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3
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0442289994
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On the Currents Carried by Electrons of Uniform Initial Velocity
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A more general study was given by
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(1944)
Physical Review
, vol.65
, pp. 91
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Jaffe, G.1
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4
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84927426381
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See, e.g., R. B. Miller, An Introduction to the Physics of Charged Particle Beams (Plenum, New York, 1982), p. 44;
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5
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84927426380
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J. D. Lawson, Physics of Charged Particle Beams (Oxford Univ. Press, Oxford, 1988), 2nd ed., p. 108;
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9
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84927426379
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The other scales, ns and Js, may be deduced as follows. Associated with the gap width D is a characteristic wave number ks= 1/ D, from which we may construct a velocity scale vs= hbar ks/ m = hbar / mD and a frequency scale ωs= vs/ D = hbar / mD2. If we identify ωs as the scale for the plasma frequency ( e2ns/ m curlep0)1/2, we obtain the number-density scale ns, and the current-density scale Js= ensvs, apart from the numerical coefficients of order unity. Thus, ns is the electron number density above which the electrostatic energy may no longer be ignored, Js is the classical space-charge-limited current density when the applied voltage is of order Vs, and eVs is the minimum kinetic energy of a particle localized to D, as required by the uncertainty principle.
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10
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84927426378
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Numerically, we integrate Eqs. (3) and (4) backward, from x bar = 1 to x bar = 0. The boundary conditions (7) and (8) specify the potential imposed on the gap, whereas (9) and (10) follow from the requirement that the solution at x bar = 1 matches a (preassigned) transmitted wave ψt(x). To integrate (3) and (4), we use (8)–(10) and, in addition, assume a value for V bar prime (1) as initial conditions at x bar = 1. The value V bar prime (1) is adjusted so that condition (7) is satisfied, after integrating (3) and (4) back to x bar = 0. The incident-wave and the reflected-wave amplitudes can then be inferred from the numerical solutions.
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11
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84927426377
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For λ < λq, we find two solutions of Eqs. (3) and (4) satisfying (7)–(10), for specified values of E bar, φg, and lambda. For various reasons, we argue that the one with higher potential energy is inaccessible. In this paper, we focus only on the solutions with lower potential energy. When λ = λq, the two solutions merge. These properties are also shared by the classical theory.
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12
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84927426376
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The present paper essentially treats a nonlinear scattering problem of beam injection into a gap by an external source. It is conceivable that a different physical situation would require different boundary conditions on psi at x bar = 0 that would lead to the interesting possibility of quantization of both E bar and lambda. The existence of such states, and their stability, will be the subjects of a future publication. We should add, however, that regardless of the boundary conditions on psi that would be imposed at x bar = 0, the normalized critical current λq calculated in this paper is still the upper limit.
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