8. Introduction of Particles with Matter

Question 1

What are alpha, beta and gamma rays all examples of?

Answer 1

-ionising radiation

Question 2

Ionising Radiation

Definition

Answer 2

-interacts with matter as it passes through it, ionising nearby atoms

Question 3

What are the two mechanisms by which charged particles can lose energy?

Answer 3

1) Ionisation

2) Bremsstrahlung

Question 4

Ionisation

Definition

Answer 4

• as a charged particle moves through bulk matter, its electric field ionises atoms in the material
• this slows the particle

Question 5

Ionisation

Momentum Transfer Parallel to v

Answer 5

-momentum transfer parallel to v is negligible

Question 6

Ionisation

pe

Answer 6

pe = 2Ze² / 4πεovb

Question 7

Ionisation

Energy Transferred to a Single Electron

Answer 7

E = pe²/2m = 2/m (Ze²/4πεov)² 1/b²

Question 8

Ionisation

Number of Electrons in a Cylindrical Shell

Answer 8

nZmat2πb db dx

Question 9

Ionisation

Rate of Energy Loss

Answer 9

• dE/dx = 4πnZmat/mv² (Ze²/4πεo)² ln(bmin/bmax)

- where bmin and b max are the minimum and maximum impact parameters that can cause ionisation

Question 10

Maximum Impact Parameter

Answer 10

-if the particle is too far away, the amount of energy it can transfer is less than the ionisation energy
-to a good approximation, consider the Coulomb force to act only over a distance where |x|<b>
bmax = hv/πI
-where I is the mean ionisation energy</b>

Question 11

Minimum Impact Parameter

Answer 11

-b is only well defined up to uncertainty in position (of order of the de Broglie wavelength):
bmin = ℏ/2mev

Question 12

Bethe Formula

Answer 12

• subbing in expressions for bmin and bmax, we arrive at the Bethe Formula for stopping power due to ionisation;
• dE/dx = 4πnZmat/mev² * (Ze²/4πεo)² * ln(2me*v²/I)

Question 13

Bethe Formula Generalised for Relativistic Formula

Answer 13

-for relativistic (i.e fast moving) particles:
-dE/dx ∝ Z² Zmat/A 1/β² [1/2 ln(2mec²β²γ²*Tmax/I²)-β²-𝛿(β)/2]
-where:
β = v/c
γ = 1/√[1-v²/c²]
and Tmax is the maximum kinetic energy that the charged particle can transfer to a single (stationary) electron

Question 14

What is Tmax?

Answer 14

Tmax is the maximum kinetic energy that the charged particle can transfer to a single (stationary) electron
Tmax = [2mec²β²γ²] / [1 + 2γ me/M + (me/M)²]

Question 15

What is x?

Answer 15

-x is not a physical distance, we define x such that:
x = ρl
-where ρ is the density and l is the physical path length
-note that x has units of g/cm²

Question 16

Minimum Stopping Power

Answer 16

-occurs at βγ≈3

Question 17

Stopping Power vs βγ Graph

Answer 17

• minimum at βγ≈3
• below this, S(E) increases sharply
• above this there is a logarithmic increase

Question 18

Lorentz Contraction

Answer 18

-on a larger scale, relativistic effects become important, in particular the amount of Lorentz contraction

Question 19

Bremsstrahlung

Description

Answer 19

-Bremsstrahlung o braking radiation is the radiation emitted by an energetic charged particle as it decelerated in the presence of an electromagnetic field, e.g. due to motion through a medium

Question 20

Bremsstrahlung

Energy Loss

Answer 20

• energy loss is proportional to energy:

- dE/dx ∝ E

Question 21

Bremsstrahlung

Radiation Length

Answer 21

-get exponential decay, define radiation length Xo such that:
E = Eo*exp(-x/Xo)

Question 22

Ionisation Losses and Radiation Losses With Energy

Answer 22

• ionisation losses increase logarithmically with energy
• radiative losses increase linearly with energy
• there must be a critical energy above which radiative losses dominate and below which ionisational losses will dominate

Question 23

Bremsstrahlung

Total Energy Loss

Answer 23

-dE/dx = Eionisation + Eradiation

= [aln(E)+b]/E + kE

Question 24

Bremsstrahlung

Bragg Peak

Answer 24

• we find that stopping power peaks suddenly just before the particle comes to rest with the particle depositing the majority of its energy in a short distance
• this is known as the Bragg peak

Question 25

Cerenkov Radiation

Description

Answer 25

• another way that charged particle interact with their surroundings
• this is the electromagnetic analogue of a bow wave in water surfaces or a sonic boom in air
• when an object capable of causing disturbances in a medium travels through that medium, it causes waves to propagate outwards
• of the object travels faster than the waves can propagate, then constructive interference of successive waves creates a plane wave that propagates at a particular angle away from the objects direction of motion