Abstract
Full Text
UDC 548.4
CRYSTALLOGRAPHY
B. Ya. SUKHAREVSKII, B. G. ALASHIN, I. I. VISHNEVSKII
THERMODYNAMICS OF THE DISTRIBUTION OF CATION VACANCIES AMONG SUBLATTICES IN SOLID SOLUTIONS OF FERRITES—SPINELS
(Presented by Academician N. V. Belov on 31 III 1969)
The principal question in solving thermodynamic problems connected with analysis of the defect structure of nonstoichiometric compounds is the choice of a correct model that takes into account the degree of nonstoichiometry, the presence of vacancies of both types, and the distribution of structural elements over crystallographic positions. Such a model determines the form of the thermodynamic potential and, consequently, the final result of the calculation.
In complex multisublattice structures, which include solid solutions between ferrites with the spinel structure, the choice of such a model is not unambiguous. As an example, let us consider a solid solution between some ferrite and magnetite
\[ \mathrm{Me}_{\lambda}^{\prime\prime}\mathrm{Fe}_{1-\lambda}^{\prime\prime\prime} \left[\mathrm{Me}_{1-a-\lambda}^{\prime\prime}\mathrm{Fe}_{1+\lambda}^{\prime\prime\prime}\mathrm{Fe}_{a}^{\prime\prime}\right]\mathrm{O}_{4}, \tag{1} \]
where \(\lambda = f(a,T)\) is the degree of inversion with respect to the divalent metal Me; \(a\) is the molar concentration of magnetite.
When the ratio \(\mathrm{Fe}^{\prime\prime}/\mathrm{Fe}^{\prime\prime\prime}\) deviates from the stoichiometric value \((a/2)\), the following cases are possible:
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\(\mathrm{Fe}^{\prime\prime}/\mathrm{Fe}^{\prime\prime\prime} < a/2\) (oxidation). According to the generally accepted point of view, in this case the number of cation positions increases while the number of cations remains unchanged, i.e., cation vacancies are formed.
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\(\mathrm{Fe}^{\prime\prime}/\mathrm{Fe}^{\prime\prime\prime} > a/2\) (reduction). The appearance of excess divalent atoms may lead to a relative decrease in the number of cation positions and to the displacement of part of the divalent cations into interstices uncharacteristic of the spinel structure, i.e., to the formation of an interstitial solution. Such a model was used in works \((^{1,2})\).
On the other hand, loss of oxygen by the lattice need not be accompanied by the insertion of cations, but may lead to the formation of vacancies in the oxygen sublattice \((^3)\). This point of view appears more consistent from the standpoint of crystal chemistry, providing a certain “symmetry” with respect to the possibility of vacancy formation in each of the sublattices. At the same time, the model naturally includes the necessity for the existence of thermal vacancies of both types at a strictly stoichiometric cation ratio.
We know of no direct experiments that make it possible to choose unequivocally among the indicated possibilities. Among the indirect proofs of the existence of anionic structural vacancies, in addition to the results of work \((^3)\), one may include the similarity of the concentration dependences of the additional thermal resistance obtained in the presence of cation structural vacancies \((^4)\) and in the presence of excess divalent iron atoms in reduced monoferrites of magnesium, lithium, and zinc \((^5)\).
The presence of two cation sublattices in the spinel structure permits, by analogy with the distribution of cations, a distribution of vacancies over tetrahedral and octahedral positions. An experimental study of such a distribution has been carried out only for maghemite \(\gamma\)-\(\mathrm{Fe}_{2}\mathrm{O}_{3}\), in which
the atomic concentration of cation vacancies reaches 10%. Neutron-diffraction (⁶, ⁷) and magnetic (⁸) measurements showed that these vacancies are concentrated in the octahedral sublattice.
In solid solutions of type (1), the limiting content of cation vacancies is smaller than that indicated by approximately an order of magnitude (³, ⁹), which makes the application of structural methods impossible. One may attempt to obtain information on the distribution of vacancies by methods of thermodynamic analysis.
The formula unit of solution (1), taking into account the degree of deviation from stoichiometry \(y\), the presence of anion and cation vacancies, and the distribution of the latter over the sublattices (described by the parameter \(s\)), has the form:
\[ \mathrm{Me}_{\lambda}^{\prime\prime}\mathrm{Fe}_{1-\lambda-s}^{\prime\prime\prime}\square_{v/8-s} \left[\mathrm{Me}_{1-a-\lambda}^{\prime\prime}\mathrm{Fe}_{1+\lambda+y-s}^{\prime\prime\prime}\mathrm{Fe}_{a-y}^{\prime\prime}\square_{v/4+s}\right] \mathrm{O}_{4-y/2}\square_{(v-y)/2}. \tag{2} \]
In this model the possibility of ordering and the formation of complexes of the type vacancy—cation, vacancy—electron, etc., is not taken into account.
A calculation analogous to that carried out in (³) leads to relations connecting the equilibrium values of the listed parameters with the composition of the solution and the temperature:
\[ \frac{(1+\lambda+y-s)(8+y)^{1/2}(v-8s)^{1/8}(v+4s)^{1/4}} {(8+v)^{7/8}(a-y)}=B; \tag{3} \]
\[ (v-y)^{1/2}(v-8s)^{1/8}(v+4s)^{1/4}/(8+v)^{7/8}=M; \tag{4} \]
\[ (1-\lambda+s)(2v+8s)/(1+\lambda+y-s)(v-8s)=D; \tag{5} \]
\[ \lambda(1+\lambda+y-s)/(1-\lambda+s)(1-a-\lambda)=A. \tag{6} \]
In (3)—(6) the quantities on the right-hand sides of the equations depend exponentially on temperature.
For \(A<1\), monoferrite
\[ \mathrm{Me}_{\lambda_0}^{\prime\prime}\mathrm{Fe}_{1-\lambda_0}^{\prime\prime\prime} \left[\mathrm{Me}_{1-\lambda_0}^{\prime\prime}\mathrm{Fe}_{1+\lambda_0}^{\prime\prime\prime}\right]\mathrm{O}_4 \]
is close to the inverse one. In this case it follows from (6) that
\[ \lambda \simeq \frac{A}{1+2A+y-Aa}(1-a). \]
If
\[ y \simeq y_0+Aa, \tag{7} \]
then
\[ \lambda_0 \simeq A/(1+2A+y_0)=\mathrm{const}, \tag{8} \]
\[ \lambda=\lambda_0(1-a). \tag{9} \]
The experimental data of works (², ³) confirm that the approximate equality (7) does indeed hold.
Using (5), (6), (8), and (9), we obtain
\[ s \simeq \frac{v}{8}\frac{D(1+A+y_0)-2}{D(1+A+y_0)+1}=\frac{v}{8}k, \tag{10} \]
where \(k\) does not depend on \(a\). It is easy to see that \(k=1\) corresponds to an octahedral, \(k=0\) to a uniform, and \(k=-2\) to a tetrahedral distribution of vacancies.
Relations (3), (4) provide the basis for considering the limiting case \(v>0\), \((v-y)=0\), corresponding to complete oxidation. Neglecting the quantity \(s\) in the first bracket of the numerator of (3) \((s_{\max}=v/8\ll 1)\), we obtain
\[ \lg \frac{1+\lambda_0(1-a)+y}{a-y} = \lg B+\frac{3}{8}\lg\frac{8+y}{y} -\frac{1}{8}\lg\left[(1-k)\left(1+\frac{k}{2}\right)^2\right]. \tag{11} \]
For \(A\gg 1\) the ferrite \(\mathrm{MeFe}_2\mathrm{O}_4\) is practically normal, and from (6) it follows that
\[ \lambda \simeq 1-a. \tag{12} \]
Neglecting \(s\) in the first parentheses of the numerator and denominator of (5), we obtain, taking (12) into account,
\[ s \simeq \frac{v}{8}\,\frac{D(2+y)-a(D+2)}{D(2+y)-a(D-1)}=\frac{v}{8}\,k . \tag{13} \]
Here \(k\) is already a function not only of temperature, as in (10), but also of concentration. Therefore, in treating the experimental dependence \(y=f(a)\) by formulas (11), (12), a straight line with slope \(3/8\) will be obtained only if the concentration dependence of \(k\) is used. The quantity \(D\) can be determined from the value of \(k\) at any one concentration.
Thus, relation (11) makes it possible to find the distribution of cation vacancies over the sublattices from the concentration dependence of the degree of oxidation of the solutions in those cases where the parameter of this distribution \(k\) varies with composition. As is seen from (10) and (13), this is possible for solid solutions between magnetite and a normal ferrite, whereas for solutions of magnetite with inverse ferrites a separate determination of the constant quantities \(\lg B\) and \(\lg [(1-k)(1+k/2)^2]\) from (11) is impossible.
Fig. 1. Processing of the data of Table 1 for solid solutions \(\mathrm{Me}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\), quenched from different temperatures. 1–3 — \(\mathrm{Me}=\mathrm{Mg}\) (1 — from 1670°, 2 — 1770°, 3 — 1670° according to the data of (2)); 4, 5 — \(\mathrm{Me}=\mathrm{Ni}\) (4 — 1600°, 5 — 1700°); 6 — \(\mathrm{Me}=\mathrm{Zn}\), 1700°K. Straight line 6 is shifted upward along the ordinate axis by 0.3 scale units. The dashed straight line has a slope tangent of \(3/8\).
The experiments were carried out on the systems \(\mathrm{Me}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\) (\(\mathrm{Me}=\mathrm{Mg}, \mathrm{Ni}, \mathrm{Zn}\)). The degree of oxidation of these solutions for different compositions and temperatures, determined by a chemical method, is given in Table 1. The degree of inversion \(\lambda_0\) of the magnesium and nickel ferrites was determined from magnetic measurements; zinc ferrite was assumed to be completely normal.
Table 1
Degree of oxidation of solid solutions of Mg, Ni, and Zn ferrites with magnetite, quenched from different temperatures
| \multicolumn{2}{c}{\(\mathrm{Mg}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} | \multicolumn{2}{c}{\(\mathrm{Mg}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} | \multicolumn{2}{c}{\(\mathrm{Mg}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} | \multicolumn{2}{c}{\(\mathrm{Ni}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} | \multicolumn{2}{c}{\(\mathrm{Ni}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} | \multicolumn{2}{c}{\(\mathrm{Zn}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\)} |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \multicolumn{2}{c}{1670° K} | \multicolumn{2}{c}{1770° K} | \multicolumn{2}{c}{1670° K according to data (2)} | \multicolumn{2}{c}{1600° K} | \multicolumn{2}{c}{1700° K} | \multicolumn{2}{c}{1700° K} |
| \(a\) | \(y\) | \(a\) | \(y\) | \(a\) | \(y\) | \(a\) | \(y\) | \(a\) | \(y\) | \(a\) | \(y\) |
| 0.3 | 0.026 | 0.49 | 0.040 | 0.591 | 0.052 | 0.31 | 0.068 | 0.29 | 0.028 | 0.33 | 0.025 |
| 0.4 | 0.053 | 0.59 | 0.066 | 0.640 | 0.058 | 0.40 | 0.100 | 0.31 | 0.034 | 0.36 | 0.034 |
| 0.5 | 0.080 | 0.73 | 0.094 | 0.818 | 0.108 | 0.50 | 0.170 | 0.33 | 0.048 | 0.43 | 0.043 |
| 0.6 | 0.110 | 0.84 | 0.127 | 0.882 | 0.116 | 0.61 | 0.202 | 0.60 | 0.132 | 0.54 | 0.073 |
| 0.7 | 0.148 | 1.00 | 0.180 | 0.925 | 0.141 | 0.74 | 0.283 | 0.71 | 0.172 | 0.55 | 0.082 |
| 0.8 | 0.200 | | | 0.961 | 0.159 | | | 0.79 | 0.187 | 0.62 | 0.091 |
| 0.9 | 0.240 | | | 1.000 | 0.173 | | | 0.90 | 0.246 | 0.64 | 0.106 |
| 0.10 | 0.280 | | | | | | | | | 0.70 | 0.113 |
| | | | | | | | | | | 0.72 | 0.136 |
| | | | | | | | | | | 0.88 | 0.197 |
The graphs constructed in the coordinates \(\lg \{[(1+\lambda_0(1-a)+y)/(a-y)]\}\), \(\lg [(8+y)/y]\) (Fig. 1) are straight lines, the slopes of which, calculated by the least-squares method, differ from the theoretical value \(3/8\) by no more than 8%. The exception is the system \(\mathrm{Zn}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O}_4\), for which the tangent of the slope angle is 0.53.
The most probable reasons for such a difference of the slope angle from the calculated value are the dependence \(k = k(a)\) in accordance with (13) and the partial inversion of zinc ferrite. The latter is confirmed by X-ray structural investigation data \({}^{10}\) (at \(1270^\circ\)K \(\lambda_0 = 0.9\)) and by our magnetic measurements, according to which the saturation moment of \(\mathrm{ZnFe_2O_4}\), quenched from \(1700^\circ\)K, is \(\sim 0.4\,\mu_B\). Theoretical consideration shows \({}^{11,12}\) that, for the organization of ferrimagnetic interaction, a magnetic cation in a given sublattice must have no fewer than two magnetic neighbors from the other sublattice. In the case of solutions of the type
\[ \mathrm{Zn}_{\lambda_0(1-a)}\mathrm{Fe}^{\prime\prime}_{1-\lambda_0(1-a)} \left[ \mathrm{Zn}_{(1-\lambda_0)(1-a)} \mathrm{Fe}^{\prime\prime\prime}_{1+\lambda_0(1-a)} \mathrm{Fe}^{\sim} \right]\mathrm{O_4} \]
this requirement imposes an iron-ion content in the \(A\)-sublattice \(\geq 1/3\). Measurements show that the saturation magnetic moment of the indicated solutions, quenched from \(1700^\circ\)K, increases sharply beginning with a magnetite concentration \(a^* \simeq 0.15 \div 0.20\). This makes it possible to limit \(\lambda_0\) from below by the value
\[ \lambda_0 \geq 0.67/(1-a^*) \simeq 0.8. \]
Nevertheless, introducing such a degree of inversion for \(\mathrm{ZnFe_2O_4}\) under the condition \(k \equiv 0\) (uniform distribution of vacancies over the sublattices) does not lead to the calculated value of the slope angle \(3/8\) (straight line 2, Fig. 2). A more satisfactory result can be obtained if the concentration dependence \(k\) is taken into account by putting \(k(1)=0.9\). In this case the slope angle is \(\sim 0.40\) (straight line 3, Fig. 2). Apparently, the best agreement of experiment with calculation is achieved for values of \(k\) for magnetite that are closer to 1; however, the logarithmic singularity of the term containing \(k\) in (11) leads in this case to larger errors in the calculations.
Fig. 2. Processing of data for the system \(\mathrm{Zn}_{1-a}\mathrm{Fe}_{2+a}\mathrm{O_4}\) by equation (11).
\(1\)—\(\lambda_0 = 1,\ k \equiv 0,\ \tan\alpha = 0.525;\)
\(2\)—\(\lambda_0 = 0.8,\ k \equiv 0,\ \tan\alpha = 0.488;\)
\(3\)—\(\lambda_0 = 0.8,\ k(1)=0.9,\ \tan\alpha = 0.405;\)
\(4\)—\(\tan\alpha = 0.375\)—the angle of inclination of the straight lines to the abscissa axis. For clarity, all straight lines have been shifted to the zero ordinate axis.
On the basis of the foregoing it may be concluded that in solid solutions of zinc ferrite with magnetite at temperatures up to \(1700^\circ\)K, cation vacancies are concentrated predominantly in octahedral positions. It also follows from this that, in an analogous manner, vacancies are distributed in solutions of magnetite with inverse ferrites for which the quantity \(k\) does not depend on concentration.
Physico-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Ukrainian Scientific-Research Institute of Refractories
Kharkov
Received
15 IX 1968
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