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.

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 2 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 atom Ionpair + liberated e-

=> track of ionizations and excitations in the material. Randomly distributed.

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Ionizations

Ionizations Ionization cluster

-particle

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 4 2. Inelastic collisions with a nucleus.

 Close to the nucleus => deflection by the strong Coloumb field.

Bremsstraalung losses by photon radiation

Electron X-ray Nucleus

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-

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 6 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

2 Maximal energy transfer Qo will be 2mv , where v is  -particle

velocity 2 me Qmvo  24  E M Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 8 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

dE  Total Linear Stopping Power 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 dE dE  dE • Radiative loss   dxtot dx col  dx rad

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 10 Collision Stopping Power

2  Classical theory leads to dE z  2  Z dxcol v

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

24 1 dE NA  Z z eQmax 22 C   22ln ln 1  dxCol A4200 m v I Z

NA = Avogadros number Z = atomic number for the attenuator A = mol weight for the attenuator z = Charge of the incoming particle

m0 = Rest mass for the electron v = Velocity of the incoming particle I = The average ionization potential

Qmax = Maximal transferred energy at a single collision C = Shell correction  = Polarization effect  = Incoming particle relative velocity (=v/c)

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 12 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

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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

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 14 Components in Bethe-Bloch formula

Log(S/) Relativistic z2 effects 1/v2  ln(1-2)

Log Ekin 1. A low energies S decreases (the effective charge decrease) 2. The decrease above the Bragg-peak is due to 1/v2 dependence 3. The decrease continues -  plays a role. 4. 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)

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 16 Energy Straggling

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 17

Collision Stopping-Power vs. velocity

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 18 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

dE 1  At low energies  2 dxcol v

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 20 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 dE restricted stopping-power  dx  is a energy threshold 

 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.

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 22 RCSDA – range definition

E 1 o dE  Definition of RCSDA R  dE csda   0 dx

 CSDA = Continuous Slowing-Down Approximation

 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 S real path of the particle

R: Range of the particle in the media

R 1

0.5 Rmax

Rm Sm Ro So

Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 24 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. Michael Ljungberg/Medical Radiation Physics/Clinical Sciences Lund/Lund University/Sweden 25