Cavity Theory III

1
Cavity Theory III

• The Fano Theorem
• Other Cavity Theories
• Dose Near Interfaces

2
The Fano Theorem

• In 1954 Fano pointed out that in many practical
  cases the B-G requirement for a small,
  nonperturbing cavity is ignored, and the use of
  walls and cavity that are matched in atomic
  composition is substituted, easing the size
  restriction
• He noted that this substitution had never been
  rigorously justified, and attempted to provide
  such a justification

3
The Fano Theorem (cont.)

• Unfortunately, his proof disregarded the
  influence of the polarization effect, which
  seriously undermines the validity of the Fano
  theorem for megavolt photons irradiating a
  gas-filled cavity in a matching solid wall
• For that case a more general cavity theory that
  accounts for the difference in stopping powers
  between condensed and gaseous media must be used,
  such as the B-G or Spencer theory

4
The Fano Theorem (cont.)

• Nevertheless the Fano theorem is an important
  statement in that it applies generally to
  neutrons, as well as to photons below about 1
  MeV
• Fanos theorem In an infinite medium of
  given atomic composition exposed to a uniform
  field of indirectly ionizing radiation, the field
  of secondary radiation is also uniform and
  independent of the density of the medium, as well
  as of density variations from point to point

5
The Fano Theorem (cont.)

• It follows from this that the charged-particle
  fluence at any point where CPE exists has a value
  that is independent of density variations within
  the volume of origin of the particles (assuming
  negligible polarization effect)

6
The Fano Theorem (cont.)

• A mathematical statement of the Fano theorem is
  provided by

7
Other Cavity Theories

• Spencer discussed in a general way two
  fundamentally different approaches to cavity
  theory
• The surface approach, in which one evaluates
  the total energy contribution in the cavity by
  each group of electrons that enter it
• The volume approach, in which one considers the
  energy deposition in each cavity volume element
  by electrons arriving from everywhere

8
Other Cavity Theories (cont.)

• Janssens et al. modified the Burlin theory by
  recalculating the weighting factor d with more
  detailed consideration of the penetration of wall
  electrons into the cavity
• Instead of assuming a constant spectrum and
  exponential attenuation of the electron fluence,
  they used range-energy relations applied to each
  electron energy, in the CSDA
• Electron-backscattering effects were discussed
  but not included

9
Other Cavity Theories (cont.)

• Janssens provided a modification of the Spencer
  theory in which the rate of energy loss in the
  cavity by low-energy electrons was related to
  cavity size, rather than simply assuming that
  electrons drop their energy on the spot as soon
  as T falls below the value of ?

10
Other Cavity Theories (cont.)

• Kearsley (1984) focused attention on the effect
  of electron backscattering at the cavity-wall
  interface, for electrons both entering and
  leaving the cavity
• An outstanding feature of the Kearsley theory is
  its capability of predicting dose as a function
  of depth in the cavity, so that comparisons can
  be made with the individual layers of LiF
  dosimeters

11
The average dose to a given layer of a
seven-layer stack of LiF dosimeters divided by
the equilibrium LiF dose, as calculated by the
Kearsley model (bars) and as measured by Oguleye
et al. () for Al, Cu, and Pb walls
12
Other Cavity Theories (cont.)

• Luo Zheng-Ming (1980) has developed a cavity
  theory based on application of the electron
  transport equation in the cavity and in the
  surrounding medium
• It is a very detailed theory that considers
  electron production in the cavity as well as the
  wall medium, and is applicable to all cavity sizes

13
Other Cavity Theories (cont.)

• Haider et al. (1997) included secondary-electron
  backscattering from the medium into the cavity
• They assumed that the Compton interaction was the
  dominant radiation interaction and thus the
  applicable energy range of their theory is from
  500 kV to 20 MV x-rays

14
Other Cavity Theories (cont.)

• It is arguable that the development of new and
  more complicated cavity theories may be reaching
  a period of diminishing return in competition
  with Monte Carlo computer methods, which have
  become more accessible and satisfactory and less
  expensive to run
• Simple cavity theories will continue to be useful
  for approximate solutions and estimates, but
  exact computations, especially for complex
  geometries and radiation fields, will likely be
  done by extensive use of computers and programs
  such as EGS and its later improvements

15
Dose Near Interfaces Between Dissimilar Media
Under ?-Irradiation

• Dutreix and Bernard studied the ionization
  produced by 60Co ? rays in a thin air-filled
  cavity as it was gradually moved from an
  equilibrium depth in carbon, through the
  carbon-copper interface, and to an equilibrium
  depth in the copper
• The ? rays were perpendicularly incident either
  from the carbon or the copper side of the
  interface
• The solid curves in the following diagram give
  their results

16
Variation of electron fluence with distance from
a copper-carbon interface irradiated
perpendicularly by 60Co ? rays
17
Dose Near Interfaces (cont.)

• In case A, in which the ? rays pass from copper
  to carbon, the backscatter component of electrons
  in copper is seen to decrease gradually from its
  equilibrium value of BCu as the interface is
  approached
• Its value is approximately zero at the interface
  if we assume negligible backscattering from
  carbon, so the electron flux there equals just
  the forward component, FCu

18
Dose Near Interfaces (cont.)

• In the carbon beyond the interface, this
  component gradually decays to zero, while a new
  population of forward-moving electrons is
  generated in the carbon by ?-ray interactions,
  reaching its carbon equilibrium value at the
  maximum distance to which they can penetrate from
  the interface
• Note that the decay of FCu with depth is steeper
  than the carbon buildup curve, because the
  electrons emerge from the copper nearly
  isotropically due to scattering, while the
  electrons are generated in the carbon with a
  Compton angular distribution

19
Dose Near Interfaces (cont.)

• Consequently a minimum is created in the upper
  solid curve of total electron fluence, on the
  low-Z side of the boundary
• The overall electron fluence transition, then, is
  from the equilibrium value in copper, dipping to
  a minimum on the low-Z side of the interface,
  then gradually rising to the equilibrium value in
  carbon

20
Dose Near Interfaces (cont.)

• The case of the reverse photon direction, shown
  in B, reveals a maximum instead of a minimum,
  again on the far side (now in the high-Z medium)
  of the interface
• We see the forward-moving equilibrium electron
  fluence in the carbon remaining constant until
  the interface is reached, then decaying in the
  copper

21
Variation of electron fluence with distance from
a copper-carbon interface irradiated
perpendicularly by 60Co ? rays
22
Dose Near Interfaces (cont.)

• The fluence of electrons that originate in the
  copper starts to build up at some distance inside
  the carbon, due to backscattering in the copper
• It is shown attaining the value BCu at the
  interface, then rising to its Cu-equilibrium
  value as the forward-moving fluence builds up

23
Dose Near Interfaces (cont.)

• The foregoing explanation of the processes that
  occur in case B does not take into account the
  electrons that originate in the carbon and
  backscatter from the copper
• Chapter 8, Section V.D, gives a backscattering
  coefficient of 0.43 for electrons below 1 MeV
  striking copper
• Thus, as can be seen in graph C, the forward
  fluence from the carbon is enhanced by 43 at the
  boundary, rather than remaining constant as shown
  in B

24
Variation of electron fluence with distance from
a copper-carbon interface irradiated
perpendicularly by 60Co ? rays
25
Dose Near Interfaces (cont.)

• The curve indicating the fluence of electrons
  that originate in copper is diminished
  accordingly in C, so that the sum of the Cu and C
  electrons still agrees with the experimental
  (solid) curve
• It can be seen in the diagram that the
  equilibrium fluence of electrons is about 50
  higher in copper than in carbon

26
Dose Near Interfaces (cont.)

• The equation
• implies that it should be only about 20
  higher

27
Dose Near Interfaces (cont.)

• The number of electrons produced per gram by 60Co
  ?-rays is proportional to (?/?)Cu 0.0530 cm2/g
  and (?/?)C 0.0578 cm2/g, and their mean energy
  is 1.25(?tr/?) 0.580 MeV
• The CSDA ranges for electrons of this energy are
  obtainable from Appendix E as ?Cu 0.320 g/cm2
  and ?C 0.245 g/cm2

28
Dose Near Interfaces (cont.)

• Thus the ratio of equilibrium fluences should be
• The excess observed by Dutreix and Bernard was
  probably caused by the presence of lower-energy
  scattered photons in the 60Co ?-ray beam
• Photons with an energy of, say, 0.2 MeV would
  produce more electrons and with longer ranges in
  copper than in carbon, due to the photoelectric
  effect

29
Dose Near Interfaces (cont.)

• Similar 60Co ?-ray measurements by Wall and Burke
  are shown in the following diagram, indicating
  the relative dose or electron fluence occurring
  in aluminum near an interface with gold or
  beryllium
• The same general pattern is observed as in the
  copper-carbon results a minimum is observed just
  beyond the interface when the photons go from a
  higher-Z to a lower-Z mediuma maximum is seen
  beyond the interface if the photons go from a
  lower-Z to a higher-Z medium

30
Variation of dose and electron fluence in
aluminum as a function of distance from an
interface with (a) gold, (b) beryllium. Arrows
indicate the direction of the 60Co ? rays.
31
Dose Near Interfaces (cont.)

• Comparable results should be expected at higher
  photon energies, but with an expanded scale of
  distances from the interface as the
  secondary-electron ranges increase
• At lower energies the transient effects will
  conversely be crowded closer to the interface
• At 100 keV, the transient effects in unit-density
  materials will be confined to the region within
  about 0.15 mm of the interface at 30 keV that
  distance is reduced to 20 ?m
• At larger distances from the interface the
  fluence and dose will approximate their
  equilibrium values as CPE is closely achieved