Topic 3 - Review of Fundamentals

Interactions of Radiation With Matter

Outline

• Beta Radiation
  • Range-energy relationship
  • Mechanisms of Energy Loss
• Alpha Rays
  • Range-Energy Relationship
  • Energy Transfer
• Gamma Rays
  • Exponential Absorption
  • Interaction Mechanisms
• Neutrons
  • Production/Classification/Interaction

Beta Rays (Particles)

Range-Energy Relationship

Curve of 210Bi 1.17 MeV beta particles with Al absorbers

Determination of range

• End point in absorption curve = range
• Rule of thumb:
  • Absorber half-thickness = 1/8 range of beta
• Systematic experiments have established beta range as function of material, energy:

Range-energy Curve for Beta Rays in Various Substances

Range-Energy Observations

• Required absorber thickness increases with increasing energy (!!)
• Range is largely a function of electron density ( per cm2),
  • To a lesser degree range is a function of Z
  • Result has practical implications for shielding
  • Atomic number neglected
  • Areal density is used

Density thickness

• Areal density of electrons
• Proportional to product of absorber density and linear thickness:

*Units can differ from those shown, as long as they are internally consistent

Advantage of density thickness

Range Energy Curve for Beta Particles

Beta Range – Empirical Equations

• R = 412E^{1.265 – 0.0954 ln E}
  • for 0.01 E 2.5 MeV
• ln E = 6.63 – 3.2376 (10.2146 – ln R)^1/2
  • for R 1200
• R = 530 E – 106
• for E > 2.5 MeV, R > 1200
• where
• R = range, mg/cm²
• E = maximum beta-ray energy, MeV

Beta Particles – Energy Loss Mechanisms

Ionization and Excitation

• Interaction between electric fields of beta and orbital electrons
  • inelastic collisions
  • energy loss by beta f(distance, KE)
• E_k = E_t - f
  • E_k = kinetic energy of ejected electron
  • E_t = energy lost by beta particle
  • f ionization potential of absorbing medium
• Ejected electron may produce additional ion pairs (clusters, delta rays)

Ionization Potential vs Energy Expended

Average energy lost by a beta particle in the production of an ion pair

Difference is attributed to electronic excitation

Linear Rate of Energy Loss by Beta Particle

• Called specific ionization
  • Ionization
  • Excitation
• Key
  • Understanding biological impacts
  • Detector design and response
• Number of ion-pairs formed per unit distance traveled.

Specific Ionization

Relationship of beta particle energy to specific ionization of air

Specific Ionization (S.I.)

• Number of ion-pairs created per specific distance traveled by the beta particle

• Where
  • dE/dx is the energy loss due to ionization and excitation per cm traveled (eV/cm)
  • W is the mean energy (eV) required to create an ion pair;
• High values for low-energy betas
• Decreases rapidly as beta energy increases
• Broad minimum around 1 MeV,
• Slowly increases at energies above this point

Specific Ionization - Beta Energy Loss, MeV/cm

• Where
  • q = charge on one electron
  • N = number of absorber atoms per cm³
  • Z = atomic number of the absorber
  • NZ = # of absorber electrons per cm³
  • E_m = energy equivalent of electron mass (0.511 MeV)
  • E_k = kinetic energy of the beta particle, MeV
  • b = v/c = speed of particle relative to c (light)
  • I = mean ionization and excitation potential of absorbing atoms, MeV

Mean Excitation Energies (I)

• Calculated for several elements from quantum principles
• Measured
• Approximate empirical formula:
  • I ~ 19.0 eV for Z = 1
  • I ~ 11.2 +11.7Z eV for 2 <=Z <= 13
  • I ~ 52.8 +8.71Z eV for Z > 13

Mass Stopping Power

• Another way to express energy loss
• If density thickness is used instead of length, then:

• Where S is the Mass Stopping Power ,
• r is the density of the absorber

Linear Energy Transfer

• Specific Ionization used to describe energy lost by the radiation
• In radiobiology, focus is on linear rate of energy absorption in the medium.
• This measure is Linear Energy Transfer

• dE_l is the average energy locally imparted to absorbing medium by particle traversing dl
• "Locally Imparted" refers to either
  • maximum distance from the track of the particle, or
  • maximum value of discrete energy loss
• Either case, LET refers to energy imparted within a limited volume of absorber

Energy Transferred vs Absorbed

• The initial b^-(blue) is losing energy by creation of ion pairs or delta rays. Not all its lost energy is localized.
• LET is energy deposited within the volume shown by the cylinder

Relative Mass Stopping Power

• Compares energy absorptive power of different media.
• Defined as:

• r_m is not a density
• Is relevant to dose measurements……

Bremsstrahlung

• X-rays emitted when high-speed charged particles are rapidly accelerated
  • b passes close to nucleus and is deflected from path
  • Heavy nuclei are more effective in causing deflections
• Single electron can emit X-ray up to its own kinetic energy
• Monoergetic beam of electrons produces a continuous spectrum

Estimating Bremsstrahlung Production

• f _b is the fraction of incident beta energy converted into photons (Bremmstrahlung)
• Z = atomic number of the absorber
• E_m = maximum energy of the beta particle

Importance of Bremsstrahlung

• b particles can excite and ionize atoms
• Also radiate energy via bremsstrahlung
• Relative contribution important only at high energies
  • high energy photons can be created
  • additional shielding problem from high-energy beta emitters

Stopping Power Stopping Power for Electrons in water in MeV cm²/g

Radiative to Collisional Losses

X-ray Production from Monoenergetic Electrons

• f_e is the fraction of energy in electron beam converted into X –rays
• Z = atomic number of the absorber
• E_m = maximum energy of the beta particle

ALPHA RAYS (PARTICLES)

Alpha Particles

• Least penetrating of radiations
  • in air, range ~ few cm
  • in tissue, ~ microns (10^-4cm)
• Range
  • mean range
  • extrapolated range

Alpha-particle Absorption Curve

Alpha Particles – Range

• Empirical equations for Alphas in air
• R = 0.56 E
  • R in cm
  • E in MeV, E< 4 MeV
• R = 1.24 E –2. 62
  • R in cm
• E in MeV, 4 <E< 8 MeV
• For range in air at 0⁰ C and 760 mm pressure
• Empirical equations for Alphas in other media
• R_m = 0.56A^1/3R
  • R_m is in mg/cm²
  • A = atomic mass number of the medium
  • R = range of the alpha particle in air, cm

Alpha Particles, Energy Loss

• Only significant mechanism:
  • interaction with electrons in absorbing medium
  • at very slow speeds nuclear collisions can occur
  • only small energy transfers occur at each interaction
• Result in excitation and ionization of absorber atoms
• On average, alpha loses 35 eV per ion pair
• Because of high charge and low velocity
  • (due to mass) specific ionization is very high
  • path is almost straight
  • energy loss is essentially continuous

Comparison with e^-,+

• Electrons and positrons
  • lose energy almost continuously as they slow
  • can lose large fraction of energy in single collision with atomic electron (equal mass)
  • large deflections
  • frequently scattered through large angle deflections by nuclei (result: bremsstrahlung)
• HCPs
  • straight line paths

Alpha- vs Beta –particle tracks

Alpha Particle Bragg Peak

Alpha Particles, Energy Loss

• Where
  • Z = atomic number of ionizing particle
  • q = unit electrical charge. 1.6 x 10^-19C
  • zq = electrical charge on the ionizing particle
  • M = rest mass of the ionizing particle, gms
  • V = velocity of the ionizing particle, cm/s
  • N = number of absorber atoms per cm³
  • Z = atomic number of the absorber
  • NZ = # of absorber electrons per cm³
  • C= velocity of light, 3 x 10¹⁰cm/s
  • I = mean ionization and excitation potential of absorbing atoms, for air= 1.38 x 10 ^-10erg

Notes re energy transfer equation

• Previous equation appropriate for other heavy charged particles (HCPs)
• derived by Bethe from quantum mechanics
• various versions around
• logarithmic term leads to increase in stopping power at energies
• low energies, lhs of equation dominates
• Bragg peak consequence of ln term decreasing

GAMMA RAYS

Gamma Rays

• Key differences in charged and uncharged particle interactions
  • photon electrically neutral
  • does not steadily lose energy as it penetrates
  • interaction is statistically governed by:
    • medium
    • photon energy
  • probability described by coefficient

Photon Interactions

Nature of Photon Interactions

• Absorption with disappearance of photon
• Scatter
  • direction change
  • energy change
  • no energy change
• Principal methods of energy deposition
  • photoelectric
  • Compton scattering
  • pair production
  • photonuclear reactions

Describing Photon Behavior

• Exponential Absorption Equation

I = I₀e^-mt

• I_{0_} = photon intensity with no absorbers
• I = photons transmitted through absorber
• t = absorber thickness
• e = base of natural logarithm (~2.71828183…)
• m = atteunation coeffcient (slope of absorption curve)

Determining Attenuation Coefficients

• Measurements taken under conditions of good geometry
  • monoenergetic
  • well-collimated
  • narrow beam
• Data plotted as semi-log
  • straight line
  • slope is m
  • intercept is I₀

Good Geometry Conditions

Attenuation of ¹³⁷Cs Gamma Rays under conditions of good geometry

Additional Considerations

• e^-mt
  • Exponential term – must be dimensionless,
  • m and t must have reciprocal dimensions
• m or m_l
  • Linear attenuation coefficient
  • Total attenuation coefficient
• m/r (mass attenuation coefficient)
  • also identified as mm
  • units g/cm²
  • paired with t expressed as density thickness
• Atomic attenuation coefficient, m_a
  • m_{a =} m_l / N
    

Linear, Mass, Electronic, and Atomic Attenuation Coefficients

r is the density; Ne is the number of electrons per g; Z = atomic number of the material

Miscellaneous Relationships

• Number of atoms per g = N_A/A
  • N_A = atoms per mol
  • A = gms per mol
• Number of electrons per g = N_AZ/A = N_e
• Number of electrons per kg= 1000 N_e

Photon Interaction Mechanisms

• Photoelectric effect
• Scattering
• Pair (and triplet) production
  • These 3 mechanisms result in electron emission from material interacting with the photon
• Photonuclear reactions
• This mechanism initiates a nuclear reaction and results in emission of other radiations

Summary of Photoelectric Effect

• Involves bound electrons
• Probability of ejection is greatest if photon has just enough energy to knock electron from shell
• Cross section varies with photon energy, approximately 1/hn³

Interaction Mechanism - Photoelectric Effect

Compton Scattering

• Elastic collision between photon & “free” e
• photon transfers some, not all, of its energy
• Scattered photon, and e, result

Probability of Compton interaction

• Decreases with increasing Z
• For low-Z elements, Compton interaction predominates
• every electron acts as a scattering center
• electron density is important
• scatter described wrt solid angle
• Klein-Nishina equation describes scattering coefficient

Compton relationships

• Energy of scattered photon

Klein-Nishina

• Scattering into differential solid angle dW at angle q
• a defined as
  • hn/m₀c²
  • or hn (expressed in MeV)/0.511

Summary for Pair Production

• Interaction between photon and nucleus
• Threshold is 1.02 MeV
• Increases rapidly above threshold
• Coefficient
  • per atom varies ~Z²
  • per mass Z¹
• Energy transferred to KE_particle = hn-1.022
• 2 annihilation photons, each 0.511 MeV

Photonuclear Reactions

• Photodisintegration
  • Nucleus captures photon, typically emits a neutron
  • Generally requires high energy photons
    • ~ 6 to 8 MeV common
    • Exception: ⁹Be(γ,n)⁸Be - threshold is 1.666 MeV
  • Significant for electron accelerators
    • Betatrons
    • Synchrotrons
    • Linear accelerators (common in hospitals)

Summary of Photon Interactions

Photon Interactions: Combined Effects

• Attenuation coefficients are probabilities of removal of photons under good geometry considerations.
  • Total attenuation is sum of interaction mechanisms. Principal components are:
    m_t=  m_pe + m_cs + m_pp
    where each has its own probability based on energy and absorbing material – does not account for fraction carried away via annihilation
  • m_t gives fraction of energy of a beam removed per unit distance traveled in absorber

Photon Interactions: Combined Effects

• Absorption coefficients considers only the fraction of beam energy that is transferred to the absorber by:
  • Photoelectron
  • Compton electron
  • Electron pair (from pair production)
• Doesn’t consider
• Scattered compton photon
  • Annihilation radiation after pair production
• Absorption coefficients:
  • _e=  m_pe + m_cs + m_pp(hn -1.02)/ hn

Relative Importance of Reactions

Energy Transfer and Energy Absorption

• Photon interaction with absorber complex
  • scattered photon
  • kinetic energy of electron
    • collisional losses
    • brehmsstrahlung radiation
• Calculate
  • E_tr average energy transferred
  • E_ab average energy absorbed
  • difference is bremsstrahlung

Summary

• Energy transfer from radiation field to absorbing medium is basis of radiation effects
• Charged particles
  • excite or ionize atoms
  • Have definite range in matter
  • exhibit Bragg peak
• Photon (x, gamma) differ qualitatively
  • indierctly ionizing
  • interact with atomic electrons (pe, scatter, pp)
  • stochastic events

NEUTRONS

Neutrons

• No ‘natural’ neutron emitters
• Sources of neutrons
  • Fission (reactors)
    • Pu
    • U
  • Spontaneous nuclear fission:
    • ²⁵²Cf
  • Photoneutron sources
  • Other forms of nuclear bombardment

Fission

• ²³⁵U, ²³³U, Pu
• Neutron absorbed by fissionable nucleus
• Nucleus becomes ‘compound nucleus’
• Nucleus may then fission, or decay by other emission

²³⁵Cf

• Emits alphas
• Also spontaneously fissions
  • Neutrons emitted with fission
  • 10 fissions per 313 alpha emissions
  • Most probable neutron energy is 1 MeV
  • Average neutron energy is 2.3 MeV
• Half life (T_1/2,sf) = 2.65 years
  • (both decay modes)
  • T_1/2,a= 2.73 y
  • What is T_1/2,sf?

Photoneutrons

• Photodisintegration
  • photons break up nucleus
  • protons, neutrons, and deuterons (a proton and neutron bound together) are ejected.
• Photoneutrons:
  • Photon incident on nucleus contains enough energy to overcome binding energy of nucleons
  • Neutron is emitted
  • Photons (gamma rays) are monoenergetic
  • Neutrons are monoenergetic
• Low binding energy target atoms used (Be)
• See Table 5.4 on page 151

Table 5.4 a,n Photoneutron Sources

Other Neutron Production Methods

• (a, n) : An alpha particle is absorbed by a ⁹Be nucelues
• ⁹Be + ⁴He --> (¹³C)^* --> ¹²C + ¹n
• Neutrons are poly energetic
  • Alphas lose energy prior to reaching ⁹Be
• Typically alpha emitter mixed in powder form with alpha source

Table 5.5 a,n Neutron Sources

Other Neutron Production Methods

• Accelerators may also be source of alphas
• Table 5.5 on page 152 has a list of typical a, n sources
• Typically favored over photoneutron sources since they have a significantly longer half life
• Note that all neutrons are born ‘fast’
• What is a ‘fast’ neutron?

Classification

• Neutrons are classified according to their energy
  • Neutron reactions are very dependant on energy
• Only two classifications important at this point
  • Fast neutrons
    • Generally greater than 0.1 MeV
  • Thermal
    • Have same KE as gas molecules in their environment:

Maxwell-Boltzmann Energy Distribution - As Function of Temperature

Thermal Neutrons

• Also called ‘slow’ neutrons
• Defined more precisely than fast or other types of neutrons:
  • At 293°K
    • 0.025 eV most probable energy
    • 2200 m/s most probable velocity

Maxwell-Boltzmann distribution - gas molecules

Thermal Neutrons

• Calculating most probable energy
  • Equation 5.40
  • E_mp= kT
• Average Energy:
  • Equation 5.41
  • E = 3/2 kT
• Can E = 1/2 mv² be used for neutrons?
  • Gives velocity of thermal neutrons

Neutron Interaction

• All neutrons are born fast
• Lose energy by collisions with nuclei
  • Inelastic
  • Elastic
• Collisions of fast neutrons are typically elastic
  • With low Z absorbers
  • "billiard ball" type of collision
  • KE and momentum conserved
  • Low probability for capture
• Thermal neutron collisions may result in elastic collisions or capture
• Elastic scattering and capture are most important for HP
• Capture of a neutron is typically followed by emission of a photon or other particle from nucleus
• Neutrons are removed exponentially when a material is placed in a neutron beam

Absorption

• Linear or mass attenuation coefficients are not typically used with neutrons
• Microscopic cross section or macroscopic cross sections are normally given for a material
• Microscopic cross section = s = cm² / atom
• Macroscopic cross section = S = s N = cm^-1

Cross Sections

• Absorption cross section may include activation cross section
• Absorption cross section does not include scattering cross section
• Total cross section is the sum of all cross sections, including scattering, activation and other reactions
• All cross sections are VERY energy dependent
• Typically, only thermal neutron cross sections are listed
• Be very careful in using cross sections; Tables are not always clear on how to use them properly
• Absorption cross sections are said to be about 1/3 of the total cross section, but this is only a rough estimate

Removal of Neutrons

• Equation 5.43 is used with the absorption cross section to find neutron removal from a beam of neutrons
• I = I₀e^-sNt
• I₀ = initial neutron intensity
• I = final neutron intensity
• N = number atoms absorber per cm3
• t = thickness of material in cm
• s = microscopic cross section for absorption in cm2/atom
• Note that if more than one method for removal is possible, it should be included in the cross section
• Equation 5.43 can also be used to find activation quantities

1/v Law

• Although cross sections are temperature dependant, can ‘correct’ cross sections for 0.001 to 1000 eV
• Use Equation 5.53 to find cross sections at different energies
• Note that there are some atoms that do not follow the 1/v law
• Called ‘not 1/v’ atoms
• Simply multiply by a factor to obtain the desired result
• Compilation of ‘not’ 1/v atoms in “Introduction to Nuclear Engineering” by John Lamarsh, 2nd Ed. p. 63

Cross Section Notes

• The cross sections used in calculations should be for poly energetic neutrons if reactor or (a, n) reaction
• Tables normally list cross sections for mono-energetic thermal neutrons
• Beware of extrapolating outside limits of 1/v (> 1000 eV)
• Beware of resonances and competing reactions
• Divide cross section by 1.128 to correct a monoenergetic neutron beam cross section to a poly energetic beam cross section
  • More detail on this in Chapter 12 later
• BNL is generally best source for cross sections

Neutron Absorption

• Neutron absorption may result in:
  • Formation of a stable isotope of an atom and emission of a photon
  • Fission
  • Emission of an alpha particle
  • Activation of an atom
  • Other, more exotic reactions

Activation

• When neutron absorption leads to too many neutrons
• Neutron is absorbed
• Gamma emitted (prompt)
• Atom is now radioactive
• Typically beta (negatron) emitter
• Sometimes alpha emitters formed

Activation

• Equation 5.58
• lN = fsn (1 - e^-lt)
• lN = Activity in Bq
• f = Flux, neutrons per cm² per sec
• s = Activation cross section, cm²
• n = Number target atoms
• t = Time of irradiation
• l = Decay constant, induced activity

Scattering

• Energy lost best through collisions with atoms of same mass as neutron
• "Billiard ball collisions"
• Scattering with ‘heavy’ atoms typically does not result in much energy loss
• Measure of energy loss = average logarithmic energy decrement per collision (See p. 155)

Neutron Energy Loss

• Scattering is how a neutron becomes thermal
• Hydrogen has the highest average logarithmic energy decrement per collision (almost same mass as neutron)
• Median amount of energy transferred from neutron to hydrogen in one collision is 63%

Neutron Life

• Fast Diffusion Length - average linear distance a fast neutron travels
• Thermal Diffusion length - average linear distance a thermal neutron travels
• Note that actual neutron paths are very tortuous
• Diffusion lengths are only of concern when absorption cross section is small
• Typically measured, many assumptions required for calculations

Summary

• Neutrons slowed down by scattering with light nuclei
• Absorption of neutrons more likely with thermal neutrons
• Activation is possible when neutron absorption occurs
• Exercise caution when using cross sections