The Outline
Drift of electrons and ions in gases, with and without the presence of electric and
magnetic fields
- why this is important ?
- drift of electrons and ions in electric fields
- diffusion of ions in a field free gas
- electron capture
-drift of electrons of electric and magnetic fields, diffusion
of electrons.
Anna Lipniacka , Detector techniques
L5
DELPHI construction, insertion of the Time Projection Chamber
Muon
chambers
Hadronic
calorimeter
Electro
magnetic
cal.
Tracking detector,
Time Projection
Chamber
Anna Lipniacka , Detector techniques
L5
Drift and diffusion- why this is important?
Important in gaseous detectors working in one of two modes :
- ionization detectors. Primary ionization from passing particle has to be drifted
somehow to “sense device” where it has to be collected and measured. We
would like to know what will be the spatial size of the ionization cloud, when will
it arrive, will it arrive at all ? Will it not be “recaptured” by the surrounding gas?
These detectors work often in magnetic fields
(and electric fields to collect the charge) we would like to know how magnetic
field modifies the movement of the charge. We would like to reconstruct the energy
loss of the particle from ionization measurement. If ionization is lost somehow
on our way to detection we need to correct for that.
- detectors working in proportional mode. These are proportional and drift chambers.
With larger field present electrons drift until they become ionizing. Primary and
secondary ionization has to be collected. The pulse shape is of interest, also the
drift time and diffusion of the pulse shape with the drift. From the drift time we would
like to reconstruct- for drift detectors – the position of and passing particle. The pulse
shape diffusion is important for the construction of the electronics.
Anna Lipniacka , Detector techniques
L5
TPC pads
Electrons still drift to
wires
But charge is induced
on pads next to the
wires
Provide extra
dimension
Both are readout.
Might be 150 wires crossed, less pads.
4
Proportional and drift detectors
There are typically wire chambers filled with gas.
ionize gas and electrons are attracted by anode
wires. The electric field around the wire (1/R) is high
and electrons gain enough velocity to ionize more gas.
This gives signal on anode wires- still proportional to
the number of initial ionization electrons- "proportional
counters " or MWPC. The space resolution, of the order
of a fraction of a mm typically depends on the density of
anode wires. Use drift chambers to make it cheaper
- - --8
+
-2
0
Measure drift time
of electrons to anode wire
to reconstruct particle passage
The price to pay is left-right ambiguity,
and longer drift time of electrons-> slower detectors
Field shaping wires
Tracking detectors contain as little material as possible, to not to "disturb" particles
Anna Lipniacka , Detector techniques
L5
What happens with electrons and ions in gases?
electrons and ions will drift in the electric field: drift velocity and mobility, mean free
path
electrons and ions will diffuse due to thermal energy in a field-free gas: diffusion
coefficient and mean free path
ions and electrons can be neutralized before detected: ion recombination and electron
capture.
magnetic field will modify the drift of electrons in electric field: Lorentz angle,
modification of the drift velocity
diffusion of electrons in electric and magnetic fields: Electrons thermal velocities
will spiral around B field, resulting in smaller diffusion.
Exact pattern of what happens might depend on a gas in question. Will be discussed
for most popular gas mixtures used in gaseous detectors. How do you think this
gas mixtures were chosen ?
Anna Lipniacka , Detector techniques
L5
Drift of ions and electrons in electric field
+
e
vD ,vD
Electric field will exercise a force on an ion/electron. It will
accelerate until it reaches “equilibrium velocity” at which interactions
with other particles of a gas will stop it as much as E-field accelerates
it. This velocity is called drift velocity . It will depend on properE
ties of the drifting particle, of the gas, on the electric field.
For ions we use experimental parametrization:
v
+
D
=
+
p0
E
p
+
where p-pressure, p_0 = 760 Torr and
is called ion mobility and depends
on the gas in question. Measured in cm**2/(V s)
Example : He ions in He mobility 10.2 cm**2/(V s) . For E=1KV/cm and p=p_0
we get drift velocity 10.2 *10**3 cm/s = 10.2 cm/msec ~0.1 mm/microsecond
For mixture of different n gases the mobility will be combined . Mobility of ions
i in the mixture of 1,,,n gases: c_k are relative volume concentrations of gases.
1
+
i
=
n
k =1
ck
+
ik
If several types of ions are present the ones with bigger ionisation potentials will steal
electrons from atoms with lower ionization potentials after 10**2-10**3 collisions.
Anna Lipniacka , Detector techniques
L5
Drift of electrons
“ Stopping power” of the medium can be expressed in terms of mean free path of a
particle interacting with the medium:
1
=
n
where as usual n is the concentration of interaction centers and sigma is the interaction
x-section. For electrons of kinetic energies around 1 eV mean free path will be much
longer than for ions, because atoms are nearly transparent due to long wavelength of
electrons.
We will now try to calculate drift velocity in electric field E of the “swarm” of electrons
which have thermal velocity “u” and are in a medium in which they have mean free
path
e
E In times between collisions (t) electric field E will displace thermal
motion of electrons leading to an average displacement of the
electron cloud :
< z >~ <
< t 2 >~
z
u
2
e
2
Anna Lipniacka , Detector techniques
1 2
qE 2
a t >~
<t >
2
m
thus
< z > qE e
~
vd=
< t>
m u
L5
Drift of electrons cont.
On other hand drift velocity adjusts itself in such a way that energy lost in collisions
is equal to the energy gained in the electric field
qE < z >~ qE v d < t >= qE v d
mu 2
= 2
e
u
energy gained
energy lost in a collision, assuming that an arbitrary
fraction delta is lost in a single collision
Comparing energy lost and energy gained we get another formula for drift velocity, and
we can eliminate thermal velocity u and obtain an expression for drift velocity
The dependence of drift velocity on the electric
1 2
field will depend on the fact if the fraction
E
of energy lost depends on the electron energy and
v d~
q e
if the mean free path depends on electron energy2 m
this will depend on the fact if collision x-section
depends on the electron energy.
Examples: Argon -> mean free path first increases than decreases with the energy.
The result is that the drift velocity first increases quickly with E field
then it stabilizes
In CH_4 for energies above excitation energy the fractional energy
loss decreases like 1/energy thus delta*energy = const and v_d~1/E
/
Anna Lipniacka , Detector techniques
L5
Some numbers
Typical units mm/microsecond or cm/microsecond, popular “drift” gases
like argon/methane mixtures.
Orders of magnitude :
= 10 10
17
15
cm
2
for noble gases,
might be much
bigger for other gases
for STP we have n= 2.69*10**9 molecules/cm**3
thus mean free path of the electron is
e =
1
=
10 7 10 3 cm
n
for noble gases, while might be orders of magnitude shorter for other gases.
For ions mean free path is of the order of 10**(-5) cm, and around 6 times
larger for electrons in non-noble gases
mean thermal energy at room temperature 3/2kT is around 0.035 eV.
2
mv
3
t
kT =
2
2
thus thermal velocities of ions will be of the order
10**5 cm/s=1 mm/microsecond
and thermal velocities of electrons will be
10**7 cm/s=100 mm/microsecond
Drift velocities (experimental) for electrons are of the order 10-100 mm/microsecond.
In our calculations we implicitely assumed that drift velocities are substantially smaller
than thermal velocities
Anna Lipniacka , Detector techniques
L5
Diffusion of electrons and ions
Ions and electrons will have thermal velocities with which they will disperse from
the point where they were created:
mu
3
< >T = kT =
2
2
the thermal energy distribution follows Boltzmann
statistics ( to a large extend)
2
t
P
t=0
~ exp
kT
In one coordinate (x ) the
density dN/N distribution of
electrons
or ions after time t will
probably be gaussian:
t>0
x
dN
x
=
exp
N
4 Dt
2
dx
4 D t
x
Anna Lipniacka , Detector techniques
=
2Dt
D- diffusion coefficient
L5
Diffusion coefficient
dN
x2
= exp
N
4 Dt
dx
4 D t
x=
the diffusion coefficient is
measured in cm**2/s
2Dt
Since the mean diffusion will be grow with the thermal velocity :
x~
ut~
3kT
m
the diffusion coefficient D will decrease for heavier particles- will be much bigger
for electrons than for ions for example.
Example numbers :
Hydrogen :
5
u t = 1.8 × 10 cm / s ; D
+
=
0.34 cm / s ; 2
;
cm 2
;
= 2.2
Vs
2
+
cm
= 13
Vs
=
1.8 × 10
5
1.0× 10
5
cm ;
x
t = 1s
0.7 cm
cm ;
x
t = 1s
0.35 cm
Oxygen :
5
u t = 0.46 × 10 cm / s ; D
+
=
0.06 cm / s ; 2
Anna Lipniacka , Detector techniques
+
=
L5
Electron capture and ion recombination
Ion recombination: positive ions can recombine with electrons or negative ions. The
recombination probability in a “single encounter” with an
electron or negative ion will depend on the type of ion, its ionization
potential for example. We describe it in terms of a recombination
coefficient
+
dn
dt
=
n+ n-
n -> concentrations of positive and negative particles. Recombination coefficient
is measured in cm**3/s . “Big” values of the coefficient are for “reactive gasses”
for example oxygen and C02 the recombination coefficient is 10**{-6} cm**3/s
for recombination with negative ions and 10**{-7} cm**3/s for recombination with
electrons.
Electron capture : electronegative gasses: 02, H20, NH3 etc can accumulate
electrons. For these gases probability to capture an electron in a single collision is
~ 10**{-5}, thus on average an electron will be captured after 10**{5} collisions.
How long it will take (t_a):
time between collisions:
Exercise: calculate it for 02.
ts=
s
ut
or n s =
ut
s
ta =
s
ts
=
pa ut pa
The result is of the order of 200 ns
Anna Lipniacka , Detector techniques
L5
Drift of electrons in electric and magnetic
fields
Coulomb force : qE, Lorentz force
q v×B
Simple case, magnetic field alone, or electric and magnetic field parallel :
v_t
2
v t q B
vt
q v t B = m = =
R
R
m
For electrons
= 17.6 MHz / Gauss
B
R
B
transversal velocity
unchanged.
The velocity parallel to magnetic field will not be changed if there is no electric
field, while if there is an electric field parallel to B the velocity parallel to B will
eventually reach the drift velocity characteristic for electrons in a given gas
The electron will spiral around the direction of magnetic, electric
field with frequency above, while moving along
the electric field with drift velocity
B, E
Anna Lipniacka , Detector techniques
L5
electric and magnetic field
The general equation of motion for electron in a gas in the presence of two fields
is the following:
v×B
m v= q E
mA t
stopping force due
to collision with atoms
This is Langevin equation. The stopping force will depend on the drift velocity and
time between collisions. The drift velocity will adjust itself in such a way that the
stopping force cancels the force due to EM fields- resulting acceleration will be 0.
Thus we can write :
0= q E vD × B
Solution:
m
vD
E
×
B
E
B
time between collisions
E
B B
vD =
2 2
1 B2
qB
; =
q
m
m
Anna Lipniacka , Detector techniques
2
2
L5
Perpendicular E and B
The solution for drift velocity is quite interesting for perpendicular electric and magnetic
fields:
E = E x ,0,0 , B = 0,0, B z
Ex
vx=
1 2 2
E x v y=
1 2 2
vz=0
Ex
2
2
v D= v x v y= 1 2 2
vy
tan v D , E =
=
vx
Lorentz angle
Anna Lipniacka , Detector techniques
L5
Diffusion in electric and magnetic fields
Magnetic fields will decrease diffusion perpendicular to field direction
by “curling down” thermal velocities : For B along z we have :
DZ = D
; D X = DY =
D
1
2
2
Electric fields can increase effective diffusion along E introducing the anisotropy
D_L/D
In practice we would like to decrease
diffusion
perpendicular to E-> this results
in choice of B parallel to E for drift
detectors,
whenever possible
E
B
Anna Lipniacka , Detector techniques
L5
Detecting hadrons
Charged hadrons loose energy by ionization as well and can be seen in tracking detectors.
But both charged and neutral hadrons (eg neutrons) have possibility of strong interactions in
matter. Typical cross-sections of strong interactions are of order of 10mbarn/nucleon.
We can describe mean free path λ of hadrons in matter in terms of scattering cross-section
Nn
on a nucleon σ_n
24
26 2
gram
~ ln(E)
5.63cm
7cm
E
few λ
=
0.5 10 nucleons / gram
10 mbarn
=
10
cm
200 g cm 2
N
Hadronic interaction lead to nuclear fragments
=
/
and to
hadronic cascade, which is usually contained in a few
λ 's . Cascade particles give signal in proportional
counters-the energy of the cascade is measured.
Cascade should be contained for a good measurement
200 cm of iron ~ 8 λ
Hadron calorimeter
Anna Lipniacka , Detector techniques
L5
Hadronic cascade and interaction length
Hadronic interaction length is typically much longer than radiation length for electrons
( and photons). This explains why the typical size of hadronic calorimeter is much bigger
than that of electromagnetic calorimeter.
The development of the cascade is much harder to predict than in case of EM interactions,
although Monte Carlo approaches with detailed calculations of nuclear processes make
a good progress (GEANT4). For detector planing extensive beam tests and
parametrized results of these are used. For example :
L 95
= 9.4 ln E
GeV
39
cm of Fe
The energy registered in calorimeter is usually incomplete due to several effects:
- escaping muons and neutrinos from pion decays
- nuclear collision lead to nuclear excitations, fissions and spallations. These
process result in low energy gamma rays, slow protons, nuclear fragments.
Each of them with quite different types of reactions and different “visibility”
in calorimeter.
- as a result typically the response of the same calorimeter to pions and electrons
of the same energy is ~ 30% lower for pions.
Anna Lipniacka , Detector techniques
L5
Calorimeter length
Calorimeter depth in iron needed to contain the shower in 99% and 95%
Data from two experiments and parametrisation.
Anna Lipniacka , Detector techniques
L5