Interaction: Charged Particles
Michael Ljungberg
Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 1
Introduction

Heavy charged particles (mass > e-)
• p, -particles heavy ions (Z>2)

Light charged particles e+, e• Easy to accelerate to high energy and velocities close to speed-of-light
• Dominated type of interaction for charged particles is the electromagnetic
(Coloumb interaction). Energy degraded and direction changed for a light
particle (electron).

Atoms along the track will be ionized and excitated.
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1. Inelastic collision with atomic electron

Dominating interaction type
• Ionizations and excitation due to loss of kinetic energy.
• This type results in the largest energy losses
Ionpair
atom
+
liberated e-
=> track of ionizations and excitations in the material. Randomly
distributed.
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Ionizations
Ionizations
Ionization cluster
-particle
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2. Inelastic collisions with a nucleus.

Close to the nucleus => deflection by the strong Coloumb field.
Bremsstraalung losses by photon radiation
Electron
Nucleus
X-ray
0 < hv < Ee
Important for electrons.
Less important for heavy particles
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3. Elastic collisions with a nucleus


Deflection without radiation loss and nucleus excitation
Loss of kinetic energy by incoming particle is small (keeping the
system momentum constant)

Elastic scattering

Mostly electrons e-
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4. Elastic collisions with atomic e

Charged particle interact with an energy loss less than the lowest
excitation potential by the atom (interaction with the whole atom)
Important for Ee < 100 eV
For electrons - all four types of interaction processes can occur
but for heavy charged particles the most important are the
inelastic collision with atomic e- (1)
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Maximum Energy transfer
(inelastic collision with atomic e-)

Consider an impact between an -particle with the mass M and
energy E and a electron with the mass m.
M,, E´ , v´
M,, E , v
m
Before
After
Maximal energy transfer Qo will be 2mv2, where v is  -particle
velocity
Qo  2mv 2  4 
me
 E
M
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Maximal energy transfer
(inelastic collision by atomic e-)


For a 5 MeV -particle Qo equals 2.5 keV
This means that the -particle loose its energy in small proportions
that is undergoes many collisions before coming to rest.
Well-defined range with a small statistical deviation
between different -particles of the same energy

For an electron a collision between two particles of the same mass
yields that the whole kinetic energy can be transferred. Large energy
depositions is more likely.
Less well-defined range.
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Total Linear Stopping Power

Total Linear Stopping Power
 dE 
S 

 dx tot
dE is the energy that a particle on the average loose when it passes a
range dx in a material.

Characterize the materials ability to slow-down and stop the particle.

dE includes all types of energy losses. Often separated into
• Collision loss
• Radiative loss
 dE 
 dE 

 

dx

tot  dx col
 dE 


 dx  rad
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Collision Stopping Power

Classical theory leads to
z2
 dE 

Z


2
dx
v

col
The energy loss is
• proportional to square of particle charge (z2)…
• inverse proportional to the square of the velocity of the particle (1/v2)…
• proportional to the atomic number of the material (Z)
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Bethe-Blochs quantum mechanical expression
NA  Z
z 2e4
1  dE 



A
4 02 m0 v 2
  dx Col
NA =
Z=
A=
z=
m0 =
v=
I=
Qmax =
C=
=
=
C 
 Q
  ln max  ln 1   2    2   
I
Z 2

Avogadros number
atomic number for the attenuator
mol weight for the attenuator
Charge of the incoming particle
Rest mass for the electron
Velocity of the incoming particle
The average ionization potential
Maximal transferred energy at a single collision
Shell correction
Polarization effect
Incoming particle relative velocity (=v/c)
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Components in Bethe-Bloch formula

Average ionization potential
• The energy on the average that is transferred to a bounded electron.
Experimental averaged value determined to
I ~13.5 eV
The stopping-power is proportional to the log of I and therefore varies
slowly with I.

Shell correction
• All electrons are not part of the interaction
• Electrons contribute less to the stopping-power if the velocity of the
incoming particle is in the same order as the velocity of the electrons in the
shells
• The parts in the equation that depend on  has a small impact if v<c
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Components in Bethe-Bloch formula

The density effect
• Stopping power theory based on the independence of atoms.
• Correction needed for dense materials.
• For atoms close to each others the electrical field between the particle and
the shell electron will be affected by the field from the other atoms.
i. The field reduction
ii. reduce the particles energy loss
*
+
*
-
*
+
*
-
*
+
*
-
*
+
*
-
*
+
*
-
*
+
*
*
+
*
*
+
*
*
+
*
*
+
*
• The density effect increase with energy of the particle. The correction /z
reduce the stopping-power of the particle
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Components in Bethe-Bloch formula
Log(S/)
Relativistic
effects
z2
1/v2
ln(1-2)

Log Ekin
1.
2.
3.
4.
A low energies S decreases (the effective charge decrease)
The decrease above the Bragg-peak is due to 1/v2 dependence
The decrease continues -  plays a role.
S increase at high relativistic energies due to the term ln(1- 2)
* Shell correction important for high-Z materials
* Due to  the S is reduced at high energies.
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Bragg curve for alpha particles
Ionizations per unit
of length (Mev/cm)
Range (cm)
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Energy Straggling
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Collision Stopping-Power vs. velocity
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Stopping-Power vs. energy
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The components of the Bethe-Bloch formula

Two important differences between electrons and heavy charged
particles
• e- can delived the whole energy at a collision (Qmax)
• Ee > few 100 keV result in relativistic effects.

Mass-Stopping Power S/ about the same for all materials

At low energies
1
 dE 

  2
 dx col v
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Restricted Stopping-Power

dE/dx include all energy losses along the path dx. A measure of the
energy absorbed locally along the track is the
restricted stopping-power
 is a energy threshold

 dE 


 dx 
Also denoted LET (Linear Energy Transfer), L
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Restricted Linear Collision Stopping Power

Restricted Linear Collision Stopping Power
• Defined as the energy transfer per unit length
that is caused by collision at where energy losses is less than  eV
 dE 


 dx col ,

This means that:
• -particles with higher energy than ∆ is counted as new particles.
• Secondary e- have so high energy and large range so that the cannot be
regarded as locally absorbed.
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RCSDA – range definition
1
Eo
 dE 
 
 dE
dx

0 

Definition of RCSDA

CSDA = Continuous Slowing-Down Approximation


Rcsda
Range representing the path length for a particle a an energy
loss of Eo if the energy loss per unit of length is the same as the
energy loss defined by the stopping-power value.
Differences in the ranges caused by statistical changes
(straggling) is low for  and protons)
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Range its relationsrelationer
S: Track length is the length of the
real path of the particle
S
R: Range of the particle in the media
R
1
Rmax
0.5
Rm
Sm
Ro
So
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Path lengths

R = average projected path length
• Thickness of an absorber that absorb 50% of perpendicular incoming
particles.

S = averaged path length
• Average path length for the particles. R and S about the same for heavy
charged particles but for light particles (electrons) a difference of up to 2
can be seen.

Ro = extrapolated projected path length
• represents the thickness determined by an extrapolation of the range curve.

So= extrapolated path length
• Represents the path length determined by extrapolation of the S curve in a
similar way.
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