The Astrophysical Journal, 707:539–542, 2009 December 10 C 2009.
doi:10.1088/0004-637X/707/1/539
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
TURBULENCE EFFECTS ON THE CHARGE CAPTURE PROCESS IN WEAK TURBULENT PLASMAS Sang-Chul Na1 and Young-Dae Jung1,2,3 1
2
Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426-791, Republic of Korea University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA;
[email protected] Received 2009 August 20; accepted 2009 October 28; published 2009 November 23
ABSTRACT The turbulence effects on the charge capture process are investigated in weak turbulent plasmas. The effective interaction potential taking into account the correction factor to the nonlinear dielectric function due to the fluctuation of the electric fields and Bohr–Lindhard model are employed in order to obtain the electron capture radius and electron capture cross section in turbulent plasmas. It is shown that the influence of the fluctuating electric fields in the plasma considerably decreases the electron charge capture radius and electron capture probability. Hence, we have found that the turbulence effect strongly suppresses the electron capture cross section in weak turbulent plasmas. In addition, it is found that the electron capture radius and electron cross section decrease with an increase of the projectile energy. Key words: atomic data – atomic processes – plasmas
of the random fluctuation of the electric fields in weak turbulent plasmas. The effective screened potential model (Shukla & Spatschek 1973) including the far-field term obtained from the longitudinal nonlinear plasma dielectric function is applied to describe the screened interaction in weak turbulent plasmas. The Bohr–Lindhard formalism is applied to obtain the electron capture radius and electron capture probability as functions of the impact parameter, diffusion coefficient, projectile energy, and Debye length of the plasma. In Section 2, we obtain the analytic form of the electron capture radius in turbulent plasmas. In Section 3, we discuss the electron capture probability and electron capture cross section in turbulent plasmas. We also discuss the turbulence effects on the electron capture cross section. Finally, the conclusion is given in Section 4.
1. INTRODUCTION The charge capture and exchange processes (Bransden & McDowell 1992; Ben-Itzhak et al. 1993; Kim & Jung 1998; Shokri et al. 2000; Beyer & Shevelko 2003) have received much attention since these processes in plasmas have provided useful information on various plasma parameters in astrophysical and laboratory plasmas. Among various classical, semiclassical and quantal treatments for investigating the charge capture process, the classical Bohr–Lindhard model (Bransden & McDowell 1992) has been known to be very believable to examine the energy dependence of the charge capture cross section when the relative collision velocity of the projectile ion is greater than the orbital velocity of the electron in the target ion, i.e., the intermediate-energy domain. In weakly coupled plasmas, the charge capture process has been characterized by the standard Debye shielding model (Jung & Yang 1997) of the interaction. In addition, it has been shown that the additional far-field interaction potential in collisionless plasmas falls off as the inverse cube of the distance between the projectile and target (Yu et al. 1972; Shukla et al. 2006). Moreover, it has been shown that the reaction of the random electric field fluctuations plays an important role in the binary encounters and interaction potentials in turbulent plasmas (Tsytovich 1972; Shukla & Spatschek 1973; Sitenko & Malnev 1995). Hence, it would be expected that the projectile would be scattered due to the random fluctuating electric fields in turbulent plasmas. The effective interaction potential including the additional term due to the fluctuating electric fields has been applied to investigate the elastic collision process in turbulent plasmas (Jung & Kato 2009). Therefore, it would be expected that the charge capture and exchange processes in turbulent plasmas are considerably different from those in non-turbulent plasmas due to the influence of the random fluctuation of the electric fields. Since turbulence is a widespread phenomenon in astrophysical plasmas, the investigation of the influence of the fluctuating electric field on the various atomic processes is essential for understanding the physical properties of turbulent astrophysical plasmas. Thus, in this paper, we investigate the effects of the turbulence on the charge capture process due to the appearance
2. ELECTRON CAPTURE RADIUS IN TURBULENT PLASMAS From the Bohr–Lindhard analysis (Bransden & McDowell 1992), the electron capture cross section would be written as σC (vp ) = 2π db b PC (vp , b), (1) where vp is the projectile velocity, b the impact parameter, and PC (vp , b) is the electron capture probability. In the standard Bohr–Lindhard scheme, the electron capture probability PC can be obtained by the ratio of the electron capture time tC to the electron orbital time τ in the target ion: 1 PC (vp , b) = τ
tC
dt.
(2)
− tC
In Equation (2), t = 0 is chosen as the incident at which the projectile ion with the nuclear charge Zp e makes its closest approach to the target ion with nuclear charge ZT e and the electron orbital time τ is given by τ = aZT /vZT , where aZT (= a0 /ZT ) is the Bohr radius of the hydrogenic ion with ¯ 2 /me2 ) is the Bohr radius of the nuclear charge ZT e, a0 (= h hydrogen atom, h ¯ is the rationalized Planck constant, m the electron mass, vZT (= ZT αf c) is the electron orbital velocity, αf (= e2 /¯hc ∼ = 1/137) the fine structure constant, and c the velocity of light.
3 Permanent address: Department of Applied Physics, Hanyang University, Ansan, Kyunggi-Do 426–791, Republic of Korea.
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The extremely useful analytic form of the effective screened potential (Shukla & Spatschek 1973) of a moving test charge in weak turbulent plasmas has been obtained by the longitudinal nonlinear plasma dielectric function including the additional exponential factor exp(−k 2 Dt 3 /3) (Birmingham & Bornatici 1972) due to the influence of the plasma turbulence, where k is the wave number and D is the diffusion coefficient. Using the effective interaction model (Shukla & Spatschek 1973), the screened interaction potential U (r) between the projectile ion with nuclear charge ZP e and electron including the far-field term due to the influence of the random electric-field fluctuations in weak turbulent plasmas in the limit r > λD with the condition v0 < vT is then represented by √ 2 v0 2 2 λD ZP e2 U (r) = − exp(−r/λD ) + √ cos γ r r vT π 4√ D × 1− π 3r , (3) 9 vT where r[= (b2 + z2 )1/2 ] is the distance between the projectile ion and electron, z ≡ v0 t, λD is the Debye length, v0 is the collision velocity, vT (= ωp λD ) is the thermal velocity, ωp is the plasma frequency, and γ is the angle between r and v0 . In the Bohr–Lindhard model, it is known that the electron capture process happens only when the distance between the projectile ion and released electron is smaller than charge capture radius. Hence, the charge capture radius would be obtained by equating the interaction potential between the projectile ion and electron with the kinetic energy of the released electron in the frame of the projectile ion. Therefore, the electron capture radius RC can be obtained by the following relation: √ ZP e2 ZP e2 2 2 λD 2 v0 exp(−RC /λD ) − √ cos γ RC RC RC vT π 4√ D 1 × 1− π 3 RC ∼ (4) = mv02 . 9 2 vT Since the Yukawa term exp(−r/λD )/(r/λD ) would be approximated as 0.0108/(r/λD ) − 0.1806/(r/λD )2 + 0.677/ (r/λD )3 + 0.6457/(r/λD )4 in the range of 3 r/λD 10, the useful analytic expression of the scaled electron capture radius R¯ C (≡ RC /λD ) in units of the Debye length λD is found to be 1 R¯ C (E¯ P , λ¯ D ) = A + (B + C)1/2 2 1/2 1 F D0 − C + + , (5) 2 4G(B + C)1/2 where the all parameters are, respectively, defined as follows: A ≡ 0.0054/G, G ≡ Z E¯ P λ¯ D , B ≡ G−2 (B1 + B2 ), B1 ≡ 0.0001, B2 ≡ −G(0.2408 + 1.6761MN ), M ≡ D¯ λ¯ D , N ≡ v¯0 cos γ , C ≡ H −1/3 (G−2 J − 6.5082G−1 + 0.2645H 2/3 ), H ≡ α+(α+4β 3 )1/2 , α ≡ G−3 (GK−L), K ≡ 83.0841+233.3525N + 233.7683MN + 275.0197N 2 , L ≡ 0.017 − 1.3062MN + 0.2241N − MN 2 (31.7839M 2 N + 13.6988M − 1.5598), β ≡ 3G−2 [0.0292 + 5.1656G + 0.0689N − (0.3612 + 2.5141MN )2 ], J ≡ J1 + J2 , J1 ≡ 0.0179 − 0.0868N , J2 ≡ MN (0.7627 + 2.6546MN), D0 ≡ D1 + D2 , D1 ≡ 0.0002G−2 , D2 ≡ −G−1 (0.4816 + 3.3522MN ), F ≡ F1 + F2 , F1 ≡ 10.8320 + 25.2353N , F2 ≡ 0.00001G−2 − G−1 (0.0312 + 0.2172MN ), v(≡ ¯ v0 /vT ) is the scaled collision velocity, λ¯ D (≡ λD /aZT )
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is the scaled Debye length, E¯ p (≡ μv02 /2ZP2 Ry) is the scaled projectile energy, μ is the reduced mass of the collision system, Ry(= me4 /2¯h2 ≈ 13.6 eV) is the Rydberg constant, D(≡ DaZT /vT3 ) is the scaled diffusion coefficient. As it is seen in Equation (5), the turbulence effects on the electron capture radius are explicitly indicated through the parameters B, C, D0 , and F. In the following section, we shall discuss the electron capture probability and electron capture cross section in weak turbulent plasmas. 3. ELECTRON CAPTURE PROBABILITY AND ELECTRON CAPTURE CROSS SECTION IN TRUBULENT PLASMAS Since the electron capture time tC in Equation (2) would be given by 2(RC2 − b2 )1/2 /vp for heavy ion projectile, the electron capture probability by the projectile ion with nuclear charge ZP e in turbulent plasmas is obtained by the following form:
¯ = 2E¯ p−1/2 R¯ C2 λ¯ 2D − b¯ 2 1/2 PC (E¯ p , λ¯ D , b) ⎧ ⎨ 1 = 2E¯ p−1/2 λ¯ 2D A + (B + C)1/2 ⎩ 2 ⎫1/2 1/2 2 ⎬ 1 F ¯2 D0 − C + + − b , (6) ⎭ 2 4G (B + C)1/2 where b(≡ b/aZT ) is the scaled impact parameter. Hence, the scaled differential electron capture cross section ∂b σ¯ C [≡ (dσC /db)/π aZ2 T ] in turbulent plasmas in units of π aZ2 T is then found to be ¯ = 4bE¯ p−1/2 λ¯ 2D A + 1 (B + C)1/2 ∂b σ¯ C (E¯ p , λ¯ D , b) 2 1/2 1/2 2 1 F D0 − C + + − b¯ 2 , (7) 2 4G(B + C)1/2 where the parameters A, B, C, D0 , F, and G are already given in Equation (5). In this electron capture cross section, the effects of the nonlinear damping and growing are neglected since the test particle velocity is assumed to be small compared to the thermal velocity in weak turbulent plasmas (Shukla & Spatschek 1973). It has been shown that the diffusion coefficient D is independent of t in weak turbulent plasmas (Dupree 1966). Detailed discussions on the diffusion tensor due to the nonlinear wave-particle interaction in astrophysical plasmas would be found in a recent excellent text by Kulsrud (2005). In order to specifically investigate the turbulent effects on the charge capture process, we set ZP = ZT ≡ Z. Figure 1 shows that the surface plot of the scaled electron capture radius as a function of the scaled projectile energy E¯ p and scaled ¯ It is shown that the electron capture diffusion coefficient D. radius decreases with an increase of the diffusion coefficient. Hence, it is found that the influence of the random fluctuation of the electric fields diminishes the electron capture radius in turbulent plasmas. In addition, it is found that the electron capture radius decreases with increasing projectile energy. Figure 2 represents the scaled differential electron capture cross section as a function of the scaled impact parameter b. In addition, Figure 3 shows the surface plot of the scaled
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TURBULENCE EFFECTS ON THE CHARGE CAPTURE PROCESS
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Figure 1. Surface plot of the scaled charge capture radius R¯ C as a function of the scaled projectile energy E¯ p and scaled diffusion coefficient D¯ when λ¯ D = 30, Z = 2, and v¯ = 0.5.
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Figure 2. Differential electron capture cross section ∂b¯ σ¯ C as a function of the scaled impact parameter b when E¯ p = 3, Z = 2, and v¯ = 0.5. The solid line represents the capture cross section for D¯ = 0. The dashed line represents the capture cross section for D¯ = 0.2. The dotted line represents the capture cross section for D¯ = 0.4.
differential electron capture cross section as a function of the scaled diffusion coefficient D¯ and scaled projectile energy E¯ p . As shown in these figures, the electron capture cross section in weak turbulent plasmas decreases with an increase of the diffusion coefficient. Hence, we have found that the turbulence effect strongly suppresses the electron capture cross section in turbulent plasmas. Then, it is found that the electron capture cross sections in non-turbulent plasmas would be always greater than those in turbulent plasmas. 4. CONCLUSION In this paper, we investigated the effects of the fluctuation of the electric fields, i.e., the turbulence effects on the charge capture process in weak turbulent plasmas. We have employed the Bohr–Lindhard model and effective interaction potential taking into account the additional correction factor to the non–linear plasma dielectric function due to the plasma turbulence in order to obtain the electron capture radius and electron capture cross section in turbulent plasmas as functions of the impact parameter, diffusion coefficient, projectile energy, and Debye length. From this work, it has been shown that the influence of the fluctuation of the electric fields in the plasma considerably decreases the electron capture radius and electron capture
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Figure 3. Surface plot of the differential electron capture cross section ∂b¯ σ¯ C as a function of the scaled diffusion coefficient D¯ and scaled projectile energy E¯ p for b¯ = 5, Z = 2, and v¯ = 0.5.
cross section. Moreover, it is found that the electron capture radius and electron cross section decrease with an increase of the projectile energy. Hence, we understand that the turbulence effect strongly suppresses the electron capture cross section in turbulent plasmas. Therefore, it would be expected that the electron capture cross sections in non-turbulent plasmas are always greater than those in weak turbulent plasmas. It is also found that the electron capture cross section decreases with an increase of the projectile energy. It has been shown that the charge capture process can be important in the ionization rate of some ions (Emerson 1996). In addition, the observation of charge capture line emission provides useful information on plasma parameters (Hutchinson 2002). Since the turbulence effects on the charge capture process suppresses the capture cross section, it can be expected that the charge capture photon emission rate decreases in turbulent plasmas. Hence, the charge capture photon emission rates in turbulent astrophysical plasmas would be always smaller than those in non-turbulent astrophysical plasmas. It can be expected that the effects of the plasma turbulence on the charge capture photon emission from the solar chromosphere would be smaller than those from the solar corona due to the temperature jumps (Cox 2000; Beyer & Shevelko 2003; Foukal 2004) from 104 K to 106 K. It has been found that the Landau damping effect reduces the atomic collision cross section in plasmas. In addition, it is shown that the Landau damping effect is important when the projectile velocity is equal to the electron thermal velocity (Jung & Kim 2001). Hence, it would be also expected that the Landau damping effects on the charge capture in the solar corona is less important than those in the solar chromospheres. Since the effects of the plasma turbulence reduces the charge capture photon rates, the ionization rates of ions would be decreased in turbulent plasmas. Hence, it would be expected that the effects of the plasma turbulence reduce the electron–atom bremsstrahlung emission spectrum due to the low ionization degree, i.e., the screening of the bound electrons (Jung & Lee 1995; Gould 2006). However, we can expect that the turbulence effects enhance the electron– bare ion bremsstrahlung radiation cross section since the turbulence produces the additional acceleration and electric field of the projectile electron in the bremsstrahlung process. In intermediate plasma densities, we have to consider both collisional and radiative transitions to obtain the rate equations. The rate equations for the population densities N (z) (n) of the level n of
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an ion A would be written as follows (Sobel’man et al. 1995): d (z) N (z) (m)Γmn N (n) = − N (z) (n)Γn + dt m=n + N (z+1) Rn ,
(8)
where Γn is the total decay probability, Γmn is the transition probability, and Rn is the recombination probability. Then, the ratio of the collisional recombination coefficient to the collisional ionization coefficient is proportional to N (z) (n)/N (z+1) . Since the turbulence effects on the charge capture process suppresses the capture rate, the population density of the ion strongly depends on the strength of the turbulence. For H +A+z → H + +A+(z−1) (n) capture process, the turbulence effect suppresses the population density of the ion A+(z−1) (n). In addition, the number density of H + would be reduced due to the turbulence effect. For example, when the scaled diffusion coefficient is 0.3, the capture cross section is shown to be reduced by about 20%. Hence, the capture rate would also be decreased about 20%. The radiation spectrum due to the electron-impact excitation of the ion A+(z−1) (n) in turbulent plasmas would be smaller than that in non-turbulent plasmas due to the reduction of the number density of the ion. However, it would be expected that the turbulence effects on the Coulomb focusing in the excitation process (Jung 1993) is quite small since the additional potential due to the random fluctuation of the electric field is the far-field potential. Therefore, from this work we have found that the influence of the fluctuating electric fields plays a significant role in the charge capture process in turbulent plasmas. We expect that these results would give useful information on the charge capture and exchange processes in turbulent plasmas. Y.-D.J. gratefully acknowledges Dr. M. Rosenberg for warm hospitality and useful discussions while visiting the University of California, San Diego. This paper is dedicated to the late Prof.
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R. J. Gould in memory of exciting and stimulating collaborations on atomic processes in astrophysical plasmas. The authors thank Prof. E. T. Vishniac for suggesting improvements to this text. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2009-0071573). REFERENCES Ben-Itzhak, I., Jain, A., & Weaver, O. L. 1993, J. Phys. B, 26, 1711 Beyer, H. F., & Shevelko, V. P. 2003, Introduction to the Physics of Highly Charged Ions (Bristol: Institute of Physics Publishing) Birmingham, T. J., & Bornatici, M. 1972, Phys. Fluids, 15, 1785 Bransden, B. H., & McDowell, M. R. C. 1992, Charge Exchange and the Theory of Ion–Atom Collisions (Oxford: Oxford Univ. Press) Cox, A. N. 2000, Allen’s Astrophysical Quantities (4th ed.; New York: Springer) Dupree, T. H. 1966, Phys. Fluids, 9, 1773 Emerson, D. 1996, Interpreting Astronomical Spectra (Chichester: Wiley) Foukal, P. V. 2004, Solar Astrophysics (Weinheim: Wiley-VCH) Gould, R. J. 2006, Electromagnetic Processes (Princeton, NJ: Princeton Univ. Press) Hutchinson, I. H. 2002, Principles of Plasma Diagnostics (2nd ed.; Cambridge: Cambridge Univ. Press) Jung, Y.-D. 1993, ApJ, 409, 841 Jung, Y.-D., & Kato, D. 2009, Phys. Lett., A., 373, 2351 Jung, Y.-D., & Kim, C.-G. 2001, Phys. Plasmas, 8, 3115 Jung, Y.-D., & Lee, K.-S. 1995, ApJ, 440, 830 Jung, Y.-D., & Yang, K.-S. 1997, ApJ, 479, 912 Kim, C.-G., & Jung, Y.-D. 1998, Phys. Plasmas, 5, 2806 Kulsrud, R. M. 2005, Plasma Physics for Astrophysics (Princeton, NJ: Princeton Univ. Press) Shokri, B., Tavassoli, H., & Latifi, H. 2000, Phys. Plasmas, 7, 2689 Shukla, P. K., & Spatschek, K.-H. 1973, Phys. Lett., A., 44, 398 Shukla, P. K., Stenflo, L., & Bingham, R. 2006, Phys. Lett., A., 359, 218 Sitenko, A., & Malnev, V. 1995, Plasma Physics Theory (London: Chapman & Hall) Sobel’man, I. I., Vainshtein, L. A., & Yukov, E. A. 1995, Excitation of Atoms and Broadening of Spectral Lines (2nd ed.; Berlin: Springer) Tsytovich, V. N. 1972, An Introduction to The Theory of Plasma Turbulence (Oxford: Pergamon) Yu, M. Y., Stenflo, L., & Shukla, P. K. 1972, Radio Sci., 7, 1151
