems, the electronic configuration of the transition metal(s), its/their structural location and concentration in a crystal are critical, because they together will determine the type of magnetic interaction(s). Fayalite, Fe22+SiO4, and fayaliteforsterite, Fe22+SiO4-Mg2SiO4, olivine substitutional solid solutions have received the most attention. Fayalite shows a large and relatively high-temperature magnetic transition at about 65 K, but magnetic behavior at lower temperatures down to roughly 20 K is controversial (e.g., Santoro et al. 1966; Robie et al. 1982; Lottermoser et al. 1986; Aronson et al. 2007). With increasing the forsterite component in Fe2+-Mg olivine substitutional solid solutions, the magnetic transition temperature decreases (Dachs et al. 2007; Belley et al. 2009). The common end-member silicate garnets, almandine (Prandl 1971; Murad and Wagner 1987; Anovitz et al. 1993; Dachs et al. 2014b), spessartine (Prandl 1973; Dachs et al. 2009; Lau et al. 2009), and andradite (Murad 1984; Plakhty et al. 1993; Geiger et al. 2018) have been investigated and they undergo a very low temperature (T < 12 K) spin transition. The transition in both silicate structure types of end-member composition is of the paramagnetic-antiferromagnetic type marking a disordered to a long-range ordered spin state. It is defined by the Neel temperature, TN, which in terms of experimental CP measurements is expressed by a λ-anomaly.

We undertook an analysis of the magnetic behavior of the fayalite-forsterite and three garnet binary substitutional solid solutions, namely grossular-andradite, Ca3(Fex3+,Al1–x)2Si3O12, grossular-spessartine, (Cax,Mn1x2+)3Al2Si3O12, and almandinespessartine, (Fex2+,Mn1x2+)3Al2Si3O12. A knowledge of TN behavior across a given binary join, as determined by low-temperature calorimetry or magnetic susceptibility measurements, together with an understanding of the local magnetic properties of the one or two paramagnetic end-members, as determined via neutron diffraction and/or calculations, allows the magnetic behavior as a function of composition to be analyzed. This type of study has not been done before. Furthermore, an analysis of magnetic behavior can help better understand crystal-chemical and macroscopic thermodynamic properties.

Samples and low-temperature calorimetry

The synthesis conditions or the natural localities for the various crystals of the four binary solid solutions, along with their chemical and physical characterization, have already been described in different publications. The four systems and cited descriptions, discussing the synthesis and characterization measurements, are: (1) fayalite-forsterite, Fe22+SiO4-Mg2SiO4, olivine (von Seckendorff and O’Neill 1993), (2) grossular-andradite, Ca3(Alx,Fe1x3+)2Si3O12 (Geiger et al. 2018; Dachs and Geiger 2019), (3) grossularspessartine, (Cax,Mn1x2+)3Al2Si3O12 (Geiger 2000; Rodehorst et al. 2004), and (4) almandine-spessartine, (Fex2+,Mn1x2+)3Al2Si3O12 (Geiger 2000; Geiger and Rossman 1994; Geiger and Feenstra 1997). The various samples are better than about 99% phase pure.

The low-temperature (i.e., 2 or 5 to 300 K) heat capacity, CP, of the various crystals was measured previously with the Physical Properties Measurement System constructed by Quantum Design. The calorimetric method and measurement setup have been discussed numerous times (Dachs et al. 2009, 2012, 2014a, 2014b; Geiger and Dachs 2018; Geiger et al. 2018; Dachs and Geiger 2019) and will not be repeated here.

Experimental results

Low-temperature CP behavior for synthetic olivines across the Fe22+SiO4-Mg2SiO4 binary is shown in Dachs et al. (2007). The magnetic transitions and their various TN values are shown and given, respectively, in this work. The behavior of TN across the Fe22+SiO4-Mg2SiO4 binary, as determined by the low-temperature CP and also by magnetic susceptibility (Belley et al. 2009) measurements, is shown in Figure 1 and Figure 1a, respectively. TN values are listed in Table 1.

The low-temperature CP behavior for end-member andradite and solid-solution Ca3(Alx,Fe1x3+)2Si3O12 garnets are shown in Geiger et al. (2018) and Dachs and Geiger (2019). The low-temperature CP behavior for spessartine and (Cax,Mn1x2+)Al2Si3O12 garnets are shown in Dachs et al. (2009) and Dachs et al. (2014a) and for almandine and (Fex2+,Mn1x2+)3Al2Si3O12 garnets in Dachs et al. (2012) and Dachs et al. (2014b). The behavior of TN for all three garnet binaries is displayed in Figure 2. TN values are listed in Table 1.

Analyses of the CP results, in terms of modeling the magnetic transitions and the determination of TN, are discussed at length in the cited investigations. TN is given by the peak temperature of the magnetic λ-anomaly.

Discussion

Heat-capacity measurements and brief theory on magnetism

Thermophysical properties of crystals, including magnetic behavior, can change greatly in the vicinity of the critical temperature of a transition. The subject is broad and complex and cannot be treated here (see Gopal 1966; Grimvall 1986). Suffice it to note that heat-capacity measurements, where CP = (dH/dT)P and H is the enthalpy, afford an excellent means of studying TN and magnetic behavior of crystals (e.g., Stout 1961; Gopal 1966). In the case of most silicates studied to date, magnetic transitions occur below 65 K (i.e., fayalite) and usually at much lower temperatures. Thus, the magnetic interactions are weak, but in some cases they can give rise to larger CP(T) values than those deriving from atomic vibrations (phonons) at low temperatures. When it is possible to separate the vibrational (phonon or lattice) heat capacity, Cvib, from the magnetic heat capacity, Cmag, from experimental CP measurements important information is obtained (e.g., Gopal 1966).

Experimental investigations of different types made on transition-metal-bearing olivines and garnets demonstrate that these two structure types undergo one or two magnetic or magnetic-related transitions at low temperatures. In terms of calorimetry, it is marked by a λ-peak or λ-anomaly (i.e., second-order phase transition) that describes the thermophysical changes resulting from the magnetic interactions, whereby disordered electron spins begin to interact locally and order with decreasing temperature. The start of spin ordering (short range) coincides with the onset of the high-temperature flank of the λ-peak until reaching a completely long-range ordered state at the critical temperature of TN.

According to the Heisenberg model for interacting localized spins, the effective magnetic coupling constant, Jeff (any given energy unit, for example, K), is related to TN by the relationship:

 
Jeffk=3TNzS(S+1)
(1)

where k is the Boltzmann constant, S is the total spin, and z is the number of nearest neighbor magnetic ions [z = 2 (M1) and 4 (M2) for olivine and z = 4 (dodecahedral site) or 6 (octahedral site) for garnet]. On the basis of accurate crystal-structure results, the magnetic coupling constant, J, for two weakly coupled localized spins SA and SB can be obtained from the energy difference between parallel (Smax) and antiparallel (Smin) alignment of the spins (Zherebetskyy et al. 2012 and references therein). It is given by:

 
J=E(Smax)E(Smin)Smax2Smin2
(2)

where the numerically calculated E(S) is the total energy for the spin state, S. Positive values of J correspond to parallel or ferro-magnetic and negative values to antiparallel or antiferromagnetic coupling of the two spins SA and SB.

Olivine and garnet crystal structures

Olivine.

Olivine, X2SiO4, with X = Fe2+(fayalite) and/or Mg (forsterite), is crystallographically orthorhombic with space group Pbnm, and it has 4 formula units per unit cell. The crystal structure is shown in Figure 3. The two crystallographically independent cations sites, excluding Si, are termed M1 and M2. M2, Si, O1, and O2 atoms are located on mirror planes and have m point symmetry. The M1 cation is located at the origin of the unit cell and has 1 point symmetry, while O3 and O4 occupy general positions of symmetry 1. Several structural and crystal-chemical studies investigated the nature of the long-range Mg-Fe2+ distribution over the two M1 and M2 octahedral sites in Fe22+SiO4-Mg2SiO4 solid solutions. There are contradictory results and interpretations obtained over the years. The careful, recent X-ray diffraction investigation of Heinemann et al. (2007) summarizes the situation on order-disorder.

Garnet.

The garnet crystal structure, space group Ia3d, general formula {X3}[Y2](Z3)O12, contains three different and independent cation sites (Menzer 1928; Novak and Gibbs 1971) forming a quasi-framework consisting of rigid corner-sharing ZO4 tetrahedra and YO6 octahedra (Armbruster et al. 1992). There are eight formula units per unit cell. The structure is shown in Figure 4a. The Y-cations are located at the Wyckoff site 16d of point symmetry 3. The X-cations, located at 24c of point symmetry 222, are coordinated by 8 oxygen atoms in the form of a triangular dodecahedron. All sites allow for the incorporation of various cations with or without unpaired d- or f-electrons (Winkler 1981), whereby the major cations for the common silicate garnets (Z = Si of point symmetry 4) are X = Ca, Mg, Fe2+, and Mn2+ and Y = Al, Fe3+, and Cr3+. Accordingly, magnetic interactions can occur on two different sublattices that can, furthermore, interact with each other leading to varying magnetic behavior depending on the garnet chemistry. The occurrence of solid solutions, which can be extensive, of varying compositions can lead to significant changes in the physical properties of garnet (Geiger 2013).

Magnetic and TN behavior in olivine and garnet solid solutions: Binary systems with a paramagnetic and diamagnetic end-member

Fe22+SiO4-Mg2SiO4 olivines.

Paramagnetic fayalite shows a magnetic transition at 65 K, as measured experimentally several times (e.g., Santoro et al. 1966; Robie et al. 1982; Lottermoser et al. 1986; Aronson et al. 2007). Müller et al. (1982) investigated the magnetic structure of synthetic fayalite using unpolarized neutron diffraction data recorded at 4.2, 35, and 120 K. The results show that the electronic and magnetic properties deriving from the two crystallographically independent Fe2+ atoms at M1 and M2 are complex. Magnetic interactions occur on the two different sublattices that interact, furthermore, between each other. The ab initio calculations of Cococcioni et al. (2003) for the ground state of fayalite were interpreted as showing that ferromagnetic spin ordering occurs between edge-sharing octahedra (Figs. 3b and 3d) and antiferromagnetic ordering occurs between corner-sharing octahedra (Fig. 3c) and both through oxygen-mediated superexchange.

Forsterite is diamagnetic, but all studied forsterite-containing Fe22+SiO4-Mg2SiO4 solid solutions show a “λ-anomaly.” TN decreases with increasing forsterite component in the olivine as observed via calorimetry (Dachs et al. 2007) and magnetic susceptibility measurements (Belley et al. 2009). This behavior is shown in Figure 1 and Supplemental1 Figure 1a (see data in Table 1). The intensity of the CP λ-peak also decreases accordingly. TN values obtained via magnetic susceptibility measurements on fayalite-rich olivines are in good agreement with those obtained from calorimetry. There are greater differences for Fa50Fo50 and Fa40Fo60 compositions. Belley et al. (2009) did not observe a transition in more forsterite-rich olivines. (Note: The errors in TN are considered to be larger than those in Dachs et al. (2007), Table 1).

In terms of calorimetric determinations, TN is 65 K for fayalite and TN decreases to 18.6 K for composition Fa10Fo90. TN behavior across the binary join can be described using two linear segments with a break around composition Fa50Fo50. One segment is given by the TN values from Fa100 to about Fa50Fo50, while the other segment describes TN values from about Fa50Fo50 to Fa10Fo90. For the latter, the change in TN is less compositionally dependent. All the TN data across the binary can also be fit by a third-order polynomial (Fig. 1).

Ca3(Alx,Fe1x3+)2Si3O12 garnets.

Paramagnetic andradite contains one transition-metal per formula unit, namely Fe3+, and it is located at the 16a octahedral site (Figs. 4a, 4b, and 4c). Plakhty et al. (1993) analyzed the nature of the magnons and magnetic exchange interactions in a natural nearly end-member andradite containing a small amount of Mn2+ and Al3+, as well as in isostructural synthetic Ca3Fe23+Ge3O12, from inelastic neutron scattering measurements made at 4.2 K. The strongest interactions derive from Fe3+(3d5). These workers concluded that magnetic superex-change occurred through the pσ orbitals of intermediate oxygen atoms across octahedral-dodecahedra, Fe3+-O-(Ca)-O-Fe3, bridges (Fig. 4c). Meyer et al. (2010) investigated, furthermore, the local magnetic coupling mechanisms between Fe3+ atoms in And100 using ab initio methods. They proposed that the low-temperature antiferromagnetic transition results from weak superexchange interactions via both Fe3+-O-(Si)-O-Fe3+ and Fe3+-O-(Ca)-O-Fe3+ bridges (Figs. 4b and 4c).

The two different local interactions may possibly be expressed in the CP behavior of end-member andradite (Geiger et al. 2018). Here, the “λ-peak” appears to show a shoulder on its low-temperature flank (Fig. 5), which is even more pronounced in terms of entropy behavior at these low temperatures—as given by S(T) = (CP/T)dT (Geiger and Dachs 2018). The shorter superexchange bridge [i.e., Fe3+-O-(Si)-O-Fe3+] should be marked by the higher temperature maximum intensity of the “λ-peak” at 11.3 (±0.2) K and the longer and weaker superexchange interaction [i.e., Fe3+-O-(Ca)-O-Fe3+] by the low-temperature shoulder at ~5 K. Modeling of the experimental CP data to obtain Cmag shows that the high-temperature flank of the “λ-peak” extends above 11 K (Fig. 5). Therefore, some degree of spin ordering is expected at these temperatures. More research is needed to address the precise physical nature of the λ-peak in andradite.

Grossular is diamagnetic, but all studied andradite-containing Ca3(Alx,Fe1x3+)2Si3O12 solid solutions show a “λ-anomaly”. TN decreases with increasing grossular component in the garnet from 11.3 K in And100 (Murad 1984; Geiger et al. 2018) to about 3 K for the most grossular-rich garnets roughly Gro80And20 (Fig. 2 with TN values given in Table 1). The intensity of the “λ-peak” also decreases with increasing grossular component in the garnet (Dachs and Geiger 2019). Both indicate a weakening of the local magnetic interactions. The TN data across the join can be fit with two linear segments with a break occurring around And50Gro50 (Fig. 2) or with a third-order polynomial.

(Cax,Mn1x2+)3Al2Si3O12 garnets.

Paramagnetic spessartine contains one transition-metal cation per formula unit, namely Mn2+, that is located at the 24c dodecahedral site (Fig. 4a). Prandl (1973) investigated the magnetic structure of synthetic spessartine using neutron powder data. Spessartine shows a λ-anomaly at TN = 6.2 K (Fig. 5, Dachs et at. 2009) and magnetic susceptibility measurements give a transition at 7 K (Lau et al. 2009). Short-range, but not long-range, spin ordering of Mn2+(3d5) is present above this temperature.

As stated above, grossular is diamagnetic but all studied spessartine-containing (Cax,Mn1x2+)3Al2Si3O12 solid solutions show a “λ-anomaly”. TN values for spessartine and (Cax,Mn1x2+)3 Al2Si3O12 solid-solution garnets are plotted in Figure 2 (values in Table 1). Starting from Sps100 and moving to more grossular-rich garnets, TN decreases from 6.2 K to about 2.2 K for the Sps50Gro50 composition. At grossular-rich compositions, TN shows a plateauing behavior with TN values ≤2.0 K (Table 1). A precise determination of TN for the most grossular-rich garnets is difficult due to their weak and broad λ-peaks. Moreover, our CP measurements can only be made down to 2 K. TN behavior across the join can, once again, be described using two linear segments or a third-order polynomial.

Magnetic behavior as a function of composition.

All the experimental data on olivine show a decreasing and nonlinear behavior in TN across the Fe22+SiO4-Mg2SiO4 join. TN, marking a paramagnetic-antiferromagnetic transition, decreases from Fa100 to Fa10Fo90 with a quasi-plateauing behavior for forsterite-rich compositions. The magnetic structure in the fayalite-rich solid solutions should be governed, as in Fa100 (Cococcioni et al. 2003), by superexchange interactions through oxygen among Fe2+ cations (Figs. 3b, 3c, and 3d). A decrease in the intensity of the λ-peak as a function of composition also demonstrates a weakening of the local magnetic interactions.

Analogous TN behavior is observed for Ca3(Alx,Fe1x3+)2Si3O12 and (Cax,Mn1x2+)3Al2Si3O12 garnets and the variation in magnetic properties could be similar to that in olivine. Andradite and spessartine transition to an antiferromagnetic state and this is also considered the case for andradite- and spessartine-rich solid solutions. For both binaries, TN shows a quasi-plateau-like effect, whereby TN is more strongly temperature dependent in garnets richer in paramagnetic cations compared to those richer in diamagnetic ones, namely Al3+ and Ca2+, respectively.

TN for all three solid-solution binaries appears to exhibit a change in temperature dependence roughly around the 50:50 composition region. Notably, magnetic ordering persists in paramagnetically dilute solid solutions and in the case of olivine even for the Fe2+-poor composition Fa10Fo90. Superexchange is responsible for magnetic ordering in the transition-metal-bearing end-members, and as well, we think, for the magnetic-cation-rich compositions. However, it would appear to be difficult for superexchange to persist in compositions richer in the diamagnetic component, because superexchange is a local interaction, decreasing exponentially in strength with distance. The observed magnetic ordering in diamagnetic-component-rich solutions requires long-range interactions. What are the alternatives?

The first and most obvious one is dipolar interactions between randomly distributed isolated magnetic ions. An estimate of the order of magnitude of the magnetic energy, Umag, of the dipole interaction between two free Fe2+ cations, for example, with (anti)parallel alignment is given by

 
|Umag|=2μ0μB2μ2(Fe2+)4πr3=1.731023Jμ2(Fe2+)x31.25Kμ2(Fe2+)x3
(3)

where μB = 9.28·10-24 A m2 is the Bohr magneton, μ (Fe2+) = 4.90, the magnetic moment of Fe2+ in units of μB, μo = 4π·10-7 is the permeability of the vacuum, and x is the distance in angströms between the dipoles. Since dipole-dipole interactions vary as 1/x3, they are long-range in nature. Although dipolar interactions have been shown to be significant in low-dimensional systems (Panissod and Drillon 2003), a rough estimate demonstrates that this cannot explain the observed magnetic ordering in the magnetically diluted olivine and garnet systems. For instance, in andradite with a lattice constant of 12.05 Å at 100 K (Armbruster and Geiger 1993), the assumption of randomly distributed magnetic Fe3+ ions in And20Gro80 yields average distances between about 7 and 10 Å. Substituting these values in Equation 3, estimated TN values in the range of 8 to 3×10-2 K are obtained, i.e., about two orders of magnitude smaller than the experimental ones. Similar results yield estimates of 2.75×10-2 K for And20Gro80 using mean field theory. In the case of olivine, a value of 4×10-2 K for Fa10Fo90 is calculated compared with the observed value of 18.6 K (calculations of R.J. Harrison, private communication). From this first-order analysis, it follows that magnetic dipole-dipole interactions cannot provide the dominating mechanism for spin ordering in crystals rich in the diamagnetic end-member component.

Alternatively, magnetic ordering may occur in the form of superparamagnetism as observed, e.g., in systems of magnetic nanoparticles embedded in non-magnetic matrices (Bedanta and Kleemann 2009). This implies, as the basic assumption, that the distribution of magnetic ions in dilute solid solutions is not random but that clustering is preferred. That is, in the more traditional sense, where nanoparticle-like magnetic aggregates are embedded in a nonmagnetic “matrix.” In other words, short-range-cation order should be present in the solid solutions. This proposal may get support by the fact that cation clustering is energetically favorable, in a thermodynamic sense, due to local superexchange within a nanoparticle-like aggregate compared to a nonmagnetic one.

In summary, one possible interpretation of all the data is that two different magnetic mechanisms may be operating across the Fe22+SiO4-Mg2SiO4, Ca3(Alx,Fe1x3+)2Si3O12, and (Cax,Mn1x2+)3 Al2Si3O12 joins. In terms of olivine, Belley et al. (2009) stated that “magnetic properties do not vary linearly with iron content.” It is notable that the observed TN behavior is independent of a particular chemical composition or crystal structure. In both the olivine and the two garnet systems, roughly at the 50:50 composition, the nature of the magnetic interactions changes from local superexchange to long-range interactions possibly between magnetic nanoparticle-like aggregates. If this proposal for TN behavior is correct, it is the first report of variable magnetic behavior for a silicate solid solution as well as magnetic cation ordering to the best of our knowledge.

Can short-range cation order occur in garnet or olivine solid solutions?

The question of short-range-cation order in silicate solid solutions has been addressed using 27Al and 29Si MAS NMR spectroscopy. It has been proposed to occur in diamagnetic pyrope-grossular garnets, (Mgx,Ca1–x)3Al2Si3O12 (Bosenick et al. 1995, 1999, 2000). Indeed, NMR spectroscopy is the best experimental method in terms of addressing this issue, which is by no means trivial. The experimental problem becomes even more challenging in the case of systems containing paramagnetic ions. The experiments involve the measurement of para-magnetically shifted peaks, whose position is far outside the common range of non-paramagnetic chemical shifts. The resonance assignments and their analysis are not always straightforward. The results on various garnet systems appear to be the most well understood (i.e., Palke et al. 2015; Palke and Geiger 2016). Here, at this stage of research, the spectra do not appear to show any overt or measurable short-range cation order or clustering. The NMR spectra of forsterite-rich olivines are much more complex and little can be said because the spectra show many paramagnetically shifted resonances of which nearly all cannot be assigned (McCarty et al. 2015; Stebbins et al. 2018).

Magnetic and TN behavior in the (Fex2+,Mn1x2+)3Al2Si3O12 garnet solid solution: A binary system with two paramagnetic end-members

The third garnet binary under study has two transition metals that can occur locally at the 24c position (Figs. 4a and 4d). Low-temperature single-crystal neutron (Prandl 1971) and 57Fe Mössbauer measurements (Murad and Wagner 1987) show that almandine undergoes a spin transition from a paramagnetic to an antiferromagnetic state. A λ-peak at about 9.2 K was measured via calorimetry (Anovitz et al. 1993; Dachs et al. 2012), as shown in Figure 5. The local magnetic structure of almandine in the ground state was investigated by density functional cluster calculations (Zherebetskyy et al. 2012). The interactions causing the transition are complex. The spins of the Fe2+(3d6) ions at 24c of the edge-shared dodecahedra sublattice (i.e., Fe2+-O-Fe2+, Fig. 4d) interact ferromagnetically via superexchange involving intermediate oxygen atoms. Two such separate sublattices are present and they interact further through another superexchange involving connecting SiO4 and AlO6 groups via Fe2+-O-(Si)-O-Fe2+ and Fe2+-O-(Al)-O-Fe2+ bridges. Macroscopically, the paramagnetic-antiferromagnetic transition results.

The local magnetic interactions for intermediate (Fex2+,Mn1x2+)3 Al2Si3O12 garnets are most interesting because they are totally unlike the other two garnet solid solutions discussed above. (Fex2+,Mn1x2+)3Al2Si3O12 garnets show nonlinear and negative TN behavior across the binary between Sps100 and Alm100 (Fig. 2). There is no plateauing-like behavior toward either end-member. The high-spin d-electron configurations are (d5d1) for Fe2+ and (d5) for Mn2+. If both cations are present in a solid-solution crystal, this may lead to another type of magnetic interaction known as double exchange. This mechanism was first described by Zener (1951) between Mn3+ and Mn4+ in nominal LaMnO3 perovskite, whereby some La3+ can be replaced by divalent Ca, Ba or Sr, which are then charge balanced by Mn4+ (i.e., La3+Mn3+ = [Ca,Ba,Sr]2+-Mn4+). Further analysis of the physics behind double exchange was given by Anderson and Hasegawa (1955) and de Gennes (1960). The mechanism is well known in solid-state physics and materials science, but it, as best we know, has never been reported in rock-forming minerals. It may occur in certain garnet solid solutions having two divalent magnetic cations at 24c but with different electronic configurations. For (Fex2+,Mn1x2+)3Al2Si3O12 garnets, assuming parallel alignment for the total spins of both ions, Fe2+(d5d1)-Mn2+(d5) with Ms(Fe2+) = +2 and Ms(Mn2+) = +52, the single spin-down electron of Fe2+ can delocalize toward Mn2+, thereby stabilizing the magnetic state. Indeed, electron delocalization leads to a decrease in kinetic energy in accordance with the Heisenberg uncertainty principle. This delocalization cannot occur for antiparallel alignment of spins, which is Fe2+(d5d1)-Mn2+(d5) with Ms(Fe2+) = +2 and Ms(Mn2+) = 52 and would be inconsistent with the Pauli exclusion principle. Consequently, the ferromagnetic and the stronger total antiferromagnetic interaction energy observed in Alm100 (TN = 9.2 K) and Sps100 (TN = 6.2 K) is weakened in the solid solution. Thus, TN shows negative deviations from linearity between both end-member garnets for intermediate compositions (Fig. 2).

If magnetic double exchange does occur in (Fex2+,Mn1x2+)3 Al2Si3O12 garnets, not only is the magnetic energy lowered but also the total energy of the system, albeit very slightly. It follows that there must be a thermodynamic driving force, again very slight, that maximizes the number of local Fex2+Mn1x2+ groupings (i.e., anti-clustering). In other words, there would be unfavorable energetics against forming almandine- or spessartine-like clusters.

Effect of “impurity” atoms on TN

Some of the minor scatter in TN values for almandinespessartine garnets (Geiger and Rossman 1994; Geiger and Feenstra 1997), or any garnet for that matter, may result from small amounts of “extra” cations that are not included in the ideal crystal-chemical formulas. Early indications of this are observable in the 57Fe Mössbauer spectra of almandine (Murad and Wagner 1987) and inelastic neutron scattering results on andradite (Plakhty et al. 1993). TN of synthetic almandine can be shifted to slightly lower temperatures by the presence of small amounts of octahedral Fe3+ (Dachs et al. 2012). The measurable effect of “impurity” atoms in small concentrations on TN in garnet is apparently confirmed.

This is of note because small concentrations of octahedral Fe3+ occur in many synthetic and natural almandine crystals (Murad and Wagner 1987; Geiger et al. 1988; Quartieri et al. 1993; Woodland et al. 1995). Furthermore, at high pressure there is complete solid solution between almandine and skiagite, ideally Fe32+Fe23+Si3O12 (Woodland and O’Neill 1993), and, here, the magnetic interactions can be expected to be highly complex.

Magnetic interactions and transitions: Their effect on macroscopic thermodynamic properties and the role of crystal chemistry

Compositionally end-member silicates.

Both olivines and garnets are orthosilicates, but they are fundamentally different in terms of the crystal structure. Fayalite has considerably stronger magnetic interactions than garnet, by nearly an order of magnitude. Indeed, the magnons associated with the magnetic phase transition in end-member fayalite at 65 K contribute more to CP(T) than phonons between 0 K and about 70 K (Dachs et al. 2007). The relatively energetic magnons derive from the closed-packed olivine structure in which the Fe2+ cations are relatively close to each other in M1 and M2 polyhedra, and the cations can interact magnetically in several ways (Fig. 3). The two coordination octahedra have shared edges and corners. In fayalite, magnons contribute significantly to the macroscopic thermodynamic behavior at standard conditions. The standard third-law entropy, S°, of a crystal is given by:

 
S°ST=0KJ/(molK)=0298.15KCPTdT
(4)

assuming ST=0K = 0. For fayalite Smag(298.15 K) is 26.2 J/(mol·K) and it contributes about 17% to S° that is equal to 151.4 J/(mol·K) (Dachs et al. 2007).

In the case of end-member garnet with transition-metal cations occurring just at the octahedral site, superexchange interactions are mediated through diamagnetic SiO4 and/or XO8 groups (Fig. 4). Thus, the interactions are very weak and magnons occur at very low energies. Andradite is a case in point. The modeled Smag(298.15 K) is 28.1 J/(mol·K) and it contributes about 9% to S° that is 325.0 J/(mol·K) (Geiger et al. 2018).

For garnets with transition-metal cations just at the dodecahedral site, the magnetic interactions appear to be even more subtle and complex. The total magnetic interactions involve diamagnetic SiO4 and AlO6 groups, and they do not occur directly between edge-shared dodecahedra (Zherebetskyy et al. 2012), as might be expected from a first-order crystal-chemical analysis. Thus, the corresponding magnon energies are also weaker than in fayalite. For almandine, the modeled Smag(298.15 K) is 32.1 J/(mol·K) and it contributes roughly 10% to S°, that is 336.7 J/(mol·K) (Dachs et al. 2012). For spessartine, the model Smag(298.15 K) is about 38 J/(mol·K) and S° is 335.3 J/(mol·K) (Dachs et al. 2009), thus making up about 11% of the latter. The relevant equation giving the theoretical Smag value is:

 
Smag=Rln(2S+1) per mole per cation in the formula unit
(5)

where R is the gas constant and (2S + 1) is the multiplicity, i.e., the number of electron spin orientations. Only for andradite and fayalite is the agreement between model and theoretical Smag values reasonable or good.

What can be stated, further, in terms of magnetic and CP(T) and S(T) behavior? Various purely empirical CP models, such as corresponding states formulations (Anovitz et al. 1993; Lau et al. 2009), or more “seemingly” rigorous lattice-dynamic-type calculations (Pilati et al. 1996; Gramaccioli 2002; Gramaccioli and Pilati 2003), including neutron scattering measurements (Mittal et al. 2000), have been undertaken on garnet. Their soundness, especially, in the former cases is questionable. We have been using the simplified lattice dynamic formulation of Komada (1986) and Komada and Westrum (1997) to model CP,vib(T) and Svib(T) behavior, where “vib” stands for vibrational, using experimental calorimetric CPcal(T) results as input data. If the two former functions can be modeled properly, CP,mag(T) and Smag(T) contributions can be obtained from the difference in values [e.g., CP,mag(T) = CPcal(T) – CP,vib(T); see Dachs et al. 2009, 2012, 2014a, 2014b; Geiger et al. 2018, for more detail]. An assumption of this model is that there are no or very minor phonon-magnon interactions. It turns out in some cases (i.e., almandine and spessartine) that the model Smag(298.15 K) values are less than those obtained via Equation 5. One possibility that could explain the discrepancy is that phonon-magnon coupling is occurring. Research in this direction is needed.

Substitutional solid-solution silicates.

The results of this investigation may help in yet another area involving thermodynamic properties (Geiger 2001). It involves macroscopic thermodynamic mixing behavior, namely ΔCPmix(T) and ΔSmix(T), for solid solutions containing a transition metal ion or ions (see Dachs et al. 2007, 2014a, 2014b; Dachs and Geiger 2019). In short, a precise determination of ΔCmagmix(T) and ΔSmagmix(T) behavior, obtained from an application of the Komada and Westrum (1997) model, can be problematic if they are small in magnitude. However, TN behavior for a solid solution can help qualitatively in this question, because it can be measured precisely, and it is not in any respect model dependent. Consider the system olivine (Fig. 1). Dachs et al. (2007) argued that ΔSmagmix(298.15 K) behavior shows slight negative deviations from ideality across the Fe22+SiO4-Mg2SiO4 join (i.e., ΔSmagmix < 0). TN behavior shows as well negative deviations from linearity (Fig. 1) between Fa100 and Fa10Fo90. It must be noted, on the other hand, that a similar relationship does not appear to exist for andraditegrossular or spessartine-grossular garnets, where in both cases ΔSmagmix(298.15 K) = 0.

Implications

An understanding of the magnetic behavior of silicates, and especially their solid solutions, in both a solid-state physical and mineralogical context, is in its infancy. Little is known, and much research remains to be done. Several significant implications can be drawn from this first investigation on olivine and garnet.

First, we conclude that the observed λ-anomaly in the low-temperature CP(T) results on synthetic uvarovite, Ca3Cr23+Si3O12, and knorringite, Mg3Cr23+Si3O12 (Klemme et al. 2005; Wijbrans et al. 2014) is caused by a paramagnetic-antiferromagnetic transition. It must be expected that most, if not all, transition-metal-bearing silicate and germanium garnets will have very low-temperature magnetic spin transitions. This may be true for other silicates as well. A determination of their heat-capacity and magnetic behavior will require measurements down to the lowest possible temperatures. This was not always done in the past, and it led to incorrect results (see the case for andradite, Geiger et al. 2018).

Second, it can be proposed that double-exchange interactions may occur among other magnetic ions than just between Fe2+ and Mn2+. In terms of garnet, it may occur, for example, between Fe2+ at 24c and Fe3+ at 16a in garnet. For example, double exchange may possibly occur in certain andradites and almandines, where CP results show small variations in TN and λ-anomaly behavior among different crystals (Dachs et al. 2012; Geiger et al. 2018). Furthermore, several rock-forming silicate systems show an exchange between Fe2+ and Mn2+ and, here, magnetic double exchange may occur. This goes, for example, for the fayalite-tephroite (Mn2SiO4) join (Burns and Huggins 1972). Marked exchange of Mn2+-Fe2+-(Mg) cations occurs in pyroxenes, amphiboles, and micas. In all these silicates, Mn2+ and Fe2+ can be found in corner- and edge-shared octahedral sites and, thus, d-electron delocalization could be expected.

Finally, and almost needless to say, the precise magnetic behavior of many solid-solution silicates, containing two or more different transition-metal cations, may prove to be complex in nature. Their low-temperature CP and magnetic behavior can be expected to be complicated by virtue of the range of possible chemistries and structural sites. The number of different local-electron-spin interactions is expected to be large.

Funding and Acknowledgments

E.C. Ferré (Lafayette, Louisiana) kindly supplied the data from the magnetic susceptibility measurements on olivine. This study was supported by a grant to C.A.G. from the Austrian Science Fund (FWF: P 30977-NBL). C.A.G. also thanks the “Land Salzburg” for financial support through the initiative “Wissenschafts- und Innovationsstrategie Salzburg 2025”. We thank the two referees, and especially R.J. Harrison (Cambridge, U.K.), whose keen review encouraged us to consider more fully the possible role of superparamagnetism instead of dipole-dipole interactions in the solid solutions. The editor S. Speziale (Potsdam, Germany) also provided useful remarks on improving the clarity of the manuscript.

I (C.A.G.) first met John Valley at the University of Michigan years ago, where I was an undergraduate and John a graduate student in the Department of Geological Sciences. The electron microprobe was located in the Engineering Department and I had technical difficulties with the machine working completely alone late one night (lowly students only received measuring time on the night shift). I ran, almost in a panic, back to the Geology Department hoping to find someone there who could help me out. John was working in his office, as often was the case, and he came to the rescue. I dedicate, now, this manuscript to him.

Endnote:

1
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