Cavity Theory I

1
Cavity Theory I

• Bragg-Gray Theory
• Averaging of Stopping Powers

2
Bragg-Gray Theory

• The basis for cavity theory is contained in the
  following equation

3
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
• and on the w side,

4
(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.
5
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

6
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

7
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

8
(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.
9
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

10
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

11
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

12
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

13
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

14
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

15
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

16
Bragg-Gray Theory (cont.)

• Likewise, for a thin layer of wall material w
  that may be inserted in place of g,

17
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

18
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
• 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)

19
Bragg-Gray Theory (cont.)

• By substitution, we obtain the B-G relation
  expressed in terms of cavity ionization
• 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

20
Bragg-Gray Theory (cont.)

• So long as 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

21
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

22
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

23
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

24
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
• and the dose in gas g2 as

25
First B-G Corollary (cont.)

• The ratio of charges therefore becomes
• which reduces to the first B-G corollary

26
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

27
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

28
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

29
Second B-G Corollary (cont.)

• The absorbed dose in the wall of the first
  chamber, adjacent to its cavity, can be written
  as
• A similar expression can be written for the
  absorbed dose in the second chamber

30
Second B-G Corollary (cont.)

• The ratio of the two ionizations in the two
  chambers is
• where the constancy of (W/e)g for electron
  energies above a few keV allows its cancellation

31
Second B-G Corollary (cont.)

• A further simplification of the final factor to
• 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

32
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
• 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

33
Spencers 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

34
Spencers 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
• where 1 MeV/g 1.602 ? 10-10 Gy

35
Spencers 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
• where (dT/?dx)w has the same value as the
  mass collision stopping power for w, in the
  absence of bremsstrahlung generation

36
Spencers Derivation of the B-G Theory (cont.)

• The value of ?eT that satisfies the integral
  equation formed by setting these two equations
  equal is
• 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

37
Spencers 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

38
Example of an equilibrium fluence spectrum, ?eT
N/(dT/?dx), of primary electrons under CPE
conditions in water, assuming the
continuous-slowing-down approximation
39
Spencers 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

40
Spencers Derivation of the B-G Theory (cont.)

• The ratio of the dose in the cavity to that in
  the solid w is then
• which is the same as the B-G relation,
  considering Spencers added assumptions of
  monoenergetic starting energy T0,
  charged-particle equilibrium, and zero
  bremsstrahlung

41
Spencers Derivation of the B-G Theory (cont.)

• The equivalence of , as employed here, to
  the reciprocal of as defined in
• may not be immediately obvious, and will be
  explained in the next section

42
Spencers 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
• where (Kc)w is the collision kerma and Yw(T0)
  is the radiation yield for medium w

43
Spencers Derivation of the B-G Theory (cont.)

• The equations for the doses are changed to
• and
• where (dT/?dx)c,w and (dT/?dx)c,g are the
  mass collision stopping powers in media w and g,
  respectively

44
Spencers Derivation of the B-G Theory (cont.)

• The equilibrium fluence, as given by
• remains unchanged hence one can rewrite
  Spencers statement of B-G theory in the
  following form to take account of bremsstrahlung

45
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
• The evaluation of in
• is seen to be a simple average of the ratio
  of stopping powers throughout the energy range 0
  to T0, apparently unweighted by ?eT

46
Averaging of Stopping Powers (cont.)

• In fact the fluence weighting is implicit, as can
  be seen by applying Spencers assumption to

47
Averaging of Stopping Powers (cont.)

• Setting Tmax T0 for the upper limit of
  integration, assuming CPE and the absence of
  bremsstrahlung, this equation becomes

48
Averaging of Stopping Powers (cont.)

• A similar equation can be written for mSw
• The mean mass-stopping-power ratio can
  then be obtained as shown in
• through the application of
• which clearly depends on the existence of an
  equilibrium spectrum

49
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

50
Averaging of Stopping Powers (cont.)

• The absorbed dose in w is given by
• while the dose in the cavity medium g is

51
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
• where the double bar on signifies
  integration over the T0 distribution, as well as
  over T for each T0-value

52
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

53
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

54
Averaging of Stopping Powers (cont.)

• For example, one may first determine the average
  energy T of the charged particles crossing the
  cavity
• and then look up the tabulated mass collision
  stopping powers for the media in question at that
  energy

55
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