Medical Imaging Physics 11April 29, 2008

Medical Imaging Physics Spring Quarter

Week 6-1

X-Rays

Davor Balzarbalzar@du.edu

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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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