Mantle values of thermal conductivity and the geotherm from phonon lifetimes

Science

Volumn 283, Issue 5408, 1999, Pages 1699-1706

  Hofmeister, A M ^a  

^a WASHINGTON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FERROUS ION; SILICATE;

GEOTHERMAL GRADIENT; LITHOSPHERE; MANTLE; THERMAL CONDUCTIVITY;

ARTICLE; GEOLOGY; HEAT TRANSFER; INFRARED RADIATION; PHYSICS; PRESSURE; PRIORITY JOURNAL; TEMPERATURE; THERMAL CONDUCTIVITY;

EID: 0033548504     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.283.5408.1699     Document Type: Article

References (124)

1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1959).
2. R. G. Ross, P. Andersson, B. Sundqvist, G. Bäckström, Rep. Prog. Phys. 47, 1347 (1984).
3. H. Kanimori, N. Fujii, H. Mizutani, J. Geophys. Res. 73, 595 (1968).
4. Y. Koyabashi, J. Phys. Earth 22, 359 (1974).
5. G. Bäckström, in Proceedings of the 7th Symposium on Thermophysical Properties, A. Cezairliyan, Ed. (American Society of Mechanical Engineers, New York, 1977), pp. 169-180; U. Brydsten, D. Gerlich, G. Bäckström, J. Phys. C Solid State Phys. 16, 143 (1983); O. Alm and G. Backström, J. Phys. Chem. Solids 35, 421 (1974).
6. G. Bäckström, in Proceedings of the 7th Symposium on Thermophysical Properties, A. Cezairliyan, Ed. (American Society of Mechanical Engineers, New York, 1977), pp. 169-180; U. Brydsten, D. Gerlich, G. Bäckström, J. Phys. C Solid State Phys. 16, 143 (1983); O. Alm and G. Backström, J. Phys. Chem. Solids 35, 421 (1974).
7. G. Bäckström, in Proceedings of the 7th Symposium on Thermophysical Properties, A. Cezairliyan, Ed. (American Society of Mechanical Engineers, New York, 1977), pp. 169-180; U. Brydsten, D. Gerlich, G. Bäckström, J. Phys. C Solid State Phys. 16, 143 (1983); O. Alm and G. Backström, J. Phys. Chem. Solids 35, 421 (1974).
8. M. Chai, J. M. Brown, L. J. Slutsky, Phys. Chem. Miner. 23, 470 (1996).
9. J. Zaug, E. Abramson, J. M. Brown, L. J. Slutsky, in High-Pressure Research: Application to Earth and Planetary Sciences, Y. Syono and M. H. Manghnani, Eds. (Terra, Tokyo, 1992), pp. 157-166.
10. E. H. Abramson, L. J. Slutsky, J. M. Brown, J. Chem. Phys. 104, 5424 (1996).
11. M. D. Collins, J. M. Brown, E. H. Abranson, L. J. Slutsky, Eos 79, F753 (1997).
12. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).
13. P. Debye, Vortrage über die kinetische Theorie der Materie und der Electrizität (Teuber, Berlin, 1914), pp. 19-60.
14. ij as defined is computed per mole, reporting κ in units of watts per meter-kelvin requires multiplication of Eq. 2 by the density p and division by the molecular weight of the compound.
15. For example, J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Clarendon, Oxford, 1962); G. Liebfried and E. Schlömann, Nachr. Ges. Wiss. Goettingen Math. Phys. K1, 71 (1954); M. Roufosse and P. Klemens, Phys. Rev. B 7, 5379 (1973); P. Sindzingre and M. J. Gillan J. Phys. Cond. Matter 2, 7033 (1990).
16. For example, J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Clarendon, Oxford, 1962); G. Liebfried and E. Schlömann, Nachr. Ges. Wiss. Goettingen Math. Phys. K1, 71 (1954); M. Roufosse and P. Klemens, Phys. Rev. B 7, 5379 (1973); P. Sindzingre and M. J. Gillan J. Phys. Cond. Matter 2, 7033 (1990).
17. For example, J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Clarendon, Oxford, 1962); G. Liebfried and E. Schlömann, Nachr. Ges. Wiss. Goettingen Math. Phys. K1, 71 (1954); M. Roufosse and P. Klemens, Phys. Rev. B 7, 5379 (1973); P. Sindzingre and M. J. Gillan J. Phys. Cond. Matter 2, 7033 (1990).
18. For example, J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Clarendon, Oxford, 1962); G. Liebfried and E. Schlömann, Nachr. Ges. Wiss. Goettingen Math. Phys. K1, 71 (1954); M. Roufosse and P. Klemens, Phys. Rev. B 7, 5379 (1973); P. Sindzingre and M. J. Gillan J. Phys. Cond. Matter 2, 7033 (1990).
19. For example, C. L. Julian, Phys. Rev. 137, A128 (1965); P. G. Klemens, Solid State Phys. 7, 1 (1958); M. Roufosse and R. Jeanloz, J. Geophys. Res. 88, 7399 (1983); G. K. White, High Temp. High Pressures 21, 233 (1989).
20. For example, C. L. Julian, Phys. Rev. 137, A128 (1965); P. G. Klemens, Solid State Phys. 7, 1 (1958); M. Roufosse and R. Jeanloz, J. Geophys. Res. 88, 7399 (1983); G. K. White, High Temp. High Pressures 21, 233 (1989).
21. For example, C. L. Julian, Phys. Rev. 137, A128 (1965); P. G. Klemens, Solid State Phys. 7, 1 (1958); M. Roufosse and R. Jeanloz, J. Geophys. Res. 88, 7399 (1983); G. K. White, High Temp. High Pressures 21, 233 (1989).
22. For example, C. L. Julian, Phys. Rev. 137, A128 (1965); P. G. Klemens, Solid State Phys. 7, 1 (1958); M. Roufosse and R. Jeanloz, J. Geophys. Res. 88, 7399 (1983); G. K. White, High Temp. High Pressures 21, 233 (1989).
23. S. W. Kieffer, J. Geophys. Res. 81, 3025 (1976).
24. D. G. Cahill, S. K. Watson, R. O. Pohl, Phys. Rev. B 46, 6131 (1992).
25. J. M. Brown, Geophys. Res. Lett. 13, 1509 (1986).
26. J. F. Schatz and G. Simmons, J. Geophys. Res. 77, 6966 (1972).
27. The linear dependence derived in (18) for κ(T) results from the form assumed for the lattice contribution. Further, the laser used to heat the sample has a frequency similar to the Si-O stretching modes of the silicates. The presence of an undesirable resonance is suggested by the different behavior seen for Fo and enstatite than for corundum and periclase, in that the latter two materials are not directly stimulated because Al-O and Mg-O fundamentals lie below the laser frequency.
28. S. P. Clark Jr., Trans. Am. Geophys. Union 38, 931 (1957).
29. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992), pp. 229-230, 374-382, 385-386, and 424-426; R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), pp. 468ff and 487.
30. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992), pp. 229-230, 374-382, 385-386, and 424-426; R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972), pp. 468ff and 487.
31. T. Goto, T. J. Ahrens, G. R. Rossman, Y. Syono, Phys. Earth Planet. Inter. 22, 277 (1980).
32. T. J. Shankland, U. Nitsan, A. G. Duba, J. Geophys. Res. 84, 1603 (1979).
33. T. Katsura, Geophys. J. Int. 122, 67 (1995).
34. This method is commonly used to extract peak parameters from solids [for example, W. G. Spitzer, R. C. Miller, D. A. Kleinman, L. E. Howarth, Phys. Rev. 126, 1710 (1962); A. M. Hofmeister and A. Chopelas, Phys. Chem. Miner. 17, 503 (1991)].
35. This method is commonly used to extract peak parameters from solids [for example, W. G. Spitzer, R. C. Miller, D. A. Kleinman, L. E. Howarth, Phys. Rev. 126, 1710 (1962); A. M. Hofmeister and A. Chopelas, Phys. Chem. Miner. 17, 503 (1991)].
36. i, combine equations 13-19 and 13-27, and compare to equation 9-13.
37. F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972); A. M. Hofmeister, in A Practical Guide to Infrared Microspectroscopy, H. J. Humecki, Ed. (Dekker, New York, 1995), pp. 377-416.
38. F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972); A. M. Hofmeister, in A Practical Guide to Infrared Microspectroscopy, H. J. Humecki, Ed. (Dekker, New York, 1995), pp. 377-416.
39. This is commonly applied to minerals; see, for example, A. M. Hofmeister, Phys. Chem. Miner. 24, 535 (1997).
40. i.
41. 2 [J. G. Traylor, H. G. Smith, R. M. Nickow, M. W. Wilkinson, Phys. Rev. B 10, 3457 (1975)] is one of the few minerals with complete INS data.
42. For example, N. Choudhury, S. Ghose, C. P. Chowdhury, C. K. Loong, S. L. Chaplot, Phys. Rev. B 58, 756 (1998) for enstatite; D. L. Price, S. Ghose, N. Choudhury, S. L. Chaplot, K. R. Rao, Physica B 174, 87 (1991) for Fa; K. R. Rao et al., Phys. Chem. Miner. 16, 83 (1988) for Fo.
43. For example, N. Choudhury, S. Ghose, C. P. Chowdhury, C. K. Loong, S. L. Chaplot, Phys. Rev. B 58, 756 (1998) for enstatite; D. L. Price, S. Ghose, N. Choudhury, S. L. Chaplot, K. R. Rao, Physica B 174, 87 (1991) for Fa; K. R. Rao et al., Phys. Chem. Miner. 16, 83 (1988) for Fo.
44. For example, N. Choudhury, S. Ghose, C. P. Chowdhury, C. K. Loong, S. L. Chaplot, Phys. Rev. B 58, 756 (1998) for enstatite; D. L. Price, S. Ghose, N. Choudhury, S. L. Chaplot, K. R. Rao, Physica B 174, 87 (1991) for Fa; K. R. Rao et al., Phys. Chem. Miner. 16, 83 (1988) for Fo.
45. S. W. Kieffer, Rev. Geophys. Space Phys. 17, 20 (1979) and references therein.
46. See examples in V. C. Farmer, The Infrared Spectra of Minerals (Mineralogical Society, London, 1974), and in (25, 27, 28).
47. This technique is closely related to Raman spectroscopy. See A. Chopelas, H. J. Reichmann, L. Zhang, in Mineral Spectroscopy: A Tribute to Roger G. Burns, M. D. Dyar, C. McCammon, M. W. Schaeffer, Eds. (Geochemical Society, Houston, TX, 1996), pp. 229-242, and references therein.
48. A. M. Hofmeister, Phys. Rev. B 56, 5835 (1997).
49. -1 for olivine (6).
50. G. Burns and B. A. Scott, Phys. Rev. B 7, 3088 (1973); A. M. Hofmeister, Eos 79, S163 (1998).
51. G. Burns and B. A. Scott, Phys. Rev. B 7, 3088 (1973); A. M. Hofmeister, Eos 79, S163 (1998).
52. The radiative term should be pressure independent, because Γ is pressure independent (35), and thus was not integrated over pressure. The change in frequency with pressure is immaterial because the rate of change for the absorption and emission bands would be essentially the same (discussed further below).
53. B. Reynard, Phys. Chem. Miner. 18, 19 (1991).
54. J. L. Servoin and B. Piriou. Phys. Status Soldi B 55, 677 (1973); K. Iishi, Am. Mineral. 63, 1198 (1978).
55. J. L. Servoin and B. Piriou. Phys. Status Soldi B 55, 677 (1973); K. Iishi, Am. Mineral. 63, 1198 (1978).
56. C. Clausner and E. Huenges, in Rock Physics and Phase Relations: A Handbook of Physical Constants, T. J. Ahrens, Ed. (American Geophysical Union, Washington, DC, 1995), pp. 105-126.
57. A. M. Hofmeister, unpublished IR data. The spectral parameters are close to other garnets in the middle of the pyrope-almandine binary measured by A. M. Hofmeister, T. J. Fagan, K. M. Campbell, and R. B. Schaal [Am. Mineral. 81, 418 (1996)].
58. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 251-253.
59. The blackbody radiation is associated with vibronic transitions, implying that high levels must be populated in order for visible light to be given off. The continuum exists because of the large number of overtones and finite widths.
60. For example, at room temperature the first overtone of the Si-O bands is intense at submillimeter thickness (23), even though these are expected to be populated at less than 1% of the fundamental population, and the second overtone is also seen but poorly resolved.
61. Because bandwidths on average are about the same for all silicates, the result can be generalized. Oxides have broader mid-IR bands (33) and therefore broader overtones than silicates, and could not have higher b values. Furthermore, this value for b equals the average of the two curves derived for MgO (22) by assuming an average absorbance (20), which suggests that the above representation is also appropriate for oxides.
62. R. G. Burns, Mineralogical Applications of Crystal Field Theory (Cambridge Univ. Press, Oxford, 1970), in particular pp. 80-81.
63. rad low at high temperature. Furthermore, because widths do not change across solid solution series [for example, (47)], the radiative conductivity will not be affected by Fe content as long as enough is available to produce at least a small absorption in an appropriate grain size. Thus, minerals with Fe contents below perhaps 1 or 2 weight % should probably be treated as Fe-free.
64. rad (78, 23) are inconsistent with total K.
65. Y. Fei, in Mineral Physics and Crystallography: A Handbook of Physical Constants, T. J. Ahrens, Ed. (American Geophysical Union, Washington, DC, 1995), pp. 29-44.
66. K. Horai, J. Geophys. Res. 76, 1278 (1971).
67. C. B. Agee, Annu. Rev. Earth Planet. Sci. 21, 19 (1993).
68. The pressure dependence obtained from Eq. 10 using the data in Table 1 with q = 0 [see A. M. Hofmeister, J. Xu, H. K. Mao, P. M. Bell, T. C. Hoering, Am. Mineral. 74, 281 (1989)] gave negligibly different values from Eq. 11. For silicate Pv, q is unknown and γ and K′ have large uncertainties. Thus, results for Eq. 11 are given.
69. A. M. Hofmeister, unpublished data. Fe end-member spectra are in P. F. McMillan and A. M. Hofmeister, Rev. Mineral. 18, 99 (1988).
70. A. M. Hofmeister, unpublished data. Fe end-member spectra are in P. F. McMillan and A. M. Hofmeister, Rev. Mineral. 18, 99 (1988).
71. H. Fujisawa, N. Fujii, H. MiZutani, H. Kanamori, S. Akimoto, J. Geophys. Res. 75, 4727 (1968).
72. M. Manga and R. Jeanloz, ibid. 102, 2999 (1997).
73. H. Yutatake and M. Shimada, Phys. Earth Planet. Inter. 17, 193 (1978).
74. W. R. McPherson and H. H. Schloessin, ibid. 29, 58 (1982).
75. S. Andersson and G. Bäckström, Rev. Sci. Instrum. 57, 1633 (1986).
76. T. Katsura, Phys. Earth Planet. Inter. 101, 73 (1997).
77. M. Osako and E. Ito, Geophys. Res. Lett. 18, 239 (1991).
78. N. L. Ross and R. M. Hazen, Phys. Chem. Miner. 16, 415 (1989).
79. R. Lu, A. M. Hofmeister, Y. Wang, J. Geophys. Res. 99, 11795 (1994).
80. C. A. Stein and S. Stein, Nature 359, 123 (1992); B. Parsons and J. G. Sclater, J. Geophys. Res. 82, 803 (1977).
81. C. A. Stein and S. Stein, Nature 359, 123 (1992); B. Parsons and J. G. Sclater, J. Geophys. Res. 82, 803 (1977).
82. M. L. Renkin and J. G. Sclater, ibid. 93, 2919 (1988); J. G. Sclater and L. Wilson, in The Geology of North America: The Western North Atlantic, P. R. Vogt and B. E. Tucholke, Eds. (Geological Society of America, Boulder, CO, 1986), pp. 257-270.
83. M. L. Renkin and J. G. Sclater, ibid. 93, 2919 (1988); J. G. Sclater and L. Wilson, in The Geology of North America: The Western North Atlantic, P. R. Vogt and B. E. Tucholke, Eds. (Geological Society of America, Boulder, CO, 1986), pp. 257-270.
84. This approach is consistent with solutions of steady-state heat flow between stacked parallel planes [see section 3.2 in (1)]. It is reasonable because the pressure dependence is weak (κ doubles over 760 km).
85. A. M. Dziewonski and D. L. Andersen, Phys. Earth Planet. Inter. 25, 297 (1981).
86. T. Inoue and H. Sawamoto, in (7), pp. 323-331. The 1-atm value is the same as the dry solidus [see A. E. Ringwood, Composition and Petrology of the Earth's Mantle (McGraw-Hill, New York, 1975)].
87. T. Inoue and H. Sawamoto, in (7), pp. 323-331. The 1-atm value is the same as the dry solidus [see A. E. Ringwood, Composition and Petrology of the Earth's Mantle (McGraw-Hill, New York, 1975)].
88. J. B. Gaherty, T. H. Jordon, L. S. Gee, J. Geophys. Res. 101, 22291 (1996).
89. Y. Xu and D. A. Weins, ibid. 102, 27439 (1997).
90. E. Takahashi, ibid. 91, 9367 (1986).
91. D. L. Turcotte and G. Schubert, Geodynamics (Wiley, New York, 1982).
92. E. Ito and T. Katsura, Geophys. Res. Lett. 16, 425 (1989).
93. E. Ito and E. Takahashi, J. Geophys. Res. 94, 10637 (1989); E. Ito and T. Katsura, in (7), pp. 315-322.
94. E. Ito and E. Takahashi, J. Geophys. Res. 94, 10637 (1989); E. Ito and T. Katsura, in (7), pp. 315-322.
95. D. C. Presnall, Y. H. Weng, C. S. Milholland, M. J. Walter, Phys. Earth Planet. Inter. 107, 83 (1998).
96. E. Ohtani, K. Moriwaki, T. Kato, K. Omurna, ibid., p. 75.
97. A. Zerr, A. Diegeler, R. Boehler, Science 281, 243 (1998).
98. T. Katsura and E. Ito, J. Geophys. Res. 94, 15663 (1986).
99. F. D. Stacey, Phys. Earth Planet. Inter. 15, 341 (1977).
100. J. M. Brown and T. J. Shankland, Geophys. J. R. Astron. Soc. 66, 579 (1981).
101. R. Boehler, Earth Planet. Sci. Lett. 111, 217 (1992).
102. _, Annu. Rev. Earth Planet. Sci. 24, 15 (1996); F. D. Stacey, in preparation.
103. _, Annu. Rev. Earth Planet. Sci. 24, 15 (1996); F. D. Stacey, in preparation.
104. F. D. Stacey and D. E. Loper, Phys. Earth Planet. Inter. 33, 45 (1983).
105. D. L. Anderson, Theory of the Earth (Blackwell Scientific, Boston, 1989).
106. J. Verhoogen, Phys. Earth Planet. Inter. 7, 47 (1973); F. D. Stacey, Physics of the Earth (Brookfield, Brisbane, Australia, 1992).
107. J. Verhoogen, Phys. Earth Planet. Inter. 7, 47 (1973); F. D. Stacey, Physics of the Earth (Brookfield, Brisbane, Australia, 1992).
108. O. L. Anderson, Phys. Earth Planet. Inter. 109, 179 (1998).
109. S. Zhong and M. Gurnis, J. Geophys. Res. 99, 15683 (1994); B. H. Hager, Eos 71, 1567 (1990).
110. S. Zhong and M. Gurnis, J. Geophys. Res. 99, 15683 (1994); B. H. Hager, Eos 71, 1567 (1990).
111. Y. Sumino and O. Anderson, in Handbook of Physical Properties of Rocks III, R. S. Carmichael, Ed. (CRC Press, Boca Raton, FL, 1984), pp. 139-280; J. D. Bass, in (50), pp. 45-63; E. Knittle, ibid., pp. 98-142.
112. Y. Sumino and O. Anderson, in Handbook of Physical Properties of Rocks III, R. S. Carmichael, Ed. (CRC Press, Boca Raton, FL, 1984), pp. 139-280; J. D. Bass, in (50), pp. 45-63; E. Knittle, ibid., pp. 98-142.
113. Y. Sumino and O. Anderson, in Handbook of Physical Properties of Rocks III, R. S. Carmichael, Ed. (CRC Press, Boca Raton, FL, 1984), pp. 139-280; J. D. Bass, in (50), pp. 45-63; E. Knittle, ibid., pp. 98-142.
114. H. Cynn and A. M. Hofmeister, J. Geophys. Res. 99, 17717 (1994).
115. T. Ashida, S. Kume, E. Ito, A. Navrotsky, Phys. Chem. Miner. 16, 239 (1988).
116. A. M. Hofmeister, in (34), pp. 215-227.
117. A. E. Beck, D. M. Darba, H. H. Schloessin, Phys. Earth Planet. Inter. 17, 35 (1978).
118. G. H. Scharmli, in High Pressure Research in Geosciences, W. Schreyer, Ed. (Schweizerbartsche, Stuttgart, Germany 1982), pp. 349-373.
119. H. H. Schloessin and Z. Dvorak, Geophys. J. R. Astron. Soc. 27, 499 (1972).
120. M. Osako and Y. Kobayashi, Phys. Earth Planet. Inter. 18, P1 (1979).
121. S. W. Kieffer, I. C. Getting, G. C. Kennedy, J. Geophys. Res. 81, 3018 (1976).
122. K. Horai and J. Sasaki, Phys. Earth Planet. Inter. 55, 292 (1989).
123. A. Franson and R. G. Ross, J. Phys. C Solid State Phys. 12, 219 (1983),
124. I thank R. Boehter for suggesting this problem, the Humboldt Foundation for making our interaction possible, and the David and Lucile Packard Foundation for funding the IR spectroscopy that formed the basis of this study. The input of O. L. Anderson, A. Chopelas, R. E. Criss, R. F. Dymek, S. A. Hauck, R. Phillips, P. Shore, M. Simons, D. Weins, M. Wyssession, and four anonymous reviewers is greatly appreciated. The project was partially supported by NSF grant EAR712311.