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The high temperature bulk modulus of aluminium: an assessment using experimental enthalpy and thermal expansion data
PERGAMON
Solid State Communications 122 (2002) 671±676
www.elsevier.com/locate/ssc
The high temperature bulk modulus of aluminium: an assessment using experimental enthalpy and thermal expansion data S. Raju a,*, K. Sivasubramanian b, E. Mohandas a a
Physical Metallurgy Section, Materials Characterisation Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India b Safety Engineering Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India Received 12 October 2001; accepted 3 December 2001 by J.F. Sadoc
Abstract There exists a certain ambiguity with regard to the actual high-temperature bulk modulus of aluminium. In particular, there is a considerable disparity between various single crystal elastic constant measurements. This point has not been addressed on the theoretical front as well. In view of this situation, we seek to assess the existing bulk modulus data for their internal thermodynamic consistency and also obtain a reliable estimate using experimental data on thermal expansion, enthalpy and speci®c heat. The procedure adopted for this purpose makes use of a thermodynamic framework that relates thermal and elastic properties through Gruneisen's hypothesis. The present analysis suggests that the oft-cited data of Gerlich and Fisher and the older one due to Sutton are not fully consistent with the existing thermal property data. The more recent data of Tallon and Wolfenden, although in better consonance with the requirements of thermodynamic consistency, are also found to be less reliable. A fresh calculation of the bulk modulus is made such that the estimated values exhibit a high degree of thermodynamic legitimacy with the selected thermal property data. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 62.20.Dc; 65.40.-b; 65.40.Ba; 81.05.Bx Keywords: A. Aluminium; D. Thermodynamic properties; D. Heat capacity; D. Thermal expansion; D. Bulk modulus
1. Introduction It has been widely considered that aluminium is archetypal of simple sp-bonded metals in view of the fact that there exists an extensive database on its physical and chemical properties, and further that they are reasonably well understood in terms of appropriate conceptual frameworks. In the light of such a scenario, it is rather surprising to note that there is some controversy with regard to the high temperature bulk modulus of aluminium. In brief, the issue at hand may be addressed as follows. From Refs. [1,2] we could identify three major experimental studies on the determination of single crystal elastic constants of aluminium in the moderately low to hightemperature regime. The prevailing situation is graphically shown in Fig. 1. The oft-quoted measurement by Gerlich and Fisher [3] reported data on all the three second order * Corresponding author. Tel.: 191-4114-80306; fax: 191-411480081. E-mail address: [email protected] (S. Raju).
elastic constants cij, (ij: 11, 12 and 44) of high purity single crystal aluminium, in the temperature range 293±925 K. The adiabatic bulk modulus (BS) obtained in this study exhibits a smooth variation with temperature. In addition, it also shows a smooth continuity with the low-temperature single crystal data of Kamm and Alers [4]. A latter study by Tallon and Wolfenden [5] on single crystal aluminium in the temperature range 273±913 K, revealed a rather distinctly different behaviour for the temperature dependence of BS : The bulk modulus values reported in this study are higher than that of Gerlich and Fisher's [3], and further it appears, that their quoted c12 values in particular, apart from their inexplicable temperature dependence 2c12 =2Tp $ 0; are somewhat doubtful. Tallon [5,6] offers an apparent justi®cation for their observed bulk modulus versus temperature data, in terms of a general model for the isobaric volume dependence of bulk modulus. The third data set originating from the study of Sutton [7] using the composite oscillator technique in the temperature range 77±773 K, exhibits an altogether different trend. The BS values of Sutton in the low temperature range were questioned by Kamm and Alers [4];
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00517-8
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S. Raju et al. / Solid State Communications 122 (2002) 671±676
Fig. 1. The prevailing bulk modulus data on aluminium and their diverse temperature dependencies are shown. Note in particular, that both Gerlich±Fisher [3] and Tallon±Wolfenden [5] data, while exhibiting reasonable continuity with the low temperature data of Kamm and Alers [4]; are drastically different at high temperatures. The totally discordant behaviour of Sutton's data [7], as questioned by Kamm and Alers [4] is fully evident.
besides, they are not in agreement with other low-temperature results as well [6,8±10]. In addition, Wawra [11] has estimated the effective bulk modulus values from sound velocity data on polycrystal using Kroner's averaging procedure. This data suggests a sharper decrease with temperature than that advocated by Gerlich±Fisher, or by Tallon±Wolfenden, but less than that of Sutton. Further, McLellan and Ishikawa [12] have investigated the temperature dependence of shear moduli by measuring the longitudinal and transverse sound velocities using the pulse±echo technique in a thin MARZ grade aluminium wire. However, these authors have not tabulated their data to enable a comparison to be made with other values. Thus in summary, it can be said that the bulk modulus of aluminium in the high-temperature regime has not been established unequivocally. The present study is aimed at providing a critically assessed and internally consistent bulk modulus data in the temperature range 300±933 K. Our assessment procedure makes use of certain thermodynamic relationships between bulk modulus and other thermo-physical properties like enthalpy and thermal expansion [13,14]. The relevant theoretical framework needed for this purpose is brie¯y outlined below. For further details, the readers are referred to our recent papers [13,14]. 2. Assessment procedure Let us begin with the de®nition of isothermal bulk modulus BT BT 2V 2p=2VT :
1
BT V 22 F=2V 2 T :
2
Fig. 2. The temperature dependence of l aV =Cp ; as obtained from the selected thermal expansivity and speci®c heat data is graphically shown. Although, l exhibits a mild temperature dependence, it is rather small and hence ignored. It is also worth noting that by adopting the Cp recommendation of Brooks and Bingham [20] in place of Ditmars [16], the increase seen at high temperatures can further be minimised.
In Eqs. (1) and (2), p 2 2F=2Vp ; V and F stands for thermodynamic pressure, volume and Helmholtz free energy, respectively. Thus, a straightforward calculation of bulk modulus calls for an explicit formulation of the F; VT equation of state [15]. Although, such a direct method is highly desirable, we plan to adopt in the present study a different strategy, namely, the calculation of compressibility from thermal properties. This procedure is based on a simple thermodynamic framework, which in turn rests on an assumption proposed by Gruneisen in his pioneering work on thermal physics. Initially, it was assumed that the ratio (l ) of volume thermal expansivity (a V) to isobaric speci®c heat Cp is temperature independent. This assumption is obeyed to a good measure by a number of solids irrespective of their bonding diversities over a wide temperature range, starting from the Debye temperature (u D) to melting point Tm (see Fig. 2). Starting from this assumption, it is possible to derive thermodynamically consistent approximations relating thermal expansion and enthalpy with bulk modulus. A detailed account of this thermodynamic formalism has been published recently [13,14].
l aV =Cp :
3
VT V0 exp lDH:
4
BS T {gG = lV0 }exp 2lDH:
5
In the above expressions, l stands for the ratio of volume thermal expansivity (a V) to isobaric speci®c heat Cp ; and is taken to be temperature independent. BS T represents the adiabatic bulk modulus at temperature T, VT is the (molar)volume at T, V0 is the corresponding volume at a reference temperature T0. DH is the difference in enthalpy, HT 2 HT0 ; between the actual value HT and that at the reference
S. Raju et al. / Solid State Communications 122 (2002) 671±676
temperature H0 : g G is the thermal Gruneisen parameter, again taken to be temperature independent. The application potential of Eqs. (3)±(5) is as follows. For those systems in which l happens to be fairly a constant over a wide range of temperature, it is in principle possible to obtain a unique value for l by ®tting the experimental thermal expansivity, speci®c heat and bulk modulus data to Eqs. (3)±(5). In reality however, there will invariably be a small spread among the diverse estimates of l owing to uncertainties of various kind in the experimental data. Nevertheless, if the input data are of suf®ciently high quality, then one can estimate l to within a narrow margin. In the event of the availability of only partial or con¯icting data sets, such as the one encountered in the compressibility measurements of aluminium, the combined use of Eqs. (3)±(5) serves to obtain a fully optimised thermodynamic and elastic property data set. This requires a ®ne-tuning of the value of l from its initial or trial estimate through an iterative procedure. As a measure of ensuring internal consistency, one can back calculate the enthalpy using the computed bulk modulus and the experimental volume versus temperature data. The relevant expression is given below [13,14] HT H0 2 k{1 2 h12d };
6
where k B0 V0 {gG 1 2 d}21 ; d aV BS 21 2BS =2Tp and h VT =V0 : 3. Application to aluminium The thermodynamic properties of aluminium have been extensively investigated, analysed and assessed as well [16± 21]. In general, there is a good agreement among different estimates on speci®c heat, except at temperatures close to the melting point. This increase could have arisen from the proliferation of thermal vacancies at these temperatures [22]. In any case, for moderate to fairly high temperatures, there is very little to choose between different Cp recommendations. This point emerges clearly from the most comprehensive experimental study of the relative enthalpy of aluminium by Ditmars et al. [16]. In any case, the Cp data of Ditmars et al. [16] did not deviate by more than ^4% from rest of the literature data for a good part of the stability range of aluminium (see ®g. 5 of Ref. [16]). As for the measured enthalpy values are concerned, the quoted experimental uncertainty was even less # ^ 4 for the entire stability range of solid aluminium. In view of this, we have adopted in the present study, the tabulated enthalpy and speci®c heat data of Ditmars et al. [16]. It is also gratifying to note that this data set has subsequently been recommended by Desai as well [17]. The thermal expansion of aluminium has also been investigated extensively, both by X-ray and dilatometric techniques [23±28]. In the present study, we ®tted these data to obtain an optimised estimate of the thermal expansivity data
673
in the temperature range 300±933 K. By this procedure, we could identify and discard certain data set [27] that exhibited a signi®cant deviation from the common behaviour seen in other data sources. In addition, we also ensured the smooth continuity of the chosen high-temperature data with the corresponding assessed low-temperature (0±300 K) estimate of Kroger and Swenson [28]. The reference temperature (T0) is taken as 300 K and the corresponding volume is calculated from the accurate lattice parameter data [29,30]. The reference bulk modulus at 300 K is likewise taken as the average of a few standard recommendations [1,2]. The input data are listed in Table 1. In Fig. 2, the ratio l aV =Cp ; is plotted for the temperature range 300±933 K. As can be seen from this ®gure, this ratio is fairly temperature independent, save a small region near the melting point. If we choose the slightly higher estimates of Cp recommended by Brooks and Bingham [20] in place of Ditmars data [16], the increase in l seen at high temperatures can be minimised. Since, the variation in itself is rather small, it is justi®able to take this quantity as reasonably temperature independent. In the present study, we take the average, 3:1617 £ 1026 mol/J, as the representative of the direct experimental estimate. In Fig. 3, we have plotted the ln VT versus HT 2 H300 data. A remarkable linear correlation, in accordance with Eq. (4) is readily apparent. The value of l obtained from this correlation is 3:07645 £ 1026 mol/J. It is interesting to note that this value is well within the range of the direct experimental estimate (cf. Fig. 2). This fact certi®es to the internal consistency of our selected enthalpy and thermal expansion data. In the next step, we proceed to assess the two con¯icting experimental bulk modulus data [3,5] by ®tting them to Eq. (5) and judging the relative closeness of the resulting l values with the corresponding estimate obtained from Eq. (4). This analysis is shown in Fig. 4. It is clear from this ®gure, that both sets of data evince a good linear correlation with HT 2 H300 ; but the corresponding l values differ signi®cantly. While the l value of 3:8847 £ 1026 mol/J, obtained from the ®t of Tallon±Wolfenden's data is close to the direct experimental, as well as to the estimate from Eq. (4), the corresponding value of 10:459 £ 1026 mol/J, obtained from Gerlich±Fisher's data evinces very poor agreement. This conclusion holds inspite of all possible uncertainties in the concerned experimental data used for estimating l . The extent of disagreement between the l value of Gerlich±Fisher and the experimental ones is simply too much to be accounted for by any possible sources of random and systematic errors. It must also be mentioned that the Gruneisen parameter (g G) estimated from the intercept gG =lV0 of the ®t of Gerlich±Fisher BS to Eq. (5) turns out to be 7.98, an unusually high value. Therefore, we conclude that the compressibility data of Gerlich±Fisher is thermodynamically inconsistent with our selected speci®c heat and thermal expansion data. In a similar vein, it also emerges that Sutton's data likewise suffer from serious
7:62 £ 10 7:58 £ 10110 7:55 £ 10110 7:51 £ 10110 7:46 £ 10110 7:42 £ 10110 7:37 £ 10110 7:33 £ 10110 7:29 £ 10110 7:25 £ 10110 7:21 £ 10110 7:17 £ 10110 7:14 £ 10110 7:12 £ 10110 7:61 £ 10 7:53 £ 10110 7:44 £ 10110 7:34 £ 10110 7:25 £ 10110 7:14 £ 10110 7:04 £ 10110 6:93 £ 10110 6:83 £ 10110 6:72 £ 10110 6:61 £ 10110 6:50 £ 10110 6:40 £ 10110 6:33 £ 10110 7:52 £ 10 7:49 £ 10110 7:46 £ 10110 7:43 £ 10110 7:40 £ 10110 7:37 £ 10110 7:34 £ 10110 7:31 £ 10110 7:27 £ 10110 7:24 £ 10110 7:21 £ 10110 7:17 £ 10110 7:14 £ 10110 7:11 £ 10110 2:878 £ 10 2:900 £ 10206 2:931 £ 10206 2:965 £ 10206 3:000 £ 10206 3:033 £ 10206 3:067 £ 10206 3:108 £ 10206 3:159 £ 10206 3:226 £ 10206 3:315 £ 10206 3:418 £ 10206 3:538 £ 10206 3:621 £ 10206 24.25 25.11 25.78 26.34 26.84 27.35 27.89 28.47 29.10 29.80 30.56 31.50 32.61 33.50 300 350 400 450 500 550 600 650 700 750 800 850 900 933
a V (1/K) Cp (J/mol K)
6:978 £ 10 7:284 £ 10205 7:557 £ 10205 7:809 £ 10205 8:052 £ 10205 8:295 £ 10205 8:553 £ 10205 8:847 £ 10205 9:192 £ 10205 9:612 £ 10205 1:013 £ 10204 1:077 £ 10204 1:154 £ 10204 1:213 £ 10204
4584.30 5819.30 7092.50 8388.50 9725.20 11075.50 12460.00 13882.50 15308.00 16804.50 18289.00 19818.50 21430.00 22521.00
1:0001 £ 10 1:0036 £ 10205 1:0074 £ 10205 1:0113 £ 10205 1:0153 £ 10205 1:0194 £ 10205 1:0237 £ 10205 1:0282 £ 10205 1:0328 £ 10205 1:0377 £ 10205 1:0428 £ 10205 1:0483 £ 10205 1:0541 £ 10205 1:0582 £ 10205
BS (Pa) Tallon±Wolfenden
110 110
BS (Pa) Gerlich±Fisher BS (Pa) Present study
110 206
thermodynamic inconsistency. This point has already been brought out clearly in the study of Kamm and Alers [4]. It must also be emphasised that a somewhat better agreement seen with the data of Tallon±Wolfenden should not be taken to imply that, this data is in perfect accord with the requirements of thermodynamic consistency. As mentioned earlier, the temperature variation of c12 of Tallon and Wolfenden gives rise to some controversy. According to Tallon and Wolfenden their c11 and c44 variations are in reasonable agreement with Sutton's disputed data [7]; but however, no attempt is made to compare them with the immediately previous measurements of Gerlich and Fisher. Thus, in the ®nal analysis, we may conclude that there is an alarming degree of mismatch between the temperature dependencies of individual stiffness values of Gerlich± Fisher and Tallon±Wolfenden. Sometime ago, there was a suggestion by Hearmon [31], the noted compiler and assessor of elastic properties that much of the discrepancy between various measurements on aluminium stems from errors associated with the conversion of stiffness to compliance values or vice versa, depending on the type of measurement technique employed. While this is certainly one of the reasons, the magnitude of the mismatch in the stiffness values implies that there could be other contributing factors. In the light of foregone discussion, we recommend that it may be of some interest to measure afresh the elastic properties of single crystal aluminium.
205
H (J/mol)
V (m 3)
205
a V/Cp (mol/J)
Fig. 3. A plot of ln VT versus DH HT 2 H300 ; yielding a straight line and thus attesting the validity of Eq. (4).
T (K)
Table 1 A listing of thermal property data on aluminium taken from standard sources together with the estimated bulk modulus values
0.00 21.19 21.83 22.03 22.17 22.04 21.87 21.68 21.19 20.82 0.01 0.99 2.03 2.82
S. Raju et al. / Solid State Communications 122 (2002) 671±676 dH (%)
674
4. Estimation of bulk modulus Meanwhile, we proceed with our task of obtaining a thermodynamically consistent estimate of BS using Eq. (5). The input parameters required for this calculation are: gG 2:3; V300 1:00072 £ 1025 m 3; l 3:06956 £ 1026 mol/J. The g G value used here corresponds to 300 K and is based on experimental speci®c heat, bulk modulus and thermal expansivity data (see Table 1). It is also close to the value obtained in recent lattice dynamical study of Zoli and
S. Raju et al. / Solid State Communications 122 (2002) 671±676
Fig. 4. A plot of ln BS versus DH HT 2 H300 ; is presented for Gerlich±Fisher [3] and Tallon±Wolfenden [5] elastic constant data. Note that both these data display a good linear behaviour; however, with signi®cantly differing slopes.
Bortolani [15]. The selected value for l is based on ®ne tuning of its initial estimate, 3:07645 £ 1026 ; obtained from a ®t of experimental volume versus temperature data to Eq. (4). The calculated BS values are listed in Table 1. In Fig. 5, a comparison is made between our calculated values and that of Gerlich±Fisher and Tallon±Wolfenden. As can be seen from this ®gure, the calculated BS values fall in between these two experimental estimates at the lower end of the temperature scale and approach Tallon±Wolfenden data, at high temperatures. In the ®nal step, we use the estimated BS values to back calculate enthalpy using Eq. (6). The percentage error between the calculated enthalpy and its experimental counterpart dH 100 £ Hcal 2 Hexp =Hexp is plotted as a function of temperature in Fig. 6. Since dH is within a few percent, it is readily apparent, that our BS values satisfy the criterion of internal thermodynamic consistency. 5. Discussion The present analysis is based on a thermodynamic formalism, which is rigorous when the Gruneisen's assumption
Fig. 5. A comparison of estimated BS is made with the corresponding experimental measurements [3,5].
675
Fig. 6. The error dH 100 £ { Hcal 2 Hexp =Hexp }; between the calculated enthalpy using estimated BS in Eq. (5), and the assessed experimental values is plotted as a function of temperature. Note that not withstanding the limitations of our assessment procedure, the percentage error is rather small.
namely, the thermal expansivity to speci®c heat ratio (l ) is temperature dependent is valid. That this is approximately so for temperatures higher than the Debye characteristic temperature has amply been borne out by experimental data for a number of solids ([13], see in particular ®g. 1 of Ref. [14]). As mentioned earlier, the deviation from strict constancy observed at high temperatures owes predominantly to point defects. Nevertheless, as seen in the case of aluminium, the extent of this deviation is rather small and no more than the normal variation encountered in the case of conventional thermal Gruneisen parameter. Assuming an extremely improbable ^100% margin for l , which for aluminium falls in the range 3:6 £ 1027 to 3:6 £ 1025 ; we can certainly rule out both the Gerlich±Fisher and Sutton's measurements, purely for the lack of thermodynamic consistency with the selected thermal property data. We have explored the possibility of possible uncertainties in the thermodynamic data of aluminium. As mentioned earlier, even by taking into account, the somewhat higher (14% at 900 K) Cp values of Brooks and Bingham, the resulting estimate for l is not changed much to warrant a revised conclusion. As for our selected thermal expansion data are concerned, it is based on secure experimental data that is unlikely to suffer from any extraordinary sources of error. Therefore, we believe that our analysis, based on sound input thermal property data and assessment strategy certainly brings out the inadequacy of the existing elastic property measurements. Further, it is also heartening to note that our present estimate of BS compares favourably with the full ¯edged lattice dynamical estimate of Zoli and Bortolani [15] and the recent ab initio pseudo potential combined with the quasiharmonic formalism based results of Debernardi et al. [32]. This is shown in Fig. 7. In order to complete the assessment, we also include in this ®gure, the result of recent molecular dynamics simulation by Alper and Politzer [33]. While, the lattice dynamical values of Zoli and Bortolani [15] support a
676
S. Raju et al. / Solid State Communications 122 (2002) 671±676
modulus near the melting point, as suggested for example by Sutton.
References
Fig. 7. The estimated BS values are compared with some recent theoretical results [15,32,33]. Note that there is a reasonable degree of agreement with the lattice dynamical [15] and the pseudopotential-quasi-harmonic formalism based estimates [32]. The results of molecular dynamics simulation [33] exhibits however, a poor match with our results. Refer to text for details.
linear variation with temperature, the present thermodynamic estimate shows a mild curvature. Besides, by the way of comparing our estimated BS with that of Alper and Politzer [33], we note that their BS, while exhibiting a marked curvature with temperature deviates appreciably from the experimental data. The study of Debernardi et al. [32], on the other hand exhibits a certain curvature at high temperatures. It is rather unfortunate that in all these theoretical studies [15,32,33], a concerted attempt towards a detailed comparison of the calculated result with the available experimental data have not been made, with the result, that the actual high temperature bulk modulus behaviour of aluminium remains yet to be resolved. It is amidst this reality that we present our theoretical assessment with the hope that, its utility lies in providing a reliable ®rst order approximation to actual experimental values, an accurate reinvestigation of which is rather long due. 6. Conclusions In the present study an assessment of the prevailing contradictory estimates for the bulk modulus of aluminium has been made. By adopting an integrated treatment of thermal and elastic properties by means of a proven thermodynamic framework, we have calculated the adiabatic bulk modulus, for temperatures 300±933 K. The present results are closer to, but somewhat smaller than Tallon and Wolfenden data. Further, they do not exhibit a sharp drop in bulk
[1] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, Second ed, MIT Press, Cambridge, 1971 pp. 8±11. [2] R.F.S. Hearmon, in: K.H. Hellwege, A.M. Hellwege (Eds.), Elastic Constant Data From Landolt±Bornstein: New Series, Springer, Berlin, 1979. [3] D. Gerlich, E.S. Fisher, J. Phys. Chem. Solids 30 (1969) 1197. [4] G.N. Kamm, G.A. Alers, J. Appl. Phys. 35 (1964) 327. [5] J.L. Tallon, et al., J. Phys. Chem. Solids 40 (1979) 831. [6] J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837. [7] P.M. Sutton, Phys. Rev. 91 (1953) 816. [8] J. Vallin, et al., J. Appl. Phys. 35 (1964) 1825. [9] N.J. Trappeniers, et al., Physica 85B (1977) 33. [10] A. Ramakanth, et al., Ind. J. Phys. 49 (1975) 373. [11] H. Wawra, Z. Metallkd. 69 (1978) 518. [12] R.B. McLellan, T. Ishikawa, J. Phys. Chem. Solids 48 (1987) 603. [13] S. Raju, et al., Scripta Mater. 44 (2001) 269. [14] K. Sivasubramanian, et al., J. Eur. Ceram. Soc. 21 (2001) 1229. [15] M. Zoli, V. Bortoloni, J. Phys.: Condens. Matter 2 (1990) 525. [16] D.A. Ditmars, et al., Int. J. Thermophys. 6 (1985) 499. [17] P.D. Desai, Int. J. Thermophys. 8 (1987) 621. [18] O. Knacke, et al. (Eds.), Second ed, Thermochemical Properties of Inorganic Substances, vol. 1, Springer, Berlin, 1991, p. 19. [19] K. Wang, R.R. Reeber, Philos. Mag. 80A (2000) 1629. [20] C.R. Brooks, et al., J. Phys. Chem. Solids 29 (1968) 1553. [21] T.E. Pochapsky, Acta Metall. 1 (1953) 747. [22] Y.A. Kraftmakher, Phys. Rep. 299 (1998) 79. [23] R.O. Simmons, R.W. Balluf®, Phys. Rev. 117 (1960) 52. [24] B. Van Guerard, et al., Appl. Phys. 3 (1974) 37. [25] W.B. Pearson, A Handbook of Lattice Spacings and Structure of Metals and Alloys, Pergamon Press, London, 1958 p. 311, see many early references cited there in. [26] P.D. Pathak, R.G. Vasavada, J. Phys. C3 (1970) L44. [27] H. In-Kook-Suh, Y. Ohta, Waseda, J. Mater. Sci. 23 (1988) 757. [28] F.R. Kroeger, C.A. Swenson, J. Appl. Phys. 48 (1977) 853. [29] A.J.C. Wilson, Proc. Phys. Soc. 54 (1942) 487. [30] T. Heumann, Z. Naturforsch. Wiss. 32 (1944) 296 as cited in Ref. [25]. [31] R.F.S. Hearmon, Solid State Commun. 37 (1981) 915. [32] A. Debernardi, M. Alouani, H. Dreysse, Phys. Rev. B63 (2001) 064305. [33] H.E. Alper, P. Politzer, Int. J. Quant. Chem. 76 (2000) 670.
Solid State Communications 122 (2002) 671±676
www.elsevier.com/locate/ssc
The high temperature bulk modulus of aluminium: an assessment using experimental enthalpy and thermal expansion data S. Raju a,*, K. Sivasubramanian b, E. Mohandas a a
Physical Metallurgy Section, Materials Characterisation Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India b Safety Engineering Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India Received 12 October 2001; accepted 3 December 2001 by J.F. Sadoc
Abstract There exists a certain ambiguity with regard to the actual high-temperature bulk modulus of aluminium. In particular, there is a considerable disparity between various single crystal elastic constant measurements. This point has not been addressed on the theoretical front as well. In view of this situation, we seek to assess the existing bulk modulus data for their internal thermodynamic consistency and also obtain a reliable estimate using experimental data on thermal expansion, enthalpy and speci®c heat. The procedure adopted for this purpose makes use of a thermodynamic framework that relates thermal and elastic properties through Gruneisen's hypothesis. The present analysis suggests that the oft-cited data of Gerlich and Fisher and the older one due to Sutton are not fully consistent with the existing thermal property data. The more recent data of Tallon and Wolfenden, although in better consonance with the requirements of thermodynamic consistency, are also found to be less reliable. A fresh calculation of the bulk modulus is made such that the estimated values exhibit a high degree of thermodynamic legitimacy with the selected thermal property data. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 62.20.Dc; 65.40.-b; 65.40.Ba; 81.05.Bx Keywords: A. Aluminium; D. Thermodynamic properties; D. Heat capacity; D. Thermal expansion; D. Bulk modulus
1. Introduction It has been widely considered that aluminium is archetypal of simple sp-bonded metals in view of the fact that there exists an extensive database on its physical and chemical properties, and further that they are reasonably well understood in terms of appropriate conceptual frameworks. In the light of such a scenario, it is rather surprising to note that there is some controversy with regard to the high temperature bulk modulus of aluminium. In brief, the issue at hand may be addressed as follows. From Refs. [1,2] we could identify three major experimental studies on the determination of single crystal elastic constants of aluminium in the moderately low to hightemperature regime. The prevailing situation is graphically shown in Fig. 1. The oft-quoted measurement by Gerlich and Fisher [3] reported data on all the three second order * Corresponding author. Tel.: 191-4114-80306; fax: 191-411480081. E-mail address: [email protected] (S. Raju).
elastic constants cij, (ij: 11, 12 and 44) of high purity single crystal aluminium, in the temperature range 293±925 K. The adiabatic bulk modulus (BS) obtained in this study exhibits a smooth variation with temperature. In addition, it also shows a smooth continuity with the low-temperature single crystal data of Kamm and Alers [4]. A latter study by Tallon and Wolfenden [5] on single crystal aluminium in the temperature range 273±913 K, revealed a rather distinctly different behaviour for the temperature dependence of BS : The bulk modulus values reported in this study are higher than that of Gerlich and Fisher's [3], and further it appears, that their quoted c12 values in particular, apart from their inexplicable temperature dependence 2c12 =2Tp $ 0; are somewhat doubtful. Tallon [5,6] offers an apparent justi®cation for their observed bulk modulus versus temperature data, in terms of a general model for the isobaric volume dependence of bulk modulus. The third data set originating from the study of Sutton [7] using the composite oscillator technique in the temperature range 77±773 K, exhibits an altogether different trend. The BS values of Sutton in the low temperature range were questioned by Kamm and Alers [4];
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00517-8
672
S. Raju et al. / Solid State Communications 122 (2002) 671±676
Fig. 1. The prevailing bulk modulus data on aluminium and their diverse temperature dependencies are shown. Note in particular, that both Gerlich±Fisher [3] and Tallon±Wolfenden [5] data, while exhibiting reasonable continuity with the low temperature data of Kamm and Alers [4]; are drastically different at high temperatures. The totally discordant behaviour of Sutton's data [7], as questioned by Kamm and Alers [4] is fully evident.
besides, they are not in agreement with other low-temperature results as well [6,8±10]. In addition, Wawra [11] has estimated the effective bulk modulus values from sound velocity data on polycrystal using Kroner's averaging procedure. This data suggests a sharper decrease with temperature than that advocated by Gerlich±Fisher, or by Tallon±Wolfenden, but less than that of Sutton. Further, McLellan and Ishikawa [12] have investigated the temperature dependence of shear moduli by measuring the longitudinal and transverse sound velocities using the pulse±echo technique in a thin MARZ grade aluminium wire. However, these authors have not tabulated their data to enable a comparison to be made with other values. Thus in summary, it can be said that the bulk modulus of aluminium in the high-temperature regime has not been established unequivocally. The present study is aimed at providing a critically assessed and internally consistent bulk modulus data in the temperature range 300±933 K. Our assessment procedure makes use of certain thermodynamic relationships between bulk modulus and other thermo-physical properties like enthalpy and thermal expansion [13,14]. The relevant theoretical framework needed for this purpose is brie¯y outlined below. For further details, the readers are referred to our recent papers [13,14]. 2. Assessment procedure Let us begin with the de®nition of isothermal bulk modulus BT BT 2V 2p=2VT :
1
BT V 22 F=2V 2 T :
2
Fig. 2. The temperature dependence of l aV =Cp ; as obtained from the selected thermal expansivity and speci®c heat data is graphically shown. Although, l exhibits a mild temperature dependence, it is rather small and hence ignored. It is also worth noting that by adopting the Cp recommendation of Brooks and Bingham [20] in place of Ditmars [16], the increase seen at high temperatures can further be minimised.
In Eqs. (1) and (2), p 2 2F=2Vp ; V and F stands for thermodynamic pressure, volume and Helmholtz free energy, respectively. Thus, a straightforward calculation of bulk modulus calls for an explicit formulation of the F; VT equation of state [15]. Although, such a direct method is highly desirable, we plan to adopt in the present study a different strategy, namely, the calculation of compressibility from thermal properties. This procedure is based on a simple thermodynamic framework, which in turn rests on an assumption proposed by Gruneisen in his pioneering work on thermal physics. Initially, it was assumed that the ratio (l ) of volume thermal expansivity (a V) to isobaric speci®c heat Cp is temperature independent. This assumption is obeyed to a good measure by a number of solids irrespective of their bonding diversities over a wide temperature range, starting from the Debye temperature (u D) to melting point Tm (see Fig. 2). Starting from this assumption, it is possible to derive thermodynamically consistent approximations relating thermal expansion and enthalpy with bulk modulus. A detailed account of this thermodynamic formalism has been published recently [13,14].
l aV =Cp :
3
VT V0 exp lDH:
4
BS T {gG = lV0 }exp 2lDH:
5
In the above expressions, l stands for the ratio of volume thermal expansivity (a V) to isobaric speci®c heat Cp ; and is taken to be temperature independent. BS T represents the adiabatic bulk modulus at temperature T, VT is the (molar)volume at T, V0 is the corresponding volume at a reference temperature T0. DH is the difference in enthalpy, HT 2 HT0 ; between the actual value HT and that at the reference
S. Raju et al. / Solid State Communications 122 (2002) 671±676
temperature H0 : g G is the thermal Gruneisen parameter, again taken to be temperature independent. The application potential of Eqs. (3)±(5) is as follows. For those systems in which l happens to be fairly a constant over a wide range of temperature, it is in principle possible to obtain a unique value for l by ®tting the experimental thermal expansivity, speci®c heat and bulk modulus data to Eqs. (3)±(5). In reality however, there will invariably be a small spread among the diverse estimates of l owing to uncertainties of various kind in the experimental data. Nevertheless, if the input data are of suf®ciently high quality, then one can estimate l to within a narrow margin. In the event of the availability of only partial or con¯icting data sets, such as the one encountered in the compressibility measurements of aluminium, the combined use of Eqs. (3)±(5) serves to obtain a fully optimised thermodynamic and elastic property data set. This requires a ®ne-tuning of the value of l from its initial or trial estimate through an iterative procedure. As a measure of ensuring internal consistency, one can back calculate the enthalpy using the computed bulk modulus and the experimental volume versus temperature data. The relevant expression is given below [13,14] HT H0 2 k{1 2 h12d };
6
where k B0 V0 {gG 1 2 d}21 ; d aV BS 21 2BS =2Tp and h VT =V0 : 3. Application to aluminium The thermodynamic properties of aluminium have been extensively investigated, analysed and assessed as well [16± 21]. In general, there is a good agreement among different estimates on speci®c heat, except at temperatures close to the melting point. This increase could have arisen from the proliferation of thermal vacancies at these temperatures [22]. In any case, for moderate to fairly high temperatures, there is very little to choose between different Cp recommendations. This point emerges clearly from the most comprehensive experimental study of the relative enthalpy of aluminium by Ditmars et al. [16]. In any case, the Cp data of Ditmars et al. [16] did not deviate by more than ^4% from rest of the literature data for a good part of the stability range of aluminium (see ®g. 5 of Ref. [16]). As for the measured enthalpy values are concerned, the quoted experimental uncertainty was even less # ^ 4 for the entire stability range of solid aluminium. In view of this, we have adopted in the present study, the tabulated enthalpy and speci®c heat data of Ditmars et al. [16]. It is also gratifying to note that this data set has subsequently been recommended by Desai as well [17]. The thermal expansion of aluminium has also been investigated extensively, both by X-ray and dilatometric techniques [23±28]. In the present study, we ®tted these data to obtain an optimised estimate of the thermal expansivity data
673
in the temperature range 300±933 K. By this procedure, we could identify and discard certain data set [27] that exhibited a signi®cant deviation from the common behaviour seen in other data sources. In addition, we also ensured the smooth continuity of the chosen high-temperature data with the corresponding assessed low-temperature (0±300 K) estimate of Kroger and Swenson [28]. The reference temperature (T0) is taken as 300 K and the corresponding volume is calculated from the accurate lattice parameter data [29,30]. The reference bulk modulus at 300 K is likewise taken as the average of a few standard recommendations [1,2]. The input data are listed in Table 1. In Fig. 2, the ratio l aV =Cp ; is plotted for the temperature range 300±933 K. As can be seen from this ®gure, this ratio is fairly temperature independent, save a small region near the melting point. If we choose the slightly higher estimates of Cp recommended by Brooks and Bingham [20] in place of Ditmars data [16], the increase in l seen at high temperatures can be minimised. Since, the variation in itself is rather small, it is justi®able to take this quantity as reasonably temperature independent. In the present study, we take the average, 3:1617 £ 1026 mol/J, as the representative of the direct experimental estimate. In Fig. 3, we have plotted the ln VT versus HT 2 H300 data. A remarkable linear correlation, in accordance with Eq. (4) is readily apparent. The value of l obtained from this correlation is 3:07645 £ 1026 mol/J. It is interesting to note that this value is well within the range of the direct experimental estimate (cf. Fig. 2). This fact certi®es to the internal consistency of our selected enthalpy and thermal expansion data. In the next step, we proceed to assess the two con¯icting experimental bulk modulus data [3,5] by ®tting them to Eq. (5) and judging the relative closeness of the resulting l values with the corresponding estimate obtained from Eq. (4). This analysis is shown in Fig. 4. It is clear from this ®gure, that both sets of data evince a good linear correlation with HT 2 H300 ; but the corresponding l values differ signi®cantly. While the l value of 3:8847 £ 1026 mol/J, obtained from the ®t of Tallon±Wolfenden's data is close to the direct experimental, as well as to the estimate from Eq. (4), the corresponding value of 10:459 £ 1026 mol/J, obtained from Gerlich±Fisher's data evinces very poor agreement. This conclusion holds inspite of all possible uncertainties in the concerned experimental data used for estimating l . The extent of disagreement between the l value of Gerlich±Fisher and the experimental ones is simply too much to be accounted for by any possible sources of random and systematic errors. It must also be mentioned that the Gruneisen parameter (g G) estimated from the intercept gG =lV0 of the ®t of Gerlich±Fisher BS to Eq. (5) turns out to be 7.98, an unusually high value. Therefore, we conclude that the compressibility data of Gerlich±Fisher is thermodynamically inconsistent with our selected speci®c heat and thermal expansion data. In a similar vein, it also emerges that Sutton's data likewise suffer from serious
7:62 £ 10 7:58 £ 10110 7:55 £ 10110 7:51 £ 10110 7:46 £ 10110 7:42 £ 10110 7:37 £ 10110 7:33 £ 10110 7:29 £ 10110 7:25 £ 10110 7:21 £ 10110 7:17 £ 10110 7:14 £ 10110 7:12 £ 10110 7:61 £ 10 7:53 £ 10110 7:44 £ 10110 7:34 £ 10110 7:25 £ 10110 7:14 £ 10110 7:04 £ 10110 6:93 £ 10110 6:83 £ 10110 6:72 £ 10110 6:61 £ 10110 6:50 £ 10110 6:40 £ 10110 6:33 £ 10110 7:52 £ 10 7:49 £ 10110 7:46 £ 10110 7:43 £ 10110 7:40 £ 10110 7:37 £ 10110 7:34 £ 10110 7:31 £ 10110 7:27 £ 10110 7:24 £ 10110 7:21 £ 10110 7:17 £ 10110 7:14 £ 10110 7:11 £ 10110 2:878 £ 10 2:900 £ 10206 2:931 £ 10206 2:965 £ 10206 3:000 £ 10206 3:033 £ 10206 3:067 £ 10206 3:108 £ 10206 3:159 £ 10206 3:226 £ 10206 3:315 £ 10206 3:418 £ 10206 3:538 £ 10206 3:621 £ 10206 24.25 25.11 25.78 26.34 26.84 27.35 27.89 28.47 29.10 29.80 30.56 31.50 32.61 33.50 300 350 400 450 500 550 600 650 700 750 800 850 900 933
a V (1/K) Cp (J/mol K)
6:978 £ 10 7:284 £ 10205 7:557 £ 10205 7:809 £ 10205 8:052 £ 10205 8:295 £ 10205 8:553 £ 10205 8:847 £ 10205 9:192 £ 10205 9:612 £ 10205 1:013 £ 10204 1:077 £ 10204 1:154 £ 10204 1:213 £ 10204
4584.30 5819.30 7092.50 8388.50 9725.20 11075.50 12460.00 13882.50 15308.00 16804.50 18289.00 19818.50 21430.00 22521.00
1:0001 £ 10 1:0036 £ 10205 1:0074 £ 10205 1:0113 £ 10205 1:0153 £ 10205 1:0194 £ 10205 1:0237 £ 10205 1:0282 £ 10205 1:0328 £ 10205 1:0377 £ 10205 1:0428 £ 10205 1:0483 £ 10205 1:0541 £ 10205 1:0582 £ 10205
BS (Pa) Tallon±Wolfenden
110 110
BS (Pa) Gerlich±Fisher BS (Pa) Present study
110 206
thermodynamic inconsistency. This point has already been brought out clearly in the study of Kamm and Alers [4]. It must also be emphasised that a somewhat better agreement seen with the data of Tallon±Wolfenden should not be taken to imply that, this data is in perfect accord with the requirements of thermodynamic consistency. As mentioned earlier, the temperature variation of c12 of Tallon and Wolfenden gives rise to some controversy. According to Tallon and Wolfenden their c11 and c44 variations are in reasonable agreement with Sutton's disputed data [7]; but however, no attempt is made to compare them with the immediately previous measurements of Gerlich and Fisher. Thus, in the ®nal analysis, we may conclude that there is an alarming degree of mismatch between the temperature dependencies of individual stiffness values of Gerlich± Fisher and Tallon±Wolfenden. Sometime ago, there was a suggestion by Hearmon [31], the noted compiler and assessor of elastic properties that much of the discrepancy between various measurements on aluminium stems from errors associated with the conversion of stiffness to compliance values or vice versa, depending on the type of measurement technique employed. While this is certainly one of the reasons, the magnitude of the mismatch in the stiffness values implies that there could be other contributing factors. In the light of foregone discussion, we recommend that it may be of some interest to measure afresh the elastic properties of single crystal aluminium.
205
H (J/mol)
V (m 3)
205
a V/Cp (mol/J)
Fig. 3. A plot of ln VT versus DH HT 2 H300 ; yielding a straight line and thus attesting the validity of Eq. (4).
T (K)
Table 1 A listing of thermal property data on aluminium taken from standard sources together with the estimated bulk modulus values
0.00 21.19 21.83 22.03 22.17 22.04 21.87 21.68 21.19 20.82 0.01 0.99 2.03 2.82
S. Raju et al. / Solid State Communications 122 (2002) 671±676 dH (%)
674
4. Estimation of bulk modulus Meanwhile, we proceed with our task of obtaining a thermodynamically consistent estimate of BS using Eq. (5). The input parameters required for this calculation are: gG 2:3; V300 1:00072 £ 1025 m 3; l 3:06956 £ 1026 mol/J. The g G value used here corresponds to 300 K and is based on experimental speci®c heat, bulk modulus and thermal expansivity data (see Table 1). It is also close to the value obtained in recent lattice dynamical study of Zoli and
S. Raju et al. / Solid State Communications 122 (2002) 671±676
Fig. 4. A plot of ln BS versus DH HT 2 H300 ; is presented for Gerlich±Fisher [3] and Tallon±Wolfenden [5] elastic constant data. Note that both these data display a good linear behaviour; however, with signi®cantly differing slopes.
Bortolani [15]. The selected value for l is based on ®ne tuning of its initial estimate, 3:07645 £ 1026 ; obtained from a ®t of experimental volume versus temperature data to Eq. (4). The calculated BS values are listed in Table 1. In Fig. 5, a comparison is made between our calculated values and that of Gerlich±Fisher and Tallon±Wolfenden. As can be seen from this ®gure, the calculated BS values fall in between these two experimental estimates at the lower end of the temperature scale and approach Tallon±Wolfenden data, at high temperatures. In the ®nal step, we use the estimated BS values to back calculate enthalpy using Eq. (6). The percentage error between the calculated enthalpy and its experimental counterpart dH 100 £ Hcal 2 Hexp =Hexp is plotted as a function of temperature in Fig. 6. Since dH is within a few percent, it is readily apparent, that our BS values satisfy the criterion of internal thermodynamic consistency. 5. Discussion The present analysis is based on a thermodynamic formalism, which is rigorous when the Gruneisen's assumption
Fig. 5. A comparison of estimated BS is made with the corresponding experimental measurements [3,5].
675
Fig. 6. The error dH 100 £ { Hcal 2 Hexp =Hexp }; between the calculated enthalpy using estimated BS in Eq. (5), and the assessed experimental values is plotted as a function of temperature. Note that not withstanding the limitations of our assessment procedure, the percentage error is rather small.
namely, the thermal expansivity to speci®c heat ratio (l ) is temperature dependent is valid. That this is approximately so for temperatures higher than the Debye characteristic temperature has amply been borne out by experimental data for a number of solids ([13], see in particular ®g. 1 of Ref. [14]). As mentioned earlier, the deviation from strict constancy observed at high temperatures owes predominantly to point defects. Nevertheless, as seen in the case of aluminium, the extent of this deviation is rather small and no more than the normal variation encountered in the case of conventional thermal Gruneisen parameter. Assuming an extremely improbable ^100% margin for l , which for aluminium falls in the range 3:6 £ 1027 to 3:6 £ 1025 ; we can certainly rule out both the Gerlich±Fisher and Sutton's measurements, purely for the lack of thermodynamic consistency with the selected thermal property data. We have explored the possibility of possible uncertainties in the thermodynamic data of aluminium. As mentioned earlier, even by taking into account, the somewhat higher (14% at 900 K) Cp values of Brooks and Bingham, the resulting estimate for l is not changed much to warrant a revised conclusion. As for our selected thermal expansion data are concerned, it is based on secure experimental data that is unlikely to suffer from any extraordinary sources of error. Therefore, we believe that our analysis, based on sound input thermal property data and assessment strategy certainly brings out the inadequacy of the existing elastic property measurements. Further, it is also heartening to note that our present estimate of BS compares favourably with the full ¯edged lattice dynamical estimate of Zoli and Bortolani [15] and the recent ab initio pseudo potential combined with the quasiharmonic formalism based results of Debernardi et al. [32]. This is shown in Fig. 7. In order to complete the assessment, we also include in this ®gure, the result of recent molecular dynamics simulation by Alper and Politzer [33]. While, the lattice dynamical values of Zoli and Bortolani [15] support a
676
S. Raju et al. / Solid State Communications 122 (2002) 671±676
modulus near the melting point, as suggested for example by Sutton.
References
Fig. 7. The estimated BS values are compared with some recent theoretical results [15,32,33]. Note that there is a reasonable degree of agreement with the lattice dynamical [15] and the pseudopotential-quasi-harmonic formalism based estimates [32]. The results of molecular dynamics simulation [33] exhibits however, a poor match with our results. Refer to text for details.
linear variation with temperature, the present thermodynamic estimate shows a mild curvature. Besides, by the way of comparing our estimated BS with that of Alper and Politzer [33], we note that their BS, while exhibiting a marked curvature with temperature deviates appreciably from the experimental data. The study of Debernardi et al. [32], on the other hand exhibits a certain curvature at high temperatures. It is rather unfortunate that in all these theoretical studies [15,32,33], a concerted attempt towards a detailed comparison of the calculated result with the available experimental data have not been made, with the result, that the actual high temperature bulk modulus behaviour of aluminium remains yet to be resolved. It is amidst this reality that we present our theoretical assessment with the hope that, its utility lies in providing a reliable ®rst order approximation to actual experimental values, an accurate reinvestigation of which is rather long due. 6. Conclusions In the present study an assessment of the prevailing contradictory estimates for the bulk modulus of aluminium has been made. By adopting an integrated treatment of thermal and elastic properties by means of a proven thermodynamic framework, we have calculated the adiabatic bulk modulus, for temperatures 300±933 K. The present results are closer to, but somewhat smaller than Tallon and Wolfenden data. Further, they do not exhibit a sharp drop in bulk
[1] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, Second ed, MIT Press, Cambridge, 1971 pp. 8±11. [2] R.F.S. Hearmon, in: K.H. Hellwege, A.M. Hellwege (Eds.), Elastic Constant Data From Landolt±Bornstein: New Series, Springer, Berlin, 1979. [3] D. Gerlich, E.S. Fisher, J. Phys. Chem. Solids 30 (1969) 1197. [4] G.N. Kamm, G.A. Alers, J. Appl. Phys. 35 (1964) 327. [5] J.L. Tallon, et al., J. Phys. Chem. Solids 40 (1979) 831. [6] J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837. [7] P.M. Sutton, Phys. Rev. 91 (1953) 816. [8] J. Vallin, et al., J. Appl. Phys. 35 (1964) 1825. [9] N.J. Trappeniers, et al., Physica 85B (1977) 33. [10] A. Ramakanth, et al., Ind. J. Phys. 49 (1975) 373. [11] H. Wawra, Z. Metallkd. 69 (1978) 518. [12] R.B. McLellan, T. Ishikawa, J. Phys. Chem. Solids 48 (1987) 603. [13] S. Raju, et al., Scripta Mater. 44 (2001) 269. [14] K. Sivasubramanian, et al., J. Eur. Ceram. Soc. 21 (2001) 1229. [15] M. Zoli, V. Bortoloni, J. Phys.: Condens. Matter 2 (1990) 525. [16] D.A. Ditmars, et al., Int. J. Thermophys. 6 (1985) 499. [17] P.D. Desai, Int. J. Thermophys. 8 (1987) 621. [18] O. Knacke, et al. (Eds.), Second ed, Thermochemical Properties of Inorganic Substances, vol. 1, Springer, Berlin, 1991, p. 19. [19] K. Wang, R.R. Reeber, Philos. Mag. 80A (2000) 1629. [20] C.R. Brooks, et al., J. Phys. Chem. Solids 29 (1968) 1553. [21] T.E. Pochapsky, Acta Metall. 1 (1953) 747. [22] Y.A. Kraftmakher, Phys. Rep. 299 (1998) 79. [23] R.O. Simmons, R.W. Balluf®, Phys. Rev. 117 (1960) 52. [24] B. Van Guerard, et al., Appl. Phys. 3 (1974) 37. [25] W.B. Pearson, A Handbook of Lattice Spacings and Structure of Metals and Alloys, Pergamon Press, London, 1958 p. 311, see many early references cited there in. [26] P.D. Pathak, R.G. Vasavada, J. Phys. C3 (1970) L44. [27] H. In-Kook-Suh, Y. Ohta, Waseda, J. Mater. Sci. 23 (1988) 757. [28] F.R. Kroeger, C.A. Swenson, J. Appl. Phys. 48 (1977) 853. [29] A.J.C. Wilson, Proc. Phys. Soc. 54 (1942) 487. [30] T. Heumann, Z. Naturforsch. Wiss. 32 (1944) 296 as cited in Ref. [25]. [31] R.F.S. Hearmon, Solid State Commun. 37 (1981) 915. [32] A. Debernardi, M. Alouani, H. Dreysse, Phys. Rev. B63 (2001) 064305. [33] H.E. Alper, P. Politzer, Int. J. Quant. Chem. 76 (2000) 670.