Passage of Charged Particles through Plasma

8

Passage of Charged Particles through Plasma

ι. Field of Charge in a Plasma. Shielding A peculiarity of the electromagnetic properties of a plasma is also manifest in the analysis of the field and dynamics of a charged particle moving in the plasma. Polarization, manifested in the spatial shielding of the field of charged particles, occurs because of the interaction between the charged particle and the surrounding particles. Moreover, the interaction leads to the origin of a retarding force acting on the moving particle and associated with the excitation of electromagnetic waves in the plasma. Let us first consider the field generated by the charged particle moving in the plasma. This field is determined by the system of Maxwell equations (2.1) in which the additional charge and current densities associated with the moving particle must be taken into account: (8.1) Jo = ?v8(r - vi), Po = Φ(* - ν/), where q is the charge and ν is the velocity of the particle. Expanding the field and currents into Fourier integrals and eliminating the magnetic induction from the equations, we obtain the following equation for the electric field intensity: ,) + „€ ( ω ,

k)j E,(k, ω) = 123

4^7

At

(8.2) ω),

124

8. Passage of Charged Particles through Plasma

where / / ( k , ω) is the Fourier component of the current density, connected with the moving charge. In the case of an isotropic plasma for which the dielectric permittivity tensor is determined by (2.7), the solution of (8.2) is

£(k,' ω Ι-Λτν + -t—k 2 (*« IV - ΐ)Ι >' At ω)· ') = ίω4^j(ω, k) k) — f ,v

2

€t(œ,

k

€ι

2

η \

13

k

2

3V

(8.3) If the velocity of charge motion is constant, the Fourier component of the current density (8.1) equals

j(k,*)=-^v8(«-kv). 0

(8.4)

Substituting (8.4) into (8.3), and using the inverse Fourier transform, we obtain the following expression for the electric field of a point charge moving with constant velocity in the plasma: k(kv)

E(r,0=^jV 2

k et(œyk)

'

Γ

ω

ÄVli -^-J j

k) (8.5) ω = kv. The first member in (8.5) describes the potential (longitudinal) part of the field of the moving charge, the second member in (8.5) corresponds to the vortex (transverse) portion of the field. The vortical portion of the field is directly connected with the charge motion. If the charge is at rest (v = 0), the vortical part in (8.5) is missing, and the field is completely potential. Introducing the scalar field potential φ in this case Ε = — grad φ, we find according to (8.5)

Substituting the static value of the longitudinal dielectric per-

2. E n e r g y Losses in Charged-Particle Motion in Plasma 2

125

2

mittivity of the plasma €t(0, k) = 1 + (l/a k ) into (8.6) and integrating, we finally obtain the following expression for the potential of the field of a charge at rest in a plasma

=ler*i*.

ψ{γ)

(8.7)

(The charge is assumed to be at the origin.) As we see, the presence of spatial dispersion in the plasma leads to shielding of the point-charge field. The Debye radius a determines the range at which essential attenuation of the field of a charge at rest in a plasma will occur.

2 . Energy Losses in Charged-Particle Motion in a Plasma Upon passing through a plasma, a charged particle loses part of its energy because of the interaction with surrounding particles, resulting in polarization of the plasma.* These energy losses are simply computed if it is assumed that the energy lost by the particle per unit time is small as compared with the energy of the particle itself, and therefore, the change in velocity of the particle during motion may be neglected. The energy losses of a charged particle are determined by the work of the retarding forces acting on the particle in the plasma from the electromagnetic field generated by the particle itself while moving. Evidently the energy losses of the particle per unit time are —

= qvE \ r = v t,

(8.8)

where the field is taken at the location of the particle. Using (8.5) for the field, we find dE_ _ _Φ_ C „ ( dt

2τΑ J

Κ

ω

,

2

2

2

v -(œ jk )

} 2

j # € ι( ω , k) ^ ω[€«(ω, k) - (kW/ω )]\ ' ™

KV

*

Integration with respect to k in (8.9) should be up to some maximum value k 0 , for which the macroscopic analysis is still valid. * Passage of a charged particle through a plasma is considered in Reference 1.

8. Passage of Charged Particles through

126

Plasma

Taking into account that the real parts of et and et are even functions of the frequency, and the imaginary parts are odd functions, (8.9) may be rewritten as: çW=J*_r

dt

f

2

2ττ

«

(

^

K

J

j

*

•

+

Α*€,(ω, k)

ω^ω,

k)

2v 2 / * )

ω

-

(

-

2

I

2

2

(Ä ί: /cü )]i

t

,

ω

(8.10) The first member in (8.10) takes account of the particle interaction with the longitudinal field and determines the polarization energy losses of the particle. The second member in (8.10) takes account of particle interaction with the transverse field, and corresponds in the general case to energy losses connected with radiation of transverse waves (Cerenkov radiation). Since the phase velocities of the transverse waves are greater than the velocity of light for the isotropic plasma, the Cerenkov-radiation condition is not satisfied, and the second member in (8.10) vanishes in this case. Let us note that both the polarization energy losses (if recoil of the particle is neglected) as well as the losses associated with the Cerenkov radiation are independent of the mass of the moving particle. In particular, these losses will differ from zero even for particles with infinite mass. 3 . Polarization Energy Losses of a Fast Particle in a Plasma Let us evaluate the polarization energy losses of a charged particle moving with velocity v, significantly greater than the mean thermal velocity of the plasma electrons s. Noting that dk = 2nk dk dto/v, where k is the component of k perpendicular to v, let us rewrite the expression for the polarization losses in the form: im

§=t

ω J

>

*

j

>

e

2

/|* +S+i

f

1 t

-

*

*

>

+

^ ~i-

(8.11) The expression (8.11) diverges as k — > • 0 0 . This is associated with the fact that by describing the plasma by the macroscopic dielectric permittivities we actually take into account only the far interaction,

=

3· Polarization E n e r g y Losses

127

and do not take account of the nearby collisions of the particle with the plasma electrons. These collisions cannot be taken into account if the collision parameter b is considerably greater than ν/Ω; if ν/Ω> then collisions of the charged particle with the individual b electrons of the plasma play the principal part, and the macroscopic description of the interaction between the particle and the plasma becomes meaningless. Large k correspond to small b} hence the integration with respect to k in (8.11) should be carried out to a maximum value k0 . If the particle velocity ν is considerably greater than the mean thermal velocity of the plasma electrons s> then values ζ ^> 1 are effective. Using the asymptotic expansion (3.28), the energy losses due to long-range collisions may be represented as

dt TTV

3E

2

φΩ

^

I m

(8

2i*yy ·

&k

J 0 l_m[» + w*)w-Q* + where y is the damping decrement of the plasma waves (3.34). The integral with respect to ω along the real axis is equal to the residue relative to the single pole ω = ikv, in the upper halfplane, hence

no 2

2 2

2

2

νν JJ ο, k + (Ω 1ν ) qHP

dt 3E

'

dkk

2

Assuming k0 v ^> Ω , we obtain the following expression for the energy loss due to the long-range collisions:

dt

| £

_ 4

=

Ä

V mv

l

n

^ . Ω

(8.13) '

Let b0 denote the critical value of the collision parameter separating the domains of short and long-range collisions. It is easy to see that the maximum value k0 is connected with the critical collision parameter b0 . Indeed, the energy losses due to the long-range collisions may be defined as the energy flux of the electromagnetic field passing through a cylindrical surface of radius b0 surrounding the trajectory of the charge: f

=

Β

]

Λ

.

(8.14)

v

1

128

8. Passage of Charged Particles through Plasma

Taking account of axial symmetry of the field we may rewrite this equality as: PsF

1

Λ

00

- ä r — ä ^ L * - * * -

·

1

)5

For v^> s the spatial dispersion is insignificant. In this case Cerenkov radiation is absent, hence, the energy losses ( 8 . 1 5 ) will be associated only with the longitudinal portion of the electric field. Using ( 8 . 5 ) , the component of the longitudinal electric field and the magnetic field may be represented as Mr.

B(r,

0 = - £

£

r) exp [fc, (? -

Ko

t)=-^-Ç kK (kr) x

,)] ω Λ ,

exp [ζω ^ - *)] da>9

(8.16)

il/2

. - i f i [ . - ^ . , r

Substituting ( 8 . 1 6 ) into ( 8 . 1 5 ) , and noting that e(œ) = 1 — Ω /ω when spatial dispersion is neglected, we find as a result of integration 2

q*a>b

3E _

0 v

(Ωολ

2

„ /-Oft,

Putting (Ω/ν)ο0 <^ 1, and using the asymptotic expressions for the MacDonald functions (K0(x) ~ In 2/γχ9 γ = 1.78107... and /^Χ(Λ;) ~ l/x for χ <^ 1), we obtain 3E

_

2 2

_ 4nn0e q

^i

mz;

^ 2 ρ

y ί2ό0

(8.18)

Comparing with ( 8 . 1 3 ) we see that 2 J_ *ο = - τ - ,

2 - = 1.123...

(8.19)

In order to determine the energy losses due to the short-range collisions (domain of large k)> the collisions of the particle with the individual electrons of the plasma should be considered. If n0 is the density of the plasma electrons and b the collision para-

3· Polarization E n e r g y Losses

129

meter, the mean number of collisions within the time dt for which the quantity b will be between b and b -f- dby is 2 π η ^ db dt. The energy lost by a particle during a collision with a plasma electron equals 1 +

2βψ

+ M) f

-

'

·

2 0) ( 8

where q is the charge and M the mass of the particle. Multiplying this expression by lirn^vb db dt, we find the energy losses due to short-range collisions between the particle and the plasma electrons dE

dt

_

b

4ττη0βΥ

ro

dbb

mv

where integration with respect to b is up to a certain value b0 , for which the plasma electrons may be considered as free electrons. we obtain Assuming that b0 ^>

dt

mv

eq(m + M)

Combining (8.18) and (8.22), we obtain /ehe following formula for the total energy losses of a moving particle per unit time for v^>s (1)* 2 2

3

dE _ _ 4nnQe g ^ 2 Mmv dt mv γ (m + M)eqQ

(8.23)

This formula may be obtained directly from (8.13) by formally 2

taking (2ly)[Mmv l(m

+

+ M)eq] as k0 . 2

Such an analysis is valid if [Mmv /(m + M)eq] < l/λ, where λ is the wavelength of the plasma electron in a reference system connected with the moving particle (λ = [n(m + M)]/Mmv). Otherwise, quantum-mechanical effects play an essential part. The condition of applicability of the classical analysis may be written in the form ν < eqfi.

(8.24)

* Formula (8.23) corresponds to Bohr's classical retardation formula (2). + The reciprocal Debye radius is taken as the maximum value k0 in Pines and Böhm (3).

130

8. Passage of Charged Particles through Plasma

We see that for sufficiently high particle velocities this condition is violated. If ν > eq/fi, the upper limit of integration b in (8.13) should be connected with the maximum transfer of momentum of the moving particle to the electron of the medium. For sufficiently high transfers the electrons may be considered free, hence, according to (8.20) the maximum momentum transfer equals M

o = g -

(8-25)

Substituting χ0 = 2eq/nbQv into (8.13) and combining with (8.22), we obtain the total energy losses as [Lindhard (4)] dE_ dt

2 2

__ 4nn0e q mv

2

2Mmv (m + M)W

K }

Formula (8.26) may be obtained directly from(8.10)if the quantummechanical expression is used for the plasma dielectric permittivity € l and recoil of the particle is taken into account. Formulas (8.23) and (8.26) determine the total energy losses of a moving particle only if the particle mass considerably exceeds the electron mass. The interaction between the particle and the fluctuating field in the plasma plays an essential role in the motion of a light particle (electron) through a plasma.

4 . Change in Energy of a Moving Charge due to Field Fluctuations in the Plasma Let us consider the change in energy of a charge in motion in a plasma, which is due to electromagnetic field fluctuations (5). The equation of motion of the charged particle is written as v(i) = ^E(r(i), i),

(8-27)

where E(r(i), t) is the electric field at the point where the particle is (q and M are the particle charge and mass). Integrating the motion equation (8.27) we find v(t)=v0

+ jJf

dt'E(r(t'),

f),

(8.28)

4· E n e r g y Changes due to Field Fluctuations

r(t) = r0 + v 0(r - t0) + ± j | dt' || dt" E(r(i"), Ο

131

(8.29)

( r 0 and v 0 are the particle radius-vector and velocity at the intitial time t0). Evidently, the mean change in energy of the moving particle per unit time equals d E

dt

=*,

(8.30)

where the brackets <···) denote the statistical averaging operation. Let us select a sufficiently large time segment At as compared with the period of the random fluctuations of the electric field in the plasma, but small as compared with the time during which the particle motion changes in an essential manner. Since the particle trajectory differs only slightly from a straight line during this time segment, the particle velocity and the field acting on the particle at time t = t0 + At may be represented approximately as

v(0-v

M

E(r(i), t) ~

C

+ ^

o

J

t

°

E(r(0, t) + ± o

A ' E ( r 0( O ,

0>

J* dt' j|' dfE^t"),

f) ^-E(r0(t),

(8.31) t),

where r0(t) is the particle radius-vector during uniform and straight-line motion r

v

r0(*) = o + o(^ - *o).

(8-32)

Substituting (8.31) into (8.30), we obtain the following expression for the mean change in energy of the moving particle per unit time, to the accuracy of quadratic terms in q: 3 E

dt

o

M

o

(f ΛΈ(Γ (0,ΟΕ(Γο(0.θ)· t

0

(8-33)

132

8. Passage of Charged Particles through Plasma

Evidently, the mean value of the fluctuating part of the field equals zero, hence
dE

dt

POL

Let us note that the first member in (8.33), as well as the second and third, is proportional to the square of the charge of the moving particle since the mean value of the field is proportional to q. The second and third members in (8.33) characterize the statistical change in the energy of a moving particle, associated with the fluctuations of both the electric field in the plasma, and the velocity of the particle itself under the effect of this field. The second member in (8.33) determines the dynamic friction of the particle due to the presence of space-time correlations between the fluctuating electric fields in the plasma. The presence of such correlations leads to additional losses (in addition to the polarization losses) in the energy of the moving particle. Inverting the order of integration with respect to dt' and dt" in the second member of (8.33), integrating with respect to dt' and introducing the new variable ξ = t — t", we obtain M

dE I _ φ_ r

dt

~~

MJ„

di ξ (E,{r0{t - f ), t - ξ) - A - ^ E , ( r 0 ( i ) , *))•

(8-35)

Since the correlation functions for the field fluctuations is exponentially small for large At, the upper limit of integration in (8.35) may be extended to infinity (At —>• 00). Representing the field Ei(r, t) as space-time Fourier integrals, we find for the statistical energy losses of guided particle motion: dE I _

dt

q

2

3

16TT M

j dkœ A <£7>KAJ , ω = kv 0 . (8.36) The third member in (8.33) determines the statistical change in the particle energy associated with the correlations between the fluctuating change in the velocity of the particle itself and the

4· Energy Changes due to Field Fluctuations

133

fluctuating electric field in the plasma. The presence of such correlations leads to a statistical magnification of the energy of the moving particle. It is easy to obtain the following expression for the statistical magnification of the particle energy: dE dt

Λ

Ε

=kv

\ - 1G*M S < »*» " m

Combining (8.36) and (8.37), we obtain

dk

{ω<Ε

ω

kv

-f- L = τα™ ί -L ^ = ° for the total statistical change in the energy of the moving particle. In the case of an isotropic plasma, the correlation function of the longitudinal field (Ε?\ω depends only on the modulus k. Integrating with respect to the angles in (8.38), the statistical change in the energy of the moving particle may be reduced to dE dt

st

WMvf

\"ΑΩΩ\Ε?>ω^ω,

(8.39)

where k 0 is the maximum value of the wave vector determining the domain of applicability of the macroscopic analysis. is Since the spectral density of the field fluctuations (β?\ω positive by definition, then according to (8.39) the particle energy in an isotropic plasma will grow because of the particle interaction with the fluctuating longitudinal field. When charged particles move in an anisotropic plasma (plasma permeated by charged particles; plasma in a magnetic field, etc.), interaction with the fluctuating field may lead to diminution in the particle energy. In contrast to the polarization energy losses, which are independent of the mass of the moving particle, the statistical change in the energy is inversely proportional to the mass of the moving particle. Hence, statistical effects may be neglected in the motion of a heavy particle, with a mass considerably greater than the electron mass, in a plasma. In the case of the motion of a light particle (electron), the statistical effects will be of the same order of magnitude as the polarization effects. Let us evaluate the polarization losses and the statistical magnification of the energy of a moving particle in the case of

134

8. Passage of Charged Particles through P l a s m a

an electron plasma for which the spectral distribution of the field fluctuations is determined by the formula: 2

<^ >κω =

δ-π-Γ Im ej(o>, k) ω I €j(co, k)| 2

Utilizing (3.23) for the longitudinal dielectric permittivity and integrating with respect to the absolute value of the wave vector k between zero and some maximum value k0 for which the macroscopic analysis is still valid in (8.37) and (8.38), we obtain the following general formulas: dE dt dE ~df

2 2

16 VTTHQ e q

poi

2

J

»ra>o

Z2

f dz ζ e~ L(zy ak0),

ο

(8.40)

2 2

st

_ 8 Vnn0 e q ~ Μυ0

(8.41)

w here Ζ = YÏ V0/S and L(Z, ak0) = In ak0 - ^ In {[1 - φ(ζ)] + πζ e~ } 2

1

1-

2

\Z_

arctan

2

2z2

1 — Φ) z2 Vnz e~

(8.42)

(The reciprocal of the minimum parameter of long-range collisions 2 k0 ~ Mmv /[(M + m)eq] may be taken as k0 .) For sufficiently high temperature and low electron density in the plasma ak0 ^> 1. Limiting ourselves in (8.40) and (8.41) to taking account of the first main terms ( ~ l n ak0), we find* dE dt dE dt

2 2

4nn0e q

13 Φ(Ζ) - ζΦ'{ζ) In akn ζ

pol

(8.43)

2 2

4nnQe q Ms

-

ζ2

β~

2V77

In akr,,

(8.44)

* Kaiman and Ron (6) obtained analogous results. Tsytovich (7) considered the question of the passage of charged particles through a plasma when radiation is present in the plasma.

4· E n e r g y Changes due to Field Fluctuations

135

where Φ(ζ) is the customary error function 2 ri Φ(ζ) =-L dzeJ V π 0

f

In the limiting cases of low and high particle velocities, approximate expressions dE dt dE dt

2 2

POL

4nn 0e q ms

POL

4nn0e q mv0

-2-

In ak , 0

v0 < sy

(8.45)

2 2

(8.46)

may be obtained for the polarization energy losses. The polarization losses, statistical magnification, and total 2 2 energy change per unit time [in the units (4nn0e q /ms) \ / | In ak0] are presented in Fig. 16 as a function of the velocity for an electron moving in a plasma (M = m). As we see, in the case of particle motion in an equilibrium plasma

FIG.

16.

ι 36

8. Passage of Charged Particles through Plasma

at a velocity considerably exceeding the mean thermal electron velocity (v0 ^> s)> the influence of the fluctuating field may be neglected. Hence, the total change in particle energy reduces to polarization losses in the energy of the directed motion. If the velocity of the charged particle motion is comparable to the thermal electron velocity, the role of the fluctuating interaction then becomes significant. In particular, if the particle velocity is less than the thermal velocity of the plasma electrons, the energy of the particles moving in the plasma increases because of the fluctuating interaction. Let us note that (8.43) and (8.44) are a good approximation if the large quantity ak0 ^> 1 is in the logarithm. This condition is satisfied if the energy of particle interaction in the plasma is small as compared with the kinetic energy. Comparing (8.46) with (8.13), we see that (8.13) differs from (8.34) by the factor v0/s in the logarithm. This indicates that taking account of just the principal term in (8.42) is not a good enough approximation at high particle velocities.

5. Influence of Ions on the Energy Change of a Particle Moving in a Plasma Let us examine how taking account of the ion motion in a plasma affects the change in energy of a moving charged particle (5). Using (5.52) for the spectral distribution of the fluctuations and (5.8) for the dielectric permittivity, the following formulas for the polarization losses and the statistical magnification of the energy of a moving particle in a nonisothermal electron-ion plasma may be obtained on the basis of (8.34) and (8.38): dE dt I pol

2

'ψ 4nn0e q' ms

β \Φ(ζ)-ζΦ'(ζ) ζ

V 2 |

t %

Φ(μζ)-μζΦ'(μζ) IN ak. ο> μζ

(8.47) (8.48) Here ak0 >

1, t =

TJTi

2

and μ

=

ί(Μ^ηι).

6. E n e r g y Losses of a Charged Particle

137

If the velocity of the charged particle is considerably greater than the thermal electron velocity v0 ^> s, the change in energy of the moving particle is due primarily to particle interaction with the electrons. The interaction between the charged particle and the ions may be neglected here and (8.47) and (8.48) go over into (8.43) and (8.44). As the particle velocity diminishes v0 < s its interaction with the plasma electrons diminishes relatively, and its interaction with -1 the plasma ions becomes essential. If the condition 1 ^> ζ ^> μ, is satisfied, the polarization energy losses are determined by the expression 3E dt

2 2

47RN0e q

ms

pol

VI

,_?^A

£2

3V:

Μ,

In akn

(8.49)

ζ

3

In particular, for ζ ^> m\Mi we have dE dt

_ ^ V

l

n

ö

*

o

(8.50)

pol

For ζ <^ μ~ we find for the polarization losses χ

dE dt

2 2

4NN0e q pol

ms

16 π s*

ίμ In ak0 .

(8.51)

In this case the statistical change in energy equals dE ~df

2 2

4NN0e q ms

In akn

(8.52)

Hence, at low velocities the change in energy of a particle moving in a plasma is determined primarily by particle interaction with the ions. 6. Energy Losses of a Charged Particle in a Nonequilibrium Plasma When a charged particle moves in a nonequilibrium plasma (especially if the plasma is near the boundary of the stability domain), the change in particle energy is determined mainly by

ι 38

8. Passage of Charged Particles through Plasma

the interaction with the fluctuating field. As an example, let us consider a plasma through which a neutral beam of charged particles passes. If the beam density is sufficiently small, the polarization energy losses of a particle moving through the plasma do not change substantially. Interaction of the charged particle with the fluctuating longitudinal field will lead to additional energy losses associated with the excitation of Langmuir oscillations in the plasma. Utilizing (8.38) and (5.96), the following expression may easily be found for the particle energy losses:

dt

TTM

Τ =

Ω*) δ'(ω - kv),

(8.53)

v

J

T'l\ 1

ω

- cos # u

Μ

=

=

Τ ·

In the simplest case, when ν || u we find by integrating in (8.53) BE ¥

-

L

.....

«V

fT =

N

-

W

P

'

-

V

>

U

'

5 4)

Formulas (8.53) and (8.54) describe the charged-particle interaction with fluctuating Langmuir waves in a plasma. According to (8.54) the intensity of Langmuir wave excitation by the moving charge grows anomalously if the charged particle velocity ν approaches the beam velocity u. The magnitude of the energy losses due to particle interaction with the fluctuating field may hence considerably exceed the losses associated with binary collisions. , REFERENCES 1. A. I. Akhiezer and A. G. Sitenko,

ZETF

23, 161 (1952).

2. N. Bohr, Dan. Mat. Fys. Medd. 18(8) (1948). 3. D. Pines and D. Böhm, Phys. Rev. 85, 338 (1952).

4. J. Lindhard, Dan. Mat. Fys. Medd. 28(8) (1954). 5. A. G. Sitenko and Tsien Yu-Tai, ZTF 32, 1325 (1962) (Sov. Phys.—Tech.

Phys.).

6. G. Kaiman and A. Ron, Ann. Phys. 16, 118 (1961).

ZETF 42, 803 (1962); 44, 846 (1963) (Sov. Phys. JETP); Izv. Vuz, Radiofizika 5, 1078 (1962) (CFSTI).

7. V. N. Tsytovich,

( 8