98
H. J. Hilke and W. Riegler
Due to the cut-off, this relation applies not only to heavy particles but also to
ionization by electrons [19]. The minimum dE/dx deposited by a minimum ionizing
particle (mip) still lies around γ = 3 − 4, with δ = 0. For β → 1, the density
correction δ approaches
δ → 2 ln
hv p γ /I
− 1,
(4.9)
hv p being the quantum energy of the plasma oscillation of the medium. The
restricted energy deposit then reaches a constant value, the Fermiplateau, the δ-term
compensating the lnγ term:
dE/dx restricted → P
2 (Z/A) 0.5 ln
2mc
2 T cut /
hv p
2
.
(4.10)
In Ar one obtains for the ratio R of energy deposit on the Fermi plateau to the
minimum deposit R = 1.60, 1.54, and 1.48 for a cut-off T cut = 30,150 and 1000 keV,
respectively [19]. A precise determination of R requires a good estimate of T cut .
To use the β-dependence of dE/dx for particle identification, one has to measure
many samples and take their truncated mean, e.g., the mean of the lowest 50% pulse
heights, to be insensitive to the long tail and to obtain an approximation to the most
probable value. See Chap. 2 for details.
4.2.2 Transport of Electrons and Ions
4.2.2.1 Drift Velocities
On the microscopic scale, electrons or ions drifting through a gas are scattered on
the gas molecules. In a homogenous electric field E they will acquire a constant
drift velocity u in the E field direction or, in the presence of an additional magnetic
field B, in a direction determined by both fields. Their drift velocities are much
smaller than their instantaneous velocities c between collisions. Electrons and ions
will behave quite differently because of their mass difference.
In the chapters on drift velocities and diffusion we shall follow the argumentation
developed in [19]. A relatively simple derivation brings out the main characteristics
and does describe a number of experimental results with good approximation. The
main approximation of the simple models is to take a single velocity c to represent
the motion between collisions. In reality, these velocities c are distributed around
a mean value. The shape of the distribution depends on the variation of crosssection and energy loss with the collision velocity. The rigorous theory takes these
distributions into account. For lowest velocities there is only elastic scattering, for
higher energies various inelastic processes contribute. The elastic and the inelastic
spectrum may be described by a single effective cross-section σ (c) combining the
various processes, sometimes called momentum transfer cross-section, and by the
average fractional energy loss (c) per collision.
H. J. Hilke and W. Riegler
Due to the cut-off, this relation applies not only to heavy particles but also to
ionization by electrons [19]. The minimum dE/dx deposited by a minimum ionizing
particle (mip) still lies around γ = 3 − 4, with δ = 0. For β → 1, the density
correction δ approaches
δ → 2 ln
hv p γ /I
− 1,
(4.9)
hv p being the quantum energy of the plasma oscillation of the medium. The
restricted energy deposit then reaches a constant value, the Fermiplateau, the δ-term
compensating the lnγ term:
dE/dx restricted → P
2 (Z/A) 0.5 ln
2mc
2 T cut /
hv p
2
.
(4.10)
In Ar one obtains for the ratio R of energy deposit on the Fermi plateau to the
minimum deposit R = 1.60, 1.54, and 1.48 for a cut-off T cut = 30,150 and 1000 keV,
respectively [19]. A precise determination of R requires a good estimate of T cut .
To use the β-dependence of dE/dx for particle identification, one has to measure
many samples and take their truncated mean, e.g., the mean of the lowest 50% pulse
heights, to be insensitive to the long tail and to obtain an approximation to the most
probable value. See Chap. 2 for details.
4.2.2 Transport of Electrons and Ions
4.2.2.1 Drift Velocities
On the microscopic scale, electrons or ions drifting through a gas are scattered on
the gas molecules. In a homogenous electric field E they will acquire a constant
drift velocity u in the E field direction or, in the presence of an additional magnetic
field B, in a direction determined by both fields. Their drift velocities are much
smaller than their instantaneous velocities c between collisions. Electrons and ions
will behave quite differently because of their mass difference.
In the chapters on drift velocities and diffusion we shall follow the argumentation
developed in [19]. A relatively simple derivation brings out the main characteristics
and does describe a number of experimental results with good approximation. The
main approximation of the simple models is to take a single velocity c to represent
the motion between collisions. In reality, these velocities c are distributed around
a mean value. The shape of the distribution depends on the variation of crosssection and energy loss with the collision velocity. The rigorous theory takes these
distributions into account. For lowest velocities there is only elastic scattering, for
higher energies various inelastic processes contribute. The elastic and the inelastic
spectrum may be described by a single effective cross-section σ (c) combining the
various processes, sometimes called momentum transfer cross-section, and by the
average fractional energy loss (c) per collision.