- Email: [email protected]
Thermal expansion of whiskers of Ti(C,N) solid solutions
Journal of Alloys and Compounds 264 (1998) 223–227
L
Thermal expansion of whiskers of Ti(C,N) solid solutions K. Wokulska* Institute of Physics and Chemistry of Metals, University of Silesia, Bankowa 12, 40 -007 Katowice, Poland Received 29 April 1997; received in revised form 22 May 1997
Abstract In this investigation we present results of lattice parameter measurements of whiskers and needle-like crystals of TiNx ,TiCx and Ti (C, N) solid solutions of varying composition. Precise values of the lattice parameters were determined for the temperature range 293–600 K. For the stoichiometric crystals the real values were obtained by introducing different corrections of systematic errors. The thermal expansion coefficients in the whole range of TiN-TiC compositions were determined and also a polynomial fitting a 5f [C / C1N] is given. 1998 Elsevier Science S.A. Keywords: Lattice parameters; Thermal expansion coefficient; Solid solution crystals; TiC x ; TiN x ; Ti(C,N)
1. Introduction Refractory compounds of transition elements from group IV with carbon and nitrogen have been a subject of studies for many years [1–3]. This interest is caused by their unusual properties connected with the type of bonds and the electronic structure [4–8]. On one hand, they demonstrate metallic character such as metallic lustre, thermal and electrical conductivity, and even in some cases (e.g. TiN) superconductivity. On the other hand, they have properties typical for covalent materials such as high hardness and brittleness, high melting temperature and their crystal structure is typical for ionic crystals (B1, space group Fm3m.). Strength properties of these materials as well as their resistance to aggressive environments make them interesting from the application point of view. It is especially important to elucidate the relationship between the physical and chemical studies results and the material structure because there are still unknown aspects of the heterodesmic character of their bonds [7–11]. Since typical feature of these compounds is their non-stoichiometry and easiness in creation solid solutions by replacing carbon by nitrogen and oxygen, it is worth to have well-characterised single crystals to carry out studies. Whiskers of nitrides and carbides could be such model materials. In this work the relationship between the chemical composition and lattice parameters a of the Ti(C,N) *E-mail: [email protected] 0925-8388 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 97 )00260-0
whiskers and their coefficients of thermal expansion a have been determined.
2. Material The TiN x , TiC x and Ti(C,N) whiskers were grown by the CVD (Chemical-Vapour-Deposition) technique [12– 14]. They crystallised at temperatures between 1473 to 1723 K from the gas phase due to reduction of TiCl 4 by hydrogen with coexisting N 2 and / or CCl 4 . The whisker growth directions were mainly [100], [111] and [011] and thus they had low-index, almost atom-smooth faces (Fig. 1). Whiskers of 10–500 mm in diameter and 1–5 mm length were used for our studies. By this growth method and when using pure substrates one can obtain crystals of a very high purity. Most of the crystals were defect-free which was proven by Lang X-ray topography and TEM studies. In the case of TiN crystals the critical temperature T c of transition into the superconductivity state was about 6 K [15,16]. In the whole population there were also crystals that had non-stoichiometric compositions. Determination of the chemical composition of the crystals was based on a precision measurement of lattice parameters with a relative precision da /a51 4 9310 26 . A lattice parameter is, as it is proven by many examples [17,18], a linear function of composition and fulfils the Vegard law. In the literature it is difficult to find exact values of lattice parameters for carbides and nitrides corresponding
224
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
Fig. 1. SEM images of TiC (a) and TiN (b) whiskers.
to fully stoichiometric compositions. In most cases studies were carried out on polycrystalline materials and polycrystalline X-ray diffractometry was used in the studies. Only in few investigations relatively precise values for TiN were obtained: Dunand et al. [4] found a50.424129633 10 26 nm, Kato et al. [19] reported a50.4240065310 25 ¨ nm and Hochst et al. [11] gave a50.42366610 25 nm. However, these are not absolute values. No corrections were included for systematic errors and the values were not related to any measurement temperature. In the literature there are no data on values of lattice parameters for TiC crystals of a stoichiometric composition. This is mainly due to bed quality of the crystals. Only Dunand et al. [4] quoted a precise value a50.432965(13) nm for a titanium carbide single crystal of a composition close to stoichiometric i.e. TiC 0.94 . However, this value cannot be regarded as an accurate one. In the present study a proper Vegard relation was determined for whiskers of stoichiometric composition using the Bond method [21].
3. Absolute lattice parameters of TiN and TiC whiskers For an absolute measurement it should be carried out in a metric system or be based on an emission length of the X-rays, that is known with a high accuracy Dl /l#10 26 .
The maximum accuracy for the absolute measurements is Dd /d510 27 –10 28 . A measurement with such an accuracy can be carried out only using one of the following methods: simultaneous X-ray and laser interferometry [20] or the Bond method [21]. X-ray interferometry, although most accurate, requires expensive and complicated instruments and is rarely used. The Bond method is experimentally much less complicated but is creates some interpretation problems. The measurement is carried out on a precise X-ray goniometer of the Bragg type with an angular precision of about 10. To obtain a precision not lower than the required accuracy of the measurement it is necessary to use crystals without a mosaic structure and without dislocations. It is possible to determinate the lattice distances and the lattice parameters in one point of a crystal during a single measurement (for crystals of cubic symmetry). According to the Bond measurement a given high u angle scan (u .708) reflection is repeated for two different symmetrical positions of the crystal. The values of u are burdened with many systematic errors, due to differences of interpretation of diffraction on an ideal crystal and due to applying the Bragg kinematic equation. To make the reflex position more accurate the Du angle difference between the measured uM value and the real Bragg value uB is determined (uB 5uM 2Du ). Based on the dynamic X-ray scattering theory it is possible to determinate very accurately the influence of the factors that shift the peak position: refraction, horizontal and vertical divergence of the beam, absorption in the crystal and in air, collimator tilt. In the most accurate measurements also the polarisation of Xrays, dispersion in the crystal and absorption in the anode material is taken into account [22]. Also aberrations caused by a convolution of an asymmetrical spectral line, reflection curve and the function of collimator divergence should be considered. This is important in correcting the horizontal divergence, dispersion, coefficient of integrated reflectivity in the crystal and X-ray absorption in the anode material and in air. In the present measurements the CuKb 600 and 440 as well as the CuKa 1 333 reflexes were used and thus it was necessary to know the spectral line profiles of CuKa 1 and Kb [23]. Recently, an analytical description of some other basic emission lines (Co, Cr, Ni, Fe) [24] has become known as a superposition of symmetrical functions. The value Du of the shift of the diffraction line I(v ) is interpreted as a sum of all the aberrations i.e. the total systematic error. Absolute values of the particular corrections are differentiated and depend on the quality and sizes of the crystal, goniometer precision, beam geometry, reflex type and even on the position of the X-ray source. Examples of corrections of the most important aberrations calculated for the TiN whiskers are given in the Table 1. The corrections for the TiC and solid solution crystals are similar. The horizontal divergence of the beam plays an important role in the determination of the total aberration.
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
225
Table 1 The most important aberrations of the u angle and the corresponding corrections of the lattice parameters Da for TiN whiskers; lCuKa 1 50.1540562 nm * , lCuKb50.1392218 nm * ( a), electrical susceptibility /x0 / 52.669310 25 , collimator tilt angle k 55.5’, slits: vertical s 1 50.28 mm, s 2 50.05 mm (whisker diameter), horizontal h 1 5h 2 51 mm Corrections
Du [0]
Reflections
333 Ka 1 (u 570,712 0 )
Refraction Duref Horizontal divergence DuHD Vertical divergence DuVD Dispersion and integrated reflectivity DuS Absorption within crystal DuAC Other absorptions DuA Collimator tilt DuCT
210.67 24.58 20.46 20.04
0.000 007 8 0.000 003 4 0.000 000 3 0.000 000 03
0.29 0.07 0.09
20.000 000 2 20.000 000 05 20.000 000 07
SDu (Da)
215.3
Da [nm]
Du [0]
Da [nm]
600 Kb (u 580.084 0 )
0.000 011 2
216.43 23.97 20.93 20.169 0.52 0.22 0.05 220.7
0.000 0.000 0.000 0.000
005 001 000 000
9 4 3 06
20.000 000 19 20.000 000 08 20.000 000 017 0.000 007 5
a
nm * -non-metric unit according to Bearden used until today, 1nm * 51nm65 ppm. The wavelength in metric units l 5CuKa 1 50.154 052 922 (45) nm [25,26] and l 5CuKb50.139 224 1 nm [23].
This aberration causing the second biggest systematic error is often omitted. In case of small crystals of which the diameters are smaller than the beam width, this correction is especially important and the crystal diameter plays the role of an exit slit, which was taken into consideration in the present study. The described method was used to determine accurate values of lattice parameters for TiN a50.423909 nm and for TiC a50.432650 nm. Similar values but obtained with smaller precision were obtained in Ref. [27]. The above values of a were used to establish the linear relation: a50.42390910.00874 x, where x5C / [C1N]. This relation allows the determination of the composition of solid solution crystals. It was found that several of the crystals grown under the same conditions had lattice parameters slightly different from the above ones. This could be due to traces of carbon or oxygen introduced to the TiN x crystals and nitrogen or oxygen in the TiC x crystals. Also vacancies in a non-metal position can change the lattice parameters. Table 2 contains results obtained for crystals of composition close to stoichiometric. Also crystals of Table 2 Lattice parameters of TiN x and TiC x whiskers Composition
Lattice parameter a [nm]
Standard deviations 6s 3[10 26 ]
Ti (C 0.02 N 0.98 ) Ti (C 0.03 N 0.97 ) Ti (C 0.03 N 0.97 ) Ti (C 0.055 N 0.945 ) Ti (C 0.067 N 0.933 ) Ti (C 0.08 N 0.92 ) Ti (C 0.087 N 0.913 ) Ti (C 0.09 N 0.91 ) Ti (C 0.95 N 0.05 ) Ti (C 0.96 N 0.04 ) Ti (C 0.98 N 0.02 )
0.424 068 7 0.424 135 3 0.424 156 8 0.424 393 6 0.424 493 6 0.424 622 7 0.424 674 1 0.424 729 0 0.432 195 1 0.432 291 4 0.432 475 5
1 15 9 10 9 3 15 9 10 10 10
different compositions of the TiC x N 12x solid solutions (where x50.21, 0.52, 0.56, 0.68, 0.82) were studied. It can be seen that the amount of carbon in TiN x did not exceed 9 at % and nitrogen in TiC x did not exceed 5 at % and that even small composition changes are detectable by this precise method of lattice parameter measurements.
4. Results and discussion The thermal expansion coefficient reacts much stronger to compositional changes than the lattice parameter. Even single atoms substituted in the parent lattice cause a disturbance in the electronic structure, as each distortion in the non-metal position due to the co-ordination number (6) leads to a six times stronger change in the electronic structure. These distortions will influence the thermal expansion as a consequence of the change of the interaction potential between the atoms and affect the anharmonic lattice vibrations. The thermal expansion coefficients were determined from the relation a5f(T ) by using the precision measurements of the lattice parameters. The relative measurements of the lattice parameters was carried out in the temperature range 275–630 K. The measurements were not continued to higher temperatures because there was a possibility for the random oxidation process to occur above 700 K. The presence of traces of oxygen can cause substitution of these atoms into the non-metal positions in the lattice which leads to difficulties in interpretation. The results for the solid solution crystals are presented in the Table 3. Examples of plots of the lattice parameter changes for Ti(C,N) whiskers are shown in Fig. 2. The determined relation a 5f(C / [C1N]) is shown in Fig. 3. The changes can be described by the following polynomial:
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
226
Table 3 Results of thermal expansion measurements for Ti(C,N) whiskers Whisker
1. 2. 3. 4. 5. 6. 7. 8. 9.
Composition
TiC 0.03 N 0.97 TiC 0.05 N 0.95 TiC 0.21 N 0.79 TiC 0.52 N 0.48 TiC 0.56 N 0.44 TiC 0.82 N 0.18 TiC 0.68 N 0.32 TiC 0.95 N 0.05 TiC 0.96 N 0.04
a 310 26 K 21 613[10 27 K 21 ]
a(T )5a 0 1a 1 T [nm]
9.06 9.35 8.10 7.94 7.44 8.55 7.11 8.17 8.45
a0
a 1 310 26
a1Da kor [nm] at (293 K)
0.42292 0.42313 0.425032 0.427613 0.427776 0.428172 0.428926 0.431134 0.431179
3.73264 3.955 3.443 3.3954 3.1919 3.6686 3.0506 3.5298 3.6423
0.424 0.424 0.426 0.428 0.428 0.431 0.429 0.432 0.432
396 394 296 642 770 053 849 195 291
Fig. 2. Plot of a lattice parameters versus compositions for the TiC 12x N x whiskers.
H
F
G
F
C C a 5 9.40 2 6.47 3 ]] 1 5.72 3 ]] C1N C1N 3 10 26 [K 21 ] .
Fig. 3. Plot of thermal expansion coefficients versus compositions TiC 12x N x whiskers.
GJ 2
from which the thermal expansion coefficients were determined for the stoichiometric compositions. These are a 59.40310 26 K 21 for TiN, and a 58.64310 26 K 21 for TiC. Recent studies of the thermal expansion by Aigner et al., Lengauer et al. [17,18] were carried out for polycrystalline titanium carbonitrides. They determined the relation a5 f(C / [C1N],T ) for a large temperature range (298–1473 K) and for a few different compositions. There is a good agreement between our and their results of lattice parameters measurements in comparable ranges of temperatures. The parameters values were not published but they could be calculated from the equations given. On the other hand in [17] the correlation between the thermal expansion coefficients is weaker. In the presently investigated range of temperatures a linear character of a5f [T ] was observed
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
and thus a 5f(T )5const. A deviation from linearity which was especially well visible above 700 K in the data of [17]. On the other hand in [17] the values of the thermal expansion coefficients for different compositions are not much differentiated in the temperature range 293–500 K. These values can be extrapolated to the stoichiometric compositions and to 293 K. Thus a 57.0310 26 K 21 for TiN and a 56.88310 26 K 21 for TiC. One can suggest that such a small difference in the coefficient of thermal expansion a for different compositions results from a large temperature range and the small number of measurement points. The discrepancies increase for higher temperatures. The obtained polynomial relation a 5f(C / [C1N])has a gentle minimum for the composition TiN 0.4 C 0.6 which can suggest that the degree of anharmonicity of lattice vibration in the solid solution crystals decreases compared with stoichiometric crystals and thus the bonds become stronger. When more than half of the interstitial positions are occupied by carbon the strong covalent bonds become dominating while the fraction of ionic bonds decreases. Also the tendency to vacancy formation is smaller and the screening of metal-metal bonds becomes stronger. The formation of vacancies in non-metal positions is responsible for the increase of the thermal expansion coefficient for compositions close to TiC. Davis [28] claims that the ideal stoichiometry in TiC x and TaC x carbides does not exist and that the carbon deficiency can be higher than 3 at %. The decrease in the screening of metal-metal bonds causes the creation of weak bonds through the vacancies positions in the second co-ordination zone. Consequently all these factors lead to weakening of the bonds which is demonstrated by the increase in value of the thermal expansion coefficient a. The TiC x N 12x carbonitrides show also some extreme behaviour of their physical and chemical properties for the compositions x50.5–0.8. The Seebek coefficient [29] has a maximum for TiC 0.5 N 0.45 and there is a minimum of the magnetic susceptibility for TiC 0.8 N 0.2 . Even the melting point and hardness of TiC x N 12x are higher than for TiC and TiN crystals. Thus we can confirm the earlier suggestion that the Ti-C bonds are responsible for the bond power in these materials.
5. Conclusions 1. Accurate values of the lattice parameters for stoichiometric TiN and TiC whiskers have been determined by using the Bond method.
227
2. It was shown that in the temperature range 293–600 K the thermal expansion coefficient a depends strongly on the solid solution composition. The obtained polynomial relation a 5f(C / [C1N]) has a minimum for the TiC 0.6 N 0.4 composition which indicates that the Ti-C bonds dominate in the crystals of Ti(C,N) solid solutions.
References [1] L.E. Toth, Transition Metal Carbides and Nitrides, Academic Press, N.Y., 1971. [2] G.W. Samsonov, Nitridy, Kiev, Naukova Dumka, 1969. [3] E.K. Storms, The Refractory Carbides, Academic Press, N.Y., 1967. [4] A. Dunand, H.D. Flack, K. Yvon, Phys. Rev. B 31 (1985) 2299. [5] P. Blaha, J. Redinger, K. Schwarz, Phys. Rev. B 31 (1985) 2316. [6] P. Blaha, K. Scharz, F. Kubel, K. Yvon, J. Solid State Chem. 70 (1987) 199. [7] A. Neckel, Inter. J. Quantum Chem. XXIII (1983) 1317. [8] E. Gustenau-Michalek, P. Herzig, A. Neckel, J. Alloys Comp. 219 (1995) 303. [9] W.S. Williams, in: Progress in Solid State Chemistry, Vol. 6, H. Reiss, J.O. McCaldin (Eds.), Pergamon Press, N.Y., 1971, p.57. [10] K. Winzer, J. Reichelt, Solid State Comm. 49 (1984) 527. ¨ [11] H. Hochst, R.D. Bringans, P. Steiner, Th. Wolf, Phys. Rev. B 25 (1982) 7183. [12] Z. Bojarski, K. Wokulska, Z. Wokulski, J. Crystal Growth 52 (1981) 290. [13] Z. Wokulski, K. Wokulska, J. Crystal Growth 62 (1983) 439. [14] Z. Wokulski, J. Crystal Growth 82 (1987) 427. [15] C. Sułkowski, Z. Wokulski, K. Wokulska, Phys. Stat. Sol. (a) 95 (1986) K67. [16] Z. Wokulski, K. Wokulska, in: Surface Modification Technologies II, The Minerals, Metals and Materials Society, USA, 1989, p. 193. [17] K. Aigner, W. Lengauer, D. Rafaja, P. Ettmayer, J. Alloys Comp. 215 (1994) 121. [18] W. Lengauer, S. Binder, K. Aigner, P. Ettmayer, A. Guillou, J. Debuigne, G. Groboth, J. Alloys Comp. 217 (1995) 137. [19] A. Kato, N. Tamari, J. Crystal Growth 29 (1975) 55. [20] D. Windisch, P. Becker, Phys. Stat. Sol. (a) 118 (1990) 379. [21] W.L. Bond, Acta Cryst. 13 (1960) 814. ¨ [22] J. Hartwig, S. Grosswig, Phys. Stat. Sol. (a) 115 (1989) 369. ¨ ¨ ¨ [23] J. Hartwig, G. Holzer, E. Forster, K. Goetz, K. Wokulska, J. Wolf, Phys. Stat. Sol. (a) 143 (1994) 23. ¨ ¨ ¨ [24] G. Holzer, M. Deutsch, M. Fritsch, J. Hartwig, E. Forster, submitted to Phys. Rev. ¨ ¨ ¨ [25] J. Hartwig, G. Holzer, J. Wolf, E. Forster, J. Appl. Crystallogr. 26 (1993) 539. ¨ [26] J. Hartwig, S. Grosswig, P. Becker, D. Windisch, Phys. Stat. Sol. (a) 125 (1991) 79. [27] H. Schachner, G. Horlavill, P. Fontain, J. Phys., Suppl. 50 (1989) C5–219. [28] R.F. Davis, in: Advances in Ceramics, Vol. 23: Nonstochiometric Compounds, 1987, p. 529. [29] A.S. Borukhovich, Phys. Stat. Sol. (a) 46 (1978) 11.
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Thermal expansion of whiskers of Ti(C,N) solid solutions K. Wokulska* Institute of Physics and Chemistry of Metals, University of Silesia, Bankowa 12, 40 -007 Katowice, Poland Received 29 April 1997; received in revised form 22 May 1997
Abstract In this investigation we present results of lattice parameter measurements of whiskers and needle-like crystals of TiNx ,TiCx and Ti (C, N) solid solutions of varying composition. Precise values of the lattice parameters were determined for the temperature range 293–600 K. For the stoichiometric crystals the real values were obtained by introducing different corrections of systematic errors. The thermal expansion coefficients in the whole range of TiN-TiC compositions were determined and also a polynomial fitting a 5f [C / C1N] is given. 1998 Elsevier Science S.A. Keywords: Lattice parameters; Thermal expansion coefficient; Solid solution crystals; TiC x ; TiN x ; Ti(C,N)
1. Introduction Refractory compounds of transition elements from group IV with carbon and nitrogen have been a subject of studies for many years [1–3]. This interest is caused by their unusual properties connected with the type of bonds and the electronic structure [4–8]. On one hand, they demonstrate metallic character such as metallic lustre, thermal and electrical conductivity, and even in some cases (e.g. TiN) superconductivity. On the other hand, they have properties typical for covalent materials such as high hardness and brittleness, high melting temperature and their crystal structure is typical for ionic crystals (B1, space group Fm3m.). Strength properties of these materials as well as their resistance to aggressive environments make them interesting from the application point of view. It is especially important to elucidate the relationship between the physical and chemical studies results and the material structure because there are still unknown aspects of the heterodesmic character of their bonds [7–11]. Since typical feature of these compounds is their non-stoichiometry and easiness in creation solid solutions by replacing carbon by nitrogen and oxygen, it is worth to have well-characterised single crystals to carry out studies. Whiskers of nitrides and carbides could be such model materials. In this work the relationship between the chemical composition and lattice parameters a of the Ti(C,N) *E-mail: [email protected] 0925-8388 / 98 / $19.00 1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 97 )00260-0
whiskers and their coefficients of thermal expansion a have been determined.
2. Material The TiN x , TiC x and Ti(C,N) whiskers were grown by the CVD (Chemical-Vapour-Deposition) technique [12– 14]. They crystallised at temperatures between 1473 to 1723 K from the gas phase due to reduction of TiCl 4 by hydrogen with coexisting N 2 and / or CCl 4 . The whisker growth directions were mainly [100], [111] and [011] and thus they had low-index, almost atom-smooth faces (Fig. 1). Whiskers of 10–500 mm in diameter and 1–5 mm length were used for our studies. By this growth method and when using pure substrates one can obtain crystals of a very high purity. Most of the crystals were defect-free which was proven by Lang X-ray topography and TEM studies. In the case of TiN crystals the critical temperature T c of transition into the superconductivity state was about 6 K [15,16]. In the whole population there were also crystals that had non-stoichiometric compositions. Determination of the chemical composition of the crystals was based on a precision measurement of lattice parameters with a relative precision da /a51 4 9310 26 . A lattice parameter is, as it is proven by many examples [17,18], a linear function of composition and fulfils the Vegard law. In the literature it is difficult to find exact values of lattice parameters for carbides and nitrides corresponding
224
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
Fig. 1. SEM images of TiC (a) and TiN (b) whiskers.
to fully stoichiometric compositions. In most cases studies were carried out on polycrystalline materials and polycrystalline X-ray diffractometry was used in the studies. Only in few investigations relatively precise values for TiN were obtained: Dunand et al. [4] found a50.424129633 10 26 nm, Kato et al. [19] reported a50.4240065310 25 ¨ nm and Hochst et al. [11] gave a50.42366610 25 nm. However, these are not absolute values. No corrections were included for systematic errors and the values were not related to any measurement temperature. In the literature there are no data on values of lattice parameters for TiC crystals of a stoichiometric composition. This is mainly due to bed quality of the crystals. Only Dunand et al. [4] quoted a precise value a50.432965(13) nm for a titanium carbide single crystal of a composition close to stoichiometric i.e. TiC 0.94 . However, this value cannot be regarded as an accurate one. In the present study a proper Vegard relation was determined for whiskers of stoichiometric composition using the Bond method [21].
3. Absolute lattice parameters of TiN and TiC whiskers For an absolute measurement it should be carried out in a metric system or be based on an emission length of the X-rays, that is known with a high accuracy Dl /l#10 26 .
The maximum accuracy for the absolute measurements is Dd /d510 27 –10 28 . A measurement with such an accuracy can be carried out only using one of the following methods: simultaneous X-ray and laser interferometry [20] or the Bond method [21]. X-ray interferometry, although most accurate, requires expensive and complicated instruments and is rarely used. The Bond method is experimentally much less complicated but is creates some interpretation problems. The measurement is carried out on a precise X-ray goniometer of the Bragg type with an angular precision of about 10. To obtain a precision not lower than the required accuracy of the measurement it is necessary to use crystals without a mosaic structure and without dislocations. It is possible to determinate the lattice distances and the lattice parameters in one point of a crystal during a single measurement (for crystals of cubic symmetry). According to the Bond measurement a given high u angle scan (u .708) reflection is repeated for two different symmetrical positions of the crystal. The values of u are burdened with many systematic errors, due to differences of interpretation of diffraction on an ideal crystal and due to applying the Bragg kinematic equation. To make the reflex position more accurate the Du angle difference between the measured uM value and the real Bragg value uB is determined (uB 5uM 2Du ). Based on the dynamic X-ray scattering theory it is possible to determinate very accurately the influence of the factors that shift the peak position: refraction, horizontal and vertical divergence of the beam, absorption in the crystal and in air, collimator tilt. In the most accurate measurements also the polarisation of Xrays, dispersion in the crystal and absorption in the anode material is taken into account [22]. Also aberrations caused by a convolution of an asymmetrical spectral line, reflection curve and the function of collimator divergence should be considered. This is important in correcting the horizontal divergence, dispersion, coefficient of integrated reflectivity in the crystal and X-ray absorption in the anode material and in air. In the present measurements the CuKb 600 and 440 as well as the CuKa 1 333 reflexes were used and thus it was necessary to know the spectral line profiles of CuKa 1 and Kb [23]. Recently, an analytical description of some other basic emission lines (Co, Cr, Ni, Fe) [24] has become known as a superposition of symmetrical functions. The value Du of the shift of the diffraction line I(v ) is interpreted as a sum of all the aberrations i.e. the total systematic error. Absolute values of the particular corrections are differentiated and depend on the quality and sizes of the crystal, goniometer precision, beam geometry, reflex type and even on the position of the X-ray source. Examples of corrections of the most important aberrations calculated for the TiN whiskers are given in the Table 1. The corrections for the TiC and solid solution crystals are similar. The horizontal divergence of the beam plays an important role in the determination of the total aberration.
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
225
Table 1 The most important aberrations of the u angle and the corresponding corrections of the lattice parameters Da for TiN whiskers; lCuKa 1 50.1540562 nm * , lCuKb50.1392218 nm * ( a), electrical susceptibility /x0 / 52.669310 25 , collimator tilt angle k 55.5’, slits: vertical s 1 50.28 mm, s 2 50.05 mm (whisker diameter), horizontal h 1 5h 2 51 mm Corrections
Du [0]
Reflections
333 Ka 1 (u 570,712 0 )
Refraction Duref Horizontal divergence DuHD Vertical divergence DuVD Dispersion and integrated reflectivity DuS Absorption within crystal DuAC Other absorptions DuA Collimator tilt DuCT
210.67 24.58 20.46 20.04
0.000 007 8 0.000 003 4 0.000 000 3 0.000 000 03
0.29 0.07 0.09
20.000 000 2 20.000 000 05 20.000 000 07
SDu (Da)
215.3
Da [nm]
Du [0]
Da [nm]
600 Kb (u 580.084 0 )
0.000 011 2
216.43 23.97 20.93 20.169 0.52 0.22 0.05 220.7
0.000 0.000 0.000 0.000
005 001 000 000
9 4 3 06
20.000 000 19 20.000 000 08 20.000 000 017 0.000 007 5
a
nm * -non-metric unit according to Bearden used until today, 1nm * 51nm65 ppm. The wavelength in metric units l 5CuKa 1 50.154 052 922 (45) nm [25,26] and l 5CuKb50.139 224 1 nm [23].
This aberration causing the second biggest systematic error is often omitted. In case of small crystals of which the diameters are smaller than the beam width, this correction is especially important and the crystal diameter plays the role of an exit slit, which was taken into consideration in the present study. The described method was used to determine accurate values of lattice parameters for TiN a50.423909 nm and for TiC a50.432650 nm. Similar values but obtained with smaller precision were obtained in Ref. [27]. The above values of a were used to establish the linear relation: a50.42390910.00874 x, where x5C / [C1N]. This relation allows the determination of the composition of solid solution crystals. It was found that several of the crystals grown under the same conditions had lattice parameters slightly different from the above ones. This could be due to traces of carbon or oxygen introduced to the TiN x crystals and nitrogen or oxygen in the TiC x crystals. Also vacancies in a non-metal position can change the lattice parameters. Table 2 contains results obtained for crystals of composition close to stoichiometric. Also crystals of Table 2 Lattice parameters of TiN x and TiC x whiskers Composition
Lattice parameter a [nm]
Standard deviations 6s 3[10 26 ]
Ti (C 0.02 N 0.98 ) Ti (C 0.03 N 0.97 ) Ti (C 0.03 N 0.97 ) Ti (C 0.055 N 0.945 ) Ti (C 0.067 N 0.933 ) Ti (C 0.08 N 0.92 ) Ti (C 0.087 N 0.913 ) Ti (C 0.09 N 0.91 ) Ti (C 0.95 N 0.05 ) Ti (C 0.96 N 0.04 ) Ti (C 0.98 N 0.02 )
0.424 068 7 0.424 135 3 0.424 156 8 0.424 393 6 0.424 493 6 0.424 622 7 0.424 674 1 0.424 729 0 0.432 195 1 0.432 291 4 0.432 475 5
1 15 9 10 9 3 15 9 10 10 10
different compositions of the TiC x N 12x solid solutions (where x50.21, 0.52, 0.56, 0.68, 0.82) were studied. It can be seen that the amount of carbon in TiN x did not exceed 9 at % and nitrogen in TiC x did not exceed 5 at % and that even small composition changes are detectable by this precise method of lattice parameter measurements.
4. Results and discussion The thermal expansion coefficient reacts much stronger to compositional changes than the lattice parameter. Even single atoms substituted in the parent lattice cause a disturbance in the electronic structure, as each distortion in the non-metal position due to the co-ordination number (6) leads to a six times stronger change in the electronic structure. These distortions will influence the thermal expansion as a consequence of the change of the interaction potential between the atoms and affect the anharmonic lattice vibrations. The thermal expansion coefficients were determined from the relation a5f(T ) by using the precision measurements of the lattice parameters. The relative measurements of the lattice parameters was carried out in the temperature range 275–630 K. The measurements were not continued to higher temperatures because there was a possibility for the random oxidation process to occur above 700 K. The presence of traces of oxygen can cause substitution of these atoms into the non-metal positions in the lattice which leads to difficulties in interpretation. The results for the solid solution crystals are presented in the Table 3. Examples of plots of the lattice parameter changes for Ti(C,N) whiskers are shown in Fig. 2. The determined relation a 5f(C / [C1N]) is shown in Fig. 3. The changes can be described by the following polynomial:
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
226
Table 3 Results of thermal expansion measurements for Ti(C,N) whiskers Whisker
1. 2. 3. 4. 5. 6. 7. 8. 9.
Composition
TiC 0.03 N 0.97 TiC 0.05 N 0.95 TiC 0.21 N 0.79 TiC 0.52 N 0.48 TiC 0.56 N 0.44 TiC 0.82 N 0.18 TiC 0.68 N 0.32 TiC 0.95 N 0.05 TiC 0.96 N 0.04
a 310 26 K 21 613[10 27 K 21 ]
a(T )5a 0 1a 1 T [nm]
9.06 9.35 8.10 7.94 7.44 8.55 7.11 8.17 8.45
a0
a 1 310 26
a1Da kor [nm] at (293 K)
0.42292 0.42313 0.425032 0.427613 0.427776 0.428172 0.428926 0.431134 0.431179
3.73264 3.955 3.443 3.3954 3.1919 3.6686 3.0506 3.5298 3.6423
0.424 0.424 0.426 0.428 0.428 0.431 0.429 0.432 0.432
396 394 296 642 770 053 849 195 291
Fig. 2. Plot of a lattice parameters versus compositions for the TiC 12x N x whiskers.
H
F
G
F
C C a 5 9.40 2 6.47 3 ]] 1 5.72 3 ]] C1N C1N 3 10 26 [K 21 ] .
Fig. 3. Plot of thermal expansion coefficients versus compositions TiC 12x N x whiskers.
GJ 2
from which the thermal expansion coefficients were determined for the stoichiometric compositions. These are a 59.40310 26 K 21 for TiN, and a 58.64310 26 K 21 for TiC. Recent studies of the thermal expansion by Aigner et al., Lengauer et al. [17,18] were carried out for polycrystalline titanium carbonitrides. They determined the relation a5 f(C / [C1N],T ) for a large temperature range (298–1473 K) and for a few different compositions. There is a good agreement between our and their results of lattice parameters measurements in comparable ranges of temperatures. The parameters values were not published but they could be calculated from the equations given. On the other hand in [17] the correlation between the thermal expansion coefficients is weaker. In the presently investigated range of temperatures a linear character of a5f [T ] was observed
K. Wokulska / Journal of Alloys and Compounds 264 (1998) 223 – 227
and thus a 5f(T )5const. A deviation from linearity which was especially well visible above 700 K in the data of [17]. On the other hand in [17] the values of the thermal expansion coefficients for different compositions are not much differentiated in the temperature range 293–500 K. These values can be extrapolated to the stoichiometric compositions and to 293 K. Thus a 57.0310 26 K 21 for TiN and a 56.88310 26 K 21 for TiC. One can suggest that such a small difference in the coefficient of thermal expansion a for different compositions results from a large temperature range and the small number of measurement points. The discrepancies increase for higher temperatures. The obtained polynomial relation a 5f(C / [C1N])has a gentle minimum for the composition TiN 0.4 C 0.6 which can suggest that the degree of anharmonicity of lattice vibration in the solid solution crystals decreases compared with stoichiometric crystals and thus the bonds become stronger. When more than half of the interstitial positions are occupied by carbon the strong covalent bonds become dominating while the fraction of ionic bonds decreases. Also the tendency to vacancy formation is smaller and the screening of metal-metal bonds becomes stronger. The formation of vacancies in non-metal positions is responsible for the increase of the thermal expansion coefficient for compositions close to TiC. Davis [28] claims that the ideal stoichiometry in TiC x and TaC x carbides does not exist and that the carbon deficiency can be higher than 3 at %. The decrease in the screening of metal-metal bonds causes the creation of weak bonds through the vacancies positions in the second co-ordination zone. Consequently all these factors lead to weakening of the bonds which is demonstrated by the increase in value of the thermal expansion coefficient a. The TiC x N 12x carbonitrides show also some extreme behaviour of their physical and chemical properties for the compositions x50.5–0.8. The Seebek coefficient [29] has a maximum for TiC 0.5 N 0.45 and there is a minimum of the magnetic susceptibility for TiC 0.8 N 0.2 . Even the melting point and hardness of TiC x N 12x are higher than for TiC and TiN crystals. Thus we can confirm the earlier suggestion that the Ti-C bonds are responsible for the bond power in these materials.
5. Conclusions 1. Accurate values of the lattice parameters for stoichiometric TiN and TiC whiskers have been determined by using the Bond method.
227
2. It was shown that in the temperature range 293–600 K the thermal expansion coefficient a depends strongly on the solid solution composition. The obtained polynomial relation a 5f(C / [C1N]) has a minimum for the TiC 0.6 N 0.4 composition which indicates that the Ti-C bonds dominate in the crystals of Ti(C,N) solid solutions.
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