A random walk to a simple stationary electron energy distribution

Physica 40 (1968) 139-149

o North-Holland

Publishing

A RANDOM TO A SIMPLE

STATIONARY

Co., Amsterdam

WALK

ELECTRON

ENERGY

DISTRIBUTION

F. J. DE HOOG Afdeling

Natuurkunde,

Technische

Hogeschool,

Eindhouen,

Nederland

Received 11 April 1968

The solution by Markoff of the random flight problem is applied to the calculation of an electron energy distribution. This distribution develops in a gas when a homogeneous constant electric field is present. The electron-electron interaction is neglected and some other classical conditions are supposed to be fulfilled. The introduction of a so called energy-orientation space makes it possible to describe changes in energy and direction of the electron in geometric terms. It is also possible to take into account the elastic losses and by doing so a stationary electron energy distribution is obtained. From the result it can be seen that the form of the exponential factor in the Druyvesteyn distribution can be interpreted as to originate from a combination of diffusion and elastic friction in the energy-orientation space.

1. Introduction. The calculation of the electron energy distribution in a gas discharge under various conditions has drawn considerable attention. An excellent review of the work in this field up to 1955 was given by Loebi). The present contribution deals with the development of the electron energy distribution in a simple case. This development is discussed in terms of a random flight method. This method can be applied to the development of an electron energy distribution in a homogeneous, constant electric field under the following simplifying assumptions : 1) The electron density is so small that mutual interaction of electrons can be neglected. 2) When colliding with a gas atom an electron loses a fraction f of its energy. 3) The mean free path of the electron 2&Vis constant and independent of energy. 4) There are only elastic collisions and isotropic scatterers; in that case f = 2m/lM, where m is the mass of the electron and 1M that of the gas atom. In section 2 the method of Markoff as described by Chandrasekhars) is reviewed and an extension of this method is introduced. In section 3 the 139

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F. J. DE

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application of the extended method to the problem mentioned is treated in the case that no loss of energy occurs and in section 4 the problem of obtaining a stationary distribution is solved. A discussion of the result and some conclusions concerning the origination of classical electron energy distributions are given in section 5. 2. Markoffs method. The position rl(i = 1, ..., N) in three dimensional

R =

of a particle after N displacements space is given by

R

2 rf.

P-1)

i=l

Let the probability placement between T&)

that the particle in the ith displacement ri and rt + drt be

suffers a dis-

dri,

(2.2)

then the probability WN(R)dR that after N displacements arrived in the interval between R and R + dR is WN(R) dR = J (rl)

e. ., rN) I1 T&)

the particle has

drl.

i=l

The function A must satisfy the following

conditions:

A=lwhenR-$:dR
(2.3)

i=l

A = 0 in all other cases. There exists written as

an expression

for A(rl, . - ., rN) and

1 a &V(P) exp[-_jp*Rl = __(2X)3 J

WN(R)

dp

W&R)

which

may

be

(2.4)

(P)

with 1V &(p)

=

II i=l

J 4-O (r0

expljp-nl h.

P.5)

In order to show that the method of solution of the random flight problem just described is applicable in the case of the interaction of an electron with a gas under the influence of a homogeneous and constant electric field, it is necessary to introduce an operation working on a once established probability distribution IV,{ R) . This operation is a displacement of the particle that after the Nth displacement has arrived in the point (jR/, 8,+) in such a way, that the (N + 1)th displacement starts from a point (IRI, 8', 4'). It is assumed that there is no correlation between (13,4) and (O’, 4’). In case the distribution

STATIONARY

ELECTRON

ENERGY

141

DISTRIBUTION

WN(R) possessses spherical symmetry this operation will not change it. In general the result of this operation is spherical symmetry for any distribution W&R). Mathematically the operation WN(R) over the solid angle: IN

= --&j

can be performed

by integration

s WN(R) sin 8 df3d$.

of

(2.6)

Another operation which can be performed on a probability distribution belongs in principle to the class of displacements T((~z), but because of its importance it is defined more precisely. It is a translation of the probability distribution WN(R) as a whole along a fixed distance XO. The resulting distribution is WN(R) = WN(R -

xo).

(2.7)

The result of these two operations, performed in the same order as they were introduced, on a probability distribution with spherical symmetry W&(R) is identical with the result obtained when a series of N displacements ~a(Yt)(i = 1, e-e, N) leading to that spherically symmetric probability distribution, is extended with a displacement 1 cv+1(m+1)

=

~

4x lxo13

wrlv+1/3

-

lxo12).

To prove this theorem the Markoff method ments W&(R) and ~+i(r~+i). The resulting

d(R’, YN+I)WS,(R’) This can be written

NQJ+112

is applied to the two displacedistribution is

- Ixo12)

dR,

dyN+l

(2.9)

47c Ix013

as

s

w” (R _ r) WI2 - lxo12)dr N

4x jxajs

(2.10)

*

(r)

Because becomes 1 44x

of the spherical

W$(R

-

symmetry

of the functions

involved

eq. (2.10)

x0) sin eded+,

which is the same result as that originating from the successive of the two operations on the distribution W&(R).

application

3. Case withotit energy loss. In order to make it possible to apply the above mentioned method to the gain and loss of energy of electrons, a space must be

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F. J. DE

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chosen in which the anticipated random flight can take place. An obvious choice is to take the three-dimensional velocity space. The operation of directional randomizing is then suited to describe isotropic scattering. Because of the following argument, however, velocity space seems not suitable to describe the random flight of the electron. Between two collisions the gain of velocity in the direction of the field Av, is dependent on the absolute velocity v the electron possessed after the first of these two collisions according

to

eF& Av, = p, mv where F is the electric field strength. Therefore the elementary displacement in velocity space depends on its starting position and for the displacement r~ no distribution function ~g(ri) can be assigned a priori. From Chandrasekhar’s contributions) one could conclude that the Markoff method of solution for the problem of random flight in that case does not apply. Although in sections 4 and 5 it is shown that this limitation is not valid in all cases, for the sake of clearness Chandrasekhar’s point of view is adopted for the time being. Therefore, another space is chosen for the description of the electron flight. This space will be called the energy-orientation space. In it the electron is represented by a vector of a length equal to the energy just after the collision at the start of each particular flight and with the direction of the velocity vector of the electron at the same time. Each point of this space corresponds to one and only one point of the usual velocity space. Now the possibility of uniquely defining in this space the gain and loss of the electron and the changes of direction of its velocity has to be investigated. First this point will be considered without regard to elastic losses. An electron having an energy E just after its last collision with an atom and moving under an angle 8 with respect to the direction of the electric field, will gain approximately an amount of energy eFAavcos 13and retain its original direction if the curvature of its path is neglected. After the next collision the direction will be arbitrary and the total amount of energy will be (see fig. 1) E + eFA,,

cos 0.

The same situation would occur when the electron was displaced along a distance Eo in the direction of the field and, subsequently, the operation of directional randomizing would take place. This displacement Eo then has to satisfy (see fig. 1): (E + eFA,, cos6)2 = E2 + Ei + 2EEo cos 13. Now by taking Eo constant and equal to eFjZav an error 8(AE) is introduced

STATIONARY

ELECTRON

ENERGY

DISTRIBUTION

-

Fig. 1. The approximate

143

Er

step of an electron in energy-orientation

space.

into the change of energy. The size of this error is &.(AE) =

(eF.,P

si;se

.

(3.1)

An estimation of the error introduced by the neglection of the curvature of the path of the electron in configuration space shows that this neglection gives a contribution to the total error of the same order of magnitude as (3.1). One has to conclude from this that only for E > eF& the mathematical model represents physical reality. In that case irrespective of original energy and direction of velocity the interaction of each electron with the field between two collisions can be represented in energy-orientation space by a displacement over a distance eFA av in the direction of the field and an application of the operation of directional randomizing. According to the theorem derived in the preceding chapter the probability distribution of an electron in energy-orientation space starting with energy zero after N steps W&E), can be constructed from the N elementary displacements

7$(Q)=

1 4x(eF&,)3

W42 - W%12)

(i = 1, . . . , N).

(34

There is one important point still to be discussed. The assumption that every electron covers the same distance 1 BV in configuration space before making a collision must be replaced by the assumption that the probability that an electron covers a distance between r and r + dr is $(r) dr, where

F. J. DE

144

HOOG

So a fraction p(r) dr of the electrons with the probability suffering a displacement 1

TN+l(EN+l)

=

density WN(E) is

W~N+II~ - leW2).

4n(eFr)3

This results in a fractional

probability

density

dWN+r = Sj d(E’, W+I) WN(E’) p(r) dr w+~(cv+~)

dE’

d&n;+1 ;

like eq. (2.10) this can be written in the following way

’

WN(E-P)

4x{eF lrl}3

’

x VA2 - W IrIP9 0. Integration

over all the fractions

yields

Co WN+l(E)

=

s

dir] “““[4~[~

0

WN(E -

eFr) sin 8 d6 d+,

fS

* SJ

which can be written as

WN+@) =

exp[--eF

A (E’, eFr) WA@‘)

4xeF&{eF

lrl/eFM

dE’ d(eFr).

Irl}2

From this result can be concluded that the construction of the distribution WN(E) can be performed by applying Markoff’s method to the series of displacements

+i)

=

1

exp[-I~~l/eF~avl.

1-%12

WeFk)

This series leads up to an AN(~) of the form (2.5) : exp[Is (8)

1~1/eF&d

4xeFil,,

l&l2

N

exp[jp.c]

ds

I

.

For not too small N (see section 4) this can be written as

AdP) = An interesting

AN(p) =exP

case occurs when N is very larges), then

N IpI2(eF&d2 -

3

1

STATIONARY

and

ELECTRON

ENERGY

% WA@) = 1[ 3

47CN(eFit~v)2

exp

-

DISTRIBUTION

3 IE12 4N(eF&)2

145

1’

From this result it can be seen that the probability distribution expands radially in energy-orientation space. Consequently the mean energy of the electron is continually increasing with the average number of collisions. 4. Obtaining a stationary distribution. In order to get a stationary distribution it is necessary to take into consideration the elastic loss of energy of the electrons in collisions with atoms. To handle this process mathematically in energy-orientation space the following reasoning may be followed. When a particle has reached a distribution WN(R) after N steps, this probability density is compressed in such a way that a point r is transformed into a point Kr with k < 1. This linear reduction results in a new distribution

During the generation of a distribution in N steps by displacements of spherical symmetry it is possible to perform the operation of linear reduction followed by the operation of directional randomizing and still retain the possibility to apply Markoff’s method. The probability distribution T:(q), defining the first step, finally has been reduced N times. Consequently as the probability for the first step must be taken

Tl(rl) =

,&

~~(rlP).

In general the probability

+rz/kN+l-“). (ks):+i-c

Q(Q)=

The final distribution

s

(n)

...

s

for the ith step is

d(ri,...,rN)

WN(R) becomes 1 S ,&I (k3)N+l-l

T:(ri/kN+l-i)

drt.

(of)

The solution

can be written

as eq. (2.5) with

The considerations given are directly applicable to the problem of elastic energy loss. This follows from the fact that WN(E) is generated by displace-

146

F. J. DE

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ments (3.3). The process of elastic loss after a displacement is represented by a linear reduction and successive directional randomizing. According to the assumption for the energy loss the factor k can be written k =

1 -

2mlM.

For the probability

WN(E)=

-

1 (2n)2

distribution

WN(E) it follows again that

AN(P) exp[--jp*E] s (P)

dp

(4-l)

with AN(P) =

x

exp

s

i

i=i

-

b1)

1

4x(eF&)

(eFA,,)

kN+l-gIs#

kN+r_”

The change of variables

dst* 1 ‘d exp[ip

n

i=l

l

and integration

N arctan[(eF&)

AN(P) =

’

W’&)

yields

kb lpl] ke

IA

(4.4

*

If (eF&)

kr IpI <

(4.3)

1,

eq. (4.2) can be written

(4.4) For not too small values of N there is no interest in changes of probability density in intervals of energy of the order of magnitude (eF&) k”. So, to stay in the language with

IPI >

of Fourier

transforms,

frequency

components

1

eF&,ki

are not relevant for the final distribution and every continuous monotonously decreasing choice of AN(~) for \p) > l/[(eF&) kg] would yield the same picture. For convenience eq. (4.4) is therefore declared to be valid in regions where JpI does not satisfie eq. (4.3). For N + co (4.4) becomes (eFW2 3

IpI2

1’

k2

1

STATIONARY

ELECTRON

ENERGY

DISTRIBUTION

147

With neglection of terms of the order ms/Ms with respect to 1, AZ/( 1 can be written as M/4m. Substitution of A&) in eq. (4.1) gives 3m -___ M

W(E) =

PI2 (eFA,#

k2)

1*

The probability f(E) dE for an electron that has taken a great many steps in energy orientation space, to have an energy between E and E + dE is now

(4.5) This distribution is sketched in fig. 2. 5. Discussion and conclusions. From the result (4.5) it can be seen that the exponential factor in the derived distribution has exactly the same form as the exponential factor of the Druyvesteyn distributions). Nevertheless, the factor E2 is different from the factor E* obtained by Druyvesteyn and other authors. From (3.3) it can be seen that the largest error introduced by assuming Eo to be constant and equal to eF& occurs when 8 = x/2. In this case the error is equal to i(eF&)2/E. Then the error introduced by the assumption of straight paths is equal to $(eF&)2/E and negative. So the net error intro-

eFtt,,,sc#a eV o Druywsieyn

distribution

+ present

‘f-u a5

50

75

Fig. 2. A comparison between the Druyvesteyn

100

z

E(eV)

distribution and the present one.

F. J. DE

148

HOOG

duced by the two assumptions together comes down to the fact that too large an amount of energy is put into the electron in the preliminary stages of its random walk and that, consequently, too many high energy electrons are found. A correction of this error could be achieved by introducing a mean free path A(E) < il,, for small energies. Although, according to the remarks made in section 3, one would not expect the Markoff method of solution to be applicable, ye? it is possible once the function ii(E) is known to be of spherical symmetry, to transform the chosen space in such a way that all the steps taken have the same absolute length. After the random flight the derived probability distribution is transformed back to reality with the inverse transformation. In this manner it is possible to apply the following transformation to (4.5) E

E* =

s

za

WI)

dlE’I,

(5.1)

0 where E* is the variable in the transformed space and E the one in real energy orientation space. Attention must be drawn to the fact that the whole setup of a random walk in energy space is an illustration of the transformation introduced. Assuming that for E < eF& the error is of the order (eFA,,)2/E it is readily visible that the displacement from E*- to E-space does not affect the distribution essentially. So the distribution is not transformed into the Druyvesteyn distribution when the errors introduced are taken into account. A distribution of the same appearance was derived by Yarnold4) also by a Monte Carlo method. His results were interpreted by Wijsmanl), who concluded that too high an amount of energy was put into the electrons during their random walk. It is remarkable that in the present case the correction cannot explain the difference between the Druyvesteyn distribution and the one derived here. An illustration of the difference between both distributions is provided by the calculation

of the mean energies

Irrespective of the difference between the distribution obtained from the present approach and the ones obtained previously a calculation can be made of the number of collisions an electron has to make before fitting into the distribution. With use of eq. (4.4) an expression may be obtained for the average energy after N collisions N.

STATIONARY

It follows that

ELECTRON

ENERGY

DISTRIBUTION

149

N is

00 3 4x1eFilav)2 z1 2 -4 -/ \’ .lC

4Z ___ eF&, 3

where Z = ; P. i=l If this problem is stated in terms of finding a value of No in order that N

-


< 5%

N

for N < No, the number satisfying this condition is No LZ-:5 x 10+4. From this rather high value of the “relaxation number” it can be seen that the value of distributions obtained under the conditions that only elastic collisions occur is only theoretical. Even in many discharges where these conditions may be satisfied approximately, it is unlikely, because of this high relaxation number that an equilibrium distribution would ever occur. Of course by taking inelastic collisions and ionization into account this relaxation number would be strongly lowered. This results from the fact that a sphere in energy orientation space with a radius equivalent to the ionization energy cannot be surpassed and even causes surpassers to start again at energy zero. So a continuous stream of electrons is walking from zero to that sphere and it is clear that a stationary distribution is obtained much quicker. From the derivation finally it can be seen that the factor E2 in the exponent is due to the fact that the electron diffuses in energyorientation space. So by using the random walk approach for the gain and loss of energy of an electron under the conditions mentioned, an insight is obtained ponent.

into the origin of this previously

rather

obscure

form of the ex-

REFERENCES

1) Loeb, L. B., Basic processes of gaseous electronics (Univ. of Calif. Press, 1960). 4 Chandrasekhar, S., Rev. mod. Phys. 15 (1943) 1. 3) Druyvesteyn, M. J., Physica 1 (1934) 1003. 4) Yarnold, G. D., Phil. Mag. 36 (1945) 185.