Low-Temperature Region

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/L_{293.15 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, C_p, 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.

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 (N_A) 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 N_A (√3 a² c)/4 which is equivalent to atomic weight divided by density. Thermal expansion is α_avr = (2 α_a + α_c)/3 and length change is δavr/avr_{293.15 K} = (2 δa/a_{293.15 K} + δc/c_{293.15 K})/3 (avr = average).

Summary

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.