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Platinum Metals Rev., 2013, 57, (2), 127

doi:10.1595/147106713x665030

Crystallographic Properties of Ruthenium

Assessment of properties from absolute zero to 2606 K

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Article Synopsis

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.

Thermal Expansion

High-Temperature Region

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.

Table I

High-Temperature Crystallographic Properties of Ruthenium

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, %
293.15 5.77 8.80 6.78 0 0 0
300 5.79 8.83 6.80 0.004 0.006 0.005
400 6.09 9.29 7.16 0.063 0.097 0.074
500 6.40 9.77 7.52 0.126 0.192 0.148
600 6.72 10.25 7.90 0.191 0.292 0.225
700 7.05 10.76 8.28 0.260 0.398 0.306
800 7.39 11.27 8.68 0.333 0.509 0.391
900 7.73 11.80 9.09 0.409 0.625 0.481
1000 8.09 12.34 9.51 0.488 0.746 0.574
1100 8.46 12.90 9.94 0.571 0.873 0.672
1200 8.83 13.47 10.38 0.658 1.006 0.774
1300 9.22 14.05 10.83 0.749 1.145 0.881
1400 9.61 14.65 11.29 0.844 1.291 0.993
1500 10.02 15.26 11.76 0.943 1.442 1.110
1600 10.43 15.88 12.24 1.046 1.600 1.231
1700 10.85 16.51 12.74 1.154 1.765 1.358
1800 11.28 17.16 13.24 1.266 1.936 1.489
1900 11.71 17.82 13.75 1.382 2.115 1.627
2000 12.16 18.49 14.27 1.503 2.300 1.769
2100 12.61 19.17 14.80 1.629 2.493 1.917
2200 13.08 19.86 15.34 1.760 2.693 2.071
2300 13.55 20.56 15.89 1.895 2.901 2.231
2400 14.03 21.28 16.44 2.036 3.117 2.396
2500 14.51 22.00 17.01 2.182 3.340 2.568
2600 15.01 22.74 17.58 2.333 3.571 2.746
2606 15.04 22.78 17.62 2.342 3.586 2.756

 a avr = average

Table II

Further High-Temperature Crystallographic Properties of Ruthenium

Temperature, K Lattice parameter, a, nma Lattice parameter, c, nm c/a ratio Interatomic distance, d1, nm Atomic volume, 10−3 nm3 Molar volume, 10−6 m3 mol−1 Density, kg m−3
293.15 0.27058 0.42816 1.5824 0.26502 13.574 8.174 12364
300 0.27059 0.42819 1.5824 0.26503 13.576 8.175 12363
400 0.27075 0.42857 1.5829 0.26524 13.604 8.193 12337
500 0.27092 0.42898 1.5834 0.26547 13.634 8.211 12310
600 0.27110 0.42941 1.5840 0.26570 13.666 8.230 12281
700 0.27128 0.42986 1.5845 0.26595 13.699 8.250 12251
800 0.27148 0.43034 1.5851 0.26620 13.734 8.271 12220
900 0.27169 0.43083 1.5858 0.26647 13.770 8.293 12188
1000 0.27190 0.43135 1.5864 0.26676 13.809 8.316 12154
1100 0.27213 0.43190 1.5871 0.26706 13.849 8.340 12118
1200 0.27236 0.43247 1.5878 0.26737 13.891 8.366 12082
1300 0.27261 0.43306 1.5886 0.26769 13.936 8.392 12043
1400 0.27286 0.43369 1.5894 0.26803 13.982 8.420 12003
1500 0.27313 0.43434 1.5902 0.26838 14.030 8.449 11962
1600 0.27341 0.43501 1.5911 0.26875 14.081 8.480 11919
1700 0.27370 0.43572 1.5919 0.26913 14.134 8.512 11874
1800 0.27401 0.43645 1.5929 0.26953 14.189 8.545 11828
1900 0.27432 0.43722 1.5938 0.26995 14.247 8.580 11780
2000 0.27465 0.43801 1.5948 0.27038 14.307 8.616 11731
2100 0.27499 0.43883 1.5958 0.27083 14.369 8.653 11680
2200 0.27534 0.43969 1.5969 0.27130 14.434 8.692 11627
2300 0.27571 0.44058 1.5980 0.27178 14.502 8.733 11573
2400 0.27609 0.44150 1.5911 0.27229 14.572 8.776 11517
2500 0.27648 0.44246 1.6003 0.27281 14.646 8.820 11459
2600 0.27689 0.44345 1.6015 0.27335 14.722 8.866 11400
2606 0.27692 0.44351 1.6016 0.27338 14.727 8.869 11396

 a a = d2

 

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.

Fig. 1.

The difference between length change values derived from the measurements of Hall and Crangle (9), Schröder et al. (11) and Shirasu and Minato (13)

The difference between length change values derived from the measurements of Hall and Crangle (9), Schröder et al. (11) and Shirasu and Minato (13)

 

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/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.

Table III

Low-Temperature Crystallographic Properties of Ruthenium

Temperature, K Lattice parameter, a, nma Lattice parameter, c, nm c/a ratio Interatomic distance, d1, nm Atomic volume, 10−3 nm3 Molar volume, 10−6 m3 mol−1 Density, kg m−3
0b 0.27028 0.42742 1.5814 0.26462 13.520 8.142 12414
10 0.27028 0.42742 1.5814 0.26462 13.520 8.142 12414
20 0.27028 0.42743 1.5814 0.26462 13.520 8.142 12414
30 0.27028 0.42743 1.5814 0.26462 13.520 8.142 12414
40 0.27028 0.42743 1.5814 0.26462 13.520 8.142 12413
50 0.27028 0.42744 1.5815 0.26462 13.521 8.142 12413
60 0.27028 0.42745 1.5815 0.26463 13.521 8.143 12412
70 0.27029 0.42746 1.5815 0.26464 13.522 8.143 12411
80 0.27030 0.42748 1.5815 0.26465 13.524 8.144 12410
90 0.27031 0.42750 1.5815 0.26466 13.525 8.145 12409
100 0.27031 0.42752 1.5816 0.26467 13.527 8.146 12407
110 0.27033 0.42755 1.5816 0.26468 13.529 8.147 12406
120 0.27034 0.42757 1.5816 0.26470 13.531 8.148 12404
130 0.27035 0.42760 1.5817 0.26471 13.533 8.150 12402
140 0.27036 0.42763 1.5817 0.26473 13.535 8.151 12400
150 0.27037 0.42766 1.5817 0.26475 13.537 8.152 12398
160 0.27039 0.42769 1.5818 0.26476 13.539 8.154 12396
170 0.27040 0.42772 1.5818 0.26478 13.542 8.155 12394
180 0.27041 0.42776 1.5819 0.26480 13.544 8.156 12391
190 0.27043 0.42779 1.5819 0.26482 13.547 8.158 12389
200 0.27044 0.42782 1.5820 0.26484 13.549 8.159 12387
210 0.27045 0.42786 1.5820 0.26485 13.552 8.161 12385
220 0.27046 0.42789 1.5820 0.26487 13.554 8.162 12382
230 0.27048 0.42793 1.5821 0.26489 13.557 8.164 12380
240 0.27050 0.42796 1.5821 0.26491 13.559 8.166 12377
250 0.27051 0.42800 1.5822 0.26493 13.562 8.167 12375
260 0.27053 0.42804 1.5822 0.26495 13.565 8.169 12373
270 0.27054 0.42807 1.5823 0.26497 13.567 8.170 12370
280 0.27056 0.42811 1.5823 0.26499 13.570 8.172 12368
290 0.27058 0.42815 1.5824 0.26501 13.573 8.174 12365
293.15 0.27058 0.42816 1.5824 0.26502 13.574 8.174 12364

 a a = d2

 b Since all values below 293.15 K are estimated they are given in italics

Table IV

Further Low-Temperature Crystallographic Properties of Ruthenium

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, %
0a 0 0 0 −0.113 −0.172 −0.132
10 0.04 0.06 0.05 −0.113 −0.172 −0.132
20 0.09 0.16 0.12 −0.113 −0.172 −0.132
30 0.32 0.48 0.37 −0.112 −0.171 −0.132
40 0.70 1.07 0.83 −0.112 −0.171 −0.131
50 1.25 1.91 1.47 −0.111 −0.169 −0.130
60 1.85 2.82 2.17 −0.109 −0.167 −0.129
70 2.39 3.66 2.56 −0.107 −0.163 −0.126
80 2.88 4.40 3.39 −0.105 −0.159 −0.123
90 3.30 5.04 3.88 −0.102 −0.155 −0.119
100 3.66 5.58 4.30 −0.098 −0.149 −0.115
110 3.95 6.03 4.65 −0.094 −0.144 −0.111
120 4.20 6.42 4.94 −0.090 −0.137 −0.106
130 4.42 6.74 5.19 −0.086 −0.131 −0.101
140 4.60 7.02 5.40 −0.081 −0.124 −0.096
150 4.75 7.25 5.58 −0.077 −0.117 −0.090
160 4.88 7.44 5.73 −0.072 −0.109 −0.084
170 4.98 7.61 5.86 −0.067 −0.102 −0.079
180 5.08 7.76 5.97 −0.062 −0.094 −0.073
190 5.17 7.89 6.07 −0.057 −0.086 −0.067
200 5.25 8.01 6.17 −0.052 −0.079 −0.061
210 5.32 8.12 6.25 −0.046 −0.070 −0.054
220 5.38 8.22 6.33 −0.041 −0.062 −0.048
230 5.45 8.31 6.40 −0.036 −0.054 −0.042
240 5.50 8.40 6.47 −0.030 −0.046 −0.035
250 5.58 8.48 6.55 −0.024 −0.037 −0.029
260 5.63 8.56 6.61 −0.019 −0.029 −0.022
270 5.68 8.63 6.66 −0.013 −0.020 −0.016
280 5.71 8.70 6.71 −0.008 −0.011 −0.009
290 5.75 8.77 6.76 −0.002 −0.003 −0.002
293.15 5.77 8.80 6.78 0 0 0

 a Since all values below 293.15 K are estimated they are given in italics

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).

Table V

Lattice Parameter Values at 293.15 Ka

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
Černohorský (1960) (22) 295 Å 0.27059 0.42812
Savitskii et al. (1962) (23) rt kX 0.27059 0.42819
Schröder et al. (1972) (11) 284 Å 0.27056 0.42826

 a Selected values for the present paper are: a = 0.27058 ± 0.00002 and 0.42816 ± 0.00007

 b rt = room temperature

[iii] Notes to Table V

[iv] (a) For information only – not included in the average

[v] (b) Lattice parameter values given by Touloukian et al. (14)

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.

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)

High-Temperature Specific Heat Equation (298.15 K to 2606 K)

Cp (J mol−1 K−1) = 23.1728 + 7.28378 × 10−3 T − 2.703021 × 10−6 T2 + 1.50844 × 10−9 T3 − 97572.6/T2 (vii)

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.

Table VI

Low-Temperature Specific Heat Values for Ruthenium

Temperature, K Specific heat, J mol−1 K Temperature, K Specific heat, J mol−1 K Temperature, K Specific heat, J mol−1 K
10 0.0438 50 3.682 130 17.130
15 0.0955 60 5.838 140 18.050
20 0.186 70 7.991 150 18.837
25 0.359 80 10.000 160 19.509
30 0.731 90 11.839 170 20.093
35 1.233 100 13.455 180 20.607
40 1.877 110 14.854 190 21.066
45 2.707 120 16.071 200 21.480
210 21.857 250 23.047 290 23.889
220 22.200 260 23.277 293.15 23.950
230 22.514 270 23.490 298.15 24.046
240 22.796 280 23.693 300 24.071

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  24.  J. W. Arblaster, CALPHAD, 1995, 19, (3), 339 LINK http://dx.doi.org/10.1016/0364-5916(95)00031-9

The Author

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.

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