Crystallographic Properties of Platinum
Crystallographic Properties of Platinum
Crystallographic and bulk properties of platinum from absolute zero to the melting point are assessed from a review of the literature covering the period 1901 to date. Selected values of thermal expansion are used to calculate the variation with temperature of length changes, lattice parameters, inter-atomic distances, atomic and molar volumes, and density. The crystallographic properties are based mainly on the precise dilatometric thermal expansion data, including corrections to account for thermal vacancy effects at the highest temperatures. Literature values are compared graphically with the selected values, and an Appendix is included to explain some of the terms used.
Platinum exists in a face-centred cubic structure (Pearson symbol cF4) up to the melting point, which is a proposed secondary fixed point on the International Temperature Scale, ITS-90, at 2041.3 K (1). High precision thermal expansion data for platinum is available up to 85 K and from 246 to 1900 K which allows the use of this metal as a thermal expansion standard above room temperature. In the temperature interval from 85 to 246 K thermal expansion values are estimated from a relationship between thermal expansion and specific heat developed by the present author from earlier Russian work (see Appendix). The close agreement between experimental length change measurements in this region and those obtained by integrating the estimated thermal expansion values gives validity to this approach to interpolation.
Only the lattice parameter at 293.15 K (20°C) has been used in deriving the crystallographic properties, all other values are based on the precision dilatometric measurements and include the thermal vacancy corrections which apply above 1300 K. The large scatter in the lattice parameter measurements, especially at high temperatures, is surprising for an element which is inert and obtained in very high purity. This scatter precludes comparison with the dilatometric data to obtain an independent estimate of the thermal vacancy parameters.
Of the 22 sets of thermal expansion data tabulated by Touloukian and co-workers (2) in 1975, Kirby (3) considered that only those of: Holborn and Day (4) from 273 to 1273 K, Austin (5) from 273 to 1173 K, White (6) 3 to 85 and 283 K and Hahn and Kirby (7) 293 to 1900 K, as well as the more recent measurements of Edsinger, Reilly and Schooley (8) 246 to 841 K, satisfy the necessary criteria of sensitivity of method and purity of the materials used, and of these only those of Edsinger, Reilly and Schooley were reported in sufficient detail to allow an unambiguous temperature scale correction (from IPTS-68 to ITS-90).
In the low temperature region the thermodynamic thermal expansion coefficient, α, is based on the measurements of White (6) and Edsinger, Reilly and Schooley (8). The derivation of Equations (ii) and (iii) is explained in the Appendix.
The values of Cp, are taken from the recent review by the present author (9). Equation (i) is accurate to ± 3 × 10−10 K−1 and Equations (ii) and (iii) to ± 2 × 10−8 K−1. Although Kirby (3) used a different approach to interpolate within the range 85 to 245 K, calculated values of α* in this region agree to within 7 × 10−8 K−1. Since Table II is given at relatively wide temperature intervals at low temperature, a knowledge of Cp, is required in order to use Equations (ii) and (iii). For interpolation this requirement has been overcome by fitting smoothed values of α calculated from Equations (ii) and (iii) to a series of spline-fitted polynomials, Equations (iv) to (vii).
|Low Temperature Thermal Expansion Data|
|(Spline-Fitted Equations above 25 K)|
|High Temperature Dilatometric Thermal Expansion|
|High Temperature Lattice Parameter Thermal Expansion|
In the high temperature region selected values of α*, defined in the Appendix, are based on the measurements of Holborn and Day (4) (to 1023 K), Austin (5) (to 873 K), Hahn and Kirby (7) and Edsinger, Reilly and Schooley (8), see Equation (viii).
Equation (viii) for α* has an overall accuracy of ± 2 × 10−8 K−1 and agrees with the interpretation of the same data by Kirby (3) to within 5 × 10−8 K−1.
Because of the scatter in the lattice parameter data, Equations (i) to (ix), were used to represent both the crystallographic and bulk properties of platinum up to the melting point, with the crystallographic properties in the high temperature range calculated from the dilatometrie data by taking into account thermal vacancy effects.
Unfortunately, there are no absolute measurements of the thermal vacancy concentration, cv (that is simultaneous measurements of dilatometric and lattice thermal expansion) but such measurements for copper (10), silver (11), gold (12) and aluminium (13) suggest a common value of (7 ± 2) × 10−4 at the melting point. The enthalpy of monovacancy formation in platinum, Hvf, is now well established as 1.51 eV (14–16) and adopting the above value of cv leads to a value for the entropy of monovacancy formation, Svf, of (1.32 ± 0.3)k (k is the Boltzmann Constant), which is in excellent agreement with values of 1.3k, calculated by Schumacher, Seeger and Härlin (17) from a re-analysis of the self-diffusion experiments of Kidson and Ross (18) and Cattaneo, Germagnoli and Grasso (19), and 1.45 k estimated by Heigl and Sizmann (15) from results using a millisecond heat pulse technique. Therefore a value for Cv of 7 × 10−4 at the melting point is adopted for platinum, and the lattice thermal expansion can be related to the dilatometric thermal expansion as in Equations (x) and (xi).
|Authors||Reference||Lattice constant at 293.15 K, nm||Original temperature, K|
|Stenzel and Weerts||23||0.39232||293|
|Owen and Yates||24||0.39240||291|
|Owen and Yates||25||0.39244||288|
|Esch and Schneider||27||0.39239||rt*|
|Goldschmidt and Land||28||0.39242||rt*|
|Grube, Schneider and Esch||29||0.39238||rt*|
|Swanson and Tatge||30||0.39232||298|
|Evans and Fischer||32||0.39236||293|
|Schröder and co-workers||33||0.39234||927|
|Waseda, Hirata and Ontani||34||0.39241||298|
|Stankus and Khairulin||35||0.39240**||293|
|Selected||0.39236 ± 0.00006|
Lattice Parameter at 293.15 K
A combination of those values selected by Donohue (20) as well as more recent determinations were corrected from kX and ångström units to nm using conversion factors recommended in the 1986 review of the fundamental constants (21). Lattice parameters were corrected to 293.15 K using thermal expansion coefficients selected in this review.
Density values in Table II were calculated using an atomic weight of 195.078 (36) and 6.0221367 × 1023 mol−1 for Avogadro’s Constant (21). Crystallographic and bulk properties, Tables II and III, respectively, would be identical below 1300 K.
This Table would be identical to Table II below 1300 K
Comparison with Other Data
Figure 1 shows those dilatometric measurements which differ notably from the selected values – the measurements of Vertogradskii (37) 356–1589 K and the direct density measurements of Stankus and Khairulin (35) 293–2042 K. The finer scale graph, see Figure 2, shows the dilatometric measurements of Cowder and colleagues (38) 293–1773 K and the AGARD (Advisory Group for Aerospace Research and Development) pooled data, as reported by Weisenburger (39) and 573–1173 K (five participants), Fitzer (40) 573–1273 K (average of ten participants) and Fitzer and Weisenburger (41) 373–1873 K (five participants in all).
Not shown are the measurements of Dorsey (42) 93–293 K which, on the basis of:
agree with the selected values to within 0.001; and those of Scheel (43) 83–373 K, Scheel and Heuse (44) 90–289 K, Henning (45) 82–289 K, Nix and MacNair (45) 85–368 K, Branchereau, Navez and Perroux (46) 573 K, Amatuni and co-workers (47) 373–1273 K, and the interferometric measurements of participant no. 7 (373–1073 K) noted by Fitzer and Weisenburger (41), which all agree with the selected curve to within 0.002. The measurements of Esser and Eusterbrock (49) 273–1273 K agree to within 0.001 up to 973 K, then deviate up to 0.007 low. The single measurement of Masumoto and Kobayashi (50) at 313 K is 0.006 low.
The comparison of lattice parameters, see Figure 3, includes measurements by: Owen and Yates (25) 288–873 K, Shinoda (51) 288–1373 K, Esser, Eilander and Bungardt (52) 293–1373 K, Eisenstein (53) 1148 K, Edwards, Speiser and Johnston (54) 291–2005 K, Mauer and Bolz (55) 273–1663 K, Brand and Goldschmidt (56) 673–1607 K, Lui, Takahashi and Bassett (57) 286–701 K, Evans and Fischer (32) 293–1783 K, Schroder, Schmitz-Pranghe and Kohlhaas (33) 85–1933 K, and Waseda, Hirata and Ohtani (34) 298–2029 K. The scatter in the data, especially at high temperatures, indicates the difficulties in reaching conclusions concerning thermal vacancy effects by trying to match unrelated lattice and dilatometric measurements.
Figure 4 shows deviations of the thermal expansion coefficients reported by Valentiner and Wallot (58) 96–284 K, which are up to 3 per cent low, and those of Kraftmakher (59) 1000–1900 K which rise sharply upwards reaching 25 per cent higher than the selected value at 1900 K. The low temperature measurements of Andres (60) 2–13 K are reported in equation form, which gives a value at 10 K of 1.0 × 10−8K−1 (14 per cent) higher than the selected value.
The Definition of Thermal Expansion
The thermodynamic thermal expansion coefficient, α, is the instantaneous change in length per unit length per degree or (1/L)(∂L/∂T). In practice it is determined as a change in length over a very short temperature interval and over this length α is defined as ((L2 – L1)/L1))/(T2 -T1) at the mean temperature (T2 + T1)/2. In the higher temperature region length changes are measured relative to a reference temperature such as 293.15 K (20°C) and are fitted to a polynomial as (LT – Lref)/Lref. Differentiation of this equation leads to the instantaneous thermal expansion coefficient relative to the reference temperature α* = (l/Lref)(∂L/∂T); α and α* are related by the Equation:
The kX Lattice Parameter Unit
The X unit (X.U.) was introduced by Seigbahn (61, 62) to be exactly equal to the milliångström unit (10−13 m) and is defined as being equal to a lattice spacing of a calcite crystal. Throughout the 1930s and early 1940s X-ray wavelengths were defined in terms of this scale, and lattice spacing values were reported in 1000 X.U. (kX units). However when the lattice spacings of different calcite crystals were shown to be variable and it was realised that the kX unit was 0.2 per cent too large because of an error in the fundamental constants used in its original derivation, then by international agreement the kX unit was defined as being exactly equal to 1.00202 ångström units (63). Continual improvement in the determination of this ratio led in the 1986 revision of the fundamental constants (21) to a derivation that the kX unit was equivalent to 0.100207789 ± 0.000000070 nm on the scale in which the primary X-ray wavelength, Cu Kα1, is defined as being exactly 1537.400 X.U. All relevant lattice parameter values reported in the literature as being in kX units have been corrected using this ratio and those reported in conventional ångström units have been corrected using the ratio 0.100207789/1.00202.
Precision Relationship between Thermal Expansion and Specific Heat
In the specific heat at constant pressure (CP) version of the Grüneisen equation:
where α is the linear thermal expansion coefficient, Γ is the Grüneisen parameter, Bs is the adiabatic bulk modulus and V is the molar volume, α and CP increase rapidly below room temperature, whereas Γ, Bs and V vary only slowly and can be lumped together as a weak function of temperature. Mel’nikova and co-workers (64–67) developed this relationship to:
where A and B are constants and T is temperature, in order to represent the sparse thermal expansion coefficient data for the alkali metals lithium to rubidium. However the bulk of the thermal expansion data for these elements is in the form of length change measurements which are poorly represented by the constants chosen by Mel’nikova and colleagues, especially for sodium and potassium below 200 K.
An independent assessment of this equation for certain metals for which high quality thermal expansion and specific heat data is available below room temperature indicates that it gives a reasonable correlation down to about 150 K but is unsatisfactory below this temperature. However it can be shown (68) that the addition of further terms to this basic equation does lead to a precision relationship between α and CP in the form:
which, when n = 2, leads to a fit of the thermal expansion coefficient data for aluminium and tungsten below 300 K to within 1.6 x 10 −8 K −1 and 1.1 ×10−8 K−8 respectively, and for copper using either n = 1 or 2 to within 8 × 10−9 K−1. Therefore the behaviour of this equation suggests that it can be used for interpolation for the surprising number of metals for which high quality thermal expansion data is available only below 90 K and at room temperature. The order of the equation used depends on the quality of the available data and for platinum it is found that the best compromise is to use two equations both with n = 1 rather than a single equation or one of higher order.
The Thermal Vacancy Effect
The generation of thermal vacancies close to the melting point leads to a difference between macroscopic or bulk thermal expansion (dL/dT) and microscopic or lattice thermal expansion (da/dT) and this difference can be exploited in order to obtain an absolute value of the vacancy concentration parameter (cv) as:
where L0 and a0 are referred to a temperature at which Cv is zero but which can be assumed to be the reference temperature 293.15 K without any loss in accuracy. Considered only in terms of monovacancies and neglecting the effects of divacancies or interstitials, Cv can then be related to the entropy (Svf) and enthalpy (Hvf) of vacancy formation as: exp (Svf/k – Hvf/k T) where k is Boltzmann’s Constant, 8.617385 x 10 −5 eV K−1(21) and T is temperature.
Therefore since the above two equations are equal, Svf and Hvf can be determined by a measurement of the thermal expansion difference. Of other techniques which could be used these parameters can also be determined with a fair degree of accuracy from a combination of electrical resistivity measurements on specimens with quenched-in vacancies and diffusion experiments (17). These measurements and a comparison with actual determinations for other metals allow an accurate estimate of the thermal vacancy parameters for platinum even though absolute values have not yet been determined.
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J. W. Arblaster, in preparation