Chapter 2
Interactions of Charged Particles
- With Focus on Electrons and Positrons -
Calorimeters Chapter 2
1
Interactions of Particles with Matter
Outgoing Particle (p’)
Incoming Particle (p)
Scattering Center:
Nucleus or Atomic Shell
Detection Process is based on Scattering
of particles while passing detector material
Energy loss of incoming particle: E = p0 - p’0
Calorimeters Chapter 2
2
Energy loss due to collisions
à la Leo:Partially taken from lecture at
NIKHEF by unknown author
E = cos  Z1 e / r2 = Z1 e b / r3
Movement particle 2 in S
Z1 e
2
r

b
(energy of particle is
assumed to be high,
-> b is assumed to
be constant)
1
x  v1t
t = 0: distance between
1 and 2 is minimal
In the laboratory system S' particle 2 is in rest, so there is no effect
from the (time dependent) magnetic field caused by particle 1.
In S': E '  E   Z1eb /r 3,b /r  cos  time in S' !
electric field
strength
r 2  b 2  x 2  b 2  v12 t 2  b 2   2v12 t'2
Calorimeters Chapter 2
E   Z1eb /b 2   2 v12t'2 
3/2
3
p  

 Fdt  

Z 2e  Z1edt
b
2
 v t
2
2 2
1
3
2






t
2Z1Z 2e 2
 p   p   Z 2e  Z1e

1 
v1b
2
2
2 2 2 2


b
b


v
t


1


p
E 
2
2Z12 Z 22e 4
 2 2
2m
b v1 m

(Particle 2 is in rest -> non-relativistic
calculation of energy loss possible)
Interactions with electron:
m = me, Z2 = 1
 E(e)
2Z12e 4

 E(nucleus ) b2 v12me
Interactions with nucleus with
mass number A and atomic number Z:
m = Amp ≈ 2Zmp, Z2 = Z
2mp
2Z12 Z 2e 4

 4000 /Z
b2 v12 2Zmp Zme
-> Energy loss due to collisions is dominated by interactions with the electrons
(NB: we are comparing interactions with 1 electron to interactions with the nucleus of an atom,
anon-ionized atom has Z electrons)
Calorimeters Chapter 2
4
So far interactions with single electrons
Material consists of many electrons and nuclei
N. Bohr: Particle passes through center of thin shell
Number of electrons in shell: ne 2bdb dx
with ne = number of atoms per cm3
e
4Z12e 4 n edbdx
E 
bv12m
b
r
db
Z1e
dx
2Z12e 4 b max db 4neZ12e 4
dE
bmax
 2 ne

ln
2  
dx 
b
mev12
bmin
 mev1  b
min
Value for bmax

Calorimeters Chapter 2
5
Determination of Energy Loss
Transversal Field Strength:
(b2   2v 2t 2 ) 3 / 2
1
b/v1
‘Interaction’ Time:

Interaction Time < Orbital
Frequency of Electrons such that
binding effects can be neglected (adiabatic invariance)

 bmax = v1/
Maximal Energy Transfer for m >> me(see above): 2me
Using:
2Z12e 4
E  2 2
b v1 m e
it follows: bmin
Z1e 4
 2 so:
v1 me
dE 4 n e Z12e 4
me v13  2

ln
2

dx
me v1
Z1e 2

Detailed discussion follows

Calorimeters Chapter 2
6
‘Real’ Bethe-Bloch Formula
After consistent quantum mechanical calculation
Valid for particles with m0 >> me
E = 0: Rutherford Scattering
E 0: Leads to Bethe-Bloch Formula
2 2 2


dE
Z
1
1
2m
e
c
  Tmax  2  
2
2 2
 4NAre mec z
 ln
   
2
dx
A  2 
I
2

z
- Charge of incoming particle
Z, A - Nuclear charge and mass of absorber
re, me - Classical electron radius and electron mass
NA
- Avogadro’s Number = 6.022x1023 Mol-1
I
- Ionisation Constant, characterizes Material
typical values 15 eV
Fermi’s density correction
Tmax - maximal transferrable energy (later)
Calorimeters Chapter 2
7
Discussion of Bethe-Bloch Formula I
Describes Energy Loss by Excitation and Ionisation !!
We do not consider lowest energy losses
‘Kinematic’ drop
~ 1/
Scattering Amplitudes:
f i () 1 (p  p')2,( p  p')2 v 2
Large angle scattering
becomes less probable
with increasing energy of
incoming particle.
Drop continues until  ~ 4
Calorimeters Chapter 2
8
Discussion of Bethe-Bloch Formula II
Minimal Ionizing Particles (MIPS)
dE/dx passes
broad Minimum @
  4
Contributions from
Energy losses
start to dominate
kinematic dependency
of cross sections
typical values in Minimum
Lead
Steel
O2
[MeV/(g/cm2)] [MeV/cm]
1.13
20.66
1.51
11.65
1.82
2.6·10-3
Role of Minimal Ionizing Particles ?
(See Chapter 10)
Calorimeters Chapter 2
9
Discussion of Bethe-Bloch Formula III
‘Visible’ Consequence
of Excitation and Ionization
Interactions.
Logarithmic Rise
-
100
Interesting question:
Energy distribution of
electrons created by
Ionization.
Nuclear
Losses
1
0.001
Without 
0.01
1
Relativistic: |p1| ≈M
2
m
p
4
 1
M Abs. 2M Abs.
Tmax
Radiative
Losses
10
0.1
1
10
100
[Me V/ c]
Non relativistic: E1 ≈M
Tmax
Bethe Bloch
Ec
0.1
-Electrons
Ra d ia t ive
AndersonZiegler
Dominate over kinematic
drop
2mc 2 2 2

2
m m 
1 2
  
M M 

10
100
1000
1
10
100
[Ge V/ c]
10 4
1
10 5
10 6
10
100
[Te V/ c]
In the relativistic case an incoming
particle can transfer
(nearly) its whole energy to an
electron of the Absorber
These -electrons themselves
can ionize the absorber !
Calorimeters Chapter 2
10
Discussion of Bethe-Bloch Formula IV
Fermi’s Density Correction - ‘Tames’ logarithmic rise
Interaction with absorber
medium
-
100
Ra d ia t ive
AndersonZiegler
Energy losses not treated
as statistically independent
processes
 Single  elm. wave
Basic Effect: Amplitude felt by
Atom at P is shielded by other
Atoms  Damped wave !!!
P
P does not contribute to
energy loss !
Bethe Bloch
Ec
Radiative
Losses
10
Nuclear
Losses
1
0.001
0.1
Without 
0.01
1
0.1
1
10
100
[Me V/ c]

10
100
1000
1
10
100
10 4
[Ge V/ c]
1
10 5
10 6
10
100
[Te V/ c]
Density Correction more important
in Solids than in Gases
Remark: Density Correction tightly
coupled to Čerenkov-Effect
Calorimeters Chapter 2
11

Bethe-Bloch for Ultrarelativistic Particles
v ~ c, i.e. electrons and positrons
See e.g. E.A. Uehling 1954, Ann. Nucl. Sci. 4, 315 Sect. 1.1
Results for Electrons:
2 3 / 2 


dE
Z
1
1
2m
e
c

1  
2
2 2
 4NAre mec z
ln
  
2 
2
dx
A  2  2I  16 2
Results for Positrons:
1 2 2mec 2 3 / 2  23  
dE
Z
1
 4NAre2 mec 2 z 2
 
 ln

2
2
dx
A  
I
 24 2
2 

These particles are already ultrarelativistic at E ≈100 MeV
Calorimeters Chapter 2
12
Discussion of Bethe-Bloch Formula V
Radiative Losses - Not included in Bethe-Bloch Formula
Particles interact
with Coulomb Field
of Nuclei of
Absorber Atoms
-
100
Ra d ia t ive
AndersonZiegler
Bethe Bloch
Ec
Radiative
Losses
10
Nuclear
Losses
1
0.001
0.1
Energy loss due to
Bremsstrahlung
Without 
0.01
1
0.1
1
10
100

[Me V/ c]
10
100
1000
1
10
100
[Ge V/ c]
10 4
1
10 5
10 6
10
100
[Te V/ c]
Important for e.g. Muons with E > 100 GeV
Dominant energy loss process for
electrons (and positrons) (m/me)2 ~ 40000
Calorimeters Chapter 2
13
Critical Energy c
dE 
dE 

1) Point where 
dx 
Brems 
dx 
ion
dE 
dE 

2)   (E  C )    (E  C )   C X0 = Radiation Length (see later)
dx ion
dx Brems
X0

e- in Cu
(from PDG)
610 MeV
c 
Z + 1.24


c 

For materials
in solid and liquid
phase
710 MeV For gases
Z + 0.92
Emperical values
Based on fits to data
c is a characteristic parameter of a material
e.g. c for Uranium 6.75 MeV
Calorimeters Chapter 2
14