R696 Philips Res. Repts 24,231-240, 1969 TRANSVERSE-VELOCITY DISTRIBUTION FOR SPACE-CHARGE-LIMITED BEAM CURRENT IN AN ELECTRON GUN WITH ROTATIONAL SYMMETRY by J. HASKER Abstract In most cathode-ray tubes the current density at the cathode is a decreasing function of the distance to the axis. Since then the potential field in front of the cathode is curved, velocity selection by this field may occur and we need not find a Maxwellian transverse-velocity distribution with cathode temperature for the beam electrons at emission. This will affect the paraxial-imaging properties which are based on a Maxwellian distribution. In the present paper the distribution function is calculated for a space-charge-limited beam current with the aid of an approximation for the field in front of the cathode. The distribution found is Maxwellian and its temperature is almost equal to the cathode temperature. Moreover, the results obtained can be used to calculate the axial- and the total-velocity distribution at the cathode of the beam electrons. These distributions are of interest for camera tubes and electron microscopes. 1. Introduetion When considering the paraxial-imaging properties of cathode-ray tubes 1.2) the transverse-velocity distribution at the cathode of the beam electrons must be known. Since in most of these tubes the current density at the cathode is a decreasing function of the distance to the axis, the potential field in front of the cathode is curved. Hence, velocity selection by this field may occur and we need not find a transverse-velocity distribution which is Maxwellian with the cathode temperature. The distribution function could be calculated in a rather simple way for small beam currents, i.e. when space charge can be neglected, and we have found a considerable deviation from the usual Maxwellian distribution with cathode temperature in this case 3). Under normal operating conditions, however, we are interested in the velocity distribution for a spacecharge-limited beam current. This problem was left because the potential field in front of the cathode was not known. The calculation of the transversevelocity distribution of the beam electrons at the cathode in the case of a space-charge-limited beam current is the subject of this paper. For this purpose we shall first consider the potential field in front of the cathode. 2. The field in front of the cathode For the calculation of the field in front of the cathode Poisson's equation must be solved, taking into account the initial velocities of the emitted electrons
232 J.HASKER and the finite value of the saturation current density of the cathode 4). To avoid this very complicated problem we employ an approximation of this field, which can be obtained from the approximate beam-current calculation described in ref. 4. We shall consider the gun examined in that paper. For a space-chargelimited beam current of about 100 (J.A it was shown that the current density distribution of the beam electrons at the cathode, calculated with the aid of the model of concentric diodes and Langmuir's theory, can be brought into good agreement with experiments ifthe equivalent-diode distance is chosen correctly. In fig. 1 this calculated current density distribution is plotted against r". Integration of j(r) over the emitting area yields a beam current of 97 fj.a. In a planar diode the potential along a line perpendicular to the cathode can be calculated with the aid of Langmuir's theory when the current density at the cathode of the electrons reaching the anode, the cathode temperature and the saturation current density of the cathode are known. It should be noted that in this calculation the value ofthe diode distance does not matter. Now, starting 0 4- j{ajcm2j i0'3 r. '\ 0'2 0 1 r-, "- r-. r-, 1'-.1:::. o o 80 200.10-6 Fig. 1. Calculated current density distribution at the cathode of a rotation-symmetrical electron gun for a beam current of 97!:LA. The circles are the values of j according to the calculated transverse-velocity distributions which are shown in fig. 7. from the calculated values of j(r) shown in fig. land applying Langmuir's theory for the planar diode, we calculate the potential cp(r,z) 'for various values of r for the above-mentioned system of concentric planar diodes, which are assumed to be independent of each other. Since we use L-cathodes, Tc = l350 "K and I, = 10 AJcm 2 in these calculations. Instead of the actual potential field we shall use cp(r,z) to solve the transverse-velocity problem. Figure 2 'shows cp(r,z) for r 2 = 14.1O- 6 cm 2 U(r) = 0 34Ajcm2)andfig. 3 for r2 = 162.1O-6cm2
TRANSVERSE-VELOCITY DISTRIBUTION IN AN ELECTRON GUN 233 1- _ r=3 72.10 3cm I II I 0 2 V o ZrQ.. ZO... _zeem) 1\, 0 2 0 6,1 21 3 4-5 :1J-iJ 1\, LI, ' I'-CPm Fig. 2. Calculated potential in the vicinity of the cathode for r 2 = 14.10-6 cm 2 (j(r) = 0.34 A/cm2). The dashed curve shows the approximation used in the calculation of the transverse-velocity distribution. 0 4- ()'4- cp (V) t 0 2 o J. 1_3 r=12 72.10 cm Zo" zlll,. 2,4-6 8 10./12 1/;..10"13,, zeem) 0 2 04 0-6,,,,W I /,y \, I, / I:" V f;:-- m Fig. 3. Calculated potential in the vicinity of the cathode for r 2 = 162.10-6 cmê (j(r) = 0.03 A/cm2). The dashed curve shows the approximation used in the calculation of the transverse-velocity distribution. U(r) = 0 03 Afcm 2 ). In fig. 4 the functions zm(r), zo(r) - Zm(r) (see figs 2 and 3) and CPm(r)= cp{zm(r)} are plotted versus r. The calculation of the transverse-velocity distribution for the field cp(r,z) must be carried out with a digital computer. To simplify these calculations, the functions CPm(r), zm(r) and zo(r) - zm(r) are approximated by CPm(r)= CPm(O)+ hl r 2 + kl r B, zm(r)= zm(o) + h2 r 2 + k2 r B (1) (2) and (3) with properly chosen constants. The values of CPm(r), zm(r) and zo(r) - zm(r)
234 J.HASKER 0 3 -ft' -pl 1 V IZo-zm,..."...V l-f.,f /,_ /',... L _,,- i-- ;m,z;,-z, (cm). 4.10- J Î 12 o o 0 o 2 4 6 8 10 12 14 16.1()3 -r(cm) Fig. 4. Calculated functions -rpm(r), zm(r) and zo(r) - zm(r) for 1= 97 fla. The circles give the values according to eqs (1), (2) and (3) used in the calculation of the transverse-velocity distribution. for various values of r according to eqs (1), (2) and (3) are shown by the circles in fig. 4. Next the potential rp(r,z) is approximated by when 0 z z",(r) rp(r,z) = -cp",(r) [{z",(r) - z }/z",(r)] + cp,lr) (4) and by rp(r,z) = -CPm(r) [{z-zm(r)}/{zo(r)-zm(r)}]1i + CPIII(r) (5) when z z",(r). With 0: = 3 5 and fj = 1 6 we obtain the approximate function rp(r,z) shown by the dashed curves in figs 2 and 3. Since rp",(r) and z",(r) are slowly varying functions near the axis, and because z",(o) is small with respect to the distance between the cathode and the crossover (this being about 60.10-3 cm) the calculated potential rp(r,z) must be in good agreement with reality in the vicinity of the centre of the cathode. Beyond this region deviations may occur. With the aid of the calculations described in sec. 3 it will be shown that the influence of these deviations on the calculated transverse-velocity distribution is small. 3. Calculation of the transverse-velocity distribution According to ref. 3, eq. (9), the current of electrons passing the potential barrier and emitted from the cathode with transverse velocities between (2 e (/Jtlm)l/2 and {2 «(/Jt+ d(/jt)lm p/2 is given by 8 6 4 2 dl = [Us elk Tc) {exp (-e 00 " (/Jtlk Tc)} f dr 2 f exp (-e (/Jztfk Tc) d#] d(/jt, (6) 00'
TRANSVERSE-VELOCITY DISTRIBUTION IN AN ELECTRON GUN 235 where (/>::!(r, (/>"{}) is the potential corresponding to the axial emission velocity of an electron emitted at a distance r from the centre of the cathode at an angle {} (fig. 5) and just passing the barrier. "Centre of the cathode z axis Fig. 5. The coordinates used in the calculations. The problem which remains is to calculate (/>::!(r,(/>t,{}) for thefield given by eqs (1) to (5). This is done with the aid ofthe well-known general ray equations in cylindrical coordinates 5): d 2 r/dt 2 = (e/m) è)cp/è)r+ G/r3, d2z/dt 2 = (e/m) è)cp/è)z, (7) where G is the initial value of'r" (d8/dt)2 (fig. 5). The angular velocity at emission can be expressed in the initial transverse velocity Vt: (d8/dt)! = (vt sin {})/r = {(2 e (/>t/m)1!2 sin {}}/r. (8) Since the initial values of r, zand dr/dt (= -Vt cos {}) are known, we can calculate the initial value of dz/dt for which the electronjust passes the barrier; (/>z! is the potential corresponding to this axial velocity. The function (/>Z!(r,(/>t,{)) was determined by means of a digital computer. The calculations were performed for r = 0, 7.10-3, 9.10-3, 11.10-3, 12.10-3 and 13.10-3 cm at each time for (/>t = 0,0'05,0'10,0'20,0'30,0'40 and 0 54 V. The values of (/>!(r,(jjt,{)) = (/>Z!(r,(/>,,{}) + (/>t for the various values of (/>t and r are shown in fig. 6. Next we have calculated dj/d(/>t = Us e/n k Tc) {exp (-e (/>t/k Tc)} J exp (-e (/>ztfk Tc) d{}. (9) o Figure 7 shows dj/d(/>t versus (/>t for the various values of r, The accuracy of the points shown in this figure is about 1 %. For r = 0, 7.10-3, 9.10-3 and 11.10-3 cm the distribution functions are Maxwellian with a temperature which 7t
236 J.HASKER r 1 0 0 9 0 8 0 7 0 6 0 7 0 6 0 5 r=ocm,_- r-.-._ -- 0 3 0 7(/3 27(/3 H 1 3 r=9.1o- i 1 2 3cm 0'54-V (V) 1-1 V V./ / It' V V ).-" r )...-' r: ;>-- --- t-- 1-- t- 1-- 0 4-0V 0'30V 0 20V 0'10V.J..--1-3 r =7.10 t1jj -<\.--< 7"'" - ;>-- -.-.-_ r- t= 0 54.V 0 4-0V 0 30V 0 20V 0 10V OV 7(/3 2H/3 7( r= l'.1o- 3 cm 0 54-V V /... / / VV.Y v.v... Y""' r--r-.- t- r- r =12.1O- 3 cm JV / If!t' /v V 0 54-V 2 2 0 4-0V 2 0 0 5 0 4-1-0,...- 0 9,...-. 0 8 2 4-0 30V 1 8 1 6 JVJV... 0 20V 1 4- V V r- v: >- 0'10V 1 2 r-- r-- V 1-0 OV 0 8 is almost equal to cathode temperature. But it can be seen from fig. 7 that for r = 12.10-3 cm and 13.10-3 cm the distribution functions are not Maxwellian. By integrating dj/dept over cj>twith boundaries 0 and 00 we find the current density at the cathode of the beam electrons for the various values of r. These
TRANSVERSE-VELOCITY DISTRIBUTION IN AN ELECTRON GUN 237 (}1 (Jo02 (}01 (}()07 (}0()4 0-002 r-._ '\. -,,\,,_ ",,\ -, l\_'\ 0_'\ <; r-._"\ <, r-..."\. I\."\. I'\. Tc = 1350 0 K :. ktc/e=(}115v Î'r-..."\'" Î'-'\ '\'iri- "\ l'0 '\ 7 Mdxwellion 1I 9 12 not 13}Maxwellian 0 2 0 4 0 5 0 8 -t(v) Fig. 7. Transverse-velocity distribution at the cathode of the electrons which pass the potential barrier, for various values of r. values of j are shown by the circles in fig.1.except for r = 13.10-3 cm they are in good agreement with the curve in this figure. This shows that the approximation of the field according to eqs (1) to (5) may be used. The distribution function di/difjt according to eq. (6) has been calculated for ifjt = 0, 0,05, 0,10, 0,20, 0,30, 0 40 and 0 054V by graphical integration of dj/difjt over the emitting cathode area (up to r = 13.4.10-3 cm, this being the geometric radius of the emitting area; the rest of the tail gives a negligible contribution). The result of this calculation is shown in fig. 8. The distribution function found is Maxwellian and its temperature is 2 % lower than cathode temperature. Finally, it should be noted that eq. (6) is valid only when all electrons emitted with a transverse velocity Vt = (2 e ifjrfm)1/2 and with an axial velocity Vz > Vz1 = (2 e ifjzi/m)1/2 do pass the potential barrier, while all electrons with V z < Vzl return to the cathode. To check this we have to consider two cases: (1) Vz < Vzl> while r > 0 and -& = O. In this case the tangent of the electron trajectory for z = 0 may be directed towards a point of the potential minimum with a smaller value of I!Pml than when V z = Vzl' Thus it might be possible that, in spite of the smaller value of the velocity at emission, this electron will pass the barrier. However,
238 J.HASKER 1Ö J T I 7 " di/drjt Tc -1350 o li (AV- 4- ---- K"i t(1/e)::: 0 114V T :::0 98Tc Î 2 \ 1Ö* I\. "\. 7 4-2 10-5 7 4- -, '\ '\ -, Fig. 8. Transverse-velocity the screen. 0 20 0-40 0 60 _jft(v) Q.80 distribution at the cathode for the electrons creating the spot on calculation of trajectories for the various values of rand ({Jt with V z = n.o Ol Vzl> where n = 1, 2,..., 99, showed that none of these electrons passes the barrier. (2a) Vz > Vzl> while r > 0 and f) = O. Now the tangent of the electron trajectory for z = 0 may be directed towards a point of the potential minimum with a greater value of IIPml than when Vz = Vzl' Thus it might be possible that, in spite of the greater velocity at emission, this electron does not pass the barrier. (2b) Vz > Vzl> while r > 0 and f} = n. Due to the focussing action of the curved field, it may be possible that these electrons do not pass the barrier. For both (2a) and (2b) we have calculated trajectories of electrons for 100 values of Vz > Vz1 up to tat value of Vz for which the integrand in eq. (6) is one per cent of its value at Vz = Vz1' All these electrons appeared to pass the barier. Taking into account the desired accuracy of the integral, it is not necessary to consider greater values of VZ' From these calculations we may conclude that all electrons emitted with a transverse velocity (2 e ({Jt/m)1/2 and an axial velocity greater than (2 e ({Jztfm)1/2 will be the only ones to pass the potential barrier. This means that eq. (6) may be used.
TRANSVERSE-VELOCITY DISTRIBUTION IN AN ELECTRON GUN 239 4. Discussion of the results The calculations in this paper have been carried out for an approximation of the potential field in front of the cathode. We have found in sec. 3 that this approximation gives the correct value of the current density to within a few per cent except near the edge ofthe emitting area. However, because the current density is very low in this region, this hardly influences the calculated transversevelocity distribution shown in fig. 8. The calculated distribution appeared to be Maxwellian with a temperature which is almost equal to cathode temperature. This result for a space-chargelimited beam current differs from what was found for the retarding-field region of the beam-current characteristic 3). Since the velocity selection was found to have no effect on the transverse-velocity distribution for one value of the beam current in the space-charge-limited situation, this result was considered satisfactory. There is no special reason to suppose that a different result would have been obtained at higher beam currents. Hence, transverse-velocity selection only occurs at very low beam currents. That means that the conclusions of ref. 2, where velocity selection was not considered, remain valid for practically all values of the beam current. In the calculations we have assumed that the motion of a single electron in the cathode region is determined by the electrode potentials and the resultant space charge of all other electrons, i.e. individual Coulomb interactions have been left out of consideration. The ratio of the average Coulomb force and the average space-charge force acting on the electron can be calculated from the average density and velocity of the electrons in the cathode region. This ratio is of the same order of magnitude as for an electron of the accelerated beam in the equipotential space (see ref. 2, fig. I), namely about 1 : 100, thus being small enough to neglect Coulomb interactions. From the results obtained in this paper we may not conclude that the axialand the total-velocity distribution at the cathode of the beam electrons are Maxwellian with cathode temperature. These distributions, which are of interest for camera tubes and electron microscopes, can also be calculated with the aid of fig. 6. This will be the subject of a separate paper. Acknowledgement The author is indebted to Messrs W. P. J. Fontein and D. Bous ofthe Philips Computing Centre for their help in the programming of the computer calculations. Eindhoven, May 1969
240 J.HASKER REFERENCES 1) H. M oss, J. Electronics and Control 10, 341, 1961. 2) J. Hasker and H. Groendijk, Philips Res. Repts 17, 401, 1962. 3) J. Hasker, Philips Rès. Repts 20, 34, 1965. 4) J. Hasker, Philips Res. Repts 21, 122, 1966. 5) R. Courant and D. Hilbert, 1931, Vol. I. Methoden der mathernatischen Physik, Springer, Berlin,