Crystallographic Properties of Ruthenium
Crystallographic Properties of Ruthenium
Assessment of properties from absolute zero to 2606 K
The crystallographic properties of ruthenium at temperatures from absolute zero to the melting point at 2606 K are assessed following a review of the literature published between 1935 and to date. Selected values of the thermal expansion coefficients and measurements of length changes due to thermal expansion have been used to calculate the variation with temperature of the lattice parameters, interatomic distances, atomic and molar volumes and densities. The data is presented in the form of Figures, Equations and Tables.
This is the sixth in a series of papers in this Journal on the crystallographic properties of the platinum group metals (pgms), following two papers on platinum (1, 2) and one each on rhodium (3), iridium (4) and palladium (5). Ruthenium exists in a hexagonal close-packed (hcp) structure (Pearson symbol hP2) up to the melting point which is a secondary fixed point on ITS-90 at 2606 ± 10 K (6).
The thermal expansion is represented by five sets of lattice parameter measurements, those of Owen and Roberts (7, 8) (from 323 K to 873 K), Hall and Crangle (9) (from 799 K to 1557 K), Ross and Hume-Rothery (10) (from 1793 K to 2453 K), Schröder et al. (11) (from 84 K to 1982 K) and Finkel’ et al. (12) (from 80 K to 300 K) and one set of dilatometric measurements, those of Shirasu and Minato (13) (from 323 K to 1300 K). The measurements of Hall and Crangle, Ross and Hume-Rothery and Finkel’ et al. were only shown graphically with actual data points as length change values being given by Touloukian et al. (14). Because there is a certain degree of incompatibility between the high-temperature measurements, and those obtained at low-temperature by Finkel’ et al., the high- and low-temperature data were initially treated separately. Available thermal expansion data covers the range from 293.15 K to 2453 K with estimated values below the lower limit whilst in the high-temperature region the derived equations are extrapolated to the melting point.
Length change values derived from the measurements of Owen and Roberts (7, 8) and Ross and Hume-Rothery (10) agree satisfactorily and were combined to give Equations (i) and (ii) to represent the thermal expansion from 293.15 K to the melting point. On the basis ± 100δL/L293.15 K Equation (i) for the a-axis has an accuracy of ± 0.009 and Equation (ii) for the c-axis an accuracy of ± 0.025. Crystallographic properties derived from Equations (i) and (ii) are given in Tables I and II.
|Temperature, K||Thermal expansion coefficient, α a, 10−6 K−1||Thermal expansion coefficient, αc, 10−6 K−1||Thermal expansion coefficient, αavr, 10−6 K−1||Length change, δa/a293.15 K × 100, %||Length change, δc/c293.15 K × 100, %||Length change, δavr/avr293.15 K × 100a, %|
On the basis of the expression:
100 × (δL/L293.15 K (experimental) − δL/L293.15 K (calculated))
where δL/L293.15 K (experimental) is the experimental length change relative to 293.15 K and δL/L293.15 K (calculated) is the selected length change value, then length change values derived from the measurements of Hall and Crangle (9) deviate continuously from selected values and both axes are 0.14 low at the experimental limit 1557 K. Above room temperature the a-axis values of Schröder et al. (11) initially trend to be 0.080 low at 1300 K before increasing to 0.089 high at 1982 K. The c-axis values behave similarly, initially trending to 0.072 low at 1100 K before increasing sharply to 0.35 high at 1982 K. The dilatometric measurements of Shirasu and Minato (13) trend to 0.10 low. The deviations of these three sets of values are shown in Figure 1.
The lattice parameter measurements of Finkel’ et al. (12), given as length change values by Touloukian et al. (14), were fitted to cubic Equations (v) and (vi) for the a- and c-axes respectively. Derived thermal expansion coefficients at 293.15 K of 6.5 × 10−6 K−1 for the a-axis and 11.5 × 10−6 K−1 for the c-axis are notably higher than those derived from Equations (i) and (ii) as given in Tables II and III and indicate the degree of incompatibility between the high- and low-temperature data. Various manipulations of subsets of the low-temperature measurements to try and reconcile the differences proved to be unsatisfactory and the measurements of Finkel’ et al. were rejected. Therefore in order to extrapolate below room temperature Equations (i) and (ii) were differentiated and derived values of the thermal expansion coefficient relative to 293.15 K, α*, were converted to thermodynamic thermal expansion, α, using α = α*/(1 + δL/L293.15 K). The α values obtained at 293.15 K and over the range 300 K to 800 K at 50 K intervals were then fitted to Equations (iii) and (iv) where the values of the specific heat used, Cp, are given by Equation (vii). Equations (iii) and (iv) were then extrapolated below the room temperature region using specific heat values given in the Appendix in order to represent the thermal expansion to absolute zero, although a-axis thermal expansion coefficients above 240 K were slightly adjusted in order to give a smooth continuity with the high-temperature selected values. Crystallographic properties derived from Equations (iii) and (iv) are given in Tables III and IV.
|Temperature, K||Thermal expansion coefficient, αa, 10−6 K−1||Thermal expansion coefficient, αc, 10−6 K−1||Thermal expansion coefficient, αavr, 10−6 K−1||Length change, δa/a293.15 K × 100, %||Length change, δc/c293.15 K × 100, %||Length change, δavr/avr293.15 K × 100, %|
There is the possibility of significant uncertainty in this procedure but it is noted that in comparison, using the same procedure as for the high-temperature data, the measurements of Finkel’ et al. (12) show a maximum deviation of only 0.006 low at 80 K for the a-axis and then converge towards the selected values. For the c-axis, there is initially agreement with the selected values and a maximum deviation of only 0.010 low at 220 K. These small differences would actually suggest agreement between the high- and low-temperature data; however, the fitting procedure is so sensitive that these differences represent incompatibility. The low-temperature measurements of Schröder et al. (11) are initially 0.027 low at 84 K for the a-axis and then converge towards the selected values, whilst for the c-axis the value is initially 0.026 low but there is agreement to better than 0.001 above 210 K.
Normally, as an alternative method of calculation, Equations (iii) and (iv) would be fitted to a series of spline fitted equations; however as there are two axes this could involve a significant number of equations and therefore the much simpler procedure has been adopted of substituting values of Cp from the Appendix into the equations.
The Lattice Parameter at 293.15 K
The values of the lattice parameters, a and c, given in Table V represent a combination of those values selected by Donohue (15) and more recent measurements. Values originally given in kX units were converted to nanometres using the 2010 International Council for Science: Committee on Data for Science and Technology (CODATA) Fundamental Constants (16, 17) conversion factor for CuKα1, which is 0.100207697 ± 0.000000028 whilst values given in angstroms (Å) were converted using the default ratio 0.100207697/1.00202 where the latter value represents the old conversion factor from kX units to Å. Lattice parameter values were corrected to 293.15 K using the values of the thermal expansion coefficient selected in the present review. Density values given in Tables II and III were calculated using the currently accepted atomic weight of 101.07 ± 0.02 (18) and an Avogadro constant (NA) of (6.02214129 ± 0.00000027) × 1023mol−1 (16, 17). From the lattice parameter values at 293.15 K selected in Table V as: a = 0.27058 ± 0.00002 nm and c = 0.42816 ± 0.00007 nm, the derived selected density is 12364 ± 3 kg m−3 and the molar volume is (8.1743 ± 0.0018) × 10−6 m3 mol−1. In Tables II and III the interatomic distance d1 = (a2/3 + c2/4)½ and d2 = a. The atomic volume is (√3 a2 c)/4 and the molar volume is calculated as NA (√3 a2 c)/4 which is equivalent to atomic weight divided by density. Thermal expansion is αavr = (2 αa + αc)/3 and length change is δavr/avr293.15 K = (2 δa/a293.15 K + δc/c293.15 K)/3 (avr = average).
|Authors (Year)||Reference||Original temperature, K||Original units||Lattice parameter, a, corrected to 293.15 K, nm||Lattice parameter, c, corrected to 293.15 K, nm||Notes|
|Owen et al. (1935)||(18)||291||kX||0.27044||0.42818||(a)|
|Owen and Roberts (1936)||(7)||291||kX||0.27042||0.42819||(a)|
|Owen and Roberts (1937)||(8)||293||kX||0.27040||0.42819||(a)|
|Ross and Hume-Rothery||(10)||303||Å||0.27042||0.42799||(a), (b)|
|Finkel’ et al. (1971)||(12)||293||Å||0.27062||0.42815||(a), (b)|
|Hellawell and Hume-Rothery (1954)||(19)||298||kX||0.27058||0.42817|
|Swanson et al. (1955)||(20)||300||Å||0.27059||0.42819|
|Hall and Crangle (1957)||(9)||rtb||Å||0.27058||0.42805|
|Anderson and Hume-Rothery (1960)||(21)||293||kX||0.27058||0.42814|
|Savitskii et al. (1962)||(23)||rt||kX||0.27059||0.42819|
|Schröder et al. (1972)||(11)||284||Å||0.27056||0.42826|
Because there is disagreement between the high- and low-temperature measurements for ruthenium, satisfactory thermal expansion data is only available above 293.15 K with a novel approach being used to extrapolate below this temperature to derive values which must be considered to be tentative. Clearly further measurements are required for this element.
High-Temperature Thermal Expansion Equations for Ruthenium (293.15 K to 2606 K)
δa/a293.15 = −1.56642 × 10−3 + 4.93471 × 10−6 T + 1.34455 × 10−9 T2 + 1.69158 × 10−13 T3 (i)
δc/c293.15 = −2.39045 × 10−3 + 7.52727 × 10−6 T + 2.06251 × 10−9 T2 + 2.61425 × 10−13 T3 (ii)
Low-Temperature Thermal Expansion Equations for Ruthenium (0 K to 293.15 K)
αa (K−1) = Cp (1.92207 × 10−7 + 8.09046 × 10−11 T + 7.16082 × 10−6 / T) (iii)
αc (K−1) = Cp (2.93088 × 10−7 + 1.24609 × 10−10 T + 1.09421 × 10−5 / T) (iv)
Thermal Expansion Equations Representing the Measurements of Finkel’ et al. (12)
δa/a293.15 = −1.40337 × 10−3 + 3.25082 × 10−6 T + 4.63332 × 10−9 T2 + 2.07266 × 10−12 T3 (v)
δc/c293.15 = −1.87652 × 10−3 + 3.44170 × 10−6 T + 2.91501 × 10−9 T2 + 2.44946 × 10−11 T3 (vi)
Appendix: Specific Heat Values for Ruthenium
Because of the large number of spline fitted equations that would be required to conform to Equations (iii) and (iv), a simpler approach is used for the non-cubic metals in that specific heat values are directly applied to these equations. However this would require that the Table of low-temperature specific heat values originally given by the present author (24) has to be more comprehensive and the revised Table is given as Table VI. The high-temperature specific heat values corresponding to the above reference is given as Equation (vii) and is derived by differentiating the selected enthalpy equation.
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John W. Arblaster is interested in the history of science and the evaluation of the thermodynamic and crystallographic properties of the elements. Now retired, he previously worked as a metallurgical chemist in a number of commercial laboratories and was involved in the analysis of a wide range of ferrous and non-ferrous alloys.