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Topic 3 - Review of Fundamentals

Interactions of Radiation With Matter

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  • 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

Absorber density thickness, mg/cm2
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 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:

Density thickness formula

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

Advantage of density thickness

Advantage of density thickness

Range Energy Curve for Beta Particles

Range Energy Curve for Beta Particles

Beta Range – Empirical Equations

  • R = 412E1.265 – 0.0954 ln E
    • for 0.01 less than equal to E less than equal to 2.5 MeV
  • ln E = 6.63 – 3.2376 (10.2146 – ln R)1/2
    • for R less than equal to 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)
  • Ek = Et - f
    • Ek = kinetic energy of ejected electron
    • Et = 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

Potential, eV
w, Mean energy expended per ion Pair, eV
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
    • Em = energy equivalent of electron mass (0.511 MeV)
    • Ek = 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

  • dEl 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:

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


  • 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
  • Em = 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

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


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
  • Rm = 0.56A1/3R
    • Rm 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

  • 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

  • I0_ = 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 ml
    • 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, ma
    • ma = ml / 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 = NA/A
    • NA = atoms per mol
    • A = gms per mol
  • Number of electrons per g = NAZ/A = Ne
  • Number of electrons per kg= 1000 Ne

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


  • 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 KEparticle = 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:
      mt=  mpe + mcs + mpp
      where each has its own probability based on energy and absorbing material – does not account for fraction carried away via annihilation
    • mt 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=  mpe + mcs + mpp(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
    • Etr average energy transferred
    • Eab average energy absorbed
    • difference is bremsstrahlung


  • 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



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


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


  • 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 (T1/2,sf) = 2.65 years
    • (both decay modes)
    • T1/2,a= 2.73 y
    • What is T1/2,sf?


  • 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?


  • 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
    • Emp= 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


  • 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


  • 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


  • 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


  • 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


  • 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


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