Charged Particle Tracking and Momentum Resolution
Why do we need charged particle tracking in an experiment?
FDetermine the number of charged particle produced in a reaction.
FDetermine the identity of a charged particle (e.g. p, K, p ID using dE/dx).
FDetermine the momentum of a charged particle.
We measure the momentum of a charged particle by determining its trajectory in a
known magnetic field.
Simplest case: constant magnetic field and p^B trajectory is a circle with p=0.3Br (GeV/c, T, m)
We measure the trajectory of the charged particle by measuring its coordinates
(x, y, z or r, z, f, or r, q, f) at several points in space.
Simplest case: determine radius of circle with 3 points
We measure coordinates in space using one or more of the following devices:
Wire Chamber
Drift Chamber (or TPC)
Silicon detector
low spatial resolution (1-2 mm)
moderate spatial resolution (50-250mm)
high spatial resolution (5-20 mm)
Better momentum resolution better mass resolution better physics
Many particles of interest are observed via their decay products: Z0e-e+,D+K-p+ p+, K0p+ p+
By measuring the momentum of the decay products we measure the mass of the parent.
m  m1+m2  m2=(E1+E2)2-(p1+p2)2 = m12+ m22 +2[(m12+ p12 )1/2 (m22+ p22 )1/2 - p1p2 cosa]
For fixed a :  m2 / m 2   p / p
880.P20 Winter 2006
a
1
2
Richard Kass
1
Momentum and Position Measurement
(L/2, y2)
y

Trajectory of
charged particle
s=sagitta
x


(0, y1)
(L, y3)
z
Assume:we measure y at 3 equi-spaced measurements in (x, y) plane (z=0)
each y measurement has precision y
have a constant B field in z direction so p^=0.3Br
Note: The exact
The sagitta is given by:
expression for s is:
y1  y3 L2
L2
0.3BL2
s  y2 



2
8r 8 p^ /(0.3B)
8 p^
L2
sr r 
4
2
The error on the sagitta, s, due to measurement error is (using propagation of errors):
 s  3 / 2 y
Thus the momentum (^to B) resolution due to position measurement error is:
 p^  s
3 / 2 y
8 p^ 3 / 2 y
p^ y



 32.6 2 (m, GeV/c, T)
2
2
p^
s (0.3L B) /(8 p^ )
0.3L B
LB
880.P20 Winter 2006
Richard Kass
2
Momentum and Position Measurement
From previous page the momentum resolution due to measurement error is:
 p^  s
3 / 2 y
8 p^ 3 / 2 y
p^ y



 32.6 2 (m, GeV/c, T)
2
2
p^
s (0.3L B) /(8 p^ )
0.3L B
LB
Typical numbers for BaBar (or CDF) are: B=1.5T, L=0.8m, y=150mm
p
1.5  10 4 p
3
^
p^
 32.6
^
2
0.8 (1.5)
 5.1  10 p ^
Thus for a particle with transverse momentum (p^) = 1GeV/c:  p  0.5%
^
The above momentum resolution expression can be generalized for the case of n
position measurements, each with a different y. The expressions are worked out
in Gluckstern’s classic article, NIM, 24, P381, 1963. A popular formula is
for the case where we have n>>3 equally spaced points all with same y resolution:
 p^
720  y p^

(m, GeV/c, T)
2
p^
n  4 (0.3BL )
Note: The above expression tells us that the best way to improve this component
of momentum resolution is to increase the path length (L).
For the model with 3 points the factor is (96)1/2 Vs (720/(n+4))1/2 . For n=100 we get 9.8 Vs 2.6
880.P20 Winter 2006
Richard Kass
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More on Momentum Resolution
On the previous page we calculated the position measurement contribution to the
momentum resolution. This is only part of the story. We also have contributions
from multiple scattering (MS) and angular resolution.
Previously we saw that the momentum resolution contribution to MS was given by:
p
T
pT
p
T
pT


s
L 13.6  10 3
z L / Lr
p
4 3

with L  L / sin q , p ^  p sin q
0.3BL2 z /(8 p ^ )
rms
plane
sB
52.3  10
z, B
q
3
y
B LLr sin q
x
MS depends on the total path length (L) and momentum (p).
Bending in the magnetic field depends on p^(=psinq) and projected path length (L).
The MS contribution is independent of the position resolution contribution so the
combined resolution is the two added in quadrature:
  p^

 p^



2
 52.3  10 3 
 720  y p^ 

 
 
2 


 n  4 (0.3BL ) 
 B LLr sin q 
2
2
( m, GeV/c, T)
Technically speaking, the above is only the transverse momentum (p^) resolution.
We want an expression for the total momentum resolution!
880.P20 Winter 2006
Richard Kass
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Even More on Momentum Resolution
We can get an expression for the total momentum (p) resolution using:
p  p^ / sin q  p^ 1  cot 2 q
& treating qand p^ as independent variables. Using propagation of errors we find:
Often detectors measure the r-f
coordinate independently of
the z coordinate. In these cases
p^ and q are independent.
  p    p^ 
  cotq q 2
   
 p   p^ 
2
2
Putting it all together we have for the total momentum resolution:
 p

 p
2
 52.3  10 3 
 720  y p sin q 


  cot q q 2

  

2 
 B LL sin q 

r
 n  4 (0.3BL ) 


2
Position resolution
2
Multiple scattering
GeV/c, T, m, radians
Angular resolution
While the above expression is only approximate it illustrates many important features:
a) p^ resolution improves as B-1 and depends on p as L-2 or L-1/2.
b) For low momentum (0), MS will dominate the momentum resolution.
c) Improving the spatial resolution (y) only improves momentum resolution if the first term is dominate.
d) Angular resolution is not usually the most important term since qmin30-45o and q10-3 rad.
For more detailed information must do a Monte Carlo simulation (GEANT+detector).
Include: hit efficiencies, discrete scattering off of wires, non-gaussian tails, etc, etc….
880.P20 Winter 2006
Richard Kass
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Still More on Momentum Resolution
Let’s examine the momentum resolution equation for a BaBar/CDF-like system:
B=1.5T
y=1.5x10-4 m
N=50
q=10-3 radians
L=0.8 m
Lr=166.7 m (gas+wires)
 p 

  2.85  103 p sin q
 p 
2


2
2
 3.0  103 
  3.0  103 cotq
 
  sin q 
momentum resolution at 900

2
momentum resolution at 450
0.012
0.012
spatial resolution
MS
combined
0.01
angular resolution
spatial resolution
MS
combined
0.01
0.008
0.008
p/p
p/p

0.006
0.006
0.004
0.004
0.002
0.002
0
0
0
0.5
1
1.5
2
p(GeV/c)
880.P20 Winter 2006
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
p (GeV/c)
Richard Kass
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Mass Resolution and Physics
Discovery of the Upsilon at Fermilab in
1977 using a double arm spectrometer. Had
to do an elaborate fit to find 3 resonances:
U(1S), U(2S), U(3S)
pBemmX
1977
PRL 39, 252 (1977)
PRL 39, 1240 (1977)
Double arm spectrometer (E288)
mm
1986
Upgraded double
arm spectrometer
(E605) clearly
separates the 3 states:
improved mass resolution
and particle ID (RICH)
Better fit
880.P20 Winter 2006
Richard Kass
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Wire Chamber Operation
A wire chamber is just a gas tight container with a wire inside.
The gas is the medium that gets ionized by a passing charged particle.
The wire helps define an electric field and “collects” ionization, part of signal path.
A typical cylindrical wire chamber has:
a wire (anode) held at +V
outside of cylinder (cathode) held at ground
Charged particle passing through cylinder creates ions
movement of ions creates a voltage or current pulse
signal pulse travels down wire to “outside world”
usually to preamplifier
Location of charged particle is measured relative to wire
Operating characteristics depend on the applied voltage (E-field)
recombination: no signals
ionization: signals, but no gas gain
proportional: “big” signals due to gas gain
Geiger-Muller: gas gain so large it produces sparks (discharge)
Have to run in proportional or geiger mode
to detect single particles like e’s, p’s, K’s, p’s.
880.P20 Winter 2006
Richard Kass
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Signal Production
First consider a very simple case: A capacitor in a gas tight box.
+Vo
anode R
d
--gas
cathode
+++
c
The electric field inside the chamber is E=Vo/d.
signal The parallel plates of the chamber have capacitance C
with stored charge Qo=CVo. If N ions are produced by
r a charged particle passing through the gas then the
electrons will drift to the anode and the positive
ions will drift to the cathode. Assuming that the e-’s
and positive ions make it to the plates long before the
power supply can recharge the plates back to Vo (RC very large)
the charge on each plate will be diminished by N|q|, with
|q| the charge of an electron. Therefore the voltage across the
plates will drop by DV=N|q|/C and we would see this voltage
drop as our signal.
C =Chamber capacitance
c =pulse shaping (and HV decoupling) capacitor
R =power supply series resistor (large)
r =pulse shaping resistor
880.P20 Winter 2006
Richard Kass
9
Pulse Formation in a Cylindrical Wire Chamber
In a cylindrical chamber the electric potential, E-field and capacitance are given by:
 (r )  
CV0
ln( r / a)
2pL
wire radius= a, tube radius=b, length of tube= L
E (r ) 
CV0 1
2pL r
C
2pL
ln(b / a)
Note: I put the L dependence in , E, and C.
Leo’s C is my C/L.
The potential energy stored in the electric field is W=1/2CV02.
Assume a charged particle goes through the cylinder and ionizes the gas.
As a charge, q, moves a distance dr there is a change in the potential energy (dW):
d (r )
q d (r )
dW  q
dr and dW  CV0 dV  dV 
dr
dr
CV0 dr
The total induced voltage from electrons produced at r is:
 q a d (r )
 q a (CV0 ) dr (q ) a  r 

V 
dr 

ln


CV0 a  r  dr
CV0 a  r  2pL r 2pL
a
The total induced voltage from positive ions produced at r is:
 q b d (r )
 q b (CV0 ) dr
q
b
V 
d
r


ln


CV0 a  r  dr
CV0 a  r  2pL r 2pL a  r 
Note: the total induced voltage is: DV=V++V- = -q/C

880.P20 Winter 2006
Richard Kass
10
Pulse formation in a cylindrical wire chamber
Note: the positive ions and electrons do not contribute equally to the DV if there is
multiplication in the gas. Since the avalanche takes place near the wire (r=1-2mm)
and the electrons are attracted to the wire the positive ions travel a much greater distance.
For typical values of a (10mm) and b (1cm) we find:
b
3
V
ln
10

 ar 
 75


a

r
ln(
11
/
10
)
V
ln
a
We can find the voltage vs time by looking at V(t) for the positive ions:
r (t )
dV (r )
q
r (t )

V (t )  V (t )  
dr 
ln
2pL
a
r ( 0 )  a dr
The problem now is to find r(t).

ln
By definition, the mobility, m, of a gas is the ratio of its drift velocity to electric field.
1 dr
E (r ) dt
For cylindrical geometry we have:
dr
CV0 1
CV0
 mE (t )  m
 rdr  m
dt
dt
2pL r
2pL
m  v / E (r ) 
880.P20 Winter 2006
Richard Kass
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Pulse Formation in a Cylindrical Wire Chamber
From previous page we had: rdr  m
CV0
dt
2pL
CV0 t
 2 mCV0 
t
 rdr  m
 dt  r (t )   a 
2
p
L
p
L


r (0)  a
0
1/ 2
r (t )
q
r (t )  q
mCV0
q
t
V (t ) 
ln

ln(1 
t
)

ln(
1

)
2
2pL
a
4pL
4pL
t0
pLa
a 2 ln(b / a )
With: t0 
2 mV0
t0 2
b2
2
The total drift time is:
T  2 (b  a )  2 t0
a
a
Typical gas mobilities are m=1-2 cm2s-1V-1.
Example: Let m=1.5 cm2s-1V-1, V=1500V, a=10mm, b=1cm
then: t0=1.5x10-9 s and T=1.5x10-3 s.
t
0
t0
10t0
102t0
103t0
T
ln[1+t/t0]
0
0.69
2.4
4.6
6.9
13.8
pulse shape determined
by electronics
Time development of voltage pulse
880.P20 Winter 2006
Richard Kass
12
Gas Gain
In many gases (e.g. argon) ion multiplication occurs as the “original” electrons
get close to the wire. If the electric field is high enough the electrons will be
accelerated to the point where they have enough kinetic energy to liberate electrons
in collisions with other atoms/molecules.
A cartoon of the
multiplication process.
Over a limited range of electric field the final amount of ionization is proportional to
the amount of primary ionization.
The total amount ionization, n, is related
to the primary ionization by, np,:
r2
n=Mnp where: M  exp[  a ( x)dx]
a is the first Townsend coefficient. r1
Model by Rose and Korff gives: a / p  A exp[
 Bp
]
E
A and B are gas dependent
Gas gains up to M=108 are possible. Above this limit breakdown (sparking)
occurs. This limit (ax<20) is the Raether limit.
880.P20 Winter 2006
Richard Kass
13
Example of a Wire Chamber: BaBar LSTs
Recently completed a project where we assembled several hundred wire chambers.
“Limited Streamer Tubes”
Ionizing particle produces a “streamer”
a controlled spark-> ~same charge for all tracks
Big signal induced on wire
Tubes are made out of extruded PVC
“easy” to assemble, low cost….
LSTs are used to detect muons
8 cells per tube
Extruded PVC
sleeve and profile
Endcap (HV, Gas Inlet)
basement of Smith Lab
fground
full size tube
full scale grad student
880.P20 Winter 2006
plane
Wire
Graphite
coating
Richard Kass
Wire
holder
14
LSTs in BaBar
August-October 2004: Two sextants of LSTs installed in BaBar
August 2006: Install remaining 4 sextants worth of LSTs
cosmic ray
880.P20 Winter 2006
e+e-m+m-
Richard Kass
15
Multiwire Proportional Chamber (MWPC)
In late 1960’s early 1970’s techniques were developed that allowed many
sense wires (anodes) to be put in the same gas volume. The MWPC was born!
The spatial resolution () of an MWPC is determined by the sense wire spacing (Dx):
Dx

12
Typical wire spacings are several mm, but MWPC with 1mm spacing have been built.
cathode
. . . . .
Dx
anodes
sense wires
Gas volume
Charpak wins 1992
Nobel Prize for developing
MWPCs
Advantages of MWPC:
Disadvantages of MWPC:
can cover large area
poor spatial resolution
elaborate electronics
need low noise premps
systems with thousands of wires
planar or cylindrical geometry
can get pulse height info
dE/dx
position info along wire using charge division
easy to get a position measurement (digital)
can handle high rates
works in magnetic field
ease of construction
880.P20 Winter 2006
miniaturization of electronics
elaborate gas system
must understand electrostatics
forces on wires
Richard Kass
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