NASA/TMm2001-210890

0

Preliminary Study of Electron Emission

for Use in the PIC Portion of MAFIA

Jon C. Freeman

Glenn Research Center, Cleveland, Ohio

June 2001

i il ii ̧

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NASA/TM--2001-210890

Preliminary Study of Electron Emission

for Use in the PIC Portion of MAFIA

Jon C. Freeman

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

June 2001
This report contains preliminary

findings, subject to revision as

analysis proceeds.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs._rc.nasa.gov/GLTRS
Table of Contents

Abstract ...................................................................................................................... 1

Introduction ............................................................................................................... 3

Existing Electron Gun Software ................................................................................ 6

Analysis Summary ..................................................................................................... 8

Chapter 1 Emission Modeling ................................................................................ 10

Chapter 2 Low Voltage Diode ................................................................................ 30

Chapter 3 Particle Enumeration ............................................................................. 41

Beam Velocity Profile ............................................................................................. 52

Conclusions ............................................................................................................. 55

Appendix I ............................................................................................................... 60

References ............................................................................................................... 63

List of Symbols ........................................................................................................ 65

NASA/TM--2001-210890 i

Preliminary Study of Electron Emission for use in the PIC Portion of MAFIA

Jon C. Freeman

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

This memorandum summarizes a study undertaken to apply the program

MAFIA to the modeling of an electron gun in a traveling wave tube (TWT). The

basic problem is to emit particles from the cathode in the proper manner. The

electrons are emitted with the classical Maxwell-Boltzmann (M-B) energy

distribution; and for a small patch of emitting surface; the distribution with angle

obeys Lambert's law. This states that the current density drops off as the cosine of

the angle from the normal. The motivation for the work is to extend the analysis

beyond that which has been done using older codes. Some existing programs use

the Child-Langmuir, or 3/2 power law, for the description of the gun. This means

the current varies as the 3/2 power of the anode voltage. The proportionality

constant is termed the perveance of the gun. This is limited, however, since the

3/2 variation is only an approximation. Also, if the cathode is near saturation, the

3/2 law definitely will not hold. In most of the older codes, the electron beam is

decomposed into current tubes, which imply laminar flow in the beam; even

though experiments show the flow to be turbulent. Also, the proper inclusion of

noise in the beam is not possible. These older methods of calculation do, however,

give reasonable values for parameters of the electron beam and the overall gun,

NASA/TM--2001-210890
and these values will be used as the starting point for a more precise particle-in-cell

(PIC) calculation.

To minimize the time needed for a given computer run, all beams will use

the same number of particles in a simulation. This is accomplished by varying the

mass and charge of the emitted particles (macroparticles) in a certain manner, to be

consistent with the desired beam current.

Chapter 1, Emission Modeling, gives a general overview of how an emission

calculation should proceed, and develops equations that adhere to the basic

constraints of the emission process. It ends by stating the number of particles for

a simulation using this technique would be prohibitive. The remaining sections

show alternate paths that will be pursued, to obtain outputs that will be similar to

published experimental reports. Chapter 2, Low Voltage Diode, gives the

analytical result for a 1-dimensional planar diode with only 1.172 volts applied to

the anode. This example demonstrates the magnitudes of the numbers involved.

Then Chapter 3, Particle Enumeration, shows how to modify the particle mass

and charge to reduce the run time to a reasonable value. Finally, the probability

density for a reasonable model of an electron beam in other parts of the TWT is

developed for analysis purposes. Appendix I gives a proof of constraints used in

earlier sections.

NASA/TM--2001-210890 2
Introduction

This memorandum summarizes the study that was undertaken to determine

if the electromagnetic solver, MAFIA, could be used to model the electron gun of

a traveling wave tube. The preliminary conclusion is that it can be used; and the

outline of how it will be executed is explained herein.

A critical element of an electron gun is the thermionic cathode. The

emission properties of any small portion of it may be viewed as the emitting

surface of a 1-dimensional diode. The solution for this 1-dimensional diode

model may be found in many references; and it is called the 3/2 power, or ChildLangmuir law. The simple solution is an elegant display of solving Poisson's

equation, but it does have some assumptions that make it difficult to implement

in a program that uses a particle-in-cell (PIC) approach.

To obtain a clearer understanding of the modeling difficulties, the actual

process of the emission details should be considered first. The physical

characteristics of the emission may be thought of in the following manner. At the

specified cathode temperature, the electrons are emitted with a classic MaxwellBoltzmann (M-B) energy distribution. As the anode potential is increased, the

collected current increases and will eventually saturate. This condition is denoted

as the "thermal limited regime"; See the figure below.

NAS AffM--2001-210890

\
I=KpV 3/2

J

I ThermalSpace-Charge ]

Limited Regime I Limited

V3__ T=T_

Regime

J_:

' V

Space-Charge

Region

(Repelling Electric Field)

/q

E

Accelerating

Regionof Electric

Field E

Cathode Anode

(a) (b)

+V

Figure 1 - (a) The J-V family of curves showing the space-charge-limited and the

thermal-limited regimes. The parameter T is the Cathode temperature; T] > To.

(b) Schematic of the 1-dimensional diode. Near the cathode is the dense spacecharge layer with its resulting decelerating field.

As the anode potential is reduced, the collected current will drop, and the

diode is in its 3/2 law regime, which is called the space charge limited region. In

this case the reduced electric field at the cathode does not drain off the thermally

emitted particles fast enough, and a space-charge layer forms adjacent to the

cathode. The space-charge layer acts to repel some of the thermally emitted

electrons back into the cathode surface. This has the effect of retarding the slower

particles, and only allowing the more energetic ones to climb the potential hill due

to the space charge layer, and then proceed on to the anode. Experimentally [1],

[2] it is known that the energy distribution of the ensemble that overcomes the

retarding space-charge field is still M-B, but with a lower effective temperature.

The space-charge layer causes a retarding electric field at the cathode

surface, which decays rapidly to zero as one moves toward the anode. Beyond

this point, the field is reversed in direction and begins to increase, and serves to

accelerate the electrons toward the anode. The plane where the field is zero is also

the point of the potential minimum (the virtual cathode). It is at this plane where

NASA/TM--2001-210890 4
the classic boundary conditions are applied. Physically, the electric field is zero

there, and the potential is just a few tenths of a volt below zero, but the carrier

velocity is not zero; it is of the order of the thermal speed. The electrons leave the

cathode with velocities around 10 5 m/sec (their thermal speeds). Since they are

generally accelerated up to speeds of 10 6 or 10 7 m/sec in typical applications, their

initial speeds at the real and virtual cathodes may be neglected. The potential

minimum is very close to the actual cathode, and it is only a few tenths of a volt

below the grounded cathode; so at this plane, the electric field, potential, and

velocity are set to zero. These boundary conditions yield a solution that is very

close to measured results. One problem is the fact that the charge density must

be infinite (since the velocity is zero) at the virtual cathode, since the current must

be constant in a 1-dimensional problem. In a real diode, the charge density near

the cathode is extremely large, but its value is not of particular importance; only

the current-voltage relationship is relevant in device use.

To model a cathode with a PIC code, one could emit particles at the cathode

with a M-B energy distribution and a net current equal to the measured saturated

thermal current density. Then for any applied anode potential, the space-charge

layer would form appropriately and the collected current should match that

measured to within numerical and experimental error. Unfortunately this straightforward recipe is impossible to implement. A large fraction of the thermally

emitted electrons are repelled back into the surface and do not contribute to the

collected current, and are wasteful of computer resources. Also, the maintenance

of the space-charge layer would be an overbearing tax on the computation time.

The total number of particles to use gets to be unreasonable, and the run time

would be prohibitive. Finding a way, or several ways, to circumvent this dilemma

constitutes the discussion of this memorandum.

NASA/TM--2001-210890 5
Existing Electron Gun Software

While an exhaustive study of all available electron gun simulators has not

been attempted, one of the most popular ones has been researched in some depth.

The program is called EGUN, and was developed by Hermannsfeldt [3]. The

program uses a "starting surface" which is several mesh cells away from the actual

cathode. Current filaments are formed on the surface, and their values are varied

until the net cathode current and anode voltage satisfy the 3/2 power law. That

is the current is proportional to the 3/2 power of the anode voltage. The

proportionality constant is the perveance of the gun, which is a function of the

geometry of the electrodes. Notice this approach completely sidesteps the problem

of forming the space-charge layer. In a real electron gun the electric field across

the cathode surface varies due to the electrode geometry and applied potentials.

This results in a variation of the space charge limited current across the surface,

which is called variable cathode loading. This variation may change by 20%, with

the largest emission density near the edge of the cathode, and the lowest emission

density near the center. The program EGUN does calculate this variable loading

phenomena. The electric field and potential are held to zero on the starting

surface, and the iteration proceeds. The current filaments act on one another in

a reasonable approximation, and magnetic focusing is incorporated. One level

of approximation used on the initial EGUN calculation is the following. The

documentation of the program dated in November 1979 (p. 85), says the program

has difficulty determining the space charge on the beam axis. The calculated value

is multiplied by the empirical factor of 5.5 to agree with some measured results

(the details were not given). Notice the beam is "cold" (no thermal velocity), and

thus noiseless. To simulate the noise in the beam, each filament is then subdivided

into rays of varying strength and they are emitted at specified angles from the

NASAITM--2001-210890 6
normal to the starting surface. These filaments are quickly curved back toward

the direction of the normal, as no crossing of the filaments is allowed in the

calculation. It has also been stated [4] that the program determines the perveance

to within a few percent, but has a somewhat harder time predicting the beam

diameter. Unfortunately, for our analysis effort, the beam diameter determination

is critical. This basic scheme is used by others [5], and is also incorporated in

MAFIA as a macro entitled SCLE (Space Charge Limited Emission).

NASA/TM--2001-210890 7
Analysis Summary

With the physical concepts of the emission process in mind, the approach to

model an electron gun using the program MAFIA is outlined here. The first step is

to use the SCLE module to get the appropriate variation of cathode loading for the

chosen electrode configuration. Next the cathode will be partitioned into segments

(patches) where each segment is assumed to emit a uniform current density. Using

this uniform current density in the patch, we will decompose it into the proper

surface carrier density and velocity distribution to model a M-B distribution.

Notice this step is emitting only those particles that have overcome the potential

hill, (virtual cathode) and will be collected at the anode. The formation of the

M-B distribution will be developed in the very long chapter entitled "Emission

Modeling".

MAFIA requires the specification (in a patch) of the number of emitted

particles, their charge, mass, energy (speed), position, and emission directions;

this must be kept in mind in the "Emission Modeling" section. Notice the cathode

is at temperature T1, with the appropriate M-B distribution for that temperature.

At the voltage minimum (virtual cathode, and the location of the starting surface)

the effective temperature is Tefy. We must determine Teff, and the correct M-B

distribution at this plane. A procedure is developed to provide this information.

The emitted particles (in a patch) are grouped into 14 discrete energy bins,

and the variation in the number emitted as a function of angle from the normal is

determined exactly. After much trial and error it was decided to always emit the

same number of particles for every simulation. The variation in current for

different conditions is handled by varying the charge of the emitted particles

(called macroparticles). In a M-B distribution the number of particles in a given

energy (speed) bin is dependent on that energy. By altering the charge and mass

NASA/TM--2001-210890 8
of the particles, we may use the same number of particles in each bin. The

adjustments conserve both energy and momentum, and the change in mean-freepath is not severe. For example, we consider a low voltage diode later to get a

feel for the magnitude of numbers involved in a calculation. If the diode is solved

using electrons, their mean free path is about 5.89 microns. When we use our

derived macroparticles, the path length is 15.67 microns. This variation will be

assumed to be acceptable. The motivation for this plan is to keep each run to about

the same time duration. From previous work it is known that using about 300,000

particles per run requires about 4-6 hours of computing time.

To verify the outlined scheme, a series of MAFIA computations for a known

solution for a 1-dimensional diode will be carried out. The details of the known

solution are in the chapter "Low Voltage Diode".

NASA/TM--2001-210890 9
Chapter 1 Emission Modeling

We know the energy distribution for the emitted particles is M-B both at

the metallic cathode, and more importantly, at the virtual cathode (plane of the

potential minimum). The program MAFIA requires the specification of: a) area

of the emitting surface, b) the number of particles and their positions, c) the

particle speed, and d) the particle direction. We may also specify the time step

for successive bunch emissions. Given the temperature of a cathode T1, and the

material, we can use Figure 2.3-1 of Gewartowski [6] to determine the saturation

emission current density Jo (Amps/cm2). The value Jo is the limit in the thermal

limited regime. (see [6] pages 42, 59, and 615). The sketch below indicates the

typical J-V plot for a thermionic cathode.

J0(0)

I I

Figure 2 - The J-V curve for a single temperature, along with the J0(0) for a small emitting

patch.

If we choose T1 = T = 1100 ° K, we can assume the typical value of Jo = 0.5 A/cm 2.

In the above sketch we have indicated an emitting patch and the current density

in the direction 0 from the normal. Let np (0, c_)and v (0, ¢) be the number of

particles emitted and their velocities in the (0, q_)direction, respectively. Here 0

is the polar angle from the normal, and 4)is the azimuthal measure. The average

velocity, <v>, of an emitted particle (averaged over all directions) may be obtained

from

NASA/TM--2001-210890 10
--l m(v} 2 -,,- 2WT

2

where m is the mass of a particle, and 2W T is their thermal energy. W T is

kT T

- - eV

WT q 11,600

where k is Boltzmann's constant. For our example,

- 2kT--2(13s×10 looo )

Then

Hence

= 3.036 × 10 -20Joule

3.036 × 10 -_-°J

• = 0.1895eV

1.602× 10 -'9 J/

/eV

"WT=.O95eV

(v)=_-F .6'072×10--'°J ])_ L9.108xlO-_'Kg

(l-l)

(v) = 2.582× l0 sm/sec

We arbitrarily choose 1,4 (W_) for energy of the slowest particle and 5.75 (W_) for

the energy of the fastest particle; then

VsLow = 1.291 x 105 m / sec

v_Asr = 6.191×105m/sec

so we have a sense of the order of magnitudes with which we are dealing. Observe

the ratio of fastest to slowest is only about 4.8.

Assume no ¢ dependence, then

Jo (0) = q_ n p (O)v(O) (1-2)

where q - the particle charge.

From reference [7] we have:

NASA/TM--2001-210890 11
NR = total number of ions emitted per unit area per unit of time in the joint interval

du dO dd? is

NR = --u2h _-m2 3e_h,,,__sin0 cos0 du dO d_ (1-3)

where m = mass of ion (here just an electron), h = 1/2kT (see pg 156), and u is a

particular velocity. Note that the average energy of an emitted particle is 2kT

(see eq. 1-1). This is in contrast to the value of 3kT/2 for a particle in the classical

M-B gas. This is due to the fact that the group that has escaped, must have enough

energy to overcome the retarding binding force at the cathode surface. For the

particles remaining in the metal, their average energy is 3kT/2. See Appendix I for

the derivation of this result. Thus, the mean kinetic energy (KE) in the metal may

be expressed as 3/(4h). Integrate over _;

N R(u,O)= 4h'-m2u3e -h'''_ sin0 cos0 du dO (1-4)

Observe this is eq. (2.4-8) of Gewartowski [6]. In eq. (1-4), u ranges from 0 to oo,

but we know the major contribution is obtained during the interval

V SLOW _ U _ V FAST

Consider a group of particles with the same velocity (speed, call it ul); then

their distribution with 0 is

N_R(u',O )= (const.). sinO cosO du'dO (1-5)

Then the number per unit solid angle (U.S.A.) is

sin0cos0

N'R(u',O )u s.A = const du'dO (1-6)

2_ sin 0 dO

N_R(u',O )_,.S.A.= constl "cos0 du _ (1-7)

We interpret eq. (1-7) as stating "for a given particle speed, those with that

speed are distributed as cos0." The units of N' R(u_,O) are [particles/unit

NASA/TM--2001-210890 12
area]/[unit of time/U.S.A.] which equals [particle flux/U.S.A.]. Equation (1-7) is

similar to the result of Gewartowski as follows. Start with his eq. (2.4-8)

dP(u,O)=sinOcos _,kT ue t, dudO (1-8)

where dP(u, 0) is the probability that the emission velocity lies in the range u to u +

du and makes an angle with the normal in the range 0 to 0 + dO.

Integrate over all velocities (then the velocity dependence is removed).

_ _2

fodP(u,O)du=sinOcosO(c)- _u3e-Tdu

=slnOcosO(c)- • = 2sinOcosO dO (1-9)

.'. dP(O )= 2sin0 cos0 dO

Which is eq. (2.4-11) of Gewartowski. Now we say

dP(O)

J(O) - [total emission current averaged over all directions] x

dP(O) 2sinOcosOdO

=J ---Jo

o U.S.A. 2zr sin 0 dO

J

= ---&°COSO

R"

U.S.A.

emitted current density/U.S.A., which is Gewartowski's eq. (2.4-

12).

For a given T, we know the range of emission energies. The fastest .5% of

particles will have energy about 6Wr. The distribution of particles as a function of

2Nror [ E e-_T dE

dUe-E__E r

(1-10)

energy is

NASA/TM--2001-210890 13
where Ev = Wr and E = W. Notice the symbol for energy will be either W or E; we

do this to conform to whichever author we are paralleling at a certain point in the

analysis. Let p, = dN do where 77= E / E r .

NTov = particle density (particles/m3). We will subdivide the range of energy into

B energy bins. For now B is arbitrary. The velocity of a particle in binj is

Where the o_

J

J

Vj --

Let 17i be the particle density of those with energy Ej.

I pATCH= JoApArCH

where ApATCH is the patch area. Write Jo in Amps/m 2 and ApATCH in m 2.

It, ArCH "- JoAezrcH =QApArCH (nyl + n2v2 +" "+ rt_vB )

where nj = density of particles in binj j = 1,2...,B

vj = velocity of particles in binj

We will show

n I =o¢ NTOT

n 2 -oc 2 Nro r

n_ =o¢ 8 Nro r

are known constants•

Then

IpArC H =qAeaTcttNT.or(Oct Vl-F-¢,c 2 V z "i-'"+°c1_ V.)

but the vj and ocj are known, so

N ( particles ) IpArC_ ror( _ _=qAeArCH]_°¢JV j

Once Nvor is found, ni are computed from eq. (1-13a).

The current from a patch is

(1-12)

(1-13)

(1-13a)

j=I,...,B (1-14)

NASA/TM--2001-210890 14
From Millman and Seely [8], their section 5-15, we have

KE No_._L = 2KErANOEmva _

which states the KE normal to the surface is twice the tangential value, when

calculated over the ensemble. See Appendix I for its verification. Let nj; represent

the particle density in energy binj and emitted at angle Oi. Let vj be the velocity of

particles in binj. For calculation purposes we will choose 4 energy bins (j = 1 to 4)

and 5 (i = 0 to 4) emission angles. The sketch below illustrates the model.

(1-15)

00 01

/

/ ,02

/ /

/ /

/ /

/

/ 03

/ / I.

/ / t,

/ / //

I//,11/ 04

,;.-/_...---

Figure 3 - The case for emission at 5 specific angles from a small patch. Emission occurs at

0 ° as well as 01 through 04.

Eq. (1-15) becomes

--1 {(n,oV_+ n:oV22 + n3oV32 + n_ov,2)+ (//11121 ' .J[- _,_11222 .4- //31123 2 + //41124 2)cos: 0, +

2

hi2 22 + n22122 2 "_ //32]23 2 '']- //42]24 2) cOs2 02 "]- (n13127 , + n23 ]22 "_ //33 ]22 "4- _'/43124 z )COS 2 03 +

(]//14127 , "_ //24]22 , -[- _v/341232 "31-]"/44V4 2)COS204}=RH S

Where RHS means the right hand side of eq. (1 - 15).

But we know the velocities at the various angles are related by constants, i.e.

]22 = K'2VI, F3 -" K'3VI, ]24 "- K'4121

NASA/TM--2001-210890 15
?/tl = nlo COS0j

//12 =//10 COS0:

n13 = to c°s03

i_14 -- 1110cos0 4

//31 = 1/30 c°s01

tl32 -- 1130c0s02

//33 =//30 c°s03

//34 = ll30 c°s04

1121 = 1120 COS01

n22 =//20 COS02

1123 -- 1120 COS0 3

n24 = n2o cos0 4

//41 _-- //40 COS01

//42 =//4o COS02

I'/43 =//40 COS03

//44 -"//40 COS04

So

Where

2

+ (n _oV_+ n.,or,'v.. _"+//3or3 5'-V 1 "Jf-n40tC4V? "_)c OS3O,

+ (ntoV_ +//20K'2Vl2 +//30K'3 Vl,.+//40K'4Vl,_ +( )cos O 

lv_X{I+cos30, + cos30, +cos303 +cos304} +( )cos O4}= 

2 2

X = //10 dr. n20 _2 ..[_ ]//30K- 3 --F rl40K" 4

Now the right hand side (RHS) of eq. (1-15) is

{(n,,v_ + n2,v _ + n3_v_ + n4,v 4 )sin 20, + (nl._V_+//j_ + n3.v 3 + n42v_ )sin _-05

+ (//_3v_+n23v _ +n_v_ +n43vj)sin'-O 3 + (n,4v_ +n,.4v _ +n_v_ +n44v_)sin"04}

Or

{(nloV_ + n.otC_v _ + n3otC_v_+ naotC_v_)cos0, sin 20_ q:-

( )cos0'- sin" 0'- + ( )cos03 sin s 03 + ( )cos04 sin" 04 }

= v_ {X (cos0, sin 201 + cos0'- sin" 05 + cos0_ sin" 03 + cos04 sin" 04 )}

Then

(1-16)

(1-16a)

NAS A/TM--2001-210890 16
Then

Or

Now define

1 2

_-V, X{1JI-COS30, "_ COS3 02 "31-COS3 03 Jl-COS3 04}= _I-X(cosO, sin2 el

+ cos02 sin" 02 + cos03 sin e 03 + cos04 sin 204)

1 + cos 301 + cos _02 + cos 303 + cos 3 04 = 2cos0, sin 20, + ....

_-1 + (}cos 1 q01 _ cosO, sin 2O,)+(1⁄2cos 3oa - cos02sin " 02)

+ (1⁄2cos' 0_- cosO, sin e 0,)+_ (+cos ' _0 4 --COS0, sin" 04 )=0

which is sketched below.

f(O) = __cos30 - cos0 sin: 0

(1-17)

(1-18)

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

-0.3

-0.4

f(o)

10 20 30 _90

Figure 4 - A sketch of the function f(0) in eq. (1-18).

r

NAS A/TM--2001-210890 17
Rearrange eq. (1 - 17) as

f(O,)+ f(02)+ f(03) + f(04) =-4 "

So we must choose the 0,,0,.,03,04 to satisfy eq. (1-17a)

(1-17a)

o° f(o)

1 .499

2

3

6

7

O

22

.498

.495

4 .492

5 .487

•481

.474

8 --i 6g- ......

9 .458

10 .448

11 .437

12 .426

13 .413

14 .400 35

15 .386

16 .371

17 .356

18 .339

19 .322

2O

23

24

25

26

28

29

.268

.249

.230

.210

.190

21

0

43

44

45

46

47

f(o)

-.206

0

64

65

66

67

68

69

f(o)

-.312

-.309

-.306

-.301

-.296

27 .170 48 -.220 -.289

J

•150 49 -.232 70 -.282

30

5O

51

52

53

54

55

56

57

58

59

60

61

31

32

-.244

-.255

-.266

-.275

.129

.108

.O88

.O67

.046

.026

-.283

71

72

73

74

75

-.291 76

-.274

-.265

-.255

-.244

-.233

-.221

.0053 -.297 77 -.208

-.015 -.302 78 -.194

-.035 -.307 79 -.180

80

.305

.287

33

34

-.054

-.073

-.310

81

-.092 82

36

37

38

39

40

41

42

-.110 62 -.314

-.128 63 -.314

83

84

-.166

-.151

-.135

-.119

-.103

NASA/TM--2001-210890 18
85 -.086 87 -.052 89 -.017

86 -.069 88 -.035 90 0.0

The following 17 sets of angles that satisfy eq. (1-17a) will be listed for

reference purposes.

chosen randomly.

Set

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

15,60,66,71

Notice that every angle from 1 to 90 is used. The sets were

Angles

10,20,30,40,50,52,60,70,80

5,15,25,40,45,50,55,60,70,81,88

15,20,30,43,48,53,58,68,87

7,14,28,35,36,37,40,60,65,70,75,80,85

2,12,31,42,47,51,56,62,74,86

11,19,63,67,73,77,79

3,8,54,57,63,67,72

1,59,66,71,84

4,61,73,76,77

23,64,69,82,89

22,24,26,41,44,46,49,78,79,83

6,18,34,38,39,45,46,57,61,75

9,52,64,73,77,85,88

13,32,68,71,73,81

16,33,48,64,74,84,88

17,29,42,62,68,76,89

21,27,34,44,56,67,77

Some 4 angle sets are: (noticetheirsumshould be-0.5)

theirsum=-.506

NASA/TM--2001-210890 19
20,48,57,70

25,62,75,80

30,68,77,84

Recall

Or

their sum = -.499

their sum = -.503

their sum = -.499

tllo q- tlli 4- nx: q- nl. _ -t- t/14 )= n_

n,o(1+ cos0, + cos02 + cos03 + cos04)= n,

" /lto ----

n!

1+ _cos0_

And 1120, n30,/140 can be expressed similarly. Thus all 16 bunches

lift, i,j=1,2,3,4 are determined. For each angle set, we will distribute them

uniformly in phi (the azimuthal angle).

Now we calculate the nj. Recall from eq. (1-13a)

n I '-'oc I Nro r

llj -'-ocj Nro r

The energy distribution function is shown in the plot below

0.4 .....

2NTorO.2 oo!i/ j

0 1 2 3 4 5 6

11

Figure 5 - The Maxwell-Boltzmann energy distribution function plotted versus the

normalized variable q=F-JI_T.

(1-19)

NASA/TM--2001-210890 20
Consider a small patch which emits Nrov particles over some specified

time interval. Subdivide the emitted particles into 14 groups (now we will use

14 energy bins), with the particles in each bin having the same energy, (and thus

speed). By graphically making the partitions, we may calculate the number of

particles in binj

2N_-o_-

n;= _ f_,,,i_il-_e-'dr/

Evaluating eq. (1-20) in general terms gives

n = _,,:x/-_e-' dr/

Where we have suppressed the coefficient 2Nro r/-vl--_

Let

x = ._/-_, x 2 = r/, 2xdx = dr/

So

Let

Or

Then

n = 2 Iill 2x2e-X_dx

f _ e -x2

f' = -2xf

f" = -2xf' -2f =4x'f -2f

x2f =-_,(f" +2f)

(1-20)

(1-21)

NAS A/TM--2001-210890 21
So

Thus

2 x'e -x dx= " +2f

X2

}fi +fl-e- dx--[f'(x2)- f (x,)]+Sg"'-e--"'dx

- SO"'e-_"dx= 1⁄2[2x, f(x,)- 2x2f(x2)]+ "f_-[erf(x 2) -erf(x,)]

2

n = x,f (x,) - x:f (x:) + --_---[e rf (x 2) - erf (x,)]

(1-22)

NASA/TM--2001-210890 22
Our values are:

77

0

.5

1.5

2.0

2.5

3.0

3.5

4.0

X

0

.70711

1.2247

1.4142

1.5811

1.7321

1.8708

2.0

e--X"

.60653

.36788

.22313

.13534

.082085

.049787

.030197

.018316

2

xe-X

0

.42888

.36788

.27327

.1914

.12978

.086236

.056493

.036632

erf (x)

0

.68260

.84270

.91679

.95449

.97464

.98569

.99185

.99532

4.5 2.1213 .011109 .023566 .9973

5.0 2.2361 .0067379 .015067 .99843

5.5 2.3452 .0040868 .0095844 .99909

6.0 2.4495 .0024788 .0060718 .99947

6.5 2.5495 .0015034 .0038329 .99969

7.0 2.6458 .00091188 .0024127 .99982

Then

NASA/TM--2001-210890 23
As a check,

Thus the final table is:

n_ = Nror {-_(-.4288)+.6826} =.19866 Nro T

n2=NTo_{ }= .2289 N_o_

n 3 =. 18085Nmr

n, =. 13008NmT

n 5 = .089682Nro _

n 6 - .060185NTo_-

n 7 = .039722Nro _

n 8 = .025881Nro r

n 9 = .016724N_o _

nl0 = .01072Nro _-

n,, = 6.8466 x 10 -_ N_o _

n_,_= 4.3436 x 10 -3 Nro r

n_3 = 2.7464 x 10 -_ Nro r

nt, = 1.7326 x 10 -3Nro r

i4

y__nj = .9971Nro r

j=i

(1-23)

Bin 0i _/NToT

1 .25 .19866

2 .75 .22893

3 1.25 .18085

4 1.75 .13008

5 2.25 .089682

NASA/TM--2001-210890 24
6 2.75 .06O185

7 3.25 .039722

8 3.75 .025881

9 4.25 .016724

10 4.75 .01072

11 5.25 6.8466x10 -3

12 5.75 4.343x10 -3

13 6.25 2.7464x10 -3

14 6.75 1.7326x10 -3

m

Where r/j

The emission velocity is developed as follows.

1 2

v_mvj = qEj; write Ej = 2Er_ j

Where E_ = ,,IT/(1,600 ; the factor of 2 is to account for those that have escaped.

or

For our case, T = 11O0 ° K, thus vj

is the normalized energy parameter at the center of the partition.

4- mv:.J = 2-_J( 11_00 T )= 2-__kT

]4kT- [ 4(1.38 × 10-23) ] _ X/-_iT

vj=_---m-rl; =L _1-b-_3, j

= 7.785 × 10__-_--_;Tm/sec

= 2.582× 10'_;

(1-24)

(1-25)

(1-26)

NASA/TM--2001-210890 25
i

Bin 7"/j nj/NTo T vj (rrdsec)

i -- .25 .19866 1.29x 105

2

3

4

5

6

7

8

9

10

11

12

13

14

.75

1.25

1.75

2.25

2.75

3.25

3.75

4.25

4.75

5.25

5.75

625

6.75

.22893

.18085

.13008

.089682

.060185

.039722

.025881

.016724

.01072

6.8466x10 -3

4.3436x10 -3

2.7464x10 -3

1.7326x10 -3

2.2361x105

2.8868x105

3.4157x105

3.873 "

4.282 "

4.655 "

5.0x105

5.32 "

5.627"

5.92 "

6.19 "

6.46 "

6.71x105

Note VFAST ] VSLOW -- 5.2

Gewartowski shows

j(o)=JOcosO

IV

Assume all particles are identical, then

Amps/cm 2

U.S.A

J(O)=qZ[n,(O,O)v, +n_(O,O)v_-Ji- •• --1-_14(0,_/_)_14]

Where nl (0, ¢) = the number of particles in direction (0, ¢) with speed Vl

Then

j_Ion, (O,r/))dOd¢ = n, =/¢,Nror

(1-27)

(1-28)

NASAITM--2001-210890 26
In general

Iolo nj(O'_)dOdcp=nj=_J NTor (1-29)

It is shown in Richardson [7] that for a fixed velocity, the angular distribution is

uniform in ¢ and cosine in theta (0). Thus

nj(O,¢)= Aj cos0

Then

l¢,Nro r = _o (2*r)4, cos0 dO = 2_A,

• A = tCjNr°r

' 2_

Observe eq. (1-30) satisfies a fundamental constraint; namely

KEorma_= 2 x KEt_e,,,,a,

For a given velocity

m JocA O(v, cosO_dO rnA, v_. /

KE,or,._, =--_ Aj cos - ;:cos'/9dO 2

(1-30)

(1-31)

(1-32)

(1-33)

mAjv_ mA ff _

KE,.._e,,_,., - -_ _o/"cosOsin"OdO - -_ 1(_3) (1-34)

Observe eq. (1- 33) is twice eq. (1- 34), which validates the procedures used. Now

combining eqns. (1-27) and (1-28)

J(O) = J°cosO = q{(A, cos0)v, + (A_ cos0)v 2 +..-+ (A14cos0 )Via}

7r

Or

NASA/TM--2001-210890 27
J--_°=-_ {tc'NT°rv':rc + t'c'-NT°rv2 +'"+ tC'4Nr°rV'4 }

• Jo - qNr°r 14

2 _tCjvj

Now we can determine Nror given Jo and T.

A typical example might use Jo = 5 Amp/cm 2 = 5 x 104 Amp/m 2

14

Evaluating _ tcjvj

1

h-,v, = 2.565 x 104

_;.,v_, =5.119x104

x'y 3 =5.221x104

x'4v 4 =4.443x104

K'sv5 = 3.473 x 104

_'6V6 "- 2.577x 104

teTv 7 = 1.849x104

tfsV 8 = 1.294x 104

From eq. (I-35)

rgV9 = 8.902 x 103

h',oV_o= 6.032 x 103

_'llV_l =4.051X103

h'j_,VL_,= 2.689X l0 s

lq3vx _ = 1.773X l0 3

tq4v14 = 1.162 x 10'

14

_]tcjvj = 2.9x10 _

j---i

= 2.15x10'8 fi/m3

(1-35)

NASAITM--2001-210890 28
A_ = 6.798x10 '6

A 2 = 7.834x 1016

A 3 =6.188×10 '6

A, = 4.451 x 10 '6

= 3.069 x 1016

A 6 = 2.059x 10 _6

A 7 =1.359x10 '6

A_ = 8.856x 1015

A 9 =5.723X1015

A,o = 3.668 x 10 '5

Aj, = 2.343x10 '5

A_2 = 1.486 x 1015

A,3 = 9.398x 10 '4

Al4 = 5.929 x 10 '4

While the above is without approximation (except for the finite number of

partitions), the numbers are too large for MAFIA. There are many constraints on

the emitted particles when using the program; some of them are: There is no easy

way to incorporate the cosine distribution with 0. Instead, we'll use a parabolic

distribution that closely matches a cosine. The number of particles/bin cannot be

too large, or the run time will become prohibitive. The number of patches may be

large, especially when modeling near the edge of the cathode. The question as to

whether one should use systematic emission versus random emission is a very

perplexing problem. The systematic case develops lower computation noise, but

a random process seems to be what one needs. However, how does one produce

such a condition, yet guarantee the emission is theoretically correct? Another

constraint is the use of 4 to 5 particles per mesh cell to obtain good results. These

along with other conditions will be explored and overcome in the succeeding work.

NASA/TM--2001-210890 29
Chapter 2 Low Voltage Diode

To obtain a clearer perspective of the magnitudes of the parameters for a

thermionic cathode, an analytical study of a typical 1-dimensional diode was

undertaken. The following example is close to that given by Beck [9], pg. 173

(the example has some errors). The diode has plates with surface area of 0.2 cm 2

and separated the distance d, as shown below.

.447 cm

1

T= IIOOK

d = .5 mm

Figure 6 - The geometry of the 1-dimensional diode.

The current drawn is 1.0 mA, the current density is 5 x 10 .3 A/cm 2 , and the thermal

limited value Jo is 5.0 A/cm 2. At the cathode the parameter 77is

77--r/c - ln(J °/J) = lnl03 = 6.9078

From the table on page 527 of [9], we see this corresponds to the parameter

-_c = 2.509. Where _ is defined by

: 9.174 × 10'T -_-,ff(x - X m ) (2-1)

NASA/TM--2001-210890 30
where x, Xm are in cm, J is in A/cm 2, and T is in Kelvins. The voltage minimum is

located at x,,. This equation is taken from the original paper by Langmuir [ 10]. At

the cathode, _ = _, and x = 0. Then

- 2.509 = -3.3963 x lO'-x m

or

x m = 7.4 x 10 -5m = 0.074mm

Then the voltage at the minimum is

At the anode

1100 In 10 -_= -0.65471 volts

11,600

_a = 3.396×102(.05-.0074) = 14.468

From the table in [9] we read rL = 19.27

Now

11,606

T

19.27= 11,606(V +.65471)

1100

.'. V = 1.172 volts

Where Va is the required anode voltage.

With this information we may generate a plot of the potential through the

diode versus x. The ruble is:

x(cm)

0

.000625

.00125

.0025

_ V(voI_)

-2.509 6.9078 0

-2.301 3.5 -.323

-2.089 2.286 -.438

-1.6642 1.12 -.549

NAS A/TM--2001-210890 31
•005 -.8151 .2 -.636

•006 -.4755 .065 -.649

•007 -. 1359 .0045 -.6543

•008 +.204 .001 -.6546

•009 +.543 .067 -.648

•010 +.883 .17 -.6386

.020 +4.279 2.8 -.389

•030 +7.676 7.2 +.028

.040 +11.07 ....i2.7 +.549

.050 +14.47 19.27 +1.172

The plot of the voltage is shown below.

V(Volts)

1.2"

1.0-

.8

.6

.4

.2"

-2  ̧

_4 o

-.6-

-.8

I I I I I

,05 .I .15 .2 .25J.3 .35 .4 .45

/

=-.655

m

1.172

x(mm)

Figure 7 - A sketch of the voltage V(x) through the diode. The value at the minimum

position (Xm) is about -0.655 volts.

NAS A/TM--2001-210890 32
We could numerically differentiate V(x) to obtain the electric field, E but to

check for consistency between various treatments of the 1-dimensional diode, [ 11 ],

we will use an analytic formula for E from [9]. For x < Xm the field is given by

-E'-=K[e o _ 1 + eOerf v/-_ _ 2(_)'_ ]

For y > Xm, we have

E2 = K[eO - l - e" erf _/-_ + 2(_ )_ ]

Where E is in V/cm, K = 6250J _ = 1.0364 x 103 V2/cm 2. The table is:

x(cm) 0 G erf E(V / cm)

0 6.9078 2.6283 .9998 1.4382x103

.000625 3.5 1.8708 .9918 255

.00125 2.286 1.512 .967 131.3

.0025 1.12 1.058 .865 60.4

.005 .2 .447 .466 17.17

.006 .065 .255 .281 9.07

.007 .0045 .067 .075 104

.010 .17 .4123 .438 -11.67

.030 7.2 2.683 .99985 -48.06

.040 12.7 3.56 1 -55.96

.050 19.27 4.3898 1 -64

The first and last E-field entries are those at the cathode E c and anode E a. The

magnitude of the field at the anode is 64 V/cm, and from the graph of the potential,

NASA/TM--2001-210890 33
the slope at the anode is about 1 volt over .015 cm, or 66.7 V/cm. The agreement

is well within tolerance.

Next we may determine the charge density p(x) as follows. For

convenience we first determine it in the accelerating part of the field;

For x > X m

-E'- = K[eO - l - e" erf _/_ + 2(_ y _-]

Let f,(rl)=eOerf(_/-_)

o( 2 _0"] 2

f_'(q)=e t-_--e )+erf,q/-_ e"

f(r/)= e_ - 1- f_ (r#)-t-_r# _

211

f ' (rl) = e" - f " (rl ) + _ 2 x/-_

=e" 2 eOerf_/-_ + ___L_I 4; dr4

+ e"erf_/-_

2 dE=

But

drl _ dr# dV _ __ drl

dx dV dx dV

2-E dE (rl i _- d,TdV

• 2p___= _K[e _ 2 eo

where e ° is the permittivity of free space.

Or

NASA/TM--2001-210890 34
p(x) - ,, e _ e_erf _-_ +

2 "v@ dV

,, e _ eOerfff-_ +

2 _ 1100

Note

p (x) - Ke° I:'< )il-66 (11606 "_

Where

K = 1.0364x 10' =1.0364x10 v _,

• p(x)= -4.8409xlO-4[f'(rltC°UJ//m3

=3.3962xlO'-(x-x )

Our table is below.

x(cm) _ rl _ erf _f_ f_(rl) P(C/m3 )

.0075 .03396 .00027 1.64E-2 ......

.0076 .0679 .001 3.16E-2 .338 17.38

•008 .2038 .0096 9.8E-2 ....

.009 .5434 .067 .259 .284 1.0993

.010 .883 .17

.020 4.28 2.8

.030 7.68 7.2

.040 11.07 12.7

.045 12.77 15.7

.050 14.47 19.27

-84.1x10 -4

-4.59x104

-5.32x10 _

.412 .438 .906 -4.39x10 a

1.67 .982 .49522 -2.4x10 _

2.68 .99985 ....

3.56 1 ....

3.96 1 ....

4.39 1 .99988 -4.84x10 a

NASA/TM--2001-210890 35
Now for x < Xm.

So

Now we estimate

The final table is

l

X_oF. 2 eO_,f_

p(x) =---2--[e + _ +

x(cm) _ ri f2'(17)I p(coul/m s)

0 -2.509 6.9078 2.001x103 .9687

p from the slope of E.

/

p = e ° --dE= (8.854x 10 -1-"

dx

= 8.854 x 10-'2 (--_- _°u_jn3

x(cm)

3.125x10 4

9.375x10 -4

1.875x10 -3

3.75x10 -3

5.5x10 3

8x10 -3

2x10 -2

-.1673

-.01753

-.005

-1.531x10 -3

-7.17x10 _

-4.59x10 _

-1.61x10 -5

NA SA/TM--2001 -210890 36
3.5x10 2 -6.9947x10 -5

4.5x10 -2 -7.119x10 -5

So

Now we determine the total space charge between the plates.

e --=d_ p(x) co<m,

° dx

a'a dx

¢oul /

7,,2

Where x a and xb are arbitrary points, and A = cross-sectional area.

For x < x,,

+

x=0, E(0)=l.4382X102Y_m; x=x,, E(xm)=0

So the charge between x = 0 and x = Xm is

eo[-1.4382x lO3 V_cm XlO2 C_mm] = Q_/_A

A = .2cm: = 2x 10-Sm 2

So Q, =(2X10-Sm2)(8.854X10-12_/m)(--1.4382X10_)

=--2.547X10 TM coul _ 1.59x108e -

For x > Xm

NASA/TM--2001-210890 37
At x = x m

m

Hence E(x ) = 0 as before.

2 :KD-l-o+o]:o

At x = xa (the anode) E(xo ) = -64

[64 , cm°l:O m

" Q2 =-.l133xlo-_'coul

Then the total charge is

Qror = Q, + Q2 = -2.66 × 10-" coul

We may now ascertain the velocity field:

The table is:

J = 5oAmP_m2 = p(x)v(x)

• v(x)- 50 m/sec

p(x)

7.08 x 106 e

1.66xlO_e -

x(cm)

3.125x10 4

9.375x!04

1.875x10 _

3.75x10 -3

5.5x10 -3

8x10 -3

2x10 -3

3.5x10 "2

4.5x10 2

v(m/sec)

2.99x102

2.85x103

9.956x103

3.27x104

6.97x104

1.089x10 s

3.1x105

7.148x105

7.02x105

NASA/TM--2001-210890 38
Note the thermal emission velocity is about 4.0E+5 m/sec. So the values in the

above table are very reasonable. Notice the velocity varies by about a factor of

1000.

The magnitude of the electric field E(x), across the diode is shown below. The

cathode value is 1,438.2 volts/cm, while that at the anode is -64 volts/cm. The

field passes through zero at 0.074 mm from the cathode.

1000

CATHODE

100

10-4

ANODE

1 I

10-3 I 10-2

DISTANCE (CM) _ .074 mm

Figure 8 - A plot of through the diode.

NASAfrM--2001-210890 39
Below is the charge density p(x) and the velocity field v(x) across the diode.

10 o

10 I

10-2

10-3

t0-_

10-5

Charge Density

m/see

10 5

I0 _

10 3

10:

Velocity Field

Position of

Voltage Minimum

T t 2 3 4 5 T 2 3

x= x (cm) x 10" t ANODE 1 x (cm) x 10 2

x m

4 5

Figure 9 - The charge density p(x) and velocity field v(x) for the problem at hand.

This completes the calculation of a 1-dimensional diode in a regime where

the 3/2 power law is not applicable. That regime is when J/Jo is between very

roughly 0.01 to 0.5, rather than our case of 0.001. It is a useful example as the

voltage minimum is away from the plane of the cathode and the charge density is

not extremely large near the cathode. Here the density drops 3 orders of magnitude

from the cathode to the voltage minimum (the virtual cathode), and the applied

voltage is only 1.172 volts. We are drawing 1 mA, which is-not t0ofar from some

actual TWTs (at least order of magnitude)i Reference [111, by Kirstein, Kino, and

Waters, has a good treatment of the problem, albeit in a notation that is unique.

See refs. [12] through [15] for more information on this problem.

NASA/TM--2001-210890 40
Chapter 3 Particle Enumeration

We now attempt to determine the way to model the 1-dimensional diode

using a finite number of particles emitted over some specified time interval. First

we divide the space between the plates into equal segments A d, and over the

segments we use At = Ad/Av, where Av is the change in the velocity field over

a segment. By using 9 such equal length segments we ascertain the transit time

is about 27 nsec (for the Low Voltage Diode studied earlier). Gewartowski [6],

pg. 629 gives the time of flight for the 3/2 law regime as

Fa fl

Lq J J

which for our case is 1.45 nsec. The difference of more than an order of

magnitude shows how far we are from the normal regime.

Recall, we would like to emit particles at the virtual cathode (plane of the

voltage minimum). Help in this task is provided by [11 ] on pg.275, eq. (2.44).

This expression gives the variation of potential with distance near the voltage

minimum

J in Amp/m 2

V - V = 4_-1"82× 10-4 J(x" - x,, )2 (3-1)

2E o

dV _/_-1.82 x 10 -4

dx 2eo ff

J[2(x- x )] (3-2)

The next derivative gives the charge density, which we see is not a function of

position about the virtual cathode.

d2V ,f_-1.82 x 10-"

dx 2 Eo

J (3-3)

NASA/TM--2001-210890 41
Observe eq. (3-2) shows that the electric field varies linearly through the virtual

cathode plane. Then

p(x,,)=-_l.82xlO-4J

Notice the analytical result in the section "Low Voltage Diode" gave the charge

density of -7 x 10 -4 coul/m _, so our results show remarkable consistency. Since

p(x )=-_/_1.82 × 10-_ [(- p(x))(v(x))]

• (v(x°,))=_5.49×lO3m/sec

Which states the average velocity at Xm is independent of all other parameters

except the cathode temperature. For typical ranges of T we find

T(K) V(Xm) rn/sec

1000 9.79x104

1050 1.0x 105

1100 1.03x 105

1150 1.05xl05

1200 1.07x 105

(3-4)

So _V (Xm) > is nearly 1.0 x 105 rn/sec for all temperatures and applied voltages,

and currents. For reference, the velocity values in bins 1 and 14 are 1.29 x 105 and

6.71 x 105 rn/sec.

Observe that the energy needed to overcome the retarding field of the space

charge layer is

NASAfrM--2001-210890 42
KE = q(O.655volts) = 1.048 x 10 -'° Joules

The velocity required is

-mv- = 1.048x 10 -19

2

• v =4.8xl0_m/sec

So only those particles in bins 8-14 can escape the region and proceed on to the

collector. This amounts to only 0.07 of the total number emitted per unit time.

Now consider a diode carrying the current i. Then the current in a given

patch is

i --------i _AQp (3-5)

P N At

P

Where Np = the number of patches chosen for the entire cathode. Then

AQp = ipAt = _,n AQ (3-6)

Where np = the number of particles per patch, and _ AQ is their net charge. Here

is the total charge emitted in a patch in the specified time interval At.

where

AQp = AQ1 + AQ2 +... + AQI 4

AQ_ = nlq I ,..., AQ_ 4 = r/,4ql 4

Since there are 14 energy bins. Here qj is the charge value in a given bin.

Consider binj; its total kinetic energy (KE) is

(3-7)

(3-7a)

(3-8)

NASA/TM--2001-210890 43
Where nj, Mj, vj are the number of particles, their mass, and their common speed

respectively. If the particles in all of the bins had the same mass m (this is the

actual case in a real device), then

KEI4 = hi4 m ,

re-arrange

KE s _ n_ (/vj =l'lJ

KEI4 n,4 _ v_4)

Now let us use n14 particles in binj, but keep its KE the same.

We retain the original KE by assigning the particles a new mass Mj

From eqns. (3-1 1) and (3-12)

Or

Or

F/j j mI4V24F/t4 = n_4 mj 's

V/t4 V_4

n.

Mj = -'---L-J ml4

//14

Now the ratios Fl/Fll4 are known from the M-B distribution, and for our chosen

energy bins, they are:

(3-9)

(3-1o)

(3-11)

(3-12)

(3-1 3a)

(3-13b)

(3-14)

NAS AffM--2001-210890 44
n,/n,4 = 114.66

n2/nl4 =132.13

n3/n_4 = 104.38

tl4 /nj4 -- 75

n5/n,4 = 51.76

n6/n,4 = 34.74

n7/11,4 -- 22.93

n_ In,4 = 14.94

n_ I nj4 = 9.65

n,o In_ = 6.19

n_ In,, = 3.95

n,_/nj4 = 2.51

/l,3///,4 -'-1.59

Since MAFIA keeps the charge to mass ratio constant (equal to that of an electron),

q___L_J = q____4

M s rn,4

Hence

Returning to eq. (3-7) we observe

M .

3

qJ --_ql4

m,, (3-16)

q_4n'4 (3-16a)

ml4

And so forth; Thus

- +"- +1 ,4n,4 (3-I7) m_4 m,4

Where we have set the number of particles in each bin to be n14, as stated after eq.

(3-11). Now from eq. (3-14) _,_, _ _, so the term in parentheses in eq. (3-17) is

known

+... + n,_ + 1 =575.43

f/14

Where we have used eq. (3-15). Thus eq. (3-17) becomes

(3-18)

AQp = 575.43q,,n,4 = ipAt (3-19)

NASAITM--2001-210890 45
Now eq. (3-19) is our basic working equation. The current from the patch ip

is known, the time interval At may be chosen compatible with MAFIA's

constraints on run time, etc. The number of carriers may be chosen, then the

charge on the particles (macroparticles in MAFIA) is determined. Eq. (3-19)

determines the charge on the fast particles, which are in bin 14. The mass and

charge for the remaining 13 bins are found using eqns. (3-14), (3-15), and (3-16a).

Before q14 is determined we must choose 1114. The total number of charges

emitted during the interval At is

NToTc = hi4 (14)Np (3-20)

Where Nrorc is the total number emitted for all patches. For a given energy bin

we know the particles are distributed as cosO, so we choose the following recipe.

Let 1 particle emerge at 85 ° and let the number increase as 0 ---)0. The table is:

0

85

80

75

70

65

60

55

50

45

40

35

# particles

2

3

4

5

6

cosO

0.0872

0.1737

0.2588

0.342

0.4226

0.500

7 0.574

7 0.643

8

9

9

0.707

0.766

0.8192

NASAfrM--2001-210890 46
30 10 0.866

25 10 0.906

20 11 0.9397

15 11 0.9659

10 11 0.9848

5 11 0.99619

0 11 1.000

Then hi4 = 136 particles/energy bin. The number of particles processed in a

complete MAFIA run is

NTOT_RUN = (136)(14)Np N_,c

Where Ncyc is the total number of emission cycle intervals.

We can apply this to our low voltage diode problem as follows. The total

charge between the cathode and the voltage minimum is

QI =-2.5468x 10-" C _ 1.5917x108e -

While that between the virtual cathode and the anode is

Q2 =--1133x 10 -1' C _ 7.0833x 106e -

QTo_-= Q, + Q2 = -2.66 x 10-" C _ 1.6625 x 10Se -

Since the time of flight is about 27 nsec, choose At = 2.7 nseconds. Choose

10 patches, then ip = 10 -4Amp, and from (3-19)

AQp = 10-4(2.7 × 10 -9) = 2.7 × 10-'3 C

So the amount of charge developed/emission cycle is

10AQp = 2.7 × 10 -L"C

The total charge between the plates is then generated in

(3-21)

NASA/TM--2001-210890 47
2.66x10 -_'

2.7 x 10 -P

Also from (3-19), the charge q14 is

2.7x10-'C

= 9.85 cycles

= 3.45 x 10-'8 C _ 21.56eqJ4 = (575.43X136)

If we use 50 cycles, the total number of particles processed is

Nror_RvN = (136)(14)(10)(50) = 9.52 x 10' particles/run

So 9.52 x 105 particles emitted during 135 nsec. This value corresponds with that

used by C. Kory and K. Vaden using the program. C. Kory uses about 2.8 x 105

particles per run.

At this point we have all the information to emit particles from the virtual

cathode. We now must determine the correct M-B energy distribution at this

plane. We know the saturation current is 103 times larger than that collected. Only

10 .3 of that emitted by the actual cathode is actually collected; the remainder is

pushed back into the cathode, only to be re-emitted again. In other words, as we

move from the cathode toward the voltage minimum, the carrier density decreases

rapidly, but the imbalance of carriers moving in opposite directions still maintains

the net 1 mA current toward the anode. At the virtual cathode and beyond,

particles are only moving toward the anode and they constitute the 1 mA current

at every plane. ......

..... 7 - " -

The velocity distribution for each bin was given in the Emission Modeling

section, eq. (1-26) and subsequent table. Eq. (3-4) above shows the average

velocity at the minimum to be

(v(x,,,))= _T--5.49 x 103 m/sec (3-22)

And for our temperature, T - 1100 ° K, we have

NASA/TM--2001-210890 48
(v(x)} = 1.03 x 10_m/sec

From [8], the most probable speed occurs when 17i = 1, (see eq. 8.32, pg 224).

Then from eq. (1-26) of the Emission Modeling section we find the most probable

speed = vc = 2.582 x l0 s m/sec. Then the average speed v is 1.128 vc - 2.9125 x

l0 s m/sec. The rms value is 1.224 vc ---3.16 x 105 m/sec. Also from eq. (1-26)

vc = 7.785 x 103

So the average velocity is

(since _i = 1) (3-23)

v = (1.128)(7.785 x 103)_/-T (3-24)

The above is for the particles leaving the cathode, what we need, however, is

the velocity distribution at the plane of the voltage minimum (the virtual cathode).

It is calculated as follows. We know the energy in a particular bin is 2r/jkT_, and

for bin 14 for our temperature it is 1.2808 eV. The factor of two occurs since we

are considering those electrons that have escaped. The potential hill each particle

must overcome is 0.655 eV (the voltage at the minimum is 0.655 volts). Thus, the

fastest particles beyond the minimum have energy 1.2808 - .655 = .6258 eV. Now

from experiments, we know the distribution is still Maxwell-Boltzmann, so this

energy must correspond to those in bin 14 at the new, but lower, effective

temperature. Bin 14 is characterized by r/j = 6.75. Thus our new temperature is

defined by

.6258 = (6.75)k Teff =77 jkTeff

.'. Teff = 1075 °K

Notice here the factor of 2 does not appear, as now we treat all emitted electrons

as constituting a normal M-B gas. Thus our new temperature is 1075 ° K. We can

always determine the effective temperature since the value of the voltage minimum

(3-25)

NASA/TM--2001-210890 49
is known once the total current is chosen, and the saturation current is known for

the specific cathode. The minimum voltage is given by

Vo, - 11,605

Where T Once the new temperature Teff is known, the is that for the cathode.

velocities for the bins become

vj = 3.893 x 10_- Oj/-_ _] " l j- eff

(3-26)

m/sec (3-27)

The new table for the velocities (at the voltage minimum) as compared to those

in the Emission Modeling section (which were at the cathode), is:

m

Bin 71j v (m/sec)

1 .25 3.191x104

2 .75 9.573x104

3 1.25 1.596x 105

4 1.75 2.234x105

5 2.25 2.872x105

6 2.75 3.5101x105

7 3.25 4.148x 105

8 3.75 4.787x105

9 4.25 5.425x105

10 4.75 6.063x105

11

12

13

14

5.25

5.75

6.25

6.75

6.7x105

7.34x105

7.98x105

8.616x105

NASA/TM--2001-210890 50
To simplify the modeling, we may use a built-in feature of MAFIA; that

being a parabolic distribution with chosen parameter. We will approximate the

cosine distribution with the first two terms of its Taylor expansion, which is a

parabolic representation.

cos0 = 1+ a:O 2

• p(O) = 1 -.49670"-

A

a 2=- .4967

0<0<_/_ 2

(3-28)

NAS A/TM--2001-210890 51
Beam Velocity Profile

After a careful study of the experimental results on the beams created by a

typical electron gun, we may make the following tentative statements. Firstly,

the reported results are somewhat ambiguous, and secondly, they do not give the

information in the form useful for our modeling effort. We desire the particle

density and velocity at least over a specified plane normal to the axis of the beam.

The reports give approximations that yield either velocity or charge density vs.

radial value, but not both. Some studies have stated the current density falls off

in a Gaussian manner from the beam center to the edge. Stating that the current

density is Gaussian only tells us that the product of particle density and velocity

are such. For our efforts, we need both quantities, not just their product. Some

papers however, state that the beam is far from laminar (or Gaussian) and that tings

of charge start at the edge and move through the beam to the center. It is not clear

if the reported results are due to the particular electron gun used, or are typical of

all guns.

With the state of affairs as such, a reasonable approximation for the beam

velocity profile chosen by C. Kory is as follows. At the emission plane initialize

particles such that their angle with the normal 0 is given by

tan0 =u, (1)

/g

Z

Where u, and uz are the transverse and longitudinal velocities, respectively. The

longitudinal velocity is constant, due solely to the anode potential. The transverse

component is assumed to be Gaussian. Writing the above in more familiar

notation

 :tan

NASA/TM--2001-210890 52
In essence we are seeking the probability density function for 0 (or y), given the

density for x (the transverse component u,).

When y > 0

x = a tan 3'

Let g(x) = y, then

a I

g (x)=

a +x 2

Then the density function for y (y = O) isfr 0'),

f,. atan y) _ a 2 + X 2

f"(Y)=a/(a 2 + x 2) a

f,. (a tan 3')

wherefx(x) is the density of the random variable x (x = ut)

If x is normally distributed

1 e_X:/,.o,. f x

Then

fr(Y) =

a 2 "_L"X2 1

__ e-a2tan2y/2a z (3)

When y < 0 x=atany

So eq. (3) is the same as before. In eq. (3)

X 2 = a 2 tan 2 y

Hence

1 e_a2tanayl2a2 f (y)=asec:y (4)

In most guns

NASA/TM--2001-210890 53
u___L_ 10 -2

H_

X 0_ 2

.'.--=1 =tany

a

• y = 0 =.573 °

The standard deviation is

o = _ - 1.29 x 102

c= r - _ =28.44

.'.6 =6.88x10_m/sec

For typical values o" - 2 × 10 _m / sec, a - 10 7m / sec

fr (0 =.573°)= 17.65

Therefore eq. (4) is the probability density function for the emission angle for

a given small patch, for a typical electron gun under the assumption of eq. (1).

A sketch offr(0) is below.

a sec'- 0 e_O_tan20/20 .2

f,,(O) =

15

10

5

I

'1

a =u. = 107m/sec

0"= 1"3x105( )!r9r--__

= 2xlO'm/sec

Figure 10 - The distributionf_O) for the emission angle 0 from the normal.

NAS AITM--2001-210890 54
Conclusions

It appears that MAFIA can be used to model electron guns with a MaxwellBoltzmann energy distribution of the emitted electrons. To keep the run-time to a

minimum, a scheme was devised that uses the same number of particles for any

cathode current. The charge and mass of the particles are varied to conform to the

given current. Similar to the older program EGUN, the calculation starts at a plane

close to the actual cathode. In EGUN it is called the "starting surface" and is

chosen somewhat arbitrarily (a few mesh cells from the cathode). EGUN sets up

current tubes on the surface and varies them until the solution (forced to obey the

Child-Langmuir 3/2 power law) is determined. We, however, emit charged

particles (macroparticles) of varying charge and mass, which have the M-B energy

distribution, and the proper distribution in emission angle from the cathode. Our

emission plane is close to the virtual cathode whose position is known once the

current is chosen. The actual virtual cathode varies with distance from the true

cathode; it depends on the local cathode loading on a given patch. The loading

varies by as much as 20% in some cases. The effective temperature of the

macroparticles is calculated, given the known temperature of the cathode. The

modeling of the beam should be closer to reality than any published scheme to

date. The solution is not constrained to obey the 3/2 power law, so all regimes of

emission are able to be examined. Specifically, if some region of the cathode is

in saturation, this scheme can handle it easily. Also the non-laminar (noise)

properties of the beam should be modeled completely. The use of particles will

determine the true beam diameter as it exits the gun, as well as the spread of

trajectories at the beam waist (near the point where the PPM stack applies the

confining magnetic field).

NASA/TM--2001-210890 55
The following bulletized list summarizes the accomplishments that are

distributed throughout the memorandum.

• Explained how the program EGUN operates, and some of its shortcomings.

Fundamentally, the procedure bypasses the establishment of the spacecharge layer, from which the anode current is developed. Instead, it models

the beam as current tubes. The tubes cannot cross one another; so laminar

flow is implied, even though experiments suggest otherwise. The noise

properties of the beam are therefore lost completely. The calculation

assumes the 3/2 law, which eliminates the possibility of portions of the

cathode being in saturation. A consequence of the assumptions is its

difficulty in correctly determining the beam diameter along the gun, as well

as finding the correct value of the space-charge along the axis. An empirical

factor of 5.5 is used to correct the value found by the calculation. It does,

however, find the perveance to a few per cent.

• Started with Lambert's law as given in Gewartowski [6]

J(O )= J° cos0 = qn(O )v(O )- q_,, nj (O)v j (0)

/17

and showed that nj(O) is distributed as cos0 for a fixed value of velocity vj.

This result followed from the statement that the kinetic energy normal to the

surface is twice that tangential to the surface. This was proven in Appendix

I, and developed in Chapter 1, Emission Modeling.

• With particle use, we will be able to model all parts of the I-V plane, as well

as noise, _and non-laminar flow. - _ _-_ _,

• Chapter 1, Emission Modeling, indicates the basic way the problem should

be handled. It ends by showing the numbers needed are too large for

MAFIA. It demonstrated the difficulties encountered in trying to establish

NASA/TM--2001-210890 56
the space-charge layer. It is actually this layer which serves to emit particles

that form the beam.

• It was recognized that the emission maybe viewed as follows. There is a

M-B gas in the cathode, and there is another M-B gas, at a different

temperature, which forms the space-charge layer. Then we can define a

third temperature Te_, which characterizes those electrons that have traversed

the space charge layer and are going toward the anode. Those that are

collected are also M-B which was determined from the experimental results

of references [1], and [2]. Appendix I also showed the escaping electrons

to have a temperature such that the average energy was 2kT vs the 3kT/2 of

those that remain in the metal. Here T is the cathode temperature. However,

in that calculation, the repelling force of the space-charge layer at the surface

was not considered.

• Chapter 2 detailed the solution for the 1-dimensional diode that used the

thermal velocity of the emitted charge as a boundary condition. The

equations are non-linear, and tables have been established to facilitate the

voltage, charge, electric field, and velocity field between the plates. We

followed the lead of Beck [9] in this regard.

• We stated the boundary conditions for the 3/2 law require the charge density

to be infinite at the virtual cathode plane, which is not possible to be

modeled using particles.

• In principle, one should use particles in the following manner. With the

cathode temperature given, the thermal limited current density is established

for the particular cathode material. Emit particles with the M-B distribution,

and which satisfy the current density constraint. The space-charge layer

should form in the simulation, and steady state should occur. However, in

NASA/TM--2001-210890 57
the middle range of the I-V characteristic, about 1⁄2 of those emitted will

return to the cathode, only to be re-emitted. This would be a waste of

computational time, just to establish this "bias condition" to allow the actual

current to the anode to be established. Thus one must somehow sidestep the

formation of the space-charge layer.

• Chapter 3 outlined a method to sidestep the formation of the space-charge

layer. The idea is to only emit those that reach the anode. The

determination of their effective temperature Teg was presented.

• Chapter 1 presented some bunches (17 angle sets) to be emitted if a

distribution function in the program MAFIA was found to not be

appropriate. A function f(0) was developed to determine the correct makeup

of a given bunch.

• We plan to model the cosine distribution for each energy bin group with a

parabolic distribution function which is available in MAFIA.

• Chapter 2 gave a solution for 1 mA of current and 1.172 volts. There,

representative numbers were determined. In an actual gun, say the one in

the Cassini Mission TWT, the current is 15 mA and the voltage is 5kV.

Thus we must verify the model using the 1.172 case.

• For the example we found the charge density varies by 4 orders of

magnitude through the diode. The electric field changed from + to -, and

varied by a factor of 22 from cathode to anode. The velocity field changed

by a factor to 1000. Finally, the total number of charges between the plates

in steady state is 166 million. Of these, 159 million are between the cathode

and the plane of the voltage minimum.

• Chapter 3 explained the method to keep the total number of particles in a

given run constant. The variation in current is handled by adjusting the

NASA/TM--2001-210890 58
charge and mass of the particles in the fixed 14 energy bins. This will

dovetail into the program's requirement of 4-5 macroparticles per mesh cell.

We used [11] to realize the average velocity near the voltage minimum is

nearly constant for all current levels; it varies only slightly with cathode

temperature. Looking back at the numerical solution obtained in Chapter 2

we found complete compliance with this result!

With all of the compression done on the number of particles used, the runs

may take 12-18 hours. We hope we have over-estimated the run times.

We have developed the probability distribution of the angle 0 from the beam

axis, fv(O), for a reasonable approximation for defining an electron beam for

other uses in parts of TWT simulations. If the transverse velocities are

gaussian, while the normal component is constant, we find 0 is distributed as

f r (O)= fo (O)= u: sec- O crv_-/

NASATYM--2001-210890 59
Appendix I

Section 5-15 of [8], page 141, concerns itself with the energies of escaping

electrons from a hot cathode. This is an important result, as most formulas, results,

concepts, etc. from electron gas theory in solids is for the case where the carriers

remain in the metal. While we know the statistics of the ensemble for the electrons

in the metal, what are they for the portion that escapes? It is more convenient to

attack this problem as a function of velocity, thus

1

qE = - mv"

2

Notice the notation that electric potential is given the symbol E (a peculiarity of

Millman and Seely), while in Gewartowski it is given the label W. We use this

notation since the development to follow is from Millman and Seely. Our goal is

to show

KE°o,_a_ = 2 × KE,a . gential (I-l)

The calculation proceeds by determining the average energy of those electrons in a

metal that are capable of escaping. This is eq. (5-42)

lm(v_+v2+ )dN]

_ • Y V 2

_=",, _=-_=- 2 q

E = (1-2)

After some calculations the following expression is found

_ lmv3xe-aV_dv _ /_Bvxe -_''_dv x

2q /_

m

Evaluating the integral in the numerator yields; let -- = k,

2q

T 3 -_,v_

Btq vxe dx let u = v _ du = 2vx dv x'

then

NASA/TM--2001-210890 60
________ • .._

22 ""

m

.,..-

2qEr

Their expression in the denominator is

-xv_dv

_ vxe x

(1-3)

and its evaluation proceeds as (a standard form)

1 -E s I E r

=--e

22

E 8 = Min. of E

The ratio is

Now

2v_B = m v2 . xB

2qEr

if we assume

1 2

E B =--mvxR

2q

then

e -EBIEr+ES/Er =1

Then the ratio is

(I-4)

(1-5)

(1-6)

NASA/TM--2001-210890 61
as stated.

k 2 k/-ln____l z +

v,B +/J,- 2 vxB

q

The second term of eq. (5-42) is

j.j.j,(_m qVy 2 y2m___3er_,_ Jt h_ ,

1 m 2qE_ = EB + E_ (I-7)

2q m

-_vT-_v_-av2" , " . .

"1_flvxaVyaV z

f f f2m3 eE"/_T -_"_-_"_-z"_v dv dv dv

aa h 3 e - , x y z

-.a.v{ 2 -,h'7.

_.Te vxdv,_ klvye 7dvy

= *[Fe_a,,_vxdv _e_a,,dv

IdvaB . .___ _'

ff__v;e-a"Tdvy (2)(_)_11⁄4 '%:

= kt e ° -_,'7- = ki k_

joe av (2X1⁄2_)- 2-;t,

(I-8)

as stated. The final statements are: In leaving the metal, the average X-directed

energy of each electron is reduced by an amount Es, so that the average X-directed

energy of each electron after it has left the metal is (EB + Ev) - EB = Er electron

volts. Hence the total average energy of each electron that escapes from the metal

is Er + 1/2Er + 1/2Er = 2Er electron volts. In other words the normal is ET

whereas the Y-directed (which is tangential here) is (1/2)Er. This is our first

equation in this Appendix.

NASA/TM--2001-210890 62
References

[1] P. Kirstein, "On the Effects of Thermal Velocities in Two-Dimensional and

Axially Symmetric Beams," IEEE Trans. Elec. Devices, ED-IO, March 1963,

pp.69-80.

[2] E. Wheatcroft, "The Theory of the Thermionic Diode, J. Institute Electrical

Engineers, Vol. 86, pp 473-484, 1940.

[3] W.B. Herrmannsfeldt Electron Trajectory Program National Technical

Information Service, U.S. Department of Commerce, Springfield, VA. September

1973.

[4] W. B. Herrmannsfeldt, Numerical Design of Electron Guns and Space Charge

Limited Transport Systems, Nuclear Instruments and Methods, Vol. 187, pp245-

253, 1981.

[5] R. True, "Electron Beam Formation, Focusing, and Collection in Microwave

Tubes," Chap. 14 of Handbook of Microwave Technology

Vol. l, Academic Press, 1995.

[6] J. Gewartowski and H. Watson, Principles of Electron Tubes, D. Van

Nostrand, 1965.

[7] O. Richardson, Emission of Electricity From Hot Bodies, Longmans, Green &

Co., 1921. p.178.

[8] J. Millman and S. Seely, Electronics, McGraw-Hill, 1951.

[9] A. H. Beck, Thermionic Valves, Cambridge University Press, 1953.

[10] I. Langmuir, "The Effect of Space Charge and Initial Velocities on the

Potential Distribution and Thermionic Current Between Parallel Plane Electrodes",

Physical Review, Vol. 21, 1923.

NASA/TM--2001-210890 63
[ 1 l] P. Kirstein, G. Kino, and W. Waters, Space-Charge Flow, McGraw-Hill,

1967.

[ 12] N. van der Vaart and A. Leeuwestein, "Full 3D Simulation of Thermionic

Emission in Electron guns for CRTs", SID O0 Digest, pp. 956-959.

[13] M. van den Broek, "Electron-optical simulation of rotationally symmetric

triode electron guns, "J. Applied Phys. Vol 60, No.11, December 1986, p. 3825

[ 14] L. Page and N. Adams, "Diode Space Charge for Any Initial Velocity and

Current," Phys. Rev. Vol. 76, No 3, Aug. 1, 1949, pp. 381-388.

[15] T. Frye, "The Thermionic Current Between Parallel Plane Electrodes;

Velocities of Emission distributed According to Maxwell's Law," Phys. Rev.

Vol. 17, No. 4, April 1921, p. 441.

NASA/TM--2001-210890 64
List of Symbols

I

Kp

J

V

E

T1

Teff

M-B

Jo

0

V

< >

<V>

WT

m

k

q

np (e)

v(O)

U.S.A

dP (u, 0)

dNz

d.c. gun (cathode) current

perveance of electron gun

current density (amps/m 2)

voltage (or potential) along gun or 1-dimensional diode

electric field in 1-dimensional diode

cathode temperature

effective temperature of the M-B ensemble that constitutes the anode

current

Maxwell-Boltzmann energy distribution

saturation current density at a given temperature for a specific cathode

material

angle from the normal to an emitting patch

azimuth angle

particle velocity

average of quantity over the ensemble

average velocity of the particles over the ensemble

thermal equivalent of energy

electron mass

Boltzmann's constant

magnitude of electron charge

number of particles emitted vs. 0 (number/m 3)

velocity of particles emitted vs. 0 (m/sec)

unit solid angle ( differential solid angle = 2rt sin0 dO)

joint probability of emission velocity u and direction 0

number of particles with energy in the range E + dE

NASA/TM--2001-210890 65
NTOT

ET

PO

T1

E

IpATCH

ApATCH

nj

ocj

B

_normal

KEtangentiai

nji

K'j

fro)

err

r/j

q

_lc

_c,a

d

total number of particles in ensemble (or a density:/m 3)

WT energy (in Millman's [8] notation). Note [8] uses E for both

energy and voltage, a very confusing choice, even though they attempt

to justify it in the beginning chapter

dN,/d_7 Particle density with respect to the normalized energy 7"/

E/ET normalized energy parameter

a constant - kT (Joules) or kT/q electron volts; for a temperature T

current from a patch of cathode

area of patch

particle density in energy binj

constant, j - 1,...,B

number of energy bins

total kinetic energy normal to the surface

total kinetic energy parallel to the surface

particle density in binj with velocity at angle 0i

a constant

I 3 2

__cos 0 - cos0 sin 0

error function

E/ET normalized energy parameter for binj

normalized parameter in the 1-dimensional diode; not to be confused with

TI for normalized energy

q parameter at cathode

normalized distance parameter defined by Langmuir [ 10]

normalized parameter at cathode or anode

separation of planes in 1-dimensional diode

NASA/'rM--2001-210890 66
VITI

Va

K

p(x)

Eo

A

Xa

V(X)

TWT

tf

ip

Np

AQp

AQj

np

qj

Mj

m

nj

Ut

Uz

EGUN

PPM

voltage at minimum (virtual cathode plane)

voltage at anode

constant in analytical expression for

charge density (Coulombs/m 3)

permittivity of free space

cross-sectional area

position of anode

velocity field in the 1-dimensional diode

traveling wave tube

time of flight of a particle in the 3/2-law regime

patch current

number of patches

total charge emitted from a patch during At

net charge in binj

number of particles/patch emitted during At

charge of particles in binj

mass of particles in binj

mass (generally an electron)

number of particles (per m 3) in binj

transverse velocity in electron beam

longitudinal velocity in electron beam

popular electron gun program (used for last 25 years)

periodic permanent magnet (stack)

NASA/TM--2001-210890 67
REPORT DOCUMENTATION PAGE FormApproved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing Instructions, searching exisling data sources,

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

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Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

June 2001 Technical Memorandum

4. TITLE AND SUBTITLE

Preliminary Study of Electron Emission for Use in the PIC Portion of MAFIA

6. AUTHOR(S)

Jon C. Freeman

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

5. FUNDING NUMBERS

WU-755-1B-O(O0

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-12766

10. SPONSORING/MONITORING

AGENCY REPORTNUMBER

NASA TM--2001-210890

11. SUPPLEMENTARY NOTES

Responsible person, Jon C. Freeman, orgamzationcode5620,216-433-3380.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Categories: 31, 32 and 33 Distribution: Nonstandard

Available electronically at http://eltrs.grc.nasa._ov/GLTRS

This publication is available from the NASA Center for AeroSpace Information, 301-621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This memorandum summarizes a study undertaken to apply the program MAFIA to the modeling of an electron gun in a traveling wave

tube (TWT). The basic problem is to emit particles from the cathode in the proper manner. The electrons are emitted with the classical

Maxwell-Boltzmarm (M-B) energy distribution; and for a small patch of emitting surface; the distribution with angle obeys Lambert's

law. This states that the current density drops off as the cosine of the angle from the normal. The motivation for the work is to extend

the analysis beyond that which has been done using older codes. Some existing programs use the Child-Langmuir, or 3/2 power law,

for the description of the gun. This means the current varies as the 3/2 power of the anode voltage. The proportionality constant is

termed the perveance of the gun. This is limited, however, since the 3/2 variation is only an approximation. Also, if the cathode is near

saturation, the 3/2 law definitely will not hold. In most of the older codes, the electron beam is decomposed into current tubes, which

imply laminar flow in the beam; even though experiments show the flow to be turbulent. Also, the proper inclusion of noise in the beam

is not possible. These older methods of calculation do, however, give reasonable values for parameters of the electron beam and the

overall gun, and these values will be used as the starting point for a more precise particle-in-cell (PIC) calculation. To minimize the

time needed for a given computer run, all beams will use the same number of particles in a simulation. This is accomplished by varying

the mass and charge of the emitted particles (macroparticles) in a certain manner, to be consistent with the desired beam current.

14. SUBJEC{" TERMS

TWT gun modeling; PIC code; MAFIA

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE

Unclassified Unclassified

NSN 7540-01-280-5500

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OFPAGES

74

16. PRICE CODE

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescribedby ANSI Std, Z39-18

298-102