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# Cavity Theory I by 1fv6p5p

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```									    Cavity Theory I

Bragg-Gray Theory
Averaging of Stopping Powers
Bragg-Gray Theory
• The basis for cavity theory is contained in
the following equation:
dT / dx c t     dT 
D                      dx 
                (MeV/g)
t                 c
10     dT 
 1.602 10           dx  Gy
     
     c
Bragg-Gray Theory (cont.)
• If a fluence  of identical charged particles of
kinetic energy T passes through an interface
between two different media, g and w, as shown in
A in the following diagram, then one can write for
the absorbed dose on the g side of the boundary
 dT  
 dx  
Dg        

      c, g T

and on the w side,
 dT  
 dx  
Dw        

      c,w T

(A) A fluence  of charged particles crossing an interface
between media w and g. (B) A fluence  of charged particles
passing through a thin layer of medium g sandwiched between
regions contain medium w.
Bragg-Gray Theory (cont.)
• Assuming that the value of  is continuous
across the interface (i.e., ignoring
backscattering) one can write for the ratio of
absorbed doses in the two media adjacent to
their boundary
Dw dT / dxc,w

Dg dT / dxc, g
Bragg-Gray Theory (cont.)
• Bragg and Gray applied this equation to the
problem of relating the absorbed dose in a probe
inserted in a medium to that in the medium itself
• Gray in particular identified the probe as a gas-
filled cavity, whence the name “cavity theory”
• The simplest such theory is called the Bragg-Gray
(B-G) theory, and its mathematical statement, the
Bragg-Gray relation, will be developed next
Bragg-Gray Theory (cont.)
• Suppose that a region of otherwise homogeneous
medium w, undergoing irradiation, contains a thin
layer or “cavity” filled with another medium g, as
in B in the following diagram
• The thickness of the g-layer is assumed to be so
small in comparison with the range of the charged
particles striking it that its presence does not
perturb the charged-particle field
• This assumption is often referred to as a “Bragg-
Gray condition”
(A) A fluence  of charged particles crossing an interface
between media w and g. (B) A fluence  of charged particles
passing through a thin layer of medium g sandwiched between
regions contain medium w.
Bragg-Gray Theory (cont.)
• This condition depends on the scattering
properties of w and g being sufficiently similar
that the mean path length (g/cm2) followed by
particles in traversing the thin g-layer is
practically identical to its value if g were replaced
by a layer of w having the same mass thickness
• Similarity of backscattering at w-g, g-w, and w-w
interfaces is also implied
Bragg-Gray Theory (cont.)
• For heavy charged particles (either primary, or
secondary to a neutron field), which undergo little
scattering, this B-G condition is not seriously
challenged so long as the cavity is very small in
comparison with the range of the particles
• However, for electrons even such a small cavity
may be significantly perturbing unless the medium
g is sufficiently close to w in atomic number
Bragg-Gray Theory (cont.)
• Bragg-Gray cavity theory can be applied whether
the field of charged particles enters from outside
the vicinity of the cavity, as in the case of a beam
of high-energy charged particles, or is generated in
medium w through interactions by indirectly
• In the latter case it is also assumed that no such
interactions occur in g
Bragg-Gray Theory (cont.)
• All charged particles in the B-G theory must
originate elsewhere than in the cavity
• Moreover charged particles entering the
cavity are assumed not to stop in it
Bragg-Gray Theory (cont.)
• A second B-G condition, incorporating these
ideas, can be written as follows:
– The absorbed dose in the cavity is assumed to be
deposited entirely by the charged particles crossing it
• This condition tends to be more difficult to satisfy
for neutron fields than for photons, especially if
the cavity gas is hydrogenous, thus having a large
neutron-interaction cross section
Bragg-Gray Theory (cont.)
• The heavy secondary charged particles (protons,
-particles, and recoiling nuclei) also generally
have shorter ranges than the secondary electrons
that result from interactions by photons of
quantum energies comparable to the neutron
kinetic energies
• Thus we see that the first B-G condition is the
more difficult of the two to satisfy for photons and
electrons, while the second B-G condition is the
more difficult to satisfy for neutrons
Bragg-Gray Theory (cont.)
• For a differential energy distribution T (particles
per cm2 MeV) the appropriate average mass
collision stopping power in the cavity medium g is
Tm ax       dT 

0               dx  dT
ΦT 


c, g
m   Sg                    Tm ax
0
 T dT
1 Tm ax  dT        Dg
  T   dx  dT  

 0          c, g
Bragg-Gray Theory (cont.)
• Likewise, for a thin layer of wall material w that
may be inserted in place of g,
Tm ax       dT 

0               dx  dT
ΦT 


c,w
m   Sw                    Tm ax
0
 T dT
1 Tm ax  dT        Dw
  T   dx  dT  

 0          c,w
Bragg-Gray Theory (cont.)
• Combining these two equations gives for
the ratio of absorbed dose in w to that in g,
which the B-G relation in terms of absorbed
dose in the cavity:
Dw m S w
      m S g
w

Dg m S g
Bragg-Gray Theory (cont.)
• If the medium g occupying the cavity is a gas in
which a charge Q (of either sign) is produced by
the radiation, Dg can be expressed (in grays) in
terms of that charge as
Q W   
Dg       
m e


g
where Q is in coulombs, m is the mass (kg) of gas
in which Q is produced, and (̅W/e)g is the mean
energy spent per unit charge produced (J/C)
Bragg-Gray Theory (cont.)
• By substitution, we obtain the B-G relation
expressed in terms of cavity ionization:
Q W     
Dw          m S gw
m e


g
• This equation allows one to calculate the absorbed
dose in the medium immediately surrounding a B-
G cavity, on the basis of the charge produced in
the cavity gas, provided that the appropriate values
of the various parameters are known
Bragg-Gray Theory (cont.)
• So long as m S gw is evaluated for the charged-
particle spectrum T that crosses the cavity,
the B-G relation requires neither charged-
particle equilibrium (CPE) nor a
• However, the charged-particle fluence T
must be the same in the cavity and in the
medium w where Dw is to be determined
Bragg-Gray Theory (cont.)
• If CPE does exist in the neighborhood of a point
of interest in the medium w, then the insertion of a
B-G cavity at the point may be assumed not to
perturb the “equilibrium spectrum” of charged
particles existing there, since by definition a B-G
cavity satisfies the B-G requirements
• Thus a B-G cavity approximates an evacuated
cavity in this respect
• The presence of an equilibrium spectrum of
charged particles allows some simplification in
estimating T
Bragg-Gray Theory (cont.)
• The medium w surrounding the cavity of an
ionization chamber is ordinarily just the
solid chamber wall itself, and one often
refers to the B-G theory as providing a
relationship between the doses in the gas
and in the wall
Corollaries of the Bragg-Gray
Relation
• Two useful corollaries of the B-G relation
can be readily derived from it
• The first relates the charge produced in
different gases contained in the same
chamber, while the second relates the
charge in the same gas contained by
different chamber walls
First Bragg-Gray Corollary
• A B-G cavity chamber of volume V with wall
medium w is first filled with gas g1 at density 1,
then with gas g2 at density 2
• Identical irradiations are applied, producing
charges Q1 and Q2, respectively
• The absorbed dose in gas g1 can be written as
Q1     W    
D1  Dw m S   g1
        
 e   

1V
w
     1
and the dose in gas g2 as
Q2     W    
D2  Dw m S   g2
        
 e   

 2V
w
     2
First B-G Corollary (cont.)
• The ratio of charges therefore becomes
Q2  2V W / e1 m S w 2
g
            
Q1 1V W / e2 m S w 1
g

which reduces to the first B-G corollary:
Q2  2V W / e 1
              m S gg12
Q1 1V W / e 2
First B-G Corollary (cont.)
• This equation does not depend explicitly upon the
wall material w, implying that the same value of
Q2/Q1 would be observed if the experiment were
repeated with different chamber walls
• This is true as long as the spectrum T of charged
particles crossing the cavity is not significantly
dependent on the kind of wall material
• For example, the starting spectrum of secondary
electrons produced in different wall media by -
rays is the same if the -energy is such that only
Compton interactions can occur
First B-G Corollary (cont.)
• Although different wall media modify the
starting electron spectrum somewhat
differently as the electrons slow down, the
resulting equilibrium spectrum that crosses
the cavity in different thick-walled ion
chambers is sufficiently similar that Q2/Q1
is observed to be nearly independent of the
wall material in this case
Second Bragg-Gray Corollary
• A single gas g of density  is contained in two
B-G cavity chambers that have thick walls
(exceeding the maximum charged-particle range),
penetrating x- or -rays, producing CPE at the
cavity
• The first chamber has a volume V1 and wall
material w1, the second has a volume V2 and wall
w2
Second B-G Corollary (cont.)
• The absorbed dose in the wall of the first chamber,
adjacent to its cavity, can be written as
  en 
Dw1  K c w1   
CPE

    
       w1
Q1  W
 D1 m Sw1
       m S g
w1

V1  e
g
 g
• A similar expression can be written for the
absorbed dose in the second chamber
Second B-G Corollary (cont.)
• The ratio of the two ionizations in the two
chambers is
V2 en /  w
w1
Q2                             S
                
2       m       g

Q1    V1 en /  w
1       m   S   w2
g

where the constancy of (̅W/e)g for electron
energies above a few keV allows its
cancellation
Second B-G Corollary (cont.)
• A further simplification of the final factor to
w1
m   S   w2
can be made only if the charged-particle spectrum
T crossing the cavity is the same in the two
chambers
• If such a cancellation of stopping powers thus
eliminates g from the equation, the same value of
Q2/Q1 should result irrespective of the choice of
gas
Second B-G Corollary (cont.)
• A similar expression can be obtained for neutron
irradiations in place of photons by substituting
kerma factors Fn for the mass energy-absorption
coefficients:
Q2 V2     Fn w2 m S gw1 W / e1
                   
Q1 V1 Fn w1 m S g W / e 2
w2

• The ratio ̅W/e may have to be retained here if w1
and w2 differ sufficiently to produce heavy
charged-particle spectra that have somewhat
different ̅W/e values even in the same gas
Spencer’s Derivation of the
Bragg-Gray Theory
• Consider a small cavity filled with medium g,
surrounded by a homogeneous medium w that
contains a homogeneous source emitting N
identical charged particles per gram, each with
kinetic energy T0 (MeV)
• The cavity is assumed to be far enough from the
outer limits of w that CPE exists
• Both B-G conditions are assumed to be satisfied
by the cavity, and bremsstrahlung generation is
assumed to be absent
Spencer’s Derivation of the B-G
Theory (cont.)
• The absorbed dose at any point in the
undisturbed medium w where CPE exists
can be stated as
CPE
Dw  K w  NT0   (MeV/g)

where 1 MeV/g = 1.602  10-10 Gy
Spencer’s Derivation of the B-G
Theory (cont.)
• An equilibrium charged-particle fluence spectrum
eT (cm-2 MeV-1) exists at each such point, and the
absorbed dose can be written in terms of this
spectrum as
T0      dT 
Dw          dx  dT
 
e
T      
0
     w
where (dT/dx)w has the same value as the mass
collision stopping power for w, in the absence of
bremsstrahlung generation
Spencer’s Derivation of the B-G
Theory (cont.)
• The value of eT that satisfies the integral equation formed
by setting these two equations equal is
N
 
e
T
dT / dxw
• The equilibrium spectrum for an initially monoenergetic
source of charged particles is directly proportional to the
number released per unit mass, and is inversely
proportional, at each energy T  T0, to the mass stopping
power in the medium in which the particles are allowed to
slow down and stop
Spencer’s Derivation of the B-G
Theory (cont.)
• The following diagram is a graph of the
equilibrium spectrum of primary electrons that
result for this equation when it is applied (twice)
to the example of two superimposed sources of N
electrons per gram each, one emitting at T0 = 2
MeV and the other at T0 = 0.2 MeV, in a water
medium
• This is not a realistic spectrum, however, as -ray
production has been ignored
Example of an equilibrium fluence spectrum, eT = N/(dT/dx), of
primary electrons under CPE conditions in water, assuming the
continuous-slowing-down approximation
Spencer’s Derivation of the B-G
Theory (cont.)
• Since the same equilibrium fluence
spectrum of charged particles, eT, crosses
the cavity as exists within medium w, the
absorbed dose in the cavity medium g can
be written as
T dT / dx g
e  dT 
T
Dg   T   dx  dT  N 0 dT / dx dT
0                   0


0
     g                     w
Spencer’s Derivation of the B-G
Theory (cont.)
• The ratio of the dose in the cavity to that in the
solid w is then
Dg  1      T0dT / dxg
       0 dT / dxw dT  m S w
g

Dw T0
which is the same as the B-G relation, considering
starting energy T0, charged-particle equilibrium,
and zero bremsstrahlung
Spencer’s Derivation of the B-G
Theory (cont.)
• The equivalence of S , as employed here,
m w
g

w
to the reciprocal of S as defined in
m g

Dw m S w
      m S g
w

Dg m S g
may not be immediately obvious, and will
be explained in the next section
Spencer’s Derivation of the B-G
Theory (cont.)
• The foregoing Spencer treatment of B-G theory
can be generalized somewhat to accommodate
bremsstrahlung generation by electrons and its
subsequent escape
• The dose in the medium w can be rewritten as

Dw  K c w  NT0 1  Yw T0 
CPE

where (Kc)w is the collision kerma and Yw(T0) is
the radiation yield for medium w
Spencer’s Derivation of the B-G
Theory (cont.)
• The equations for the doses are changed to
T0      dT 
Dw             dx  dT
 
e
T      
0
     c,w
and
T0         dT 
Dg             dx  dT
 
e
T      
0
     c, g
where (dT/dx)c,w and (dT/dx)c,g are the mass
collision stopping powers in media w and g,
respectively
Spencer’s Derivation of the B-G
Theory (cont.)
• The equilibrium fluence, as given by
N
 e
T
dT / dxw
remains unchanged; hence one can rewrite
Spencer’s statement of B-G theory in the
following form to take account of bremsstrahlung
Dg         1          T0 dT / dx c , g

Dw T0 1  Yw T0  0 dT / dxw
                                       dT  m S w
g
Averaging of Stopping Powers
• For the special case treated by Spencer, the
spectrum of primary charged particles crossing the
cavity is known, being given by
N
 
e
T
dT / dxw
• The evaluation of m S w in
g

Dg 1 T0 dT / dxg
                dT  m S w
g

Dw T0 0 dT / dxw
is seen to be a simple average of the ratio of
stopping powers throughout the energy range 0 to
T0, apparently unweighted by eT
Averaging of Stopping Powers
(cont.)
• In fact the fluence weighting is implicit, as
can be seen by applying Spencer’s
assumption to
Tm ax       dT 

0               dx  dT
ΦT 


c, g
m   Sg                    Tm ax
0
 T dT
1 Tm ax  dT        Dg
  T   dx  dT  

 0          c, g
Averaging of Stopping Powers
(cont.)
• Setting Tmax = T0 for the upper limit of
integration, assuming CPE and the absence
of bremsstrahlung, this equation becomes
T0      dT 
         
e
 dT
  dx  g
T
0
1       T0      dT 
Sg                               e            
e
 dT
T                dx  g
m                        T0                          T

         dT
e           0
0        T

N             T0 dT /  dx  g    Dg
 e
            0  dT /  dx  dT   e
w
Averaging of Stopping Powers
(cont.)
• A similar equation can be written for m̅Sw
• The mean mass-stopping-power ratio m S w can then
g

be obtained as shown in
Dg 1 T dT / dxg
                dT  m S w
0                   g

Dw T0 0 dT / dxw
through the application of
CPE
Dw  K w  NT0   (MeV/g)

which clearly depends on the existence of an
equilibrium spectrum
Averaging of Stopping Powers
(cont.)
• Since the Spencer B-G treatment was limited to only a
single starting energy (T0) of the charged particles, it will
be useful to extend it to distributions of starting energies,
such as are generated by photons in a statistically large
number of Compton events
• Consider a homogeneous source of charged particles
throughout medium w, emitting a continuous distribution
of starting energies: Let NT0 charged particles of energy T0
to T0 + dT0 be emitted per gram of w and per MeV interval,
where 0  T0  Tmax
• Assume that CPE exists, and that bremsstrahlung may be
produced and it escapes
Averaging of Stopping Powers
(cont.)
• The absorbed dose in w is given by
CPE
Dw   Kc w              NT0 T0 1  Yw T0  dT0
Tmax

T0 0                      
while the dose in the cavity medium g is
Tmax       dT 
T0
Dg   dT0             dT
e

  dx c , g
T0  0 T 0            T

Tmax         dT /  dx c, g
T0
  NT dT0                        dT
T 0
0            
T  0 dT /  dx
0
w
Averaging of Stopping Powers
(cont.)
• Thus for a continuous distribution of charged-particle
starting energies the ratio of absorbed doses in cavity and
wall is given by

Tmax          dT /  dx c, g
T0

Dg     0 NT dT0 0  dT /  dx  dT
0

                             w
 m Sw
g

0 NT T0 1  Y T0  dT0
T
Dw             max

     0     
g
where the double bar on m Sw signifies integration over the
T0 distribution, as well as over T for each T0-value
Averaging of Stopping Powers
(cont.)
• Where CPE does not exist in the vicinity of the
cavity, mean stopping powers can be calculated as
an average weighted by the differential charged-
particle fluence distribution T crossing the cavity
• Thus in general the mean stopping-power ratio for
a B-G cavity can be expressed as
1 Tmax   dT 
 0 T   dx c, g dT
                        Sg            Dg
             mS 
m           g

 dT 
w
1 Tmax                            Sw            Dw
 0 T   dx c,w dT
      
m
Averaging of Stopping Powers
(cont.)
• Since collision stopping powers for
different media show similar trends as a
function of particle energy, their ratio for
two media is a very slowly varying function
• This allows the preceding equation to be
reasonably well approximated through
simple estimation
Averaging of Stopping Powers
(cont.)
• For example, one may first determine the average
energy ̅T of the charged particles crossing the
cavity:
Tmax

T 
 0
 TTdT    1 Tmax
  T TdT
Tmax
 0
 0
 T dT

and then look up the tabulated mass collision
stopping powers for the media in question at that
energy
Averaging of Stopping Powers
(cont.)
• For an equilibrium spectrum resulting from
charged particles of mean starting energy ̅T0, the
stopping powers may be looked up at the
energy ̅T0/2 for a crude (but often adequate)
estimate of the mean stopping-power ratio
required for the B-G relation
• The average starting energy ̅T0 of Compton-effect
electrons can be calculated from
 tr
T  h  e

e

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