ChargedParticle Interactions in Matter II

1
Charged-Particle Interactions in Matter II

• Stopping Power (cont.)
• Range

2
Polarization or Density-Effect Correction

• The polarization effect influences the soft
  collision process, which is an energy-transferring
  interaction between a passing charged particle
  and a relatively distant atom
• In gases the atoms are spaced widely enough so
  that they undergo interactions independently of
  one another
• In condensed media the density is increased by a
  factor of 103 104 over that of a gas at
  atmospheric pressure, and the average atomic
  spacing is less than 1/10 as great as in the gas

3
Density Correction (cont.)

• In this situation the dipole distortion of the
  atoms near the track of the passing particle
  weakens the Coulomb force field experienced by
  the more distant atoms, thus decreasing the
  energy lost to them
• Because of this, the mass collision stopping
  power is decreased in condensed media

4
Density Correction (cont.)

• The correction term, ?, is a function of the
  composition and density of the stopping medium,
  and of the parameter
• for the particle, in which p is its
  relativistic momentum mv, and m0 is its rest mass
• ? may be taken as zero below a threshold value ?0
  in a given nonmetal
• A small nonzero value of ? ? 0.1 exists in metals
  even for very low-energy particles, because of
  the conduction electrons

5
Density-effect correction ? as a function of ?
and electron kinetic energy T
6
Density Correction (cont.)

• The figure shows that ? increases almost linearly
  as a function of ? above ? ? 1 for a variety of
  condensed media, being somewhat larger for low-Z
  than for high-Z media at a given ?-value
• ? only begins to become important above the
  rest-mass energy of the particle

7
Density Correction (cont.)

• The size of the polarization effect for
  electrons, expressed as a percentage decrease in
  mass collision stopping power in solids or
  liquids compared with gases of the same Z, is
  shown in the following table
• It increases roughly as the logarithm of T above
  a few MeV of electron energy, and decreases
  gradually with increasing Z

8
Polarization Effect for Electrons
9
Density Correction (cont.)

• Appendix E contains tables of electron stopping
  powers, ranges, radiation yields, and
  density-effect corrections ?, for a variety of
  elements and compounds
• The following table relates mass collision
  stopping powers for positrons to those tabulated
  for electrons
• The positron stopping power is evidently somewhat
  greater than that for electrons below 0.5 MeV,
  the reverse being true above that energy

10
Ratio of Mass Collision Stopping Powers for
Positrons to that for Electrons
11
Density Correction (cont.)

• The following diagram illustrates the influence
  of the polarization effect on electron (or
  positron) stopping powers vs. kinetic energy
  above 0.5 MeV
• The same trends previously discussed for heavy
  particles are also followed for electrons and
  positrons
• The steep rise for ? lt m0c2 is not shown, but the
  minimum at ? 3 m0c2 is evident, as is the
  continuing rise at still higher energy

12
Mass collision stopping power for electrons in
anthracene, Al, Li, AgCl, and Au, with (solid
curves) and without (dashed curves) correction
for polarization effect
13
Density Correction (cont.)

• The polarization effect is particularly relevant
  to radiological physics measurements in which
  ionization chambers are used in electron or
  photon beams above 2 MeV
• Relating the absorbed dose in the gas to that in
  the solid surrounding medium through the
  application of cavity theory requires knowledge
  of the stopping powers, which are influenced by
  the polarization effect in the solid

14
Mass Radiative Stopping Power

• Only electrons and positrons are light enough to
  generate significant bremsstrahlung, which
  depends on the inverse square of the particle
  mass for equal velocities
• The rate of bremsstrahlung production by
  electrons or positrons is expressed by the mass
  radiative stopping power (dT/?dx)r, in units of
  MeV cm2/g, which can be written as
• where the constant ?0 1/137(e2/m0c2)2
  5.80 ? 10-28 cm2/atom, T is the particle kinetic
  energy in MeV, and Br is a slowly varying
  function of Z and T

15
Mass Radiative Stopping Power (cont.)

• Br has a value of 16/3 for T ltlt 0.5 MeV, and
  roughly 6 for T 1 MeV, 12 for 10 MeV, and 15
  for 100 MeV
• BrZ2 is dimensionless
• The mass radiative stopping power is proportional
  to NAZ2/A, while the mass collision stopping
  power is proportional to NAZ/A, the electron
  density
• Thus their ratio would be expected to be
  proportional to Z

16
Mass Radiative Stopping Power (cont.)

• The equation also shows proportionality to T
  m0c2, or to T for T gtgt m0c2
• The corresponding energy dependence of the
  collision stopping power is not obvious from its
  formula, but can be seen in the following diagram
• Above T m0c2 it varies only slowly as a
  function of T
• Thus the ratio of radiative to collision stopping
  powers will be roughly proportional to T at high
  energies

17
Mass radiative and collision stopping powers for
electrons (and approximately for positrons) in C,
Cu, and Pb
18
Mass Radiative Stopping Power (cont.)

• The ratio of radiative to collision stopping
  power is often expressed in the form
• in which T is the kinetic energy of the
  particle, Z is the atomic number of the medium,
  and n is a constant variously taken to be 700 or
  800 MeV

19
Mass Radiative Stopping Power (cont.)

• The figure shows the stopping power trends vs.
  energy and Z
• The collision stopping power is relatively
  independent of Z, so any ratio of (dT/?dx)c for
  one medium to that for another is only weakly
  dependent on T
• Also, above 1 MeV the variation of (dT/?dx)c
  itself vs. T is very gradual, and becomes even
  flatter in condensed media when the polarization
  effect is corrected for

20
Mass Radiative Stopping Power (cont.)

• The total mass stopping power is the sum of the
  collision and radiative contributions
• Along with its parts, dT/?dx is tabulated as a
  function of T for a given stopping medium and
  type of charged particle in Appendix E, for
  electrons
• For heavier particles (dT/?dx)r ? 0, so (dT/?dx)
  (dT/?dx)c almost exactly

21
Radiation Yield

• The radiation yield Y(T0) of a charged particle
  of initial kinetic energy T0 is the total
  fraction of that energy that is emitted as
  electromagnetic radiation while the particle
  slows and comes to rest
• For heavy particles Y(T0) ? 0
• For electrons the production of bremsstrahlung
  x-rays in radiative collisions is the only
  significant contributor to Y(T0)
• For positrons, in-flight annihilation would be a
  second significant component, but this has
  customarily been omitted in calculating Y(T0)

22
Radiation Yield (cont.)

• If we define y(T) as
• for an electron of instantaneous kinetic
  energy T, then the radiation yield Y(T0) for the
  electron of higher starting energy T0 is an
  average value of y(T) for T varying from 0 to T0,
  as given by
• The amount of energy radiated per electron is
  simply Y(T0) T0

23
Radiation Yield (cont.)

• In Chapter 2 the concept of W was discussed and
  defined in terms of a quantity gi
• Its mean value g appears in the relation ?en/?
  (?tr/?)(1 g) in Chapter 7
• g is also the average value of Y(T0) for all of
  the electrons and positrons of various starting
  energies T0 present

24
Radiation Yield (cont.)

• Assuming that only Compton interactions occur,
  given a photon energy E?,
• in which ? is the Compton (Klein-Nishina)
  interaction cross section (e.g., in cm2/e) and
  (d?/dT0)E? is the differential cross section
  (cm2/e MeV), and Tmax is the maximum electron
  energy

25
Stopping Power in Compounds

• The mass collision stopping power, the mass
  radiative stopping power, and their sum the mass
  stopping power can all be well approximated for
  intimate mixtures of elements, or for chemical
  compounds, through the assumption of Braggs Rule
• It states that atoms contribute nearly
  independently to the stopping power, and hence
  their effects are additive
• This neglects the influence of chemical binding
  on I

26
Stopping Power in Compounds (cont.)

• In terms of the weight fractions fZ1, fZ2, of
  elements of atomic numbers Z1, Z2, etc. present
  in a compound or mixture, the mass stopping power
  (dT/?dx)mix can be written as
• where all stopping powers refer to a common
  kinetic energy and type of charged particle

27
Stopping Power in Compounds (cont.)

• A rough approximation to the polarization
  correction ? can also be gotten from the Bragg
  rule as

28
Restricted Stopping Power

• The mass collision stopping power (dT/?dx)c
  expresses the average rate of energy loss by a
  charged particle in all hard, as well as soft,
  collisions
• The ?-rays resulting from hard collisions may be
  energetic enough to carry kinetic energy a
  significant distance away from the track of the
  primary particle
• If one is calculating the dose in a small object
  or a thin foil traversed by charged particles,
  the use of the mass collision stopping power will
  overestimate the dose, unless the escaping ?-rays
  are replaced (i.e., unless ?-ray CPE exists)

29
Restricted Stopping Power (cont.)

• The restricted stopping power is that fraction of
  the collision stopping power that includes all
  the soft collisions plus those hard collisions
  resulting in ? rays with energies less than a
  cutoff value ?
• The mass restricted stopping power in MeV cm2/g,
  will be symbolized as (dT/?dx)?
• An alternative and very important form of
  restricted stopping power is known as the linear
  energy transfer, symbolized as L?

30
Restricted Stopping Power (cont.)

• The usual units for L? are keV/?m, so that
• If the cutoff energy ? is increased to equal
  T?max T/2 for electrons, T for positrons, then
• and

31
Restricted Stopping Power (cont.)

• The calculation of (dT/?dx)? for heavy particles
  gives (in MeV cm2/g)
• For electrons and positrons this quantity is
  given by the following equation, in which ? ?
  T/m0c2 and ? ? ?/T

32
Restricted Stopping Power (cont.)

• For electrons
• and for positrons, substituting ? ? (? 2)-1,

33
Range

• Range may be defined as follows
• The range ? of a charged particle of a given type
  and energy in a given medium is the expectation
  value of the pathlength p that it follows until
  it comes to rest (discounting thermal motion)
• A second, related quantity, the projected range,
  is defined thus
• The projected range lttgt of a charged particle of
  a given type and initial energy in a given medium
  is the expectation value of the farthest depth of
  penetration tf of the particle in its initial
  direction

34
Illustrating the concepts of pathlength p and
farthest depth of penetration, tf, for an
individual electron. p is total distance along
the path from the point of entry A to the
stopping point B. Note that tf is not
necessarily the depth of B.
35
CSDA Range

• Experimentally the range can be determined (in
  principle) for an optically transparent medium
  such as photographic emulsion by microscopically
  following each particle track in three
  dimensions, and obtaining the mean pathlength for
  many such identical particles of the same
  starting energy

36
CSDA Range (cont.)

• A closely similar but not identical quantity is
  called the CSDA range, which represents the range
  in the continuous slowing down approximation
• In terms of the mass stopping power, the CSDA
  range is defined as
• where T0 is the starting energy of the
  particle
• If dT/?dx is in MeV cm2/g and dT in MeV, then
  ?CSDA is thus given in g/cm2

37
CSDA Range (cont.)

• For all practical purposes ?CSDA can be taken as
  identical to the range ? as defined earlier
• Their small and subtle difference is due to the
  occurrence of discrete and discontinuous energy
  losses
• The effect is expected to make the CSDA range
  slightly underestimate the actual range, by 0.2
  or less for protons and by a somewhat greater
  (but undetermined) amount for electrons

38
CSDA Range (cont.)

• The following figure gives the CSDA range ?CSDA
  for protons in C, Cu, and Pb
• (?CSDA) for carbon can be approximately
  represented (?5) in g/cm2 by
• for proton kinetic energies 1 MeV lt T0 lt 300
  MeV
• Because of the decrease in the stopping power
  with increasing atomic number, the range (in
  mass/area) is greater for higher Z

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40
CSDA Range (cont.)

• The range of other heavy particles can be
  obtained from a proton table by recalling that
• All particles with the same velocity have kinetic
  energies in proportion to their rest masses
• All singly charged heavy particles with the same
  velocity have the same stopping power
• Consequently the range of singly charged heavy
  particles of the same velocity is proportional to
  their rest mass, since a proportional amount of
  energy must be disposed of

41
CSDA Range (cont.)

• In general the procedure for finding the CSDA
  range of a heavy particle of rest mass M0 and
  kinetic energy T0 is to enter proton CSDA range
  tables at a proton energy T0P T0M0P/M0, where
  M0P is the protons rest mass
• If the tabulated proton CSDA range is ?PCSDA, the
  other particles range ?CSDA is then obtained
  from

42
Projected Range

• The projected range lttgt is most easily visualized
  in terms of flat layers of absorbing medium
  struck perpendicularly by a beam of charged
  particles
• One counts the number of incident particles that
  penetrate the slab as its thickness is increased
  from zero to ? (or to a thickness great enough to
  stop all the incident particles)

43
Projected Range (cont.)

• lttgt may be defined in that case as
• where N0 is the number of incident particles
  minus those that undergo nuclear reactions, N(t)
  is the number of particles penetrating a slab of
  thickness t, and tf(t) dN(t)/dt is the
  differential distribution of farthest depths of
  penetration tf

44
Projected Range (cont.)

• The following figure shows typical graphs of the
  number of particles penetrating through slabs of
  varying thickness t
• All particles are assumed to be monoenergetic and
  perpendicularly incident

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46
Projected Range (cont.)

• In pane a practically no reduction in numbers of
  particles is observed until the projected range
  lttgt is approached, where a steep decrease to zero
  occurs
• The value of t beyond which no particles are
  observed to penetrate is called tmax, the maximum
  penetration depth
• For a proton or heavier particle this is only
  slightly less than the maximum pathlength, since
  tmax represents those particles which happen to
  suffer little scattering
• The range ? (the mean value of the pathlength) is
  generally not more than 3 greater than lttgt for
  protons

47
Projected Range (cont.)

• In pane b we see a steady decline of N with
  increasing t from its initial value N0 to N0,
  which is equal to N0 minus the number of
  particles undergoing nuclear reactions, and is
  approximately the number reaching the knee of the
  curve
• Note that the equation calculates the projected
  range lttgt on the basis of N0, not N0
• Likewise the CSDA range, which closely
  approximates the range ?, is customarily
  calculated without including nuclear
  interactions, which are usually (but not always)
  negligible

48
Straggling and Multiple Scattering

• One can see from panes a and b that there is
  typically a distribution of farthest depths of
  penetration, tf, by individual particles, giving
  rise to an S-shaped descending curve
• This results from the combination of two effects
  multiple scattering (which is predominant), and
  range straggling a consequence of stochastic
  variations in rates of energy loss
• Range straggling alone also affects pathlengths,
  giving rise to a less-pronounced distribution
  than is observed in tf

49
Straggling and Multiple Scattering (cont.)

• A related effect, energy straggling, is the
  spread in energies observed in a population of
  initially identical charged particles after they
  have traversed equal path lengths
• It will be somewhat exaggerated if the particles
  have passed through a layer of material, since
  multiple scattering then causes individual
  differences in path length as well
• Multiple scattering in a foil also spreads an
  initially parallel beam of heavy charged
  particles into a conical angular distribution

50
Electron Range

• The electron CSDA range and the projected range
  are calculated the same as for heavy charged
  particles
• However, it should be evident from pane c that
  these quantities are of marginal usefulness in
  characterizing the depth of penetration of
  electrons (or positrons)
• Scattering effects, both nuclear and
  electron-electron, cause the particles to follow
  such tortuous paths that tf(t) is smeared out
  from very small depths up to t tmax ? 2lttgt

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52
Electron Range (cont.)

• For low-Z media, tmax is comparable to ? (or
  ?CSDA), which is a convenience in the practical
  application of range tables
• ? increases as a function of Z, similar to the
  that seen for protons
• However, a corresponding increase in the
  incidence of nuclear elastic scattering also
  takes place and tends to make lttgt and tmax
  roughly independent of Z for electrons and
  positrons

53
Comparison of Maximum Penetration Depth tmax with
CSDA Range for Electrons of Energy T0
54
Electron Range (cont.)

• The final column in this table gives the ratio
  tmax/?CSDA, which decreases from about 0.85 to
  0.48 as Z goes from 13 to 79
• This ratio shows very little energy dependence in
  the range 50 ? T0 ? 150 keV
• This trend is continued at higher energies as
  well, judging from the calculations of Spencer,
  which are excerpted in the following table

55
tmax/?CSDA for a Plane Perpendicular Source of
Electrons of Incident Energy T0
56
Electron Range (cont.)

• Spencer predicts (tmax/?CSDA) 0.95 for carbon
  at all energies
• This is consistent with the statement earlier
  that tmax is comparable to (i.e., probable ? 5
  less than) ?CSDA for electrons in low-Z media
• The following figure is a graph of ?CSDA vs. T0
  for electrons in carbon
• Note the proportionality to T0 above 2 MeV and to
  T02 below 0.1 MeV

57
CSDA range (? 1.05tmax) of electrons in carbon
58
Photon Projected Range

• For comparison with the charged-particle
  penetration curves, pane d gives a corresponding
  curve for monoenergetic ?- or x-rays, where
  scattered photons are ignored
• It is exponential vs. depth, with tmax at t ?
• The concept of projected range lttgt is even less
  useful here than it is for electrons as an
  indication of how far an individual ray will
  penetrate

59
Photon Projected Range (cont.)

• Nevertheless if the equation for lttgt is applied
  to the photon penetration curve, employing N(t)
  N0e-?t, one obtains lttgt 1/ ?, which is known as
  the mean free path or relaxation length of the
  photons in the medium
• This is the mean distance traveled by the
  individual photons in a large homogeneous
  population
• When t lttgt, N N0/e