Advances in Materials Physics and Chemistry, 2011, 1, 14-19
doi:10.4236/ampc.2011.12003 Published Online September 2011 (http://www.SciRP.org/journal/ampc)
Acoustic Wave Propagation in Nanocrystalline
RuCo Alloys
Pramod Kumar Yadawa1*, Devraj Singh1, Dharmendra Kumar Pandey2,
Giridhar Mishra3, RajaRam Yadav3
1Department of Applied Physics, AMITY School of Engineering and Technology, New Delhi, India
2Department of Physics, P. P. N. College, Kanpur, India
3Department of Physics, University of Allahabad, Allahabad, India
Received June 27, 2011; revised July 27, 2011; accepted August 9, 2011
Abstract
The ultrasonic properties like elastic constant, ultrasonic velocity in the hexagonal structured nanocrystalline
RuCo alloys have been studied along unique axis at room temperature. The second and third order elastic
constants (SOEC & TOEC) have been calculated for these alloys using Lennard-Jones potential. The orien-
tation dependent ultrasonic velocity has been also evaluated to study the anisotropic behaviour of these al-
loys. The velocities VL and VS1 have minima and maxima, respectively at 45° with unique axis of the crystal,
while VS2 increases with the angle from unique axis. The inconsistent behaviour of angle-dependent veloci-
ties is associated to the action of second order elastic constants. Debye average ultrasonic velocities of these
alloys are increasing with the angle and has maximum at 55° with unique axis at room temperature. Hence,
when a ultrasonic wave travels at 55° with unique axis of these alloys, then the average ultrasonic velocity is
found to be maximum. Elastic constants and density are mainly the affecting factor for anomalous behaviour
of ultrasonic velocity in these alloys. The mechanical and ultrasonic properties of Co0.75Ru0.25 alloy will be
better than the other compounds due to their high SOEC, ultrasonic velocity and low ultrasonic attenuation.
Co0.75Ru0.25 alloy is more suitable for industrial and other uses, as it has the highest elastic constants and
lowest ultrasonic attenuation in comparison to other of these alloys. The results of this investigation are dis-
cussed in correlation with other known thermophysical properties.
Keywords: Alloys, Elastic Properties, Ultrasonic Properties
1. Introduction
The structure of ruthenium (Ru) is hexagonal close pack-
ed (hcp) whereas cobalt (Co) has three phases those are
ferromagnetic; body centered cubic (bcc), face centered
cubic (fcc) and hcp. The structure of Co strongly de-
pends on th e grain size. For small grain size, fcc is stable
and for large grain size, the stab le structur e is hcp an d for
a distribution of grain sizes, a mixture of fcc and hcp ph-
ases exists. Huang et al. reported that the mixture of fcc
and hcp phases transforms to either fcc or hcp single ph-
ase by ball milling process which introduces structural
defects [1]. Alloys of 3d tran sition metals such as Ni, Co,
and Fe exhibit fascinating magnetic properties. In par-
ticular, their alloys with the 4d transition metals Ru,
Rh, and Pd are the subjects of experimental and theo-
retical investigations. Theoretical investigations suggest
an increasing ferromagnetic order in Pd, Rh, and Ru sim-
ilar to their analog 3d-metals Ni, Co, and Fe [2-4]. In
addition, it has b een discovered that there is an tiferroma-
gnetic interlayer exchange coupling and enhanced mag-
netoresistance in the metallic superlattices of Co/Ru [5,6].
Because the thin layer of Ru strongly antiferromagneti-
cally couples magnetic moments of the two Co layers in
an antiparallel configuration, the superlattice of Ru/Co
are finding applications in magnetic random access me-
mory devices [7].
It has been shown that nanocrystalline particles of Fe,
Co, PdFe, and RuFe alloys were prepared using organo-
metallic precursors and followed by pyrolysis [5,6,8,9].
Using the same chemical synthesis procedure and start-
ing with organic pr ecursor mixtures of Ru and Co, RuCo
alloys were synthesized over the entire co mpositional ra-
nge. Qadri et al. studied the structural and magnetic pr-

15 P. K. YADAWA ET AL.
operties of RuCo alloys and show the ex istence of either
the hexagonal phase or the fcc phase depending on the
composition and the particle size. Also Qadri et al. repor-
ted the structural and magnetic properties of PdFe and
RuFe alloys synthesized through organometallic route
[10].
Ultrasonic non-destructive testing (NDT) is a useful
technique that can be applied to a rang of materials for the
characterization of their microstructures, the appraisal of
defects and the determination of physical properties such
as density, thermal conductivity and electrical resistivity.
Ultrasonic measurements taken during the fabrication and
heat treatment of materials can be used to ensure that the
preferred microstructure is obtained and to prevent the
formation of defects, including defects in welds between
two different alloys. Information about the microstructure
can also be used in material description studies, such as
non-destructive determination of grin size Wave propaga-
tion velocity is key parameter in ultrasonic characteriza-
tion and can provide information about crystallographic
texture. The ultrasonic velocity is directly related to the
elastic constants by the relationship

VC

, where
C is the relevant elastic constants and
 is the density of
that particular material. The elastic constants of a solid
provide valuable insight into nature of atomic bonding
forces and also related hardness [11,12].
There are three types of acoustic mode of lattice vibra-
tions: one longitudinal acoustical and two transverse ac-
oustical for hexagonal structured materials. Hence, there
are three types of ultrasonic wave velocities for each dir-
ection of propagation of wave, which are well related to
second order elastic constants. But all the three types of
ultrasonic velocities and elastic constants of these alloys
are not reported in the literature. Therefore in this paper,
we have calculated the three types of ultrasonic sound ve-
locities for the alloys Co0.00Ru1.00: alloy-1; Co0.25Ru0.75: al-
loy-2; Co0.40Ru0.60: alloy-3; Co0.50 Ru0.50: alloy-4; Co0.60Ru0.40:
alloy-5; Co0.75Ru0.25: alloy-6; for each direction of propa-
gation of wave using second order elastic constants that
are important for surface and structural study of these all-
oys. The six second order elastic constants and ten third
order elastic constants are calculated using Lenard-Jones
Potential that is a many body interaction potential. The
results obtained are interesting for the characterization of
these alloys.
2. Theory
2.1 Higher Order Elastic Constants
The second (CIJ) and third (CIJK) order elastic constants
of material are defined by following expressions.
2
IJ IJ
U
C; or J1,,6
ee


  I 
(1)
3
IJK IJ K
U
C; I or J or K1,,6
eee


  (2)
where, U is elastic energy density , eI = eij (i or j = x, y, z;
I = 1,,6) is component of strain tensor. Eqs. (1)-(2)
leads six second and ten third order elastic constants
(SOEC and TOEC) for the hexagonal closed packed
structured materials [13].
44
11 12
68
13 33
44
44 66
C24.1 C C5.918 C
C1.925 C C3.464 C
C2.309C C9.851 C
pp
pp
pp


 



 



 


(3a)
where
5
Cap 
; 33
Bap  ;
 



4
0
18 n χnbnma 




;




2
6 6amn   ;
m, n = integer quantity; b0 = Lennard Jones parameter.
p = c/a: axial ratio; “c” is the height of the unit cell and
“a” be the basal plane distance.
2.2 Ultrasonic Velocity
The anisotropic behaviour of the material can be under-
stood with the knowledge of ultrasonic velocity because
the velocity is related to the second order elastic constants
[13]. On the basis of mode of atomic vibration, there are
three types of velocities (longitudinal, quasi shear and
shear) in acoustical region [14] . These vel ocities vary with
the direction of propagation of wave from the unique
axis of hexagonal structured crystal [15]. The ultrasonic
velocities as a function of angle between direction of
propagation and unique axis for hexagonal structured
materials are [16]:
24 2
111 112
46 4
113 123
64
133 155
4
144 344
C126.9B8.853C C19.168B1.61C
C1.924B1.155C C1.617B1.155C
C3.695B C1.539B
C2.309B C3.
pp pp
pp pp
pp
p
4
6


 

  


6
24 8
222 333
464 B
C101.039B9.007C C5.196B
p
pp p








  
(3b)

P. K. YADAWA ET AL.
16






2
22222 22
L331144 113344
1/2
2
2213 44
VCcosCsinCC sinCcosCcossin
4 cos sin CC2
 
 


  



(4)






2
22222 22
S1 331144113344
12
2
2213 44
VCcosC sinCCsinCcosCcossin
4 cos sin CC2
 
 


  


 (5)

222
S2 4466
VCcosCsin



 (6)
where VL, VS1 and VS2 are longitudinal, quasi shear and
pure shear wave ultrasonic velocities. Variables

and
 represent the density of the material and angle with the
unique axis of the crystal respectively. The Debye tem-
perature (TD) is an important physical parameter for the
characterization of materials, which is well related to the
Debye average velocity (VD).

1/3
2
D
D
V 6
Ta
B
n
k

 (7)
here
–1/3
D33 3
LS1S2
11 11
V3VVV




 




 
(8)
where is quantum of action and is equal to Planck’s
constant divided by ; kB is Boltzmann Constant; na is
atom concentration.

2
The above formulae have been used fo r the evaluation
of ultrasonic velocity and related parameters for the se-
lected materials.
3 Results
The unit cell parameters “a” for these six alloys (1, 2, 3,
4, 5 and 6) are 2.705 Å, 2.682 Å, 2.655 Å, 2.606 Å,
2.595Å and 2.565 Å respectively and axial ratio “p” for
these alloys are 1.583, 1.589, 1.590, 1.613, 1.616 and
1.623 respectively [10]. The value of m, n and b0 for
these alloys are 6, 7 and 9.9 × 10–64 erg cm7 correspond-
ingly. The second and third order elastic constants (SOE
C and TOEC) have been calculated for RuCo alloys us-
ing Eqs. (3a) and (3b) an d are presented in Table 1. The
calculated oriented depend ent ultrasonic velocities at 300
K are shown in Figures 1-4.
4. Discussions
The elastic constants are important since they are related
to hardness and are used for the determination of the
ultrasonic velocity. It is obvious from Table 1 that, th ere
is good agreement between the present values of SOEC
and TOEC with other of alloy: 1 (i.e. Ru). Hence present
values of elastic constants are justified. The bulk
modulus (B) for these alloys can be calculated with the
formula B = 2(C11 + C12 + 2C13 + C33/2)/9. The evaluated B
for these alloys is presented in Table 1. It is obvious from
Table 1, that there is good agreement between the calcu-
lated values from this study and the previously re ported
values for “B” for Ru [17]. Thus our theoretical approach
for the calculation of second order elastic constants
Table 1. SOEC, TOEC and Bulk modulus (B) (in the unit of 1011N·m–2) of RuCo alloys at room temperature.
Alloys C11 C
12 C 13 C 33 C 44 C 66 B
1 6.28 1.54 1.26 5.67 1.51 2.46 2.91
2 6.83 1.68 1.38 6.26 1.65 2.68 3.19
3 7.53 1.85 1.52 6.92 1.82 2.95 3.52
4 8.95 2.20 1.86 8.71 2.23 3.51 4.27
5 9.34 2.29 1.95 9.15 2.34 3.66 4.46
6 10.432.56 2.19 10.402.63 4.09 5.02
Ru[17] 6.28 1.54 1.26 5.65 1.51 2.46 2.91
Alloys C111 C
112 C 113 C 123 C 133 C 344 C 144 C 155 C 222 C 333
1 –102.44 –16.24 –3.22 –4.10 –19.14 –17.95 –4.77 –3.18 –81.05 –67.44
2 –111.37 –17.66 –3.53 –4.49 –21.15 –19.82 –5.23 –3.48 –88.12 –75.09
3 –122.77 –19.46 –3.90 –4.96 –23.38 –21.92 –5.78 –3.85 –97.14 –83.15
4 –145.93 –23.14 –4.77 –6.06 –29.41 –27.57 –7.06 –4.71 –115.47 –107.61
5 –152.25 –24.14 –4.99 –6.35 –30.91 –28.98 –7.40 –4.93 –120.46 –113.52
6 –170.08 –26.97 –5.63 –7.15 –35.14 –32.95 –8.34 –5.56 –134.57 –130.20
Ru[17] –102.40 –16.24 –3.22 –4.09 –19.10 –17.91 –4.77–3.18 –8.03 –67.21

17 P. K. YADAWA ET AL.
3
3.5
4
4.5
5
5.5
0 102030405060708090
angle
V
L
(10
3
m/s)
alloy1alloy2 alloy3
alloy4alloy5 alloy6
Figure 1. VL vs angle with unique axis of crystal.
1.5
2
2.5
3
3.5
10 20 304050 60 7080 90
angle
V
s1
(10
3
m/s)
alloy1alloy2 alloy3
alloy4alloy5 alloy6
Figure 2. VS1 vs angle with unique axis of crystal.
1.5
2
2.5
3
3.5
0 10203040506070809
angle
V
s2
(10
3
m/s)
0
alloy1 alloy2alloy3
alloy4 alloy5alloy6
Figure 3. VS2 vs angle with unique axis of crystal.
for hexagonal structured alloys at room temperature is
well justified. Hence applied theory for the evaluation of
higher order elastic constants at room temperature is
justified. All the SOEC and TOEC for these alloys are
found to be higher than those of Mo-Ru-Rh-Pd alloys
[13]. It can be also seen from Table 1, that the SOEC
and TOEC are found to be increasing from alloys: 1-6
due to large grain size of hcp Co metals.
Figures 1-3 show that the velocities VL and VS1 have
minima and maxima respectively at 45˚ with the unique
axis of the crystal while VS2 increases with the angle
from the unique axis. Anomalous behaviour of angle
dependent velocities is correlated to the behaviour of
second order elastic constants. The nature of the angle
dependent velocity curves in the present work is the
same as the nature of angle dependent v elocity curve for
others hexagonal wurtzite structured materials [13,15-
18]. Thus our angle dependency of the velocities for
these nanocrystalline wurtzite structured alloys is justi-
fied.
Figures 1-3 indicate that the magnitude of acoustic
velocity is larger for alloy: 6 and smaller for alloy: 1.
The respective smaller magnitude of acoustical velocity
in alloy: 1 is due to its higher gravitational density. The
larger longitudinal acoustical velocity along the [001]
direction (θ = 0˚ with unique axis) for alloy: 6 are due to
the highest value of C33 second order elastic constants.
The shear wave is also called the surface wave. There-
fore the acoustical velocities VS1 and VS2 are the surface
wave velocity. VS1 and VS2 have the same value for ac-
oustic wave propagation along θ = 0˚ while variation is
obtained between them for other directions of propaga-
tion (Figures 2 and 3). This implies that the [001] direc-
tion is the direction of symmetry for these alloys. Debye
average velocities (VD) of these alloys are increasing
with the angle and have maxima at 55˚ at 300 K (Figure
4). Since VD is calculated using VL, VS1 and VS2 with
Equation (8), therefore the orientation variation of VD
follows the combined effect of temperature variation of
VL, VS1 and VS2. The maximum in VD at 55˚ is due to a
1.5
2
2.5
3
3.5
0 10203040506070809
angle
V
D
(10
3
m/s)
0
alloy1 alloy2
alloy3 alloy4
alloy5 alloy6
Figure 4. VD vs angle with unique axis of crystal.

P. K. YADAWA ET AL.
18
significant increase in longitudinal and pure shear wave
velocities and a decrease in quasi-shear wave velocity.
Thus it can be concluded that when a sound wave travels
at 55˚ with the unique axis of these crystals then the av-
erage sound wave velocity is maximum. The orientation
dependent ultrasonic velocity VL, VS1 and VS2 and Debye
average velocity VD in alloy: 1 is same as pure hcp Ru
[17]. Since the Debye average velocity is calculated us-
ing the constituent velocities VL, VS1 and VS2, hence a
good resemblance in VD implies that our calculated ve-
locities are correct.
It can be seen that from Figures 1-4 alloy: 6 has
maximum velocity and alloy: 1 has least velocity for all
angles of the crystals. Since ultrasonic attenuation A
and velocity is the largest for alloy: 6 among oth-
ers thus the attenuation A should be smallest and mate-
rial should be most ductile. The minimum ultrasonic at-
tenuation for alloy: 6 justify its quite stable hcp structure
state. Also alloy: 6 has maximum elastic constants and
bulk modulus among others. Hence alloy: 6 (Co0.75Ru0.25)
is more ductile, stable and contain few defects in the cry-
stal structure in comparison to other alloys.
3
V 
The pulse echo technique (PET) can be used for the
measurement of ultrasonic parameters, because it avoids
heat loss and scattering loss. Elastic constants and other
ultrasonic parameters of binary Fe-Co and Fe-Ru alloys
have been determined by a pulse echo technique [19].
5. Conclusions
Based on the above discussion it is worthwhile to state
that:
1) Present method to evaluate second and third-order
elastic constants involving many body interaction poten-
tial for hexagonal wurtzite crystal structured materials is
correct.
2) Elastic constants and density are mainly the affect-
ing factor for anomalous behaviour of acoustical velocity
in these alloys.
3) When a sound wave travels at 55˚ with the unique
axis of these crystals then the average sound wave veloc-
ity is maximum. Since the Debye average velocity is
calculated using the constituent velocities VL, VS1 and
VS2, hence a good resemblance in VD implies that our
calculated velocities are correct.
4) The [001] direction is the direction of symmetry for
these alloys as they have the same quasi-shear and pure
shear wave velocities.
5) Co0.75Ru0.25 (alloy-6) is more suitable for industrial
and other uses, as it has the highest elastic constants as
well as wave velocity and lowest attenuation in com-
parison to other chosen alloys.
6) The mechanical and ultrasonic properties of Co0.75Ru0.25
alloy will be better than the other compounds due to their
high SOEC and low ultrasonic attenuation.
The results obtained in this investigation can be used
for further study of these alloys. Our whole theoretical
approach can be applied to the evaluation of ultrasonic
velocities and related parameters to study the microstr-
uctural properties of h.c.p. structured materials.
6. References
[1] J. Y. Huang, Y. K. Wu and H. Q. Ye, “Response to Com-
ment on Phase Transformation of Cobalt Induced by Ball
Milling,” Applied Physics Letters, Vol. 67, No. 13, 1995,
pp. 1944-1945. doi:10.1063/1.114577
[2] J. Crangle and W. R. Scott, “Dilute Ferromagnetic Al-
loys,” Journal of Applied Physics, Vol. 36, No. 3, 1965,
pp. 921-928. doi:10.1063/1.1714264
[3] J. Neuwenhuys, “Magnetic Behaviour of Cobalt, Iron and
Manganese Dissolved in Palladium, ” Advances in Physics,
Vol. 24, No. 4, 1975, pp. 515-591.
doi:10.1080/00018737500101461
[4] P. H. Dederich, R. Zeller, H. Akai and H. Ebert,
“Ab-Initio Calculations of the Electronic Structure of
Impurities and Alloys of Ferromagnetic Transition Met-
als,” Journal of Magnetism and Magnetic Materials, Vol.
100, No. 1-3, 1991, pp. 241-260.
doi:10.1016/0304-8853(91)90823-S
[5] S. S. P. Parkin, N. More and K. P. Roche, “Ab-Initio Cal-
culations of the Electronic Structure of Impurities and
Alloys of Ferromagnetic Transition Metals,” Physical
Review Letters, Vol. 64, 1990, pp. 2304-2307.
doi:10.1103/PhysRevLett.64.2304
[6] Y. Y. Huang, G. P. Felcher and S. S. P. Parkin, “Anti-
Ferromagnetic and Ferromagnetic Order in Co/Ru Multi-
layers,” Journal of Magnetism and Magnetic Materials,
Vol. 99, No. 1-3, 1991, pp. L31-L38.
doi:10.1016/0304-8853(91)90044-B
[7] S. S. P. Parkin, K. P. Roche, M. G. Sa mant, P. M. Ric e, R.
B. Beyers, R. E. Scheueriein, E. J. O’Sullivan, S. L.
Brown, J. Bucchigano, D. W. Abraham, Y. Lu, M. Rooks,
P. L. Trouilloud, R. A. Warner and W. J. Gallagher, “Ex-
change-Biased Magnetic Tunnel Junctions and Applica-
tion to Nonvolatile Magnetic Random Access Memory,”
Journal of Applied Physics, Vol. 85, No. 8, 1999, pp.
5828-5833. doi:10.1063/1.369932
[8] W. Koester and E. Horn, “Characterization of Cobalt,
Rhenium, Ruthenium, Osmium, Rhodium and Iridium,”
Journal of Materials Chemistry, Vol. 43, No. 12, 1952,
pp. 444-449.
[9] T. M. Keller and S. B. Qadri, “Ferrocenylethynylbenzenes
as Precursors to in Situ Synthesis of Carbon Nanotube
and Fe Nanoparticle Compositions,” Chemistry of Mate-
rials, Vol. 16, No. 6, 2004, pp.1091-1097.
doi:10.1021/cm030624e
[10] S. B. Qadri, T. M. Keller, M. Laskoski, C. A. Little, M. S.
Osofsky and H. R. Khan, “Structural and Magnetic Prop-

P. K. YADAWA ET AL.
19
erties of Nanocrystalline RuCo Alloys,” Applied Physics
Letters, Vol. 91, 2007, pp. 214101: (1-3).
[11] D. Singh and P. K. Yadawa, “Effect of Platinum Addition
to Coinage Metals on Their Ultrasonic Properties,”
Platinum Metals Review, Vol. 54, No. 3, 2010, pp.
169-176. doi:10.1595/147106710X500602
[12] P. K. Yadawa, “Ultrasonic Wave Propagation in Cadmium
Chalcogenides Compounds,” Multidiscipline Modeling in
Materials and Structures, Vol. 7, No. 1, 2011, pp. 63-72.
doi:10.1108/15736101111141449
[13] A. K. Yadav, R. R. Yadav, D. K. Pandey and D. Singh,
“Ultrasonic Study of Fission Products Precipitated in the
Nuclear Fuel,” Materials Letters, Vol. 62, 2008, pp.
3258-3261. doi:10.1016/j.matlet.2008.02.036
[14] C. Oligschleger , R. O. Jones, S. M. Reimann and H. R.
Schober, “Model Interatomic Potential for Simulations in
Selenium,” Physical Review B, Vol. 53, 1996, pp.
6165-6173. doi:10.1103/PhysRevB.53.6165
[15] P. K. Yadawa, “Acoustic Wave Propagation in Ni3R (R =
Mo, Nb, Ta) Compounds,” Pramana, Vol. 76, No. 4,
2011, pp. 613-619. doi:10.1007/s12043-011-0066-7
[16] P. K. Yadawa, “Ultrasonic Characterization of Ceramic
Material Titanium Diboride,” Ceramics-Silikaty, Vol. 55,
No. 2, 2011, pp. 127-133.
[17] D. K. Pandey, D. Singh and P. K. Yadawa, “Ultrasonic
Study of Osmium and Ruthenium,” Platinum Metals Re-
view, Vol. 9, No. 2, 2009, pp. 91-97.
doi:10.1595/147106709X430927
[18] D. K. Pandey, P. K. Yadawa and R. R. Yadav, “Ultra-
sonic Properties of Hexagonal ZnS at Nanoscale,” Mate-
rials Letters, Vol. 61, 2007, pp. 5194-5198.
doi:10.1016/j.matlet.2007.04.028
[19] G. R. Speich, A. J. Schwoeble and W. C. Leslie, “Elastic
Constants of Binary Iron-Base Alloys,” Metallurgical
and Materials Transctions B, Vol. 3, 1972, pp.
2031-2037.