24
Progress of Theoretical Physics, Vol. 10, No.1, July 1953
Ionization of Gas by Electrons
Julian K. KNIPP
Institute for Atomic Research and Department of Physics,
Iowa State College, Ames, Iowa, U.S.A.
Tetsuo EGUCHI, Masao OHTA and Shozo NAGATA
Department of Physics, Faculty of·Scienc.e,
Kyushu University, Fukuoka, Japan
(Received' May 15, 1953)
A statistical analysis is made for the number of ions produced by an electron when it is absorbed
in a hydrogen gas. The average number and the mean square fluctuation of the ion pairs as functions
of the incident energy of electron satisfy inhomogeneous linear integral equations which yield Fano's
results in the first approximation.
§ 1. Introduction
A charged particle in its passage through a gas produces a number of ion pairs
along its path. The fact that the number is proportional to the energy lost in the gas
by the incident particle is well known and is often used for the measurement of the
energy of charged particles. Theoretical considerations on this subject have been tried by
several authors1) ,2) ,3) ,4) ,5), but the linearity mentioned above has not been deduced 'very
clearly. Recently a few experiments6),7) have been carried out which seem to indicate a
slight deviation of the number of ion pairs from the linear relation to the energy lost
by the incident particle producing these ions.
In the present paper the authors try to analyse the above problem by a statistical
method. For the sake of simplicity we consider the case in which an electron is absorbed
in a gas of atomic hydrogen. "Our formalism, however, can be easily generalized for the
case of more complex nature without much modification.
The probability distribution function for an electron with a given energy to produce
a given number of ion pairs in the gas is defined which satisfies a set of nonlinear integral
'equations. Instead pf solving these equations the linear integral equations for the average
number of ion pairs and their mean square fluctuation are obtained, whose approximate
solutions are estimated for high values of the incident energy.
§ 2.
Fundamental equations
When an electron collides with a gas atom it will either ionize or excite the atom.
The elastic collision can be ruled out in this case because it changes neither the energy
Ionization,
0/ Gas
by Electrons
25
of the. electron nor the number of ions. We neglect the electron exchange effect and
the recoil of the gas atom hit by the electron as well as the higher order processes sud~
as radiative collision and recombination of ion pairs etc. Thus we can define the following
collision probabilities: q",(E) is the probability per collision that an electron with an
incident energy E excites a gas atom to its n-th level with the excitation energy En, and
dE q(E, E) is the probability per collision that the same electron ionizes the atom losing
its energy by an amount between E and E+dE. In the latter case the secondary electron
has a kinetic energy E - EJ , where f 1 is the ionization potential of the gas atom.
The above defined q.. (E) and q (E, E) are normalized in the sense that
'the actual forms for the q's are obtained with the corresponding cross sections by
(2)
q(E E)=.
aCE. E) _ __
~2 a,,(E) + fdEa(E, E)
,
-
J
In the following .we assume that these a's and consequently q's have been given by
quantum-mechanical calculations.
Let PC E, I). be the probability that an electron with an incident energy E produces
exactly 1 pairs of io~s in the gas in which it is entirely absorbed. P( E, I) is normalized
with respect to I:
CD
2J PCE, 1)=1.
(3)
1=0
Considering the change of PC E, I) on one collision between the electron and a gas atom
we obtain the following set of non-linear integral equations for peE, I) ;
CD
P(E, I) = 2::; q,,(E)P(E - e_l)
,,=2
+~ JEdE q(E,
£=0
1,
E)P(E-E, i)P(E-E I-i-l),
']
for
(4)
1=1,2,3···
with
PCE, 0)=
.
2J q,.(E)P(E-e", 0),
for
"=2
and
for
0< E<
E1 •
E>
E1 ,
J.
26
1<. Knipp, T. Eguchi, M. Ohta anJ S. Nagata
Note that the above equations take into account all contributions from the secondary,
tertiary etc., electrons as well as that of the primary.
We apply a Laplace transformation to PCE, I) by
co
QCE,).) =e-}. L]e-).} PCE, I),
(5)
}=o
and obtain the Laplace transform of eqs. (4) as
co
Q(E, i.)
= n=2
~ qn(E)Q(E-En, A)
+Jli. dE q(E, E)QCE-E, A)Q(E-Ej, ).),
(6)
'1
which should be solved with the initial condition
for
0
<E<
Ej
•
The PCE, I) is given in terms of Q(E, A) by
peE, I) = ~J-(/(E, A)eW
2n
+1)
dA.
0
§ 3. A v:erage number and mean square fluctuation
Instead of solving eq. (6) or (4) directly we obtain the equations for the.moments
of the number of ion pairs by expanding eq. (6) into the power series of A and regarding
the equation as an identity with respect to A. Thus the first and second moments
co
00
I(E) ='fJ P(E, I)I, and
J2(E)=2J P(E, I)J2
1=0
j=O
satisfy the following equations,
co
I(E) =q(E) + 2J .q,,(E )I(E-En)
71=2
+JEdE qCE, E) {ICE-E) +I(E-Ej)},
(7)
'1
and
J2(E) =q(E)
+2JEdE q(E, E) {1(E- E) +I( E-
1)}
Ej) +I(E- E)I( E- E
'1
(8)
where
q(E)=JEdE q(E, E).
<1
From eqs. (7) and (8) the equation for the mean square lluctuati:on
Ionisation
of Gas
by Eltetrolls
27
co
D(E) =::E [/-/(E)J 2P(E, 1) =12(E) -[/(E)J2
(9)
1=0
is obtained as
IX)
D(E)=r(E)
+~
q,.(E)D(E-E n )
7>=2
+JEdE q(E, E) {D(E-E) +D(E-E1)},
(10)
'1
where
r(E)=}J q.. (E) {/(E)-I(E-E,.)}2
.. =2
fli dE q(E, E) {I(E-E) +I(E-E z) +1-/(E)}2.
+
(11)
• fZ
Botheqs. (7) and (10) are inhomogeneous linear integral equations of similar type
which should be solved with the initial conditions
I(E) =0,
and
D(E )=0
for
0
<E <
E1 •
(12)
We can easily prove from eq. (7) that
0< I(E) < EjEz,
which has an obvious physical interpretation.
§ 4. Formal and approximate solutions
Because of the initial condition (12) eqs. (7) and (10) can be expressed as sums
of finite series. Let K .. (E, E) be defined by
with
.
q(E, E)=2:;q.. (E)J(E-E,,)
for
"=2
and
n=2, 3, ....
Then I(E) and D(E) are given by
I(E)=q(E)
+,.~f~;"K.. (E'
E)q(E),
(13)
E)r(E),
(14)
and
D(E)=r(E) +..~
J~;"K,,(E,
J.
28
K. Knipp, T. Eguchi, M. Ohta and S. Nagata
where N is an integer which satisfies
El+NE2<E< E1 +(N+l)E 2.
The above solutions are exact, but the complex nature of the Kn(E, E) makes it hard to
estimate I(E) or D(E) for h~gh values of E by the above formulas.
We know experimentally that the average number of the ion. pairs I(E) is rou;ghly
proportional to the energy E,
I(E) =xoE,
In order to determine the value of
Xo
=
Xo
Xo ~
(15)
const.
we insert (15) into eq. (7) and .obtain
q(E)
(16)
--co-----''--'--~----
2J q.. (E) E,.+ EJq(E)
'16=2
which varies very slowly with E and becomes constant as E increases, thus coiifirming the
assumption (15). The energy loss per ion pair is given by l/xo with (16), which is
essentially the same expression as the one obtained by Fan04} by means of the consideration
that the energy loss greater than Ez contributes to Xo only by an amount Ez because the
secondary electrons can also produce ion pairs along their paths. In our formalism eq.
(16) is only the first approximation and the better result would be obtained by solving
eq. (8) starting with the first approximation.
Likewise. if we assume that
D(E) =rol(E),
(17)
ro ~ const,
we obtain from eqs. (10), (n) and (15),
co
ro= {xo2 2J q,.(E) E!+ (I- XoE/)2q(E)} I q(E),
(18)
n=2
which
IS
again essentially the Fano's expression4) obtained by his ingenuous consideration.
§ 5. Numerical evaluation and comparison with experiment
Our q's are evaluated using Bethe'sR) formulas for u's in (2).
_
q,,(E) -
2 8 n7 (n-l
1
(/)0-
They are approximately,
)2.. -5
(Ell E) ·3· (1t+ 1)2"+5
'
q(E, E) =Ql(E, E) +q2(E, E),
ql(E
,
E) =
1
(/)0-
(Ezi E)
o
qiE, E)= {
.~{4(~)4
ge4
E
_( El)&}~
E
E/
E$5Ez,
1
(Ez)21
-
fPo-(E/IE)e
Ez
E ~ 5E z •
In the above expressions ql(E, e) and q2(E, e) refer respectively to the soft and violent
Ionization of Gas by Electrons
29
collisions of the electron with an atomic electron; (/)0 comes from the normalization ( 1)
and (/)0= 1.2.
Using -these expressions the values of Xo and ro are obtained by eqs. (16) and (18),
whicbare shown in Fig. 1. For high values of E, "0 and 0 are constants and
r
ro=0.47.
The corresponding value of the average energy loss per ion pair is 28 eV. The discrepancy
out. numerical value of Xo from that· of Fano's is probably due to the fact that he
has neglected the effect of the violent collision, which contributes more to the numerator
than to the denqminator of eq. (16).
Instead of assuming (15) we can put
of
l(E) =xlE-Eo).
with
Xl
(15')
and Eo constants, then
q(E)
X]=--------~~~---------
.
(16')
i]q,,(E)En+(Ez+Eo)q(E)'
n,...2
and assuming (17) again (and writing
rl in the place of ro)
rl=[x12iJ q.. (E)E~+ {1-X](E z +Eo)}2q (E)J(q(E),
,,=2
(18')
,.. ~
~~-===================~
.,x,(E)
~-----------------------~
~
rl(E)·------------------,.;.....---l
.O.2'-------1o-------;~------;;;:;;__----___,J.
10
10'
W'
E/ez- 10'
Fig, 1. The results of numerical evaluation of eqs. (16), (16 / ), (18) Ilnd (18 / ) are shown.
One will see that they. are practically constant for E>10€z=136eV.
J.
30
K. Knipp, and T. Eguchi, M. Ohtaand S. Nagata
It would be a tedious calculation to find the value of Eo from eq. (7) but we can
estimate it by comparing (16') with the experimental value* 1/x1 ,,-39 ey9) and get
E o.-l1 eY. Using these values of Xl and Eo, h becomes 0.25'. That the value of rl
is smaller than unity indicates that the distribution P(E,J) is narrower than the Poisson
distribution. h~ introduced above is analogous to the ionization defect defined by KniJw
and lingO in their calculation on the ionization yields of heavy charged particles. xland
rl are also shown in Fig. 1.
The estimations made above are only very crude and the better approximations are
now under consideration.
This work was initiated and partly completed while one of the authors (T.E.) was
staying at the Department of Physics, IowaState College, U.S.A. and it is his great honour
to thank for the kindness and hospitality with which he was' received there.
References
1)
2)
3)
4)
5)
6)
7)
8)
9),
H. A. Bethe, Ann. d. Phys. 5 (1930), 325.
E. J. Williams, Proc. Roy. Soc. A. 135 (1932), 108.
E. Bagge, Ann. d. Phys. 30 (1937), 72.
U. Fano, Phys. Rev. 70 (1946), 44; 72 (1947), 26.
J. K. Knipp and R. C. Ling, Phys. Rev. 82 (1951), 30.
R. B. Leachman, Phys. Rev. 83 (1951), 17.
J. Rhodes, W. Franzen and W. E. Stephens, Phys. Rev. 87 (1952), 141.
H. A. Bethe, Hb. d. Phys. 24/1, Kap. 3, J. Springer, Berlin (1933).
J. B. Johnson"Phys. Rev. 10 (1917),609.
* The
molecular structure of the actual- hydrogen _gas is disregarded.