# 8. Drift and Diffusion in Materials, Wire Chambers

"*Time is a sort of river of passing events,\...and this too will be
swept away*."\...Marcus Aurelius

Wire chambers are perhaps the most commonly used detection devices in
present high energy physics experiments. A chamber consists,
conceptually, of a gas volume in which the gas is ionized by the passage
of a charged particle. Ionization has been discussed in Section 6. The
ionization then drifts (and diffuses) in an electric (and perhaps
magnetic) field toward an electrode (wire). Subsequent collection and
amplification of the signal charge on the anode and the charge induced
on the cathode creates a detectable signal. Since the radius of
curvature of a charged particle in a magnetic field depends on momentum
(Section 7), measurement of points on the trajectory determines
$\overrightarrow{p}$ the vector momentum. We can then make a second
redundant momentum measurement using "destructive" measurement
techniques (see Sections 11-13).

In this Section we develop a general treatment suitable for "unity gain"
devices, e.g. liquid argon calorimetry, or devices with gain, e.g.
proportional wire chambers. We first introduce the drift velocity,
mobility and the diffusion coefficient. The signal capacitively induced
on the two electrodes is then considered both for a unity gain device
and for a device with gain. The methods used here are later applied, in
the following Section, to silicon devices.

## 8.1 Thermal and Drift Velocity

We begin by considering the drift of the produced ionization in a
uniform electric field and the accompanying diffusion of the localized
ionization. First, we note that gas molecules of mass M have a thermal
velocity ![](media/image1.emf) due to the thermal kinetic energy T~T~,
which is of order kT, where k is Boltzman\'s constant and T is the
absolute temperature. The numerical value of k is given in Table 1.1.

$\begin{array}{r}
T_{T}\text{} = \text{Mv}_{T}^{2}\frac{}{}2\text{}\text{\textasciitilde}\text{}\frac{3}{2}\text{kT} \\
v_{T}\text{}\text{\textasciitilde}\text{}\sqrt{\frac{3\text{kT}}{M}}
\end{array}$ (8.1)

The thermal velocity goes like $\sqrt{T}$. Numerically, we find that the
thermal velocity of N~2~ at room temperature is of order 105 cm/sec or
0.1 cm/μsec. Other light molecular gas velocities at standard
temperature and pressure (STP) are in the range of (0.1 to 1.0) cm/μsec.
In comparison, escape velocity on earth is \~ 1.1 cm/μsec. This fact
explains why there are no light gases such as helium in our atmosphere.
Fluctuations in the thermal velocity allow light gases to attain escape
velocity and they are lost to the atmosphere.

For an electron, given the $1\sqrt{M}$scaling, we expect that typical
thermal velocities would be of order \~(4 - 40) cm/μsec. Note that the
thermal energy scale is well below the scale of the binding energies
(see Section 1) which are relevant to atomic excitations, so we can
consider the atoms to be inert "billiard balls".

Let\'s consider next the situation where we have an applied electric
field,$\overrightarrow{E}$, in addition to the random thermal motion.
Between collisions, ions and electrons are accelerated by
$\overrightarrow{E}$. The integral of the acceleration
$\overrightarrow{a}$ is the drift velocity *v~d~*. The parameter
![](media/image2.emf) is defined to be the mean time between collisions.
Since the mean free path ![](media/image3.emf) is the mean distance
between collisions, ![](media/image4.emf) is approximately
$\frac{\left\langle L \right\rangle}{v_{T}}$. The average drift velocity
is $\left\langle v_{d} \right\rangle$.

$\begin{array}{r}
\overrightarrow{a}\  = \ e\frac{\overrightarrow{E}}{M} \\
\left\langle v_{d} \right\rangle \ \text{\textasciitilde}\ a\tau \\
\text{}\ \text{}\text{\textasciitilde}\ \left( \frac{\text{eE}}{M} \right)\ \left( \frac{\left\langle L \right\rangle}{v_{T}} \right)
\end{array}$ (8.2)

The drift velocity ![](media/image5.emf), which is imposed on top of the
randomly oriented thermal velocity, is proportional to the applied
field. We assume that the main velocity components are caused by the
random thermal motion and not by the applied electrical field, which is
assumed to be only a small perturbation to the thermal motion.

As we discussed in Section 1, the inverse mean free path is proportional
to the collision cross section, ![](media/image6.emf). For atomic
collisions at low energy scales, the collision cross section is of order
the geometric cross section ![](media/image7.emf). The total elastic
cross section for e on CH~4~ as a function of e kinetic energy is shown
in Fig. 8.1.

![](media/image8.png){width="5.529166666666667in"
height="3.651388888888889in"}

Fig. 8.1: Cross section for *e* on *CH~4~* as a function of the *e*
energy in *eV*.

The scale is set by the geometric cross section, ![](media/image9.emf)
with ![](media/image10.emf). Since the signal current induced on an
electrode is proportional to the drift velocity (see Eq. 8.7) and since
the drift velocity is larger for electrons than ions, we concentrate
mostly on electrons in what follows.

$\begin{array}{r}
\left\langle L \right\rangle^{- 1}\  \simeq \ N_{o}\frac{\text{ρσ}}{A} \\
\text{}\sigma\text{}\text{\textasciitilde}\ \pi a_{o^{2}} \\
\left\langle v_{d} \right\rangle\text{}\text{\textasciitilde}\ \left( \frac{\text{eE}}{m_{e}} \right)\left( \frac{\frac{A}{N_{o}}\text{ρσ}}{v_{T}} \right) = \frac{\text{eEA}}{N_{o}\text{ρσ}\ \sqrt{3\text{kT}\ m_{e}}}
\end{array}$ (8.3)

## 8.2 Mobility

The drift velocity is proportional to the applied electric field and
inversely proportional to the collision cross section and the thermal
velocity. Therefore, in this very simple-minded picture, we expect that
the drift velocity per unit electric field divided by density would be a
constant which is called the mobility. The electric field per unit of
pressure, P, or per unit of density, ![](media/image11.emf), with
respect to STP is called the reduced electric field. The mobility
![](media/image12.emf), is defined to be the drift velocity
![](media/image13.emf) per unit reduced electric field
![](media/image14.emf).

$\begin{array}{r}
\mu\text{}\ \text{} \equiv \ \frac{\left\langle \text{}v_{d} \right\rangle}{\frac{E}{\rho}}\ \text{\textasciitilde}\ \text{const},\ \left\langle \text{}v_{d} \right\rangle\text{} \equiv \text{}\mu E(\frac{P_{o}}{P}) \\
P_{o} = \text{}1\text{ATM}\  = \ \text{760}\text{Torr}\text{}(\text{STP})
\end{array}$ (8.4)

The measured ion mobilities for different gases are reasonably constant
over most of the periodic table and are ![](media/image15.emf)

$\begin{array}{r}
\mu\text{}\text{\textasciitilde}\ \text{}(\frac{e}{M})\ \left\lbrack \frac{A}{N_{o}{\sigma v}_{T}} \right\rbrack\ ,\ \text{Eq}\text{.}\ 8\text{.}3\text{} \\
\mu_{\text{Ar}}\text{} = \ 1\text{.}7(\frac{\text{cm}^{2}}{V} \cdot \text{sec}) \\
\mu_{\text{CO}_{2}} = \ 1\text{.}1(\text{cm}^{2}V \cdot \text{sec})
\end{array}$ (8.5)

In methane the electron drift velocity in a field of 1 kV per cm is
about 12 cm/![](media/image16.emf)sec. For comparison, the speed of
sound in air at STP is about 0.033 cm/![](media/image17.emf)sec. Note
that the drift velocities are, in fact, less than the electron thermal
velocities, something we had initially assumed.

Shown in Fig. 8.2 are measurements of the electron drift velocity in
cm/![](media/image18.emf)sec as a function of a reduced electric field
for gaseous argon - methane gas mixtures. We see the expected linear
rise of the drift velocity with electric field. The values of the drift
velocity at a 1 kV/cm applied field are \~10
cm/![](media/image19.emf)sec. For comparison, the electron drift
velocity in liquid argon, as shown in Fig. 8.3, at the same field is
\~0.15 cm/![](media/image20.emf)sec indicating a lower mobility in the
liquid.

We also see, looking at Fig. 8.2, another phenomenon. The linear rise
with field no longer holds at high electric fields. At those fields the
drift velocity becomes comparable to the electron thermal velocity,
which we said, is of order (4-40) cm/![](media/image21.emf)sec. Our
assumptions are then no longer valid, because we assumed that the drift
velocity was a small perturbation on the random thermal motion. Of some
practical use is the experimental fact that a 50:50 mixture of argon:
ethane has a "saturated" drift velocity. By this we mean that the
electron drift velocity ![](media/image22.emf) is essentially
independent of the field when E \> 0.5 kV/cm. This gas mixture is widely
used in "drift chambers" since, if E is kept high, the velocity is a
constant, so that a drift time measurement is a linear measure of the
distance from the point of particle passage to the anode wire. (See Fig.
8.7.)

![](media/image23.png){width="4.038194444444445in"
height="6.741666666666666in"}

Fig. 8.2: Drift velocity in cm/μsec as a function of \"reduced\" field,
E/P. a) A + CH4 gas mixtures, with fraction of CH4 ≤ 30%. b) A + CH4 gas
mixtures, with fraction of CH4 \> 40%. (From Ref. 2, with permission.)

![](media/image24.wmf){width="3.1305555555555555in"
height="4.559722222222222in"}

Fig. 8.3: Plot of electron drift velocity as a function of electric
field in liquid argon. (From Ref. 9, with permission.)

## 8.3 Pulse Formation in "Unity Gain" Detectors

Let us now turn to the signal formation in a "unity gain" detector. The
ionization produced by the passage of a charged particle is drifted to,
and collected on, the electrodes which apply the electric field. There
is no charge gain, as there is in a photomultiplier tube (see Section
2). For a parallel plate configuration with spacing d and applied
voltage V~0~, a field, E = V~0~/d, exists in the gas or liquid located
between the electrodes. Assume, initially, that a high energy charged
particle passes perpendicular to the electrodes and uniformly ionizes
the medium between them.

Using the relationship between capacity, C, voltage, V, and electrode
charge, Q, Q = CV, the energy stored in the electric field is U =
CV^2^/2 = Q^2^/2C \~ QV. The motion of the ionization charge in the
field with time, q(t), capacitively induces a current pulse on the
electrodes, I(t). We ignore the ions, because the size of the current
pulse is proportional to the drift velocity. The change in field energy
dU due to the drift of the ionization is equal to the work done, Fdx, by
the field on the drifting charges.

$\begin{array}{r}
\text{dU}\text{} = \ Q_{o}\text{dQ}(t\frac{)}{C}\  = \ \text{Fdx} \\
\text{}\ \text{} = \ \lbrack q(t)E\rbrack\ \lbrack\left\langle v_{d} \right\rangle \text{dt}\rbrack
\end{array}$ (8.6)

The charge Q(t) induced on the electrodes is proportional to the
ionization charge q(t) in the electrode gap and the drift velocity
![](media/image25.emf). Note that this treatment is quite general,
having used only energy conservation. Thus it can be used later in the
discussion of wire chambers and Si detectors. The expression for I(t) is

$\begin{array}{r}
\text{dQ}(t)\text{}\  = \ q(t)\left\langle v_{d} \right\rangle \ \text{dt}\ \lbrack\frac{E}{V_{o}}\rbrack\  = \text{}\ \left\lbrack q(t)\mu \frac{E^{2}}{V_{o}} \right\rbrack\ \text{dt} \\
\frac{\text{dQ}(t)}{\text{dt}}\  \equiv \ I(t)\  = q(t)\frac{{\mu E}^{2}}{V_{o}}
\end{array}$ (8.7)

The charge/length initially in the gap is q~s~/d = Ne/d where q~s~ is
the total signal charge. Note that N is the number of independent
"events", just as, for example, N is the number of photo-electrons which
is statistically distributed (Section 2). Therefore, the fractional
spread in the output signal due to the statistical variation of N, dN,
is given by ![](media/image26.emf) \~ ![](media/image27.emf). The
characteristic time for all the charge to be swept up by the electrodes
is ![](media/image28.emf). For example, for a 1 cm gap filled with
argon-methane and a 1 kV applied voltage, this time is 100 nsec.

$\begin{array}{r}
q(t) = q_{s}\left( 1 - \frac{t}{t_{d}} \right),t < t_{d} \\
0,t > t_{d} \\
t_{d} = \frac{d}{\left\langle v_{d} \right\rangle}
\end{array}$ (8.8)

The charge q(t) in the gap is q~s~ at t = 0 and decreases linearly to
zero at t = ![](media/image29.emf). The induced current behaves
similarly, I(0) = q~s~/![](media/image30.emf), I(![](media/image31.emf))
= 0. The total induced charge is Q(0) = 0, Q(![](media/image32.emf)) =
q~s~/2, since half the induced charge appears on the other electrode,
which we have assumed to be grounded.

$\begin{array}{r}
I(t) = (\frac{q_{s}}{t_{d}})(1 - \frac{t}{t_{d}}) \\
\int_{}^{}{I(t)\text{dt}ºQ(t)} \\
q_{s}\lbrack y - \frac{y^{2}}{2}\rbrack,y = \frac{t}{t_{d}}
\end{array}$ (8.9)

Sometimes all the charge is located at a single point, x~o~, in the gap,
for example, due to a short range ![](media/image33.emf) decay which
ranges out almost immediately or a Compton recoil electron due to a
scattering of an x ray from Fe^55^ decay (see Section 10). In that case
Q(t) is different from the case where charge is spread throughout the
gap.

In the "point" ionization case, with total charge q~s~ = e.

${\begin{array}{r}
\text{dQ}\text{} = \ \frac{q(t)E}{V_{o}}\ (\left\langle v_{d} \right\rangle\text{}\text{dt}) \\
q(t)\text{} = \ q_{s}\text{}\text{for}\ \text{}t\  < \text{ \{}\tau
\end{array}_{d}^{'},\text{}\  = \ \text{}0,\text{}t\  > \ \tau_{d}^{'},\text{ \{}\tau}_{d}^{'}\  \equiv \ \ \frac{x_{o}}{\left\langle v_{d} \right\rangle}$
(8.10)

The current in this case is constant in time, and the charge collected
on the electrodes increases linearly with time up to $t_{d}^{'}$, i.e.
![](media/image34.emf)(*x~o~/d*).

$\begin{array}{r}
I(t)\text{}\  = \ \text{}\frac{q_{s}}{\tau_{d}}\text{},\ \text{}t\  < \tau_{d}^{'} = 0,\ t\  > \tau_{d}^{'} \\
Q(t)\text{}\  = \ q_{s}\frac{t}{\tau_{d}} \\
\text{}\ \text{} \leq \ q_{s}\ \left( \frac{x_{o}}{d} \right)
\end{array}$ (8.11)

For example, a d = 1 mm gap in liquid argon has a collection time for a
typical electric field of ![](media/image35.emf) = 0.25
![](media/image36.emf). The collected charge, Q(t) as a function of time
is shown in Fig. 8.4 for point ionization, provided by short range
![](media/image37.emf) irradiation, and for line ionization provided by
traversal of minimum ionizing particles. Clearly linear and quadratic
behavior is seen, respectively, with a time scale set by
![](media/image38.emf). The signal noise, which is seen on the traces of
Fig. 8.4, will be discussed in Section 9.

![](media/image39.wmf){width="2.5284722222222222in"
height="2.838888888888889in"}

Fig. 8.4: Collected charge in a liquid argon detector as a function of
time, a) point\
ionization, 100 nsec/division b) line ionization, 200 nsec/division.
(From Ref 9, with permission.)

## 8.4 Diffusion and the Diffusion Limit

Diffusion of the drifting charge is due to multiple scattering. This
diffusion process sets a fundamental limit to the accuracy of position
measurements in ionization chambers. Consider charge drifting with a
velocity ![](media/image40.emf) in field E for time t or distance *x*.
The distribution, dN, of transverse coordinate, *x~T~*, is defined by
the diffusion coefficient D, with units of cm^2^/sec, \[D\] =
\[L^2^/T\].

The diffusion equation is well known in classical physics in the study
of heat. For a density $\rho(x,t)$, the equation without external forces
being applied is
$\partial\frac{\rho}{\partial}t\  = \  - D\ \partial^{2}\frac{\rho}{\partial}x^{2}$.
A one dimensional solution in the case that particles appear
instantaneously and tightly localized spatially at x = 0 at t = 0 is:

$\begin{array}{r}
\frac{\partial\rho}{\partial t}\  = \  - D\ \partial^{2}\frac{\rho}{\partial}x^{2} \\
\rho(x,t)\ \text{\textasciitilde}\ \frac{1}{\sqrt{\text{Dt}}}\ e^{\frac{- x^{2}}{4}\text{Dt}}
\end{array}$ (8.12)

The solution is a Gaussian distribution of particles with a width;
$\sigma\ \text{\textasciitilde}\ \sqrt{2\text{Dt}}$ which grows with
time as $\sqrt{t}$. In the case where there is a uniform applied
electric field the mean moves with a drift velocity brought about by the
replacement,
$x\  \rightarrow \ x\  - \ \left\langle v_{d} \right\rangle t$ in Eq.
8.12. This is a stochastic process, as indicated by the fact that the
rms of the distribution goes as $\sqrt{t}$. Recall the similar multiple
scattering behavior,
$\left\langle \theta_{\text{MS}} \right\rangle\text{\textasciitilde}\sqrt{t}$.

We assert that dimensional arguments lead to the conclusion that D \~
![](media/image41.emf) \<L\> showing the relation between diffusion,
thermal velocity, and collision mean free path. The rms of the diffusion
distribution is

$\begin{array}{r}
\sigma_{x_{T}}\text{\textasciitilde}\ \sqrt{2\text{Dt}}\text{}\ \text{\textasciitilde}\ \sqrt{2v_{T}\ \left\langle L \right\rangle\ (\frac{x}{\left\langle v_{d} \right\rangle})} \\
\sigma_{x_{T}}\ \text{\textasciitilde}\ \ \sqrt{\frac{v_{T}^{2}\text{}x}{a}},\ \left\langle v_{d} \right\rangle\text{\textasciitilde}\ a\frac{\left\langle L \right\rangle}{v_{T}}
\end{array}$ (8.13)

We use the relationship between *v~T~* and the thermal energy *kT* and
between *a* and the applied field *E*. This "thermal limit" for
diffusion shows the competition between random thermal energy,
![](media/image42.emf), and the imposed drift field with energy eV~o~.

$\begin{array}{r}
\sigma_{x_{T}}\text{} \simeq \sqrt{\left( \frac{2\text{kT}}{\text{eE}} \right)x\text{}} \\
\sigma_{x_{T}}\text{} \simeq \left\lbrack \sqrt{\frac{2\text{kT}}{\text{eV}_{0}}} \right\rbrack x
\end{array}$ (8.14)

A plot of ![](media/image43.emf) as a function of reduced electric field
is shown in Fig. 8.5. The expected behavior, ![](media/image44.emf), is
roughly followed by the data at low electric fields. At higher electric
field our assumption about small drift velocity,
$v_{T}\ \text{>>}\ \left\langle v_{d} \right\rangle$ breaks down. We
ignore here the difference between transverse and longitudinal
diffusion. A typical value of ![](media/image45.emf) at a drift field of
1 kV/cm is \~(100-300) ![](media/image46.emf) for a 1 cm drift distance.
Therefore a drift chamber with 1 cm spacing between wires cannot achieve
a position resolution better than \~100 μm. This limitation is
intrinsic.

An example of the geometry of a drift tube, with "field shaping"
electrodes which are used to provide a drift field of \> 0.5 kV/cm, is
shown in Fig. 8.6. Also shown are the equipotentials. By design there is
a reasonably uniform electric field across most of the drift space. The
relationship of drift distance to drift time for this structure is shown
in Fig. 8.7 for cosmic ray traversals at two angles of incidence, 0o and
45o. In this example a mixture of argon and carbon dioxide is used as
the drift gas. The phenomenon called "saturation" also roughly obtains
in this gas mixture for drift fields greater than about 0.5 kV/cm. This
implies a linear relationship between the drift distance and the
measured drift time (the time between particle passage and the
collection of the ionization on the wire). We see from Fig. 8.7 that
this is roughly the case for drift distances up to 5 cm with a maximum
drift time of about 800 nsec. The average drift velocity is about 6.2
cm/![](media/image47.emf).

![](media/image48.png){width="3.417361111111111in" height="3.91875in"}

Fig. 8.5: Measured values of the longitudinal diffusion coefficient
![](media/image49.emf) as a function of applied reduced electric field
for different commonly used gases compared to the thermal limit,
$\frac{s_{x_{T}}}{x}\text{\textasciitilde}\sqrt{2\frac{\text{kT}}{\text{eV}_{o}}}$,
indicated by the dashed line. (From Ref. 5, with permission.)

![](media/image50.png){width="4.447222222222222in"
height="3.276388888888889in"}

Fig. 8.6: Transverse dimensions and equipotentials for a large area
drift tube with field shaping. (From Ref. 11, with permission.)

As a numerical example, electrons in gaseous Ar in a drift field of 1
kV/cm have a drift velocity of ![](media/image51.emf) \~ 5
cm/![](media/image52.emf). In a 1 cm drift distance the transverse
diffusion distance is \~1000 ![](media/image53.emf), while in pure C0~2~
it is only 100 ![](media/image54.emf).  In the mixture Ar:C0~2~ = 80:20
it is 300 ![](media/image55.emf) (see Fig. 8.5). The width of the lines
in Fig. 8.7 gives some indication of the spatial resolution. Diffusion
limits the resolution for the longest drift distances. The observed
resolution is \~ 100 ![](media/image56.emf) near the wire and \~ 400
![](media/image57.emf) at the maximum 5 cm drift distance.

![](media/image58.png){width="3.9097222222222223in"
height="3.948611111111111in"}

Fig. 8.7: Relationship of drift distance to drift time for the structure
shown in Fig. 8.6 for particles incident at 0o and 45o. (From Ref. 11,
with permission.)

## 8.5 Motion in $\overrightarrow{E}$ and $\overrightarrow{B}$ Fields, with and without Collisions

So far we have only considered random thermal motion and motion in an
applied electric field. In Section 7 we have also discussed free motion
in a purely magnetic field. We concluded that the path was circular with
a "cyclotron frequency", ![](media/image59.emf) (Table 1.1). Now we
consider the more complex case of free motion in a region where there
are simultaneous electric and magnetic fields. The dynamics in the
absence of collisions is controlled by the Lorentz force equation. (CGS
units here.)

$\begin{array}{r}
\overrightarrow{F}\text{} = q(\overrightarrow{E} + \overrightarrow{\beta} \times \overrightarrow{B}), \\
{\overrightarrow{\beta}}_{d} = (\overrightarrow{E} \times \overrightarrow{B}\frac{)}{\left| \overrightarrow{B} \right|^{2}}
\end{array}$ (Eq. 8.15)

The general case of non-relativistic motion in a vacuum with an
arbitrary angle *θ* between $\overrightarrow{E}$ and
$\overrightarrow{B}$ is treated in Appendix G. In the special case that
the electric and magnetic fields are transverse to one another we can
see that in a direction transverse to both the electric and magnetic
fields, the force is 0 if ![](media/image60.emf). A particle can thus
move at a constant drift velocity, ![](media/image61.emf), perpendicular
to crossed electric and magnetic fields in vacuum. This is the principle
of electrostatic separation where we prepare a beam of particles of
fixed momentum by bending in a magnet, focussing parallel, and then
collimating (see Section 7). We then put the beam through a "separator"
which has crossed electric and magnetic fields. Only particles of the
right mass, and therefore the right velocity, ![](media/image62.emf),
will be undeflected whereas the unwanted particles will be physically
deflected and subsequently can be removed, again by collimation. This
technique is a form of "destructive" particle identification.

![](media/image63.emf)![](media/image64.wmf){width="1.0833333333333333in"
height="0.9256944444444445in"}

Fig. 8.8: Photo of an electrostatic beam separator. The plates are
charged to establish the electric field. The vacuum tank surrounding the
plates is made of soft iron and acts as the pole tips of the magnet
supplying the magnetic field. The line of sight is the beam axis,
![](media/image65.emf). (From Ref. 10, with permission.)

A photo of a working
"$\overrightarrow{E} \times \overrightarrow{B}\text{}\text{separator}$"
is shown in Fig. 8.8. It is hard to maintain a voltage greater than a
megaVolt in air due to corona and other discharge effects. Typical
operating parameters are E \~ 1 MV/cm and β = 0.5. To achieve that drift
velocity a magnetic field of 7 kG is required.

In the case where there are collisions, in a medium such as a gas, they
can be approximated as an effective frictional force that is
characterized by a mean time between collisions ![](media/image66.emf).
The non-relativistic relationship between the frictional force
![](media/image67.wmf) and momentum ![](media/image68.wmf) follows
because force is the time rate change of momentum.

$\begin{array}{r}
{\overrightarrow{F}}_{F}\  = \ \frac{\overrightarrow{p}}{\tau},\text{ \{}\overrightarrow{F}\ \text{} = \ \text{}q(\overrightarrow{E} + \overrightarrow{\beta} \times \overrightarrow{B})\  + \ \text{mc}\frac{\overrightarrow{\beta}}{\tau}\ \text{} = \ \text{}m\overrightarrow{a}
\end{array}\overrightarrow{a}\  = \ 0\text{}\ \text{when}\ \overrightarrow{v}\text{}\  = \text{} - \ \frac{q\tau}{m}\ (\overrightarrow{E} + \overrightarrow{\beta} \times \overrightarrow{B})$
(8.16)

We can solve Eq. 8.16 for the situation where the force is zero and we
have a uniform drift velocity. The drift velocity ![](media/image69.emf)
is now a function of the charge, mass, fields and
![](media/image70.emf).

$\begin{array}{r}
\left\langle {\overrightarrow{v}}_{d}^{'} \right\rangle \  = \ \text{}\frac{\frac{- e\tau}{m}}{\lbrack 1 + (\omega_{c}\tau)^{2}\rbrack}\ \text{}\lbrack\overrightarrow{E} + (\omega_{c}\tau)\hat{B} \times \overrightarrow{E} + (\omega_{c}\tau)^{2}(\hat{B} \cdot \overrightarrow{E})\hat{B}\rbrack \\
\omega_{c}\text{} = \ \text{}\frac{\text{eB}}{m}
\end{array}$ (8.17)

The drift velocity has, in general, components along the
$\overrightarrow{E}$, $\overrightarrow{B}$, and
$\overrightarrow{B}x\overrightarrow{E}$ directions.

The vectors appropriate to drifting in perpendicular electric and
magnetic fields are shown in Fig. 8.9. The "Lorentz angle", *φ~L~*, is
defined to be the angle which the drift velocity
$\left\langle {\overrightarrow{v}}_{d}^{'} \right\rangle$ makes with the
applied electric field. In the vacuum case ![](media/image71.emf) is 90o
as it is perpendicular to both
$\overrightarrow{E}\text{}\text{and}\text{ \{}\overrightarrow{B}$.
In general the drift velocity for perpendicular $\overrightarrow{E}$ and
$\overrightarrow{B}$ fields is in the $\overrightarrow{E}$ and
$(\overrightarrow{B}x\overrightarrow{E})$ plane.

$\begin{array}{r}
\overrightarrow{E}\bot \overrightarrow{B} \\
\text{tan}\varphi_{L} = \omega_{c}\tau
\end{array}$ (8.18)

The quantity ![](media/image72.emf) is a measure of the rotation angle
of the momentum in the B field between collisions. Therefore if
![](media/image73.emf) we expect small collisional effects. If
$t \rightarrow ¥$, the angle ![](media/image74.emf) becomes 90o and we
recover the case of the vacuum drift velocity. For a mean free path
between collisions of 10 μm (gas) with a 10 cm/μsec drift velocity
(argon-methane), τ is \~ 1psec and ωτ is 0.17 or φ~L~ is \~ 11 degrees.
Note that there are serious practical consequences. Wire chambers
operated in magnetic fields often have their electrode structure
modified to follow the Lorentz angle.

![](media/image75.wmf)

Fig. 8.9: Vectors for drift in perpendicular electric and magnetic
fields. The Lorentz angle φL is in the ![](media/image76.emf) plane.

In general, the drift velocity ![](media/image77.emf) is less than the
velocity ![](media/image78.emf) appropriate to the
$\left| \overrightarrow{B} \right| = 0$ case because of the
additional helical path length.
$\left( \left\langle {\overrightarrow{v}}_{d} \right\rangle = e\overrightarrow{E}\frac{\tau}{m},\ \text{Eq}\text{.}\ 8\text{.}2 \right)$.

$\begin{array}{r}
\left\langle {\overrightarrow{v}}_{d}^{'} \right\rangle\  = \ \frac{\lbrack{\overrightarrow{v}}_{d}\  + \ \omega_{c}\tau \ \left( \hat{B}\  \times \ {\overrightarrow{v}}_{d} \right)\rbrack}{\lbrack 1\  + \ \left( \omega_{c}\tau \right)^{2}\rbrack} \\
\left\langle v_{d}^{'} \right\rangle \  = \ \frac{\left\langle v_{d} \right\rangle}{\sqrt{1\  + \ \left( \omega_{c}\tau \right)^{2}}}
\end{array}$ (8.19)

The effective diffusion coefficient in the direction perpendicular to B
is lowered by a factor ![](media/image79.emf) due to the tight helical
orbits. This effect is sometimes used to improve the position resolution
for long drift distances where diffusion errors dominate.

## 8.6 Wire Chamber Electrostatics

We now consider a simplified description of a "wire chamber". We
consider only the simplest case of a cylindrical tube with outer radius
b coaxial with a small diameter wire of radius a, Fig 8.10. A particle
is localized to a position resolution \~*b* if ionization is detected on
a given wire assuming a linear array of such tubes as is shown in Fig.
8.10. This is a simple problem in two dimensional electrostatics. We use
Gauss\' law (CGS units). In terms of λ, the charge per unit length on
the wire, we find the electric field and the potential V.

![](media/image80.wmf){width="2.5972222222222223in" height="2.125in"}

Fig. 8.10: Cylindrical proportional tube of outer radius b at voltage
V~o~ and inner (wire) of\
radius a at voltage zero.

$\begin{array}{r}
E\ \  = \ 2\frac{\lambda}{r} \\
V_{o}\  = \ 2\lambda \text{ln}(\frac{b}{a}),\ V(r)\  = \ V_{o}\ \text{ln}(\frac{r}{a})\frac{}{}\text{ln}(\frac{b}{a}) \\
E(r)\  = \ \frac{(\frac{V_{o}}{r})}{\lbrack\text{ln}\text{}(\frac{b}{a})\rbrack}
\end{array}$ (8.20)

The passage of a charged particle causes ionization. The electrons then
drift towards the anode. Near the wire the high electric fields cause a
multiplication of the electrons by collisions since at a small radius
the energy gain can exceed the ionization potential, I, of the gas (see
Section 1). The newly created electrons in turn will be accelerated and
cause ionization. This is a runway process similar to what we already
discussed in the operation of a photomultiplier tube. Note that the "gas
gain" follows from the change in the number of electrons dN(r) at a
given position *r* in the multiplication process. This change depends on
the number of electrons present N(r) and the multiplication factor,
![](media/image81.emf) per unit length. The parameter
![](media/image82.emf) is the inverse of the distance over which a
multiplication occurs. It is called the first Townsend coefficient and
is identifiable as the inverse mean free path for ionization,
![](media/image83.emf).

$\begin{array}{r}
\text{dN}(r) = N(r)\alpha\text{dr} \\
N(r)\text{} = N_{o}e^{\alpha r} \\
\left\langle I \right\rangle_{A}\text{}\text{\textasciitilde}26\text{}\text{eV}
\end{array}$ (8.21)

The amplification or "gas gain" is $e^{\alpha r}$. A typical gas gain
might be 10^5^. Above that level we approach the "Geiger region" which
is a true runaway process where the whole tube is discharged. We
concentrate in what follows on the "proportional region" where the
output signal is \~ proportional to the input signal. The transition
region between these operational modes occurs experimentally for
![](media/image84.emf) \~ 20 pairs or for gas gain \~ 10^8^.

## 8.7 Pulse Formation in a Wire Chamber

The time development of the signal is different from that of the "unity
gain" ionization chamber we discussed previously. The electrons from the
primary ionization are multiplied by the high fields near the wire, E(r)
\~ 1/r, and are collected on the anode. The ions remain, "sheathing" the
wire and moving slowly toward the cathode. The motion of all the charges
in the applied field causes a change of system energy and a resultant
capacitively induced signal, dV.

Suppose the multiplication takes place at N wire radii, i.e. r = Na. The
voltage signal on the anode is due to the motion of the electrons,
![](media/image85.emf), and the ions, ![](media/image86.emf). If q~s~ is
the source charge and C is the capacity, then dU = C V~o~ dV = q~s~Edr
(see Eq. 8.6). The induced voltage due to electron and ion motion is;

$\begin{array}{r}
\text{}\text{dV}\text{}\  = \ \frac{q_{s}}{\text{CV}_{o}}\ E(r)\text{dr} \\
\text{}V^{-}\text{}\  = \ \int_{}^{}{\text{dV},\text{}\ \text{}V^{+}} = \ \int_{}^{}{( - \text{dV})} \\
\text{}V^{-}\text{}\  = \ \frac{q_{s}}{C}\ \frac{\text{ln}(N)}{\text{ln}(\frac{b}{a})},\ \text{}V^{+} = \text{}\ \frac{q_{s}}{C}\ \frac{\text{ln}(\frac{b}{\text{Na}})}{\text{ln}(\frac{b}{a})}\ \text{}\text{>>}\ V^{-}
\end{array}$ (8.22)

Clearly, in the case of the wire chamber the signal due to the ions,
V^+^, dominates, basically because they move all the way to the cathode.
Therefore, the ion motion is responsible for the majority of the anode
signal. The electron contribution will be ignored from now on.

The signal can be found using the technique already applied to unity
gain devices. Assuming a constant ion mobility, ![](media/image87.emf),
dr = ![](media/image88.emf)E(r)dt. Thus, dr =
![](media/image89.emf)$\lbrack 2\lambda\rbrack$dt/r (t). Assume that the
ions all appear at r = a at t = 0 due to the final multiplication.

$\begin{array}{r}
\int_{}^{}\text{rdr}\  = \ \mu\lbrack 2\lambda\rbrack \int_{}^{}\text{dt} \\
r\  = \ a\sqrt{1\  + \ 4\text{μλ}\frac{t}{a^{2}}} \\
\text{} \equiv \ a\sqrt{1\  + \ \frac{t}{\tau_{o}}}\text{},\text{}\tau_{o}\  = \ \text{}\frac{a^{2}}{4}\text{μλ}
\end{array}$ (8.23)

The fact that the fields goes as 1/r implies that the ion position r
goes as \~ ![](media/image90.emf). Using the general equation, Eq. 8.7,
we can then find the current induced on the wire.

$\begin{array}{r}
I(t) = q_{s}\frac{{mE}^{2}}{V_{o}} \\
\frac{q_{s}m\left( 2\frac{\lambda}{r^{2}} \right)}{\text{ln}\left( \frac{b}{a} \right)},(\text{Eq}\text{.}8\text{.}\text{20})
\end{array}$ (8.24)

Substituting in the expression for *r*(*t*) we find I(t) in terms of
time t.

$\begin{array}{r}
I(t) = \frac{q_{s}\left( 2\frac{\text{λμ}}{a^{2}} \right)}{\text{ln}\left( \frac{b}{a} \right)\left( 1 + \frac{t}{\tau_{o}} \right)} \\
I(t) = \left\lbrack \frac{\frac{q_{s}}{2}\tau_{o}}{\text{ln}\left( \frac{b}{a} \right)} \right\rbrack\frac{\text{}}{\left( 1 + \frac{t}{\tau_{o}} \right)}\  = \ I(0\frac{)}{\left( 1\  + \ \frac{t}{\tau_{o}} \right)}
\end{array}$ (8.25)

The key parameter $t_{o}$ is the characteristic time for the pulse to
form. It can be thought of as roughly the time it takes for the ions to
move away one wire radius, a, towards the cathode under the influence of
the electric field, *E*(*a*) = 2![](media/image91.emf)/*a*, which exists
near the surface of the anode wire.

$\begin{array}{r}
\mu\text{}\  = \ \frac{\left\langle v_{d} \right\rangle}{E}\ \text{\textasciitilde}\ \frac{a}{(}\tau_{o}E(a)) \\
\tau_{o}\text{}\ \text{\textasciitilde}\ \frac{a}{\mu E}(a)\  = \ \frac{a^{2}}{2}\text{μλ}\text{}
\end{array}$ (8.26)

We define [C]{.underline} to be the capacity per unit length of the
anode while the capacity is C and the capacity/unit length =
[C]{.underline}. The relationship of voltage, charge and capacity is V =
Q/C = *λ* /[C]{.underline}. From Eq. 8.20 we find [C]{.underline} =
1/\[2 ln(b/a)\] so that [C]{.underline} is dimensionless in CGS units.
It can be thought of as *ε* in *ε~o~* units (Table 1.1). Typically for
wire chambers, [C]{.underline} \~ (0.01 - 0.1) pF/cm,
[C]{.underline}V~o~ \~ *λ* \~ 5 x 10^-10^ Coulomb/cm = 500 pC/cm. Recall
that (Table 1.1), *ε~o~* = 8.85 pF/m in MKS units.

The charge Q is derived by integrating Eq. 8.25. The signal voltage
follows using V = Q/C.

$V(t) = \lambda\left\lbrack \text{ln}\left( 1 + \frac{t}{\tau_{o}} \right) \right\rbrack$
(8.27)

As a numerical example consider a 1 cm radius tube strung with a 20
micron radius wire, b = 1 cm, a = 20 *μ*m. The field is about 160 kV/cm
at the surface of the wire, *λ* = 180 pC/cm, and the pulse formation
time is a few nsec for argon ions. This is the kind of speed that we
observe in the rise time of pulses from proportional chambers. (See
photo Section 9.)

$\begin{array}{r}
E(a)\  = \ \text{160}\frac{\text{kV}}{\text{cm}} \\
\mu_{A}\ \text{\textasciitilde}\ 1\text{.}5\frac{\text{cm}^{2}}{(}V \cdot \text{sec}) \\
\tau_{o}\  = \ 8\text{.}3\ n\text{sec}
\end{array}$ (8.28)

## 8.8 Mechanical Considerations

A photograph of a real proportional wire chamber, PWC, not operated as a
simple linear array of coaxial tubes but as a planar series of detecting
wires is shown in Fig. 8.11. This particular chamber has a 1 mm spacing
between the wires and the wires are about 20 cm long. The cathode is a
plane of aluminum foil. This chamber has been used to measure the
position of particles to an accuracy \~$\pm$`<!-- -->`{=html}0.5 mm in
prepared beams at Fermilab. A photograph of a large area drift chamber
is shown in Fig. 8.12. In this case there is a large spacing, about 2
centimeters, between the anode detection wires. If we want precise
position localization with this "drift chamber" we need to accurately
measure the drift time (Fig. 8.7).

![](media/image92.wmf)

Fig. 8.11: Photo of a PWC with 1 mm wire spacing and without wire
supports. The connectors for the readout electronics are evident.
(Photo - Fermilab.)

![](media/image93.png){width="5.027083333333334in"
height="6.059722222222222in"}

Fig. 8.12: Photograph of a large area drift chamber. Note the large
spacing between anode detection wires. (Photo - Fermilab.)

Constructing proportional wire chambers with parallel wires, we need to
keep track of the fact that the wires are charged to a typical value of
2 x 10-10 C/cm. The wires all being charged the same, will repel. There
will be a deflection unless the wires are maintained under a certain
tension, T. The tension has to be greater than the repulsive force
between the wires if the PWC is to be "stable". Crudely, for a wire
spacing, d, wire length L and a charge per unit length λ the stability
condition is;

$T³\left( \frac{L\lambda}{d} \right)^{2}$ (8.29)

For example with a 20![](media/image94.emf) diameter wire, a PWC built
with a 1 mm spacing, d, having a typical charge per unit length
$\lambda$ of 2 x 10-10 C/cm has a maximum unsupported wire length L of
about 13 cm for a wire tension T yield strength of about 50 grams. The
natural limit to T for a given wire diameter comes at or below
elongation or breaking tension, T~max~. Clearly there is a competition
between high spatial resolution, d small, rapid pulse formation time
$\tau_{o}$, (large ![](media/image95.emf) - small a), and the ability to
make a stable large area chamber, (small ![](media/image96.emf) - large
d - large a). Typically, the existence of these incompatible
requirements is evaded by supporting the wires mechanically in some
fashion at a separation between supports which is less than the maximum
length allowed for stability. In this way we can build a large area,
finely spaced, high speed, proportional wire chamber.

A schematic of the spatial pattern for typical wire chamber instability
is shown in Fig. 8.13. For tension too small, the wires repel one
another, and the uniform plane of wires which is desired is not a stable
configuration. The distorted plane is a configuration with reduced
energy (charges farther apart) so that unless sufficiently restrained,
the system will adopt this configuration. These wires in motion are
unstable, and such a chamber cannot be operated reliably.

![](media/image97.wmf)

Fig. 8.13: Schematic of a wire chamber wire array for both an
unperturbed and an unstable condition.

A typical fast amplifier configuration used to process the signal pulse
which forms on the anode is shown in Fig. 8.14. The detector current
source signal i~s~ appears as a voltage output V~o~ proportional to the
source current i~s~, V~0~ = i~s~ R. Some algebra for this and other
operational amplifier circuits is given in Appendix I. Front end
electronics is relegated to this Appendix, and references are given in
Section 13 while front end noise is discussed in Section 9.

![](media/image98.wmf){width="4.722222222222222in"
height="2.388888888888889in"}

Fig. 8.14: Schematic of a "transimpedence amplifier". The voltage output
is proportional to the source current.

## 8.9 The Induced Cathode Signal

So far we have not discussed the cathode signal. The almost
instantaneous appearance of charge near the anode wire capacitively
induce charge on the cathode. Therefore, we have a second detectable
signal. Note that we are free to cut the cathode into whatever shape we
wish, as long as all parts of it are maintained at the same voltage.
This freedom of design has made cathode readout popular.

The geometric situation is shown in Fig. 8.15 where a source charge,
q~s~, appears effectively on the anode wire located at a perpendicular
height a above the cathode pad. The sudden appearance of the source
charge induces a surface charge, ![](media/image99.emf), on the cathode
pad. We can evaluate σ at an arbitrary location labeled by radius vector
r by using the "method of images", placing an image charge -qs at the
virtual symmetric point below the pad and evaluating the resultant
dipole-like 1/r^3^ behavior of ![](media/image100.emf). The image charge
technique is a method often employed to satisfy the electrostatic
boundary condition that there be no electric field parallel to the
conductor.

$\begin{array}{r}
\text{}E_{\text{11}} = 0 \\
E_{T}(r)\text{\textasciitilde}(\frac{q_{s}}{r^{2}})(2\frac{a}{r})\text{\textasciitilde}\sigma(r)
\end{array}$ (8.30)

The resulting field is perpendicular to the pad surface by construction.
By Gauss' law the perpendicular field (CGS) is also the induced surface
charge density, ![](media/image101.emf). The surface charge density, if
sampled on several different cathode electrodes, allows us to infer the
centroid location

![](media/image102.wmf){width="2.6944444444444446in"
height="3.1527777777777777in"} a

![](media/image103.wmf){width="2.763888888888889in"
height="2.6805555555555554in"} b

Fig. 8.15: a) Geometry for deriving the induced surface charge on a pad,
σ, for source charge qs at height = a above the pad. b) Geometry for
infinitely long strip electrode of width d.

of the instantaneous charge which has appeared on the anode. This
technique offers a rather precise second coordinate measurement if pulse
amplitudes are well measured. For example anode wires may have
associated cathode strips at right angles. Wire pulses then localize in
1 coordinate, while strip centroids yield the second orthogonal
coordinate.

As an example, we can use the geometry shown in Fig. 8.15 and integrate
over a strip which is infinite in the z direction and is of width d in
the y direction. We find that the charge induced on the pad, q~p~, is
simply related to the source charge, q~s~, and the angles of the wire
subtended by the pad boundaries. In the limit that we come very close to
the pad, a\<\<d, the ratio of q~p~ to q~s~ approaches 1/2, as expected
by geometry, since half the charge is induced below the wire, half
above. In general, the induced charge is approximately proportional to
the angle subtended by the pad \~ (*θ~2~* - *θ* *~1~* ).

$\begin{array}{r}
\text{}q_{p}\text{} = \int_{\frac{- d}{2}}^{\frac{d}{2}}\int_{\infty}^{\infty}(2q_{s}\frac{a}{r^{3}})\text{}\text{dy}\text{dz} \\
\text{}\ \text{} = \text{}\ q_{s}(\theta_{1} - \theta_{2}\frac{)}{\pi},\text{}\  \pm \text{}\ \frac{d}{2}a\ \text{} = \ \text{tan}(\theta_{1,2}) \\
\frac{q_{p}}{q_{s}}\  = \ \ \frac{\text{tan}^{- 1}(\frac{d}{2}a)}{\pi}\  \rightarrow \ \frac{1}{2}
\end{array}$ (8.31)

An example of an application of "pads" is to use the induced charge on
two different pad electrodes to infer the location of the charge along
the wire. We use the drift time to infer the location transverse to the
wire. We can therefore achieve full three dimensional readout in a
single plane. A concrete example of such a detector is shown in Fig.
8.16 where the inner, Q~B~, and outer pad, Q~A~, charges are both
measured. The charge ratio is a measure of the position along the wire.
Data taken with an external high resolution device yields a precise
value for the position along the wire. When this is compared to the
charge ratio, as in Fig. 8.16, we see that we can get quite good spatial
resolution by the use of induced pad charges. As a rule of thumb the
technique is noise limited to a resolution \~1% of the geometric feature
size of the pads. That limit is evident in the line width visible in
Fig. 8.16.

Other uses of the induced charge exploit the complete freedom to
configure the cathodes and their geometry. For example, in gas
calorimetry we use the cathode signals so as to easily make projective
"towers" which match the outgoing paths of particles from a collision
(see Fig. I.1). Because layers of calorimetric readout (see Section 11 -
13) occur at different depths, each tower layer has a different size,
which is easily achieved by cutting a different pad on the readout
circuit board.

![](media/image104.png){width="3.7597222222222224in"
height="4.6409722222222225in"}

Fig. 8.16: Relationship of the pad charge ratio to the position along
the wire for the structure shown in Fig. 8.6. (From Ref. 11, with
permission.)

## References

Section 8: Drift and Diffusion in Materials, Wire Chambers

\[1\] Drift and Diffusion of Electrons in Gases: A Compilation, A.
Peisert, F. Sauli, CERN 84-08 (1984).

\[2\] Principles of Operation of Multiwire Proportional and Drift
Chambers, F. Sauli, CERN 77-09 (1977).

\[3\] Signal, Noise and Resolution in Position-Sensitive Detectors, V.
Radeka, IEEE Transactions on Nuclear Science, [21]{.underline} 51
(1974).

\[4\] [Spark, Streamer, Proportional and Drift Chambers]{.underline}, P.
Rice-Evans, RicheLieu (1974).

\[5\] High Resolution Electronic Particle Detectors, G. Charpak, F.
Sauli, Annual Review of Nuclear and Particle Science, [34]{.underline}
285 (1984).

\[6\] [Basic Data on Plasma Physics]{.underline}, S.C. Brown, Wiley
(1959).

\[7\] [Atomic and Molecular Radiation Physics]{.underline}, L.G.
Christophorou, Wiley (1971).

\[8\] [Electrons in Gases]{.underline}, J. Townsend, Hutchinson (1947).

\[9\] [Experimental Techniques in High Energy Physics]{.underline}, T.
Ferbel, Addison-Wesley Publishing Co., Inc. (1987).

\[10\] L. Yuan and C.W. Wu, [Methods of Experimental
Physics]{.underline}.

\[11\] S. Abachi et al., The D0 Detector, Nucl. Inst. and Meth.
[A338]{.underline} 185 (1994).

Table of Contents

8\. Drift and Diffusion in Materials, Wire Chambers
[1](#drift-and-diffusion-in-materials-wire-chambers)

8.1 Thermal and Drift Velocity [1](#thermal-and-drift-velocity)

8.2 Mobility [3](#mobility)

8.3 Pulse Formation in "Unity Gain" Detectors
[6](#pulse-formation-in-unity-gain-detectors)

8.4 Diffusion and the Diffusion Limit
[9](#diffusion-and-the-diffusion-limit)

8.5 Motion in $\overrightarrow{E}$ and $\overrightarrow{B}$ Fields, with
and without Collisions
[12](#motion-in-overrightarrowe-and-overrightarrowb-fields-with-and-without-collisions)

8.6 Wire Chamber Electrostatics [15](#wire-chamber-electrostatics)

8.7 Pulse Formation in a Wire Chamber
[17](#pulse-formation-in-a-wire-chamber)

8.8 Mechanical Considerations [19](#mechanical-considerations)

8.9 The Induced Cathode Signal [22](#the-induced-cathode-signal)

References [26](#references)