moderne teilchendetektoren - theorie und praxis 2
TRANSCRIPT
Gaseous Detectors
Bernhard KetzerUniversity of Bonn
XIV ICFA School on Instrumentation in Elementary Particle PhysicsLA HABANA
27 November - 8 December, 2017
Plan of the Lecture
1. Introduction2. Interactions of charged particles with matter3. Drift and diffusion of charges in gases4. Avalanche multiplication of charge5. Signal formation and processing6. Ionization and proportional gaseous detectors7. Position and momentum measurement / track reconstruction
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2.3 Fluctuations of Energy Loss
Consider single particle: statistical fluctuations of
• number of collisions• energy transfer in each collision range straggling (if stopped in medium) energy straggling (if traversing medium)
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[H. Bichsel, NIM A 562, 154 (2006)]
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Fluctuations of Energy Loss
Important quantity in order to understand response of detector:
probability density function for energy loss ∆ in material of thickness x,determined by• collision cross section d𝜎𝜎/d𝐸𝐸• 𝑛𝑛𝑒𝑒𝑥𝑥
Calculation of energy loss distribution: two approaches• Convolution method• Laplace transform method[Allison, Cobb, Ann. Rev. Nucl. Part. Sc., 253 (1980)]
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[H. Bichsel, NIM A 562, 154 (2006)]
Straggling functions
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2.3.1 Convolution Method
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In each collision, the probability to transfer an energy 𝐸𝐸 is given by
Energy loss Δ for exactly 𝑁𝑁𝑐𝑐 collisions 𝑁𝑁𝑐𝑐-fold convolution of 𝐹𝐹(𝐸𝐸)
with and
Convolution Method
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Number of collisions 𝑁𝑁𝑐𝑐 in layer of thickness 𝑥𝑥
Linked to CCS through mean free path
Mean value
Standard deviation
Relative width
Convolution Method
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Pdf for total ionization energy loss ∆ in material slice of thickness 𝑥𝑥= sum of all 𝐹𝐹𝑁𝑁𝑐𝑐(∆), weighted by their Poissonian probability for
exactly 𝑁𝑁𝑐𝑐 collisions
Straggling functions
• Poissonian contribution dominant for very small number of collisions (very thin absorbers)
• Peak structure vanishing for larger 𝑁𝑁𝑐𝑐
Convolution Method
Solution for thickness 𝑥𝑥:• Iterative application of convolution integral (numerical)
[Bichsel et al., Phys. Rev. A 11, 1286 (1975)]
• Monte-Carlo method [Cobb et al., Nucl. Instr. Meth. 133, 315 (1976)]
• calculate mean number of collisions 𝑚𝑚𝑐𝑐 from integrated cross section• for each trial (particle penetration) choose actual number of collisions
from Poisson distribution with mean 𝑚𝑚𝑐𝑐
• total energy loss = sum of energy losses in single collisions, takenfrom normalized dσ/dE distribution 𝐹𝐹(𝐸𝐸)
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Straggling Functions
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[H. Bichsel, NIM A 562, 154 (2006)]
Straggling Functions
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[H. Bichsel, NIM A 562, 154 (2006)]
Bethe-Bloch mean energy loss: ∆ = 400 eV
2.3.2 Laplace Transform Method[L. Landau, J. Phys. USSR 8, 201 (1944)]
Change of energy-loss distribution 𝑓𝑓(Δ; 𝑥𝑥) as a result of the particlepassing through a thin elemental layer δx:
• 1st term: probability that the energy loss in x was (∆−E ), and a collisionwith energy transfer E occurred in δx, which makes the total energy loss equal to ∆ (particle scattered into ∆)
• 2nd term: probability that the energy loss in x was already equal to ∆before entering δx, where a further collision increased the energy loss beyond ∆ (particle scattered out of ∆)
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Laplace Transform Method
Put in form of a transport equation:
upper integration limit 𝐸𝐸 → ∞for 1st term ok, since
( ), 0 for 0f x ∆ = ∆ <
Solution: Laplace transform of both sides
+ solve for 𝑓𝑓(𝑠𝑠; 𝑥𝑥)
+ inverse Laplace transform
Exact solution, but numerical integration necessary in most cases!
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0 < 𝑐𝑐 ≪ 1
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∆
( ) ( ),
s
F x s
•
∆ =L f x,
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2.3.3 Straggling Functions
Remarks to both methods:
• result determined by d𝜎𝜎/d𝐸𝐸
• given the same cross section d𝜎𝜎/d𝐸𝐸, both methods are equivalent
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Straggling Functions
Different approximations, depending on thickness of absorber
Characteristic parameter: ,
𝜉𝜉 = scaling parameter (1st term of Bethe-Bloch eq.)
Thin absorbers: κ ≤ 10• possibility of large energy transfer in single collisions: δ-electrons• long tail on high-energy side, strongly asymmetric shape
∆m
∆
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Landau Distribution
Very thin absorbers: κ→0 (i.e. Tmax→∞)• single energy transfers sufficiently large to consider e- as free
Rutherford
• particle velocity remains constant
Landau distribution [Landau, J. Phys. USSR 8, 201 (1944)]
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dσ(E
’)/dE
’/ σ
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Landau Distribution
Very thin absorbers: κ→0 (i.e. Tmax→∞)• single energy transfers sufficiently large to consider e- as free
Rutherford
• particle velocity remains constant
Analytical approximation: Moyal distribution [Moyal, Phil. Mag. 46, 263 (1955)]:
1 ( )2
Mm( ,,
ef x e
λλλ
π ξ
−− +1∆)
∆ − ∆==
2
Landau distribution [Landau, J. Phys. USSR 8, 201 (1944)]
L ( ,f x φ λξ1
∆) = ( )
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𝜆𝜆 = universal parameter, see next page for relationto Δ𝑚𝑚 and 𝜉𝜉
Note: 𝜆𝜆 different fromparameter in Landau distr. above
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Landau Distribution
Properties of φ(λ):• asymmetric: tail up to 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 → ∞• Maximum at λ=-0.223• FWHM=4.02•λ• numerical evaluation
Universal Landau distribution:
Landau distribution in ROOT:
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Comparison Landau - Moyal
Landau:p1=1.p2=500.p3=100.
Moyal:• coarse shape correct• tail too low!
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But: tails are important for detector resolution!
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Landau Distribution
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Landau distribution:• Uses Rutherford cross section• Does not reproduce straggling functions based on more realistic
models for CCS for very small 𝑁𝑁𝑐𝑐• Narrower width also for higher 𝑁𝑁𝑐𝑐 related to mean free path 𝜆𝜆
− Rutherford CCS underestimates 𝜆𝜆 (overestimates 𝑁𝑁𝑐𝑐)− Poisson contribution to straggling function leads to broadening
Symon-Vavilov Distribution
Thin absorbers: 0.01 < κ < 10• use correct expression for Tmax
• use Mott cross section instead of Rutherford• reduces to Landau distribution for very small κ• less asymmetric shape for larger κ
[S.M. Seltzer, M.J. Berger, Nucl. Sc. Ser. Rep. No. 39 (1964)]
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Realistic Straggling Functions
[H. Bichsel, Rev. Mod. Phys. 60, 663 (1988)]
Iterative Solution of convolution integral
[Blunck, Leisegang, Z. Phys, 128, 500 (1950)]
PAI Model vs Landau• Histogram: data• Landau + corrections
• PAI model
Ar/CH4 (93/7) Ar/CH4 (93/7)
[Maccabee, Papworth, Phys. Lett. A 30, 241 (1969)]
[Allison, Cobb, Ann. Rev. Nucl. Part. Sci. 30, 253 (1980)]
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2.4 Measurement of Energy Loss
Gas Detectors
Restricted energy loss:
( )cut
2 4 2e cut cut
2 2 2max
2d 4 1 ln 1d 2 2T T
z e n mc T TEx mc I T
β γπ β δβπε
2 2 2
2< 0
− = − + − 24
approaches normal Bethe-Bloch equation for 𝑇𝑇cut → 𝑇𝑇max
Transforms into average total number of e- ion pairs nT along path length x:
TddEx n Wx
=
But: actual energy loss fluctuates with a long tail (Landau distribution)mean value of energy loss is a bad estimator use truncated mean of N pulse height measurements along the track:
1
1 tN
itit
A AN =
= ∑ [ ]1 for 1, ,
, 0,1i i
t
A A i NN t N t
+≤ =
= ⋅ ∈
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Measurement of Energy Loss
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Measurement of Energy Loss
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Measurement of Energy Loss
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Resolution (empirical):
N = number of samplesΔ𝑥𝑥 = sample length (cm)p = gas pressure (atm)
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Example: ALICE TPC
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Example: FOPI GEM-TPC
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• PID using dE/dx:• Resolution ~ 14-17%• No density correction
First dE/dx measurementwith GEM-TPC!
• Sum up charge in 5 mm steps• Truncated mean: 0-70%• Momentum from TPC + CDC
[F. Böhmer et al., NIM A 737, 214 (2014)]
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Example: FOPI GEM-TPC
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• PID using dE/dx:• Resolution ~ 14-17%• No density correction
First dE/dx measurementwith GEM-TPC!
• Sum up charge in 5 mm steps• Truncated mean: 0-70%• Momentum from TPC + CDC
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[F. Böhmer et al., NIM A 737, 214 (2014)]
Energy Loss of Charged Particles
[Review of Particle Physics, S. Eidelmann et al., Phys. Lett. B 592, 1 (2004)]
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Z2 electrons, q=-e0
Electromagnetic Interaction of Particles with Matter
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Interaction with the atomic electrons. The incoming particle loses energy and the atoms are excited or ionized.
Interaction with the atomic nucleus. The particle is deflected (scattered) causing multiple scattering of the particle in the material. During this scattering a Bremsstrahlung photon can be emitted.
In case the particle’s velocity is larger than the velocity of light in the medium, the resulting EM shockwave manifests itself as Cherenkov Radiation. When the particle crosses the boundary between two media, there is a probability of the order of 1% to produce an X ray photon, called Transition radiation.
M, q=Z1 e0
Slide courtesy of W. Riegler
2.5 Multiple Scattering
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In addition to inelastic collisions with atomic electrons elastic Coulomb scattering from nuclei Rutherford cross section for single scattering (no spin, recoil)
( )( )
22
2 2 4 4Rutherford0
d 1d 4 4 sin sin
2 2
zZe
E
σθ θπε
= ∝ Ω ⋅
small angular deflection of incident particle random zigzag path, 0θ =
For hadronic projectiles also the strong interaction contributes to multiplescattering
contribution to momentum resolution!
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Multiple Scattering
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Theory:• Single scattering: very thin absorber such that probability for more than one
Coulomb scattering small angular distribution given by Rutherford formula
• Plural scattering: average number of scatterings N<20 difficult!
• Multiple scattering: N >20, energy loss negligible statistical treatment Gaussian distribution via Central Limit Theorem non-Gaussian tails due to less frequent hard scatterings probability distribution for net angle of deflection as function
of thickness of material traversed: Molière distribution (up to )30θ ≈
[G. Molière, Z. Naturforsch. 3a, 78 (1948), H.A. Bethe, Phys. Rev. 89, 1256 (1953)]
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Multiple Scattering
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Gaussian approximation: ignore large-angle (>10°) single scatterings
2plane
plane plane plan00
e21( ) exp
22P d d
θθ θ
θθ
πθ
= −
T T0
0 0
13.6MeV 1 0.038lnL Lzcp X X
θβ
= +
central limit theorem
rms 2 20 plane plane
12
θ θ θ θ= = =
X0 = radiation length of absorberLT = track length in medium (w/o multiple scattering),
i.e. straight line or helix
Accurate to < 11% for 3 T
0
10 100LX
− < <
from fit to Molière distribution
[V.L. Highland, NIM 129, 497 (1975)][G.R. Lynch et al., NIM B 58, 6 (1991)]
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Multiple Scattering
2plane
plane plane plane200
1( ) exp22
P d dθ
θ θ θθπθ
= −
T T0
0 0
13.6MeV 1 0.038lnL Lzcp X X
θβ
= +
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Literature
W. Blum, L. Rolandi, W. Riegler: Particle Detection with Drift Chambers. Springer, 2008.
H. Spieler: Semiconductor Detector Systems. Oxford 2005.C. Leroy, P.-G. Rancoita: Principles of Radiation Interaction in Matter and
Detection. World Scientific, Singapore 2012.Specialized articles in Journals, e.g. Nuclear Instruments and Methods A, Ann. Rev. Nucl. Part. Sci.
C. Grupen, B. Shwartz: Particle Detectors. Cambridge Univ. Press, 2008.K. Kleinknecht: Detektoren für Teilchenstrahlung. Teubner, Stuttgart 1992.G. F. Knoll: Radiation Detection and Measurement. J. Wiley, New York
1979.W. R. Leo: Techniques for Nuclear and Particle Physics Experiments.
Springer, Berlin 1994.
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