ChargedParticle Interactions in Matter I

1
Charged-Particle Interactions in Matter I

• Types of Charged-Particle Coulomb-Force
  Interactions
• Stopping Power

2
Introduction

• Charged particles lose their energy in a manner
  that is distinctly different from that of
  uncharged radiations (x- or ?-rays and neutrons)
• An individual photon or neutron incident upon a
  slab of matter may pass through it with no
  interactions at all, and consequently no loss of
  energy
• Or it may interact and thus lose its energy in
  one or a few catastrophic events

3
Introduction (cont.)

• By contrast, a charged particle, being surrounded
  by its Coulomb electric force field, interacts
  with one or more electrons or with the nucleus of
  practically every atom it passes
• Most of these interactions individually transfer
  only minute fractions of the incident particles
  kinetic energy, and it is convenient to think of
  the particle as losing its kinetic energy
  gradually in a frictionlike process, often
  referred to as the continuous slowing-down
  approximation (CSDA)

4
Introduction (cont.)

• Charged particles can be roughly characterized by
  a common pathlength, traced out by most such
  particles of a given type and energy in a
  specific medium
• Because of the multitude of interactions
  undergone by each charged particle in slowing
  down, its pathlength tends to approach the
  expectation value that would be observed as a
  mean for a very large population of identical
  particles

5
Introduction (cont.)

• That expectation value, called the range, will be
  discussed in a later lecture
• Note that because of scattering, all identical
  charged particles do not follow the same path,
  nor are the paths straight, especially those of
  electrons because of their small mass

6
Types of Charged-Particle Coulomb-Force
Interactions

• Charged-particle Coulomb-force interactions can
  be simply characterized in terms of the relative
  size of the classical impact parameter b vs. the
  atomic radius a, as shown in the following figure
• The following three types of interactions become
  dominant for b gtgt a, b a, and b ltlt a,
  respectively
• Soft collisions
• Hard (or knock-on) collisions
• Coulomb-force interactions with the external
  nuclear field

7
Important parameters in charged-particle
collisions with atoms a is the classical atomic
radius b is the classical impact parameter
8
Soft Collisions (b gtgt a)

• When a charged particle passes an atom at a
  considerable distance, the influence of the
  particles Coulomb force field affects the atom
  as a whole, thereby distorting it, exciting it to
  a higher energy level, and sometimes ionizing it
  by ejecting a valence electron
• The net effect is the transfer of a very small
  amount of energy (a few eV) to an atom of the
  absorbing medium

9
Soft Collisions (cont.)

• Because large values of b are clearly more
  probable than are near hits on individual atoms,
  soft collisions are by far the most numerous
  type of charged-particle interaction, and they
  account for roughly half of the energy
  transferred to the absorbing medium
• In condensed media (liquids and solids) the
  atomic distortion mentioned above also gives rise
  to the polarization (or density) effect, which
  will be discussed later

10
Hard (or Knock-On) Collisions (b a)

• When the impact parameter b is of the order of
  the atomic dimensions, it becomes more likely
  that the incident particle will interact
  primarily with a single atomic electron, which is
  then ejected from the atom with considerable
  kinetic energy and is called a delta (?) ray
• In the theoretical treatment of the knock-on
  process, atomic binding energies have been
  neglected and the atomic electrons treated as
  free

11
Hard Collisions (cont.)

• ?-rays are of course energetic enough to undergo
  additional Coulomb-force interactions on their
  own
• Thus a ?-ray dissipates its kinetic energy along
  a separate track (called a spur) from that of
  the primary charged particle

12
Hard Collisions (cont.)

• The probability for hard collisions depends upon
  quantum-mechanical spin and exchange effects,
  thus involving the nature of the incident
  particle
• Hence, as will be seen, the form of
  stopping-power equations that include the effect
  of hard collisions depend on the particle type,
  being different especially for electrons vs.
  heavy particles
• Although hard collisions are few in number
  compared to soft collisions, the fractions of the
  primary particles energy that are spent by these
  two processes are generally comparable

13
Hard Collisions (cont.)

• It should be noted that whenever an inner-shell
  electron is ejected from an atom by a hard
  collision, characteristic x rays and/or Auger
  electrons will be emitted just as if the same
  electron had been removed by a photon interaction
• Thus some of the energy transferred to the medium
  may be transported some distance away from the
  primary particle track by these carriers as well
  as by the ?-rays

14
Coulomb-Force Interactions with the External
Nuclear Field (b ltlt a)

• When the impact parameter of a charged particle
  is much smaller than the atomic radius, the
  Coulomb-force interaction takes place mainly with
  the nucleus
• This kind of interaction is most important for
  electrons (either or -) in the present context,
  so the discussion here will be limited to that
  case

15
Interactions with the External Nuclear Field
(cont.)

• In all but 2 3 of such encounters, the
  electron is scattered elastically and does not
  emit an x-ray photon or excite the nucleus
• It loses just the insignificant amount of kinetic
  energy necessary to satisfy conservation of
  momentum for the collision
• Hence this is not a mechanism for the transfer of
  energy to the absorbing medium, but it is an
  important means of deflecting electrons

16
Interactions with the External Nuclear Field
(cont.)

• It is the principle reason why electrons follow
  very tortuous paths, especially in high-Z media,
  and why electron backscattering increases with Z
• In doing Monte Carlo calculations of electron
  transport through matter, it is often assumed for
  simplicity that the energy-loss interactions may
  be treated separately from the scattering (i.e.,
  change-of-direction) interactions

17
Interactions with the External Nuclear Field
(cont.)

• The differential elastic-scattering cross section
  per atom is proportional to Z²
• This means that a thin foil of high-Z material
  may be used as a scatterer to spread out an
  electron beam while minimizing the energy lost by
  the transmitted electrons in traversing a given
  mass thickness of foil

18
Interactions with the External Nuclear Field
(cont.)

• In the other 2 3 of the cases in which the
  electron passes near the nucleus, an inelastic
  radiative interaction occurs in which an x-ray
  photon is emitted
• The electron is not only deflected in this
  process, but gives a significant fraction (up to
  100) of its kinetic energy to the photon,
  slowing down in the process
• Such x-rays are referred to as bremsstrahlung,
  the German word for braking radiation

19
Interactions with the External Nuclear Field
(cont.)

• This interaction also has a differential atomic
  cross section proportional to Z², as was the case
  for nuclear elastic scattering
• Moreover, it depends on the inverse square of the
  mass of the particle, for a given particle
  velocity
• Thus bremsstrahlung generation by charged
  particles other than electrons is totally
  insignificant

20
Interactions with the External Nuclear Field
(cont.)

• Although bremsstrahlung production is an
  important means of energy dissipation by
  energetic electrons in high-Z media, it is
  relatively insignificant in low-Z (tissue-like)
  materials for electrons below 10 MeV
• Not only is the production cross section low in
  that case, but the resulting photons are
  penetrating enough so that most of them can
  escape from objects several centimeters in size
• Thus they usually carry away their quantum energy
  rather than expending it in the medium through a
  further interaction

21
Interactions with the External Nuclear Field
(cont.)

• In addition to the foregoing three modes of
  kinetic energy dissipation (soft, hard, and
  bremsstrahlung interactions), a fourth channel is
  available only to antimatter (i.e., positrons)
  in-flight annihilation
• The average fraction of a positrons kinetic
  energy that is spent in this type of radiative
  loss is said to be comparable to the fraction
  going into bremsstrahlung production

22
Nuclear Interactions by Heavy Charged Particles

• A heavy charged particle having sufficiently high
  kinetic energy ( 100 MeV) and an impact
  parameter less than the nuclear radius may
  interact inelastically with the nucleus
• When one or more individual nucleons (protons or
  neutrons) are struck, they may be driven out of
  the nucleus in an intranuclear cascade process,
  collimated strongly in the forward direction

23
Nuclear Interactions by Heavy Charged Particles
(cont.)

• The highly excited nucleus decays from its
  excited state by emission of so-called
  evaporation particles (mostly nucleons of
  relatively low energy) and ?-rays
• Thus the spatial distribution of absorbed dose is
  changed when nuclear interactions are present,
  since some of the kinetic energy that would
  otherwise be deposited as local excitation and
  ionization is carried away by neutrons and ?-rays

24
Nuclear Interactions by Heavy Charged Particles
(cont.)

• One special case where nuclear interactions by
  heavy charged particles attain first-order
  importance relative to Coulomb-force interactions
  is that of ?- mesons (negative pions)
• These particles have a mass 273 times that of the
  electron, or 15 of the proton mass
• They interact by Coulomb forces to produce
  excitation and ionization along their track in
  the same way as any other charged particle, but
  they also display some special characteristics

25
Nuclear Interactions by Heavy Charged Particles
(cont.)

• The effect of nuclear interactions is
  conventionally not included in defining the
  stopping power or range of charged particles
• Nuclear interactions by heavy charged particles
  are usually ignored in the context of
  radiological physics and dosimetry
• Internal nuclear interactions by electrons are
  negligible in comparison with the production of
  bremsstrahlung

26
Stopping Power

• The expectation value of the rate of energy loss
  per unit of path length x by a charged particle
  of type Y and kinetic energy T, in a medium of
  atomic number Z, is called its stopping power,
  (dT/dx)Y,T,Z
• The subscripts need not be explicitly stated
  where that information is clear from the context
• Stopping power is typically given in units of
  MeV/cm or J/m
• Dividing the stopping power by the density ? of
  the absorbing medium results in a quantity called
  the mass stopping power (dT/? dx), typically in
  MeV cm2/g or J m2/kg

27
Stopping Power (cont.)

• When one is interested in the fate of the energy
  lost by the charged particle, stopping power may
  be subdivided into collision stopping power and
  radiative stopping power
• The former is the rate of energy loss resulting
  from the sum of the soft and hard collisions,
  which are conventionally referred to as
  collision interactions
• Radiative stopping power is that owing to
  radiative interactions

28
Stopping Power (cont.)

• Unless otherwise specified, radiative stopping
  power may be assumed to be based on
  bremsstrahlung alone
• The effect of in-flight annihilation, which is
  only relevant for positrons, is accounted for
  separately
• Energy spend in radiative collisions is carried
  away from the charged particle track by the
  photons, while that spent in collision
  interactions produces ionization and excitation
  contributing to the dose near the track

29
Stopping Power (cont.)

• The mass collision stopping power can be written
  as
• where subscripts c indicate collision
  interactions, s being soft and h hard
• The terms on the right can be rewritten as

30
Stopping Power (cont.)

• T is the energy transferred to the atom or
  electron in the interaction
• H is the somewhat arbitrary energy boundary
  between soft and hard collisions, in terms of T
• Tmax is the maximum energy that can be
  transferred in a head-on collision with an atomic
  electron, assumed unbound

31
Stopping Power (cont.)

• For a heavy particle with kinetic energy less
  than its rest-mass energy M0c2,
• which for protons equals 20 keV for T 10
  MeV, or 0.2 MeV for T 100 MeV
• For positrons incident, Tmax T if annihilation
  does not occur
• For electrons, Tmax ? T/2

32
Stopping Power (cont.)

• Tmax is related to Tmin by
• in which I is the mean excitation potential
  of the struck atom, to be discussed later
• Qsc and Qhc are the respective differential mass
  collision coefficients for soft and hard
  collisions, typically in units of cm2/g MeV or
  m2/kg J

33
The Soft-Collision Term

• The soft-collision term was derived by Bethe, for
  either electrons or heavy charged particles with
  z elementary charges, on the basis of the Born
  approximation which assumes that the particle
  velocity (v ?c) is much greater than the
  maximum Bohr-orbit velocity (u) of the atomic
  electrons
• The fractional error in the assumption is of the
  order of (u/v)2, and Bethes formula is valid for
  (u/v)2 (Z/137?)2 ltlt 1
• This appears to be a rather severe restriction,
  but the formula is found to be practically
  applicable even where this inequality is not well
  satisfied

34
The Soft-Collision Term (cont.)

• The Bethe soft-collision formula can be written
  as
• where C ? ?(NAZ/A)r02 0.150Z/A cm2/g, in
  which NAZ/A is the number of electrons per gram
  of the stopping medium, and r0 e2/m0c2 2.818
  ? 10-13 cm is the classical electron radius

35
The Soft-Collision Term (cont.)

• We can further simplify the factor outside the
  bracket by defining it as
• where m0c2 0.511 MeV, the rest-mass energy
  of an electron
• The bracket factor is dimensionless, thus
  requiring the quantities m0c2, H, and I occurring
  within it to be expressed in the same energy
  units, usually eV

36
The Soft-Collision Term (cont.)

• The mean excitation potential I is the
  geometric-mean value of all the ionization and
  excitation potentials of an atom of the absorbing
  medium
• In general I for elements cannot be calculated
  from atomic theory with useful accuracy, but must
  instead be derived from stopping-power or range
  measurements
• Appendices B.1 and B.2 list some I-values
  according to Berger and Seltzer

37
The Soft-Collision Term (cont.)

• Since I only depends on the stopping medium, but
  not on the type of charged particle, experimental
  determinations have been done preferentially with
  cyclotron-accelerated protons, because of their
  availability with high ?-values and the
  relatively small effect of scattering as they
  pass through layers of material
• The paths of electrons are too crooked to allow
  their use in accurate stopping power
  determinations

38
The Hard-Collision Term for Heavy Particles

• The form of the hard-collision term depends on
  whether the charged particle is an electron,
  positron, or heavy particle
• We will treat the case of heavy particles first,
  having masses much greater than that of an
  electron, and will assume that H ltlt Tmax
• The hard-collision term may be written as

39
The Hard-Collision Term for Heavy Particles
(cont.)

• The mass collision stopping power for combined
  soft and hard collisions by heavy particles
  becomes
• which can be simplified further by
  substituting for Tmax

40
Dependence on the Stopping Medium

• There are two expressions influencing this
  dependence, and both decrease the mass collision
  stopping power as Z is increased
• The first is the factor Z/A outside the bracket,
  which makes the formula proportional to the
  number of electrons per unit mass of the medium
• The second is the term ln I in the bracket,
  which further decreases the stopping power as Z
  is increased

41
Dependence on the Stopping Medium (cont.)

• The term ln I provides the stronger variation
  with Z
• The combined effect of the two Z-dependent
  expressions is to make (dT/?dx)c for Pb less than
  that for C by ?40-60 within the ?-range
  0.85-0.1, respectively

42
Dependence on Particle Velocity

• The strongest dependence on velocity comes from
  the inverse ?2 (outside of the bracket), which
  rapidly decreases the stopping power as ?
  increases
• That term loses its influence as ? approaches a
  constant value at unity, while the sum of the ?2
  terms in the bracket continues to increase
• The stopping power gradually flattens to a broad
  minimum of 1-2 MeV cm2/g at T/M0c2 ? 3, and then
  slowly rises again with further increasing T

43
Mass collision stopping power for singly charged
heavy particles, as a function of ? or of their
kinetic energy T
44
Dependence on Particle Velocity (cont.)

• The factor 1/?2 implies that the stopping power
  increases in proportion to 1/T without limit as
  particles slow down and approach zero velocity
• Actually the validity of the stopping-power
  formula breaks down for small ?
• However, the steep rise in stopping power that
  does occur accounts for the Bragg peak observed
  in the energy-loss density near the end of the
  charged particles path

45
Dependence on Particle Charge

• The factor z2 means that a doubly charged
  particle of a given velocity has 4 times the
  collision stopping power as a singly charged
  particle of the same velocity in the same medium
• For example, an ?-particle with ? 0.141 would
  have a mass collision stopping power of 200 MeV
  cm2/g, compared with the 50 MeV cm2/g shown in
  the figure for a singly charged heavy particle in
  water

46
Dependence on Particle Mass

• There is none
• All heavy charged particles of a given velocity
  and z will have the same collision stopping power

47
Relativistic Scaling Considerations

• For any particle, ? v/c is related to the
  kinetic energy T by
• The kinetic energy required by any particle to
  reach a given velocity is proportional to its
  rest energy, M0c2
• The rest energies of some heavy particles are
  listed in the following table

48
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49
Shell Correction

• The Born approximation assumption, which
  underlies the stopping-power equation, is not
  well satisfied when the velocity of the passing
  particle ceases to be much greater than that of
  the atomic electrons in the stopping medium
• Since K-shell electrons have the highest
  velocities, they are the first to be affected by
  insufficient particle velocity, the slower
  L-shell electrons are next, and so on
• The so-called shell correction is intended to
  account for the resulting error in the
  stopping-power equation

50
Shell Correction (cont.)

• As the particle velocity is decreased toward that
  of the K-shell electrons, those electrons
  gradually decrease their participation in the
  collision process, and the stopping power is
  thereby decreased below the value given by the
  equation
• When the particle velocity falls below that of
  the K-shell electrons, they cease participating
  in the collision stopping-power process
• The equation underestimates the stopping power
  because it contains too large an I-value
• The proper I-value would ignore the K-shell
  contribution

51
Shell Correction (cont.)

• Bichsel estimated the combined effect of all i
  shells into a single approximate correction C/Z,
  to be subtracted from the bracketed terms
• The corrected formula for the mass collision
  stopping power for heavy particles then becomes

52
Shell Correction (cont.)

• The correction term C/Z is the same for all
  charged particles of the same velocity ?,
  including electrons, and its size is a function
  of the medium as well as the particle velocity
• C/Z is shown in the following figure for protons
  in several elements
• A second correction term, ?, to account for the
  polarization or density effect in condensed
  media, is sometimes included also
• It is negligible for all heavy particles within
  the energy range of interest in radiological
  physics

53
Semiempirical shell corrections of Bichsel for
selected elements, as a function of proton energy
54
Mass Collision Stopping Power for Electrons and
Positrons

• The formulae for the mass collision stopping
  power for electrons and positrons are gotten by
  combining Bethes soft collision formula with a
  hard-collision relation based on the Møller cross
  section for electrons or the Bhabha cross section
  for positrons
• The resulting formula, common to both particles,
  in terms of ? ? T/m0c2, is

55
Mass Collision Stopping Power for Electrons and
Positrons (cont.)

• For electrons,
• and for positrons,
• Here C/Z is the previously discussed shell
  correction and ? is the correction term for the
  polarization or density effect