Thermophysical properties of FeAl (Fe-40 at.%Al)

Thermophysical properties of FeAl (Fe-40 at.%Al)

Intermetallics 8 (2000) 1369±1376 Thermophysical properties of FeAl (Fe-40 at.%Al) B.V. Reddy *, S.C. Deevi Researc...

238KB Sizes 0 Downloads 1 Views

Intermetallics 8 (2000) 1369±1376

Thermophysical properties of FeAl (Fe-40 at.%Al) B.V. Reddy *, S.C. Deevi Research Development and Engineering Center, 4201 Commerce Road, Philip Morris USA, Richmond, VA 23234, USA Received 6 April 2000; accepted 17 May 2000

Abstract The thermophysical properties Ð electrical resistivity, thermal conductivity, thermal expansion, and speci®c heat, of a B2 ironaluminide (Fe-40 at.% Al) alloy are measured. The measured values of electrical resistivity indicate three distinct regions. An initial sharp rise below 400 C is followed by a gradual increase to near saturation around 900 C. Resistivity above this temperature exhibits an anomalous negative temperature dependence. The thermal conductivity displays a continuous rise as a function of temperature for T<800 C, beyond which it saturates to a value of 0.17 W/cm- C. The relation between electrical resistivity and thermal conductivity obeys the Wiedemann-Franz law signifying the dominance of electrons in the heat transport. The measurements of speci®c heat indicate a complex behavior suggesting inseparable contributions of various temperature dependent phenomena arising from phonons, conduction electrons and magnons. Both the thermal expansion and mean coecient of thermal expansion (MCT) exhibit a rising trend with temperature. The temperature dependence of the various modes of lattice, electronic, and magnetic excitations is invoked to explain the observed variations in properties. The role of the inherent electronic and magnetic structure on physical properties is highlighted. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Iron-aluminides, based on FeAl; B. Thermal properties; B. Electrical resistance and other electrical properties

1. Introduction Iron aluminides have been a subject of renewed attention in the recent past. The desired oxidation and corrosion resistance, high melting point, low material cost and low density characterize these materials as potential candidates for several high temperature structural applications in heat treating, automotive and power generation industries [1±5]. Although the ironaluminum phase diagram [6] exhibits several intermetallic phases at varying compositions and temperatures, two ordered phases, namely the B2 and DO3 predominate the ®eld with stability over a wide range of composition. The DO3 phase of stoichiometric Fe3Al undergoes a gradual transition towards the crystalline B2 phase with increasing Al composition attaining the FeAl stoichiometry. In past years, considerable e€orts have been devoted to understanding the processing and mechanical properties of these ordered phases with an aim to increase strength, ductility, and corrosion

* Corresponding author. Tel.: +1-804-274-1961; fax: +1-804-2744778. E-mail address: [email protected] (B.V. Reddy).

resistance. While these e€orts imparted signi®cant knowledge on their mechanical behavior [2], an understanding of their physical properties is still limited due to lack of substantial attention. In recent papers, Deevi et al. [3±5, 7] prepared and studied the electrical properties of Fe1-xAlx alloys as a function of increasing Al composition. The investigations revealed a monotonic increase of resistivity with increasing Al content for x<0.33. Beyond this limiting value, a negative slope of resistivity with Al concentration was observed. We explained the anomaly by the use of a phenomenological model [7] based on the original proposition of Mott and Jones [8], and also validated its limitations using ®rst-principles calculations [9]. Ecient utilization of iron-aluminides requires a good understanding of other thermophysical properties such as heat capacity, thermal expansion, thermal di€usivity, and thermal conductivity for high temperature applications and for ®nite element modeling of their behavior in the temperature range of interest. For example, the machinability of a material with low heat conductivity greatly hinders heat dissipation, leading to high temperatures in the cutting zone which in turn leads to extensive tool wear [10]. A thermophysical property such as speci®c heat also yields useful information

0966-9795/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0966-9795(00)00084-4


B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376

about other temperature dependent phenomena in materials relating to lattice vibrations, conduction electrons and magnetic excitations [11]. Besides, parameters such as thermal expansion and conductivity are critical for determining the thermal shock behavior and other properties of concern for use at high temperatures [12,13]. Despite this importance, studies involving the thermophysical properties of iron-aluminides are scant. In the past, this could be attributed partly to lack of interest in these properties for structural applications and partly to the tedious processing techniques and measurements required to prepare these materials. Suzuki et al. [14] studied the room temperature thermal conductivity of aluminides close to the AB and A3B stoichiometries as a function of Al composition and crystal structure. Using an ab-initio force-constant method and the frozen-phonon calculations, Meyer et al. [15], studied the extent to which the lattice vibrations in¯uence the various physical properties in B2±FeAl. Such vibrational properties as a function of temperature for the b-phase FeAl, NiAl and CoAl were also carried out by the extended X-ray- absorption-edge-®ne-structure (EXAFS) [16] and other techniques [17]. Vogl and co-workers [18,19], on the other hand, investigated the elementary di€usion jump mechanism in B2 ordered FeAl single crystals employing the quasielastic MoÈssbauer spectroscopy at 57Fe. The electrical resistivity of the iron-aluminides as a function of composition and temperature has been studied by a number of authors [20±27]. The electrical resistivity in (Fe1-x Mx)3Al with M=Ti, V, Cr and Mn show a resistance maximum near Tc and decreases dramatically with increasing temperature. This negative resistivity slope leads to the break down of Mattheisson's rule [28], which predicts no variation in the slope of resistivity-temperature curves of dilute alloys with respect to that observed in their base elements. The origin of the negative temperature dependence in these alloys induced by the substitution of elements to the left of Fe in the periodic table is attributed to the 3d-heavy fermion-like or semiconductor-like behavior [20,21]. In this paper, we present our investigations on the speci®c heat, thermal expansion, thermal conductivity and electrical resistivity of Fe-40 at.% Al as a function of temperature. In Section 2, we present our experimental methodology used to study the various thermophysical properties. In Section 3, we present our results and discussion. In the ®nal section, we present our conclusions. 2. Experimental methodology The technical details for the processing of the FeAl samples are provided in our previous papers [7,29±31].

Here we refer only to the details of the procedures employed to investigate the thermophysical properties. A dual push-rod dilatometer of Theta instruments was used to measure the linear thermal expansion from room temperature to 1100 C. The di€erential expansion between the sample and a known standard reference material alumina (sapphire) was measured as a function of temperature. The expansion of the sample was computed from this di€erential expansion and the expansion of the standard. For the purposes of calibration, one NIST standard was measured against another NIST standard. The speci®c heat was measured using a standard Perkin-Elmer Model DSC- 2 (di€erential scanning calorimeter) using sapphire as a reference material. The standard and sample both, encapsulated in pans, were subjected to the same heat ¯ux, and the di€erential power required to heat the sample at the same rate was recorded using the digital data acquisition system. From the mass of the sapphire standard, pans, the di€erential power, and the known speci®c heat of sapphire, the speci®c heat of the sample was computed. All measured quantities are directly traceable to NIST standards. A Netzsch Model 404 di€erential scanning calorimeter (DSC) was used to measure variation of speci®c heats with temperature in vacuum. The thermal conductivity was measured using two complimentary techniques. (1) A direct technique based on the modi®ed Kohlrausch method [32] and (2) ¯ash di€usivity method used to measure thermal di€usivity which in turn is used to calculate thermal conductivity with the help of known values of speci®c heat and density of the sample. The Kohlrausch method involves the determination of the product of the thermal conductivity  and the electrical resistivity . Since the electrical resistivity is also measured at the same time,  can be calculated. The method involves passing constant direct current through the specimen to heat the sample while the ends are kept at a constant temperature. In the ¯ash method, the front face of a small diskshaped sample is subjected to a very short (1 ms or less) burst of radiant energy using a laser or a xenon ¯ash lamp. The resulting temperature rise of the rear surface of the sample is measured and thermal di€usivity values are computed from the temperature rise versus time data. For a detailed description of the two techniques, the reader is referred to Ref. [33]. 3. Results and discussion 3.1. Electrical resistivity The variation of resistivity with temperature of materials is a function of the scattering of electrons due to phonons and imperfections. The magnetic ¯uctuations

B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376

also play a signi®cant role in alloys where magnetic elements like Fe are involved. In Fig. 1, we present the variation of electrical resistivity of Fe-40 at.% Al with temperature. We divide the curve into three regions as a matter of convenience for discussion and interpretation of data. We note (1) a sharply rising resistivity with temperature for T4400 C represented as region A (2) a weakly varying resistivity with temperature tending towards saturation represented as region B (3) a third region having a negative temperature coecient of resistivity beyond 900 C represented as region C. As mentioned earlier, electrical resistivity of materials is intimately linked to their electronic structure. The observed features could be understood based on a body centered cubic cluster of 15 atoms as represented in Fig. 2. The dark shaded atoms represent the Fe atoms and the light shaded atoms represent the Al atoms. Note that this model cluster represents a schematic of the bulk crystal structure of Fe-40 at.% Al. Note here that the central Fe (anti-site Fe atom) has eight Fe neighbors, while each of the corner Fe atoms in the cube have one Fe and other Al neighbors. Our electronic structure calculations [9] on such a model cluster reveals two important features. (1) An expansion of the lattice as compared to that of the bulk Fe. This is expected since, the ionic radius of Al (1.43 AÊ) is larger than that of the corresponding value for Fe which is equal to 1.27 AÊ [28]. (2) Co-existence of ferromagnetic coupling between Fe±Fe neighbors and an antiferromagnetic coupling between the corresponding Fe±Al neighbors. This is also consistent with the experimental observations of Arrot et al. [34,35]. Region A constitutes a typical behavior of metallic alloys, where the phonons and impurity scatterings play


a major role in enhancing the resistivity of conduction electrons with temperature [36]. In region B, as depicted, there is a weak rise of resistivity with temperature which seems to be reaching a near saturation with increase in temperature. As seen in Fig. 2, iron-aluminide of Fe-40 at.% Al, being the Fe rich phase contains anti-site Fe atoms still possessing a considerable number of Fe neighbors with a positive Fe±Fe exchange coupling J (ferromagnetic). As the temperature increases, the ferromagnetic coupling is weakened. This weakening is also favored by the increased distance between Fe±Fe because of an inherent expansion of the lattice. This forces the Fe-atoms to behave like near dilute magnetic impurities with localized moments involving weak or no direct exchange. Furthermore, the negative exchange coupling (antiparallel spins) between the Fe atoms and the conduction electrons o€ers an ideal situation for the observation of some kind of an extended Kondo e€ect [37]. Note that the Kondo e€ect as originally predicted for dilute magnetic alloys leads to a shallow minimum in electrical resistivity at low temperatures due to the scattering of the conduction electrons by the localized magnetic moments. We believe that this e€ect negates the expected rise in resistivity with temperature in this region. With further increase in temperature i.e. in region C, the ferromagnetic coupling is completely destroyed. Hence in this region, we believe that the dominant contribution comes from the extended or accrued Kondo e€ect, leading to the observed anomaly in the resistivity. Note, however, that even though the Kondo temperatures for dilute alloys do not typically exceed 10±20 K, it might not be the case for this kind of an extended Kondo e€ect, where weak d±d interactions still persist amongst the Fe-atoms.

Fig. 1. Electrical resistivity of FeAl as function of temperature.


B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376

marked by a Al-3p band lying just above the Fermi energy [9]. At higher temperatures, the signi®cant lattice expansion experienced by the FeAl alloy might lead to a modi®ed electronic structure with a reduced gap between the valence and the conduction bands. We believe such modi®cations in the electronic structure could also contribute to the observed decrease in resistivity at higher temperatures. 3.2. Speci®c heat of FeAl

Fig. 2. A model 15- atom cluster of a body-centered-cubic crystal depicting the schematic of the bulk structure of Fe-40at.% Al. The dark circles represent Fe atoms, while the light shaded circles represent the Al atoms.

Secondly, it is known that at such high temperatures thermally generated vacancies could be signi®cant. Though the impact of such imperfections on resistivity is not clearly understood, the only indication we have from earlier work is an increasing trend of resistivity with increasing imperfections. For example, Ishizawa et al. [38] have studied the electrical resistivity in NbCx and TiCx single crystals. The resistivity increase due to the carbon vacancy in NbCx was found to be 8 m -cm/ at.% Vc and 24 m -cm/at.% Vc in the TiCx single crystal, where Vc is the carbon vacancy concentration. Alternatively, one should note that the resistivity has an inverse relationship to the density of states at the Fermi energy. The electronic structure of Fe-40 at.% Al is

In Fig. 3, we plot the speci®c heat of FeAl as a function of temperature. In principle, any temperature dependent phenomenon can contribute to the speci®c heat of a material. Hence, the complexity of the contributions can be represented quite generally as: Cv ˆ Cl ‡ Ce ‡ CM ‡ Cother


Where, Cl=lattice speci®c heat due to phonon excitations; Ce=contributions from conduction electrons; CM=contributions from magnetic excitations; Cother= contributions from other terms such as lattice vacancies, order-disorder transformations, etc. We divide the observed curve of the sample in to three temperature zones. Moderate (T4300 C), high (300 4T4650 C), and very high (beyond 650 C) temperature zones. The rise of the curve in the low to moderate temperature regions seems to deviate from the expected T3 behavior. Note that the Debye model predicts this kind of behavior for the lattice speci®c heat at low temperatures [15]. We believe that the contributions to the speci®c heat in this region is a combination of several competing e€ects in addition to the phonon speci®c

Fig. 3. Variation of speci®c heat of FeAl with temperature.

B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376


Fig. 4. Variation of thermal expansion of FeAl with temperature in the parallel and the transverse directions.

heat. Though the conduction electron contributions are estimated to be directly proportional to temperature over all the temperature regions, we believe that the free-electron like behavior is signi®cantly altered in this system because of the presence of the transition element Fe with localized d-electrons leading to an increased s-d scattering. In addition, as mentioned earlier, the electronic structure calculations [9] as well as experiments by Arrot et al. [34,35], suggest the coexistence of ferromagnetic Fe±Fe and antiferromagnetic Fe±Al couplings in Fe-40 at.% Al. This further complicates the scenario by contributing the magnon terms proportional to T3

(antiferromagnetic) and T3/2 ( ferromagnetic) [11]. In zone II, the increase in speci®c heat seems to exhibit a moderate rise or near constant (300±650 C) behavior. We believe that in this region, the dominant contribution comes from the phonon excitations with the harmonic expansion yielding a constant term of order R (Gas constant R=8.31 J molÿ1 Kÿ1 ) when evaluated to the ®rst order [15]. The anharmonic lattice contribution is expected to increase with temperature, but in any case is expected to be small in the present temperature range. In addition, one should also note that the intermetallics such as FeAl in the iron-rich phase are marked by a

Fig. 5. Variation of `mean coecient of thermal expansion(MCT)' of FeAl with temperature in the parallel and the transverse directions.


B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376

Fig. 6. Measured thermal conductivity of FeAl as a function of temperature.

large density of electronic states near the Fermi energy because of the d-electron rich transition element [38]. Hence, the electron contribution to the speci®c heat at these temperatures is expected to signi®cantly depart from the free electron behavior, which as mentioned earlier contribute terms proportional to T in the ®rst order. At very high temperatures, beyond 650 C (Zone III), we notice the beginning of an anomalous rise in the speci®c heat curve. It should be noted that this feature is typically observed in refractory metals like Ti, Nb and W before melting [39]. Such a rise followed by a l-type singularity is also observed in certain magnetic materials [40±42]. This anomalous rise in speci®c heat at high temperatures in this intermetallic FeAl could be attributed to an accrued e€ect of several temperature dependent phenomena. In addition to the higher order contributions of phonon and electronic speci®c heats, other modes of excitations also become active. For example, in the vicinity of the stoiciometric FeAl, Vogl and co-workers [20,21] predict a continuous decrease of short-range order at elevated temperatures, which in turn could also lead to increased generation of vacancies thus a€ecting the total speci®c heat. Further, at very high temperatures, we also believe that the magnetic couplings start breaking apart leading to a paramagnetic state of the intermetallic; an order±disorder transition.

between the two directions indicating the isotropic behavior of the linear expansion. A careful examination of the curves also reveals a monotonic increase in slope with temperature. The corresponding variation of the slope or the mean coecient of thermal expansion (MCT) is plotted in Fig. 5. The Fe±Al bond strength is intermediate to the Fe±Fe and Al±Al bond strengths. This is seen from the higher cohesive energy and melting point of Fe (4.28 eV and 1808 K) as compared to that of Al (3.39 eV and 933 K) [14]. Also, note that the MCTs, aFe=1.2 10ÿ5 and aAl=2.2 10ÿ5 at room temperature [12,13]. The thermal expansion of Fe-40 at.% Al is a complicated function of thermal and elastic properties of the individual components. However, at least in the vicinity of room temperature, one could still justify the observed values of MCT for Fe-40 at.% Al (Fig. 5), based purely on the di€erences in chemical bond strengths and melting points of Fe ad Al. With the increase in temperature, the phonon vibrations increase, accompanied by an increased domination of the anharmonic term in the asymmetric potential curve. At very high temperatures beyond 1100 C, mixed phases and vacancy formation also begin to a€ect the thermal expansion [6,20,21].

3.3. Thermal expansion of FeAl

In Fig. 6, we present the data obtained for thermal conductivity for the Fe-40 at.%Al sample. Thermal conductivity K=1/3 CVL Where, C=speci®c heat per unit volume;

In Fig. 4, we present the thermal expansion of FeAl, in both the parallel and transverse directions to the rolling direction of the sheet. Note the similarity

3.4. Thermal conductivity

B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376


Fig. 7. A plot of the product of thermal conductivity() and electrical resistivity() against temperature to verify the Wiedemann±Franz law.

V=average velocity of the particle (particle referring to phonon for insulators and electron in the case of metals); L=mean free path of the phonon+mean free path of electrons. In general, in pure metals the main contribution to thermal conductivity comes from electrons. The electron heat transport is directly proportional to temperature and the relaxation time. Depending on the kind of alloy, one could also expect signi®cant contributions from the phonons to the heat transport. It is also interesting to note that while the thermal conductivity in general, experiences an increase with temperature, the electrical conductivity has an opposite behavior. To determine the main component of thermal conductivity in the current sample, we plot the product of thermal conductivity () and electrical resistivity () against temperature in Fig. 7. Note the reasonably good representation of the linearity in the relationship validating the Wiedemann±Franz law [15]. The curve of thermal conductivity of Fe-40 at.% Al is not a purely linear variation with temperature. This could be because of three reasons. (1) The electron mean free path is decreased considerably because of the impurity scattering as opposed to that observed in a pure metal. A clear evidence of this is the considerably decreased value of thermal conductivity at 300 k of the Fe-40 at.% Al sample as compared to the individual thermal conductivity's of Fe ( 0.8 W/cm-K) or Al (2.37 W/cm-K). (2) Phonon scattering with imperfections also contributes to the slight negating e€ect to the expected linear variation. (3) The e€ective mass of the electron in

these systems is not the same as the free electron mass due to the increased s-d scattering [22,25]. A second feature noteworthy in the conductivity curve is the decrease or saturation of thermal conductivity beyond 800 C. The decreasing feature in thermal conductivity beyond this temperature is attributed to considerable decrease in the phonon mean free path due to the increasing number of phonons which is directly proportional to T at higher temperatures. 4. Conclusions In conclusion, we have measured the thermophysical properties Ð electrical resistivity, speci®c heat, thermal conductivity and thermal expansion of iron aluminides (Fe-40 at.%Al) as a function of temperature. Several interesting observations were noted. 1. Electrical resistivity as a function of temperature indicates three distinct regions of interest. A sharp rise below 400 C is followed by a weak rise to near saturation below 900 C. Beyond this range an anomalous negative dependence of resistivity on temperature is observed. The electronic and magnetic structure seem to play a key role in the observed trends. 2. Thermal conductivity displays a monotonic rise with temperature below 800 C and saturates beyond it. The Wiedemann±Franz law seems to hold good for these samples, indicating the role of electrons as carriers of heat.


B.V. Reddy, S.C. Deevi / Intermetallics 8 (2000) 1369±1376

3. The contributions to speci®c heat at various ranges of temperatures are marked by complex temperature dependent phenomena arising from phonons, conduction electrons and magnons. At very high temperatures beyond 650 C, the curve exhibits a sharp rise which is typical of some refractory metals like Nb and W just before melting. 4. Both the thermal expansion and mean coecient (MCT) of thermal expansion exhibit a monotonic rise with temperature. However, the rising trend of both the quantities seems to fall in the intermediate range relative to the pure elements Fe and Al.

Acknowledgements The authors are thankful to Dr. A. C. Lilly of Philip Morris USA for his keen interest, encouragement and permission to publish this work. We also greatly appreciate the valuable discussions with Professor Jena of Virginia Commonwealth University. We also thank F. Logan for the technical support. The authors also acknowledge the support of Dr. R. Taylor of TPRL, Indianapolis.

References [1] Liu CT, George EP, Maziasz PJ, Schneibel JH. Mat Sci & Eng 1998;A258:84±98. [2] McKamey CG, Devan JH, Tortorelli PF, Sikka VK. J Mater Res 1991;6:695. [3] Deevi SC, Sikka VK. Intermetallics 1996;4:357. [4] Deevi SC, Sikka VK, Liu CT. Prog Mater Sci 1997;42:177 [5] Deevi SC, Maziasz PJ, Sikka VK, Cahn RW, editors. Proceedings of the International Symposium on nickel and iron aluminides: processing, properties, and applications. Metals Park, OH: ASM International, 1997. [6] Massalski TB. Binary alloy phase diagrams. Metals Park, OH: ASM International, 1986, p. 112. [7] Lilly AC, Deevi SC, Gibbs ZP. Mat Sci & Eng 1998;A258:42±49 [8] Mott NF, Jones H. The properties of metals and alloys. Clarendon Press, Oxford (1936). Reprinted by Dover publications, Inc. New York 1958. Mott NF, Stevens KWH, Philos Mag 1957;2:1364. [9] Reddy BV, Lilly AC, Deevi SC, Jena P. submitted to Phys Rev B. [10] Poduraev V.N. Cutting of dicult-to-machine materials (in Russian), Moscow, Vyssh, Shkola, 1974.

[11] Kittel C. Quantum theory of solids. New York: Wiley, 1963. [12] Ho C.Y. Thermal expansion of solids, CINDAS Data Series on Material Properties, Vols. I±4, ASM International 1998. [13] Klemens PG, Williams RK. Int Met Rev 1986;31:197±215. [14] Suzuki T, Terada Y, Mohri T. In: Deevi SC, Maziasz PJ, Sikka VK, Cahn RW, editors. Proceedings of the International Symposium on nickel and iron aluminides: processing, properties, and applications. Metals Park, OH: ASM International, 1997. [15] Meyer B, Schhott V, FaÈhnle M. Phys Rev B, 1998;58:R14673. [16] Brewe D, Pease DM, Budnick JI, Law CC. Phys Rev B 1997;56: 11449. [17] BuÈhrer W., Zolliker M., Vogl G. Annual Report 1995, Labor fuÈr Neutronenstreuung, ETH ZuÈrich and Paul Scherrer Institut, Report No. LSN-181 p. 51±2 (unpublished). [18] Feldwisch R, Sepiol B, Vogl G. Acta Metall Mater 1995;43:2033± 9. [19] Vogl G, Sepiol B. Acta Metall Mater 1994;42:3175±81. [20] Nishino Y, Kato M, Asano S, Soda K, Hayasaki M, Mizutami O. Phys Rev Lett 1997;79:1909. [21] Nishino Y. Electrons in solids: experiment and theory, 10th international summer school of condensed matter physics. Bialowieza, 1996. [22] Inoue K, Nakamura Y, Kunitomi NJ. Phys Soc Japan 1969; 27:1159. [23] Inoue K, Nakamura YJ. Phys Soc Japan 1972;32:441. [24] Onuki Y, Shimizu Y, Komatsubara TJ. Phys Soc Japan 1984; 53:1210. [25] Nishitami SR, Masaki K, Shirai Y, Yamaguchi M. Scr Metall Mater 1991;25:1921. [26] Cahn RW, Feder R. Philos Mag 1960;5:451. [27] Feder R, Thornton PH, Cahn R. Philos Mag 1961;6:1093 [28] Kittel C. Introduction to solid state physics. 7th ed. New York: John Wiley & Sons, 1996. [29] Hajaligol MR, Deevi SC, Sikka VK, Scorey CR. Mat Sci & Eng 1998;A258:249±57. [30] Mistler RE, Sikka VK, Scorey CR, McKerman JE, Hajaligol MR. Mat Sci & Eng 1998;A258:258±65. [31] Deevi SC, Hajaligol MR, Sikka VK, McKermon J, Scorey CR. Mat Res Soc Sympo Proc 1999;552:KK4.6.1. [32] Kohlrausch F. Sitz Berlin Akad 1899;38:711±8. [33] Maglic KD, Cezairliyan A, Peletsky VE, editors. Compendium of thermophysical property measurement methods. Plenum Press, 1984. [34] Arrott A, Sato H. Phys Rev 1959;114:1920. [35] Sato H., Arrott A. Phys Rev 1959: 1427. [36] Pollock D. D. Electrical conduction in solids Ð an introduction. American Society for Metals, 1985. [37] Kondo J. Progress of Theoretical Physics 1964;32:37. [38] Ishizawa Y, Otani S, Nozaki H, Tanaka TJ. Phys Condens Matter 1992;4:8593. [39] Kulikov NI, Postnikov AV, Borstel G, Braun J. Phys Rev B 1999;59:6824. [40] Leadbetter AJJ. Phys 1968;C 1:1481. [41] Brooks CRJ. Phys Chem Solids 1968;29:1377. [42] Cezairliyan A. High Temp Sci 1980;13:117.