Nonthermal and geometric effects on the symmetric and anti-symmetric surface waves in a Lorentzian dusty plasma slab
Myoung-Jae Lee and Young-Dae Jung
Citation: Physics of Plasmas 22, 022125 (2015); doi: 10.1063/1.4913355 View online: http://dx.doi.org/10.1063/1.4913355
View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/22/2?ver=pdfcov Published by the AIP Publishing
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Nonthermal and geometric effects on the symmetric and anti-symmetric surface waves in a Lorentzian dusty plasma slab
Myoung-Jae Lee1and Young-Dae Jung2,3,a)
1Department of Physics and Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea
2Department of Electrical and Computer Engineering, MC 0407, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0407, USA
3Department of Applied Physics and Department of Bionanotechnology, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea
(Received 26 January 2015; accepted 9 February 2015; published online 19 February 2015) The nonthermal and geometric effects on the propagation of the surface dust acoustic waves are investigated in a Lorentzian dusty plasma slab. The symmetric and anti-symmetric dispersion modes of the dust acoustic waves are obtained by the plasma dielectric function with the spectral reflection conditions the slab geometry. The variation of the nonthermal and geometric effects on the symmetric and the anti-symmetric modes of the surface plasma waves is also discussed.VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4913355]
I. INTRODUCTION
The propagation of surface waves on the bounded and semi-bounded has been of great interests in numerous areas such as laser physics, materials science, nanotechnology, plasma spectroscopy, and space science due to the impor- tance of dispersion properties of plasmas in laboratory and space environments.1–9Recently, there has been a great in- terest in investigating physical characteristics and proper- ties of plasmas containing charged dust grains including collective effects and strong electrostatic interactions in order to obtain the information on various plasma parame- ters in dusty plasmas since the charged dust grains are ubiq- uitous in astrophysical compact objects as well as in various laboratory nano-scale objects such as nano-wires, quantum dots, semiconductor devices, and laser produced dense plasmas.10–25It would be expected that the theoreti- cal calculation on the dispersion relation of the surface wave in bounded dusty plasmas can be used as a useful tool for investigating the structure and physical properties of such plasmas. However, to the best of our knowledge, the dispersion properties of the surface dust acoustic plasma wave in bounded dusty plasmas have not been investigated as yet. Thus, in this paper, we investigate the dispersion properties of the low frequency surface dust acoustic waves in a thin Lorentzian dusty plasma slab with the kinetic dis- persion model.
This paper is composed as follows: In Sec.II, we obtain the plasma dielectric function for the Lorentzian dusty plasma and also discuss the specular reflection condition for the slab geometry of the dusty plasma. Then the dispersion relation for the symmetric and anti-symmetric modes of the surface dust acoustic waves is obtained. In Sec.III, we dis- cuss the dispersion relations for the symmetric and the anti-
symmetric modes for various cases to investigate the non- thermal and geometric effects on the surface dust acoustic wave. Finally, the conclusions are given in Sec.IV.
II. DISPERSION RELATIONS
We shall consider a complex plasma slab which occu- pies the region between x¼0 andx¼L, bounded by vac- uum (x<0 and x>L). We employ here the specular reflection boundary condition on fa according to which the charged particles undergo a mirror reflection such that
faðx¼0;y;z;vx;vy;vz;tÞ ¼faðx¼0;y;z;vx;vy;vz;tÞ; and
faðx¼L;y;z;vx;vy;vz;tÞ ¼faðx¼L;y;z;vx;vy;vz;tÞ; wherefais the perturbed plasma distribution function of spe- ciesa(a ¼e,i,d; electron, ion, and dust). Then the disper- sion relation for surface waves propagating at the interface of the plasma slab and the vacuum, i.e., at x¼0 and L, is given by the formula9,26
pþ ð1
1
dkx
k2 kz
eL
17eikxL 16eikxL
¼0; (1) wherekxandkzare thex-andz-components of the wave vector k, respectively,k2¼k2xþk2z, andeL is the longitudinal com- ponent of the plasma dielectric function. The y-component of wave vector has a translational invariance and can be ignored in our geometry without loss of the generality. The surface waves on a plasma slab have two modes as represented by the double signs in Eq.(1): the upper and the lower signs corre- spond to symmetric and anti-symmetric modes, respectively. If L! 1, the exponential factor in Eq.(1)vanishes due to rapid oscillations and both modes reduce to the well known surface wave dispersion relation for surface waves in a semi-infinite plasma.2
a)Permanent address: Department of Applied Physics and Department of Bionanotechnology, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea. Electronic mail: [email protected]
1070-664X/2015/22(2)/022125/4/$30.00 22, 022125-1 VC2015 AIP Publishing LLC
Since the plasmas encountered in space are often non- Maxwellian, we seek the dust-acoustic surface wave on the plasma slab containing the superthermal electrons and ions in the high energy tail.26,27 Such plasmas can be well described by the Lorentzian distribution function. Therefore, we adopt the longitudinal dielectric function for the Lorentzian complex plasma given by28
eL¼1þ 2 ljk2
x2pe v2Te þx2pi
v2Ti
! x2pd
x2 ; (2) where x is the wave frequency, xpa¼ ð4pnaZ2ae2=maÞ1=2 is the plasma frequency of species a (¼e, i, d), vTa
¼ ð2Ta=maÞ1=2 is the thermal speed of species a, and lj
¼ ð2j3Þ=ð2j1Þis the parameter governed by the spec- tral index j of the Lorentzian distribution function. Putting Eq.(2)into Eq.(1), we have
pþ kz
1x2pd x2
ð1 1
dkx
k2xþw2
17eikxL 16eikxL
¼0; (3)
where
w2¼k2z þ 2 x2pe
v2Te þx2pi v2Ti
!
lj 1x2pd x2
: (4)
After some mathematical manipulations using the residue of the integral in Eq.(3), we obtain the dispersion relations for the dust acoustic surface waves in the following dimension- less forms
1 1
x2þ 1þ 1 ljk2z 1 1
x2
2 64
3 75
12
tanh kzL
2 1þ 1
ljkz2 1 1 x2
2 64
3 75
1 2
8>
><
>>
:
9>
>=
>>
;
¼0; ðsymmetric modeÞ (5)
and
1 1
x2þ 1þ 1 ljkz2 1 1
x2
2 64
3 75
12
coth kzL
2 1þ 1
ljk2z 1 1 x2
2 64
3 75
1 2
8>
><
>>
:
9>
>=
>>
;
¼0; ðanti-symmetric modeÞ; (6) wherex x=xpd,kz kzkD, andL L=kDare the scaled frequency, the scaled wave number, and the scaled slab
thickness, respectively. The symbol kD is the effective Debye length defined bykD¼ ð2x2pe=v2Teþ2x2pi=v2TiÞ1=2. III. DISCUSSIONS
Figure 1compares the wave frequencies for the sym- metric and the anti-symmetric modes in the case ofL¼10.
We assume that the plasma is Maxwellian and put the spec- tral indexjto infinity in the dielectric function. As can be seen from the figure, the phase velocity of the symmetric mode is always higher than the case of anti-symmetric mode, especially in the range ofkz<1. Forkz>1, the dif- ference in phase velocity becomes negligible. However, the group velocity of the anti-symmetric mode is found to be larger than that of the symmetric mode. Figures 2 and 3 show the geometric effects on the dust acoustic surface wave. In Fig. 2, the dispersion relation of the symmetric mode is depicted for the three different slab thicknesses:
L¼5;10;and 20. We observe that the phase velocity is enhanced as the slab thickness is decreased. The enhance- ment is even boosted as the scaled wave number kz is decreased (or as the wavelength is increased). It is interest- ing that the phase velocity of anti-symmetric mode
FIG. 1. The symmetric (blue) and the anti-symmetric (red) modes of disper- sion relation (x¼x=xpdÞfor the dust acoustic surface wave as a function of the scaled wave number (kz¼kzkD) in the case of Lð¼ L=kDÞ ¼10.
Maxwellian distribution is assumed.
FIG. 2. The symmetric modes of dispersion relation (x¼x=xpdÞfor the dust acoustic surface wave as a function of the scaled wave number (kz¼kzkD) in the case ofLð¼ L=kDÞ ¼5 (blue solid line), 10 (blue dashed line), and 20 (blue dotted line). Maxwellian distribution is assumed.
022125-2 M.-J. Lee and Y.-D. Jung Phys. Plasmas22, 022125 (2015)
decreases only slightly as the slab thickness decreases, as can be seen in Fig. 3. The degree of variation in the phase velocity is not as much as big compared to the case of the symmetric. Hence, it would be expected that the spectral investigation of the symmetric mode can be used as the diagnostic tool for the geometric effect on the dust acoustic surface wave. When the slab thickness goes to infinity, the two modes merge into one dispersion curve because the split two modes vary oppositely as the slab thickness changes. This is obvious because as the slab thickness L goes to infinity, the dispersion relation should agree with the case of semi-bounded plasma.29 Figures4and5repre- sent the nonthermal effects on the dust acoustic surface wave. Figure 4 compares the cases of j¼2, 5, and Maxwellian for the symmetric mode. We see that as the population of superthermal particles in the high energy tail increases, the phase velocity increases. However, the group velocity is found to be the largest for the Maxwellian plasma. The anti-symmetric mode reveals similar nonther- mal effects on the wave as the symmetric mode. Hence, we have found that the nonthermal character of the Lorentzian
plasma suppresses the group velocity of the dust acoustic surface wave.
IV. CONCLUSIONS
In this work, we have investigated the nonthermal and geometrical effects on the dust acoustic surface waves propa- gating at the interface of the Lorentzian dusty plasma slab and the vacuum. The dispersion relation is kinetically derived and obtained for the symmetric and the anti- symmetric modes of the surface plasma wave. The symmet- ric mode is found to have higher phase velocity than the anti-symmetric counterpart, especially, for small wave num- bers. The symmetric mode is highly dependent on the slab thickness. The wave frequency of the symmetric mode is enhanced as the slab thickness is decreased, whereas the anti-symmetric mode behaves oppositely. Both the symmet- ric and the anti-symmetric modes are also highly affected by the superthermal electrons and ions. As the population of superthermal particles increases, i.e., as the spectral indexj decreases, the phase velocities of both modes increase corre- spondingly. From this work, we have found that the influ- ence of nonthermal character and geometric configuration play crucial role in the propagation of the surface dust acous- tic waves in a Lorentzian dusty plasma slab. These results will be useful for understanding the physical characteristics and properties of the dust acoustic surface waves in the non- thermal dusty plasma slab.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Professor H. J. Lee for useful comments and discussions. One of the authors (Y.-D.J.) gratefully acknowledges Dr. M. Rosenberg for useful discussions and warm hospitality while visiting the Department of Electrical and Computer Engineering at the University of California, San Diego. This work was supported by National Research Foundation of Korea funded by Korean government (NRF-2010-0007196) and also by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of
FIG. 3. The anti-symmetric modes of dispersion relation (x¼x=xpdÞ for the dust acoustic surface wave as a function of the scaled wave number (kz¼kzkD) in the case of Lð¼ L=kDÞ ¼ 5 (red solid line), 10 (red dashed line), and 20 (red dotted line). Maxwellian distribution is assumed.
FIG. 4. The symmetric modes of dispersion relation (x¼x=xpdÞfor the dust acoustic surface wave as a function of the scaled wave number (kz¼kzkD) in the cases ofj¼2 (blue solid line),j¼5 (blue dashed line), and Maxwellian (blue dotted line). The scaled slab thickness is given by Lð¼ L=kDÞ ¼10.
FIG. 5. The symmetric modes of dispersion relation (x¼x=xpdÞfor the dust acoustic surface wave as a function of the scaled wave number (kz¼kzkD) in the cases ofj¼2 (red solid line),j¼5 (red dashed line) and Maxwellian (red dotted line). The scaled slab thickness is given by Lð¼ L=kDÞ ¼10.
Education, Science and Technology (Grant No. 2012- 001493).
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