THE INFLUENCE OF INITIAL VELOCITIES ON THE BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS

R 585 Philips Res. Repts 21, 122-150, 1966 THE INFLUENCE OF INITIAL VELOCITIES ON THE BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS by J. HASKER Abstract It is shown that some properties of the cut-off voltage and the drive factor of.an electron gun do not agree with the known approximate methods of calculating the beam current. This means that the cut-off voltage Va, corresponding to zero Laplace field strength at the centre of the cathode and necessary for a correct interpretation of measuring results in electron-gun research, cannot be determined from the experimental beam-current characteristic. In the present paper the initial velocities at emission and the finite value of the 'saturation-current density of the cathode are introduced in the approximate beam-current calculation. This leads to an explanation of the experimental behaviour of cut-off voltage and drive factor, to an accurate determination of the cut-off voltage and the "durchgriffs" of anode and final anode (tetra de gun) from experimental beam-current characteristics, and to the conclusion that if the beam current were calculated by means of a numerical solution of Poisson's equation with the aid of a computer, the total velocity distribution at emission and the finite value of the saturationcurrent density would have to be taken into account. A method of measuring the current-density distribution at the cathode for spacecharge-limited beam current is described. At a beam current of 100 ij.a the measured distribution fits the calculated one rather well. Our accurate experimental determination of the cut-off voltage Va makes it possible to examine gun properties at visual cut-off. The current-density distribution at the cathode and the beam current at visual cut-off arecalculated. This current-density distribution in the retarding-field region is essentially different from the current-density distribution at the cathode in the space-charge region. Simple equations for the dependence of the visual cut-off voltage at grid drive and at cathode drive on the anode potential are derived. A method is described to determine the anode "durchgriff" from simple measurements of the visual cut-off voltage. This method may be of interest to relate the spread in the production of cathode-ray tubes to deviations from the face values of the gun dimensions. 1. Introduction In this paper we shall examine some properties of the beam-current characteristic of electron guns and in particular the influence of the initial velocities of the electrons emitted by the cathode on these properties. For this purpose we first consider the schematic representation of an electron gun as shown in fig. 1. The gun can be driven by a. positive voltage Vc on the cathode (cathode drive, Vg = 0) or by a negative voltage -Vg on the grid (grid drive, Vc = 0). A schematic representation of the beam-current characteristic of this gun is shown in fig. 2.. The cut-off voltage Vo ofthe gun is defined as that value of Vc or Vg at which

BEAM CURRENT CHARACTERISTIC OF ELECTRON GUNS 123 Fig. L Schematic representation of the Philips AW 59-91 gun. J Î Vv Va I{: (\Y~ I reta~l ~-charge'region region I. I idrive voltage: \(J i I~ Ilî Fig. 2. Schematic representation of the beam-current characteristic of an electron gun. the field strength, according to Laplace's equation, i.e, when no current is drawn, at the centre of the cathode is zero. If the gun dimensions are known Vo can be calculated from potential measurements on a resistance-network analogue. It is well known that different values are found for cathode drive and grid drive. These values will be denoted by Voe and VOg, respectively. The cut-off voltage can oe written as Vo = Dl Va, + D2 Ve, where Va, and Ve are the potentials and Dl and D2 the "durchgriffs" of anode and final anode, respectively. The values of D, which can also be determined from analogue measurements on a resistance board, will be denoted by DIe, D2e, DIg and D2g for cathode drive and grid drive, respectively. The visual-cut-offvoltage Vv is determined from experiments as follows: at a beam current of a few!lathe beam is focussed in a static spot on the screen

124 J. HASKER in a "dark" room; next Vc or Vg is increased up to the value Vv at which there is just no visible light output from the screen. Different values of Vv a~e found for cathode drive and grid drive which will be denoted by Vvc and VVg, respectively. Due to the fact that the emitted electrons do not leave the cathode with zero initial velocities, Vv > Vo. The maximum beam current, Im, is obtained at zero bias. Obviously, I m has the same value for grid drive and cathode drive. The drive factor k is defined as the ratio" Im/Vo3/2. lts values for cathode drive and grid drive will be denoted by kc and kg, respectively. Clearly, we want to choose the gun dimensions so that k is as great as possible. For electron-gun research and mass production of cathode-ray tubes the following are of interest: (1) Calculation of the beam-current characteristic as a function of gun geometry. (2) Current-density distribution at the cathode in the retarding-field region and the space-charge region of the beam-current characteristic (see fig, 2). (3) Determination of Vo, Dl and D2 from experimental beam-current characteristics. To relate for instance the spread in the production of display tubes to deviations from the face values of the gun dimensions, this experimental value of Vo or Dl can then be compared with the value found from potential measurements on a resistance board. (4) Gun properties at visual cut-off and the relation between Vv and Vo. lt must be possible to determine the value of Vo or Dl from simple measurements of the visual-cut-off voltage. To calculate the space-charge-limited beam current of an electron gun Poisson's equation must be solved. As will be known, there is no simple solution of this problem. In principle a solution can be obtained with the aid of a computer, but in this complicated manner it is not possible to obtain any quick insight into the dependence ofthe beam-current characteristic on gun geometry. Approximative methods must therefore be used. Approximate calculations of the space-charge-limited beam-current characteristic have been carried out by Ploke 1), Francken 2), Moss 3) and Gold and Schwartz 4). Their calculations disregarded the influence of the initial velocities of the emitted electrons and the fact that the saturation-current density of the cathode has a finite value., Comparing the electron gun with a planar diode. Ploke derived an approximate relation for the beam current: where I is the beam current in amperes, Vo is the cut-off voltage in volts as defined above, Vel is the drive voltage in volts (see fig. 2), R is the radius of the grid hole in cm, (1)

BEAM CURRENT CHARACTERISTIC OF ELECTRON GUNS 125 de -is the equivalent-diode distance in cm, Cl is a constant depending on gun geometry. Writing 1= CIVdY we find from eq. (1) that y = 2 5 over the whole characteristic. Francken described a method by which the maximum cathode loading can be' calculated. Use was made of Ploke's equation for the beam-current and analogue measurements on a resistance board. It was found from experiments by Moss that the beam current is proportional to Vd 7 / 2 : (2) From eq. (2) we find that y = 3 5 over the wholecharacteristic.. Gold and Schwartz derived relations for the beam current, y and drive factor of an electron gun. Basics are the equivalent-diode concept introduced by Ploke and analogue measurements on a resistance board. In their relations for beam current and drive factor a constant occurs to account for space-charge effects. In fact this means that beam current and drive factor cannot be calculated. The method can be used to calculate the dependence of y on the drive voltage Vd. and the ratio (cathode-drive factor)j(grid-drive factor), since in these calculations the space-charge constant does not occur. As to y, it is found that y = 2'5 at Vd = 0 and increases with increasing values of. Vd. In practice y depends on Vd and therefore the approximation used by Gold and Schwartz seems to be the most generalone. In sec. 2 we shall give a survey of the Laplace field near the cathode and of a beam-current calculation which principally' agrees with the Gold-and-Schwartz method. This will give rise to a discussion of eqs (1) and (2). Furthermore some properties of the cut-off voltage Vo and of the drive factor k are derived from these calculations. In sec. 3 we show that the properties of cut-off voltage and drive factor derived in sec. 2 do not agree with experiments. In sec. 4 the initial velocities of the electrons at emission and the finite value of the saturation-current density of the cathode are introduced in the beam-current calculations to account for the discrepancy found in sec. 3.. In sec, 5 it is shown that the experimental behaviour of cut-off voltage and drive factor agrees with the theoretical results obtained with the aid of sec. 4. It js found that Vo, Dl and D2 can be determined accurately from experimental beam-current characteristics. In addition the experimental beam-current characteristic and current-density distribution at the cathode are compared with the calculated ones. In sec. 6 the gun properties at visualcut-off areexamined and the current-density distribution at the cathode in the retarding-field region of the beam-current

126 J. HASKER characteristic is calculated. This can be done thanks to the accurate experimental determination of Vo described. in sec.. 5. In sec. 7 we shall derive equations for the visual-cut-off voltage with the aid of the results obtained in sec. 6. The relation between VVDand VVg appears to be quite simple. In sec. 8 a method is described to determine the anode "durchgriff" Dl from simple measurements of the visual-cut-off voltages VVDand VVg. 2. Electrostatic field near the cathode; calculation of beam current The Laplace field near the cathode, i.e. the field when no current is drawn, will be called the electrostatic field in this paper. The electrostatic-field strength at the cathode of an axially symmetric electrode system is a linear function of the electrode potentials. Thus for grid drive we obtain E(r) = -/l(r) Vg +!z(r) Va +/3(r) Ve, (3) where 0, -Vg, Va and Ve are the potentials of cathode, grid, anode and ~al anode, respectively. The functions/l,!z andfa can be determined with the aid of a resistance-network analogue. It has already been shown that the electrostatic-field-strength distribution at the cathode is parabolic over nearly the whole beam-current characteristic 5). Hence we may write /l(r) = pr 2 + q, /2(r) = -sr2 + t, /3(r) = -ur2 + v. (4) The constants p, q, s, t, u and?j can be determined from the analogue measurements. In virtue of eqs (4), eq. (3) becomes E(r) = -(pr 2 + q) Vg + (-sr 2 + t) Va + (-ur2 + v) Ve. (5) For fixed values of Va and Ve we find the geometrical cut-off voltage VOg as that value of Vg for which E(O) = O. Thus where DIg and D2g are the "durchgriff" of anode and final anode, respectively. The geometric radius of the emitting area, Rog, is given by the condition that at the edge of the emitting area the field strength is zero. From eq. (5) and E(Rog) = 0 we obtain (6) and ROg2 = Vdg/{(SVa + uve)/q + (P/q) Vg} = Vdg/(ag + bgvg) (7) E(r) = (Eo/Vog) Vdg (1-: r2/rog2), (8)

" BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 127 where Vag = Vog - Vg and Eo is the field strength at the centre ofthe cathode for zero bias. Now we consider a fixed value VVgof Vg with VVg > Vog. In this case the field strength at the cathode is negative everywhere. Since farther away from the cathode the potential rises again, the potential must have a minimum in front of the cathode. The depth of this minimum on the axis is denoted by Vm(O). Now we calculate Vm(O). The potential in the vicinity of the centre of a flat cathode in an axially symmetric electrode system is given by If Z = 0 we get: (1) (C) V/C)z)r=o = E(O), and hence a = E(O), (2) (C) V/C)Z)r=Ro g = 0, and hence a2 = -E(0)/Ro~2, (3) from Laplace's equation then follows aa = 2E(0)/3 ROg2, (9) so that eq. (9) may be written as V = E(O) z{l + (t z2 - r2)/rog2}. Thus the electrostatic potential on the axis is given by V = E(O) Z (1 + t z2/rog2). (10) At the minimum, C)V/C)z = O. From this condition and eq. (10) we find. zm(o) = (_ROg2/2)1!2. (11) In virtue of eqs (10) and (11) Vm(O) = IV{Zm(O)}1 = -te(o) (_ROg2/2)1!2. Using eq. (8) we obtain Vm(O) = t (Eo/Vog) (VVg - Vog) (_ROg2/2)1!2. (12) It will be clear that zm(o) and Vm(O) can be calculated from the analogue measurements. It must be noted that for VVg > Vog, ROg2 < O. In the case of cathode drive eq. (3) is altered into Er = -{/1(r) +12(r) +la(r)} Vc +12(r) Va +la(r) Ve, where Vc is the cathode potential. We then find and VOe= (Vat + Vev)/(q + t + v) = D1cVa + D2eVe, (6a) Roe 2 = Vac/{(s Va + U Ve)/(q + t + v).n- (p - s - U) Vc/(q + t + v)} = Vac/(ac + heve) (7a) E(r) = (Eo/Voc) Vac (1 - r 2 /Roe 2 ), - (8)

128 J. HASKER where Vac = VO C - so that Vc. In virtue of eqs (6) and (6a), = Dlu/(l + DIU + D2g), + DIU + D2g), DIC D2c = D2g/(1 (13) VOg = Voc (1 + DIg + D2u). (14) The depth of the electrostatic-potential minimum in front of the cathode, existing when Vc = Vvc > Voc, and the distance of this minimum to the cathode can be calculated in the same manner as for grid drive. We find and z~(o) = (-Roc2/2)1/2 (lla) Vm(O) = t (Eo/Voc) (Vvc - Voc) (~Roc2/2)1/2. (12a) In the calculations and experiments which are to follow in this section and secs 3-7 cathode drive will be used. It is remarked that we should find similar results in the case of grid drive. In order to obtain an approximation for the beam-current characteristic of an electron gun, we first consider a planar diode, consisting of two parallel plates with an area of A cmê. Thè distance between the plates is d cm. One of the plates, the cathode, is kept at earth potential and the other plate, the anode, has a potentialof Va volts. Neglecting the initial velocities of the emitted electrons and the fact that the saturation-current density of the cathode has a finite value, the space-charge-limited current in the diode is given by the wellknown Child law I = 2.33.10-6 A Va3/2/d2 amperes. (15) The electrostatic-field strength E in the diode, i.e. the field strength when no current is drawn, is given by E= Va/do (16) Thus, in virtue of eqs (15) and (16), the current density j at the cathode of the diode 'is given by. j = I/A = 2.33.10-6 E3/2/d l /2 (17) In the case of an electron gun the electrostatic-field strength E is known from analogue measurements on a resistance board. Now, following Ploke, eq. (17) is applied to an electron gun. For this purpose the gun is replaced by a system of concentric diodes, all with the same diode distance de. Again following Ploke, this equivalent-diode distance is determined as follows.

BEAM CURRENT CHARACfERISTIC OF ELECTRON GUNS 129 In virtue of eq. (8a) the field strength at the centre of the cathode of the gun is given by E(O) = (Eo)fVoc) VdC; (18) then eq. (18) is compared with eq. (16) and Vdc is identified with Va and Voc/Eo with d and this means that de is given by de = Voc/Eo. Thus, de can be determined with the aid of analogue measurements. Now we write for the anode potentialof the system of concentric diodes as a function of 1': (20) so that the electrostatic-field strength at the cathode of the system of concentric diodes, Vdc(r)/de, equals the electrostatic-field-strength distribution at the cathode of the gun and is given by Clearly, it would be correct to conclude from the comparison of eq. (18) with eq. (16) that de is proportional to Voc/Eo. In fact this is the reason for.the occurrence of the so-called space-charge constant in the relation for the beam current derived by Gold and Schwartz. In our calculations we shall take the proportionality constant equal to unity, but bear in mind that in practice this constant is a function of gun geometry. Now, applying eq. (17) to the system of concentric diodes, the beam current is given by Rac Rac 1= J j(r). 2nr dl' = J 2.33.10-6 {E(r)3/2/de 1 2 / } 2nr dr. o 0 When use is made of eq. (21) we find Comparison of eq. (22) with eqs (1) and (2) shows that in Ploke's approximation Roc2 oc Vdc while in the approximation according to Moss Roc 2 oc Vdc 2. For the Philips television-display tube, type AW 59-91 (fig. I), Roc 2 versus Vdc, calculated with the aid of analogue measurements and eq. (7a), is shown in fig. 3. It can be seen from this plot that Roc2 is not proportional to Vdc nor to Vdc2. A similar result was obtained for other types of guns. Maximum beam current is obtained at zero bias, i.e..for Vc = 0 (Vdc = Voc). Thus, in virtue of eq. (22), (19) (21) (22)

130 J. HASKER x1ötn-z R~ t 80 60 40 Voc=53V _,/ V /./ v V / 1/ 10 20 30 40 50 60 -Vdc(V) Fig. 3. Roc 2 versus Vdc for the AW 59-91 gun with V" = 450 V and V. ",; 18 kv, calculated from analogue measurements on the resistance board and eq. (7a). where Rm is the geometric radius of the emitting area at zero bias. Since the drive factor kc is defined as the ratio Im/Voca/2, Thus, according to the calculation presented in this section kc is a function only of gun geometry since Rm is practically independent of the electrode potentials and also Ca is a geometry constant. Clearly, a deviation of the electrostaticfield-strength distribution from the parabolic approximation used in the calculation will not affect this conclusion. In virtue of eq. (14) we find (23) (23a) From I = CaVacY we find that y is the slope of the double logarithmic plot of the beam current versus the drive voltage: Thus, in virtue of eqs (22) and (7a), y = blog I/b log Vac = (Vac/I) bi/b Vac. y = 2 5 + hcroc2. (24) This means that y = 2 5 at Vac = 0 and increases with increasing values of Vac. For the AW 59-91 gun it was found from analogue measurements and eq. (24) that y = 2 68 at I/Im R::I 0 05 and 3 14 at I/Im = 1. The principal dimensions of this gun are shown in fig. 1.. From the definition of VOc and the calculation of the drive factor presented in this section we conclude that when all electrode potentials are multiplied by a factor n, the cut-off voltage Voc is also multiplied by n while the drive factor remains constant. In sec. 3 these properties will be examined with the aid of experiments on the AW 59-91 gun. 3. Experimental behaviour of cut-off voltage and drive factor For a correct interpretation of measuring results in electron-gun research it is necessary to be able to determine the cut-off voltage VOc, corresponding to zero

- _. ~_._--~--------------. BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 131 electrostatic-field strength at the centre of the cathode, from the experimental beam-current characteristic. As examples we mention the comparison of calculated and measured values of y (ref. 4) and the examination of gun behaviour at visual cut-off presented in sec. 6 of this paper. Obviously, due to retardingfield current caused by the initial velocities.of the emitted electrons, VOc is not the value of Vc for which the beam current is "zero". Therefore we must deal first with the question of how to determine Voc from the experimental beamcurrent characteristic. This determination must be consistent with the calculations and definitions in the preceding section. Since for small beam currents (l/lm -;:; 0 05) Y ~ 2 5, the relationship between 1 4 and Vc is a linear one. lt should be remarked that retarding-field currents are left out of consideration. Then, Voc must be found from extrapolation of 1 4 versus Vc towards the Vc àxis. Figure 4 shows this plot for the AW 59-91 gun with Va = 450 V Voce!y // -: -:./V,/ V -'" / " g ~ $ ~ ~ ~ ~ ~ ~ ~ ~ -lf(v) Fig. 4. Experimental values of 1 4 versus Vc for the AW 59-91 gun with Va = 450 V and V.= 18 kv. and Ve = 18 kv. Thus in fig. 4, Vel V«= 40. The value of the cut-off voltage found from the extrapolation is denoted by VOce. So, if the approximations presented in sec. 2 are used, Voce must be equal to Voc. Then VOce must be multiplied by n when Va and Ve are multiplied by n, i.e. VOce.must equal zero when Va = Ve = O. To check this behaviour the ratio Vd V«was kept constant (= 40) and VOce was also determined for Ve = 12, 6 and 3 kv. Figure 5 shows VOce versus Ve (Ve/Va = 40) and it can be seen that Voce(O) =1= 0, i.e. the value of the cut-off voltage found from the extrapolation of the beamcurrent characteristic is not equal to Voc. This is due to the fact that initial velocities were left out of consideration in sec. 2. In fig. 6 the drive factor lm/voc3/ 2 *) is plotted versus Ve and it can be seen 5 3 2 o *) Voc was determined with the aid of the method described in sec.-s.l.

132 J. HASKER 60 l6ce(v) t 50 40 30 20 10 V 3 Voce (O)=3 07V / 1/ V / L 6 9 12 15 18 - Ve (kv) Fig. 5. Experimental values of Vac.versus V. obtained by scaling of Va and V. with V.I Va = 40 for the AW 59-91 gun. that kc is anything but constant. Thus, this experimental result does not agree either with the theoretical results of the preceding section. Introduetion of initial velocities and tbe finite value of the saturation-current density of the cathode in the calculations wi1llead to an explanation for this behaviour of the drive factor. -, r--- - --. ~ 3 2 o o 3 6 9 m ffi ~ - Ve (kv) Fig. 6. Experimental values of the drive factor kc versus V. obtained by scaling of Va and V. with V.IVa = 40 for the AW 59-91 gun. 4. Approximate calculation of beam current including the influence of initial velocities In tbis calculation the equivalent-diode concept is applied to an electron gun in the same way as was done in sec. 2. Therefore we first consider the spacecharge-limited emission in a planar diode including the influence of the initial velocities of the emitted electrons and the finite value of the saturation-current density of the cathode. Tbe electrostatic-potential distribution in a planar diode

BEAM CURRENT CHARACTERISTIC OF ELECTRON GUNS 133 is shown in fig. 7a, while the potential in the case of electron emission under, space-charge conditions, including the influence of the initial velocities and the I, finite value of the saturàtion-current density, is shown in fig. 7b. The spacecharge minimum in front of the cathode is caused by the fact that electrons are emitted with certain finite velocities. Only electrons with initial ~elocities in the ol/1~-----~~---l d d cathode anode {U SI I ---:positi!>'! of the ttvmmum.l1. Fig. 7. Potential distribution ina planar diode. (a) Potential distribution according to Laplace's equation. (b) Potential distribution under space-charge conditions, including the influence of initial velocities and the finite value of the saturation-current density of the cathode. z-direction greater than or equal to the velocity corresponding to the depth of this minimum reach the anode. When the cathode temperature Tc, the saturationcurrent density js, the diode distance d and the current density reaching the anode are known, the value Va of the anode potential can be calculated with the aid of Langmuir's theory and the well-known g-1] tables 6). This means that the current density reaching the anode at a :fixed value of Va must be determined with the aid of an iteration process. It follows from Langmuir's theory that both Vm and Zm are independent of the diode distance d but are determined by js, Tc and the current density reaching the anode; Vm and Zm increase when the current density reaching the anode decreases, that is when Va decreases. The change-over from the space-charge region of the beamcurrent characteristic to the retarding-field region is reached when Vm = Va; then the space-charge minimum is situated on the anode. When applying Langmuir's theory for the planar diode to an electron gun, transverse-velocity selection in the cathode region 7) is left out of consideration. Again the gun is replaced by a system of concentric diodes, all with the same cathode, and the distance from this cathode to the fictive anode is de (eq. (19)). The anode potential as a function of r is given again by Vac(r) = Vac (1 - r 2 /Roc 2 ), (20) so that Vac(r)/de equals the electrostatic-field-strength distribution at the cathode. From eq. (7a) Roc 2 (Vac) can be calculated with the aid of analogue measurements on the resistance board. Then; for a certain value of Vac, the current density at the cathode within an annular area with radii rand r +. dr can be calculated by means of the iteration process described above. Next, the

134 J. HASKER ------------------------~----------------------------- beam current is found by integration of the current density over the emitting area. At this integration the upper limit rb for r is given by the condition that for r = r» the space-charge minimum is situated on the fictive anode, thus r» > Roe. The described calculation of the beam current can be carried out by hand with the aid of the ~-'YJ tables. Because of the fact that this is a very laborious process the calculation was programmed for the Philips computer Stevin. The results of these calculations will be discussed in the following section. 5. Results of beam-current calculations including the influence of initial velocities We shall consider the beam-current calculations, in accordance with the method described in sec. 4, for the AW 59-91 gun shown in fig. 1. In the experiments we used L-cathodes, hence j, = 10 AJcm 2 and Tc = 1350 "K in the calculations. 5.1. Cut-off voltage With Vel V«_:: 40 the beam-current characteristics were calculated for Ve = 18, 12,6 and 3 kv. From the calculated characteristics, VOce was found by extrapolation in the manner described in sec. 3. Figure 8 shows the calculated values of 1 4 versus Vc for Ve = 18 kv, while the calculated values of the cut- Voee V ~/ 16c= 53 04V V / V 1/ V V _,v o ~ 53 S ~ ~ ~ a ~ ~ ~ -!'c(v) Fig. 8. Calculated values of 1 '4 versus Vc for the AW 59-91 gun with Va = 450 V and V. = 18 kv.. off voltage found from extrapolation, Voce, versus Ve are shown in fig. 9. As in fig. 5, the extrapolated value of the cut-off voltage for Va = Ve = 0 is not equal to zero, so that this effect must be ascribed to the influence of initial velocities. However, it should be remarked that the calculated value of VOce(O) is smaller than the value found from the experiments in sec. 3. This may be due to retarding-field currents near the edge of the emitting area which could 3 2

BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 135 60 20 V ~ A ~ % ~ Vac / liv 1/ Voce (O)~1-41Y o o J 6 9 ~ ~ ~ - Ve{kV) Fig. 9. Calculated values of VOceversus V. obtained by scaling of Va and Vc with Vd V«= 40 for the AW 59-91 gun. not be taken into account in the calculations *). Figure 9 also shows the cut-off voltage Voc, corresponding to zero electrostatic-field strength at the centre of the cathode and determined with the aid of analogue measurements on the resistance board, versus Ve (Ve/Va =40). Itcan be seen that we.obtain Vocversus Ve to a good approximation by shifting VOceversus Ve over the amount VOce(O) towards the origin. Consequently, because of the fact that the curve in fig. 5 is to a good approximation parallel with curve (1) in fig. 9, we shall obtain Voc versus Ve from the experiments in sec. 3 by means of an analogous shift of the curve in fig. 5: In this way the value of the cut-off voltage Voc at Va = 450 Vand Ve = 18 kv was determined for five guns. Before the guns were assembled all principal dimensions were measured. Then the network value of Voc was corrected for deviations from the face values of grid thickness, gridhole diameter, grid-anode spacing, anode-hole diameter and anode thickness with the aid of analogue measurements. In addition it was made clear by separate experiments that the uncertainty in the hot value of the cathode-togrid distance may give rise to a maximum deviation of ± 5 % in the experimental value of Voc. The results obtained for the five guns are listed in table I, where TABLE I gun no. (VOcb - VOcn)/Vocn (%) 1 2 3 4 5 +3 4 ---'6 7 +6 3-2 1 +3 5 *) Correction for the influence of contact potentials would increase tlie discrepáncy by a few tenths of a volt.

136 J. HASKER VOeb is the value of VOe found from beam-current characteristics and VOen the value found from analogue measurements. It is clear that the deviations listed in the table are substantially due to deviations in the' hot value of the cathode-togrid distance. For other types of guns similar results were obtained. Obviously an average- deviation of about zero as shown in the table would not have been found if instead of VOeb the extrapolated cut-off voltage VOceor any visual-cutoff voltage were used. In sec. 5.2 another illustration of the accuracy of our experimental determination of the cut-off voltage will be given. 5.2. "Durchgriff" of anode andfinal anode According to eq. (6a) the cut-offvoitage of an electron gun can be written as where Di«and D2e are the "durchgriff" and Va and Ve the potentials of anode and final anode, respectively. We considered one of the five guns mentioned before and ascribed the deviation of the experimental value of VOewith respect to the value found from analogue measurements to a deviation from the face value of the cathode-to-grid distance. Then, taking into account all deviations from the face values, Die and D2e were determined from analogue mea~urements. Next VOe was determined experimentally for Ve = 18 kv and Va = 750, 450, 225 and 120 V in the manner described in sec. 5.1. Since the value of Voee(O) depends on the ratiovejva this was done by scaling down Ve and Va for VejVa = 24, 40, 80 and 150.In fig. 10 VOeversus Va at Ve = 18 kv, found from the experiments, is shown. From fig. 10 and eq. (6a) the experimental values of Die and D2e Ve were determined. The results are listed in table 11. (6a) tod Vaé(V) 1 80 60 2fJ V /v 7 jl / V o o 2fJO 400 s» f1y) tcxxj -Lti(V) Fig. 10. Experimental values of Vac versus Va for the AW 59-91 gun with VB = 18 kv. It can be seen from the table that this determination of Di» and D2e based on our experimental determination of Vo leads to accurate results. For other types of guns similar results were obtained.

BEAM-fURRENT CHARACTERISTIC OF ELECTRON GUNS 137 TABLE 11 experimental determination from beam-current characteristics DIe = 0 111 D2eVe = 3 31 V analogue measurements DIe = 0 111 D2eVe = 3 43 V 5.3. Drive factor It was remarked in sec. 3 that the drive factor depends on the electrode potentials. As average values for the five guns under examination we found Ve = 18 kv, Va = 450 V : ImjVoc3/2 = 4'99; Ve = 3 kv, Va = 75 V : Im/Voc 3/2 = 6 98. Thus at Ve = 3 kv the value ofthe drive factor is 40% greater than at 18 kv. The calculated values of the drive factor also showed this increase of 40 %. This means that this behaviour of the drive factor is a specific initial-velocity effect. 5.4. Beam-current characteristic and current-density distribution at the cathode I We again consider the selected gun discussed in sec. 5.2. For this gun the constants ae en be occurring in eq. (7a) were corrected for the deviations from the face values with the aid of analogue measurements. Next, ROe 2 (Vdc) being known from the analogue measurements on the resistance board, the beamcurrent characteristic was calculated with the aid of the method described in sec. 4. The result of this calculation for Ve = 18 kvand Va = 450 V is shown in fig. 11, together with the beam-current characteristic found from experiments. Moreover, the deviations of the calculated values of beam current with respect to the measurements are shown in the figure. It can be seen that there is good agreement between the calculated and the experimental values. In fact this means that the choice of the equivalent-diode distance de according to eq. (19) was right for this gun. Nevertheless, we may not conclude that this will be the case also for other gun configurations. The current-density distribution at the cathode was calculated for a beam current of 100!LA.The result ofthis calculation is shown in fig. 12. The abscissa in this figure was chosen so that at x 2 = 100 the value of j is equal to the lje value ofthe maximum current density.it is clear from the figure that the currentdensity distribution at the cathode is not Gaussian. Now we want to compare this calculated distribution function with measurements. This can be done by imaging the cathode on the screen, followed by a measurement of the light-. intensity distribution along the diameter of the cathode image. The measuring

138 J. HASKER Jf 7% IL j +o.",?,. ;k96,t -O.O%V 1- +3.7J11-6.6% ~-e.9% 1% 60 50 4(} 30 2IJ la 0 IfM- 2000 I(J.lA) IBOOi 16001 1400 1200 fooo BOO 600 400 200 o Fig. 11. Experimental and calculated values of beam current for the AW 59-91 gun with! Va = 450 V and Ve = 18 kv; 0: experimental values, X: calculated values. 100 5,( 10 J f 5 - ~- normalized Gaussianf'-e-, -- ~it\tioo function - ~ f... '1 Z, =20mA l- t; If =25 " 17 o If =30 ~ normalized calculated )( Ir =35 " distribution function cif_45, 2 1 o ID 4() ro 00 m w ~ ~ _x 2 Fig. 12. Demonstration of the measurement of the current-density distribution at the cathode for a beam current of 100!lA.

BEAM CURRENT CHARACTERISTIC OF ELECTRON GUNS 139 set-up for such measurements was described in ref. 5, sec. 3.2. The only difference is that the slot of about 40 microns wide in the plate in front of the screen must be replaced by a small hole. It was observed by Moss 8) that a sharp cathode image can be obtained only by running the cathode under temperaturelimited conditions. This was also the visual impression of the present author. However, when the cathode temperature is decreased the original currentdensity distribution at the cathode, i.e. at the cathode image, will be disturbed. Therefore we consider the paraxial-imaging properties of a c.r.t. as described in ref. 5. Going from the cathode to the screen, we first meet the cross-o~er (situated in the gun region), then the cathode image (situated in the equipotential space of the tube) and finally the spot on the screen. The currentdensity distribution in the cross-over is Gaussian, in the cathode image it must deviate from the Gaussian distribution according to fig. 12, and in the spot it is Gaussian again. Thus, going from the cathode to the screen the current-density distribution in the equipotential space must change from Gaussian to a distribution according to fig. 12and then again to Gaussian. This means that we must find the current-density distribution in the cathode image from a measurement of the current-density distribution across the beam at various places in the equipotential space. This was done by variation ofthe current If through the image coil (for If = 0 the spot is focussed on the screen). The measured distributions were normalized in the same way as the calculated distribution function. The results of these measurements are shown in fig. 12. It can be seen from the figure that the cathode image is passed by within the range of imaging currents used. Closer investigation shows that the best fit to the calculated distribution is found at If -::-30 ma. ' ', Finally the normalized calculated and experimental distributions (ft = 30 ma) are shown in fig. 13. For comparison the normalized Gaussian distribution is also shown in this figure. It can be seen that the calculated points fit the exper- 1oo -. ~ r~ ior~/}zed e1p.erim'ental distribution 80 normalized I/~~, 60 - GaussiondistrIbution, j ~ t 40 7J) ~ o o Z 4 6 8 m ~ M m -x Fig, 13. Calculated and measured current-density distributions at the cathode for a beam current of 100 [LA; X : calculated values.

140 J. HASKER imental curve rather well. The smallest values' of j were left outof consideration because with the aid of the equivalent-diode concept we cannot expect to find a correct value for j near the edge of the emitting area. We used the rather small beam current of 100(.Lt,\. to avoid as much as possible the disturbing influence of spherical aberration and space-charge blowing-up on the imaging of the cathode. 6. Gun properties at visual cut-off From the results obtained in' the preceding section it is clear that the cut-off voltage Vo'e can be determined accurately from experiments. This opens the possibility of examining gun properties at visual cut-off. The visual-cut-off voltage Vve of a gun is determined from experiments as follows: at a beam current of a few (.LAthe beam is focussed in a static spot on the screen in a."dark" room; next Ve is increased up to the value Vve at which there is just no visible light output from the screen. We find from experiments that Vvo> Voc, Therefore there exists an electrostatic-potential minimum in front of the cathode at Vc = Vvc (see sec. 2). We consider the ~un discussed in secs 5.2 and 5.4 and first direct our attention to the depth of this minimum on the axis of the gun. Further the current-density distribution at the cathode and the beam current in the case of visual cut-off will b~ calculated. 6.1. Depth of the electrostatic-potential minimum in front of the cathode For Ve = 18 kv the visual-cut-off voltage was measured at various values of the anode potential. Since Eo, Voc and Roc 2 (Vvo) can be calculated with the aid of the analogue measurements on the resistance board, the depth of the electrostatic-potential minimum on the axis in front of the cathode can be determined with the aid of eq. (I2a). The results are listed in table Ill. TABLE III Va CV) Vvc CV) VOo(V) from fig. 10 Vm(O) CV) 750 97 86 2 1 52 450 62 5 53 1 1 60 225 36 28 3 1 55 120 23, 16 5 1 54 Taking into account the accuracy of the measurements of Vvc we may conclude that Vm(O) is constant; its average value is 1 55.volts. Considering the depth of the potential minimum this behaviour is clear. Since Te == 1350 "K, ktc/e =

BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 141 0-116 V SO that ktc/e «Vm(O) and this means that space-charge effects will be absènt, i.e. the current reaching the screen is determined by the depth of the electrostatic minimum in, front of the cathode. In fact, because the emitting cathode area is a function of Va, the depth of the minimum depends on Va. However, the effect of this variation on Vm(O) is small because the transmitted - current is roughly proportional to A exp {-e Vm(O)/kTc} where A is the emitting area. As ktc/e «Vm(O) a very small variation in Vm(O) is sufficient to compensate the variation in A. 6.2. Current-density distribution at the cathode and beam current at visual cut-off We shall calculate the current-density distribution at the cathode of the electrons passing the electrostatic-potential barrier existing in front of the cathode at visual cut-off. Because of the velocity selection due to the focussing action of the curved field in front of the cathode, this is a rather complicated problem. However, this problem has already been solved in ref. 7 for Vm(O) = 0 5 Y while the distance between the cathode and the minimum on the axis, zm(o), was 1.6.10-3 cm. In the calculations we used the properties of an L- cathode. Thus i«= 10 A/cm 2 and ktc/e = 0 116 V. We calculated in that paper the current density, djt, of electrons passing the potential barrier and emitted with transverse velocities between (2ecpt/;n)1/2 and {2e(cpt +' dcpt)/m}112 for various values of the starting point reo) on the cathode and as.a function of CPt.The results of this calculation are given once more in fig. 14. It should be remarked that je' in this figure is equal to stj«: From the results shown in fig. 14 the current density at the cathode can be calculated for the values of reo) shown in the figure by integrating djt over CPtfrom 0 to co. The results of this integration are listed in table IV. ', TABLE IV reo) (cm) o 1.11.10-3 1.57.10-3 2.22.10-3 95.0.10-3 29.2.10-3 8.1.10-3 0.6.10-3 In fig. 15,jt is plotted versus r 2 (0), and it can be seen from this figure that the current-density distribution at the cathode is Gaussian, In the case of visual cut-offfor Va = 450 V and Vc = 18 kv, Vm(O) = 1 55 V and it can be found from' the known value of Vvc - Voc and eq. (lla) that zm(o) = 6.27.10-3 cm. Now, the current-density distribution at the cathode of

142 J. HASKER 4 -, Tc-1J.5Q -«I. ~. --~ I- 911 (f/e) = o.08v :.t: 9J.Q0/{ = "\. ~ -, 5 0-01 I"\. 0.0 ~ "\ O.QO 5 o 040 0020 (Jo30 (}04() 0 0 10 (}o21j!no ().4() 1 1... Î'- 0-2 O 0.05 Tc 1350 /(1== 1== - -....-91t (1/e) 0 095 V - l- "\. :. T -1108 ok I- 1 -, Q: r(o)=o -flit(volts) 12.: r(0)=1.f1.1o- 3 cm -91t(volts) Tc -135QoK ~t&/e)- 0.126 V--=..vt: 1468 K= <, <, <, r-, oo4 0.0~ Tc=1350oK 0.00,... ~twe) -0 320V... T=3725oK (}oooo ().(}O t "\. -, 5 0.000 0.002 (}{)OW o 0 10 Q.20 0 30 ().40 0 0 10 0 20 (Jo30 0-40. : r(0)=1.57.to-.3cm -~t(volts) d: r(0)=2.22.10-acm-91t(volts) Fig. 14. Transverse-velocity distribution for the current density at the cathode of the electrons passing the electrostatic potential barrier; V",(O) = O 5 V and Zm(O) = 1.6.10-3 cm. the electrons passing the potential barrier can be calculated with the aid of the results summarized above and the scaling properties of the problem discussed in sec. 4 of ref. 7. The results of this calculation are listed in table V. TABLE V reo) (cm) o 4.35.10-3 6,15.10-3 1100.10-8 24.1,0-8 0,3.10-8 In fig. 16 the calculated values oîj, at visual cut-off are plotted versus r 2 (0) and it can be seen that at visual cut-off the current-density distribution at the

BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 143 - ~ s=',\ z.~. 2 1Ö \ 1Ö 5 \ z \ J 1\ S 2 'Kj 4 o Fig. 15. Current-density distribution at the cathode of the electrons passing the electrostaticpotential barrier; V1l1(O) = 0 5 V and z",(o) = 1.6.10-3 cm. cathode for the transmitted electrons is Gaussian. It was made clear in sec. 6.1 that at visual cut-off space-charge effects do not play any 'role. Thus in that part ofthe retarding-field region ofthe beam-current characteristic where spacecharge effects may be neglected the current-density distribution at the cathode is Gaussian, i.e. essentially different from the current-density distribution in the space-charge region (sec. 5.4). It is found with the aid of the curve in fig. 16 that jt {reo)} = 1100.10-8 exp {-r 2 (0)j5 03.1O- 6 }. Thus, the beam current ft at visual cut-off is given by It = jjt {r(o)} 2nr(0) dr(o) = 17.10-11 A. o On the other hand this beam current can be estimated from the current density on the screen at which there is only just a visible light output. Because of (1) background illumination in the room during the measurement of Vvc, (2) background illumination during the measurement of Vvc caused by electron emission from the grid,

144 J. HASKER 5 \ Z \ 6 1\ 5 a 7 5 \ 1\ z 1\ Bi- \ 5 10- o 10 '20 JO 4O.1CPcm 2 r2(0) Fig. 16. Current-density distribution at the cathode of the AW 59-91 gun for the electrons passing the electrostatic-potential barrier at visual cut-off; Va = 450 V, V. = 18 kv. (3) subjective and physiological effects in the determination of V vc, (4) uncertainty concerning the thickness of the metal backing, (5) influence of cathode structure on the value of i, to be used in the calculation, we must content ourselves with a comparison of the orders of magnitude. When the beam current is 18amperes the power impinging on the screen at Ve = 18 kv is 1818.10 3 watts. If we suppose that the efficiency of the metal backing is 6 % we find for the light output from the screen 18.103 watts = 518.103 lumens. At small- beam currents the diameter of the focussed spot is about 1 mm, i.e. the area is about 1 mmê. According to 'the theory of paraxial-image formation in a c.r.t. 5) the spot diameter is determined only by cathode temperature. This means that at visual cut-off the area of the screen hit by the beam will be of the order of 1 mmê. Because of the reasons for deviations mentioned above, the current-density distribution over this area is left out of consideration. Hence, the light intensity radiated by the screen is 51 8.10 7 1umensfcm 2 À light intensity of 10-3 lumens/ems is just visible and so we find 18 = 2.10-11 A. This value is of the same order of magnitude as ft.

i BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS 145- Finally we remark that a great variation in It is obtained by a small variation in the value of Vvc - Voc; It varies by a factor of about 3 due to a variation of about 8 % in Vvc - Voc' 7. Visual cut-ofl' as a function of Va for cathode drive and grid drive.. It was shown -in sec. 6.1 that the depth of the Laplace mini'mum existing in front of the cathode at visual cut-off does not depend on Va. Consequently, in virtue of eqs (12a) and (7a) we obtain Vm(O) = t (Eo/Voc) (Vvc - Voc) {(Vvc - Voc)/(ac + bcvvc) 2}1/2. When bcvoc takes the place of bcvvc this equation can be rewritten as Vvc - Voc = {3 VocVm(O)/Eo 2 1 / 2 }2/3 (ac + bcvoc)i/3. (25) For the AW 59-91 gun considered in sec. 6.1 the relative error in Vvc - calculated from eq. (25) is-4 % at Va = 120 V and -2 % at Va = 450 V. This means that eq. (25) is a good approximation for Vvc - Voc' First, we shall simplify eq. (25). For this purpose we consider ac + bcvoc: ac + bcvoc = VOc(ac/Voc + bc). In virtue of eq. (7a), ac/voc = I/Rm 2, where Rm is the radius of the emitting area at zero bias. In practice Rm is independent of Va. Thus, According to eqs (?5) and (26) Vvc - VOc= {3 VOcVm(O)/Eo 2 1 / 2 }2/3 {(l + bcrm 2 )/Rln 2 p/3 Vocl/3 = ccvoc l / 3, (27) where Cc is a constant dependent on gun geometry and cathode properties. In virtue of eqs (27) and (6a) Voc (26) When Voc(Va) and Cc are known, Vvc(Va) can be calculáted. We calculated Cc with the aid of analogue measurements, eqs (5), (6a), (7a), (27) and the value for Vm(O) obtained in sec. 6.1 (1 55 V) and found: Cc = 2 50. Then Vvc was calculated for various values of Va with Ve = 18 kv from eq. (27) and the values of Voc(Va) listed in table Ill. The result of this calculation is shown in table VI.

146 J. HASKER TABLE VI Va (V) Vvc (V) Vvc (V) measured calculated 750 97 97 2 450 62 5 62'5 225 36 35 9 120 23 23 0 Obviously, eq. (28) is a good approximation for Vvc versus Va. Since for grid drive the value of Vm(O) will be the same as for cathode drive, we find, in the manner as described above for cathode drive, in the case of grid drive: Vvo - Voo = {3 VooVrn(O)/Eo 21/2}2/3 {(1 + bor1n2)/rm2}i/3 Vool/3 = cgvool/3 (27a) and In virtue of eqs (27a), (27) and (14) (Vvo - VOo) / (Vvc- Voc) = (1 + DIO + D2o) {(I + boem2) / (l + bcrm2) }I/3. (29) In order to obtain a relation between cathode drive and grid drive, we consider the ratio (1 + borm2)/(1 + bcrm2). When use is made of eqs (6), (7) and (7a), we find where (30) and Rml1 is the radius of the emitting area at zero biás for Va = Ve = 1. Substitution of this result in eq. (29) gives (Vvo - VOo)/(Vvc - VOc) = (l.+ Dlo + D2o)4/3f {I + (l -'-(3) (Dlo + D2o) }1/3. It can be seen from eq. (30) that f3 R::I 1. For the AW 59-91 gun with Va == 45,0V and Ve = 18 kv we found f3 ~ 0 995. Since D20 «DIo'and Dlo is ofthe order, of magnitude of 0'1, the term (1 - (3) (Dlo + D2o) can be neglected with respect to unity and we obtain (Vvo -' Voo)/(Vvc - Voc) = (1 + DIO + D2o)4/3, (31)

BEAM-CURRENT CHARACTERISTIC OF ELECTRON GUNS' 147 In virtue of eqs (14), (28) and (31), the visual-cut-off voltages' VVg and. (Ve is constant) can be written as Vve Vve -. [(DIgVa + D2gVe) + Cg{(DIgVa + D2gVe)/(1 + DIg + D2g)}l/3]/ (1 + DIg + D2g) = [VOg + Cg{Vog/(l +,DIg + D2g)}1/ 3 ]/(1 + DIg + D2g), (32) while VVg = DIgVa + D2gVe + Cg (DIgVa + D2gVe)1/3 = VOg + CgVOi/3. (28a) 8. Determination of the anode "durchgriff" Dl with the aid of visual-cut-off voltages The deviations in Vac, VOg or Dl due to deviations from the face values of the gun dimensions can be determined with the aid of measurements on a resistance board. To relate, for instance, the spread in the production of display tubes to deviations with respect to the face values of the gun dimensions, we must be able to determine VOe, Vog or Dl from experiments on a tube. An accurate method of determining these quantities from measured beam-current characteristics is described in sec. 5 of this paper. However, this method is rather laborious and therefore it is a question of whether these quantities can be determined from simple measurements of the visual-cut-off voltage. To.answer this question we again consider eq. (28) and the results listed in table Ill. Differentiation of eq. (28) with respect to Va gives. dvve/dva = DIe (1 + ce/3 Voc2/3). With Ce = 2'50 (see sec. 7) we find Va=120V: Va = 750V: dvve/dva dvve/dva = 1'13 DIe, = 1 04 DIe. Considering the accuracy of the measurements the difference between these two values of dvvc/dva is too small to enable us to distinguish the measured relationship between Vvc and Va from a linear one. This implies that the constants Ce, Dl and D2, occurring in eq. (28), cannot be determined from the measured values of Vve. According to the values of d Vve/d Va mentioned above, the tangent of the best straight line through the measured points will be about 8 % greater than DIe for values of Va between 120 V and 750 V. However, this deviation cannot be calculated from the measured values of Vvc. Nevertheless Dl can be determined from the measurement of VVg and Vve versus Va. For this purpose both series of measurements are appoxrimated by the best straight lines. The tangents of these lines are denoted by dvvg/dva and dvve/dva

148 J. HASKER Writing dvvu/dva = DIU (1 + LI), we obtain in virtue of eqs (32) and (28a) (33) (dvvu/dva)/(dvvc/dva) = {I + LI/(l + DIu + D2!1)I/3} (1 + DIU + D2u). Since DIu «1 and D2u «DIU this equation gives to a good approximation (dvvu/dva)/(dvvc/dva) - 1 = DIU (1 + t LI). (34) Now DIU can be calculated from the experimental values of dvvu/dva and --- dvvc/dva with the aid of eqs (33) and (34). Figure 17 shows Vvu and V1'( 100 Vv(V) Îeo 6(J 40 20 / V /' V V\l!1 -y' t- I-Vvc.:~/ -: [/ :.-'.) / V / l/ / /' o o 60 120 100?AD 300 360 420 480-16(V) Fig. 17. Experimental values of the visual-cut-off voltage versus Vafor V. = 18 kv; X: cathode drive, 0: grid drive. versus Va (Ve = 18 kv) and the best straight lines through the measured points. In the manner discussed above, we find from the tangents of these lines DIU = 0 142. A determination of DIU from beam-current characteristics, as discussed in sec. 5, delivered DIU = 0 139. Thus there is good agreement between the two methods of experimental de- "