THE ELASTIC CONSTANTS of Mn2sio4, Fe2sio4 and Co2sio4, and the ELASTIC PROPERTIES of OLIVINE GROUP MINERALS at HIGH TEMPERATURE

J. Phys. Earth, 27, 209-238, 1979

THE ELASTIC CONSTANTS OF Mn2SiO4, Fe2SiO4 AND Co2SiO4, AND THE ELASTIC PROPERTIES OF OLIVINE GROUP MINERALS AT HIGH TEMPERATURE

Yoshio SUMINO

Departmentof Earth Sciences,Nagoya University,Nagoya, Japan (ReceivedOctober 14,1978)

Elastic constants of single-crystal Mn2SiO4, Fe2SiO4 and Co2SiO4 olivine

are measured by the rectangular parallelepiped resonance (RPR) method at

temperatures between 20° and 400℃. Elastic constants Cij (Mbar) and their

temperature derivatives ∂Cij/∂T (kbar/deg) are:

The isotropic properties, density ρ, adiabatic bulkmodulus K3, rigidity

μ, the Poisson's ratio σ, and their temperature derivatives calculated by the Voigt-Reuss-Hill scheme are:

By combining the present results with the previous data on a magnesium rich olivine, the effect of cation substitution on the elastic properties of olivine group minerals is discussed and the elastic constants of olivine at high temperature in the earth's mantle are clarified.

209 210 Y. SUMINO

1. Introduction

The olivineis one of the major constituentsin the upper mantle of the earth,and an accuratedetermination of itselastic constants is of basicim- portancein the interpretationof structureof the upper mantle. In nature olivineoccurs mostly as the solidsolution of forsteriteMg2SiO4 and fayalite Fe2SiO4. The six-coordinatedcations (Mg and Fe) are substitutedby small amounts of Ca, Mn, Co, and Ni. The known end members of olivinesolid solutionare tephroiteMn2SiO4, Co-olivineCo2SiO4, and Ni-olivineNi2SiO4 in additionto forsteriteand fayalite. The elasticconstants of forsteriteMg2SiO4 have been determined with fairlygood accuracy by KUMAZAWA and ANDERSON (1969),GRAHAM and BARSCH (1969),and SUMINO et al.(1977) by usingsingle crystalline samples. The olivinespecimens for which a setof nine elasticconstants has been reported so far are naturalperidot (Fe/(Mg+Fe)=0.072-0.083) (VERMA, 1960; KUMAZAWA and ANDERSON, 1969;OHNO, 1976). On the other hand, a set of elasticconstants of singlecrystal fayalite Fe2SiO4has not yet been reported.The reportedelastic properties of fayalite are limitedto threedilatational wave velocitiesalong the a, b and c axes of imperfect fayalitecrystal (Furuhashi, see SUWA, 1964; MIZUTANI et al., 1970), the ultrasonicvelocities in polycrystallinesample (MIZUTANI et al., 1970; CHUNG, 1970, 1971;AKIMOTO, 1972;revised data of MIZUTANI et al., 1970),and thecompressibilities determined by a volumetrictechnique (ADAMS, 1931),and an X-ray diffractiontechnique (TAKAHASHI, 1970; YAGI et al., 1975). The elasticproperties of otherolivine type silicateminerals have never been reported. In thispaper, the elasticconstants of fayalite,tephroite Mn2SiO4, and Co-olivine Co2SiO4, Which are analogue of forsterite and fayalite, are determined at temperatures between 20° and 400℃ by means of the RPR method

(rectangular parallelepiped resonance method). By combining the present results with previous results of forsterite, the elastic parameters of olivine

group minerals are summarized and the elasticity systematics relating to cation substitution is discussed. The temperature variation of seismic wave velocities

of olivine at high temperature in the earth's mantle is investigated .

2. Specimens The singlecrystal specimens of Mn2SiO4,Fe2SiO4, and Co2SiO4olivine were provided as a loan from Prof. H. Takei of Tohoku University, who grew them for his own work (TAKEI, 1976, 1978). Severalpieces of single crystallineCo2SiO4 olivine were providedby Prof.S .Naka, Dr. H. Furuhashi, and Prof.T. Noda of Nagoya University(NAKA et al.,1968). The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 211

Mn2SiO4olivine Singlecrystal of Mn2SiO4was grown by the Czoch- ralski pulling technique, and the specimen for the present purpose was the same one as sample T-6 of TAKEI (1976) . The as-grown boule of this crystal was brownish dark under a room light. The polished thin section was clearly transparent to visible light and looked bluish gray. It does not contain any inclusion or any flaw (see the photograph of T-6 specimen in TAKEI, 1976).

The boule was cut along {100} plane to yield a rectangular parallelepiped specimen (designated sample Mn2SiO4 (T)). The prescribed orientation were determined within 0.5° using a Laue back-reflection X-ray camera. The surface of the specimen was polished by abrasive powder #3,000.

Fe2SiO4 olivine Single crystal of Fe2SiO4 was grown by the floating- zone method, and the specimen for the present purpose was sample number

Fa-15 (TAKEI, 1978). Although the as-grown boule of this crystal was opaque under a room light, the polished thin section was clearly transparent (brown in transmitted light) to visible light. From an original boule, two rectangular parallelepiped specimens were prepared in the same way as Mn2SiO4 (designated sample Fe2SiO4 (TA) and Fe2SiO4 (TB)). The sample Fe2SiO4 (TA) was perfect, but in the case of sample Fe2SiO4 (TB), a few minutes cracks

(〓10-2mm) had developed at edge of specimen along (010) plane during the preparation. Co2SiO4 olivine Singlecrystal of Co2SiO4 obtainedfrom Takei (un- published)was grown by the floating-zonemethod. The as-grown boule of thiscrystal was opaque under a room light,and the polishedthin section was transparentto visiblelight. The colorof the thin sectionwas pale purple in transmittedlight. From the originalboule, a specimen was cut out and designatedCo2SiO4 (T). Singlecrystals of Co2SiO4 obtainedfrom NAKA et al.(1968) were grown by the Bridgman technique,and the specimen used for the presentpurpose was the sample E-I-3 describedin NAKA et al.(1968). From the original boule,a specimen was cut and designatedCo2SiO4 (F). The colorof the thin section,lattice constants, and refractiveindex of thiscrystal Co2SiO4 (F) are the same as those of Co2SiO4(T) as listedin Table 1. However, the bulk densityof Co2SiO4(F) was largerthan thatof Co2SiO4(T) and the X-ray densityof Co2SiO4 (F) by 0.9%, probably as a resultof deviationfrom Co2SiO4 stoichiometry.The singlecrystal Co2SiO4 (F) was directlygrown from the compositionwith an excessCoO (1.4mol%) to stoichiometricCo2SiO4 (NAKA et al.,1968). The presenceof a small amount of CoO phase in the single crystalCo2SiO4 (F) was recognizedby X-ray diffraction,and alsoit could be detectedby E.P.M.A. analysisas an inclusion(less than a few microns in length).When theCo2SiO4 (F) is regarded as themixture of CoO and Co2SiO4, the amount of CoO includedin Co2SiO4(F) iscalculated to be 2.3mol% from 212 Y. SUMINO

Table 1. Basic descriptions of olivine Mn2SiO4, Fe2SiO4 and Co2SiO4.

comparison of bulk densities. Table1 showsthe basic descriptions of the specimens; the three edge lengthsof the specimen,density, lattice constants ,mean atomicweight, molar volume, refractiveindex, and chemical composition. The bulk densityof each specimen is in good agreement with the X-ray densitywith the exception of Co2SiO4(F). The measurements forelastic constants were carriedout on the fivespecimens described above .

3. Measurement and Results

3.1 Thermal expansion Thermalexpansivity data of the specimen are necessary forreducing the The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 213 temperature derivatives of elastic constants from measured temperature variation of resonance frequencies. The coefficients of linear thermal expansion of Mn2SiO4 have been reported by OKAJIMA et al. (1978), and those of Fe2SiO4 by SUWA (1964), SUZUKI et al. (1977), and TAKEI (1978) by using a diatometric method. All the expansivity data of Fe2SiO4 are consistent within 17%.

In the present data analysis of Mn2SiO4 and Fe2SiO4, the values reported by OKAJIMA et al. (1978) for Mn2SiO4 and those of SUZUKI et al. (1977) for

Fe2SiO4 were employed, since the specimens measured by Okajima et al. and

Suzuki et al. were the same ones as used in the present work.

Although the linear thermal expansivity of Co2SiO4 has been determined by SATO (1970) at temperatures between 20° and 400℃ by using the X-ray diffractometer, the data are presumed to be not reliable. Since the volume expansivities of olivine group minerals are almost the same within 10%

(SUZUKI, 1975; SUZUKI et al., 1977; OKAJIMA et al., 1978), the size correc- tions for temperature for the determination of the elastic constants of Co2SiO4

were made by using the thermal expansivity data of Mn2SiO4. It is noted that this procedure is justified because the thermal expansion affects the temperature derivatives of elastic constants only several percent (SUMINO et al., 1977).

3.2 Elastic constants

The elastic constants were determined by an RPR method. The theory and the technical details of this method have been reported by OHNO (1976),

SUMINO et al. (1976), and the details of measuring the temperature variation

of elastic constants have been described in SUMINO et al. (1977).

Measurement of resonance frequencies was made at temperatures between room temperature and 410℃ on the lower ca. 30 vibrational modes for each

specimen as listed in Table 2 ((a), (b), and (c)). In this table, type nomen-

clature of the modes follows those of OHNO (1976), and the observed values

(Fobs)of resonance frequencies, the calculated most probable values (Fcal), and the normalized residuals [ΔF=(Fobs-Fcal)/Fobs] are presented. At high

temperatures up to 410℃, the resonance frequencies were measured at ir- regular temperature intervals. Several examples of temperature variation of

the resonance frequency are shown in Fig. 1 (A, B, C, and D) for each speci-

men. As seen in Fig. 1, the resonance frequency data at high temperatures showed scattering (up to 2%) in several modes, presumably due to the changes

in the mechanical coupling between the specimen and transducers. In the presentdata analysis,the nine independentelastic constants K1, K2, K3, Cs1, Cs2,Cs3, C44, C55, and C66[Ki=(Ci1+Ci2+Ci3)/3 and Csi=(Cjj+ Ckk-2Cjk)/4]were derivedfirst. Elastic constants C11, C22, C33, C23, C31, and C12were computed from Ki and Csi (i=1,2,3). All the data of the elastic 214 Y. SUMINO

Table 2 (a). Comparison of observed and calculated RPR frequencies in Mn2SiO4 (T)

at 30℃ and 400℃.

constantsand theirtemperature derivatives for each olivineare listed in Table 4 ((a)and (b)).In thistable, the statederrors of the elasticconstants are the root mean square deviationscalculated by the leastsquares method . The isotropicproperties calculated by the Voigt, Reuss, and Hill scheme (e.g., KUMAZAWA, 1964) are also summerized in Table 4 and are illustratedin Fig. 2. The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 215

Table 2 (b). Comparison of observed and calculated RPR frequencies in Fe2SiO4 (TA)

and Fe2SiO4 (TB) at 24℃ and 400℃.

Mn2SiO4 olivine Measurement of resonance frequencies was made on the lower 32 vibrational modes at temperatures between 30° and 400℃. From

Fig. 1B, the temperature variations of resonance frequencies were found to be almost linear for all modes. Therefore, the temperature variations of resonance frequencies for Mn2SiO4 were fitted by straight lines and the elastic constants were reduced at 30° and 400℃ by using 32 free vibrational modes listed in Table 2 (a). The relative deviations |O-C| lie within 0.2% except 216 Y. SUMINO The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 217

Fig. 1. Resonance frequencies plotted against temperature for Fe2SiO4, Mn2SiO4

and Co2SiO4. The frequencies were normalized to the values at 25℃. A,

Fe2SiO4 (TA); B, Mn2SiO4 (T); C, Co2SiO4 (F); and D, Co2SiO4 (T). Type nomen- clature of the modes is the same as in Table 2. 218 Y. SUMINO

Fig. 2. Temperature variation of elastic parameters in olivine group minerals.

Lower: A and B are adiabatic bulk modulus and rigidity respectively, computed by the Voigt-Reuss-Hill scheme. The values of Mg2SiO4 are from SUMINO et al.

(1977). Upper: Values of ∂K8/∂T and ∂μ/∂T as calculated from the data.

for EV-1 mode (0.6% at 30℃ and 0.7% at 400℃) and EX-4 mode (-0.5% at 30℃ and -0.7% at 400℃). The results of elastic constants and their temperature derivatives, analyzed by these 32 resonance modes, are listed in

Table 4 ((a) and (b)).

Fe2SiO4 olivine Measurement of resonance frequencies was made at room temperature on the lower 30 vibrational modes for two specimens

Fe2SiO4 (TA) and Fe2SiO4 (TB). The higher temperature measurement up to

410℃ was made on the lower 28 vibrational modes for Fe2SiO4 (TA) only.

The data of resonance frequencies are listed in Table 2 (b). The relative de-

viations |O-C| are almost within 0.2% over the whole range of temperature.

As seen from Fig. 1A, the temperature variation of resonance frequencies

for Fe2SiO4 was found to be non-linear in this temperature range, particularly

for the OX-1 mode which is identified as a C44-dominated mode. Therefore,

the data reduction was made at every 20 degrees of temperature using 28

resonance modes for Fe2SiO4 (TA). All the data were listed in Table 3. The The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 219 220 Y. SUMINO

Table 4 (a). Elastic parameters in Mn2SiO4, Fe2SiO4 and Co2SiO4 at 25℃.

resultsof elasticconstants determinations and theirtemperature derivatives are listedin Table4 ((a)and (b)). The elasticconstants of two specimensFe2SiO4 (TA) and Fe2SiO4(TB) arein good agreementwith one anotherwithin 0.3%, exceptfor the constants C22,C23 and K2 (linearincompressibility), which are relatedto thedilation parallelto the crystallographicb axis of olivine.The C22,C23 and K2 of Fe2SiO4(TB) are smaller than those of Fe2SiO4 (TA) by 1.1%, 1.9%,and 1.2%, respectively.The differencebetween these constants for the two specimensis presumed to be due to theminute cracks (or cleavage) developed along the (010) planein Fe2SiO4(TB) as noted previously.Therefore, the recommended valuesof the elasticconstants for Fe2SiO4 olivine are thoseof specimen Fe2SiO4(TA). Co2SiO4olivine Measurement of resonancefrequencies was made at The Elastic Constants of Mn2SiO4 , Fe2SiO4 and Co2SiO4 Olivine 221

Table 4 (b). Temperature derivatives of elastic parameters in M n2SiO4, Fe2SiO4 andC o2SiO4 at the temperatures between 25° and 400℃ .

room temperature on the lower 32 vibrational modes for a specimen Co 2SiO4 (T) (pure cobalt olivine) and 31 vibrational modes for a specimen Co2SiO4 (F)

(cobalt olivine+2.3mol% CoO). The higher temperature measurement was made at 400℃ on selected 25 modes for Co2SiO4 (T) and selected 28 modes for Co2SiO4 (F). The results of resonance frequencies were listed in Table

2 (c). The temperature variation of resonance frequencies was almost linear for all modes with an exception of OX-1 , which is the same C44-dominated mode as in Fe2SiO4. The non-linearity of OX-1 mode is quite slight both in

Co2SiO4 (T) and Co2SiO4 (F) when compared with that of Fe 2SiO4 as observed in Fig. 1 (A, C and D). Therefore, the temperature variation of resonance 222 Y. SUMINO

frequencies for Co2SiO4 was fitted by straight line except for the OX-1 mode, and the elastic constants were reduced at 30℃ and 400℃ by using the modes listed in Table 2 (c). The relative deviations |O-C| in both specimens lie almost within 0.2% for Co2SiO4 (T) and within 0.3% for Co2SiO4 (F) at the temperatures 30℃ and 400℃. The results of elastic constants and their temperature derivatives are listed in Table 4 ((a) and (b)). When the elastic constants of Co2SiO4 (T) and of Co2SiO4 (F) are compared, we note the following features. First, the Ki(i=1,2,3) of Co2SiO4 (T) are systematically higher than those of Co2SiO4 (F) up to 4%, although the six independent shear constants Cii(i=4,5,6) and CSi(i=1,2,3) are almost the same within 1.2% (mean deviation is 0.6%). Second, the degree of an- isotropy of several elastic constants in Co2SiO4 (F) is smaller than that of

Co2SiO4 (T). For example, the ratios C44/C66(=0.743) and CS1/CS3(=0.750) of

Co2SiO4 (F) are larger than C44/C66(=0.721) and CS1/CS3(=0.743) of Co2SiO4 (T) by 1.8% and 0.9%, respectively. These differences between the two specimens Co2SiO4 (T) and Co2SiO4 (F) originated possibly from the inclusion of 2.3mol% CoO in Co2SiO4 (F). Therefore, the recommended values of elastic constants for Co2SiO4 olivine are those of specimen Co2SiO4 (T).

4. Discussion

4.1 Anomalous temperaturevariation of rigidityfor Fe2SiO4and Co2SiO4 The elasticconstants are usuallyapproximated by a linearfunction of temperatureparticularly above the Debye temperature.The deviationfrom the linearityis smallfor ordinarysilicates and oxideseven at room temperature. The elasticwave velocitiesas well as resonancefrequencies of these mineralsalso vary almost linearlywith temperature. However, a largedeviation from a lineartemperature variation of resonance frequenciesis found in allmodes of Fe2SiO4olivine (Fig. 1A). A small deviationis also found in the OX-1 mode alone of Co2SiO4olivine (both T and F specimens),although the resonance frequencies in Mn2SiO4 and Mg2SiO4 (SUMINO et al.,1977) vary almostlinearly with temperature. The most ab- normal mode is an OX-1 mode (C44-dominatedmode) in both Fe2SiO4and Co2SiO4as illustratedin Fig. 1. This anomaly isdirectly related to the anoma- loustemperature variation of the shearconstants C44, C55, and C66for Fe2SiO4, and C44 for Co2SiO4. The behaviorof the elasticconstants of olivinegroup mineralsare summarized in Table 5, and alsothose of relatedrock saltoxides are summarized in the same table,for comparison. As shown in Table 5, the type of anomalous temperaturedependence of elasticconstants in olivinegroup issimilar to thoseof rock saltoxides in that The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 223

Table 5. Anomaly of temperature derivatives of elastic constants for olivine group

minerals at temperatures between 20° and 400℃, and for simple oxides at tem-

peratures between the Neel temperature and room temperature.

the bulk modulus of both olivinegroup and rock saltoxide group shows a normal variationwith temperaturewhile the shear constantC44 (or CS for CoO) is anomalous. Further,anomalous behavior does not occur in magnesium compounds. This impliesthat the anomaly is relatedto the divalent transitionmetal ions;Fe, Mn, and Co. In thosetransition metal oxides,the shearconstant C44 (or CS forCoO) shows a drasticchange at and around the Neel temperatureand the non-lineartemperature dependence occurs even at temperatures200 degreesabove Neel point,while another shearconstant CS (or C44for CoO) changesalmost linearly with temperatureeven in the vicinity of the Neel point(SUMINO et al.,1979). In olivinegroup minerals,the temperaturesof the antiferromagnetic- paramagneticphase transition(NOMURA et al.,1964; SANTORO et al.,1966) are around 50K, which is considerablylower than those of the rock salt oxidesas listedin Table 5. However, itis likely that the magnetictransition has an influenceupon the elasticconstants up to high temperatures,and that non-lineartemperature variation of shearconstants for Fe2SiO4and Co2SiO4 iscaused by the second-ordertransition mentioned above, althoughthe shear constantsof Mn2SiO4 did not show any detectableanomalous temperature dependence in the presentmeasurements.

4.2 Summary of elasticconstants of olivinegroup minerals 4.2.1 The relationbetween elastic constants and Fe/Mg ratio The effectof Mg-Fe substitutionon acousticvelocities of the olivine solid-solutionsystem has been investigatedby CHUNG (1970,1971), by using polycrystallinesamples. With the refineddata of the elasticproperties of olivineby single-crystalmeasurement, the effectof Mg-Fe substitutionon elasticproperties is discussed here again.The complete setof elasticconstants of singlecrystal Mg-Fe olivinehas been reportedby KUMAZAWA and ANDERSON 224 Y. SUMINO

(abbreviated to KA) (1969), GRAHAM and BARSCH (GB) (1969), and SUMINO et al. (SE) (1977) for pure end-member Mg2SiO4, and by VERMA (VE) (1960),

KUMAZAWA and ANDERSON (KA) (1969), and OHNO (OH) (1976) for natural

peridot with a ratio Fe/(Mg+Fe) between 7.2% and 8.3%. The measured values of elastic parameters of peridot are compared with

the calculated values by the weighted average of the corresponding elastic

parameter by mole fraction of two end-members; Mg2SiO4 and Fe2SiO4. The values of end-member Mg2SiO4 are given by averaging the data of KA, GB

and SE, with an exclusion of KA's C44 and C12, which deviate slightly from the other data (SUMINO et al., 1977). The actual deviations of the observed value in peridot from the calculated value obtained from the molar average of the two end-members are illustrated in Fig. 3 (A and B) for nine independent elastic constants (compliance Sij and stiffness Cij), and two isotropic

parameters: Ks and μ.

Fig. 3. The deviation of the observed

elastic constant from the calculated value of peridot. The calculated

value is obtained by a molar average of corresponding parameters

of two end-members Mg2SiO4 and

Fe2SiO4. A: Elastic compliance constants. Isotropic parameters

K-18 and μ-1 are given by the Reuss

value. B: Elastic stiffness constants. Ks and μ are by the Voigt

value. The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 225

In Fig. 3B, the mean value of the deviations (Cobsij-Ccalij)/Ccalij for diagonal

terms Cii(i=1,…,6) is 0.2% for data of KA, 1.3% and 0.8% for two

peridot data of OH, and 2.2% for VE. That of C23, C31, and C12 is 0.7% for KA (with the exception of C12, 3.6%), 9.2%, and 4.4% for OH, and 10.2%

for VE. These results show that the deviations |Cobsij-Ccalij| of KA's data are

notably smaller than those of the others. For comparison, the deviation

(Sobsij-Scalij)/Scalij for nine elastic compliance constants Sij is also shown in Fig. 3A. The addition law of shear modulus by molar average in olivine solid-

solution appears to hold better for elastic stiffness rather than for elastic com-

pliance, although there is not much difference for bulk modulus and other anisotropicparameters. In Fig.4 (A and B), severalexamples of the relationbetween elasticpa- rametersand chemicalcomposition (or density)are shown. The elasticcon- stantsof KA are very closeto the straightline between Mg2SiO4 and Fe2SiO4, although the valuesof C12only, in both Mg2SiO4 and peridotof KA are systematicallybelow the straightline by 4% as illustratedin Fig.4A. In this figure,it is also pointed out that the bulk modulus seems to increaselinearly with increaseof Fe-content in the olivineas isfound in Mg-Fe aluminate spinel (WANG and SIMMONS, 1972; CHANG and BARSCH, 1973), pyrope- almandine garnet (BABUSKA et al.,1978) and MgO-FeO (JACKSON et al., 1978; SUMINO et al.,1979). The variationaltrend of bulk modulus with Mg-Fe substitutionreported by CHUNG (1970) is opposite.From the four data set of the above solid-solutionseries, we concludedthat, as a general rule,in the Mg-Fe solidsolution the bulk modulus increaseswith Fe-content. The elasticwave velocitiesof peridotlie almost on a straightline con- nectingthe end-members as shown in Fig.4B. However, when we look more closely,the elasticwave velocitiesof peridotappear systematicallybelow the straightline and liecloser to the broken lineindicating the linearaddition law of elasticstiffness constants. This difference,due to the differentaddition laws,is small but significant,as shown in Fig.4A. This factimplies that the molar averageaddition law shouldbe appliedto the elasticstiffness constants rather than to the elasticwave velocities. 4.2.2 The effectsof pressure,temperature and cationsubstitution on the

elastic constants

For an interpretation of seismic data and a better understanding of physi- cal state in the earth'sinterior, it is important to find how the elastic constants depend upon pressure, temperature, structural and atomic parameters. The

effects of pressure, temperature and cation substitution on the elastic constants in olivine group minerals are discussed in terms of the quantity w(=

-∂ln Cij/∂ln V) . The effects on the elastic constants are essentially expressed by three parameters; 226 Y. SUMINO The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 227

(1) (2) (3) where P, T, and M represent pressure, temperature and composition (cation species Mp=Mg, Co, Fe and Mn), respectively, KT is the isothermal bulk modulus, and α is the coefficient of volume thermal expansion.

The parameters wP, wT and wM are calculated by using the present elastic data combined with the data of Mg2SiO4 reported by KUMAZAWA and ANDERSON

(1969), GRAHAM and BARSCH (1969), and SUMINO et al. (1977), and the results are listed in Table 6. The several examples of the quantity w and of the relation between the elastic constant and the molar volume are also illustrated in Fig. 5 (A and B). From Fig. 5 and Table 6, we can illustrate the empirical rules described below. (A) Generaltrends The quantitieswP, wT, and wM in olivinegroup mineralsfor each elasticconstant have generallythe same orderof magnitude, scarcelydepending upon the parametersof pressure,temperature and cationsubstitution.Their valueslie usually between 1 and 8. Especiallyin the case of constantC11, the values of wP, wT, and wM are mostly confined in the narrow range between 3 and 5. However, there are severalexceptions; (a)The wM relatingto the substitutionbetween magnesium and transition metals vary widely from -20 to 20, and (b)the wT relatedto the shearcon- stantsof Fe2SiO4influenced by magnetic transitiondeviate greatly from the common valuesof wP, wT, and wM.

(B) wT value The wT in olivine group minerals is virtually constant at 5.5±1.5, being independent of the respective elastic constants and the divalent cation. An exception is wT related to the shear constants of Fe2SiO4, as noted above.

(C) wM value Among Mn2SiO4, Fe2SiO4, and Co2SiO4, the wM of the elastic constants relating to dilation [Cii and Ki(i=1,2,3), and Cij(ij=

23,31,12)] are virtually constant at 1.6±0.9. However, those wM values between Mg2SiO4 and the other olivines are scattered between -20 and 3.

Further the wM related to the shear constants [Cii(i=4,5,6) and CSi(i=1,2, 3)] are largely scattered between -5 and 20. However, all the wM values for shear constants change almost linearly with volume. The variation of wM value with volume for C44 is shown as an example in Fig. 5A.

(D) wP, wT and wM values of isotropic parameters Ks and μ

(a) Bulk modulus The wP and wT values of Ks are virtually constant at 5.0±0.5 for all olivines, being independent of pressure and temperature as illustrated in Fig. 5B. This value 5.0 is close to those of ordinary oxide 228 Y. SUMINO

Table 6. The variation of elastic constants with molar volume as a function of temperature T, pressure P and cation substitution M.

Table 6. (continued) The Elastic Constants of Mn2SiO4 , Fe2SiO4 and Co2SiO4 Olivine 229 230 Y. SUMINO compounds. The values of wP and wT are usually between 4 and 6 for oxide compounds, regardless of their crystalline structure and chemical component

(e.g., ANDERSON et al., 1968). The wM value of K8 among Co2SiO4, Fe2SiO4 and Mn2SiO4, or between Mg2SiO4 and Mn2SiO4 lies between 0 and 2 or wM=

1±1, which is close to the value 4/3 obtained from the assumption that the major bonding force is provided by electrostatic energy (ANDERSON and NAFE,

1965). The wM value is 〓 1 for alkali halides and flourides which are the typical ionic compounds (ANDERSON and NAFE, 1965), and the empirical relationship K・V=constant(wM=1) also holds very well for many oxides

(ANDERSON and ANDERSON, 1970). However, the wM values of K8 between Mg2SiO4 and the other olivines with transition metal ions Co2+ and Fe2+, which have a crystal field effect, deviate greatly from the usual value wM〓l as is seen in Table 6. Therefore, the large diviation of wM value appears to be due to the contribution of the crystal field stabilization energy (CFSE) of

Co2+ and Fe2+.

The effects of the CFSE of Fe2+, Co2+, and Ni2+ ions on the bulk modulus K of divalent monoxides were investigated by OHNISHI and MIZUTANI (1978).

They showed that the deviation of the bulk modulus from the "K・V=constant" law in rock salt oxides is well explained by considering the CFSE.

The variational pattern of K8 and V in olivine group minerals is quite similar to that in rock salt oxides. The samne pattern is also observed in garnets

(BABUSKA et al., 1978). Therefore, the contribution of CFSE to the elasticity of olivines and garnets is definitely important. Discussion of these problems will be reported in a forthcoming paper. However, it is noted here that the large deviation of the bulk modulus of olivines with transition metal ions from that of Mg2SiO4 may be too large to be accounted for by CFSF.

(b) Rigidity The wT value for rigidity of olivine group minerals is virtually constant 6±1, and is almost one half of wP for rigidity, although the values wP and wT for bulk modulus are almost the same. The wM values for rigidity vary almost linearly with volume as is seen in Fig. 5B, however, generally they are considerably scattered. In other words, the variation of rigidity with pressure, temperature and cation-substitution does not follow any conceivable systematics. This is an important difference between bulk modulus and rigidity.

4.2.3 The estimated values of pressure derivatives of isotropic properties in Fe2SiO4

The pressure derivatives of isotropic parameters in Fe2SiO4 are discussed by using some empirical relations obtained in the previous section and the relations of elastic parameters to the Gruneisen constant proposed by SUZUKI and KUMAZAWA (1979).

The pressure derivative of bulk modulus in Fe2SiO4 was determined by The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 231

CHUNG (1971) by measurement of ultrasonic velocities in polycrystalline samples up to 7.5kbar. He reported ∂Ks/∂P=5.92 for Fe2SiO4. The bulk modulus and its pressure derivative of Fe2SiO4 was also investigated by TAKAHASHI

(1970) up to 150kbar, and YAGI et al. (1975) up to 70kbar by means of the X-ray diffraction technique. The reported values of Fe2SiO4 are KT=1.35±

0.15 assuming ∂KT/∂P=4.5 (Takahashi), and KT=1.24±0.02 assuming

∂KT/∂P=5 (Yagi et al.). Their values of KT(=1.35-1.25) are in good agreement with the present data of KT=1.37, within 9%. Therefore, ∂KT/∂P is

presumed to be around 5. Then, ∂Ks/∂P has also the same value, since the difference between ∂KT/∂P and ∂Ks/∂P is usually within 1% in oxide compounds.

On the other hand, the ∂Ks/∂P(〓value of wP for Ks in Eq. (1)) is usually close to the wT value of Ks in Eq. (2) and these values lie between 4 and 6 as

mentioned in a previous section. For example, wP is 5.14-5.08 for Ks of

Mg2SiO4 and peridot, whereas wT is 4.2-4.8. Since wT=5.3 for Fe2SiO4,

∂Ks/∂P is supposed to be around 5. Combining all the information above,

we may conclude that the value of ∂Ks/∂P in Fe2SiO4 must be around 5, or

∂Ks/∂P=5.0±1.0, which is quite normal for ordinary solids.

The pressure derivatives of rigidity in Fe2SiO4 are also predicted from the relation between Gruneisen constant and pressure derivatives of elastic

parameters, proposed by SUZUKI and KUMAZAWA (1979). Based on this relation, the value ∂μ/∂P of Fe2SiO4 is estimated to be ∂μ/∂P=0.7±0.2 (per-

sonal communication from Kumazawa), which is close to ∂μ/∂P=0.62 for

polycrystalline Fe2SiO4 (CHUNG, 1971) and is quite different from the value ∂μ/∂P=1.79-1.82 of Mg2SiO4 and peridot (KUMAZAWA and ANDERSON, 1969;

GRAHAM and BARSCH, 1969). The marked difference of ∂μ/∂P in Mg2SiO4

from that in Fe2SiO4 may be related to the large difference in the value of

Poisson's ratio (0.336 for Fe2SiO4 and 0.240 for Mg2SiO4). However, Poisson's

ratio of Fe2SiO4 tends to decrease with temperature and thus the value of

∂μ/∂P would not be so low at high temperatures.

The pressure derivatives of isotropic seismic wave velocities ∂Vp/∂P, ∂Vs/∂P and ∂VΦ/∂P are calculated by using the predicted values of ∂Ks/∂P

Table 7. Pressure derivativesof isotropicproperties in Fe2SiO4. 232 Y. SUMINO and ∂μ/∂P. All the pressure derivatives of isotropic parameters in Fe2SiO4 are listed in Table 7, and compared with the data of CHUNG (1971).

4.3 Elastic parameters of Mn2SiO4, Fe2SiO4 and Co2SiO4 at high temperature

The numerical values of elastic parameters for Mn2SiO4, Fe2SiO4 and Co2SiO4 olivine are calculated up to very high temperatures by using empirical methods based on the theory of the Mie-Gruneisen equation of state.

The Gruneisen constant γ and the Gruneisen-Anderson parameter δs are defined by

γ=αVKs/Cp (4)

δs=-(∂1nK8/∂T)/α (5) where Cp is the specific heat at constant pressure, V is the specific volume, and

α is the coefficient of volume thermal expansion. Constancy of the quantities

γ and δ8 with respect to temperature was theoretically given by GRUNEISEN

(1926) and ANDERSON (1966), and was experimentally supported by many investigators [e.g., SOGA and ANDERSON (1966) for MgO and Al2O3, and

SUMINO et al. (1977) for Mg2SiO4]. As far as the experimental data are con- cerned, constancy for δ8 was assumed to be better than that for γ at high temperature (SUMINO et al., 1977), so that the calculation of the elastic parameters of Mn, Fe, and Co-olivines up to high temperatures is made by assuming the constancy of δ8.

The values of γ and δ8 listed in Table 8 were calculated by Eqs. (4) and (5) from the present elasticity data combined with α reported by OKAJIMA et al. (1978) and Cp by JEFFES et al. (1954) for Mn2SiO4, and α of SUZUKI et al. (1977) and Cp of ORR (1953) for Fe2SiO4. In the case of Co2SiO4, the computation of γ and δ8 were made by using Cp reported by WATANABE and

KAWADA (1979) and α of Mn2SiO4, The γ is virtually constant, γ=1.1±0.1, for all olivine group minerals studied here, while the values of δ8 for Mn2SiO4,

Table 8. Gruneisen constant γ and Gruneisen-Anderson parameter δ8 at temperatures between 300 and 700K, and Debye temperatures of olivine group minerals, The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 233

Fe2SiO4 and Co2SiO4 are between 4.7 and 5.6, that is higher than that of

Mg2SiO4 by 20-30%.

From Eq. (5), the bulk modulus Ks at temperature T is given by

Ks(T)=Ks0[V0/V(T)]δs (6) where the subscript (0) indicates the quantity at reference temperature T0

(SUMINO et al., 1977). The value of Ks at high temperatures are calculated by Eq. (6) and are listed in Table 9 ((a), (b) and (c)). Since the constancy of

δs is expected to be good at high temperatures, the adopted reference temperature is 500K, which is in the vicinity of the acoustic Debye temperature θD=549K for Mn2SiO4,526K for Fe2SiO4, and 559K for CO2SiO4 as listed in Table 8. The conversion of Ks to isothermal bulk modulus KT is made by using γ=1.1 for three olivine minerals.

The values of rigidity μ at high temperatures are calculated by the following equation (SUMINO et al., 1977):

μ(T):(3/2)[(1-2σ(T))/(1+σ(T))]Ks(T) (7) where

σ(T)=σ0+(∂ σ/∂T)(T-T0). (8)

Table 9(a). Observed and calculated values of elastic parameters of Mn2SiO4 as a function of temperature. 234 Y. SUMINO The Elastic Constants of Mn2SiO4 , Fe2SiO4 and Co2SiO4 Olivine 235

Table 9(c).Observed and calculatedvalues ofelastic parameters of Co2SiO4as a f unctionof temperature.

Equation (7) combined with Eq. (8) is a good approximation at high temper - atures because of the weak dependence of Poisson's ratio σ(T) on temperature .

In the case of Fe2SiO4, however, it is obvious that even in the limited temperature range of this experiment ∂σ/∂T is not a constant as it is influenced by magnetic transition. Therefore , the extrapolation of σ to high temperature in Fe2SiO4 is made by using the two different methods . One is the linear extrapolation of σ to high temperatures by using the lower temperature data of σ and ∂σ/∂T. The adopted value ∂σ/∂T is 0 .93×10-5 deg-1, which is the mean value at 300-500K and is close to the value of Mg 2sio4 [∂σ/∂T=0 .63-0.91×10-5 deg-1, KUMAZAWA and ANDERSON (1969) , GRAHAM and BARSCH (1969), and SUMINO et al . (1977)] and is also close to ∂σ/∂T= 0.64×10-5 of Mn2SiO4. The other way is the linear extrapolation of σ by using the higher temperature data of σ and ∂σ/∂T . The adopted value is ∂σ/∂T=-0.26×10-5 deg-1, which is the mean value for 500 -700K and is close to ∂σ/∂T=-0.12×10-5 of Co2SiO 4. The rigidity of Fe2SiO4 at high temperatures is calculated by using these two different values of ∂σ/∂T , and the results are compared in Table 9(b). The actual value of rigidity at high temperatures for Fe2SiO4 is presumed to lie between the two calculated values . 236 Y. SUMINO

At the melting temperature, the difference between the two rigidities calculated here amounts to 8%.

The calculated values of adiabatic and isothermal bulk moduli, Ks and

KT, rigidity μ and other isotropic elastic parameters are summarized for

Mn2SiO4 in Table 9(a), for Fe2SiO4 in Table 9(b), and for Co2SiO4 in Table

9(c). Any elastic parameter of natural olivine, which is a solid solution be-

tween Mg2SiO4 and Fe2SiO4 with a trace of Mn2SiO4, is given by the average

of the corresponding parameters by mole fractions of these end-members;

Mg2SiO4 (SUMINO et al., 1977), Fe2SiO4 and Mn2SiO4.

The author expresses his sincere gratitude to Prof. Y. Shimazu and Prof. M. Kumazawa for their support and guidance in the present work. He would like to thank Prof. H. Takei of Tohoku University for providing single crystals of Mn2SiO4, Fe2SiO4 and Co2SiO4, Dr. H. Furuhashi, Prof. S. Naka and Prof. emeritus T. Noda of Nagoya University for providing single crystals of Co2SiO4, Dr. T. Agata of Nagoya University for determining the refractive index, Dr. K. Suzuki of Tokyo Gakugei University for the EPMA analysis, Dr. T. Takahashi of Toshiba Research and Development Center for providing TPM-600 transducers, Dr. O.

Nishizawa of Geological Survey of Japan for providing a goniometer, and Mr. S. Yogo of

Nagoya University for his help in sample preparation.

He would like to thank Prof. H. Mizutani of Nagoya University, Prof. Y. Ida of Tokyo University, and Prof. T. Hirasawa of Tohoku Universityfor critical readlng of the manu- script. He also is indebted to Prof. S. Akimoto of Tokyo University, and Dr. I. Ohno of Ehime University for their encouragements during the course of the present study. This work is supported by the Japanese Geodynamic Committee, and also by a grant-in-aid of the Ministry of Education. The numerical computation was carried out by FACOM 230-60 at the Computation Center of Nagoya University (problem No. 4001 ENO 106 and 4001

FNO 106).

REFERENCES

ADAMS, L. H., The compressibility of fayalite, and the velocity of elastic waves in peridotite

with different iron-magnesium ratio, Beitr. Geophys., 31, 315-321, 1931. AKIMOTO, S., The system MgO-FeO-SiO2 at high pressures and temperatures-phase equillbria

and elastic properties, Tectonophysics, 13, 161-187, 1972. ANDERSON, D.L and O.L. ANDERSON, The bulk modulus-volume relationship for oxldes,

J. Geophys Res., 75, 3494-3500, 1970. ANDERSON, O.L., Derivation of Wachtman's equation for the temperature dependence of

elastic moduli of oxide compounds, Phys. Rev., 144, 553-557, 1966. ANDERSON, O.L. and J.E. NAFE, The bulk modulus-volume relationship for oxide com-

pounds and related geophysical problems, J. Geophys. Res., 70, 3951-3963, 1965. ANDERSON, O.L., E. SCHREIBER, R.C. LIEBERMANN, and N. SoGA, Some elastic constant dataon mineralsrelevant to geophysics,Rev. Geophys. Space Phys., 6, 491-524,1968. BABUSKA, V., J. FIALA, M. KUMAZAWA , I. OHNO, and Y. SUMINO, Elasticproperty of garnetsolid-solution series, Phys. Earth Planet, Inter., 16, 157-176, 1978.C HANG, Z.P. and G.R. BARSCH,Pressure dependence of single-crystalelastic constants and anharmonicproperties of spinel,J. Geophys.Res., 78, 2418-2433, 1973. CHUNG, D.H., Effectsof iron/magnesiumratio on P-and S-wavevelocities in olivine,J. Geophys.Res., 75, 7353-7361, 1970. The Elastic Constants of Mn2SiO4, Fe2SiO4 and Co2SiO4 Olivine 237

CHUNG, D.H., Elasticity and equations of state of olivines in the Mg2SiO4-Fe2SiO4 system,

Geophys. J.R. Astron. Soc., 25, 511-538, 1971. GRAHAM, E.K. Jr. and G.R. BARSCH, Elastic constants of single-crystal forsterite as a func-

tion of temperature and pressure, J. Geophys. Res., 74, 5949-5960, 1969. GRUNEISEN, E., State of a solid body, NASA (Natl. Aeronaut . Space Adm.) Publ., No. RE2-

18-59W (translation of Hardbuch der Physik, 10, 1-52, 1926). JACKSON, I., R.C. LIEBERMANN, and R.E. RINGWOOD, The elastic properties of (MgxFe1 -x)O

solid solution, Phys. Chem. Miner., 3, 11-31, 1978. JEFFES, J.H.E., F.D. RICHARDSON , and J. PEARSON, The heats of formation of manganous

orthosilicate and manganous sulphide, Trans. Faraday Soc., 50, 364-370, 1954.K ITTEL, C., Introduction to Solid State Physics , 4th ed., Chap. 16, pp. 568, Wiley-Maruzen,

Tokyo, 1974. KUMAZAWA, M., The elastic constants of rocks in terms of elastic constants of constituent

mineral grains, petrofabric and interface structures, J. Earth Sci. Nagoya Univ., 12, 147- 176, 1964. KUMAZAWA, M. and O.L. ANDERSON, Elastic moduli, pressure derivatives, and temperature

derivatives of single-crystal olivine and single-crystal forsterite, J. Geophys. Res., 74, 5961-5972, 1969. MIZUTANI, H., Y. HAMANO, Y. IDA, and S. AKIMOTO, Compressional-wave velocities of

fayalite, Fe2SiO4 spinel, and coesite, J. Geophys. Res., 75, 2741-2747, 1970. NAKA, S., H. FURUHASHI, and T. NODA, Crystal growth of cobalt olivine (Co2SiO4), J. Ceram.

Assoc. Japan, 76, 179-183, 1968. NOMURA, S., R. SANTORO, J. FANG, and R. NEWNHAN, Antiferromagnetism in cobalt or-

thosilicate, J. Phys. Chem. Solids, 25, 901-905, 1964. OHNISHI, S. and H. MIZUTANI, Crystal field effect on bulk moduli of transition metal oxides,

J. Geophys. Res., 83, 1852-1856, 1978. OHNO, I., Free vibration of a rectangular parallelepiped crystal and its application to deter-

mination of elastic constants of orthorhombic crystal, J. Phys. Earth, 24, 355-379, 1976. OKAJIMA, S., I. SUZUKI, K. SEYA, and Y. SUMINO, Thermal expansion of single-crystal

tephroite, phys. Chem. Miner., 3, 111-115, 1978. ORR, R.L., High temperature heat contents of magnesium orthosilicate and ferrous or-

thosilicate, J. Am. Chem, Soc., 75, 528-529, 1953. SANTORQ, R.P., R.E. NEWNHAN, and S. NOMURA, Magnetic properties of Mn2SiO4 and

Fe2SiO4, J. Phys. chem. Solids, 27, 655-666, 1966. SATO, Y., Modified spinel structure and geophysical significance, Master Thesis, University

of Tokyo, pp. 74-75, 1970. SoGA, N. and O.L. ANDERSON, High temperature elastic properties of polycrystalline MgO

and Al2O3, J.Am. Ceram. Soc., 49, 355-359, 1966. SUMINO, Y., O. NISHIZAWA, M. KUMAZAWA, and W. PLUSCHKELL, The elastic constants of

MnO, FeO and CoO between the Neel temperature and 30℃, 1979 (in preparation). SUMINO, Y., O. NISHIZAWA, T. GOTO, I. OHNO, and M. OZIMA, Temperature variation of

elastic constants of single-crystal forsterite between -190° and 400℃, J. Phys. Earth, 25, 377-392, 1977. SUMINO, Y., I. OHNO, T. GOTO, and M, KUMAZAWA, Measurement of elastic constants and internalfrictions on single-crystalMgO by rectangularparallelepiped resonance, J. Phys. Earth,24, 263-273, 1976. SUWA, K., Mineralogyof fayalite,with special reference to itsthermal and thermodynamical properties,J.Earth Sci., Nagoya Univ.,12, 129-146, 1964. SUZUKI,I., Thermal expansionof periclaseand olivine,and theiranharmonic properties, J. Phys.Earth, 23, 145-159,1975. 238 Y. SUMINO

SUZUKI, I. and M. KUMAZAWA, Dependence of Gruneisen'sparameters on shear properties of minerals,1979 (inpreparation). SUZUKI, I.,S. OKAJIMA, and Y. SUMINO, Thermal expansion of fayalite,Programme and Abstracts,Seismol. Soc. Japan,No. 2, 186, 1977. TAKAHASHI, T., Isothermalcompression of fayaliteat room temperature,EOS (Trans.Am. Geophys. Uniom),51, 827, 1970. TAKEI, H., Czochralskigrowth of Mn2SiO4 (tephroite)single crystal and itsproperties, J. Cryst.Growth, 34, 125-131,1976. TAKEI, H., Growth of fayalite(Fe2SiO4) single crystals by the floating-zonemethod, J. Cryst. Growth,43, 463-468, 1978. VERMA, R.K., Elasticityof some high-densitycrystals, J. Geophys.Res., 65, 757-766,1960. WANG, H. and G. SIMMONS, Elasticityof some mantle crystalstructures; 1. pleonasteand hercynitespinel, J. Geophys.Res., 77, 4379-4392, 1972. WATANABE, H. and K. KAWADA, Heat capacitiesof the major constituentminerals in the earth'smantle at high temperatures,1979 (inpreparation). YAGI, T., Y. IDA, Y. SATO, and S. AKIMOTO, Effectof hydrostaticpressure on the lattice parametersof Fe2SiO4 olivineup to 70kbar, Phys. Earth Planet.Inter., 10, 348-354, 1975.