APPROXIMATE EXPRESSION OF THE "THREE HALVES LAW" FOR A BOUNDED CATHODE IN A UNIFORM FIELD

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1 A translation from Atomic Energy of Canada Limited APPROXIMATE EXPRESSION OF THE "THREE HALVES LAW" FOR A BOUNDED CATHODE IN A UNIFORM FIELD (INTRODUCED BY ACADEMICIAN N.N. SEMENOV 3 JULY 1952) by I.I. LEVINTOV translated by J.A. HULBERT Translated from Doklady Acodemii Nauk, S.S.S.R. 85{6): , Chalk River Nuclear Laboratories Chalk River, Ontario February 1973 AECL-4371

2 Approximate expression of the "three halves law" for a bounded cathode in a uniform field* (Introduced by Academician N.N. Semenov 3 July 1952) by I.I. Levintov Translation by J.A. Hulbert (November 1972) Chalk River Nuclear Laboratories Chalk River, Ontario February 1973 AECL-4371 Translated by Atomic Energy of Canada Limited from Doklady Academii Nauk, S.S.S.R. 85(6): , 1952.

3 Approximate expression of the "three halves law" for a bounded cathode in a uniform field* (Introduced by Academician N.N. Semenov 3 July 1952) by I.I. Levintov Translation by J.A. Hulbert (November 1972) ABSTRACT (prepared by the translator) Limiting values are derived, by a qualitative argument, for the modifications that must be made to Langmuir's theory of space-charge-limited conduction in a semi-infinite plane diode, to account for the properties of a real bounded electron stream in a practical diode. For a sufficiently narrow stream the current density r,iay be enhanced by nearly two orders of magnitude over the value for the semi-infinite diode. Chalk River Nuclear Laboratories Chalk River, Ontario February 1973 AECL-4371 Translated by Atomic Energy of Canada Limited from Doklady Academii Nauk, S.S.S.R. 85(6): , 1952.

4 Expression approximative de la "loi des trois moitiés' pour une cathode limitée dans un champ uniforme* (Présenté par l'académicien N.N. Semenov, le 3 juillet 1952) par I.I. Levin to v Traduit en anglais par J.A. Hulbert (Novembre 1972) Résumé du traducteur Des valeurs limites sont établies, par un argument qualitatif, pour les modifications devant être apportées à lu théorie de Langmuir sur la conduction limitée par la charge d'espace dans une diode plane, semi-infinie pour tenir compte des propriétés d'un véritable flux électronique limité dans une diode de type pratique. Pour un flux suffisamment étroit, la densité du courant peut être accrue de près de deux ordres de grandeur au-delà de la valeur établie pour la diode semi-infinie. L'Energie Atomique du Canada, Limitée Laboratoires Nucléaires de Chalk River Chalk River, Ontario Février 1973 AECL-4371 Traduction de l'eacl d'un rapport ayant paru dans Doklady Academii Nauk. S.S.S.R.. 85(6):

5 Approximate expression of the "three halves law" for a bounded cathode in a uniform field* As is known, the "three halves law" (J. Langmuir, Phys. Rev. 2: 450 (1913)) gives an expression for the currewt density, with bounded volume charge, for an infinite plane cathode placed in the field of an infinite plane anode. Below is given an approximate expression for the magnitude j r of the current density on the axis of the beam, in the case of a current with bounded volume charge, with cathode, radius r, having infinite emission capability and being in the uniform field of an anode (voltage) V a at a distance d (see Fig. 1). r i Figure 1 V At the beginning, the case of a beam without space charge repulsion is considered, then by including volume charge and, consequently, r = constant, a practical case is realised including the effect of magnetic focussing. Furthermore we will compute the current density j r and the volume charge p(x) for a uniform beam cross section and electron velocity leaving the cathode equal to zero. Then, the strength of the field on the axis of the beam at a distance x from the cathode will be -2TTJ X * Trans, note: suffix "r" refers to the finite radius r of the beam not a radial variation. -1-

6 where 0 r (x) - potential of the field: the first integral determines the field generated on the axis at the point x by volume charge distributed according to the rule p^rj) with magnitude TJ < x; the second integral determines the field at the point x generated by volume charge distributed with magnitude 17 >x; V a /d determines the field at the anode; K r (x) gives the effect on the field at the anode which is partially screened by charges induced at the cathode and the anode by the volume charge of the beam. For x = 0, according to the limitation of the current by volume charge, Eg = 0 which gives fd.d _ x = K r (0) - 2n *1' p (x) dx = 0 (2) 0 J 0 r L -yr + x z J It may be shown that for > r >0, 2/3 > K r (0) > 1/2 The physical significance of relation (2) is that, for a space charge limited current the normal component of the field on the axis created by volume charge at the surface of the cathode must be equal in magnitude and opposite in direction to the axial field created near the cathode by the anode plane. Moreover, it is necessary for the relation for current continuity to be satisfied: = Jr/r- = constant (3) for which (4) From (2), (3) and assuming ^(x) = = x/d, 0 = r/d, is derived (5) where 2/3 > K^ > 1 /2 for» > 0 > 0.

7 Considering (5) we notice that the solution of our problem comes down to an investigation of the character of the variation of potential along the axis of the beam. For r = <», it is ^^ 2/3 4/3 X 1 /ZeW elementary physical considera- where J<» = 9^ rfny jt ' tions it follows that j ^ <J r wnence from Below it will be shown that 0 r (x) lies between the magnitude of the x undisturbed potential and the potential corresponding to r =, i.e. V a > > consequently the actual current density a: In the investigation of the character of ^r(x) we enquire qualitatively without calculation of the effect of induced charge, i.e. we assume in (1), (2), K r (x) = 1, because such simplification does not change the estimates of ip r (x) derived below. Y? r (x) has the following properties: I. It increases monotonically with distance (C,d). II. Near zero, < r (x) increases as x n, where n>l, which follows from the relation (d0 r /dx)q = 0; n <2 from the requirement of the convergence of the integral in (2). III. I/'J.CX) in the interval (0,d) may have not more than one point of inflection (d 2 0/dx 2 = 0). This follows thus, the field strength d r /dx may have a second extremum for uniform surface field, but with the present distribution of volume charge condition I would then contradict the equation of continuity. IV. For x «r, breaking down the left side of (1), limiting the first term to order x and using (3) we have <% V a -x dr?

8 ^J,, «..if C?TW (7) In view of (2) and II, for x «r - hr " "'% (Tj)dTJ + O(x) (8) By solution of (8) is found " 2/3 4/3 (9) Comparing (9) with (6) and since J M < J f, we note that for x «r, 4> T > ^ for any r/d. In other words for x «r, volume charge disturbs the potential weakly for a bounded cathode, more for an unbounded (cathode). V. We examine the case r/tl «1 and we estimate the form of (d<p/dx) x=c j. From (1), (2), (3) and assuming K r (x) = 1, we have _/d0a _ X ^a r d \dx/ x=d ~ d + d J o $ (10) Assuming 0(x) = Cx n, 1 < n < 2, over the interval (0,d), it is possible to show that for "jj «1 4

9 Comparing ( I dx/ x=d with the appropriate expression for an infinite cathode V dx /x=d " 3d we find that for /3«1 \ dx /x=d \ dx /x=c and since 0 r (d) = 0«,(d) = V a, then it is possible to conclude that also for x» r, volume charge disturbs the potential more weakly for a bounded cathode than for unbounded. For 0 -* 0, the potential for x» r tends to the unperturbed value VI. Reckoning that for small x, 0 r (x) ~ x n, 1 < n < 2, it is possible to show that (d 2 0 r /dx 2 ) is reduced to zero for x ~ r; moreover (d 2 0 r /dx 2 ) x<r >0 and (d 2 0 r /dx 2 ) x>r <O. Consequently (V, III): / d *r\ \ dx /x~r V a d / d a 4 v M 1 * Comparing (11) with I I = - 3 d I ~ I \ dx /x=r we see that, for 0 «1, for example, <0.1, W I 1 >l 1, and, consequently, \ dx /x~r \ dx /x~r for x~r again 0 r<<:d (r) > 0^) From III, IV, V, VI it follows that for 0 «1 volume charge perturbs the potential more weakly for a bounded cathode than for unbounded, over the whole range (0,d). -5

10 Fitting a picture to the distribution of potential for j3 «1, Fig. 2 gives (curve) 2 shown for comparison with the unperturbed potential 1 and 3. Figure 2 o r VII. Considering the case r» d and assuming, according to IV, that in the first approximation <t> T ~ x 4/3, it is possible to show that even in this case v J r>d( x ) > <AJO( X ) in an" the interval (0,d), during which V In the basis of the exposition we produce the hypothesis that ip r always lies between the values of the unperturbed potential and of the potential corresponding to 3 =. We showed that (to be true) for 0 «l./j»l for any j3 while x «r. Thus V a T d (12) consequently, from (5) we have: 27 /2e\ J fl (Jo 1 - l-l (13) where 2/3 >Kp>\/2 while

11 Resulting from IV and V, as an approximate expression, J* for all values of 0 is suggested (to be) It is assumed that 0 < 1; for 0 «1 it follows that only the first integral should be considered and evaluated in the range (0,1). In Table 1 are given estimates of Jg/Joo derived from (13). For the calculated data in Table I it was assumed that K^ = 2/3 for 0» 1 and K^ = l/2for0<l. TABLE I IO- 1 lo" 2 10' 3 V J Expressions (13), (14) remain approximately correct even in the absence of focussing; in that case (13), (14) will give the current density on the axis of the beam near the cathode. Actually, the integral in relation (2) reflects the magnitude of the force of the field at the cathode, which gives the volume charge, and may be written in the following form: where \ x is the linear density of volume charge in a ring element of the beam and S2 X is the solid angle which that element subtends at the centre of the cathode. The physical meaning of this expression is that the size of the

12 component of the field normal to a uniformly charged sheet with charge X x dx is equal to the product of X x dx with the solid angle, J2 X, subtended at the point of observation by the contour of the sheet. Thus, the supposition concerning the applicability of expressions (13), (14) for the present analysis is not violated by the first approximation to relation (2) as it leaves invariant the product X x fi x dx, and is consequently justified. I offer thanks to G.I. Barenblatt and to M.I. Podgoretskii for discussion. Submitted 3 June 1952 Literature cited. 1. J. Langmuir, Phys. Rev. 2: 420 (1913)

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