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Medical Imaging Physics 11 April 29, 2008 Medical Imaging Physics Spring Quarter Week 6-1 X-Rays Davor Balzar [email protected] www.du.edu/~balzar
Medical Imaging Physics 11April 29, 2008
Medical Imaging Physics Spring Quarter
Week 6-1
X-Rays
Davor [email protected]
www.du.edu/~balzar
Medical Imaging Physics 11April 29, 2008
Outline• Quiz• Crystal structure determination• Interaction with Matter, Attenuation• Radiology, mammography…
• Reading assignment:CSG D 16; http://www.sprawls.org/ppmi2/
• HomeworkPostedDue Tuesday, May 6
• Midterm Tuesday, May 6
Medical Imaging Physics 11April 29, 2008
Quiz 2Quiz 2
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140
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
Student Number
Perc
ent S
core
103%
Medical Imaging Physics 11April 29, 2008
Crystal Structure
• Long-range order (periodic arrays of atoms)
• Structure factor from a single unit cell (basis) F and lattice points S:
Basis + Lattice = Crystal structure
SFfc =
Medical Imaging Physics 11April 29, 2008
Scattering Over Lattice Points S
d2≤λλθ ndhkl =sin2
Bragg equation:
Medical Imaging Physics 11April 29, 2008
Crystal Planes
• Orientation of a crystal plane:3 non-collinear points in terms of the lattice constantsAlternatively, integer (Miller) indices are used:
• Intercepts on 3 axes in terms of lattice constants (a,b,c)• Reciprocals are reduced to 3 integers having the same ratio• Result is called the the Miller index of the plane (hkl)
Medical Imaging Physics 11April 29, 2008
Crystal Planes
Medical Imaging Physics 11April 29, 2008
Crystal Planes and Directions
{100}
[uvw]
<100>Directions
[001]
[010][111]
Medical Imaging Physics 11April 29, 2008
Example-Cubic Lattice
λθ =sin2 hkld222 lkh
adhkl++
=
Medical Imaging Physics 11April 29, 2008
Geometric Structure Factor F
• Geometrical (basis) structure factor:
• x,y,z - fractional atom coordinates in the basis (unit cell)• h,k,l – Miller indices of a crystal plane
∑ ++=m
lzkyhxim
mmmefF )(2π
Medical Imaging Physics 11April 29, 2008
Geometric Structure Factor
• bcc latticex1 = y1 = z1 = 0x2 = y2 = z2 = ½
• In general: Different structure -> different reflections missing -> identify it!Different kinds of atoms => more complicated!
]1[ )( lkhiefF +++= π
⎩⎨⎧
=++=++
=integereven2
integerodd0lkhflkh
F missing,...300,111,100
∑ ++=m
lzkyhxim
mmmefF )(2π
Medical Imaging Physics 11April 29, 2008
Cubic Structures
2
222
2
1a
lkhdhkl
++=
Medical Imaging Physics 11April 29, 2008
Experimental Techniques
• Something is fixed, something varied
• “Diffraction of Matter” Web pagehttp://www.matter.org.uk/diffraction/
λθ ndhkl =sin2
Medical Imaging Physics 11April 29, 2008
Single-Crystal Technique
• Laue (polychromatic beam)
Medical Imaging Physics 11April 29, 2008
Powder Technique
• Debye-Scherrer
Medical Imaging Physics 11April 29, 2008
Favorite Powder Technique
• Bragg-Brentano
Medical Imaging Physics 11April 29, 2008
Synchrotron X-Rays
• Bremsstrahlung
Good collimationGood resolutionParallel beam
Medical Imaging Physics 11April 29, 2008
Synchrotrons
E (keV)
Medical Imaging Physics 11April 29, 2008
Crystal-Structure Determination
• Any technique:Positions of diffraction lines (Bragg law => d spacings) Indexing diffraction pattern (guessing Miller indices)
• Lattice parameters• Absent reflections => crystal structure
Intensities of diffraction lines• Atom types and their arrangement in the unit cell
Model• Calculate a diffraction pattern given by the model and compare it to
the experimental patternRefine the model
• Rietveld refinement
λθ ndhkl =sin2
222 lkh
adhkl++
=
Medical Imaging Physics 11April 29, 2008
Experimental
Medical Imaging Physics 11April 29, 2008
Experiment
λθ ndhkl =sin2
Medical Imaging Physics 11April 29, 2008
Data
• Cu Kα λ = 0.15406 nm
hkl 2θ (°) dhkl (nm) a (nm) 43.33 50.55 74.27 90.04 95.33
a = ???(?) nm
HW: Calculate a and its standard uncertainty
Medical Imaging Physics 11April 29, 2008
Next Time
• Interaction with Matter, Attenuation• Radiology, mammography
