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On the melting of alkali halides at high pressures
1074
TECHNICAL
Table 1. Parameters
C,, 2,
in analytical for triv&t
n
Cl
4 6 8 IO 12 14
1251~81 1676.72 2188.66 2797.67 351346 4345.79
ca 194,289 278.194 385.383 520-930 690.919 902.486
representations
= s C, emzJ of 4f radial functions I-1 ions with n 4f eIectrons
la&&de
cs 27.6555 37.3965 49.2579 63.5658 80.9179 102.2024
P&r)
c4
Zl
Z!d
Za
Z
0.798327 l-23048 I.89821 2.84272 4.07972 s-59927
1 l-183 12.095 13m7 13.919 14.831 15.743
6.314 6.800 7.286 7.772 8.258 8.744
4.209 4.533 4.857 5.181 5*505 S-829
2.307 2.489 2.671 2.853 3.035 3.217
not considered by FW. A second measure of the accuracy of the present method is the small discrepancy between expectation values calculated by FW((Y-~), (ta), (r4), (Y*)) and those obtained from our renormalized wave functions employing the fitted C, and 2,. The maximum relative deviations from the FW values for (rq3), (r2), (r4), (r6) are found to be, respectively, 0.28 (for ti = 7), 3*0,2*1 and 3.2 per cent (all for lt = 5). In this note, a set of near-Hartree-Fock 4f orbitals for all of the trivalent lanthanide ions was obtained by a numerical method. These orbitals yield charge distributions that correspond very closely to those obtained by Freeman and Watson for Ce+3, Pr+3, Nd+“, Sm+3, Gd+3, DY+~, Er+3 and Yb i3. Even for operators that magnify deficiencies in charge distributions, such as r6, the expectation values differ by only a few percent or less from those given by the FW Hartree-Fock functions. This fact, as well as the smooth variation of orbital parameters and expectation values along the lanthanide series, indicates that the 4f orbitals listed in Table 1 are close to restricted HartreeFock functions and certainly more accurate than those obtained from the prescriptions of Slater or Burns for the other members of the series: Pm+3, Eu+~, Tb+3, Hof3, Tm+3 and Lu+~. General Telephone & Electronics Laboratories, Inc. Bayside, New York
NOTES
0. J. SOVERS
References 1. See, for example, KhumsANN W., Quantum Chemistry, p. 332. Academic Press, New York (1957). 2. BURNSG., J. Chem. Phys. 41, 1521 (1964). 3. F-m A. J. and WATSON R. E., Phys. Rev. 127, 2058 (1962).
4. WYBOURNZB. G., Spectroscopic Properties of Rare Eatths. __ TIP. 117, 165. Interscience, New York (1965).
5. RAYCHAUDHUR A. K. and RAY D. K., Proc. Rays. Sot. 86,891 (1965). 6. See, for -&m&e, ~~IIJMACHERD. P. and HOLLINGSWORTH~C. A., J. Phys. Gem. Solids 27, 749 (1966).
J. Phys. Chem. Solids
Vol. 28, pp. 1074-1077.
On the melting of alkali halides at high pressures (Received 17 October 1966)
THE ALKALI halides have long been model solids for many experimental and theoretical studies. The discovery of polymorphic transitions in many of them from a study of the melting at high pressures has shown that the understanding of the thermal behaviour of these solids is far from complete. In the past it has been usual to correlate the melting temperature T,,, with the melting pressure P through Simon’s equation (P-Po)/a
= (T,JT,,,“)c-l
involving
adjustable parameters.(l) Recently has suggested a linear correlation of T,,, with the volume compression lAv/vol of the solid. A relation of this type is easilv derived(3*4) from Lindemann equatioi_T,,, = C.&I. 82v2’3 and Gtineisen relation (d8/dv) = -r(Q). It follows that tiNNEDY’2’
--= dT, dv
(1)
TECHNICAL
If A&-, is small and y strictly a constant
T,
3
[1+2(y-$3]. (24
Tmo
If there were to be a phase transition in the solid, equation (1) can be integrated from the melting of the new phase
Tm= T~c~~+2(~‘-~)I(v--er,)~vcllPI Here T,,,c, v, and y’ refer to the new phase. Equation (2) gives a linear variation of T,,, with [ Ar.&,l. It contains no arbitrary constant and it becomes possible to relate melting phenomena to other solid state studies, for the Griineisen
0
Ok.4 VOLUME
NOTES
1075
constant y = v@/k, * C, can be calculated independently from the measured coefficients of cubical expansion /3, compressibility and specific heat. The relationship is plotted in Fig. 1 for a number of alkali halides and the linearity of fit is good. However the values of y calculated from equation (2) are systematically higher than the thermal values as shown in Table 1. It is possible that the compressions lAv/vol near the melting point must be used in making a quantitative estimate of y(oal,.). In absence of such data, the room temperature compressions may be approximately corrected in the following manner. For T 9 8, y = Vj9/kpC, is essentially constant and the variation of V/C’” is
0.08
0.12
COMPRESSION
Fm. 1. Melting curves for alkali halides. The numbers near the experimental points denote the pressure in kb.
1076
TECHNICAL
independent check is an advantage as compared to the other methods of correlating the melting data of alkali halides based on Simon’s equation(l) or on the principle of corresponding states.(a0) Ytcorr.,+YtTn.rmti)~Yrahookt:Equation (2) of course, carries the limitations of rrve, the simple equations of Lindemarm and Griineise~. Firstly, at large compressions y is no longer strictly 1.78 1.76s I.72 a constant and correspondingly a polynomial in l-708 l-74 JAW/W,/gives a better fit than the linear approxi1.45s 1.81 mation. Secondly some metals melt at lower 190 temperatures on the application of high pressures I.6611 1.60 which by equation (2) implies y < 4, in serious 1.58rs 1.58 disagreement with the thermal values of y.
Table 1. Cbqarison of G%Geken consmeWgkwwith~undshoekwaveydues
Yw..,* LiCl NaF NaCl NaBr NaI KC&I KC&II KBr-I KBr-II KI-I KI-II RbCI-II Ag AU Cd Gd Nd Pb Pr Tb Zn
4.~S.6 2.998.’ 2.9os.r 2446.7
2.68ev7 2.826.10
2.767.10 1 .gp10 2.537.10 *.gg6*10
1.56’s
I.72
2.191’ 3.3714 2.2216 0.7318 0.601* 2*3Ss 0.34ls 0.9918 2.27ls
244 3.06 2.28 0.52 0.74 2.74 0.49 0.83 2.03
front
Note added in proof-Recently
2.647.10 2.97O.7 3.007.13 4.387.13 2.797.15 1.~7.17 (p(j87.17 2.737.18
0.357.18 1.427.17 2.647,‘s
NOTES
2.29 2.22 2.33
2.38
2.24
* Superscript numbers indicate references for compression and melting data. t Corrected by using fiR.T/&.P ratio. Superscripts indicate references for thermal expansion data. $ Taken from Ref. 21.
less than a few per cent. So jAr@e] is nearly pro-
portional to /3. Using p values near room temperature and near T,,, the correction can be effected. Suitable j3 values are not available for some materials while no estimate of y values is available for the high pressure phases. But where a comparison with the corrected values is possible in Table 1, the agreement is substantially improved. Incidentally such a rough correction for compression data improves the agreement among the y values in a number of other solids also. A few examples are included in Table 1. In the case of metals, Griineisen constants may be deduced from shock wave experiments also and these values are included in Table 1. In conclusion equation (2) appears to account well for the observed melting phenomenon of solids. The values of y calculated from it agree with those deduced from other phenomena.
This
it has been found that equation (1) gives sn excellent account of the melting of a number of solidified gases also. (J. Indian Inst. Sci., 1967.in press).
thank Prof. R. S. KRISHNANfor continued encouragement and support.
Acknowledgement-We
Department of Physics Indian Institute of Science Bangalore 12, India
S. N. VAIDYA E. S. R. GOPAL
References 1. BABBS. E., Rev. Mod. Pkys. 35, 400 (1963). 2. KRAUT E. A. and KENNEDYG. C., Phys. I&v. L&t. 16,608 (1966); Phys. Rev. 151, 668 (1966). 3. VAIDYAS. N. and GOPALE. S. R., Phys. Rev. Lett. 17,635 (1966). 4. GILVARRY 1. Phvs. Rew. Lett. 16. lb89 (1966); LIBBY W. F., >hys. Rev. Lett. i7, 423 (1966): 5. S~ATHRJ. C., Proc. Am. Acad. Arts Sci. 61, 135 (1926). 6. CLARKS. P., J. Chem.Phys. 31,1526 (1959). 7. BRIDGMAN P. W., Proc. Am. Acad. Arts Sci. 66,255 (1931); 74, 21, 435 (1940); 76, 1, 8, 187 (1945); 77, 187 (1949); 83, 1, 149 (1954). V. T., Acta CrystaZZogr. 14,794 (1961). 8. DHSHFANDE 9. ENCK F. D. and DOMM~L J. G., J. at&. __ Phys. _ 36, 839 (1965). 10. PISTOFUUS C. W. F. T., J. Phys. Chem. Solids 26, 1543 (1966). 11. ENCK F. D., ENGLE D. G. and MARKS K. I., J. appl. Phys. 33,207O (1962). 12. PATHAKP. S. and PANDYAN. V., Indian J. Phys. 34,416 (1960). 13. Comr~ L. H., KLEMENTW. and KENNEI)YG. C.,
Phys. Rev. i45, 592(1966).
14. LBKSINAI. E. and NOV~KOVAS. I., Sowiet Phys. Solid St. 5,798 (1963). E. G. and SHAKHOS15. BUTUZOVV. P., PO~WATOV~KII KOIG. P., Dokl. Akad. Nauk SSR 109,519(1956). 16. Upp~~aslw F. R., PkiZ. Mag. 10, 648 (1930). 17. JAYAWUAN A., Pkys. Rev. 139,A690 (1965).
TECHNICAL 18. GOLDSMITHA., WATBRMANT. E. and HIRCHOORN H. J., Handbook of Tkermo#ky&al Pqwtiee of Metals, Vol. 1, MacMillan, New York (1961). 19. BUTUZOV V. P. and GONI~BBROM. G., Do&l. A&ad. Nauk, SSR. 91,1083 (1963). 20. Owws B. B., J. Chnn. Phys. 44,3144 (1966). 21. KAIXIUVA C. M. and SAXENAS. C., J. Chn P!zys. 44, 986 (1963) for alkali halides. GSCHEIDN~R K. A., Solid St. Phys. 16,275 (1944) ; for elements.
J. Phys. Ckem. Solids Electronic
NOTES
1077
relationship a = PO, -liz for a point defect in a single predominant state of ionization, these results are consistent with a defect structure consisting of doubly ionized oxygen vacancies, x=6, andthismodelwillbeassumedinthe determination of electron mobility. Figure 1 is a
Vol. 28, pp. 1077-1079.
mobility in rutile (TiO,) temperatures
at high -’
(Received7 October 1966; inreviredform 2 December 1966)
THE ELECTRON mobility in rutile may be calculated by combining electrical conductivity and thermogravimetric data taken over the same range of temperature and partial pressure of oxygen. The concentration of charge carriers, n, is obtained from the thermogravimetric data by assuming a model for the defect structure and combined with the conductivity data in the relation u=neu, where p is the electron mobility. The objective of thii note is to report the results of such calculations in order to establish (1) the magnitude of the electron mobility at elevated temperatures and (2) the temperature dependence of the mobility. Nonstoichiometric rutile (TiO,_.J may be classified as a metal excess, n-type semiconductor on the basis of experimental observations.(l*a) This metal excess may arise from the presence of a defect structure involving either anion vacancies or cation interstitials. Recent investigations(3-5) have favored a defect model consisting of quasifree electrons and titanium interstitials in one or more states of ionization. Current studies in this laboratory@) are consistent with a model involving quasi-free electrons and both triply and quadruply ionized titanium inter&&. The first determination of mobility is based on the weight change data of KOFSTAD”)and the c direction electrical conductivity data of BLUMENTHAL et uZ.@)which overlap at an oxyp partial pressure of lO_lsatm in the range 135&15WK. From Fig. 2 of Kofstad, the plot of log x vs. log PO, yields straight-line relations with a slope of - 116 in this region. Since the pressure dependence of conductivity obeys the
P 1
1350
1400 TEMPEP.Am
lb50
15m
VW
FIG. 1. Mobility vs. temperature at Pot= lo-“atm obtained by combining the data of KOFSTAD(~)and BLUMENTHAL et oZ.(e’
plot of the mobility as a function of temperature at P =lO-la obtained by combining the data of K:fstad and Blumenthal et al. It can be seen that the order of magnitude of the mobility is 10-l cma/V-set and it is independent of temperature. The second determination is based on the weight change data of MOSERet uZ.(*)and the conductivity data used in the above calculation which overlap at PO, =lO-la atm in the range 1373-1573%. From Fig. 1 of Moser et al., the plot of log x vs. log PO, yields straight limes at 1373”, 1473”, and 1573°K. Since the values of x calculated from the slopes are 4.5, 5.1 and 4.2, respectively, this data cannot be interpreted in terms of a defect model consisting of only triply ionized titanium interstitials, x = 4, or only quadruply ionized titanium interstitials, x = 5. Therefore, both modela were assumed individually and the mobilities determined. Figure 2 is a plot of the electron mobility as a function of temperature assuming a predominance of quadruply ionized titanium interstitials, x = 5. It can be seen that the order of magnitude of the mobility is 10-l cn?/v_sec in agreement with the Grst determination and the mobiity decreases with increasing temperature. For the other model,
TECHNICAL
Table 1. Parameters
C,, 2,
in analytical for triv&t
n
Cl
4 6 8 IO 12 14
1251~81 1676.72 2188.66 2797.67 351346 4345.79
ca 194,289 278.194 385.383 520-930 690.919 902.486
representations
= s C, emzJ of 4f radial functions I-1 ions with n 4f eIectrons
la&&de
cs 27.6555 37.3965 49.2579 63.5658 80.9179 102.2024
P&r)
c4
Zl
Z!d
Za
Z
0.798327 l-23048 I.89821 2.84272 4.07972 s-59927
1 l-183 12.095 13m7 13.919 14.831 15.743
6.314 6.800 7.286 7.772 8.258 8.744
4.209 4.533 4.857 5.181 5*505 S-829
2.307 2.489 2.671 2.853 3.035 3.217
not considered by FW. A second measure of the accuracy of the present method is the small discrepancy between expectation values calculated by FW((Y-~), (ta), (r4), (Y*)) and those obtained from our renormalized wave functions employing the fitted C, and 2,. The maximum relative deviations from the FW values for (rq3), (r2), (r4), (r6) are found to be, respectively, 0.28 (for ti = 7), 3*0,2*1 and 3.2 per cent (all for lt = 5). In this note, a set of near-Hartree-Fock 4f orbitals for all of the trivalent lanthanide ions was obtained by a numerical method. These orbitals yield charge distributions that correspond very closely to those obtained by Freeman and Watson for Ce+3, Pr+3, Nd+“, Sm+3, Gd+3, DY+~, Er+3 and Yb i3. Even for operators that magnify deficiencies in charge distributions, such as r6, the expectation values differ by only a few percent or less from those given by the FW Hartree-Fock functions. This fact, as well as the smooth variation of orbital parameters and expectation values along the lanthanide series, indicates that the 4f orbitals listed in Table 1 are close to restricted HartreeFock functions and certainly more accurate than those obtained from the prescriptions of Slater or Burns for the other members of the series: Pm+3, Eu+~, Tb+3, Hof3, Tm+3 and Lu+~. General Telephone & Electronics Laboratories, Inc. Bayside, New York
NOTES
0. J. SOVERS
References 1. See, for example, KhumsANN W., Quantum Chemistry, p. 332. Academic Press, New York (1957). 2. BURNSG., J. Chem. Phys. 41, 1521 (1964). 3. F-m A. J. and WATSON R. E., Phys. Rev. 127, 2058 (1962).
4. WYBOURNZB. G., Spectroscopic Properties of Rare Eatths. __ TIP. 117, 165. Interscience, New York (1965).
5. RAYCHAUDHUR A. K. and RAY D. K., Proc. Rays. Sot. 86,891 (1965). 6. See, for -&m&e, ~~IIJMACHERD. P. and HOLLINGSWORTH~C. A., J. Phys. Gem. Solids 27, 749 (1966).
J. Phys. Chem. Solids
Vol. 28, pp. 1074-1077.
On the melting of alkali halides at high pressures (Received 17 October 1966)
THE ALKALI halides have long been model solids for many experimental and theoretical studies. The discovery of polymorphic transitions in many of them from a study of the melting at high pressures has shown that the understanding of the thermal behaviour of these solids is far from complete. In the past it has been usual to correlate the melting temperature T,,, with the melting pressure P through Simon’s equation (P-Po)/a
= (T,JT,,,“)c-l
involving
adjustable parameters.(l) Recently has suggested a linear correlation of T,,, with the volume compression lAv/vol of the solid. A relation of this type is easilv derived(3*4) from Lindemann equatioi_T,,, = C.&I. 82v2’3 and Gtineisen relation (d8/dv) = -r(Q). It follows that tiNNEDY’2’
--= dT, dv
(1)
TECHNICAL
If A&-, is small and y strictly a constant
T,
3
[1+2(y-$3]. (24
Tmo
If there were to be a phase transition in the solid, equation (1) can be integrated from the melting of the new phase
Tm= T~c~~+2(~‘-~)I(v--er,)~vcllPI Here T,,,c, v, and y’ refer to the new phase. Equation (2) gives a linear variation of T,,, with [ Ar.&,l. It contains no arbitrary constant and it becomes possible to relate melting phenomena to other solid state studies, for the Griineisen
0
Ok.4 VOLUME
NOTES
1075
constant y = v@/k, * C, can be calculated independently from the measured coefficients of cubical expansion /3, compressibility and specific heat. The relationship is plotted in Fig. 1 for a number of alkali halides and the linearity of fit is good. However the values of y calculated from equation (2) are systematically higher than the thermal values as shown in Table 1. It is possible that the compressions lAv/vol near the melting point must be used in making a quantitative estimate of y(oal,.). In absence of such data, the room temperature compressions may be approximately corrected in the following manner. For T 9 8, y = Vj9/kpC, is essentially constant and the variation of V/C’” is
0.08
0.12
COMPRESSION
Fm. 1. Melting curves for alkali halides. The numbers near the experimental points denote the pressure in kb.
1076
TECHNICAL
independent check is an advantage as compared to the other methods of correlating the melting data of alkali halides based on Simon’s equation(l) or on the principle of corresponding states.(a0) Ytcorr.,+YtTn.rmti)~Yrahookt:Equation (2) of course, carries the limitations of rrve, the simple equations of Lindemarm and Griineise~. Firstly, at large compressions y is no longer strictly 1.78 1.76s I.72 a constant and correspondingly a polynomial in l-708 l-74 JAW/W,/gives a better fit than the linear approxi1.45s 1.81 mation. Secondly some metals melt at lower 190 temperatures on the application of high pressures I.6611 1.60 which by equation (2) implies y < 4, in serious 1.58rs 1.58 disagreement with the thermal values of y.
Table 1. Cbqarison of G%Geken consmeWgkwwith~undshoekwaveydues
Yw..,* LiCl NaF NaCl NaBr NaI KC&I KC&II KBr-I KBr-II KI-I KI-II RbCI-II Ag AU Cd Gd Nd Pb Pr Tb Zn
4.~S.6 2.998.’ 2.9os.r 2446.7
2.68ev7 2.826.10
2.767.10 1 .gp10 2.537.10 *.gg6*10
1.56’s
I.72
2.191’ 3.3714 2.2216 0.7318 0.601* 2*3Ss 0.34ls 0.9918 2.27ls
244 3.06 2.28 0.52 0.74 2.74 0.49 0.83 2.03
front
Note added in proof-Recently
2.647.10 2.97O.7 3.007.13 4.387.13 2.797.15 1.~7.17 (p(j87.17 2.737.18
0.357.18 1.427.17 2.647,‘s
NOTES
2.29 2.22 2.33
2.38
2.24
* Superscript numbers indicate references for compression and melting data. t Corrected by using fiR.T/&.P ratio. Superscripts indicate references for thermal expansion data. $ Taken from Ref. 21.
less than a few per cent. So jAr@e] is nearly pro-
portional to /3. Using p values near room temperature and near T,,, the correction can be effected. Suitable j3 values are not available for some materials while no estimate of y values is available for the high pressure phases. But where a comparison with the corrected values is possible in Table 1, the agreement is substantially improved. Incidentally such a rough correction for compression data improves the agreement among the y values in a number of other solids also. A few examples are included in Table 1. In the case of metals, Griineisen constants may be deduced from shock wave experiments also and these values are included in Table 1. In conclusion equation (2) appears to account well for the observed melting phenomenon of solids. The values of y calculated from it agree with those deduced from other phenomena.
This
it has been found that equation (1) gives sn excellent account of the melting of a number of solidified gases also. (J. Indian Inst. Sci., 1967.in press).
thank Prof. R. S. KRISHNANfor continued encouragement and support.
Acknowledgement-We
Department of Physics Indian Institute of Science Bangalore 12, India
S. N. VAIDYA E. S. R. GOPAL
References 1. BABBS. E., Rev. Mod. Pkys. 35, 400 (1963). 2. KRAUT E. A. and KENNEDYG. C., Phys. I&v. L&t. 16,608 (1966); Phys. Rev. 151, 668 (1966). 3. VAIDYAS. N. and GOPALE. S. R., Phys. Rev. Lett. 17,635 (1966). 4. GILVARRY 1. Phvs. Rew. Lett. 16. lb89 (1966); LIBBY W. F., >hys. Rev. Lett. i7, 423 (1966): 5. S~ATHRJ. C., Proc. Am. Acad. Arts Sci. 61, 135 (1926). 6. CLARKS. P., J. Chem.Phys. 31,1526 (1959). 7. BRIDGMAN P. W., Proc. Am. Acad. Arts Sci. 66,255 (1931); 74, 21, 435 (1940); 76, 1, 8, 187 (1945); 77, 187 (1949); 83, 1, 149 (1954). V. T., Acta CrystaZZogr. 14,794 (1961). 8. DHSHFANDE 9. ENCK F. D. and DOMM~L J. G., J. at&. __ Phys. _ 36, 839 (1965). 10. PISTOFUUS C. W. F. T., J. Phys. Chem. Solids 26, 1543 (1966). 11. ENCK F. D., ENGLE D. G. and MARKS K. I., J. appl. Phys. 33,207O (1962). 12. PATHAKP. S. and PANDYAN. V., Indian J. Phys. 34,416 (1960). 13. Comr~ L. H., KLEMENTW. and KENNEI)YG. C.,
Phys. Rev. i45, 592(1966).
14. LBKSINAI. E. and NOV~KOVAS. I., Sowiet Phys. Solid St. 5,798 (1963). E. G. and SHAKHOS15. BUTUZOVV. P., PO~WATOV~KII KOIG. P., Dokl. Akad. Nauk SSR 109,519(1956). 16. Upp~~aslw F. R., PkiZ. Mag. 10, 648 (1930). 17. JAYAWUAN A., Pkys. Rev. 139,A690 (1965).
TECHNICAL 18. GOLDSMITHA., WATBRMANT. E. and HIRCHOORN H. J., Handbook of Tkermo#ky&al Pqwtiee of Metals, Vol. 1, MacMillan, New York (1961). 19. BUTUZOV V. P. and GONI~BBROM. G., Do&l. A&ad. Nauk, SSR. 91,1083 (1963). 20. Owws B. B., J. Chnn. Phys. 44,3144 (1966). 21. KAIXIUVA C. M. and SAXENAS. C., J. Chn P!zys. 44, 986 (1963) for alkali halides. GSCHEIDN~R K. A., Solid St. Phys. 16,275 (1944) ; for elements.
J. Phys. Ckem. Solids Electronic
NOTES
1077
relationship a = PO, -liz for a point defect in a single predominant state of ionization, these results are consistent with a defect structure consisting of doubly ionized oxygen vacancies, x=6, andthismodelwillbeassumedinthe determination of electron mobility. Figure 1 is a
Vol. 28, pp. 1077-1079.
mobility in rutile (TiO,) temperatures
at high -’
(Received7 October 1966; inreviredform 2 December 1966)
THE ELECTRON mobility in rutile may be calculated by combining electrical conductivity and thermogravimetric data taken over the same range of temperature and partial pressure of oxygen. The concentration of charge carriers, n, is obtained from the thermogravimetric data by assuming a model for the defect structure and combined with the conductivity data in the relation u=neu, where p is the electron mobility. The objective of thii note is to report the results of such calculations in order to establish (1) the magnitude of the electron mobility at elevated temperatures and (2) the temperature dependence of the mobility. Nonstoichiometric rutile (TiO,_.J may be classified as a metal excess, n-type semiconductor on the basis of experimental observations.(l*a) This metal excess may arise from the presence of a defect structure involving either anion vacancies or cation interstitials. Recent investigations(3-5) have favored a defect model consisting of quasifree electrons and titanium interstitials in one or more states of ionization. Current studies in this laboratory@) are consistent with a model involving quasi-free electrons and both triply and quadruply ionized titanium inter&&. The first determination of mobility is based on the weight change data of KOFSTAD”)and the c direction electrical conductivity data of BLUMENTHAL et uZ.@)which overlap at an oxyp partial pressure of lO_lsatm in the range 135&15WK. From Fig. 2 of Kofstad, the plot of log x vs. log PO, yields straight-line relations with a slope of - 116 in this region. Since the pressure dependence of conductivity obeys the
P 1
1350
1400 TEMPEP.Am
lb50
15m
VW
FIG. 1. Mobility vs. temperature at Pot= lo-“atm obtained by combining the data of KOFSTAD(~)and BLUMENTHAL et oZ.(e’
plot of the mobility as a function of temperature at P =lO-la obtained by combining the data of K:fstad and Blumenthal et al. It can be seen that the order of magnitude of the mobility is 10-l cma/V-set and it is independent of temperature. The second determination is based on the weight change data of MOSERet uZ.(*)and the conductivity data used in the above calculation which overlap at PO, =lO-la atm in the range 1373-1573%. From Fig. 1 of Moser et al., the plot of log x vs. log PO, yields straight limes at 1373”, 1473”, and 1573°K. Since the values of x calculated from the slopes are 4.5, 5.1 and 4.2, respectively, this data cannot be interpreted in terms of a defect model consisting of only triply ionized titanium interstitials, x = 4, or only quadruply ionized titanium interstitials, x = 5. Therefore, both modela were assumed individually and the mobilities determined. Figure 2 is a plot of the electron mobility as a function of temperature assuming a predominance of quadruply ionized titanium interstitials, x = 5. It can be seen that the order of magnitude of the mobility is 10-l cn?/v_sec in agreement with the Grst determination and the mobiity decreases with increasing temperature. For the other model,