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Electrodynamics of effective charge
Volume 104A, number 3
PHYSICS LETTERS
20 August 1984
ELECTRODYNAMICS OF EFFECTIVE CHARGE
James L. COX Jr.
Department of Physics, Old Dominion University, Norfolk, VA 23508, USA Received 27 July 1983 Revised manuscript received 18 April 1984
A portion of the charge induced in an isotropic plasma by an injected charged particle is shown to combine with the particle to form a charged quasi-particle, the effective charge, that is different from the conventional "dressed" charge. The remainder of the induced charge is found in a wake behind the effective charge. Properties of effective charge are described, and the application of this concept to coherent ion acceleration and stopping power is discussed.
A fast charged particle passing through an isotropic medium such as an unmagnetized plasma has its electromagnetic fields screened by charge induced in the medium in response to perturbation by the particle. A conventional approach for taking this screening into account is to treat the "bare" particle charge plus the induced charge that surrounds it as a single charged quasi-particle, i.e. as a "dressed" charge [1]. Such a dressed charge, however, is not a well-defined quasiparticle: Its current density is not the product of the bare particle velocity and the total (bare plus induced) charge density. A consistent definition is possible, however, in terms of the concept of effective charge * 1 introduced in this letter. The effective charge of a particle is found to consist of its bare charge plus a portion of the induced charge; it is seen to be the physical entity that acts like a moving quasi-charge when a bare charge is injected into matter. Areas of research in which this description should provide fundamental insight include coherent ion acceleration and coherent stopping power, as discussed briefly below. A particle with velocity u and (Fourier-transformed) true charge and current densities p(k, 60) and J(k, 60) acts like a quasi-particle with effective charge density ,1 The effective charge defined in ref. [2] differs by a factor of e(w) from the effective charge in the present work. The present ~eeifieation is required ff the effective charge is to behave as a quasi-particle. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Pe(k, 60) and velocity tJ in an isotropic medium only if relations of the following kind are satisfied:
Pe(k, w) = p(k, w) [1 + A ( k , w)] .
(1)
Je(k, 60) = J(k, 60) [1 + A(k, 60)] .
(2)
[The variables (x, k) and (t, 60) are Fourier-transform pairs; x and t are space-time variables; the function A(k, 60) is identified below.] The required condition that Je(k, 60) = UPe(k , 60) is satisfied since J(k, 60) = up(k, 60). Furthermore, Pe(k, 60) and Je(k, 60) satisfy the continuity equation. These relations can be given physical content by incorporating them into the Maxwell equations for the medium:
- i k " D(k, 60) = 41rp(k, 60). - i k × E(k, 60) = - ( i w / c ) B ( k , w ) . - i k . B(k, w) = O.
(3) (4) (5)
- i k X H(k, w) = (47r/c)J(k, 60) + (i60/c)D(k, 60). (6) By invoking the constitutive relations D(k, 60) = e(60)E(k, 60) and tt(k, 60) = B(k, 60) (for a medium in which/~ = 1) and by making the unique choice ,2 *2 The function A(k, to) corresponding to general dielectric and permeability functions is obtained simply by replacing e(to) by the product e(k, to)9(k, oJ). The choice of • = e(to) and t~ ffi 1 is made for simplicity; the theory develops along similar lines for e = e(k, to) and t~ = 9(k, to). 149
Volume 104A, number 3 A(k, CO) = 6o2 [e(CO)
PHYSICS LETTERS 1] [k2c 2
CO2e(CO)l- 1
(7)
Eqs. (3) - (6) can be reduced to the following form:
- i k . Ee(k , CO) = 47rPe(k , CO).
(8)
- i k X Ee(k, CO) = -(iCO/c)Be(k, CO).
(9)
- i k . Be(k , CO) = 0.
(10)
- i k X Be(k, CO)= ( 4rr/C)Je(k, CO)+ (iCO/c)Ee(k, co) .(1 1) The connections between the effective fields, Ee(k, co) and Be(k , co), and the ordinary fields, E(k, CO) and B(k, CO), are as follows:
Ee(k , CO)= E(k, CO) + A(k, CO)(kc/CO)2E~(k, CO), (12) and
Be(k , CO) = O(k, CO),
(13)
where E~(k, CO)= [k" E(k, ¢o)] k/k 2 is the longitudinal part of E(k, co). This reduction establishes the effective charge defined by eqs. (1), (2) and (7) as a charged quasi-particle: Eqs. (8)-(1 1) show that it, alone, creates electric and magnetic fields in exactly the same way as a bare charge in vacuum does. Once these effective fields are determined by solving eqs. (8)-(1 1), the connections given by eqs. (12) and (13) completely determine the total electric and magnetic fields, E(k, co) and B(k, co). Eqs. (12) and (13) also show that the total electric and magnetic fields decompose into the fields produced by the effective charge plus a purely longitudinal electric field that can be shown to have the form of a wake in the medium behind the effective charge. Since eqs. ( 8 ) - ( 1 3 ) are fully equivalent to the Maxwell equations ( 3 ) - ( 6 ) and the linear constitutive relations, this decomposition has physical significance. Moreover, this decomposition is unique since the A(k, CO)given by eq. (7) is the only function that will allow the Maxwell equations to be reduced to the form ofeqs. (8)-(1 1). The picture that emerges, then, is that a portion of the charge induced in the medium by the bare charge comoves +s with the bare charge to form a charged quasi-particle, the effective charge, while the remainder of the induced charge forms a wake in the medium behind the effective charge. ,3 Individual plasma electrons, however, do not comove with the bare charge. Rather, their cooperative motion produces a comoving induced charge. 150
20 August 1984
This description differs from the conventional approach in which the bare charge plus all of the induced charge is treated as a dressed charge. A better understanding of effective charge can be obtained by calculating the effective charge of a charged point particle in matter, both in the limit when the particle is moving rapidly and in the limit when it is nearly at rest. Such a particle with charge q and velocity o has a bare charge density p(x, t) = qf(x - or), where 6(x - ~t) is the Dirac delta. In the former limit, is much greater than a characteristic velocity for particles in tire medium, such as the Fermi velocity in a degenerate electron gas or the mean thermal velocity in a plasma. In that case, the electrodynamic fields mainly contain phase velocities that are also much greater than the characteristic velocity. The dielectric function that describes the dominant behavior of virtually all homogenous isotropic media [3] in the limit of large phase velocities [4,5] is e(co) = 1 - 602/6o 2, where COp = (47rne2/m)l/2 is the plasma frequency of the electrons in the medium; n is the density of those electrons; e and m are the electronic charge and mass. By use of eqs. (1) and (7), it is found that the effective charge density of this particle is
Pc(X, t) = q6(x - o t ) - (q7CO2/ 4rw2 R ) exp(-COpR /c ) , (14) whereR = ( r 2 + 72~-2)1/2,7 -2 = 1 v2/c2,~=z vt, and r is the radial coordinate normal to the z-axis. This charge distribution consists of the point charge and a shielding cloud that is spherically symmetric ,4 about the point charge in its rest frame but is distorted by Lorentz contraction along the z-axis in the rest frame of the medium. The potentials (in the Lorentz gauge) produced by this effective charge are of the Yukawa type:
¢Pe(X, t) = (q~//R)exp(-COpR/C), Ae(X, t) = p¢~e(X, t) ,
(15)
where Ee(X, t) = - V~e(X, t) - (1/c) 3Ae(X, t)/3t, and Be(X, t) = V × Ae(X, t). These fields are short-range and, when quantized, have a field quantum [6] with ,4 The entire induced charge density consisting of the symmetric cloud about the bare charge and the wake, however, is asymmetric.
Volume 104A, number 3
PHYSICS LETTERS
"mass" mp = h60p/C2. In the opposite limit of slow phase velocities, the dielectric function has the approximate [4,5] form e(k, w) ~ 1 + m602/k20, where 0 depends on temperature in a thermal plasma or on the Fermi velocity in a degenerate electron gas. For quasistatic conditions, this implies that Pc(x, t) ~ p(x, t), i.e. that the effective charge is nearly the same as the bare charge. Consecluently, Ee(x, t) is nearly the same as the electric field produced by a bare charge in vacuum. The ordinary electric field, on the other hand, has the longitudinal part
e~(k, 60) = [1 + A(k, 60)(kc/60) 21 - i Ee~(k , co), which, for quasi-static conditions, implies that
E(x, t) ~ - X7[(q /r) exp(-r/?~d) ] ,
(1 6)
where Xd = 60pl (O/m)l/2. This is the Debye-Hiickel screening of the ordinary electric field that one expects in a plasma for quasi-static conditions. It is clear, therefore, that effective charge is created by electromagnetic screening [7] rather than by electrostatic screening as a bare charge moves through a medium faster than the characteristic velocity for the medium. The short-range nature of the effective fields can also be demonstrated in a general way. In order to do this, it is noted that Pc(k, 60) and Ye(k, 60) defined by eqs. (1), (2) and (7) can be expressed as follows:
Pc(k, 60) = p(k, 60) +(602/ 41rc2) [e(60) - 1] ~be(k,co), (17) Je(k, 60) = J(k, 60) + (602/41rc) [e(60) - 1] Ae(k, 60). (18) By substituting these expressions into eqs. (8) - (1 1), one obtains the following set of field equations:
-ik" Ee(k , 60) = 4rtp(k, w) + (602/c2) [e(60) - 1] ¢e(k, 60).
(19)
-ik X Ee(k , 6o) = -(i60/c)Be(k , 60).
(20)
-ik "Be(k, 60)= 0.
(21)
20 August 1984
quency 60 has rest mass mp = (60/i/c 2) [1 - e(60)] 1/2. For large phase velocities, the dominant part of this expression for mp is ~60p/C2, as found above. The relativistic invariance of these results is made clear by writing eqs. (19) - (22) in covariant form:
ikal~e (k, w) = (4n/c)fl(k, 60) -{(60/c) 2 [1 - e(60)] )a~(k, 60),
kaF~e (k, 60) +k#FSea(k, 60) +k 6F~#(k, 60) = 0,
(23) (24)
Feat~ has the form of the usual electromagnetic field tensor except that E e and B e replace E and B, respectively; Jq(k, 60) = (CPe(k, 60),Je(k, 6o)), and A~ (k, 60) = (¢e(k, 60), Ae(k, 60)). Consistency requires that (602/ c 2) [1 - e(60)] be a scalar. That it is a scalar can be verified by noting that kaka = (602/c2) [1 - e(60)] is the disperion relation for a medium with dielectric function e(60), where k a = (60/c, k) is a four-vector in consequence of the phase invariance of plane waves under Lorentz transformation. The validity of the concept of effective charge can be illustrated by calculating the stopping power of an isotropic medium on a bare particle of charge q and velocity t~. According to the picture developed above, one can think equivalently in terms of an effective charge moving at velocity ~ and the wake disturbance in the plasma behind it. The electromagnetic fields can be considered, therefore, to consist of the self-fields of the effective charge and the fields of the wake. If this picture is correct, the stopping power (including the density effect) should be due entirely to the force of the wake fields on the effective charge and should not involve the self-fields. A calculation of the force on the effective charge by the longitudinal wake field (the second term in eq. (12)) yields * s S = (q2/Tro2) f 0
(Im e(60) log F
+ [I e(w)l 2 _ Re e(60)] tan-1
G) 601e(60)1-2 d60, (25)
where
-ik × Be(k, 60) = (4~r/c)Y(k, ~) + ( ~ 2 / c 2 ) [e(60) - 1 ] Ae(k, ~). (22) Eqs. (19)-(22) are generalizations of the Proca equations [8] which imply that the field quantum of fre-
e - ( o 2 / J b 2 m ) [1 - t32e(60)] -1, G = [1 - i~2Re e(60)] -1 ~2im e(60), ,s This will be shown in a subsequent publication. 151
Volume 104A, number 3
PHYSICS LETTERS
and bmin is the usual minimum value of the impact parameter required by the classical calculation. This result is essentially the same as that given by Crispin and Fowler [9] for stopping power including the density effect. One can, therefore, think of the effective charge as being stopped by the wake fields. It would appear, then, that the possibility of coherently enhancing stopping power would depend on having more than one bare charge within the volume occupied by the effective charge of a single bare charge. In that case, the effective charge created by N bare particles of charge q would be approximately the same as the volume occupied by the effective charge of one of them, and the total wake would have approximately the same spatial distribution as it would for only one bare charge. In that case, the stopping power would be nearly the same as given by eq. (25), except thatNq would replace q. The conclusion that coherent stopping power would result from putting N bare charge within a small volume of radius ~ c/~p was also reached by McCorkle and Iafrate [10] on the basis of other considerations. To the extent that coherent ion acceleration by a beam of electrons can be treated as being due to stopping power on ions in the rest frame of the electron beam, it is clear that one needs to consider a clump of ions within a volume approximately equal in size to that of the effective charge created by a single ion in the electron beam plasma. The division of the electromagnetic fields into those produced by the effective charge and those produced by the wake is also advantageous in describing the emission of Cherenkov radiation. Since the electric field of the wake is purely longitudinal, it can not contribute to (transverse) electromagnetic radiation. Rather,
152
20 August 1984
it produces the well-known Cherenkov modes [ 11 ] of plasma oscillation. All of the transverse fields are produced by the effective charge. It is clear, therefore, that all Cherenkov modes emitted as electromagnetic radiation originate within the volume occupied by the effective charge. It can be shown [8] that the sum of the energy lost by the wake to Cherenkov modes in the plasma plus the Cherenkov losses by the effective charge yield the usual expression for Cherenkov energy losses. The author gratefully acknowledges beneficial discussions with Dr. R.A. McCorkle during the course of this work.
References [ 1] M.N. Rosenbluth and N. Rostoker, Phys. Fluids 5 (1962) 776. [2] R.A. McCorkle and J.L. Cox Jr., Phys. Lett. 83A (1981) 440. [ 3 ] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1975), pp. 310-312. [4] J. Lindhard, Kgl. Danske Vidensk., Mat. Fys. Medd. 28 (1954) No. 8. [5] M.G. Calkin and P.J. Nicholson, Rev. Mod. Phys. 39 (1967) 361. [6] A. Bergstrom, Phys. Rev. D8 (1973) 4394. [ 7 ] J. Neufield and R.H. Ritchie, Phys. Rev. 99 ( 1955) 1125. [8] A.S. Goldhaber and M.M. Nieto, Rev. Mod. Phys. 43 (1971) 277. [9] A. Crispin and G.N. Fowler, Rev. Mod. Phys. 42 (1970) 290. [10] R.A. McCorkle and G.J. lafrate, Phys. Rev. Lett. 39 (1977) 1263, 1691. [ 11 ] A.I. Akhiezer et al., Plasma electrodynamics, Vol. 1 : Linear theory (Pergamon, Oxford, 1975).
PHYSICS LETTERS
20 August 1984
ELECTRODYNAMICS OF EFFECTIVE CHARGE
James L. COX Jr.
Department of Physics, Old Dominion University, Norfolk, VA 23508, USA Received 27 July 1983 Revised manuscript received 18 April 1984
A portion of the charge induced in an isotropic plasma by an injected charged particle is shown to combine with the particle to form a charged quasi-particle, the effective charge, that is different from the conventional "dressed" charge. The remainder of the induced charge is found in a wake behind the effective charge. Properties of effective charge are described, and the application of this concept to coherent ion acceleration and stopping power is discussed.
A fast charged particle passing through an isotropic medium such as an unmagnetized plasma has its electromagnetic fields screened by charge induced in the medium in response to perturbation by the particle. A conventional approach for taking this screening into account is to treat the "bare" particle charge plus the induced charge that surrounds it as a single charged quasi-particle, i.e. as a "dressed" charge [1]. Such a dressed charge, however, is not a well-defined quasiparticle: Its current density is not the product of the bare particle velocity and the total (bare plus induced) charge density. A consistent definition is possible, however, in terms of the concept of effective charge * 1 introduced in this letter. The effective charge of a particle is found to consist of its bare charge plus a portion of the induced charge; it is seen to be the physical entity that acts like a moving quasi-charge when a bare charge is injected into matter. Areas of research in which this description should provide fundamental insight include coherent ion acceleration and coherent stopping power, as discussed briefly below. A particle with velocity u and (Fourier-transformed) true charge and current densities p(k, 60) and J(k, 60) acts like a quasi-particle with effective charge density ,1 The effective charge defined in ref. [2] differs by a factor of e(w) from the effective charge in the present work. The present ~eeifieation is required ff the effective charge is to behave as a quasi-particle. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Pe(k, 60) and velocity tJ in an isotropic medium only if relations of the following kind are satisfied:
Pe(k, w) = p(k, w) [1 + A ( k , w)] .
(1)
Je(k, 60) = J(k, 60) [1 + A(k, 60)] .
(2)
[The variables (x, k) and (t, 60) are Fourier-transform pairs; x and t are space-time variables; the function A(k, 60) is identified below.] The required condition that Je(k, 60) = UPe(k , 60) is satisfied since J(k, 60) = up(k, 60). Furthermore, Pe(k, 60) and Je(k, 60) satisfy the continuity equation. These relations can be given physical content by incorporating them into the Maxwell equations for the medium:
- i k " D(k, 60) = 41rp(k, 60). - i k × E(k, 60) = - ( i w / c ) B ( k , w ) . - i k . B(k, w) = O.
(3) (4) (5)
- i k X H(k, w) = (47r/c)J(k, 60) + (i60/c)D(k, 60). (6) By invoking the constitutive relations D(k, 60) = e(60)E(k, 60) and tt(k, 60) = B(k, 60) (for a medium in which/~ = 1) and by making the unique choice ,2 *2 The function A(k, to) corresponding to general dielectric and permeability functions is obtained simply by replacing e(to) by the product e(k, to)9(k, oJ). The choice of • = e(to) and t~ ffi 1 is made for simplicity; the theory develops along similar lines for e = e(k, to) and t~ = 9(k, to). 149
Volume 104A, number 3 A(k, CO) = 6o2 [e(CO)
PHYSICS LETTERS 1] [k2c 2
CO2e(CO)l- 1
(7)
Eqs. (3) - (6) can be reduced to the following form:
- i k . Ee(k , CO) = 47rPe(k , CO).
(8)
- i k X Ee(k, CO) = -(iCO/c)Be(k, CO).
(9)
- i k . Be(k , CO) = 0.
(10)
- i k X Be(k, CO)= ( 4rr/C)Je(k, CO)+ (iCO/c)Ee(k, co) .(1 1) The connections between the effective fields, Ee(k, co) and Be(k , co), and the ordinary fields, E(k, CO) and B(k, CO), are as follows:
Ee(k , CO)= E(k, CO) + A(k, CO)(kc/CO)2E~(k, CO), (12) and
Be(k , CO) = O(k, CO),
(13)
where E~(k, CO)= [k" E(k, ¢o)] k/k 2 is the longitudinal part of E(k, co). This reduction establishes the effective charge defined by eqs. (1), (2) and (7) as a charged quasi-particle: Eqs. (8)-(1 1) show that it, alone, creates electric and magnetic fields in exactly the same way as a bare charge in vacuum does. Once these effective fields are determined by solving eqs. (8)-(1 1), the connections given by eqs. (12) and (13) completely determine the total electric and magnetic fields, E(k, co) and B(k, co). Eqs. (12) and (13) also show that the total electric and magnetic fields decompose into the fields produced by the effective charge plus a purely longitudinal electric field that can be shown to have the form of a wake in the medium behind the effective charge. Since eqs. ( 8 ) - ( 1 3 ) are fully equivalent to the Maxwell equations ( 3 ) - ( 6 ) and the linear constitutive relations, this decomposition has physical significance. Moreover, this decomposition is unique since the A(k, CO)given by eq. (7) is the only function that will allow the Maxwell equations to be reduced to the form ofeqs. (8)-(1 1). The picture that emerges, then, is that a portion of the charge induced in the medium by the bare charge comoves +s with the bare charge to form a charged quasi-particle, the effective charge, while the remainder of the induced charge forms a wake in the medium behind the effective charge. ,3 Individual plasma electrons, however, do not comove with the bare charge. Rather, their cooperative motion produces a comoving induced charge. 150
20 August 1984
This description differs from the conventional approach in which the bare charge plus all of the induced charge is treated as a dressed charge. A better understanding of effective charge can be obtained by calculating the effective charge of a charged point particle in matter, both in the limit when the particle is moving rapidly and in the limit when it is nearly at rest. Such a particle with charge q and velocity o has a bare charge density p(x, t) = qf(x - or), where 6(x - ~t) is the Dirac delta. In the former limit, is much greater than a characteristic velocity for particles in tire medium, such as the Fermi velocity in a degenerate electron gas or the mean thermal velocity in a plasma. In that case, the electrodynamic fields mainly contain phase velocities that are also much greater than the characteristic velocity. The dielectric function that describes the dominant behavior of virtually all homogenous isotropic media [3] in the limit of large phase velocities [4,5] is e(co) = 1 - 602/6o 2, where COp = (47rne2/m)l/2 is the plasma frequency of the electrons in the medium; n is the density of those electrons; e and m are the electronic charge and mass. By use of eqs. (1) and (7), it is found that the effective charge density of this particle is
Pc(X, t) = q6(x - o t ) - (q7CO2/ 4rw2 R ) exp(-COpR /c ) , (14) whereR = ( r 2 + 72~-2)1/2,7 -2 = 1 v2/c2,~=z vt, and r is the radial coordinate normal to the z-axis. This charge distribution consists of the point charge and a shielding cloud that is spherically symmetric ,4 about the point charge in its rest frame but is distorted by Lorentz contraction along the z-axis in the rest frame of the medium. The potentials (in the Lorentz gauge) produced by this effective charge are of the Yukawa type:
¢Pe(X, t) = (q~//R)exp(-COpR/C), Ae(X, t) = p¢~e(X, t) ,
(15)
where Ee(X, t) = - V~e(X, t) - (1/c) 3Ae(X, t)/3t, and Be(X, t) = V × Ae(X, t). These fields are short-range and, when quantized, have a field quantum [6] with ,4 The entire induced charge density consisting of the symmetric cloud about the bare charge and the wake, however, is asymmetric.
Volume 104A, number 3
PHYSICS LETTERS
"mass" mp = h60p/C2. In the opposite limit of slow phase velocities, the dielectric function has the approximate [4,5] form e(k, w) ~ 1 + m602/k20, where 0 depends on temperature in a thermal plasma or on the Fermi velocity in a degenerate electron gas. For quasistatic conditions, this implies that Pc(x, t) ~ p(x, t), i.e. that the effective charge is nearly the same as the bare charge. Consecluently, Ee(x, t) is nearly the same as the electric field produced by a bare charge in vacuum. The ordinary electric field, on the other hand, has the longitudinal part
e~(k, 60) = [1 + A(k, 60)(kc/60) 21 - i Ee~(k , co), which, for quasi-static conditions, implies that
E(x, t) ~ - X7[(q /r) exp(-r/?~d) ] ,
(1 6)
where Xd = 60pl (O/m)l/2. This is the Debye-Hiickel screening of the ordinary electric field that one expects in a plasma for quasi-static conditions. It is clear, therefore, that effective charge is created by electromagnetic screening [7] rather than by electrostatic screening as a bare charge moves through a medium faster than the characteristic velocity for the medium. The short-range nature of the effective fields can also be demonstrated in a general way. In order to do this, it is noted that Pc(k, 60) and Ye(k, 60) defined by eqs. (1), (2) and (7) can be expressed as follows:
Pc(k, 60) = p(k, 60) +(602/ 41rc2) [e(60) - 1] ~be(k,co), (17) Je(k, 60) = J(k, 60) + (602/41rc) [e(60) - 1] Ae(k, 60). (18) By substituting these expressions into eqs. (8) - (1 1), one obtains the following set of field equations:
-ik" Ee(k , 60) = 4rtp(k, w) + (602/c2) [e(60) - 1] ¢e(k, 60).
(19)
-ik X Ee(k , 6o) = -(i60/c)Be(k , 60).
(20)
-ik "Be(k, 60)= 0.
(21)
20 August 1984
quency 60 has rest mass mp = (60/i/c 2) [1 - e(60)] 1/2. For large phase velocities, the dominant part of this expression for mp is ~60p/C2, as found above. The relativistic invariance of these results is made clear by writing eqs. (19) - (22) in covariant form:
ikal~e (k, w) = (4n/c)fl(k, 60) -{(60/c) 2 [1 - e(60)] )a~(k, 60),
kaF~e (k, 60) +k#FSea(k, 60) +k 6F~#(k, 60) = 0,
(23) (24)
Feat~ has the form of the usual electromagnetic field tensor except that E e and B e replace E and B, respectively; Jq(k, 60) = (CPe(k, 60),Je(k, 6o)), and A~ (k, 60) = (¢e(k, 60), Ae(k, 60)). Consistency requires that (602/ c 2) [1 - e(60)] be a scalar. That it is a scalar can be verified by noting that kaka = (602/c2) [1 - e(60)] is the disperion relation for a medium with dielectric function e(60), where k a = (60/c, k) is a four-vector in consequence of the phase invariance of plane waves under Lorentz transformation. The validity of the concept of effective charge can be illustrated by calculating the stopping power of an isotropic medium on a bare particle of charge q and velocity t~. According to the picture developed above, one can think equivalently in terms of an effective charge moving at velocity ~ and the wake disturbance in the plasma behind it. The electromagnetic fields can be considered, therefore, to consist of the self-fields of the effective charge and the fields of the wake. If this picture is correct, the stopping power (including the density effect) should be due entirely to the force of the wake fields on the effective charge and should not involve the self-fields. A calculation of the force on the effective charge by the longitudinal wake field (the second term in eq. (12)) yields * s S = (q2/Tro2) f 0
(Im e(60) log F
+ [I e(w)l 2 _ Re e(60)] tan-1
G) 601e(60)1-2 d60, (25)
where
-ik × Be(k, 60) = (4~r/c)Y(k, ~) + ( ~ 2 / c 2 ) [e(60) - 1 ] Ae(k, ~). (22) Eqs. (19)-(22) are generalizations of the Proca equations [8] which imply that the field quantum of fre-
e - ( o 2 / J b 2 m ) [1 - t32e(60)] -1, G = [1 - i~2Re e(60)] -1 ~2im e(60), ,s This will be shown in a subsequent publication. 151
Volume 104A, number 3
PHYSICS LETTERS
and bmin is the usual minimum value of the impact parameter required by the classical calculation. This result is essentially the same as that given by Crispin and Fowler [9] for stopping power including the density effect. One can, therefore, think of the effective charge as being stopped by the wake fields. It would appear, then, that the possibility of coherently enhancing stopping power would depend on having more than one bare charge within the volume occupied by the effective charge of a single bare charge. In that case, the effective charge created by N bare particles of charge q would be approximately the same as the volume occupied by the effective charge of one of them, and the total wake would have approximately the same spatial distribution as it would for only one bare charge. In that case, the stopping power would be nearly the same as given by eq. (25), except thatNq would replace q. The conclusion that coherent stopping power would result from putting N bare charge within a small volume of radius ~ c/~p was also reached by McCorkle and Iafrate [10] on the basis of other considerations. To the extent that coherent ion acceleration by a beam of electrons can be treated as being due to stopping power on ions in the rest frame of the electron beam, it is clear that one needs to consider a clump of ions within a volume approximately equal in size to that of the effective charge created by a single ion in the electron beam plasma. The division of the electromagnetic fields into those produced by the effective charge and those produced by the wake is also advantageous in describing the emission of Cherenkov radiation. Since the electric field of the wake is purely longitudinal, it can not contribute to (transverse) electromagnetic radiation. Rather,
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it produces the well-known Cherenkov modes [ 11 ] of plasma oscillation. All of the transverse fields are produced by the effective charge. It is clear, therefore, that all Cherenkov modes emitted as electromagnetic radiation originate within the volume occupied by the effective charge. It can be shown [8] that the sum of the energy lost by the wake to Cherenkov modes in the plasma plus the Cherenkov losses by the effective charge yield the usual expression for Cherenkov energy losses. The author gratefully acknowledges beneficial discussions with Dr. R.A. McCorkle during the course of this work.
References [ 1] M.N. Rosenbluth and N. Rostoker, Phys. Fluids 5 (1962) 776. [2] R.A. McCorkle and J.L. Cox Jr., Phys. Lett. 83A (1981) 440. [ 3 ] J.D. Jackson, Classical electrodynamics (Wiley, New York, 1975), pp. 310-312. [4] J. Lindhard, Kgl. Danske Vidensk., Mat. Fys. Medd. 28 (1954) No. 8. [5] M.G. Calkin and P.J. Nicholson, Rev. Mod. Phys. 39 (1967) 361. [6] A. Bergstrom, Phys. Rev. D8 (1973) 4394. [ 7 ] J. Neufield and R.H. Ritchie, Phys. Rev. 99 ( 1955) 1125. [8] A.S. Goldhaber and M.M. Nieto, Rev. Mod. Phys. 43 (1971) 277. [9] A. Crispin and G.N. Fowler, Rev. Mod. Phys. 42 (1970) 290. [10] R.A. McCorkle and G.J. lafrate, Phys. Rev. Lett. 39 (1977) 1263, 1691. [ 11 ] A.I. Akhiezer et al., Plasma electrodynamics, Vol. 1 : Linear theory (Pergamon, Oxford, 1975).