New England Particle Physics Student Retreat (NEPPSR)
Front End Electronics for Particle Detection
Case study : ATLAS Muon Spectrometer
August 20, 2003
John Oliver
• Signal formation in gas detectors
• Basic electronic components & noise sources
• Noise calculations
• Amplifier/Shaper/Discriminators (ASDs) for ATLAS Muon Spectrometer
Disclaimer
This will be a fairly analytical approach. The idea is to develop a tool-box of methods you will
need to analyze similar applications. If you find the need, you can explore any of these
subjects in more detail later. You don’t need to memorize this stuff!!
Homework
There will be numerous examples given as homework. Go as far with them as you can. We’ll
go over the solutions Wednesday/Thursday evening.
(1)
Signal formation in gas detectors
Simple example : Uniform electric field in drift gas (eg Ar/CO2)
D
x (10 mm)
Anode
Cathode
A
Vhv
(1 kV)
Q
• Electric field  E = 1e 5 Volts/meter
• Particle track ionizes N electron/ion(Ar) pairs with total charge N*e = Q
• Electrons/ions drift toward Anode/Cathode with velocity given by their mobilities
Electrons : ve   e  E
Positive ions : vion   ion  E
met 2
 e  0.4
V s
met 2
ion  6 105
V s
• Question : What signal current i(t) is seen by ideal ammeter in series with battery ?
(2)
• Electrons drift to Anode, ions to Cathode
• First the electrons…
In time Dt, electrons move distance
Dx  vdrift  Dt
DV  E  vdrift  Dt
through potential
Work done by field is
DW  Q  DV  Q  vdrift  E  Dt
Work done by battery is
DW  Vhv  i(t )  Dt
In general 
where vdrift   e  E
i (t ) 
In this example 
Q  vdrift  E
Vhv
i (t )  Q 
-------------------- (1)
vdrift
D
 const
-------------------- (1b)
Total signal charge collected in arbitrary time t,
q (t )  Q 
DV (t )
Vhv
-------------------- (2)
(3)
Example :
• Q = 1fc , x = 2 mm
 electron signal current i(t) = 4 nA
 velocity = E =4*10^4 met/sec
 time to hit anode = 50 ns
 total electron signal charge collected in 50 ns
q = 4 nA * 50 ns = 0.2 fc ( from eq #2,  Q*200V/1kV)
i(t)
4 nA
50 ns
t
 What happened to the rest of the ionization charge?
Homework : Liquid argon ionization chamber (or calorimeter)
• Liquid argon filled gap, horizontal track
• e = 0.01 met^2/(V-s) (neglect ion drift)
• Ionization density = 7000 electron-ion pairs per mm of track (MIP)
• Vhv = 5kV
10 mm
 Find signal current i(t)
 Total signal charge
Vhv
(5 kV)
(4)
Signals in circular drift tube (ATLAS Muon Spectrometer)
• Electric field 
Vhv
1
E (r )  
r ln( R / r0 )
-------------------- (3)
• Primary electrons drift to Anode wire
• High field at wire surface (~ 2e7 V/m) causes avalanche “centered”
very close to wire, typically ~ 10V from wire surface[1]
• Each primary electron liberates Qtot ~ 2e4 secondary electrons
• Must analyze both electron and ion signal response to single primary
electron
• Wire radius : r0=25 microns
• Tube radius : R= 15 mm
• Gas : Ar/C02 (93%/7%)
• Vhv = 3080 Volts
• Gas gain: G = 2e4
• ion = 6e-5 m^2/V-s
• e = 0.4 m^2/V-s
Electron signal
• Charge centroid very close to wire  short collection time , < 1 ns
• i(t)=qelectron * d(t)
• qelectron= Qtot* DV/Vhv =Qtot * (~10V/3080V) ~ (1/3 % ) Qto t~ 130
e
(5)
Ion signal
• Ion velocity
v(t ) 
 ion Vhv
dr
  ion  E (r (t )) 
dt
r (t )  ln( R / r0 )
-------------------- (4)
• Simple diff-e in can be integrated to give r(t)
r 2 (t )  r02 (t ) 
t  t0
t0
-------------------- (5)
where t0 is a constant of integration given by,
1 r 2 ln( R / r0 )
t0   0 
 11 ns
2  ion
Vhv
-------------------- (6)
• From (1)
i (t )  Q   ion 

i (t )  Q 
 ion  Vhv
[ln( R / r0 )]
2
E 2 (r (t ))
Vhv

1 t0

r02 t t 0
-------------------- (7a)
and integrated signal charge :
qion (t )  Q 
t  t0
1
 ln(
)
ln( R / r0 )
t0
-------------------- (7b)
(6)
Pulse properties
• Initial current
1 Q 
i (0)   tot 2 ion  23nA
2
r0
-------------------- (8)
ion pulse shape
normalized current
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
20
40
60
80
100
tim e (ns)
• Extremely long pulse. (See eq #5) Pulse terminates when ions arrive at cathode at time
2
Tmax
R
 t0     4ms
 r0 
• Question: What proportion of charge is collected in time t0 (11ns) ?
qion 1 ln( 2)
 
 5%
Qtot 2 ln( R / r0 )
-------------------- (9)
-------------------- (10)
Conclusions:
• Ion charge is not much but its a lot more then the electron signal charge. For ATLAS MDT
its about 1000 electrons for each primary electron.
• Electron charge can generally be neglected in simulations & calculations
HW: Go through this analysis to make sure you understand it. It will come in handy some day!
(7)
Basic electronic components & noise sources
Part – I : Mathematical note
• Circuit analysis is almost always done by means of Laplace (not Fourier) transforms
• More convenient than Fourier when
 circuit is considered quiescent before t = 0
 stimulus occurs after t = 0
• Steady state (fixed frequency) response is obtained by sjwj2pf
t-domain
definition
time derivative
s-domain

f (t )
df (t )
dt
F (s)   f (t )  e  st  dt
0
s  F (s)
delay
f (t  T )
Delta function
d (t )
1
Unit step
u (t )
1
s
exponential
e t / T
e  sT  F (s)
T
s T  1
unipolar pulse
t t / T
e
T
T
( s T  1) 2
bipolar pulse
t
t
 (1 
)  e t / T
T
2 T
s T 2
( s  T  1)3
Universal method of inverting Laplace transforms
• Look them up in a table or use Maple or Mathematica
(8)
Basic electronic components and noise sources
Part -II
• Inductor
 Stores energy in magnetic field
 E = ½ L i2
 Impedance Z(s)= sL
 Lossless, noiseless
• Capacitor
Stores energy in electric field
E = ½ C v2
 Impedance Z(s) = 1/(sC)
 Lossless, noiseless
• Ideal transmission line
 Considered infinite sequence of series inductors & parallel capacitors
 Results in wave equation with phase velocity :
v
1
 
 and a constraint between voltage and current :
V


 Z 0 characteristic impedance
I

 noiseless
 Note that an infinitely long transmission line is indistinguishable from a resistor of value Z0
 As corollary, a finite line with a resistor of value Z0 at its end, is also indistinguishable from a resistor
of value Z0.
 In other words, if you launch a pulse into a terminated line, it never comes back (no reflection).
(9)
• Resistor
 Electric field pushes conduction electrons through a lattice
 Dissipates power = I*V
 Conduction electrons (generally) in thermal equilibrium with environment
 Noisy  Noise characterized by “noise power density” p(f) [watts/hz]
p( f )  df  power ( watts) radiated in frequency range f to f  df
 p(f) is frequently expressed as a voltage (in series) or current density (in parallel)
2
e (f)
p( f )  n
R
p( f )  in ( f )  R
2
 en(f) & in(f) given in volts/sqrt(hz) & amps/sqrt(hz)
 To get values of en(f) & in(f) ; [Nyquist, Phys Rev, Vol 32, 1928, p. 110]
Length L, total time L/v, Impedance Z0
½
Z0
in
½
in
Z0
Total noise current in cable
i 
2
tot
in2( left )
4

in2( right)
4

in2
2
(10)
• Disconnect resistors trapping noise power in cable
• Total energy in cable, in frequency band Df is
Energy  Df 
in2
L
R
 Df
2
ccable
• Open circuit cable requires current at ends to be zero, and voltage at ends to be nodes
• Standing waves (integer number of half-waves)
n

L
2
• In frequency band Df , number of modes is
Dn 
2 L
 Df
ccable
• Energy equipartition requires that each mode (one for voltage, one for current) corresponds to
energy kT/2
Energy  Df  k  T  Dn
in2
L
2 L
R
 Df  k  T 
 Df
2
ccable
ccable

from which we get
in 
4  k T
R
&
en  4  k  T  R
Independent of frequency  “white” noise
}
---------------------------- (11)
(11)
Basic electronic components and noise sources
Part III : General circuit analysis primer
Objective
• To find response of arbitrary circuits to arbitrary stimulus in both time and frequency domains.
• Use this to get expected response from MDT Amp/Shaper to
o calibration pulses injected into drift tubes
o real ion-tail pulses in chambers
Method - A
• Define ratio of “output” variable to “input” variable : Transfer function: H(s)
• Write node and loop equations and solve for transfer function
• Delta function response of circuit is just h(t), the inverse transform of H(s)
• To get response to other inputs g(t)
• Get transform of g(t)  G(s)
• Transform of response is  F(s)=G(s)H(s)
• Invert to get f(t)
G(s)
G(s)H(s)
H(s)
Simple example
Find step response of simple low-pass filter
H ( s) 
C
V1
V2
1
s C
R
V1 ( s ) 
V2 ( s ) 
R
1
R
s C

s T
1 s T
where T  R  C
1
s
T
1 s T
   v2 (t )  e t / T
(12)
Basic electronic components and noise sources
Part III : General circuit analysis primer
Objective
• To find response of arbitrary circuits to arbitrary stimulus in both time and frequency domains.
• Use this to get expected response from MDT Amp/Shaper to
o calibration pulses injected into drift tubes
o real ion-tail pulses in chambers
Method – B
• What if method A fails?  ie the transforms or inverse transforms cannot be found in closed form
• Resort to the time domain solution  Convolution integral and use numerical solution if necessary
t
Vout (t )   g ( )  h(t   )d
0
• Simulation (SPICE)
(13)
General circuit analysis - Homework
An ideal integrator is followed by a 2-pole “RC-CR shaper” as follows
Unity gain buffers
C
f (t )

f (t )  dt
R
1x
R
1x
C
• Show that transfer function of system is :
1
s
s T
(1  s  T ) 2
(T  R  C  15ns)
a) With delta function input, what do the signals look like throughout the circuit?
b) Is this circuit suitable for use in “high rate” (eg LHC/ATLAS) environment?
Extra credit part
c) Replace ideal integrator (1/s) with “leaky” integrator, 1/(sT+1) with T=15ns
• Same questions as (a) and (b)
(14)
Part-IV : Noise calculations
Example # 1
• What is the rms terminal voltage of the following simple circuit?
vrms
in
Solution
1)
R 10k
Add noise current source,
C = 1 pf
in 
4kT
R
2)
Solve for circuit “transfer function” This gives you output voltage density
•
3)
vn ( s )
R

in
1 s  R C
set s = jw2pjf
Integrate over frequencies (quadrature)

2
vrms
  | vn2 ( f ) | df 
0
2
vrms


4  k T 1
R2


 dw
R
2  p 0 (1  R 2  C 2  w 2 )

4  k T 1
1


dx
C
2  p 0 (1  x 2 )
where x  w  RC
• In general, such integrals can either be looked up or done by contour integration.

1
1 
1
1 2pi p
dx


 

0 (1  x 2 )


2
( x  i)( x  i) 2 2i
2
(15)
• In this case,
Vrms 
kT
C
---------------------------- (12)
• At room temperature kT ~ 4e-21 
Vrms  63V
• Called “kT over C” noise, or kTC noise (when dealing with charge instead of voltage).
• Example (Homework) :
 Suppose we want to use an array of 100 femtofarad capacitors to build an integrated “voltage
waveform recorder” in a 3.3Volt CMOS process.
 Note that the “switches” are CMOS devices and are actually voltage controlled resistors which
switch between some finite resistance in the “On” state to “near infinite” resistance in the “Off”
state
ADC
Vin
 Assume maximum signal swing inside the circuit is limited to about half the supply voltage, or
1.6 volts
 What is the maximum possible dynamic range (in bits) of this device? (Dynamic range is max
signal divided by rms noise).
 If we plan to digitize this signal, should we pay more money for a 16 bit ADC, or will a 14-bit
ADC do the job?
(16)
CMOS components : Field effect transistors (nfets)
• In undoped (intrinsic) silicon, electron and hole densities are the same
• n-doped (arsenic, phosphorous, antimony,..) : electron density increases, hole density decreases
• p-doped (boron, aluminum, gallium, ..) : vice-versa
• Strength of doping denoted by + sign  n, n+, p, p+
• + sign indicates higher doping, lower resistivity
Conductive gate electrode
Dielectric “gate oxide”: Si02
n+ source and drain implants
gate
source
n+
n+
drain
p substrate (~ 10kW/sq)
• With Vgate = 0, structure is non-conductive (back to back diodes)
• Increasing Vgate in positive direction, attracts electrons from substrate
• When Vgate > Vthreshold, “channel” becomes conductive. Conductance increases as Vgate increases.
• As Vdrain is made more positive than Vsource, current starts to flow
• Voltage gradient appears from drain to source.
• Electric field is strongest near source, weakest near drain..
(17)
CMOS components : Field effect transistors (nfets)
• In undoped (intrinsic) silicon, electron and hole densities are the same
• n-doped (arsenic, phosphorous, antimony,..) : electron density increases, hole density decreases
• p-doped (boron, aluminum, gallium, ..) : vice-versa
• Strength of doping denoted by + sign  n, n+, p, p+
• + sign indicates higher doping, lower resistivity
Conductive (polysilicon) gate electrode
Dielectric “gate oxide”: Si02
n+ source and drain implants
gate
source
n+
n+
drain
p substrate (~ 10kW/sq)
• With Vgate = 0, structure is non-conductive (back to back diodes)
• Increasing Vgate in positive direction, attracts electrons from substrate
• When Vgate > Vthreshold, “channel” becomes conductive. Conductance increases as Vgate increases.
• As Vdrain is made more positive than Vsource, current starts to flow
• Voltage gradient appears from drain to source.
• Electric field is strongest near source, weakest near drain.
• Channel charge density “tilts” toward source.
• Drain current increases with Vdrain
• When Vdrain comes within a “threshold voltage” of Vgate, (Vdrain = Vgate – Vthreshold) current “saturates”
• Saturation region also called “pinch-off
(18)
CMOS components : Field effect transistors
pfets
nfet
gate
source
n+
n+
gate
drain
drain
source
p+
p+
n well
p substrate (~ 10kW/sq)
• Generally, pfet drain current constant, Kp, is about 1/3 that of nfets due to lower hole mobility
• For same size transistor at same current, pfet transconductance is smaller by ~sqrt(3)
(19)
FET properties (simplified model)
Idrain
Idrain vs Vds for several
values of Vgs
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Vds
Linear, resistor,
or triode region
“Saturation” region
Drain current properties in saturation region
• Id increases quadradically with Vgs
I d  (Vgs  Vth ) 2
• Increases linearly with transistor channel width, W
• Decreases linearly with transistor gate length, L
Id 
1
W
 K n   (Vgs  Vth ) 2
2
L
---------------------------- (13)
• Definition: “Transconductance” = ratio of change in drain current
per unit change in gate voltage.
gm 
I d
W
 2  I d  Kn 
Vgs
L
"Typical " value : K n  100 A / V 2
---------------------------- (14)
(20)
FET properties (con’t)
Terminal “impedances”
• Gate: No dc current flow, just gate capacitance Cgate
(of order ~tens of femtofarads to pf)
• Drain:
“Wiggle” the drain voltage a little, what’s the
change in drain current?
 Ans: no (or very little) change so 
V
Z drain  ds  
I d
• Source
 “Wiggle” the source voltage a little bit.
 This changes Vgs and thus drain current (and source current)
by (Vgs) x (gm).
Zs 
Vs
1

I d g m
 Typical numbers depending on application;
.001  g m  .01
or
1kW  Z source  100W
 Example;
 Idrain = 1 ma, L=0.5u, W=100u
gm  2  I d  Kn 
or
W
 2  (103 )  (100 10 6 )  200  6.3 103
L
Z s  160W
(21)
Some common FET circuits
A) Source follower
C) Current mirror
d
Iin
g
Vin
Iout
s
Vout
Used as (almost) unity gain buffer.
Hi-Z in, Lo-Z out.
d
d
g
g
s
Voltage gain  1
s
I out  I in
B) Common source amplifier
D) Differential amplifier
Gain  g m  Rload
R
d
Vin
g
Vout
d
d
g
g
s
s
s
DI out  g m  DVin
(22)
Some common FET circuits – con’t
E) Fancier differential amplifier
F) Common gate or “cascode”
Z1
d
d
g
g
s
Z2
d
Iout
g
s
s
Iin
• “Boxes” Z1 & Z2 can be whatever you need
• example:
• Z1 is parallel RC
Current gain  1
Used as current buffer.
Lo-Z in, Hi-Z out.
• Z2 is series RC
• Transfer function is
H (s) 
Z1 ( s )
sRC

Z 2 ( s ) (1  sRC ) 2
 Implements a “Bipolar” shaping function (See table of Laplace transforms)
(23)
Noise in Fets
• Fet channel is resistive and will thus have a thermal noise current component, in
in 
4  k T
 4  k T  gm
Reffective
• Channel is not a single resistor, but rather a series of increasing resistances (from source to drain)
• A fudge factor will be needed! (ff = 2/3)
in 
8
 k T  gm
3
R
d
Vin
Vout
g
en
in
s
• Noise current in drain is equivalent to noise voltage in gate with en 
en 
8 k T

3 gm
in
gm
---------------------------- (15)
(24)
Noise in Fets
HW
• What is the noise voltage density (nV/sqrt(Hz)) of a 100W resistor?
• How much transconductance is required of a fet to have this same noise voltage density ?
• Assuming we have a power budget of 1 ma (drain current) for this transistor and that the
transistor has minimum allowed gate length of L = 0.5 microns, how wide (W) does the
transistor have to be?
• How much improvement in en do we get by doubling the drain current? …or the transistor
width?
(25)
Front end amplifiers (eg ATLAS MDT)
• “Open loop” gain
Cstray
gm
G( s) 
s  Cstray
1x
d
Vin
• Example
Cstray = 500ff
Gm = 0.006
• What is “unity gain frequency”?
g
s
|G(f)|
f
ft 
gm
 2  GHz
2p  Cstray
• What’s the gain at 1 Hz?
(26)
Front end amplifiers (eg ATLAS MDT)
• “Open loop” gain
R0 Cstray
gm
G( s) 
s  Cstray
1x
d
Vin
• Example
Cstray = 500ff
Gm = 0.006
• What is “unity gain frequency”?
g
s
|G(f)|
<100
f
ft 
gm
 2  GHz
2p  Cstray
• What’s the gain at 1 Hz?
• Answer: 2x10^9 !!!!
• This can’t be right! Actually, gain rarely exceeds 100.Why is this?
• Ans: Output (drain) impedance of fet is not actually infinite… same thing for current source
G( s) 
gm
g m  R0
G0


s  Cstray  1 / R0 1  s  Cstray  R0 1  s  T
(27)
Front end amplifiers (eg ATLAS MDT)
To make this a practical amplifier  use feedback
Rf=15k
Cf=1pf
G(s)
“Transimpedance” amplifier
• For high rate environment we want RC to be small ~ 15 ns is optimal for MDTs
(electronics + chamber simulations)
• Important question is input impedance. In general, it is given by
Z in 
Z feedback
Gain
•We’d like it to be
 real or resistive (alternatives are capacitive or inductive)
 low ( a hundred W or so)
 as constant over frequency as possible
• HW Questions:
o Why the above criteria?
o How might you go about assuring this?
• Second important concept is “Sensitivity” or Peak voltage output per unit charge input
(typically in mv/fc)
• HW Question: What is the Sensitivity (in mv/fc) of the above circuit?
(28)
Front end amplifiers (con’t)
Now attach this to an ATLAS Muon tube
T
( s T  1)
T  R  C  Tpeak
Rterm=390W
and add a shaper
Question : Why do we need the terminating resistor?
For extra credit (LOTs of extra credit!)
• What’s the delta function response at preamp output, shaper output?
• Characterize system gain in peak shaper signal output per unit charge input. (volts/coul or mv/fc)
• How might you get the response to the actual ion signal from the gas tube? (Don’t actually do it!)
• What is the rms noise voltage at the shaper output produced by the terminating resistor?
• How much charge is this equivalent to (equivalent noise charge, or enc)?
Ans:
enc  e1 
kT
 Tpeak  5,000 electrons rms
2  Rterm
Hint: For the purposes of this exercise, you can assume input impedance of amplifier is zero.
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MDT-ASD topology
1 of 8 channels
Transimpedance
preamps
(delta response)
Discriminator
Bipolar
shaper
Dummy
Wilkinson
ADC
DACs
Calibration
Mode
Control logic
Signal
LVDS
Deadtime
Serial string register
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MDT-ASD topology
Transimpedance Preamplifier
Vdd
Vb[1:4] are bias
voltages supplied from
elsewhere
Vb4
M4
Vb3
M3
Two transistor current
source (Better than single
fet current version)
M2
Common gate
“cascode”
IN
Common source
amplifier
Source
follower
R1
Vb2
Cf
OUT
R2
M1
Vb1
Current sink
(half of a
current mirror)
Feedback R & C
Detailed circuit descriptions can be found in the following documents
• ATLAS-MUON-2002-003
Extra conductance on Hihttp://doc.cern.ch//archive/electronic/cern/others/atlnot/Note/muon/muon-2002-003.pdf
• ATLAS-MUON-080
Z node to tailor gain vs
http://doc.cern.ch/tmp/convert_muon-95-080.pdf
frequency, & Zin
• muon-96-127
http://doc.cern.ch/cgi-bin/extractFigs.check.sh?check=/archive/electronic/cern/others/atlnot/Note/muon/muon-96-127.ps.gz
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MDT-ASD preamp
layout
180u
300u
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MDT-ASD topology
Shaper differential amplifiers
Vdd
Vref
M5b
M5
M5a
R
Vb2
M4b
M4a
OUTb
M3b
M4
OUTa
Z1(s)
INa
M3
M3a
INb
Z2(s)
M2c
M2b Vb1
M2a
M2
M1c
M1b
Bias network
M1
M1a
“Fancy”
differential
amplifier
(pg 23)
GND
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MDT-ASD topology
Shaper differential amplifiers
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Summary & conclusions
Analysis/design methodology
1.
Understand requirements
• Noise, dynamic range,..
• Impedances
• Signal shapes
• etc
2.
Hand calculations will get you close and/or guide design
• Noise contributions of worst offenders
• Transfer functions, response shapes, etc
• Transistor sizing for CMOS circuits
3.
SPICE modelling
• Vendor SPICE models can be very accurate but very complicated
• Produce best analysis at expense of intuitive understanding
To learn more
Attend IEEE Nuclear Science Symposia and take “Short course” programs in
Front End Electronics (IEEE NSS 2003 Portland, Oregon, Oct 19-25)
Bibliography & lending library
1)
“Particle Detection with Drift Chambers” W.Blum, L. Rolandi Springer Verlag, pg 134, 155-158
2)
“Tables of Laplace Transforms” Oberhettinger & Badii, Springer Verlag
3)
“Complex Variables and Laplace Transform for Engineers” LePage, Dover
4) “Electronics for the Physicist” Delaney, Halsted Press
5) “Noise in Electronic Devices and Systems” Buckingham, Halsted Press
6) “Low-Noise Electronic Design” Motchenbacher & Fitchen, Wiley Interscience
7) “Processing the signals from solid state detectors” Gatti & Manfredi, Nuovo Cimento
8) “Analog MOS Integrated Circuits for Signal Processing” Gregorian & Temes, Wiley Interscience
9) “Detector Physics of the ATLAS Precision Muon Chambers” Viehhauser, PhD thesis, Technical
University Vienna.
10) “MDT Performance in a High Rate Background Environment” Aleksa, Deile, Hessey, Riegler –
ATLAS internal note, 1998
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