A theory of the pressure-induced high-spin—low-spin transition of transition-metal oxides

130

Physics of the Earth and Planetary Interiors, 17 (1978) 130—139 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

A THEORY OF THE PRESSURE-INDUCED HIGH-SPIN-LOW-SPIN TRANSITION OF TRANSITION-METAL OXIDES SHUHEI OHNISHI Geophysical Institute, Faculty of Science, University of Tokyo, Tokyo (Japan)

(Received August 23, 1977; revised and accepted September 13, 1977)

Ohnishi, S., 1978. A theory of the pressure-induced high-spin—low-spin transition of transition-metal oxides. Phys. Earth Planet. Inter., 17: 130—139. The pressure-induced high-spin—low-spin transition of the transition-metal ions in octahedral coordination is studied theoretically. The relation between the crossover point and the transition point is discussed and the formula to determine the transition point isgiven in terms of the crystal-field splitting ~ and the spin-pairing energy [I as ~ = all. The numerical coefficient a is determined by the ratio of the change of the interatomic distances between the transition-metal ion and ligands. It is proved that a is generally less than 1 and takes a value of about 0.95—0.80 for the transition-metal oxides. Based on the discussions on the F— V relation of the low-spin oxides, transition pressures are estimated to be about 700—1300 kbar for MnO, CoO and Fe 203, and about 250—400 kbar for FeO. The magnitude 1 for transition-metal ofthe shifts of oxides. transition It ispressures discussed due whether to entropy the transition changes isatroughly high temperature evaluated to takes be about place con0.1 tinuously kbar Kor not. For MnO, FeO and Fe 2 03, the high-spin—low-spin transition will take place gradually at temperatures higher than about 2000 K. For CoO, the gradual transition will take place at above 300 K.

1. Introduction It is expected that transition-metal ions in oxides and silicates will take a low-spin state under2~ion high presin sure. Especially, it is considered Fe will be octahedral coordination with six that 02 the ligands in a low-spin state in the lower mantle of the earth, Several discussions concerning the possibility of the spin-pairing of the Fe2~ion have been made. Fyfe (1960) suggested that the pressure-induced high-spin—low-spin transition has important consequences in geochemistry and geophysics. Since then, Strens (1969, 1976), Burns (1970), and Gaffney and D.L. Anderson (1973) made some related studies on the basis of semi-experimental evidence. However, the physical meaning of this transition has not been clanfled and discussions on the transition point have not yet been done thoroughly. Theoretical considerations on the transition point are necessary to study the

pressure-induced high-spin—low-spin transition. The criterion that determines which spin state the transition-metal ion prefers is given by the crossover point, the state pointofatthe which Hund’s rule on electronic i.e., ground transition-metal ionthe breaks down. The crossover point is given by the vertical lines for d4—d7 systems in the Tanabe-Sugano diagram (Tanabe and Sugano, 1954) and is expressed by using the spin-pairing energy H as: 11(V) = (1) where 11 is the crystal-field splitting and Vc is the volume. The formula for Hun terms of Racah parameters B and Cwas given by Orgel (1955), Griffith (1956) and Jç6rgensen (1962) by averaging the electronic energy differences of the two limiting cases, namely weak and strong fields. However, it must be noted that at the crossover point the electronic energies of both states become equal and, hence, is not an equi-

131

librium point of both phases. Transition pressures will not be given by eq. 1 in the strict sense. In this paper, I will give the relation between the crossover point and the transition point, and I will investigate transition pressures of the transition-metal oxides, MnO, FeO, CoO and Fe203. The temperature effect on the high-spin—low-spin transition, which involves interesting problems to be solved theoretically and experimentally, is investigated by taking the entropy change into consideration. Transition pressure is shifted by terms due to the change of vibrational frequences of phonons and that of the electronic configuration of d-electrons in t2g and eg orbitals. Strens (1969, 1976) estimated the pressure shift approximately. In this paper, I will re-evaluate it by correcting the electronic entropy term of Strens. The electronic transition from high to low spin may undergo a gradual transition at high2~ temperature as the undergoes divalent toa trivalent iontransition in Sm and Sm3~systems continuous as discussed by P.W. Anderson and Chui (1974) (see also Slichter and Dnickamer, 1972). Therefore, it is expected that the high-spin-—low-spin transition of transition-metal oxides may also take place gradually at high temperature; high- and low-spin ions at high pressure and temperature will coexist randomly in a crystal. In this paper, I will discuss the possibility of the continuous transition and evaluate the temperature at which it happens. 2. The crossover point and the transition point It is well known that the d-electrons of a transitionmetal ion are localized around a nucleus and in a crystal the transition-metal ion with its nearest-neighbour anions (six 02_ ions are considered in this place) shows molecular-like behaviour in its electronic properties. This system is called the transition-metal cluster (TMC). Transition-metal oxides are approximately regarded as an aggregate of TMC’s connected by cornmon ligands, e.g., 02 ions. Hence, the contribution from the electronic energy of d-electrons to the total electronic energy of the crystal will be given by N times the electronic energy in the TMC (N is the number of transition-metal ions in a crystal). We express the total energy of the crystal as: =

~( V) +

Unuci( 1”)

(2)

where Vis the volume of the crystal. The subscript k represents an electronic ground state, either the highspin or the low-spin state. Ufluci is the electrostatic interaction energy between nuclei in all lattice points. The total electronic energy, Ei’, is given by the adiabatic potential energy if we follow the Born-Oppenheimer adiabatic approximation. The change of the electronic ground state of d-electrons will induce the change of the charge distribution in a crystal. However, if the lattice configuration were kept unchanged, the change of the charge distribution in a TMC will not cause a serious disturbance on ions outside the cluster due to the localizability of d-electrons. Then, we assume that the electronic energy difference between high- and low-spin states is given by the energy change in the TMC under fixed lattice configuration. In the first-order approximation, the energy change in theenergy TMC is(CFSE) given by the crystal-field stabilization andthat theofelectron—electron interaction energy. Then: (TA

T

(TA

—

‘4’H~~)

~‘)

VelrrA V)

— “Lv

LCIfTA ~ H’~ ~‘)

—

S

.

.

.

— —

(3) .

4

where s is the number of the spin-pairing, I for d and d systems, and 2 for d 5 and d 6 systems in octahedral coordination. Then, eq. 1 is given by: .~

.

sN[fl

—

11(V~)}=

WL(VC)

—

WH(VC) =

0

(4)

Therefore, the crossover point is not a true transition point because an equilibrium is not achieved sirnultaneously for both states at the same volume V~under the same pressure. Equilibrium volumes Vj~,V~in the two states are determined by the equation of state as: (dWHIdV)v=v~= _(dWLIdV)v=vL

~

(5)

In what follows, a symbol attached to a quantity means the value in equilibrium. The transition point should be determined by the zero-point of the Gibbs free-energy difference between both states. The Gibbs free-energy difference is given by: *

—

—

5 W = WL(VL)

—

WH(VH)

(6)

—

where Wk = Wk + Pl7k is the Gibbs free energy at zero temperature. The equilibrium value, S W~’,at pressure Pis given by putting V~ineq. 5 into Vkin eq. 6. We decompose the above equation into the following two

132

parts as:

ENERGY

S W = EWL(V~) Wu(VL)I —

+

[~H(VL)

—

~u(VH)]

~ A=rr

(7) The first part of the above equation is reexpressed by using the relationship represented in eq. 3 as follows: SW=sN[ll—11(VL)] + [~u(VL)—

WH(VH)]

o I I I

(8)

—

WL(VH)1

I I

I

Or, in the same manner, S W is also expressed as SW=sN[H—11(VH)] + [i~L(VL)

W~

V~Vc V~ (8a)

Strictly speaking, 11 and H in eqs. 8 and 8a are quantities to be evaluated at the low- and high-spin states, respectively. However, within a perturbation treatment in quantum mechanics, the common wave functions for both states are used to obtain electronic energies. Hence, the difference in 11 and in eqs. 8 and 8a will be approximately given by considering only the difference of the volume. We express the volume dependence of 11 of the transition-metal oxides as: 11(V) = v—~°I~ (9)

LATTICE CONFIGURATION

Fig. 1. Gibbs free-energy curves of the transition-metal oxide in high- and low-spin states at the transition point. Both states are under the same pressure condition. V~and V~.are equilibrium volumes. V~represents a volume corresponding to the crossover point.

Eliminating V~K~ in eq. 10 with this equilibrium value x~,5 W~is approximately re-expressed as follows: = SNEH 11(V~) (m/6) 11(V~)x + ...] (12)* —

—

*

At the transition point, 11(V~)and H are related by the condition of S W’~= 0.

Within the treatment of simple point-charge crystalfield theory, m is equal to 5. Experimentally, this r5 law seems to be a rather good approximation (Stephens and Drickamer, 1961; Drickamer, 1967) (r is the interionic distance between the transitionmetal ion and ligands). At the same time the spinpairing energy iwifibe assumed to be almost constant with the variation of the lattice constant because His the intra-ionic quantity being given by the Racah parameters. If the P—V relation of the high-spin form is given, the equilibrium volume of the low-spin form is determined by minimizing 514/with respect to VL. Expanding eq. 8 around V = V~with the ratio of the volume change:

a = 1/(1 + *16) (14) In Fig. 1, the relation between the crossover point and the transition point is illustrated. It is not difficult to see from eqs. 1, 9, 13 and 14 that V~in Fig. 1 corresponds to the middle point of V~and V~approximately. Eq. 14 shows that the coefficient a is smaller than 1. It also shows that the high-spin—low-spin transition occurs when the crystal-field splitting 11 of the high-spin state takes a smaller value than the spinpairing energy H. In order to see the value of a, we

x

estimate x’ from the linear relationship between the

=

(Vii

—

VL)IV~

11(V~)= aH

(13)

where

SWis expressed by: SW = sN[H —

—

11(V~

1)— (m/3) 11(V~)X 2 + ...] + V~K~x2/2 + {m(m + 3)/I 8} 11(V~) x (10)

where K~is the bulk modulus of the high-spin oxide, V~is given by the equation: (35 ~i’/ax)~~*

0

(11)

*

We can solve eq. 11 with a perturbation method. If we write the solution of eq. 11 as x~+ 4 + ..., where x~is given by considering terms up to the second order ofx in eq. 10 and where 4 is the residual term (if we take thirdorder terms of x in eq. 10 into consideration), x~in eq. 12 should be substituted with xc~+ (2/3)4. But, 4/3 is negligible becausex~is the order of x~2.Hence, eq. 12 holds approximately as it is.

133

in Table I. Although these results are not sufficiently accurate to determine the transition pressure, we

6C

// 50

M

cannot neglect a in eq. 13. Ifm in eq. 9 is greater than 5, a takes a smaller value than that of the model of m = 5, and f will become larger.

203(corundum)

Fe

-

Cr Iii ~

-J

40

-

M02(Rutite)

/AL

o

3. P— V relation of the low-spin oxides

>

< -J

30

P’Tj

In order to determine the transition pressure, we must know the P—Vrelation of the low-spin form. For the high-spin transition-metal monoxides, Hush and Pryce (1958) explained a periodic variationbyofconthe interatomic distances with the atomic number

~Cr

-

MO(NaCl)

U

W

_...—

-J

0

20

Mn

-

-~~NjMg

10

I

I

I

I

A3)

3( 0.4 r Linear relationship between 0.2

0

I

I

I

0.6

I

0.8

Fig. 2. the volume of oxides and the ionic volume. Ionic radii are those of Shannon and Prewitt (1969). Mrepresents a transition-metal ion.

ionic volume of transition-metal ions and the molecular volume of oxides as shown in Fig. 2, where we employ the ionic radii given by Shannon and Prewitt (1969). Assuming m = 5 in eq. 9, values of a are listed TABLE I

sidering the effect of CFSE with the assumption of m = Sin eq. 9 on the normal ionic crystal. The variation of bulk moduli of the high-spin transition-metal monoxides is also explained by developing their treatment (Ohnishi and Mizutani, 1978). These results show that the assumption of m = 5 in eq. 9 is acceptable and that the effect of the change of the electronic ground state of the transition-metal ion is approximately given by that of the change of CFSE, because the electronic ground state changes by replacing ions with other transition-metal ions of the same charge. Then we assume that the P-- V relation of the low-spin form is given by superposing the pressure Pc(V) which comes from the change of CFSE at the high-spin— low-spin transition to that of the high-spin one:

Values of a estimated from Fig. 2 PL(V)

=

PH(V) + Pc(V)

(15)

rj 00 (A)

Putting m = 5 in eq. 9, P~(V)is expressed by: Pc(V) =Pc(Vo)(V/Vo)813

LS 0.73 0.58

1’ x (0.13) * (0.08)

a

HS 0.82 0.65

0.90 0.94

where V

(16)

d4

Cr2~ Mn3~

5

Mn2~

0.82 0.65

0.67 0.55

0.23 0.14

0.84 0.90

0 is thea zero-pressure zero-pressure value volumeof of high-spin form. Writing thethe crystal-field splitting as 11~,Fc(Vo) is given by:

d6

Fe2~ Co3~

0.77 0.61

0.61 0.53

0.20 (0.12)

0.86 0.91

Pc(Vo) = —(5/3) sN11o/Vo

d7

Co2~

0.74

0.65

0.11

0.92

Equilibrium

Ni3~

0.60

0.56

(0.06)

0.95

d

Ionic radii are those of Shannon and Prewitt (1969). HS and LS represent high- and low-spin values respectively. * Values in () are hypothetical quantities estimated from ionic radii in Fig. 2. x’1’ and a of divalent ions are estimated from the transitionmetal monoxide MO, and those of trivalent ions are estimated from the corundum structure M 2O3.

(17)

volumes ofboth states are determined by the following equations p

=

PH(V~)= P~(V~) + Pc(V~)

(18)

Values of 11~used in the calculation are listed in Table II. For FeO, 11~is calculated from the coefficient of the crystal-field potential term which was used

by Clendenen and Drickamer (1966). For MnO

134 TABLE II Data used in the calculation and transition pressures MnO 3) V0 (10~2erg) (A ~o Ho(10~2 erg) s N K 0 (Mbar) HS LS *1 K~ HS LS ~ P*(kbar) (a) 2 (b) (a) a* (b) x~

a*3 *1 *2 *3

FeO

CoO

21.97 1.94 4.36 2 1

20.32 2.34 3.63 2 1

19.30 1.76 3.49 1 1

1.47 1.99 4.0 3.87 1310 6600.87 0.91 0.16 0.88

1.74 2.44 4.0 3.86 360 230 0.83 0.87 0.20 0.86

1.81 2.04 4.0 3.92 1450 720 0.96 0.96 0.064 0.95

Fe

2 03 50.27 2.68 5.07 2 2

2.07 2.93 4.53 4.55 1390 7200.91 0.93 0.12 0.91

Calculated values. K~,= 4.0 is the assumption. (a) and (b) are models with n Determined from Fig. 4 atP = Determined from eq. 14.

and CoO, observed values obtained from the optical spectra are used (Pratt and Coelho, 1959). (11o of CoO is not so different from the value calculated in the same manner as that of FeO.) For Fe 203, 11~is derived from data in aqueous solution (Tanabe and Sugano, 1954). As to the P—V relation of the highspin form, we adopt the Birch-Murnaghan equation of state such as: 713 (V/V 5131 K PH(V) = (3/2)[(V/V0) 0) 0 213 1}] (19) x [1 +(3/4)(K~—4)~(V/V0) For the bulk modulus K 0, we employ elastic data obtained from the ultrasonic measurements (FeO, Mizutani et at., l972;MnO and CoO, Uchida and

=

0 and n

=

3, respectively.

4. Transition pressure We can determine transition pressure P~from eqs. 8 and 8a, or the approximated equation (13). Since the energy difference WH(V~) i~~(V~) is obtamed by integrating the P—V relation with V as: —

*

VL

14/H(VL)

—

WH(VH)

=

the pressure derivative of the bulk modulus K’0, there Saito, 1972; Fe2O3, O.L. Anderson et a!., 1968). For are no available data determined by the ultrasonic measurements except Fe203 (O.L. Anderson et al., 1968). Then, we assume K’~= 4 for all transitionmetal monoxides. In Table II, data used in the calculation are listed. Calculated results of the volume change ratio x~are shown in Fig. 3. They are almost constant with pressure. Therefore, we may consider that a in eq. 14 is a constant parameter.

f

*

PH(V) d V + P(V~— V~~)

VH

—

—

—

(20) we use eq. 8 and calculate the pressure change of 0.2

0.2

‘I

~ > 0.1•

CoO

0

I

I 200

I

I 400

I

I 500

I

I 800

I 1000

P(kb) Fig. 3. Volume change ratio due to the high- to low-spin transition.

135 rr(nO) —‘

4

TQ

3

n~0) Tr(n3)

~

w z Ui

I

—I

—2

-

400

800

4

~0

3

Ui

0)

-1-

MnO

_

I ~00

200

FeO

__________________________________________________

I

600P(kb)

3)

—2 10

©

5

—.

z

P(kb) j600

I 1200

©

8 ,~

-

&

6

0

Tr(n~3)

z

(!) U) Lii

C2

~—-

400

800—.~~1200

£

I P(kb) i~acr-

-1-

0 Ui w z

0

2

~IIi

~-P(kb) 1200 ç~-lQOO

-

400

‘~800

-2.

CoO -2

—4

Fe203

Fig. 4. Pressure changes of energies of (A) MnO; (B) FeO; (C) CoO; and (D) Fe203. and correspond to models of n = 0 and n = 3, respectively.

~

~

(i),

As to the spin-pairing energy, we employ the

formula for H given (1956). the Racah parameters B by andGriffith C, H’s are givenWith by (15/2)8 ÷ SC, (S/2)B + 4C, 4B + 4C, and (1 5/2)B + SC for MnO, FeO, CoO and Fe 203, respectively. For values of B and C, we use data given by Tanabe and Sugano (1954) for FeO and Fe203, and those given by Pratt and Coelho (1959) for MnO and CoO. The Racah parameter will decrease with increasing pressure because of the delocalization of d-electrons. According to experimental evidence, the decreasing rate of B of the M06-type and100 silicate seems to TMC be at of most several in peroxides cent per kbar. Since we can calculate the bulk modulus and its pressure derivative of the low-spin form from eqs. 18 and 19, we can obtain the term WL(VL) — WL(Vi-i) in eq. 8a in the same manner as eq. 20 and calculate the pressure change of 6 W~. However, calculated results of P~from eq. 8a gave almost sanie values as those from eq. 8. Therefore, we used eq. 8 in the present calculation.

From the optical spectra of the spin-allowed transi2~in tion transition-metal impurities, Cr~and Ni the MgO,ofMinomura and Drickamer (1961) estimated change in B with pressure to be about 3% per 100 kbar. While, from the observation of the spin-forbidden bands being dependent only on the Racah parameters, Abu-Eid (1976) concluded that the change in B with pressure was negligible. Gaffney and D.L. Anderson (1973) evaluated the upper and lower bounds of P~ of FeO from the crossover point with the model of n = 3 and n =1.6,However, where they the variability the expressed upper bound of P~ of B as B ~ r’ should be given by the model of n = 0. If we use P—V relations of transition-metal oxides, the model of n = 3 corresponds to the assumption that B decreases by about 5—7% per 100 kbar. To see the effect of the pressure change in B on P~,we adopt the model of n = 3 together with that of n = 0, and will compare P~‘s with their estimate of FeO. In Fig. 4, energy differences between both states are illustrated together

136

with pressure changes of 11 and H. Zero points of curves of S ~ give transition pressures. P~of MnO, CoO, and Fe203 are very high on account of the large spin-pairing energy. Values ofP~and x” of Fe2O3 are consistent with the estimates from shock-wave experiments (Davies and Gaffney, 1973; Syono et al., 1977). The upper bound of P~of FeO becomes 360 kbar without assuming the variability of B with pressure. Comparing with the estimate from the crossover critenon given by Gaffney and D.L. Anderson (1973), P~ decreases about 200 kbar (K’0 = 3.4 was assumed in the present calculation also). Coefficients a determined from values of 11 and H at the transition point in Fig. 4 are listed in Table II. Values of a calculated from eq. 14 usingx* in Fig. 3, are also listed in Table II. They are almost equal to those given from Fig. 4. Therefore, it is very useful to evaluate the high-spin— low-spin transition point from the approximated eq. 13.

S. Temperature effects At a finite temperature, the entropy difference between the high- and low-spin states should be taken into consideration. At first, we consider the case that all transition-metal ions in a crystal become low-spin states. The entropy change is given by the sum of two terms, SSei and SS,,,, which are due to the change of of the electronic configuration of d-electrons and that of vibrational frequences of the lattice, respectively, The transition pressure is shifted upwards by these terms. Strens (1969, 1976) discussed the temperature effect on the transition point by estimating SSei and 5S~approximately. bSei is given by the change of the degeneracy, m(2a + 1), of multiplets (a and m are the total spin and orbital degeneracy, respectively) as: bSet

=

NkB ln E(20L + I )/(2a~+ 1)]

(21)

where kB is the Boltzmann constant. SS,,, will be approximated by applying the thermodynamical relation such as: = —

X”IthCV

(22)

where 7th is the Grüneisen constant and Cvis the specific heat. Assuming that 7th and C~,are constants with pressure and temperature, we can roughly evalu-

ate the magnitude of shifts of P~’by displacing a horizontal axis in Fig. 4 by —T(SS~+ SSei) downward (Tis the temperature). Results are listed in Table III. Roughly speaking, p* is shifted by these entropy terms by 100 kbar per 1000 K. Since it depends on the local condition around the TMC whether the transition-metal ion takes the high-spin state or not, transition-metal oxides contaming low-spin ions as impurities may be stable at high temperature. If the system in which high- and low-spin ions coexist randomly is in equilibrium, the mixing entropy SSmix and the interaction energy between low-spin impurities, S W~,,t,should be taken into account. SSmix is given, with the low-spin concentration c, by: SSmix = —NkB[c ln c + (1 — c) ln(l — C)] (24) Since the energy difference between TMC’s in both states becomes very small near the transition point, the mixing entropy term TSSmix added to Gibbs free-energy change SG will become dominant as temperature rises. Then, at very high temperatures, we can expect that the system is stabilized by this entropy contribution and that a configurational mixing of high- and low-spin ions will occur. Since an interaction between volume defects caused by the displacement field around small low-spin ions is attractive, the critical temperature Tc is analogous to the Curie temperature at which ferromagnetism exists (Slichter and Drickamer, 1972). If the temperature T is greater than Tc, SG will have one minimum with respect to c. Otherwise, it will have more than two minima. Therefore, it is determined by the condition of T> T~or T < T~whether the transition takes place continuously or not. In the case of dilute solutions of transition-metal ions in oxides, T~may be small because of the weak interaction between low-spin impurities of low concentration. In the case of transition-metal oxides themselves, there would be strong interactions between impurities and Tc will take a large value. The value of T~is effectively determined by the strain interaction energy caused by the volume mismatches of low-spin ions. S W1,~is calculated by applying the sphere-in-hole model, as was described by Friedel (1954), Eshelby (1956) and P.W. Anderson and Chui (1974). Following the assumption made by them that impurities distributing randomly in a crys-

137 TABLE III Temperature effects

20H+l)mH (20L+l)mL ( SSei (10—16 erg deg~) GrUneisen constant *1 6S~(l0~6 erg deg’) (a) (b) Pressure shift (a) (kbarK~) (b) ~ (Mbar) R*2 (Mbar) y 6 V (A3) (a) (b) Tmax(K) (a) (b)

MnO

FeO

CoO

6 6 0.0 1.52 —1.44 —1.54 0.063 0.036 0.783 1.73 1.60 2.35 2.88 1700 1500

15 1 —3.74 1.67 —2.42 —2.61 0.20 0.12 0.543 2.09 1.35 3.55 3.93 1300 1250

12 4 —1.52 1.65 —0.58 —0.73 0.26 0.15 0.803 1.92 1.56 0.84 1.04 250 215

-

Fe

2O3 6 6 0.0 1.99 —5.90 —6.51 0.14 0.082 0.91 2.50 1.49 2.10 2.63 1230 1170

(a) and (b) are models of n = 0 and n = 3, respectively. ~ For Fe 203, the value given by 0.L. Anderson eta!. (1968) is used. For others, they are calculated from data of Clendenen and Drickamer (1966). *2 Calculated from Table II.

tat displace the lattice uniformly and linearly, the harmonic strain interaction energy in S ~ per formula is given in terms of the volume change of the whole crystal S V resulting from the volume mismatch of one transition-metal impurity: 2NF (25) Swhere Wint = c

high-spin form and evaluate the high-pressure value with the assumption that the Poisson ratio is constant with pressure. Then, 7 becomes also constant with pressure. Since the additivity rule on the total volume change due to the volume mismatch of inclusions, Vegard’s law,volume is assumed in our between model, S the V will be and given by the difference highlow-spin oxides which are regarded as the end-members of the solid solution of both ions. If we take only

F = (2~/3)(SV)2/(

2-term inS Wjnt into consideration, SG has at the mostc double minima and two points of inflection with respect to c. (The contribution from other terms except S W~and SSmjx to SG are considered to be a linear function of c because of the additivity assumption of S V and the localized nature of the TMC.)

7V1)

(26)

in which 7 = [I + (4j1/3K)J —

K and ~ are respectively the effective bulk modulus and rigidity of the crystal containing both ions. V 1 is the total volume of formula per transition-metal ion, viz., V~= V/N. In the above eq. 26, the difference in the modulus between the low-spin inclusion and the bulk matrix is neglected. Instead, we estimate K with the geometric average of bulk moduli in both states. The value of V~is also given by the mean value of the high- and low-spin forms. Because we shall make an order-of-magnitude estimate in this place, we can approximate the effective rigidity ~ with that of the

Therefore, the critical temperature T~is determined by the condition that a minimum point coincides with the point of inflection at T= Tc. = 0 2SG/ac2)~_~~ (27) (3bG/3c)~~~ T=T = (a TT C

C

being a function of c~,takes a maximum value at c~= ~. Denoting it as Tmax, Tmax is given by the equation. Tc,

Tmax

=

r/2kB

(28)

138 In Table III, values of Tmax at P = P~are listed for models of n = 0 and n = 3. Although some inaccuracy Ofl Tmax cannot be avoided due to the uncertainty of ~at high pressure, we can understand the qualitative nature of this transition from the knowledge of a value of Tmax. In general, the high-spin—low-spin transition of Co2~(and probably Cr2~)in octahedral coordination of which number of spin-pairing is 1 (s = I) will take place gradually at high temperature more than at about room temperature because S V could not 2~, take large value such as the casecoordination, of s = 2. For Mn Fe2~aand Fe3~ions in octahedral the high-spin—low-spin transition will take place discontinuously at temperature lower than 1000—2000 K. However, the temperature at the depth where the spin-pairing transition transition-metal ions such as 2~and Fe3~ may takeofplace in the earth’s mantle is Fe probably higher than 2000 K, so the spin-pairing transition in the earth’s mantle may be rather gradual. Therefore, it could be very difficult to detect such a transition seismologically in the earth.

6. Discussion As was discussed in Section 2, the transition point of the high-spin—low-spin transition should be determined by taking the value of the parameter a into consideration. For other pressure-induced electronic transitions, such as the metal—insulator transition, the transition point might not be given by considering only the electronic energy difference. It will be a future problem to give a criterion for determining the transition point of other pressure-induced electronic transitions. The transition pressure of the high-spin—low-spin transition is greatly influenced by the choice of the values of 11~and no• In particular, data for FeO Will have some uncertainties due to non-stoichiometry. 11~of FeO used in the present calculations (11,800 cm’), which was used by Clendenen and Drickamer (1966) and was originally evaluated by Hush and Pryce (1958) from the thermodynamical data, is somewhat larger than that determined opti2~in aqueous solution from [11~=the 10,300 cal spectra of Feand Sugano (1954)]. [For CoO, there cm~in Tanabe is not such a large discrepancy between the thermodynamic data (11~= 8,850 cm~in Clendenen and

Drickamer, 1966, and the optical data (11~= 8,880 cm_i in Pratt and Coelho, 1959).] The change ~f~* of FeO caused by the reduction of 11~to 10,300 cm_i is easily evaluated with the aid of eqs. 13 and 19 with a’s in Table II. Re-evaluated values of P~become about 690 kbar (n = 0) and 470 kbar (n = 3). Thus, in FeO, the reduction of 11~by about 13% produces an increase of P~by about 200—300 kbar. Concerning a value of H 0,the same discussion holds. The value the be Racah parameter of evaluated TMC in the low-spin stateofmay smaller than that at the high-spin state on account of the covalency of d-electrons with ligands. The value of H 0 will be somewhat smaller than those used in the present calculalions. p* will decrease with decreasing H~in the same manner the case of 11g. We need more accurate data on theseasparameters. The nature of the high-spin—low-spin transition at high temperature is characterized by the magnitude of the critical temperature. Compared with the value for MnO, Tmax of FeO seems to be somewhat small. The rigidity of FeO employed here which is obtained from data for a polycrystalline specimen is small, compared with those of MnO and CoO whose Poisson ratios are about 0.3 while that of FeO is 0.36. If we assume the Poisson ratio of FeO to be 0.3, Tmax takes a value about 1700 K. In any case, we can roughly consider that the critical temperature of MnO, FeO, and Fe 2O3 of which numbers of spin-pairing are equal to 2 are lower than 2000 K. Temperature effects on the high-spin—low-spin transition and also other pressure-induced electronic transitions will become an important problem for the future studies of the earth’s interior. Acknowledgement I am grateful to Dr. H. Mizutani for giving constant support over the long period from the first study. And, I would like to thank Profs. S. Akimoto, R.GJ. Strens andY. Ida for reading my paper and giving many suggestions. I would like to acknowledge to Prof. S. Sugano for supporting this research. References Abu-Eid, R.M., 1976. Absorption spectra of transition metalbearing minerals at high pressures. In: R.G.J. Strens (Edi~

139 tor), The Physics and Chemistry of Minerals and Rocks. Wiley, London, pp. 641—675. Anderson, O.L., Shreiber, E., Liebermann, R.C. and Soga, N., 1968. Some elastic constant data on minerals relevant to geophysics. Rev. Geophys., 6: 491—524. Anderson, P.W. and Chui, S.T., 1974. Anharmonic strain effects in crystals and mixed valence states. Phys. Rev., 9: 3229—3236. Burns, R.G., 1970. Mineralogical Applications of Crystal Field Theory. Cambridge University Press, London. Clendenen, R.L. and Drickamer, H.G., 1966. Lattice parameters of nine oxides and sulfides as a function of pressure. J. Chem. Phys., 44: 4223—4228. Davies, G.F. and Gaffney, E.S., 1973. Identification of highpressure phases of rocks and minerals from Hugoniots data. Geophys. J.R. Astron. Soc., 33: 165—183. Drickamer, H.G., 1967. Effect of interionic distance on the ligand field in NiO. J. Chem. Phys., 47: 1800. Eshelby, J.D., 1956. The continuum theory of lattice defects. In: F. Seitz and D. Turnbull (Editors), Solid State Physics, Vol. 3. Academic Press, New York, N.Y., pp. 79—144. Friedel, J., 1954. Electronic structure of primary solid solutions in metals. Adv. Phys., 3: 446—507. Fyfe, W.S., 1960. The possibility of d-electron coupling in olivine at high pressure. Geochim. Cosmochim. Acta, 19: 141—143. Gaffney, J.S. and Anderson, D.L., 1973. Effect of low-spin 2~on the composition of the lower mantle. J. Geophys. Fe Res., 78: 7005—7014. Griffith, J.S., 1956. On the stabilities of transition metal complexes, l;II. J. Inorg. Nucl. Chem., 2: 1—10; 229—236. Hush, N.S. and Pryce, M.H.L., 1958. Influence of the crystalfield potential on interionic separation in salts of divalent iron-group ions. J. Chem. Phys., 28: 244—249. J~rgensen,C.K., 1962. Chemical bonding inferred from visible and ultraviolet absorption spectra. In: F. Seitz and D. Turnbull (Editors), Solid State Physics, Vol. 13. Academic Press, New York, N.Y., pp. 375—462. Minomura, S. and Drickamer, H.G., 1961. Effect of pressure on the spectra oftransition metal ions in MgO and A12O3. J. Chem. Phys., 35: 903—907.

Mizutani, H., Hamano, Y., Akimoto, S. and Nishizawa, 0., 1972. Elasticity of stishovite and wüstite. EOS (Trans. Amer. Geophys. Union), 53: 527 (abstract). Ohnishi, S. and Mizutani, H., 1978. Crystal-field effect on the bulk moduli of transition-metal oxides. J. Geophys. Res. (in press). Orgel, L.E., 1955. Electronic structure of transition-metal complexes. J. Chem. Phys., 23: 1819—1823. Pratt, G.W. and Coelho, R., 1959. Optical absorption of CoO and MnO above and below the Néel temperature. Phys. Rev., 116: 281—286. Shannon, R.D. and Prewitt, C.T., 1969. Effective ionic radii in oxides and fluorides. Acta Crystallogr., Sect. B, 25: 925—946. Slichter, C.P. and Drickamer, H.G., 1972. Pressure-induced electronic changes in compounds of iron. J. Chem. Phys., 56: 2142—2160. Stephens, D.R. and Drickamer, HG., 1961. Effect of pressures on the spectrum of ruby. J. Chem. Phys., 35: 427—429. Strens, R.G.J., 1969. The nature and geophysical importance of spin-pairing of iron(II). In: S.K. Runcorn (Editor), The Application of Modern Physics to the Earth and Planetary Interiors. Academic Press, New York, N.Y., pp. 213—220. Strens, R.G.J., 1976. Behaviour of iron compounds at high pressure. In: R.G.J. Strens (Editor), The Physics and Chemistry of Minerals and Rocks. Wiley, London, pp. 545 —554. Syono, Y., Goto, T. and Nakagawa, Y., 1977. Phase-transition pressures of Fe 304 and GaAs determined from shockcompression experiments. In: M.H. Manghnani and S. Akimoto (Editors), High-Pressure Research. Academic Press, New York, N.Y., pp. 463—476. Tanabe, Y. and Sugano, S., 1954. On the absorption spectra of complex ions, I; II. J. Phys. Soc. Jpn., 9: 753—766; 766—7 79. Uchida, N. and Saito, S., 1972. Elastic constants and acoustic absorption coefficients in MnO, CoO and NiO single crystals at room temperature, J. Acoust. Soc. Am., 51: 1602—1605.