Ultrasonic Study of Osmium and Ruthenium - Platinum Metals Review

C 11 = 24.1 p 4 C'
C 13 = 1.925 p 6 C'
C 44 = 2.309 p 4 C'
C 111 = 126.9 p 2 B + 8.853 p 4 C'
C 113 = 1.924 p 4 B + 1.155 p 6 C'
C 133 = 3.695 p 6 B
C 144 = 2.309 p 4 B
C 222 = 101.039 p 2 B + 9.007 p 4 C'
C 12 = 5.918 p 4 C'
C 33 = 3.464 p 8 C'
C 66 = 9.851 p 4 C'
C 112 = 19.168 p 2 B – 1.61 p 4 C'
C 123 = 1.617 p 4 B – 1.155 p 6 C'
C 155 = 1.539 p 4 B
C 344 = 3.464 p 6 B
C 333 = 5.196 p 8 B
(ii)
where p = c/a (the axial ratio); C' = χ a/p 5 ;
B = Ψ a 3 /p 3 ; χ = (1/8)[{n b 0 (n – m)}/{a n + 4 }];
Ψ = – χ/{6 a 2 (m + n + 6)}; and c is the height of
the unit cell.
The basal plane distance ‘a’ and axial ratio ‘p’ of
Table I
Basal Plane Distance, a, and Axial Ratio, p, for
Osmium and Ruthenium *
Value
Parameter Units Os Ru
a Å 2.729 2.700
p – 1.577 1.582
*All values obtained from Reference (6)
Os and Ru are given in Table I (6). The calculation
of higher-order elastic constants was carried out
using the set of Equations (ii), taking appropriate
values of m, n and b 0. The values of the second- and
third-order elastic constants are shown in Table II.
The Young’s modulus ‘Y’ and bulk modulus
‘B’ were evaluated using the second-order elastic
constants presented here. It is obvious from
Table II that there is good agreement between
the calculated values from this study and the previously
reported values for the Young’s modulus
and the bulk modulus (6, 12, 13). It is observed
that the higher-order elastic constants of Os and
Ru are analogous to those of gallium nitride and
indium nitride, respectively (14). All the second-
Table II
Second- and Third-order Elastic Constants for Osmium and Ruthenium
Second-order Value, 10 11 Nm –2 Third-order Value, 10 11 Nm –2
elastic constant Os Ru elastic constant Os Ru
C 11 8.929 6.279 C 111 –145.620 –102.400
C 12 2.193 1.542 C 112 –23.088 –16.236
C 13 1.774 1.255 C 113 –4.550 –3.220
C 33 7.938 5.654 C 123 –5.783 –4.093
C 44 2.128 1.506 C 133 –26.815 –19.098
C 66 3.166 2.463 C 344 –25.139 –17.904
B 4.119 2.911 C 144 –6.738 –4.768
B * 3.800 2.920 C 155 –4.491 –3.178
B ** 4.110 – C 222 –115.219 –8.026
Y 5.525 3.971 C 333 –93.777 –67.211
Y * 5.600 4.300 – – –
Y *** 5.620 4.220 – – –
*Reference (12); **Reference (13); ***Reference (6)
Platinum Metals Rev., 2009, 53, (2) 92