Revista Brasileira de Física , Vol. 6, N? 2, 1976 Electron Charge Distribution in Orbitrons M. PORTO PATO Departamento de Física Nuclear", Instituto de Física da Universidade de São Paulo, São Paulo SP and L. C. GOMES Centro Brasileiro de Pesquisas Físicas **, Conselho Nacional de Desenvolvímento Científico e Tecnológico (CNPq), Rio de Janeiro RJ Recebido em 14 de Junho de 1976 The theoretical assumptions that lead to the determination of the s in- gle electron phase space distribution functions in orbitrons are dis- cussed. The models of Hooverman, Feaks et aí. and Deichelbohrer are analyzed and compared wi th the model used by Ol i vei ra, Pato and Rogé- rio. The ion energy spectrum measurement of Cybulska and Douglas is interpreted on the basis of the model by Oliveira et aZ.. As hipóteses teóricas que levam à determinação da função de distribui- çãono espaço das fases do elétron em órbi trons são discutidas. Os mo- delos de Hooverrnan, Feaks :*t aZ. e Deichelbohrer são analisados e com- parados com o modelo usado por Oliveira, Pato e Rogério. A medida do espectro de energia de ions, de Cybulska e Douglas, é interpretada em base ao modelo de Oliveira et aZ. * Postal address: Caixa Postal 20516, 01000-São Paulo , SP. ** Postal address: Av. Wenceslau Braz, 72, 20000-Rio de Janeiro , RJ. 1. INTRODUCTION Electrostatic electron containment was proposed, in 1963, by Herb, Pauly and Fi sher' using a device consisti ng essential ly of a cyl indri - cal condenser, where the outer cylinder, the cathode, is grounded and the inner cylinder, the anode, is kept at a positive potential,common- ly of the order of 103v. Electrons are injected, in the region between the cylinders, by a filament geometrically arranged in such a way that an appreciable proportion of the electrons leave the i njector region, with sufficient angular momentum to avoid hitting the anode. The fi- lament is maintained at a positive bias for the electrons to be repel- led by the cathode and end plates. This device was cal led an crbi tron, and proposed both as an ionization gauge2 and an icn vacuum pump3. The attractive feature of orbi trons is the large mean free path obtained for the electrons: in a container of 1.2 cm cathode radius, 0.08 cm anode radius and 5 cm height, electron mean free paths of 103 cm are reportedl. The advancement on the construction and performance of operational am- plifiers and the ease of construction of conventional ionization gau- ges have greatly contributed to diminish the concern for the orbi tron as a vacuuni gauge. On the other hand, its use as a vacuum pump has been stabl ished4 due to i ts lower power consumption and faci 1 i ty of i ts construction as compared with conventional magnetically confining de- vices for getter-ion pumps. As a pump, the performance of an orbi tron improves the larger as the total space charge is increased and its dis- tribution more uniformily distributed in space. This fact leads na- turally to the study of the space charge distribution, both from the theoretical as well as experimental, points of view. The earliest theoretical description of the orbitron's electron densi- ty was given by ~ooverman~as a superposi tion of noninteracting charge particle orbits in a logarithmic potential. In his description, it was assumed that the electrons moved in plane orbits, perpendicular to the axis of the orbitron, which slowly drifted in the axial direction. Under these assumptions, one is able to conclude, after investigating the geometry of the orbi ts, and assuming a sharp distribution in angu- lar mornentum, that the radial distribution of the electronic chargehas two maxima corresponding roughly to the apogees and perigees of the plane orbits. This has been the major conclusion of Feaks et aZ6. Continuing in a sornewhat similar reasoning, ~eichelbohrer~has discus- sed the effect on the charge density of different velocity distribu- tions after the electrons leave the injector region. He showed that the electronic distribution is strongly dependent on assumptions made for the axial velocity as well as for the angular momenturn distributi- ons. In particular, the introduction of a spread in the axial veloci- ty had the effect of lowe ring the electronic charge distribution near the cathode, and eventual ly e1 iminating the outer peak shown by the previous authors5. Direct measurement of the electronic charge distribution is a difficult task, and so far has not been reported. Cybulska and ~ou~las'have gotten evidences for the electronic charge distribution by measuring the energy spectrum of posi tive ions that reach the cathode, for ' low pressures, and small electronic total charge. -+ -+ Let us call, f(r,p), the single particle distribution function in pha- se space, and let us suppose no to be the densi ty of neutral gas insi- de the orbitron. The yield, dT/dQ, of ions per unit volume, and time, can be wri tten as where v is the speed of the electron and a the ionization cross sec- tion for the neutral gas under consideration. de are assuming the gas atoms to have negligible speed as compared wfth the electron speed,~. Def ining and n (r') = -eui d3pf -r where n(r) is the electron space charge distribution, we have: Fig. 1 shows schematical ly the arrangement of the different parts of an orbitron. Cybulska's measurement of the ion currents were restricted to the region of the orbitron which is distant from the filament to diminish the effect due to the collection of primary electrons. In the central zone, we may assume the electrostatic field to have cylindri- ca! symmetry, and if we further assume the ions to be produced with negligible kinetic energy, the energy of the ion hitting the cathode can oe equated to the electrostatic potential energy acquired by the ion at the moment of its ionization. Therefore, the ions hitting the cathode, wi th energy between E and E + dE, come from the volume dS2 = 2.rrrZd.r of the cylindrical shell of radius r, length I, of the central zone, where the potential ~(r)satisfies e~(r)= E. We have, therefore, Using Eq. (1.41, one obtains: One sees from Eq. (1.6) that the measurement of dI/dE is di rectly rela- -+ ted to n(r). The potential ~(r)may be considered logari thmic in the limit of very low electronic charge in the interior of the orbitron , and therefore known and control led by the experimental setup. On the other hand, <ov> is sensitive on the model assumed for the electronic distribution in phase space. This fact makes it difficult the inter- pretation of Cybulska's measurement as the derived space charge is mo- de1 dependent. In this paper, we will discuss the theoretical background that led to -+ + the deterrnination of f(rYp), and we consider one model for the expli- cit calculation of f(;,G) which differs substantially from the previous considerations of Hooverman, Feaks and Deichelbohrer. This mdel is applied to the interpretation of Cybulska's experiment. 2. THEORETICAL CONSIDERATIONS Let us consider electrons interacting with an externa1 potential (r), and among themselves by thei r Coulomb interactions. The - 'ext particle Hamiltonian can be written as -+ where r i j = /;i - rjl. The evolution in time of the N-particle system can be described by the -+ + + -+ distribution function fN(pl ,..,,pN ,rl ,...,r ,t ) which satisfies g N Liouvi 1 le's equation The one- and two-particle distribution functions are defined as: Taking into consideration the fact that H involves snly one- and two- particle interactions, and fN is symnetric with respect to exchange of particles, one easily obtains We now make the independent part'icle approximation, writing from where, we obtain, substituting Eq. (2.5) into Eq. (2.4) : whe re + We observe that n(r,t) is the electronic charge densi ty, and V($&) the self-consistent electric potential, that is, the externa1 e lec- trostatic field V ($) togetherwith the fieldproduced by ali the ext other electrons on the electron under consideration. Eq. (2.6) is a particular case of Vlasov's equation for plasmas10. Let us define the single particle Hamiltonian by Then, Eq.(2.6) can be written as af, ah af, ah af, - +- ' - --.-- -o, at a; al; a$ a; , which is Liouville's equation in the single particle phase space 250 We wi 11 now discuss the possible solutions, for E~.(2.101, which des- cribes the orbitron's space charge. Let us begin observing that the -+ -+ dependence of ~(r,t),on fl(G,P,t), given by Eq.(2.8), makes Vlasov's equation nonlinear. If q, is the charge per unit of length in the anode, for maintaining the electrostatic field in the absence of space charge, and q the space charge per uni t of length of the orbi tron, the dirnensionless parameter, is a good measure for the influence of the space charge on V(r) . The charge q, is given by where V. is the anode potential, and b and a the cathode and anode ra- di i, respectively. Assuming Cybulska's orbi tron parameters: V,= 800V, b = 4.5 cm and a = 0.08 cm, and expressing q, in uni ts of the electron charge we get Cybulska's reported value for X is A = 0.08, which gives for the space charge per uni t of length. the low value for 1, in Cybulska's experiment, justifies the linear approximation for Vlasov's equation, what greatly simplifies the cal- culations. -+ + Let us now observe that any function of the Hami 1 tónian h(r,p) is a stationary solution of Eq.