st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Coupled Elasto-Electromagnetic Waves in Magneto-Active Piezo-Plasma Media
S. KH. Alavi1
1
Department Of Physics, Science Faculty, University Of Mazandaran, Babolsar, Iran
Abstract: Considering a piezo-plasma-like medium with the hexagonal symmetry and approximating it by an isotropic medium the coupling of the elastic waves with electromagnetic
waves is shown in the presence of the strong external magnetic field.
Keywords: Piezo-plasma-like medium, elasticity wave, coupled-elasto-electromagnetic wave
1. Introduction
The Piezoelectricity is the ability of certain material to
produce a voltage when subjected to mechanical stress. In
a piezoelectric crystal, the positive and negative charges
are separated but symmetrically distributed so that the
crystal overall is neutral electrically. When a stress is applied, this symmetry is destroyed and such charge asymmetry produces a voltage. Recently, wave propagation in
piezoelectric solids attarcts many attention due to its potential application in the field of non-destructive evaluation of piezoelectric materials[1-3]. In earlier works coupled elasto-electromagnetic (CEE) waves were studied in
isotropic plasma media [4-6] but there is no magnetic
field in the medium. Now we study this problem in the
presence of an external magnetic field.
2. Method and formulation
In order to investigate the elasto-electromagnetic waves
in
piezo-plasma
like
medium
such
as
piezo-semiconductors we should obtain the dispersion
relation of these kind of waves. The dispersion relation
for CEE waves is obtained from the following equation[4]
2
( m)
ikjl k k k l
ij
where,
lik k k

(
,
k
)
ij
1
lm
mjnk n
4
c
(1)
ij
(m)

(
,
k
),
ij
2
 c
ij ( , k ) are the lattice density,
, ikjl , lik and
elastic modulus tensor, piezoelectric tensor and dielectric
permittivity of the medium, respectively[7]. The analysis
of dispersion equation (1) is very complicated due to the
presence of several oscillation branches. Therefore, we
confine our study only in low frequency waves which are
more strongly coupled
 to the elasticity lattice vibration..
Thus we can use E
and as a result Eq. (1) can be
written as
2
( m)
ij
link l k s
where,
ikjl k k k l
4
k2
sjmk m

(2)
0,
 ( ,k)

( , k ) k i k j ij ( , k ) / k 2 is the longitudinal
1
xy
1
yy
0
where,
0
0 ,
1
ij
0
2
k2
zz
1
xx
1
yy
2
c2
1
yx
xx
c2
2
k
1
xy
(3)
1
zz
1
c
2
c2
,
4
2
xx
4
xy c
2
2
(k
2
0,
ki k j
1
xx
1
yx

1
(
,
k
)
ij
2
2
k2
dielectric permittivity of the magneto-active plasma. It
should be noted that, slow electromagnetic waves (EM) in
solid state magneto-active plasma are assumed to be potential. However, it is possible to have purely transverse
slow waves such as Alfven waves, helicon, fast and slow
magnetosonic waves in such a plasma. In order to investigate the coupling of such waves with elasticity vibration
of crystal lattice, we analyze the dispersion
Eq. (1) for
 
purely longitudinal propagation k B z without use of
potential approximation. In this case the tensor
1
ij ( , k ) get the following simple form
2
xy
.
4
xx )
2
c4
2
xy
Considering a crystal lattice with hexagonal symmetry
under the condition that the forth order symmetry axis is
along the z-axis, i.e., parallel to the magnetic field, the
tensor ijl has the following form[myself]
1
xxz
2
xzx
zxx
Moreover, the tensor
tr
ikjl
ijkl
(
yzy ,
yyz
zyy ,
zzz .
3
has the form
ki k j
ij
2
)k 2
l
ki k j
because, a crystal with ak hexagonal symmetry is
verysimilar to an isotropic medium. The quantities
tr
and l are related to the elasticity coefficient of the
medium[8].
Under these assumptions Eq(2) find the following form
(
2
k2
2
l ,tr )

( ,k)
8
2 2
3,1 k
( m)
,
(4)
l ,tr
where, vl .tr
/ ( m) is the longitudinal and
transverse velocity of acoustic waves. This equation cou-
st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
ples elasticity acoustic waves with longitudinal oscillation
of charge carrier plasma. Let us now consider transverse
waves. Equation (1) for one dimensional transverse wave
propagating along the magnetic field is simplified as the
following system of two independent equations
2
(
2
(
k2
2
tr )
k
2
k2
2
l ) zz
c
2
(
i
xx
2 2 2
3k
2 ( m)
4
4
xy )
c
c
2 2 2
1 k
2 (m)
.
,
(5)
The latter equations describe the coupling of elasticity
acoustic vibrations of the crystal lattice with normal and
abnormal EM waves in the solid state plasmas of free
charge carriers. From Eqs. (4) and (5) it is clear that in
piezo-semiconductors, elasticity acoustic vibration of the
crystal lattice are coupled to the EM waves. The coupling
of elasticity acoustic vibrations of the crystal lattice with
longitudinal EM waves is described by Eq. (4) and with
transverse EM waves by Eq. (5).
It is well-know that the strong coupling between elasticity acoustic waves and EM waves appears when their
frequency are close. Taking into account this condition,
we can write Eqs. (4) and (5) in the following form.
2
(
k 2 v 2 )(
i
)
A,
(6)
where A is the coupling parameter given by
2 2
3k
( m)
8
A
(
4
A
2
k2
)
2
k 2 c32 2
(
2
1
2
2
l
,
,
2
2
tr
2
2
l
(8)
(9)
indicates one of the eigen frequency
of plasma waves such as ion-acoustic waves, Alfven
waves and etc.
From Eq. (6) it is clear that the strong coupling between
elasticity waves and EM waves takes place under the resonance condition
(k ) k . In this case if we
assume the solution of Eq. (6) to be
2
for Eq. (5). cHere,
i
Then for
k
A
2
0.
Since EM oscillations are assumed to be weakly damping
(
) we can conclude that the imaginary part of
the dielectric permittivity is small. Therefore, it can be
neglected in the coupling parameter A and consequently
parameter A can be assumed to be a purely real quantity.
Under this condition when the coupling of piezoelectric is
k
0,
pe
2
Te
(1 i
/2
)
k
Te
(12)
the ion-acoustic frequency spectrum
2
pi
2
m
8M
,
2
pe
1
2 2 is
Here,
k TeTe
,
(13)
is obtained.
the electron thermal velocity.
When the piezoelectric coupling is weak, the above frequency spectrum in the first approximation dose not
changes. However, elasticity acoustic waves have the following damping decrement
2 4 2
3k
Te
2
(m)
pi
.
(14)
The damping decrement of these waves is stipulated by
Cherenkov dissipation mechanism on electrons.
Now we can consider the coupling between elasticity
waves with helicons in the strongly collisional solid state
plasma. The frequency spectrum of helicons waves propagating along the magnetic field is
k 2c 2
2
pe
e
c2k 2
,
2
pe
(15)
where
eff for strongly ionized non-degenerate
plasmas, and
en for weakly ionized plasmas [9].
Moreover,
is
the
Langmuire frequency of electron in
e
the magnetic field. In this case, coupling parameter “A”
gets the the following form
A
(10)
2
2
pe
2
i
we find the following equation
2
( ,k) 1
(7)
i1 xy )
xx
zz
In the latter equations 1 describes the damping decrement
of EM waves which has not changed in the weak piezoelectric coupling approximation. In addition 2 describes
the damping decrement of elasticity acoustic waves which
generally vanishes when the coupling is not taken into
account and as a consequence has plasma character only.
Now we consider Eqs. (4-11) in the presence of a strong
external magnetic field. In the ion-acoustic frequency
region for a classical collisionless strongly magnetized
2
plasmas with dispersion
equation

2
,
2
c2
4
2
1
for Eq. (4) and
A
weak, we can find the damping decrement of aforementioned wave by
A
(11)
,
.
1
2
2 2 2
k
1
2
(m)
pe
4
e
.
(16)
Considering Eqs. (15) and (16), we find the relation
2
3
2 2 2
1 k
e
2
(m)
pe
,
(17)
for the damping decrement of the acoustic waves. This
acoustic damping is caused due to the collisional absorption of helicon waves, coupled to the crystal acoustic os-
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21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
cillation according to the Eq. (6), by plasma electron.
3. Conclusion
We considered a piezoelectric of hexagonal symmetry
in which the principle symmetry axis is parallel to the
z-axis. Based on these assumptions the crystal can be regarded as an isotropic medium in the plane perpendicular
to the main symmetry axis and the elastic moduli can be
approximated by an isotropic tensor with two nonzero
components. In this case, we obtained the dispersion relations show the coupling of the elasticity acoustic waves
with longitudinal oscillation of the charge carrier and also
such coupling with the electromagnetic (EM) waves in the
crystal lattice in the presence of external magnetic waves.
Then we discussed about the strong coupling between the
different eigen frequency of the plasma waves and the
elasticity waves and their related damping decrements.
3. References
[1] R. Peach, IEEE Trans. Ultrason. Ferroelectr. Freq.
Control., 48, 1308 (2001).
[2] M. Romero, Int. J. Eng. Sci., 42, 753(2004).
[3] M. Romero, Wave Motion, 39, 93 (2003).
[4] B. Shokri, S. KH, Alavi and A. A. Rukhadzeh, Physica
Scripta, 73, 1(2006).
[5] B. Shokri, S. KH. Alavi and A. A. Rukhadzeh, Waves
in Random and Complex Media, 16, 2(2006).
[6] S. KH. Alavi, B. Shokri, Waves in Random and Complex Media, 18, 4(2008).
[7] L. D. Landau and E. M. Lifshitz, Electrodynamic of
Continuous Media, Oxford: Pergamon (1960)
[8] L. D. Landau and E. M. Lifshitz, Theory of Elasticity
2nd Edn., Oxford: Pergamon (1970).
[9] A. F. Alexadrov, L. S. Bogdenkovich, A. A. Rukhadze,
Principle of Plasma Electrodynamics, Heidelberg:
Springer(1984).