The design of rotationally symmetric triode electron guns

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T H E BESIGN OF ROTATIONALLY SYMMETRIC

TRIODE ELECTRON GUNS

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THE DESIGN OF ROTATIONALLY SYMMETRIC

TRIODE ELECTRON GUNS

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Universiteit Delft, op gezag

van de rector magnificus, Prof. dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan van het College van

Dekanen op donderdag 20 november 1986 te 16.00 uur

door

MARTINUS HYACINTHUS LAURENTIUS MARIA VAN DEN BROEK

Natuurkundig doctorandus geboren te 's-Gravenhage

TR diss

1511

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Dit proefschrift is goedgekeurd door de promotoren Prof. Dr. Ir. K.D. van der Mast

en

Prof. Dr. Ir. H.L. Hagedoorn j

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Preface

The work described in this thesis was carried out, while I worked for the Nederlandse Philips Bedrijven B.V.. It started in august 1979, when I joined the company in the Development Department for transmitting tubes. The basis for the simulation of electron guns was laid by the development of a simulation method for transmitting tubes which incorporated a number of the features of the method described in this thesis. At the end of 1980, I joined the group of Dr. J. Verweel, later of Dr. W.M. van Alphen, at the

Natuurkundig Laboratorium to continue my research in electron optics. The investigated devices changed from transmitting tubes to electron guns for display tubes, but design methods remained the key item. I shared my room with Aart van Gorkum and together we learned a lot about the design of electron guns.

I am greatly indebted to the management of the Natuurkundig Labo ratorium that gave me the opportunity to combine experimental work with theoretical and numerical investigations. They also allowed me to publish

/ the results of my work as a thesis.

A large part of the thesis has been or will be published elsewhere also. The text of the thesis may deviate slightly from the publications. Chapter 3 was published in the Journal of Physics D 19,1389(1986); Chapter 4 was published in the Journal of Physics D .19,1401(1986); Chapter 5 was written by Aart van Gorkum and myself and appeared in the Journal of Applied Physics 58,2902(1985); Chapter 6 was published as a Communication in the Journal of Applied Physics 59,3923(1986); Chapter 7 was published in Optik (Stuttgart) 67,69(1984); and Chapter 8 is scheduled for publication in the 1 november 1986 issue of the Journal of Applied Physics.

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1.0 Introduction

Electron guns are applied in many devices of our technological age. For example, every television set contains an electron gun, and measurement equipment like oscilloscopes could not operate without an electron gun. Although new technologies, for instance Liquid Crystal Displays, are emerging, cathode ray tubes are expected to remain the dominant technol ogy for the display of pictures for at least one or two decades. A thorough understanding of the physics of electron guns is therefore essential.

The theory of electron optics has undergone considerable development since about 1900. However, application of this theory to the actual design of electron guns proved to be difficult. A quotation from Hilary Moss1 may

serve as an illustration: "In 1939, fresh from the university he (Hilary Moss) joined an industrial team engaged in the design of a wide variety of cathode ray tubes. Obviously here was a wonderful opportunity to mount the White Charger and thereby bring the analytic beauty of electron optics to bear on Empirical Procedures. Alas! after about a year, the White Charger had vanished, Empirical Procedures remained in firm possession of the battle field, and a much chastened Hilary Moss might have been observed con templating less ambitious adventures." In 1968, nearly twenty years ago, these sentences were written in the preface of Moss' book on the design of narrow angle electron guns. Moss restricted himself to the design of guns in which geometrical aberrations could be ignored. The guns used in oscil loscope tubes belong to this class, but the guns in picture tubes do not. The design of electron guns in picture tubes is also more complicated than the design of guns in oscillope tubes by the action of the space-charge density as a negative lens with spherical aberration.

This thesis is concerned with the design of rotationally-symmetric tri-ode electron guns with a flat cathtri-ode. Spherical aberration and the lens ac tion of the space-charge density are taken into account. Its purpose is the same as that of Moss' book; to form a bridge between electron-optical the ory and the actual design of electron guns. The 'analytic beauty' of electron optics is still not sufficient to design electron guns, but this thesis shows that the possession of the battlefield by purely empirical methods has become less firm. The maturing of two developments allowed us to continue where Moss stopped. First, phase-space or emittance diagrams are now frequently used to describe electron beams. In a phase-space diagram, the angle of a trajec tory with the beam axis is plotted as a function of its radial position. In an emittance diagram, the electron density is given as a function of the position and the velocity or the angle with the beam axis. Phase-space or emittance diagrams are an excellent tool to analyze a beam propagating through lenses with or without spherical aberration, or drifting under influence of its own space-charge density, since they focus the attention to the properties of an electron as a whole. Moss made this point already in his book, when he

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re-G R D 3 re-GRID 4 TARre-GET

PREFOCUSNQ MAN LENS LENS

Fig. 1: Schematic drawing of the trajectories in a typical triode electron gun.

marked that electron optics was concerned more with the properties of in dividual rays than with the properties of the beam, whereas the properties of the beam as a whole are important for the designer.

The second important development has been the increase in computer power during the past decades. Tedious analytic calculations can be done rapidly by a computer and numerical techniques for the solution of ordinary differential equations or the calculation of definite integrals allow to obtain results, where analytic solutions fail. Moreover, the simulation of electron guns became possible. We can calculate currents, spot sizes, and focusing potentials with an accuracy better than 10% and simulation can replace a large number of experiments, allthough not all. A second advantage of a simulation is that it allows the designer to look into aspects of his design that are inaccessible to an experiment. A lot of understanding can be gained from simulations in this way. In the past, an engineer made a design based on empirical data and rules of thumb, and he checked it by means of ex periments. Nowadays, the designer still has to do his job, but he starts using a combination of analytic and numerical methods to make a design. Next he checks his design by means of simulations and, if he is content with the results, he does a few experiments. As a consequence, the design time can be reduced significantly.

We will now briefly discuss the regions of the triode gun which deter mine the imaging characterics. Figure 1 shows a schematic drawing of the trajectories in a triode electron gun used in display tubes. We can distinguish five different regions in such a gun. The first is the beam-forming region in which the beam is generated. It is called a triode and consists of a flat ca thode, a first grid biased negatively with respect to the cathode, and a sec ond grid biased positively. The current is modulated by changing the potential difference between the cathode and the first grid. The potential

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field in front of the cathode acts as a positive lens, the cathode lens, and focuses the electrons in a crossover. The size of the crossover is determined by the spherical aberration of the cathode lens and the initial velocities of the electrons at the cathode. A divergent electron beam emerges from the triode. The second region is formed by the prefocusing lens, that adapts the aperture angle of the beam emerging from the triode to the main lens. In the third region, the beam travels from the prefocusing lens to the main lens. The space-charge density acts as a negative lens that has positive spherical aberration if the space-charge density is inhomogeneous with a maximum at the axis. The fourth region is formed by the main lens that images the crossover at the target. The spherical aberration of the main lens often contributes significantly to the spot size at the target. Finally, in the fifth region the beam travels from the main lens to the target. Here, the space-charge density again acts as a negative lens and sets a lower limit to the spot size that can be attained at the target.

The different regions briefly discussed above are treated in Chapters 3 to 7, but first Chapter 2 treats in detail a few topics from electron optical theory: Langmuir's law in a planar diode, linear electron optics (the law of Helmholtz-Lagrange and the use of linear transformations to describe lenses), and lenses with spherical aberration. The use of phase-space dia grams is explained also. Chapter 2 is meant to provide background to the subsequent chapters. Readers familiar with electron optical theory can skip this chapter.

Chapters 3 and 4 are concerned with the triode. Chapter 3 describes a combination of analytic and numerical methods to calculate the radius of the emitting area and the current. In the calculation of the current, the initial velocities of the electrons are taken into account (Langmuir's approach) or ignored (Child's approach). A comparison is made between measured and calculated currents. Chapter 4 describes the measurement of emittance dia grams for different geometries, currents, and current densities at the cath ode. The beam aperture angle, the crossover size caused by the spherical aberration of the cathode as well as that caused by the initial velocities of the electrons can be extracted from the emittance diagrams. Phenomeno-logical relationships are presented from which the beam aperture angle and the crossover sizes can be derived as a function of the geometry and the current. Chapters 3 and 4 present a coherent framework to describe the properties of the triode over a wide range of geometries and potentials.

Chapter 5 is devoted to the prefocusing lens. It describes in detail the properties of the so-called selective prefocusing lens that acts close to the crossover and refracts the trajectories selectively. The spot size is shown to be reduced 46% by the use of a selective prefocusing lens instead of an 'in tegral' prefocusing lens which acts at beam diameters that are large com pared to the diameter of the crossover. The beam is described by phase-space diagrams. The prefocusing lens and the main lens are repres ented by transformations of these phase-space diagrams. Chapter 5 shows the power of the phase-space diagrams for the description of electron beams in an imaging electron-optical system.

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Chapters 6 and 7 treat the influence of the space-charge density on a drifting beam. In the former, perturbation theory and the description of the beam with phase-space diagrams are used to estimate the increase of the crossover size caused by the spherical aberration of the space-charge lens. In the latter, we show numerically calculated graphs that give the minimal spot size at the target for beams with a homogeneous or an inhomogeneous space-charge density distribution and with initially a finite or an infinite brightness.

Chapter 8 is devoted to the simulation of triode electron guns. It treats in detail a new method to calculate self-consistently the potential field in front of the cathode that takes into account the initial velocities of the electrons. The space-charge density is calculated with a fitting technique. Data of a few trajectories are used to estimate interpolating polynomials. The space-charge density can be calculated rapidly by the substitution of a large number of initial conditions in these polynomials. Emphasis in this chapter is put on a sound mathematical derivation of the applied methods and on the minimazation of the computer time. The simulation yields the current, the spot size, and the focusing potential. The calculated values are compared with measured ones. The simulation is sufficiently accurate to replace a large number of experiments.

Finally, Chapter 9 summarizes the design rules encountered in the preceding chapters.

References

1. H.L. Moss, "Narrow angle electron guns and cathode ray tubes", in Advances in Elec tronics and Electron Physics, Suppl. 3, Academic Press, New York and London, 1968

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2.0 Elementary electron optics

2.1 Introduction

This chapter serves two purposes. First, it presents the theoretical background to and an explanation of a few methods used in the following chapters of this thesis. Second, it supplies some important formulas for the design of electron guns that are not treated elsewhere in this thesis. This chapter treats the different topics rather concise. Readers who are not fa miliar with electron optics can consult the textbooks given in Refs. 1 to 4.

Section 2.2 describes the theory of thermionic emission that takes into account the finite saturation current density of the cathode and the Max wellian velocity distribution of the electrons leaving the cathode. This the ory was developed by Langmuir5. In electron-optical design, a more simple

model of the thermionic emission is often used that ignores the Maxwellian velocity distribution and the finite saturation current density (Child's ap proach)6 . Child's approach will be shown to yield erroneous results.

Section 2.3 presents some results from the theory of linear electron optics. The Helmholtz-Lagrange relation between the spot size, the aperture angle, the potential at the target, the radius of the emitting area, and the cathode temperature is derived. The linear transformations for the propa gation of a beam in an equipotential space and for a lens acting between two equipotential spaces are given. Phase-space diagrams, expressing the angles of the trajectories with the beam axis as a function of the position, are in troduced. The use of the linear transformations and the phase-space dia grams is illustrated.

Section 2.4 discusses the most important defect of the electron-optical lenses used in triode electron guns: spherical aberration. The phase-space diagrams are applied to beams with spherical aberration. A useful result that gives the minimum spot size for given brightness of the beam and spherical aberration of the final lens is presented. The spot size is shown to be equal within 10% for bipotential, unipotential, and magnetic lenses with the same diameter over a certain range of magnifications. These results were previ ously presented in Ref. 7.

2.2 Thermionic emission

The electron source in many electron-optical systems is a hot cathode of about 1100 K. The electron density/, at the surface of a flat cathode placed in the origin is given by

2 -2

fc(x,y, Ó = Tjsat e x p ( — — ) , (1)

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where jsal is the saturation current density that depends on the physical

properties and the temperature T of the cathode.8 The electrons cause a

space-charge density p which contributes through the Poisson equation

AV=-ple0 (2)

to the potential field V.

Langmuir5 discussed the solution of the Poisson equation in a one-di

mensional diode with p calculated from Eq. 1. Earlier, Child6 derived a so

lution of Eq. 2 in a one-dimensional diode, assuming that the cathode emits the electrons with zero velocity. The current density settles itself such that the fieldstrength in front of the cathode vanishes. The resulting expression for the current density jch is

Jch (3)

where d is the distance from the cathode to the anode and Vd is the potential

difference between the anode and the cathode.

Let us return to the solution based on Eq. 1. Three distinct curves that give the potential as a function of z can be distinguished and they are shown in Fig. 1. The first curve decreases monotonously between cathode and an ode. Most electrons are reflected back and only a fraction

• ■ reVd, Jla=JiataP(-j-jr) _Cathode (4) Anode re >

Z(a.u.)-Fig. 1: The potential in a one-dimensional diode for three different types of emission: blocked emission (solid), space-charge limited emission (dashed), and saturation-limited emission (dot ted). Unlike the two other types of emission, the space-charge limited emission reveals a poten tial minimum.

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reaches the anode. The set of Rvalues for which we find curves like this is called the blocked region. The second curve decreases, reaches a minimum at z = zM with a potential value Vu, and then increases monotonously. The

current density reaching the anode is determined by the depth of the mini mum and is given by

eVM

Jla=JsatexP(-£jr)- (5)

The emission is space-charge limited for the values of Vd for which a po

tential minimum occurs. The third curve increases monotonously between cathode and anode. All emitted electrons are able to reach the anode, the current density is equal to jsa„ and the emission is saturation limited. Most

display tubes operate with a space-charge limited emission for reasons that will become clear after a more detailed treatment in the remainder of this section.

The space-charge density at an arbitrary point in the diode is given by

p(l)=jdr/\r,r), (6)

where ƒ is the electron density at an arbitrary point. Liouville's theorem re lates the electron density at the point r to that at the cathode

/ ( r , r ) = /cM r , r ) , i ( r , r ) ) , (7)

where r, and t are the position and velocity at the cathode, respectively, of the trajectory with the velocity r at the point r. Liouville's theorem may be applied when the force on the electrons caused by their charge can be de rived from a potential field, in which case the statistical fluctuations in the force are ignored. At the cathode, the electron density does not depend on the position, so that £. is irrelevant. According to Eq. 1 fc depends on g,

which is related to r2 by the conservation of energy. Thus Eq. 6 can be

written as

P(l) = ™ j m e x p ( | £ ) f dr e x p ( - ^ - ) , (8)

2n(kJT kT J 2kT

where V is the potential at the point r. The integrations of x and y extend from — oo to oo. For the integration of i , we have to treat separately the region between the cathode and the potential minimum, and that beyond the potential minimum. Between the cathode and the potential minimum, there are electrons running from the cathode to the minimum with velocities ranging from 0 to oo, and electrons that are reflected back by the minimum with velocities ranging from zM = - Jl-jpriV— KM) t o z e r o- The

space-charge density p. between the cathode ana the potential minimum is given by

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2 jr foo poo poo «2

P-(r)= m 27 ,a te x p ( ^ ) x rfi dx\ dy e x p ( - ^ r )

27r(/cr) — -ZM

Evaluation of Eq. 9 yields

P - t ó = ( ^ r )1 / 2^ e x p ( - | ^ ) 1+<D|

where O(x) is the error function

<S>(x)=-j=\Xdy exp(-j2 )-•V/TT Jft (9) 6 ( K - FM)

&r

(10) (ii) V7I •'O

Beyond the potential minimum, the values of z range from Ev i ^~m^~ ^ ^

or ele c t r o n s w't n z e r o velocity at the minimum to oo.

valuation of Eq. 8 yields for p+, the space-charge density beyond the min

imum, — / W7C AJ2; , v / »»«. \ l / 2 . , « ) ' , P+ fe) = ( - ^ r ) AW e xP(l^r) A:7" 1 -

»(7

e(V-VM) kT (12)

For the solution of the Poisson equation in a one-dimensional diode, the dependence on x and y can be ignored. It is convenient to transform z and

V to the dimensionless coordinates £ and r\ given by

^ = (2,)

1/

V

1/2

(^)-

3/

U)-

1

/

4

^

2

(z-z^)

and

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"

=

I T

( K

~

F M )

-If we substitute Eq. 5 for j , the Poisson equation reduces to ,2 * 2 R ' t x rt»rM'/2\ = ^ - [ l + 0 ( ^ ) ] (14) (15) where + and - denote the regions beyond the potential minimum and be tween the cathode and the minimum, respectively. Multiplication of Eq. 15 by

drj dt]±

dt, and subsequent integration yields

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which vanishes for r/± = 0 as should be the case in the potential minimum.

Equation 16 cannot be integrated analytically, but its solution has been ta bulated, for example in Ref. 9.

The current density for given diode distance d and anode potential Vd

can be obtained from the tables with the following procedure. An estimate of the current density j gives through Eq. 5 the depth VM of the potential

minimum and thus the value of r\_ at the cathode. The table yields the cor responding value of £, and thus the distance zM between the minimum and

the cathode. Subsequently, the distance from the minimum to the anode and the corresponding value of ^ are calculated. The table yields the value of f/+, from which the potential of the anode can be calculated. The calcu

lated anode potential will not be equal to Vd for the first estimate of ƒ We

estimate a new value ofy' and repeat the whole procedure until the calculated value of the anode potential agrees with Vd.

Instead of the tables, we can also use the asymptotic series expansions of £ in terms of rj_ and q+. Beyond the potential minimum, we find

3/2

É = 4r-7r1/4>7+/4 + 1.66854>;!/4 + 0.50880 - 0.1677»?;1'4 + - . (17)

Substitution of the first two terms in Eq. 13, followed by squaring and combination with Eq. 14 yields

IT (v

d

- v

M

)

3

'

2

f, . _ / IT \

7=

^^-^rrrl

{d

_

ZM

f \ V e(V

1+2

-

6 V-

d

-V

M

) )

The first term of Eq. 18 is Child's law (Eq. 3) if we ignore VM and zM. In

Chapter 3, we will use Eq. 18 to estimate the current in triode electron guns, ignoring VM and zM with respect to Vd and d.

Between the cathode and the potential minimum, the series expansion is

{ = -2.55389 + y/Ye~r'-12 - 0.0123e~ *- +

_Lj./»p

+

,y^-_. e»

3vrv

v

" /

Substituting rj = log(/ra,/y') and using only the first three terms, we find an

approximate expression for zM, that is valid for values of jljsal that are not

too close to one.

2 . 5 5 3 8 9 . / ^ _V2~ + 0.0123., U— V J V Jsai

'

M= {

2^%>i\J£

r

ii\^r

ll4

j!l

2

'

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The treatment of the space-charge limited emission is rather theore tical up to this point. Let us therefore look at an example. We consider a planar diode with a diode distance of 100 urn. The cathode has a

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temper-10°

' 10

10

2 I I I I 1 0 ~31 . i i i i

0.0 20.0 40.0 60.0

V

D

(V)

Fig. 2: The current density versus the potential in a one-dimensional diode with a diode distance of 100 urn. Solid line: Langmuir's approach with T = 1160 K andyM,= 10 A/cm2, dashed line:

Child's approach (Eq. 3), encircled plus signs: Eq. 18, crosses: Langmuir's approach with r = 1160 K and ./„„ = 20 A/cm2, and plus signes: Langmuir's approach with T=800 K and jsat

= 10 A/cm2.

ature of 1160 K and a saturation current density of 10 A/cm2. Figure 2

shows the current density based on Langmuir's and Child's approach as a function of the anode potential. The difference between both curves in creases with decreasing current and amounts to a factor of two at

jjjsa,~0.04. Since up to such values the current density still contributes sig

nificantly to the current in triode electron guns, Langmuir's approach, tak ing into account the initial velocities and the finite saturation current density, has to be used to obtain correct values. Moreover, even close to jsa,

a difference of about 10% exists between Child's and Langmuir's approach. The initial velocities of the electrons lower the space-charge density in Langmuir's approach. Figure 2 shows that Eq. 18 is a good approximation to Langmuir's approach (encircled plus signs), and that j depends only weakly on^a/ (crosses) and T (plus signs). Since the dependence of; onytfl,

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Fig. 3: The depth of the potential minimum (dashed) and the distance of the potential minimum from the cathode (solid) in a one-dimensional diode with/,<,,= 10 A/cm2 and T- 1160 K. The

plus signs give the distance of the potential minimum from the cathode calculated with Eq. 20.

of the cathode if the emission is space-charge limited. This is the reason why most electron guns are used with a space-charge limited emission.

Figure 3 shows the distance between the potential minimum and the cathode and the depth of the potential minimum as a function of j/jsal. The

plus signes are calculated with Eq. 20, which turns out to approximate zM

very well. The distance of the minimum to the cathode increases rapidly with decreasing current density to values as large as 30 to 50 fim. The po tential field in front of the cathode calculated with Langmuir's approach will differ greatly from that calculated with Child's approach. It was shown experimentally that Langmuir's approach yields the correct potential field8

. Chapter 8 describes a numerical method to calculate self-consistently the potential field in front of the cathode, in which the initial velocities of the electrons (Eq. 2) are taken into account.

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2.3 Linear electron optics

In this section, we shall linearize the equations of motion of the elec trons in a potential field and derive some useful results of this linearization. We use Cartesian coordinates with the z-axis along the axis of rotational symmetry. The potential V can be expanded in a series of powers of

x2 +y2

OO

F (

W

) = £V

2

,.(z)(*

2

+/)''• (21)

If V obeys the Laplace equation, V2i{z) is given by

2 (il)

where PJ2,,(z) is the 2i'-th derivative of V6(z). Using in the equations of mo

tion only the first relevant term of the series expansion, we find

mx = 2exV2(z), (23a)

my = 2eyV2(z), (23b)

mz = eV0(z), (23c)

where a dot denotes differentiation with respect to time and a prime with respect to z. The equations in the x- and ^-directions are independent of each other and the equation in the z-direction is independent of the equations in the x- and _y-directions. In the remainder of this section we will discuss only the motion in the x- and z-directions. Equation 23c can be in tegrated separately and its solution can be substituted in Eq. 23a, which becomes the linear equation

x =f(t)x. (24)

The function ƒ depends on the initial z-velocity. As the initial kinetic energy of the electrons is of the order of 0.1 eV, this dependence will be very weak in most triode electron guns.

Usually, one first eliminates / to obtain a differential equation for the trajectory with z as independent variable. Subsequently, one linearizes the trajectory equation using the series development of Eq. 21 for the potential and assuming small angles with the beam axis. In contrast to the usual as sumption of limitation to small angles with the beam axis, we assumed the run time of all trajectories to a certain z-plane to be independent of the ini tial conditions in x- and ^-direction. Our assumption is violated if kinetic energy in the z-direction is converted into kinetic energy in the transversal direction along the trajectory. However, the increase in run time will be

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small when the angles with the beam axis are small. The advantage of li nearization of the equation of motion with respect to time is that angles caused by the initial velocities in the transversal direction do not spoil the linearization, as is the case with the linearization of the trajectory equation. This fact allows an elegant derivation of the law of Helmholtz-Lagrange.

2.3.1 The law of Helmholtz-Lagrange

Liouville's theorem, the conservation of the electron density along a trajectory, is equivalent to the fact that the Jacobian J of the canonical transformation from the initial coordinates and impulses at time t$ to those at time / equals unity10

J =

ex

dx dpx dx dZ dx SP2 dX 8Px dpx 8Px dZ 8Px 8PZ dX dz dPx dz dZ dz 8PZ

ex

dpz dpx dpz dZ dpz 8PZ dx dpx dpz (25)

Capital letters denote quantities at the time t and small letters denote quantities at the initial time f0. As the values of Z and P, are independent

of x and px according to Eq. 23c, the Jacobian J reduces to

dX dX dZ dZ dx dp„ dz dp. J = J*Jz _dPr dP8Px r dx dpx 8PZ dP^ dz dpz = 1. (26)

Since Liouville's theorem applies separately to the z-direction and J, is equal to one, Jx should also equal one. Equation 24 is linear, so X and Px can be

written as the linear combination of two special solutions. The first solution, denoted by Xi0 and PxW, is based on the initial conditions x= 1 and px = 0.

The second solution, denoted by Xm and Pm, is based on the initial condi

tions x = 0 and px = 1. The general solution for X and Px can be written as

X — xXl0 + PXXQX,

Px = xPxi0 + pxPxo\- (27)

Two kinds of special planes, adjoint to the initial plane, can be dis tinguished. First, the planes in which Xm equals zero are called image planes.

All trajectories starting in one point in the initial plane pass again through one point in an image plane. An image of the initial plane is formed, whence its name. Second, the planes in which Xl0 is equal to zero are called

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crosso-ver planes. All trajectories with equal initial values of px intersect each other

at one point in a crossover plane. The trajectories starting with px = 0 cross

the axis in a crossover plane, whence its name. Evaluation of Jx in an image plane yields

V * 0 l = 1. (28) from which the law of Helmholtz-Lagrange can be derived. Consider the

trajectory starting without initial velocity at the boundary RQ of the emitting area, and the trajectory starting with initial transverse velocity s7kT\m at the axis. In the image plane, the position of the former trajectory is R and the angle with the beam axis of the latter trajectory is a. Equation 28 can now be written as

**«oV^F=*aVF, (29)**

where V is the potential at the axis in the image plane. In the crossover

plane, R and a correspond to \2kTjm and RQ, respectively, but we find the same relation except for the sign.

This shows the law of Helmholtz-Lagrange to be a special case of the conservation of the electron brightness, valid between adjoint planes. The law of Helmholtz-Lagrange gives a lower limit for the spot size that can be obtained in an electron-optical system. Usually, this lower limit is not at tained, since aberrations of the lenses and the space-charge density set a lower limit to the spot size.

2.3.2 Linear transformations

In this section, we assume the trajectory equation to be linearized. The transformation of the position and the angle with the beam axis between an initial plane z = z, and a final plane z — zf is described by a matrix A

(£)=A(J), (30)

where capital letters denote the quantities in the final plane and small letters

those in the initial plane. An example of such a transformation is the transformation T(Az) for a drift over a distance Az

T(Az) = (1 \z). (31)

The angle of a trajectory does not change, whereas its position changes as X = x + AZJC'.

Another example is the ideal electron-optical lens, which is usually described by its two principal planes if, and H2, and the corresponding focal

distances F{ and F2 (Fig. 4). The corresponding transformation L from the

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Fig. 4: Definition of the principal planes H^ and H2, and the focal planes F\ and F2 of an elec

tron-optical lens. The asymptote of two trajectories that approach the lens parallel to the axis, one from each side, are drawn as solid lines.

1 F, (32)

From the law of Helmholtz-Lagrange, one easily derives

(33) where Vx and V2 are the potentials at the object and image side of the lens,

respectively, corresponding to the kinetic energy of the electrons.

These transformations can be visualized in so-called phase-space dia grams which show the angle of the trajectories as a function of their radial position. Let us consider as an example the imaging of a finite object which

object H5 H i image

_M'.:iS

Fig. 5: Imaging of an object with radius RM emanating in the half angle R'M by an ideal lens

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®

$ --R m "Rn % 'm "Rm

/1

0 'm

Fig 6: Phase-space diagrams corresponding to the imaging in Fig. 5. (a) the object plane, (b) the object principal plane, (c) the image principal plane, and (d) the image plane.

emits between -RM and RM trajectories with angles between -R'M and R'M .

Figure 5 illustrates the imaging of the trajectories and Fig. 6 shows a few phase-space diagrams. The distance between the object and the object prin cipal plane is P. In the figures, we show imaging with a magnification of minus one and for equal values of F, and F2, but the formulas are general.

The object (Fig. 6a) can be parametrized as

r0 = RMx ( - 1 £ * £ ! ) > (34)

The drift matrix T{P) transforms Eq. 34 to

rm = RMX + R'\tpy> (35)

r'm = R,My

in the object principal plane (Fig. 6b). The transformation L between the two principal planes yields

(23)

r#2 = RMX + R'\ipy>

r

H2-RMx + R'MPy , F, (36)

in the image principal plane (Fig. 6c). Finally, the drift transformation T(ö) to the image plane (Fig. 6d) yields

r, = -MRMx,

RMx + R'MPy Fx

J

2 +

-^

RMy

'

(37)

where Q is determined by the equation

P Q

-+ + -É-=\

and the magnification M is given by

M = - - = -Q Fx

P F,

(38)

(39) The phase-space diagrams show clearly the positions and angles of the rays in each stage of the imaging. Comparing Fig. 6a with Fig. 6d, we see for example that the object is reversed with respect to the original.

Phase-space diagrams are a powerful tool for analyzing electron-opti cal devices. However, the restriction to linear transformations is too severe. The lenses used in the electron guns in display tubes have aberrations, of which spherical aberration is the most important. In the next section, we

Fig. 7: Imaging of a point axial object emanating in the half-angle OQ by a lens with spherical aberration. Ar is the radius of the image in the Gaussian image plane.

(24)

will discuss a simple method to incorporate spherical aberration in the phase-space diagrams.

2.4 Spherical aberration

Spherical aberration, illustrated in Fig. 7, is the most important defect of the electron-optical lenses used in triode electron guns. Paraxial rays form a focus at the Gaussian image plane, whereas non-paraxial rays come to a focus before this plane. The point axial object therefore forms a disc at the Gaussian image plane. The radius Ar of this disc is given, to the lowest or der, by

Ar = MCs{M)<xl, (40)

where M is the linear magnification defined by Eq. 39, <x0 is the maximum

half-angle of the rays emanating from the object, and C, is the third order spherical aberration coefficient. It can be shown that CS(M) is a fourth order

polynomial in M~l. " Values of C, have been tabulated for a number of

lenses in Ref. 11. Spherical aberration cannot be avoided in electron optics. Scherzer12 showed that every potential field between two equipotential

spaces acts as a positive lens with positive spherical aberration, if the field is rotationally symmetric, if no electrodes intersect the axis, and if no charges influence the field.

We will now describe an approach to spherical aberration, introduced by Verster13 , that fits well with the use of phase-space diagrams. This sec

tion is not meant as a thorough introduction to the theory of Verster, but as an illustration of the use of phase-space diagrams. In general, the third order aberrations of a lens can be taken into account by the addition of four third-order terms (r*3, r'2r, rV, and r3) to the transformation of r as well as

r' between the two principal planes (Eq. 32). Only five of the eight additional coefficients are independent. The values of the coefficients depend on the focal distance of the lens.

We will describe the imaging of an axial point object taking into ac count only one of the eight terms. The transformation between the two principal planes is given by

rm = rm>

r HI- jr + r m - KV—p-) r Hb

where D is the diameter of the lens, F its focal distance, and k a constant determining the amount of spherical aberration. We anticipate a result of van Gorkum and Spanjer7 with this particular choice of the third order

term. They used a slightly different definition of P and Q, defining P and

Q with respect to the centre of the lens instead of to the principal planes.

Figure 8 shows the phase-space diagrams in the object principal plane (a), the image principal plane (b), and the plane of minimum cross section in the

(25)

Fie 8- Phase-space diagrams corresponding to the imaging in Fig. 7. (a) the object principal plane, (b) the image principal plane, and (c) the plane of minimal cross section.

image space (c). A parametrized form of the phase-space diagram in the object principal plane is

r'm = aoP> ( - 1 <;/»£!)

(42) where a0 is the maximum half-angle of the trajectories emanating from the

object. Application of the lens transformation (Eq. 41) yields in the image principal plane rH2 = «0Pp, , _ XQPP r HI- T~ + a0p-kct -Hr/T. ..3 P' . 3 (43) Dx

Application of a drift transformation over a distance L yields for the posi tion rL in the image space

(26)

rL = H{P + L-^)p-k-?-^y = ap-bp\ (44)

* D2

The beam diameter is minimal when the minimum value of ap + bp* for — 1 < p < 1 is equal to its maximum value. By substitution, one can easily verify that this condition is fulfilled when 4a equals 3b. The phase-space diagram then adopts an 'S' shape. The value of L derived from this condi tion is

PF , (45)

F+

i^^o

4 D2

For k = 0, the value of L reduces to the usual paraxial image distance Q (Eq. 38). The radius RM of the minimum cross section is given by

RM = \b = l.k-P^Locl (46)

Usually, the paraxial image distance Q differs little from L. Substi tuting L = Q and a^P = /JQ, where /? is the maximum half-angle in the image space, we find

R

M=T

k

-^rf- <

47

>

4 PD2

Van Gorkum and Spanjer7 found that Eq. 47 describes well the spherical

aberration in bipotential and unipotential lenses for 1.7 < PjD < 8.3 and 5 <Q/D <, 17.5, and in magnetic lenses for 1.4 <, P\D <, 5.6 and 4.2 < QID < 11.7. They found as values of A: =5.6±1.5 for the bipotential lens, k = 6.4+1.0 for the decelerating unipotential lens, and A: = 4.4±0.7 for the magnetic lens. Note that they used a slightly different definition of P and

Q mentioned above. The third order term introduced in Eq. 41 appears to

be larger than the other third order terms in this range of P and Q values. Moreover, the coefficient of this term is almost independent of the focal distance of the lens.

The magnification and the spherical aberration both contribute to the spot size when a finite object is imaged. The contribution rm of the object

magnification can be found from the Helmholtz-Lagrange product HL at the object side of the lens as

HL (48)

NV2

The contribution rs of the spherical aberration is given by

(27)

where C is the spherical aberration constant related to CS(M) by

C(M) = M4Cs(M)(V2IVif12. (50)

The expression kQ*l(PD2) found in Eq. 47 is a special form of C.

The sum of both contributions can be minimized with respect to /?. The minimum spot size rm is given by

. _ / ^ _ « t t _ V

/ 4

c ^ (51)

\3y/V

2

/

and the optimum value /?m of /? by

0

m

= ( ^ U

1 / 4

. (52)

Over the range for which Eq. 46 is valid, the values of rsm agree within 10%

for bipotential, unipotential, and magnetic lenses. This remarkable result was presented earlier by van Gorkum and Spanjer7 . Equations 51 and 52

can be used to calculate the optimum aperture angle at the object side of the lens and the minimum spot size that can be obtained with a lens having spherical aberration.

References

1. W. Glaser, Grundlagen der Elektronenoptik, Springer Verlag Wien, 1952 2. P. Grivet, Electron Optics, Pergamon Press Oxford, 1965

3. O. Klemperer, M.E. Barnett, Electron Optics, Cambridge University Press, 1971 4. A.B. El-Kareh, J.C.J. El-Kareh, Electron Beams, Lenses, and Optics (2 Volumes), Aca

demic Press New York, 1970

5. I. Langmuir, Physical Review, 41,419(1923) 6. CD. Child, Physical Review, 32,492(1911)

7. A.A. van Gorkum, T.G. Spanjer, Optik, 72,134(1986) 8. P.A. Lindsay, Advances Electronics Physics, .13,181(1960) 9. P.H.J.A. Kleynen, Philips Research Reports, 1,86(1946)

10. H. Goldstein, Classical Mechanics, Addison and Wesley New York, 1964 11. E. Harting, F.H. Read, Electrostatic Lenses, Elsevier Amsterdam, 1976 12. O. Scherzer, Z. Phys. 101,593(1936)

(28)

(29)

3.0 Calculation of the radius of the emitting area and

the current in triode electron guns

Abstract

Analytic formulas are presented for the radius of the emitting area and the current in rotationally symmetric electron guns. They are based on an approx imation of the field-strength in front of the cathode by a series expansion up to fourth order in the radial position. The coefficients of the series expansion are calculated with bicubic spline interpolation from the geometry. The initial velocities of the electrons may be taken into account (Langmuir's approach) or ignored (Child's approach).

The calculated currents are compared with experimental ones for various drives, cathode loadings, and geometries. The accuracy, using Langmuir's ap proach is better than 10 % for currents between 0.05 and S mA. The calculated currents, using Child's approach, are systematically too low for currents smaller than 0.5 mA.

Also presented is a method to derive the distance between cathode and first grid from a measurement of the visual cut-off potential as a function of the potential of the second grid. It is based on the series expansion of the field-strength and similar to a method presented earlier by Haskerl .

3.1 Introduction

The cathode loading and the emitted current are important properties of the triode electron gun applied in for instance picture tubes, camera tubes, and oscilloscope tubes. A designer may calculate the radius of the emitting area and the current with numerical simulation or analytic formu las. Simulation2 yields accurate results but is expensive. Analytic calcu

lations or simple numerical approximations are less accurate but much used in the first design phase. Moss3 presented a semi-empirical formula based

on numerous experiments. Hasker4 presented a simple numerical method to

calculate the current, taking into account the initial electron velocities. This chapter presents analytic formulas for the radius of the emitting area and the current in rotationally symmetric electron guns with a flat ca thode. It uses the equivalent diode concept introduced by Ploke5 . The

formulas are based on the approximation of the field-strength in front of the cathode with a series expansion up to the fourth order in the radial position. The coefficients of the series are calculated for a large number of geometries. Interpolation yields the values of the coefficients for arbitrary geometries. The combination of interpolating bicubic splines with analytic formulas en ables the calculation of the radius of the emitting area and the current for the distance between cathode and first grid and the thickness of the first grid varying over a broad range. The initial velocities of the electrons may be either taken into account (Langmuir's approach6 ) or ignored (Child's ap

proach). The calculated currents are compared to experimental ones to judge the accuracy and validity of our approach.

(30)

Cathode grid 1

S

R

Fig. 1: Idealized model for the triode electron gun.

The distance between cathode and first grid usually can not be meas ured directly. An accurate method, similar to that given by Hasker1 , is

presented to derive it from the visual cut-off potential as a function of the potential of the second grid. The method is based also on the series ex pansion of the field-strength in front of the cathode.

In Section 3.2 the series expansion is introduced and the interpolation with bicubic splines is shortly discussed. In Section 3.3 the determination of the distance between cathode and first grid is discussed and in Section 3.4 formulas for the radius of the emitting area and the current are derived. In Section 3.5 the calculated currents are compared to experimental ones. Finally, in Section 3.6 the conclusions are drawn.

3.2 Series expansion of the field-strength in front of the

cathode

The field-strength in front of the cathode is important, because the calculation of the distance between cathode and first grid, of the current, and of the radius of the emitting area are based upon it. Therefore, we need a simple method - either analytic or numerical - that gives the field-strength as a function of the geometry and the potentials of the grids. The triode electron gun must be idealized to obtain a tractable method and Fig. 1 shows the geometry of the idealized triode. The parameters R , L , and

S characterize the geometry ( R is the radius of the hole in the first grid,

S the distance between cathode and the first grid, and L is equal to S plus the thickness of the first grid. As the Laplace equation is scale invariant, two dimensionless parameters ( L/R and S/L ) suffice to determine the field-strength distribution in front of the cathode. Two parameters describe the idealized triode electrically: the potential Vgl of the first grid and the

field-strength E beyond the first grid. We assume E to be homogeneous. In stead of Vgl we shall use the relative drive

(31)

grid 1 grid 2

S12 T2

R2

E92

Fig. 2: Idealized model for the first and second grid to calculate the field-strength E . The distance between first and second grid is St2, the radius of the second grid is R2, its thickness

is T2 , and the field-strength applied to the second grid is E„2

-n = \-v

g\l ' CO' gX

\v

c (1)

where Vco is the potential of the first grid to obtain zero field-strength at the

centre of the cathode.

The field-strength E depends on the geometry and the potentials of the grids. As a uniform field-strength applied to the first grid is disturbed by the hole in it, we replace the first grid by a plate without a hole in it. Exactly the same model can be applied to the plate and the second grid as to cathode and first grid, see Fig. 2. We take for E the axial field-strength at the plate which is given by

E 2

^ 2 - ^ , y s

1 2 >12 R,

+ Eg2\ 1 - 2B

^12 +T2

Ro (2)

The various symbols are defined in Fig. 2. The electron optical function B is defined as

B(z) = i f

°° „ sin(fcz)

**ak-M*)**

(3)

Durand7 gives details about the derivation of Eq. 2 and the function B. The

same method may be applied to find the value of Eg2, if Eg2 contributes sig

nificantly to E.

The field-strength in front of the cathode is the superposition of two independent components (£gl and EE) which give the field-strength for Vgl

= 1 V with E = 0 and for E = 1 V/m with Vgl = 0, respectively. We can

(32)

E

gl

(rlR) = - ^ ( 4 , + M-jf + Mjï)

4

), (4)

truncated after the third term, and for EE we do the same

EE(rlR) = E^B0 + B2(-^)2 + BA(-^)4y (5)

The coefficients AQ, A2, A4, B0, B2, and B4 can be calculated analytically

or numerically. To apply analytic formulas the potential at r = R has to be known for every value of z . One usually assumes linearly varying po tentials between cathode and first grid and beyond the first grid. The coef ficients A0 etc. can then be expressed in the electron optical function B and

its derivatives up to fifth order, but the formulas obtained are not simple8 .

Numerical calculation is a better approach. Given the geometry, the components Egï and EE are obtained from a numerical solution of the La

place equation. As boundary condition a linearly varying potential at r = 2 R between cathode and first grid and beyond the first grid is chosen. The boundary is closed at a distance 4 R from the cathode. A least squares fit of the field-strength for r smaller than R yields the coefficients AQ,

A2, At, B0, B2, and B4. They are calculated for L/R varying from 0.55 to

1 in steps of 0.05 and for S/L varying from 0.25 to 0.95 in steps of 0.1. The deviation between the series expansion and the calculated field-strength is less than a few percent for S/L greater than 0.35. For smaller S/L the fit is worse, since the small distance between cathode and first grid causes a rapid increase of Egl for r approaching R . It is important to take into

account the fourth power in r as the following example shows. For L/R = 0.6 and S/L =0.3 we obtain ^40= 1.30, A2 = 0A7, AA =3.11, 50 = 0.37,

B2 = -0.39 and 54 = 0.034.

We want to calculate the six coefficients for arbitrary values of L/R and S/L and we use interpolation with four bicubic splines. The lines S/L

= 0.6 and L/R =0.775 are chosen as knots. The NAG routine E02DAF9

calculates the 25 coefficients that determine the bicubic splines. The absolute deviation between the original coefficients and those obtained from the ap proximation is typically less than 5 % of the largest of A0, A2, A4, B0, B2 ,

and 2?4. The largest errors occur for small values of S/L in the approxi

mation of A0, A2, and A4. Values of A0, A2, AA, B0> B2, and B4 calculated with

the bicubic splines are used in this chapter.

3.3 The distance between cathode and first grid

The current emitted in a triode depends strongly on the distance S between cathode and first grid. The value of S is about 100 /an and may change up to about 10 /mi, when the cathode is heated. Direct measurement of S is usually impossible and we thus have to determine S from a pro perty of the gun that depends on S and can be measured easily and accu rately. The Durchgriff D - the ratio of Vgl and Vg2 for which the

(33)

field-strength in the centre of the cathode is just zero

D can be expressed in the coefficients AQ and B0 as

is such a property.

D =

Wo

AolR-pB

0

'

(6)

where /? is the factor that transforms Vg2 in the corresponding field-strength

E . The term /?i?0 appears in the denominator to correct for the contribution

of Vgl to E . Sometimes a more accurate value of D is necessary, which can

be obtained from a numerical solution of the Laplace equation.

The Durchgriff can be determined experimentally in a CRT or CRT-like set-up by a measurement of the potential VK of the first grid for

which the light on the phosphor screen just extinguishes. A plot of VK versus

Vg2 yields a straight line with a slope dVvJdVg2, which is about equal to D .

If a very accurate value of D is necessary, then dVJdVg2 has to be corrected

for the initial velocities of the electrons. Hasker1 showed, that extinction of

the light does not correspond to a field-strength zero in front of the cathode, but to a potential minimum with a depth VM of -1 to -1.5 V. A derivation

like that given by Hasker expresses D in dVvcldVg2, the assumed value of

VM, the Laplace cut-off potential Vco, and the coefficients A0, A2, B0, and

Bi. D 1 dVm 1 + C dV_g2t where (7) (8) Deriving Eq. 7 we used a series expansion of the potential field up to fifth order in z and fourth order in r , whereas Hasker applied third order in z and second order in r . Furthermore, we assumed that Vco is about equal

to V„, and that

- 1 AHB, 0-°4 A4B0

(A0B2 - A2B0Y

■AQB0 < < 1. (9)

It was not necessary to assume that AA and BA are much smaller than 1.

Nevertheless Eq. 7 is the same as the formula obtained by Hasker. The de crease of C with increasing Vco is due to the shift of the potential minimum

(34)

3.4 The radius of the emitting area and the current 3.4.1 The radius of the emitting area

The field-strength distribution Ec in front of the cathode can be cal

culated up to fourth order in r for arbitrary values of VgX and Vg2 from the

coefficients A0 etc. In this section we will express Ec as

Ec(r) = c + br2 + arA. (10)

The coefficients a , b , and c depend on L/R , S/L , Vgl, and E. Solution

of the equation E^RQ) = 0 yields the radius RQ of the emitting area:

Ro =

-b + Jb

T

-

4ac

la (11)

The root with the smallest absolute value has to be chosen. The value of

RQ is independent of E , a common factor in a , b , and c .

In Fig. 3, RQ/R is given as a function of the drive r\ for five values of

L/R . The thickness of the first grid is equal to 0.43 R for all curves, so that S/L decreases for decreasing L/R . The maximum value of RQ/R is one,

since the coefficients AQ etc. were calculated for r < R. For L/R =0.99,

where S/L is relatively large, RQ equals R already for r\ about 0.87. The 1.0 0.8 Q6

I

_0.4 0.2 "% ■ " /A

Y

i

é^

' ^ / # ■ l

///fi

//.tf

^ * L/R = 99 L/R = 86 L/R = 77 L/R = 66 ■ L/R=.57 0 0.2 0.4 06 0.8 1.0 Drive H

Fig. 3: The radius of the emitting area RQ as a function of the relative drive r\ for five values of L/R .

(35)

Table 1: L/R , S/L , the coefficients of the series development for Egi ( A ) and EE( B), and

the durchgriff Dc as calculated with Eq. 6 of the triode with large diameter second grid (Fig. 4). L/R 0.99 0.86 0.77 0.66 0.57 S/L 0.57 0.50 0.45 0.35 0.25

4>

1.08 1.16 1.20 1.26 1.37 A2 0.589 0.906 1.10 0.978 -0.519 A, -0.049 0.021 0.256 1.60 5.27 B0 0.160 0.213 0.259 0.329 0.397 B2 -0.194 -0.262 -0.312 -0.371 -0.399 B4 0.065 0.078 0.080 0.061 0.011

A

0.0110 0.0136 0.0160 0.0195 0.0217

value of rj for which RQ equals R , increases with decreasing S/L . The curve for L/R =0.57 deviates from the other curves because A4 is rather large and

A2 is negative. This deviation shows that it is important to take into account

the fourth order in r . The curves of RQ given in Fig. 3 are those for the first kind of triode gun discussed in Section 3.5. The values of the coefficients are given in Table 1.

3.4.2 Formulas for the current

Two different formulas have been derived for the current density j in a planar one-dimensional diode. The simplest formula is Child's law, which ignores the initial velocities of the electrons and the possible satu ration of the cathode. Child's law expresses the current density j in the potential difference V and the distance d between anode and cathode as

y = 2.33x 10"6 VL5ld2. (12)

Langmuir6 has taken into account the initial velocities and the finite

saturation current density of the cathode. A potential minimum in front of the cathode determines the current reaching the anode. The resulting for mulas for the current density as a function of V and d are rather compli cated. However, j can be developed in a series and truncation after the second term yields

j = 2.33 x 10-

6

ZzJ^lIi

+

\J-^— 1 (13)

(d-d

M

)

2

\

2

V e(V-V

M

) J

where VM is the potential of the minimum, dM the distance between the mi

nimum and the cathode, k Boltzmann's constant, e the electron charge, and T the temperature of the cathode. VM and dM are complicated func

tions of V and d. VM is of the order of -0.1 to-0.5 V and dM is of the order

of 1 to 10 fim. By neglecting VM and dM with respect to V and d we can

rewrite Eq. 13 as the sum ofy' given by Child's law, Eq. 12, and a correction term

(36)

A/ = 2.33 x 10"

U

nkT V

e d1'

(14) The current ƒ in a rotationally symmetrie triode can be found as / = dr2nrj(r).

Jo (15)

The values of V and d in Eqs. 13 and 14 depend on r . For d we assume

d — a/?o l0 • Moreover, Ec d replaces V and Eq. 10 is substituted for Ec.

The integration of Eq. 15 with Eq. 12 substituted for j then yields for Child's approach = =2 . 3 3 x l O -6, r _ / ^ _ + ^ \ W 7 A2 ( 2aR% + b

128aVö~ \2-Ja7 + b

(16a) for a > 0 and I = 2.33 x 10'67T f _ / _ c _ 3A \

JÏK L " V

8 a

k +

64a

^ ^

2

/

+ ■ 3A' 128aV-a arcsin — arcsin 2aR$+b I-A (16*) for a < 0, where A = 4 a c - . 62. (17)

Similar to that derivation we find a correction of the current for the Lang-muir approach based on Eq. 14:

A/ = 2.33 x 10 _a

-6

**±y/^[co + ±è + ±Z]**

(18)

The unknown a in the Eqs. 16a,b and 18 has to be determined from a comparison with experimental currents. A / / / i s about 0.1 for currents of 4 mA and about 0.4 for currents of 0.1 mA.

3.5 Comparison with experimental currents

For two different kinds of triode guns the currents are calculated and compared to the experimental values. The first triode gun is shown in Fig.

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Cathode grid!

S12 grid 2

Rg2

JL

1.

Fig. 4: Triode gun with large diameter second grid. The distance S between cathode and first grid is variable. The thickness T of the first grid is 0.15 mm, the radius R of the first grid is 0.35 mm, the distance Sl2 between first and second grid is 1.8 mm, and the radius R2 of the

second grid is equal to 6 mm.

4. In this gun the currents were measured for five values of S , for four values of Vg2, and for five values of»; (0.3, 0.5, 0.65, 0.8, and 0.9). The ratio

of the largest and smallest value of Vg2 is about 2.5. The accuracy of the data

is 5 % for currents greater than 0.1 mA and 10 % for the smaller currents. The value of S was determined with a Laplace calculation from the corrected value of the slope 3VJdVg2. Table 1 gives the values o[L/R , S/L

, the coefficients A0, A2, AA, B0, B2, and BA, and the Durchgriff D calculated

with Eq. 6. Table 2 shows the values of S , dVJdVg2> the correction factor

C, and the Durchgriff D as found from the Laplace calculation. With the exception of L/R =0.57 we find a very good agreement between the Dur chgriff calculated with the coefficients and the Laplace calculation. The discrepancy is due to the inaccuracy of the bicubic spline interpolation for small values of S/L .

The constant a in Eqs. 16 and 18 is adjusted in such a way that the relative standard deviation between the measured and the calculated cur rents is minimal. We find a = 0.59 for Eq. 16 (Child's approach) and a = 0.84 for the sum of Eqs. 16 and 18 (Langmuir's approach). The mean Table 2: The slope dVvcldVg2, the correction factor {, the Durchgriff D as found from a Laplace

calculation, and the distance S between cathode and first grid of the five triodes with large diameter second grid (Fig. 4).

dVJdVg2 0.0114+0.002 0.0140+0.002 0.0164+0.003 0.0200+0.003 0.0235+0.003

c

0.03 0.03 0.025 0.02 0.01 D 0.0111 0.0136 0.0160 0.0196 0.0232 5 (/mi) 195+3 151+3 121+3 82+3 50+3

(38)

10° 10-,J o IQ2 '10'2 10"1 10° 10' UxpImA)

Fie 5: The current calculated with (Eqs. 16 and 18) and without (Eq. 16) the correction for the initial velocities versus the experimental current. The currents were measured in the mode drawn in Fig. 4. The drawn lines are the lines lcaic = /exp

-relative standard deviation between the calculated and the measured cur rents is 0.13 and 0.11 respectively. Figure 5 shows the calculated currents as a function of the experimental currents. The upper curve ('Child') was calculated from Eq. 16. The drawn line is the line 4,,c = 7exp. The calculated

currents deviate systematically from this line: They are too low for the small currents and too high for the large currents. The lower curve ('Langmuir') was calculated from the sum of Eqs. 16 and 18. No systematic deviation from the drawn line is visible. The largest errors occur for L/R =0.57 and are probably due to the inaccuracy of the bicubic spline interpolation.

The second type of triode gun is drawn in Fig. 6. It is a conventional triode and the holes in the first and second grid are equal in diameter. Figure 6 and Table 3 give the values of the distances, thicknesses, and diameters for five slightly different geometries. The distance between cathode and first grid is determined as for the former triode gun. The potentials of the second and third grid are 600 and 8500 V, respectively. Table 4 gives the values of the coefficients and of the Durchgriffs calculated with Eq. 6 and the nu merical solution of the Laplace equation. Generally we find a good agree ment. The discrepancy of triode T5 is probably caused by the small distance between first and second grid in comparison to the radius of the second grid. The field-strength E is calculated in the idealized triode on the axis, where the field-strength assumes its lowest value. As the field-strength increases about 60 % for r approaching Rg, the contribution of the second grid to the field-strength in the centre of the cathode will be underestimated and the Durchgriff as calculated with Eq. 6 will be too low.

i u 10' -iio°L 10" Child m

The design of rotationally symmetric triode electron guns

T H E BESIGN OF ROTATIONALLY SYMMETRIC

TRIODE ELECTRON GUNS

_

—

_

VAN DEN BROEK

-p I

lb

•4

«IS l ' « '

THE DESIGN OF ROTATIONALLY SYMMETRIC

TRIODE ELECTRON GUNS

TR diss

1511

CONTENTS

Preface

1.0 Introduction

2.0 Elementary electron optics

2.1 Introduction

2.2 Thermionic emission

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20.0 40.0 60.0

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