22
H. Bichsel and H. Schindler
2.3.4.1 Inverse Mean Free Path
In the relativistic first-order Born approximation, the inverse mean free path for
ionising collisions has the form [1, 2]
M 0 =
2πz 2 (α ¯
hc)
2
mc 2 β 2 N
M
2
ln
β
2 γ
2
− β
2
+ C
,
(2.26)
where
M
2
=
1
E
df (E)
dE
dE,
C = M
2
ln ˜
c + ln
4
α 2
,
and ˜
c is a material-dependent parameter that can be calculated from the generalised
oscillator strength density. Calculations can be found, for example, in Refs. [53, 54].
As in the Bethe stopping formula (2.28) discussed below, a correction term can be
added to Eq. (2.26) to account for the density effect [55].
The inverse mean free path for dipole-allowed discrete excitations is given by [2]
M
(n)
0 =
2πz 2 (α ¯
hc)
2
mc 2 β 2 N
f n
E n
ln
β
2 γ
2
− β
2
+ ln ˜
c n + ln
4
α 2
.
We can thus obtain a rough estimate of the relative frequencies of excitations and
ionising collisions from optical data. In argon, for instance, the ratio of
f n /E n
and M 2 is ∼20% [25].
For gases, M 0 can be determined experimentally by measuring the inefficiency
of a gas-filled counter operated at high gain (“zero-counting method”). Results
(in the form of fit parameters M 2 , C) from an extensive series of measurements,
using electrons with kinetic energies between 0.1 and 2.7 MeV, are reported in
Ref. [52]. Other sets of experimental data obtained using the same technique can
be found in Refs. [56, 57]. Table 2.1 shows a comparison between measured and
calculated values (using the FVP algorithm) of M 0 for particles with βγ = 3.5
at a temperature of 20 ◦ C and atmospheric pressure. The inverse mean free path is
Table 2.1 Measurements
[52] and calculations (using
the FVP algorithm as
implemented in HEED [49])
of M 0 for βγ = 3.5 at
T = 20 ◦ C and atmospheric
pressure
M 0 [cm −1 ]
Gas
Measurement FVP
Ne
10.8
10.5
Ar
23.0
25.4
Kr
31.5
31.0
Xe
43.2
42.1
CO 2
34.0
34.0
CF 4
50.9
51.8
CH 4
24.6
29.4
iC 4 H 10 83.4
90.9
H. Bichsel and H. Schindler
2.3.4.1 Inverse Mean Free Path
In the relativistic first-order Born approximation, the inverse mean free path for
ionising collisions has the form [1, 2]
M 0 =
2πz 2 (α ¯
hc)
2
mc 2 β 2 N
M
2
ln
β
2 γ
2
− β
2
+ C
,
(2.26)
where
M
2
=
1
E
df (E)
dE
dE,
C = M
2
ln ˜
c + ln
4
α 2
,
and ˜
c is a material-dependent parameter that can be calculated from the generalised
oscillator strength density. Calculations can be found, for example, in Refs. [53, 54].
As in the Bethe stopping formula (2.28) discussed below, a correction term can be
added to Eq. (2.26) to account for the density effect [55].
The inverse mean free path for dipole-allowed discrete excitations is given by [2]
M
(n)
0 =
2πz 2 (α ¯
hc)
2
mc 2 β 2 N
f n
E n
ln
β
2 γ
2
− β
2
+ ln ˜
c n + ln
4
α 2
.
We can thus obtain a rough estimate of the relative frequencies of excitations and
ionising collisions from optical data. In argon, for instance, the ratio of
f n /E n
and M 2 is ∼20% [25].
For gases, M 0 can be determined experimentally by measuring the inefficiency
of a gas-filled counter operated at high gain (“zero-counting method”). Results
(in the form of fit parameters M 2 , C) from an extensive series of measurements,
using electrons with kinetic energies between 0.1 and 2.7 MeV, are reported in
Ref. [52]. Other sets of experimental data obtained using the same technique can
be found in Refs. [56, 57]. Table 2.1 shows a comparison between measured and
calculated values (using the FVP algorithm) of M 0 for particles with βγ = 3.5
at a temperature of 20 ◦ C and atmospheric pressure. The inverse mean free path is
Table 2.1 Measurements
[52] and calculations (using
the FVP algorithm as
implemented in HEED [49])
of M 0 for βγ = 3.5 at
T = 20 ◦ C and atmospheric
pressure
M 0 [cm −1 ]
Gas
Measurement FVP
Ne
10.8
10.5
Ar
23.0
25.4
Kr
31.5
31.0
Xe
43.2
42.1
CO 2
34.0
34.0
CF 4
50.9
51.8
CH 4
24.6
29.4
iC 4 H 10 83.4
90.9