- Email: [email protected]
The analysis of high pressure experimental data
1. Phys. C&m. Soiid~ Vol. 52, No. 4, pp. 635437, Printed in Great Britain.
1991
0022.3697191 $3.00 + 0.00 Pefopmoa PM plc
TECHNICAL NOTE THE ANALYSIS OF HIGH PRESSURE EXPERIMENTAL DATA HERBERT Scmossmt Physics Dept., Cleveland State University, Cleveland, OH 44115, U.S.A. JOHN FERRANTE National Aeronautics and Space Ad~nistration, Lewis Research Center, Cleveland, OH 44135, U.S.A. (Received 5 Jury 1989; accepted in revised form I4 November 1990)
Abstract-This letter is concerned with the analysis of high pressure experimental data. We demonstrate that ln H plots based on the Vinet et al. universal equation of state (EOS) are a simple sensitive means for identifying anomalous P-V data in high pressure experiments, and for detecting structural and phase transitions in solids subjected to high pressure. Keywords: Equation of state, phase transition, data analysis, glass, manganese oxide, calcium.
1. INTRODUCTION
occurs there is either a discontinuity or a pronounced change in slope of the In H plots, and that the data points above and below the transition may be fit by two straight line segments.
A universal binding energy-distance relationship [l] of the form, E(R) = A.&??*@*), describes the adhesion and cohesion of metals, the chemisorption of gas atom on metal surfaces [2], as well as the binding of certain diatomic molecules [3,4], where E*(u*) is a universal function of the scaled length a* and AE is the total energy at eq~~b~urn. The universal energy relation leads to the Vinet et al. universal EOS [5,6], P(V) = 3(1 -X)
B,[expn(l
-X)1/X2,
2. ANALYSiS OF EXP~~~~
(1)
which accurately represents the experimental data for all classes of solids (metals, covalently bonded, ionic and Van der Waals) in compression, where X I (V/Vo)“3, B, is the isothermal bulk modulus and Y, is the volume at zero pressure, tf = 3[& - 1]/2, and B; = (~~/~~)~=~. For many classes of solids the function, G*(a*) = (X2/3(X - 1)) P(V)/&,, is a universal function of a* [5,6]. Now defining the function H(V) E -E,G*(a*) In H(V)=In&-tn(l
= B,exp[n(l -X)], -X).
(2a) (2b)
Vinet et al. [7] found that in the absence of phase transitions, plots of In H vs (1 - X) for most solids, are to a good approximation straight lines (least square fits in in H vs (1 - X) yield correlation coefficient values, t > 0.999). Recently (81, we showed that this theory also applies to liquid metals and to rare gas liquids [9], and yields results with accuracy comparble to that in solids. In this letter we demonstrate that the universal EOS and In H plots are useful for the detection of anomalous high pressure experimental data, and for the detection of phase or structural transitions in solids subjected to high pressure. We observe that when a phase or structural transition t Supported in part by a NASA-ASEE Summer Faculty Fellowship, and by a Ohio Board of Regents Research Challenge Grant.
DATA
(A) Detection of spurious experimental data Now we demonstrate that the In H plots provide a sensitive means of detecting spurious high pressure experimental data, i.e. falls well &side the stand&d deviation of the data. Consider the In H nlot for MnO in Fia. 1. The raw experimental P and V data-of Jeanloz and R&y [lo] were utilized to obtain the In H vs 1 - x plot in Fig. 1, along with a P-Y plot for the same data. A least square fit to the entire set of In H points yields B, = 181.6 GPa, and B; = 3.682 with a correlation coetEcient r = 0.6752. Inspection of Fig. 1 readily indicates that, with the exception of the first point, all of the data points cluster about a straight line. Now if the first point is eliminated, a least square fit yields B, = 156.94 GPa, and B; = 5.489 with a correlation coefficient r = 0.9238. These values are close to the experimental values reported for ultrasonic measurements [ll-131, namely B, = 155.5 + 3.5 GPa [l l], and Be = 149.2 f 0.5 GPa [12] and B&= 5.5 f 0.1 [13]. Thus, by simple inspection, we may con&de that the first data point is anomalous, as indeed did Jeanloz and Rudy [lo]. In analyzing the experimental data we have not explicitly weighted the data to reflect the lower experimental accuracy at higher pressures [14], since Vinet et al. (Ref.7, Table 6) found that the values of B,, and Bi obtained from curve fitting eqn 2b are relatively insensitive to the number of high pressure points included in the linear least square fits. In addition, we have observed that the &, and B; values obtained from the In H fits agree to within experimental error with the values obtained from nonlinear fits to eqn 1. We have not included experimental error bars in the P-Y data since most of the data papers do not report them. The power of the In H plots is that data points which are well outside the rms deviation of the data can be readily observed by inspection. Thus, they provide a simple method for data analysis during an experiment.
635
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0f.M
loId~u[ aq$Jo adop u! aSu8qa a[qehlasqo @pea1 aql aIoN &I] zoIu8ar pue apwagqyo sep @luauyadxa aw Su!zgyn zO!S pasrg ‘OJ X1 SA H IIIpallold aAEq aM ‘5:27!d III suoy~suv~~asvqd puv pkn~wws Jo uoymlaa
'19 arenbs 1slla1 :aug pgos t[oI Jaa] oum-_x-
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a[q!slaAay aql 01 anp d[qeurnsaJd s! puo~as aql pue Q-gs u! aSueq3 Iem]3ruls aql 01 spuodsaJIoa ls.xyau .saSuaqct Is~nlan~ls 0~~a~~a~aq~ss8~Saq~10~~8q~aas aM.z 23g mog zo!s pasnJ JOJ 838paq qi~MSuo~8'zo~u8a~pu8 apeaM uxolJ OSI~‘SS@ ~~-%&4-83-loJ loId~u18wasaJdaM E ‘8~ 1x1 7.9 r 9’5- =:a pu8 ‘5'sT 0.1~ =Og VodaJ ZOIU8af pU8 apEaM aI!qM ‘fjE'OTE6'S- =!a pU8 ‘8df) 51.0 T EL'gf = Og aI8 samssard JaMoI JE sari@@@ 3!uoseq[n as8IaA8 au wmssald MO1 18 yg 'OJ afqeh aAp8sau 8 ID!pad OqMZOIUEafpU8apEa~q~!MpUlrE~8p3!UOS8J]Inaq~ ql!M luatUaaJ88 u! s! sg~, '(Z~Oc X) 818~ alnssad la@q am JOJ LLS'I = %I PuE =d!.J 65’6Z = Of7 PUE ‘(ZO’O > X) 838p amssaId MO1 aq] JOJ 566’9= :a pU8 8&-J Z1’6f = “a splati slu!od H tq aql 01 sly aJ8nbs lsaa[ aw ‘lu!od Ialuamuadxa sy~ Su~~euyg~ ‘aug lg am&s jsaal Jaql!a UIOlJSUO!$Ewp pJ8pU8lS 18laAaSSn83sw!ode]Epaq)JO aU0 ]eq] ]uaJEddE @p8aI s! 11 'SlU!Odmap aqr q%nOJqJ UM8Jp uaaq aA8q sauy ly aranbs js8aI leaug OMJ, ‘[91] UO!l!SU8ll ~8Jn]XLl)S 8 JO aAQ83!pU! S! qXqM ‘ZO’O= X 30 kI!U!a!A aql U!
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Technical Note coefficient from 0.3142 to 0.5448, the value of B, changes from 9.067 f 0.5371 x 1OroPa to 8.932 f 0.357 x 10’” Pa and a change in Bh from 2.586 f 1.021 to 2.947 f 0.67. As is evident, elimination of these two points which could be identified by inspection in the In H plots cause a significant change in B;. (C) Conclusion
We have demonstrated that In H plots are a simple, powerful tool for detecting anomalous P-V data in high pressure experiments, and for detecting structural and phase transitions in solids subjected to high pressure. Acknowledgement-The authors would like to thank the reviewer for pointing out that the data point identified as spurious for SiO, in Fig. 2 was a typographical error, thus illustrating the utility of the technique.
REFERENCES 1. Rose J. H., Ferrante J. and Smith J. R., Phys. Rev. Lat. 47, 675 (1981); Ferrante J. and Smith J. R., Phys. Rev. B 31, 3427 (1985). 2. Smith J. R., Ferrante J. and Rose J. H., Phys. Rev. B 25, 1419 (1982).
637
3. Ferrante J., Smith J. R. and Rose J. H., Phys. Rev. Lat. 50, 1385 (1983). 4. Rose J. H., Smith J. R. and Ferrante J., Phys. Reu. B 28, 1835 (1983). 5. Vinet P., Ferrante J., Smith J. R. and Rose J. H., J. Phys. C: Solid Stare Phys. 19, 1467 (1986). 6. Vinet P., Ferrante J., Rdse J. H. and Smith J. R., J. aeophvs. Res. 92. 9319 (1987). 7. V&et-P, Rose J. H., Ferrantk J. and Smith J. R., J. Phys.: Condensed Mutfer 1, 1941 (1988). 8. Schlosser H. and Ferrante J., Phys. Rev. B 40, 6405 (1989). 9. Schlosser H., Phys. Rev. B41, 1173 (1990). 10. Jeanloz R. and Rudy A., J. geophys. Res. 92, 11,433 (1987). 11. Sumino Y., Kumazawa M., Nishizawa 0. and Pluschkell W., J. Phys. Eurth 28, 475 (1980). 12. Webb S. L., Jackson I. and McCammon C. A., EOS Trans. AGU. 64, 847 (1983). 13. Webb S. L. and Jackson I., unpublished, private communication quoted in Ref. 8 (1986). 14. Meade C. and Jeanloz R., Phys. Reu. B 35, 236 (1987). 15. Jeanloz R., Nature 332, 207 (1988). 16. Jeanloz R., Geophys. Res. Letf. 8, 1219 (1981). 17. Leger J. M. and Redon A. M., J. less-common Mar. 156, 137 (1989).
1991
0022.3697191 $3.00 + 0.00 Pefopmoa PM plc
TECHNICAL NOTE THE ANALYSIS OF HIGH PRESSURE EXPERIMENTAL DATA HERBERT Scmossmt Physics Dept., Cleveland State University, Cleveland, OH 44115, U.S.A. JOHN FERRANTE National Aeronautics and Space Ad~nistration, Lewis Research Center, Cleveland, OH 44135, U.S.A. (Received 5 Jury 1989; accepted in revised form I4 November 1990)
Abstract-This letter is concerned with the analysis of high pressure experimental data. We demonstrate that ln H plots based on the Vinet et al. universal equation of state (EOS) are a simple sensitive means for identifying anomalous P-V data in high pressure experiments, and for detecting structural and phase transitions in solids subjected to high pressure. Keywords: Equation of state, phase transition, data analysis, glass, manganese oxide, calcium.
1. INTRODUCTION
occurs there is either a discontinuity or a pronounced change in slope of the In H plots, and that the data points above and below the transition may be fit by two straight line segments.
A universal binding energy-distance relationship [l] of the form, E(R) = A.&??*@*), describes the adhesion and cohesion of metals, the chemisorption of gas atom on metal surfaces [2], as well as the binding of certain diatomic molecules [3,4], where E*(u*) is a universal function of the scaled length a* and AE is the total energy at eq~~b~urn. The universal energy relation leads to the Vinet et al. universal EOS [5,6], P(V) = 3(1 -X)
B,[expn(l
-X)1/X2,
2. ANALYSiS OF EXP~~~~
(1)
which accurately represents the experimental data for all classes of solids (metals, covalently bonded, ionic and Van der Waals) in compression, where X I (V/Vo)“3, B, is the isothermal bulk modulus and Y, is the volume at zero pressure, tf = 3[& - 1]/2, and B; = (~~/~~)~=~. For many classes of solids the function, G*(a*) = (X2/3(X - 1)) P(V)/&,, is a universal function of a* [5,6]. Now defining the function H(V) E -E,G*(a*) In H(V)=In&-tn(l
= B,exp[n(l -X)], -X).
(2a) (2b)
Vinet et al. [7] found that in the absence of phase transitions, plots of In H vs (1 - X) for most solids, are to a good approximation straight lines (least square fits in in H vs (1 - X) yield correlation coefficient values, t > 0.999). Recently (81, we showed that this theory also applies to liquid metals and to rare gas liquids [9], and yields results with accuracy comparble to that in solids. In this letter we demonstrate that the universal EOS and In H plots are useful for the detection of anomalous high pressure experimental data, and for the detection of phase or structural transitions in solids subjected to high pressure. We observe that when a phase or structural transition t Supported in part by a NASA-ASEE Summer Faculty Fellowship, and by a Ohio Board of Regents Research Challenge Grant.
DATA
(A) Detection of spurious experimental data Now we demonstrate that the In H plots provide a sensitive means of detecting spurious high pressure experimental data, i.e. falls well &side the stand&d deviation of the data. Consider the In H nlot for MnO in Fia. 1. The raw experimental P and V data-of Jeanloz and R&y [lo] were utilized to obtain the In H vs 1 - x plot in Fig. 1, along with a P-Y plot for the same data. A least square fit to the entire set of In H points yields B, = 181.6 GPa, and B; = 3.682 with a correlation coetEcient r = 0.6752. Inspection of Fig. 1 readily indicates that, with the exception of the first point, all of the data points cluster about a straight line. Now if the first point is eliminated, a least square fit yields B, = 156.94 GPa, and B; = 5.489 with a correlation coefficient r = 0.9238. These values are close to the experimental values reported for ultrasonic measurements [ll-131, namely B, = 155.5 + 3.5 GPa [l l], and Be = 149.2 f 0.5 GPa [12] and B&= 5.5 f 0.1 [13]. Thus, by simple inspection, we may con&de that the first data point is anomalous, as indeed did Jeanloz and Rudy [lo]. In analyzing the experimental data we have not explicitly weighted the data to reflect the lower experimental accuracy at higher pressures [14], since Vinet et al. (Ref.7, Table 6) found that the values of B,, and Bi obtained from curve fitting eqn 2b are relatively insensitive to the number of high pressure points included in the linear least square fits. In addition, we have observed that the &, and B; values obtained from the In H fits agree to within experimental error with the values obtained from nonlinear fits to eqn 1. We have not included experimental error bars in the P-Y data since most of the data papers do not report them. The power of the In H plots is that data points which are well outside the rms deviation of the data can be readily observed by inspection. Thus, they provide a simple method for data analysis during an experiment.
635
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loId~u[ aq$Jo adop u! aSu8qa a[qehlasqo @pea1 aql aIoN &I] zoIu8ar pue apwagqyo sep @luauyadxa aw Su!zgyn zO!S pasrg ‘OJ X1 SA H IIIpallold aAEq aM ‘5:27!d III suoy~suv~~asvqd puv pkn~wws Jo uoymlaa
'19 arenbs 1slla1 :aug pgos t[oI Jaa] oum-_x-
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a[q!slaAay aql 01 anp d[qeurnsaJd s! puo~as aql pue Q-gs u! aSueq3 Iem]3ruls aql 01 spuodsaJIoa ls.xyau .saSuaqct Is~nlan~ls 0~~a~~a~aq~ss8~Saq~10~~8q~aas aM.z 23g mog zo!s pasnJ JOJ 838paq qi~MSuo~8'zo~u8a~pu8 apeaM uxolJ OSI~‘SS@ ~~-%&4-83-loJ loId~u18wasaJdaM E ‘8~ 1x1 7.9 r 9’5- =:a pu8 ‘5'sT 0.1~ =Og VodaJ ZOIU8af pU8 apEaM aI!qM ‘fjE'OTE6'S- =!a pU8 ‘8df) 51.0 T EL'gf = Og aI8 samssard JaMoI JE sari@@@ 3!uoseq[n as8IaA8 au wmssald MO1 18 yg 'OJ afqeh aAp8sau 8 ID!pad OqMZOIUEafpU8apEa~q~!MpUlrE~8p3!UOS8J]Inaq~ ql!M luatUaaJ88 u! s! sg~, '(Z~Oc X) 818~ alnssad la@q am JOJ LLS'I = %I PuE =d!.J 65’6Z = Of7 PUE ‘(ZO’O > X) 838p amssaId MO1 aq] JOJ 566’9= :a pU8 8&-J Z1’6f = “a splati slu!od H tq aql 01 sly aJ8nbs lsaa[ aw ‘lu!od Ialuamuadxa sy~ Su~~euyg~ ‘aug lg am&s jsaal Jaql!a UIOlJSUO!$Ewp pJ8pU8lS 18laAaSSn83sw!ode]Epaq)JO aU0 ]eq] ]uaJEddE @p8aI s! 11 'SlU!Odmap aqr q%nOJqJ UM8Jp uaaq aA8q sauy ly aranbs js8aI leaug OMJ, ‘[91] UO!l!SU8ll ~8Jn]XLl)S 8 JO aAQ83!pU! S! qXqM ‘ZO’O= X 30 kI!U!a!A aql U!
‘qy
am&s
:saJenbs
pgos f[p~gas] ssep3EN-SJ+-E~ :sa@I8g--_X - 1 SA H IIT 'f %A
ls8aI :sauq
fzo!s
pasnj
‘H UI 01 $IJaJ8nbs Isea uxog paA!.tap sanpA yg $0~ Su!sn I uba uxog paApap ahIn :auy pgos f[o~ .Jax] OU~---/~ SA d '~1 %d
X-l
ocw 3 OC’tL =
OSfZ
9f9
Technical Note coefficient from 0.3142 to 0.5448, the value of B, changes from 9.067 f 0.5371 x 1OroPa to 8.932 f 0.357 x 10’” Pa and a change in Bh from 2.586 f 1.021 to 2.947 f 0.67. As is evident, elimination of these two points which could be identified by inspection in the In H plots cause a significant change in B;. (C) Conclusion
We have demonstrated that In H plots are a simple, powerful tool for detecting anomalous P-V data in high pressure experiments, and for detecting structural and phase transitions in solids subjected to high pressure. Acknowledgement-The authors would like to thank the reviewer for pointing out that the data point identified as spurious for SiO, in Fig. 2 was a typographical error, thus illustrating the utility of the technique.
REFERENCES 1. Rose J. H., Ferrante J. and Smith J. R., Phys. Rev. Lat. 47, 675 (1981); Ferrante J. and Smith J. R., Phys. Rev. B 31, 3427 (1985). 2. Smith J. R., Ferrante J. and Rose J. H., Phys. Rev. B 25, 1419 (1982).
637
3. Ferrante J., Smith J. R. and Rose J. H., Phys. Rev. Lat. 50, 1385 (1983). 4. Rose J. H., Smith J. R. and Ferrante J., Phys. Reu. B 28, 1835 (1983). 5. Vinet P., Ferrante J., Smith J. R. and Rose J. H., J. Phys. C: Solid Stare Phys. 19, 1467 (1986). 6. Vinet P., Ferrante J., Rdse J. H. and Smith J. R., J. aeophvs. Res. 92. 9319 (1987). 7. V&et-P, Rose J. H., Ferrantk J. and Smith J. R., J. Phys.: Condensed Mutfer 1, 1941 (1988). 8. Schlosser H. and Ferrante J., Phys. Rev. B 40, 6405 (1989). 9. Schlosser H., Phys. Rev. B41, 1173 (1990). 10. Jeanloz R. and Rudy A., J. geophys. Res. 92, 11,433 (1987). 11. Sumino Y., Kumazawa M., Nishizawa 0. and Pluschkell W., J. Phys. Eurth 28, 475 (1980). 12. Webb S. L., Jackson I. and McCammon C. A., EOS Trans. AGU. 64, 847 (1983). 13. Webb S. L. and Jackson I., unpublished, private communication quoted in Ref. 8 (1986). 14. Meade C. and Jeanloz R., Phys. Reu. B 35, 236 (1987). 15. Jeanloz R., Nature 332, 207 (1988). 16. Jeanloz R., Geophys. Res. Letf. 8, 1219 (1981). 17. Leger J. M. and Redon A. M., J. less-common Mar. 156, 137 (1989).