Elastic properties from acoustic and volume compression experiments

Physics of the Earth and Planetaiy Interiors, 25 (1981)140—158 Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

140

Elastic properties from acoustic and volume compression experiments Jay D. Bass

~,

Robert C. Liebermann I, Donald J. Weidner I and Stephen J. Finch2 Department of Earth and Space Sciences and

2

Department ofApplied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794 (U.S.A.) (Received February 2, 1980; revised and accepted October 2, 1980)

Bass, J.D., Liebermann, R.C., Weidner, D.J. and Finch, S.J., 1981. Elastic properties from acoustic and volume compression experiments. Phys. Earth Planet. Inter., 25: 140—158. Hydrostatic compression data for a number of high-pressure phases of oxides and silicates, which have been studied independently by acoustic techniques, have been analyzed by least-squares fitting of the Birch— Murnaghan equation of state to determine the zero-pressure bulk modulus K 0 and its pressure derivative K~for each material. The standard deviations of K0 and K& so determined are generally underestimated unless the experimental errors in the measurements of volume and pressure are explicitly included. When the values of K0 determined from the acoustic and compression techniques are consistent, test results for quartz and rutile demonstrate that constraining K0 to be equal to the acoustic value significantly improves both the accuracy and the precision of K~obtained from the compression data. Similar analyses for high-pressure phases (e.g., pyrope garnet and silicate spinels) indicate that by combining the acoustic and P—V data, the standard deviation of K~is typically reduced by a factor of three. Thus, we conclude that this approach does allow precise deternunations of K~even when neither technique alone is able to resolve this parameter. For some materials, however, the p—V and acoustic experiments do not define mutually consistent values of K0, invalidating any combination of these data. The compression data for stishovite clearly exhibit run to run effects, and we infer that systematic errors are present in some of the P—V data which are responsible for many of the interlaboratory inconsistencies. Such systematic biases in the P—V data can at least be partially compensated for by performing several duplicate experimental runs.

1. Introduction Interpretations of seismic Earth models require an accurate knowledge of the elastic moduli and their pressure dependence for silicate minerals and their high pressure polymorphs. Laboratory methods to determine these properties fall under two broad categories: physical acoustics and volume compression techniques. Acoustic methods measure various mode velocities and yield directly the adiabatic bulk modulus, K3, the shear modulus ~u, and the isothermal pressure derivatives (8Kg /~P)~ and (a,.I/aP)T from experiments performed at P ~ 10 kbar. Isothermal compression experiments measure the volume and hence density of a material at

elevated hydrostatic pressures up to approximately 100 kbar. Since the isothermal bulk modulus KT is defined as V [aP/W]~, the derivative of a pressure—volume curve determines KT. In cases where the P—V data are sufficiently precise, higher order derivatives, such as K~ (aKT/OP)T, may be resolvable. In the absence of systematic errors in either technique, acoustic and compression experiments should provide compatible results for K and K’ (with appropriate corrections to convert K3 to KT). For phases which are not stable at atmospheric pressure, acoustic experiments to date have been able to provide only determinations of the zero-pressure bulk modulus, K0 (e.g., Liebermann, —

-

0031-920l/8l/0000—0000/$02.50 © 1981 Elsevier Scientific Publishing Company

141

1975; Weidner and Carleton, 1977). Isothermal compression experiments on similar materials have generally yielded compatible values of K0 but have

independent of the analytical form of the equation of state (e.g., Macdonald and Powell, 1971; Sato,

For the purposes of interpreting the pressure— volume data, we have used the Birch— Murnaghan equation of state (Birch, 1952) which is based on an expansion of the free energy to third order in terms of the Eulerian finite strain

1977). Therefore, we have concluded that no senous systematic bias is introduced by our choice of the Birch—Murnaghan equation for interpolating and smoothing the data. The question of model adequacy would be of greater concern if we were attempting to extrapolate the data beyond the experimental P range, but this is a separate problem which will not be addressed here. Our choice of the Birch— Murnaghan equation was also influenced by its widespread use in the high-pressure geophysics literature. Each P—V data set was fit to eq. 1 yielding values for K0 and K~by the least-squares techmque. The standard deviations of these parameters are estimated in two ways. The first assumes that the scatter of the data from the resulting curve correctly reflects the uncertainty of the data and hence uses the mean square residual for error. The second uses reported uncertainties of each data point. Then in both cases these uncertainties are used with eq. 1 to define the parameter (K0, K~)uncertainties. If the uncertainties vary within a data set, then a weighted least-squares fit is used in this second case to estimate K0 and K~,with a weighting factor of 1/e,~(for details see the appendix). If the experimental uncertainties have been realistically assessed then both solutions should yield approximately equal uncertainties for K0 and K~.This approach will only reflect the random component of the uncertainties with no information about the systematic errors.

P

Central to thisofpaper is the determination of reliable estimates the uncertainties of K0 and K~

not been able to resolve precise (or sometimes even reasonable) values for K~. The objective of this study is to explore the possibility of extracting geophysically significant values of K~for high-pressure phases by combining the complementary acoustic and compression data when neither set alone is sufficient to resolve this parameter. We first examine the compatibility of results for materials that have been well studied by both acoustic and compression techniques (e.g., Si02 -quartz and Ti02-rutile). We then analyze all of the available elasticity data for high-pressure phases of oxide and silicate compounds, and condude that this approach does indeed permit more precise estimates of the pressure derivative of the bulk modulus for these materials. In the course of our investigation, we have observed an incompatibility in the respective elasticity data provided by the two techniques for certain materials and suggest some guidelines for future experiments to resolve these ambiguities.

2. Method of analysis

~K 0\~x

= 2

X [1

—

7/3

—

3/4 (4

x

—5/3 ~

—

K~)(x—2/3

—

I)]

(I)

where x = V/V0 is the volume at pressure P normalized to the volume at zero-pressure (1 bar). Other two-parameter equations of state (e.g., Murnaghan, Lagrangian finite-strain, Keane equations) could have been used to extract K0 and K~ from a fit to the P—V data. However, pressures attained in the studies we have examined are small relative to 0.1%, ~K and GK~ are virtually independent of a~,while for a~, 0.510. kbar and An approach insensitivity of GK to and problem of determining

\~ ~

isothermal elastic parameters compression fromdata curve-fitting is to constrain analysis one of the parameters (K 0, K~) and to adjust only the one remaining parameter to fit the data. This method has been used extensively by the research group of Bassett and Takahashi (e.g., 1974) who constrained K~and obtained solutions for K0. In view of our interest in high-pressure phases for

• 10 0.5

2

5 10 IS 2025 ________________________

I

1000

0K kbars 00

which K~is virtually unknown but for which we have reasonable estimates of K0 for certain materi-

~

~ 30 kbar. However, the relationship between the parameter uncertainties is clear. Note that even at aK 0, K~is not exactly determined. When systematic errors are significant, then the uncertainty estimated in the above manner is not

the appropriate measure. The presence of systematic errors can thus be indicated by an incom-

143

3.0

-

• •

/

• 20 K

.

1.0

-

/

-

-

/

/

/

/

/

/

/

//

/

Smith, 1966, pp. 64—67; Wonnacott and Wonnacott, 1970, pp. 256—259). Two confidence ellipses, corresponding to the 50 and 95% probability levels, are shown in Fig. 3 for the synthetic data set and conditions used in Fig. 2. The meaning of the confidence ellipses is as follows: pairs of (K0, K~) values which lie outside a given confidence ellipse are considered incompatible with the data set, with

/

50

strongly coupled and do not vary independently. This may be seen most clearly by means of confidence effipses in the K0 —K~plane (Draper and

100

kbars

a probability given by the confidence level. For example, it is 95% probable that the K0 —K~pair given by point P is not consistent with the data. Note, however, that the K0, K~coordinates for P are within one standard deviation, a (shown by the error bars about K0 = 1.5 mbar and K~= 4.0), of the least-squares best estimate from the twoparameter fit. This demonstrates a potential danger

Fig. 2. The uncertainty in K~,given a priori knowledge of K0, vs. the uncertainty in K0 for the same material as in Fig. 1(b) and n = 10. The circled cross indicates the values of UK and 0~from a two parameters fit of the P—V data alone. 6-

patibility of parameters for two data sets. We measure compatibility of results from acoustic and compression data with a model test which may be described briefly as follows. The best estimates of K0 and K~from the P—V measurements are those which minimize the sum of the squares of the residuals (SSR) of the data points from the fitted curve. If either of these values is fixed at a value other than the best estimate, then the SSR will increase. Depending upon the number of data points and the error of each (as measured by the residual from the best estimate fit or the pressure and volume uncertainties), one can calculate the probability that the SSR corresponding to any K0—K~pair is inconsistent with the best estimate from the compression data only (see Appendix). Thus, the model test defines the confidence level at which the acoustic and compression data parameter estimates are incompatible, When comparing two sets of results, it is necessary to apply the appropnate model test, rather than to inspect only for overlapping standard deviations, since the uncertainties in K0 and K~are

P

K0 4

-

2o~

I •

50%

-

2

-

400 I

K0, kbars Fig. 3. Confidence ellipses for IC~and K~at 50 and 95% levels, showing the covariance of the parameters. The data is the same as used in Fig. 2. Error bars are ±one standard deviation, from a two parameter fit of the P—V data. ar-. (= ± 0.3) shows the uncertainty in K6 for this data given a pnon knowledge that K0 = 1500 ±10 kbar. Note that although the K0—K~ values given by point P are within one standard deviation of the best estimates, they are not consistent with the data.

144

of interpreting the uncertainties in K0 and K~ separately. The angle which the axes of the ellipse make with the K0 —K~axes reflects the coupling, or covariance, of K0 and K~.As a consequence of this interdependence, if either parameter could be more accurately measured, as shown by the a~ value for K0, then the uncertainty of the other parameter can be reduced markedly, as illustrated in Fig. 3 by ~ In the event that two sets of results are not compatible, then combining the data is futile. The statistical procedure is violated by forcing a curve through the data which does not fit the data. Hence, any K~which results may well be less meaningful than the K~for the compression data alone,

3. Results Most of the general features of our analysis are illustrated by the results for Si02 -quartz and Ti02 -rutile. These materials have been chosen as test cases because both have been studied exten-

sively in several laboratories by ultrasonic techniques as well as by volume compression methods. Therefore, by assuming thit the values of K0 and K~determined from the ultrasonic experiments are correct, we can adopt these values as standards with which to compare the results of our P—V data analysis. These two materials are also particularly interesting because they differ by more than a factor of five in compressibility. In Tables II and III we give the results of our analyses of two separate compression experiments on quartz and on rutile. For each data set the results of four different cases are listed. In both cases I and 2, K0 and K~are determined solely from the compression data. The case 1 errors are estimated from the scatter of the data about the least-squares curve and in case 2 they are estimated from the quoted uncertainty of the measured data. In cases 3 and 4, K~ and aK6 are determined from compression data after fixing K0 and aK0 at acoustic values. aK& for case 3 is estimated in the same manner as for case I, and case 4 is similar to case 2. The ultrasonically determined values of K0 and K~are also given for comparison.

TABLE I Ultrasonic data for the bulk modulus and its pressure derivative Compound

Structure

K~(kbar)

(~K~/~P)

Reference

Si02 Ti02 Fe2 Si04 Ni 2SiO4 MgGeO3 Pyrope

Quartz Rutile Spinel Spinel Ilmenite Gamet

371± 2 2090± 10 1920 ± 40 2150 ± 40 1970± 80 1750± 10

6.3±0.3 6.9±0.3

McSkiniinetal.(1965) Manghnani (1969) Liebermann (1975) Liebermann (1975)

4.8±0.2*3

Leitner et al. (1980) Bonczar et al. (1977)

Almandite Diopside Si02

Garnet Pyroxene Stishovite

1760± 10 1070± 10 2430±300

5.5±0.3

Soga(1967)

*1 *

2

*1,2

Liebermann (1974)

Levien, Weidner and Prewitt (1979) Liebermann et al. (1976)

All bulk moduli are isothermal moduli calculated from the adiabatic single-crystal elastic moduli and from KT = K, (I + Ta,,’y)~ where ~‘a,,K,/pC~ Note that it is the Reuss-averaged value of the bulk modulus which should govern the hydrostatic compression of an anistropic

material since this value is calculated on the assumption of homogeneity of stress. For the materials with cubic symmetry (spinels and gasnets), this distinction is not relevant since the Voigt and Reuss averages are degenerate. For Si02-quartz, TiO2-rut~1eand diopside the Reuss moduli have been obtained directly from the single-crystal elastic moduli. For MgGeO3-ilmenite, the Reuss value is estimated to be equivalent to the Voigt value by analogy with corundum. For Si02 -rutile, the polycrystal K0 (adiabatic and presumed equal to the Voigt—Reuss—Hill average) is corrected to the Reuss value by analogy with the isostructural Ti02. Uncertainties are ±0.5%for the single-crystal data and ±2.4%for the polycrystalline data. All (~K/~P) values are ±5%.These uncertainties reflect both the precision of the individual experiments and, where available, the reproducibility between experiments. *3 Pressure derivative data for pyrope-rich sample with composition [Mg0 ,Fe036 Ca 002 JAI25i 30,2 (Bonczar et al., 1977).

145 TABLE II Results of Ti0

2-rutile analyses. Acoustic values: K0

=

2090 ± 10 kbar, K~= 6.9 ±0.3

Analysis

Case

K0 (kbar)

MingandManghnani(1979) I run, 17 points 2,a~=3% o,,variable*

1 32

4

2160±110 2100±130 2090± 10 2090± 10

Sato (1977) All data 2 runs, 20 points = 0.2%; a~,= 1.5%

1 2 3 4

1880 ± 50 1880 ±180 2090± 10 2090 ± 10

Sato(1977) run No. Til6, 11 points

1 2 3 4 1 2

Sato (1977) run No. Ti19, 9 points

3 4 *1

*

2

K6

SSR

Model test

188.6 4.35 194.6 4.49

A A

2.2 8.1 0.8 1.8

23.09 1.30 47.66 2.68

R A

1950± 40 1950 ±250 2090 ± 10 2090 ± 10

8.4± 1.6 8.3 ± 10.1 3.2 ± 0.6 3.2 ± 2.2

4.21 0.24 9.84 0.55

R A

1850 ± 100 1850 ±280 2090± 10 2090± 10

10.6 ± 5.0 10.7 ± 15 0.1± 1.5 0.1± 3.4

14.37 0.81 26.84 1.52

R A

5.0± 3.0 7.1 ± 0.9 4.0 7.0± 7.3± 1.3 10.6 ± 10.6 ± 2.4± 2.3 ±

*

Model test column indicates whether the approach of constraining K 0 to be equal to the acoustic value in cases 3 and 4 can be accepted (A) or must be rejected (R) at the 95% confidence level. Uncertainty varies for each data point but is not a simple percent error. See original paper for individual errors.

The sources of these parameters are described in Table I. In two of the studies (Olinger and Halleck, 1976; Sato, 1977), data were collected from two distinct runs. In addition to analyzing the data TABLE III Results of Si02 -quartz analyses. Acoustic values: K0

=

from both runs together, as was done by the original authors, we have analyzed each run separately. Finally, for each least-squares solution we give the sum of the squares of the residuals (SSR)

371 ±2 kbar, K~= 6.3 ±0.3

Analysis

Case

K0 (kbar)

K~

SSR

Model test

Olinger and Halleck (1976) All data 2 runs, 15 points a,, and o~,variable

1 2 3 4

380±11 384± 8 371 ± 2 371 ± 2

5.8±0.4 5.6 ±0.3 6.1 ±0.1 6.2 ±0.1

23.96 20.65 25.17 23.45

A A

OlingerandHalleck(l976) Data obtained with Al pressure scale. 8 points

1 2 3 4

361±13 371±10 371 ± 2 371± 2

6.5±0.5 6.0±0.4 6.1±0.1 6.1±0.1

9.14 6.35 10.04 6.35

A A

OlingerandHalleck(l976) Data obtained with NaF pressure scale. 7 points

1 2 3 4 1 2

404±16 408±13 371 ± 2 371± 2 379±33 404±12

5.0±0.5 4.9±0.4 6.2 ±0.1 6.2±0.1 5.7±1.8 4.5±0.6

6.39 5.84 12.11 14.46 10.61 17.75

A R

3

371 ± 2

6.2±0.3

10.78

A

Levienetal.(l980) I run, 6 points a,, and a~,variable *

4 371± 2 6.3±0.1 25.85 R * Measured volumes of the pressure standard were converted to pressure via the Birch—Nurnaghan equation of state (I) and the ultrasonic values of K0 and K6 for Al (Thomas, 1968) or NaF (Miller and Smith, 1964).

146

of the data from the regression line, and for cases 3 and 4, the results of a model test at the 95% significance level,

creased by a factor of 3 from the corresponding uncertainties of the two-parameter fits. Again, we choose the weighted solution for K~= 7.3 ± 1.3 as

Consider first the Ti02 data of Ming and Manghnani (1979). The results of case I indicate

being more realistic than the unweighted value. Since the values of K~ for most rock-forming minerals fall in the range 3.5—7.0 (Anderson et al., 1968) the uncertainty of ±1.3 is small enough to make the value of K~for rutile distinguishable from that of other materials.

that, within the mutual experimental errors, their data are consistent with the ultrasonically determined values of K0 and K~.We remind the reader that this solution (case 1) gives equal weight to each data point and the uncertainties in K0 and K~are determined solely from the scatter of the

data points. On the other hand, case 2 takes into account the fact that some data are better determined than others, and weights each data point accordingly. Also the uncertainties on K0 and K~ from case 2 are determined from an a priori knowledge of the errors in P and V. Comparing the results of cases 1 and 2, two significant facts should be noticed. Firstly, the values of K0 and K~ from case 2 agree with the ultrasonic values (within one standard deviation) and are consistent with the unweighted results of case I. This indicates that no serious systematic errors are present in the experiment. Secondly, the uncertainties on K0 and K~from case 2 are slightly larger than those from case 1. There are two possible reasons for this discrepancy: the data set may be uncharactenstically good, yielding fortuitously small uncertainties for case I; or the experimental uncertainties may have been overestimated resulting in unrealistically large uncertainties in case 2. If the precisions with which P and V can be measured have been properly assigned by calibration of the instrument, then we suggest that the weighted results of case 2 be considered the best and most precise estimates of K0 and K~that can be determined from the compression data alone. It is clear from the results of case 2 that K~= 7.1 ±4.0 is not sufficiently well resolved to be of interest in geophysics. In cases 3 and 4, we show the effect of constraining K0 to the ultrasonic value, which is much more precise than the value obtained from cases 1 or 2, and then using the P V data to determine K~alone. Note first that for both cases 3 and 4 the values of K~are in agreement with the correct value. Equally significant is the fact that the uncertainties on Kj from cases 3 (unweighted) and 4 (weighted) are de—

•

Two factors are responsible for the decreased uncertainty in K~for cases 3 and 4. Firstly, only one parameter is being fitted, resulting in an additional degree of freedom over cases 1 and 2. In other words, the same amount of data is being used to determine only one physical parameter rather than two, and the data can thus determine the single parameter with greater certainty. More important, however, is the effect of using an assumed value of K0 which has a much lower uncertainty than the value from the two parameter solution. This is due to the strong coupling, or covariance, of K0 and K(~ as discussed in the previous section. Therefore, the uncertainty in K~ directly reflects the error with which K0 has been determined. If we turn our attention now to the data of Olinger and Halleck (1976) for quartz (Table III), some other important effects are illustrated. As in the above example, the unweighted 2 parameter fit (case 1) gives results which agree with the ultrasonic values, considering their mutual uncertainties. Although the weighted results of case 2 are

virtually identical to the results of case I, the weighted uncertainties on K0 and K~are a bit smaller. Assuming that the relative weights on the data are correct, the weighted values of K0 = 384 kbar and K~= 5.6 are preferred. From the discrepancy in aK6 between cases 1 and 2, it would appear that either the absolute values of the experimental uncertainties have been underestimated, or that due to the random nature of the errors the dispersion of the data is simply worse than one would normally expect from P and V uncertainties. For this particular case, the difference is small and we infer that the random experimental errors have been more or less realistically estimated. In general, however, the larger uncertainties should be more appropriate. There-

147

fore, the best estimates of K0 and K~from these data alone are K0 = 384 ±11 kbar and K~= 5.6 ±0.4. Using the a priori knowledge of K0 = 371 ±2 kbar from ultrasonic experiments, the results of cases 3 and 4 are obtained. The values of K~are closer to the true values, and the uncertainties are much reduced from cases 1 and 2. However, the weighted value of K~,= 6.2 ±0.1 from case 4 does not appear to be consistent with the value from the result of case 2. This is because the assumed value of K0 = 371 ±2 is outside the range of K0 = 384 ±11 from case 2. Therefore, at the con-

fidence level given by one standard deviation (‘—‘ 70%), it is meaningless to impose a value of K0 on the data which is outside of the limits on K0 from the two-parameter fit. At a 95% confidence level (about two standard deviations), the independent value of K0 is within the bound on K0 from case 2, and hence the resulting value of K~is as well. The compression data of Olinger and Halleck were collected from two different runs: one with Al, and one with NaF as the pressure standard. These runs are analyzed separately in Table III. The results for cases I and 2 show that the NaF run is not in agreement with the ultrasonic values of K0 and K~.When the value of K0 is constrained to the ultrasonic value, the residuals for cases 3 and 4 are increased by 90% over cases I and 2, and

the model tests indicate that the ultrasonic value of K0 is just barely compatible with the P—V data at a 95% confidence level (i.e. there is a 90% probability that these values are inconsistent but not 95%; compare with Al run). Furthermore, the Al and NaF data are not mutually consistent, implying that a systematic run to run effect may be present; that is, a larger variability is observed between different runs than would be expected from the variability observed within any individual run. However, the run conditions were not identical, inasmuch as two different pressure standards

were used. In an attempt to rationalize these inconsistencies, the pressures corresponding to measured values of the unit-cell volume of the standard material have been recalculated by the Birch Murnaghan equation (1), using the ultrasonically determined values of K0 and K~for NaF (K0 = 457 kbar, K~=5.25; Miller and Smith, 1964) and Al (K0 = 759 kbar, K~= 4.417; Thomas, 1968). Analysis of these revised data yield the results in Table IV. While the differences between the original and modified results are not large for the Al run, the NaF results change by more than one standard deviation. Both sets of results change in a direction to improve the consistency between runs, which now agree within the mutual uncertainties. Thus, if the modified pressure scales are accepted, there is no evidence of systematic run to run —

TABLE IV

Analysis of Olinger and Halleck’s (1976) compression study of Si02 using ultrasonic pressure scales ~. Acoustic values: K0 = 371 ± 2kbar, K6=6.3±0.3

Analysis

Case

K0 (kbar)

K6

SSR

All data 2 runs, 15 points

1 2 3 4 1 2 3 4 1 2 3

378± 9 382± 8 371± 2 371± 2 367±13 378±10 371 ± 2 371 ± 386±13 392±13 371± 2

5.8±0.3 5.6 ±0.3 6.0±0.1 6.1±0.1 6.1±0.5 5.7±0.4 6.0±0.1 6.0±0.1 5.5 ±0.5 5.3 ±0.5 6.1±0.1

17.08 13.81 17.74 15.67 8.72 6.36 8.86 6.79 4.62 4.55 5.81

4

371± 2

6.1±0.1

7.18

Dataobtainedwith Al pressure scale 8 points Data obtained with NaF pressure scale 7 points

*

Model test

A A

A A

A A

Measured volumes of the pressure standard were converted to pressure via the Birch— Murnaghan equation of state (I) and the ultrasonic values of K,, and K~,for Al (Thomas, 1968) or NaP (Miller and Smith, 1964).

148

effects. Although the modified NaF results are still not within one standard deviation of the ultrasonic values of K0 and K~,the discrepancies are smaller

results of case 2 do not agree. This data set shows an appreciable difference between the weighted

than those obtained using the original pressure scale. In addition, the residuals in Table IV are consistently lower than the corresponding residuals in Table III, indicating that the fit has improved. Therefore, we contend that the results in Table IV are a more accurate representation of the data of Olinger and Halleck than those in Table III. Let us turn our attention now to Table II and the Ti02 data of Sato (1977), who also made two separate runs using the same pressure standard (NaCl). Considering all of the data together, cases 1 and 2 are not in agreement with the ultrasonic values. Also, the discrepancy between the uncertainties of cases 1 and 2 imply that the experimen-

of the data points has twice the error of the other five, and hence 17% of the data is weighted in case 2 with only one fourth of the weight that it had in case 1. The model tests for cases 3 and 4, which reject the combination of the weighted results with acoustic values while accepting the unweighted fit, reflect this discrepancy at a 95% probability level. An explanation for the large standard deviations for case 1, relative to case 2, is that the scatter of the data is atypically large, or that the errors have been underestimated. Either

tal errors may have been overestimated, as in the example of Ming and Manghnani, but more pronounced, If we force K0 to be consistent with the ultrasonic value, we obtain solutions for K~in cases 3 and 4 at the expense of greatly compromising the quality of the fit of the equation to the data, as reflected in the large increase in SSR in cases 3 and 4 relative to cases I and 2. We may

We suggest, therefore, that the best estimates of K0 and K~from the P—V data are 404 ± 33 kbar and 4.5 ±1.8, respectively, and for the combined P—V and acoustic experiments K~= 6.3 ±0.3. Due to the larger uncertainties, these values are within the 95% confidence bounds. Although the main focus of the experiments of Levien et al. (1980) was to refine the crystal structure of quartz at high pres-

state these conclusions more quantitatively in terms of the model tests which imply, for the unweighted case, that the hypothesis that K0 = 2090 ±10 kbar is consistent with the data can be rejected with 95% confidence. Consequently, one would condude that it is of no use (and probably of no physical significance) to combine these ultrasonic and P—V results. This is precisely what led Ming and Manghnani to repeat the experiment. It is interesting to note, however, that the two separate runs show no indication of a systematic run to run effect. Also, if we assume that the true uncertainties in the data are those reported for P and V, and not the residuals, then the ultrasonic value cannot be rejected with 95% confidence. The tradeoff here is that these weighted results are of such low precision as to make them of limited utility. The last example is the P—V data set of Levien et al. (1980) on quartz (Table III). Case I gives unweighted results for K0 and K~in very good agreement with the ultrasonic values, while the

sure, the discrepancy discussed above should serve as a signal that more data are required and/or that the uncertainty of the measurements needs to be reevaluated. The conclusions of our analyses of SiO2 -quartz and TiO2 -rutile may be summarized as follows: (1) Both weighted (case 2) as well as unweighted (case 1) solutions of K0 and K~should be obtained for a collection of P—V data. The values of the parameters from the weighted case are preferred,

and unweighted values of K0 and K~because one

way, the larger uncertainty must be assumed. Also,

if the relative magnitudes of the weights are correct, then the results of case 2 should be, in principie, a more appropriate representation of the data.

although these will not usually differ significantly

from the unweighted solutions; (2) If the uncertainties in the parameters from the weighted (unweighted) solution are larger, the experimental errors in P and V may have been overestimated (underestimated) or the data set may be of uncharacteristically good (bad) quality. Irrespective of the cause, the larger uncertainties for K0 and K~should be adopted unless supplementary experimental details suggest otherwise; (3) When the value of K0 determined from a two-parameter fit of the P—V data is consistent

149

with an independent but more precise measurement of K0 from ultrasonic data, then the two

experiments may be combined to refine the value of K~and improve its precision. We now proceed to examine some minerals for which K~is poorly known or completely undetermined. Our results for these materials are given in Tables V—X.

vious determinations of K~for garnets in the pyrope— almandite series (e.g. Soga, 1967) indicate that K~is approximately 5, we can find no statistical or physical reason to reject the value of K~=

1.4 ±0.4 for the pure almandite end member, obtained in cases 3 and 4. Pyrope

hydrostatic conditions is that of Sato et al. (1978)

We will consider two investigations of the compression of pyrope under hydrostatic conditions, which have been made by independent X-ray diffraction techniques: Sato et al. (1978) on powder samples and Levien, Prewitt and Weidner (1979) on a single crystal. [The P—V relationships of py-

and we present our analysis of their data in Table V. Cases 1 and 2 are consistent with each other and with independent values of K0. Separate analyses of run 56 and the sum of runs 57 and 59 give no indication of run effects. Although pre-

rope and grossular have been measured by Hazen and Finger (1978). However, Levien, Prewitt and Weidner point out that unusually large discrepancies between these results and those of other investigators indicate a systematic bias in the measure-

Garnets (Table V) Almandite The only P—V study of almandite garnet under

TABLE V Results of analyses for garnets. Acoustic values: K0 Analysis Almandite Sato et al. (1978). All data 3 runs,2%,a~=O.O6% 12 points * a,,rO.l Sato et al. (1978) Run 56,6 points

Satoetal.(l978) Runs 57 and 59,6 points

Pyrope Satoet al.(l978) All data 2 runs, 10 points a~O.O6%,o~=0.06% Levien, Prewitt and Weidner (1979) I run,5 points a~variable,a~=0.5kbar

*

=

1760±10 kbar for almandite: K0 = 1750±10 kbar for pyrope

Case

K0 (kbar)

K~,

SSR

1 23 4 1 2 3 4

1750± 70 1750± 80 1760± 10 1760± 10 1720± 50 1720±100 1760± 10 1760± 10

1.5±1.6 1.5±1.9 1.4±0.4 1.4±0.4 2.4±1.2 2.4±2.5 1.4±0.4 1.4±0.6

35.77 35.80 8.02 8.02 4.24 0.95 5.02 1.13

1 2 3 4

1810±160 1810±120 1760± 10 1760± 10

0.3±3.5 0.3 ±2.7 1.4±0.6 1.4±0.5

30.13 6.75 30.78 6.90

1 2 3 4 1 2 3 4

1710± 30 1710± 40 1750± 10 1750± 10 1750± 10 1760±100 1750± 10 1750± 10

1.8±0.7 1.8±0.9 0.9±0.3 0.9±0.4 4.4±0.4 4.3±4.9 4.5±0.5 4.5±1.0

5.48 4.97 6.67 6.05 0.019 0.022 0.020 0.026

Model test

A A

A A

A A

A A

A A

Uncertainty in pressure was not specified. We have assumed it to be the same as the uncertainty with which the volume of the pressure standard was measured.

150

ments. To avoid redundancy, we refer the reader to these papers for a detailed discussion of these P—V data.] As in the case of almandite, the data of Sato et al. (1978) yield a value of K0 (1710 ±40

modified the NaF pressure scale of Sato (1977) as was done for the quartz data of Olinger and Halleck (1976). The results of case 1 for the P—V

kbar, case 2) which is in agreement with the acoustically determined value, but again an unusually low value for K~. Sato made two different runs, but unfortunately one contains only three P—V points, making a separate analysis of these runs of little value, The data of Levien, Prewitt and Weidner (1979) for cases I and 2 are consistent with the acoustic value of K0, and the value of K~is reasonable

(1976), are both in agreement with the ultrasonic value of K0 = 1920 ±40 kbar but not with each other for K0 and barely within the mutual uncertainties for K~.All parameters agree for case 2,

based upon ultrasonic measurements for other silicate gamets (e.g., Bonczar et al., 1977). How-

ever, a marked discrepancy exists between the uncertainties for cases 1 and 2, implying that the errors are overestimated or that the data are fortuitously good. The latter explanation is very possible in this example because so few data were collected. Assuming that the results of case 2 mdicate the true uncertainty of these parameters, case 4 shows that by constraining the value of K0 to be 1750 ±10 kbar, the uncertainty of K~is reduced by a factor of five to ±1.0. Spinels (Table VI) Fe2 Si04

For the sake of internal consistency, we have TABLE VI Results for spinels. Acoustic values: K0 Analysis Fe2 SiC)2 Sato (1977) I run, 8 points 5%,a~=2.S% a,, O.O Wjlburn and Bassett(l976) I run, 9 points a,, O.36%,o~, 0.75kbar

=

data of Sato (1977) and of Wilburn and Bassett

which we assume to be the more representative solution. In the case of Sato et al. (1977), the constraint that K0 = 1940 ±40 kbar increases K~ from 2.4 ±2.7 to 6.3 ±1.6. The important feature to point out is the substantial increase in the precision of K~.Although the P—V data of Wilburn and Bassett yield larger uncertainties in K0 and K~

than those of Sato et al., a comparable increase in the precision of K~is obtained going from case 2 to 4. Note also that while the unweighted results (case I) do not appear to agree, including the P and V uncertainties brings the values of K0 and K~ from the two experiments within one standard deviation of each other. N12S1O4

For the Ni2SiO4-spinel data of Sato (1977), the results of case 1 do not agree with the ultrasomcally determined value of K0 =2150 ±40 kbar. As in many of the previous examples, the results are compatible when the P—V errors are explicitly included and the data points are weighted (case 2).

1920 ±40 kbar for Fe2SiO4 K0

=

2150 ±40 kbar for Ni2SiO4

Case

K,, (kbar)

K~

SSR

1 23 4 I 2 3 4

2040 ± 50 2040 ± 40 80 1920± 1920± 40 1770±140 1770 ±400 1920± 40 1920 ± 40

2.3 ± 1.6 2.4 ± 2.7 6.0± 1.4 6.3± 1.6 7.4± 5.5 7.4 ± 16 2.0± 1.9 2.0 ± 3.7

4.57 1.68 8.56 3.85 39.88 0.83 46.58 0.97

1 32 4

2050± 50 2040±130 2150± 40 2150± 40

11.9± 2.2 12.1 7.9± ± 5.8 1.5 7.7± 1.9

4.93 7.41 1.06 1.75

Model test

A A

A A

Ni 2SiO4 Sato(1977) I run, 10 8%,a~ points= 2.5% a~, O.O

A A

151

When K0 is constrained to be 2150 ±40 kbar (cases 3 and 4), the values obtained for K~,decrease significantly and become more precisely determined at the expense of greatly increasing the residuals (but remain marginally acceptable at the 95% confidence level). Although the residuals and the large standard deviation on K~indicate that more measurements are needed, from those data the best estimates are K0 = 2150 ±40 kbar and = 7.7 ±1.9. MgGeO3-ilmenite (Table VII)

The case I results for the compression data of Sato (1977) on the germanate analog of MgSiO3 ilmemte agree very well with the ultrasonic value of K0. Since the ultrasonic value of K0 is of comparable precision to those from cases 1 and 2, the values of K~obtained in cases 3 and 4 are neither different nor more precise than those from cases 1 and 2. If we consider the two runs for MgGeO3 separately, we see that the unweighted results of case 1 are not in agreement with each other for K0 and are barely so for K~. On the other hand, the weighted values of case 2 are consistent for both parameters. This indicates that the random variations in any one run are not always sufficient to explain run to run variations and the true uncertainties are closer to the weighted values. The -

TABLE VII Results of MgGeO3-ilmenite analyses. Acoustic values: K0

=

pair of two parameter solutions for run 1L44 is

one of the few examples where the unweighted and weighted errors agree. Diopside (Table VIII)

Several aspects of the diopside analyses are worthy of comment. Although the values of K0 from cases 1 and 2 are mutually consistent and of relatively high precision, the difference in standard deviations may indicate that too few data have been collected to reflect the true variability of the measurement technique. Also, it might appear from the standard deviations that the acoustic and cornpression values of K0 are inconsistent. Yet for

cases 3 and 4 the results of the model tests do not reject the acoustic value of K0 (and, hence, the values of K~)with 95% confidence. This is due to the small number of data, which has the effect of making the confidence intervals (ellipses) large, despite the low residuals. Therefore, given that the acoustic value is more accurate than the compression value, the best estimate of K~,is 7.9 ±0.6. Stishovite (Tables IX and X) This mineral has received considerable attention in the past and its elastic properties have been studied by a variety of techniques (see discussion in Liebermann and Ringwood, 1977). We will

1970 ± 80 kbar

Analysis

Case

K0 (kbar)

K~

SSR

Model test

Sato et al. (1977) All data 2 runs, 14 points a,, O.2%,o,, 0.1 kbar*

1 2 3 4

1960 ± 70 1960 ± 100 1970± 80 1970± 80

2.1 ± 1.3 2.1 ± 1.9 2.0± 1.6 2.0± 1.6

93.43 5.92 93.49 5.93

A A

Sato et al. (1977) Run 1L44,6points

1 2 3 4 I 2 3 4

1850 ± 160 1850±160 1970± 80 1970± 80 2060 ± 40 2060 ± 150 1970± 80 1970± 80

4.2 ±3.2 4.2±3.2 1.9±1.7 1.9±1.7 0.5 ±0.7 0.5 ±2.4 1.9±1.8 2.0± 1.7

69.94 4.18 78.94 4.78 8.10 0.49 13.71 0.87

Sato et al. (1977) Run 1L47, 8 points

*

A A

A A

Uncertainty in pressure and volume of pressure standard were not specified. We have assumed an uncertainty of ±0.1 kbar.

152 TABLE VIII Result of diopside analysis. Acoustic value: K,,

=

1070 ± 10 kbar

Analysis

Case

K

K~

SSR

Levien, Weidner and Prewitt (1979) I run,Spoints o,, and c~,variable

1 2

1130 ± 20 1140±40

4.9 ± 1.0 4.5±1.8

0.35 0.89

3

1070 ±10

7.9 ±0.6

1.27

A

4

1070±10

7.9±0.6

3.95

A

0(kbar)

focus our attention on the two most recent compression studies of Sato (1977) and Olinger (1976),

which were performed under hydrostatic conditions. In Table IX we have listed the results of the two parameter fits of both studies using the pressures as given by the authors; the values of K0 and K~ are seen to be in good agreement. However, in contrast to the data of Olinger, the uncertainties for case 2 of Sato’s data exceed those for case 1. Since Sato’s data set consists of 44 measurements, there is little possibility of the dispersion being fortuitously small relative to the inherent uncertainty in the measurements. Thus the standard deviations may be somewhat overestimated in case 2. Sato (1977) and Kuzio (1977) have demonstrated ‘that the values of K~from these data become larger and more typical of silicates if the pressure is calibrated using a standard which has been characterized by an ultrasonic technique. Both

Model test

Sato (1977) and Olinger (1976) used NaF as an internal standard. Following the example of Sato (1977), we have redetermined the pressures as we did with the quartz data of Olinger and Halleck (1976) and the spinel data of Sato (1977), by assuming the values of K0 = 457 kbar and K~= 5.25 for NaF (Miller and Smith, 1964) and converting the measured volumes of NaF to pressure via the Birch—Murnaghan equation (1). When this revised pressure scale is used to interpret the data, the least-squares solutions for K0 and K~of the different data sets are in better agreement (Table IX). However, despite the internal consistency of these investigations, acoustic measurements of K0 by Liebermann et al. (1976; 2430 ± 300 kbar) and Mizutam et al. (1972; 3390 ±240 kbar) are neither sufficiently close to the results of the compression studies nor precise enough to better constrain K~,. The experiment of Sato consisted of five runs, with 8 or 10 points each. We have analyzed each of these runs separately using the revised pressure

TABLE IX Results of stishovite analyses Analysis

Case

K0 (kbar)

K~

SSR

Original pressure scales Olin~er(l976) I run, l6points a,, and c~,variable

1 2

3070± 130 3040±140

1.3±2.9 1.8±3.1

133.20 12.85

1 2

2990± 50 2980± 130

0.6± 1.1 0.7±3.5

153.72 4.24

1 2

2980±130 2940±140

2.7±3.1 3.2±3.3

133.16 12.78

1 2

2940± 50 2940±160

3.0± 1.2 3.1±3.7

161.25 4.45

Sato(1977), All data Sruns,44points a,, = 0.2%, u~= 2.5% Ultrasonic NaP pressure scale Olinger(1976) Sato(l977) All data

153 TABLE X Individual run and average results of the stishovite compression data of Sato (1977) using ultrasonic NaF pressure scale

served in the five separate experiments and not a result of the internal consistency of any given experiment. It is important to distinguish between the standard deviations of mean values of the

Run

Case

K

ST- 14, 8 points

1

2980 ± 40

2

2980±590

1

3120± 60

2 1 2 1

parameters (K0, K~)and the standard deviations, or root-mean-square errors, of the distributions of K0 (±200)and K~(±4.0).The latter uncertainties indicate the ranges about the mean values which are expected to contain about 68% of the individual experimental determinations (i.e. single runs) of K0 or K~,.In other words, the results of any one run have a 68% chance of being within ±200kbar and ±4.0of the true values of K0 and K~,respectively. On the other hand, if hve more runs were made, the resulting averages of the parameters, K0 and K~,would have about a 68% probability of being within the standard deviations of the means given in Table X. A comparison of the average results with the analyses from each run indicates that: (I) the data within each run scatter much less than do the data

ST-IS, 10 points ST-28, 8 points ST-35, 8 points ST-40, 10 points

0 (kbar)

K~

SSR

1.7 ± 1.5

1.11

1.7±20

0.03

—1.1± 1.4

7.23

3130±390 3060 ± 80 3060±490 28 10± 50

1.2± 8.9

0.21 5.47

2

2810±400

7.1± 9.3

3.21 0.09

1

2630±100

8.7± 2.6

28.43

2

2620±330

8.9± 7.9

0.77

Average values for the five runs — Case I: K,, = 2920 ±90 — K0 3.8± 1.8

2.6 ± 2.3 2.5±14.0 7.0± 1.2

0.16

. .

202 kbar RMSerror4.0 RMS error

=

—

Case 2:

K,, = 2920 ±93 K~= 3.8± 1.8

RMS error = 209 kbar RMS error4.I

.

.

.

from run to run; (2) the standard deviations in Combination of Sato s and Olmger s data —

K0’=2930 ±80kbar K’ = 37-1- 1.6 0

—

scale (Table X). The important feature to note is the low standard deviations for both K0 and K~ obtained in case 1 for each run and the large variability in the parameters between runs. In other words each experimental run is self-consistent and the data define a precise P—V trajectory; however, the individual runs do not provide mutually consistent values of K0 and K~.It is thus clear that the case 1 standard deviations in Table V underestimate the real uncertainties in K0 and K~based on Sato’s data. A more realistic measure of the uncertainty in K0 and K~for stishovite may be obtained by averaging the results of the individual runs. These average values, K0 and K~are given at the bottom of Table X. The mean values of K0 and K~and their standard deviations obtained from the case 1 and case 2 values differ negligibly. The standard deviations for these mean values (±90 kbar for K0 and ±1.8 for K~)are a consequence only of the distribution of the values of K0 and K,~,ob-

.

.

case 1 underestimate the uncertainties in K and K~,while those in case 2 may overestimate them; (3) each run is systematically biased, and the observations within each run are not independent. If we assume that the bias of a run varies randomly from run to run, then on the basis of comparisons (1)— (3) above it is proposed that each run is a single, random observation of K0 and K~, and that over a large number of runs the average of K0 and K, will converge to the true values of these parameters. Thus, the best estimate of K0 and K~,from Sato’s data are 2920 ±90 kbar and 3.8 ±1.8, respectively. Inasmuch as the data of Olinger (1976) are consistent with those of Sato, their results can be combined to yield values of K,, = 2930 ±80 kbar and K~= 3.7 ±1.6. Kuzio (1976) has corrected the data of Sato (1977), Olinger (1976), and Liu et al. (1974) to a common pressure scale based on MgO and obtamed K0 = 3070 ±50 kbar and K~= 3.26 ±0.9 with a Birch— Murnaghan equation of state, in agreement with our results. However, the standard deviations reported by Kuzio are probably underestimated since all of the data of Sato were analyzed as one run. Furthermore, we feel that a common pressure scale based on NaF is more

154

appropriate since it was used in all of the experiments of Sato and Olinger. The data of Liu et al. (1974) have not been incorporated into our analysis because they were obtained in an ungasketed diamond-anvil pressure cell using NaCl as the pressure-transmitting medium. This is a demonstrably non-hydrostatic environment, and will thus bias K0 on the high side of the correct value (Ruoff, 1975; Kinsland and Bassett, 1977). Indeed, Liu et al. obtain values of K0 = 3350 ±50 kbar and K~= 5.7 ±1, which are significantly different from the results in Tables IX and X. Even after employing a correction for non-hydrostatic stress, Kuzio (1976) could not bring the results of Sato, Olinger, and Liu et al. into mutual agreement.

4. Sununary and conclusions We have analyzed the hydrostatic compression data for a number of oxides and silicates which have been studied independently by acoustic methods. For each material, we have obtained values for the zero-pressure bulk modulus K0 and its pressure derivative by fitting a Birch Murnaghan equation of state to the observed P—V —

data. We have also examined the consequences of constraining K0 to be equal to the acousticallydetermined value with the objective of obtaining more precise estimates of K~than were warranted by the P—V data alone. The elasticity parameters (K0, K~) obtained from these two analytical approaches are given in Table XI. Where two compression studies were available for a particular material, we have listed both sets of results, except for stishovite. The impact of combining information from the cornplementary acoustic and compression data is clearly illustrated by comparing the two columns for K~.When K0 is constrained for both quartz and rutile, K~increases markedly and becomes indistinguishable from the ultrasonic value (compare Table I). Furthermore, the uncertainty in the K~values so obtained are reduced by a factor of three from the two-parameter determinations. This dramatic improvement in the precision with which K~maybe determined from the P—V data when K0 is constrained is also realized (see Table XI) for the other materials (except MgGeO3 -ilmemte and stishovite). We remind the reader that the estimates of the errors on K~in Table XI are from the weighted least-squares solutions and, hence, most

TABLE XI

Elastic parameters obtained by combined analysis of acoustic and hydrostatic compression data in this paper Compound

Compression data alone

K0 (kbar)

Combined acoustic and compression data K,, (kbar)

K~

Ti02-rutile

2100±130

7.1± 4.0

2090±10

7.3±1.3

Si02-quartz

382± 8 404± 33

5.6± 0.3 4.5± 1.8

371± 2 371 ± 2

6.0±0.1 6.3 ±0.3

1760±100

4.3± 4.9 1.8± 0.9

1750±10 1750±10

4.5±1.0 0.9±0.4

Pyrope-garnet

1710± 40

K~

Almandite-garnet

1750±80

1.5± 1.6

1760±10

1.4±0.4

Ni2SiO4-spinel

2040±130

12.1± 5.3

2150±40

7.7±1.9

Fe2SiO4-spinel

1770±400 2040± 80 1960±100

7.4±16 2.4± 2.7 2.1± 2.0

1920±40 1920±40 1970±80

2.0±3.7 6.3±1.6 2.0±1.6

Diopside

1140± 40

4.5± 1.8

1070±10

7.9±0.6

SiO2-stishovite *

2930± 80

3.7±1.6

MgGeO3-ilmenite

*

A weighted average of the results of Olinger (1976) and Sato (1977).

155

of them are probably rather conservative. Our analysis of the data of Sato (1977) for stishovite demonstrated that the uncertainty in K~or K0 from a given set of compression data was approximately mid-way between the standard deviations obtained from the unweighted and weighted fits of the data. Thus, the true uncertainty on K~ from, for example, the pyrope data of Levien, Prewitt and Weidner (1979) is most likely between 0.5 (unweighted) and 1.0 (weighted). In the case of stishovite, the data of Olinger (1976) and Sato (1977) have been reduced to a common pressure scale based on ultrasonic measurements of the elastic properties of NaF, and in addition we have compensated for systematic run to run effects present in Sato’s data. These two studies gave internally consistent results and we suggest therefore, that although the P—V data are not in agreement with acoustic measurements of K0, the values of K0 and K~for stishovite in Table XI are the best estimates available at present from the compression data alone, Two criteria have been used to determine whether a particular analysis has yielded an improved value of K~: (1) An analysis of the compression data alone must yield a value of K0 which is compatible with the acoustically-determined value, considering the mutual uncertainties. When this condition is not met, we must conclude that one or both of the values are incorrect (e.g. rutile data of Sato, Table II). Note that the same arguments apply if a value of K~is imposed in order to elicit a value of K0 from the P—V data (e.g., Sato’s (1977) discussion of the Ni2 SiO4 -spinel data). Such an exercise is only valid when K~is constrained to a value which is consistent with the two-parameter solution. (2) The uncertainty in the acoustic value of K0 must be less than the uncertainty in K0 obtained from the P—V data alone. Otherwise, the precision on Kj will not be improved by combining the data (e.g., MgGeO3 -ilmenite). For some materials, simple unweighted regression analyses of the P—V data implied interlaboratory inconsistencies in determinations of the elastic properties. However, these discrepancies were for the most part resolved by taking into account the experimental pressure and volume uncertainties.

It should be emphasized that we do not claim in this paper to have necessarily determined accurate values of K~for all of the compounds listed in Table XI. Even after combining the P—V and acoustic data, the uncertainties are still quite large, interlaboratory discrepancies remain (e.g. pyrope) and some of the values of K~may appear contrary to typical values obtained by acoustic methods for silicates and other oxides (‘-= 3.0—7.0). Rather, our intent has been to examine a portion of the existing data base in a critical and often unflattering manner, and to demonstrate that it is possible to make a precise estimate of K~. Our hope is to provide a stimulus for experimental effort in this direction. This study has demonstrated several points which can yield improved accuracies of K0, K~ and UK6. In addition to those illustrated in Fig. 2, we found that repeated runs, as opposed to simply collecting more measurements in a given run, can prove quite valuable. A good example of this is given by the stishovite data of Sato (1977). Accurate estimates of UK0 and GK6 in this study depended upon accurate assessments of measurement uncertainties. Specifically, we required accurate estimates of precision as well as an assessment of the effects of systematic errors. We conclude that volume compression data may be used to determine geophysically significant values of K~when independent acoustic information is available on K0. It is therefore just as necessary to improve the existing acoustic values of K,,, and suggest that a concerted effort to obtain complementary acoustic and volume compression data will be far more valuable than either experiment alone. The potential of this approach has been clearly demonstrated by the experiments of Leitner et a!. (1980) and Levien, Prewitt and Weidner (1979) who studied the elasticity of the same synthetic pyrope single-crystal by acoustic techniques based on Bnllouin scattering and by hydrostatic compression techniques utilizing X-ray diffraction.

Acknowledgments We would like to thank L. Levien, C.T. Prewitt, L.C. Ming and M.H. Manghnani for supplying us

156

with preprints of their papers and allowing us to discuss their data prior to publication. Discussions with J. Scott Weaver, T. Takahashi, T. Yagi, L. Levien, and S. Akimoto were extremely helpful, as were comments by R.M. Hazen, J.C. Jamieson, R. Jeanloz, and Y. Sato. Special thanks are due to H.H. Demarest, Jr. for his comments and criticisms. This research was supported by NSF grants EAR 76-84066 and EAR 78-12941.

squares solution for A is given by

(A4) A = (Ftw ‘Fl ‘F’W ‘P where W is a weighting matrix. When the regression is unweighted, W ~‘is simply the identity matrix. For a weighted regression, w~have used the squares of the uncertainties in the individual data points, a,~,as weighting factors. Thus W is given by -

—

‘

0 Appendix

w=I The following is a brief description of the data reduction methods applied to the pressure—volume data. Our intent in this Appendix is only to outline the procedures by which calculations were performed and to show how various errors are accounted for and reflected in the final results. For detailed discussions on least-squares methods, the reader is referred to any text on statistics (e.g., Draper and Smith, 1966; Bevington, 1969; Wonnacott and Wonnacott, 1970). The Birch—Murnaghan equation is an isotherrnal equation of state, P( V), which is based on a third-order expansion of the free energy of a solid in terms of Eulerian strain (Davies, 1973) 7”3—x5/3) P(x)3/2K,,(x x [1 —3/4(4—K~)(x2/3 1)] (Al)

.

\0 When K 0 is a known input parameter, K~is given by 2/o,~) ~ (P, .1, /a12) = ~ (L1 .1, /a12) + A3 ~ (J where A 3 = 4— K~,, J ~j K0F2(x,) and L = ~ K0F,(x1), and the summation runs over the number of data, n. The uncertainty in the ith data point, a,, may be calculated most simply by propagating the volume error into the dependent parameter, pressure. Assuming that the pressure and volume errors are2 uncorrelated, then2 2 a~2 ~

U,

+~~(.~_-)

—

But

where x= V/V 0, and K,, = V(OP/aV) bulk modulus, K~—(aK/aP)~,both evaluated at zeropressure (P = 1 bar). This can be linearized into the form PA,F,(x)+A2F2(x) (A2) where A, ~ K,,, A2 7/3= —~K0(4— x573), K~,)A,[~(4 and F K~)}, F1(x) 5/3)(x = (x 2(x) I) F (x 7/3 x 1(x) X (x -2/3 Equation A2 may be given in matrix form as —

-2/3

—

P=AF

—

—

~p

a~~2

~)

2~’

—

Ux,~

2 = UV/V[

1~

I

2

______

a(V/v0)]

= U~

vow)

=U~(_#K) Since

—

= —

I)

(A3)

where P and A are column vectors, and F is an n X 2 matrix, for a set of n observations, with elements = F(x1). Denoting the transpose by a superscript “t” and the inverse by “—1”, the least-

2 / a~= ~ we get

2

V

~)

{(a~/V2) + (a~/~2)]

a~+[(a~./v2)+(a~/v

a

02)]K2

and if a~,is given as a percent error in volume (PEV) 2

a?

= Up

~

+

PEV 2

2 + (a~o/ V,,2)] K

157

If the relative uncertainties of the individual data points are not known, the standard deviation of a data point, a,, may be estimated by the root-mean-square of the residuals, or ~ [P,(meas.) a2

—

P1(calc.)j 2

i

=

(A6)

n—N

where N = 2 for the two parameter fit of K,, and K~(eq. A3), or I if only one parameter is to be found (eqs. A4, A5). It is this uncertainty which is used in the “unweighted cases” referred to in the above text. Uncertainties of the parameters K,, and K~are determined as follows: The variance—covariance matrix V for the coefficients in eq. 3 is a symmetric 2 X 2 matrix and is given by V =(FIW_1F)’a2 where

2

=

Wa

(a~

0

(A7)

)

0.~ a,~/

For the unweighted cases all of the variances in W are equal and given by eq. A6. The diagonal terms of V are equal to the conventional variance of the coefficients, and the off-diagonal terms yield the covariance between A, and A 2. Theofvariance of any parameter, y, which is a function A, and/or A 2 is given by 2 = ~ J’~(~y/aA ay 1)(ay/aA1) (A8) i

i

Because K,, is determined only from A,, the standard deviation of K,, is given simply as ~K = (2/3)V,,. However, UK~will reflect also the covariance terms. For the one parameter fit of K~,the standard deviation will reflect the uncertainty in the independent input parameter and the observations as follows =

[~

2

a~(aA3/aF1)2]+ GK~,

~K

\2

-~-k-)

Here the derivatives may be obtained from eq. AS. For every one parameter solution where K0 was fixed at a value determined by an acoustic experi-

ment, a model test was performed in order to determine quantitatively whether the resulting fit of the data was consistent with the two-parameter solution at a 95% confidence level. When the scatter of the data was used to estimate the variance of K~ (case 3) an F-test was used for this purpose. An F-value compares the increase in the sum of the squares of the residuals (SSR), with the meansquare residual for the two parameter fit. Thus the F-value is F= [ssR(l) SSR(2)J/[i~(l) SSR(2)/v(2) —

—

where the numbers in parentheses refer to the one-parameter (1), or two-parameter (2) fits and v is the number of degrees of freedom (number of data points minus number of fitted parameters). Since the same data are used to fit both models, v(l)— v(2) 1. The F-value can be compared with critical values of the F distribution (with 1 and 2 degrees of freedom) to decide if the one-parameter F-test also used tolarger determine a cate lesscan solution is agreement to be be rejected; between the models. values whether of The F same indigiven value of K~is consistant with the data. Where the variance of the data has been specified (cases 2 and 4), instead of simply summing the squares of the residuals, the residual is normalized to the standard deviation of the data point. Thus, the appropriate sums to compare in cases 2 and 4 2/a,2), which are listed in Tables II—X. are ~(, The model test then consists of comparing the difference between the above sums for cases 2 and 4 with the critical value of chi-square at the desired probability (in our case 95%) with one degree of freedom. If the difference is greater than the critical value, then the acoustic and compression data are deemed incompatible at the specified probability level. References Anderson, O.L., Schreiber, E., Liebermann, R.C. and Soga, N., 1968. Some elastic constant data on minerals relevant to geophysics. Rev. Geophys. Space Phys., 6: 491—524. Bassett, W.A. and Takahashi, T., 1974. X-ray diffraction studies up 300 kbar.Research. In: R.H.Vol. Wentorf, Jr. (Editor), in Highto Pressure 4. Academic Press,Advances New York, NY, 297 pp.

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