4 Gaseous Detectors
99
Collision cross-sections σ have in some cases been measured directly. Often,
however, σ as well as (c) have to be deduced from measurements of u(E), the
dependence of on E, and of diffusion, based on some assumptions on the excitation
functions. The consistency of the methods, when applied to other gas mixtures, has
improved over the years and is presently very good in a number of practical cases, in
particular for the Magboltz simulation [13]; for a comparison of experiments with
various models see e.g. [26].
Drift of Electrons
Because of their small mass, electrons will scatter isotropically in a collision and
forget any preferential direction. They will acquire a drift velocity u given by the
product of the acceleration eE/m and the average time τ between collisions
u = eEτ/m.
(4.11)
Instead of, the notion of mobility μ is often used, with μ defined by
u = μE → μ = eτ/m.
(4.12)
Over a drift distance x there will be a balance between the collision loss ε E and
the energy picked up:
(x/u) (1/τ ) Δε E = eEx.
(4.13)
Here ε E is the energy gained between collisions, the average fraction of the
energy lost in a collision, and (x/u)(1/τ ) the number of collisions on a distance x.
For an instantaneous velocity c, the mean time τ between collisions is related to
the collision cross- section σ and the number density N of gas molecules by
1/τ = Nσ c.
(4.14)
The total energy ε of the electron is given by
(m/2) c
2
= = E + (3/2) kT ,
(4.15)
including the thermal energy.
In the approximation e >> (3/2)kT, which is often fulfilled for drift of electrons
in particle detectors, one obtains
u 2 = (eE/mNσ )
√
(Δ/2) , and
c 2 = (eE/mNσ )
√
(2/Δ) for ε − ε E >> (3/2) kT .
(4.16)
99
Collision cross-sections σ have in some cases been measured directly. Often,
however, σ as well as (c) have to be deduced from measurements of u(E), the
dependence of on E, and of diffusion, based on some assumptions on the excitation
functions. The consistency of the methods, when applied to other gas mixtures, has
improved over the years and is presently very good in a number of practical cases, in
particular for the Magboltz simulation [13]; for a comparison of experiments with
various models see e.g. [26].
Drift of Electrons
Because of their small mass, electrons will scatter isotropically in a collision and
forget any preferential direction. They will acquire a drift velocity u given by the
product of the acceleration eE/m and the average time τ between collisions
u = eEτ/m.
(4.11)
Instead of, the notion of mobility μ is often used, with μ defined by
u = μE → μ = eτ/m.
(4.12)
Over a drift distance x there will be a balance between the collision loss ε E and
the energy picked up:
(x/u) (1/τ ) Δε E = eEx.
(4.13)
Here ε E is the energy gained between collisions, the average fraction of the
energy lost in a collision, and (x/u)(1/τ ) the number of collisions on a distance x.
For an instantaneous velocity c, the mean time τ between collisions is related to
the collision cross- section σ and the number density N of gas molecules by
1/τ = Nσ c.
(4.14)
The total energy ε of the electron is given by
(m/2) c
2
= = E + (3/2) kT ,
(4.15)
including the thermal energy.
In the approximation e >> (3/2)kT, which is often fulfilled for drift of electrons
in particle detectors, one obtains
u 2 = (eE/mNσ )
√
(Δ/2) , and
c 2 = (eE/mNσ )
√
(2/Δ) for ε − ε E >> (3/2) kT .
(4.16)