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Nonlinear effects on bremsstrahlung emission in dusty plasmas
Physics Letters A 333 (2004) 426–432 www.elsevier.com/locate/pla
Nonlinear effects on bremsstrahlung emission in dusty plasmas Young-Woo Kim, Young-Dae Jung ∗ Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea Received 23 September 2004; received in revised form 26 October 2004; accepted 27 October 2004 Available online 10 November 2004 Communicated by F. Porcelli
Abstract Nonlinear effects on the bremsstrahlung process due to ion–dust grain collisions are investigated in dusty plasmas. The nonlinear screened interaction potential is applied to obtain the Fourier coefficients of the force acting on the dust grain. The classical trajectory analysis is applied to obtain the differential bremsstrahlung radiation cross section as a function of the scaled impact parameter, projectile energy, photon energy, and Debye length. The result shows that the nonlinear effects suppress the bremsstrahlung radiation cross section due to collisions of ions with positively charged dust grains. These nonlinear effects decrease with increasing Debye length and temperature, and increase with increasing radiation photon energy. 2004 Elsevier B.V. All rights reserved. PACS: 52.20.-j Keywords: Nonlinear effects on dust bremsstrahlung emission in dusty plasmas
The bremsstrahlung process [1–10] in plasmas has received much attention since this process has been widely used in plasma diagnostics in laboratory and space plasmas. In recent years, there has been a considerable interest in the dynamics of plasmas containing dust grains including collective effects and strong electrostatic interaction between the charged components in dusty plasmas. Various physical processes in dusty plasmas have been investigated in order to obtain the information of plasma parameters in dusty plasmas [11–13]. The particle interaction potentials in dusty plasmas have been represented as the standard Debye–Hückel potential obtained by the linearization of the Poisson equation with the Boltzmann distribution function. However, in the case of highly charged dust grains the description of the interaction potential would not be described by the Yukawa type Debye–Hückel potential due to strong nonlinear electrostatic interactions. Hence, the ion–dust grain bremsstrahlung spectrum would be affected by these nonlinear correction effects. To the best of our knowledge, the nonlinear effects on the bremsstrahlung process due to collisions of ions with charged dust grains in dusty plasmas have not been investi* Corresponding author.
E-mail addresses: [email protected], [email protected] (Y.-D. Jung). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.10.065
Y.-W. Kim, Y.-D. Jung / Physics Letters A 333 (2004) 426–432
427
gated as yet. Thus, in this Letter we investigate the nonlinear correction effects on the bremsstrahlung process due to ion–dust grain collisions in dusty plasmas. The interaction potential between the projectile ion and the charged dust grain is introduced by the nonlinear screened potential [14]. The classical straight-line trajectory analysis [1,15,16] is applied to obtain the differential bremsstrahlung radiation cross section as a function of the scaled impact parameter, projectile energy, radiation photon energy, and Debye length. The classical expression of the differential bremsstrahlung cross section [15] dσb can be represented as dσb = 2π db b dwω (b), (1) where b is the impact parameter and dwω is the differential probability of emitting a photon of frequency ω within dω when a projectile particle changes its velocity in collisions with a static target system. For all impact parameters, dwω is given by the Larmor formula for the emission spectrum of a nonrelativistic accelerated charge: 8π d 2 d(t) 2 dω , dwω = (2) 3h¯ c3 dt 2 ω ω where dω is the Fourier coefficient of the dipole moment d of the collision system. For the ion–dust grain system, the term [d 2 d(t)/dt 2 ]ω can be presented as 2 ze Ze d d(t) − = (3) Fω , dt 2 ω M mi where M and mi are the dust mass and the ion mass, respectively, and Fω is the Fourier coefficient of the force F(t): 1 Fω = 2π
∞ dt eiωt F(t).
(4)
−∞
Since the dust grain is much massive than the projectile ion, we can assume to a good approximation that the center of mass of the ion–dust grain system is at the position of the dust grain. Using the straight-line trajectory analysis, the position of the projectile ion can be represented as r(t) = b + vt with b · v = 0, where v is the velocity of the projectile ion and then, |Fω |2 = |F⊥ω |2 + |Fω |2 where the Fourier coefficients F⊥ω and Fω are the components of force perpendicular and parallel to the projectile velocity, respectively. Using the standard theory, we can obtain the electrostatic potential φ(r) produced by the charged dust grain (Q) with the surrounding plasma as the Debye–Hückel form [17] by solving the Poisson equation with the help of the Boltzmann distribution function: Q exp(−r/λD ), (5) r where λD is the Debye length. Here, for the sake of simplicity, the dust grain is assumed to be a spherical shape with the radius a. For typical laboratory and space dusty plasmas, it has been known that λD a [11]. Recently, the modified screening interaction potential in dusty plasmas was given by Vranjes et al. [14] by using the nonlinear corrections in the expanded Boltzmann distribution. Using the modified potential, the interaction potential energy V (r) between the projectile ion (ze) and the charged dust grain (Ze) can be represented as φDH (r) =
V (r) =
Ze2 Zze2 Zze2 exp(−r/λD ) − exp(r/λD ) Ei(−3r/λD ) r 4λD T r Ze2 Zze2 + exp(−r/λD ) Ei(−r/λD ), 4λD T r
(6)
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Y.-W. Kim, Y.-D. Jung / Physics Letters A 333 (2004) 426–432
∞ where T is the temperature and Ei(−r)[= − r dt e−t /t] is the exponential integral function [18]. After some algebra using the straight-line trajectory impact-parameter analysis and Eq. (4), we obtain the analytic formulas for the Fourier coefficients of the force as follows: Fµω =
Zze2 ¯ Fµω πva
(µ = ⊥, ).
(7)
¯ β, λ¯ D , η) and F¯ω (b, ¯ β, λ¯ D , η) are the Here, the scaled Fourier coefficients including the nonlinear effects F¯⊥ω (b, components of force perpendicular and parallel to the projectile velocity, respectively, and found to be as follows: ¯ β, λ¯ D , η) = F¯⊥ω (b,
∞
¯ τ, λ¯ D , η), dτ b¯ cos βτ G(b,
(8)
0
¯ β, λ¯ D , η) = i F¯ω (b,
∞
¯ τ, λ¯ D , η), dτ τ sin βτ G(b,
(9)
0
¯ b/a) is the scaled impact parameter, β ≡ ωa/v, τ (≡ vt/a) is the scaled time, λ¯ D (≡ λD /a) is the scaled where b(≡ 4 Debye length, η ≡ Z(Ry/2kB T )(a0 /a)λ¯ −1 = 13.6 eV is the Rydberg constant, m is the electron ¯ 2) ∼ D , Ry(= me /2h 2 2 ¯ τ, λ¯ D , η) is given rest mass, kB is the Boltzmann constant, a0 (= h¯ /me ) is the Bohr radius, and the function G(b, by 2 1/2 ¯ 2 1/2 ¯ ¯2 ¯2 ¯ τ, λ¯ D , η) = exp[−(b + τ ) /λD ] + exp[−(b + τ ) /λD ] G(b, (b¯ 2 + τ 2 )3/2 (b¯ 2 + τ 2 )λ¯ D exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 2 1/2 ¯ +η − Ei −3 b + τ / λ D (b¯ 2 + τ 2 )3/2 1/2 exp[(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 Ei −3 b + τ 2 /λ¯ D + 2 2 ¯ ¯ (b + τ )λD
+ +
1/2 exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 Ei − b + τ 2 /λ¯ D (b¯ 2 + τ 2 )3/2
exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 2 1/2 ¯ Ei − b + τ / λ . D (b¯ 2 + τ 2 )λ¯ D
(10)
(b, ¯ β, λ¯ D ) and F¯ (b, ¯ β, λ¯ D ) becomes If we neglect the nonlinear effects, the scaled Fourier coefficients F¯⊥ω ω ¯ β, λ¯ D ) = (b, F¯⊥ω
∞
¯ τ, λ¯ D ), dτ b¯ cos βτ G (b,
(11)
0 ¯ F¯ω (b, β, λ¯ D ) = i
∞
¯ τ, λ¯ D ), dτ b¯ sin βτ G (b,
(12)
0
and the function
¯ τ, λ¯ D ) G (b,
¯ τ, λ¯ D ) = G (b,
is
exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] + . (b¯ 2 + τ 2 )3/2 (b¯ 2 + τ 2 )λ¯ D
(13)
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Then, the differential bremsstrahlung radiation cross section [19] is defined as dσb dχb ≡ h¯ ω, dε h¯ dω
16z2 α 3 a02 µ Zm zm 2 d b¯ b¯ |F¯⊥ω |2 + |F¯ω |2 , = − m M mi 3E¯
(14) (15)
¯ E/Z 2 Ry) is where ε(≡ h¯ ω) is the radiation photon energy, α(= e2 /h¯ c ∼ = 1/137) is the fine structure constant, E(≡ 2 the scaled collision energy, E(≡ µv /2) is the collision energy, and µ[= Mmi /(M + mi )] is the reduced mass of the collision system. In nonrelativistic limit, the parameter β can be expressed as β = Z(µ/m)1/2(a/a0 )(¯ε/2E¯ 1/2) where ε¯ (≡ h¯ ω/Z 2 Ry) is the scaled photon energy. Then, the scaled doubly differential bremsstrahlung radiation cross section including the nonlinear effects, i.e., the scaled differential bremsstrahlung radiation cross section per scaled impact parameter, can be presented as d 2 χb d ε¯ d b¯ πa02
=
z m 2¯ 16Z 2 z2 α 3 µ m − b 3π E¯ m M Z mi ¯ ε¯ , E, ¯ λ¯ D , η)2 + F¯ω (b, ¯ ε¯ , E, ¯ λ¯ D , η)2 , × F¯⊥ω (b,
(16)
where the scaled Fourier coefficients F¯⊥ω and F¯ω are already given in Eqs. (8) and (9), respectively. It is obvious that the bremsstrahlung cross section for the electron–dust grain collision is greater than that for the ion–dust grain collision due to the large mass difference between the electron and the ion. However, it has been know that the electron density is much smaller than the ion density for a dusty plasma with negatively charged dusty grains and the electron temperature is greater than the ion temperature [12]. Then, the low-energy ion–dust bremsstrahlung process is more frequent than the electron–dust bremsstrahlung process since the power density [9] of bremsstrahlung is proportional to the density function of the projectile. In addition, the energy loss of ions due to the ion–dust grain bremsstrahlung is quite important for determining the force acting on dust grains since the ion drag force is related to momentum exchange between positive ions and a dust grain. Thus, the nonlinear effects should be properly accounted for in analysis of the ion–dust bremsstrahlung process in dusty plasmas. The formation of regular structures of dust grains was observed in a laminar flow of a thermal plasma, in which case the grain charge was positive due to thermal electron emission [20]. Thus, in order to investigate the nonlinear correction effects on the bremsstrahlung radiation cross section due to collisions of ions and positively charged dust grains, we set a = 0.01 µm, Z = 500, z = 1 (proton projectile), and the density of the dust grain is ρ ∼ = 2 g cm−3 . Here, we choose E¯ = 0.9 since the bremsstrahlung radiation cross section, Ref. [18], is known to be valid for low collision ¯ energies [15] (E < Z 2 Ry). Figs. 1 and 2 show the ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neincluding the nonlinear effects (Eqs. (8) and (9)) to the bremsstarhlung radiation cross section (d 2 χb /d ε¯ d b) glecting the nonlinear effects (Eqs. (11) and (12)) as a function of the scaled Debye length (λ¯ D ) and the temperature (kB T ), respectively. Fig. 3 shows the three-dimensional plot of the ratio as a function of the scaled Debye length and the temperature. As we see in these figures, the nonlinear effects suppress the bremsstrahlung radiation cross sections due to collisions of ion with positively charged dust grains. It is found that the ratio approaches the unity with increasing Debye length and temperature. In other words, the nonlinear effects are decreased with increasing Debye length and temperature. It can be also found that the nonlinear effects are significant for small Debye lengths since exp(r/λD ) Ei(−3r/λD ) − exp(−r/λD ) Ei(−r/λD ) is exponentially decreasing with the distance. However, even for large Debye lengths, for example, λ¯ D = 150, the nonlinear effects are found be more than 10%. Fig. 4 shows the variation of the nonlinear correction effects on the bremsstrahlung radiation cross section with changing radiation photon energy. As we see in this figure, it is also found that the nonlinear effect increases with increasing radiation photon energy. The variation of the nonlinear effect is found to be more significant for small Debye lengths. Hence, it is important to note that the nonlinear correction plays an important role in bremsstrahlung processes in dusty plasmas. It can be also expected that the nonlinear screening effects play important roles in
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Y.-W. Kim, Y.-D. Jung / Physics Letters A 333 (2004) 426–432
¯ including the nonlinear effects to the bremsstrahlung radiation Fig. 1. The ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neglecting the nonlinear effects as a function of the scaled Debye length (λ¯ D ) when kB T = 2Ry, E¯ = 0.9, and cross section (d 2 χb /d ε¯ d b) ε¯ = 0.18 at b¯ = 2.
¯ including the nonlinear effects to the bremsstrahlung radiation Fig. 2. The ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neglecting the nonlinear effects as a function of the temperature (kB T ) when λ¯ D = 50, E¯ = 0.9, and ε¯ = 0.18 at cross section (d 2 χb /d ε¯ d b) b¯ = 2.
Fig. 3. The three-dimensional plot of the ratio as a function of the scaled Debye length (λ¯ D ) and the temperature (kB T ) when E¯ = 0.9, and ε¯ = 0.18 at b¯ = 2.
Y.-W. Kim, Y.-D. Jung / Physics Letters A 333 (2004) 426–432
431
Fig. 4. The nonlinear correction effects on the bremsstrahlung radiation cross section as a function of the scaled Debye length (λ¯ D ) when kB T = 2Ry, E¯ = 0.9 at b¯ = 2. The dotted line is the case of ε¯ /E¯ = 0.1. The solid line is the case of ε¯ /E¯ = 0.2.
investigating the ion drag force in dusty plasmas since the ion drag force is related to the collision frequency determined by the collision cross section between the ion and the dust grain. Since the bremsstrahlung radiation cross section including the nonlinear correction effect due to collisions of ions with positively changed dust grains is found to be smaller than that neglecting the nonlinear effect, the nonlinear correction effect would be measured using a comparison of the continuous bremsstrahlung spectra due to collisions of ions with the conducting dust grains and with the nonconducting dust grains. These results provide useful information on the bremsstrahlung process due to ion–dust grain collisions in dusty plasmas.
Acknowledgements One of the authors (Y.-D. Jung) greatly acknowledges Profs. R.J. Gould and M. Rosenberg for useful discussions on bremsstrahlung processes in dusty plasmas and warm hospitality while visiting the University of California, San Diego. The authors would like to thank the anonymous referees for suggesting improvements to this text. This work was supported by the research fund of Hanyang University (HY-2004).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
G.R. Blumenthal, R.J. Gould, Rev. Mod. Phys. 42 (1970) 237. R.J. Gould, Astrophys. J. 238 (1980) 1026. R.J. Gould, Astrophys. J. 243 (1981) 677. H. Totsuji, Phys. Rev. A 32 (1985) 3005. V.P. Shevelko, Atoms and Their Spectroscopic Properties, Springer-Verlag, Berlin, 1997. Y.-D. Jung, Phys. Rev. E 55 (1997) 21. V. Shevelko, H. Tawara, Atomic Multielectron Processes, Springer-Verlag, Berlin, 1998. Y.-D. Jung, Phys. Lett. A 313 (2003) 296. H.F. Beyer, V.P. Shevelko, Introduction to the Physics of Highly Charged Ions, Institute of Physics, Bristol, 2003. A. Fridman, L.A. Kennedy, Plasma Physics and Engineering, Taylor & Francis, New York, 2004. P. Bliokh, V. Sinitsin, V. Yaroshenko, Dusty and Self-Gravitational Plasma in Space, Kluwer Academic, Dordrecht, 1995. P.K. Shukla, A.A. Mamum, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol, 2002. P.K. Shukla, Dust Plasma Interaction in Space, Nova Science Publishers, New York, 2002. J. Vranjes, M.Y. Tanaka, B.P. Pandey, M. Kono, Phys. Rev. E 66 (2002) 037401. R.J. Gould, Am. J. Phys. 38 (1970) 189. Y.-D. Jung, Phys. Plasmas 2 (1995) 332.
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M. Rosenberg, G. Kalman, Phys. Rev. E 56 (1997) 7166. G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, fifth ed., Academic Press, San Diego, 2001. J.D. Jackson, Classical Electrodynamics, third ed., Wiley, New York, 1999. V.E. Fortov, A.P. Nefedov, O.F. Petrov, A.A. Samarian, A.V. Chernyschev, Phys. Lett. A 219 (1996) 89.
Nonlinear effects on bremsstrahlung emission in dusty plasmas Young-Woo Kim, Young-Dae Jung ∗ Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea Received 23 September 2004; received in revised form 26 October 2004; accepted 27 October 2004 Available online 10 November 2004 Communicated by F. Porcelli
Abstract Nonlinear effects on the bremsstrahlung process due to ion–dust grain collisions are investigated in dusty plasmas. The nonlinear screened interaction potential is applied to obtain the Fourier coefficients of the force acting on the dust grain. The classical trajectory analysis is applied to obtain the differential bremsstrahlung radiation cross section as a function of the scaled impact parameter, projectile energy, photon energy, and Debye length. The result shows that the nonlinear effects suppress the bremsstrahlung radiation cross section due to collisions of ions with positively charged dust grains. These nonlinear effects decrease with increasing Debye length and temperature, and increase with increasing radiation photon energy. 2004 Elsevier B.V. All rights reserved. PACS: 52.20.-j Keywords: Nonlinear effects on dust bremsstrahlung emission in dusty plasmas
The bremsstrahlung process [1–10] in plasmas has received much attention since this process has been widely used in plasma diagnostics in laboratory and space plasmas. In recent years, there has been a considerable interest in the dynamics of plasmas containing dust grains including collective effects and strong electrostatic interaction between the charged components in dusty plasmas. Various physical processes in dusty plasmas have been investigated in order to obtain the information of plasma parameters in dusty plasmas [11–13]. The particle interaction potentials in dusty plasmas have been represented as the standard Debye–Hückel potential obtained by the linearization of the Poisson equation with the Boltzmann distribution function. However, in the case of highly charged dust grains the description of the interaction potential would not be described by the Yukawa type Debye–Hückel potential due to strong nonlinear electrostatic interactions. Hence, the ion–dust grain bremsstrahlung spectrum would be affected by these nonlinear correction effects. To the best of our knowledge, the nonlinear effects on the bremsstrahlung process due to collisions of ions with charged dust grains in dusty plasmas have not been investi* Corresponding author.
E-mail addresses: [email protected], [email protected] (Y.-D. Jung). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.10.065
Y.-W. Kim, Y.-D. Jung / Physics Letters A 333 (2004) 426–432
427
gated as yet. Thus, in this Letter we investigate the nonlinear correction effects on the bremsstrahlung process due to ion–dust grain collisions in dusty plasmas. The interaction potential between the projectile ion and the charged dust grain is introduced by the nonlinear screened potential [14]. The classical straight-line trajectory analysis [1,15,16] is applied to obtain the differential bremsstrahlung radiation cross section as a function of the scaled impact parameter, projectile energy, radiation photon energy, and Debye length. The classical expression of the differential bremsstrahlung cross section [15] dσb can be represented as dσb = 2π db b dwω (b), (1) where b is the impact parameter and dwω is the differential probability of emitting a photon of frequency ω within dω when a projectile particle changes its velocity in collisions with a static target system. For all impact parameters, dwω is given by the Larmor formula for the emission spectrum of a nonrelativistic accelerated charge: 8π d 2 d(t) 2 dω , dwω = (2) 3h¯ c3 dt 2 ω ω where dω is the Fourier coefficient of the dipole moment d of the collision system. For the ion–dust grain system, the term [d 2 d(t)/dt 2 ]ω can be presented as 2 ze Ze d d(t) − = (3) Fω , dt 2 ω M mi where M and mi are the dust mass and the ion mass, respectively, and Fω is the Fourier coefficient of the force F(t): 1 Fω = 2π
∞ dt eiωt F(t).
(4)
−∞
Since the dust grain is much massive than the projectile ion, we can assume to a good approximation that the center of mass of the ion–dust grain system is at the position of the dust grain. Using the straight-line trajectory analysis, the position of the projectile ion can be represented as r(t) = b + vt with b · v = 0, where v is the velocity of the projectile ion and then, |Fω |2 = |F⊥ω |2 + |Fω |2 where the Fourier coefficients F⊥ω and Fω are the components of force perpendicular and parallel to the projectile velocity, respectively. Using the standard theory, we can obtain the electrostatic potential φ(r) produced by the charged dust grain (Q) with the surrounding plasma as the Debye–Hückel form [17] by solving the Poisson equation with the help of the Boltzmann distribution function: Q exp(−r/λD ), (5) r where λD is the Debye length. Here, for the sake of simplicity, the dust grain is assumed to be a spherical shape with the radius a. For typical laboratory and space dusty plasmas, it has been known that λD a [11]. Recently, the modified screening interaction potential in dusty plasmas was given by Vranjes et al. [14] by using the nonlinear corrections in the expanded Boltzmann distribution. Using the modified potential, the interaction potential energy V (r) between the projectile ion (ze) and the charged dust grain (Ze) can be represented as φDH (r) =
V (r) =
Ze2 Zze2 Zze2 exp(−r/λD ) − exp(r/λD ) Ei(−3r/λD ) r 4λD T r Ze2 Zze2 + exp(−r/λD ) Ei(−r/λD ), 4λD T r
(6)
428
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∞ where T is the temperature and Ei(−r)[= − r dt e−t /t] is the exponential integral function [18]. After some algebra using the straight-line trajectory impact-parameter analysis and Eq. (4), we obtain the analytic formulas for the Fourier coefficients of the force as follows: Fµω =
Zze2 ¯ Fµω πva
(µ = ⊥, ).
(7)
¯ β, λ¯ D , η) and F¯ω (b, ¯ β, λ¯ D , η) are the Here, the scaled Fourier coefficients including the nonlinear effects F¯⊥ω (b, components of force perpendicular and parallel to the projectile velocity, respectively, and found to be as follows: ¯ β, λ¯ D , η) = F¯⊥ω (b,
∞
¯ τ, λ¯ D , η), dτ b¯ cos βτ G(b,
(8)
0
¯ β, λ¯ D , η) = i F¯ω (b,
∞
¯ τ, λ¯ D , η), dτ τ sin βτ G(b,
(9)
0
¯ b/a) is the scaled impact parameter, β ≡ ωa/v, τ (≡ vt/a) is the scaled time, λ¯ D (≡ λD /a) is the scaled where b(≡ 4 Debye length, η ≡ Z(Ry/2kB T )(a0 /a)λ¯ −1 = 13.6 eV is the Rydberg constant, m is the electron ¯ 2) ∼ D , Ry(= me /2h 2 2 ¯ τ, λ¯ D , η) is given rest mass, kB is the Boltzmann constant, a0 (= h¯ /me ) is the Bohr radius, and the function G(b, by 2 1/2 ¯ 2 1/2 ¯ ¯2 ¯2 ¯ τ, λ¯ D , η) = exp[−(b + τ ) /λD ] + exp[−(b + τ ) /λD ] G(b, (b¯ 2 + τ 2 )3/2 (b¯ 2 + τ 2 )λ¯ D exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 2 1/2 ¯ +η − Ei −3 b + τ / λ D (b¯ 2 + τ 2 )3/2 1/2 exp[(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 Ei −3 b + τ 2 /λ¯ D + 2 2 ¯ ¯ (b + τ )λD
+ +
1/2 exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 Ei − b + τ 2 /λ¯ D (b¯ 2 + τ 2 )3/2
exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] ¯ 2 2 1/2 ¯ Ei − b + τ / λ . D (b¯ 2 + τ 2 )λ¯ D
(10)
(b, ¯ β, λ¯ D ) and F¯ (b, ¯ β, λ¯ D ) becomes If we neglect the nonlinear effects, the scaled Fourier coefficients F¯⊥ω ω ¯ β, λ¯ D ) = (b, F¯⊥ω
∞
¯ τ, λ¯ D ), dτ b¯ cos βτ G (b,
(11)
0 ¯ F¯ω (b, β, λ¯ D ) = i
∞
¯ τ, λ¯ D ), dτ b¯ sin βτ G (b,
(12)
0
and the function
¯ τ, λ¯ D ) G (b,
¯ τ, λ¯ D ) = G (b,
is
exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] exp[−(b¯ 2 + τ 2 )1/2 /λ¯ D ] + . (b¯ 2 + τ 2 )3/2 (b¯ 2 + τ 2 )λ¯ D
(13)
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Then, the differential bremsstrahlung radiation cross section [19] is defined as dσb dχb ≡ h¯ ω, dε h¯ dω
16z2 α 3 a02 µ Zm zm 2 d b¯ b¯ |F¯⊥ω |2 + |F¯ω |2 , = − m M mi 3E¯
(14) (15)
¯ E/Z 2 Ry) is where ε(≡ h¯ ω) is the radiation photon energy, α(= e2 /h¯ c ∼ = 1/137) is the fine structure constant, E(≡ 2 the scaled collision energy, E(≡ µv /2) is the collision energy, and µ[= Mmi /(M + mi )] is the reduced mass of the collision system. In nonrelativistic limit, the parameter β can be expressed as β = Z(µ/m)1/2(a/a0 )(¯ε/2E¯ 1/2) where ε¯ (≡ h¯ ω/Z 2 Ry) is the scaled photon energy. Then, the scaled doubly differential bremsstrahlung radiation cross section including the nonlinear effects, i.e., the scaled differential bremsstrahlung radiation cross section per scaled impact parameter, can be presented as d 2 χb d ε¯ d b¯ πa02
=
z m 2¯ 16Z 2 z2 α 3 µ m − b 3π E¯ m M Z mi ¯ ε¯ , E, ¯ λ¯ D , η)2 + F¯ω (b, ¯ ε¯ , E, ¯ λ¯ D , η)2 , × F¯⊥ω (b,
(16)
where the scaled Fourier coefficients F¯⊥ω and F¯ω are already given in Eqs. (8) and (9), respectively. It is obvious that the bremsstrahlung cross section for the electron–dust grain collision is greater than that for the ion–dust grain collision due to the large mass difference between the electron and the ion. However, it has been know that the electron density is much smaller than the ion density for a dusty plasma with negatively charged dusty grains and the electron temperature is greater than the ion temperature [12]. Then, the low-energy ion–dust bremsstrahlung process is more frequent than the electron–dust bremsstrahlung process since the power density [9] of bremsstrahlung is proportional to the density function of the projectile. In addition, the energy loss of ions due to the ion–dust grain bremsstrahlung is quite important for determining the force acting on dust grains since the ion drag force is related to momentum exchange between positive ions and a dust grain. Thus, the nonlinear effects should be properly accounted for in analysis of the ion–dust bremsstrahlung process in dusty plasmas. The formation of regular structures of dust grains was observed in a laminar flow of a thermal plasma, in which case the grain charge was positive due to thermal electron emission [20]. Thus, in order to investigate the nonlinear correction effects on the bremsstrahlung radiation cross section due to collisions of ions and positively charged dust grains, we set a = 0.01 µm, Z = 500, z = 1 (proton projectile), and the density of the dust grain is ρ ∼ = 2 g cm−3 . Here, we choose E¯ = 0.9 since the bremsstrahlung radiation cross section, Ref. [18], is known to be valid for low collision ¯ energies [15] (E < Z 2 Ry). Figs. 1 and 2 show the ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neincluding the nonlinear effects (Eqs. (8) and (9)) to the bremsstarhlung radiation cross section (d 2 χb /d ε¯ d b) glecting the nonlinear effects (Eqs. (11) and (12)) as a function of the scaled Debye length (λ¯ D ) and the temperature (kB T ), respectively. Fig. 3 shows the three-dimensional plot of the ratio as a function of the scaled Debye length and the temperature. As we see in these figures, the nonlinear effects suppress the bremsstrahlung radiation cross sections due to collisions of ion with positively charged dust grains. It is found that the ratio approaches the unity with increasing Debye length and temperature. In other words, the nonlinear effects are decreased with increasing Debye length and temperature. It can be also found that the nonlinear effects are significant for small Debye lengths since exp(r/λD ) Ei(−3r/λD ) − exp(−r/λD ) Ei(−r/λD ) is exponentially decreasing with the distance. However, even for large Debye lengths, for example, λ¯ D = 150, the nonlinear effects are found be more than 10%. Fig. 4 shows the variation of the nonlinear correction effects on the bremsstrahlung radiation cross section with changing radiation photon energy. As we see in this figure, it is also found that the nonlinear effect increases with increasing radiation photon energy. The variation of the nonlinear effect is found to be more significant for small Debye lengths. Hence, it is important to note that the nonlinear correction plays an important role in bremsstrahlung processes in dusty plasmas. It can be also expected that the nonlinear screening effects play important roles in
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¯ including the nonlinear effects to the bremsstrahlung radiation Fig. 1. The ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neglecting the nonlinear effects as a function of the scaled Debye length (λ¯ D ) when kB T = 2Ry, E¯ = 0.9, and cross section (d 2 χb /d ε¯ d b) ε¯ = 0.18 at b¯ = 2.
¯ including the nonlinear effects to the bremsstrahlung radiation Fig. 2. The ratio of the bremsstrahlung radiation cross section (d 2 χb /d ε¯ d b) ¯ neglecting the nonlinear effects as a function of the temperature (kB T ) when λ¯ D = 50, E¯ = 0.9, and ε¯ = 0.18 at cross section (d 2 χb /d ε¯ d b) b¯ = 2.
Fig. 3. The three-dimensional plot of the ratio as a function of the scaled Debye length (λ¯ D ) and the temperature (kB T ) when E¯ = 0.9, and ε¯ = 0.18 at b¯ = 2.
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Fig. 4. The nonlinear correction effects on the bremsstrahlung radiation cross section as a function of the scaled Debye length (λ¯ D ) when kB T = 2Ry, E¯ = 0.9 at b¯ = 2. The dotted line is the case of ε¯ /E¯ = 0.1. The solid line is the case of ε¯ /E¯ = 0.2.
investigating the ion drag force in dusty plasmas since the ion drag force is related to the collision frequency determined by the collision cross section between the ion and the dust grain. Since the bremsstrahlung radiation cross section including the nonlinear correction effect due to collisions of ions with positively changed dust grains is found to be smaller than that neglecting the nonlinear effect, the nonlinear correction effect would be measured using a comparison of the continuous bremsstrahlung spectra due to collisions of ions with the conducting dust grains and with the nonconducting dust grains. These results provide useful information on the bremsstrahlung process due to ion–dust grain collisions in dusty plasmas.
Acknowledgements One of the authors (Y.-D. Jung) greatly acknowledges Profs. R.J. Gould and M. Rosenberg for useful discussions on bremsstrahlung processes in dusty plasmas and warm hospitality while visiting the University of California, San Diego. The authors would like to thank the anonymous referees for suggesting improvements to this text. This work was supported by the research fund of Hanyang University (HY-2004).
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