interactions of fast particles in a medium electromagnetic interaction: –ionization –radiative...
TRANSCRIPT
Interactions of Fast Particles in a Medium
• Electromagnetic interaction:– Ionization– Radiative processes– Cerenkov radiation
• Hadronic interactions
Energy Loss Processes
• First part considers “soft” interactions between a particle and the medium it is travelling through (atomic excitation and ionization)
• Discuss two approaches 1) Energy loss as a succession of scatters
A) Classical approximationB) Relativistic treatment
2) Moving particle as source of virtual
Energy Loss by Scattering
• As a particle goes through matter it suffers many “soft” or “glancing” collisions.
• In each collision the particle loses energy and changes direction slightly.
• Consider a single collision….
Energy Loss as a Succession of “Soft” Electromagnetic
Scatterings.
Classical EM Scattering – Impact parameter, b
• Consider small angle scattering:– Force F(b) = (ze)e/b2
– Assume force actsfor t = 2b/v
• Momentum changept ~ F(b)t = 2ze2/bv = 2z/bv(fine structure constant: = e2/40h`c)
– Scattering angle, ~ pt /p = 2z/pbv– In terms of K.E. of incoming particle, T ~ (z/b)/T
eme , p
M , ze
b
2b
Classical EM Scattering – Cross-section
• Have just “derived” the relationship between impact parameter and scattering angle.
• In practice can’t measure impact parameter – so we need to find the relationship between the scattering probability and the scattering angle.
Scattering Probability vs. Scattering Angle
• All areas equally likely to be hit.dP ~ db = d ( dP = prob of scattering)
• Every particle will have some scattering angle
• d ~b db d• Use
• and
• Want to get d()/d ….
b.db.d
b
d
ddd
ddSind ..
Classical EM Scattering – Cross-section
• b ~ 2z/pv , Sin ~ • db/d = z/2pv
• Which gives the Rutherford Scattering Formula
ddb
Sinb
dd
4
21
Tz
dd
Energy Loss by Scattering –Approximate,
Non-Relativistic Model
• Energy transferred to target:E = (pt)2/M ~ z22/b2v2
• Integrating over impact parameter will add a constant, but not change dependence on z,,v. So for a single scatter:E ~ z22/v2
Scattering – approx. model
• Almost all energy loss will be to electrons ( me << mN )
• Number of electrons that a projectile passes per unit length ~ elec= Zatoms
• atoms= mass/A , so elec= Z mass/A
• So, energy loss per unit length:
22 1
~A
ZKz
dx
dE
Kinematics of energy loss via scattering
• NB. Not necessarily elastic
• Substitute for E’ and p’:
– (Taylor expand in terms of /E to 2nd order, valid for small /E )
(E,p)(E,p)
Energy loss =E–E
EE '
3
222
2
2
2
22
2
21
21
21
p
M
p
Ep
pp
Ep
MEp
Energy Loss via Scattering• Use -q2=2M2-2EE’+2pp’Cosand our
approximation for p’
• For small angles, Cos ~ 1 – ½ 2
• Hence, for small , /E:
• At finite angles, the first term dominates. The second term defines a minimum q2 for a given energy loss .
22222
pq
Towards a Model of Energy Loss by ionization
• Energy loss of charged particle through scattering will mainly be from scattering from electrons– me << Mnucleus , maximum energy exchange is
higher for electrons than nuclei.
• We are interested in particles of mass M ( normally M >> me ), charge Z, scattering from stationary electrons.
• However, cross-section the same as for electrons scattering from stationary particle.
Large Energy Loss Scatters.
• “Close” collisions (~ large q2 , small b) Large angle and/or Large q2 can resolve electrons in atom Minimum energy exchange to consider
a collision as “close” , min ,set by some ionization energy scale of medium ( 10-100eV)
• Maximum energy loss ( Tmax ) set by kinematics
Small Energy Loss Scatters.
• “Distant” collisions ( ~ low q2 , large b)Low energy loss Low q2 photons will interact with atom as
a whole.
• Maximum energy loss set by boundary with “close” collisions
• Minimum energy loss set by smallest available excitation energy of medium
Rutherford Cross-section• Consider the electron moving and the
projectile stationary.(“Close” collisions) • Earlier we “derived” the Rutherford
scattering cross-section:– z = electron charge=1
– mp= electron mass
– T = projectile kinetic E
• The complete result is:
4
2
4
2121
cm
zTz
dd
e
)2/(sec2
4
2.
Cop
czd
d Ruth
Relativistic Corrections – Mott Scattering Formula
• As 1, Rutherford formula becomes inaccurate. Simplest modifications give Mott formula.– First order perturbation theory
– Assumes no recoil of target
– No spin or structure effects.
2
sin2
cosec4
224
22
222
Mott
1
epcz
dd
Mott Cross-Section as a Function of q2
• Can write Mott formula in terms of q2, rather than – put q2=4pe
2sin2(/2)
• Hence:
2
22
24
222
2 41
4
ep
q
q
cz
dq
d
Back to Stationary Electron and Moving Projectile
• Eliminate electron momentum– pe= me
• Have Mott Scattering formula for moving electron. Now move back to lab frame, where electron is stationary. – For free electron at rest q2=2me – Hence
eee mm
cz
d
d
mdq
d222
222
2 21
)2(
4
2
1
Energy Loss from “Close” collisions
• In summary, cross-section for transfer of energy in the range to + d
dmm
czd
d
d
ee
222
222
21
12
Energy Loss Per Unit Distance
• So the energy –dE lost by a particle passing through distance dx in a material is:
• The number density,n, is
n = NA / A
(=density, NA=Avogadro’s number,A= molar mass )
max
cut
dd
dnZdxdE
de/dx From “Close” Interactions
• Putting in the differential cross-section and integrating gives
• Recall expression for max :
• For small compared to M/me
e
cut
cute
A
mA
Zz
m
cN
dx
dE2
maxmax2
2
22
close 2ln
12
2
22
max21
2
MmMm
mm
s
p
ee
ee
em2max 2
de/dx From “Close” Interactions
• Putting in expression for max and assuming that max- cut ~ max
2max
22
22
close
ln12
cute
A
A
Zz
m
cN
dx
dE
dE/dx for “Distant” collisions
• Can no longer treat electrons as free.
rather more complicated (but result looks rather similar)
• Get a term ln(q2min/ q2
cut)
• q2min=I2/(
• Lower energy limit I, represents some typical ionisation energy – ( I ~ 10Z eV for Z>10)
dE/dx for “Distant” collisions
• q2max= 2memin (any energy exchange
larger than cut is counted as “close”)
• Get:
• The “” term models the “density effect” (polarization of the medium)
22
cut2
22
22
distant
2ln
12
I
m
A
Zz
m
cN
dx
dE e
e
A
21ln)/ln(22 I
p
dE/dx – the Bethe-Bloch Equation
• Adding contributions from “close” and “distant” collisions we get the Bethe-Bloch equation:
• Where:
2
2ln
2
114 22
max22
22
22
I
m
A
Zz
m
cN
dx
dE e
e
A
1
31
22
mol g
cm g cm MeV 307.0
4
AAm
cN
e
A
Approximate Bethe-Bloch Equation
I
m
A
Zz
m
cN
dx
dE e
e
A22
22
22 2ln
14
• There are different versions of the formula.
• At moderate energies can approximate by ignoring and putting max=()2me
Bethe-Bloch References
• References:– Rossi, “High-Energy Particles”, chap. 2
– Jackson, “Classical Electrodynamics”, chap. 13
– Fano, Ann. Rev. Nucl. Sci., Vol. 13 (1963), p.1
– Particle Data Group (PDG), in the WWW edition of “Review of Particle Properties”, via link from the course Web site
dE/dx – 1/
• At low velocity, energy loss falls steeply with increasing energy (1/
• nteracting particles have less time to “see” each other at higher speeds.– (Curve is actually better modelled by
Virtual Interaction Range vs. and
• Recall (from “Quarks and Leptons”) that the “range” of a virtual particle is proportional to 1/|q|
– Range ~ c/|q2|1/2 ( large q2 short distances )
• q2 small for zero angle : |q| /
– Hence virtual photon range c/
• I.e. range increases with
• Range decreases with
• Range won’t increase indefinitely – polarisation of medium.
Maximum Range for “Close” Interactions
• The lower limit for energy exchange, min , sets an upper limit on virtual photon range (for “close” interactions)
• |qmin| min/• Rmax 1/ |qmin| / min
• Bigger Rmax more target particles to couple to – rate of interactions will increase.
• Recall that
• Since qmin2 decreases with increasing …..
Cross-section (and hence prob. of interaction) increases with increasing (“relativistic rise”)
4
2
2
EM
dE/dx – “Minimum Ionising Particles”
• As 1 curve flattens off to a minimum– Minimum reached at
• “Minimum ionising” value is roughly 2 MeV g-1cm2
dE/dx – Relativistic Rise
• Above minimum dE/dx rises with momentum “Relativistic Rise”– ln( term in Bethe-Bloch
• From “derivation” can see two factors, each ln( – max rises with more “close” collisions– Decrease in q2
min for “distant” collisions – increase in range of virtual photons.
dE/dx – “Density” / “Screening”
• Rise flattens off in solids (and to a lesser extent gasses) due to “density effect”– Modeled by 2 in Bethe-Bloch
• Comes from bulk effects such as polarization ( virtual photons are “screened” from distant atoms )– Stronger in solids (e.g. copper – plotted above) than in
gases.
dE/dx – “Femi Plateau”
– If high-energy knock-on electrons ( -rays) are excluded, measured dE/dx reaches a constant value less than 1.5 “minimum ionising” (“Fermi Plateau”)
– At very high energy, radiative energy loss processes (bremsstrahlung) become important.
dE/dx Data
• Data from gaseous track detector.– Each point from a
single particle
– Several energy loss samples for each point
– “Averaged” to get energy loss
– Fluctuations easily seen ( see later in course)
dE/dx (keV/cm)
p (GeV/c)
p
K
e
dE/dx in Different Materials• Z/A similar for
most nuclei• Effective
ionization, I, varies only slowly with Z
• Min. ionization occurs for ~ same value of
Scattering of a Charged Particle by Exchange of
Virtual Photons
Charged Particles and Virtual Photons
• EM interaction is mediated by exchange of photons.
• “On-shell” photons have zero mass, but from uncertainty principle:
• p xh/2– The more localized the photon near a charge the
larger the uncertainty in its momentum
• E th/2– The shorter the life of a photon (more confined
to source charge) the larger the uncertainty in its energy.
Charged Particle as Photon Source
• View charged particle as surrounded by cloud of virtual photons.
• Higher energy virtual photon (more “off-shell”) clustered more tightly round charge.
• As charged particle passes through matter the virtual photons will interact with medium.
• Hence, study interaction of photons with matter:
Interaction of Photons with Matter
• Can not study interaction of virtual photons directly, but can study interaction of real photons.
• Photon Interactions are interesting in their own right.
• Different processes important at different energy ranges.
Coherent Elastic Scattering• Two sorts:a) Rayleigh scattering
from atomic electronsb) Thompson scattering
from nuclear charge.- Does not excite atoms
or cause energy loss.- Doesn’t leave a
detectable signal in medium
Photo-excitation• Photon absorbed by
atom – exciting it into a higher energy state.
• Strong absorption peaks for photon energies corresponding to atomic transitions.
• Mainly in low energy (<10eV ) region– Not labelled on plot
Photoelectric Effect
• Photon absorbed by atom which expels and electron.
• Cross-section depends on atomic charge Z– At high energies varies
approx. as Z5
( ZC=6, ZPb= 82 )
Compton Scattering
• In Rayleigh/Thompson scattering, photon scatters off all the electrons in an atom coherently.
• In Compton scattering the photon interacts with a single electron– Needs a shorter wavelength (higher
energy) photon
• Scattered electron usually has enough energy to leave atom ionization
Compton Scattering• Atomic cross-section
proportional to the number of electrons in the atom– Pb /C = 82/6
• KE of scattered electron (incoming photon E, scattered through ):
)cos1(
)cos1(2
2
Ecm
ET
e
Pair Production
• When a photon has an energy greater than twice the energy of an electron it can convert to an electron and a positron
• A photon in free space can not create a e+/e- pair ( conservation of p,E )
• Can convert if third body to transfer momentum to pair production near nuclei
Pair Production
• Heavy nucleus takes less energy for a given momentum exchangeThreshold energy higher
for carbon than lead
• Can also get pair production near atomic electrons (energy threshold higher)
• Cross-section ~ Z2
Photonuclear Absorption• Photons ( ~ >10MeV)
can excite resonant states in nuclei– Nuclear analogue of
atomic photo-excitation.– Cross-section small
overall, but with peaks in region of nuclear “giant resonance”.
• (Photons can cause nuclear photo-disintegration)
Photon Propagation in a Medium
• The effect of the medium a photon travels through can be characterised by the dielectric constant.
• In an dense medium a photon will interact with many atoms simultaneously. ( Depending on wavelength/energy of photon)
Dielectric Constant
• Elastic collisions at zero angle do not change the energy of a photon, but they do change its phase.– Described by the real component of
• Interactions resulting in photon absorption or energy loss are described by the imaginary component of
Dielectric Constant – Phase Velocity
• Dielectric constant describes the way that the interaction of photons with the medium modifies their phase velocity u–the phase velocity is not necessarily
the speed at which a signal is transmitted.
)()(
c
u
Refractive Index and Dielectric Constant
• The refractive index for an optical medium and its dielectric constant are related by
• The dielectric constant (refractive index) is a function of the photon energy.
)(n
Dielectric constant as function of energy
Im
R e - 1
1 10 100 1000 photon energy(eV)
optica lregion
resonanceregion
X-rayregion
Dielectric Constant – Optical Region
Im
R e - 1
1 10 100 1000 photon energy(eV)
optica lregion
resonanceregion
X-rayregion
• Photon energy below threshold for exciting atomic transitions.
• Interaction modifies the phase velocity of the light.
• Absorption low – medium transparent
• Dielectric constant real and greater than 1– Phase velocity < c
Dielectric Constant – Resonance Region
Im
R e - 1
1 10 100 1000 photon energy(eV)
optica lregion
resonanceregion
X-rayregion
• Photon energy comparable with atomic excitation energy.
• Absorption high – medium opaque
– Imlarge
Dielectric Constant – X-ray Region
Im
R e - 1
1 10 100 1000 photon energy(eV)
optica lregion
resonanceregion
X-rayregion
• Photon energy well above ionization energy.– Photon “sees” atomic
electrons as free particles.
– Scattered electrons can have high energy: -rays
• Absorption low•
– Phase velocity > c
EM Field from a Moving Charge
• For a high energy particle traveling through a medium individual interactions will not affect energy significantly– View particle as traveling at constant
velocity through an infinite medium.
• In rest-frame of particle, EM field is a static coulomb field.
EM Field from Moving Charge
• Write the EM field in the lab frame of the moving charge as a sum of plane waves:
))(.()(),( tkiedkt rkkr
Field from Moving Charge
• The field at a given position relative to the particle remains the same:Components of field must propagate
with velocity of particle: v . k
Effect of the Medium
• In a vacuum and c related by = c k
• The effect of the medium is to modify the phase velocity of the photons
So we have: nccu
)()(
222 ck
A simple 2-D model
• Limit ourselves to two-dimensions to keep things simple. (no Bessel functions)
x
y
Particle track
v=c
Wave-number, k, in 2-D
• Components of wave-number k
• Rearrange to get ky :
vkx
2
2
2
2222
cukkk yx
2/1
2
22/1
2
2
1)(
1
c
v
vu
v
vk y
Moderate velocity – virtual photons• If particle velocity, v, through the
medium is less than the phase velocity:ky is imaginary
I.e. virtual photons rather than free propagation.
• Define a transverse range, y0 :
– y0 = i / ky
• k.r = kxx + kyy …. So:
0y
yvtx
viykitxkitykxkiti eeeeee yxyx
rk
Range of Virtual Photons
• EM field shows an exponential fall with transverse distance from track ( e-y )
• (= Decrease in density of cloud of virtual photons away from their source)
• Transverse range, y0 :2/1
2
2
0
)(1
c
vv
k
iy
y
1
1
1
122
20
ccy
Range of transverse EM field
• Photon energy,
• Range of virtual photons decreases with increasing photon energy
11
1
11
1
2222
0 Ey cc
E
Dielectric constant,
• In optical region of photon energies, phase velocity , u < c
• Fast particles can have a velocity, v>uky real , I.e. photons are real and can
propagate away from particle track
Cerenkov radiation
Cerenkov Radiation in Popular Culture…
• Picture of Cerenkov radiation from the core of a water-cooled nuclear reactor.
• given off by fast electrons emerging from fission reactions
Dielectric constant,
• If e.g. in X-ray region of photon energies) phase velocity u>c– No particle can travel faster than u
– ky always imaginary ( y0 always real )
– No Cerenkov radiation.
Relativistic Rise
• Photon range:
• Range rises with momentum ( p = m )
• If y0=c/relativistic rise
11
1
22
0 Ey c
Density Effect• Relativistic rise will not continue
indefinitely due to (1- term• Range of virtual photons at high photon
energies tends to:
• As tends to 1 (material acts like the vacuum) , range tends to infinity
• Classical view: polarisation of medium screening of charge
1
10
cy
The Allison-Cobb Expression for Energy Loss
• References:– Kleinknecht “Detectors for Particle Radiation”,
chap. 1– Allison and Cobb, Ann. Rev. Nucl. Part. Sci,
Vol. 70 (1980), p. 253– Allison and Wright in “Experimental Techniques
in High Energy Physics”, ed. T. Ferbel, p. 371
Steps in Allison & Cobb Derivation
1) Solve Maxwell’s equation in the medium to obtain a field (produced by polarisation of the medium) through which the particle moves.
2) that the particle loses energy by doing work against this field. (This is the same as the energy transfer by virtual particles, but in classical terms)
Steps in Allison&Cobb3) Express energy loss as integral over
angular frequency • Interpret energy loss as energy transfer by
photons of energy .
(Quantum picture)
(photon)
time
space
A Batom
fastchargedparticle
E
Steps in Allison&Cobb
• Integral over and k becomes integral over E ( = h`and p ( = h`k ) of exchanged photons
• (Photons are virtual so E,p not related by E=pc )
v dEdp
dEdpdEn
dx
dE/
2
0
Steps in Allison&Cobb
4) Don’t have direct data for interactions of virtual photons.use cross-sections for real photons,
dispersion relations and assume that high energy virtual photons interact with free electrons
- This step is the same as calculating k,
Steps in Allison&Cobb
5) Previous step allows us to integrate over virtual photon momentum, p
Get energy loss per unit length as integral over photon energy:
0
dEdE
dEn
dx
dE
Allison&Cobb – ddE• Differential cross-section per unit
energy loss:
• i• phase ( 1 - i
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE• Terms in ddE represent contributions
to generic Feynman scattering diagram– Factor z2 from coupling of photon to fast
particle
– As before, all terms contain 1/slower particles spends more time in vicinity of atoms in medium higher probability of interaction
– Coupling to medium at “B” described by photo-absorption cross section (E)
Allison&Cobb – ddE• First two terms in
ddE correspond to exchange of transversely polarized photons.
• Only significant at high speed (
• (Real photons have to be transversely polarized)
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE
• Last two terms in ddE correspond to exchange of longitudinally polarized photons.
• Adding transversely polarized photons ~ RutherfordMott
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE
• First term has ln(behaviour
- (relativistic rise found in 2D model)
- Eventual saturation at Fermi plateau
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE• For low energy
photons in optical region, soonly second term contributes.
• In this regime, term describes Cerenkov radiation (see later)
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE• Third term
corresponds to photoelectric emission of electrons.
• Photon energy in the resonance region.
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Allison&Cobb – ddE• Fourth term
describes Compton scattering of energetic photons from atomic electrons.
• production of -rays.
E
EdZ
E
E
z
E
mc
EZ
Ez
cn
z
EZ
Ez
dE
d
022
2
22
2
2
212
2
2
22
42
12
2
2
1
2ln
1
1ln2
1
Features of Allison&Cobb dE/dx
• Overall features the same as Bethe-Bloche (of course!)
• Better fit to data than Bethe-Bloche
• … now the model gives: visible-wavelength photons produced by fast particles in a transparent medium.
dEdx
1 10 100 1000 10000
minimumionising
relativisticrise
Fermiplateau
Cerenkov Radiation
• For photons with energy in visible region, refractive index of transparent materials, n>1Phase velocity, u < cTransverse component photon momentum
( ky in our 2D model) can be real (if charged particle velocity v>u)Passage of charged particle through medium
produces real rather than virtual photons.
Cerenkov radiation
• Photons emitted at angle c:
• Cerenkov radiation (optical light) produced when >1/n
wavefro
nt
light,
velocity
=c/n
Fast particle, velocity=c
1tan2
2
u
v
k
k
x
yc nv
c
v
uc 1
cos
Cerenkov Light Used to Detect Neutrinos -SNO
Cerenkov Angle• High refractive index low threshold for Cerenkov production
• Can either use threshold or angle for particle identification
n=1.5 n=1.1
c
Intensity of Cerenkov Radiation• Intensity of Cerenkov radiation described by
second term in Allison and Cobb formula:
– (only describes Cerenkov radiation in optical region)
– In this region so– Multiply ddE by NEdE to get energy loss due to
Cerenkov radiation per unit length.
2
122
2 1
cn
z
dE
d
Intensity of Cerenkov Radiation
• Have found dE/dx from Cerenkov• Since energy of photon E = h’can
calculateflux of photons per unit length:
• ( Phase angle jumps from 0 for to when ie. At threshold )
cccdd
Nd 22
sin1
1x
Intensity of Cerenkov Radiation
• Number of photons per unit length increases as increases.
• Over range of visible light photon energies: c
dNdx(arb units)
24 sin105L
N
Intensity of Cerenkov Radiation
• Spectrum almost flat over optical region.
• Photons only emitted in the optical region.– In the resonance region there is too much
absorption
– In the X-ray region so u>c, so particle velocity always less than phase velocity
Transition Radiation
• Transition radiation is closely related to Cerenkov radiation.
• Occurs when a charged particle crosses the boundary between materials of different refractive indices.
Transition Radiation
• Can think of as diffractively broadened Cerenkov radiation (thin source)– Broadening of Cerenkov angle causes
radiation at angles that would otherwise be unphysical e.g. in X-ray region
• Can also be thought of as apparent acceleration of the charge as it passes through the boundary– (Think of an object underwater breaking
through surface)
Intensity of Transition Radiation• Total energy flux of
transition radiation:
• “Typical” photon energy:
• Number of photons ~ many “foils”
• Plasma frequency:• In x-ray region:
ptot 3
ptypE 4
1
m
eN
ep
0
2
2
2
1 p
Intensity of Transition radiation• The energy given off in transition
radiation is proportional to – Compare this to Cerenkov radiation where
the threshold and the intensity are a function of
– Transition radiation can be used to discriminate between particles of high momenta – where very close to 1 and Cerenkov detectors can not discriminate.
• TR intensity saturates at high values of destructive interference between faces of radiator)
Transition Radiation
Bd0J/ψ Ks
0
~1 TR hit
~7 TR hits
Electrons with radiator
Electrons without radiator
Two threshold analysis
MIP threshold 0.2 keV – precise tracking/drift time determination
TR threshold 5.5 keV – electron/pion separation
Transition radiation is produced when a charged ultra-relativistic particle crosses the interface between different media, PP (fibers or foils) & air for the TRT.
TR photons are emitted at very small angle with respect to the parent-particle trajectory.
Energy deposition in the TRT is the sum of ionization losses of charged particles (~2 keV) and the larger deposition due to TR photon absorption (> 5 keV)
dE/dx for Electrons
• Assumption that individual interactions transfer energy much less than particle energy is not true for electrons (much lighter than other charged particles)
• Quantum effects – incident particle identical to atomic electrons it is interacting with.
• Different behaviour of energy loss as function of momentum.
dE/dx for Electrons
• Modified Bethe-Bloch equation:
• T = electron energy• ( c.f. simplified Bethe-Bloch for other
particles: )
I
m
A
Zz
m
cN
dx
dE e
e
A2
22
222
ln14
222
2
2
22 1122ln
2ln
12
I
Tm
A
Z
m
cN
dx
dE e
e
A
electron
Energy Loss Processes other than Ionization
• Bremsstrahlung– Electromagnetic showers
• Hadronic Interactions– Hadronic showers
• Charged Particles in a Magnetic Field… Ok, so this isn’t an energy loss process
• Synchrotron Radiation
Energy Loss due to Radiation• Have assumed that energy exchanged per
interaction is small compared to particle energy.• Particle can transfer a large amount near to
atomic nuclei– Nuclei are surrounded by very high energy virtual
photons
• The particle is momentum is significantly changed by interaction – i.e. particle is accelerated
• Particle radiates photons • Process called Bremsstrahlung
– Lit. “braking radiation”
Bremsstrahlung• Simplest Feynman diagrams have
three vertices– Two for exchange of virtual photon
between nucleus (charge Z) and particle (charge z)
– One for emission of real photon
• Cross section:– Factor Z2 from nucleus
– Factor from vertices
– Can view as separating off one of the cloud of virtual photons surrounding particle
N u c l e u s
32 Z
Bremstrahlung
• The lighter the particle the greater its acceleration for a given momentum exchange – cross-section has a factor of 1/m2
• If the energy of the emitted photon is the differential cross-section is
2
32
m
Z
2
232
em
cZ
d
d
dE/dx from Bremsstrahlung• Integrate over photon energy to get
energy lost by particle per unit length:
– Where max~ particle kinetic energy T andmin~0. ( n is the density of nuclei)
– Put in factors missed out by hand-waving (eg. screening) and get (for electrons)
)('' minmax2
232max
min
eRAD m
cnZd
d
dn
dx
dE
3/12
232 183ln
4
ZT
m
cnZ
dx
dE
eRAD
Bremsstrahlung
Critical Energy
• All high energy charged particles can loose energy through bremsstrahlung.
• For a given material the energy at which bremsstrahlung becomes significant relative to energy loss by ionization is dependant on the particle mass
• Particle energy at which bremsstrahlung overtakes ionization called the “critical energy”, Ec (~m2)– ( Other definitions as well )
Electron Bremsstrahlung• For (current) particle detectors in high
energy physics, the only particle that undergoes significant bremsstrahlung is the electron.
• Critical energy for electrons– Heavy metals (Pb,W,U) ~ 10MeV– Hydrogen ~ 300MeV
• Critical energy for muons ~ 100’s GeV• High energy hadrons – hadronic
interactions much more important.
Radiation Length• The rate of energy loss due to
bremsstrahlung is proportional to the kinetic energy:
• I.e.• So…
… where X0 is the radiation length– Distance in which energy reduced by
factor of e
Tdx
dT
dx
dE
0X
dx
T
dT
00 X
x
eTT
1
3/12
232
0
183ln
4
Zm
cnZX
e
Radiation Length• Number density of nuclei, n , Avagadro’s
number NA and density, related by
– n NA A
– E.g. Pb has X0 for electrons of 6.37g/cm2 (0.56cm)
• Z/A ~ constant, so
X0 • Use high Z materials to shield against
photons (and electrons)
Radiation Length• Units: X0 is often (indeed usually) given in
terms of g/cm2
– Multiply by density, , to get Xo as a distance.
• Radiation length of mixture/compound ( wj = weight fraction)
• Radiation length depends on incident particle ( ~ m2 ).
XwX jj01
236mmuon
0.56cmelectron
X0 in leadParticle
Bremsstrahlung Energy Spectrum• Energy spectrum ~ flat up to maximum
energy max = T
– Cross-section for a photon of energy – Photon energy E = h’– dE = dN d ~ constant
• Spectrum doesn’t actually have a sharp cut-off at =0 and = T : rolls off from e.g. screening of nuclear charge.
Bremsstrahlung Energy Spectrum
• Plot of scaled bremsstrahlung cross-section as a function of y =
Electromagnetic and Hadronic Showers
Electromagnetic Showers
• High energy electron produces photon through bremsstrahlung
• Photon produces e+ e- through pair production
Electromagnetic Showers
• Shower of electrons/positrons and photons continues until the energy of the particles is too low for further multiplication ( E<Ec ) – after which the shower dies away
• Electrons/photons will deposit energy by ionization/excitation of the medium.
• If energy deposited gives a signal can measure total energy of incoming particle
Shower Profile• Shower profile for a Lead/Scintillator
calorimeter:
EM-Shower Profile
• Maximum number of shower particles ( “shower maximum” ) occurs at a depth of roughly 5X0 into the absorber.
• Depth of shower maximum depends logarithmically on particle energy.
• In transverse direction energy deposited falls off (very approximately) exponentially.– Length scale (the Moliere radius) ~ (7g/cm2)(A/Z)– 90% of energy contained in ~ 1 RM
– RM= 1.5cm in lead
EM-Shower Transverse Profile• Results from a test Fe calorimeter with
500MeV electrons:
EM-Shower Profile
• Example: 3.2 GeV shower in lead has 400 particles in the cascade at shower maximum, which occurs at a depth of 6X0 ( ie. 3.36cm in lead)
• A total absorber thickness of 25-30X0 is
enough to absorb most of the energy in a shower.
Electromagnetic Showers
• Total energy deposited is the same for electron and photon of equal energy
• Total energy deposited depends linearly on the energy of incoming particle.
• Maximum number of shower particles ( “shower max” ) occurs at a depth of roughly 5X0 into the absorber.
• Depth of shower max depends logarithmically on particle energy.
Hadronic Showers• High-energy hadrons give hadronic
showers.• Hadron interacts with nucleus by the strong
interaction.• Number of particles produced in each
collision ln(E)• Length scale :
– I is nuclear cross-section for strong interaction– I = hadronic interaction length
IAI N
A
n
1I
Hadronic Showers – Length Scale
• I is approximately independent of particle energy and type – nucleus behaves like a black ball
• In terms of g/cm2 : I (35 g cm-2)A1/3
• Depth of shower maximum depends logarithmically on incident particle energy:
• xshower-max/I 0.2 ln ( E/1GeV ) + 0.7
Hadronic Showers – Length Scale
• At typical HEP energies need roughly 9 I to contain average of 99% of energy in hadronic shower.
Hadronic and EM length scales• For materials other than hydrogen I is
several times larger than X0
6.37194Pb
12.86134.9Cu
61.2850.8H2
X0
(g/cm2)I
(g/cm2)Material
Hadronic Shower Development• On average about 1/3 of the particles
produced in each hadronic interaction are neutral. Mainly pions, 0
• 0 rapidly decays to photons.
• Photons initiate EM showers.• For an energetic hadronic shower most
of the detectable energy deposited is from e+/e- in EM-shower from decaying 0
Fluctuations in Hadronic Showers• Fluctuation in detectable ( ionization, atomic
excitation) energy deposited is greater than for electromagnetic showers:
• Fluctuation in “neutral fraction” of shower ( 0,n,etc. )– In general response to hadron and electron of
same energy is not the same (non compensating)
• Energy in nuclear binding effects is not “detectable”.
Fluctuations in Hadronic Showers
• Fraction of undetectable energy changes with particle energy:
Total visible energy deposited does not depend linearly on the energy of incoming particle.
Detectable energy deposited in detector depends on type of particle as well at its energy
Charged Particle Motion in a Magnetic Field
• Lorentz Force
• Radius of Curvature
Lorentz Force
• A particle of charge ze , and velocity v moving in a magnetic field B feels a force F given by:
• Force is at right angles to particle’s path – so direction of velocity changes but not its magnitude.
• In general the path is a helix, with the axis along the field lines.
Bvp
F edt
d
Particle Trajectory• Particle travelling a small distance through a
magnetic field.• Look in plane perpendicular to the field:
– Compt. of velocity,momentum perp. to field =
d
d
m o m e n t u m t r i a n g l e
p a r t i c l e p a t h
p
p
d m a g n e t i c
f i e l d
p
vdt
(into page)
pv ,
Particle Trajectory• Force: • Change in momentum: • Angular deflection:
• Path is part of a circle, so d = (vdt)/
• Hence • And
– pt in GeV , B in Tesla, in metres
d
d
m o m e n t u m t r i a n g l e
p a r t i c l e p a t h
p
p
d m a g n e t i c
f i e l d
p
vdt
(into page)
BvzeF dtBvzepd )(
pdpd
dtvp
dp
eBp 3.0 Bp
Synchrotron Radiation
Synchrotron Radiation
• Accelerated charged particles radiate.• High energy particles in a high magnetic field
can radiate energetic photons: synchroton radiation
• In particle centre of mass frame the magnetic field in the lab frame transforms to have an electric component.– 4-vector for EM field A = ( , A )– Look at acceleration caused by this E field:
Synchrotron Radiation
• For a particle moving with velocity v in a magnetic field B:
– (particle moving along x-axis. B field at an angle relative to x-axis)
• Resulting acceleration:
– Here me is the rest energy
sinBvEy
2
sin
cm
vBe
m
Fv
e
Synchrotron Radiation
2
223
sinsin)( c
m
Bcev
e
• Since p = mc and p=eBc2then
me= eBc/– Again, is the radius of curvature in
field)
Synchrotron Radiation - Intensity• From classical electrodynamics, energy
radiated from an accelerating charge:
– Since p/mc at a given momentum dE/dt (p/m)4
– dE/dt calculated in CoM frame but same in lab. frame.
30
22
6 c
ve
dt
dE
4
2
4
0
422
sin3
2
6
sin)( cc
ce
dt
dE
Synchrotron Radiation – Angular Distribution
• Accelerated charge Dipole field• In laboratory frame the boost distorts
this shape ( “headlights effect” ) v
v
o r b i t c e n t r e
C M
L a b .
Synchrotron Radiation – Frequency Distribution
• In classical electrodynamics frequency of the emitted radiation is the same as the frequency of the accelerating force.– E.g. Accelerate electrons in an antenna in
simple harmonic motion, frequency , get an EM wave of frequency
• Relativistic boost increases this by
• In relativistic circular motion get a distribution of frequency (photon energy)
Synchrotron Radiation – Frequency Distribution
• Characteristic frequency c=c/2a