Effect of pressure on equilibrium isotopic fractionation

Geochimiea et Cosmochimica Acta, Vol. 58, No. 21, pp. 4739-4750, 1994 Copyright 0 1994 Ekvier Science Ltd

Pergamon

Printed in the USA.Ali rightsreserved 00 i&7037/94$6.00+ .oo 0016-7037( 94)00250-9

Effect of pressure on equilibrium isotopic fractionation V. B. POLYAKOVand N. N. KHARLASHINA V. 1. Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Science,Moscow B-334, Russia (Received May 18, 1993; accepted in revised form May 10, 1994)

Abstract-The effect of pressure on the equilibrium isotope fractionation is examined theoretically. The calculation technique is developed, using the quasi-harmonic approximation for solids. The formula connoting the pressure derivative of the @-factor creduced isotopic partition function ratio) and its temperature derivative is obtained. The application of the technique to some silicates (quartz, albite, enstatite, pyrope, grossular, forsterite), rutile and calcite ( 160- I80 fractionation), graphite-calcite-diamond ( “C- 13Cfractionation), and brucite-water (H-D fractionation) shows that pressures of tens of kbars may produce a measurable effect on the equilibrium isotope fractionation and even change the sign of the isotopic shift. Estimation of the pressure effect is important for correct interpretation of high pressure experimental isotopic data and for correct calculation of equilibrium isotopic constants.

INTRODUCITON THE EFFECTSOF PRESSUREon the equilibrium isotope fractionation are known to be small. These effects are usually neglected in isotope geothermometry, in laboratory isotope exchange experiments, as well as in investigations of stable isotope fractionation processes in nature. Nevertheless, an accurate estimate of the pressure effect on the equilibrium isotope fractionation is important for determination of pressure limitations in the experiments and for correct interpretation of isotopic distribution in natural objects and in experimental run products. There are a few papers on this subject. The first semiquantitative estimation of the pressure effect was done by JOY and LIBBY( 1960). They concluded that the effect may be measurable. However, more recent theoretical and experimental investigations (CLAYTONet al., 1975; CLAYTON, 198 1) of the calcite-water system on 160- I80 fractionation led these authors to the conclusion that at pressures of 20 kbar or less pressure-induced corrections should be less than 0.1 b and thus, pressure effects may be ignored in the application of oxygen isotope fractionation to thermometry of crustal rocks. The pressure influence on the isotope fmctionation was estimated by HAMANN et al. ( 1984), with the aid of the Debye model for some crystals of cubic symmetry (LiH, MgO, CaO, ZnS, PbS, FeS2). The pressure corrections turned out to be about 0.1%0 for 10 kbar at ambient temperature for sulfides. This is too small to be significant for isotope geothermometry. The method of calculation in the papers mentioned above reduced to the estimation of the volume change on isotopic substitution, AV, followed by the application of the therm~ynamic formula alna

A?’

ap =RT'

(1)

where cyis the equilibrium isotope fractionation factor, P is the pressure, T is the absolute temperature, and Ii is the universal gas constant. The authors had to use some crude approximations and assumptions, like application of the Debye model to complex substances ( HAMANN et al., 1984) or estimation of A V for calcite by calculating the difference in the mean C-O bond length, only (CLAYTONet al., 1975). Thus,

the theoretical results cited above are more likely to be rough estimations than exact calculations. In addition to the experimental results of CLAYTON et al. ( 1975 ) mentioned above, data on mineral-water oxygen isotope equilibrium at different pressures were presented by CASHEWS et al. ( 1983). Their data do not point (within experimental error of about several tenths permil) to the notable pressure effect in the systems albite-water ( T = 500DC, P = 2-15 kbar) and quartz-water (T = 4OO”C, P = 2-22 kbar and T = 6OO”C, P = 2-20 kbar). There is some evidence for the pressure effect on equilibrium isotope fractionation. GARLICK et al. ( I97 1) were first to suppose the pressure dependence of oxygen Fractionation in the system crystal-melt to explain the isotope distribution in kimberlite xenoliths. MINEEV et al. ( 1989) suggested the pressure dependence of hydrogen isotope equilibrium fractionation factor in the serpentine-water system to explain a significant disparity between data obtained by WENNERand TAYLORf 1974) for ocean bottom samples and values measured by SAKAI and TSUTSLJMI( 1978) in laboratory experiments at pressures from 1 to 3.8 kbar. To verify their suggestion, MINEEVet al. ( 1989) conducted experiments at normal pressure and obtained excellent agreement with WENNER and TAYLOR(1974). Up to now, the effects of pressure on equilib~um isotope fractionation have been known to be small and are usually neglected. A fully consistent theory of this subject is absent, however. The aim of this paper is to develop a calculation technique for the estimation of the pressure effect on the equilibrium isotope fractionation, to evaluate quantities of the pressureinduced corrections, and to specify for which systems and at what pressures the effects are nonnegligible. Some aspects of this problem were studied in our previous papers ( POLYAKOVand KHARLASHINA,1989, 199 1; I&ARLASHINA and POLYAKOV,1992). THE EFFECT OF PRESSURE ON @-FACTOR OF SOLIDS (GENERAL CASE ) The equilibrium fractionation factor for an isotope exchange reaction between two substances A and B may be written in terms of p-factors of these substances ( VARSHAVSKY and VAISBERG, 1957):

4739

V. B. Polyakov and N. N. Kharlashina

4740

a=-.

PA

(2)

Pa

So, to examine the effect of pressure on isotopic fractionation, it is enough to estimate the pressure dependence of the p-factors of the substances. The &factor is expressed in terms of the reduced isotopic partition function ratio fas follows ( BIGELEISENand MAYER, 1947; SINGHand WOLFSBERG, 1975):

p

=fw

(Q*;Qcl1I”

=

Q Q3

(3)

Here, Q is the quantum mechanical partition function, Q,, is the partition function calculated in accord with classical mechanics, r is the number of atoms of the element undergoing exchange, and the asterisk * refers to an isotopically substituted form. Consider a perfect crystal consisting of n atoms. In the harmonic approximation, the motion of atoms in the crystal is described as 3N-6 uncoupled harmonic oscillators. In this case, the reduced isotopic partition function ratio may be expressed by the Urey formula ( UREY and GREET, 1935): (4)

Here, II, = $

For the pressure derivative, it can be found that

aln V -I, ISthe isothermal bulk modulus. ap 7 Formula I 1, with the definition of y given by Eqn. 10, is exact in the quasi-harmonic approximation and provides a calculation of the pressure effect on fl-factors of minerals. The dependence p(T) is usually known from experiments or calculations. Calculation of the parameter y is a more complicated problem. Below, we discuss it in detail. We note that quantities y, and 7: are not independent. It can bc shown (see Appendix A) that C y, = C -r: As a rule, we do not know the real vibrational spectrum of the crystal completely, so the application of some model is needed. The most popular and the simplest Debye and Einstein models require a unique value of the Griineisen parameter; consequently, y, = 7: = y. This equality is obviously valid in the approximation yi = const which is ofien used in theoretical analyses (BORN and HUANG, 1954). In all these approximations, the quantity y is equal to the thermal Griineisen parameter: where Br = -

1

i

- dimensionless frequency of i mode, v, refers to i

harmonic normal mode frequency, T is absolute temperature, h is the Planck constant, and k is the Boltzmann constant. The harmonic approximation does not describe the pressure or volume dependence of vibrational frequencies. These factors can be accounted for with the quasi-harmonic approximation ( LEIBFRIEDand LUDWIG, 196 1) In this approximation, vibrational frequencies vi are assumed to be volume dependent with the following formula: aln v,

” = -

i-1

(5)

a In V r’

au,

T =

C, = 2 C,,, = R

,

14;exp( u;) , [exp(u,) - 11”

z:

(13)

C, is the molar heat capacity of a crystal; C,, is the heat capacity referring to ith mode. This relationship is important, because the thermal Grilneisen parameter can be obtained from experimental data: nVBT

where yi is the mode Grtineisen parameter of the ith lattice vibrational mode at frequency vi. In this approximation, the volume derivative of a frequency is

(-1 a*n”

Here,

-y’u’.

Y’h = -c-

(14)

’

where (Yis the thermal expansion and V is the molar volume. The quantity y defined by Eqn. 10 is equal to yth in the case of single-element solids too. Having in mind the application of this method to the system diamond-graphite, let us examine this case more carefully.

On the other hand,

au,

u,

(aT1v=-r

(7)

Comparing Eqns. 6 and 7, we obtain

THE PRESSURE EFFECT ON THE EQUILIBRIUM ISOTOPE FRACTIONATION IN SINGLE-ELEMENT SOLIDS: CALCULATION OF THE EQUILIBRIUM ISOTOPE CONSTANT IN THE SYSTEM DIAMOND-GRAPHITE

(8a) In the first order of thermodynamic perturbation theory, the following expression for the P-factor is valid ( POLYAKOV,

The similar expression for isotopically substituted form is

(&),= rt T(g),.

1989):

(8b) B-l+%

Differentiating Eqn. 4, with respect to volume and taking into account Eqns. 8a and 8b, one can obtain (9) where C [r,u, cth (O.~U,) - y:u:

cth (0.5~: )]

-Y= ’ C [u, cth (O.~U,) - u: cth (0.5u:)]

I

(10)

/

*iK, -

(Kjhl,

(15)

where “/ is the average value of the kinetic energy of the jth atom, (K& is the average kinetic energy calculated classically, j is the number of the isotopically substituted atom, m and m * are masses of light and heavy isotope atoms, respectively, and Am = m* - m. The derivation of Eqn. 15 and the discussion of its applicability are given in Appendix B. The index j is unnecessary in the case of single-element solids. From the virial theorem, the mean value of kinetic energy of harmonic oscillations is a half of the whole vibration en-

Effect of pressure on isotope fractionation

4741

ergy, so one can write:

p=1+

&j%

t&b

-

(&ib)cl).

(16)

The classical vibrational energy of a solid is 3RT, the quantum mechanical expression of&b is

E”ib=~(i=~(~+hui(exp~-l~), (17) and ti is the vibrational energy referring to the ith mode. Using the expression of the P-factor (Eqn. 16) one can derive a@ Taking into account the definitions of the thermal ( ap 1r’ Griineisen parameter modulus, we obtain

(Eqn.

=

12) and the isothermal

-!_ (YE&b BT

~thTCu).

bulk

(18)

Here, c Yi% # YE = Evib ’

FIG. 1. The isotopic shift in the system diamond-graphite Ano along the phase equilibrium curve: ( I ) calculated in harmonic ap proximation (without pressure corrections); (2) calculated in quasiharmonic approximation (taking into account the effect of pressure), and (3) the phase equilibrium curve for the system graphite-diamond in (P, T)-coordinates, the pressure axis is on the left and pointing down.

(19)

are negligible, from Eqns. 12 and 19 it follows that 7th = yE. So ( 18 ) may be rewritten

as

(S),=:[ l)fpg

-2.0

(p-

1

.

(20)

This equation expresses the derivative of the B-factor in terms of thermodynamic properties of a crystal. Writing the anala@ one can obtain Bqn. 11, in which ogous expression for aT ( 1“’ -,, = 7th = YE. Consequently, to calculate the pressure dependence of the P-factor in the case of single-element solids, it is necessary to know either BT, yth, or @(T) for the application of Eqn. 11 or @,C,, Tth, and BT for the use of Eqn. 20. These results can be directly applied to the calculation of the isotope fractionation in the system graphite-diamond. The dependences @(T) for graphite and for diamond at normal pressure were calculated by BOTTINGA ( 1969). With results of these calculation and with handbook data (graphite: Yfh = OS, B7 = 338.0 + 6.0 P kbar, diamond: 7th = 1.0, B7 = 4423 + 4.2 P kbar), B-factors of graphite and diamond were calculated from Eqn. 11. The results of these calculations along the phase equilibrium curve are shown in Fig. 1. It was found that the pressure derivative of the graphite /3factor exceeds that of diamond by several times. It produces a change of sign of the equilibrium isotopic shift between diamond and graphite at elevated pressure. According to the calculations of BOTTINGA( 1969)) the isotopic shift A of the system diamond-graphite is more than zero at any temperature (Fig. 1). We have calculated the isotopic shift along the phase equilibrium curve diamond-graphite (Fig. 1). One can see that graphite is isotopically heavier than diamond

under P-T conditions (interesting for geochemical and experimental applications). In light of these results, it is necessary to revise the interpretation of some experimental data suggested without consideration of the pressure effect. For example, measuring the equilibrium isotopic shift in the system diamond-graphite, GALIMOV ( 1984) observed negative A(diamond_mphite) values -0.2 to -0.4% at temperatures T = 1050-1200” and pressures 41-50 kbar. On the basis of Bottinga’s calculations ( BOTTINGA, 1969) this isotopic effect was considered as kinetic. Taking into account the pressure influence, this effect should be considered as equilibrium. One can note an excellent accord between experimental and theoretical results (the theory gives A(diamon,+graphite) = -0.45 to -0.65%). CALCULATIONS OF THE EQUILIBRIUM OXYGEN ISOTOPIC FRACTIONATION FACTORS FOR SILICATE SYSTEMS In the general case of a compound solid, the parameters y and 7th are not equal. To calculate /?( T,p),it is necessary to know the spectrum of the unsubstituted solid, the spectrum of the completely substituted form, and yi and y t for every mode. All these data may be obtained if the theoretical vibration problem of the unsubstituted and isotopically substituted crystal is solved. These computations are a complicated problem that have been resolved for only a limited number of minerals. Therefore, a simplifying model is needed. To check the applicability and the exactness of the approach, calculations by four different models have been done for “‘O“‘0 exchange. Schematic frequency distributions are shown in Fig. 2 (a,b,c,d). The corresponding spectrum of the isotopically substituted form is obtained by shifting every mode or group of modes to lower frequencies. In all the models, the vibrational unit of a crystal is taken as the Bravais or primitive unit cell. The primitive unit cell containing s atoms is described by 3s degrees of freedom,

4142

V. B. Polyakov and N N. Kharlashina

1

d

I.2 FIG. 2. Schematic

vibrational

spectra used for minerals:

(a) model

I. (b) model 2, (c) model 3, and (d) model 4. Corresponding spectra of isotopically substituted forms can be obtained by shifting every mode or group of modes to lower frequencies.

three of them are acoustic, and 3s3 are optical. Following KIEFFER ( 1982), three acoustic modes are assumed to obey a sine wave dispersion relation. The acoustic frequency distribution is the same for all the models, because acoustic modes are not significant in isotopic calculations. The considered model spectra differ from each other in their optic mode distributions. In model 1, all optic modes are represented by Einstein oscillators at frequencies corresponding to fundamental vibrations (long-wavelength vibrations with a wave vector k = 0, at which physical observations are frequently made). So it is possible to use the IR- and Raman spectroscopic information on the pressure dependence of u, and isotopic shift Y: /vi directly. The spectroscopic model ignores the dispersion v, (k), which limits its accuracy. On the other hand, in this model, we have the most detailed description of the pressure dependence of a spectrum. Unfortunately, the experimental IR- and Raman spectroscopic data are not complete for the majority of minerals. So the calculations by this model were done only for quartz and forsterite. We have a complete set of fundamental frequencies vi, their statistical weights qi, the mode Grtineisen parameters yi, and isotopic frequency shifts s, = Y: /v, for quartz. In the case of fotsterite, y, of eighteen optical frequencies, only, are known (HOFMEISTER, 1987, 1989). The primitive unit cell of forsterite contains twenty-eight atoms, so the complete set of funda-

mental frequencies includes eighty-four modes. The calculations have been done on the basis of the known frequencies. The isotopic shifts and the corrections of the statistical weights have been taken in accord with the model of forsterite suggested by KIEFFER ( 1982). The parameters of model 1 are given in Table 1. Model 2 was suggested by KIEFFER ( 1979a,b) on the basis of elastic, structural, and spectroscopic data analysis. The model was applied to the prediction of the thermodynamic functions Cl (heat capacity), S (entropy), E (internal energy), and F (Helmholtz free energy). The oxygen isotope equilibrium factors of silicates, calcite, and rutile and the pressure dependence of thermodynamic properties C, and S were calculated by KIEFFER ( 1982) as applications of the model. There are three groups of modes in model 2 (Fig. 2b). Acoustic modes of a spectrum are assumed to obey a sine wave dispersion relation, as well as in the other models discussed. The other (3n-3) optic modes are divided into internal and external modes. Internal optic modes (so-called stretchmodes) correspond to v, and vj vibrations of Si-0, AI-O, or C-O bonds and are presented as Einstein oscillators. The external modes are presented by one or more frequency intervals with uniform density of states-optic continua. Frequency ranges, isotopic shifts s, , and statistical weights y, are taken from KIEFFER ( 1982), but the values of y, have been corrected. Note that in KIEFFER ( 1982) calculations of thermodynamic properties C, and S at high pressures, the parameters y, of acoustic and low-frequency optic modes are more significant than y, of high-frequency optic modes. This fact allowed KIEFFER ( 1982) to set yi = 0 for Einstein stretchmodes. For our purposes, this approximation is not valid. So, the model has been completed by the experimental parameters y1 of high-frequency stretch-modes, which are the most important in the calculation of the equilibrium isotope fractionation. Using y, of acoustic modes from KIEFFER ( 1982) and y, of stretch-modes from IR- and Raman high pressure spectroscopic data, one can recalculate y, of optic continua: 1 (Co% YOPl= C u,o!Jt

-

zkYkCu,k).

(21)

Here index k numbers acoustic and stretch-modes. Model 2 data for quartz and forsterite are listed in Table 2. Models 3 and 4 are modified versions of model 2. In model 3 internal stretch-modes are represented by a uniformly distributed frequency region, instead of a set of Einstein oscillators in model 2. An attempt to detail model 2, using high pressure spectroscopic data has been made in model 4. Keeping as a whole the shape of the vibrational spectrum, the number of optic continua has been increased, and some averaged experimental values for yI have been taken for each region. Models 3 and 4 data for quartz and forsterite are presented in Table 2. As is evident from the foregoing, the parameter yi refers to a large number of modes of the same group (optic continuum, stretch-mode), On the other hand, the next equality is fulfilled (see Appendix A):

4743

Effect of pressure on isotope fractionation Table 1. The parameters of model 1 for quartz and forsterite.

r

i' -1 cm

T

a

Quartz 'i

Forsterited

1

s. 1

'ii"-l/ qi

'i

/ si

'i

102.0t 0.03704 0.96825

0.02:I

98.0b

0.0119, 0.9727

0.88

122.0t 0.03704 0.96825

0.02:I

99.0b

0.0119

0.9725

0.88

164.0t 0.03704 0.96825

.I.18 I.71.0b 0.0119

0.9725

1.65

128.0

0.07407 0.93828

1.65

142.0

0.0595

0.9736

1.52

205.6

0.03704 0.93677

3.58

201.0

0.0595

0.9736

1.91

263.1

0.07407 0.95173

0.49

246.0

0.0595

0.9736

1.04

354.3

0.03704 0.97883 -0.13

275.0

0.0595

0.9736

1.37

363.5

0.03704 0.97387

0.00

275.0

0.0595

0.973C

1.69

393.8

0.03704 0.96470 -0.01

295.0

0.0595

0.9731

1.25

401.8

0.03704 0.95744 -0.01

295.0

0.0595

0.9736

1.47

450.0

320.0

0.0595

0.973C

1.23

463.6

0.07407 0.9426 0.38 ci 0.64 0.03704 0.95513

362.0

0.0595

0.9736

0.85

509.0

0.03704 0.9426'

0.38

421.0

0.0476

0.95

0.65

697.4

0.07407 0.98179

0.37

474.0

0.0476

0.95

0.58

796.7

0.03704 0.98795

0.27

545.0

0.0476

0.95

0.88

808.6

0.07407 0.98417

0.27

601.0

0.0476

0.95

0.39

066.1

0.07407 0.96398

0.04

644.0

0.0476

0.95

0.39

083.0

0.03704 0.96602

0.05

837.0

0.0476

0.943

0.44

160.6

876.0

0.0476

0.977

0.39

231.9

925.0

0.0476

0.977

0.30

986.0

0.0476

0.977

0.79

'ibrationalfrequencies vi ~MLEY

(19871,

isotopic

and

parameters ;T~ for quartz:

are

f‘i-on

shifts si are from !&To et al. (19871,

mode

weights qi correspond to K~EFFER (1982). b The upper frequency limit of an acoustic branch, from K~EFFER

'Assumed to coincide &values

with K~EFFER (1982) and to obey the

product rule. d. VIbrational frequencies vi and parameters xi for forsterite are from HQFMEI~TER et al. (1989). isotopic shifts si and mode weights qi correspond to K~EFFER (1982.1.

where the summation is over all the modes of a spectrum. The same equality is true for the summation over the modes of the same type of symmetry. From the experimental data, one may conclude that closely spaced modes differ less in yi values than widely spaced ones. On these grounds, we set yi = r’ , which seems to be reasonable for a model. All four models give similar &factor values at normal pressures. The difference does not exceed 0.6%0 at 298 K and 0.1%0 at 1173 K. It is interesting to compare the parameters y and the pressure-induced corrections to &factor calculated by all the models. The corrections to @-factorsand the dependency y( T, P) have been calculated at temperatures T = 273-1273 K and pressures P = O-50kbar. It turns out that the parameter y is effectively independent of pressure. The change in y is less than 0.05 at pressures up to P = 50kbar. The dependence

y(T) for the discussed models is shown in Fig. 3. One can note that models 2 and 3 do not differ greatly in y, as well as models 1 and 4. But the values of y calculated for models 2,3 and 1,4 differ markedly. Comparing the parameters of the models one can see that the difference is caused by the large values of yi of the upper part of the optic continuum in models 2 and 3. The reason is that in models 2 and 3, yi of the optic continuum is defined as the average value, the averaging being made with respect to the statistical distribution with weight CVi,but not by formula 10. The corrections to quartz and forsterite @-factors at a pressure P = 10kbar calculated by different models are shown in Fig. 4. All four models exhibit a strong dependence of the corrections on temperature. For example, at T = O‘C, the pressure-induced correction to lo3 In @ of quartz is about 0.7%0for models 1 and 4 or 1.4%0for models 2 and 3, whereas,

4744

V. B. Polyakov and N. N. Kharlashina Table The

2.

parameters

of

the

models

2,3,and

4 for

I Frequency_l

3,and

Acoustic

4

Model

v4

s.

qi

‘i

1

0.03704

0.96825

0.023

0

- 98

0.0119

0.9727

0.88

0.03704

0.96825

0.023

0

- 99

0.0119

0.9727

0.88

0 -

164

0.03704

0.96825

1.18

0

-

171

0.0119

0.9727

1.65

90

-

550

0.48170

0.94975

0.80

128

- 420

0.5363

0.9736

1.37

697

-

809

0.18520

0.99

420

- 650

0.2380

0.95

1.37

1080

0.03704

0.96

0.0480

0.943

0.5

1117

0.07407

0.96

0.1420

0.977

0.5

1162

0.03704

0.96

1200

0.07407

0.96

c

0.80

~ 837

Ii

90

-

550

0.48170

0.94975

0.80

128

- 420

0.5363

0.9736

1.38

697

-

809

0.18520

0.99

0.80

420

- 650

0.2380

0.95

1.38

1080

-

1200

0.22222

0.96

0.02

700

- 840

0.0480

0.9430

870

- 990

0.1420

0.977

External, v4

cm

122

v3

v2and

nterva1,

102

Internal

3,

Frequency_l

‘i

1

2,

vland

Model

s.

0 -

optic modes

FORSTERITE

0-

External, v2and

forsterite

__‘i

cm

interval,

2,

and

QUARTZ

=requencies

lodels

quartz

optic Internal

modes

vland

v3

External, Model

4.

v2and

v4

optic

90

-

265

0.18326

1.60

128

- 320

0.3528

0.9736

1.45

265

-

450

0.19373

.0.04

320

- 420

0.1835

0.9736

0.75

420

- 650

0.2380

0.95

0.56

450

-

550

0.10472

0.50

697

-

809

0.18520

0.31

modes Internal vland

v3

i

1080

0.03704

0.96

0.05

837

0.0480

0.943

0.5

1117

0.07407

0.96

0.04

930

0.1420

0.977

0.5

0.96

-0.10

0.96

0.04

1162

0.03704

1200

0.07407

I

at T = 5OO"C,this correction is equal to 0.1 %Ofor models I and 4 or 0.2%0 for models 2 and 3. It can be seen that the pressure-induced corrections to the p-factor vary significantly in their values for different models. However, the pressureinduced correction to the isotopic fractionation factor is much lower and approximately the same for different models. For example, the corrections to IO3 In cr~quanr_rOors,en,ej are 0.1%0 (models 1 and 4) or -0. I%0 (models 2 and 3) at T = 0°C and 0.01 %Oor -0.01 %Oat T = 500°C. In other words. an

agreement between the pressure-induced corrections to the isotopic fractionation factor calculated by different models is good, especially for geochemical applications. By and large, the model approach is found to be applicable to the calculations of the pressure effect on the equilibrium isotopic fractionation in minerals. In order to obtain a consistent set of pressure-induced corrections to the P-factors

‘emperature,

10

“C

For2.3

^,

orz,.

E

0 w c 1.0 Fori For4 Qtzl Qtr4

I 6 f: 0

i

lb-,:

'GO.5 c v "0 0.0

0

200

400

600

800

1000

T."C

FIG. 3. The parameters y of quartz and forsterite calculated by different models. Forsterite: the curve For 1 corresponds to model I : the matched curves For2, 3, to models 2 and 3; the curve For4, to model 4. Quartz: the curve Qtzl corresponds to model 1; the curve Qtz2,3 corresponds to models 2 and 3; the curve Qtz4, to model 4.

10

lo6

T--12

14

Tm2

FIG. 4. The pressure-induced corrections to the value IO3 In @ for quartz and forsterite under a pressure P = 10 kbar as a function of temperature (oxygen 160- “0 fractionation ). The calculations have been made by different models. The notations are the same as in Fig. 3.

Effectof pressure on

isotope fractionation

4745

As a rule, it is more important to know the equilibrium isotope fractionation factor between two substances in a system than the value of the b-factor itself. According to formulae 2 and 11, the effect of pressure on the equilibrium isotope fractionation factor is caused, first, by the difference in parameters y of minerals, second, by the difference in values W of the derivatives - , and third, by the difference in isoalthermal bulk moduli. Let us consider all these factors.

and thus, to get a reliable estimate of the effect of pressure on a mineral-pair fractionation factor, we start from single model 2, in the following consideration. This model is the most advanced and provides a good accuracy in estimation of the isotope fractionation factors for minerals (KIEFFER, 1982; CLAYTONand KIEFFER, 1989 ) . At present time, models 1 and 4 can not be used for a large number of minerals, because of lack of complete high pressure spectroscopic data. The dependences ,13( T, p) and y( T, p) have been calculated at temperatures 298-1273 K and pressures O-50 kbar for the set of minerals: quartz, calcite, albite, enstatite, pyrope, grossular, forsterite, and t-utile for ‘%- “0 substitution. The parameters of the model were taken from KIEFFER ( 1982); some corrected values, source references, and compressibility data are listed in Table 3. Results of our calculations are presented in Figs. 5 and 6.

The derivative ar a6 does not differ markedly for different minerals, except for hydrogen fractionation, but this factor a@ causes the derivative dp to depend strongly on temperature. For example, corrections to the B-factor at a pressure P = 10 kbar exceed 0.8% at T = 298 K for most solids under in-

Table 3. The compressibility (V -VI/V =Ap-Rp'. thermal Grtineisenparameters= and mode GriineisenPar%aeter~ of minerals used in the aodel 2.

1 Mineral

Quartz

A,

T

B . _>

Acoustic

-I-

rbar

'th

11=T2

xi-- opt

2.697

20.4

0. 6Y4t

0.023

1.18

0.x

2.02

21.6

o.4sd

0.46

3.46

0.46

1.01

l.lf

1.16

0.583

1.24h

0.597

_1

nbar

astr

a at t=2s0c

0.05( 0.04 -0.10 0.04

0.29

0.3e

0.36

1.16

1.16 0.23'

0.64

1.13

1.95

1.29 0.5Si

0.96

1.22h

1.25

1.25

1.25

O.f?

1.06

0.82

1.0

1.2gh

0.88

1.65

1.36

0.5j

0.98

0.48

0.92

1.5k

0.7

3.7

1.81

1.56' 0.95

1.32

1.367

3.9

o.4sd

0.49

I.49

0.49

0.23m -0.04 0.40 0.46

0.34

a The compressibility data are taken from BIRCH (1966) or O'NEIL et al. (19891, the acoustic mode Griineisen parameters are from K~EFFER (1982) or calculated by the formula (22) as well as optic ones. b From SGGA (1968).

’ The values 0.05, 0.04, -0.10, and 0.04 correspond to the modes 1080, 1117, 1162, and 1200 cm-l, from &.&~y

(1987).

d Calculated by (14) using the thermal expansivity from SI
(1966)

e The mean of the data CCGuTy and VELDE,1Y86) for stretch-modes. f From ARAGGt et.al. (1987). g The mean of the data CXu et al.,1983) and (DIETRICHand ARmT,1982). h From ANDERSON et al. (1992). i The mean of GILLET et al. (1992) for stretch-modes. J The mean of HOFMEISTER et

al. (1989) for stretch-modes.

k (SHANER, 19731. 1 Using the data of ty@,&$3k~~ et al. ~$324~;~

M.95,

~(610~;

(1980) it was

calculated

)=1.56.

m From &L_FT et al.(1993) for the modes 712; 881; 1070; and 1460 cm-l correspondingly.

4746

V. 8. Polyakov and N. N. Kharl~~ina

FIG. 5. The dependence y(T) calculated by model 2 for the minerals: (Qtz) quartz, ( Alb) albite, (Cal-O) calcite, (Ens) enstatite, (Pyr) pyrope, (For) forsterite, (Gras) grossular. (Rut) Mile; fractionation in all the minerals listed is on oxygen ‘60- ‘6: Cal-C-calcite. carbon ‘*C-3.Z fractionation.

whereas, at T = 1000 K they are about 0.1 L (see Fig. 6). It turns out that the parameter y depends only shghtly on tempemture and is essentially independent of pressure. The temperature dependence of y for minerals under consideration is shown in Fig. 5. One can see that y has values ranging from 0.2 to 1.4 for minerals under consideration. Examination of definition 10 shows that the position and the mode Gruneisen parameters of the high-frequency optic and especially stretch-modes are the most signifi~nt for the y value. As was shown by KIEFFER ( 1979a,b, 1982), the position of these modes in silicate spectra varies systematically with the degree of polyme~zation of silicon-oxygen tetrahedra. One can note from Table 3 that the stretch-mode Griineisen parameters reveal the same regularity. These facts appear to be the reason for the systematic change of y with crystal structure (see Fig. 5). The application of the model gives the following order in y: quartz < albite = calcite < enstatite < pyrope < forsterite < grossuiar < rutile. Note. that by the calculations of KIEFFER ( 1982), the model gives the following order in p-factor values at 298 K and zero pressure: quartz > calcite 2 albite > enstatite > pyrope > grossular > forsterite 2 rutile. Except for the position of grossular relative to forsterite, one can note that minerals have an inverse order in y, relative to p-factor values. Obviously, this regularity depends on the regularity in behavior of vibrational modes with degree of polymerization of Si-0 bonds. It is interesting to compare y and the thermal Griineisen parameter yth. The values of these two parameters at 7’ = 298 K are presented in Table 3. The difference between +ythand y varies within wide limits for different minerals. It is caused by the fact that yth is determined essentially by yi of the acoustic and low-frequency optic modes. Since the pressure-induced corrections depend on several

vestigation,

factors [y, I&-, $).

it is difficult to reveal some general

regularity in their values. One can see in Table 3 that compressibihty of minerals varies in the following order: quartz > albite > calcite > enstatite > forsterite > grossular = pyrope > rutile. The greater the value of y, the lower the com-

pressibility of a mineral. This tendency cause the pressure effects on the isotope equilib~um f~~ionat~on factor of a mineral-pair to decrease. However, the pressure-induced corrections may be notable for systems of minerals with different crystal structure and composition, for example: rutilesilicate, calcite-silicate. These results allow us to evaluate the limitations on pressure in oxygen isotope exchange experiments and to estimate the pressure effect on oxygen geothermometry of natural rocks. CLAYTON and KIEFFER ( 1989)useda combination of laboratory experiment and statistical thermodynamic calculation to obtain the function IO3In p over a wide temperature range for several rock-forming minerals. They represent IO3 tn fi in the polynomial form 103 In p = :1.Xi- IL?.\-* + cX.3,

(23)

where x = lOhI’-‘. The majority of experimental data were obtained at pressure P = I5 kbar and temperatures from 600 to 1200°C (CHIBA et al., 1991; CUYTON et al., 1989). For convenience, we approximate the pressure-induced corrections at P = 10 kbar by similar expressions (Table 4). For exampie, for the caicite-albite system. the fractionation factor from CLAYTONand KIEFFER ( 1989) is lo3 In Ntcc_&)= 0.647.~ - 0.094,~~ + 0.0054.x’.

(24)

From Table 4, the pressure-induced correction for this system atPIOkbaris 103[ln cuf10 kbarf - In (Y(I bar)] = -0.0294.v - 0.0002.\-2.

(25)

Errors in the expe~mental calcite-minerai fmct~onation were estimated by CHIBAet al. ( 1989), based on the internal scatter of the data and the accuracy of isotopic measurements. They obtained the following linear approximation for the calcitealbite system: 1O3 In (Y~,,.,~~, = (0.56 +- 0.06)~.

0.0

0

2

4

6

B

(26)

10

10" T-.'

FIG;.h. The pressure-induced corrections to the value 10’ In $ at a pressure P = 10 kbar relative to ambient pressure. The abbreviations are the same as in Fig. 5.

Effect of pressure on isotope fractionation

One can see that the dominant term for the pressure-induced correction at P = 10 kbar, -0.0294x, is about 5% of the dominant term for the fractionation factor and is about a half of its experimental uncertainty. Since the corrections are approximately linear with pressure, a pressure of twenty kilobars and higher may produce a measurable effect on oxygen isotope fractionation in this system. Similar reasoning shows, that at pressures from 20 to 30 kbar, the pressure-induced corrections become comparable with experimental errors for other pairs of minerals represented in Table 4. This estimate is in agreement with that of CLAYTON( 198 1). As noted above, the order of In /3( 10 kbar) - In P( 1 bar) values more or less follows the order of the &factor values calculated by KIEFFER ( 1982). So the largest values of the corrections correspond to mineral pairs with the largest magnitude of the fractionation (the most sensitive isotopic geothermometers). For example, the fractionation factor of the quartz-t-utile system may be expressed as (MATTHEWS, 1993) A = 103’ In ‘Y~~_,,,~) = 5.03 106T2.

can be written

6T = - 6AT310-6/ 10.06,

(28)

where GTand 6A are errors in temperature and A, respectively. The pressure-induced corrections at P = 10 kbar are 0.17%~~ at T = 300°C and 0.05L at T = 800°C. Substitution ofthese values in Eqn. 28 gives changes in temperature about 4°C at 300°C and 24’C at 800°C. These numbers are comparable with the effect of analytical uncertainty in the rock data to which the geothermometer may be applied. The foregoing shows that a pressure higher than 20 kbar may produce a measurable effect on results of laboratory oxygen isotope exchange experiments as well as on oxygen isotope geothermometry in rocks. The system calcite-graphite-diamond (carbon fractionation) is an example of a system of minerals with different crystal structure. The calculation of graphite and diamond has been described above. The model spectrum has been used for calcite. Calculations show that the parameters y of calcite for 12C-j3C and 160- “0 isotope substitutions are close, but not equal (see Fig. 5 ) . In general case, for a given substance, Table 4. P=lOkbar relative Mineral

corrections

to the p-factor

to ambient pressure

for

Quartz

d = 0.1268x

- 0.0018~’

Forsterite

6 = 0.1212x

- 0.0019x2

Albite

6 = 0.1298x

- 0.0034~’

Enstatite

6 = 0.1052x

- 0.0017x2

Grossular

d = 0.1035x

- 0.0020x2

Pyrope

6 = 0.0914x

- 0.0016x2

Calcite

6 = 0.1004x

- 0.0032x2

Rutile

6 = 0.0705x

- 0.0012x2

lX =

106T-2.

values

individual

d = 1n3[ln(3(10kbar)-ln@(latm)l

0

0

I

,

200

I

,

,

400

600

I

E IO

T,“C FIG. 7. The pressure-induced variation of 10’ In j3 for graphite, calcite, and diamond under a pressure P = 10 kbar relative to ambient pressure as a function of temperature (carbon ‘2C-‘3Cfractionation).

(27)

From this expression, errors in temperature as

The pressure-induced

4741

at

minerals:

the parameters y relating to different elements may differ markedly. The pressure corrections to In fi of calcite, graphite, and diamond at P = 10 kbar are presented in Fig. 7. One can see that the pressure effect is comparatively large and achieves a few permil at low temperatures. The calcite-graphite isotopic geothermometer is well suited to estimating temperatures in high-grade metamorphic rocks. Quantitative application of the thermometer, however, requires a precise knowledge of the position of the calcitegraphite equilibrium fractionation curve. This is extremely important, because of a significant disagreement between experimental (SHEELE and HOEFS, 1992) and combined experimental-theoretical ( CHACKO et al., 199 1) calibrations, on the one hand, and cation exchange calibrations, on the other hand (see a comprehensive discussion by SCHEELEand HOEFS( 1992 ) and references therein). From our results (Fig. 7), it is clear that the pressure effect may be significant for the calcite-graphite isotope thermometer and should be taken into account in comparing the geothermometer calibrations mentioned above.* SCHEELE and HOEFS( 1992) were first to obtain the calcitegraphite equilibrium isotope fractionation factor in laboratory isotope exchange experiments. They determined the carbon isotope equilibrium fractionation curves for calcite and CO* and for graphite and CO2 by direct isotope exchange and then combined them to obtain the calcite-graphite fractionation curve. Unfortunately, the authors used various pressures ( 1- 15 kbar) in constructing each of the experimental curves and in evaluating the calcite-graphite equilibrium curve. Consequently, in order to estimate the pressure effects from these data, the calculations involving CO2 at high pressures

* For example, the equilibrium curve by CHACKO et al. ( 199 1) lies higher by about 0.5% at temperatures 600-800°C than the empirical equilibrium curve obtained by VALLEY and O’NEIL ( 198 1) using the cation exchange calibration. Since the VALLEY and O’NEIL ( 198 1) data came from marbles in a granulite terrain, with metamorphic pressures of about 7-8 kbar, the main part of the observed discrepancy may be an equilibrium pressure effect. This possibility was suggested by one of our reviewers.

4748

V. 8. Polyakov and N. N. Kharlashina

are required. Nevertheless, the Sheele-Hoefs method allows experimental verification of our calculations. For this purpose, the experiments at several pressures for the both calciteCO2 and graphite-CO* systems should be conducted.

THE CALCULATION OF HYDROGEN ISOTOPE FRACTIONATION IN THE SYSTEM BRUCITE-WATER

Suppose that hydrogen isotopes form ideal mixtures in the case of minerals as well as water. The model spectra of the isotopic forms of brucite have been developed by analogy to oxygen isotopic and the~~ynamic calculations for silicates ( K~EFFER, 1982). Thermodynamic data on B?, yz, and 7th have been taken from SAXENA (1989). The corrections to IO3 In fi of brucite under pressure P = 1 kbar were found to be about 2.5 at T = 273 K and 0.4 at T = 1273 K. Water corrections have been calculated using thermodynamic formula 1. The data on p( T) ( SINGHand WOLFSBERG, 1975 ) and molar volumes of Hz0 and DzO ( RABINOVICH, 1968) are well known. It is interesting that, for liquid phase, the molar volume of I&O is greater than that of D20, so that the b-factor of water decreases with increasing pressure as opposed to the P-factor of solids. The pressure derivative along the phase equilib~um pour-liquid

curve water va-

water is presented in Fig. 8. One can see that has a maximum

value -0.6

kbar-’

at T

= 1lO”C, down to -2 kbar-’ at lower, and to -3 kbar-’ at higher temperatures, and is not determined in the neighbourhood of the critical point. Thus, in the brucite-water system the pressure correction to A~~~~,~~~~~~~~ is about 3-4% per kbar. CONCLUSIONS A calculation technique for the estimation of the pressure effect on equilibrium isotope fractionation has been developed, using the quasi-harmonic approximation. Equation 1 1 expresses the pressure-induced corrections to the isotopic

partition function ratio (&factor) in terms of thermodynamic and isotopic constants of a mineral. To describe the anharmonicity effect on the P-factor of a mineral, the parameter y (Eqn. 10) has been suggested. The parameter y is equal to the thermal Grilneisen parameter in the case of elemental substances, and is expressed in terms of the mode Griineisen parameters and vibrational frequencies of isotopic forms in the case of compounds. The effect of pressure on @-factors has been found to be strongly temperature dependent. The pressure effect for the system diamond-graphite has been presented as an example of calculations in the case of elemental substances. The great difference in compre~ibilities of the substances causes the isotopic shift of the system to change sign at high pressure. The application of the technique to a set of silicates (quartz, albite, enstatite, pyrope, grossular, forsterite), rutile and calcite ( 160- ‘*O fractionation), calcite ( “C- i3C fractionation), and brucite-water (H-D fractionation) has been presented. The model approach has been used for compounds and discussed. The most significant pressure effects may be expected for systems of minerals with different crystal structure. It has been shown that pressure of ten kilobars and higher may produce a measurable effect on the equilibrium isotope fractionation (e.g., calcite-graphite). In this connection, it may be possible to test ex~~men~ly the calculated pressure effect on the calcite-graphite carbon isotope fractionation. In the case of the oxygen isotope equilibria, the effect of pressure is rather small, and becomes measurable only at pressures above twenty kilobars. However, even small corrections may be important, because they introduce a systematic error into calculations or interpretation of experimental data. It is especially important for the purposes of extrapolation. In the case of H-D substitution for mineral-water system, the pressure-induced corrections to the p-factor of a mineral is of opposite sign to that of water, so that corrections do not counteract, but add together. For the system brucite-water, the corrections to AD reach 3-4%~ per kbar. it is clear that all the models presented in this paper may be not correct in detail, because of a lack of experimental data on vibrational spectra and mode Grilneisen parameters. At the present time, high-pressure IR- and Raman-spectros-

copy is progressing rapidly, as are theoretical calculations of vibrational spectra of minerals. All these data can be used to %f3/vHlo /----

/I

1010

refine the calculations for the minerals discussed and to consider others. On the basis of the investigations presented, it can be concluded that an estimate of the effect of pressure on equilibrium isotope fractionation is necessary in high-pressure isotopic research. An unjustified neglect of this effect may introduce errors into the findings of investigations.

thank greatly Dr. A. Matthews, Dr. T. Chacko, and an anonymous reviewer for the helpful and constructive reviews; their comments and su~estions significantly improved the manuscript. Support by the Russian Fond of Fundamental Researches (grant #95-059215) and the International Science Foundation established by Mr. George Soros (grant #ES I- 15990925 ) are gratefully acknowledged. Acknowlrdgm~nts-We

FIG. 8. The effect of pressure on the p-factor of water in the case of hydrogen fractionation. Curve A exhibits a temperature dependence of liquid phase molar volume ratio VW/V,,, (right axis) (RABINOVICH, 1968). The data are taken at pressures of saturated water vapour. Curve B is the pressure derivative of lo3 In 6 as a function of temperature.

Editoriul

handling:

B.

E. Taylor

Effect of pressure on isotope fractionation REFERENCES AKAOGI

M., NAVROTSKY A., YACI T., and AKIMOTOS. ( 1987) Pyroxen-garnet transformation: thermochemistry and elasticity of garnet solid solution and application to a pyrolite mantle. In Highpressure Research in Mineral Physics (ed. M. H. MANGHNANI and Y. S~ONO), pp. 25 I-260. Terra. ANDERSON0. L., ISAAKD., and ODA H. ( 199 I ) The~~l~tic parameters for six minerals at high temperature. J. Geophys. Rex 96, 18037-18046. BIGELEISEN J. and MAYERM. Cl. ( 1947) Calculation of equilibrium constants for isotope exchange reactions. J. Chem. Phys. 15,26 l267. BIRCHF. ( 1966) Compressibility; elastic constants. In Handbook of Physical Constanfs (ed. S. P. CLARKJR .); GSA Mem. 97,97- 174. BORN M. and HUANG K. ( 1954) dynamical Theory ofCrystaL Lattices. Clarendon Press. BOTTINGAY. ( 1969) Carbon isotope fractionation between graphite, diamond and carbon dioxide. Earth Planet Sci. Lett. 5, 301-307. CHIBAT., MAYEDAT. K., CLAYTONR. N., and GOLDSMITHJ. R. ( 1991) Oxygen and carbon isotope fractionation between CO2 and calcite. Geochim. Cosmo~h~m. Acta S&2867-2882. CLAYTONR. N. ( I98 I ) Isotopic thermomet~. In Thermodynamics of Minerals and Melts (ed. R. C. NEWTON et al.), Adv. Phys. Geochem., Vol. I, pp. 85- 109. Springer-Verlag. CLAYTONR. N. and KIEFFERS. W. (1989) Oxygen isotopic thermometer calibrations. In Stable Isotope Geochemistry: A Tribute to Samuel Epstein (ed. H. P. TAYLORJR. et al.): Geochem. Sot.. Spec. Pubi. No. 3, pp. 3-10. Lancaster Press. CLAYTONR. N., GOLDSMITHJ. R.. KAREL K. J., MAYEDA‘I. K., and NEWTON R. C. ( 1975) Limits on the effect of oressure on isotopic fractionation. ‘Geochim. Cosmochim. Acta 39, ‘1l97- 120I. CLAYTON R. N., GOLDSMITHJ. R., and MAYEDAT. K. ( 1989) Oxygen isotope fractionation in quartz, albite, anorthite and calcite. Geochim. Cosmochim. Acta 53, 725-733. COUTYR. and VELDEB. ( 1986) Pressure-induced band splitting in infrared spectra of sanidine and albite. Amer. Mineral. 71, 99104. DIETRICHP. and ARNDTJ. ( 1982) Effect of pressure and temperature on the physical behavior of mantle-relevant olivine, orthopyroxene and garnet: 2. Infrared absorption and microscopic Gtineisen parameters. In High-pressure Research in Geoscience (ed. W. SCHREYER ) , pp. 307-3 19. E. Schweizerbart’sche Verlag-buchhandlung, Stuttgart. GALIMOVE. M. ( 1984) Variation of isotopic composition of diamonds and their relation to conditions of diamond-fo~ation. Geokhimiya N8, 1091-l 117 (in Russian). GARLICKG. D., MACGREGORI. D., and VOGELP. E. ( 1971) Oxygen isotope ratios in eclogites from kimberlites. Science 172, 10251027. GILLETPH., FIQUETG., MALEZIEUXJ. M., and GEIGERC. A. ( 1992) High-pressure and high-temperature Raman spectroscopy of endmember garnets: Pyrope, grossular and andradite. Eur. J. Mjnerai. 4,65 l-664. GILLETPH ., BIELLMANN C., REYNARDB., and MCMILLANP. ( 1993) Raman spectroscopic studies of carbonates. Part 1: High-temperature behaviour ofcalcite, magnesite. dolomite and aragonite. Phys. Chem. Minerals 20, I - 18. HAMANNS. D., SHAW R. M., LUSK J., and BATTSB. D. ( 1984) Isotopic volume differences: The possible influence of pressure on the dist~bution of sulfur isotopes between sulfide minerals. Australian .I. Chem. 37, 1979-1989. HEMLEYR. J. ( 1987) Pressure dependence of Raman spectra of SiOZ polymorphs: a-quartz, coesite, and stishovite. In High-pressure Research in Mineral Physics (ed. M. H. MANGHNANIand Y. SYONO), pp. 347-359. Terra Sci. HOFMEISTERA. M. ( 1987) Single crystal absorption and reflection infrared spectroscopy of forsterite and fayalite. Phys. Chem. Minerais 14,499-5 13. HOFME~STER A. M., Xv J., MAO H.-K., BELLP. M., and HOERING T. C. ( 1989) Thermodynamics of Fe-Mg olivines at mantle pressures: Mid- and far-infrared spectroscopy at high pressure. Amer. Mineral. 74, 28 l-306.

4749

JOY H. W. and LIBBYW. F. ( 1960) Size effect among isotopic molecules. J. Chem. Phys. 33, 1276. KHARLASHINAN. N. and POLYAKOVV. B. ( 1992) The effect of pressure on the equilibrium isotopic fractionation: Silicates, calcite, futile. Geokhimiya 2, 189-200 (in Russian). K~EFFERS. W. (1979a) Thermodynamics and lattice vibrations of minerals: 3. Lattice dynamics and an approximation for minerals with appli~tion to simple substances and framework silicates. Rev. Geophys. Space Phys. 17, 35-59. KIEFFERS. W. ( 1979b) Thermodynamics and lattice vibrations of minerals: 4. Application to chain and sheet silicates and orthosilicates. Rev. Geophys. Space Phys. 18, 862-886. KIE!=FERS. W. ( 1982) Thermodynamics and lattice vibrations of minerals: Application to phase equilibria, isotopic fractionation, and hip-pr~ure the~~yn~ic properties. Rev. Geophys. Space Phys. 20, 827-849. LANDAUL. D. and LIFSHITSE. M. ( 1976) Statisticheskaja Fizika, Part 1. Nauka Press (in Russian). LEIBFRIEDG. and LUDWIGW. ( 196 I ) Theory ofAnharmonic E’cts in Crystals; Solid State Physics, 12, 275-444. Academic Press. MAMMONEJ. F., SHARMAS. K., and NIC~L M. ( 1980) Raman study of rutile (TiOz) at high pressures. Solid State &mm. 34, 799802. MATTHEWSA., GOLDSMITHJ. R., and CLAYTONR. N. ( 1983) On the mechanisms and kinetics ofoxygen isotope exchange in quartz and feldspars at elevated temperatures and pressures. GSA Bull. 94, 396-4 12. MINEEVS. D., GRINENKOV. A., and DEVIRTSA. A. ( 1989) Influence of pressure on stable isotope fmctionation. 1. “Direct” way. In Isoloaes in Nature: Abstr. 5th Working Mta. Leipzig. DV. 66-67. O’NEI~B., BASSJ. D., SMYTH J. R., and V&CHAN M: T. f 1989) Elasticity of grossular-pyrope-almandine garnet. J. Geophys. Rex 94, 17819-17824. POLYAKOVV. B. (1989) Quantum statistical development of the method of isotopic bond numbers. In Isofopes in Nature; Proc. 5th Working Mtg. Leipzig, pp. 747-758. POLYAKOVV. B. and KHARLASHINAN. N. (1989) The effect of pressure on the ~uilib~um isotopic fractionation in solids. In Isntopes in Nature; Proc. 5th Working Mtg. Leipzig, pp. 735-745. POLYAKOVV. B. and KHARLASHINAN. N. C1991) The effect of pressure on the equilibrium isotopic fractionation’in solids. Geekhimiya 11, 1605-1612 (in Russian). RABINOVICH I. B. ( 1968) Viiyaniye Izotopii na Fiziko-Khimicheskie Svoystva Zhidkostey. Nauka Press. (in Russian). SAKAI H. and TSUTSUMIM. ( 1978) D/H fractionation factors between serpentine and water at 100°C to 500°C and 2000 bar water pressure and the D/H ratios of natural serpentines. Earth Planet. Sci. Lett. 40, 23 l-242. SATO R. K. and MCMILLAN P. F. (1987) Infrared spectra of the isotopic species of alpha quartz. J. Phys. Chem. 91, 3494-3498. SAXENAS. K. ( 1989) Assessment ofbulk modulus, thermal expansion and heat capacity of minerals. Geochim. ~osm~him. Acta 53, 785-789. SCHEELEN. and HOEF~J. ( 1992) Carbon isotope fractionation between calcite, graphite and COr: an experimental study. Contrib. Mineral. Petrol. 112, 35-45. SHANERJ. W. ( 1973) Griineisen y of rutile (TiOr). Phys. Rev. B 7, 5008-50 10. SINGH G. and WOLFSBERGM. ( 1975) The calculation of isotopic partition function ratios by a ~~urbation theory technique. J. Chem. Phys. 62,4 165-4180. SKINNERB. J. ( 1966) Thermal expansion. In Handbook ofPhysica/ Constants (ed. S. P. CLARKJR.); GSA Mem. 97, 75-96. SOGA N. ( 1968) Temperature and pressure derivatives of isotropic sound velocities of a-quartz. J. Geophys. Res. 73, 827-829. UREY H. C. and GREIFF L. J. ( 1935) Isotopic exchange equilibria. J. Amer. Chem. Sac. 57,321. VALLEYJ. W. and Q’NEIL J. R. ( 1981) ‘3C/‘2C exchange between calcite and graphite: A possible geothermometer in Grenville marbles. Geochim. Cosmochim. Acta 45.4 I I-4 19. VARSHAVSKYYA.M. and VAISBERGS. E. (1957) Termodinamicheskie i kineticheskie osobennosti reakciy izotopnogo obmena. Uspekhi Khimii 26, 1434-1468 (in Russian).

4750

V. B. Polyakov

and N. N. Kharlashina

WENNER D. B. and TAYLOR H. P., JR. (1974) D/H and ‘80-160 studies of serpentinisation of ultramathic rocks. Geochim. Co.s-

And so the sum of y, is independent L, or 1.: Thus.

of the system of eigenvectors

mochim. Acfa 38, 1255- 1286. WILSON E. B., DECIUS J. C., and CROSS P. C. (1955) Molecuiur Vibrations. McGraw-Hill. Xu J., MAO H. K., WENG K., and BELL P. M. (1983) Preliminary data on the Fourier-transform infrared frequency shifts in hypersthene at high pressure. Carnegie Inst. Year Book 1982-1983,

c Yz = c r:. One can obtain the same conservation with the same type of symmetry.

(A7)

law for any group of frequencies

Geophys. Lab., 352-354. APPENDIX APPENDIX

A

The Mode Griineisen Parameters of Isotopically Unsuhstituted Forms

Substituted and

X = L-‘T-‘UL.

If = 2 -

Pi

I -Tm,

+ C’(r,, r,,

a In V, ai, yI - = 0.5 x-l aln V ’ dInV’

tj*

= 2 pz

+ p:

+, 2m,

f Li(r,,

r2, . . . rN),

(Bib)

2mT

Here, p, is the momentum operator of ith atom, U( r, , rz. . I~), of the potential energy of atomic interaction. It is clear that a perturbed operator 1/ is

the operator

(~42)

and inversion

of matrices

one is the kinetic energy operator

of

thejth atom in unsubstituted for;. In the first order of the perturbation theory the difference between the partition functions of substituted and unsubstituted forms may be written as (LANDAUand

:

LIFSHITS, 1976)

y, = ; 2 L,’ C’-‘TLkL;‘T-’

aci aln

!i

V

L,.

(A3)

The expression for yI is written as a sum. It is clear that all terms with i # k cancel out because of the orthogonality of eigenvectors L, Writing as a sum allows one to obtain the expression for y, ,

And for the isotopically

(Bla)

and

where dm, = m,* ~ m,; K, = 2

for y,

. 1. rN)

(Al)

Here, X is the eigenvalue of the matrix T-‘U, L is the columneigenvector of the matrix T-‘Cl, L-’ - the row-eigenvector of the matrix T-‘CT, T-’ is the matrix of kinetic coefficients, and I/ is the harmonic force constant matrix. The eigenvalues of this equation are squares of frequencies: X, = Y:, so that the Griineisen parameter of i mode may be expressed as

Consequently,

Perturbation

The Hamiltonian operators for unsubstituted and isotopically substituted forms (j atom is substituted by heavy isotope) are:

It is known that the vibrational eigenvalue equation for a system of interacting particles may be written in matrix form (WILSON et al.. 1955):

Using the rules of differentiation can write

B

The Application of the First-Order Thermodynamic Theory to the Calculation of &Factors

substituted

(83) I

where K, is the average kinetic energy of the light isotope atom which is to be substituted by the heavy one. The averaging in Eqn. B3 has been made using the quantum distribution function. The similar formula is obviously valid in the classical case too. Inserting Eqn. B3 into the B-factor definition (3) gives @-It-

au * *-I lJ_’ -L: Y, =- L, These expressions are exact brational problem is solved. It is clear that L, = L: in pletely substituted form). In One should note that the

-$$!&

form,

I

2

Q* -Qzz

dln

&$

[K, ~ (&I,

(B4)

i

V

and allow one to calculate

y, if the vi-

the caSe of single-element solids (comthis caSe yz = y : sum of y, is equal to the spur of the

where (K,),, is the average kinetic energy, calculated classically. Formula B4 is correct in the first order of the perturbation theory for any systems: solid, liquid, or gas. It was derived and examined by POLYAKOV ( 1989) and POLYAKOV and KHARLASHINA ( 1991). Except for the case of hydrogen isotopes the discrepancy between the @-factor values calculated by formula B4 and by the Urey formula (4) does not exceed 0.001 for 0°C and decreases with increasing temperature. One can note that the higher order corrections of the perturbation theory are nonlinear in Am/m*. So these corrections are connected with nonlinear isotopic effects. As a rule, these effects are negligible, excepting hydrogen isotopes and very low temperatures.