Cavity Theory I

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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
  ionizing radiation
• 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
  homogeneous field of radiation
• 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),
  and that receive identical irradiations of
  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
  Spencer’s added assumptions of monoenergetic
  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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