Isostructural phase transitions in spinels

0038-1098/85 $3.00 + .00 Pergamon Press Ltd.

Solid State Communications, Vol. 56, No. 10, pp. 905-908, 1985. Printed in Great Britain.

ISOSTRUCTURAL PHASE TRANSITIONS IN SPINELS V.M. Talanov and G.V. Bezrukov Department of Chemical Technology, Novocherkassk Polytechnical Institute, Novocherkassk, Rostovskaya obl., USSR

(Received 28 June 1984 by E.F. Bertaut) Equations are studied, which describe cation distribution in spinels. All types of the dependence of the inversion parameter on temperature are determined. The possibility of isostructural phase transitions in spinels is shown and the phase diagram is built. TEMPERATURE DEPENDENCE OF cation distribution in spinels has been studied both theoretically [1-7] and experimentally [ 8 - 1 1 ] . Six types of curves characterizing the dependence of the inversion parameter ~ on temperature T have been determined [7]. The present work deals with the new types of ;k(T), the possibility of isostructural phase transitions in spinels is shown. Let us consider a spinel (AxBl_x)[Al_xBl÷x]X4 , where tetrahedrally situated ions are denoted by enclosure in parentheses, and octahedrally situated ions by enclosure in square brackets. It is known from the experimental works that cation distribution is well described in Bragg-Williams approximation if the ~2. members are allowed for in the energy expression (cation-cation interaction is taken into account). The free energy per a formula unit can be written in the form [11]: F = 0.{60k--),~/2 + t [~,ln), + 2(1 --)01n(1 --),) + (1 + X)ln(1 + X)-- Xlnx]},

(1)

where 0, 60 and X are phenomenological parameters, × =/= 1 if one wants to take into account the entropy of ion oscillations; t = T/O, T is the absolute temperature in energy units (in equation (1) the ), independent members are omitted). The equilibrium value of X is defined from the equation ~F/O~, = 0, which can be written in the form: [ (1--70 2 ] (60 - X)/t = in X ~ ] ~ j-

(2)

Let us analyse the state equation (2), denoting the right-hand side of equation (2) as ~o~(~), the lefthand one as ~o~(;~). ¢1(X) function has a single zero ~0~(X**)= 0, where x~. =

vq+sx-l-2x 2 --2X

and a single inflexion point d2~(~,)/dk2, = O, where ~, is defined from a cubic equation: 6~,a,+3~,2.--1 = 0;

(3)

(4)

~l(X) function is illustrated in Fig. 1 where ¢2(X) function is represented by a straight line passing through ;~ = 60 point. It is convenient to analyse equation (2)graphically. Its solutions are intersection points of ~o2(h) with ¢1(~). The ~o2(X) straight line slope varies with temperature. Thus, the solution ;~(t) of equation (2) can be plotted by observation of the intersection points of straight line beam, its centre being in ~, = 60 point, with ~1(~) curve. Changing the position of the beam centre according to variation of 60 and shifting the ~, axis up and down with respect to ~l(h) function diagram in Fig. 1, which corresponds to X variation, it is possible to plot ;~(t) for all values of 60 and X parameters. Note first that at 0 < 0 for any 60, X, t there is only one point of intersection of ~0~(X) with ¢l(X) since in this case ~2(X) at T > 0 gives straight lines with a positive slope. Therefore at 0 < 0 there are no phase transitions in the system. In the future we shall assume 0 > 0. In this case at T > 0 t is positive and the straight lines ~02(~,) have a negative slope. There are not more than three points of intersection of ~o2(X) with ~1(~,). Two terminal roots should correspond to the minimum of the F function and the middle one to the maximum. It follows from the fact that ~4F/~?~4 > O. Consider an example. Let the inequalities ~ , < ~** < 60 < ;k~ < 1 be satisfied where hA is introduced in the following way. Draw a tangent to ~I(X) at its inflexion point. The intersection point of the tangent line with ;~ axis in Fig. 1 is X~ point: Xk = X, + t,~t(X,),

,

k, ~ 0.42457.

where 905

(5)

906

ISOSTRUCTURAL PHASE TRANSITIONS IN SPINELS ,X

Vol. 56, No. 10

X

>,

7 t=

h

k

XK<~
;~

"'~

0

X

trots, t ~

~

0

~o>O, X K < W < h ®

t~t m

~

k

~

-

-

~m I) X ~ < t a < a , X~ ~ X . 2) X . < X ® < X K < w < a

to = X ~ < h .

k

'

®

0

-t

2) O
k

Fig. 1. Plot of ~ol(h) function and characteristic points of the problem. X~

-

~ r

X,(1 t, = , /

~

-

l+3h,

(6)

0

~

0

~>~a, tO>XK

At t = 0 (~02(h) is a vertical line). Equation (2) has three solutions hi = 0, h2 = ~ , ha = 1. With the increase of t (one rotates the straight line with h = co centre anti-clockwise), h2 and h3 begin to converge and at some value of t = t m they merge. At t = t m ~02(h)touches ~ol(h). At t > t m only one solution is left h~ < h... Go on rotating the straight line. At a temperature tsl > tra t.02(h) touches ~ol(h) for h < h~. In a temperature range tsl < t < t,2 there are again three solutions: hi, h2, ha < h**. At t > ts2 only one solution of equation (2) is left, that tends to h** at t ~ ~ (horizontal straight line t.02(h)). The plot of h(t) see in Fig. 2(h). Other curves h(t) are plotted similarly for various relationships amongst the parameters (Fig. 2). Curves analogues to those in Fig. 2 were obtained by Gurevich and Kharkats for temperature dependence of cation distribution in superionic crystals [ 12]. The fact that the free energy has two minima is responsible for the possibility of phase transitions between them. The model parameters (6o, t, X) do not change during the transition, so the transformation is isostructural. Figure 2 defines temperatures tin, tst and t,2. At these temperatures one of the phases loses its stability. The condition under which stability loss occurs has the form a2F/~h 2 = 0. Together with the state equation (2) it gives an equation system

I'" i '

t~ "%, tsa 0 X®<~o < XK, ~0<~

X k®

(7)

to. ~m t" X . < X = = ~0

X ___

k~

/o /

~s,~

o

,.

a<~o < XK

t

X® = X. =to

Fig. 2. All possible types of h(t) curves. Continuous line corresponds to the minimum of the free energy a n d . . , to the maximum. Inequalities define relationships amongst the parameters for each curve.

t

O

XK~k® 0 5

I tO

0

~k O5

a

%

.....

O

t tO

8

I

t.

too

h(1 -- h 2) t l+3h ' = h + h(1 -- h2)~0~(h)/(l + 3h).

X.~ 1 t..

......

"t,o

05

I

I uo

O

OSh**;kK

I to

b

which can be regarded as a parametric representation of curves of stability loss on (to, t)-plane with h parameter ranging from 0 to 1.

Fig. 3. Phase diagram on (co, 0-plane. The phase equilibrium curve (thick line) and the curves of phase stability loss (thin line) are shown.

Vol. 56, No. 10

ISOSTRUCTURAL PHASE TRANSITIONS IN SPINELS

The form of the curves )`(t) calculated from formulae (7) at different values of × parameter (and hence )`**) is shown in Fig. 3. The curves of phase stability loss has a resetting point at w = ),k, t = tk. At (w = )`**, t = t**) where t,. = )`**(1 -- )`2~)/(1 + 3)`**),

(8)

Idt/du~l = ~ is satisfied. Figure 2 shows that (w = )`**, t = t**) point in Fig. 3 corresponds to the bifurcation point of X(t) curve. Now let us examine the phase transition temperature by plotting the curve of phase equilibrium. Substituting equation (2) into (1) we obtain free energy value at the bend point. F()`) = 0{)`2/2 + tin [(1 -- ),)2(1 + )`)] }.

(9)

If the values of inversion parameter in two isostructural phases are )`1 and X2 then the equality F()`,) = F()`2),

(10)

should be satisfied at the transition point. Combining equations (9) and (10) with state equation (2) for Xl and )`2 we obtain t = (Xl -- )`5)/[~(X2) -- ~0()`,)l, co = [)`,~0(x2)-x2~0(Xl)l/[~(x2)-~0()`0l,

(11) (12)

(1 -- Xl)2(1 + )`1) z1()`1 + )`2)[~0()`2) -- ~0()`~)] + In (1 -- X2)2(1 + )`2) = 0.

(13) Equation (13) determines the relationship between the values of inversion parameters in the phases at equilibrium. It does not include co, X and t parameters and can be solved numerically. Subject to this relationship equations (11) and (12) in a parametric form give a curve of phase equilibrium on (w, 0-plane. The form of this curve at different values of X parameter is shown in Fig. 3. Thus, a phase diagram of isostructural transitions in a spinel is plotted for (co, t)-variables. In many models go is monotonically dependent on pressure p. In this case Fig. 3 is equivalent to the phase diagram for (p, T)variables to an accuracy of coordinate substitution. Call the state that is the general one on the left of the phase equilibrium curve Phase I and the general state on the right Phase II. On the right Phase I is metastable and it disappears on the right stability loss curve. The same is with Phase II ont he left. From the potential (1) analysis we obtain in particular that at t = 0 )` = 1 in Phase I (a normal spinel) and )` = 0 in Phase II (an inversed spinel). At low temperatures isostructural phase are clearly discernible. If e.g. a crystal contains magnetic cations, the phases will have different magnetic characteristics.

907

The same is true for other characteristics dependent on cation distribution over sublattices. At the increase of temperature the difference between )`1 and X2 values in isostructural phases decreases disappearing at (co = )`k, t = t.) point. This is the critical point of phase diagram. Transition from one phase into another can occur continuously by the way round the critical point. This situation is analogous to that in the "liquid-gas" system. In a special case when )`** = )`. the critical point becomes the point of second order phase transition. It is easy to estimate the heat capacity jump AC stipulated by a curve break )`(t) at the transition point (Fig. 2). Simple calculations yield AC = -

1 + 3)`.

3)`.

(14)

In the cases shown in Fig. 2(e) there is also a bifurcation point. (w = )`**, t = t**) point which lies on the curve of phase stability loss corresponds to the bifurcation point. Thus a continuous branch in Fig. 2(e) loses its stability at the bifurcation point and the system transforms to a high temperature state by the first order phase transition. Parallel examination of Figs. 2 and 3 facilitates full determination of cation distribution thermodynamics in spinels. Figure 2 at fixed co and × gives the form of )`(t), and Fig. 3 shows which phase is the main and determines the phase transition temperature. It should be noted that in the case illustrated in Fig. 3(c) two phase transitions take place for some range of values w. Isostructural phase transitions in MgGa204 spinel has been experimentally detected [11]. Lattice parameter jump was recorded at temperature ~ 770°K. Isostructural phase transition seems to be possible in NiA1204. According to the results of our analysis of the experimental data from [9, 10] the second phase is likely to appear in the system on the decrease of temperature to ~ 700°K. That phase (Phase I in our notation) becomes the general at low temperatures. Phase transition temperature is approximately 400°K. Phase diagram in Fig. 3(a) at w ~- 0.43, )`0~~ 0.32 and )`(t) in Fig. 2(f) correspond to NiAI204. The authors believe that isostructural phases in spinels and other multisublattice crystals are met with fairly often. Low ion mobility hinders from the establishment of equilibrium cation distribution and makes difficult the experimental detection of isostructural phases in spinels. In other cases step-like cation redistribution is accompanied by a jump of magnetic characteristics. Then the phase transition is considered to result from magnetic interaction. It follows from our investigation that a reverse phenomenon is possible, when the

908

ISOSTRUCTURAL PHASE TRANSITIONS IN SPINELS

jump of magnetic characteristics results from the transition of mentioned above type. REFERENCES 1. 2. 3. 4. 5.

L. Neel, Compt. Rend. 230, 190 (1950) and ibidem E.F. Bertaut, 230, 213 (1950; 231, 88 (1950). J.S. Smart,Phys. Rev. 94, 847 (1954). H.B. Callen, S.E. Harrison & C.J. Kriessman, Phys. Rev. 103,851 (1956). S. Borghess, J. Phys. Chem. Solids, 29, 2225 (1967). A.N. Men, FTT, 3, 1054 (1961).

6. 7.

8. 9. 10. 11. 12.

Vol. 56, No. 10

A.N. Men, FTT, 4, 14 (1962). Yu.N. Kurushin et al., Ordering of atoms and its influence on properties of alloys (Uporyadochenie atomov i ego vliyanie na svoistva splavov), p. 71, Kiev (1968). C.T. Kriessman & S.E. Harrison, Phys. Rev. 103, 857 (1956). R.K. Datta & Rustum Roy, J. Amer. Ceram. Soc. 50, 578 (1967). R.F. Cooley & J.S. Reed, J. Amer. Ceram. Soc. 55,395 (1972). B.I. Pokrovskii, V.F. Kozlovskii, J. neorg, khimii (in Russian), 24, 544 (1979). Yu.Ya. Gurevich & Yu.I. Kharkats, J. Phys. Chem. Solids, 39, 751 (1978).