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```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
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.
We measure coordinates in space using one or more of the following devices:
Multiwire Proportional Chamber
low spatial resolution (1-2 mm)
Drift Chamber
moderate spatial resolution (50-250mm)
Silicon detector
high spatial resolution (5-20 mm)
Most common
Simplest case: determine radius of circle with 3 points
Better momentum resolution better mass resolution
Many particles of interest are observed through their decay products:
D0K-p+,D+K-p+ p+,pp-, K0p+ p+
By measuring the momentum of the decay products we measure the mass of the parent.
1
m  m1+m2  m2=(E1+E2)2-(p1+p2)2 = m12+ m22 +2[(m12+ p12 )1/2 (m22+ p22 )1/2 - p1p2 cosa]
a
For fixed a: m2 / m2   p / p
2
Richard Kass
880.A20 Winter 2002
Momentum and
Position
Measurement
(L/2, y )
y
2
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 found using propagation of
errors to be:
 s  3 / 2 y
Thus the momentum (^to B) resolution due to position measurement error is:
 p^
p^

s
s

880.A20 Winter 2002
3 / 2 y
2
(0.3L B) /(8 p^ )

8 p^ 3 / 2 y
2
0.3L B
 32.6
p^ y
2
LB
(m, GeV/c, T)
Richard Kass
Momentum and Position Measurement
From previous page the momentum resolution due to measurement error is:
 p^
p^

s
s

3 / 2 y
2
(0.3L B) /(8 p^ )

8 p^ 3 / 2 y
2
0.3L B
 32.6
p^ y
2
LB
(m, GeV/c, T)
Typical numbers for CLEO (or BELLE or BABAR) are: B=1.5T, L=0.8m, y=150mm
p
^
p^
 32.6
1.5  10 4 p ^
0.82 (1.5)
 5.1  103 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 NIM 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^
p^
720  y p^

(m, GeV/c, T)
2
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).
880.A20 Winter 2002
Richard Kass
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
p

s rms
plane
sB
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^ )
p
52.3  10 3

p B LLr sin q
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
  p^

 p^



2
 52.3  10 3 
 720  y p^ 

 
 
2 
 B LL sin q 
r
 n  4 (0.3BL ) 


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.A20 Winter 2002
Richard Kass
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
and treating qand p^ as independent variables. Using propagation of errors
(L) we find:
2
2
Often detectors measure the r-f
  p    p^ 
  cotqq 2
   
 p   p^ 
coordinate independently of
the z coordinate. In these cases
p^ and q are independent.
Putting it all together we have for the total momentum resolution:
 p

 p
2
 52.3  10

 720  y p sin q 


  cot q q 2

  

2 
 B LL sin q 
n

4
(
0
.
3
BL
)

r



2
2
Position resolution
3
Multiple scattering
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.A20 Winter 2002
Richard Kass
Still More on Momentum Resolution
Let’s examine the momentum resolution equation for a CLEO-like system:
B=1.5T
y=1.5x10-4 m
N=50
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.A20 Winter 2002
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
p (GeV/c)
Richard Kass
3.5
4
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
arm spectrometer
(E605) clearly
separates the 3 states:
improved mass resolution
and particle ID (RICH)
880.A20 Winter 2002
Better fit
Richard Kass
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.A20 Winter 2002
Richard Kass
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.A20 Winter 2002
Richard Kass
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
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.A20 Winter 2002
Richard Kass
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.A20 Winter 2002
Richard Kass
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 
2mV0
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
Time development of voltage pulse
880.A20 Winter 2002
Richard Kass
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
1st Townsend coefficient for argon+organic vapors
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
Gas gains up to M=108 are possible. Above
this limit breakdown (sparking) occurs. This
limit (ax<20) is the Raether limit.
880.A20 Winter 2002
Richard Kass
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
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.A20 Winter 2002
miniaturization of electronics
elaborate gas system
must understand electrostatics
forces on wires
Richard Kass
Drift Chambers
Drift Chambers are MWPCs where the time it takes for the ions to reach the
sense wire is recorded. This time info gives position info:
ts
x   v(t )dt t0= start time, ts=stop time=time electrons reach sense wire
t0
For some gases the drift velocity is constant (independent of E-field): x=v(ts-t0)
A gas with almost constant drift velocity is
50-50 Argon-Ethane, drift velocity  50mm/nsec
By using the drift time information we can improve our spatial resolution by a
factor of 10 over MWPCs (1mm 100 mm).
Hex-cell
drift chamber
880.A20 Winter 2002
drift times are
circles around the
sense wires
Richard Kass
Drift Chambers Resolution
The spatial resolution of a drift chamber is limited by three effects:
Statistics of primary ionization
location of the primary ionizations (a few 100mm apart)
Diffusion of the electrons as they drift to the wire
1 2 Dx

n mE
N=# of primary ions
D=gas diffusion constant
m=mobility
x=drift distance
E=electric field
magnetic field changes alters drift path:
drift path depends on “lorentz angle”, ExB
How well the electronics measures time
must measure time to < 1nsec,
must know start time (t0)
Contributions to spatial resolution
880.A20 Winter 2002
Richard Kass
Drift Chambers
Drift chambers come in all sizes, shapes and geometries:
planar  fixed target
cylindrical  colliding beam
Time information gives a “circle” of constant distance around the sense wire (more complicated in B field)
In almost all cases, wires in different layers are staggered to resolve the left-right ambiguity
Typical cylindrical DC:
Many wires in same gas volume.
Use small angle stereo for z.
Usually use single hit electronics.
Sense (anode) and field wires.
CLEO, CDF, BELLE, BABAR
Tube Chamber:
Single sense wire in a cylinder
Can make out of very thin wall tubes.
 very little material
Small drift cell  single hit electronics
Good cell isolation
 broken wire only affects one tube
CLEO’s PTL detector
880.A20 Winter 2002
Jet chamber: optimized to resolve two tracks in a “jet”.
Drift direction roughly perpendicular to wire plane.
Single track gives multiple hits on several wires.
Use multi-hit electronics so two tracks on a wire can be resolved.
Lorentz angle must taken into account
wires are “slanted”
Richard Kass
Time Projection Chamber
TPC measures all 3 space coordinates
Many hits per track (>100)  good dE/dx measurement
Used at LEP, RHIC
Very complicated electric field shaping: E||B to reduce effects of diffusion
Long drift times  complicated gas system
Lots of electronic channels  complicated electronics
880.A20 Winter 2002
Richard Kass
Silicon Strip Detectors
SSD’s are solid state proportional chambers
Approximately 1000X more
ionization in silicon compared
to a gas. Not necessary to have
charge multiplication to get
useable signals.
• silicon strip detector measures position to ~10mm.
• silicon detector has many thin metal strips on top and (sometimes)
bottom surface of silicon wafer
• charged particle ionizes the silicon as it passes through
• electric field in silicon guides ions to top/bottom
• ions are collected on one or or two (or 3) strips
• knowing which strip has signal gives position of charged track relative
to silicon detector
880.A20 Winter 2002
Richard Kass
Silicon Strip Detectors
Dx
Resolution is mainly determined by strip pitch:  
12
Dx=3.5
 need strips every 50mm to get 15 mm resolution 00strips per cm
Strips can only be 5 cm long (technological limit)
Modern silicon strip detectors have 105-106 strips!
CLEO III hybrid (one of 122)
Require custom electronics
electronics must be small
low power dissipation
wire bond connections (105-106)
Mechanical Structure
must be rigid/strong
must be low mass to minimize MS
mechanical tolerances ~mm
preamps
Much more engineering involved with silicon
detectors compared to drift chambers!
880.A20 Winter 2002
Digital
capacitors
Richard Kass
Double sided silicon detector (CLEO)
Put orthogonal (x,y) strips on top and bottom surface.
Allows 2 coordinate measurements per silicon wafer
minimizes amount of material  less MS
Problems in high rate environments  poor two track separation
Pixel detector (BTEV, LHC)
Get position location (x,y) from hit pad (50mm x 50mm)
minimizes amount of material  less MS
Quick response time
Small detector capacitance good s/n with thin detector  less MS
Good two track resolution
880.A20 Winter 2002
Richard Kass
CLEO III Silicon Detector
Installation of
CLEO III silicon
detector
1.25x105 strips
Each strip has its own:
cables
880.A20 Winter 2002
hybrids
Silicon wafers
(layer 4)
Drift chamber
Richard Kass
CLEO II.V charged particle tracking
3 layer silicon detector
10 layer drift chamber (VD)
51 layer drift chamber (DR)
All in a 1.5T B field
DR
Si
VD
880.A20 Winter 2002
Richard Kass
```