spinel structure

American Mineralogist, Volume 6E, pages 1E1-I%, 1983

Simple spinels: crystallographic parameters, cation radii, lattice energies,and cation distribution

Hucn Sr. C. O'NerLr nno AlexeNone NevnorsKy

Department of ChemistryArizona State University

Tempe, Arizona 85287

Abstract

Electrostatic lattice energy calculations and consideration of the structure of the spinelgroup of minerals show that the size of a cation is important in determining its sitepreference. In oxide spinels containing 2+ and 3* ions (2-3 spinels) there is a tendency forthe larger ion to prefer the tetrahedral site; the reverse holds for 2-4 spinels. A set of cationradii optimized to best fit spinel lattice parameters is presented. Based on lattice energycalculations, the enthalpy change accompanying cation disordering is not a linear functionof the degree of cation disorder, but a quadratic, with the two parameters approximatelyequal in magnitude and opposite in sign. Most published data on the change of cationarrangement with temperature fit a thermodynamic model with this non-linear change inenthalpy, but those for Fe3Oa may require an additional non-configurational entropy aswell, and the data for MgAl2Oa cannot be described adequately by any simple model.

Introduction

The oxide spinels comprise a large group ofternary compounds, significant not only as natural-ly occurring minerals, but also in many branches ofsolid state science. Despite their deceptively simplestructure, many spinels exhibit complex disorderingphenomena involving the two cation sites, whichhave important consequences both for their thermo-chemical and for their physical properties. Recent-ly, there have been a number of studies of spinelsolid solutions, (Porta et al., 1974:' Jacob and Al-cock, 19771' Mason and Bowen, 1981) which haveemphasized the necessity of understanding the cat-ion distribution in these more complex systems.

A quantitative treatment of the thermodynamicsof cation distribution in simple spinels was givensome time ago (Navrotsky and Kleppa, 1967).Inthis paper we present a modification to this model,necessitated by some new insights into the energet-ics of the process, and by some new, more precisedata on experimentally determined cation distribu-tions. In a subsequent paper we will apply this newmodel to explain some features of the thermody-namic and structural properties of spinel solid solu-tions.

t Present address: Research School ofEarth Sciences, Austra-lian National University, Canberra, A.C.T. 2600, Australia.

Structure

An understanding of the structure of the spinelgroup is fundamental to any theory of cation distri-bution, and so a brief description of the salientdetails will be given here. More complete reviewsare available elsewhere, (Blasse, 1964; Hill et al.,1979). The spinel unit cell is face-centered cubic,and contains 32 anions. The space group is Fd3m(Ol, number 227 in the International Tables) withthe cations occupying the special positions 8a and16d. The anions occupy the general positions 32e,which require an additional parameter, generallydesignated u and known in oxide spinels as theoxygen parameter, for their complete description. Ifthe origin of the unit cell is taken at the center ofsymmetry, then u lies between about 0.24 (for thesilicate spinels) and 0.275. For a value of u equal to0.250 the anions form an exactly cubic close packedarray, and define a regular tetrahedral coordinationpolyhedron about the 8a sites (point symmetry 43m)and a regular octahedron about the 16d sites (m3m).The octahedral cation-anion distance (bond length)in this case is 1.155 times larger than the tetrahedralbond length. As u increases, the oxygens displacealong a [1ll] direction, causing the tetrahedral siteto enlarge at the expense of the octahedral, whilethe symmetry of the latter degenerates to 3m.However. all six octahedral bond distances remain

m03--(M)V83/0 102-018 I $02.00 lE l

tE2

the same. Taking a to be the length of a unit cell, thecation to anion distances, R, are given by:

Rt"t : ar@ ( l)

Ro"t : a(3uz - 2u + 318)t/2 (2)

Thus the octahedral and tetrahedral bond lengthscan be used to determine the two structural parame-ters, a and u.

A majority of simple oxide spinels have thestoichiometry AB2Oa, with the cations A and Bhaving the formal charges required by the conditionfor electrical neutralitv:

Q e + 2 Q s * 4 Q o : 6 (3)

where Qa and Qs are the charges on the A and Bcations, and Qe that on the anions (-2 for oxidespinels). Commonly, A:2 and B : 3, (defining a2-3 spinel) or A : 4 and B : 2 defining a 4-2spinel). A perfectly normal spinel is one in whichthe single A cation of the formula unit occupies thetetrahedral site, and the two B cations the twoequivalent octahedral sites. If we designate theparentheses ( ) and [ ] to denote these two typesof sites, then this distribution may be written(A)[82]O4. However, as Barth and Posnjak (1932)first suggested for MgAl2Oa the alternative distri-bution (B)tABlO4 is also possible, and this arrange-ment is referred to as being perfectly inverse.In factall distributions falling between these two extremesmay be realized, and to describe this situation it isconvenient to define an additional parameter, x, thedegree of inversion, which is the fraction of tetrahe-dral sites occupied by B ions. Thus x may varybetween 0 for the perfectly normal case and Ifor the perfectly inverse case. Of special note isthe completely random arrangment, (ArnBzn)IA4B473'}O4. In general, any two ions A and B willhave different sizes, so that a change in x will resultin a corresponding adjustment of the two structuralparameters a and u. This observation, which issupported by the few studies that have been under-taken on the matter, (see, for example, Tellier(1967) on MgFe2O+) will provide the basis for ourtreatment of cation distribution.

Thermodynamics of cation distribution

A thermodynamic treatment of cation distribu-tions in simple spinels was presented some yearsago by Callen et al. (1956), and is also summarizedin Navrotsky and Klepp a (1967) . We shall present abriefreview here.

O'NEILL AND NAVROTSKY: SIMPLE SPINELS

thus:

1( _ tAl(B)(A)tBl

The configurational entropy of a substance isgiven, assuming completely random mixing of ionson each site by:

Sc : -Rlb'Ni ln Ni (4)

where Ni is the fraction of species i in site s, and b'is the number of sites of type "s" per formula unit.For an AB2O4 spinel this may be evaluated as:

Sc : -Rlxlnx + (1 - x) ln ( l - x) * x ln(xl2)

+ (2 - x) ln (1 - xlz)l (5)

Thus at X : 0, Sc : 0. The configurational entropythen rises sharply to a maximum at the randomarrangement of x : 213, and then decreases to avalue of 2R ln 2 for an inverse spinel (x : l). Here itis convenient to define as our standard state theperfectly normal spinel at the temperature andpressure of interest. If we write the changes inenthalpy and volume on disordering as Aflo andAVp, and the change in the nonconfigurationalentropy as ASp then the change in free energy ondisordering, AGp, is:

AGo : LUo - (ASc + ASD) + PAVD (6)

where AUo is the internal energy change, AVp thevolume change, and P and T are pressure andtemperature.

At equilibrium

aAGo/ax : 0 (7)

Substituting Equations (5) and (6) into Equation(7) and carrying out the differentiation where possi-ble, we obtain:

ln [x2(1 - i-t(2 - x)-t] : -(RD-t(aA,unlax)

+ R-t(da^tD/ax) - P(RT)-t(aavolax) (8)

As will be demonstrated later, AVo is generallysufficiently small so that the last term in the abovewill be negligible at low or moderate pressures.

Navrotsky and Kleppa (1967) have shown thatthe term inside the logarithm on the left hand side ofEquation (8) is equivalent to an equilibrium con-stant, K, for the interchange reaction:

(A) +tBl : (B) +tAl (e)

xz

( l - xX2 -x )(10)

A plot of ln K versus x is shown in Figure 1, andmay be used to illustrate two points. First, for 0.2 <


x

Fig. 1. The logarithm of the equilibrium constant (K) for thesite exchange reaction in spinels as a function of the degree ofinversion, x.

x < 0.8 this function is nearly linear (with a slope ofabout 7.7). Therefore, provided that the right handside of Equation (8) takes a reasonably simple formfor its dependence on x, and has a slope less thanthe above value, there will be a unique solution forx. Conversely if the right hand side of Equation (8)has a steeper slope, there will be (typically) threeapparent solutions, two of which will generally bemetastable with respect to the third.

Second, as T -+ co, the enthalpy terms on theright hand side of Equation (8) become negligible,and x approaches a value determined by the non-configurational entropy term, ASp. As will be notedlater, the majority of experimental studies of thetemperature dependence of the cation distributionin spinels show that x tends towards the randomdistribution (x:213) at very high temperatures, thusimplying the ASp is usually small or even negligible.(However, it is to be expected that in those spinelscontaining crystal field ions, there will be a smallbut definite electronic contribution to ASp.) Thevalues of x : 213 then provides a useful point atwhich to divide spinels with intermediate cationdistributions, into those that are "largely normal"(x -+ 0 as T -+ 0K) and those that are "largelyinverse" (x -+ I as T -+ 0K).

If for the moment, we assume the ASp is indeedsmall, the question then reduces to what the formand magnitude of AUp and AVp are likely to be.

The change in internal energy on disordering

Dunitz and Orgel (1957) and, independently, Mc-Clure (1957) showed that a number of the observed

cation distributions in spinels could be explained bycrystal field theory. A transition metal ion willgenerally have a crystal field stabilization energy(CFSE) in octahedral coordination in excess of thatin tetrahedral coordination, and so, in this model,AUp is given by the difference in this excess CFSEbetween the two ions involved, and will be a simplelinear function of x, i.e., 6(A,Up)/dx : ACFSE(A) -

ACFSE(B). In fact, in about half of the cases wherethis model is applicable, it does not predict theobserved cat ion distr ibut ion correct ly (seeGlidewell (1976) for a fairly comprehensive tabula-tion on this point). Furthermore, many of the ionsthat occur commonly in the spinel structure do nothave any CFSE, and therefore, in cases where bothions are of this type, the model has nothing at all tosay about which ion prefers which site. It is possibleto conceive of ad hoc extensions to this model. Forexample, in the case of zinc spinels one couldpostulate an excess energetic stabilization for thezinc ion in the tetrahedral site. derived from itsformation of sp3-type covalent bonds, which wouldexplain why all 2-3 zinc spinels are normal. But thisapproach could not be extended to many othercations, including those in the type mineral of thegroup, MgAl2Oa, in which the apparent preferenceof Al over Mg for the octahedral site may seemsomewhat surprising to mineralogists who are fa-miliar with the opposite site preferences in suchsilicates as pyroxenes and amphiboles. (It was justthis that led Barth and Posnjak (1932) to originallypropose the inverse arrangement.)

A somewhat different empirical approach wasused by Navrotsky and Kleppa (1967). Using Equa-tion (8), they assumed that ASp was negligible, andthat AUe was proportional to the degree of disor-der, x and to the difference in the "site preferenceenergies", assumed constant for all spinels, of thetwo ions involved. Values for these site preferenceenergies were derived from a small number ofobserved cation distributions at known tempera-ture, and could then be applied to predict cationdistributions in other, less well-studied, spinels.The cation interchange energies calculated fromthis set of site preference energies was encouraging-ly similar to one obtained by Miller (1959) from amore theoretical treatment.

At this point we note that the precision withwhich it is possible to measure x as a function oftemperature for a single stoichiometric spinel isseldom sufficient, and the total range of x seldomlarge enough, to justify the taking of a more compli-

lE4 O'NEILL AND NAVROTSKY: SIMPLE SPINELS

cated form for the right hand side of Equation (8).However, the variation to be expected in x fromone end of a solid solution series to the other can bemuch larger, especially when the two end membershave widely different cation distribution and so themodel used becomes of much more consequencewhen dealing with solid solutions than with simplespinels. We therefore propose to take a closer lookat the form of AUD than a simple theory for theenergetics of the cation distribution might suggest.

Both Dunitz and Orgel and McClure dismissedthe change in lattice energy with cation distributionas being relat ively unimportant. However,Glidewell (1976) has shown that in many instancesthe change in the electrostatic part of the latticeenergy with disordering is likely to be of muchgreater magnitude than any CFSE. For a crystalcomposed of point charge ions the internal energy,U, is given by:

U : U e * U p * U y ( t 1)

where Ue is the electrostatic energy that accompa-nies the movement of the constituent ions of thecrystal from an infinite separation to their observedpositions in the crystal lattice, and depends only onthe geometry of the lattice, and may thus be evalu-ated fairly straightforwardly. Up, as written, is anamalgam of other, non-coulombic, terms, includingdipole-dipole and higher order "Van der Waal's"attractions, and a repulsion term which is usuallygiven an inverse power or exponential form. Thesum of these terms is opposite in sign to, andtypically about 10-207o of , Un. Uy is the vibrationalenergy, which for the sake of argument, we shalltake as being given by a simple Debye model (i.e.,the harmonic approximation). This implies that Uvis independent of a, u and also x (and, of course,gives ASp : 0). At 0K, AUp, in this model, is thengiven by the difference in lattice energy between xat the value of interest, and x : 0 (our standardstate). At finite temperature, the AUy term will beassumed negligible in accord with the above ap-proximation.

The electrostatic term is both the largest and themost tractable, and will be considered first. It maybe expanded into the form:

U n :Ne2M

kJ mol-r (12)4Tesa

where M is the Madelung constant, which in thiscase will be a function of the charges on each ion,

and also of the oxygen parameter, u. Values of Mfor completely normal and inverse spinels were firstcomputed for diferent values of u by Yerwey et al.(1948). Subsequent calculations by Hermans et al.(1974) and Thompson and Grimes (1977) have re-vealed some errors in this earlier work, but agree-ment between the latter two works is excellent. Wehave chosen the formulation offered by Thompsonand Grimes as it allows for the calculation of M forintermediate cation distributions. Here M is givenby:

M: crrQ6 * azQoQr + ":Qi

* a+QoQeN + osQi" + aoQrQeN (13)

where Qo, Qr and Qay are the average charges onthe tetrahedral, octahedral and anion sites; dt, e2and a3 are constants; and d4, ds and a6 are, to agood approximation, quadratic functions of u. Notethat a1 to a6 are not uniquely determined parame-ters, but depend on the method of summation.However, all such sets of parameters are related bythe condition of electrostatic neutrality in the crys-tal (Equation 3) and produce the same value of M.

In Figure 2 we have plotted M against u for theextreme cases of a completely normal 2-3 spinel, aninverse 2-3 spinel, and a normal 4-2 spinel. Sincean inverse 4-2 spinel has the same average chargesin each site as a normal 2-3 spinel, it will also havethe same Madelung constant. The cross-over pointfor the normal and inverse curves for a2-3 spinel isat u : 0.2555. For values of u less than this, theelectrostatic energy would favor the inverse distri-bution, and vice versa. The corresponding cross-

r28

t24

M

ozt|o 0.243

Fig. 2. Madelung constant (M) versus the crystallographicoxygen parameter (u) for normal and inverse 2-3 spinels andnormal 4-2 spinels. Inverse 4-2 spinels have the same chargedistribution and therefore the same Madelung constant as normal2-3 spinels.

0.255

u: 1389L

a

O,NEILL AND NAVROTSKY: SIMPLE SPINELS 185

over point for a 4-2 spinel is at u : 0.2625 (whichalso is the value for u at which the tetrahedral andthe octahedral bond lengths are equal). Table Ishows the observed values of u and the knowncation distribution for a number of spinels. Nearlyall the 2-3 and most of the 4-2 spinels conform tothe expectation of this model, but there are a fewnotable exceptions, e.9., Zn2TiOa, which will bediscussed later.

That many spinels show an oxygen parameterappropriate to their cation distribution does notprove that the electrostatic energy controls thiscation distribution: for, as stated earlier, u, and to alesser extent, a, may be expected to vary with thecation distribution.

Both a and u are uniquely determined by acombination of the octahedral and tetrahedral bondlengths (Equations (l) and (2)). In their recent studyof spinel systematics ,Hill et al. (1979) showed that,by using Shannon's (1976) ionic radii, 96.9Vo of thevariation in a in a sample of 149 oxide spinels could

Table l. Oxygen parameters and cation distributions for somespinels. Data from Tabulation of Hill er al. (1979). Normal 2-3spinels are expected to have an oxygen parameter greater than0.2555, inverse 2-3 spinels less than this; for 4-2 spinels thecorresponding point is at u = 0.2625.In the table N means x <

2/3. I means x> 213.

spinel Cation DistrLbution

Table 2. Effective ionic radii in the oxide spinels. The radius forthe 02- ion is assumed to be 1.384 (Shannon. 1976).

Tetrahedral

Coord ina t ion

ocrahedral

Coordinationkt io

{tet loct)fris worl

h d i u s ( A ) R a d i u s ( A )mis work Shannon fris sork Shannon

2+ Cd

Co

FE

Xg

b

Ni

3+ AI

CE

FE

M

4 + e

si

Sn

Ti

0 . 7 8

0 . 5 8

0 . 5 1 5

o . 5 8 5

o . 6 5 5

0 . 5 6 5

0 . 3 9

0 . 4 5

o . 4 4 5

o . 4 1

0 . 3 8

o . 2 7 5

o . 7 8

0 . 5 8

0 . 6 3

o . 5 7

0 . 5 5

0. 60

o . 3 9

o . 4 9

o . 4 7

o . 3 9

0 . 5 5

o . 4 2

- o . 9 5 0 . a 2 r

o . 7 2 0 . 7 4 5 0 . 4 0 6

o . 7 4 0 . 7 4 o . 8 3 r

0 . 7 1 5 0 . 1 2 0 . A 2 4

o . 8 0 0 . 8 3 0 . g 1 9

0 . 5 9 0 . 6 9 0 . 8 1 9

o . 7 3 0 . 1 4 0 . 7 9 5

0 . 5 3 0 . 5 3 5 0 . 7 3 6

0 . 5 3 0 . 5 4 5 r

0 . 6 1 5 0 . 6 1 5

0 . 6 4 5 0 . 5 4 5 0 . 7 6 0

0 . 6 1 5 0 . 6 2 0 . 7 6 4

o . 5 8 0 . 6 6 5

0 . 6 4 5 0 . 6 4

o . 5 2 0 . 5 3 0 . 7 3 r

- 0 . 4 0 0 . 6 7 5

0 . 1 4 5 0 . 5 9 0 . 7 3 8

0 . 6 0 0 . 6 0 5 0 . 7 0 0

0 . 5 5 0 . 5 8

MgAI2oq

Znca2Ob

l,lgFe2Ot+

ZnIe2OI

DrgCr2Ol

NiCr2Ob

M9v2Ob

NiFe2O|{

M9Fe2O{

NiAI2O'r

Fe3Of

MgGa2O4

Fe2SiO\

Co2SiO\

Ni2s io4

Mg2G€O4

Za2TLO4

Ugv2O,,

Fe2TiO4

Zn2SnOI

o.2624

o.261 7

o .2615

o.26' t5

o . 2 6 1 2

o .260

o .2599

o.2573

o.25' l

o.256

o.254A

0. 254

o.2409

o.2423

o.2439

o .2508

o.261

o.265

N

N

N

N

N

N

N

I

T

T

I

I

N

N

N

N

I

Y

i

I

rLd sp in s ta te

be accounted for. This remarkably good statisticalcorrelation led Hill et al. to state that up to 40Vo ofthe published u parameters, which were not so wellaccounted for, may be in error. In view of the fargreater difficulties in determining u as compared toa, this may well be the case.

If one assumes that Ro", and Rtet are determinedcompletely by a spinel's composition, site occupan-cy, and the ionic radii, one can compute directly thevariation of a and u, and hence of the electrostaticenergy, with the degree of disorder, x, for 4nyspinel.

Interionic distances in oxide spinels

Shannon's ionic radii (Shannon and Prewitt,1969; Shannon, 1976) were derived from a largenumber of diferent structures, rather than confinedto spinels only, and therefore it is not surprising that}Iill et a/.'s tabulation contains certain systematicdifferences between the observed and the calculat-ed values of a and u. Accordingly, we have refinedShannon's radii specifically for the spinel structure,so as to eliminate these anomalies. The results arelisted in Table 2. We have used Shannon's value of1.38A for 02- in tetrahedral coordination for easycomparison. Our data base was largely similar tothat of Hill et al.,but we used only those spinels in


which the oxidation states of the cations are unam-biguously known, and for which the cation distribu-tion was close to either the normal or the inverseextreme. Less weight was given to the value of uthan to a, particularly for some of the older struc-tural determinations, in which the accuracy in eval-uating u was not high. For the inverse spinels, someassumption needs to be made for averaging the radiiof the two different ions in the same site. Wedecided on the simplest approach (like Hill et al.),and assumed that these distances could be averagedin a simple linear fashion, so that:

Rt"t : Rt"t,a(l - x) + Rt.t,s(x) (14)

Ro"t : R."t,a(0.5 x) * Ro"t,e(l - 0.5x) (15)

The radii were then obta-ined by a process ofadjustment (in steps of 0.005A) until the optimum fit(as judged from histograms such as that shown inFig. 3) was achieved. This histogram shows thedifference (Aa) between the calculated and theexperimentally determined values of a. From Equa-tions (1) 4d (2) it may be estimated that a variationof 0.005A in bond length will cause an uncertaintyin a of about 0.014 (and in u of -0.001). Over 75%of the spinels fall within these limits. Much of theremaining variation comes from the silicate andgermanate spinels.

These radii have a number of applications. First,they may be used to identify some of the anomaliesin the literature, which may be simply due toincorrect measurements, or, more significantly, ac-curate measurements on poorly characterized sam-ples. Many spinels contain cations that occur inmore than one oxidation state, and others have atendency to form defect structures with cation

-.05 -.04 -.o3 -.02 -.ol o .0r .o2 .03 .o4 .05aa (observed a - measured a (in A))

Fig. 3. Histogram showing the differences between themeasured lattice constant (a) and that calculated from the ionicradii of Table 2 for 66 simple oxide spinels. The measured valuesare from the tabulation of Hill er al. (19791.

vacancies, particularly at high temperatures(Schmalzried, 1961). A combination of these twofactors may easily produce substantial variations inlattice parameters in a spinel of nominally fixedcomposition with different conditions of prepara-tion. These variations cannot be ascribed to cationdisordering. For example, NiFe2Oa has been shownby Robertson and Pointon (1966) to have an almostcompletely inverse cation distribution over a widerange of temperatures. Yet Subramanyan andKhare (1979\ found that the measured value of aincreases with the temperature of sintering in air,which was inferred to be caused by oxygen lossfrom the sample and the accompanying formation ofsome Fe2*. In this case the calculated value for afrom this work may be compared with values re-ported in the literature to enable one to distinguishbetween stoichiometric and nonstoichiometric sam-ples. Similarly, the calculated values of a and u maybe used to disriminate between various alternativearrangments that could be postulated for cationdistribution and oxidation state. This will be partic-ularly useful in describing spinel solid solutions.For example, consider a spinel with the formulaCoFe2Oa. There are four possible extremes of cat-ion and valence arrangements, as shown below,which, together with their calculated values of aand u are:

l. (Co2+XFez3*lO+

a(A) u

8.4091 0.2596

2. Ge3+)[Co2*Fe3*]Oo 8.3702 0.2536

3. (Co3+XFe2*Fe3*lOo 8.3439 0.2516

4. (Fe2+)[Co3*Fe3*1Oo 8.3004 0.2638

Blasse (1964) lists measured values of a :8.35 andu : 0.256. A more accurate determination of c isgiven by Roiter and Paladino (1962) as8.382=0.001A for CoFe2Oa prepared at high tem-peratures. These values are nearest to option 2above, tending toward option l. Sawatzky et al.(1969) have in fact shown by Mcissbauer spectros-copy and magnetic measurements that the cationarrangement in CoFe2Oa is nearest to the secondoption, with a temperature-dependent degree ofdisorder toward the first. For an actual high tem-perature arransement of (CoAjFeAl)lCgAiFe?1lOo,calculated values of a and u are 8.38264 and 0.2554respectively, in excellent agreement with the ob-served values.

Second, the change of a with x gives the volume

tnq)tr

o

o

a)

E

z

O'NEILL AND NAVROTSKY: SIMPLE SPINELS r87

change of the spinel on disordering (AVp), andhence the dependence of x on pressure (see Equa-tion (8)). There have been relatively few studies inwhich both a and x have both been measured for aspinel equilibrated at different temperatures. Anexception is magnesium ferrite, (which, as a compli-cation, also tends to be slightly nonstoichiometric)and in Figure 4 the calculated values of a are shownas a function of x, together with the experimentaldata obtained by Tellier (1967) on "MgFe2O4," andby Mozzi and Paladino (1963) on Mg1.s6Fe r.s+Ot.gt.The former used saturation magneti zation measure-ments to determine x, the latter a combination ofthis technique and X-ray diffraction. The data ofAllen (1966) also show the same trend, but withconsiderably more scatter.

The slight decrease in lattice parameter withincreasing degree of inversion seen in magnesiumferrite is generally to be expected for 2-3 spinels,with cations of sizes similar to Mg2+ and Fe3+. Fora 4-2 spinel such as Mg2SiOa, a calculation usingthe radii of Table 2 shows that an increase in x from0 to 0. 1 causes an increase in the molar volume ofabout 0.4Vo.

This point is of particular importance in Mg2SiOa,which may be an abundant mineral in the lower partof the upper mantle. Navrotsky (1977) has shownthat the small amounts of cation disordering whichmight occur in this spinel at the very high tempera-tures (2000K and above) of this region may have animportant influence on the slopes of the phaseequilibrium boundaries defining the stability field ofMg2SiOa spinel. The predicted increase in molar

'wtgF'ero;

o o o M & po o

i l t Tx x $

l.o o.9 0.t *

o.t 0.6

Fig. 4. Calculated lattice constant (a) for MgFe2Oa as afunction of the degree of inversion (x). Shown also are theexperimental data of Tellier (1967) (crosses labelled T) andMozzi and Paladino (1963) (circles labelled M and p) on"Mgy.66Fe1 soOs.sti' Note that the effect of impurity Fe2* ionswould be to increase a.

volume with disordering, although small, will tendto decrease any tendency toward disordering atpressures of several hundred kilobars.

Finally, as both a and u can be calculated as afunction of x, the change in the electrostatic part ofthe lattice energy with x may be evaluated accurate-ly.

The change in enthalpy on disordering:revised model

The constant ionic radii of the previous sectionpredict that a and u are linear functions of x.Substituting this into Equations (12) and (13), to-gether with the appropriate expressions for theaverage charges on each site, shows that the elec-trostatic contribution to the lattice energy will havea quadratic dependence on x, which we may writein the form:

AtlE: aBx * Bsx2 (16)

This quadratic dependence was also recognizedempirically by Kriessman and Harrison (1956) whotried to justify it in terms of electrostatic energiesbut did not consider the simultaneous variation of aand u with x.

In order to demonstrate the behavior of thiselectrostatic term. we have calculated the coefr-cients aB and BB for a number of specific spinelstructures, as given in Table 3. A number of pointsworthy of further discussion arise:

Table 3. Calculated values for the change in the electrostaticenergy on disordering given in the form A (Js : apx + Brx2 , Lu,-

= Us (x : 0) - Ue (x = 1). Units are kJ/mol.

69n9s

MgAl2O4

UoGa.O, .

MgFe2Oq

Mg h2Oq

NiFe2Oq

UgFe2O!

ZnFe2Dq

Fe 3O!

UnFe2O{

cdFe2Oq

Mg2Sio{

M92GeOl

u92T iO{

Ug2SnO4

Ni2GeOq

M92GeOq

Zn2ceo\

Fe2GeO4

Mn2GeO\

1 66?

1 093

a 6 l

- 1 a r

750

461

958

1042

1 3 6 1

1 903

6 4 1 7

4 7 9 2

4417

2461

4639

4 7 9 2

4792

5070

5 4 5 9

-1320

- 9 0 3

-806

- ! e r

-a06

-420

- 9 3 1

- l l l l

- 1 5 2 a

-4764

- 3 5 0 0

-3042

- r 8 0 6

- 3 4 0 3

- 3 5 0 0

- 3 5 1 4

- 3 ? 3 6

-4042

341

1 4 1

5 6

- 3 6 1

1 4

5 6

1 2 5

1 1 1

250

r 6 5 3

1 2 1 A

1 3 7 5

r o 5 6

1 250

1 2 7 I

1218

1 4 1 7


(l) AUE is very large and would swamp anyCFSE term. Even at temperatures near 2000K,AUB is about two orders of magnitude larger thanthe product T56, the contribution to the free energyof disordering from the configurational entropy.

(2) LUE is consistently larger for 4-2 than for 2-3spinels. This implies that, all other things beingequal, 2-3 spinels are more likely to show interme-diate cation distributions. Most 4-2 spinels ob-served are in fact either completely normal orcompletely inverse.

(3) The term in x2 (B) is approximately equal inmagnitude, and opposite in sign, to that in x (a6).

(4) Both terms increase with the difference in theradii of the two ions.

(5) MgInzOr is the only example given for whichAUB is negative at x: 1, and hence the only ex-ample for which the electrostatic energy wouldfavor the inverse arrangement.This conclusion isquite different from that reached by Glidewell(1976) in his consideration of the relative effects ofthe electrostatic versus CFSE terms. The differencearises because Glidewell did not take into accountthe change of a and u with x.

(6) In the 2-3 spinels there is a tendency for theelectrostatic energy to favor putting the larger ioninto the tetrahedral site (e.9., compare the observedcation distributions in the series MgB2Oa where theradius of the B ion increases in the order Al < Ga <Fe ( In, and the series AFe2O4, where the radius ofthe A ion increases in the order Ni < Mg - Zn IFe2* < Mn < Cd). Inthe 4-2 spinels, the effect actsoppositely, so as to favor placing the smaller ion inthe tetrahedral site.

There is considerable support for these general-izations in real spinels. Thus in the aluminates, onlyNiAlzO4 and CuAl2O4 are observed to be largelyinverse (although cpsr, would also favor this) andthe ferrites MnFe2Oa and CdFe2O4 are indeed nor-mal (although the formation of tetrahedrally sym-metric sp3 type covalent bonds would again favorthis in the latter case, as in ZnFezOq). Perhaps themost startling and emphatic illustration of this ten-dency is provided by de Sitter er al. (1977) in astudy of Ca-substituted magnetites. They showedby Mossbauer spectroscopy that all of the verylarge Ca2* ion resided in the tetrahedral site. Ca2+is normally found in eightfold coordination, andindeed Shannon (1976) does not even list a value forits ionic radius in tetrahedral coordination. That in2-3 spinels, the larger ion should seem to prBfer toassume tetrahedral coordination is an interestingand somewhat unexpected result. The variable u

parameter removes the geometric constraint which,in close-packed structures, would cause small ionsto prefer tetrahedral coordination. In the 4-2 spin-els, when the tetravalent ion is small, the spinel isalways found to be normal (e.9., the silicates andgermanates, except ZnzGeO+) and when the tetra-valent ion is large, the spinel is inverse (e.9., thetitanates and stannates).

The electrostatic energy is, of course, only onepart of the total lattice energy and since AUn is anorder of magnitude larger than what one mightreasonably expect for AUp calculated from ob-served intermediate and inverse cation distribu-tions, it is pertinent to investigate the effect of theshort-range force terms (Un in Equation 11). Themost significant of these are the repulsion termsbetween nearest neighbor cations and anions,which are usually assumed to take the exponentialform, Dexp(-Wp), or the inverse power form, DR-n,where R is the interionic separation, p or n are"hardness" parameters, and D a factor with theunits of energy. Using the former, this gives Up'(where the ' denotes that only nearest-neighborrepulsions are being considered) as:

[JR' : 4Do(l - x) exp (-R/pa)

+ 4DB x exp (-R/p'B) + 6DA x exp (-R6/p6)

+ 6DB(2 - x) exp (-Rdps) (r7)

where the subscripts A and B refer to the ions in theformula until AB2Oa, and the distances Rl and R6are the simpler linear functions of x given byEquations (14) and (15). Thus this term, too, de-pends mainly on the differences in the radii of thetwo ions. By differentiating Up' with respect to x, itmay easily be shown that to a very good approxima-tion, AUp' also takes a quadratic form in its depen-dence on x, which we may write:

AUR' : all 'x * 9*' x' (18)

The p quantities may in theory be evaluated fromsuch sources as the equilibrium interionic distancesin, and compressibilities of constituent binary ox-ides. Unfortunately all the required data are notavailable. Nevertheless, by assuming physicallyreasonable values for pa and ps (e.g., this hardnessparameter may be taken as 0.35A for both A and Bby analogy with alkali halides, (Tosi, 1964) anddivalent metal oxides, (Ohashi and Burnham,1972),we can derive approximate values of Da and Dgfrom the equations for equilibrium in the spinelstructure:

O'NEILL AND NAVROTSKY: SIMPLE SPINEZ,S 189

dU,du :0a ta :ao

(19)

dU,

d r r : 0 a t u : u o Q 0 )

where a6 and u6 &r€ the lattice constant and oxygenparameter calculated for the equilibrium cation dis-tribution, and U' : fJB * fIn'. If independentvalues for D and p could be found, as suggestedabove, then the interionic distances could be calcu-lated from the lattice energy model, and wouldprovide an interesting insight into our ionic radiiand the assumptions we have made about theirconstancy from one spinel structure to another.

Such calculations as are possible with the presentmodel suggest that the coefficients cp' and pq' areagain approximately equal in magnitude, and oppo-site in sign to each other. Moreover, they are bothopposite in sign to the analogous coefficients ap andpB derived from the electrostatic energy equation.Thus the dependence of U on x will result from asmall difference between the relatively large quanti-ties Us and Up'. Under such conditions otherminor terms that appear to be of negligible signifi-cance in the lattice energy equation may exert anappreciable effect, such as next-nearest-neighborrepulsions, oxygen-oxygen interactions, and the"Van der Waals" terms. These also may be shownto take on a quadratic dependence in x as a firstapproximation. Thus, clearly the true form to beconsidered for the change in enthalpy on disorder-ing (AUp in Equation (8)) if one neglects the pVterm) is also a quadratic one, which we will simplywrite as:

LHD - AUD: ax * pxz (2r)and in which o and B are generally expected to be ofapproximately equal magnitude and opposite sign.The effect of crystal field stabilization energies maybe included in this model: since this term is as-sumed to depend linearly on x, it may be added ontothe o term. Also, since both AUe and AUp' havebeen shown to depend mainly on the difference inradii of the two ions, so too will the coefficients aand p.

One may question the validity of applying atraditional lattice energy approach (Coulombic plusshort range terms) to oxides in which much of thebonding can be described as being covalent incharacter. Indeed the lattice energy calculationsconsiderably overestimate the magnitude of disor-dering energies, compare Tables 3 and 5. This may

suggest that the actual charge on an ion in a crystalis some fraction (0.2 to 0.5) of its formal charge, ashas been suggested for albite feldspar by Senderov(1980) on the basis of lattice energy calculations forthe order-disorder reaction. The point ofsuch elec-trostatic lattice energy calculations is not to calcu-late the actual energy of disordering but to predictthe relative stability of various possible configura-tions. As long as the degree of covalency (howeverdefined) and the nature of each metal-oxygen bondare not strongly affected by changing next-nearestneighbor configurations, this approach will correct-ly predict the most stable structure, though it willnot give an accurate prediction of energetics. Mo-lecular orbital calculations also predict configura-tions more readily than absolute energies, and theirapplication to molecular clusters as big as the spinelunit cell is very difficult. Thus, at present, theenthalpy of transformation or of disordering re-mains an empirical parameter, though its form canreceive theoretical justification from calculationssuch as those above.

Effect of short-range order

Before leaving the topic of lattice energies, wewish to point out one more effect that may beexplained, at least qualitatively, by this model.Earlier, it was noted that the observed cation distri-bution for most 2-3 spinels is compatible, from theelectrostatic point of view, with the measured uparameter, in that the Madelung constant for this uis larger for this distribution than for the alternativelimiting distribution. The same does not alwayshold for the 4-2 spinels, however, in that themeasured u values for many of the inverse spinels,while in agreement with those calculated from theionic radii, are still less than the cross-over point forthe Madelung constant at u : 0.2526 (see Fig. 2).

There are, however, two lines of evidence indi-cating that such inverse 4-2 spinels may possess aconsiderable amount of short-range order of the twoions in the octahedral site. It has been shown that.with decreasing temperature, a number of thesecompounds undergo a phase transition to a deriva-tive structure (tetragonal, space group P\22) inwhich the two octahedral cations are ordered intotwo non-equivalent sites. This phenomenon hasbeen reviewed recently by Preudhomme and Tarte(1980) who suggest that the temperature of thetransition decreases with decreasing diference inthe radii of the two ions. The highest temperaturefound so far is -770'C for Mn2TiOa (Hardy et al.,1964). In addition, Jacob and Alcock (1975) have

argued, from thermodynamic evidence, thatZn2TiOashows alarge degree of residual short-rangeordering at higher temperatures (900-ll00K), wellabove the transition point.

Hermans et al. (1974) have shown that the order-ing of two ions with different charges into anordering scheme like that in magnetite results in alarge increase of about 4.0 units in the Madelungconstant. This increase would enhance the stabilityof an inverse 4-2 spinel with ae : 8.5A by about 650kJ/mol. The type of ordering and distortion found inthese titanate spinels is not identical to that inmagnetite, and so the precise magnitude of theeffect for titanates is not known. A similar increaseof 4.0 units in the Madelung constant would, how-ever, move the cross-over point down to u: 0.257(see Fig. 3). This effect is therefore of great poten-tial importance in stabilizing the inverse structure.Note too that any appreciable short-range order willcause a deviation from the ideal configurationalentropy, and may therefore lead to apparent excessentropies of disordering (ASo) if the simple ap-proach of Equation (8) is used.

The effect of electronic entropy in transition-metalspinels

Another hitherto neglected factor that may influ-ence the cation distribution and its change withtemperature in certain spinels is that crystal fieldtheory, in addition to predicting an excess octahe-dral crystal field stabilization enthalpy for manytransition metal cations, also suggests that therewill be a change in electronic entropy for these ionsin going from octahedral to tetrahedral coordina-tion. In crystal field theory, this entropy is given by,(Wood, 1981):

S"r : -R)P1 ln P; (22)

where Pl is the probability of the ith state occurring.In Table 4 we have listed S"1 for perfectly regulartetrahedral and octahedral coordination for somecations of interest, and also AS"1, which would bethe electronic contribution to any change in entropywith cation distribution. In fact, the octahedral siteis generally slightly distorted, which leads to re-moval of the degeneracy of the relevant d-orbitalsand a lowering of the octahedral site electronicentropy. However, at the high temperatures en-countered in spinel studies this effect may not besignificant.

Compared to the CFSE, the electronic entropyeffect is small. But note for example, that AS"1 forV3+ reinforces the relatively small (-50 kJ/mol

O,NEILL AND NAVROTSKY: SIMPLE SPINELS

Cat ion

according to Dunitz and Orgel (1957)) excess octa-hedral CFSE for this ion, thereby keeping all V3*spinels exclusively normal despite this ion's largeionic radius. Conversely AS"r for Niz* favors thetetrahedral site, thus contributing towards the ran-domization of NiAlzO+.

Application of the model

Substituting Equation (21) into Equation (8), weobtain a form similar to that suggested empiricallyby Kriessman and Harrison (1956), n4mely:

: l n K :( l - x X 2 - x )

_(a + 2Fx) _ P-d|Vollx *

?dASe/dx e3)

R T R T R

In Table 5 we present the results of least squaresregression analyses for o and B obtained from thevariation of x with temperature as reported in theliterature. All these studies are at atmosphericpressure, and hence the AVp term is negligible. Theaccuracy of the data does not warrant fitting morethan two parameters, so we have elected to ignoreany possible excess entropy (ASp) term. Despitethe variation found in the values for nominally thesame spinel in different studies, crand B are approx-imately equal in magnitude and opposite in sign toeach other for all these substances, which is inexcellent agreement with the predictions of thelattice energy model. However, these data shouldbe treated with due caution, as: (1) Many of thesespinels are known to be slightly nonstoichiometric(e.g. MgFe2Oq) or to contain ions in variable oxida-

Table 4. Change of electronic entropy with disorder for sometransition metal ions

Elec t !on ic

conf igurac ion ager(JK-Inol- l )

x2ln

Ti e-

v3+

c r 3 +

h 3 +

Fe2 +

c o 3 + t s . s .

r . s .

n i 2 *

cu2 +

do

d t

d 3

d 5

d 6

0 0

R l n 2 R I n 3

0 R 1 n 3

R l n 3 0

R 1 n 3 R l n 2

o o

R l n 2 R l n 3

R I n 2 R I n 3

- o

O R t n 3

R I n 3 0

' o

- 9 . 1 3

9 . 1 3

3 . 3 7

o

- 9 . 1 3

9 . 1 36 8

d9

h.s . = h igh sp in , L .s . = lov sp in

O,NEILL AND NAVROTSKY: SIMPLE SPINELS

Table 5. Values of a and B in Equation (22) found by regression analysis of experimental data on the change of x with temperature

l9l

spinel

a p

(kJmoI-I ) (kJmol-I )

Method ofDetermination

o f x Reference

UgGa2O4

MgFe2Ob

1 , t 9 1 . q 5 F e 1 . 9 4 0 3 . 9 7

NiAI2Oq

NiMn2Ob

CuFe2O4

1 9 . 2

1 5 . 9

- 5 . 9

3 . 8

1 5 . 5

25.9

3 2 . 2

z ) . I

4 . 6

1 3 . a

32.2

- 2 1 . 8

- 1 7 . 6

- 1 S . 0

o . o- 8 . 8

- 1 8 . 4

- 2 3 . 0

- 2 0 . 1

- 1 0 . 5

- 1 6 . ?

- 2 5 . 1

XRD

SM

sl{

Sl't

SI.{, XRD

XRD

XRD

ND

SM

SM

XRD

schna lz r ied (1961 )

Kriessmann and

H a r r i s o n ( 1 9 5 6 )

Epstein and

Frankiewicz ( 1958)

T e l l i e r ( 1 9 6 7 )

Pauthenet and

Boch i ro l (1951 )

Dtozzi and

Pa lad ino (1 963)

Schna lz r ied (1961 )

Coo ley and Reed (1972)

Boucher e t a t . (1959)

l reSt ( rgso)

Pauthenet and

B o c h i r o l ( 1 9 5 1 )

Ohnishi and

T e r a n i s h i ( 1 9 6 1 )

(SM = saturation nagnetization, XRD = powder x-ray diffraction, ND = neutron

d i f f rac t ion ) .

tion states (e.g. NiMn2O+). (2) CuFe2Oa transformsto a tetragonal structure on quenching. (3) There isalways the possibility that the kinetics of the changein cation distribution are so slow at low temperatureas to prevent equilibritrm or, conversely, are suffi-ciently fast at high temperature that the cationdistribution cannot be quenched. None of thesestudies contain reversal experiments. Nevertheless,the data seem consistent and support the model.

A recent study by Wu and Mason (1981) using thevariation of the Seebeck coefficient as a function ofx in Fe3Oa deserves separate consideration. Herethe range of temperatures investigated was largeenough to be sure ofa substantial variation in x, andthe precision of the determinations was far betterthan in any of the above studies. Moreover, the rateof attainment of equilibrium was demonstrated tobe rapid, and the measurements were carried out atthe temperature of interest, thereby obviatingquenching problems. An unusual feature of the datais that, unlike the other studied spinels, x in Fe3Oaappears to actually reach the random value of2l3 athigh temperatures, and perhaps go slightly beyond

it. This suggests that an excess entropy term isnecessary. Using a linear form for ASo, we ob-tained from a nonlinear regression analysis: d :24.284+ 19.15 kJ/mol, F: -22.995! 19.15 kJ/mol,ASo : -3.27!4.47 J/K mol, P : 0.986. Wu andMason themselves chose to analyze their data ac-cording to a simple linear enthalpy plus linearentropy model, which, using the same regressionprogram, gave: d : -23.643!0.78 kJ/mol, F:0,ASD : -13.83+0.61 J/K mol, f : 0.982. A non-linear enthalpy-only model gave; q. : 4l.l4l+'1.65kJ/mol, F : 31.195-10.90 kJ/mol, ASp : 0, f :0.980, which represents, however, a metastablesolution at temperatures below about 1000K. Thedramatically larger standard deviations of the pa-rameters in the three-parameter fit reflect a highdegree ofapparent correlation. It therefore does notseem possible to distinguish between the relativemerits of these models on statistical grounds aloneby using the quality of fit to the data as a criterion.This is one of the reasons we feel it is so importantto show that a non-linear enthalpy is to be expectedfrom theoretical considerations.

O'NEILL AND NAVROTSKY: SIMPLE SPINEI,S

It is not clear why Fe3Oa alone should require anexcess entropy term. Moreover, the sign of this isopposite to what one would expect for the electron-ic entropy effect in Fe2* (see Table 4). Neverthe-less, an essentially random cation distribution inFe3Oa at high temperature is consistent with analy-ses of defect concentrations and transport proper-ties, although nonstoichiometry is again a complica-tion (Dieckmann and Schmalzried, 1977). Thequestion remains whether Fe2* and Fe3* are distin-guishable ions on the octahedral sublattice andshow a configurational entropy of mixing, or wheth-er they are essentially indistinguishable and anentropy arises from partial electron delocalization,despite the fact that Fe3Oa does not exhibit metallicproperties. In the latter case, the models developedabove would clearly be inapplicable.

All the spinels listed in Table 4 are largely inverse(x > 213). Much less data exist on the change ofcation distribution with temperature for the normal2-3 spinels. CoAlzO+ has been examined a numberof times (Schmalzried, 1961; Greenwald et al.,1954; Furuhashi et al., 1973), but the results aresomewhat contradictory (apart from all authorsagreeing that CoAl2Oa is a basically normal spinel)and no one study covers a sufficient temperaturerange to apply anything but a one-parameter model.Similar conclusions hold for other 2-3 normal spin-els, with the possible exception of MgAlzO+.Schmocker et al. (1972) and Schmocker andWaldner (1976) have used the electron spin reso-nance of impurity Cr3* ions to measure x fromabout 700'C to 1000'C with a reported high degreeof precision. The two samples used were naturalspinels, and therefore not chemically pure, and,rather disconcertingly, each one gave somewhatdifferent results. Nevertheless. the form of thecurves of x against T is very similar to that obtainedby Navrotsky and Kleppa (1967) for the change inheat of solution for another natural sample of"MgAl2Oa" annealed at various temperatures. Thisis illustrated in Figure 5.

None of the models used for Fe3Oa are able toreproduce the form of the curves shown forMgAl2Oa. Indeed, the apparent sudden and rapidincrease in x (and in the heat of solution) over ashort temperature interval followed by no furtherchange at higher temperatures suggests to us thepossibility of a second-order phase transition in thisspinel, which has been wisely postulated and hotlydebated in the literature (see Smith (1978) for arecent review). Observation ofa second-order tran-

sition has been claimed by Mishra and Thomas(1977) at -450'C on a synthetic spinel grown by theCzochralski method, and anomalous thermal ex-pansion has been measured on another syntheticspecimen at 660"C by Suzuki and Kumazawa(1980). The wide range in temperatures reported forthese phenomena may be related to the thermalhistory of the spinel, impurities, or kinetic factors.Indeed, the shapes of the curves describing thevariation of cation distribution and enthalpy withtemperature may be controlled by kinetic factors,since the changes have not been shown to bereversible. Nevertheless, it does appear thatMgAlzOa, despite lending its name to the wholegroup of structures, is a rather unusual spinel.

Conclusions

The unit cell parameters, a, and oxygen parame-ters, u, of simple spinels can be predicted accurate-ly using a set of ionic radii which are modifiedslightly from those of Shannon and Prewitt. Both aand u depend on the degree of inversion, x. Latticeenergy calculations have been performed whichtake into account the depencence of u and a on x.They predict a general quadratic form for the en-thalpy of disordering, AIlp : ax * 9x2.

Most of the data on the change in cation distribu-tion of 2-3 spinels with temperature can be fittedsatisfactorily to this simple quadratic "enthalpyonly" model. We therefore believe that any vibra-tional entropy accompanying the disordering proc-ess is probably small, and that it is generally not

700 &n 9(x) ldxt 1100 r2d, ls(xt

Temperature of heat treatment ("C)

Fig. 5. Degree of disorder (x) versus temperature of heattreatment ('C) for two natural samples of MgAl2Oa fromSchmocker and Waldner (1976), and the change in heat ofsolution with temperature of heat treatment for another naturalsample of MgAl2Oa from Navrotsky and Kleppa (1967). Note thesimilarity in the forms of the curves.

E

t

F

o

E

T

O'NEILL AND NAVROTSKY: SIMPLE SPINELS t93

necessary to assume a temperature dependence forthe free energy of disordering which would benecessitated by such a nonconfigurational entropyterm, except possibly for Fe3Oa where electroniceffects may play a role.

For the 4-2 spinels the change in lattice energywith cation distribution is calculated to be muchlarger than for the 2-3 spinels. This is confirmed bythe fact that the former do not generally showintermediate cation distributions, but show only thenormal or the inverse extremes. Short-range orderin the octahedral sites of these spinels is probablyan important factor in stabilizing the inverse distri-bution. The volume change on disordering a normal4-2 spinel is predicted to be slightly positive.

The nonlinear nature of the enthalpy term meansthat the apparent "site preference energy" ofan iondepends on the structure of the spinel into which itis substituting. This will have an important effect oncation anangements in spinel solid solutions, andwill be discussed in a subsequent paper.

AcknowledgmentsWe thank T. R. Mason for providing data prior to publication

on the cation distribution in magnetite and J. Stuart for help withstatistical analysis. This work was supported by the NationalScience Foundation, Grants DMR 78-1003E and 8t-06027.

ReferencesBarth, T. F. W., and Posnjak, E. (1932) Spinel structures, with

and without variate atom equipoints. Zeitschrift fi.ir Kristallo-graphie, 82, 325-341.

Blasse, G. (1964) Crystal chemistry and some magnetic proper-ties of mixed metal oxides with spinel structures. PhillipsResearch Reports Supplement, 3, l-139.

Boucher, B., Buhl, R., and Perrin, M. (l%9) Etude crystallogra-phique du manganite spinelle cubique NiMn2Oa par difractionde neutrons. Acta Crystallographica, B.25, 2326-2333.

Callen, H.8., Harrison, S. E., and Kriessman. C. J. (1956)Cation distributions in ferrospinels: Theoretical. Physical Re-view, 103, 851-856.

Cooley, R. F., and Reed, J. S. (1972) Equilibrium cation distribu-tions in NiAl2O4, CuAl2Oa and ZnAl2Oa. Journal of theAmerican Ceramic Society, 55, 395-398.

de Sitter, J., Govaert, A., de Grave, E., Chambaere, D., andRobbrecht, G. (1977) A Mcissbauer study of Ca2* containingmagnetites. Physica Status Solidi (a), 43, 619-624.

Dieckmann, R. and Schmalzried, H. (1977) Defects and cationdifusion in magnetite (II). Berichte der Bunsen-Gesellschaftftir Physikalische Chemie, 81, 414419.

Dunitz, J. D., and Orgel, L. E. (1957) Electronic properties oftransition metal oxides II. Journal of the Physics and Chemis-try of Solids, 3, 318-323.

Epstein, D. J., and Frankiewicz, B. (1958) Some properties ofquenched magnesium ferrites. Journal ofApplied Physics, 29,376-37',r.

Furuhashi, H., Inagaki, M., and Naka, S. (1973) Determination

of cation distribution in spinels by X-ray diffraction method.Journal of Inorganic and Nuclear Chemistry, 35, 3009-3014.

Glidewell, C. (1976) Cation distribution in spinels. InorganicaChimica Acta. 19. L45-48.

Greenwald, S., Pickart, S. J., and Grannis, F. H. (1954) Cationdistribution and g factors of certain spinels containing nickel,manganese, cobalt, aluminum, gallium and iron. Journal ofChemical Physics, 22, 1597-1ffi.

Hardy, A., Lecerf, A., Cault, M., and Villers, G. (1964) Pr€para-tion, propri6t6s crystallines et magndtiques de I'orthotitanatede mangandse Mn2TiOa. Comptes Rendus de Academie Sci-ence, Paris, 259, 3462-3465.

Hermans, L., Weenk, J., and Van Gool, W. (1974) Electrostaticenergy of cubic spinel structures. Zeitschrift fiir PhysikalischeChemie, Neue Folge, EE, 15-22.

Hill, R. J., Craig, J. R., and Gibbs, G. V. (1979) Systematics ofthe spinel structure type. Physics and Chemistry of Minerals,4,317-340.

Jacob, K. T., and Alcock, C. B. (1975) Evidence of residualentropy in the cubic spinel Zn2TiOa. High Temperatures-HighPressures. 7. 433439.

Jacob, K. T., and Alcock, C. B. (1977) Activities and theirrelation to cation distribution in NiAl2Oa-MgAl2Oa spinel solidsolutions. Journal of Solid State Chemistry,20,79-BE.

Kriessman, C. J., and Harrison, S. E. (1956) Cation distributionsin ferrospinels ; magnesium-manganese ferrites. Physical Re-view. 103. E57-860.

Mason, T. O., and Bowen, H. K. (1981) Cation distribution anddefect chemistry of iron-aluminate spinels. Journal of theAmerican Ceramic Society, 64, 86-m.

McClure, D. S. (1957) The distribution of transition metal cationsin spinels. Journal ofPhysics and Chemistry ofSolids, 3, 3ll-3r7 .

Miller, A. (1959) Distribution of cations in spinels. Journal ofApplied Physics (Suppl), 30, 245-255.

Mishra, R. K., and Thomas, G. (1977) Structural phase transitionin the spinel MgAlzO+. Acta Crystallographica, A33, 67E.

Mozzi, R. L., and Paladino, A. E. (l%3) Cation distributions innonstoichiometric magnesium ferrite. Journal of ChemicalPhysics, 39,435439.

Navrotsky, A. (1977) Calculation ofeffect ofcation disorder onsilicate spinel phase boundaries. Earth and Planetary ScienceLetters. 33.437-442.

Navrotsky, A., and Kleppa, O. J. (1967) The thermodynamics ofcation distributions in simple spinels. Journal oflnorganic andNuclear Chemistry, 29, 2701-27 14.

Ndel, L. (1950) Aimantation 6 saturation de certains ferrites.Comptes Rendus de Academie Science, Paris, 230, 190-192.

Ohashi, Y., and Burnham, C. W. (1972) Electrostatic andrepulsive energies of the M1 and M2 cation sites in pyroxenes.Journal of Geophysical Research, 77, 5761-5766.

Ohnishi, H., and Teranishi, T. (1961) Crystal distortion in copperferrite-chromite series. Journal of the Physical Society ofJapan, 16,35-43.

Pauthenet, R., and Bochirol, L. (1951) Aimantation spontaneedes ferrites. Journal de Physique et de Radium, 12,249-251.

Porta, P., Stone, F. S., and Turner, R. G. (1974) The distributionof nickel ions among octahedral and tetrahedral sites inNiAlzOr-MgAl2Oa solid solutions. Journal of Solid StateChemistry, ll, 135-147 .

Preudhomme, J., and Tarte, P. (1980) Studies of spinels, VII.Order-disorder transition in the inverse germanate spinels.Journal of Solid State Chemistry , 35, 272J77 .

194 O,NEILL AND NAVROTSKY: SIMPLE SPINELS

Robertson, J. M., and Pointon A. J. (l%6) The cation distribu-tion in nickel ferrite. Solid State Communications, 4,257-259.

Roiter, B. D., and Paladino, A. E. (1962) Phase equilibria in theferrite region of the system Fe{o-O. Journal of the AmericanCeramic Society, 45, l2E-133.

Sawatzky, G. A., Van der Wande, F., and Mimrsh, A. H. (1969)Mdssbauer study of several ferrimagnetic spinels. PhysicalReview, 187 ,747-757.

Schmalzried, H. (l%l) Rcintgenographische Untersuchung derKationenverteilung in Spinellphasen. Zeitschrift fiir Physika-lische Chemie, Neue Folge, 2E,203-219.

Schmocker, U., Boesch, H. R., and Waldner, F. (1972) A directdetermination of cation disorder in MgAl2Oa spinel by E.S.R.Physics Letters, 40A, 237 -218.

Schmocker, U., and Waldner, F. (1976) The inversion parameterwith respect to the space group of MgAlzOr spinels. Journal duPhysique, C9, 1235-1237 .

Senderov, E. E. (1980) On the theory of Al, Si ordering in albite.Physics and Chemistry of Minerals, 6,251-268.

Shannon, R. D. (1976) Revised effective ionic radii and systemat-ic studies of interatomic distances in halides and chalcogen-ides. Acta Crystallographica, A32, 7 5l:l 67 .

Shannon, R. D., and Prewitt, C. T. (1969) Effective ionic radii inoxides and fluorides. Acta Crystallographica, B25, 925-946.

Smith, P. P. K. (197E) Note on the space group of spinelminerals. Philosophical Magazine, 3E, 99-102.

Subramanyan, K. N., and Khare, L. R. (1979) The efect of heat

treatment on the lattice parameter of nickel ferrite. ActaCrystallographica, 83 5, 269.

Suzuki, I., and Kumazawa, M. (1980) Anomalous thermal ex-pansion in spinel MgAl2Oa, a possibility for a second orderphase transition. Physics and Chemistry of Minerals, 5,279-284.

Tellier, J-C. (1967) Sur la substitution dans le ferrite de magn6-sium des ions ferriques par les ions trivalents, t€travelents etpentavalents. Revue de Chimie Minerale, 4, 325-365.

Thompson, P., and Grimes, N. W. (1977) Madelung calculationsfor the spinel structure. Philosophical Magazine, 36, 501-505.

Tosi, M. P. (1964) Cohesion of ionic solids in the Born model.Solid State Physics, 16, l-120.

Verwey, E. J. W., de Boer, F., and Van Santen J. H. (1948)Cation arrangements in spinels. Journal of Chemical Physics,16, 1091-t092.

Wood, B. J. (1981) Crystal field electronic efects on the thermo-dynamic properties of Fe2* minerals. In R. C. Newton, A.Navrotsky and B. J. Wood, Eds., Thermodynamics of Miner-als and Melts, p. 63-84. Springer-Verlag, New York.

Wu, C. C. and Mason, T. R. (1981) Thermopower measurementof the cation distribution in magnetite. Joumal of the AmericanCeramic Society, 64, 23'7 -242.

Manuscript received, January 26, 1982;accepted for publication, August 23 , 1982.

spinel structure

Documents