Cavity Theory by gjjur4356


									Cavity Theory III

 The Fano Theorem
Other Cavity Theories
Dose Near Interfaces
           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
• He noted that this substitution had never been
  rigorously justified, and attempted to provide such
  a justification
      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
      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:
     Fano’s 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
     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
        The Fano Theorem (cont.)
• A mathematical statement of the Fano
  theorem is provided by
          T0       T0       NdT
      dT  
    e          e
                                        N CSDA
          0    T   0
                         dT /  dx w
                                     N
          Other Cavity Theories
•    Spencer discussed in a general way two
     fundamentally different approaches to cavity
    a. The “surface” approach, in which one evaluates the
       total energy contribution in the cavity by each group
       of electrons that enter it
    b. The “volume” approach, in which one considers the
       energy deposition in each cavity volume element by
       electrons arriving from everywhere
   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
   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 
   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
 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
   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
  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
   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
     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
• 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
Variation of electron fluence with distance from a copper-
carbon interface irradiated perpendicularly by 60Co  rays
    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
• 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
    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
    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
   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
Variation of electron fluence with distance from a copper-
carbon interface irradiated perpendicularly by 60Co  rays
    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
    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
Variation of electron fluence with distance from a copper-
carbon interface irradiated perpendicularly by 60Co  rays
   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
   Dose Near Interfaces (cont.)
• The equation
        T0         T0       NdT
      dT  
    e        e
                                        N CSDA
        0    T   0
                         dT /  dx w
                                     N
 implies that it should be only about 20%
   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
• 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
    Dose Near Interfaces (cont.)
• Thus the ratio of equilibrium fluences should be
           e    0.0530  0.320
                                1.20
           C 0.0578  0.245

• 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
    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 medium;a maximum is seen
  beyond the interface if the photons go from a
  lower-Z to a higher-Z medium
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.
    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

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