Effects of magnetic field and temperature on the nonrelativistic bremsstrahlung process in magnetized anisotropic plasmas

Physics Letters A 373 (2009) 1959–1961

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Physics Letters A www.elsevier.com/locate/pla

Effects of magnetic field and temperature on the nonrelativistic bremsstrahlung process in magnetized anisotropic plasmas Young-Dae Jung a,b,∗ , Daiji Kato a a b

National Institute for Fusion Science, Toki, Gifu, 509-5292, Japan Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea

a r t i c l e

i n f o

Article history: Received 20 December 2008 Accepted 3 April 2009 Available online 9 April 2009 Communicated by F. Porcelli PACS: 52.20.-j Keywords: Magnetized anisotropic plasmas Nonrelativistic bremsstrahlung

a b s t r a c t The magnetic field and thermal effects on the nonrelativistic electron–ion bremsstrahlung process are investigated in magnetized anisotropic plasmas. The effective electron–ion interaction potential is obtained in the presence of an external magnetic field. Using the Born approximation for the initial and final states of the projectile electron, the bremsstrahlung radiation cross section and bremsstrahlung emission rate are obtained as functions of the electron energy, radiation photon energy, magnetic field strength, plasma temperature, and Debye length. It is shown that the effects of the magnetic field enhance the bremsstrahlung radiation cross section for low plasma temperatures and, however, suppress the bremsstrahlung cross section for high plasma temperatures. It is also shown that the magnetic field effects diminish the bremsstrahlung emission rate in magnetized high temperature plasmas. © 2009 Elsevier B.V. All rights reserved.

The bremsstrahlung process [1–5] has been of a great interest since the continuum emission spectrum due to the electron–ion collisions in plasmas has provided useful information on various plasma parameters. In weakly coupled isotropic unmagnetized plasmas, the screened electron–ion interactions have been represented by the Debye–Hückel potential model [2]. However, it would be anticipated that the physical processes in anisotropic plasmas may differ from those in isotropic plasmas since the standard Debye–Hückel would not be reliable to elucidate the interaction potentials in anisotropic plasmas. Therefore, it is necessary to comprehend the field around a test charge in the presence of the external strong magnetic field since the increase of the magnetic field strength produces the local density fluctuations. Recently, the understanding the numerous physical properties of anisotropic plasmas due to the interaction with external magnetic fields has considerably increased. Moreover, the electromagnetic properties of magnetized anisotropic plasmas have been extensively investigated by the modified Debye–Hückel potential using the plasma dielectric response function [6–8]. Hence, it can be expected that the nonrelativistic electron–ion bremsstrahlung emissions in magnetized anisotropic plasmas would be considerably different from those in standard Debye–Hückel plasmas. Thus, in this Letter we investigate the magnetic field and thermal effects on

the nonrelativistic electron–ion bremsstrahlung process in magnetized anisotropic plasmas. The effective screened interaction model [8] using the plasma dielectric function is employed to describe the electron–ion interaction potentials in magnetized anisotropic plasmas. The nonrelativistic Born approximation [9] is utilized to both the initial and final states of the projectile electron in order to obtain the electron–ion bremsstrahlung cross section including the magnetic field and thermal effects. In addition, the magnetic field effects on the bremsstrahlung emissivity are also investigated in magnetized anisotropic plasmas. Using the second-order nonrelativistic perturbation analysis [9], the differential electron–ion bremsstrahlung cross section dσb would be written as dσb = dσC dW ω , where dσC is the differential elastic scattering cross section: d σC =

1 2π

*

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.001

h¯ v 20

   V˜ (q)2 q dq,

(2)

h¯ is the rationalized Planck constant, v 0 is the initial velocity of the projectile electron, the Fourier transformation V˜ (q) of the electron–ion interaction potential V (r) is given by V˜ (q) =

Corresponding author at: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, South Korea. Tel.: +82 31 400 5477; fax: +82 31 406 1777. E-mail address: [email protected] (Y.-D. Jung).

(1)



d3 r e −iq·r V (r),

(3)

q(= k0 − k f ) is the momentum transfer, and k0 and k f are the wave vectors of the initial and final states of the projectile electron, respectively. Here, dW ω represents the differential probability

1960

Y.-D. Jung, D. Kato / Physics Letters A 373 (2009) 1959–1961

ω and ω + dω in the

of emitting a photon of frequency between differential solid angle dΩ : dW ω =

α 2 dω Λ |ˆe · q|2 dΩ, ω 4π 2

(4)

eˆ

where α (= e 2 /¯hc ∼ = 1/137) is the fine structure constant, c is the velocity of the light, Λ(= h¯ /mc ) is the Compton wave number, m is the mass of the electron, and eˆ is the unit photon polarization vector. By integrating over the directions of the photon radiation, we may write the bremsstrahlung cross section in the following manner: dσb =

c2

   V˜ (q)2 q3 dq dω ,

α

3π 2 v 20 (mc 2 )2

(5)

ω

since the summation over polarizations provides the angular distribution factor sin2 θ , where θ is the angle between k0 and q. Recently, a useful analytic form of the effective screened potential of the charged particle in a magnetized anisotropic plasma has been obtained by the plasma dielectric response function [8]. Using the effective screened interactions [8], the interaction potential between the projectile electron and the target ion with nuclear charge Z e in the magnetized anisotropic plasma for r > λ D > a Z can be represented by

 Z e 2 λ2D  Z e 2 −r /λ D V (r) = − −β 1 − 3 cos2 χ , e r r3

(6)

where λ D is the Debye length, β ≡ (αa0 /2λ D )2 [η−1 − (sinh η × cosh η)−1 ], a0 (= h¯ 2 /me 2 ) is the Bohr radius of the hydrogen atom, η ≡ h¯ ωL /2k B T , ωL (= e B /mc ) is the Larmor frequency, k B is the Boltzmann constant, T is the plasma temperature, and χ is the angle between the radius vector r and the external magnetic field B. This effective screened interaction potential has an additional anisotropic term apart from the standard Debye–Hückel potential (− Z e 2 /r )e −r /λ D . Using the cylindrical coordinates, i.e., r = ρ ρˆ + zzˆ with ρˆ · zˆ = 0, the Fourier transformation of the electron–ion interaction potential is then represented by V˜ (q) = 4π λ2D

∞ 0



∞

d z¯ cos(qλ D z¯ ) −

dρ¯ ρ¯ 1

− β Z e2

1

(ρ¯ 2 + z¯ 2 )3/2 

= −4π λ2D Z e 2

+ 3β Z e 2

1

1 + (qλ D )2

Z e2

(ρ + z¯ 2 )1/2 ¯2

z¯ 2

2 2 1/ 2 e −(ρ¯ +¯z )



16 α 3 a20 3

E¯

¯ 4D Z 2λ



(ρ¯ 2 + z¯ 2 )5/2

 − β(qλ D ) K 1 (qλ D ) ,

1

¯ D )2 1 + (¯qλ

dχb dε¯

=

16 α 3 a20 E¯

3

¯ 4D Z 2λ

¯ max q



dq¯ q¯ 3

2 − β(¯qλ¯ D ) K 1 (¯qλ¯ D ) ,

1

¯ D )2 1 + (¯qλ

q¯ min

(9)



where q¯ min [≡ (k0 − k f )a0 ] = E¯ /2 − ( E¯ − ε¯ )/2 is the scaled minimum momentum transfer and q¯ max [≡ (k0 + k f )a0 ] = E¯ /2 + ( E¯ − ε¯ )/2 is the scaled maximum momentum transfer. If, however, we neglect the additional magnetic field part in the interaction potential, i.e., using the standard Debye–Hückel potential, the bremsstrahlung radiation cross section dχb /dε¯ is then found to be dχb dε¯

=



8 α 3 a20 E¯

3



+ ln

Z2

1

¯ D )2 1 + (¯qmax λ

−

 . ¯ D )2 1 + (¯qmin λ

1

¯ D )2 1 + (¯qmin λ

¯ D )2 1 + (¯qmax λ

(10)

In addition, the bremsstrahlung emission rate P ε¯ , i.e., the spectral photon radiation rate, is given by P ε¯ =



dE rad

dne ni ε¯

=

dV dt dε¯

dσb dε¯

(11)

,

where dE rad is the radiation energy, dV the volume element, dt is the time interval, dne is the differential electron density, and ni is the total ion density. For Maxwellian plasmas, the energy distribution dne ( E¯ ) of plasma electrons can be represented by dne ( E¯ ) = 2π ne



1

3/2

π T¯

¯ ¯

E¯ 1/2 e − E / T d E¯ ,

(12)



where ne (= dne ) is the total electron density and T¯ (≡ k B T /Ry) is the scaled thermal energy. Hence, the bremsstrahlung emission rate P ε¯ for the electron–ion interaction in magnetized anisotropic plasmas is found to be 32 ¯ 4D ni ne T¯ −3/2 P ε¯ ( T¯ ) = √ Z 2 α 4 a20 c λ 3 π

(7)

where ρ¯ ≡ ρ /λ D , z¯ ≡ z/λ D , and K 1 (qλ D ) [10] is the first-order modified Bessel function. It should be noted that the lower-cutoff in the integration z¯ in Eq. (7) has been introduced as z¯ = 1 since the additional term due to the magnetic field in the effective potential is known to be reliable for the region of r > λ D . Then, the bremsstrahlung cross section can be obtained by the following form: dσb =

as dχb /dε¯ dq¯ ≡ (dσb /¯h dω dq¯ )¯hω , where ε¯ (≡ ε /Ry) is the scaled radiation photon energy, and ε (= h¯ ω) is the photon energy. The bremsstrahlung radiation cross section dχb /dε¯ due to the electron–ion interaction in magnetized anisotropic plasmas is then obtained by

2 dω − β(¯qλ¯ D ) K 1 (¯qλ¯ D ) q¯ 3 dq¯ ,

ω

(8)

where E¯ (≡ mv 20 /2 Ry) is the scaled energy of the projectile

electron and Ry(= me 4 /2h¯ 2 ≈ 13.6 eV) is the Rydberg constant, λ¯ D (≡ λ D /a0 ) is the scaled Debye length, and q¯ (≡ qa0 ) is the scaled momentum transfer. In the bremsstrahlung process, it has been shown that the emission spectrum would be investigated through the bremsstrahlung radiation cross section [11] defined

√

∞

¯ ¯ d E¯ e − E / T

× ε¯

−

α ¯D 2λ

√

√



E¯ /2+ ( E¯ −¯ε )/2

√

E¯ /2−

2  ¯ −1 B

T¯

dq¯ q¯

3

( E¯ −¯ε )/2

− sinh

¯ B

T¯

1

¯ D )2 1 + (¯qλ

cosh

¯

−1  B

T¯

2 × (¯qλ¯ D ) K 1 (¯qλ¯ D )

,

(13)

where B¯ ≡ B h¯ 3 /m2 ce 3 . In order to investigate the magnetic field and thermal effects on the nonrelativistic electron–ion bremsstrahlung process in magnetized anisotropic plasmas, we set Z = 1 and consider high-energy projectiles. Fig. 1 represents the scaled bremsstrahlung radiation cross section ∂ε¯ χ¯ b [≡ (dχb /dε¯ )/π a20 ] in units of π a20 as a function of the scaled photon energy ε¯ for low and high plasma temperatures. It is shown that the temperature effects enhance the bremsstrahlung radiation cross section for the weak magnetic field case. Fig. 2 shows the scaled bremsstrahlung radiation cross sec-

Y.-D. Jung, D. Kato / Physics Letters A 373 (2009) 1959–1961

Fig. 1. The scaled bremsstrahlung radiation cross section ∂ε¯ χ¯ b as a function of the ¯ D = 0.1, and B¯ = 1. The solid line represents scaled photon energy ε¯ when E¯ = 10, λ the case of T¯ = 1. The dotted line represents the case of T¯ = 10.

Fig. 2. The scaled bremsstrahlung radiation cross section ∂ε¯ χ¯ b as a function of the ¯ D = 0.1, and T¯ = 10. The solid line represcaled photon energy ε¯ when E¯ = 10, λ sents the case of B¯ = 1. The dotted line represents the case of B¯ = 10.

1961

Fig. 4. The scaled bremsstrahlung emission rate P¯ ε¯ as a function of the scaled pho¯ D = 0.1 and T¯ = 10. The solid line represents the case of B¯ = 1. ton energy ε¯ when λ The dashed line represents the case of B¯ = 5. The dotted line represents the case of B¯ = 10.

surface plot of the bremsstrahlung radiation cross section as a function of the scaled magnetic field B¯ and scaled temperature T¯ . From this figure, it is also found that the effects of the magnetic field enhance the bremsstrahlung radiation cross section for low plasma temperatures and, however, suppress the bremsstrahlung cross section for high plasma temperatures. Fig. 4 represents the scaled bremsstrahlung emission rate P¯ ε¯ (≡ P ε¯ /a20 cni ne ) for high plasma temperatures as a function of the scaled photon energy ε¯ for various strengths of the magnetic field. It is also shown that the magnetic field effects reduce the bremsstrahlung emission rate in magnetized high temperature plasmas. These results provide useful information on the electron–ion bremsstrahlung spectra in magnetized thermal plasmas. Acknowledgements One of the authors (Y.-D.J.) gratefully acknowledges the Director–General Professor O. Motojima, Director Professor M. Sato, Director Professor Y. Hirooka, and Professor I. Murakami for warm hospitality while visiting the National Institute for Fusion Science (NIFS) in Japan as a long-term visiting professor. The authors are also grateful to NIFS for supporting the research. This work was done while Y.-D.J. visited NIFS. References

Fig. 3. The surface plot of the scaled bremsstrahlung radiation cross section ∂ε¯ χ¯ b as a function of the scaled magnetic field B¯ and scaled temperature T¯ when E¯ = 10, ε¯ = 1, and λ¯ D = 0.1.

tion ∂ε¯ χ¯ b as a function of the scaled photon energy ε¯ for weak and strong magnetic field cases. It is also shown that the magnetic field effects suppress the bremsstrahlung radiation cross section in high temperature plasmas. In addition, Fig. 3 shows the

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