Gamma and XRay Interactions in Matter I

1
Gamma- and X-Ray Interactions in Matter I

• Compton Effect

2
Introduction

• There are five types of interactions with matter
  by x- and ?-ray photons which must be considered
  in radiological physics
• Compton effect
• Photoelectric effect
• Pair production
• Rayleigh (coherent) scattering
• Photonuclear interactions

3
Introduction (cont.)

• The first three of these are the most important,
  as they result in the transfer of energy to
  electrons, which then impart that energy to
  matter in many (usually small) Coulomb-force
  interactions along their tracks
• Rayleigh scattering is elastic the photon is
  merely redirected through a small angle with no
  energy loss
• Photonuclear interactions are only significant
  for photon energies above a few MeV, where they
  may create radiation-protection problems through
  the (?, n) production of neutrons and consequent
  radioactivation

4
Introduction (cont.)

• The relative importance of Compton effect,
  photoelectric effect, and pair production depends
  on both the photon quantum energy (E? h?) and
  the atomic number Z of the absorbing medium
• The following figure indicates the regions of Z
  and E? in which each interaction predominates

5
Relative importance of the three major types of
?-ray interactions. The curves show the values
of Z and E? for which the two types of effects
are equal.
6
Introduction (cont.)

• The photoelectric effect is dominant at the lower
  photon energies, the Compton effect takes over at
  medium energies, and pair production at the
  higher energies
• For low-Z media (e.g., carbon, air, water, human
  tissue) the region of Compton-effect dominance is
  very broad, extending from ?20 keV to ?30 MeV
• This gradually narrows with increasing Z

7
Compton Effect

• A description of the Compton effect can be
  conveniently subdivided into two aspects
  kinematics and cross section
• The first relates the energies and angles of the
  participating particles when a Compton event
  occurs the second predicts the probability that
  a Compton interaction will occur
• In both respects it is customary to assume that
  the electron struck by the incoming photon is
  initially unbound and stationary

8
Compton Effect (cont.)

• These assumptions are certainly not rigorous,
  inasmuch as the electrons all occupy various
  atomic energy levels, thus are in motion and are
  bound to the nucleus
• The resulting errors remain inconsequential in
  radiological physics applications, because of the
  dominance of the competing photoelectric effect
  under the conditions (high Z, low h?) where
  electron binding effects are the most important
  in Compton interactions

9
Compton Effect - Kinematics

• The following figure shows a photon of energy h?
  colliding with an electron
• The photons incident forward momentum is h?/c,
  where c is the speed of light in vacuum
• The stationary target electron has no initial
  kinetic energy or momentum

10
Kinematics of the Compton effect
11
Compton Effect Kinematics (cont.)

• After the collision the electron departs at angle
  ?, with kinetic energy T and momentum p
• The photon scatters at angle ? with a new, lower
  quantum energy h?? and momentum h??/c
• The solution to the collision kinetics is based
  upon conservation of both energy and momentum
• Energy conservation requires that

12
Compton Effect Kinematics(cont.)

• Conservation of momentum along the original
  photon direction (0) can be expressed as
• or
• Conservation of momentum perpendicular to the
  direction of incidence gives the equation

13
Compton Effect Kinematics (cont.)

• pc can be written in terms of T by invoking the
  law of invariance
• in which m0 is the electrons rest mass
• This equation can be derived from the following
  three relativistic relationships

14
Compton Effect Kinematics (cont.)

• As a result of the substitution for pc, we have a
  set of three simultaneous equations in these five
  parameters h?, h??, T, ?, and ?
• These equations can be solved algebraically to
  obtain any three of the variables we choose in a
  single equation
• Of the many equations that may be thus derived,
  the following set of equations, each in three
  variables, provides in convenient form a complete
  solution to the kinematics of Compton interactions

15
Compton Effect Kinematics (cont.)
16
Compton Effect Kinematics (cont.)

• It will be seen from the first of these equations
  that for a given value of h?, the energy h?? and
  angle ? of the scattered photon are uniquely
  correlated to each other
• The second equation then provides the kinetic
  energy T of the corresponding scattered electron,
  and the third equation gives its scattering angle
  ?

17
Compton Effect Kinematics (cont.)

• The following figure is a simple graphical
  representation of the kinematic relationships
  between h?, h??, and T
• It can be seen that for h? smaller than about
  0.01 MeV, all the curves for different ?-values
  converge along the diagonal, indicating that h??
  h? regardless of photon scattering angle
• Consequently the electron receives practically no
  kinetic energy in the interaction
• This means that Compton scattering is nearly
  elastic for low photon energies

18
Graphical representation of the kinematic
relationship of h?, h??, and T in the Compton
effect
19
Compton Effect Kinematics (cont.)

• An earlier theory of ?-ray scattering by Thomson,
  based on observations only at low energies,
  predicted that the scattered photon should always
  have the same energy as the incident one,
  regardless of h? or ?
• This is shown in the figure by the extension of
  the diagonal line to high energies
• This curve also applies to the Compton effect for
  the trivial case of straight-ahead scattering, ?
  0

20
Compton Effect Kinematics (cont.)

• The failure of the Thomson theory to describe
  high-energy photon scattering necessitated the
  development of Comptons theory
• For high-energy incident photons the
  backscattered photon (? 180) has an energy h??
  approaching 0.2555 MeV, while side-scattered (?
  90) have h?? ? 0.511 MeV

21
Compton Effect Kinematics (cont.)

• The kinetic energy of the recoiling electron is
  given graphically in the figure as the vertical
  distance of the curve for the appropriate ? below
  the diagonal line, in terms of energy on the h??
  scale
• Thus for the example shown by the arrow (h? 10
  MeV and ? 90), T 10 0.49 9.51 MeV

22
Compton Effect Kinematics (cont.)

• For backscattering of photons, the electron is
  projected forward (? 0) with an energy equal to
  h? - h??, which approaches h?-0.2555 MeV for very
  large h?
• The photon is evidently able to transfer most of
  its energy to the electron in that case, but can
  never give away all of its energy in a Compton
  collision with a free electron

23
Compton Effect Kinematics (cont.)

• The following figure contains graphs of the
  relationship between ?, ?, and h? for several
  values of h?
• When ? 0, ? 90, and when ? 180, ? 0,
  for all photon energies
• Obviously the electron can only be scattered in
  the forward hemisphere by a Compton event
• The dependence of ? upon ? is a strong function
  of h? between the angular extremes

24
Relationship of the electron scattering angle ?
to the photon scattering angle ? in the Compton
effect
25
Compton Effect Kinematics (cont.)

• For low photon energies ? ? 90 - ?/2 the
  electron scattering angle gradually decreases
  from 90 to 0 as the photon angle increases from
  0 to 180
• At high photon energies the major variation in ?
  is concentrated at small ?-values, and vice versa

26
Compton Effect Kinematics (cont.)

• It is important to remember that these figures
  and equations tell us nothing about the
  probability of a photon or an electron being
  Compton-scattered in any particular direction
  that is a separate matter
• The foregoing figures and equations only state
  how the various parameters must be related to
  each other if a Compton interaction does occur

27
Interaction Cross Section for the Compton Effect
Thomson Scattering

• J.J. Thomson provided the earliest theoretical
  description of the process by which a ?-ray
  photon can be scattered by an electron
• In this theory the electron was assumed to be
  free to oscillate under the influence of the
  electric vector of an incident electromagnetic
  wave, then promptly to reemit a photon of the
  same energy
• The electron thus retains no kinetic energy as a
  result of this elastic scattering event
• This agrees with the kinematic predictions of the
  later relativistic Compton treatment quite well
  up to about h? 0.01 MeV, for which h?? 0.0096
  MeV

28
Thomson Scattering (cont.)

• Thomson also deduced that the differential cross
  section per electron for a photon scattered at
  angle ?, per solid angle, could be expressed as
• in typical units of cm2 sr-1 per electron
• r0 e2/m0c2 2.818 ? 10-13 cm is called the
  classical electron radius
• The value of this equation is 7.94 ? 10-26 cm2
  sr-1 e-1 at ? 0 and 180, and half of that at ?
  90

29
Thomson Scattering (cont.)

• Thus the angular distribution of scattered
  photons for a large number of events is predicted
  to be front-back symmetrical, according to
  Thomson
• If the beam of photons is unpolarized, there will
  also be cylindrical symmetry around the beam axis
• The angular distribution of Thomson-scattered
  photons is approximated by the uppermost curve in
  the following figure, which was drawn to show the
  corresponding distribution of Compton-scattered
  photons for h? 0.01 MeV
• When h? approaches zero, the two theories
  converge, as relativistic considerations become
  irrelevant

30
Differential Klein-Nishina cross section, de?/d??
vs. angle ? of the scattered photon, for h?
0.01, 0.1, 1.0, 10, 100, and 500 MeV
31
Thomson Scattering (cont.)

• The total Thomson scattering cross section per
  electron, e?0, can be gotten by integrating over
  all directions of scattering
• This will be simplified by assuming cylindrical
  symmetry and integrating over 0 ? ? ? ?, noting
  that the annular element of solid angle is given
  in terms of ? by d?? 2? sin ? d?

32
Thomson Scattering (cont.)

• This cross section (which can be thought of as an
  effective target area) is numerically equal to
  the probability of a Thomson-scattering event
  occurring when a single photon passes through a
  layer containing one electron per cm2
• It is also the fraction of a large number of
  incident photons that scatter in passing through
  the same layer, e.g., approximately 665 events
  for 1027 photons
• So long as the fraction of photons interacting in
  a layer of matter by all processes combined
  remains less than about 0.05, the fraction may be
  assumed to be proportional to absorber thickness
• For greater thicknesses the exponential relation
  must be used

33
Klein-Nishina Cross Sections for the Compton
Effect

• In 1928 Klein and Nishina applied Diracs
  relativistic theory of the electron to the
  Compton effect to obtain improved cross sections
• Thomsons value of 6.65 ? 10-25 cm2/e,
  independent of h?, was known to be too large for
  h? gt 0.01 MeV
• The error reached a factor of 2 at h? 0.4 MeV
• The Klein-Nishina (K-N) treatment was remarkably
  successful in predicting the correct experimental
  value, even though they assumed unbound
  electrons, initially at rest

34
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• The differential cross section for photon
  scattering at angle ?, per unit solid angle and
  per electron, may be written in the form
• in which h?? is given by the Compton equation
• For low energies, as was previously pointed out,
  h?? ? h? hence this equation becomes
• which is identical to the classical cross
  section

35
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• The following figure is a graphical
  representation of the K-N differential cross
  section for six values of h?
• The forward bias of the scattered photons at high
  energies is apparent

36
Differential Klein-Nishina cross section, de?/d??
vs. angle ? of the scattered photon, for h?
0.01, 0.1, 1.0, 10, 100, and 500 MeV
37
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• The total K-N cross section per electron (e?) can
  be gotten by an integration over all photon
  scattering angles ?
• where ? h?/m0c2

38
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• The K-N cross section is shown graphically as the
  upper curve of the following figure
• As expected, it is almost equal to the Thomson
  scattering cross section (6.65 ? 10-25 cm2/e) at
  h? 0.01 MeV
• It decreases gradually for higher photon energies
  to approach a e? ? (h?)-1 dependence

39
Klein-Nishina (Compton-effect) cross section per
electron (e?) and corresponding energy-transfer
cross section (e?tr) as a function of primary
photon quantum energy h?
40
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• It is important to remember that e?, which is
  tabulated in Appendix D.1, is independent of the
  atomic number Z since the electron binding energy
  has been assumed to be zero
• Thus the K-N cross section per atom of any Z is
  given by

41
Klein-Nishina Cross Sections for the Compton
Effect (cont.)

• The corresponding K-N cross section per unit
  mass, ?/?, which is also called the Compton mass
  attenuation coefficient, is obtained from

42
Energy-Transfer Cross Section for the Compton
Effect

• The total K-N cross section, multiplied by a unit
  thickness of 1 e/cm2, also may be thought of as
  the fraction of the incident energy fluence,
  carried by a beam of many monoenergetic photons,
  that will be diverted into Compton interactions
  in passing through that layer of material
• In each interaction the energy of the incident
  photon (h?) is shared between the scattered
  photon (h?) and the recoiling electron (T)

43
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• It is of interest to know the overall fraction of
  h? that is given to electrons, averaged over all
  scattering angles, as this energy contributes to
  the kerma and thence to the dose
• That is, we would like to know the value of
  T/h?, where T is the average kinetic energy of
  the recoiling electrons

44
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• This can be obtained through first modifying the
  differential K-N cross section to obtain a
  quantity referred to as the differential K-N
  energy-transfer cross section, de?tr/d??

45
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• Integrating this over all photon scattering
  angles ? from 0 to 180 yields the following
  statement of e?tr, the K-N energy-transfer cross
  section

46
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• This cross section, multiplied by the unit
  thickness 1 e/cm2, represents the fraction of the
  energy fluence in a monoenergetic photon beam
  that is diverted to the recoil electrons by
  Compton interactions in that layer
• The Compton (or K-N) energy-transfer cross
  section is plotted in the following figure (lower
  curve)

47
Klein-Nishina (Compton-effect) cross section per
electron (e?) and corresponding energy-transfer
cross section (e?tr) as a function of primary
photon quantum energy h?
48
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• The vertical difference between the two curves
  represents the K-N cross section for the energy
  carried away by the scattered photons, e?s
• Thus

49
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• The average fraction of the incident photons
  energy given to the electron is simply
• and one can obtain the average energy of the
  Compton recoil electrons generated by photons of
  energy h? as

50
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• The mean fraction of the incident photons energy
  given to the recoiling electron is plotted in the
  following figure
• At low energies the average fraction of h? given
  to the electron approaches zero for h? 1.6 MeV
  the electrons get half, or T 0.8 MeV

51
Mean fraction (T/h?) of the incident photons
energy given to the recoiling electron in Compton
interactions, averaged over all angles
52
Energy-Transfer Cross Section for the Compton
Effect (cont.)

• The contribution of the Compton effect to the
  photon mass attenuation coefficient ?/? is ?/?
• The corresponding contribution to the mass
  energy-transfer coefficient is
• The contributions of the several kinds of
  interactions to ?/?, ?tr/?, and ?en/? will be
  summarized in a later section

53
Alternative Forms of the K-N Cross Section

• Before proceeding to discussions of other types
  of interactions, it will be helpful to show and
  explain two other useful forms of the
  differential K-N cross section
• The first is de?/d??, the differential K-N cross
  section for electron scattering at angle ?, per
  unit solid angle and per electron
• Note that the solid angle referred to in this
  case means that through which the electron
  scatters at angle ?

54
Alternative Forms of the K-N Cross Section (cont.)

• For de?/d?? the solid angle is that through which
  the photon scatters at angle ?
• The relationship between these two differential
  cross sections is
• in which ? 2 tan-1(cot ?)/(1 ?)
• Integration over all electron scattering angles
  from ? 0 to 90 must again give e?

55
Alternative Forms of the K-N Cross Section (cont.)

• The following figure displays de?/d?? graphically
  for several values of h?, plotting de?/d?? vs. ?
• The probability of electrons being scattered at ?
  90 approaches a constant value (zero) for all
  h?, while de?/d?? 7.94 ? 10-26 cm2/sr e for all
  h? at ? 0
• As the cross section decreases to de?/d?? 2 ?
  10-29 cm2/sr e for backscattered electrons at h?
  500 MeV, the corresponding cross section for
  0-scattered electrons is seen from the figure to
  reach de?/d?? 7.78 ? 10-23

56
Differential Klein-Nishina cross section de?/d??
vs. angle ? of the scattered electron for h?
0.01, 0.1, 1, 10, and 500 MeV
57
Alternative Forms of the K-N Cross Section (cont.)

• This is an indication of how very strongly
  forward-directed the electrons become at high
  incident photon energies, while at the same time
  it becomes relatively unlikely that photons will
  be 180-backscattered
• The high forward momentum in the collision causes
  most of the electrons and most of the scattered
  photons to be strongly forward-directed when h?
  is large

58
Alternative Forms of the K-N Cross Section (cont.)

• The second additional form of differential K-N
  cross section that deserves mention is de?/dT,
  typically in cm2 MeV-1 e-1
• This is the probability that a single photon will
  have a Compton interaction in crossing a layer
  containing one e/cm2, transferring to that
  electron a kinetic energy between T and T dT
• Thus de?/dT is the energy distribution of the
  electrons, averaged over all scattering angles ?

59
Alternative Forms of the K-N Cross Section (cont.)

• It is given by the relation

60
Alternative Forms of the K-N Cross Section (cont.)

• The following figure is a graphical
  representation of this equation for several
  values of h?
• It is evident that the distribution of kinetic
  energies given to the Compton recoiling electrons
  tends to be fairly flat from zero almost up to
  the maximum electron energy, where a higher
  concentration occurs

61
Differential Klein-Nishina cross section de?/dT
expressing the initial energy spectrum of Compton
recoiling electrons
62
Alternative Forms of the K-N Cross Section (cont.)

• As mentioned earlier, the maximum electron energy
  Tmax resulting from a head-on Compton collision
  (? 0?) by a photon of energy h? is (h? -
  h??min), which is equal to
• This approaches h? - 0.2555 MeV for large h?
• The higher concentrations of electrons near this
  energy is consistent with the high probability of
  electron scattering near ? 0?
• Both trends become more pronounced at high
  energies

63
Alternative Forms of the K-N Cross Section (cont.)

• It should be remembered that the energy
  distributions shown in the diagram are those
  occurring at production
• The spectrum of Compton-electron energies present
  at a point in an extended medium under
  irradiation is generally degraded by the presence
  of electrons that have lost varying amounts of
  their energy depending on how far they have
  traveled through the medium
• Under charged-particle equilibrium conditions
  this degraded electron energy distribution is
  called an equilibrium spectrum