The computation of the growth of a gaseous discharge in space-charge distored fields

The computation of the growth of a gaseous discharge in space-charge distored fields

COMPUTER PHYSICS COMMUNICATIONS 3 (1972) 322-333. NORTH-HOLLAND PUBLISHING COMPANY THE COMPUTATION OF THE GROWTH OF A GASEOUS DISCHARGE IN SPACE-CHAR...

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COMPUTER PHYSICS COMMUNICATIONS 3 (1972) 322-333. NORTH-HOLLAND PUBLISHING COMPANY

THE COMPUTATION OF THE GROWTH OF A GASEOUS DISCHARGE IN SPACE-CHARGE DISTORTED FIELDS A.J. DAVIES and C.J. EVANS Department of Physics, University College of Swansea, Singleton Park, Swansea £42 8PP, UK

Received 8 March 1972

PROGRAM SUMMARY Title of program: SPARK71 Catalogue number: ABUD Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: ICL 1905E. Installation: Computer Centre, University College of Swansea, ICL ATLAS. Atlas Computer Laboratory Chilton, Didcot, U.K. Operating system: GEORGE 2 Programming language used: FORTRAN high speed storage required: 18600 words. No. of bits in a word: 32 Is the program overlaid? No No. of magnetic tapes required: 2 at most What other peripherals are used ? Card Reader; Line Printer No. of cards in combined program and test deck: 1075 Card punching code: BCD Keywords: Plasma, Atomic, Electrical Discharge, Electrical Breakdown, Space-charge, Plane-parallel Electrodes, Positive Ion, Electron, Photon, Cathode.

Nature of physical problem SPARK71 computes the growth of ionization currents flowing between parallel plane electrodes. The effects which are incorporated are primary ionization, secondary cathode emission due to the incidence of photons and ions, spacecharge distortion of the applied field, and the properties of the external electrical circuit. Method of solution The hyperbolic differential equations describing the growth of ionization in one dimension are integrated along the two characteristic directions in the x—t plane [1,2]. Iteration in the t direction allows the change in the coefficients of the

equations (due to field distortion) to be followed. The electric field is found from an approximation to the correct threedimensional distribution [3]. The x axis is divided into equallength meshes, and the time step is chosen to minimise the number of iterations needed. Restrictions on the complexity of the problem As given in the listing, the program will accommodate up to 80 meshes in the x direction, and up to 4 photon secondary processes, each with its own characteristic delay time. (These numbers may be increased by changing the DIMENSION statements only.)

A.i Davies and Cf. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields Typical running time This is of course a function of the number of meshes in the x direction, the number of time steps, and the number of iterations per time step. On ATLAS, one iteration with 80 meshes takes 230 instruction interrupts (about 1.44 see). A typical run may contain 100 to 200 time steps, each having 3 to 10 iterations, and running times of the order of 20 to 30 mm are usually obtained. The ICL 1905E takes about 7.1 sec per iteration with 80 meshes. Unusual features of the program For some combinations of the input data, the non-linearity

323

of the equations becomes so strong that an impossibly high order of accuracy is needed to ensure convergence. In such cases, the program will eventually fail due to instability leading to overflow.

References [1] A.J. Davies, C.J. Evans and F. Llewellyn Jones, Proc. Roy. Soc. A281 (1965) 164. [2] A.J. Davies, CS. Davies andC.J. Evans, Proc. lEE 118 (1971) 816. [3] A.J. Davies and C.J. Evans, Proc. lEE 114 (1967) 1547.

324

A.J. Davies and C.J. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

LONG WRITE-UP 1. Introduction The purpose of this paper is to make generally available a method of computing the development of ionization currents between plane-parallel electrodes, when the field is distorted by the electrons and ions. A formal solution to this problem has been given by Davidson [1], for the case where the space-charge distortion can be neglected. The program SPARK71 computes the very rapid growth of an electrical discharge when space-charge effects predominate and no formal

positive ions per unit length of the discharge, and We and W~,are the electron and positive ion drift velocities. In the case where the field is distorted by spacecharge, eqs. (1) and (2) are non-linear, since a, We and W~are functions of the electric field, E, and hence of x and t. At the cathode, the boundary condition to be satisfled is t

Ie(0~t)

‘o(~)+ ~

+

The following ionization processes are incorporated: (a) Primary ionization of gas molecules by electrons. The Townsend primary ionization coefficient a is defined as the mean number of ionizing collisions per unit distance travelled by an electron. (b) Secondary electron emission from the cathode due to the incidence of positive ions. The secondary coefficient ‘y~represents the mean number of secondary electrons leaving the cathode for each ionizing collision in the gas. (c) Secondary electron emission from the cathode 7ph is the mean due to the incidence ofphotoelectrons photons. number of secondary released at the cathode, due to the incidence of photons of species k, per ionizing collision in the gas. The photons are emitted from excited atoms after a characteristic delay time Tk. (d) A photoelectric current 1 0(t), produced at the cathode by external means. The breakdown voltage, V~,is defined as the value of the applied voltage for which the breakdown critenon ~T(e



1)

=

1 7T

isthesatisfied, d being the electrodey~ separation, total secondary coefficient + ~k~Yph and The continuity equations describing the growth of ionization between parallel-plate electrodes are [lj 8net=aWene_Wene)/Zix

,

3n~Iat= aWene + a(W~n~)!ax ,

where ~e and np are the number

of electrons and

X exp

[(t’



t)/TkJ

while at the anode (x

=

Yp~t~(X,t)Tj~

o o

k

solution exists.

d

fJ’

~

dxdt’

(3)

,

d),

I~(d~t) = 0.

(4)

In the above,Ie andI~are the electron and ion currents (=~eWe and n~,W~respectively). An arbitrary charge distribution fle(X,0), n~(x,O)may be specified at time t = o, and the current 1 0(t)the(generally by ultra-violet irradiation of cathode)obtained may be any specified function of time. In addition, the electric field distribution must at all times satisfy the condition d

f E(x,t) dx

=

V(t)

(5)

,

0

where V is the potential difference between the electrodes. In the present work, the external circuit is taken to consist of a resistance R connecting the gap capacitance Cg to the supply capacitance C. Many practical be approximated by this arrangement. Thecircuits circuitcan equations are dr/a V~ V = iCR, —

~

‘c ~

dV (11c) ~j-~=

(6)

——~

(1) (2)

where I and I~are the currents in the and the external circuit, and Vc(t) is

the supply capacitor.

discharge gap the voltage on

A..!. Davies and Cf. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

2. Method of solution [2,3] Eq. (1) may be written a/Ic ——

at

+

an W e eax

aWe \

/

aW ‘ e

=



In

axle

or Dn

aw

/

(~aW~ —

=

(7)

n~,

where D/Dt represents the derivative evaluated in a frame of reference moving with the electrons that is, along the characteristic curve x Wet = constant, Eq. (7) may now be integrated to give —



=

t1eO

awe/ax) dt,

exp f(aWe C

(8)

integral is taken along the characteristic curve C, tieO is the electron density at the beginning of the curve, and t~eis the required density at some other point of the curve. There is an analogous expression for ni,. Let us suppose that (1) and (2) have been integrated up to a time T (which will be referred to as the “present” time), and that tables of all the relevant quantities ~e’n~and E have been listed at a set of mesh points x = 0, h, 2h, (N—I)h(= d). We wish to calculate the quantities ~e’n~,E at the same mesh points at some later time T+ z~t(referred to as the “future” time). Fig. 1 shows the mesh points at the two instants of time, where the

. .

. ,

325

To calculate tie at the point D’, for example, we must follow the characteristic curve through D’ back to the present time T. As a first approximation, (since none of the variables has been obtained at D’) we assume that the field at D’ is the same as that at D, and hence obtain an approximate value for W at D’. It e would be possible to trace the characteristic curve back as a straight line to the point F. If it reaches the line CC’, however, it is better to calculate a new value for We(C”) (equated to We at C for the first approximation), and hence to allow for any possible curvature of the characteristic. This process can be continued to the point G. It is then necessary to interpolate for ~e at G, and to integrate aWe awe/ax in the exponent of (8) along the sequence of straight lines which approximate the characteristic curve. The value offle at G is found by a four-point interpolation routine (this being replaced by a three-point routine at each electrode). The positive ion characteristic is treated similarly, but since the ions move only about 1/100th of the distance moved by the electrons, there is no need to allow for the curvature of their characteristics. When tie and n~have been calculated at all the pivotal points A’, B’, C’ the electric field at the future time, T+ ~t, may be found. The electric field is calculated by regarding the discharge as being made up of discs of thickness equal to the mesh spacing, and summing the contributions to the field at a point due to each of these discs. For greater accuracy, the total charge density p, is assumed to vary linearly from one mesh point to the next. The field at a point due to a disc whose faces are at distances x and x + h from the point is then given by an integral of the form —

. . . ,

x+h ~

B

C

D

f

E

(Ax+B)[1

_x(x2+r2)~]~

I

—‘

which is conveniently expressed in the form C(x+h)[p(x)+p(x+h)]+C

_~

1(x+h)[p(x)—p(x+h)]. FAG

cathode

B

C

DH

(9)

E

anode

Fig. 1. Characteristic curves traced out by electrons and ions reaching point D’.

The coefficients C and C1 may be evaluated at the beginning of the calculation, since they will be functions of the radius of the discharge, r, and of the distance x + h, which will always be a multiple of the mesh length h. The calculation of the field then reduces to

326

A..!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

one of multiplication and summation. The presence of plane metal electrodes at the ends of the gap introduces an infinite series of image charges, but it has been shown [4] that sufficient accuracy can be obtained by considering only those images within a distance d of the point at which the field is required. This means that the number of coefficients C and C1

Cg,

C and R, the gap voltage will fall at a much faster rate than that on the supply capacitor, eqs. (6) may be integrated over the time step &, to give

may be limited toN(the number of mesh points). Provision is also made for calculating the field from Poisson s equation in one dimension

Vc(T+

IC(T + ~t)

=

+I(T+~t)[l —exp(—&/RC)] ~t)

=

V~(T)~c(T+

~t)&/C

(11)

and V(T+~st) =

aE/ax=—4~p

Ic(T) exp (—&/RC)

Vc(T+~t)_RIc(T+1~t)

(10)

but care should be exercised in using this option as considerable errors (of several orders of magnitude) may be introduced if the radius of the discharge is small compared with the electrode separation [4]. The electron current at the cathode at the future time (T + tie’ ~t) n~, may be evaluated fromintegration the approximate andE, by numerical of(3). values of 2.1 Iteration and testing for accuracy11e~n~and E at approximations to the above theOnce time the T + first L~thave been determined, procedure may be repeated in an iterative manner to obtain more accurate approximations for these quantities. On subsequent iterations, the values of a, We, etc., at a point such as C” (in fig. 1) are found by linear interpolation between the values at C and C’ which, in general, will be different. The iterative procedure is repeated until a self-consistent set of values of iie~np and E is found for the future time. It is convenient to use the electron current at the cathode to test for convergence, since it depends on the currents and fields at every point of the gap. At large values of the total current, the calculation may fail to converge (probably due to the error in evaluating awe/ax in the neighbourhood of a sharp maximum). The rate convergence is otherwise determined mainly by the length of the step &, and & is halved in the program if more than ten iterations are required to obtain the desired accuracy in Ie(0,t). To economise on computer time, ~t is doubled if fewer than six iterations are required, but the maximum value of z~t is restricted to h/We (evaluated at the total applied voltage J7~)so as to ensure stability, Since for practical values of the circuit components

It would be possible to allow for an arbitrary distribution of charge between the electrodes at time zero, and any functional dependence for the externallygenerated photocurrent 10(t). The present program incorporates the following two cases (a)]’0 constant, and the initial charge distribution calculated steady an applied voltage (which mayinbethezero) lessstate thanwith the breakdown voltage. (b)1 0(t) a gaussian distribution in time, and the charge density initially zero everywhere. In both cases, the calculation starts with the field uniform. Other situations can be covered by writing a suitable subroutine in place of UVCUR.

3. Program description The detailed working of the program will not be described here as it is set out on COMMENT cards incorporated into the program deck. The program consists of a large main segment containing the integration routines, accuracy tests, and input and output instructions, together with a number of short subroutines and functions which will be listed below. The structure of the main segment is illustrated in the block diagram (fig. 2), which should require little explanation. The re-start option using tape (MT3) is provided so that a long run may be continued without wastage of computer time in the event of a machine fault, or a miscalculation of running time. In addition to dumping the current variables on to tape 3 at selected intervals, another tape (MT4), which stores details of the light output from the discharge, may be created.

A..!. Davies and Cf. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

Start

New Data?

Yes

Read cards.

No

Read current variables

Calculate

chargetime distribution zero. at

(

from mag tape 3

Use “future” values ITSR

=

0

as new “present” values of variables.

Integrate equations over one time step to give next approximation to values of charges and fields at future time.

_______________

Use

boundary condition to find electron current at cathode. ITSR = ITSR + 1.

N

1~~ficient

accuracy?

Yes

0utputto and tomaglineprinter tapes 3 and ~ equired.

ITSR<6?

Yes =

of

run?

At/2

Reset variabl =

to values at

2At

present time. ~4~o?Yes~, Fig. 2. Block diagram for main

segment.

Stop

No

327

328

A..!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

4. List of subroutines and functions 4.1. Subroutine FIELD(IA,IB)

Computes the electric field distribution corresponding to a specified charge distribution. Variables are transferred to and from the main segment via COMMON.

DELT(I,X) GAMMA(X)

TERPF(A,X,I,N)

TERPL(A,X,1,N)

UVCUR(T)

WI’vIIN(X) WPLU(X)

KPLR

CARDS 2—Il

4.2.Functions

ALPH(X)

KNR

Evaluates the primary ionization coefficient a for an electric field X. Evaluates the Ith photon secondary coefficient. Evaluates the positive ion secondary coefficient; X is the field at the cathode. Interpolates between four successive values of an array A of dimension N. X is the distance from the point Ito the interpolated point, expressed as a fraction of the mesh spacing. Similar to TERPF except that the vector A contains the logarithms of the variables to be interpolated, and it includes precautions to prevent a negative result. Defines the externally maintained photocurrent at the cathode as a function of time, Evaluates the electron drift velocity. Evaluates the positive ion drift velocity.

Y(l) Y(2)—Y(7)

Writing on TAPE 3 commences at dump no. KNR+l. If KMT = 1, starting values are read from dump no. KNR. KPLR is the no. of the last plotting record to be retained on TAPE 4; writing commences at record KPLR+1. If KPLR = —1, no plotting information is written, and TAPE 4 need not be allocated. FORMAT (3E15.8) contain values of Y(l)—Y(30). Gas pressure in torr. Coefficients in the formula for Townsend’s primary ionization coefficient a. E Y5 a = 0, < p / = const exp ~ E
.

k—





E

> Y(4) Y(8)—Y(I 1)

Coefficients in formula for the positive ion drift velocity. E/ E W = Y(9) (1 Y(l0)—

p~

p



p


4.3. Common variables A list and description is given on COMMENT cards in the deck.

IE\~ Wp Y(l l)~) 2 const—

~>Y(8). 5. Input CARD 1

KMT

P FORMAT (315) contains values of KMT, KNR, KPLR. = 0 for starting from cards with new data. = 1 for re-starting from magnetic tape (TAPE 3)

Y(12)—Y(13)

Coefficients in formula for electron drift velocity.

E We Y(12)—+ Y(l3)

-

A.f. Davies and C’..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

Y(14)—Y(15)

Coefficients in formula for positive ion secondary coefficient, =

Y( 16)

Y(17) Y(18)

Y(19) Y(20)

/ F~ const exP~Y(l4)_)+Y(l5).

Externally-maintained photocurrent at cathode (in the case of constant illumination), or maximum current (in the case of flash illumination with a gaussian pulse). Radius of discharge, cm. If positive, eq. (9) is used for the calculation of the field distortion. If negative, Poisson’s equation in one dimension (10) is used. Approach voltage applied before time zero, volts, Applied voltage at time zero, volts.

Y(2 1)

Breakdown potential, volts,

Y(22)

If positive, represents four times

Y(23)

Y(24)

329

NPP

Number of photon secondary processes.

NextNPPCARDS

FORMAT(3El5.8)eachcontains values of FDP(I), TD(l), B(I), I1, NPP (the case NPP = 0 is permitted). Fraction of the total secondary effect due to the Ith photon

FDP(I)

process.

TD(1) B(l)

Time delay associated with the Ith process. Coefficient in formula for Yph(’) 1)

~~h(

=

/ const . exp (,,—B(I)

-~

IfB(I) < 0~-~(I)is assumed proportional to the primary ionization coefficient a. CARD 13+NPP onwards FORMAT (315) Contain

values of NT,NPR,L. NT

No. of detailed print-outs between

the standard deviation of the gaussian pulse, and the computation starts at time —2Y(22). If negative, steady illumination is

successive dumps on TAPE 3. If NT = 0, the program is started afresh with new data from CARD 2 onwards, and with TAPES 3

assumed, and the calculation starts at time zero. Electrode separation, cm. ~ at the breakdown potential, as

and 4 left in their current posi-

tions. NPR

No. of time steps between detailed

print-outs. (NTXNPR is therefore the no. of time steps between

a fraction of the total secondary Y(25)

effect. Resistance in external circuit,

L

Y(26)

ohms. Capacitance of external circuit,

is written at every Lth mesh point. This card may be repeated as many times as desired.

farads.

LAST CARD

Y(27) Y(28)

dumps). In a detailed print-out, information

Capacitance of discharge gap, farads. Fractional accuracy required in

A blank card, or three zeros in FORMAT (315), terminates the run.

iterations for Te(O,t),

Y(29)

Fraction of the mesh spacing

Data for restarting from tape When restarting from TAPE 3, (KMT = 1), only CARD I and CARD 13 + NPP onwards are required.

Y(30) CARD 12

travelled by electrons during the first time step. Normally set to 1.0. Minimum allowed time step, sec. FORMAT (215) contains values of N,NPP, Number of mesh points (including the two boundaries).

N

6. Output Lineprinter As can be seen from the test run, the input data is

330

A..!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

first listed, followed by a summary of the steady-state

from the authors.) Reference to TAPE 4 may be sup-

calculations, namely the time, the total voltage and the current. Next the current values of NT, NPR and L are printed, At the end of each time step, a summary of the results is given, comprising the current values of the time, the time step, the gap voltage, the total current, the no. of iterations, and the current record no. on TAPE 4. Every NPR time steps, the following information is printed for every Lth mesh point, in addition to the above summary: (a) electric field, volt/cm 3 (b) electron density, coulomb/cm (c) positive ion density, coulomb/cm3 (d) light output, arbitrary units. After NPRXNT time steps, the record no. of the dump on TAPE 3 is printed, followed by the next set of values of NT, NPR and L. The above output is repeated until a data card is encountered with NT or NPR = 0. TAPE 3 All the information necessary to restart the program is dumped on TAPE 3 after NPRXNT time steps, in the form of an unformatted record. TAPE 4 At each time step, the program outputs the logarithm

pressed by putting KPLR = -—1. The same data could also be used for drawing graphs, simulating still photographs, or photomultiplier records.

of

the light intensity at each mesh point. This data may

be used to simulate a streak photograph of the discharge

[3]. (A listing of a program suitable for running on the SC4020 plotter at the ATLAS lab., Chilton, is available

7. Description

of test run

The test run calculates the growth of ionization currents in nitrogen, for the case where the discharge is initiated by a pulse of light incident on the cathode. Two sections of the output have been selected, showing the results obtained (a) at the beginning of the calculation, and (b) at a later time, when the field has become appreciably distorted. Note that in (a) the time is initially negative, since time zero is arbitrarily fixed at the maximum of the (gaussian) pulse of light falling on the cathode. The test run took approximately 500 seconds of CPU time on an ICL l905E. This corresponds to approximately 90 seconds of ATLAS CPU time.

References [1] P.M. Davidson, Brit.J.Appl.Phys. 4 (1953) 170. Evans and F. Liewellyn Jones, Proc. Roy. Soc. A281 (1965) 164. 131 A.J. Davies, (1971) 816. CS. Davies and C.J. Evans, Proc. lEE 118 141 A.J. Davies and C.J. Evans, Proc. lEE 114 (1967) 1547.

[21 A.J. Davies, C.J.

A..!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

TEST RUN OUTPUT SECTION (a) TEMPORAL GROWTH OF IONIZATION

INPUT

KMT



DOUBLE CHARACTERISTIC METHOD

DATA

_0.

KNR

0.

KPLR

0

PLOTTING INFORMATION WRITTEN TO TAPE 4 FOLLOWING RECORD

0

2.00000000E 02

GAS PRESSURE

2.10000000E—01 4.40000000E 01 1.76000000E 02 2.76000000E 02 1.17000000E—04 3.2200000DB 01

) 1 1 ) ) 1

8.00000000E 01 2.00000000E 03 4.00000000E03 1 .25000000E 04

) ) ) 1

2.44000000E 05 3.10000000E 06

) )

CONSTANTS IN FORMULA FOR ELECTRON DRIFT VELOCITY

0.00000000E—01

I

CONSTANTS IN FORMULA FOR POSITIVE ION SECONDARY COEFFICIENT

C.00000000B—01

)

t.14000000E—09

MAXIMUM (OR CONSTANT) VALUE OF EXTERNAL PHOTOCURRENT AT CATHODE

1.50000000E—01

RADIUS OF DISCHARGE (CM)

~.00000000E

+1.0 DENOTES DISC METHOD.

00

(bAR)

CONSTANTS IN FORMULA FOR PRIMARY IONIZATION COEFFICIENT

CONSTANTS IN FORMULA FOR POSITIVE ION DRIFT VELOCITY

—1,0 DENOTES POISSONS EQUATION

s.00000000E—01

APPROACH VOLTAGE

2.92560000E 04

TOTAL APPLIED VOLTAGE

2.54400000E 04

BREAKDOWN VOLTAGE

4.?0000000E—08

4

3.0000000DB 00

GAP LENGTH

0. 00000000E—01

FRACTION OF TOTAL

1.00000000E 00

RESISTANCE

I .00000000E—06

CAPACITANCE OF EXTERNAL CIRCUIT (FARADS)

.00000000E—11 ‘i.00000000E—06

*

STANDARD DEVIATION OF LIGHT AULSE (SEC).

IF NEGATIVE ILLUMINATION IS CONSTANT

SECONDARY COEFFICIENT DUE TO POSITIVE

IONS

IN EXTERNAL CIRCUIT (OHMS)

CAPACITANCE OF DISCHARGE GAP (FARADS) FRACTIONAL

ACCURACY REQUIRED IN CATHODE CURRENT

1.00000000E 00

FRACTION OF MESH TRAVELLED BA ELECTRONS IN FIRST TIME STEP

2.00000000E—10

MINIMUM ALLOWED VALUE

41

MESH POINTS.

FRACTION OF SECONDARY C0EF F IC I ENT 1.00000E 00

1

OF TIME STEP

SECONDARY PROCESS(ES)

TIME DELAY (SEC)

CINSTANT B(I)

O.00000E—01

—1.0000DB 00

NUMBER OF SECONDARY COB FF1 CIENT 1

331

332

A..!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

START OF CALCULATION

INITIAL CONDITIONS~ TIME ~—9.40000E—08 SEC

NT

2

NPR

TIME SEC —8.89991E—08

S

GAP VOLTAGE



L

2,925e0E T4,

TOTAL CURRENT

0.00000E—01 AMP

1

TIME STEP SEC 5.00085E—09

GAP VOLTAGE VOLTS 2.9256)R 04

ELECTRIC FIELDS (VOLTS/CM) AT TIMF—8.89991E—08 SEC 9.?5198B 03 9.75198E 03 9.75198E 03 9~7S198E 9. 75198E 03 9. 7SICRE 03 9.7519dB 03 9,751 ORE 9. 15198E 03 03 9. 75198E 03 03 9. 7519dB 03 03 9, 75198E 9.75198E 9J519AE 9.75198E 9,75198E 9.75198E 03 9.75198E03 9.75198E 03 9.75198E 9.75198E 03

TOTAL CURRENT NUMBER AMPS ITERATIONS 1.86591f—23 3

03 03 03 03 03

9.)’5198B 9. 15198E 9. 1519dB 9,15198E 9.1519RE

03 03 03 03 03

9.75195E 9. 15195E 9. 15196E 9.75199E 9.7519dB

PLOT RECORD 1

03 03 03 03 03

9.1519dB 9. 15198E 98E 9. 151 9.1519dB 9.1519dB

03 03 03 03 03

9.1519dE 9..75198E 1519dR 9 9.?5198E 9.7519dB

03 03 03 03 03

ELECTRON DENSITIES (COUL/CC) 7.03164E—28 1.07894E—47 1,07R94E—47 1.07894E—47 1.07B~4E—4? 1.O7AO4E—47 1.0?B94E—47 1.07894E—47 1.07894E—47 1.07894E—47 1.07894E—47 1.07894E—47 1. 07894E—47 1. 07894E—47 1 - 07894E—47 I .07894E—47

1.07896E—47 1.07804E—4? 1.07894E—47 1.07N94E—47 1 . 07694E—47

1.01894E—47 1.01B94E—47 1.01b94E—4? 1.U1A94E—47 I• O1AVNE—4?

1.07d94E—4? 1.07d94E—41 1.Omd94E—47 1.07694E—47 1, 07e94E—41

1.07894E—47 1.07b94E—47 1.07894E—47 1.07694E—47 1~07694E—47

1.07894E—41 1.07d94E—47 1.07894E—41 1.0?894E—41 1. 07894E—41

POSITIVE ION DENSITIES (COOL/CC) 1.69206B—28 4.20086E—48 4.200R6E—48 4.20086E—48 4.20086E—48 4.20086E—48 4.20086E—48 4.2O086E~48 4.20086E—48 4.20086E48 4.20086E48 4.20086E—48 4.20086E—48 4.20086E—48 4.20086E—48 0 • 00000E—01

4.20086E—48 4.20086E—48 4,20086E—48 4.20086E—48 4.20086E—48

4.20086E—48 4.20056E—48 4.~0OA6E—48 4.200d6e—48 4.20086E—48

4.200dOE—48 4.20086E—4b 4.ZOOEoE—4B 4.20086E—46 4.20086E—4d

4.20086E—48 4.20086E—48 4.~0Ob6E—48 4.20086E—48 4.20086E—48

4.200d6E—4d 4.20086E-48 4.~O0E6E—4B 4.20086E—4d 4.20086E—48

LIGHT OUTPUT (ARBITRARY UNITS) 2.1 3608E—24 1. 00000E—40 1 . 00000E—40 1. 00000E—40 1. 00000E—40 1, 00000E—4O 1 .00000E—40 1 .00000E—40 1 .00000E—40 ‘I.OOOOOE—AO 1.000uOE—40 ‘.00000E—40 1.00000E—40 1.00000E—40 T.00000E—40 1. 00000E—40

1~00000E—40 1, 000uOE—40 1 .00000E—40 1.00000E—40 1,00000E—40

1~00000E—40 1~00000E—40 1 .00000E—40 1.00000E40 1.0000DB—At

1 . 0000(’E—40 1. 00001’EkO 1 .00000E—40 T.00000E40 1.00000E—40

1• 00000E—4O 1 . 00000E’.40 1 .00000E—40 T.(J0000E’NQ l.00000E—40

1. 00000E—40 1. 00000E40 1 .0000DB—AU T.00000EAO 1.0000DB—AU

A,.!. Davies and C..!. Evans, The computation of the growth of a gaseous discharge in space-charge distorted fields

333

SECTION (b)

TIME SEC 1,16036E—07 1.210371—07 1.26038B—07 1.31038E—07 1 .3603~9B—07

TIME STEP SEC 5.0008SE—09 5.0008SE—09 5.00085E—09 5.00085E—09 5.00085E—09

GAP VOLTAGE VOLTS 2.92560E 04 2.92560E 04 2.92560E 04 2.9256DB 04 2.~2560E 04

ELECTRIC FIELDS (VOLTS/CM) AT TIME 1.36039E—07 SEC 9.75607E 03 9.75611E 03 9.7561AE 03 9.75631E 9.?6028E 03 9.76204E 03 9.76A07B 03 9.76640E 9.79061E 03 9.80141E 03 9.dlS4SE 03 9.IA3ZGOE 9.80034B 03 9.69261E 03 9.55771B 03 9,43505E 9~7S5Q4E 03 9.88390E 03 9.93839E 03 9.92738E 9.?9824B 03 ELECTRON DENSITIES (COUL/CC) 1,84942E—12 2.OOARRE—12 2.35?39E12 2.36514B”12 2.419671—12 2.43?69E—12 8.93181E—11 2.20138E—1O 1.22179E—09 8.46098E—10 1 .74925E—13

03 03 03 03 03

9,15661E 9.16912E 9~8526dE 9.36562E 9.88932E

NUMBER ITERATIONS 4 A A 4 5

03 03 03 03 03

9.7510RE 9.11240E 9.8?090E 9,3685dB 9.85179E

PLOT RECORD 42 43 AN 45 AN 03 03 03 03 03

9.1518GB 9.17683E 9.d7d6NB 9.44839B 9.82601E

03 03 03 03 03

9.75885E 9.18269E 9.86013B 9.5904dB 9.81213E

03 03 03 03 03

2.20918E—12 2.37678E12 2.53350E—12 ?.36?94E—10 1,9865~E—10

2.~6114B—12 2.38~41B12 2.81550E—12 1.04522E—09 6,93626E—11

2.30540E—1~ 2.38b?9E—1~ 4.1643tE—12 1.30A43B—0V 2.01A48E—11

2.33029E—1~ 2.3965SE1~ 9.93441E—12 1.45494B—09 4.92134E-’14

2.34652E—12 2.40651B—12 3.04921E—11 1.43960E—09 1.01292E—12

POSITIVE ION DENSITIES (COULICC) 2.11310E—12 2.21607B—12 2.29374E—12 2.69298E—12 2.87122E—12 3.15495E—12 1,5406?E—11 2.33555E—11 3,61624B—11 5.130071—10 7.10438E.10 9.04531E—10 6.51961B—10 4.05033B—10 1.99811E—10 0. 00000E—01

2.35273E—,2 3,60971E—12 5.67?29B—11 1.05600R—09 7.d5313E—11

2.401E4E—12 A.34109E—12 8,98413E—l1 1.13919E—09 ~5~606E—11

2.4481~E—1~ ~.50326B—12 5.S1d70E—12 1.41635E—12 1.42463E—10 .?.24480E—10 1.14251E—09 1.06032E—09 b.83869R—1~ T.57141E—1i

7,51901B12 1.0A753E—11 3.46549E—10 8.92585E—1U 3.10240E—13

LIGHT OUTPUT (ARBITRARY UNITS) 5.63398E—09 6.10?72B—09 6.470971—09 7.20215E—09 7.23452E—09 7.26335E—09 7.54696E—09 7.65864E—09 7.82069E—09 2.8G415B—07 6.41866B—07 1,17363E—06 3.71937E—06 2.M0896E~06 1,58788E—06 5. 4839 71—10

6.13103E—09 7.29179E—09 8.13116E—09 1,78426E—06 6,18850E—01

6.90904E—O9 7.3~2b6E—09 9.15539E—09 2.3V9f5E—06 2.31103E—0?

9 7.02191E—0 1.35908E—09 1.3706dB—OR 3.00165E—06 6.54611E—0d

1.16194E-09 1.46538B-09 9.96886E—0d 3.90518E—06 3.205471—09

TIME SEC 1.41040E—07 1.46041E—07 1,51042E—07 1,56043E—07

2.12AOSE—12 2.3712?E’lZ 2.46580B—12 4.42988E—10 4.6148AE—10

TOTAL CURRENT AMPS 1.2A991B—04 1.99345E—04 3,15A~2E—OA A.93A91E—UA ?.605(3E—OA

TIME STEP SEC 5.00085E’-t9 5.000854—09 5.00085E—09 5.00085E—09

VARIABLES CORRESPONDING TO ABOVE

GAP VOLTAGE VOLTS 2.92560E 04 2.92560F 04 2.’~2560F. 04 2.92560E 04

TOTAL CURRENT NUMBER AMPS ITERATIONS 1.15104E—03 4 1.70941E—03 6 2.50563E—03 1 3.’70499E—03 9

TIME STEP DUMPED TO TAPE 3 AT RECORD

5

1.10?2dE—09 1.40494B-09 3.~b66lE0d 3.55665B—06 1.57~01B—08

PLOT RECORD 47 48 49 50