Radiological physics: photon interactions

It provides, in the units of cross sectional area, a microscopic measure of how likely an incoming photon is to interact with the electron (electronic cross section) or atom (atomic cross section)

What is a cross section?

The differential of electronic cross section per solid angle (i.e. has units of area per solid angle) is the likelihood of measuring a photon scattered by an electron along a path described by a given solid angle from the location of interaction

What is a differential electronic cross section?

Integrate over all solid angles to find a measure of overall scattering likelihood (in units of area)

How does an differential electronic cross section relate to the electronic cross section?

Infinite, because as long as the electron is a finite distance from the proton, an interaction occurs. Photons are much less likely to interact with an absorbing medium, so the cross sections of their different interactions are finite

Cross section of Coulomb interaction between a distant electron and proton? How does this compare to the cross sections of photon interactions?

Finite, interestingly since the cross section is infinite once integrated

Differential cross section of Coulomb interaction between a distant electron and proton?

Electronic cross section is a microscopic measure of how likely an incident photon is to interact with a single electron. Multiplying this by Z gives atomic cross section, which is the likelihood of interaction with a whole atom (via an interaction with a single electron)

How do electronic and atomic cross sections relate?

mu is the probability per unit path length that a photon interacts with the absorber (via any photon interaction: the components of mu are given their own symbol for each type of interaction). In general, it depends on the photon energy and the density and atomic number of the absorber

What is the linear attenuation coefficient symbol, meaning and general dependencies?

mu over rho is the linear attenuation coefficient divided by the mass per volume (mass density) of the material. This removes the density dependence of mu to compare behaviour of different materials regardless of their form (e.g. porous vs solid lead), commonly measured in cm2/g

What is the mass attenuation coefficient symbol, meaning and units?

Cross sections have units of area and linear attenuation coefficients have units per path length. They both measure the likelihood of an interaction between an incident photon and the body of interest, but the cross section is a microscopic measure that considers how cross sectionally large an individual body must be for a passing photon to interact, and the linear attenuation coefficient is a macroscopic measure that considers the probability of a photon interacting per unit path length in the medium

How does a cross section differ from a linear attenuation coefficient generally?

Mass linear attenuation is the atomic cross section times N sub A over atomic mass A. Linear attenuation is mass attenuation times mass density, and atomic cross section is electronic cross section times Z (when incident photon energy is higher than the electron binding energy, that is, far from the absorption edges)

How do electronic and atomic cross sections relate to linear and mass attenuation coefficients? Under what conditions?

Atomic cross section is the electronic cross section times a Compton scattering function of Z and momentum transfer x. For low momentum transfer (i.e. low energy for a given scattering angle), the scattering function approaches zero, and for high momentum transfer it approaches Z (i.e. this is the limit towards the typical relationship)

How does the electronic cross section relate to the atomic cross section when the atomic cross section when incident photon energy is comparable to an absorption edge? What are the intuitive end behaviours?

Consider the linear attenuation coefficient as the probability of interaction per unit thickness, which contains N atoms with cross section sigma in area A. Then, the probability of an interaction may be expressed as the product of N and the cross section divided by the total area. Multiply the top and bottom by dx to let the total area become an infinitesimal volume, and recall the number of atoms per unit volume is mass density times N sub A divided by A

Convenient derivation relating linear attenuation coefficient to atomic cross section?

Thomson (sigma sub Th), Rayleigh (sigma sub R), Compton (sigma sub C), photoelectric (tau), pair and triplet production (kappa sub PP or sub TP but usually given together as the sum anyway)

What are the symbols for linear attenuation coefficients for each type of photon interaction?

Electrons are loosely or tightly bound to a nucleus with respect to a given photon if the electron energy is small or comparable to the photon energy

Definition of loosely vs tightly bound electrons

The photon and a free electron (loosely) or with the atom as a whole (tightly)

Interactions between photons and loosely vs tightly bound electrons are considered to be between what?

Thomson scattering, Compton scattering

Examples of interactions between photon and loosely bound electron

Rayleigh scattering, photoelectric effect, triplet production

Examples of interactions between photon and tightly bound electron

Photodisintegration or pair production

Examples of interactions between photon and absorber nucleus

Compton scattering, photoelectric effect, pair production, triplet production

Examples of photon interactions that generate secondary charged particles?

Thomson and Rayleigh scattering

Examples of photon interactions that do not any transfer from the incident photon?

A direct interaction between a high energy photon and an absorber nucleus, can kick out neutrons (even at tx energy range), protons, and alpha particles

Photodisintegration meaning

photonuclear reaction

Photodisintegration alternative name

Photoelectric

Dominant photon interaction in all absorbers at low photon energy (i.e. 1 to 10 keV)?

Compton scattering

Dominant photon interaction in all absorbers at intermediate photon energy (i.e. ~ 1 MeV)?

The width of the Compton region is broader for lower Z absorbers and narrows with increasing Z. In tissue, the Compton region ranges ~ 20 keV to 20 MeV, making it the most important interaction in radiotherapy

How does the Compton region differ between absorbers? What is the implication?

Pair production

Dominant photon interaction in all absorbers at high photon energy (i.e. > 10 MeV)?

Rayleigh scattering when lower than electron binding energies, photoelectric near absorption edges, Compton when higher than electron binding energies, then pair production after its threshold of twice the rest energy of the electron

At what incident photon energy is each type of scattering dominant?

Pair production is more likely with higher Z (same minimum threshold but larger mass attenuation coefficient in that region)

How does the pair production region differ between absorbers?

The equation governing exponential decay of an initial intensity with increasing thickness through which the beam travels, mediated by a rate constant: here the linear attenuation coefficient

Beer-Lambert law

Consider the spatial derivative of the photon fluence as analogous to the time derivative of the number of radioactive nuclei: the fluence decreases (i.e. the derivative is negative) proportional to itself at a rate constant (i.e. the linear attenuation coefficient)

What is an elegant perspective for deriving the Beer Lambert law of photon beam attenuation?

Linear attenuation coefficient, measured by attenuation through a thin material

Most important parameter used for characterization of x-ray or gamma ray penetration into absorbing media, and how it is measured experimentally

A narrow monoenergetic photon beam travelling through material thickness x with constant linear attenuation coefficient mu, neglecting scattering and secondary radiation

Conditions under which beam intensity is reduced by a factor of exp(-mu x)

The first half value layer or x sub 1/2 is the thickness of an absorber that attenuates a photon beam to half of its intensity

HVL1 meaning

Characterizing superficial and orthovoltage x-ray beams with aluminum and copper respectively

Common use of HVL1 and associated material

Product of x sub 1/2 and mu is ln(2)

Relation between HVL1 and mu

The mean free path or x bar or relaxation length is the thickness of an absorber that attenuates a photon beam to 1/e of its intensity. This is the average distance a photon travels through the material before undergoing an interaction

MFP meaning

Product of x bar and mu is 1

Relation between MFP and mu

The tenth value layer or x sub 1/10 is the thickness of an absorber that attenuates a photon beam to 1/10 of its intensity

TVL meaning

Product of x sub 1/10 and mu is ln(10)

Relation between TVL and mu

Radiation protection in treatment room shielding calculations (given in number of TVLs)

Common use of TVL

The second half value layer or x sub 1/2 is the thickness of an absorber that attenuates a photon beam from 1/2 to 1/4 of its intensity

HVL2 meaning

Chi is the ratio of HVL1 to HVL2 for a photon beam

Homogeneity factor symbol and definition

When chi=1, the beam is monoenergetic and subsequent HVLs are identical, otherwise the beam is polyenergetic. If subsequent HVLs are thicker then the absorber is hardening the beam, and if subsequent HVLs are thinner then the absorber is softening the beam

Interpret the homogeneity factor

A material preferentially removes low energy photons (photoelectric effect region)

What does it mean to harden a photon beam and what effect causes that?

A material preferentially removes high energy photons (pair production region)

What does it mean to soften a photon beam and what effect causes that?

Through collision (ionization) loss or radiation loss via photons that leave the area

How do secondary charged particles lose energy?

Depositing energy as dose through Coulomb interactions with orbital electrons of absorber

Meaning of collision loss by secondary charged particles

Radiating energy in the form of photons through Coulomb interactions with nuclei of absorber

Meaning of radiation loss by secondary charged particles

Electrons kicked into motion via Compton scatter, photoelectric effect, or triplet production; Auger electrons emitted by filling those vacancies; electron positron pairs produced by pair and triplet production

In what form and by which processes do photon interactions produce secondary charged particles?

Photons leave the interaction site in Rayleigh, Thomson and Compton scatter, but only lose energy in Compton scatter. Characteristic (fluorescence) x rays are emitted following photoelectric effect and triplet production, which may either escape the atom or alternatively lead to Auger electrons being emitted

In what form and by which processes do photon interactions produce secondary photons?

Some of the incident photon energy is transferred to the kinetic energy of secondary charged particles (E sub tr). The transferred energy is either absorbed through charged particle collisions (E sub ab) or lost to radiative processes (E sub rad)

Trace the energy of an incident photon through a general photon interaction and cascade effects

f bar sub tr is the mean energy transferred from the photon to KE of secondary charged particles (E bar sub tr) divided by the photon energy

What is the mean energy transfer fraction symbol and meaning?

f bar sub ab is the mean energy absorbed from the photon in the absorber (E bar ab) divided by the photon energy

What is the mean energy absorption fraction symbol and meaning?

g bar is the fraction of the mean energy transferred from the photon to the secondary charged particles that is radiation loss in the form of radiated photons (E bar rad divided by E bar tr)

What is the mean radiation fraction symbol and meaning?

mu sub tr describes the energy transferred from photons to charged particles. It is the linear attenuation coefficient times the mean energy transfer fraction

What is the linear energy transfer coefficient?

mu sub ab (or sub en in some literature) describes the energy transferred from photons to charged particles then to the absorber. It is the linear attenuation coefficient times the mean energy absorption fraction

What is the linear energy absorption coefficient?

mass linear attenuation coefficient times the mean energy transfer fraction

What is the mass energy transfer coefficient?

mass linear attenuation coefficient times the mean energy absorption fraction

What is the mass energy absorption coefficient?

The difference in energy between what is transferred from the photon to released or produced charged particles (E bar tr) and what those charged particles deposit in the absorber (E bar ab) is radiation loss (E bar rad)

Relation between mean energy transfer and absorption

The mean energy absorption coefficient is the mean energy transfer coefficient times (1 - g bar)

Relation between mean energy transfer and absorption coefficients

Energy radiated from the secondary charged particles in the form of photons through bremsstrahlung and in-flight annihilation; or emitted as fluorescence photons during atomic relaxation after impulse ionization or impulse excitation of absorber atoms. This energy is carried away from the original photon interaction in the form of photons

Sources of radiation loss

Photoelectric, because the probability of photoelectric event in a given shell, fluorescence yield (i.e. probability of characteristic x ray escaping the atom), and average energy of emitted characteristic x rays all vary for different absorbers. Mean energy transfer in pair and triplet production and Compton scatter (after averaging through all scatter angles and neglecting small differences in electron binding energies) vary only with the incident photon energy (since characteristic x ray emission is negligible)

Mean energy transfer fraction depends on the absorber for which photon interactions? Why?

All of photoelectric effect, pair and triplet production, and Compton scatter have mean energy transfer fractions that increase with incident photon energy to converge to 1, although photoelectric effect does have some sharp declines following the absorption edges. Compton grows the slowest: photons with 1000 MeV only have a mean energy transfer fraction of 0.85

For which photon interactions does mean energy transfer fraction converge to 1? Which is slowest and why?

For extremely low photon energies it begins at ~1, then for high Z materials decreases abruptly at the various photoelectric absorption edges, has a peak and valley between the photoelectric and Compton transition, then increases to ~1 for large photon energies as pair production dominates. In low Z materials, the photoelectric absorption edges contribute negligibly to the total mean energy transfer fraction

What is the shape of the total mean energy transfer fraction (across all interactions) vs incident photon energy? How does it change with Z?

Elastic scatter of low energy photons (much less energy than rest energy of electron) by loosely bound electron without energy transfer to medium

What is Thomson scatter?

A free electron is accelerated by the EM field of a passing photon, and consequently behaves as a dipole that radiates its own EM field, which the Thomson differential electronic cross section characterizes with respect to the incident photon path

What gives rise to Thomson scatter?

Conceptually, the differential cross section arises from conservation of energy between the incoming photon. Energy in is given by the Poynting vector of a plane wave times the differential of electronic cross section (i.e. in what area the incident plane wave will interact with the free electron). Energy out is carried by the scattered photon, which arises from the dipole behaviour of the accelerated electron and is given by the Poynting vector of a spherical wave times the differential of the area swept out by solid angle (i.e. which direction relative to the incident photon is the scatter photon most likely to be measured)

How is the Thomson differential electronic cross section derived?

Integration of the Poynting vector over area is the power carried by an EM wave through that area. This is used when deriving Thomson's classical differential electronic cross section through conservation of energy (technically conservation of power)

Units of the Poynting vector? Why should I care?

Thomson scattering occurs when the photon energy is much less than the rest energy of the electron, so relativity and quantum mechanics are not needed. Also, relativity and quantum mechanics were in the process of being invented at the time lol

Why did Thomson not use relativity or quantum mechanics to derive his electronic cross section?

The electron forms a dipole along the direction of polarization (say the x axis) of the incident EM wave. If the wave is polarized to have electric field (say along the y axis), then there should be no measurable induced electric field along the direction the electron is moving. However, Thomson averaged over all polarization angles, so the differential electronic cross section is minimum but non-zero at a scatter angle of 90 degrees

Why is the Thomson differential electronic cross section not zero for a solid angle along a path perpendicular to the incident photon? Why might we expect that to be the case?

Squared charge of electron over 4 pi epsilon naught, divided by the rest energy of the electron

Thomson classical radius of free electron

Squared classical radius of electron times squared sin of polarization angle (which varies 0 to 2 pi about the axis of EM wave propagation)

What is the differential electronic Thomson cross section before averaging over polarization angle?

Half of squared classical radius of electron times one less squared cos of scatter angle (which varies 0 to pi about the axis of the plane containing the point of interest, i.e. gives likelihood of forward vs back scatter)

What is the differential electronic Thomson cross section after averaging over polarization angle?

8 pi by 3 times the squared classical radius of electron

What is the electronic Thomson cross section?

It doesn't, it is a constant times Z

How does Thomson atomic cross section vary with incident photon energy?

Linearly, since the electronic cross section is a constant

How does Thomson atomic cross section vary with Z?

Mass linear attenuation is the atomic cross section times N sub A over atomic mass A. Since atomic cross section is linear in Z, the mass attenuation is independent of Z as the ratio of Z to atomic mass is roughly constant

Thomson scatter mass attenuation dependencies

Inelastic scatter of a photon by a stationary (by assumption) loosely bound electron. The photon deflects by angle theta with a lower energy and the electron is ejected at angle phi with the energy difference in kinetic energy

What is Compton scatter?

Photon energy divided by c

Momentum of a photon

Total energy squared is rest energy squared plus (pc) squared

Relativistic relationship between energy and momentum

Change in photon wavelength is the "Compton electron wavelength" (h by c times mass of electron) times 1 - cos(theta)

Compton wavelength shift

Only on the angle of Compton scatter theta (with greater shift for greater deflection), no dependence on photon energy

Compton wavelength shift dependencies

Higher energy incident photons lead to smaller electron deflection angles phi

For a given Compton scatter angle theta, how does the scatter angle phi of the electron vary with photon energy?

Thomson scatter is the low incident photon energy limit of Compton scatter, which can be seen as a relativistic correction to Thomson. They are also equivalent in Compton forward scattering, as it is the limiting case when the incident photon is not deflected from it's path (theta is zero) so its energy does not change and the interaction becomes classical

How does Compton scattering relate to Thomson scattering?

Thomson scattering is elastic because the photon energy is not changed by the interaction (well handled classically), whereas Compton scattering is inelastic (and requires relativistic calculation)

Why is Thomson or Compton scattering elastic or inelastic?

The scattered photon energy saturates with increasing incident photon energy, and higher incident energies saturate to lower levels with increasing scatter angle

How does the energy of the Compton scattered photon relate to the scatter angle theta?

As the incident photon energy approaches infinity, the Compton scattered photon energy saturates to the rest energy of the electron at theta = pi/2, and to half of that at theta = pi. Therefore, there is an upper bound on the energy of back scattered photons which is practically important for shielding purposes

What is the saturation energy of a Compton photon scattered at theta = pi by 2? Of a back scattered photon? What is the implication?

The ratio of scattered to incident photon energy. For a given scatter angle, higher energy incident photons have a smaller Compton scatter fraction. For a given incident photon energy, as theta increases from 0 the Compton scatter fraction decreases from 1 to a saturation level. The higher the incident photon energy, the lower the Compton scatter fraction saturation

Compton scatter fraction meaning and trends

The ratio of electron kinetic energy to incident photon energy. For a given scatter angle, higher energy incident photons have a higher Compton scatter fraction. For a given incident photon energy, as theta increases from 0 the Compton scatter fraction increases from 0 to a saturation level. The higher the incident photon energy, the higher the Compton energy transfer fraction saturation

Compton energy transfer fraction meaning and trends

high energy photons are more efficient at transferring incident photon energy to the electron, especially in the case of backscatter

Implication of Compton energy scatter/transfer fraction trends

Strong dependence on incident photon energy: low energy photons are scatter with little change and deflect more readily, whereas high energy photons scatter with smaller deflection and a large transfer in energy to the electron

Compton energy shift dependencies

The energy transfer fraction (i.e. kinetic energy imparted to the electron divided by the incident photon energy) averaged over all scatter angles theta. It increases with increasing incident photon energy

Mean energy transfer fraction for Compton meaning and trend

Compton scattering

Incoherent scattering

We know the energy of the scattered photon as a function of scattering angle. The energy of the electron is the difference in photon energy before and after scattering. The average value can be found by weighting this by the normalized probability (i.e. the differential electronic cross section normalized by the total cross section as a function of scattering angle) and integrating over all scattering angles (and over phi, which gives a factor of 2 pi)

Derivation of average energy of Compton scattered electron?

A Klein Nishina form factor correction to the Thomson differential electron cross section gives the Compton differential electron cross section. The form factor approaches 1 for low incident photon energy (the limit where Compton becomes Thomson-like), and approaches zero for higher photon energies because energetic photons are less likely to interact so the cross section decreases

How is Compton scattering estimated from Thomson scattering? What are the intuitive end behaviours?

Calculated Compton mass coefficient levels off with decreasing incident photon energy, whereas NIST data decreases with decreasing photon energy. The difference is accounted for by assuming the Compton atomic cross section is the electronic cross section times a scattering function, which approaches zero for low incident photon energy

Why does the Compton mass coefficient calculated from electronic cross section not match NIST data? How is it corrected?

Conservation of energy implies the kinetic energy of the electron is equal to the total energy of the incoming photon, but imposing conservation of momentum violates this: the photon carries additional energy and momentum that must be imparted to the parent atom. The energy imparted is negligible to the parent atom, but the condition nonetheless requires a tightly bound electron (i.e. the photon energy is comparable to but slightly exceeds that of the electron)

Why is the photoelectric effect impossible with a free electron?

When the photon energy exceeds the K-shell binding energy of the absorber, about 80% of absorptions occur in the K-shell, and the remaining 20% with less tightly bound higher shell electrons

In which shells does the photoelectric effect occur?

A higher shell electron fills the vacancy and emits a characteristic (fluorescence) photon, or an Auger electron if the nucleus absorbs the emitted photon

What results from the vacancy left from the photoelectric effect?

The probability of true forward scatter and backward scatter are zero, and the maximum angle of scatter is around 70 degrees for 10 keV photons, and decreases to favour forward scatter with increasing energy. Essentially all scatter angles of a 10 MeV photon beam are less than 20 degrees, with a maximum of around 2 degrees

Describe the angle at which photoelectrons are emitted. How does it change with incident photon energy?