Medical Imaging Physics - University of Denvermysite.du.edu/~balzar/Imaging-11.pdf · April 29,...
Transcript of Medical Imaging Physics - University of Denvermysite.du.edu/~balzar/Imaging-11.pdf · April 29,...
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Medical Imaging Physics 11April 29, 2008
Medical Imaging Physics Spring Quarter
Week 6-1
X-Rays
Davor [email protected]
www.du.edu/~balzar
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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
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Medical Imaging Physics 11April 29, 2008
Quiz 2Quiz 2
0
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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%
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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 =
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Medical Imaging Physics 11April 29, 2008
Scattering Over Lattice Points S
d2≤λλθ ndhkl =sin2
Bragg equation:
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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)
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Medical Imaging Physics 11April 29, 2008
Crystal Planes
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Medical Imaging Physics 11April 29, 2008
Crystal Planes and Directions
{100}
[uvw]
<100>Directions
[001]
[010][111]
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Medical Imaging Physics 11April 29, 2008
Example-Cubic Lattice
λθ =sin2 hkld222 lkh
adhkl++
=
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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
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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
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Medical Imaging Physics 11April 29, 2008
Cubic Structures
2
222
2
1a
lkhdhkl
++=
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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
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Medical Imaging Physics 11April 29, 2008
Single-Crystal Technique
• Laue (polychromatic beam)
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Medical Imaging Physics 11April 29, 2008
Powder Technique
• Debye-Scherrer
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Medical Imaging Physics 11April 29, 2008
Favorite Powder Technique
• Bragg-Brentano
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Medical Imaging Physics 11April 29, 2008
Synchrotron X-Rays
• Bremsstrahlung
Good collimationGood resolutionParallel beam
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Medical Imaging Physics 11April 29, 2008
Synchrotrons
E (keV)
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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++
=
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Medical Imaging Physics 11April 29, 2008
Experimental
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Medical Imaging Physics 11April 29, 2008
Experiment
λθ ndhkl =sin2
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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
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Medical Imaging Physics 11April 29, 2008
Next Time
• Interaction with Matter, Attenuation• Radiology, mammography