Osmium (Os) and ruthenium (Ru) belong tothe platinum group metals (pgms). Both Os andRu are hard, brittle and have poor oxidation resis-tance. Both metals are mostly produced frommines in the Bushveld Complex, South Africa.Trace amounts of Os are also found in nickel-bear-ing ores in the Sudbury, Ontario region of Canada,and in Russia, along with other pgms. Os oxidiseseasily in air, producing poisonous fumes, and is therarest of the pgms. Ru is also found in ores withother pgms in Russia, and in North and SouthAmerica. Ru is isolated through a complex chemi-cal process in which calcination and hydrogenreduction are used to convert ammonium rutheni-um chloride, yielding a sponge. The sponge is thenconsolidated by powder metallurgy techniques orby argon arc welding. Ru is mostly used in harddisks and in alloys with platinum for jewellery andelectrical contacts. It also has increasing use incatalysis (1). Os and Ru were discovered in 1804(2) and 1844 (3), respectively.

Magnetic measurements of Ru and Os havebeen made by Hulm and Goodman (4). Savitskii et al. (5) investigated the microstructural, physicaland chemical properties of these metals, and theirmechanical properties have been explored by

Buckley (6). Os and Ru both possess a hexagonalclosed packed (h.c.p.) structure (6). Barnard andBennett (7) studied their oxidation states.

The investigation of ultrasonic properties canbe used as a non-destructive technique for thedetection and characterisation of a material’sproperties, not only after production but also dur-ing processing (8–10). However, none of the workreported in the literature so far is centred on theultrasonic study of Os and Ru. Thus, in the pre-sent work, we have studied the ultrasonicproperties of these metals. Ultrasonic attenuation,velocity and related parameters have been calcu-lated for use in non-destructive testing (NDT)characterisation.

Higher-Order Elastic ConstantsThe higher-order elastic constants of h.c.p.

structured materials can be formulated using theLennard-Jones potential, φ(r ), Equation (i):

φ(r) = {– (a0/r m ) + (b0/r n )} (i)

where a0, b0 are constants; m, n are integers and r isthe distance between atoms (11). The expressionsof second- and third-order elastic constants aregiven in the set of Equations (ii):

91Platinum Metals Rev., 2009, 53, (2), 91–97

Ultrasonic Study of Osmium and RutheniumDETERMINATION OF HIGHER-ORDER ELASTIC CONSTANTS, ULTRASONIC VELOCITIES, ULTRASONIC ATTENUATION AND ALLIED PARAMETERS

By D. K. Pandey*, Devraj Singh** and P. K. Yadawa***Department of Applied Physics, AMITY School of Engineering and Technology, Bijwasan, New Delhi-110 061, India;

E-mail: *pandeydrdk@rediffmail.com; **dsingh1@aset.amity.edu; ***pkyadawa@aset.amity.edu

Ultrasonic properties of two platinum group metals, osmium and ruthenium, are presentedfor use in characterisation of their materials. The angle-dependent ultrasonic velocity hasbeen computed for the determination of anisotropic behaviour of these metals. For the evaluationof ultrasonic velocity, attenuation and acoustic coupling constants, the higher-order elasticconstants of Os and Ru have been calculated using the Lennard-Jones potential. The natureof the angle-dependent ultrasonic velocity is found to be similar to those of molybdenum-ruthenium-rhodium-palladium alloys, Group III nitrides and lave-phase compounds such asTiCr2, ZrCr2 and HfCr2. The results of this investigation are discussed in correlation with otherknown thermophysical properties.

DOI: 10.1595/147106709X430927

where p = c/a (the axial ratio); C' = χ a/p 5;B = Ψ a3/p 3; χ = (1/8)[{n b0 (n – m)}/{an + 4}];Ψ = – χ/{6 a2 (m + n + 6)}; and c is the height ofthe unit cell.

The basal plane distance ‘a’ and axial ratio ‘p’ of

Os and Ru are given in Table I (6). The calculationof higher-order elastic constants was carried outusing the set of Equations (ii), taking appropriatevalues of m, n and b0. The values of the second- andthird-order elastic constants are shown in Table II.

The Young’s modulus ‘Y ’ and bulk modulus‘B’ were evaluated using the second-order elasticconstants presented here. It is obvious fromTable II that there is good agreement betweenthe calculated values from this study and the pre-viously reported values for the Young’s modulusand the bulk modulus (6, 12, 13). It is observedthat the higher-order elastic constants of Os andRu are analogous to those of gallium nitride andindium nitride, respectively (14). All the second-

Platinum Metals Rev., 2009, 53, (2) 92

C11 = 24.1 p 4 C' C12 = 5.918 p 4 C'

C13 = 1.925 p 6 C' C33 = 3.464 p 8 C'

C44 = 2.309 p 4 C' C66 = 9.851 p 4 C'

C111 = 126.9 p 2 B + 8.853 p 4 C' C112 = 19.168 p 2 B – 1.61 p 4 C'

C113 = 1.924 p 4 B + 1.155 p 6 C' C123 = 1.617 p 4 B – 1.155 p 6 C'

C133 = 3.695 p 6 B C155 = 1.539 p 4 B

C144 = 2.309 p 4 B C344 = 3.464 p 6 B

C222 = 101.039 p 2 B + 9.007 p 4 C' C333 = 5.196 p 8 B

(ii)

Table I

Basal Plane Distance, a, and Axial Ratio, p, forOsmium and Ruthenium*

Value

Parameter Units Os Ru

a Å 2.729 2.700p – 1.577 1.582

*All values obtained from Reference (6)

Table II

Second- and Third-order Elastic Constants for Osmium and Ruthenium

Second-order Value, 1011 Nm–2 Third-order Value, 1011 Nm–2

elastic constant Os Ru elastic constant Os Ru

C11 8.929 6.279 C111 –145.620 –102.400C12 2.193 1.542 C112 –23.088 –16.236C13 1.774 1.255 C113 –4.550 –3.220C33 7.938 5.654 C123 –5.783 –4.093C44 2.128 1.506 C133 –26.815 –19.098C66 3.166 2.463 C344 –25.139 –17.904B 4.119 2.911 C144 –6.738 –4.768B* 3.800 2.920 C155 –4.491 –3.178B** 4.110 – C222 –115.219 –8.026Y 5.525 3.971 C333 –93.777 –67.211Y* 5.600 4.300 – – –Y*** 5.620 4.220 – – –

*Reference (12); **Reference (13); ***Reference (6)

and third-order elastic constants for Ru arefound to be higher than those of Mo-Ru-Rh-Pdalloys (11). Thus we conclude that our calculatedhigher-order elastic constants and theory are cor-rect and justified. The bulk modulus of Os isfound to be higher than that of its nitride andcarbide (15). The values of C11 for the face cen-tred cubic (f.c.c.) structured pgms, Pd and Pt, are1.8 × 1011 Nm–2 and 2.2 × 1011 Nm–2, respective-ly (16). The high elastic modulus reveals that Osand Ru have low adhesion power and high hard-ness. Thus these metals are suitable forapplications such as sliding electrical contacts,switches, slip rings, etc.

Ultrasonic VelocitiesThe anisotropic properties of a material are

related to its ultrasonic velocities as they are related to higher-order elastic constants. There arethree types of ultrasonic velocities in h.c.p. crystals. These include one longitudinal and twoshear wave velocities, which are given by

Equations (iii)–(v) (14), where V1 is the longitudi-nal wave velocity, V2 is the quasi-shear wavevelocity and V3 is the shear wave velocity; ρ is thedensity of the material and θ is the angle withunique axis of the crystal. The densities of Os andRu are taken from the literature (6). The angle-dependent ultrasonic velocities in the chosenmetals were computed using Equations (iii)–(v)and are shown in Figure 1.

The longitudinal ultrasonic velocities of Os andRu (see Figure 1) for wave propagation alongunique axis are given in Table III, together withtheir sound velocities (17, 18). The nature of theultrasonic velocity curves in these h.c.p. metals issimilar to other h.c.p. structured materials (11, 14,19). Comparison with these sources verifies ourvelocity results. The anomalous maxima and min-ima in the velocity curves are due to the combinedeffects of the second-order elastic constants. Theangle-dependent velocities of Ru are greater thanthose of Mo-Ru-Rh-Pd alloys (11) due to the higher values of the elastic constants for Ru. Thus

Platinum Metals Rev., 2009, 53, (2) 93

V 12 = {C33 cos2θ + C11 sin2θ + C44 + {[C11 sin2θ – C33 cos2θ + C44 (cos2θ – sin2θ)]2

+ 4 cos2θ sin2θ (C13 + C44)2}1/2}/2ρ (iii)

V 22 = {C33 cos2θ + C11 sin2θ + C44 – {[C11 sin2θ – C33 cos2θ + C44 (cos2θ – sin2θ)]2

+ 4 cos2θ sin2θ (C13 + C44)2}1/2}/2ρ (iv)

V 32 = {C44 cos2θ + C66 sin2θ}/ρ (v)

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

0 10 20 30 40 50 60 70 80 90

Angle

V (

10

3 m

s-1

)

Os-V1 Ru-V1 Os-V2Ru-V2 Os-V3 Ru-V3Os – V1

Ru – V2

Ru – V1

Os – V3

Os – V2

Ru – V3

7.5

7

6.5

6

5.5

5

4.5

4

3.5

3

2.50 10 20 30 40 50 60 70 80 90

Angle, θ

Ultra

sonic

velo

city,

V, km

s–

1

Fig. 1 Plot of ultrasonic velocity ‘V’, against anglewith unique axis ‘θ’ of osmium and ruthenium

calculated results of ultrasonic velocities can beused for anisotropic characterisation of Os and Rumetals.

Ultrasonic Attenuation and AlliedParameters

The predominant causes of ultrasonic attenua-tion in a solid at room temperature arephonon-phonon interaction (Akhieser type loss)and thermoelastic relaxation mechanisms. Theultrasonic attenuation coefficient ‘(α)Akh’ due tothe phonon-phonon interaction mechanism isgiven by Equation (vi) (11, 14):

(α/f 2 )Akh = 4 π2 (3E0 < (γij ) 2 >

– < γij > 2 CVT )τ/2ρV 3 (vi)

where f is the frequency of the ultrasonic wave; Vis the ultrasonic velocity for longitudinal and shearwaves as defined in Equations (iii)–(v); E0 is thethermal energy density; T is the temperature and γi

j

is the Grüneisen number: i and j are the mode anddirection of propagation. The Grüneisen numberfor a hexagonal structured crystal along the <001>orientation or θ = 0º is a direct consequence of thesecond- and third-order elastic constants. CV is thespecific heat per unit volume of the material. Theacoustic coupling constant ‘D ’ is the measure ofacoustic energy converted to thermal energy and isgiven by Equation (vii):

D = 3(3E0 < (γij )2 >

– < γij > 2 CVT )/E0 (vii)

When an ultrasonic wave propagates through acrystalline material, the equilibrium of phonon dis-tribution is disturbed. The time taken forre-establishment of equilibrium of the thermal

phonon distribution is called the thermal relaxationtime ‘τ’ and is given by Equation (viii):

τ = τS = τL/2 = 3K/CV V D2 (viii)

where τL is the thermal relaxation time for the longitudinal wave; τS is the thermal relaxation timefor the shear wave; and K is the thermal conductiv-ity. VD is the Debye average velocity and iscalculated from the initial slopes of the threeacoustic branches, Equation (ix):

where Ω is solid angle. The integration is over alldirections and the summation is over the threeacoustic branches. The propagation of the longitu-dinal ultrasonic wave creates compression andrarefaction throughout the lattice. The rarefiedregions are colder than the compressed regions.Thus there is a flow of heat between these tworegions. The thermoelastic loss ‘(α)Th’ can be calculated with Equation (x):

(α/f 2 )Th = 4 π2 < γij >2 K T/2ρV L

5 (x)

The thermoelastic loss for the shear wave hasno physical significance because the average of theGrüneisen number for each mode and direction ofpropagation is equal to zero for the shear wave.Only the longitudinal wave is responsible for ther-moelastic loss because it causes variation inentropy along the direction of propagation.

The thermal conductivities of the selected met-als are taken from the literature (6). The values of the specific heat per unit volume ‘CV ’and energy density ‘E0’ are obtained from the American Institute of Physics Handbook (20).The thermal relaxation time ‘τ’ and Debye averagevelocity ‘VD’ are evaluated from Equations (viii)and (ix) and are shown in Figure 2. The ultrasonicattenuation coefficients are calculated fromEquations (vi) and (x). Calculated acoustic cou-pling constants ‘D ’ and ultrasonic attenuationcoefficients ‘(α/f 2)’ are presented in Table IV.

The angle-dependent thermal relaxation time varies between 6.5 ps and 9.55 ps. This indicates that after the propagation of a wave,

Platinum Metals Rev., 2009, 53, (2) 94

1 dΩ

Vi3 4π

VD = 13

3

Σ ∫i = 1

⎧⎩

–1/3

⎫⎭

(ix)

Table III

Longitudinal Ultrasonic Velocities, V1, and SoundVelocities* of Osmium and Ruthenium

Value, km s–1

Parameter Os Ru

V1 5.9 6.7Sound velocity 4.9 6.0

*The sound velocities are obtained from References (17, 18)

the distribution of phonons returns to the equi-librium position within 6.5–9.5 ps. The values ofthe thermal relaxation time in Pd and Pt are 8.8ps and 13.4 ps, respectively (16). The thermalrelaxation time of the chosen metals is directlyaffected by their Debye average velocity and thethermal conductivity of these metals.

The total attenuation is given by Equation (xi):

(α/f 2 )Total = (α/f 2 )Th + (α/f 2 )L + (α/f 2 )S (xi)

where (α/f 2 )L is the ultrasonic attenuation coeffi-cient for the longitudinal wave and (α/f 2 )S is theultrasonic attenuation coefficient for the shearwave. The value of (α/f 2 )Total is greater for Osthan for Ru. The dominant mechanism for totalattenuation is phonon-phonon interaction. The

attenuation due to phonon-phonon interaction isdirectly related to the acoustic coupling constantand thermal relaxation time. Os has a lower thermal relaxation time than Ru, while the acousticcoupling constant of Os is greater than that of Rudue to its low value of CV/E0. Thus total attenua-tion is mainly affected by the acoustic couplingconstant. Total attenuation at the nanoscale in Pdand Pt has been reported as 2.98 × 10–15 Np s2 m–1

and 7.38 × 10–15 Np s2 m–1, respectively (16).These reported values are larger than those of Osand Ru. The total attenuation in the Ru alloyMo20Ru54Rh15Pd11 has been found to be1.044 × 10–15 Np s2 m–1 (11). A comparison oftotal attenuation indicates that these metals aremore durable and ductile in alloy form than in

Platinum Metals Rev., 2009, 53, (2) 95

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70 80 90

Angle

VD

(1

03 m

s-1)

an

d R

ela

x tim

e(p

s) Os-VD Ru-VD

Os-Relax. Ru-Relax.

Os – VD

Os – τRu – VD

Ru – τ

10

9

8

7

6

5

4

3

0 10 20 30 40 50 60 70 80 90

Angle, θ

Debye a

vera

ge v

elo

city,

VD, km

s–

1

10

9

8

7

6

5

4

3T

herm

al re

laxatio

n tim

e, τ

, ps

Fig. 2 Plot of Debye average velocity ‘VD’and thermal relaxation time, ‘τ’ against anglewith unique axis ‘θ’ of osmium and ruthenium

Table IVAcoustic Coupling Constants, D, and Ultrasonic AttenuationCoefficients, (α/f 2), of Osmium and Ruthenium

Value

Parameter Units Os Ru

DL – 56.002 55.825DS – 1.532 1.473(α/f 2)Th 10–15 Np s2 m–1 0.002 0.002(α/f 2)L 10–15 Np s2 m–1 1.598 1.509(α/f 2)S 10–15 Np s2 m–1 0.158 0.144(α/f 2)Total 10–15 Np s2 m–1 1.758 1.655

their pure form. The performance of Ru as an alloycomponent for durability and ductility might bebetter than that of Os due to its lower ultrasonicattenuation.

The pulse echo technique (PET) can be usedfor the measurement of ultrasonic velocity andattenuation of materials, because it avoids heatloss and scattering loss. This method providesresults with good accuracy and would provideuseful information when combined with the the-ory presented here. For Ru, PET can be useddirectly for measurement. The high precisionindirect approaches such as resonant ultrasoundspectroscopy (RUS) and novel digital pulse echooverlap techniques can be used for the measure-ment of ultrasonic velocity in Os (21). Byultrasonic velocity measurements, the anisotropicmechanical properties of a material can be under-stood as they are related to velocity. This enablesquality assurance of products made with thesematerials in combination with other materials tobe carried out by measurements of ultrasonicattenuation, not only after production but alsoduring processing.

ConclusionsThe adopted method for theoretical study of

higher-order elastic constants is justified for thepgms. The high values of the elastic constants ofthe h.c.p. metals Os and Ru provide evidence oftheir low adhesive power and high hardness. Theresults of the elastic constants substantiate thesuperiority of h.c.p. metals over f.c.c. metals in thisregard. The anisotropic characterisation of thechosen metals can be carried out on the basis ofdirectional ultrasonic velocity and thermal relax-ation time. The phonon-viscosity mechanism is thedominant cause of total attenuation in these met-als. The acoustic coupling constant is a governingfactor for total attenuation. The mechanical behav-iour of Ru is expected to be better in terms ofdurability and ductility than that of other pgms dueto its low attenuation.

The results obtained in this investigation can beused for further study of these metals. Our wholetheoretical approach can be applied to the evalua-tion of ultrasonic attenuation and relatedparameters to study the microstructural propertiesof h.c.p. structured metals.

Platinum Metals Rev., 2009, 53, (2) 96

1 P. Ahlberg, ‘Development of the metathesis methodin organic synthesis’, Advanced information on theNobel Prize in Chemistry 2005, The Royal SwedishAcademy of Sciences, Stockholm, 2005;http://nobelprize.org/nobel_prizes/chemistry/laureates/2005/chemadv05.pdf (Accessed on 21stFebruary 2009)

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(5), 56417 Web Elements, Osmium, Physical Properties:

http://www.webelements.com/osmium/physics.html(Accessed on 16th April 2008)

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References

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Platinum Metals Rev., 2009, 53, (2) 97

The Authors

P. K. Yadawa is a Lecturer in theDepartment of Applied Physics, AMITYSchool of Engineering and Technology, NewDelhi. His major field of interest is theultrasonic characterisation of structuredmaterials.

Devraj Singh is a Lecturer in theDepartment of Applied Physics, AMITYSchool of Engineering and Technology,New Delhi. His research interest is theultrasonic non-destructive testing (NDT)characterisation of condensed materials.

D. K. Pandey is a Lecturer in theDepartment of Applied Physics, AMITYSchool of Engineering and Technology,New Delhi, India. His main activities are inthe field of ultrasonic characterisation ofcondensed materials and nanofluids.