X-Ray Physics

1. X-Ray Physics Assumptions: Matter is composed of discrete particles (i.e. electrons, nucleus) Distance between particles >> particle size X-ray photons are small particles Interact with body in binomial process Pass through body with probability p Interact with body with probability 1-p (Absorption or scatter) No scatter photons for now (i.e. receive photons at original energy or not at all.

2. N |∆x|N + ∆N The number of interactions (removals)  number of x-ray photons and ∆x ∆N = -µN∆x µ = linear attenuation coefficient (units cm-1)

3. Id (x,y) = ∫ I0 () exp [ -∫ u (x,y,z,) dz] d  Integrate over  and depth. If a single energy I0() = I0 ( - o), If homogeneous material, then µ (x,y,z, 0) = µ0 Id (x,y) = I0 e -µ0l

4. X-ray Source Accelerate electrons towards anode. Braking of electrons by nuclei creates x-ray photons Typically Tungsten Target High melting point High atomic number Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience.

5. Thin Target X-ray Formation There are different interactions creating X-ray photons between the accelerated electrons and the target. Maximum energy is created when an electron gives all of its energy, 0 , to one photon. Or, the electron can produce n photons, each with energy 0/n. Or it can produce a number of events in between. Interestingly, this process creates a relatively uniform spectrum. Power output is proportional to 02 Intensity = nh 0 Photon energy spectrum

6. Thick Target X-ray Formation We can model target as a series of thin targets. Electrons successively loses energy as they moves deeper into the target. Gun  X-rays Each layer produces a flat energy spectrum with decreasing peak energy level. Relative Intensity 0

7. Thick Target X-ray Formation In the limit as the thin target planes get thinner, a wedge shaped energy profile is generated. Relative Intensity 0 Again, 0 is the energy of the accelerated electrons.

8. Thick Target X-ray Formation Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience. ( Lower energy photons are absorbed with aluminum to block radiation that will be absorbed by surface of body and won’t contribute to image. The photoelectric effect(details coming in attenuation section) will create significant spikes of energy when accelerated electrons collide with tightly bound electrons, usually in the K shell.

9. How do we describe attenuation of X-rays by body? µ = f(Z, ) Attenuation a function of atomic number Z and energy  Solving the differential equation suggested by the second slide of this lecture, dN = -µNdx Ninx Nout µ Noutx ∫ dN/N = -µ ∫ dx Nin 0 ln (Nout/Nin) = -µx Nout = Nin e-µx

10. If material attenuation varies in x, we can write attenuation as u(x) Nout = Nin e -∫µ(x) dx Io photons/cm2 (µ (x,y,z)) Id (x,y) = I0 exp [ -∫ µ(x,y,z) dz] Assume: perfectly collimated beam ( for now), perfect detector no loss of resolution Actually recall that attenuation is also a function of energy , µ = µ(x,y,z, ). We will often assume a single energy source, I0 = I0(). After analyzing a single energy, we can add the effects of other energies by superposition. Detector Plane Id (x,y)

11. Diagnostic Range 50 keV < E < 150 keV  ≈ 0.5% Rotate anode to prevent melting What parameters do we have to play with? • Current • Units are in mA • Time • Units · sec 3. Energy ( keV)

12. Notation  I0 I = I0 e - l l Often to simplify discussion in the book or problems on homework, the intensity transmission, t, will be given for an object instead of the attenuation coefficient  t = I/Io = e-µl

13. Mass Attenuation Coefficient Since mass is providing the attenuation, we will consider the linear attenuation coefficient, µ, as normalized to the density of the object first. This is termed the mass attenuation coefficient. µ/p cm2/gm We simply remultiply by the density to return to the linear attenuation coefficient. For example: t = e- (µ/p)pl Mixture µ/p = (µ1/p1) w1 + (µ2/p2) w2 + … w0 = fraction weight of each element

14. Mechanisms of Interaction 1. Coherent scatter or Rayleigh (Small significance) 2. Photoelectric absorption 3. Compton Scattering – Most serious significance

15. Physical Basis of Attenuation Coefficient Coherent Scattering - Rayleigh • • Coherent scattering varies over diagnostic energy range as: • µ/p  1/2 •• •• • • •  

16. Photoelectric Effect Andrew Webb, Introduction to Biomedical Imaging, 2003, Wiley-Interscience. ( Photoelectric effect varies over diagnostic energy range as: ln /p   1 p 3 ln  K-edge

17. Photoelectric Effect Longest electron range 0.03 cm Fluorescent radiation example: Calcium 4 keV Too low to be of interest. Quickly absorbed Items introduced to the body: Ba, Iodine have K-lines close to region of diagnostic interest.

18. We can use K-edge to dramatically increase absorption in areas where material is injected, ingested, etc. Photoelectric linear attenuation varies by Z4/ 3 ln /p  K edge µ/p  1/ 3