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Interaction of light charged particles with matter
Ionization losses – electron loss energy as it ionizes and excites atoms
Scattering – scattering by Coulomb field of nucleus, by field of electrons
Bremsstrahlung radiation – enough high energy, accelerated motion of charged
particle → emission of electromagnetic radiation, ultrarelativistic
energies – pair production through virtual photon
Cherenkov radiation – charged particle moving faster then light at given
material emits electromagnetic radiation in the range of visible
light– minimal ionization losses
Scattering is induced by interaction with atomic nuclei ( ~ f(Z2) ) and electrons at
atomic cloud ( ~ f(Z) ) (difference from heavy particles – in this case mainly interaction
with nuclei), energy losses mainly by interaction with electrons at atomic cloud
Electromagnetic shower– very high energies
Motion of electrically charged particles in magnetic and electric fields
Energy ionization losses
Mostly relativistic ↔ electrons and positrons are light particles
They will transfer big part of their energy during ionization
Interaction of electrons – interaction of identical particles → ΔEMAX = E/2
Interaction of positrons – they are not identical particles as electrons
- anihilation on path end – production of 1.022 MeV energy
Ionization losses determination – energy losses

dE
dx
Procedure of derivation of equation for ionization losses:
1) Classical derivation for nonrelativistic heavy particles
2) Quantum derivation for nonrelativistic particles
3) Relativistic corrections and corrections on identity of particles for electrons
Bethe - Bloch formulae
Classical derivation (assumption of nonrelativistic velocity and ΔE <<E ):
Change of momentum:

p b 
 Fdt
Constant connected
to SI unit system, often
is putted equal to one

Electric force acts
on particle:
F
Z ione 2
40 x 2  b 2 
1
Impact parameter b is changed during scattering only slightly: influence of F|| on
momentum change are negated (second half negates first)
Zobrazení síly
F┴ F
b
pro elektron,
Influence has only: F  F
x b
x
F||
v případě iontu b
We express path by velocity:
dx = v·dt
je přitažlivá

2
2
If velocity v during interaction with one electron changes only slightly, transferred
momentum is:


F dx
1 Z ione 2 b
dx
pb  


v
40
v  x 2  b 2



3
Z ione 2 b  1


40
v b2
1
2


1 2Z ione 2
 
x 2  b 2   40 bv
x
Kinetic energy of electron after interaction with ionizing particle
EKINe
p   

2
e
2me
2
2 4
1  2Z ion
e

 4  b 2 m v 2
0 
e

Path of particle passage through matter Δx:
Let have thin cylinder (annulus cross-section (b,b+db):
b
b+db
dN e  ne 2  bdbx
Number of electrons at cylinder:
where ne – is electron density at material
 Ecylindr  E KINe N e
Total energy losses at cylinder:
Mention:
2
2 4
 1  2 Z ion
e
 2
EKINe  
2
4

b
m
v
0 
e

where ΔNe – number of electrons at cylinder

Energy losses in the whole roll
 E   E KINe dN e  ne x  E KINe 2bdb
0
dE  1


dx  40
2
2
 4  Z ion
e 4  db

ne 
b
me v 2

0
If charge of material atoms is Z, number of electrons ne = Z·n0, where n0 – atom density
at material. We express it by material density ρ Avogardo constant NA and atomic mass A:
n0 
and then:
dE  1


dx  40
2
  NA
A

2
 4  Z ion
e4 Z
db



N
A 
2
A
b
me v

0
 ne 
Z
  NA
A
In the case of integration limits 0 and ∞ we obtain divergent integral.
Limits for integration are not in the reality 0 and ∞ but bmin and bmax:
Maximal energy is transferred during head collision, electron obtains energy:
E MAX  E KINe ( MAX ) 
1
me (2v) 2  2me v 2
2
p MAX  p MAXe  2me v
because maximal transferred momentum
We use relation between transferred
energy and impact parameter:
 1
2me v  
 40
2
2
2
 2Z ion
 1
e4
 2


b

MIN
2
b
m
v
 MIN e
 40
 Z ion e 2

2
 me v
Minimal transferred momentum depends on mean ionization potential of electrons
at atom I, is p  vI and E  p  I
(work achieved through passage must be
2m
2m v
larger then ionization potential) and
 1  2Z e
 1  2Z e
I


corresponding impact parameter is:
 
b
 
MIN
2
MIN
2
e
e
MIN
2
2
2me v 2
We determine integral:
dE  1


dx  40
2
2
4
ion
2
2
 40  bMAX me v
Constant connected to
SI unit system, often is
expressed as equal one
MAX
 40 
2
2
 4  Z ion
e4 Z
b

  N A ln MAX
2
A
bMIN
me v

and then:
dE  1


dx  40
2
ion
where:
bMAX
bMIN
I
 1  2  Z ione 2


4

I
2 me v 2
0 


I
 1  Z ione 2


2
 40  me v
2
2
 4  Z ion
e4 Z
2me v 2

  N A ln
A
I
me v 2

Main dependency
on particle velocity
Main dependency on
material properties
Weak dependency on particle
velocity and material properties
Relativistic corrections:
Maximal transferred momentum:
p e max  2mv  p e max 
2mv
1  2
Reduction of particle electric field in the direction of flight by factor (1-β2) and in the
perpendicular direction increasing by factor 1 / 1  
2
We will obtain on the end:
dE  1


dx  40
2
2
 2me c 2  2
 4  Z ion
e4 Z
1
2



N
ln

ln




A

2 2
A
I
1  2
 me c 


dE  1


dx  40
2
2
 2me c 2  2

 4  Z ion
e4 Z

  N A ln
  2
2 2
2
A
 me c 
 I (1   )

We obtain early derived equation for v << c
This formulae is for electrons even more complex:
In the case of electron → identical particles → maximal transferred energy ΔEMAX = E/2
dE  1


dx  40
2


2
2
 4  Z ion
e4 Z
1  me c 2  2 E
1



N
 (ln 2)( 2 1   2  1   2 )  (1   2 )  1  1   2 
ln 2
A
2 2
2
A
2  2 I (1   )
8
 me c 

E ~ up to hundreds MeV → light particle losses are 1000 times lower than for heavy
E ~ GeV → ionization losses of light and heavy particles are comparable
Example of ionization losses for some particles
(taken from D. Green: The physics of particle detector)
Elastic scattering
1) Single scattering
2) Few scatterings
3) Multiple scattering
d 
d~
1
Heavy particles – important only for
scattering on atomic nuclei
n 0
1
n 0
d 
1
n 0
Light particles – important also for
scattering on electrons
Single scattering in the electric field of nucleus – described by Rutheford scattering:
2
   4πε0 mv b
cot g   
Z ionZe 2
2
1) Heavy particles – scattering to small angles → path is slightly undulated
2) Light particles – scattering to large angles → range is not defined (for „lower energies“)
Mean quadratic deviation from original direction Θ 2 depends on mean
quadratic value of scattering angle  2 :
(simplified classical derivation for „heavy particles“ – small scattering angles)
→0:

Z Ze 2
 
 tan    ion 2
2
 2  4πε0 mv b
and then 
Z Ze 

4πε mv  b
2 2
2
ion
2 2
2
0
We determine  2 :
bmax
  b2πbdb
 
2
bmin
bmax

 2πbdb
2 2
0
2 bmax
1
db
b
bmin
bmax
2π  bdb
bmin
Θ2



4πε mv 
2π Z ion Ze 2
2

 ln b
2πε mv 

π b 
π Z ion Ze 2
2
2 2
0
2 bmax
bmin
bmax
bmin
2
 Z ion Ze 2 
bmax


ln
πε0 mv2 
bmin


2
2
2 bmax
 bmin


bmin
then is determined: Θ 2  N roz 2
b
where Nroz is number of scatterings:
N roz

N A max
N
2
2
 N atom xσ  ρ
x  2πbdb  πρ A x bmax
 bmin
A bmin
A

2
Resulting value:
2
1 N A  Z ionZe 2 
bmax
1
N A xZion
Z 2e 4 bmax
 ln
Θ  ρ
xπ 

ρ
ln
2 
2
2 2
2 A
b
A
bmin
πε
mv
2
πε
p
v
min
0
 0

2
1) Strong dependency on momentum:
2) Strong dependency on velocity
1/v4
Important scattering properties: 3) Strong dependency on mass 1/m2
4) Strong dependency on particle charge: Zion2
5) Strong dependency on material Z
Z2
Bremsstrahlung radiation
Accelerated charged particle emits electromagnetic radiation

Energy emitted per time unit:
dE
~ a2
dt
Acceleration is given by Coulomb interaction:
Dependency on material charge:
and mass:


dE
~ Z2
dt
a 
FC
m
ion charge:

Z ion Ze 2 1
40
m
r2

dE
2
~ Z ion
dt
1
dE
1
~ 2
dt
m
Rozdíl v náboji iontu malý, v hmotnosti mnohem větší:
For proton and electron:
1
 dE 

 ( proton)
2
m 2p
m 2  0,511MeV 
 dt  rad
6

 e2  
  0,3  10
1
dE
m p  938MeV 



 (elektron)
2
m
 dt  rad
e
For muon and electron is same ratio 2.6·10-5
Radiation losses show itself in „normal situation“ only for electrons and positrons
For ultrarelativistic energies also for further particles
On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1:
Description is equivalent pair production description:
N 2
 dE 
2

  4 A r0   E  Z F ( E, Z )
A
 dx  rad
(it is similar calculation and result as for
pair production – see gamma ray interaction)
e2
r0 
 2,82  10 15 m
2
40 me c
where for mention:

e2
40 c

1
137
Course of function F(E,Z) depends on energy (E0 – initial electron energy)
and if it is necessary count screening of electrons:
E ≈ hν0 – eigenfrequency of atom → interaction with atom – screening has not influence
E >> hν0 – interaction with nucleus → screening is necessary count according to,
where electron interacts with nucleus:
Low energy → strong field near nucleus is necessary
High energy → weak field further from nucleus is enough – there is maximum of production
Without screening me c  E0 
2
2
me c Z
me c 2 Z 1 3

13
:
1
F ( E , Z )  ln 2  f ( Z ) 
3
F ( E , Z )  ln( 183Z 1 / 3 )  f ( Z ) 
1
18
where:
 
E0
me c 2
Complete screening E0  
:
and F(E,Z) in the case without screening depends on E only weakly and in the case
of complete screening does not depend on E:
Radiation losses linearly proportional to energy:
For radiation length :
1
 dE 
E

 
 dx  rad X 0
N
1
 4 A r02  Z 2 F ( E, Z )
X0
A
Energy losses of electron (if they are only radiation losses):
Critical energy EC:
E ( x)  E 0 e

x
X0
 dE 
 dE 
E  EC      
 dx  rad  dx  ion
For electron and positron is EC > mec2 → v ≈ c
2
 1  4  e 4 Z
N 2
 dE 
 dE 
2

  N A Fion ( E )

  
  4  A r0   E  Z Frad ( E , Z )  
2
A
 dx  rad  dx ion
 40  me c A
 dE 


F ( E, Z ) 
F ( E, Z )
 dx  rad  E

Z rad
   Z rad
2
 me c
Fion ( E )

Fion ( E )
 dE 


 dx  ion
for v → c is valid Fion(E) = f(lnE):
C 
EC
 1 Fion ( E )

2
 Z Frad ( E, Z )
me c
! Let approve!
Air
Al
Pb
EC [MeV]
80
40
7,6
Total energy losses
Total losses are given by ionization and radiation losses:

dE
 dE   dE 
 
  
dx
 dx  i  dx  r
Electron range, absorption
Protons
Electrons
Well defined range does not exist
Rextrap - extrapolated path – point fo
linear extrapolation crossing
We obtain exponential dependency for
spectrum of beta emitter
Schematic comparison of different quantities
for protons and electrons
Ultrarelativistic energies
Radiation losses by bremsstrahlung radiation and pair
production prevail for ultrarelativistic energies also for muons
(taken from D. Green: The physics of particle detector)
Electromagnetic shower creation – see gamma ray interaction
Angular and energy distributions of bremsstrahlung photons
Angular distribution:
Depends on electron (other particle) energy, does not depend on emitted photon energy
Mean angle of photon emission:
mec2
~ ΘS 

E TOT
E→ ∞  ΘS → 0
Photons are emitted to narrow cone to the direction of electron motion, preference of
forward angles increases with energy
Energy distribution:
Maximal possible emitted energy – kinetic energy of electron
Synchrotron radiation
Similar origin as bremsstrahlung – it is generated during circular motion of relativistic
charged particles on accelerators (synchrotrons). Influence of acceleration → emission of
electromagnetic radiation
Synchrotron radiation is not connected with material – lower acceleration → it has
lower energy

 
Acting force is Lorentz force:
FL  q v  B


dE 2 1 Ze   2

 

a
dt 3 4 0 c3
2
Energy losses:
γ2 
1
1  2
Classical centripetal acceleration: a=v2/R
dE 2 1 Ze  v 4

 


dt 3 4 0 c3 R 2
2
Energy losses:
Relativistic centripetal acceleration:
dE 2 1 Ze  2 v 2
2 1 Ze  4 v 4 2 1 Ze 

 
 3 γ
 
 3 γ 2  
 2 c  γ4 4
dt 3 4 0 c
R
3 4 0 c
3 4 0 R
R
2
Energy losses:
2
1 dp
γ  d( γm)v 
2 dv
2 v
a 
 
γ
γ
m  t  m  dt 
dt
R
d 
 
2
2
2
Cherenkov radiation
Particle velocity in the material v > c’ = c/n
(n – index of refraction) → emission of Cherenkov
radiation:
c
t
c
n
cos  

vt nv
cos  
1
n
Results of this equation:
1) Threshold velocity exists βmin = 1/n. For βmin emission is in the direction of
particle motion. Cherenkov radiation is not produced for lower velocities.
2) For ultrarelativistic particles cos Θmax = 1/n.
3) For water: n = 1.33 → βmin = 0.75, for electron EKIN = 0.26 MeV
cosΘmax = 0.75 → Θmax= 41.5o
Transition radiation
Passage of charged particle through boundary of materials with different index of
refraction → emission of electromagnetic radiation (discovery of Ginsburg, Frank 1946)
Creation of dipole in boundary zone → dipole,
elmg. field changes in time → emission of elmg.
radiation:
material
vacuum
+
Energy emitted by one transition material/vacuum:
1
E     P   ~ 
3
e-
+
High energy electron emits transition radiation
plasma frequency: ħωP ≈ 14 eV (for Li), 0,7 eV (air)
20 eV (for polyethylene)
Number of photons emitted on boundary (is very small, necessity of many transitions):
E
Nf 
~

Energy of emitted photons 10 – 30 keV
1 1
20eV
Nf ~ 

  ~ 0,000002 
3 137 20000eV
Emission sharply directed to the particle flight direction:
~
1

Radiators of transition radiation: material with small Z, reabsorption increases with ~ Z5
Good combination of radiators and X-ray detectors