XRay Production and Quality I

1
X-Ray Production and Quality I

• Production
• Unfiltered Energy Spectrum

2
Introduction

• The word quality as applied to an x-ray beam
  ordinarily may be taken as synonymous with
  hardness, i.e., penetrating ability
• In the earlier days of radiotherapy, before
  megavolt x- or ?-ray beams became generally
  available, the effectiveness of x-ray treatment
  of deep-seated tumors depended upon the ability
  of the orthovoltage (lt300-kV) x-rays to penetrate
  to the tumor while limiting the dose to overlying
  tissues.
• For that application, the more strongly
  penetrating the beam, the higher its quality

3
Introduction (cont.)

• The same term is still applied to x-ray beam
  hardness even in cases where penetrating power
  should not necessarily be maximized (e.g., in
  diagnostic radiology)
• Quality of radiation is also used in the more
  general sense of energy spectral distribution, or
  in the special meaning of biological effectiveness

4
Fluorescence X-Rays

• Fluorescence (also called characteristic) x-ray
  production has been discussed to some extent
  previously (in connection with electron capture
  and internal conversion, and with the
  photoelectric effect)
• It was also mentioned that when hard collisions
  occur between charged particles and inner-shell
  electrons, the filling of the resulting shell
  vacancy generates fluorescence x-rays
• Only a small fraction (?1) of the charged
  particle energy spent in collision interactions
  goes into fluorescence x-ray production, however

5
Fluorescence Yield

• The probability that a fluorescence x-ray will
  escape from the atom of its origin is called the
  fluorescence yield, symbolized by YK for a
  K-shell vacancy, and so on
• Escaping fluorescence x-rays are practically
  nonexistent for elements with atomic numbers less
  than 10, and the K-shell yield increases rapidly
  with Z to about 0.95 for tungsten (Z 74), the
  most common x-ray tube target

6
Fluorescence Yield (cont.)

• For the L-shell the yield remains relatively low,
  and since the L-shell binding energy is small
  (12.1 keV for the L1 shell in tungsten), L-shell
  fluorescence is of little practical importance as
  an x-ray production process
• Only K-shell fluorescence need be considered here

7
Initiating Event

• The initiating event in K-fluorescence x-ray
  production is the removal of a K-shell electron
  by one of the processes mentioned above
• Thus the minimum energy that must be supplied is
  the K-shell binding energy, (Eb)K
• A photon of quantum energy h? ? 69.5 keV, for
  example, can generate K-shell fluorescence in
  tungsten through the photoelectric effect

8
Initiating Event (cont.)

• An electron of kinetic energy T gt 69.5 keV can do
  likewise by ejecting a K-shell electron in a hard
  collision
• Notice that the electron is not required to have
  an incident energy exceeding twice the binding
  energy to accomplish this, even though an
  electron is conventionally supposed to be able to
  give no more than half its energy to another
  electron

9
Initiating Event (cont.)

• That formalism, as applied in the electron
  stopping-power equation, merely acknowledges that
  the incident electron and the struck electron are
  indistinguishable after the collision, and the
  one departing with the most energy is therefore
  designated, post facto, as having been the
  incident electron
• The fact that an incident electron with T gt (Eb)K
  can remove a K-shell electron proves, however,
  that kinetic-energy transfers up to T must occur
  in electron-electron collisions, as one would
  expect from momentum-conservation considerations

10
Initiating Event (cont.)

• Although electron beams are the most common means
  of generating fluorescence x-rays, they appear in
  that case against a very strong background of
  bremsstrahlung continuous-spectrum x-rays
• If it is desired to have a relatively pure
  fluorescence x-ray source with greatly reduced
  bremsstrahlung background, either heavy-particle
  excitation or x-ray excitation of fluorescence by
  the photoelectric effect may be employed

11
Initiating Event (cont.)

• When heavy particles such as protons or
  ?-particles are used to excite x-ray
  fluorescence, one might suppose from
  momentum-conservation considerations that the
  minimum energy necessary to ionize the K-shell
  would be controlled by
• where Tmax is the maximum energy that can be
  transferred by a heavy particle of rest mass M0
  and kinetic energy T to a free electron of mass
  m0 at rest

12
Initiating Event (cont.)

• Thus, on this basis, a proton (M0 1836m0) would
  have to have an energy 460 times the binding
  energy Eb to eject an electron from its shell
• However, it is found that this threshold does not
  apply for ionization of strongly bound electrons
  by heavy particles
• That is because the binding energy, in effect,
  increases the mass of the electron, thereby
  allowing larger energy transfers
• The following figure gives cross sections for
  fluorescence x-ray production by protons

13
Atomic cross sections for fluorescence x-ray
production by protons
14
K-Fluorescence Photon Energy

• Following the creation of a K-shell vacancy, an
  electron from another higher shell will fill it,
  and may emit a fluorescence photon having a
  quantum energy equal to the difference in the two
  energy levels involved
• Again citing the example of tungsten, the
  following table lists the binding energies for
  the K-, L-, M-, and N-shells, having 1, 3, 5, and
  7 subshells, respectively
• Quantum mechanical selection rules allow
  transitions to the K-shell mainly from the levels
  shown in boxes

15
Electron Binding Energies Eb in Tungsten
16
K-Fluorescence Photon Energy (cont.)

• The resulting transitions to the K-shell, the
  designation of the resulting fluorescence lines,
  and their quantum energies and relative
  frequencies of occurrence are shown in the
  following table

17
K-Shell X-Ray Fluorescence Energies in Tungsten
18
Directional Distributions of Fluorescence vs.
Bremsstrahlung

• Since fluorescence is emitted in a secondary
  transition process following a primary ionization
  event, there is no angular correlation between
  the direction of the incident particle and that
  of the fluorescence photon
• Fluorescence is emitted isotropically with
  respect to both energy and intensity, neglecting
  attenuation of rays in escaping the target
• Bremsstrahlung x rays, on the other hand, are
  emitted anisotropically, tending to go more and
  more closely in the electrons direction with
  increasing energy

19
Directional Distributions (cont.)

• In thin targets in which electron scattering can
  be neglected, bremsstrahlung production shows
  strong angular dependence and a minimum value at
  180
• The following figure compares the directional
  distributions for K-fluorescence and
  bremsstrahlung x rays generated in a thin silver
  foil by 50- and 500-keV electrons

20
Comparison of the directional distributions of K
x-rays (solid curves) and bremsstrahlung (dashed
curves) for 50- and 500-keV electrons incident on
a thin silver target
21
Directional Distributions (cont.)

• It is evident that the ratio of K-fluorescence to
  bremsstrahlung is a maximum at 180
• This is generally true irrespective of Z, T, or
  target thickness, although the angular dependence
  of bremsstrahlung x rays becomes less pronounced
  for thick targets

22
Dependence of Fluorescence Output on Electron
Beam Energy

• The energy of the incident electron beam also
  influences the intensity of fluorescence x-ray
  production
• If T is below the K binding energy, no K-lines
  appear
• For T gt (Eb)K all the K-lines are generated with
  fixed relative strengths, regardless of how much
  higher T may be
• However, the efficiency for K-fluorescence
  production increases rapidly at first for T gt
  (Eb)K, reaches a maximum, and then decreases
  slowly as T continues to rise
• The following figure shows this trend for thick
  targets

23
Dependence of K x-ray yield from thick targets of
Z 4 to 79 on incident electron energy
24
Dependence on Electron Beam Energy (cont.)

• For thin targets the maxima occur at lower
  energies, and of course the K-fluorescent x-ray
  outputs are also lower
• Since the bremsstrahlung output from a thick
  target continues to increase with T without
  limit, the ratio of K-fluorescence to
  bremsstrahlung x-ray output must also reach a
  maximum and then decrease for still higher T

25
Dependence on Electron Beam Energy (cont.)

• The following table roughly compares the
  K-fluorescence outputs and beam purities
  attainable by (a) the photoelectric effect (using
  x rays to excite the fluorescer), (b) using
  electrons with 180 geometry, and (c) using heavy
  ions
• The last method is seen to reach the
  highest-purity beams, with outputs comparable to
  those obtained with electrons
• The output with photoelectric excitation is
  evidently several orders of magnitude lower

26
Comparison of Different Excitation Sources for K
X-ray Production
27
Dependence on Electron Beam Energy (cont.)

• None of these methods for fluorescence x-ray beam
  generation is in common use for dosimetry
  applications, probably because all require
  special apparatus
• Instead, heavily filtered bremsstrahlung x-ray
  beams are usually employed
• The spectral widths for such beams are much
  greater than for K-fluorescence lines, with a
  resulting loss of energy resolution

28
Bremsstrahlung X-RaysProduction Efficiency

• The practical generation of bremsstrahlung x rays
  is done by accelerating an electron beam and
  allowing it to strike a metallic target
• We know from
• that the ratio of mass radiative stopping
  power to mass collision stopping power is
  proportional to TZ
• This means that high-Z targets convert a larger
  fraction of the electrons energy into
  bremsstrahlung x rays than lower-Z targets

29
Production Efficiency (cont.)

• Tungsten (Z 74) is a common choice, as it has
  not only a high atomic number, but a high melting
  point as well
• The energy that is not radiated as bremsstrahlung
  is of course spent in producing ionization and
  excitation by collision interactions
• This energy nearly all degrades to heat in the
  target, except for the very small fraction
  emitted as fluorescence x rays
• Thus target cooling is required

30
Production Efficiency (cont.)

• In a thin target (i.e., in the present context,
  one in which the electron beam is not appreciably
  scattered and loses so little energy that the
  stopping power is unchanged) the approximate
  fraction of the total energy lost that goes into
  bremsstrahlung x-ray production is
• where T is the electron energy in MeV, and
  the value of n for tungsten is 775 at 100 MeV,
  786 at 10 MeV, 649 at 1 MeV, 371 at 0.1 MeV, and
  336 at 0.01 MeV

31
Production Efficiency (cont.)

• The following figure shows in the upper curve the
  value of this equation
• This ratio rises roughly linearly with increasing
  T ( the incident energy T0), asymptotically
  approaching unit
• At 1 GeV, 99 of the energy lost in a thin
  tungsten foil goes into x-ray production

32
Fraction of electron energy losses that are spent
in bremsstrahlung x-ray production in thin (upper
curve) or thick (lower curve) tungsten targets
33
Production Efficiency (cont.)

• The overall energy x-ray production efficiency
  remains small in thin targets at all electron
  energies, since most of T0 is retained by the
  electron and carried out the back of the target
  foil
• Only in thick targets, in which the electrons are
  stopped, or semithick ones, in which a large part
  of the electrons energy is spent, can reasonable
  efficiencies be attained
• The lower curve in the figure expresses this,
  indicating the radiation yield, or fraction of T0
  spent in generating bremsstrahlung x-rays as the
  electron is brought to a stop in a thick tungsten
  target

34
Production Efficiency (cont.)

• It can be seen from the diagram, for example,
  that a 100-keV electron beam spends only 1 of
  its energy on bremsstrahlung production in a
  thick tungsten target
• The other 99 is spent in collision interactions,
  of which lt1 generates fluorescence x-rays and
  the rest heats the target

35
Unfiltered Bremsstrahlung Energy Spectrum T0 ltlt
m0c2

• The shape of the unfiltered bremsstrahlung
  radiant energy spectrum, generated in a thin
  target of any atomic number Z by an electron beam
  of incident energy T0 ltlt m0c2, is shown in the
  top of the following diagram
• It will be seen that the maximum photon energy
  h?max is T0, the kinetic energy of the incident
  electrons

36
Bremsstrahlung radiant-energy spectrum from (a) a
thin target, (b) a thick target irradiated by
electrons of incident energy T0 ltlt m0c2
37
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• This figure also shows that the radiant-energy
  spectrum is constant over the energy range from 0
  ? h? ? h?max
• Thus, for example, the number of photons emitted
  per unit energy interval at energy h? is twice
  the number at 2h?, assuming both energies to be
  less than h?max

38
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• It is not obvious why electrons impinging on a
  thin target should generate a spectrum of this
  simple shape, but it can be visualized
  intuitively by means of an argument based on the
  classical impact parameter
• When the impact parameter b is equal to 0, the
  electron has a direct hit on the nucleus and
  gives all of its energy T0 to create a photon
  h?max
• As the impact parameter increases, the area in an
  annulus of radius b and width db increases
  proportionately, as shown in the following figure

39
Classical explanation of the thin-target x-ray
spectrum generated by nonrelativistic electrons
40
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The differential interaction cross section
  therefore also increases in proportion to b, as
  does the number of photons generated in a given
  annulus
• However, the strength of the interaction, hence
  the quantum energy of the x-rays produced,
  certainly decreases as b increases
• If we assume that h? ? 1/b, then the number of
  photons and their quantum energy will be
  reciprocal, and a flat radiant-energy spectrum
  such as that in the diagram will result

41
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• Thick targets can be simplistically regarded as a
  stack of imaginary thin target foils, adequate in
  aggregate depth to stop the electron beam
• As the beam passes through successive foils, the
  electrons lose their kinetic energy gradually by
  many small collision interactions
• Radiative losses are negligible as a mechanism
  for reducing the beam energy for T0 ltlt m0c2

42
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The foils in the stack need not all be taken to
  be the same thickness, but instead are assumed to
  become progressively thinner with increasing
  depth, so that each one reduces the beam energy
  by the same amount
• The collision stopping power increases
  approximately as 1/T for decreasing energies
  therefore the foil thicknesses must be
  successively decreased in proportion to T to
  maintain constant energy expenditure in each one
  through collision interactions

43
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The amount of energy spent by the electron beam
  in x-ray production per foil must therefore
  decrease with depth in proportion to the foil
  thickness, since the Sommerfeld nonrelativistic
  radiative stopping power is independent of T
• These considerations explain the shape of the
  lower curve in the following figure

44
Bremsstrahlung radiant-energy spectrum from (a) a
thin target, (b) a thick target irradiated by
electrons of incident energy T0 ltlt m0c2
45
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The electron beam enters the first target foil at
  kinetic energy T0, and generates an amount of
  x-ray energy proportional to the area of the
  shaded rectangular block
• In passing through the foil it loses energy ?T,
  nearly all through collision interactions, and
  then enters the second foil with energy T1 T0 -
  ?T

46
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The amount of x-ray energy it generates in the
  second foil is represented by the area of the
  second block, which is drawn to have the same
  height as the first, but with a different maximum
  photon energy h?max T1
• Thus the x-ray energy emitted from the second
  foil is T1/T0 times that from the first, and so
  on for subsequent foils

47
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The array of rectangular areas (representing the
  x-ray outputs of all the imaginary individual
  foils comprising the thick target) can be fitted
  by a triangular envelope called the Kramers
  spectrum, having the formula
• where R?(h?) is the differential
  radiant-energy spectral distribution of
  bremsstrahlung generated in the thick target of
  atomic number Z, typically in J/MeV h?max T0
  is the maximum photon energy (MeV) C is a
  constant of proportionality and R?(h?)
  CNeZ(h?)max for h? 0

48
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• The area under the triangle represents the total
  radiant energy of the unfiltered bremsstrahlung,
  and can be seen to have the value
• The constant C/2 has a value around 1 ? 10-3
  MeV-1 when R and T0 are both expressed in MeV

49
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• It is helpful in interpreting these equations to
  observe the graphical effect of changing the
  parameters
• The first diagram shows the effect of doubling Ne
  or Z
• The second diagram shows the effect of doubling
  T0 h?max

50
Effect of doubling Ne or Z on the unfiltered
bremsstrahlung x-ray spectrum
51
Effect of doubling T0 h?max on the unfiltered
bremsstrahlung x-ray spectrum
52
Unfiltered Energy SpectrumT0 ltlt m0c2 (cont.)

• These simple triangular spectra are never
  observed experimentally, for two reasons
• Firstly, the fluorescence x-ray lines are
  superimposed, assuming the electron energy
  exceeds the shell binding energy
• Moreover, the lower-energy photons are
  preferentially removed by the photoelectric-effect
  interactions within the target material itself,
  the exit window of the x-ray tube, and such
  additional filters as may be added

53
Unfiltered Bremsstrahlung Energy Spectrum T0 ?
m0c2

• For relativistic electrons the generation of
  bremsstrahlung can no longer be adequately
  described by the Sommerfeld equation, and the
  more general Bethe-Heitler formula
• applies

54
Unfiltered Energy SpectrumT0 ? m0c2 (cont.)

• The differential cross section d?r for the
  emission of a photon with quantum energy between
  h? and h? d(h?), by an electron of kinetic
  energy T, is given in cm2/atom by
• Hence the photon output spectrum has the form

55
Unfiltered Energy SpectrumT0 ? m0c2 (cont.)

• The radiant energy spectrum is proportional to
  Br, which is a gradually decreasing dimensionless
  function having a value around 20 at h?/T 0,
  and 0 at h?/T 1
• The curve shape between these limits depends on T
• The following figure gives the energy-flux
  density spectrum (which has the same shape as the
  radiant energy spectrum) for a moderately thick
  (1.5 mm) tungsten wire target struck by 11.3-MeV
  electrons

56
Bremsstrahlung intensity (energy-flux density)
spectrum in the 0 direction for 11.3-MeV
electrons on a 1.5-mm tungsten target
57
Unfiltered Energy SpectrumT0 ? m0c2 (cont.)

• Also shown are the Bethe-Heitler theoretical
  spectra for a thin target and for 0.25- and
  0.50-mm targets, each corrected for photon
  attenuation in the target and windows
• This accounts for the low-energy decrease in all
  the curves, which would otherwise extrapolate to
  the h? 0 axis along more or less straight lines
  with slopes established by the curve trends above
  3 MeV

58
Unfiltered Energy SpectrumT0 ? m0c2 (cont.)

• It can be seen by comparing this figure with that
  for very low energy electrons that there is less
  difference between thick- and thin-target spectra
  at high generating energies such as 11.3 MeV than
  at low energies (T0 ? m0c2)
• Moreover, they both are bowed upward, in contrast
  to the straight line spectrum for low T0 and
  thick target
• This upward bowing comes from the function Br of
  Heitler