2. Physical Methods for Characterizing Solids

K 2
L
K1 K
1s
K2
2.2.2 Diffraction of X-rays
Crystalline solids- regular arrays of atoms, ions, or molecules with interatomic distance of 100 pm. The wavelength of incident light has to be of the same order as the spacing. Periodic nature of crystals act as a 3-D grating to light. Max von Laue, 1912
Neutron scattering- used to solve crystal structure, but rarely available. Neutrons have high velocity, thus short wavelength of about 100 pm -suitable for structural diffraction experiments. Scattering process - X-ray: electron, neutrons: nucleus. Scattering factor is proportional to Z, but because of the size of the atoms relative to wavelength of the X-rays, f0 decreases with sinθ/ (the scattering from different parts of the cloud is not always in phase) Neutron scattering does not depend on sinθ/ due to small nucleus radius. Neutron scattering gives true distance between the nuclei, whereas the X-ray values are distorted by the size of the electron cloud, thus are shorter.
sin2θ- 1:2:3:4:5:6:8… primitive cubic system. sin2θ- 3:4:8:11:16… face centered cubic. sin2θ- 1:2:3:4:5:6:8… primitive cubic system.
XRD pattern of fcc structure
Scattered beam by (111) plane experiences destructive interference by the beam out of phase from Na+ ions lying halfway in between Cl- ions decrease in intensity
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X-ray Diffraction Intensity
F(hkl) f iexp{2i (hx j ky j lz j )}
j
F(hkl): structure factor- phase shift and amplitude change in a specific direction according to atomic location in an unit cell. fj: atomic scattering factor-
2.3.4 Uses of Powder X-ray Diffraction
Identification of Unknowns and Phase Purity It is usefully used as a fingerprint method for detecting the presence of a known compound or phase in a product Joint Committee for Powder Diffraction Standards (JCPDS) files. Identification of compounds- qualitatively. Rough check of the purity. Metal substitution.
F(hkl) f iexp{2i (hx j ky j lz j )}[1 exp{i(h k )} exp{i (k l )} exp{i (i h)}]
j1
N 4
F(hkl) 1 exp{i (h k )} exp{i (k l )} exp{i(i h)} 0
radiation.
fcc, NaCl
fcc
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Structure factor, Fhkl- the resultant of the waves scattered by all the atoms in the unit cell, which is dependent on the atomic position and its scattering factor
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2. Physical Methods for Characterizing Solids 2.2 X-ray Diffraction
2.2.1 Generation of X-ray
M
3d 3p 3s
K 1 W filament
anode metal dependent, usually Cu of 0.154 nm 2p3/2 2p1/2 2s
2.3 Powder Diffraction
2.3.1 Powder Diffraction Patterns
A finely ground crystalline power contains a very large number of small crystals, crystallites, which are oriented randomly.
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Crystallite Size
Debye-Scherrer equation,
T
C B cos
usually C = 0.9 B: full width at half maximum (radians)
Following Reaction s and Phase Diagrams
2θ
* Diffraction occurs only when crystal plane is parallel to substrate !
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2.3.2 Absences due to Lattice Centering Primitive cubic system
Largest dhkl spacing smallest Bragg angle. In primitive cubic system, 100 planes have the largest interplannar distance.
SiO2, ZrO2, FeSO4, NaF ZrSiO4
The Rietveld Method When reflections overlap considerably, and it becomes impossible to measure the intensities of individual peaks with any accuracy Rietveld analysis. Interpretation includes line position and line intensities. Assuming each peak has a gaussian shape. Calculation for overall line profile.
Body centered cubic and face centered cubic system
Systematic absences- destructive interference for some reflections.
(200) (100)
hkl: all odd or even sum of indices must be even
unless all h, k, and l are odd or even, F(hkl) = 0 For Body-centered cubic system (bcc), equivalent position: xj, yj, zj; xj +1/2, yj +1/2, zj +1/2 ?
Indexing reflections
Body centered cubic and face centered cubic system
Systematic absences- destructive interference for some reflections.
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amplitude of wave in a specific direction from an atom amplitude of wave in a specific direction from an electron
For face-centered cubic system (fcc), equivalent position: xj, yj, zj; xj +1/2, yj +1/2, zj; xj +1/2, yj, zj +1/2; xj, yj + 1/2, zj +1/2.