Diffusion of electrons in avalanches

Diffusion of electrons in avalanches

Citation for published version (APA):

de Jonge, J. C. J. (1987). Diffusion of electrons in avalanches. Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987 Document Version:

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DEPARTMENT

o

F ELECTRICAL ENGINEERING High Voltage Laboratory (EHO)

Diffusion of el eccr'ons in avalanches

by: J.C.J. de Jonge EH.87.S.232

The Department of Electrical Engineering

of the Eindhoven University of Technology Supervisor: M.Sc. C. Wen does not iini' responsibility for the

contents of Leport.

SUMMARY

A model of physical processes in an avalanche is presented as well as an implementation on a personal computer to determine the swarm parameters from

measurements of avalanche currents using the Time-Resolved-Townsend method. Also a program to calculate an electron current waveform given its swarm parameters is presented.

The present report mainly deals with the influence of electron longitudinal diffusion on the electron current waveform.

The program base~ on the model is tested by calculated electron current waveforms with known swarm parameters and applied to measured electron current waveforms in nitrogen. The results of the program are satisfactory, but the accuracy and the calculation speed of the

CONTENTS page

1. INTRODUCTION 3

2. MODEL OF PHYSICAL PROCESSES IN AN AVALANCHE 5

2.1 The Time-Resolved-Townsend method 5

2.2 Physical model 7

2.3 Extended physical model 12

3. MEASUREMENT OF CURRENT WAVEFORMS 33

3.1 Introduction 33

3.2 The experimental setup 34

4. DETERMINATION OF SWARM PARAMETERS

FROM THE DIGITIZED WAVEFORM 36

4.1 Introduction 36

4.4 Procedure DETERMINE_Te 38

4.5 Electron longitudinal

diffuslon coefficient Dl 40

5. CALCULATED SWARM PARAMETERS

FROM THE DIGITIZED WAVEFORM 47

5.1 Accuracy of the program ANALYSE 47

5.2 Results 48

APPENDIX 1 APPROXIMATION OF THE O-FUNCTION O(X)

APPROXIMATION OF THE COMPLEMENTARY ERROR FUNCTION ERFC(X)

APPROXIMATION OF EQ.(2.34) WITH EO.(2.38)

APPENDIX 2 NUMERICAL CALCULATION OF THE COMPLEMENTARY ERROR FUNCTION ERFC(X)

APPENDIX 3 DERIVATIVE OF EO.(2.34) WITH RESPECT TO D1

APPENDIX 4 MODIFIED FOURIER-METHOD INTERVAL BOUNDS

APPENDIX 5 HEADER OF THE PROGRAM ANALYSE

APPENDIX 6 TEST RESULTS

APPENDIX 7 RESULTS

i.INTRODUCTION

In electrlcal power engineering, gases are often used as lnsulating media. To obtain more insight in pre-breaKdown phenomena in these media it is necessary to have more Knowledge about the physical processes during the early stage of an electrical discharge.

An electrical discharge starts with a number of primary electrons, which drift under the influence of the

aFplied E-field and build additional electrons and ions. In the beginning this is a stochastic process that can only be described in a microscopic way (0 < x < Xi

in Fig.t.i). AS soon as the number of electrons is large enough (e.g. 100), the electrons can be regarded

collectively, i.e. as a cloud or swarm of electrons (Xi < X < Xc in Fig.t.t). The behaviour of this

~o-called avalanche is deterministic and can be

described by macroscopic swarm parameters. When the density of the electron cloud is large enough, the resulting distortion of the E-field sets the beginning of the actual discharge (xc < X < d in Fig.i.t).

If the number of primary electrons is large enough, the first region (0 < X < Xi) is bridged. After a

negligibly small drift distance, the electron cloud reaches an equilibrium between the energy gain by the E-field and the energy loss by collisions with gas

molecules. This equilibrium condition is very important because the swarm parameters of the electron cloud in that ca~e only depend on the E-field and the density N of the gas between the electrodes.

The avalanches cause a current in the external circuit which can be measured and analyzed. A method that

facilitates a time-resolved measurement of avalanche currents is the Time-Resolved-Townsend method in which a number of primary photo-electrons are formed by a very

short UV light pulse.

E [{x,

I

" \ I I

-~---'---____________ L-~ x o d E

.-

--

--0--~.!e~.

(_ _ _

..

~\ ,

--E ~--~---2---~

___

x o x c d

Flg.1.1 The development of an electrical discharge in a homogeneous E-field. [3]

E applied E-field.

E(x): distribution of E-field in axial direction.

o

< x < xi stochastic behaviour, microscopic description. xi < x < xc: deterministic behaviour,

macroscopic behaviour.

Xc < x < d : electrical discharge under space-charge influence

2. HODEL OF PHYSICAL PROCESSES IN AN AVALANCHE

2.1 The Time-Resolved-Townsend method

In the literature this method is also known as Pulsed Townsend Discharge, Pulsed Avalanches, Temporal Growth Experiment and Electrical Method.

The main principle of the Time-Resolved-Townsend (TRT) method is g~ven in Fig.2.1. A very short UV light pulse releases a number of primary photo-electrons from the cathode (photo-electric effect), which drift to the anode under the influence of the applied E-field and build additional electrons and ions. The drift movement of the charge carriers between the electrodes as well as the change in the number of charge carriers during their drift time cause a current in the external circuit, that

can also be interpreted as the current necessary to brlng the appropriate image charges to the electrodes when the motion of the electrons causes changes in the E-field pattern. This avalanche current consists of two components, the electron current and the ion current, which can be discriminated very well in time because of

the differing order of the mobility of electrons and ions.

t

®

~

~---E •

h." x

x • 0: CatllOde

It(tl • le(tl + In(t) + Ip(t)

-• r.W + ii(t)

Fig.2.1 Principle of the TRT method. The total current it(t), induced by electrons (e), negative ions (n) and

positive ions (p). Voltage supply U. [3]

Since the measurements were performed in nitrogen only the electron current component is discussed in the following sections. For that same reason it is assumed that ionization is the only collisional process that plays a role in the formation of an avalanche.

In order to describe the physical processes in the gas by the currents in the external circuit in a

quantitative manner, a physical model has to be

formulated. By matching the currents calculated using the model with the measured currents, the swarm

parameters of the physical model can be derived.

2.2 Physical model

With a physical model the waveform of the electron current can be derived theoretically.

In this model the following assumptions are made:

- The presence of a homogeneous E-field in an infinite system of electrodes.

- The primary electrons which start the avalanche are released. instantaneously from a certain area of the cathode by an UV light pulse of negligibly short duration.

- The bacK diffusion of the electrons into the cathode is neglected. The interaction of the anode with the approaching electrons is simplified to a countins process as if the anode were a perfectly transparent grid.

- The density of the cloud of electrons is assumed to be low enough to keep the distortion of the E-field by

space-charge effects small.

- The electron has an elementary charge e.

- The ionization coefficient a and the electron drift velocity ve are constant in place and time. There is no attachment, no detachment, no conversion and no

secondary photo-emission.

- Diffusion of electrons in both longitudinal and transversal direction is neglected. The electrostatic repulsion between the electrons is also neglected.

Under the influence of the applied E-field the electrons move with the drift velocity ve to the anode as a very

thin disk. At t

=

_{Te , where Te is the transit time of}

the electrons defined as Te : d i v e in which d is the electrode distance, all electrons leave the sap at the anode. These moving electrons cause an electron component of the current in the external circuit.

To calculate the electron current we consider the work done on the ne(t) electrons during their motion over a distance dx in the direction of the E-field in a time

dt. This energy equals ne(t) e E dx and is provided by

the external circuit:

( 2 . 1 ) U i ( t ) dt ::; n e (2.2) i ( t ) ::; n ( t ) e e ( t ) e e E dx e -U dt E dx ::; n ( t ) e e v e E U

o

<

t <

T

e

where the electron drift velocity ve is defined as the average veloCity of the electrons in the direction of

the E-field.

For a homogeneous E-field and a constant voltage U, the electron drift velocity ve will be constant also and Eq.(2.2) simplifies to:

( 2 . 3a) i ( t ) ::; e n (t) e v e e d

n

( t )

e

e T e

After t ::; Te the electron current is zero:

(2.3b) i (t) ::; 0 e

o

< t <

T

e t

>

T e

Eq.(2.3) remains valid when the distortion of the E-field by space-charge effects is not negligible. [3]

When nO primary electrons are released from the cathode at t::;O under such conditions that no ionization takes place, a ::; 0, the number of electrons stays constant. ne(t) ::; nO' From Eq.(2.3) the electron current is:

(2.4a) i ( t ) ₌ e (2.4b) i ( t ) ₌ e n 0 T 0 e e

o

<

t <

T

e t > T e

In case that ionization takes place, the number of

electrons changes. When the electrons travel a distance dx in the direction of the E-field, the change in the number of electrons is given by:

(2.5) dn (x)

=

a n (x) dx

e e

where the ionization coefficient a is defined as the mean number of ionizing collisions of one electron

travelling 1 cm in the direction of the E-field.

Under the assumption that the drift velocity of the electrons is constant in place and time, Eq.(2.5) changes to:

(2.6) dn (t)

=

a n (t) v dt

e e e

With the initial condition:

(2.7) n (0) = n

e 0

the total number of electrons for 0 < t < Te is:

(2.7) n (t) = n exp(a v t)

e 0 e

With the reaction frequency Ri def1ned as:

(2.8) R :. a v

i e

Eq. (2.7) becomes:

(2.9)

The electron current can be calculated from Eq.(2.3):

n (2, lOa) i ( t ) :. e (2. tOb) i ( t ) :: ₀ e e 0 T e exp(R. t ) 1

o

< t < T e t > T e

For Ri :: 0, 1.e. a :: 0, Eq.(2.10) is identical to Eq.(2.4).

"

"

_"

" "

_{" " ...} ... "

...

_...

--_ 0:< 0

d X I -~---~---r~t t"O I )("'0 X I t=T_e I ., x

F1g.2.2 The number of electrons ne for three typical conditions: a < 0, a :. 0, a } O. [3]

Figure 2.2 shows the number of electrons ne'

Neglecting dIffusion. the electrons are constrained to a very thin disk with thickness -) O. Except for the

constant (e / Te) the plot for ie and ne is the same.

The model is also applicable in case both ionization and attachment playa role. Then the ionization coefficient a should be replaced by the effective ionization

coefficlent (a - ~). The swarm parameters a and

n.

2.3 Extended physical model

The model discussed in the previous section can be improved by taKing into account the influence of electron diffusion and the effect of the electrodes.

Consider a very thin disk of electrons at t : 0. X : O. The disK increases in diameter and thicKness by

diffusion while the center of the disK moves with the drift velocity towards the anode because of the applied E-field. ANODE

--ct0

IQ".

: Df E I • • I I

CD

• I I

,

q::> •

•

I I c::. I CATHODE

x-o

Fi€.2.3 TRT-experiment with diffusion. [3)

The diffusion process, the random-walk motion that causes a net velocity from regions with a high

concentration to regions with a lower concentration, is not isotropic because of the presence of the E-field. The dependence in longitudinal direction (parallel with the E-field) can be described by the diffusion

coefficient Dl and the dependence in transversal direction (perpendicular to the E-field) by the diffusion coefficient Dt. These coefficients can be

defined using the rms deviation of the disk of electrons with respect to the mass center ~ of the disK: (3)

2 1 d(a ) x (2.11) D

₌

1 2 dt 2 1 d(a ) r (2.12) D

₌

t 2 dt

Since in a TRT-experiment only the influence of the electron longitud~nal diffusion can be measured, therefore only this component of the diffusion is

~ncorporated ~n the model.

The drift velocity of the mass center of the electron cloud W is not always equal to the electron drift velocity ve' The latter is defined in relation to the former as: (3) (x - J;(t» R (x,t) n (x,t) i

e

n

(t) e dx

in wh~ch Ri(X,t) and ne(x,t) are microscopiC swarm

parameters, dependent on both place and time. The second term

in Eq.(2.13) results from the spatial displacement of the mass center J;(t) caused by the unequal distribution of electron energies over a drifting electron cloud.

The electrons in the front of a drifting electron cloud (x - ~(t) } 0) will have a higher average energy, because in the same time interval they in average cross a larger potential, than the average over the electron cloud. A higher average energy implicates a higher ionization

frequency, i.e. the number of electrons in the front of the electron cloud increases more rapidly than in the rest

of the cloud.

At the back of a drifting electron cloud (x - K(t) < 0) the average energy of the electrons will be lower than the electron cloud average.

Both effects result in a higher drift velocity of the mass center of the electron cloud W in comparison with

the electron drift velocity ve'

TaKing into account the influence of electron

longitudinal diffusion but neglecting the effect of the electrodes, the electron drift velocity can be

approximated by: (3) (2.14) v :::

W - D

e 1 1 on(x,t} e n (x,t) dx e

=

W - a D 1

From Eq.(2.14) and Eq.(2.8) it can be seen that the electron drift velocity ve only equals the drift

velocity of the mass center of the electron cloud W if Ri : 0 and/or Dl : 0, In all other cases the

electron drift velocity not only consists of an E-field component W but also consists of a diffusion component a Dl'

In a TRT-experiment the gap distance is known. Therefore the electron drift velocity only depends on the electron transit time: (2.15) v : : e d T e

Because of the influence of electron diffusion and the effect of the electrodes, the determination of the electron transit time can be ambiguous. A method that conslders the spread of the current is discussed in chapter 4. From this method the electron tranSit time can be determlned from the waveform of a measured electron current.

Substituting swarm parameters that can be determined in a TRT-exper1ment in Eq.(2.14), an approximation for the drift

velocity of the mass center of the electron cloud can be derived: [3] d R T D i e I (2.16) W ~ -- + T d e

If the TRT-measurements are performed over a large

E

I N-range, the product of gas density N and electrode distance d must be kept sufficiently small to avoid

space-charge effects or even breakdown (Paschen curve). This is espec1ally important in charge carrier building processes.

If the gas density decreases, however, the influence of diffus10n will increase. Therefore the following relation

exists between Dl and N (note: in a TRT-experiment

only the longitudinal diffusion component is important):

(2.17) D

1 1

N

The electrode distance cannot be made arbitrarily small because then the effect of the electrodes becomes too dominant, whereas in a realistic system of electrodes

the electrode distance must be kept small enough to ensure the E-field is homogeneous.

Taking the above, a small gas density and a bounded drift area, into account in the physical model, the following extra assumptions are made:

- During their drift time the electrons have diffusion in longitudinal direction.

- The effect of the electrodes is constituted in

boundary conditions. The primary electrons are released lnstantaneously and the back diffusion of the electrons lnto the cathode is neglected. The anode absorbs the electrons completely.

A 1-dimensional electron cloud now can be described macroscopically by the continuity equation for

ne

=

ne(x,t): [3) 2 3 on on

a

n d n e e e e (2.18)

_{= R}

n

-

W - - - +

D

+

D

+

...

i e I 3 2 3 ot ax ox ax

in which the swarm parameters Ri' the reaction frequency, W, the drift velocity of the mass center of the electron

cloud, Dl' the electron longitudinal diffusion

coefficient and D3 etc. are constant in place and time.

The swarm parameter D3 does account for the spatial asymmetry (skewness) of the electron cloud caused by large density

gradients ln the cloud (e.g. by boundary conditions at the electrodes). At some distance from the electrodes, the contribution of the higher order terms normally is negligibly small. In most cases, only the first three

terms in Eq.(2.18) are taken into account. The

continuity equation for ne : ne(x,t) then changes to: [3]

2 on on

a

n e e e (2.19) ~ R n

-

W - - - +

D

i e I 2 at ox ox 16

For Ri :: _{0, i.e.} a :: 0, Eq.(2.19) has the solution: [2 ) 2 n (X

-

W t ) 0 (2.20) n (x, t ) = exp(-e .[(Lt Tf D t ) Lt D t 1 1

This solution shows that an initially very thin disk of electrons during their drift time spreads to a Gauss-distribution. The drIft velocity W is in this case equal

to the electron drift velocity ve because the second term in Eq.(2.1Lt) disappears. The axial spread is characterIzed by (Fig.2.3):

(2.21) (J :: .[(2 D t)

X 1

WhlCh increases in time, while the maximum of the distribution decreases (Fig.2.Lt).

X:O

Fig.2.Lt The quantity ne(x,t) at t :: tl and t :: t2' [2]

17

At the electrodes the following boundary conditions for ne = ne(x,t) hold: (2.22) cathode: n (O.t)

=

n ~(t) e 0 (2.23) anode: n (d,t)

=

0 e

Condition (2.22) states that at t

=

0,

x

=

0, the nO

primary electrons have a Dirac-distribution. The back diffusion of the electrons into the cathode is

neglected. However, at low gas densities the back diffusion can be. considerable.

I

Condition (2.23) can be fulfilled by introducing an

lmage density term in Eq.(2.20), a negative density that

starts at the position x

=

2 d:

(2.24) n '(X,t)

= -

n «2 d - x),t)

e e

Appropriate weighing of this image density term gives a modified density distribution that incorporates the

effect of the anode: [3)

(2.25) n (x,t) = e n

o

.[(4 1T D t) 1 [exp(-2 (x - 'II t ) 4 D t 1 'II d (X - 2 exp( -D 4 1 d - 'II t) D t 1 2 ) J

Figure 2.5 shows that near the anode the Gauss-distribution of ne(x,t) is distorted.

Cathode Anode

- - image t.enlt

Flg. 2.5 The influence of the image density term on the denslty dlstribution due to the boundary c(mdition

(2.23), (3)

Conslderlng Eq. (2.3), the electron component of the current ln the external circuit can now be derived as:

(I.pp. 8) (2.26) i ( t ) :; e e T e d

o~

n (x,t) dx e

The calculation of the integral can be simplified by replacing the lower bound 0 by - 00, resulting in a

negligible small error: [3)

(2.27) 1 ( t ) :; e e T e d

1

- 00 n (X,t) dx e 19

Substltuting Eq. (2.25) into Eq.(2.27), the electron current in case of no ionization becomes: [3]

n

e

o

(2.28) i (t) :: e 2 T e

where. using W :: ve:

W d (2.29) a :: _exp(--) D 1 W t

-(2.30) A :: 1 [(4 D 1 W t + (2.31) ), :: 2 ..[(4 D 1 [erfC(A ) - a erfc(A )] 1 2 v d e :: exp{--) D 1 d v t

-

d e :: t ) [(4 D 1 t ) d v t + d e = t ) [(4 D t ) 1

The complementary error function erfc{x) is defined as:

2

(2.32) erfc(x): exp(- t ) dt 2

The second term in Eq.(2.28) results from the loss of the electrons at the anode because of the boundary condltion {2.25).

ChecKing Eq.(2.28) for Dl -) 0 gives, with erfc(- 00) : 2 and erfc (00) = 0: a - ) 00 A ->

-

00 erfc(). ) - ) 2 t < T 1 1 e A -> 00 ·<:>·"'fc (). ) - ) ₀ t ) _T 1 1 e A - ) ₀₀ erfc(}.. ) _{-> 0} 2 2

so for Dl - ) ₀ Eq. (2.28) equals Eq. (2.4) .

If

Ri

i

0, i.e. a

i

0, Eq.(2.19) has the solution: [3]

(2.33) n o 2 (x -

w

t) n (x, t) = e xp ( R t) [ e xp ( -e i [(4 1T D t) I 4 D 1 t 2 W d (x - 2 d -

w

t) exp( -D 1 - - - ) ] 4 D t I

Now the drift velocity W does not equal the electron drift velocity ve (Eq.(2.14».

Using Eq.(2.27) and Eq.{2.33) the electron current in the external circuit can be derived as: [3]

(2.34) i (t)

=

e

n

e

o 2 T e

exp(R t) [erfc(l , - a erfc(l )]

~ 1 2

where, us ing Eq. (2 . 14) :

(2.35) (2.36) (2.37) W d a = exp(--) D 1 W t

->..

₌

1 f(4 D 1 W t + >..

₌

2. f(4 D 1 v d e

=

e x p ( - - + a d) D 1 d (v + a D _{1 )} t e = t ) f(4 D 1 t ) d (v + a D _{1 )} t e

=

t ) ,f(4 D t ) 1 - d + d

Eq.{2.36) can be checked for Ri = 0, in which case Eq.(2.36) equals Eq.(2.30), and for Dl -> 0, with erfc(- 00)

=

2 and erfc(oo) = 0:

a -) 00 1 -) - 00 1 1 -) 00 i erfc ().. ) -) 2 1 erfc(>.. ) -} 0 1 22 t < T e t > T e

A -) CO

2 erfc(A, ) 2 -> 0

so for Dl -> 0 Eq.(2.34) equals Eq.(2.10).

Unfortunately, Eq.(2.34) cannot always be used for the calculation of the electron current ie{t). When there is little diffusion, the second term in Eq.(2.34) becomes a product of very b~g and very small numbers (only numbers 1n the interval from -10+ 38 to -10- 38 and from +10- 38 to +10+ 38 can be handled by our computer). To avoid

numerical problems, this part of Eq.(2.34) must be approximated, bounded by 1A21. Eq.(2.34) applies if

IA21 ~ 2, whereas if IA21 > 2 this equation is approximated by (App.1):

n e 0

(2.38a) i ( t ) _{::: exp(R.} t ) [erfc(A ) - b exp(-e

I

2 T 1 1 e A > 2 2 n e 0 2 A ) ] 1

(2.38b) i ( t ) ::: exp(R. t) [erfc(A )

-

2 a - b

exp{-I

e 1 2 T e A <

-

2 2

where, using Eq.(2.14):

(2.35) W d a

=

exp(--) D 1 v d e = e x p ( - - + a d) D 1 23 1 2 A ) ) 1

4- D t 4- D t 1 1 [{ [( 1T 'IT (2.39) b

₌

₌

W t + d (v ... a D _{I )} t ... d e W t

-

d (v ... a D _{1 )} t - d e (2.36) A

₌

₌

1 [(Ll D 1 t) (Ll D 1 t ) W t + d (v ... a D _{I )} t ... d e (2.37) A

₌

₌

2 ({4- D t ) ( 4 D t ) 1 1

For Dl - ) 0, with erfc(- (0) :: _{2 and erfc(oo)} :: 0:

b - ) 0 erfc (A ) - ) 2 1 A - ) - (J) t < T 1 2 e exp{- A ) - ) 0 1 erfc(A ) - ) ₀ 1 A - ) (J) t > T 1 2 e exp( - ).. ) -> 0 1

The model is also applicable in case both ionization and attachment playa role. Then the ionization coefficient a should be replaced by the effective ionization

coefficient (0 -

n).

Simulations of practical situations have shown that Eq. (2.34) is only used for extreme large values of

DI and that in most cases Eq.(2.34) is approximated by Eq.(2.3aa). Eq.(2.3ab) is only used in case that the effectLve ionization coefficient is very negative.

With Eq.{2.34) and Eq.(2.3a) the waveform of the

electron current can be derived numerically for typical values of the swarm parameters. For the determination of erfc(x) in Eq.(2.34) and Eq.(2.38) an approximation is used with a maximum error of 0.0005 (APp.2). The

reaction frequency is given three different values and the diffusion coefficient is changed five-fold each

tLme. Fig.2.6 a - f give the resulting waveforms derived wlth Eq.(2.34) and (2.38) and compared to the waveform wlthout diffusion (Eq. (2.10». Fig.2.6 d - f illustrate

the influence of diffusion for extreme large. maybe not practical values of DI (program CALC4).

Fig.2.6 g shows that part of the curve that is calculated using Eq.(2.38a); to avoid numerical

problems, the "stralght part" must be calculated using Eq.(2.10), which decreases to zero with increasing diffusion (program CALC1FEW).

Table 1 gives an indlcation under which conditions a particular formula is used for some typical values of Ri and DI' Ri [Is) Table 1 Dl aX103 4X104 -1X10 7 1,2 0 1,2 3x10 7 1 ,2 1: Eq. ( 2 . 1 0 ) 2; Eq. ( 2 . 38 a ) 1,2 1 ,2 1 ,2 [cm2 /s) 2X10 5 : 1X106 5x106 1 ,2 1,2 1,2,4 1,2 1 ,2 1,2,4 1,2 1,2 1,2 3: Eq. (2 . 38b ) 4: Eq. (2 . 34 ) 25 2.5x10 7 2,4 2,4 2.4

CALCULATED ELECTRON CURRENT [uAl

160

~"""'."""""'

.. "'.'."' ... ' ... '." ... .

152

'\

,

:

I

:

•

:

•

:

*

:

• \. : : : : ;

144 ' :\.. . . , . : . . . , . : , . , , . . .. , . . : ... : .... , . . . , .. :

136

~

... \ . )

\

j

j

i

128

. , , ,

,":"~

.. , , : .. , ... , ; . , . , , , ... : ... , , , . : , , , ... :

129

, l'

"'~:

' \ ;

_.

_.

:

_.

,

,

:

:

:

'

.

:

.

.

111e24

j' , . , , . ., "';,\/.

~

. . . , . .

~

. . . , . . .

l . . .

~

. . . :

.~. : : : : . -lilt. : : : :

96

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;

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8

129

16

29

d

=

1 em 7 R, ::: -1.10 /s 1 T

=

100 ns e 8 nO = 10 'l) D

=

0 L 3 2 2) DL

=

8.10 em /5 4 2 3} D

=

4.10 em /s L 5 2 4)

v

L = 2.10 em /s

tiMe.!:' [ns]

Fig.2.6a 26

CALCULATED ELECTRON CURRENT [uAl

11 6

52°

--..

. '\ \

... ": .. , ... : ... ,.

144

• • t~~"'lt. 1 1 • • • • • • • • '''4.\ '

l \

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:--16t' , ,

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d = 1 C1'f R.

=

0 1 T = 100 ns e 8 nO = 10 1)

n

= c

L 3 2 2} D ::: H.10 crn /s L 4 ? 3} D

=

4,10 em-/s L : ; ' ) 4) DL

= 2.10

cm·/~

tiMe [ns]

Fig.2.6b

CALCULATED ELECTRON CURRENT [uAl

~

· Mt

~

.. ' .. ' .. '.. "' ... ': ...

~

... : .. , ...

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...

i

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x_

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7 R.

= +3.10

/5 1 T == 100 ns e B nO = 10 1) DL = 0 3 2 2) DL

=

8.10 em /s 4 2 3) DL

=

4.10 em /s 5 2 4) DL

=

2.10 em /s

tiM

[ns]

Fig.2.6c

CALCULATED ELECTRON CURRENT [uAl

169

~

_,

.. ' .. ' .... ' ... , ... .

_{. ,}

152 \\

~

~

144

j'

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1

r. r ' . . . : : :

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.

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48

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-~~~~~

d = 1 em 7 R. = -1.10 /s ~ T

=

100 ns e 8 n

_o

= 10 II 0L

=

a

6 2 5) DL

=

1.10 em /s 6 2 6) 0L

=

5.10 em /5 2 5 7 2 7) 0L = , .10 em 13

tiMe

ens]

Fig.2.6d

CALCULATED ELECTRON CURRENT [uA]

16Q

\, -'\

-1-...

... ; ...

1 . . . ..

152

I, • • • • • • •

144

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56

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4

8~

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"16~

, i

'2Q~

d = 1 em R. ::: 0 1 T = 100 ns e 8 n ::: 10

o

1) DL = 0 6 2 5) DL = 1.10 em /5 6 2 6) DL

=

5.10 em /5 7 2 7) DL

=

2,5.10 em /5

tiMe (ns]

Fig.2.6e

CALCULATED ELECTRON CURRENT [uAl

3.21k

2 3 ,

~~~

: ... : .. : : : :

i : : : . : : : . : .

I : :

:/.~

: : : : I : . : .. : .. :: .::::: ... :::

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1) DL

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=

2,5.10 em /s

tiMe [n5]

Fig.2.6f 31

CALCULATED ELECTRON CURRENT [uAl

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T ...

:(\.~""'" ~

... \ ...

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1.81k

t l' t • t t : f : t .. t f l r r I . :: t • • T\' r , t f f l ; , .. , t " , II • • l' t : , , • l I , t :. t .. t ;

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71k

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61k

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t

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9

1 _{' , ,}

'4b ' , ,

'sb ' , ,

_{'12r ' ,}

'16~'

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d = 1 em 7 R. = +3.10 Is 1 T = 100 ns e 8 n = 10

o

4 2 DL = 4.10 em Is

tiMe [ns)

Fig.2.6g 32

3. MEASUREMENT OF CURRENT WAVEFORMS

3.1 Introduction

Figure 3.1 shows the experimental setup which is used for the TRT-method. The time-dependent current in the

external circuit caused by the avalanche is measured as a voltage across a resistor in the ground connection of

the low-voltage electrode.

The time resolution of the measurements is limited by the length of the light pulse and the bandwidth with which the current can be monitored. This bandwidth is not only determined by the electronics but also by the

lay-out of the setup.

The current waveforms thus obtained give information on the processes occurring and yield quantitative values for the swarm parameters which describe these processes.

digitizer 7912 AD antenna I I ! I I Rm IEEE 488

o

I;

triqgering signal I PC C - - - .- - -

f

J::--= ---

J Fig.3.1 The experimental setup. (6)

A the anode; B the central measuring part of the cathode;

C the surrounding grounded part of the cathode and the screen; D the protection diodes and the preamplifier;

3.2 The experimental setup

After a DC voltage U below the breakdown voltage has been applied to the gap, the avalanche has to be started by external means. In the theoretical derivations of the avalanche growth in chapter 2 i t is assumed that the avalanche is started by nO electrons which leave the cathode simultaneously. In practice always a certain time duration is necessary to release a number of electrons. The light source is a transversely excited atmospheric pressure (TEA) H2-laser with a pulse

duration of 0.6 ns (full width at one half maximum, FWHM) and a wavelength of 337.1 nm. Therefore an

aluminium cathode of 22 cm in overall diameter is used with a central measuring part of 4 cm in diameter, which has to be polished frequently in the presence of oxygen. The light pulse of the laser strikes the central part of

the subdivided cathode through a hole of 1.5 cm diameter in the center of the anode. A positive lens in front of this hole images the light beam so that the electrons are released over an area of approximately 1 cm2 for a gap spacing of 1 cm. The avalanche then starts from a large area and therefore a large number of electrons can be released without strong space-charge distortion. The measuring res~stor Rm should be small, in view of the

frequency response, but still large enough to retain a reasonable sensitivity. In our case it consists of four parallel resistors of 200 Q from the central measuring part of the cathode to the surrounding grounded part. After amplification, if needed, the signal is measured across the characteristic impedance of the measuring cable (50 Q), for instance by an oscilloscope or a digitizer. With this setup it is possible to analyze

current waveforms with a time resolution of about 1.4 ns, determined by both the laser and the measuring

system.

Figure 3.2 shows the equivalent circuit for high

frequencies. The avalanche is represented by the current source i g • which can be expressed in the number of

charged particles and their transit times (Eq.(2.3». The capaCitance of the central measuring part of the cathode with respect to the anode is

C

_{g . The capaCitance}

of the surrounding grounded part of the cathode with respect to the anode plus the capaCitance between the anode and the surrounding grounded vessel equals Ct. The measuring resistor Rm has a parasitic capaCitance C2

parallel to it. If C1

»

C_{g • which is satisfied}

by the setup used, the output voltage Vo is given by: [2J

( 3 . 1 ) v o = 1 2

R

i m g 1 2 R m (C g + C ) 2 dv o dt

Fig.3.2 Equivalent circuit for high frequencies.

4. DETERMINATION OF SWARM PARAMETERS FROM THE DIGITIZED WAVEFORM

4.1 Introduction

The purpose of a series of measurements in a gas, given its gas pressure and temperature, is to investigate its

behav~our over a possibly large E I N-range. Using the

TRT-method, the electrode distance normally remains fixed, whereas the quotient E I N is changed by adjusting the gap voltage U. The gas density N is

der~ved as: (3)

273

-3 16

(4. 1 ) N (cm ) = 3.54 10 p (Torr)

The dimension of the quotient E I N is Townsend, abbreviated as Td, which equals 10- 17 vcm2 .

The measured signals are digitized (Tektronix 7912 AD) and their samples stored on disc on-line with the aid of a personal computer (Philips P3102). The determination

of the swarm parameters of the measured electron current is carried ~.'.J.'~ off-line. The programs are writ~en in Turbo Pascal in an interactive way. The swarm parameters are determined separately, if necessary with use of

previously determined swarm parameters.

The off-line procedure starts with reading the data file from disc (Before starting the actual analysis by the program ANALYSE, it is necessary to rearrange the data of the measured signal produced by the on-line program DIGICON and stored in data files without extension, by a transform program called TRANSFOR after which they are stored in data files with the extension T). The measured signal is then displayed on a CRT. If needed, it can be smoothed to reduce the effects of noisy disturbances, and also the ion current component can be separated from

the electron current component (procedure IeS).

Regarding the first strong increase of the electron

current, the start of the electron current is defined by the maximum of the first derivative and assigned to the pOinter TO' The maximum of the electron current is

assigned to the pOinter T3'

The region in which the electron current shows

exponential behaviour can be defined by the pointers T11 marking the beginning, and T2' marking the end. They are standard set at:

T

-

T (4.2) T :: T ₊ 3 0 1 0 10 T

-

T 3 0 T ₌ T 2 0 (4.3) 10

If necessary, the pOinters can be changed to any desired value by hand.

Under the assumption that diffusion does not affect the first part of the the electron current waveform

(Fig.2.6), the reaction frequency Ri is derived by taking the logarithm of Eq. (2.10):

1n(i ( t » e n e

o

= 1 n ( - - ) T e +

R

t = i 37

= In(1 ) + R t

o

i

and a linear regression procedure.

Extrapolation to t

=

0 (TO) gives the value of 10. the electron current at t

=

O. It is assumed that there is no zero-offset of the measured signal.

4.4 Procedure DETERMINE_Te

Because diffusion only spreads the electrons but not affects the total number of electrons, an equivalent electron current waveform without diffusion can be derived. Its construction, using the so-called equal surface criterium is illustrated in Fig.4.1.

1

I

I.(t) ;

I

t~

Fig.4.1 Derivation of the electron drift time Te with the equal surface criterium: 01

=

02' [3]

Integration of both waveforms must result in the same value. The integral of the measured electron current waveform equals Q_{e and is determined using a numerical}

integration routine. Using Eq.(2.10), the integral of the equivalent waveform is:

T ro e (4.5)

0\

i _e ( t ) dt =

_0\

i _e ( t ) dt = n

₀

e

=

(exp(R T )

-

1 ) 1 e R T i e

Because Eq.(4.5) also equals Q_{e and the values of Ri and}

10 are Known, the electron transit time Te can be derived: Q

R

e i In( + 1 ) I 0 (4.6)

T

₌

e

R

i

If diffusion does not affect the first part of the electron current waveform, this value for Te does not depend on the amount of diffusion.

It is also possible to define the electron transit time Te by the maximum of the electron current waveform. This

so-called maximum current criterium maKes use of the difference between the pOinters TO and T3' This value for Te depends on the amount of diffusion (Flg.2.6).

It should be noted that the values of all swarm parameters that depend on Te are affected by the choice of the criterium.

4.5 Electron longitudinal diffusion coefficient Dl

Finally, Knowing the values of Ri. 10 and Te. the electron longitudinal diffusion coefficient Dl can be determined by fitting the measured electron current with the calculated one using Eq.(2.34) and Eq.(2.38).

Define the least-square function S as:

(4.7) S : E [ i ( t ) - I ( t , ) ] 2

i e i e 1

where ie(ti) is an electron current value on sample moment ti calculated with Eq.(2.34) or Eq.(2.38) and

re(ti) the measured electron current value on sample moment t_{1 .} The fitting procedure searches for a minimum of the least-square function S as function of Dl' This can be established by determining the zero-crossing of oS / oDI as a function of Dl: oS oi ( t ) e (4.8)

₌

~ {2 [1 (t )

-

I ( t . ) ) ) i e i e 1 oD aD 1 1 where (App.3): 01 ( t ) n e D ~ e 0 1 3 2 (4.9) : exp(R. t ) [ exp(- ~ ) 2 1 1 aD 2 T D ~u 1 e 1 + a v d erfc(~ ) e 2 40

a D ). 1 4 2 exp{- A

) 1

2 .[n with: v d e (2.35) a :: e x p ( - - ₊ a _d} D 1 (v + a _{D I )} t

-

d e (2.36) ).. :: 1 {(4 D t) 1 (v + a D _{1 )} t + d e (2.37) }.. :: 2 (l.! D 1 t ) (v - a D _{1 )} t - d e (4.10) }.. :: 3 ( 4 D 1 t ) (v

-

a _{D 1 )} t + d e (l.!.11) }.. :: 4 .[(4 D 1 t) 41

Eq. (4-.9) applies if 1A21 ! _{2. If P'2} 1 > 2 this equation is approximated by (App. 1 ) :

ai

( t ) n e D A e 0 1 3 2 (4-.12a)

i

exp(R. t ) [ exp(- A ) 2 1 1

aD

2 T D .flT 1 e 1 A > 2 2 2 + b v d exp(- A ) e 1 a D A 1 4- 2 exp(- A ) ] 2 .flT ai ( t ) n e D A e 0 1 3 2 (4-. 12b) N ... exp(R. t ) _[ exp(- _{A )}

I

2 1 1 aD 2 T D .flT 1 e 1 ). _<

-

₂ 2 2 + 2 a v d + b v d exp(- ). ) e e 1 a D ). 1 4- 2 exp(- A

) 1

2 .flT wi th:

v d e (2.35) a :: exp(-- _{+ a} d) D 1 4: D t 1 ( ) 'If (2.39) b :: (v + a D _{1 )} t + d e (v + a D _{1 )} t - d e (2.36)

"

:: 1 ( 4 D 1 t ) (v + a D _{1 )} t + d e (2.37) A ₌ 2 ({4: D 1 t) (v - a D _{1 )} t - d e (4.10) >.. :: 3 [(4 D 1 t ) (v - a D _{1 )} t + d e (4.11) A :: if [(4 D 1 t ) 4:3

Because the least-square function S 1S a nonlinear function of Dl' an iteration procedure is necessary to reduce a start interval for Dl' that must contain the optimal fitting value, to a smaller interval in which for several values of DI' in our case 11, the least-square function S is calculated and its minimum determined. The start interval must be given by the user.

For the iteration procedure a modified Fourier-method is

used (Fig.4.2). Given the start interval bounds Dl = K and Dl = n, new interval bounds are obtained by

calculating the least-square function value oS j oDl for

DI = m = (K + n) j 2. The zero-crossings of the straight

lines through the pOints (m,oSjoDl(m» and (K,oSjoDl(K», as well as through the pOints (m,oSjoDI(m» and

(n,oSjaDI(n», marK the new interval bounds Dl = K1 and DI

=

n1' They are calculated according to (App.4):

oS n - m (4.13) k = 1 n - - ( n ) oD oS oS I ( - ( n ) - -(m» oD aD 1 1 oS m - K (4.14 ) n

₌

m - - ( m ) 1 oD oS as 1 (-(m) - - ( k » oD aD 1 1

After each interval reduction step the following checKs are made:

(4.15) (4.16) I old I new !. 2

oS

oS

- ( : K ) (n) < 0 1 1

aD

aD

1 1

A minimum interval length for the electron longitudinaL diffusion coefficient is specified by the program as function of the given start interval length. In our

case the maximum reduction factor is 20. The part of the time-dependent electron current waveform that is fitted is given by means of the pointers T5 (begin of fitting interval) and T6 (end of fitting interval).

The accuracy of the fitting in the interval [T5,T61

is given by: (4.17) S (] ::. . f ( -D I n - 1

where n is the number of sample moments in the specified interval.

The listing of the header of the program ANALYSE is given in Appendix 5. The user of the program must :Keep in mind that the determination of the electron

longitudinal diffusion coefficient Dl of one measured electron current waveform takes at least a quarter of an hour, due to the iteration procedure and the number of

logarithms, while the other swarm parameters are determined almost immediately. Therefore the user can choose for

the option of determining all swarm parameters except for DI' The results are stored in data files with the extension RES.

'~J

O~----~~~~~-J---~---Fig.4.2 Modified Fourier-method.

5. CALCULATED SWARM PARAMETERS FROM THE DIGITIZED WAVEFORM

In thIS chapter the results using the program ANALYSE (chapter ~) are presented and compared with results available from the literature.

5.1 Accuracy of the program ANALYSE

Before running the program ANALYSE on measured electron currents, the accuracy of the program is tested. This is done by calculating electron current values with the

formulas derived in chapter 2 (program CALC1ANA) and use these "measured" electron currents, which are denoted by TEST1 to TEST~, as inputs for the program ANALYSE. The swarm parameters Ri and DI are given two typical

values. All given swarm parameters are listed in Appendix 6, that also contains the calculated swarm parameters by the program ANALYSE.

As far as i t is permitted to generalize the results, the relative deviation of the calculated values with respect to the given values of Ri' Te , v e ' a and

nO are of the order of 0.1 Z. The deviation of

Dl ranges from 1 Z to 10 Z and can be minimalized by adjusting the length of the start interval (rerun of the program) and/or by interpolation of the calculated

least-square function S and/or by increasing the number of values of DI for which the least-square function S is calculated. In our case for each measured electron current waveform 11 values of S are calculated,

whereafter the program draws a continuous line.

Given the same swarm parameters, electron current

waveforms with "zero mean noise" are calculated. In the first part of the waveform the current values are

increased and decreased by 10 Z on every other current value, in the last part by 20 Z (program CALC1ANA). The calculated electron current waveforms and swarm

parameters by the program ANALYSE are listed in Appendix 6 (TEST11 to TEST4~).

The results indicate that the addition of nOise reduces the accuracy of the calculated swarm parameters. The relative deviation of the values of Ri is still of

the order of 0.1 Z, but the relative deviation of the values of Te , v e ' a and nO are now of the