aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug Dimensional Bounds on Vircator Emission
J. I. Katz
1, 2, a)1)
Dept. of Physics and McDonnell Center for the Space Sciences1 Brookings Dr. CB1105 Washington UniversitySt. Louis, Mo. 63130 MITRE Corp. 7525 Colshire Dr., McLean, Va. 22102 (Dated: 22 September 2018)
Vircators (Virtual Cathode Oscillators) are sources of short-pulsed, high power, mi-crowave (GHz) radiation. An essentially dimensional argument relates their radiatedpower, pulse energy and oscillation frequency to their driving voltage and fundamen-tal physical constants. For a diode of width and gap 10 cm and for voltages of a fewhundred keV the peak radiated power cannot exceed O (30 GW) and the broad-bandsingle cycle radiated energy cannot exceed O (3 J). If electrons can be accelerated torelativistic energies higher powers and radiated energies may be possible.Keywords: plasma physics, Vircator, High Power Microwaves a) [email protected] and a wide variety of designs have been developed, comparativelylittle attention has been paid to fundamental limits on vircator performance and its scalingwith driving voltage. These limitsHere we demonstrate upper limits to the radiated power of a vircator in a simple, es-sentially order-of-magnitude, model. Real vircators are more complicated—their radiationpropagates in a waveguide rather than in free space as we assume, their electrons have acomplicated distribution in phase space that must be calculated numerically, rather thanour crude model described by a single length, density and velocity, the potential is also acomplicated, numerically calculated, function, rather than a single static scalar as we as-sume, radiation by semi-relativistic electrons should be calculated numerically rather thanfrom the result for dipole radiation (in the nonrelativistic limit) that we use.Our model for radiated power has only one free parameter, the driving potential, andtherefore leads to a simple result that is likely to bound the power produced in real, morecomplex, vircators. Our bound on the radiated energy also includes the size scale of the diodeas a parameter. It is not possible to derive a single bound for more complete models andcalculations of charges and currents in vacuum diodes (these authors did not calculate theemitted radiation), because they must be described by several independent parameters. Aquantitative bound would require optimization over a several-dimensional parameter space;its origin would not be apparent and its numerical value would depend on detailed geometricassumptions. However, it would scale as our bounds do; only the numerical coefficients wouldbe different. The origin and scaling of our simple bounds are transparent.2n the nonrelativistic limit our model consists of the following relations: d = Qr (1) Q = nr e (2) V = Qr (3) α ≡ eVm e c (4) ω = ω p ≡ s πne m e . (5)Here r is a length scale, Q is the charge of the electron cloud, n is its number density, V is acharacteristic electrostatic potential, d is the magnitude of the oscillating dipole moment, ω p is the nominal electron plasma frequency, which approximates the characteristic frequency ω of oscillation of the electron cloud and α is a dimensionless parameter describing thecharacteristic potential, and is of the same order as the imposed potential between cathodeand anode. The remaining variables are the fundamental physical constants e , m e and c .We assumed that there is only one length scale r , that may be taken as the distancebetween cathode and anode. If the physical cathode is smaller, the electron cloud broadensto approximately this width. If the physical cathode and anode have areas A ≫ r , as ina parallel-plate capacitor, the problem is essentially that of A/r vircators in parallel, andour results should be multiplied by this ratio.Combining these relations and using the relation P = 13 d ω c (6)for the power P radiated by an oscillating electric dipole in free space, we find P = (4 π ) α m e c e = (4 π ) α . (7)The numerical coefficient is 3.3 for α = 0 .
5, representative of vercators in practice. Theexpression for electric dipole radiation is applicable only for α ≪
1, and is expected to bewrong for α > . . As a result, the characteristic width of the pulse of radiation emitted byan instantaneous voltage pulse is ∼ /ω . The radiated energy E = 13 d ω c = (4 π ) / α / ( m e c ) (cid:18) rm e c e (cid:19) = (4 π ) / α / (cid:16) r
10 cm (cid:17) . (8)3lthough the instantaneous power is high, the radiated energy is small. The spectrum hasthe characteristic frequency ν = ν p = √ πα cr = r απ (cid:18) r (cid:19)
30 GHz . (9)These numerical values are consistent with measurements in the nonrelativistic regime .Comparing the energy (8) to the electrostatic energy Q /r of the electron cloud leads toa radiation efficiency ǫ = (4 π ) / α / , (10)valid only in the limit α ≪
1. This is greater than measured efficiencies of vircator radiationfor α ∼ .
5, that are a few percent or less. This may be attributed to use of a roughapproximation of the actual charge distribution and motion and to neglect of radiationreaction, which damps the motion of radiating charges, but which would involve a numberof well-known paradoxes .Analogous estimates are possible in the ultra-relativistic limit. The relativistic form ofthe Larmor expression for the radiation by an accelerated charge, with acceleration parallelto the velocity (the same electric field E gives the electron cloud its relativistic velocity andfurther accelerates it) is P = 23 Q a γ c , (11)where ~a is the acceleration, and γ the Lorentz factor. For ~a k ~va = eEm e γ , (12)so the nonrelativistic result P = 23 Q e c m e E (13)is recovered.Substituting Q = αm e c r/e and E = αm e c / ( er ) yields P = 23 α m e c e . (14)Aside from the numerical factor, this is the same result as Eq. 7 obtained in the nonrela-tivistic limit. The characteristic time scale of emission is now r/c , rather than 1 /ω p , andthe corresponding radiated energy is E = P rc = 23 α m e c e r. (15)4he scaling with voltage is slightly different than that of the nonrelativistic result Eq. 8,but the dimensional factor is the same. The implied efficiency would be ǫ = 23 α . (16)This would lead to the impossible result ǫ > α ≫
1, adiscrepancy that is explained by the neglect of radiation reaction.The fact that the dimensional factors in the results for P and E are the same in thenonrelativistic and relativistic limits is unavoidable. There are only three dimensional quan-tities in the problem, once the electrostatic potential has been scaled to the electron restmass, and therefore only one possible form for power and energy.These are general limits on the performance of vircators, expressed in terms of funda-mental physical constants, that are insensitive to details of design. They are consistent withmeasured performance; vircators readily emit powers of a few GW with driving potentials ofa few hundred kV ( α ≈ . REFERENCES R. A. Mahaffey, P. Sprangle, J. Golden, and C. A. Kapetanakos, Phys. Rev. Lett. , 843(1977). R. B. Miller,
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