to the Polarization Interaction in Nonideal Plasmas

to the Polarization Interaction in Nonideal Plasmas

Hwa-Min Kim^aand Young-Dae Jung^b

aDepartment of Electronics Engineering, Catholic University of Daegu, Hayang, Gyongsan, Gyungbuk 712-702, South Korea

bDepartment of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea Reprint requests to Y.-D. J.; E-mail: ydjung@hanyang.ac.kr

Z. Naturforsch.64a,49 – 53 (2009); received February 22, 2008 / revised June 30, 2008

The collective effects on the transition bremsstrahlung spectrum due to the polarization interaction between the electron and Debye shielding cloud of an ion are investigated in nonideal plasmas. The impact parameter analysis with the effective pseudopotential model taking into account the nonideal collective and plasma screening effects is applied to obtain the bremsstrahlung radiation cross-section as a function of the nonideality plasma parameter, Debye length, photon energy, and projectile energy.

It is shown that the collective effects enhance the bremsstrahlung radiation cross-section and decrease with increasing impact parameter. It is also shown that the collective effect is the most significant near the maximum position of the bremsstrahlung cross-section. In addition, it is shown that the collective effect decreases with an increase of the radiation photon energy.

Key words:Transition Bremsstrahlung; Nonideal Plasmas.

The bremsstrahlung process [1 – 3] has been of great interest since this process is one of the most funda- mental processes in many areas of physics such as astrophysics, atomic physics, and plasma physics. In addition, the inverse process to the bremsstrahlung has been also of interest, especially in astrophysi- cal plasmas [1]. The transition bremsstrahlung or so- called polarization bremsstrahlung [4 – 6] in plasmas, due to the interaction between the Debye shielding cloud and plasma particle, has been extensively inves- tigated in weakly and strongly coupled plasmas since this process has provided useful information on var- ious plasma parameters. For low-energy electron en- counters, the transition bremsstrahlung is expected to be significant due to the long collision time between the projectile electron and induced polarization charge in the Debye shielding cloud. The plasma described by the Debye-H¨uckel model has been classified as the ideal plasma since the average interaction energy be- tween charged particles is quite smaller than the aver- age kinetic energy of a particle [7]. Moreover, the mul- tiparticle correlation effects caused by simultaneous interactions of charged particles should be taken into account with increasing the plasma density. In plasmas with moderate densities and temperatures, the poten- tial energy would not be well characterized by the con- ventional Debye-H¨uckel model due to the strong col-

0932–0784 / 09 / 0100–0049 $ 06.00 c2009 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

lective effects of nonideal particle interactions [8 – 11].

Hence, it is expected that the transition bremsstrahlung processes in nonideal plasmas would be quite differ- ent from those in ideal plasmas. Thus, we now will investigate the nonideal collective effects on the tran- sition bremsstrahlung due to the polarization interac- tion between the projectile electron and Debye shield- ing cloud of an ion in a nonideal plasma using the ef- fective pseudopotential model and taking into account the collective and plasma screening effects since the- oretical atomic spectroscopy is crucial to the study of various plasma parameters. The impact parameter trajectory analysis is applied in order to obtain the bremsstrahlung radiation cross-section as a function of the nonideality plasma parameter, Debye length, pho- ton energy, and projectile energy.

In the low-energy bremsstrahlung process, the bremsstrahlung cross-section [12] dσbcan be written in the form

dσb=2π ^{db bdw}ω(b), (1) where b is the impact parameter and dw_ω(b) is the differential photon emission probability with fre- quencyω. For all impact parameters, the probability dw_ω(b)would be obtained by the Larmor formula [13]

for the instantaneous power emitted during the en-

counter of the projectile electron with the target sys- tem:

dw_ω(b) = 8π^e2

3 ¯hm²c³|F_ω(b)|²dω

ω ^{, (2)}

wheremis the electron mass andF_ω(b)is the Fourier coefficient of the forceF(t) acting on the projectile electron:

F_ω(b) = 1 2π

∞

−∞

dte^iωtF(t). (3)

Here, the absolute square|F_ω(b)|²can be decomposed into the Fourier coefficients perpendicular,F_⊥ω, and parallel,F_ω, to the direction of the projectile velocity v₀:

|F_ω(b)|²=|F_⊥ω|²+|F_ω|²=

|(v₀×F_ω)×v₀/v²₀|²+|v₀(v₀·F_ω)/v²₀|². (4) The integro-differential equation [8] for the effective potential of charged particle interactions taking into ac- count the simultaneous correlations of many particles has been obtained on the basis of a sequential solution of the Bogolyubov chain equations for the equilibrium distribution function of particles in nonideal plasmas.

In addition, the remarkably useful analytical form of the pseudopotential of the particle interaction in non- ideal plasmas has been also obtained by the application of the spline-approximation [8]. Using the pseudopo- tential model [8] and taking into account the collective and plasma screening effects, the potentialφ(r)due to

the test chargeq_Tin nonideal plasmas is given by φ(r) =q_T

r e^−r/Λ1+γ^f(r)/2

1+c(γ) , (5)

where r is the distance from the test charge, Λ ^is the Debye length, γ ⁽≡e²/Λ^kBT) is the nonideality plasma parameter,k_B is the Boltzmann constant,T is the electron temperature, f(r) = (e^−√γr/Λ−1)(1− e^−2r/Λ)/5, andc(γ)∼=0.456γ−0.108γ²+0.009γ^3is the correlation coefficient. In this pseudopotential model, the ranges of the electron number density and temperature are, respectively, known to be less than 10¹⁹cm⁻³and 10⁴– 10⁵K, and the nonideal coupling plasma parameter becomesγ≤1 [8]. Here, we assume that the plasma is fully ionized so that it is composed of electrons and ions. Thus, the interaction between the charged particles with neutral atoms is ignored in the present work. The effective potentials for the charge- atom interactions can be found in a recent excellent investigation [14]. In ideal or weakly nonideal plas- mas, i. e., in the case ofγ→0, the pseudopotential (5) goes over into the conventional Debye-H¨uckel poten- tial φ(r)→(q_r/r)e^−r/Λ. Since the electron number densityn_e(r)inside the Debye shielding cloud which contains the ion with nuclear chargeZe and plasma electrons in nonideal plasmas can be represented by

ne(r) = Z 4πΛ²

e^−r/Λ r

1+γ^f(r)/2

1+c(γ) , (6) we obtain the polarization forceF_pol(r)acting on the projectile electron due to the polarized Debye shield- ing cloud in the following form:

F_pol(r) =−



−e² r²

r≤r

d³rrn_e(r)



=−2Ze² Λ²

1 1+c(γ)

r r⁴

2Λ³−

r³

2 +r²Λ+2rΛ²+2Λ^{3 e−r/Λ}

+γ 10

2Λ³ √γ+1₃−

r³

2 +_√r²Λ

γ+1+ 2rΛ²

√γ+1₂+ 2Λ³ √γ+1₃

e⁻(^√γ+1)^r/Λ

−

2Λ³ √

γ+33−

r³

2 +_√r²Λ

γ+3+ 2rΛ² √

γ+32+ 2Λ³ √

γ+33

e(^√γ+3)r/Λ

−

2Λ³− r³

2 +r²Λ+2rΛ²+2Λ^{3 e−r/Λ}

+ 2Λ³

27 − r³

2 +r²Λ

3 +2rΛ² 9 +2Λ³

27 e^−3r/Λ

,

(7)

where r=b+v0t. After a considerable amount of some algebra using the impact parameter analysis, we, respectively, obtain the scaled perpendicular, ¯F_⊥ω, and parallel, ¯F_ω, Fourier coefficients of the polarization force as

F¯_⊥ω≡ −π^aZv₀ 2Ze²

b·F_ω b

= ∞ 0

dτ^b¯cos(ξ τ)G(r¯,Λ¯,γ), (8)

F¯_ω≡ −π^aZv₀ 2Ze²

v₀·F_ω v₀

=i ∞ 0

dττ^sin(ξ τ)G(r¯,Λ¯,γ), (9) whereτ⁽≡v₀t/a_Z) is the scaled time,a_Z (=a₀/Z) is the first Bohr radius of the hydrogenic ion with nu- clear chargeZe,a₀ (=h¯²/me²) is the Bohr radius of the hydrogen atom, ¯b(≡b/a_Z) is the scaled impact pa- rameter,ξ ≡ω^aZ/v₀, ¯r [(≡r/a_Z) = (b¯²+τ²)^1/2] is the scaled distance, ¯Λ ⁽≡Λ/a_Z) is the scaled Debye length, and the integrand functionG(r¯,Λ¯,γ)is given by

G(r¯,Λ¯,γ) = 1 1+c(γ)

1 Λ¯^2r¯4

2 ¯Λ³−

r¯³ 2 +r¯²Λ¯ +2 ¯rΛ¯²+2 ¯Λ^{3 e−¯r/Λ¯}

+ γ

10

∑

∑

q=0(−1)^p+q+1

·

2 ¯Λ³

p√γ+2q+1₃− r¯³

2 + r¯²Λ¯ p√

γ+2q+1 + 2 ¯rΛ¯²

p√

γ+2q+12+ 2 ¯Λ³ p√

γ+2q+13

·e⁻(^p√γ+2q+1)^r/¯Λ^¯ .

(10)

On substituting (2), (8), and (9) into (1), we obtain the following form of the transition bremsstrahlung cross- section:

dσb=2⁶ 3

α^3a20

E¯ dω

ω

d ¯bb¯

|F¯_⊥ω|²+|F¯_ω|² , (11) whereα ⁽=e²/hc¯ ≈1/137) is the fine structure con- stant, ¯E ≡E/Z²Ry, E (=mv²₀/2) is the kinetic en- ergy of the projectile electron, andRy(=me⁴/2 ¯h²≈ 13.6 eV) is the Rydberg constant. It has been known that the classical trajectory method is quite useful to in- vestigate the low-energy bremsstrahlung process [15].

Hence, the expression of the bremsstrahlung cross- section [equation (11)] would be reliable to investi- gate the low-energy bremsstrahlung process due to the polarization interactions in nonideal plasmas. Very recently, an excellent discussion on the Gaunt fac- tor and comparisons for the bremsstrahlung calcula- tions by using various methods such as Born, Kramers, and Sommerfeld were given by Fortmann, Redmer, Reinholz, R¨opke, Wierling, and Rozmus [16]. It has been shown that the bremsstrahlung emission spec- trum would be investigated through the bremsstrahlung radiation cross-section [13] defined as dχb/d ¯ε ≡ (dσb/hd¯ ω)h¯ω. Hence, the bremsstrahlung radiation cross-section would be extended to the soft-photon limit due to the cancellation of the factor dω/ω^{in the} photon emission probability (2). After some algebra, the scaled differential transition bremsstrahlung radia- tion cross-section in units ofπ^a20due to the polariza- tion interaction between the electron and Debye shield- ing cloud of an ion in nonideal plasmas is found to be

d²χb

d ¯ε^db/π^a20= 2⁶

3πα³ E¯

b¯



 ∞ 0

dτ^b¯cos ετ¯

2√ E¯

G(b¯,τ,Λ¯,γ)

2

+ ∞ 0

dτ τ^sin ετ¯

2√

E¯ G(b¯,τ,Λ¯,γ)

2

, (12)

where the characteristic bremsstrahlung parameterξ^is represented by ξ(ε¯,E¯) = ε¯/2√

E, ¯¯ ε ≡ ε/Z²Ry, ε⁽=h¯ω) is the radiation photon energy, and the inte- grand functionG(b¯,τ,Λ¯,γ)is written as

G(b¯,τ,Λ¯,γ) = 1 1+c(γ)

1 Λ¯²(b¯²+τ²)²

2 ¯Λ³

−

(b¯²+τ²)^3/2

2 + (b¯²+τ²)Λ¯+2(b¯²+τ²)^1/2Λ¯² +2 ¯Λ^{3 e−(b¯2+r2)1/2/Λ¯}

+ γ

10

∑

∑

q=0(−1)^p+q+1

·

2 ¯Λ³

p√γ+2q+13−

(b¯²+τ²)^3/2

2 + (b¯²+τ²)Λ¯ p√γ+2q+1 +2(b¯²+τ²)^1/2Λ¯²

p√γ+2q+12+ 2 ¯Λ³ p√γ+2q+13

·e⁻(^p√γ+2q+1)^(b¯2+τ²)^1/2/Λ¯

. (13)

Fig. 1. The surface plot of the scaled differential transition bremsstrahlung radiation cross-section (∂ε,¯b¯χ¯b) as a function of the scaled impact parameter (¯b) and nonideality plasma parameter (γ) for ¯ε=0.4, ¯E=0.8, and ¯Λ=50.

Fig. 2. The soft-photon transition bremsstrahlung radiation cross-section∂ε,¯b¯χ¯b as a function of the scaled impact pa- rameter (¯b) for ¯ε=0.08, ¯E=0.8, and ¯Λ=50. The solid line is the case ofγ=1. The dashed line is the case ofγ=0.5.

The dotted line is the case ofγ=0.2.

In order to explicitly investigate the nonideal collec- tive effects on the transition bremsstrahlung process in nonideal plasmas, we consider ¯E<1 since the polar- ization effects are expected to be significant for low- energy projectiles and the classical expression of the bremsstrahlung cross-section is known to be also reli- able forv₀<Zα^c[12]. Figure 1 represents the surface plot of the scaled differential transition bremsstrahlung radiation cross-section∂ε,¯b¯χ¯b[≡(d²χb/d ¯ε^{d ¯b})/π^a2₀^{] as} a function of the scaled impact parameter ¯band non- ideality plasma parameter γ. As shown, it is found that the collective effects significantly enhance the bremsstrahlung radiation cross-section, especially for small impact parameter domains. Figure 2 shows the soft-photon transition bremsstrahlung radiation cross- section∂ε,¯¯bχ¯bas a function of the scaled impact pa-

Fig. 3. The hard-photon transition bremsstrahlung radiation cross-section∂ε,¯b¯χ¯b as a function of the scaled impact pa- rameter (¯b) for ¯ε=0.72, ¯E=0.8, and ¯Λ=50. The solid line is the case ofγ=1. The dashed line is the case ofγ=0.5.

The dotted line is the case ofγ=0.2.

Fig. 4. The transition bremsstrahlung radiation cross-section (∂ε,¯b¯χ¯b) as a function of the scaled radiation photon en- ergy ( ¯ε) for ¯b=10, ¯E=0.8, and ¯Λ =50. The solid line is the case ofγ=1. The dashed line is the case ofγ=0.5.

The dotted line is the case ofγ=0.2.

rameter ¯bfor various values of the nonideality plasma parameter γ. It is shown that the domain of the bremsstrahlung emission is wide in the case of soft- photon emissions, and the collective effect is also found to be the most significant near the maximum position of the bremsstrahlung cross-section. Figure 3 shows the hard-photon transition bremsstrahlung radi- ation cross-section∂ε,¯b¯χ¯bas a function of the scaled impact parameter ¯bfor various values of the nonideal- ity plasma parameterγ. From this figure, it is shown that the domain of the bremsstrahlung emission is sig- nificantly reduced with an increase of the radiation photon energy. It is also found that the bremsstrahlung radiation cross-section decreases with increasing ra- diation photon energy. In addition, Fig. 4 presents the transition bremsstrahlung radiation cross-section

∂ε,¯b¯χ¯bas a function of the scaled radiation photon en-

ergy ¯εfor various values of the nonideality plasma pa- rameterγ. As it is seen, the collective effects decrease with an increase of the radiation photon energy.

Thus, we understand that the bremsstrahlung spec- tra in soft-photon regions can be used to explore the nonideal collective phenomena in nonideal plasmas.

From these results, we have shown that the collec- tive effects play very important roles in the transition bremsstrahlung process due to the polarization interac- tion between the electron and Debye shielding cloud in

nonideal plasmas. These results would provide useful information on the continuum emission spectrum due to the transition bremsstrahlung process in plasmas.

Acknowledgements

The authors are grateful for useful discussions with Professor W. Hong. This work was supported by the Catholic University of Daegu. The authors would like to thank the anonymous referees for suggesting im- provements to this text.

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