1. Solid State Physics
2. X-ray Diffraction
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2. Diffraction
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3. Diffraction
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4. Diffraction
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5. Diffraction using Light
Diffraction Grating
One Slit
Two Slits
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6. Diffraction
 The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow.
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7. Diffraction for Crystals
Photons
Electrons
Neutrons
Diffraction techniques exploit the scattering of radiation from large numbers of sites. We will concentrate on scattering from atoms, groups of atoms and molecules, mainly in crystals.
There are various diffraction techniques currently employed which result in diffraction patterns. These patterns are records of the diffracted beams produced.
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8. What is This Diffraction?
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9. Bragg Law
William
Lawrence
Bragg
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10. The characteristic lines involve the n = 1 or K shell electrons in a metal. If an electron with the appropriate energy strikes the metal target one of the K shell electrons is knocked out. Then, as an electron in an outer shell drops down, a x-ray photon is emitted.
Mo 0.07 nm
Cu 0.15 nm
Co 0.18 nm
Cr 0.23 nm
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11. Monochromatic Radiation
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12. Diffractometer
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14. Nuts and Bolts The Bragg law gives us something easy to use,
To determine the relationship between diffraction
Angle and planar spacing (which we already know
Is related to the Miller indices).
But…
We need a deeper analysis to determine the
Scattering intensity from a basis of atoms.
What we want is a set of plane waves (the incident radiation) that matches the periodicity of the lattice being studied. If we find it, we’ll see that it matches Bragg’s condition.
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15. Reciprocal Lattices Simple Cubic Lattice
The volume of the simple cubic, primitive cell is a3.
The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a.
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16. Reciprocal Lattices BCC Lattice
The reciprocal lattice is represented by the primitive vectors of an FCC lattice.
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17. Reciprocal Lattices FCC Lattice
The reciprocal lattice is represented by the primitive vectors of an BCC lattice.
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18. Drawing Brillouin Zones
Wigner–Seitz cell
The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors.
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19. Drawing Brillouin Zones
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20. Back to Diffraction Theorem The set of reciprocal lattice vectors
Diffraction is related to the electron density.
Therefore, we have a...
Theorem
The set of reciprocal lattice vectors
determines the possible x-ray reflections.
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21. The difference in phase angle is
The difference in path length of the of the incident wave at the points O and r is
The difference in phase angle is
For the diffracted wave, the phase difference is
So, the total difference in phase angle is
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22. Diffraction Conditions
Since the amplitude of the wave scattered from a volume element is proportional to the local electron density, the total amplitude in the direction k  is
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23. Diffraction Conditions
When we introduce the Fourier components for the electron density as before, we get
Constructive Interference
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24. Diffraction Conditions
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25. Diffraction Conditions
For a crystal of N cells, we can write down
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26. Diffraction Conditions
The structure factor can now be written as integrals over s atoms of a cell.
Atomic form
factor
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27. Diffraction Conditions
Let
Then, for an given h k l reflection
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28. Diffraction Conditions
For a BCC lattice, the basis has identical atoms at and
The structure factor for this basis is
S is zero when the exponential is i × (odd integer) and S = 2f when h + k + l is even.
So, the diffraction pattern will not contain lines for (100), (300), (111), or (221).
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30. Diffraction Conditions
For an FCC lattice, the basis has identical atoms at
The structure factor for this basis is
S = 4f when hkl are all even or all odd.
S = 0 when one of hkl is either even or odd.
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31. KCl
KBr
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32. Structure Determination
Simple
Cubic
When combined with the Bragg law:
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33. X-ray powder pattern determined using Cu K radiation,  = 1.542 Å
q (degrees)
sin2 q
ratios
hkl
11.44
0.0394 1
100
16.28
0.0786 2
110
20.13
0.1184 3
111
23.38
0.1575 4
200
26.33
0.1967 5
210
29.07
0.2361 6
211
34.14
0.3151 8
220
36.53
0.3543 9
300, 221
38.88
0.3940 10
310
X-ray powder pattern determined using Cu K radiation,  = Å
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34. Structure Determination (310)
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