Open Journal of Metal, 2011, 1, 34-40
doi:10.4236/ojmetal.2011.12005 Published Online December 2011 (http://www.SciRP.org/journal/ojmetal)
Sensitivity of Nanostructured Iron Metal on
Ultrasonic Properties
Alok Kumar Gupta1, Archana Gupta2, Devraj Singh3*, Sudhanshu Tripathi4
1Academic Department, National Institute of Open Schooling (NIOS),
Ministry of HRD, Govt. of India, Noida, India
2Department of Biochemistry, University of Lucknow, Lucknow, India
3Department of Applied Sciences, Amity School of Engineering and Technology, New Delhi, India
4Department of Instrumentation and Control Engineering, Amity School of Engineering and Technology,
New Delhi, India
Received October 20, 2011; revised November 14, 2011; accepted December 2, 2011
Abstract
The present investigation is focused on the influence of the nanocrystalline structure of pure iron metal on
the ultrasonic properties in the temperature range 100 - 300 K. The ultrasonic attenuation due to phonon-
phonon interaction and thermoelastic relaxation phenomena has been evaluated for longitudinal and shear
waves along <100>, <110> and <111> crystallographic directions. The second-and third-order elastic con-
stants, ultrasonic velocities, thermal relaxation, anisotropy and acoustic coupling constants were also com-
puted for the evaluation of ultrasonic attenuation in this temperature scale. The direction <111> is most ap-
propriate to study longitudinal sound waves, while <100>, <110> direction are best to propagate shear waves
due to lowest values of attenuation in these directions. Other physical properties correlated with obtained
results have been discussed.
Keywords: Iron Metal, Elastic properties, Ultrasonic Attenuation
1. Introduction
Nanocrystalline materials containing fine grains of nanome-
ter sizes exhibit the very unique physical properties, which
contrast with the corresponding polycrystalline and amor-
phous materials. The ultrasonic attenuation proved to be
a precise tool to investigate variations in the microstructure
of solids [1,2]. Ultrasonic velocity and attenuation studies
have been made in solids [3-5], liquids [6,7], and liquid
crystals [8,9]. A number of books and review articles are
available which gives experimental techniques and theo-
retical interpretation. In solids [10,11] there are several
causes of ultrasonic attenuation, the most important being
electron-phonon interaction at low temperatures [12,13]
and phonon-phonon interaction at high temperatures [14,
15]. There are several methods for theoretical evaluation
of ultrasonic attenuation, but the most accepted and re-
liable one uses second and third order elastic constants
(SOEC and TOEC) [10,11].
Iron is a very important metal for industrial and domestic
applications. We have chosen iron for the study to clarify
their nature up to room temperature (27˚C = 300˚K). The
application, quality control and assurance can be well
understood with the knowledge of ultrasonic properties
and related parameters. So, we studied the same properties
for non-destructive characterization of iron in the tempera-
ture range between 100 and 300 K.
In present investigation, we made an attempt to apply
theoretical approach to find ultrasonic properties of iron
by means of SOEC and TOEC along <100>, <110>,
<100> directions in the temperature regime 100 - 300 K.
2. Theoretical Approach
Theory is categorized in two phases: in primary phase,
we discussed the temperature dependence of SOEC and
TOEC, while the attenuation of ultrasound due to pho-
non-phonon interaction and thermoelastic relaxation me-
chanisms with allied parameters has been explained in
secondary phase.
SOEC and TOEC at absolute zero have been obtained
following Brugger’s definition [16] of elastic constant

A. K. GUPTA ET AL. 35
.
using Born-Mayer potential [17]. According to the an-
harmonic theory of lattice dynamics, the lattice energy of
the single crystal changes with temperature [18,19]. Hence
an addition of vibrational contribution to the elastic con-
stants at absolute zero provides SOEC and TOEC at de-
sired temperature.
0Vib
IJK..... IJK.... IJK......
C=C+C (1)
where and represent static and vibrational
elastic constants respectively. The expressions for
and Care given below.
0
IJK....
C
b.
K....
Vib.
IJK....
C
0
IJK....
C
Vi
IJ





22
0
11 50
4
000
0
00
21,1
00
12 4450
4
00 0
3e11 1
C2
221
2
2
3e12 1
C2
22
Sr
br rbr
r
brr b
CS r
brr br














 






 



 
(2)





23
0
111 7
4
0
0
22
0 0
0
22
0 0
00
112 166
22,1
7
4
0
0
22
0 0
000
123 144456
15 e
C2
1331
1326 22
2
2
CC
15 e
2
1326 22
2
4
CCC
S
r
r
bbr rb
r
bbrrb
S
r
r
bbrrb






















21,1,1
7
4
0
15 e
2S r


























(3)
where 0 is the short range parameter; b is hardness
parameter; is the Born-Mayer potential given by
r

0
r 
 
exp.

rb
00
rA


and


0
2 exp.2rA rb
0

;
A is the strength parameter given by



21
1
30 0
2
0
e
3 S6 exp. 122 exp. 2
r
Ab



 

,
00
rb

.
Vib.(1,1) 2(2)
111 2
Vib.(1,1) 2(2)
121 1,1
Vib. (2)
44 1,1
C=f G+fG
C=f G+fG
C=fG







(4)
Vib.(1,1,1) 3(2,1)(3)
11111 23
Vib.(1,1,1) 3(2,1)(3)
112111,1 22,1
Vib.(1,1,1) 3(2,1)(3)
123111,11,1,1
Vib. (2,1)(3)
14411,11,1,1
Vib. (2,1)(3)
16611,12,1
Vi
456
C=fG+3f GG+fG
C=fG+fG (2G+G)+fG
C=fG+3f GG+fG
C=f GG+fG
C=f GG+fG
Cb. (3)
1,1,1
=f G













(5)
where ()n
f
and are given as
n
G
 
 

23 0
3
0
1,1 2,100
32
0
2
1,1,1 00 0
322 2
0
h
f=f= cothx
8
hh
f=f=+ cothx
962kT sinhx
h(h)cothx h
f=+ +cothx
3846(kT)sinhx2kT sinhx
r
r
r


 










 
2
1000
2
00 0
G222 r
22222rH






23
20000
23
000 0
G2(66)()
(32622)(2)H
r
r


 
  






 
234
300000
23 4
00000
G23030915 22
159222)2H
r
r



 


 





23
1100 00
G32622 2r  
，H


 
2
2,10 0
34
00 0
G152215(92)2
2)2Hr





1,1,1
G0

Here, 0B x=h2KT

,2
0
+0
11 1
=+
MM Hbr


 
 



, KB
is Boltzmann’s constant,





1
000 0
H2222rr  

  and values
of lattice sums are:




 
121,1
35 5
32,1 1,1,1
777
0.58252; S1.04622; S0.23185
1.36852; S0.16115; S0.09045
S
S
 

The obtained results of SOEC and TOEC are used to
find out the ultrasonic velocities and Grüneisen parame-
ters for longitudinal and shear waves along different
crystallographic directions. Those in turn are related to
compute acoustic coupling constants and ultrasonic at-
tenuation coefficient [20-23] as discussed in secondary

A. K. GUPTA ET AL.
36
single crystal structure. The X-ray diffraction (XRD) pat-
terns of iron nanowire arrays specified that most of the
iron nanowire arrays have the obvious preferred orienta-
tion along the <200> direction [28].
phase of theory and discussion section.
The secondary objective of the present investigation is
to develop a theory for evaluation of ultrasonic attenua-
tion. Mason-Bateman Theory [11,14,15,19-27] is still
widely used successfully to study the ultrasonic attenua-
tion at higher temperature (300 K) in solids. It is more
reliable theory to study anharmonicity of the crystals as it
involves elastic constants directly through acoustic cou-
pling constant “D” in the evaluation of ultrasonic atte-
nuation (α). The thermoelastic loss and Akhiezer loss [24]
under condition 1

  is given by:
The ultrasonic attenuation due to different mechanism
at high temperatures are evaluated using nearest neighbour
distance (r0) = 1.24 Å and Born-Mayer (hardness) pa-
rameter (b) = 0.303 Å for Fe. The SOEC and TOEC are
calculated at different temperatures. The values of den-
sity are taken from the literature [29]. The microstructure
of Fe was tested by transmission electron microscopy
and X-ray diffraction. The mean size was found between
100 and 200 nm [30].
2
2j 5
th iL
=kT2V

 



(6)
The computed values of SOEC and TOEC at high
temperatures are listed in Table 1. It is clear from Table
1 that, out of nine elastic constants, five (i.e., C11, C44,
C111, C
166 and C144) are increasing and three (i.e., C12,
C112, and C123) are decreasing with the temperature while
C456 is found to be unaffected. The increase or decrease
in elastic constants is mainly due to two basic parameters
i.e., lattice parameter and hardness parameter with other
common parameters as shown in Equations (1)-(5).
2
Akh 0
=ED (6V) 


3


(7)
where

is the density;  is angular frequency of ul-
trasonic wave; k is the thermal conductivity; E0 is the
thermal energy; V is velocity of longitudinal and shear
waves; T is the temperature in Kelvin scale; L and S
represent longitudinal and shear waves. Grüneisen num-
ber i
j

(i is the mode of propagation and j is the direc-
tion of propagation) is related to SOEC and TOEC. The
thermal relaxation time for longitudinal wave is twice
that of shear wave.
This type of behaviour has already been found in ma-
terials like Ce, Yb and Th [31]. Figure 1 shows that the

2
LS VD
=0.5== 3k C V 

(8)
where CV is the specific heat per unit volume and VD is
the Debye average velocity. The acoustic coupling con-
stant, which is measure of conversion of ultrasonic en-
ergy into thermal energy, can be obtained by:

22
jj
iiV
D=93 γCT E 


0



(9)
3. Results and Discussion
The Transmission Electron Microscope (TEM) shows
that the iron nanowires are highly uniform and exhibit a Figure 1. Temperature vs. C44/C12.
Table 1. SOEC and TOEC (1011 N/m2) of Fe-metal in the temperature range 100 - 300 K.
Temp.[K] C11 C
12 C 44 C 111 C 112 C 123 C 144 C 166 C 456
100 1.8261 0.6918 0.5682 –20.6498 –0.7016 0.4734 0.4785 –0.7376 0.4441
150 1.8270 0.6903 0.5697 –20.5068 –0.6564 0.4964 0.5042 –0.7104 0.4526
200 1.8280 0.6887 0.5712 –20.3637 –0.6112 0.5195 0.5299 –0.6832 0.4611
250 1.8289 0.6872 0.5727 –20.2206 –0.5659 0.5426 0.5555 –0.6560 0.4695
300 1.8299 0.6856 0.5742 –20.0776 –0.5207 0.5657 0.5812 –0.6288 0.4780

37 A. K. GUPTA ET AL.
temperature dependence of C44/C12 for Iron. For Iron, we
have the deviation from the Cauchy relation on increase-
ing the temperature. That is the result indicates that the
dominance of ionic interaction decreases with the tem-
perature. On the other hand although still from the condi-
tion C44/C12 = 1, the result for Iron suggests that the
bonding become more ionic with increase in tempera-
ture.
The stability of cubic crystals is expressed in terms of
elastic constants as


T1112
B=C+2C3>0

, C44 > 0 and
 
S1112
C= CC2>0

, CIJ are conventional elastic con-
stants, BT is the bulk modulus. The quantize C44 and CS
are the shear and tetragonal moduli of a cubic crystal.
Estimated values of bulk and tetragonal moduli for BkY
are presented in Figure 2. These conditions are also
known as Born criterion of mechanical stability. The
Born criterion of mechanical stability in case of iron is
satisfied as shown in Figure 2. Hence our approach to
compute SOEC/TOEC is correct.
The values of SOEC at 300 K are C11 = 1.8299 × 1011
N/m2, C12 = 0.6856 × 1011 N/m2 and C44 = 0.5742 × 1011
N/m2 and the experimental values at 300 K are C11 =
2.37 × 1011 N/m2, C12 = 1.41 × 1011 N/m2 and C44 =
1.160 × 1011 N/m2 of Fe at normal grain size [32]. Al-
though SOECs of the nanosized metals are smaller than
the normal sized metal, but quanta of SOECs are same.
Thermal relaxation time ( 
th) is determined utilizing
lattice thermal conductivity values [33-35]. Specific heat
per unit volume (Cv) and energy density (E0) of crystals
have been evaluated as a function of

D/T [29]; where

D
is Debye temperature. Non-linearity constant (D) are
obtained at different temperature. The values of density
(
 ), specific heat (Cv), energy density (E0), longitudinal
(VL), shear (VS) and Debye average velocity (V) are
presented in Tabl e 2. The values of thermal conductivity
(K), thermal relaxation time (

th) and anisotropy are pre-
sented in Figure 3 and the values of non-linearity con-
stants (acoustic coupling constants D) are presented in
Table 3.
It is seen from Figure 3 that thermal relaxation time (

th)
decreases with temperature due to their thermal conductive-
ity values. It is also obvious from Figure 3 that the val-
ues of anisotropy in Fe are almost same at different tem-
Figure 2. Temperature variation of BT and CS.
Figure 3. Temperature variation of K, TS, Anisotropy.
Table 2. Density () in Kg/m3, specific heat (Cv) Joule/m3K, energy density (E0) Joule/m3, longitudinal, shear and Debye av-
erage velocity (V) 103 m/sec of Fe in the temperature range 100 - 300 K.
Temp. [K]  Cv E
0 V L V S V
100 7.96 1.449 0.4790 4.790 2.672 2.937
150 7.93 2.296 1.4430 4.800 2.680 2.947
200 7.91 2.701 2.6227 4.807 2.687 2.954
250 7.89 2.973 4.1002 4.815 2.694 2.961
300 7.87 3.113 5.6162 4.822 2.701 2.969

A. K. GUPTA ET AL.
38
Table 3. Acoustic coupling constants (DL for longitudinal wave, DS for shear wave), along <100>, <110> and <111> crystal-
lographic directions of Fe in the temperature range 100 - 300 K.
Temperature [K] 
Direction  Parameter  100 150 200 250 300
D
L 7.308 7.256 7.186 7.113 7.038
<100> D
S 0.648 0.662 0.676 0.691 0.706
D
L 12.085 12.096 12.017 11.914 11.790
DS*
DS**
0.749 0.769 0.790 0.811 0.833
<110>
12.142 11.964 11.788 11.614 11.442
DL 14.405 14.370 14.041 13.627 13.169
<111> D
S*** 15.459 15.133 14.804 14.485 14.178
*Shear wave polarized along <001> direction, **Shear wave polarized alongdirection, ***Shear wave polarized along11 0direction. 110


peratures. The ultrasonic attenuation coefficients (α/f2)Akh.
due to p-p interaction and (α/f2)th. due to thermoelastic
relaxation in nanosized Fe at 100 - 300 K along <100>,
<110> and <111> crystallographic directions are de-
termined and the temperature variation of the attenuation
is shown in Figures 4-6. A perusal of the Figures 4-6
shows that the thermoelastic loss (α/f2)th. is negligible in
comparison to the Akhiezer type attenuation (loss due to
p-p interaction).
Figure 4. [(  /f2)Th(×10–17Nps2·m–1)] vs. Temperature of Fe.
Figure 5. [(  /f2)Akh.long.(×10–16 Nps2·m–1)] vs. T emperatu re of Fe.
Figure 6. [(  /f2)Akh.Shear.(×10–16 Nps2·m–1)] vs. Temperature of
Fe.
This is due to low values of thermal conductivity and
higher values of Debye average velocities of the wave
along all the three propagating directions. There is posi-
tive temperature dependence of the ultrasonic attenuation
at high temperatures as in other metals [36,37]. This
positive temperature dependence of ultrasonic absorption
is due to the fact that p-p interaction occurs at high tem-
peratures mainly at room temperature. Order of attenua-
tion in the nanosized Fe for longitudinal wave and shear
wave is the same as in the like metals [38].
Attenuation of longitudinal wave is more than that of
shear wave along <100> and <110> direction polarized
along <001> direction. A greater value of ultrasonic at-
tenuation for longitudinal wave along <100> in com-
parison to shear wave attenuation is due to greater value
of non-linearity constants (DL). In case of wave propa-
gating along <110> direction and polarized along <001>,
although DS* is greater than the DL, the values of
(α/f2)Akh.long. are greater than the (α/f2)Akh.shear. This be-
haviour is due to smaller value of C44 (SOEC) in compare-
son to C11 affecting the wave velocity as per their relations
as given below longitudinal 11 V=C

andshear 44 V=C

,
here
 is the density of the material.
In the case of the wave propagating along <111> and

39 A. K. GUPTA ET AL.
shear wave polarized along 11 0

; shear attenuation
(α/f2)Akh.shear*** is greater than the longitudinal wave at-
tenuation (α/f2)Akh.long.. This is due to greater value of
shear wave non-linearity constants DS*** in comparison
to longitudinal wave non-linearity constants DL. Although
no experimental values of attenuation in Fe (bcc) are
available for the comparison, but order of attenuation is
found same as other bcc metals like potassium, which is
studied experimentally by Sathish et al. [39].
Since whole computation is based on only two basic
parameters i.e. lattice parameter and hardness parameter,
one may conclude that the behaviour of temperature de-
pendence of ultrasonic absorption and other allied pa-
rameters of nanosized Fe is particular one for the mate-
rial and supports our theoretical approach.
The preliminary results obtained in this work can be
used for further experimental investigation with the pulse
echo overlap (PEO) technique for ultrasonic measure-
ments. The application of this measurement is mainly in
non-destructive testing (NDT) of a material with conven-
tional analytical techniques such as polarising micros-
copy, X-ray diffraction (XRD), surface tension analysis,
solid state nuclear magnetic resonance (NMR), scanning
electron microscopy (SEM) and transmission electron
microscopy (TEM).
4. References
[1] P. A. Elmore and M. A. Breazeale, “Dispersion and Fre-
quency Dependent Nonlinearity in a Graphite-Epoxy Com-
posite,” Ultrasonics, Vol. 41, No. 9, 2004, pp. 709-718.
doi:10.1016/j.ultras.2003.11.001
[2] D. M. Profunser, J. Vollmann and J. Dual, “Determina-
tion of the Material Properties of Microstructures by La-
ser Based Ultrasound,” Ultrasonics, Vol. 42, No. 1-9,
2004, pp. 641-646.
[3] H. Ogi, A. Tsujimoto, S. Nishimura and M. Hirao, “Aco-
ustic study of Kinetics of Vacancy Diffusion toward Dis-
locations in Aluminum,” Acta Materialia, Vol. 53, No. 2,
2005, pp. 513-517. doi:10.1016/j.actamat.2004.10.007
[4] R. P. Singh and G. S. Verma, “Phonon Conductivity and
Acoustic Attenuation in Si,” Physical Review, Vol. 171,
No. 3, 1968, pp. 838-842. doi:10.1103/PhysRev.171.838
[5] K. J. Singh, Y. Matsuda, K. Hattori, H. Nakamo and S. Na-
gai, “Non-Contact Sound Velocities and Attenuation Mea-
surements of Several Ceramics at Elevated Temperatures”,
Ultrasonics, Vol. 41, No. 1, 2003, pp. 9-14.
doi:10.1016/S0041-624X(02)00392-X
[6] R. Esquivel-Sirvent, B. Tan, I., Abdelraziq, S. S. Yun and
F. B. Stumpf, “Absorption and Velocity of Ultrasound in
Binary Solutions of Poly (Ethylene Glycol) and Water,”
Journal of the Acoustical Society of America, Vol. 93,
No.2, 1993, pp. 819-820. doi:10.1121/1.405444
[7] S. K. Kor and S. C. Deorani, “Correlation between Acous-
tic and Dielectric Relaxation Times in Liquids,” Physical
Review Letters, Vol. 27, No. 5, 1971, pp. 242-244.
doi:10.1103/PhysRevLett.27.24
[8] S. K. Kor and S. K. Pandey, “Ultrasonic Investigation of
Cholesteric Liquid Crystals,” Journal of Chemical Phys-
ics, Vol. 64, No. 4, 1976, pp. 1333-1336.
doi:10.1103/PhysRevLett.27.24
[9] S. K. Kor, R. R. Yadav and D. Singh, “Ultrasonic Studies
of CTAB in Glycol,” Molecular Crystals Liquid Crystals,
Vol. 392, No. 1, 2002, pp. 75-81.
doi:10.1080.15421400390193918
[10] W. P. Mason, “Piezielectric Crystals and Their Applica-
tion to Ultrasonics,” D. Van Nostrand Co. Inc. Princeton,
New Jersey, 1950.
[11] W. P. Mason, “Physical Acoustics,” Vol. IIIB, Academic
Press, New York, 1965.
[12] D. Singh, P. K. Yadawa and S. K. Sahu, “Effect of Elec-
trical Resistivity on Ultrasonic Attenuation in NpTe,”
Cryogenics, Vol. 50, No. 8, 2010, pp. 476-479.
doi:10.1103/PhysRevLett.27.24
[13] D. Singh, D. K. Pandey, P. K. Yadawa and A. K. Yadav,
“Attenuation of Ultrasonic Waves in V, Nb and Ta at Low
Temperature,” Cryogenics, Vol. 49, No. 1, 2009, pp. 12-
16. doi:10.1016/j.cryogenics.2008.08.008
[14] D. Singh, D. K. Pandey, D. K. Pandey and R. R. Yadav,
“Propagation of Ultrasonic waves in Neptunium Mono-
chalcogenides,” Applied Acoustics, Vol. 72, No. 10, 2011,
pp. 737-741. doi:10.1016/j.apacoust.2011.04.002
[15] D. Singh and D. K. Pandey, “Ultrasonic Investigations in
Intermetallics,” Pramana-Journal of Physics, Vol. 72, No.
2, 2009, pp. 389-398. doi:10.1007/s12043-009-0034-7
[16] K. Brugger, “Thermodynamic Definition of Higher Order
Elastic Coefficients,” Physical Review, Vol. 133, 1964,
pp. A1611-A1612. doi:10.1103/PhysRev.133.A1611
[17] M. Born and J. E. Mayer, “Zur Gittertheorie der Lonenk-
ristalle,” Zeitschrift für Physik, Vol. 75, No. 1-2, 1932, pp.
1-18. doi:10.1007/BF01340511
[18] P. B. Ghate, “Third-Order Elastic Constants of Alkali Ha-
lide Crystals,” Physical Review, Vol. 139, No. 5A, 1965,
pp. 1666-1674. doi:10.1103/PhysRev.139.A1666
[19] S. Mori Y. Hiki, “Calculation of the Third- and Fourth-
Order Elastic Constants of Alkali Halide Crystals,” Jour-
nal of the Physical Society of Japan, Vol. 45, No. 5, 1975,
pp. 1449-1456. doi:10.1143/JPSJ.45.1449
[20] 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
[21] R. P. Singh and R. K. Singh, “Temperature Dependent
Physiccal Effects of Ultrasonic Waves in Beryllium Chalco-
genides,” Applied Acoustics, Vol. 71, No. 11, 2010, pp.
328-334. doi:10.1016/j.apacoust.2009.10.005
[22] J. Kumar, Kailash, V. Kumar and A. K. Choudhary, “Tem-
perature Dependent of Higher Order Elastic Constants of
TeO Crystal,” Asian Journal of Chemistry, Vol. 23, No.
12, 2011, pp. 5601-5604.
[23] J. Kumar, Kailash, S. K. Shrivastava, D. Singh and V. Ku-

A. K. GUPTA ET AL.
40
mar, “Ultrasonic Attenuation in Calcium Oxide,” Advances
in Materials Physics and Chemistry, Vol. 1, No. 2, 2011, pp.
44-49. doi:/10.4236/ampc.2011.12008
[24] A. Akhiezer, “On the Absorption of Sound in Solids,” Jour-
nal of Physics (Moscow), Vol. 1, No.1, 1939, pp. 277-287.
[25] H. E. Bömmel and K. Dransfeld, “Excitation and Atte-
nuation of Hypersonic Waves in Quartz,” Physical Re-
view, Vol. 117, No. 5, 1960, pp. 1245-1252.
doi:10.1103/PhysRev.117.1245
[26] T. O. Woodruff and H. Ehrenreich, “Absorption of Sound
in Insulators,” Physical Review, Vol. 123, No. 5, 1961, pp.
1553-1559. doi:10.1103/PhysRev.123.1553
[27] W. P. Mason and T. B. Bateman, “Relation between Third-
Order Elastic Moduli and the Thermal Attenuation of Ul-
trasonic Waves in Nonconducting and Metallic Crystals,”
Journal of the Acoustical Society of America, Vol. 40, No.
4, 1966, pp. 852-862. doi:org/10.1121/1.1910158
[28] Y. L. Sun, Y. Dai, L. Q. Zhou and W. Chen, “Single-
Crystal Iron Nanowire Arrays,” Solid State Phenomena,
Vol. 121-123, No. 3, 2007, pp. 17-20.
doi:10.4028/www.sci entifi c.net/SSP.121-123. 17
[29] D. E. Gray, Ed., “AIP Handbook,” 3rd Edition, Mc-Graw
Hill Book Co. Inc., New York, 1965.
[30] K. Pekala and M. Pekala, “Low Temperature Transport Pro-
perties of Nanocrystalline Cu, Fe and Ni,” Nanostruc-
tured Materials, Vol. 6, No. 3, 1995, pp. 819-822.
doi:10.1016/0965-9773(95)00185-9
[31] R. R. Yadav, D. Singh and A. K. Tiwari, “Ultrasonic Eva-
luation in Rare-Earth Metals,” Journal of the Acoustical
Society of India, Vol. 30, No. 1-2, 2002, pp. 59-63.
[32] L. A. Girifalco and V. G. Weizer, “Application of the Morse
Potential Function to Cubic Metals,” Physical Review,
Vol. 114, No. 3, 1959, pp. 687-690.
doi:10.1103/PhysRev.114.687
[33] Y. S. Touloukian, et al., “Thermal Conductivity: Metallic
Elements and Alloys,” TPRS Series, Vol. I, IFI/Plenum,
New York, 1970.
[34] P. Haen and G. T. Meaden, “The Thermal Conductivity,
Thermoelectric Power, and Electrical Resistivity of Tho-
rium between 5 and 100°K,” Cryogenics, Vol. 5, No. 4,
1965, pp. 194-198. doi:10.1016/0011-2275(65)90056-1
[35] M. A. Gurry, S. Legvold and F. H. Spedding, “Electrical Re-
sistivity of Europium and Ytterbium,” Physical Review B,
Vol. 117, No. 4, 1960, pp. 953-954.
doi:10.1103/PhysRev.117.953
[36] S. K. Kor, Kailash, K. Shanker and P. Mehrotra, “Behav-
iour of Acoustical Phonons in Metals in Low Tempera-
ture Region,” Journal of the Physical Society of Japan,
Vol. 56, No. 7, 1987, pp. 2428-2432.
doi:10.1143/JPSJ.56.2428
[37] S. K. Kor and R. R. Yadav, “Ultrasonic Attenuation Due
to Electron-Phonon Interaction in Palladium,” Journal of
Pure and Applied Ultrasonics, Vol. 8, No. 3, 1986, pp.
89-90.
[38] S. K. Kor and R. K. Singh, “Ultrasonic Attenuation in Al-
kali Metals,” Acustica, Vol. 79, No. 2, 1993, pp. 292-295.
[39] S. Sathish, S. Chaterjee, O. N. Awasthi and E. S. R. Go-
pal, “Electron-Electron Scattering and Ultrasonic Attenu-
ation in Potassium,” Journal of Low Temperature Physics,
Vol. 63, No. 5-6, 1986, pp. 423-429.
doi:10.1007/BF00681490