Crystal growth, structure and physical
properties of crystals

key words [paratelluride heat capacity]

The equation proposed by the authors in 1988 describes the experimental data on temperature dependences of heat capacities of crystalline substances on the basis of Lorentz distribution within the T interval from 10-30 K to phase transition point as:

T_melt

C_p = A_L[1-1/(1+C_LT₂)] + B_LT.|(1)

T_min

This equation asymptotically tends to:

T_melt

C_p = A_K +B_KT + C_K.T^-2|

T_min

that is Kelly equation, where T_min,_K>>T_min,_L˜T_max within the area C_p = f(T³);
A_L = A_K, B_L = B_K (C_VC_p), C_L = T_n^-2 = 1/C_K².

Equation (1) possesses a number of properties:

(1) lim L = 0;

T0

(2) in the limit: A_L = A_K = 3Rn;

(3) the equation has a point of inflection;

(4) B coefficient corrects the heat capacity value on transition from C_V to C_p.

The curves C_p versus T for crystalline Te approximated by Lorentz (solid line) and Kelly (dashed line) equations are shown in Fig.1 (D is the Debye area, L is the area of Lorentz description, and L==K is the area of equivalent Lorentz and Kelly descriptions).

Intricate character of the temperature dependence of heat capacity of paratelluride (Fig. 2) results from the imperfect conception about crystalline state of matter. In this case, the superposition of the following three equations is required to describe properly the available experimental data:

Kelly approximation (error is 1.5%);

(80-1006 K), T_n = 114.6 K,

Lorentz approximation of all available data using the least-squares method (error is 0.6%);

C_p =. 3.3 × 10^-3 T + L₁. (T_n,l= × 78.1K; A₁ = 38.5) + L₂. (T_n,2 = × 169K; A₂ = 6.0) + L₃ (T_n,3= × 253K; A₃ = 34.1) (30-1006 K),

the superposition of Lorentz equations (error is 0.3%).

The given example demonstrates the applicability of Lorentz equation for description of crystalline state of substances.

Figure 1.

Figure2.

^# Kotel'nikov A.R., Bychkov A.M., and Koval'skii A.M. Synthesis and refinement of the unit-cell parameters of (K,Rb)-feldspar solid solutions.

key words [feldspar init-cell parameters])

# This study was supported by the Russian Basic Reseacrh Foundation, project no. 95-05-14900.

Potassium-rubidium feldspar solid solutions were synthesized from gel mixtures at hydrothermal conditions (T = 650-700^oC, P = 1-5 kbar). The mixtures of (K,Rb)-feldspars and leucites are formed at X(Rb) > 0.5, T = 700^oC, and P = 1 kbar. The stability of the feldspars increased relative to leucite with decreasing temperature to 670^oC and increasing pressure to 5 kbar. According to microprobe studies, there was a good agreement between the real and theoretical feldspar compositions.

The (K,Rb)-feldspars obtained were analyzed by X-ray diffraction using a DRON-1.5 diffractometer at radiation Co-K₁ (=1.78892e). Silicon of spectral purity was used as an internal standard. The unit-cell parameters were verified relying on 45 - 57 reflexes in the 2q range 15 - 85^o.

The unit-cell parameters depend linearly on the solid solution composition. The following equations were derived for the concentration dependences of the unit-cell parameters:

From tentative evaluation, the (K,Rb)-feldspars obtained are assigned to the series sanidine - Rb-sanidine and characterized by zero excess mixing volume, which is the evidence for their ideality.

key words [molybdenite proton magnetic resonance]

The proton magnetic resonance (PMR) in natural sulfides (pyrite, marcasite, arsenopyrite) was first observed by G.B. Bokii and A.M. Bondar' [1]. This PMR signal was attributed to the presence of hydrogen in the anionic sublattice of the sulfides.

Figure. PMR spectra.
1 - OH^-, wt %; 2 - loose H₂O, wt%

They also suggested that the structural hydrogen in sulfides is responsible for their non-stoichiometry frequently observed. Later, Kalinichenko and co-authors [2] studied a great collection of natural pyrites by PMR. It was shown that the PMR signal intensity of pyrite correlates with its oxygen content. From these results, the assumption was made that the most of structural hydrogen in pyrite is present in the form of OH^- groups.

Molybdenite is an appropriate object for PMR studies of sulfides because its natural counterpart often shows significantly nonstoichiometric composition.

Molybdenite crystals were synthesized under hydrothermal conditions at T=350+10^oC and P=10⁸ - 0.510⁸ Pa. The run duration ranged from 14 to 368 days.

The starting solutions were prepared from NH₄Mo₇O₂₄ × H₂O with Mo concentration 5-50 mg/l. Sulfur hydrogen was bubbled through the solution for 0.5-1.5 h until the formation of thiomolybdate, the orange-red-brown color of the solution was due to MoS₄^2- ions. pH was maintained at 9-11 by adding concentrated ammonia. Saturation of the starting solution with sulfur hydrogen at a given pH value was a necessary condition of the experiment. A silver/silver sulfide electrode was used to control over the H₂S concentration. Crystalline sulfur (200-800 mg) was added to the thiomolybdate solution.

The purity of the molybdenites obtained was verified by X-ray analyses. PMR spectra for all molybdenite samples were recorded with a RYa 2031 continuous oscillation spectrometer (v_o = 16 MHz) at room temperature. The spectrum consists of two components. The minor component (figure, dashed line) (H₂ Gs) is provided by tightly bound protons.

The approximate proton contents derived from the spectra were recalculated for the equivalent OH^--groups. Ca(OH)₂ was used as a standard. The molybdenites with different OH^- content (from 0 to 2 wt % of OH^-) were obtained, i. e. the continuous series from pure molydenite to that with S^2- partially substituted for OH^-. In synthetic molybdenites the PMR signal ranges within wide limits because of significant amounts of structurally bound hydrogen (from 0 to 2 wt %), which is most likely to occur in the form of OH^--groups.

The experimental data obtained suggest that the non-stoichiometry of natural and synthetic molybdenites may result from the substitution of S^2- for OH^- in the anionic sublattice of the mineral.

1. Bokii G.B. and Bondar' A.M. (1979) Dokl. Akad. Nauk SSSR, vol. 248, no. 4, pp. 956-959.
2. Kalinichenko A.M. et al. (1981) Geokhimiya, N.5, p. 765.
3. Organova N.I. (1989) Crystal Chemistry of Disproportionated and Modulated Mix-Layered Minerals, Moscow: Nauka.

key words [pollucite thermodynamic properties]

Pollucite is a structural analog of analcime. Substitution of sodium for cesium in the analcime structure results in loss of water. Analcime and Cs-

pollucite are the end members of the solid solution series Cs_1-xNa_xAlSi₂O₆ xH₂O (0<x<1).

Natural pollucite from East Siberia of the composition Cs_0.83Na_0.17Si₂O₆×0.17H₂O was used for thermochemical studies.

Bennington and co-authors [1] measured the enthalpy of formation of the natural pollucite of the composition

Cs_0.65Na_0.185Rb_0.028(AlSi₂)[O_5.863(OH)_0.137]×0.19H₂O

by the HF-acid-dissolution calorimetry. Later, Semenov and co-authors [2] determined the enthalpy of formation of synthetic anhydrous Cs-pollucite (CsAlSi₂O₆) by high-temperature dissolution calorimetry. Correlation between these data is difficult because of great difference in composition of the pollucite samples used.

The enthalpy of pollucite dehydration was determined by the method of "repeated drop". The water loss was estimated by weighing a sample before and after the run (accuracy +2×10^-3 mg). The enthalpy of formation of pollucite was obtained by high-temperature dissolution calorimetry at 973 K using the melt solvent of 2PbO×B₂O₃ composition. Corundum and platinum samples were used as calibration standards in dehydration and dissolution experiments, respectively.

The enthalpy of pollucite dehydration was 14.1+ 4.7 kJ/mol. The ethalpy of formation was regarded as a sum of the enthalpy of dehydration and the enthalpy of formation of dehydrtated natural pollucite. The latter was calculated on the basis of two thermochemical cycles: the reaction of pollucite formation from the appropriate oxides and the exchange reaction involving the dehydrated analcime and cesium and sodium chlorides. The enthalpies of formation of dehydrated pollucite from elements are -3050.0 +14.2 and -3050.8+13.3 kJ/mol, respectively, and are almost coinsident. The enthalpies of formation of hydrous pollucite also are in good agreement: -3112.7+15.0 and -3113.5 + 14/2 kJ/mol, respectively. These results are consistent with Bennington's data [1] obtained for the natural pollucite with similar water content (-3098.5 +3.6 kJ/mol). To adjust our data to the composition of Cs-pollucite, a correction was made for the heat of dissolution of unhydrous analcime (determined previously). The obtained enthalpy of formation of Cs-pollucite (CsAlSi₂O₆) is -3068.7+13.3 kJ/mol and is in good agreement with Semenov's data [2] obtained for synthetic pollucite of the same composition (-3083.4 + 5.0 kJ/mol).

1. Bennington K.O., Beyer R.P., and Johnson G.K. (1963) Thermodynamic Properties of Natural Pollucite, Report of Invest., 8779, U. S. Bur. Mines.
2. Semenov Yu.V., Bychkov A.M., Krestina N.V., Khodakovsky, I.L., Westrum E.F., and Johnson D.A. (1993) Thermodynamic Properties of Cs-Silicates and Aluminosilicates, 4-th Intern. Conf. on Chemistry and Migration Behavior of Actinides. Charleston, USA, Dec. 12-17, , pp. 63-69.

key words [hematite entropy solution]

The proposed in the review Parker and Khodakovsky [1] standard values of the free energy and entropy of Fe³⁺ ion in an aqueous solution, fG^o =-16.23+1.1 kJ/mol, S^o =-278.4+7.7 J/mol·K, are based on the adopted values of the Fe²⁺ thermodynamic properties in aqueous solution and potential E^o Fe²⁺ /Fe³⁺ . In the present work, an attempt has been undertaken to obtain a standard entropy of the Fe³⁺ ion in aqueous solution using the measurement results of hematite solubility in aqueous solutions of acids [2-4].

1/2-Fe₂O₃(k) + 3H⁺_(sol-sol) = Fe_{3+ (sol-sol)} +3/2H₂O

The calculation of the free formation energy (Gibbs’ energy) of an ion Fe³⁺ in an aqueous solution has been performed from the data of three experimental hematite solubility studies in the range of temperatures and pH of solutions: 358K in 0.12 M HCl (Berner [2]); 373, 393, 413, and 433K, 0-0.2 M HCl (Melent'ev et al [3]) and 473K, 0.01-0.32 M HClO₄ [1].

The experimental results were processed as follows:

1) the equilibrium composition of the aqueous phase in the experimental systems was calculated using Mironenko's program [4] (using a method of free Gibbs energy minimization), therewith free formation energies for simple ions were assumed to be 0 [5]. It was supposed in the calculations that the following particles: H₂O, H⁺, O_2(sol), Cl^-, Fe³⁺, FeCl²⁺, FeOH²⁺, OH^- are present in the aqueous solutions.

2) the stability constants of the chloride complex FeCl²⁺, required for the calculation, were derived from the equation:

obtained by us from the experimental measurements by Tagirov and Zotov in the temperature range of 25-

90^oC [6], and the hydrolysis constants of an ion FeOH²⁺ from the results of the measurements by Kotova and Zotov at 25-200^oC[7].

3) the calculations for this system by Shvarov’ program (GIBBS) [8] agreed satisfactorily with the results obtained by Mironenko's program.

The equilibrium constant value of the reaction (2) at 473.15K and I=0 (lgK=-4.54+ 0.36) obtained in our work [1] agrees well with the results of [2,3].

Table 1. Equilibrium constants of the reaction (1) at I=0 (our calculation)

Using the equations of the temperature dependence of Cp for H₂O (liq.), Fe³⁺ (sol), Fe₂O₃ (c) from [9,10,11], respectively, we obtained the equation for Cp of the reaction (1) in the temperature interval from 298 to 473 K.

Cp=38.867 +0.30971T10^-3 - 3.01210^5-T ²- 21937/(T-190)
(3)

We have for rS(298.15T) of the reaction (1):

rS(298.15T) = (rH(298.15) - rG(T) + òC^opdT)/T -
-ò (C^op/T)dT (4)

Having calculated the enthalpy of the reaction (1) from the data of the works [11,12,13], one can calculate for each experimental values of lg K the corresponding value of rS(298.15) (table 2).

Table 2. Change of the entropy in the reaction (1).

Using the weighted mean of rS^o(298.15)=-212.8+8 J/mol K, we obtained for Fe³⁺ (sol): S^o(298.15) =-275.0+ 8.0 J/mol K.

Inasmuch as from the data of the work [1] fH(298.15)Fe³⁺ (sol) =- 49.0 + 1.5 kJ/mole, then

fG(298.15)Fe³⁺ (sol) =- 16.5 + 3.4 kJ/mole.

Knowing (H_298.15) and (S_298.15) for the reaction (1) one can obtain lgK (298.15) = 0.2+0.4.

1. Sergeeva E.I. et al. (1988) Terra cognita, V.8, N.2, p.182.
2. Berner R.A. (1969) Geochim. Cosmochim. Acta, V.33, p.267.
3. Melent'ev et al. (1988) In: Solubility of some ore-forming sulfides under hydrothermal conditions. Nauka (Science), Moscow, p.85.
4. Mironenko M.V. et al. (1992) Second Inter. Symp. Thermodynamics of natural processes. 13-20 Sept 1992 Novosibirsk, Russia. p.117.
5. Dorofeeva V.A., Khodakovsky I.L. (1981) Geochim., N1, p.129.
6. Tagirov B.R., Zotov A.V. (1996) Proc. Sixth Goldshmidt Conf., Heidelberg, March 31-April 14, V.1, N.1, p.613.
7. Zotov A.V., Kotova Z.Yu. (1980) Geochim., N.5, pp.768-773.
8. Shvarov Yu.V. (1992) Abstr., 2-nd Int. Symp. Thermodynamics of Natural Processes, Novosibirsk, p.51.
9. Naumov G.B.et al. (1971) In: Reference book for thermodynamic values. Atomizdat, Moscow, 240p.
10. Hovey J.K. (1988) Thermodynamic of aqueous solutions. Thesis. Univ. of Alberta, Edmonton.
11. Hemigway B.S. (1990) Amer. Mineralogist, V.75, p.781.

12. CODATA Key Vales for Thermodynamic. (1989) Ed. by Cox J.D. et al. Hemisphere Publ. Corp. N.Y.
13. Parker V.B., Khodakovsky I.L. (1996 in press).

key words [anharmonic parameter microscope model]

Taking into account anharmonic effects is needed for thermodynamic calculations at high temperatures and pressures (for example, to describe exceeding Dulong and Petit limit of heat capacity). Two types of spectroscopic parameters are commonly used for mode anharmonisity:

Gruneisen parameter (1)

and anharmonic parameter (2)

There are theoretical estimation (Sherman, 1980, 1982) and lattice dynamic calculation results (Reynard et al., 1992) for Gruneisen parameters _i , but any theoretical estimation of parameters a_i have been as yet lacking. The empirical approach suggested and developed by Gillet et al. (1989-1995) consists in inserting experimental pure-temperature dependence of frequencies w_i(T) into thermodynamic calculations. The approach gives qualitative results but is not well justified.

In this paper a simple microscope model for describing anharmonic thermodynamic and spectroscopic properties is suggested. The model is based on considering a crystal as a giant macromolecule and applying the methods of calculations developed for polyatomic molecules. If the third- and fourth-order terms are included into potential energy expansion along with the second-order terms then the energy level of a polyatomic molecule can be written as (Hezberg, 1949):

(3)

where v_i is an integer quantum number of i-th mode, w_i^qh are zero frequencies, x_ii is due to own normal mode anharmonicity and x_ij is due to coupling between modes I. and j. The transition frequency of i-th mode is:

(4)

The number of modes in a solid is of the order of 10²³, so one can not observe the fine structure of energy levels as in molecules. Let us suppose that in a crystal the visible frequency peak w_i^v corresponds to the average transition frequency in thermodynamic sense. Averaging w_i and differentiating with respect to temperature one can obtain

(5)

Here C_vk -heat capacity corresponding to k-th mode, x_ij=2x_ii was taken for convenience. Thus one can calculate all vibrational thermodynamic properties (free energy, vibrational energy, heat capacity, entropy) and spectroscopic anharmonic parameters a_i on the base of a set of vibrational frequencies w_i^qh and mode interaction coefficients x_ik.

Application of the model is shown by the examples of: 1) some polyatomic molecules (CO₂, NF₃, SO₃) considered as microcrystals, for which spectroscopic constants w_i and x_ik are known, 2) several minerals (forsterite, coesite, stishovite) using model vibrational spectra and spectroscopic values of the mean a_i and assuming x_ik/(w_i w_k)=const. The results for CO₂ and forsterite are shown in Fig.1, where mode anharmonic parameters a_i and heat capacity are presented as a function of temperature.

1. The most of calculated anharmonic parameters a_i are negative and have values of the order of 10^-5 K^-1 for molecules as well as for minerals, what is consistent with experimental spectroscopic results (Gilett et al, 1989-1995).

2. The temperature dependence of anharmonic parameters a_i is evaluated.

3. Strictly speaking, the use of the dependence w_i(T) overestimates anharmonic corrections to the heat capacity C_v, allowing estimation of the upper limit of the effect (see Fig.1). Anharmonic corrections for mineral heat capacity calculated by the presented method are somewhat lower than those obtained by Gilet et al (1991, 1992) and correspond well to experimental data.

# This study was supported by Russian Basic Research Foundation (95-05-14945)

key words [olivine meteorite zoning]

The variation of fayalite content in olivine grains of the Divnoe meteorite was previously studied by microprobe analysis [1]. The microphotographs of a thin section of one olivine grain and the diagram of spatial variation of fayalite content presented in this work testify that the clear rhythmic variation of fayalite content within the range 22-25 mol % occurs over a length of 60 . The noted rhythmic chemical zoning has a period of the order of a few micrometers and is explained by the authors [1] as a result of decomposition of olivine solid solution.

Within the present study, determination of other frequencies of the chemical rhythm by harmonic analysis methods was attempted on the basis of data [1].

As a result of reduction of the analytical data to the spatial forsterite distribution (Fig. 1), the systematic component of the function of Mg/Fe distribution in the grain bulk was derived using the linear regression method. The linear approximation of the function component gives 76.618 - 0.0122L, where L is the distance from the grain surface (in micrometers). Thus, the slope of the forsterite concentration trend is 0.0122 mol % , that is the average Mg/Fe ratio

# The authors thank M.I. Petaev for remarks and discussion of this work. The present study was supported in part by the Russian Basic Research Foundation, project no. 94-05-16405.

decreases towards the grain core (3.27 at the surface and 3.15 at a depth of 60 ).

After subtraction of the obtained linear trend from the distribution function (Fig. 2), the following algorithm was employed to determine the main frequencies of the distribution.

1. The high-frequency component of the distribution function was filtered off by the moving-average method. The period of the component proved to be equal 3 , which correlates well with [1] data. The post-filtration distribution is shown in Fig. 3 (dashed line).

2. A number of harmonic characteristics of the distribution function, such as i (harmonic number), w_i (harmonic frequency), a_i and b_i (amplitude coefficients), and l_i (harmonic period in ), were derived using the method of determination of unknown periodic components of a f(x_i) function assigned at N equidistant points [2]. This method is based on solution of an equation of the type

cos mw - _{I cos (m-1)w} - ... - 0.5w_m=0,

where m is the number of harmonics determined and _i are the coefficients fitting the linear equation system

m-1

(f_i+k-1+f_2m+i-k-1)k+ f_m+i-1m- f_i-1 - f_{2m+i-1=0, i=1..N-2m}

k=0

F(x) = _{I cos (w}i(x-11)) + b_i sin(wi(x-11))

The plot of F(x) function is shown in Fig.3.

In summary, there are reasons to conclude that the function of Mg/Fe distribution in olivine grains has the three more periods and the systematic linear trend in addition to the period noted by the authors [1]. The obtained results can not be interpreted on the basis of solid solution decomposition mechanism.

i	w	ai	bi	li,
1	.1352	.3616	-	46.5
2	.4184	-.1451	-.1840	15.
3	.6524	-.1638	-	9.7

References :

1. Petaev and J. Brearley. (1994) Exsolution in ferromagnesian olivine in the Divnoe meteorite. // Science, V.266, pp.1545-1547.
2. Korn and T. Korn. (1977) Handbook on Mathematics // Moscow: Nauka, p. 693.