# Cavity Theory by gjjur4356

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```									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
restriction
• 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
also uniform and independent of the density of the
medium, as well as of density variations from point to
point
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)
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
CSDA
 N

Other Cavity Theories
•    Spencer discussed in a general way two
fundamentally different approaches to cavity
theory:
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
• 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
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
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
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
up
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
CSDA
 N

implies that it should be only about 20%
higher
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
Dose Near Interfaces (cont.)
• Thus the ratio of equilibrium fluences should be
e    0.0530  0.320
Cu
                 1.20
C 0.0578  0.245
e

• 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
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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