442
V. Radeka
as a function of position is determined from the operating (applied) field on the
detector. This gives the induced current as a function of the position of the moving
charge,
i = −q
− →
E · − → v .
(10.1)
Third, the position of the moving charge as a function of time is determined by
solving the equations of motion. This is necessary in the case of ballistic motion of
charge, but it is simple in the case of transport by drift as the charge carriers follow
the applied electric field. If we are interested only in the total induced charge and
not in the waveforms, the induced charge is simply given by the difference in the
weighting potentials between any two positions of the moving charge,
Q s =
idt = +
− →
E w d − → x ,
Q s (1, 2) = q (V w2 − V w1 ) .
(10.2)
An example of the weighting-field (potential) profiles is illustrated by the plot of
equipotential lines for planar geometry with a strip sensing electrode. The operating
(applied field) in this case is uniform and perpendicular to the electrodes. The
weighting field map is in general quite different from that of the operating field; the
two field maps are identical only in some special cases. The minus sign in Ramo’s
equation (Eq. 10.1) for the induced current results from the arbitrary assumption of
induced current into the electrode being positive.
The sketch in Fig. 10.2a shows conceptually how the weighting field (potential) is
defined: the sensing electrode is connected to unity potential, and all other electrodes
to zero potential. The equipotential lines in Fig. 10.2b illustrate the solution for this
case, showing two strips next to the sensing electrode. A great variety of results
for the induced current and charge may arise in an electrode structure, such as this,
depending on the particle type detected (distribution of ionization) and on the ratio
of the charge observation measurement (or integration) time and the charge carrier
transit time. The current waveforms shown are drawn qualitatively for a simple
example. The operating field is assumed uniform and perpendicular to the electrode
planes. Charge q m traversing the full distance between the electrodes along line 1 is
observed as Q 1 = −q m , while the current decreases with distance from the sensing
electrode 1, as the electrostatic coupling decreases. For a charge moving along line
2, the induced charge (i.e., the difference between the weighting potentials) is zero,
if the measurement time is longer than the transit time. For a short measurement
time a net induced charge is observed. The induced current waveform (the “crosstalk
signal”) is bipolar, since the weighting field direction changes along the path.
Figure 10.3 illustrates a simple case where the real (operating) electric field and
the weighting field have the same form = 1/d. The induced current waveforms
shown are for a semiconductor detector with different electron and hole mobilities.
For extended ionization the waveforms result from superposition of the waveforms
for localized ionization, and the currents decrease as the carriers arrive at the
electrodes from different initial positions within the bulk of the detector.
V. Radeka
as a function of position is determined from the operating (applied) field on the
detector. This gives the induced current as a function of the position of the moving
charge,
i = −q
− →
E · − → v .
(10.1)
Third, the position of the moving charge as a function of time is determined by
solving the equations of motion. This is necessary in the case of ballistic motion of
charge, but it is simple in the case of transport by drift as the charge carriers follow
the applied electric field. If we are interested only in the total induced charge and
not in the waveforms, the induced charge is simply given by the difference in the
weighting potentials between any two positions of the moving charge,
Q s =
idt = +
− →
E w d − → x ,
Q s (1, 2) = q (V w2 − V w1 ) .
(10.2)
An example of the weighting-field (potential) profiles is illustrated by the plot of
equipotential lines for planar geometry with a strip sensing electrode. The operating
(applied field) in this case is uniform and perpendicular to the electrodes. The
weighting field map is in general quite different from that of the operating field; the
two field maps are identical only in some special cases. The minus sign in Ramo’s
equation (Eq. 10.1) for the induced current results from the arbitrary assumption of
induced current into the electrode being positive.
The sketch in Fig. 10.2a shows conceptually how the weighting field (potential) is
defined: the sensing electrode is connected to unity potential, and all other electrodes
to zero potential. The equipotential lines in Fig. 10.2b illustrate the solution for this
case, showing two strips next to the sensing electrode. A great variety of results
for the induced current and charge may arise in an electrode structure, such as this,
depending on the particle type detected (distribution of ionization) and on the ratio
of the charge observation measurement (or integration) time and the charge carrier
transit time. The current waveforms shown are drawn qualitatively for a simple
example. The operating field is assumed uniform and perpendicular to the electrode
planes. Charge q m traversing the full distance between the electrodes along line 1 is
observed as Q 1 = −q m , while the current decreases with distance from the sensing
electrode 1, as the electrostatic coupling decreases. For a charge moving along line
2, the induced charge (i.e., the difference between the weighting potentials) is zero,
if the measurement time is longer than the transit time. For a short measurement
time a net induced charge is observed. The induced current waveform (the “crosstalk
signal”) is bipolar, since the weighting field direction changes along the path.
Figure 10.3 illustrates a simple case where the real (operating) electric field and
the weighting field have the same form = 1/d. The induced current waveforms
shown are for a semiconductor detector with different electron and hole mobilities.
For extended ionization the waveforms result from superposition of the waveforms
for localized ionization, and the currents decrease as the carriers arrive at the
electrodes from different initial positions within the bulk of the detector.