Interaction with matter
Understanding/description of interaction between matter and high energetic particles/radiation
important:
●
●
●
●
Principles of detection of particles/radiation
Limitation of detectors
Efficiency
Position resolution
Energy resolution
Time resolution
Impact on biological systems
Radiation damage
Radiation protection
Radiation therapy
Literature
K. Kleinknecht, Teilchendetektoren
C. Grupen, Teilchendetektoren (BI Wissenschaftsverlag)
J. Ferbel, Experimental Techniques in High Energy Physics
Particle data group, chapter 26 , http://pdg.lbl.gov/pdg.html
The particle detector brief book http://rkb.home.cern.ch/rkb/titleD.html
dEdX: http://www.srim.org/SRIM/SRIMPICS/SRIM-High%20Velocity.pdf
Katharina Müller, Autumn 14
1
Interaction with matter
Energy loss per distance (dE/dx)
many different interactions, dominating processes
depend on energy and particle type
e± ,μ±, ± , ,
p,n, 
Z, A, 
z, m, E, Θ
Charged particles: defined reach
●
Ionisation and excitation of electrons on shell
→ Bethe-Bloch formula
●
●
Coulomb Scattering: Scattering in Coulomb field of nucleus
small energy loss, but deflection
Bremsstrahlung : dominant for low masses → radiation length
Photons: Absorption in matter, highly energy dependent – attenuation, no defined reach
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Photo electrical effect
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Compton scattering
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Pair production
Hadrons: inelastic scattering
●
Hadron+nucleus → π±,0 , K, p,n, fragments of nucleus
Katharina Müller, Autumn 14
2
Energy loss of charged particles in matter
Energy loss due to ionisation and excitation
Energy loss per distance (dE/dx) : many theoretical works
●
N. Bohr
Classical derivation
●
Bethe, Bloch Quantum mechanical description
●
L. Landau
Charge distribution function
●
E. Fermi
Density correction
...
Katharina Müller, Autumn 14
3
Ionisation: classical derivation (Bohr)
Energy loss via inelastic collisions with electrons on shell
→ Ionisation and excitation
Assumptions: M(particle) >> me
Shell electron at rest (collision time small wrt orbital time)
Projectile
M, v = βc , charge ze, Target: charge Ze
x
ze
r

b
b: collision parameter
Ze
Energy transfer onto target with mass mt: ΔE=Δp/(2mt)
∞
Momentum transfer:
time integral
longitudinal forces cancel
only transversal forces important
Katharina Müller, Autumn 14
 p= ∫ F Coulomb dt
−∞
FC=
1
zZe 2 /r 2
4  0
F l  x=−F l −x
b
F t =F C⋅ =F C⋅cos 
∣r∣
4
4
Ionisation: classical derivation (Bohr) I
2
r=b/cosθ
1 zZ e
1 z Z e 2 ˙3
cos˙ =
cos 
Transversal forces on projectile F C t =
2
4  0 r
4  0 b2
Momentum transfer onto target
Integration →
∞
∞
dx
1 zZe
 p b= ∫ F C t dt= ∫ F C t =
v 4   b2
−∞
−∞
0
2
b d / cos 
2 ∞
dx

v
∫ cos3 
−∞
 / 2
1 z Z e2
1 2 z Z e2
=
∫ cos  d = 4   b v
4   0 b v − / 2
0
Energy transfer onto target with mass mt
2 4
 p2
2z e
 E b=
=
2 mt
me c 2  2 b 2
1
≡1
4  0
shell electron Z = 1, mt=me
1/m: collisions with nucleus may be neglected
Now: Sum over all shell electrons
Integration over all collision parameters b
Katharina Müller, Autumn 14
5
Ionisation: classical derivation (Bohr) II
Energy loss due to collisions with electrons in tube
(length dx, thickness db) at radius b:
# of electrons: 2 π b db dx Ne electron density
−dE  b= E  b N e dV =
4  z2 e 4
2 2
me c 
Ne
b
dx
db
Ne = NA Z/A ρ
db
dx (NA Avogadro's constant, A mass number)
b
Integration over collision parameter from bmin to bmax
2 4
b max
2 4
4 z e
−dE
Z 1
dE 4  z e
Z
b=

N
db

−
=

N
ln
AA b
AA
2 2
2 2
dx
dx
b min
me c 
me c 
maximal energy transfer: central collision
2 4
2
z
e
2
2 2
 E b=2  me  c =
m e c2  2 b 2
minimal energy transfer: mean excitation
energy I (averaged excitation potential per
electron in the target)
Katharina Müller, Autumn 14
b min =
note: 1/β2
z e2
 me c 2  2
2
ze
b max =
c

2
me I
6
(dE/dx) classical derivation (Bohr) III
−dE 4  z e 
Z b max
=
N A ln
2 2
dx
A b min
m c 
2 4
e
b min =
z e2
 m e c2  2
z e2
2
b max =
 c me I

Classical formula for energy loss due to ionisation
2
−dE K z  Z 1
= 2
ln
dx
A
2

2 c 2  2 2 m e
I
K=
4 e
4
c2 me
N A =0.31 MeV cm 2 / g
Unit [dE/dx] = MeV /cm
Warning: usually energy loss is divided by density: [dE/dX] = MeV cm2/g
→ dE/dX almost independent of material
only material dependence through Z/A and ln(1/I)
Katharina Müller, Autumn 14
7
(dE/dx) Bethe Bloch Formula
full quantum mechanical derivation: Bethe-Bloch Formula
−dE
Z 1 1
=K z2
[ ln
dX
A 2 2
2 m e c 2  2  2 T max
I2
 C
−2 − − ]
2 Z
K=
4  e4
2
c me
2
N A =0.31 MeV cm / g
I mean excitation energy
Z atomic number
A mass number of absorber
Validity: 6 MeV < E < 6 GeV (π), generally 0.02< β<0.99 (1% accuracy!!)
T
max:
max kinematic energy transferred
z: Charge of particle
dE/dX ∝ z2
Corrections
●
●
δ(β)
C/Z
Density correction due to polarisation, important for high energies
Correction close to shell boundaries, relevant for small energies
1% @ βγ =0.3
Katharina Müller, Autumn 14
8
(dE/dx) Bethe-Bloch-(Sternheimer) Formula
classical result:
2 2 2
2
−dE K z Z 1 2 m e c  
= 2
ln
dX
I
 A2
full quantum mechanical derivation: Bethe-Bloch Formula
−dE
Z 1 1
=K z 2
[ ln
2
dX
Aβ 2
2 me c 2 β 2 γ 2 T max
I2
C
−β2 − δ − ]
2 Z
describes mean energy loss or stopping power
●
T
max:
max kinematic energy transferred from particle (M) onto electron
T max =
●
●
●
●
●
●
2 me c 2  2  2
2
12  me / M me / M 
≈ 2 me c 2  2  2
≈  M c2
 m e ≪M
 ∞
small energies and particles with high masses: result equals classical result
except factor 2 (imperfect description of very far collisions :
binding of electrons cannot be neglected)
independent of mass of particle ! (γm e<<M)
Unit MeV/(g/cm2) dx = ρ ds is material occupancy
dE/dX almost independent of target material
Formula needs to be modified for electrons
Bethe-Bloch formula is not valid for slow particles βγ<0.02 where dE/dX ∝ β
Katharina Müller, Autumn 14
9
9
Energy loss dE/dX
dE/dx of muons in copper
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~β
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~1/β
2
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●
~2 ln(γβ)
Titel
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●
βγ < 3 dE/dX ∝ 1/β2
βγ ≃ 3.5 minimum
dE/dX 1-1.7 MeV cm2/g
βγ > 3.5 logarithmic rise
dE/dX ≃ 2 MeV cm2/g
βγ > 1000 Bremsstrahlung
dominant
very low energy: BB not valid,
empirical models
low energies: shell corrections
high energies: density corr.
PDG http://pdg.lbl.gov/pdg.html
Minimum: βγ 3-4 minimum ionising particle (MIP)
looses ~ 1-1.7 MeV/(g/cm2)
Katharina Müller, Autumn 14
10
Shell correction
−dE
Z 1 1
=K z2
[ ln
2
dX
A 2
2 m e c 2  2  2 T max
I2
●
●
 C
−2 − − ]
2 Z
Shell corrections (C/Z) constitute a correction to
slow particles (e.g for protons in the energy range
of 1-100 MeV)
Correction for the assumption that particle velocity >> bound electron velocity
●
Assumption that electron is at rest not valid
●
Maximal about 6%
●
Calculation with various approximations
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●
J. F. Ziegler, Applied Physics Reviews / J. Applied Physics, 85, 1249-1272 (1999).
http://www.srim.org/SRIM/SRIMPICS/IONIZ.htm
Copper: about 1% at βγ=0.3 (6 MeV Pion), decreases rapidly with velocity
Katharina Müller, Autumn 14
11
Mean excitation energy I
2 2 2
2
m
c
  T max
−dE
Z
1
1
e
2
2  C
=K z
[
ln
−
− − ]
2
dX
A 2 2
2 Z
I
I characteristic for target material
PDG http://pdg.lbl.gov/pdg.html
I = 16 Z
0.9
eV for Z > 1
Mean excitation energy > ionisation E
Theoretical calculation of I has a long
history.
Summaries can be found in several
reviews :
U. Fano, Chr., Studies in Penetration of Charged
Particles in Matter, Nucl. Sci. Rpt.. 39, U. S.
National Academy of Sciences, Washington, 1-338
(1964).
J. F. Ziegler, “Handbook of Stopping Cross-Sections
for Energetic Ions in All Elements”, Pergamon Press
(1980).
I=10 eV·Z for Z>20
Example Argon: Z = 18, I = 215 eV, measured 190.8 eV (Ionisation energy 15.7eV)
Katharina Müller, Autumn 14
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Charge dependence
2 2 2
2
m
c
  T max
−dE
Z
1
1
e
2
2  C
=K z
[
ln
−
− − ]
2
dX
A 2 2
2 Z
I
Tracks of ions in emulsion
Energy loss depends quadratically
on charge of projectile
width allows estimate of charge
Iron Z = 26
Thorium Z = 90
Katharina Müller, Autumn 14
13
dE/dX discussion
−dE
2Z 1 1
=K z
[ ln
dX
A 2 2
2 m e c 2  2  2 T max
I
2
2  C
− − − ]
2 Z
depends on velocity not on mass →
●
particle identification
without 
 1/2
T max≈2 me c 2  2  2
●
●
●
βγ small dE/dx ∝ 1/β2
βγ ≅ 3.5 broad minimum
light absorbers: Z/A ≃ 0.5
dE/dx(min) ≃ 1.5 MeV /(g/cm2)
→
MIP relativistic rise
●
minimal ionising particles (MIP)
βγ > 4 dE/dx ∝ 2 ln(βγ2)
logarithmic (relativistic) rise
 ≅ 3-4
Katharina Müller, Autumn 14
14
dE/dxmin different materials
−dE
Z 1 1
=K z2
[ ln
dX
A 2 2
2 m e c 2  2  2 T max
I2
 C
−2 − − ]
2 Z
dE/dX depends on A, Z of target material
βγ≅ 3.5 broad minimum
→ minimum ionising particles (MIP)
H2
Z/A≅ 1
dE/dXmin ≅ 4 MeV /(g/cm2)
others Z/A≅ 0.5 dE/dXmin ≅ 2 MeV /(g/cm2)
dE/dXmin ≅ 1-1.7 MeV /(g/cm2)
only weak material dependence
PDG http://pdg.lbl.gov/pdg.html
Katharina Müller, Autumn 14
15
dE/dX at minimum
βγ → 3.5 broad minimum dE/dXmin ≅ 1-1.7 MeV/(g/cm2)
only small material dependence
PDG http://pdg.lbl.gov/pdg.html
dE/dX(min) for different chemical elements fitted by a straight line Z>6
Katharina Müller, Autumn 14
16
Relativistic Rise
E- field
Lorentz-contraction of field lines
transversal component of E-field grows with γ
→ larger collision distances, more collisions
Saturation for high energies→ Fermi plateau Tmax = E
Solids (dE/dX)β→1≈ 1.05-1.1 (dE/dX)MIP
Gases
(dE/dX)β→1≈ 1.5 (dE/dX)MIP
at rest
δ-correction, density effect:
E-field gets partially shielded for high densities due to polarisation δ(γ) = 2 ln γ +ζ (ζ material constant)
→ less collisions with far distant electrons, smaller dE/dX
Fermi plateau
without correction
with correction
Katharina Müller, Autumn 14
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Particle identification
−dE
Z 1
≈K z 2
[ln
2
dx
A
2 2
2 me c  
I
2
 C
− 2 − − ]
2 Z
as long
as
2  m e ≪M
dE/dx only depends on velocity not on mass of particle →
Particle identification
ALICE TPC measured energy loss
• Heavy particles
dE/dx well described by Bethe-Bloch formula
• Electrons not described by Bethe-Bloch
http://aliceinfo.cern.ch/
Katharina Müller, Autumn 14
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ALICE TPC
88 m3 cylinder filled with gas, 5.1 m long
Divided in two drift regions by the central electrode located at its axial centre.
Uniform electric field along the z-axis electrons drift towards the end plates
Signal amplification:avalanche effect near anode wires
Readout: 570132 pads in cathode of multi-wire proportional chamber
http://aliceinfo.cern.ch/Public/en/Chapter2/Chap2_TPC.html
Katharina Müller, Autumn 14
Cosmic muon in TPC
19
Restricted Bethe Bloch
Particle detectors don't measure energy loss, but energy deposited in detector.
Eg. highly energetic electrons may leave the detector → measured energy is too small
Restricted Bethe Bloch: Cut off parameter Tcut :dE with T>Tcut are neglected
−dE
Z 1 1
=K z 2
[ ln
dX
A 2 2
2 m e c 2 2  2 T upper
I2
2
− 1
T upper
T cut
 C
− − ]
2 Z
Tupper = min(Tcut,Tmax)
Katharina Müller, Autumn 14
20
Range in matter
mean range R/M (M mass of particle)
PDG http://pdg.lbl.gov/pdg.html
Formally: integration of Bethe-Bloch formula
0
R=∫
E
dX
dE
dE
statistical process → mean penetration depth
range in cm: divided by density!
Example 1 GeV particle in lead
K+
M= 493.6 MeV
βγ = 2.02
→ R/M = 800 g cm-2 GeV-1
R
= 395 g cm-2
ds = R= 35 cm in lead
Katharina Müller, Autumn 14
Myon
R/M = 7000 g cm-2 GeV-1
ds = 64 cm
Proton
R/m = 220 g cm-2 GeV-1
ds = 18 cm
21
Ionisation tracks
Energy loss largest for low energy particles → increase of energy deposits
(ionisation) towards end of range (Bragg peak)
δ-(knock-on) electrons: have high enough energy for ionisation track
Tracks of mono energetic α particles in cloud chamber: range 8.6 cm, small variation
Katharina Müller, Autumn 14
22
Range: Bragg curve
Bragg curve: describes energy loss of particle as function of penetration depth
Particle in matter
→ particle de-accelerates
→ energy loss increases as particle gets slower
Bragg-Peak: largest energy deposit at end of the track
important for radiation therapy!
Charged particles
Photons
70 MeV protons in water
Bragg peak
Penetration depth
Katharina Müller, Autumn 14
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Medicine: radiation therapy
Ionisation profile for
12
C ions in water
Application in medicine:
Concentration of energy deposit at
end of reach allows treatment of tumours
with moderate exposure dose for the
surrounding tissue, contrary to x-ray
Possible to precisely deposit dose at well defined depth by varying beam energy
Katharina Müller, Autumn 14
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Proton therapy at PSI
Spread out Bragg peaks (SOBP)
Different absorbers → almost constant
dose in region of tumour
Katharina Müller, Autumn 14
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Energy loss (Straggling)
N
Energy loss probability
Δ E=∑i=0 δ E i N: number of collisions
Energy loss is statistical process
●
ionisation loss distributed statistically
●
collisions with small dE more probable
●
large dE rare → electrons with keV (δ-electrons)
●
δ-rays have enough energy for ionisation
●
asymmetry: mean energy loss > most probable energy loss
Parametrisation: asymmetric Landau distribution
Landau-Tail: rare interactions with large
energy transfer
Tails up to the kinematic maximum
Bethe-Bloch gives mean energy loss
Thick layers or dense material: Gaussian distr.
Energy loss
dEdX: most probable value ≠mean energy loss
Katharina Müller, Autumn 14
26
Landau distribution
Measurement
ΔEmp =82 keV
ΔEmp = 56.5 keV
1
1
exp−
e− 
Landau: Probability (Δ E) =
2
2
=  E − E mp / 
ξ: material constant mean energy loss in layer x
ξ= K ρ x /β2 depends on density, thickness and velocity
Δ E mean E- loss, Δ Emp most probable E-loss
Katharina Müller, Autumn 14
27
δ Electrons
Energy distribution of secondary electrons:
F T =1− 2 T /T max
(Spin 0)
d N2 1
2 Z 1 F T 
= k z
dT d x 2
A 2 T 2
Example: 500 MeV pion in 300 μm Si: 5% produce an electron with T>166 keV
important background in Cherenkov counter
Number of δ-electrons proportional z2/β2
Katharina Müller, Autumn 14
28
δ Electrons
1969 Mc Cusker: Evidence of quarks in air shower cores (Phys Rev Lett)
Narrow tracks in cloud chambers: low ionisation→ charge < 1!
Number of δ electrons proportional to z2/β2
Katharina Müller, Autumn 14
29
Summary dE/dx through ionisation
described by Bethe-Bloch Formula
●
only small material dependence
●
Energy loss ∝ z2 (particle)
●
independent of mass
→ particle identification
most of the energy is deposited towards
the end of the range
●
●
●
●
●
●
βγ < 3 dE/dX ∝ 1/β2
βγ ≃ 3.5 minimum dE/dX 1-1.7 MeV cm2/g
βγ > 3.5 logarithmic rise
dE/dX ≃ 2 MeV cm2/g
βγ > 1000 Bremsstrahlung dominant
Bethe-Bloch is not valid for slow particles (βγ<0.05) (dE/dX ∝ β)
here only phenomenological models available
Katharina Müller, Autumn 14
30
Additional Material: Ranges
Range of electrons, protons and α particles in air and water
Katharina Müller, Autumn 14
31
Additional Material: Ionisation yield
Material Density g/cm3
Eion[eV] I[eV]
W[eV]
np [cm-1]
nt [cm-1]
H2
8.99 10-5 15.4
19.2
37
5.2
9.2
He
N2
1.78 10-4 24.5
1.25 10-3 15.6
41.8
82
41
35
5.9
10
7.8
56
Ne
Ar
Xe
C4H10
9.00 10-4
1.78 10-3
5.89 10-3
2.67 10-3
137
188
482
48.3
36
26
22
23
12
29
44
46
39
94
307
195
Ar-f
Xe-f
Si
1.4
3.06
2.33
15.4
12.1
3.6
188
482
172
29.6
15.6
98000
245000
1000000
21.6
15.7
12.1
10.8
●
W = E/nt mean energy per electron-ion pair
●
np Primary ionisation: directly produced electron-ion pairs ~ 1.45 Z
●
nT= nP +secondary ionisation: add. electron-ion pairs through primary electrons
W: mean energy for production of electron-ion pair
●
W higher than ionisation energy
●
●
Semiconductors, W small, many electron-ion pairs, good resolution, small detectors
Katharina Müller, Autumn 14
32
Additional material: dE/dX: Mixtures
Since BB applies to pure elements, direct measurements are needed in compounds for
accurate values.
Good approximation: weighted sum of loss rates of constituents
weighted according to fraction ai of electrons
a i Ai
1 dE ∑ w i dE
=


w i=
 dx
i d x i
A e ff
Z e ff =∑ a i Z i
A e ff =∑ a i A i
ai is the molar fraction of element i with atomic weight Ai,
Aeff is the atomic weight of all constituents
Tables for density and shell correction
Katharina Müller, Autumn 14
33

SWITCHBOARD PANEL METERS

Undulators to Free Electron Lasers

放射線化学, 89, 25 (2010) - 日本放射線化学会