# Energy loss in electronic plasma.

Introduction

The total energy loss of a nonrelativistic particle passing through plasma may be quantized in units of h[[omega].sub.p], where [[omega].sub.p] is plasma frequency. The length scale in plasma is divided into two regions. For dimensions large compared to the Debye screening distance [k.sub.D], the plasma acts as a continuous medium in which the charged particles participate in collective behavior such as plasma oscillations. For dimensions small compared to [k.sub.D], individual particle behavior dominates and the particles interact by the two body screened potential V (r) = [ze.sup.2] exp ([-k.sub.D]r)/r.

For close collisions, collective effects can be ignored and the two body screened potential can be used to evaluate this contribution to the energy loss.

For the distant collisions at impact parameters [bk.sub.D] > 1, the collective effects can be calculated by utilizing Fermi formula (1) with an appropriate dielectric constant for plasma.

Close Collisions

First consider the energy loss by close collisions of a fast, but nonrelativistic, heavy particle of charge ze passing through electronic plasma. Assume that the screened Coulomb interaction V (r) = [ze.sup.2] exp ([-k.sub.D]r)/r, where [k.sub.D] is the Debye screening parameter(2), acts between the electrons and the incident particle.

We may calculate the energy transfer from the impulse

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a central potential V (r) = [ze.sup.2] exp (-kdr)/r, the force is in the radial direction is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we see that the perpendicular component of the force is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

r = [square root of [b.sup.2] + [x.sup.2]] = [square root of [b.sup.2] + [v.sup.2][t.sup.2]]

The momentum-impulse theorem then gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Instead of expressing r in terms of t, we may substitute in t as a function of r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The particle's path from t = - [infinity] to t = [infinity] corresponds to taking r = [infinity] from the left, to r = b at minimum approach, and back out to r = [infinity] on the right.

Because of symmetry, we can simply double the integral for the particle to move from r = b out to infinity. This integral can be simplified by hyperbolic trig substitution r = bcosht to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [xi] = [k.sub.D]b. The integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is somewhat troublesome to evaluate. One approach is to note that this simplifies upon taking a derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, we have simply f ([xi]) = [xi] K1 ([xi]) up to a possible constant of integration.

Direct examination of f ([xi]) relation above indicates that f ([infinity]) = 0, which fixes the constant to be zero. Substituting this integral into [Delta]p[perpendicular to] then gives,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This gives an energy transfer of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To determine the energy loss per unit distance traveled for collisions with impact parameter greater than bmin, we may use the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

assuming that [k.sub.D]bmin <1, where bmin is given by the larger of the classical and quantum minimum impact parameters(3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the modified Bessel functions are exponentially suppressed at infinity, the only contribution comes from the lower limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the plasma frequency (in Gaussian units) is given by

[[omega].sup.2.sub.p] = 4[pi]N[e.sup.2]/m

Finally, using the small argument expansions of the modified Bessel functions(4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting everything together then yields the energy loss per unit distance for the case [k.sub.D]bmin <1

Distant Collisions

For a nonrelativistic particle, the Fermi formula yields the following expression for the energy loss to distances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the important frequencies in the integral turn out to be [omega] [approximately equals] [[omega].sub.p] the relevant argument of the Bessel function is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For particles incident with velocities v less than thermal velocities, this argument is large compared to unity. Because of the exponential fall off of the Bessel functions for large argument, the energy loss in exciting plasma oscillations by such particles is negligible. Whatever energy is lost is in close binary collisions. If the velocity is comparable with or greater than thermal speeds, then the particle can lose appreciable amounts of energy in exciting collective oscillations.

For a particle moving rapidly compared to thermal speeds we may use the familiar small argument forms for the modified Bessel functions. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We shall take the simple dielectric constant augmented by some damping

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The damping constant [GAMMA] will be assumed small compared to [[omega].sub.p] The necessary combination is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the limit [GAMMA] [much less than] [[omega].sub.p], the above integral leads to the simple result,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Total Energy Loss

The total energy loss of a particle passing through plasma is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where A is a number of order unity. The presence of [[omega].sub.p] in the logarithm suggests that the energy losses occur in quantum jumps of h[[omega].sub.p].

If we assume that in plasma, the electrons are completely free and feel no force other than the F = eE, which is due to the electric field, then plasma frequency, is given by(3)

[[omega].sub.p] = [([e.sup.2]N/m[[epsilon].sub.o]).sup.1/2]

in which N is the number of electrons per cubic meter. It corresponds to a frequency,

[f.sub.p] 8.98 [N.sup.1/2] (Hertz)

The common experimental situation is an electromagnetic wave propagating into spatially confined plasma, the density of which gradually increases as the wave penetrates inward. That is, the wave frequency [omega] is constant, while the plasma frequency [[omega].sub.p] increases with distance.

By inverting the first equation, we can define the critical electron density in terms of the wave frequency:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The plasma medium is transparent as long as N<[N.sub.crit]. When the wave reaches the critical layer, the wave is totally reflected (5).

The plasma frequency is basically a measure of the electron density N of the plasma.

Metallic conductors behave like plasmas at high frequencies. The alkali metals like Li, Na, K, Rb and Cs are observed to show a sharp transition from opaque to transparent at the characteristic ultraviolet wavelengths according to the following table:

On the assumption that the observed transition is a manifestation of the plasma frequency, we can calculate the effective number of free electrons per atom in each case.

Conclusions

In this article, considering both close and distant collisions, we derived an equation from which we showed that the energy losses occur in quantum jumps of h[[omega].sub.p], where [[omega].sub.p] is plasma frequency. In fact, electrons passing through thin alkali metallic foils do show this discreteness in energy loss, allowing determination of effective plasma frequencies in alkali metals. On the assumption that the observed transition is a manifestation of the plasma frequency, we can calculate the effective number of free electrons per atom in each case.

References

[1] Fano U., 1963, Ann. Rev. Nucl. Sci. 13, 1.

[2] Boyd T.J.M. and Sanderson J.J., 1963, Plasma dynamics, Nelson, London and Barnes Press, New York.

[3] Jackson, J.D., 1999, Classical Electrodynamics, 3rd Edition, John Wiley and Sons.

[4] Arfken G.B. and Weber H.J., 2005, Mathematical Methods for Physicists, 6th Edition, Academic Press.

[5] Raether H.,1965, Springer Tracts in Modern Physics, V. 38, ed. Hohler G. Springer-Verlag, Berlin.

The total energy loss of a nonrelativistic particle passing through plasma may be quantized in units of h[[omega].sub.p], where [[omega].sub.p] is plasma frequency. The length scale in plasma is divided into two regions. For dimensions large compared to the Debye screening distance [k.sub.D], the plasma acts as a continuous medium in which the charged particles participate in collective behavior such as plasma oscillations. For dimensions small compared to [k.sub.D], individual particle behavior dominates and the particles interact by the two body screened potential V (r) = [ze.sup.2] exp ([-k.sub.D]r)/r.

For close collisions, collective effects can be ignored and the two body screened potential can be used to evaluate this contribution to the energy loss.

For the distant collisions at impact parameters [bk.sub.D] > 1, the collective effects can be calculated by utilizing Fermi formula (1) with an appropriate dielectric constant for plasma.

Close Collisions

First consider the energy loss by close collisions of a fast, but nonrelativistic, heavy particle of charge ze passing through electronic plasma. Assume that the screened Coulomb interaction V (r) = [ze.sup.2] exp ([-k.sub.D]r)/r, where [k.sub.D] is the Debye screening parameter(2), acts between the electrons and the incident particle.

We may calculate the energy transfer from the impulse

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a central potential V (r) = [ze.sup.2] exp (-kdr)/r, the force is in the radial direction is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we see that the perpendicular component of the force is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where

r = [square root of [b.sup.2] + [x.sup.2]] = [square root of [b.sup.2] + [v.sup.2][t.sup.2]]

The momentum-impulse theorem then gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Instead of expressing r in terms of t, we may substitute in t as a function of r

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The particle's path from t = - [infinity] to t = [infinity] corresponds to taking r = [infinity] from the left, to r = b at minimum approach, and back out to r = [infinity] on the right.

Because of symmetry, we can simply double the integral for the particle to move from r = b out to infinity. This integral can be simplified by hyperbolic trig substitution r = bcosht to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [xi] = [k.sub.D]b. The integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is somewhat troublesome to evaluate. One approach is to note that this simplifies upon taking a derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, we have simply f ([xi]) = [xi] K1 ([xi]) up to a possible constant of integration.

Direct examination of f ([xi]) relation above indicates that f ([infinity]) = 0, which fixes the constant to be zero. Substituting this integral into [Delta]p[perpendicular to] then gives,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This gives an energy transfer of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To determine the energy loss per unit distance traveled for collisions with impact parameter greater than bmin, we may use the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

assuming that [k.sub.D]bmin <1, where bmin is given by the larger of the classical and quantum minimum impact parameters(3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the modified Bessel functions are exponentially suppressed at infinity, the only contribution comes from the lower limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the plasma frequency (in Gaussian units) is given by

[[omega].sup.2.sub.p] = 4[pi]N[e.sup.2]/m

Finally, using the small argument expansions of the modified Bessel functions(4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting everything together then yields the energy loss per unit distance for the case [k.sub.D]bmin <1

Distant Collisions

For a nonrelativistic particle, the Fermi formula yields the following expression for the energy loss to distances

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the important frequencies in the integral turn out to be [omega] [approximately equals] [[omega].sub.p] the relevant argument of the Bessel function is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For particles incident with velocities v less than thermal velocities, this argument is large compared to unity. Because of the exponential fall off of the Bessel functions for large argument, the energy loss in exciting plasma oscillations by such particles is negligible. Whatever energy is lost is in close binary collisions. If the velocity is comparable with or greater than thermal speeds, then the particle can lose appreciable amounts of energy in exciting collective oscillations.

For a particle moving rapidly compared to thermal speeds we may use the familiar small argument forms for the modified Bessel functions. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We shall take the simple dielectric constant augmented by some damping

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The damping constant [GAMMA] will be assumed small compared to [[omega].sub.p] The necessary combination is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the limit [GAMMA] [much less than] [[omega].sub.p], the above integral leads to the simple result,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Total Energy Loss

The total energy loss of a particle passing through plasma is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where A is a number of order unity. The presence of [[omega].sub.p] in the logarithm suggests that the energy losses occur in quantum jumps of h[[omega].sub.p].

If we assume that in plasma, the electrons are completely free and feel no force other than the F = eE, which is due to the electric field, then plasma frequency, is given by(3)

[[omega].sub.p] = [([e.sup.2]N/m[[epsilon].sub.o]).sup.1/2]

in which N is the number of electrons per cubic meter. It corresponds to a frequency,

[f.sub.p] 8.98 [N.sup.1/2] (Hertz)

The common experimental situation is an electromagnetic wave propagating into spatially confined plasma, the density of which gradually increases as the wave penetrates inward. That is, the wave frequency [omega] is constant, while the plasma frequency [[omega].sub.p] increases with distance.

By inverting the first equation, we can define the critical electron density in terms of the wave frequency:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The plasma medium is transparent as long as N<[N.sub.crit]. When the wave reaches the critical layer, the wave is totally reflected (5).

The plasma frequency is basically a measure of the electron density N of the plasma.

Metallic conductors behave like plasmas at high frequencies. The alkali metals like Li, Na, K, Rb and Cs are observed to show a sharp transition from opaque to transparent at the characteristic ultraviolet wavelengths according to the following table:

Element [lambda]c (nm) Li 205 Na 210 K 315 Rb 360 Cs 440

On the assumption that the observed transition is a manifestation of the plasma frequency, we can calculate the effective number of free electrons per atom in each case.

Conclusions

In this article, considering both close and distant collisions, we derived an equation from which we showed that the energy losses occur in quantum jumps of h[[omega].sub.p], where [[omega].sub.p] is plasma frequency. In fact, electrons passing through thin alkali metallic foils do show this discreteness in energy loss, allowing determination of effective plasma frequencies in alkali metals. On the assumption that the observed transition is a manifestation of the plasma frequency, we can calculate the effective number of free electrons per atom in each case.

References

[1] Fano U., 1963, Ann. Rev. Nucl. Sci. 13, 1.

[2] Boyd T.J.M. and Sanderson J.J., 1963, Plasma dynamics, Nelson, London and Barnes Press, New York.

[3] Jackson, J.D., 1999, Classical Electrodynamics, 3rd Edition, John Wiley and Sons.

[4] Arfken G.B. and Weber H.J., 2005, Mathematical Methods for Physicists, 6th Edition, Academic Press.

[5] Raether H.,1965, Springer Tracts in Modern Physics, V. 38, ed. Hohler G. Springer-Verlag, Berlin.

Saeed Mohammadi Physics Department, Payame Noor University, Mashad 91735, Iran E-mail: Mohammadi@pnu.ac.ir

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Author: | Mohammadi, Saeed |
---|---|

Publication: | International Journal of Dynamics of Fluids |

Article Type: | Report |

Geographic Code: | 7IRAN |

Date: | Jun 1, 2009 |

Words: | 1387 |

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