Advances in Materials Physics and Chemistry, 2011, 1, 44-49
doi:10.4236/ampc.2011.12008 Published Online September 2011 (http://www.SciRP.org/journal/ampc)
Copyright © 2011 SciRes. AMPC
Ultrasonic Attenuation in Calcium Oxide
Jitendra Kumar1, Kailash2, Sanjeev Kumar Shrivastava3, Devraj Singh4, Virend ra Kumar2
1Department of Physics, Govt. Girls P. G. College, Banda, India
2Department of Physics, BN PG College, Rath, Hamirpur, India
3Department of Physics, Bundelkhand University, Jhansi, India
4Department of Applied Physics, AMITY School of Engineering and Technology, New Delhi, India
E-mail: {jkumarsh, kailashrath}@gmail.com
Received July 4, 2011; revised August 15, 2011; accepted August 26, 2011
Abstract
Ultrasonic attenuation studies can be used to characterize material not only after production but during proc-
essing as well. The most important causes of ultrasonic attenuation in solids are electron-phonon, pho-
non-phonon interaction and that due to thermo elastic relaxation. The two dominant processes that will give
rise to appreciable ultrasonic attenuation at higher temperature are the phonon-phonon interaction also
known as Akhiezer loss and that due to thermo elastic relaxation are observed in calcium oxide crystal. At
frequencies of ultrasonic range and at higher temperatures in solids, phonon-phonon interaction mechanism
is dominating cause for attenuation. Ultrasonic attenuation due to phonon-phonon interaction (α/f 2)p-p and
thermo elastic relaxation (α/f 2)th are evaluated in Calcium Oxide crystal up to an elevated temperature from
100 K - 1500 K along <100>, <110> and <111> crystallographic directions. Temperature dependence of ul-
trasonic attenuation along different crystallographic direction reveals some typical characteristic features.
Keywords: Attenuation, Thermo Elastic Relaxation, Phonon-Phonon Interaction, Akheiser Loss, Gruneisen
Constants
1. Introduction
Ultrasonic velocity and attenuation parameters are well
connected to the micro structural and mechanical proper-
ties of the materials. In recent years, the ultrasonic at-
tenuation techniques [1-4] are widely used as versatile
tool in studying the inherent properties and internal
structure of solids. In several types of solids, viz. metal-
lic dielectric and semiconducting crystals, the attenuation
occurs due to various causes e.g. lattice imperfection,
ferromagnetic and ferroelectric, NMR and thermal re-
laxation and thermoelastic loss at different temperature
regions. The temperature dependent part of ultrasonic
attenuation has been explained in terms of model where
the acoustic phonon interacts with a number of thermal
phonons in the lattice. The most important causes of ul-
trasonic attenuation in solids are electron-phonon, pho-
non-phonon interaction and that due to thermo elastic
relaxation. The ultrasonic attenuation study of the mate-
rials has gained new dimensions with the progress in the
material science. In recent years, the ultrasonic attenua-
tion techniques [5-7] are widely used as versatile tool in
studying the inherent properties and internal structure of
solids. The two dominant processes that will give rise to
appreciable ultrasonic attenuation at higher temperature
are the phonon-phonon interaction also known as Akhi-
eser loss [8,9] and that due to thermo elastic relaxation
and are observed in calcium oxide crystal.
Oxides and silicates make up the bulk of the Earth’s
mantle and crust, and thus it is useful to predict their
behaviour. In this work ultrasonic attenuation due to
phonon-phonon interaction over frequency (α/f 2)Akh and
ultrasonic attenuation due to thermo elastic relaxation
over frequency (α/f 2)th are studied in calcium oxide at
an elevated temperatures (100 K - 1500 K) along <100>,
<110> and <111> crystallographic directions. For the
evaluation of the ultrasonic coefficients the second and
third order elastic constants (SOECs and TOECs) are
also calculated using Coulomb and Born Mayer [10] po-
tentials. Several investigators have given different theo-
ries; here the one given by Mason has been used. Ma-
son’s theory relates the Gruneisen constants with SOECs
and TOECs. The behaviour of ultrasonic absorption and
other parameters as a function of higher temperature
have been discussed as the characteristic features of cal-
cium oxide.
45
J. KUMAR ET AL.
2. Formulation
The calcium oxide possesses face centered cubic crystal
structure. The potential used for evaluation of second and
third order elastic constants (SOECs and TOECs) is
taken as the sum of Coulomb and Börn-Mayer potentials.
  
Qr QC+QB (1)
where Q(C) is the Coulomb potential and Q(B) is the Bӧ
rn-Mayer potential, given as
 

QC e2r and

QBA exprq
ib ib
(2)
where e is the electronic charge, r is the nearest-neigh-
bour distance, q is the hardness parameter and A is the
strength parameter.
Following Brügger’s [11] definition of elastic con-
stants at absolute zero, second and third order elastic
constants (SOECs and TOECs) are obtained. According
to lattice dynamics developed by Leibfried and Ludwig,
lattice energy changes with temperatures. Hence adding
vibrational energy contribution to the static elastic con-
stants, one gets second and third order elastic constants
(Cij and Cijk) at the required temperature.
0v
ijj ij
CCC
i
 and (3)
0v
ijkjk ijk
CCC
i

The expressions for ultrasonic attenuation coefficient
due to phonon-phonon interaction [12] over frequency
square are as
for longitudinal waves

2l
0l
2
3
Akh l
D
4πE3
f2V



(4)
for shear waves

2s
0s
2
3
Akh s
D
4πE3
f2V



(5)
where the condition

th << 1 has already been assumed.
Here E0 is thermal energy density,
is the angular fre-
quency (= 2f), f is frequency of ultrasonic waves, d is
the density and Vl, Vs are the ultrasonic velocities for
longitudinal and shear waves respectively, and the ultra-
sonic attenuation coefficient due to thermal relaxation
over frequency

2
2j
i
2
5
th
4K
f2V
T
(6)
where the condition

th << 1 has already been assumed.
Here E0 is thermal energy density, f is frequency of ul-
trasonic waves, d is the density and Vl, Vs are the ultra-
sonic velocities for longitudinal and shear waves respec-
tively.
Two relaxation times are related as
2
lsth
VD
1
2CV

 
3K
(7)
where
th is the thermal relaxation time for the exchange
of acoustic and thermal energy, K is the thermal conduc-
tivity, Cv is specific heat and D
Vis the Debye average
velocity.
D in eqs. (1) and (2) is given as

22
j
v
i
0
3C T
D9 E
j
i




(8)
where

2
j
i
and 2
j
i
are the square average Grün-
eisen parameters and average square Grüneisen parame-
ters respectively.
3. Evaluation
The SOECs (Cij) and TOECs (Cijk) for Calcium oxide has
been evaluated at different temperatures (100 K to 1500
K) following Brugger’s approach and are presented in
Table 1 at room temperature. Grüneisen parameters have
been evaluated using Mason’s Grüneisen parameters
tables [13] at different temperatures along different crys-
tallographic directions longitudinal (long.) as well as
shear with the help of SOECs and TOECs. The non-
linearity constants (D) are evaluated using the average
square Grüneisen parameters 2
j
i
and square average
Grüneisen parameters

2
j
i
. The values of

2
j
i
,
(α/f2)Akh and (α/f2)th along with V for longitudinal and
shear waves,
th and D are calculated using primary
physical data presented in Table 2 and are given in Table 3
at room temperature along different crystallographic direc-
tions. The temperature variations of all these properties are
presented graphically in Figures 1-6. All these calculations
have been done using C++ programmes developed by
Table 1. The second and third order elastic constants in 1010 N/m2 at room temperature alongwith melting point.
Melting Point (K) C11 C
12 C
44 C
111 C
112 C
123 C
144 C
166 C
456
2843 20.67 12.52 12.71 –252.88–65.1020.34 19.43 –51.26 4.71
Copyright © 2011 SciRes. AMPC
J. KUMAR ET AL.
46
Table 2. Primary Physical Data (Calculated).
Temperature (K) θD/T Cv (in 106 J/m3. K) E0 (in 106 J/m3)
100 6.480 0.3365991 9.524686
200 3.240 0.9255680 75.724882
300 2.160 1.1894222 183.37022
400 1.620 1.3075118 308.93472
500 1.296 1.3540948 444.12383
600 1.080 1.4023710 581.71147
700 0.925 1.4238437 723.34339
800 0.810 1.4379470 866.22996
900 0.720 1.4477110 1010.6013
1000 0.648 1.4547245 1155.8323
1100 0.589 1.4601010 1301.5421
1200 0.540 1.4640190 1447.8532
1300 0.498 1.4673480 1594.5340
1400 0.462 1.4697214 1741.9166
1500 0.432 1.4716960 1888.4134
Table 3. Long. and shear wave velocities in 103 m/sec,
thermal relaxation time in 10–12 sec, non-linearity constants,
thermal attenuation in 10–18 Np.sec2/m and p-p attenuation
for long and shear waves in 10–16 Np.sec2/m at room tem-
perature.
Direction of
propagation
j2
i
()
V
th D (
/f2)th (
/f2)Akh
<100>
Long. 1.947 7.5200.836 15.414 0.373 0.218
Shear 0.321 6.171 2.886 0.037
<110>
Longitudinal 1.655 9.1240.782 12.248 0.178 0.091
Shear
(pol.<001>) 0.642 6.171 5.777 0.069
(pol.<110>) 1.538 3.375 13.838 6.253
<111>
Longitudinal 1.855 9.5991.379 14.115 0.135 0.158
Shear
(pol.<110>)0.976 4.504 8.786 0.954
(pol.<11 2>) 1.503 4.504 13.529 1.469
05001000 1500
0
1
2
3
4
5
6
7
8
9
10
Tem
p
erat ur e
(
K
)
Therm al R elaxation T im e (in 1e- 12 sec.)
Al ong <100> di rection
Al ong <110> di rection
Al ong <111> di rection
Figure 1. Thermal relaxation time Vs Temp.
05001000 1500
0
2
4
6
8
10
12
14
16
18
Te m
p
erature
(
K
)
Non Linearity Constants
Dl
Ds
Dl/Ds
Figure 2. Non-linearity constants Vs Temp. along <100>.
05001000 1500
0. 05
0. 1
0. 15
0. 2
0. 25
0. 3
0. 35
0. 4
0. 45
0. 5
Temperature(K)
Therm al Ultrasonic Attenuation(in 1e-18 Nps2/m )
Along < 100> direction
Along < 110> direction
Along < 111> direction
Figure 3. Thermal ultrasonic attenuation Vs Temp.
0500 1000 1500
0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
Temperature(K)
Ultrasonic Attenuation(in 1e-16 Nps2/m )
Akh.long
Akh.Shear
Figure 4. Ultrasonic attenuation Vs Temp. along <100>.
Copyright © 2011 SciRes. AMPC
47
J. KUMAR ET AL.
author itself. The developed programmes have been veri-
fied using the known results for other FCC structured
solids. The Debye temperature (θD) is an important
physical parameter of solids, which defines a division
line between quantum and classical behaviour of pho-
nons, its value for CaO crystal is 648 K. Interpretation of
data for ultrasonic attenuation shown in figures is pre-
sented in Table 4 through curve fitting.
4. Results and Discussion
Table 1 shows that all the SOECs are positive in nature.
Among TOECs C111, C112 and C166 are positive and all
others are negative in nature. These results are in good
agreement with the results obtained for other rocksalt
structure solids. The shear waves have a relaxation time
equal to the thermal relaxation time, while longitudinal
waves have a relaxation time about twice as large. Hence,
in determining the ultrasonic attenuation due to conversion
to thermal phonons, it is necessary to weight the conver-
sion coefficients by this difference in the relaxation time.
It is clear from the Figure 1 that the value of thermal
relaxation time
th is very high at low temperature and
decreases as temperature is increased along all crystallo-
graphic directions and of the order of 10–12 sec, which is
also expected. Its value is very much at 100 K but as
temperature increases it decreases very sharply up to 500
K and after that it decreases very slowly up to 1500 K.
At any particular temperature its value is more along
<111> direction. From the values of thermal relaxation
time, we can check the validity of the condition ω
th <<1
in the ultrasonic frequency range.
0500 1000 1500
0
1
2
3
4
5
6
7
Temperature(K)
Ultrasonic Attenuation(in 1e-16 Nps2/m )
Akh.long
Akh.Shear1
Akh.Shear2
Figure 5. Ultrasonic attenuation Vs Temp. along <110>.
05001000 1500
0
0.5
1
1.5
2
2.5
3
Temperature(K)
Ultrasonic Attenuation(in 1e-16 Nps2/m )
Akh.long
Akh.Shear1
Akh.Shear2
Figure 6. Ultrasonic attenuation Vs Temp. along <111>.
Table 4. Property (P) = A0 + A1T + A2T2 + A3T3 + A4T4 + A5T5 + A6T6 + A7T7 + A8T8 + A9T9 + A10T10 fo r Ul trason ic A ttenu at ion.
P A0 A
1 A
2 A
3 A
4 A
5 A
6 A
7 A
8 A
9 A
10
(
/f2)th<100> –1.52e-02 4.59e-02 –1.90e-05 2.60e-084.23e-11 –2.21e-13 3.93e-16 –3.89e-19 2.27e-22 –7.30e-26 9.99e-30
(
/f2)Akh.l<100> –0.59 1.17e-02 –7.76e-05 2.97e-07 –7.13e-10 1.12e-12 –1.16e-15 7.85e-19 –3.28e-22 7.64e-26 –7.48e-30
(
/f2)Akh.s<100> –9.27e-02 1.91e-02 –1.31e-05 5.33e-08 –1.38e-10 2.38e-13 –2.75e-16 2.10e-19 –1.01e-22 2.80e-26 –3.36e-30
(
/f2)th<110> –7.10e-02 2.06e-02 –9.64e-06 2.47e-08 –3.93e-11 4.42e-14 –3.75e-17 2.35e-20 –9.98e-24 2.48e-27 –2.79e-31
(
/f2)Akh.l<110> –0.25 4.97e-03 –3.37e-05 1.34e-07 –3.41e-10 5.75e-13 –6.48e-16 4.81e-19 –2.25e-22 6.04e-26 –7.04e-30
(
/f2)Akh.s1<110> –0.14 3.03e-03 –1.99e-05 7.78e-08 –1.91e-10 3.10e-13 –3.32e-16 2.31e-19 –9.90e-23 2.34e-26 –2.31e-30
(
/f2)Akh.s2<110> –22.14 0.46 –3.25e-03 1.27e-05–3.18e-085.26e-11–5.82e-14 4.26e-17 –1.98e-20 5.29e-24 –6.16e-28
(
/f2)th<111> –1.16e-02 1.69e-03 –8.06e-06 2.05e-08 –3.16e-11 3.25e-14 –2.42e-17 1.35e-20 –5.36e-24 1.32e-27 –1.56e-31
(
/f2)Akh.l<111> –0.51 1.04e02 –7.30e-05 2.98e-07–7.70e-101.31e-12–1.49e-15 1.12e-18 –5.34e-22 1.44e-25 –1.70e-29
(
/f2)Akh.s1<111> –2.35 4.86e-02 –3.28e-04 1.29e-06 –3.26e-09 5.47e-12 –6.15e-15 4.57e-18 –2.15e-21 5.78e-25 –6.76e-29
(
/f2)Akh.s2<111> –3.69 0.076605 –5.19e-04 2.06e-06–5.21e-098.77e-12–9.90e-15 7.38e-18 –3.48e-21 9.40e-25 –1.10e-28
Copyright © 2011 SciRes. AMPC
J. KUMAR ET AL.
48
The temperature variation of non-linearity constant is
presented in Figure 2 along <100> direction. The value
of Dl/Ds along <100> direction is about 5.34 which is
expected for this type of crystals. We can see that ratio of
non-linearity constants Dl/Ds increases with temperature.
The D is a measure of acoustic energy converted to
thermal energy under the relaxation process, thus the
increase in Dl/Ds with temperature shows that longitudi-
nal loss increases with temperature and vice versa along
<100> direction. Along other two directions also the
values of Dl and Ds decreases with temperature in a
manner which shows that longitudinal loss increases with
temperature.
Studies of the ultrasonic absorption in crystals and its
dependence on the direction of propagation and wave
mode (polarization of the elastic displacement vector) are
important from the standpoint of the further development
of the theory of sound wave-lattice interactions. The
thermal ultrasonic attenuation along different directions
is presented in Figure 3. We can see from the figure that
thermal attenuation increases with temperature along all
these directions and of the order of 10–18 Nps2/m which is
expected for the type of crystal under inspection. At any
particular temperature the thermal attenuation along
<100> direction is more and is less along all other direc-
tions, hence <100> is more suited for the study of ther-
mal ultrasonic attenuation. Figures 4-6 represent the
ultrasonic attenuation due to phonon-phonon interaction
along <100>, <110> and <111> crystallographic direc-
tions respectively. Along all these directions the ultra-
sonic attenuation increases with increase in temperature,
the situation is little different for shear wave along
<110> direction (pol.<110>) for which the ultrasonic
attenuation decreases with temperature. Among all these
three directions the attenuation along <111> direction
changes sharply with temperature so the study along
<111> crystallographic direction is more reliable for the
characterization. From the attenuation values along dif-
ferent directions it is evident that the ultrasonic attenua-
tion is different along different directions i.e. varies with
the orientation of the crystal.
5. Conclusions
From the above results one can reach to the conclusion
that the thermal relaxation time, ultrasonic wave veloci-
ties and attenuation are the properties which depends on
the crystallographic directions or one can say that these
properties are orientation dependent which strongly sup-
ports the results given by other investigators [14,15] for
the same type of crystals that all these properties are
strongly dependent on direction of polarization. The
study of ultrasonic attenuation for longitudinal wave
along <100> is more important in characterization of the
crystal since its magnitude is more along this direction.
The rapidly increase in ultrasonic attenuation at higher
temperature is interpreted mainly due to increase of den-
sity of dislocations. The results obtained in this study can
be used for further investigations [16,17] and industrial
research and development purposes.
6. References
[1] R. P. Singh and R. K. Singh, “Theoretical Study of Tem-
perature Dependent Lattice Anharmonicity in TlCl and
TlBr,” Current Applied Physics, Vol. 10, No. 4, 2010, pp.
1053-1058. doi:10.1016/j.cap.2009.12.040
[2] J. D. Pandey, A. K. Singh and R. Dey, “Effect of Isotopy
on Thermoacoustical Properties,” Journal of Pure and
Applied Ultrasonics, Vol. 26, 2004, pp. 100-104.
[3] A. K. Upadhyay and B. S. Sharma, “Elastic Properties of
Intermetallics Compounds under High Pressure and High
Temperature,” Indian Journal of Pure and Applied Phys-
ics, Vol. 47, 2009, pp. 362-368.
[4] R. K. Singh, “Ultrasonic Attenuation in Alkaline Earth
Metals,” Journal of Pure and Applied Ultrasonics, Vol.
28, 2006, pp. 59-65.
[5] S. K. Srivastava, Kailash and K. M. Raju, “Ultrasonic
Study of Highly Conducting Metals,” Indian Journal of
Physics, Vol. 81, 2007, pp. 351-361.
[6] Kailash, K. M. Raju and S. K. Srivastava, “Acoustical
Investigation of Magnesium Oxide,” Indian Journal of
Physics, Vol. 44, 2006, pp. 230-234.
[7] T. Yanagisaa, T. Goto and Y. Nemoto, “Ultrasonic In-
vesttigation of Quadrupole Ordering in HoB2C2,” Physi-
cal Review B, Vol. 67, 2003, pp. 115129-115136.
doi:10.1103/PhysRevB.67.115129
[8] S. K. Kor, R. R. Yadav and Kailash, “Ultrasonic
Attenuation in Dielectric Crystals,” Journal of the
Physical Society of Japan, Vol. 55, 1986, pp. 207-212.
doi:10.1143/JPSJ.55.207
[9] A. Akhiezer, “On the Absorption of Sound in Solids,”
Journal of Physics-USSR, Vol. 1, 1939, p. 277.
[10] M. Born and J. M. Mayer, “Zur Gittertheorie der Lone-
nkristalle,” Zur Physics (Germany), Vol. 75, 1932, pp.
1-18.
[11] K. Brugger, “Thermodynamic Definition of Higher Order
Elastic Coefficients,” Physical Review, Vol. 133, 1964,
pp. A1611-A1612. doi:10.1103/PhysRev.133.A1611
[12] W. P. Mason, “Piezoelectric Crystals and Their Appli-
cations to Ultrasonics,” D. Van Nostrand Co, Inc., Prin-
ceton, 1950.
[13] W. P. Mason, “Effect of Impurities and Phonon-Pro-
cesses on the Ultrasonic Attenuation in Germanium,
Crystal Quartz and Silicon,” Physical Acoustics, Vol. 3B,
Academic Press, New York, 1965, p. 237.
[14] K. M. Raju, Kailash and S. K. Srivastava, “Orientation
Dependence of Ultrasonic Attenuation,” Physics Proce-
Copyright © 2011 SciRes. AMPC
49
J. KUMAR ET AL.
dia, International Congress on Ultrasonics, Santiago de
Chile, Vol. 3, 2010, pp. 927-933.
[15] Y. Hiki and J. Tamura, “Ultrasonic Attenuation in Ice
Crystals near the Melting Temperature,” Journal de Phy-
sique, Vol. C5, 1981, p. 547.
[16] J. A. Scales and A. E. Malcolm, “Laser Characterization
of Ultrasonic Wave Propagation in Random Media,”
Physical Review E, Vol. 67, 2003, pp. 046618(1) -
046618(7).
[17] T. C. Upadhyay, “Temperature Dependence of Micro-
wave Loss in ADP-type Crystals,” Indian Journal of Pure
and Applied Physics, Vol. 47, 2009, pp. 66-72.
Copyright © 2011 SciRes. AMPC