Eﬀects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves

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Effects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves

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doi:10.1209/0295-5075/102/45002

Eﬀects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves

Young-Dae Jung(a)

Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute 110 Eighth Street, Troy, NY 12180-3590, USA and

Department of Applied Physics, Hanyang University - Ansan, Kyunggi-Do 426-791, South Korea^(b) received 20 February 2013; accepted in ﬁnal form 17 May 2013

published online 11 June 2013

PACS 52.40.Db– Electromagnetic (nonlaser) radiation interactions with plasma

PACS 52.25.Os– Emission, absorption, and scattering of electromagnetic radiation

PACS 42.25.Bs– Wave propagation, transmission and absorption

Abstract – The effects of turbulence on the Thomson scattering process are investigated in turbulent plasmas. The Thomson scattering cross section in turbulent plasmas is obtained by the fluctuation-dissipation theorem and plasma dielectric function as a function of the diffusion coefficient, wave number, and Debye length. It is demonstrated that the turbulence effect suppresses the Thomson scattering cross section. It is also shown that the turbulence effect on the Thomson scattering process decreases with increasing thermal energy. The dependence of the wave number on the total Thomson scattering cross section including the turbulent structure factor is also discussed.

This paper is dedicated to the late Prof. P. K. Shukla in memory of exciting and stimulating collaborations on eﬀective interaction potentials in various astrophysical and laboratory plasmas.

Copyright cEPLA, 2013

The interaction of the electromagnetic wave by a small object has been of great interest since it is known as the one of the most common physical phenomena and has wide applications in many areas of physics [1–4]. Especially, the scattering of electromagnetic waves by electrons known as the Thomson scattering process has been the subject of considerable attention in atomic physics, astrophysics, condensed-matter physics, and plasma physics since the Thomson scattering cross section in a plasma can be used as a plasma diagnostic tool since the spectrum of scattered radiation field provides physical information on the scattering system as well as on the surrounding physical environment [5–10]. In ideal or weakly coupled plasmas, the effective interaction potential has been characterized by the Debye-Hückel shielding model since the renormalized electron charge [11] is represented by the form eexp (−r/λD), where λD is the Debye length.

In addition to the standard Debye-Hückel potential, it has been shown that the far-field potential term in collisionless plasmas falls off as the inverse cube of the

(a)E-mail:[email protected]

(b)Permanent address.

distance between the electron and the target ion [12,13].

Moreover, in turbulent plasmas the motion of the plasma electron would be affected by the fluctuating electric fields since the response of the random field fluctuations has played a crucial role in the binary collisions [14–16]. Then, the interaction potential in turbulent plasmas cannot be properly described by the standard Debye-Hückel model obtained by the Maxwellian velocity distribution due to the field fluctuations. Hence, it can be expected that the plasma dielectric function in turbulent plasmas would be different from the conventional plasma dielectric function in ideal plasmas since the influence of the random fluctuation electric field alters the plasma distribution function.

It would be then expected that the Thomson scattering cross sections by the scattering of electromagnetic waves in turbulent plasmas would be quite different from those in the standard Maxwellian plasmas since the fluctuating electric field alters the structure factor. Hence, in this paper we investigate the effects of turbulence on the Thomson scattering cross section in turbulent plasmas by the scattering of electromagnetic waves. The fluctuation- dissipation theorem and plasma dielectric function are employed to obtain the structure factor and the Thomson

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Young-Dae Jung

scattering cross section in turbulent plasmas. The inﬂu- ence of plasma turbulence and the dependence of the wave number on the Thomson scattering cross section including the turbulent structure factor are also discussed.

For the scattering of an electromagnetic wave by plasma electrons, the diﬀerential Thomson scattering cross section [5,6,9] d²σT/dωdΩ per unit scattering volume V in a frequency interval dω and also per solid angle dΩ in thekf direction is expressed by

d²σT

VdωdΩ= e²

mc² 2

(ˆηi · ηˆf)²S(q, ω), (1) where e and m are the charge and mass of the electron, c is the speed of light in vacuum, ˆη_i and ˆη_f are unit polarization vectors for the incident and scattered waves, S(q, ω) is the dynamic structure factor which produces the physical information on the density ﬂuctuation of the surrounding plasma, and q(≡k_f−k_i) is the momentum transfer,k_i is the incident wave vector,k_f is the scattered wave vector, and ω =ωf−ωi. Since the spatial Fourier componentρq(t) of the electron-density functionρ(r, t) at the positionrand time tcan be represented by

ρ_q(t) =

d³reⁱ^q^·rρ(r, t), (2) the spectral function of the density ﬂuctuations known as the dynamic structure factor [9]S(q, ω) is then expressed in the following form:

S(q, ω) = 1 2π

_∞

−∞dt e^iωtρ_q(t)ρ_−q(t+t), (3) where the bar over the symbols stands for the statistical average of the temporal correlation. Using the ﬂuctuation- dissipation theorem [2], the dynamic structure factor will then obtained by

S(q, ω) =− i 4π coth

ω 2k_BT

[Q(q, ω)−Q(−q,−ω)], (4) where is the rationalized Planck constant, k_B is the Boltzmann constant, T is the temperature, and Q(q, ω) is the linear response function:

Q(q, ω) =− q² 4πe²

1 ε(q, ω) −1

, (5)

and ε(q, ω) is the plasma dielectric function. For the coherent Thomson scattering process, the Salpeter form factor [5] can be found in an excellent work by Boyd and Sanderson [17]. By the integration over the whole frequency domain and the average over the initial polarization directions and summation over the ﬁnal polarization directions,i.e., (1/2)

ˆ ηi

ˆ ηf

,the diﬀerential Thomson scattering cross section [9] dσT/dΩ for the scattering of

the electromagnetic wave by plasma electrons will be then represented by the following expression:

dσ_T dΩ = 1

2V e²

mc² 2

ˆ ηi

ˆ ηf

dω(ˆηi ·ηˆf)²S(q, ω)

= N e²

mc² ₂

1−1 2sin²Θ

S(q), (6) where N(=nV) is the number of plasma electrons in the scattering volumeV,n is the number density of electrons, andS(q) is the static structure factor deﬁned by

S(q) = 1 n

_∞

−∞

dω S(q, ω)

= −i n

q 4πe

2 _∞

−∞

dω[ε⁻¹(−q,−ω)

−ε⁻¹(q, ω)] coth ω

2kBT

. (7)

Hence, the dynamic structure factor S(q, ω) would be interpreted as the spectral function of density ﬂuctuations decomposed in both the wave number and frequency domains [9].

The useful analytic form of the eﬀective screened potential [15] of a moving test charge in a turbulent plasma has been obtained by the longitudinal non-linear plasma dielectric function including the modiﬁcation factor [18]

e^−(1/3)q²^Dt³due to the influence of the plasma turbulence, where D is the diffusion coefficient. The diffusion coefficient will be of the form:D = ¯Dt,i.e., thet-dependence, in the Birmingham and Bornatici model [18], where ¯D is the scaled diffusion coefficient due the mean square average of the absolute square of the Fourier component of the random fluctuating electronic field|δEq|²and, however,Dwill be a constant,i.e., not-dependence, in the Dupree model [19]. Then, the Shukla and Spatschek [15]

(SS) longitudinal plasma dielectric function ε⁻_SS¹(q, ω =

−q· v0) with the condition v0 < vT including the influence of the far-field potential term due to the fluctuation field in turbulent plasmas when the energy density of the random fluctuating electric field is small compared to that of the thermal energy, has been obtained by as follows:

ε⁻_SS¹(q,−q·v0)∼=

1 + (qλD)⁻²₋₁ +i[(π/2)^1/2+ (4/27)D/qv³_T](q·v₀/qv_T)

(qλ_D)²[1 + (qλ_D)⁻²]² , (8) for D constant, i.e., the Dupree model, where v0 is the projectile velocity,vT(=λDωp) is the thermal velocity,ωp

is the plasma frequency, and ε⁻_SS¹(q,−q ·v₀)∼=

1 + (qλ_D)⁻²₋₁ +i(π/2)^1/2(q·v0/qv_T)

(qλD)²[1 + (qλD)⁻²]²+ (19/9√

2)( ¯D/qv⁴_T) (q²λD)²[1 + (qλD)⁻²]²,

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for D= ¯Dt, i.e., the Birmingham and Bornatici model.

Using the symmetric property of the response function and the plasma dielectric function ε_SS(q, ω), the Shukla and Spatschek static structure factorS_SS(q) in classical turbulent plasmas in the Birmingham and Bornatici model would be obtained by

SSS(q) = q²λ²_D

1− 1

ε_SS(q,0)

= q²λ²_D q²λ²_D+ 1

1− 19

9√ 2

Dλ¯ ²_D/v_T⁴ q²λ²_D + 1

, (10) and, however, the static structure factor in the Dupree turbulence model becomes SSS(q) =q²λ²_D/(q²λ²_D+ 1) which is identical to the case of non-turbulent plasmas.

Hence, it is important to note that the influence of plasma turbulence on the Thomson scattering process vanishes in the Dupree model,i.e., a constant D case. As is seen in eq. (10), the Shukla and Spatschek static structure factorS_SS(q) will be quite useful in the investigation of the plasma many-body problem and also in the spectrum analysis of the X-ray scattering and radiation data in turbulent plasmas since the structure factor inhibits the density fluctuations in turbulent plasmas including the plasma shielding and collective characters of plasma particles. Hence, the differential Thomson scattering cross section (dσT/dΘ) encompasses the Shukla and Spatschek static structure factor in turbulent plasmas is found to be

dσ_T(k,Θ,D)¯

dΘ = 2πN

e² mc²

₂ sin Θ

1−1

2sin²Θ

× 2(kλD)²(1−cos Θ) 2(kλ_D)²(1−cos Θ) + 1

×

1− 19 9√ 2

Dλ¯ ²_D/v_T⁴ (kλD)²(1−cos Θ) + 1

, (11) where Θ is the scattering angle, i.e., the angle between the incident wave vectork_i and the scattered wave vector k_f and k ≡ |k_i|=|k_f| since there is no change in the frequency of the radiation in the Thomson scattering process. Then, the scaled diﬀerential Thomson scattering cross section ∂_Θσ¯_T[≡d(σ_T/Nσ₀)/d(cos Θ)] in turbulent plasmas

∂Θ¯σT = 3 4

1 − 1

2sin²Θ

2¯k²(1−cos Θ) 2¯k²(1−cos Θ) + 1

×

1 − 19 9√ 2

¯λ²_D E¯_T²

D˜

k¯²(1−cos Θ) + 1

, (12)

where σ0(≡8πr₀²/3) is the Thomson scattering cross section for the scattering of the electromagnetic wave by a free electron, r0(=e²/mc²) is the classical electron radius, ¯k(≡kλD) is the scaled wave number for the elastic scattering, ¯λD(≡λD/a0) is the scaled Debye length,a0(=²/me²) is the Bohr radius of the hydrogen

atom, ˜D ≡D(ma¯ ₀Ry)² , ¯E_T ≡E_T/Ry, E_T(=k_BT) is the thermal energy, andRy(=me⁴/2² ≈13.6 eV) is the Rydberg constant. In addition, the scaled diﬀerential scattering cross section in ideal Maxwellian plasmas∂_Θσ¯_T will be

∂Θσ¯_T=3 4

1 − 1

2sin²Θ

2¯k²(1 −cos Θ)

2¯k²(1 −cos Θ) + 1, (13) since∂Θσ¯_T= lim

D˜→0∂Θσ¯T . The total scaled Thomson scattering cross section ¯σ_T in units ofNσ₀ for the scattering of the electromagnetic wave in turbulent plasmas is then found to be

σ¯T(¯k, D,˜ E¯T) = 3 4

¹

−1

dξ

1−1

2(1−ξ²)

2¯k²(1−ξ) 2¯k²(1−ξ) + 1

×

1− 19 9√ 2

¯λ²_D E¯_T²

D˜

¯k²(1−ξ) + 1

, (14) where ξ≡cosθ. Recently, the non-thermal and plasma screening effects on the Thompson scattering cross section were explored in generalized Lorentzian electron plasmas [20]. However, the influence of the plasma turbulence on the Thompson scattering process has not been investigated as yet. Hence, the expression of the Thomson scattering cross section ¯σ_T (eq. (14)) including the influence of the random field fluctuation in turbulent plasmas will be reliable when the energy density of the fluctuating electric field is small compared to the thermal energy density.

From eq. (14), it is readily found that the inﬂuence of plasma turbulence in high wave number domains vanishes and then the total scattering cross section turns out to be the free Thomson cross section,i.e.,Nσ0(=N8πr²₀/3) is the saturated cross section, since ¯σT(¯k 1, D,˜ E¯T)∼= (3/8)₁

−1dξ(1 +ξ²) = 1 and the individual-particle like fluctuations [9] due to no correlations will be the main constituent of the structure factor. On the other hand, the fluctuations will behave collectively for ¯k 1. Hence, it can be expected that the turbulence effect on the total Thomson scattering process is significant in the intermediate wave number region and also found that the diffusion caused by the plasma turbulence leads the turbulence correction on the Thomson scattering cross section in turbulent plasmas.

Figure 1 represents the scaled differential Thomson scattering cross section∂_Θσ¯_T in units ofNσ₀(=N8πr²₀/3) as a function of the scattering angle Θ for various values of the diffusion coefficient ˜D. As shown in this figure, the differential Thomson scattering cross section increases with an increase of the scattering angle. Hence, it is found that

∂Θ¯σT has the maximum value for the backward scattering, i.e., Θ =π. It is also found that the differential Thomson scattering cross section decreases with increasing diffusion coefficient. Figure 2 shows the scaled total Thomson scattering cross section ¯σT in units of Nσ0 as a function of the scaled wave number ¯kfor various values of the diffusion coefficient ˜D. As we can expect from eq. (14), it is

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Young-Dae Jung

Fig. 1: (Colour on-line) The scaled diﬀerential Thomson scattering cross section ∂Θσ¯T in units of Nσ0(=N8πr²0/3) as a function of the scattering angle Θ (in radians) when ¯k= 1, E¯T= 5, and ¯λD= 50. The solid line represents the case of D˜= 0,i.e., without the turbulent eﬀect. The dashed line represents the case of ˜D= 0.1. The dotted line represents the case of ˜D= 0.2.

Fig. 2: (Colour on-line) The scaled total Thomson scattering cross section ¯σT in units ofNσ0(=N8πr₀²/3) as a function of the scaled wave number ¯kwhen ¯ET= 5 and ¯λD= 50. The solid line shows the case of ˜D= 0,i.e., without the turbulent eﬀect.

The dashed line shows the case of ˜D= 0.1. The dotted line shows the case of ˜D= 0.2.

shown that the Thomson scattering cross section is saturated to Nσ₀ with increasing wave number. As is seen in fig. 2, the influence of plasma turbulence suppresses the Thomson scattering cross section in turbulent plasmas, especially, in the intermediate domain of the wave number. For wave numbersk >8λ_D, the turbulent effect is found to be almost negligible. Hence, we can expect that the mean free path of the photon through the turbulent plasma would be increased due to the turbulence effect since the mean free path lfree is inversely proportional to the total Thomson scattering cross section ¯σT

[21],i.e.,lfree ∝σ_T⁻¹. Figure 3 represents the surface plot of the scaled total Thomson scattering cross section ¯σT

as a function of the scaled diﬀusion coeﬃcient ˜D and scaled thermal energy ¯ET . Figure 4 shows the scaled

Fig. 3: The surface plot of the scaled Thomson scattering cross section ¯σT as a function of the scaled diﬀusion coeﬃcient ˜D and scaled thermal energy ¯ET when ¯k= 1 and ¯λD= 50.

Fig. 4: (Colour on-line) The scaled total Thomson scattering cross section ¯σT in units ofNσ0(=N8πr₀²/3) as a function of the scaled thermal energy ¯ET when ¯k= 1 and ¯λD= 50. The solid line shows the case of ˜D= 0.05. The dashed line shows the case of ˜D= 0.1. The dotted line shows the case of ˜D= 0.2.

total Thomson scattering cross section ¯σT as a function of the scaled thermal energy ¯ET for various values of the diffusion coefficient ˜D. From these figures, it is found that the Thomson scattering cross section in turbulent plasmas increases with an increase of the thermal energy. It is interesting to note that the influence of plasma turbulence on the Thomson scattering process is found to be more significant for low thermal energies. In addition, it is shown that the thermal effect on the Thomson scattering cross section gets more effective with an increase of the diffusion coefficient. Hence, the plasma diagnos- tics for the investigation of the physical characteristics of turbulent plasmas will be suggested in low-temperature plasmas since the turbulence effect on the Thomson scattering process diminishes with increasing thermal energy.

In this work, the influence of the non-thermal fluctuations has not been investigated. However, it has been known that the non-thermal effect would be investigated by the

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generalized Lorentzian non-thermal distribution function f(v, κ) [22]:

f(v, κ) = m

2πκE_κ _3/2

Γ(κ+ 1) Γ(κ−1/2)

1 + mv² 2κE_κ

_−κ−1 ,

(15) where v is the velocity of the plasma electron, κ(>3/2) is the spectral index of the Lorentzian plasma, Eκ[≡(1−3/2κ)ET] is the characteristic energy in Lorentzian plasmas, and Γ(z) represents the Euler gamma function. This generalized Lorentzian velocity distribution contains important features, first, the distribution function yields an inverse power law in the high domain such as f ∝(mv²/2κE_κ)⁻^κ–1, and second, this distribution leads to the thermal Maxwellian distribution f(v, κ→ ∞) =f_M(v)∝exp (−mv²/2E_T) at the limit of κ→ ∞, i.e., in the absence of the radiation field. In addition, it is also shown that the radiation field modifies the diffusion coefficient since the non-Columbic diffusion coefficient would be produced by the radiation-induced field and is also proportional to the square of the velocity v of the plasma particle [22]. Moreover, it is shown that the effective screening length [23] λκ of the generalized Lorenzian plasma, i.e., the critical correlation length, has been obtained by λκ=µκλD, where the parameter µ_κ≡[(κ−3/2)/(κ−1/2)]^1/2 and λ_D is the conventional Debye length in Maxwellian plasmas. This factor µ_κ represents the measure of the fraction of the non-thermal population in the Lorentzian plasma distribution. Hence, in non-thermal plasmas, the scaled wave number is increased by a factor 1/µ_κ so that the corresponding Thomson scattering cross section would be enhanced as we can see in fig. 2. In addition to this effect, the diffusion coefficient would be then increased since the total diffusion coefficient is proportional to (1 +αv²) in non-thermal Lorentzian plasmas, whereαis a constant related to the radiation intensity. Then, as is seen in eq. (14) and fig. 3, we can expect that the Thomson scattering cross section would be enhanced in non-thermal turbulent plasmas.

However, the detailed investigation for the influence of the non-thermal turbulence on the Thomson scattering process in the presence of non-thermal fluctuations will be treated elsewhere with the comparison of the measurements of turbulence by the Thomson scattering in tokamak plasmas [24–26]. Hence, we have found that the influence of plasma turbulence plays a crucial role in the Thomson scattering cross section for the scattering of the electromagnetic wave by electrons in turbulent plasmas. These results would provide useful information on the spectrum analysis of the X-ray scattering and radiation data for the scattering processes by electromagnetic waves and also on the physical characteristics of the wave propagation in turbulent plasmas. In addition, these results will be useful for the application of laser-plasma coupling in laser-matter interactions [27].

∗ ∗ ∗

The author gratefully thanks Prof. W. Roberge for useful discussions and warm hospitality while visiting the Department of Physics, Applied Physics, and Astro- nomy at Rensselaer Polytechnic Institute. The author also gratefully thanks Dr. M. Rosenberg of University of California, San Diego and Dr. S.-H.Kimof University of Iowa for delighting discussions on the collision process in turbulent plasmas. This research was initiated while the author was aﬃliated with Rensselaer Polytechnic Institute as a visiting professor. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.

2012–001493).

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Eﬀects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves

Effects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves

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Eﬀects of turbulence on the Thomson scattering process in turbulent plasmas by the scattering of electromagnetic waves