#### Interaction of Radiation with Matter, Physics tutorial

Introduction:

Radiation measurements are possible through interaction with matter either living or dead. This interaction has made of possible to be utilized in diagnosis, researchers industries X-ray and radiotherapy. Due to fundamental difference in energy transfer, radiation can be categorized into four:

i) Heavy charged particles

ii) Fast electrons particles

iii) Neutrons electrons particle

iv) Protons electrons particles

Cross section and interaction co-efficient:

Probability that interaction will occur is stated in cross sections. Cross section really explains effective area which interaction center or entities presents to radiation which if traversed by radiation, it makes sure that interaction happen.

δ = prob of interaction p/no. of conc. center area

δ = prob of interaction P/particle fluence Q

δ is stated in m2 or in barns

1 barns = 10-28m2. Linear attenuation co-efficient is provided as

dΦ/Φ = -μdl

Φ = Φ0e-μdl

Where μ = pδ that is probability of interaction per unit length. Also number of atoms per unit volume of substances

n = NAρ/M = N/V

Heavy Charged Particle Interaction:

This interaction can be separated into three broad groups that are:

i) Interaction with individual electron of atoms that leads to excitation or ionization of atoms and this collision can be:

a) Inelastic - enough energy received for excitation.

b) Elastic - energy received is less than smallest energy difference of atomic level. It may be hard or soft collision. It may also be fast or slow depending on projectile velocity and orbital velocity

ii) Interaction with nuclei if incoming particle is heavy compared with electrons.

iii) Interaction with whole coulomb field surrounding atom.

This takes place when incident particle is of low energy or heavy particle with low velocities.

Outcome of collision or interaction is measured by:

a) Velocity of collision, V

b) Distance of closest approach of participants.

c) Range of potential that governs interaction between incident particle and target.

For heavy particles like α-particles interacts with coulomb forces between positive and negative charges in matter. It then dissipates energy in succession to electron through inelastic collision which then yields in either excitation or ionization. Number of ion generated per unit distance is known as specific ionization.

Linear rate of energy loss or linear stopping power S for heavy charged particle in given absorber is

S = -dE/dX

S increase as particle velocity decreases that is S α 1/V

For non-relativistic particle,

S = 4πe4Z2NB(Z1V)/M0V2 (Beth's formula)

Where M0 - rest mass of electron

V-velocity of heavy particle

e - Electronic charge

Z - Atomic number of the absorber atom

B(Z1V) - Beth's formula

B(Z1V) = Z[ln2M0V2/I - ln(1 - V2/C2 - V02/C2)]

I = average ionization potential of absorption. While for non-relativistic charged particle

B(Z1V) = Z(lnM0V2/I), S α 1/V2 α 1/E

Beta Rays (Fast Electrons):

Energy lost by fast electrons is because of excitation and ionization also. Mainly, energy is generally lost because of:

a) Scattering of fast electrons as they are colliding with another electron in target. Energy loss per unit length D (-dE/dx)/C = collision

It occurs in form of bremsstrahlung (e-m radiation). (Generally for electrons with energy greater than rest mass energy). Thus, total linear specific energy loss is;

dE/dX = (dE/dX)/c + (dE/dX)/r and [(dE/dX)/C][(dE/dX)/r] - E8/7w

c). Cerenkov radiation that is negligible at lower energy

Photons:

Interaction of E-M radiation (i.e. X and γ rays photons). Under energy region of 0.01 - 10MeV, most interactions γ and X-rays can be described under 3 different modes

i) Photoelectric effect

ii) Compton effect

iii) Pair production

i) Photoelectric Effect:

This is a type of interaction whereby incident photon transfers all energy to electron in target and thus these electrons are emitted as photoelectrons with kinetic energy given as below;

Ee = Eγ - EB

Incident photon should have energy greater than or equal to EB of electron to nucleus of target. Thus, vacancies are created at K-shells and filled by electrons from higher shells that result into X-rays. Cross section of photoelectric effect is provided as;

σP.E = δEγ-7/2ρZ5

ρ and Z5 are density and atomic number of absorbing material or target. δ is constant

ii) Compton Effect:

This is situation whereby incident photon collides with orbital electron of target and incident photon and orbital electrons are scattered at different angles.

Observations:

a) Reduction in photon energy from hv → hv1.

b) Frequency is changed from v → v1 (reduced).

c) Wavelength of photon increases from λ → λ1.

d) Energy of scattered electron is (hv - hv1).

e) The increase in wavelength is given as

Δλ = λ1 - λ = (h/M0C)[1 - cosθ]

Where Mo is rest mass of atom of which electron is utilized.

f) Energy of scattered photon is;

Eγ1 = Eγ/(1 + [Eγ/M0C2][1 - cosθ])

g) Kinetic energy of photo electron ( EK.E. ) or ejected electron is

EK.E = [(Eγ/M0C2)][1 - cosθ]/[1 + [Eγ/M0C2][1 - cosθ]

Where MoC2 is rest energy of electron

EK.E. is at minimum when θ = 0 and EK.E. is maximum when θ =180o and this is known as Compton edge energy.

Ec = Eγ[2Eγ/(-M0C2 + 2Eγ)]

That is when photon is scattered backwards at θ=1800 that explain Compton plateau in γ spectroscopy

iii) Pair Production:

This takes place when γ rays with enough energy interacts with atom of target in coulomb field of nucleus and disappears with electron position pair in place of it.

Energy equation of process is given as hv = Ee- +Ee + 2MoC2. Pair production can only occur if greater than or equal to 2MoC2 = 1.02MeV.

Position is unstable particle once its kinetic energy = 0, it annihilates with electron to form photon that either escapes from medium or interacts with medium either photo electrically or photocompton.

σp.p = CZ2δlnEγ

Net effect of three in γ-ray passing through absorbing material or linear cross-section is exponential attenuation provided by:

I = I0e-σx

Neutrons:

The type of reaction a neutron experiences depends on energy. Neutrons are categorized according to their energy:

i) High energy neutron greater than 10MeV

ii) Thermal neutrons is same as average kinetic energy of gas molecules = 0.025eV.

All neutrons at time of their birth are fast but are slowed down (thermalizing) by colliding them elastically with atoms in their environment. After slowing down, they are now been absorbed by nuclei of absorbing material and interaction occurs. The interaction of neutron is different from past discussion as, it doesn't show variation between atomic mass and energy. Interactions instead generate:

i) Recurring nuclei

ii) Subatomic particles

iii) Photons which undergo previous processes.

Usually neutrons may collide with nuclei and experience;

1. Elastic collisions:

Fast neutrons react with low atomic number absorbers. Neutron is scattered with the reduced energy as part of energy have been transferred to recurring nuclei

e.g. 1H (n, n) 1H. (Here nucleus moves).

2. Inelastic collisions:

If neutron is of low energy, neutron may be momentarily captured by nucleus and then emitted with diminished energy leaving nucleus in excited state and may return to ground state with emission of photon e.g. 16O(n, n)16O. (Here nucleus doesn't move but brushed). If another particle is generated after interaction, it is known as non-elastic collision e.g. 16O (n,α) 13C. Thermal neutrons are captured by nuclei with reaction cross section as

σ α 1/v α 1/√E = √E0/E

Passage of neutrons through a moderating material:

The procedure of slowing down fast neutrons is called as moderation or thermalization and this is done by making use of moderators (element with low atomic mass) like graphite and heavy water, in such a way that no reaction is lost by absorption but simply have their kinetic energy being reduced by elastic collision with nuclei of moderator.

Velocity V1 of neutron after collision isV12 = V2(1 + A2 + 2AcosΦ)

And V is the velocity of neutron before collision also

V0 = V(1 + A) And V0 is velocity of neutron in real frame.

E0 = 1/2mv02 = incident energy of neutron before collision

E1 = 1/2mv12 = energy of neutron after collision.

Thus fractional energy e0 is

E1/E0 = 1/2mv12/1/2mv02 = v2(1 + A2 + 2AcosΦ)/v2(1 + A)2

E1/E0 = 1 + A2 + 2AcosΦ/(1 + A)2

Cases:

1. Glancing angle i.e. where Φ = 0

From E1/E0 = 1 + A2 + 2AcosΦ/(1 + A)2

E1/E0 = 1

2. Head-on-collision where Φ = π =180o

E1/E0 = (A - 1)2/(A + 1)2 Neutron energy loss here is maximum

Let α = [(A - 1)/(A + 1)]2

E1/E0 = α

From ΔEmax = (E0 - E1)max

= E0(1 - E1/E0)max

ΔEmax = E0(1 -  α)max

Maximum fractional energy loss can be deformed as

ΔEmax/E0 = 1-α = 1-(A-1)2/(A+1)2

For good moderator, max ΔEmax should be large and thus A should be small. Let's suppose neutron is scattered between angle Φ and Φ + dΦ and energy (that is E and E + dE) between E and E+dE. It will be observed that entire range of energy through which neutron can be scattered is between 1. E1 = E0(from glancing) and (2) E1 = αE0 (from head on collision). This means that E0 - αE0 = E0(1 - α)

Probability P(E)dE that neutron will have energy E between E0 and αEo is 1.

Probability that it will lie between E and E+dE

=P(E) = 1/E0(1-α)

αE0E0P(E)dE = ∫αE0E0 dE/(E0(1 - α)) = 1

Thus, average energy (E) of neutron after series of scattering or probability that single collision will make a neutron have energy E is:

(E) = ∫αE0E EP(E)dE/∫αE0E P(E)dE =∫αE0EEdE/E0(1-α)

= 1/E0(1-α)∫αE0E0EdE

1/E0(1-α)[E2/2]αE0E0

1/2E0(1-α)[E02 - α2E02

1/2E0(1-α)[E0(1 - α2)

1/2(1-α)[E0(1-α)(1+α)]

(1/2)E0(1+α)

Average log energy decrement:

This is utilized to get average number of collisions which a fast neutron, will make before its energy 0 E is reduced to thermal energy Et.

Take Et = E

Log energy decrement is logeEo - logEE = loge (Eo/E)

Therefore average log = [loge (Eo/E)]

ξ = ∫αE0E0 log(E0/E)P(E)dE

= ∫αE0E0 log(E0/E)dE/E0(1-α)

As ∫αE0E0 [dE/E0(1 -α)] = 1

For limits to change

For E = Eo; x = 1

And E = αEo = x = α

From x = E/E0, dE = E0dx and log E0/E = -log x

Then integral turns in

α1(-logx)(E0dx/E0(1 -α))

= - 1/1 -α∫logxdx

= 1 + (α/1-α) logα

Then substituting for α = ((A-1)2/(A+1)2)

ξ = 1+[(A-1)2/(A+1)2log(A-1)2/(A+1)2]/[1 - (A-1)2/(A+1)2]

ξ = [1-(A-1)2/2A][log(A-1/A+1)]

For A>1

ξ = 2/(A + 2/3)

Usually number of collision needed to reduce Eo + Et is provided by

n = 1/ξlogE0/Et

Distance travelled by fast neutron between its introduction in a slowing down medium and its thermalisation is known as fast diffusion length or slowing down length and square of fast diffusion length is Fermi-age.

Also, distance travelled by thermalised diffusion length and it is stated as thickness of slowing down medium.

n = n0e-t/l

n and n0 are number of neutrons before and after collision and L is thermal diffusion length. But for large absorption cross section

I = I0e-σNt

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