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Thermal expansion of β-rhombohedral boron
Journal of Alloys and Compounds 267 (1998) 54–58
L
Thermal expansion of b-rhombohedral boron a b ¨ a , *, Bertil Lonnberg ¨ Torsten Lundstrom , Josef Bauer a
˚ ¨ Laboratory, Institute of Chemistry, Uppsala University, Box 538, SE-751 21 Uppsala, Sweden Department of Inorganic Chemistry, The Angstrom b University of Rennes 1, LCSIM /CMR 6511, Av. de General Leclerc, F-35042 Rennes, France Received 24 October 1997
Abstract The cell dimensions of b-rhombohedral boron have been measured in the temperature range 10–1273 K, using X-ray diffraction. The linear thermal expansion coefficients (al 5(l2l 298 ) /l 298 (T2T 298 )) were determined along the a and c directions. It was found that brhombohedral boron is anisotropic, displaying a significantly larger thermal expansion coefficient in the c direction than in the a direction. The average linear thermal expansion coefficient is 6.4310 26 K 21 in the temperature range 293–1023 K. 1998 Elsevier Science S.A. Keywords: Thermal expansion; b-Rhombohedral boron
1. Introduction The high-temperature modification of boron, b-rhombohedral (b-rh) boron, displays very high bonding strength both within and between the boron icosahedra, which form a rigid three-dimensional boron network. The high bond strength is reflected in some physical properties, e.g. high melting point, high Debye temperature, great hardness and low thermal expansion. Thermal expansion data on b-rh boron are scarce because of the relatively great difficulties in providing high-purity boron, its great reactivity at high temperatures, as well as difficulties in fabricating the samples into shapes suitable for analysis, in particular if dilatometer methods are to be used. Furthermore, since most of the thermal expansion studies have been performed using a dilatometer no reliable information regarding the thermal expansion anisotropy of the material was available. The present investigation was initiated to study the thermal anisotropy of b-rh boron, using X-ray powder diffraction. The investigation was performed in the temperature range 10–1273 K. It was earlier reported that b-rh boron has a phase transition in the temperature range 150–180 K, resulting in discontinuities in its elastic, optical, photoelectric, magnetic properties, etc. [1]. Although it was suggested that the transition is characterised by changes in the electronic states and not an ordinary structural transition, it is interesting to establish whether *Corresponding author. Tel.: 146 501 77206; fax: 146 501 10660. 0925-8388 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 97 )00545-8
the cell dimensions and thermal expansion also display discontinuities in the temperature range 10–298 K. Therefore, in this temperature range the measurements were carried out at narrow intervals.
2. Experimental The experiments were carried out using zone-refined boron (certified purity 99.995% with carbon as main impurity) from Wacker Chemie, Munich, Germany. The measurements were performed in the temperature range 10–1273 K, using three different diffraction instruments. The lattice parameters at ambient temperature were de¨ termined using an ordinary XDC 1000 Guinier-Hagg focusing camera with Cr Ka 1 radiation and ultra-pure ˚ [2,3]) as internal calibration subsilicon (a55.430940 A stance. The cell dimensions of the b-rh boron, used for the ˚ measurements, were a510.9330(5) and c523.825(2) A, respectively, at ambient temperature. These parameters were used to calibrate the parameters obtained in the highand low-temperature studies, respectively. The linear thermal expansion coefficients, aa and ac , along the a and c axis, respectively, were calculated using the formula:
al 5 (l 2 l 298 ) /l 298 (T 2 T 298 )
(1)
where l is the lattice parameter at temperature T, and l 298 is the lattice parameter at ambient temperature. The average
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
linear thermal expansion coefficient, am , was calculated using the formula:
am 5 (2aa 1 ac ) / 3
(2)
2.1. Low-temperature studies The unit cell dimensions between 10 and 298 K were measured using a low-temperature Guinier diffractometer / camera, equipped with a low-temperature closed-cycle helium cryostat and with strictly monochromatic Cu Ka 1 radiation (Ge(111) monochromator) [4]. Silicon was used as internal calibration substance. The lattice parameters of silicon at various temperatures were calculated from the data given by Batchelder and Simmons [5] and from the NBS certificate [3]. The boron was thoroughly mixed with the internal standard silicon powder and milled in a boron carbide mortar. The 2u positions were corrected according to the parabolic function [6]:
Dcorr 5 A ? (2uobs )2 1 B ? (2u ) obs 1 C
(3)
2ucorr 5 2uobs 2 Dcorr
(4)
where 2ucorr and 2uobs are the corrected and observed peak positions, respectively. The C parameter corresponds to the position of the true ‘zero’. The corrected diffraction patterns were analysed using the Rietveld method, utilising the program DBWS-9006 as modified for Guinier geometry, described by Wiles and Young [7]. Two peaks at low angles were omitted since they were too asymmetric to be measured properly. The X-ray measurements were performed between 7 and 488 in u in steps of 0.018 with a measuring time of 10 s per step. A total time of 12 h was used for each scan. A total of 18 parameters were refined for each measurement, namely two scale factors, the A, B, and C coefficients, the a and c lattice parameters of boron, six U, V, and W parameters (three for boron and three for silicon), one asymmetry parameter for boron, two mixing parameters (one for each phase) for the pseudo-Voigt function and two overall temperature factors. The standard deviations were 0.0002 ˚ for the a and c parameter, respectively. and 0.0008 A
55
were, however, observed, which indicates that no changes regarding composition and impurities had occurred in the sample during the heating. All unit cell dimensions were least-squares refined using the local Uppsala program UNITCELL [11]. The standard deviations fell within the ˚ for the a parameter and 0.05–0.18 A ˚ range 0.002–0.005 A for the c parameter. The linear thermal expansion coefficients, aa and ac , and the average linear thermal expansion coefficient, am , were calculated between 298 and 1273 K. At temperatures above 1273 K signs of a reaction between the alumina mat and boron were detected.
3. Results and discussion The lattice parameters vs. temperature are given in Fig. 1. It is seen that a non-linear increase in lattice parameters occurs with temperature. Despite using different measuring procedures and instruments, the data obtained in the lowtemperature study agree well at ambient temperature with those measured at high temperature. A slightly larger scatter of the data occurs for the a axis at temperatures near the ambient temperature. The scatter may, at least partially, be due to the temperature calibration, which is less accurate near the ambient temperature. A magnification of the cell dimension vs. temperature curves in the temperature range 10–298 K is given in Fig. 2. The curves were obtained by fitting second-order
2.2. High temperature studies The cell dimensions above room temperature were ¨ high-temperature camera determined using a Guinier-Hagg [8]. The sample was placed on a mat of aluminium oxide whiskers. Since the thermal expansion coefficients of aluminium oxide are well known [9,10], it was used as internal calibration substance. The eight strongest lines were used for this purpose. The specimens were heated in vacuum (10 24 Pa) and the temperature measured with a Pt / Pt10% thermocouple, placed close to the sample. The room-temperature lattice parameters were determined prior to and after each run. No significant changes
Fig. 1. (a) The a axis vs. temperature (10–1273 K). (b) The c axis vs. temperature (10–1273 K).
56
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
Fig. 2. (a) The a axis vs. temperature (10–298 K). (b) The c axis vs. temperature (10–298 K).
polynomials to the experimental points. It is demonstrated that the lattice parameters increase very slowly with temperature in this temperature range. However, it is evident that the a parameter remains nearly constant between 10 and 200 K. No discontinuities in the data in the range 150–180 K could be established. The phase transition discussed in Ref. [1] is accordingly not of crystallographic nature. The thermal expansion vs. temperature is shown in Fig. 3. The curves are composites of the low- and hightemperature measurements and were obtained by leastsquares fitting to the experimental data. It is found that the
Fig. 3. Thermal expansion (Da /a 10K , Dc /c 10K ) vs. temperature (10–1273 K).
larger thermal expansion is obtained along the c axis than along the a axis. The linear thermal expansion coefficients, aa and ac , at temperatures above 298 K are displayed in Fig. 4. Naturally, the thermal expansion coefficient is also significantly larger in the c direction than in the a direction. The thermal expansion at low temperatures displayed similar behaviour. The linear thermal expansion coefficients in the temperature range 10–298 K were aa 50.98310 26 K 21 and ac 51.82310 26 K 21 . The only earlier attempt to determine the anisotropy of the thermal expansion of b-rh boron [12] is at complete invariance with the results presented above. It was reported in the previous study that, in the temperature range 400– 1473 K, the thermal expansion coefficient aa varies between 210310 26 and 125310 26 K 21 and ac between 210310 26 and 185310 26 K 21 . Furthermore, the variation of both coefficients is periodic within the temperature range mentioned. This is in itself extremely remarkable for such a rigid material as b-rh boron, and raises doubts as to the correctness of the measurements. These were performed on b-rh boron of undefined purity and density at a vacuum no better than 10 22 Pa. Only two high-angle diffraction lines (6, 4, 16) and (6, 1, 23) were utilised for the two parameters to be determined, while our cell dimension measurements are based on 25–30 diffraction lines. The use of only two diffraction lines involves a serious limitation of the accuracy due to the frequent overlaps of diffraction lines in the line-rich diffraction pattern of b-rh boron, which originates in the large unit cell, the use of copper radiation and the probably not strictly a 1 monochromated radiation used. In view of these facts. the present results seem to be considerably more reliable than those reported earlier [12]. A brief description of the crystal structure of b-rh boron [13–15] is presented here to discuss the influence of structure and interatomic distances on the anisotropy of the thermal expansion. The structure can be described in terms of B 84 , B 10 , and non-icosahedral boron atoms. The B 84
Fig. 4. Linear thermal expansion coefficients (aa , ac ) vs. temperature (298–1273 K).
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
unit consists of one central boron icosahedron and 12 half-icosahedra. The vertex atom of each half-icosahedron is directly bonded along the quasi-five-fold axis to one of the vertex atoms of the central icosahedron. The linking is such that, assuming ideal, regular icosahedra, there is a mirror plane between the two vertex atoms. The halficosahedra can be divided into two types with respect to the type of linking to other B 84 units, in which they participate. The first type is represented by the halficosahedra, whose pentagonal axes are oriented not far from the c direction, three in the positive and three in the negative c direction. These half-icosahedra are linked to the same type of half-icosahedra in neighbouring B 84 units, forming new complete boron icosahedra. The other type of half-icosahedra in the B 84 unit are those with the top atom lying nearly in the same xy plane as the central icosahedron. They have their quasi-five-fold axes only slightly inclined (about 58) to that plane. Three such halficosahedra from three different B 84 units are linked together via a B 10 unit, consisting of a condensate of three half-icosahedra, which in pairs share triangular faces. In the hexagonal unit cell there are three B 84 units and six B 10 units, which together with three non-icosahedral B(15) atoms [13] form the rigid three-dimensional network of boron atoms in b-rh boron. The non-icosahedral boron atom is situated on the c axis and co-ordinates octahedrally the partially occupied B(13) atomic position of the B 10 unit. The direction of the non-icosahedral B(15) to B(13) bond is not far from the c direction and is a relatively weak bond, since only 2 / 3 of the B(13) positions are occupied. On the other side, the ˚ are strongest bonds in the structure (B(6)–B(8)51.62 A) found from the waist boron atoms of the central icosahedron nearly exactly in the ab plane (only 58 inclined). These two facts support the occurrence of a slightly weaker average bond in the c direction than within the ab plane, explaining the observed larger thermal expansion in the c direction than in the a direction. Based on dilatometer measurements it was earlier reported [16] that anomalies occur in the thermal expansion of b-rh boron in the interval 500–600 K. It was recently found by Werheit et al. [17]. that these anomalies can be correlated to several electronic properties. These anomalies are possibly discernible in Fig. 1, although the density of measured points is too low to permit any detailed conclusions. The average linear thermal expansion coefficient, am 5 (2aa 1 ac ) / 3, is shown in Fig. 5 within the temperature range 298–1300 K. Data from earlier reports [18–20], based on the dilatometer method, are also included for comparison. It is noted from Fig. 5 that the present results agree well with those reported in Refs. [18,19], in particular within the temperature range 750–1000 K. The value reported by Dupuy and Hackspill [20] for the linear expansion coefficient (293–1023 K), 8.3310 26 K 21 , is much larger than the more recent values 6.1310 26 [18],
57
Fig. 5. Mean linear thermal expansion coefficients (am ) vs. temperature (298–1300 K). Comparison with literature data: 1, Dupuy and Hackspill [20]; 2, Tavadze et al. [19]; 3, present results; 4, Holcombe et al. [18]. Dilatometer measurement, except the present results.
6.4310 26 (present results) and 6.6310 26 K 21 [19]. Holcombe et al. [18] regarded the value of 8.3310 26 K 21 as too high for a material as hard as b-rh boron, since the thermal expansion is a measure of bond strength as well as hardness. Omitting this value, the presently most reliable value is 6.4310 26 K 21 (the average of the three most recent values) for the average linear expansion coefficient of b-rh boron in the temperature range 293–1023 K.
Acknowledgements Thanks are due to Prof. H. Werheit for discussions on Ref. [17], Financial support from the Swedish Research Council for Engineering Sciences and the Natural Science Research Council is gratefully acknowledged.
References [1] H. Werheit, R. Franz, J. Less-Common Met. 117 (1986) 163. [2] R.D. Deslattes, A. Henins, Phys. Rev. Lett. 31 (1973) 972. [3] C.R. Hubbard, National Bureau of Standards, Standard Ref. Mater. 640b, 1987. [4] J. Ihringer, J. Appl. Crystallogr. 15 (1982) 1. [5] D.N. Batchelder, R.O. Simmons, J. Phys. Chem. 41 (1964) 2324. [6] O.V. Alexandrov, M. Francois, T. Graf, K. Yvon, Physica C170 (1990) 56. [7] D.B. Wiles, R.A. Young, J. Appl. Crystallogr. 14 (1981) 149. ¨ [8] G. Hagg, N.-O. Ersson, G. Rudenholm, B. Sellberg, J. Appl. Crystallogr. 12 (1979) 221. [9] W.J. Campbell, C. Grain, US Bur. Mines, Rep. Invest., 1961, p. 5757. [10] P.J. Baldock, W.E. Spindler, T.W. Baker, UK Atomic Energy Res. Establ. Rep., AERE R5674, 1958. ¨ [11] B. Nolang, Institute of Chemistry, Uppsala University, Box 531, Uppsala (unpublished). [12] G.V. Tsagareishvili, T.G. Nakashidze, J.Sh Jobava, G.P. Lomidze,
58
[13] [14] [15] [16]
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom D.E Khulelidze, O.Sh Tsagareishvili, O.A. Tsagareishvili, J. LessCommon Met. 117 (1986) 159. J.L. Hoard, D.B. Sullenger, C.H.L. Kennard, R.E. Hughes, J. Solid State Chem. 1 (1969) 268. B. Callmer, Acta Crystallogr. B33 (1977) 1951. G.A. Slack, C.I. Hejna, M.F. Garbauskas, J.S. Kasper, J. Solid State Chem. 28 (1988) 52. G.V. Tsagareishvili, F.N. Tavadze, A.G. Khvedelidze, D.L. Gabunia, in: T. Niemyski (Ed.), Boron, vol. 3, P.W.N Polish Scientific Publishers, Warsaw, 1970, p. 295.
[17] H. Werheit, M. Laux, U. Kuhlmann, Phys. Stat. Sol. (b) 176 (1993) 415. [18] C.E. Holcombe Jr., D.D. Smith, J.D. Lore, W.K. Duerksen, D.A. Carpenter, High-Temperature Sci. 5 (1973) 349. [19] F.N. Tavadze, I.A. Bairamashvili, G.V. Tsagareishvili, D.V. Hantadze, Stud. Cercetari Metalurg. 10 (1965) 49. [20] E. Dupuy, L. Hackspill, Compt. Rend. 197 (1933) 229.
L
Thermal expansion of b-rhombohedral boron a b ¨ a , *, Bertil Lonnberg ¨ Torsten Lundstrom , Josef Bauer a
˚ ¨ Laboratory, Institute of Chemistry, Uppsala University, Box 538, SE-751 21 Uppsala, Sweden Department of Inorganic Chemistry, The Angstrom b University of Rennes 1, LCSIM /CMR 6511, Av. de General Leclerc, F-35042 Rennes, France Received 24 October 1997
Abstract The cell dimensions of b-rhombohedral boron have been measured in the temperature range 10–1273 K, using X-ray diffraction. The linear thermal expansion coefficients (al 5(l2l 298 ) /l 298 (T2T 298 )) were determined along the a and c directions. It was found that brhombohedral boron is anisotropic, displaying a significantly larger thermal expansion coefficient in the c direction than in the a direction. The average linear thermal expansion coefficient is 6.4310 26 K 21 in the temperature range 293–1023 K. 1998 Elsevier Science S.A. Keywords: Thermal expansion; b-Rhombohedral boron
1. Introduction The high-temperature modification of boron, b-rhombohedral (b-rh) boron, displays very high bonding strength both within and between the boron icosahedra, which form a rigid three-dimensional boron network. The high bond strength is reflected in some physical properties, e.g. high melting point, high Debye temperature, great hardness and low thermal expansion. Thermal expansion data on b-rh boron are scarce because of the relatively great difficulties in providing high-purity boron, its great reactivity at high temperatures, as well as difficulties in fabricating the samples into shapes suitable for analysis, in particular if dilatometer methods are to be used. Furthermore, since most of the thermal expansion studies have been performed using a dilatometer no reliable information regarding the thermal expansion anisotropy of the material was available. The present investigation was initiated to study the thermal anisotropy of b-rh boron, using X-ray powder diffraction. The investigation was performed in the temperature range 10–1273 K. It was earlier reported that b-rh boron has a phase transition in the temperature range 150–180 K, resulting in discontinuities in its elastic, optical, photoelectric, magnetic properties, etc. [1]. Although it was suggested that the transition is characterised by changes in the electronic states and not an ordinary structural transition, it is interesting to establish whether *Corresponding author. Tel.: 146 501 77206; fax: 146 501 10660. 0925-8388 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 97 )00545-8
the cell dimensions and thermal expansion also display discontinuities in the temperature range 10–298 K. Therefore, in this temperature range the measurements were carried out at narrow intervals.
2. Experimental The experiments were carried out using zone-refined boron (certified purity 99.995% with carbon as main impurity) from Wacker Chemie, Munich, Germany. The measurements were performed in the temperature range 10–1273 K, using three different diffraction instruments. The lattice parameters at ambient temperature were de¨ termined using an ordinary XDC 1000 Guinier-Hagg focusing camera with Cr Ka 1 radiation and ultra-pure ˚ [2,3]) as internal calibration subsilicon (a55.430940 A stance. The cell dimensions of the b-rh boron, used for the ˚ measurements, were a510.9330(5) and c523.825(2) A, respectively, at ambient temperature. These parameters were used to calibrate the parameters obtained in the highand low-temperature studies, respectively. The linear thermal expansion coefficients, aa and ac , along the a and c axis, respectively, were calculated using the formula:
al 5 (l 2 l 298 ) /l 298 (T 2 T 298 )
(1)
where l is the lattice parameter at temperature T, and l 298 is the lattice parameter at ambient temperature. The average
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
linear thermal expansion coefficient, am , was calculated using the formula:
am 5 (2aa 1 ac ) / 3
(2)
2.1. Low-temperature studies The unit cell dimensions between 10 and 298 K were measured using a low-temperature Guinier diffractometer / camera, equipped with a low-temperature closed-cycle helium cryostat and with strictly monochromatic Cu Ka 1 radiation (Ge(111) monochromator) [4]. Silicon was used as internal calibration substance. The lattice parameters of silicon at various temperatures were calculated from the data given by Batchelder and Simmons [5] and from the NBS certificate [3]. The boron was thoroughly mixed with the internal standard silicon powder and milled in a boron carbide mortar. The 2u positions were corrected according to the parabolic function [6]:
Dcorr 5 A ? (2uobs )2 1 B ? (2u ) obs 1 C
(3)
2ucorr 5 2uobs 2 Dcorr
(4)
where 2ucorr and 2uobs are the corrected and observed peak positions, respectively. The C parameter corresponds to the position of the true ‘zero’. The corrected diffraction patterns were analysed using the Rietveld method, utilising the program DBWS-9006 as modified for Guinier geometry, described by Wiles and Young [7]. Two peaks at low angles were omitted since they were too asymmetric to be measured properly. The X-ray measurements were performed between 7 and 488 in u in steps of 0.018 with a measuring time of 10 s per step. A total time of 12 h was used for each scan. A total of 18 parameters were refined for each measurement, namely two scale factors, the A, B, and C coefficients, the a and c lattice parameters of boron, six U, V, and W parameters (three for boron and three for silicon), one asymmetry parameter for boron, two mixing parameters (one for each phase) for the pseudo-Voigt function and two overall temperature factors. The standard deviations were 0.0002 ˚ for the a and c parameter, respectively. and 0.0008 A
55
were, however, observed, which indicates that no changes regarding composition and impurities had occurred in the sample during the heating. All unit cell dimensions were least-squares refined using the local Uppsala program UNITCELL [11]. The standard deviations fell within the ˚ for the a parameter and 0.05–0.18 A ˚ range 0.002–0.005 A for the c parameter. The linear thermal expansion coefficients, aa and ac , and the average linear thermal expansion coefficient, am , were calculated between 298 and 1273 K. At temperatures above 1273 K signs of a reaction between the alumina mat and boron were detected.
3. Results and discussion The lattice parameters vs. temperature are given in Fig. 1. It is seen that a non-linear increase in lattice parameters occurs with temperature. Despite using different measuring procedures and instruments, the data obtained in the lowtemperature study agree well at ambient temperature with those measured at high temperature. A slightly larger scatter of the data occurs for the a axis at temperatures near the ambient temperature. The scatter may, at least partially, be due to the temperature calibration, which is less accurate near the ambient temperature. A magnification of the cell dimension vs. temperature curves in the temperature range 10–298 K is given in Fig. 2. The curves were obtained by fitting second-order
2.2. High temperature studies The cell dimensions above room temperature were ¨ high-temperature camera determined using a Guinier-Hagg [8]. The sample was placed on a mat of aluminium oxide whiskers. Since the thermal expansion coefficients of aluminium oxide are well known [9,10], it was used as internal calibration substance. The eight strongest lines were used for this purpose. The specimens were heated in vacuum (10 24 Pa) and the temperature measured with a Pt / Pt10% thermocouple, placed close to the sample. The room-temperature lattice parameters were determined prior to and after each run. No significant changes
Fig. 1. (a) The a axis vs. temperature (10–1273 K). (b) The c axis vs. temperature (10–1273 K).
56
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
Fig. 2. (a) The a axis vs. temperature (10–298 K). (b) The c axis vs. temperature (10–298 K).
polynomials to the experimental points. It is demonstrated that the lattice parameters increase very slowly with temperature in this temperature range. However, it is evident that the a parameter remains nearly constant between 10 and 200 K. No discontinuities in the data in the range 150–180 K could be established. The phase transition discussed in Ref. [1] is accordingly not of crystallographic nature. The thermal expansion vs. temperature is shown in Fig. 3. The curves are composites of the low- and hightemperature measurements and were obtained by leastsquares fitting to the experimental data. It is found that the
Fig. 3. Thermal expansion (Da /a 10K , Dc /c 10K ) vs. temperature (10–1273 K).
larger thermal expansion is obtained along the c axis than along the a axis. The linear thermal expansion coefficients, aa and ac , at temperatures above 298 K are displayed in Fig. 4. Naturally, the thermal expansion coefficient is also significantly larger in the c direction than in the a direction. The thermal expansion at low temperatures displayed similar behaviour. The linear thermal expansion coefficients in the temperature range 10–298 K were aa 50.98310 26 K 21 and ac 51.82310 26 K 21 . The only earlier attempt to determine the anisotropy of the thermal expansion of b-rh boron [12] is at complete invariance with the results presented above. It was reported in the previous study that, in the temperature range 400– 1473 K, the thermal expansion coefficient aa varies between 210310 26 and 125310 26 K 21 and ac between 210310 26 and 185310 26 K 21 . Furthermore, the variation of both coefficients is periodic within the temperature range mentioned. This is in itself extremely remarkable for such a rigid material as b-rh boron, and raises doubts as to the correctness of the measurements. These were performed on b-rh boron of undefined purity and density at a vacuum no better than 10 22 Pa. Only two high-angle diffraction lines (6, 4, 16) and (6, 1, 23) were utilised for the two parameters to be determined, while our cell dimension measurements are based on 25–30 diffraction lines. The use of only two diffraction lines involves a serious limitation of the accuracy due to the frequent overlaps of diffraction lines in the line-rich diffraction pattern of b-rh boron, which originates in the large unit cell, the use of copper radiation and the probably not strictly a 1 monochromated radiation used. In view of these facts. the present results seem to be considerably more reliable than those reported earlier [12]. A brief description of the crystal structure of b-rh boron [13–15] is presented here to discuss the influence of structure and interatomic distances on the anisotropy of the thermal expansion. The structure can be described in terms of B 84 , B 10 , and non-icosahedral boron atoms. The B 84
Fig. 4. Linear thermal expansion coefficients (aa , ac ) vs. temperature (298–1273 K).
¨ et al. / Journal of Alloys and Compounds 267 (1998) 54 – 58 T. Lundstrom
unit consists of one central boron icosahedron and 12 half-icosahedra. The vertex atom of each half-icosahedron is directly bonded along the quasi-five-fold axis to one of the vertex atoms of the central icosahedron. The linking is such that, assuming ideal, regular icosahedra, there is a mirror plane between the two vertex atoms. The halficosahedra can be divided into two types with respect to the type of linking to other B 84 units, in which they participate. The first type is represented by the halficosahedra, whose pentagonal axes are oriented not far from the c direction, three in the positive and three in the negative c direction. These half-icosahedra are linked to the same type of half-icosahedra in neighbouring B 84 units, forming new complete boron icosahedra. The other type of half-icosahedra in the B 84 unit are those with the top atom lying nearly in the same xy plane as the central icosahedron. They have their quasi-five-fold axes only slightly inclined (about 58) to that plane. Three such halficosahedra from three different B 84 units are linked together via a B 10 unit, consisting of a condensate of three half-icosahedra, which in pairs share triangular faces. In the hexagonal unit cell there are three B 84 units and six B 10 units, which together with three non-icosahedral B(15) atoms [13] form the rigid three-dimensional network of boron atoms in b-rh boron. The non-icosahedral boron atom is situated on the c axis and co-ordinates octahedrally the partially occupied B(13) atomic position of the B 10 unit. The direction of the non-icosahedral B(15) to B(13) bond is not far from the c direction and is a relatively weak bond, since only 2 / 3 of the B(13) positions are occupied. On the other side, the ˚ are strongest bonds in the structure (B(6)–B(8)51.62 A) found from the waist boron atoms of the central icosahedron nearly exactly in the ab plane (only 58 inclined). These two facts support the occurrence of a slightly weaker average bond in the c direction than within the ab plane, explaining the observed larger thermal expansion in the c direction than in the a direction. Based on dilatometer measurements it was earlier reported [16] that anomalies occur in the thermal expansion of b-rh boron in the interval 500–600 K. It was recently found by Werheit et al. [17]. that these anomalies can be correlated to several electronic properties. These anomalies are possibly discernible in Fig. 1, although the density of measured points is too low to permit any detailed conclusions. The average linear thermal expansion coefficient, am 5 (2aa 1 ac ) / 3, is shown in Fig. 5 within the temperature range 298–1300 K. Data from earlier reports [18–20], based on the dilatometer method, are also included for comparison. It is noted from Fig. 5 that the present results agree well with those reported in Refs. [18,19], in particular within the temperature range 750–1000 K. The value reported by Dupuy and Hackspill [20] for the linear expansion coefficient (293–1023 K), 8.3310 26 K 21 , is much larger than the more recent values 6.1310 26 [18],
57
Fig. 5. Mean linear thermal expansion coefficients (am ) vs. temperature (298–1300 K). Comparison with literature data: 1, Dupuy and Hackspill [20]; 2, Tavadze et al. [19]; 3, present results; 4, Holcombe et al. [18]. Dilatometer measurement, except the present results.
6.4310 26 (present results) and 6.6310 26 K 21 [19]. Holcombe et al. [18] regarded the value of 8.3310 26 K 21 as too high for a material as hard as b-rh boron, since the thermal expansion is a measure of bond strength as well as hardness. Omitting this value, the presently most reliable value is 6.4310 26 K 21 (the average of the three most recent values) for the average linear expansion coefficient of b-rh boron in the temperature range 293–1023 K.
Acknowledgements Thanks are due to Prof. H. Werheit for discussions on Ref. [17], Financial support from the Swedish Research Council for Engineering Sciences and the Natural Science Research Council is gratefully acknowledged.
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