ChargedParticle and Radiation Equilibria I

1
Charged-Particle and Radiation Equilibria I

• Radiation Equilibrium
• Charged-Particle Equilibrium

2
Introduction

• The concepts of radiation equilibrium (RE) and
  charged-particle equilibrium (CPE) are useful in
  radiological physics as a means of relating
  certain basic quantities
• CPE allows the equating of the absorbed dose D to
  the collision kerma Kc, while radiation
  equilibrium makes D equal to the net rest mass
  converted to energy per unit mass at the point of
  interest

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Radiation Equilibrium

• Consider an extended volume V containing a
  distributed radioactive source
• A smaller internal volume v exists about a point
  of interest, P
• V is required to be large enough so that the
  maximum distance of penetration d of any emitted
  ray and its progeny (i.e., scattered and
  secondary rays) is less than the minimum
  separation s of the boundaries of V and v
• Radioactivity is emitted isotropically on the
  average

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Radiation equilibrium
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Radiation Equilibrium (cont.)

• If the following 4 conditions exist throughout V,
  it will be shown that radiation equilibrium (RE)
  exists for the volume v (in the nonstochastic
  limit)
• The atomic composition of the medium is
  homogeneous
• The density of the medium is homogeneous
• The radioactive source is uniformly distributed
• There are no electric or magnetic fields present
  to perturb the charged-particle paths, except the
  fields associated with the randomly oriented
  individual atoms

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Radiation Equilibrium (cont.)

• Imagine a plane T that is tangent to the volume v
  at a point P, and consider the rays crossing the
  plane per unit area there
• In the nonstochastic limit there will be perfect
  reciprocity of rays of each type and energy
  crossing both ways, since the radioactive source
  distribution within the sphere S of radius d
  about point P is perfectly symmetrical with
  respect to plane T

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Radiation equilibrium
8
Radiation Equilibrium (cont.)

• This will be true for all possible orientations
  of tangent planes around the volume v hence one
  can say that, in the nonstochastic limit, for
  each type and energy of ray entering v, another
  identical ray leaves
• This condition is called radiation equilibrium
  (RE) with respect to v

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Radiation Equilibrium (cont.)

• We can write as a consequence of radiation
  equilibrium that the following equalities of
  expectation values exist
• and
• that is, the energy carried in and that
  carried out of v are balanced for both indirectly
  and directly ionizing radiation, where the bars
  signify expectation values

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Radiation Equilibrium (cont.)

• The energy imparted can then be simplified to
• which means that under RE conditions the
  expectation value of the energy imparted to the
  matter in the volume v is equal to that emitted
  by the radioactive material in v

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Radiation Equilibrium (cont.)

• The concept of radiation equilibrium has
  practical importance especially in the fields of
  nuclear medicine and radiobiology, where
  distributed radioactive sources may be introduced
  into the human body or other biological systems
  for diagnostic, therapeutic, or analytical
  purposes
• The resulting absorbed dose at any given point in
  such circumstances depends on the size of the
  object relative to the radiation range and on the
  location of the point within the object

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Radiation Equilibrium (cont.)

• The absence of electric and magnetic fields from
  V allows the use of the simplest symmetry
  argument for proving that RE occurs, since
  radioactive point sources emit radiation
  isotropically
• The presence of a homogeneous, constant magnetic
  and/or electric field throughout V makes the
  symmetry argument more difficult to visualize,
  since the flow of charged particles past a point
  such as P will no longer be isotropic

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Radiation Equilibrium (cont.)

• Isotropicity is not a requirement for RE in the
  volume v it is merely necessary that the inward
  and outward flow of identical particles of the
  same energy be balanced for all particles present
• Even if all the particles flow in one side of v
  and out the other side, RE will still obtain so
  long as the in vs out flow is balanced
• Any source anisotropy, or distortion of
  charged-particle tracks, that is homogeneously
  present everywhere throughout V will have no
  perturbing effect on the existence of RE in v

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Radiation Equilibrium (cont.)

• Consider an elemental volume dv at point of
  interest P, and two other elemental volumes dv?
  and dv? that are symmetrically positioned with
  respect to dv
• We assume dv is located at a distance s from the
  boundary of volume V that is greater than the
  maximum range of radiation penetration, d
• Throughout V both the medium and the distributed
  source are homogeneous, but now we allow the
  presence of a homogeneous electric and/or
  magnetic field, and the source itself need not
  emit radiation isotropically, so long as the
  anisotropy is homogeneous everywhere in V

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Radiation equilibrium in homogeneous but
anisotropic fields of radiation
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Radiation Equilibrium (cont.)

• Assuming that radiation moves preferentially from
  left to right, homogeneity and symmetry
  considerations require that the particles (A)
  traveling from dv? to dv are identical to those
  (B) traveling from dv to dv?, in the
  expectation-value limit
• Likewise the lesser flow (b) of particles from
  dv? to dv is identical to that (a) from dv to dv?
  
• Consequently a B A b, that is, the flow of
  particles from dv to dv? dv? is identical to
  that from dv? dv? to dv

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Radiation equilibrium in homogeneous but
anisotropic fields of radiation
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Radiation Equilibrium (cont.)

• The pair of volumes dv? and dv? can be moved to
  all possible symmetrical locations within V, and
  their particle flows are integrated
• Locations outside of the sphere of radius d about
  point P of course neither receive particles from
  nor contribute particles to dv
• One may conclude from such an argument that each
  particle flowing out of dv is replaced by an
  identical particle flowing in
• Thus RE exists at P

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Charged-Particle Equilibrium

• Charged particle equilibrium (CPE) exists for the
  volume v if each charged particle of a given type
  and energy leaving v is replaced by an identical
  particle of the same energy entering, in terms of
  expectation values
• If CPE exists,
• is of course satisfied

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CPE (cont.)

• The existence of RE is a sufficient condition for
  CPE to exist
• The practical importance of CPE stems from the
  fact that under certain conditions it can be
  adequately approximated even in the absence of RE
• Two important cases will be considered in the
  following subsections

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CPE for Distributed Radioactive Sources

• Consider the trivial case were only charged
  particles are emitted and radiative losses are
  negligible
• Again referring to the following diagram, the
  dimension s is taken to be greater than the
  maximum range d of the particles
• If the same four conditions (1-4) are satisfied
  throughout the volume V as required for RE, then
  RE and CPE will of course both exist for the
  volume v, since they are identical in this case

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Radiation equilibrium
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CPE for Distributed Radioactive Sources (cont.)

• Consider now the more interesting case where both
  charged particles and relatively more penetrating
  indirectly ionizing radiation are emitted
• Let the distance d be the maximum range of the
  charged particles only, and let V be just large
  enough so the minimum distance s separating V
  from v exceeds d

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CPE for Distributed Radioactive Sources (cont.)

• If the indirectly ionizing rays are penetrating
  enough to escape from V without interacting
  significantly with the medium, then they will
  produce practically no secondary charged
  particles
• Only the primary charged particles then need be
  considered in the symmetry argument as before,
  where again we assume conditions 1-4 throughout
  V, as stated for radiation equilibrium
• Since the passage of identical charged particles
  in and out of v is thus seen to be balanced, CPE
  exists w.r.t. the primary charged particles

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CPE for Distributed Radioactive Sources (cont.)

• However, RE is not attained, since (Rout)u gt
  (Rin)u for the volume v
• This is evident from the fact that the indirectly
  ionizing rays that originate in v and escape from
  V are not replaced, because there is no source
  outside of V
• The equation for the expectation value of the
  energy imparted in this case becomes

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CPE for Distributed Radioactive Sources (cont.)

• Since we are assuming that the indirectly
  ionizing rays are so penetrating that they do not
  interact significantly in v, ? is equal to the
  kinetic energy given only to charged-particles by
  the radioactive source in v, less any radiative
  losses by those particles while in v
• The average absorbed dose in v is thus equal to
• divided by the mass in v, for CPE conditions

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CPE for Distributed Radioactive Sources (cont.)

• Suppose that the size of the volume V occupied by
  the source is expanded so that distance s
  gradually increases from being merely equal to
  the charged-particle range to being greater than
  the effective range of the indirectly ionizing
  rays and their secondaries
• That transition will cause the (Rin)u term to
  increase until it equals (Rout)u in value
• Thus RE will be restored, according to the
  symmetry argument applied to all rays
• The energy imparted would be transformed into
  that for RE

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CPE for Distributed Radioactive Sources (cont.)

• The calculation of the absorbed dose is evidently
  straightforward for either of these limiting
  cases (CPE or RE), but intermediate situations
  are more difficult to deal with, i.e., when the
  volume V is larger than necessary to achieve CPE
  in v, but not large enough for RE
• In that case some fraction of the energy of the
  indirectly ionizing radiation component will be
  absorbed, and it is relatively difficult to
  determine what that fraction is

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CPE for Indirectly Ionizing Radiations from
External Sources

• In the following diagram a volume V is shown,
  again containing a smaller volume v
• The boundaries of v and V are required in this
  case to be separated by at least the maximum
  distance of penetration of any secondary charged
  particle present
• If the following conditions are satisfied
  throughout V, CPE will exist for the volume v
• The atomic composition of the medium is
  homogeneous
• The density of the medium is homogeneous
• There exists a uniform field of indirectly
  ionizing radiation (i.e., the rays must be only
  negligibly attenuated by passage through the
  medium)
• No inhomogeneous electric or magnetic fields are
  present

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Charged-particle equilibrium conditions for an
external source
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CPE from External Sources (cont.)

• It is possible for CPE to exist in a volume
  without satisfying all the above conditions under
  certain geometrical conditions
• The ion-collecting region of a free-air chamber
  represents such a situation, to be discussed in
  Chapter 12
• Another example is the trivial case of a point
  source within a volume large enough so the
  radiation cannot reach the boundary surface,
  hence no replacement particles are required

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CPE from External Sources (cont.)

• Because of the uniformity of the indirectly
  ionizing radiation field and of the medium
  throughout V, one can say that the number of
  charged particles produced per unit volume in
  each energy interval and element of solid angle
  will be uniform everywhere in V
• However, the particles are not emitted
  isotropically as in the case of radioactive point
  sources

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CPE from External Sources (cont.)

• Neutron and photon interactions generally result
  in anisotropic angular distributions of secondary
  and scattered radiations
• However, this anisotropy will be homogeneous
  throughout V
• This condition, together with a uniform medium in
  which the charged particles can slow down
  throughout V (as guaranteed by the first two
  conditions) is sufficient to produce CPE for the
  volume v

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CPE from External Sources (cont.)

• For CPE conditions
• However, under those same conditions we may also
  assume that any radiative interaction by a
  charged particle after it leaves v will be
  replaced by an identical interaction inside of v,
  as shown in the following diagram

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CPE from External Sources (cont.)

• Thus
• provided that the volume v is small enough to
  allow radiative-loss photons to escape
• For that case

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CPE from External Sources (cont.)

• Reducing v to the infinitesimal volume dv,
  containing mass dm about a point of interest P,
  we can write
• and hence

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CPE from External Sources (cont.)

• The derivation of this equation proves that under
  CPE conditions at a point in a medium, the
  absorbed dose is equal to the collision kerma
  there
• That is true irrespective of radiative losses
• This is a very important relationship, as it
  equates the measurable quantity D with the
  calculable quantity Kc ( ? ?en/?)

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CPE from External Sources (cont.)

• If the same photon energy fluence ? is present in
  media A and B having two different average energy
  absorption coefficients, the ratio of absorbed
  doses under CPE conditions in the two media will
  be given by