chapter. 2 physical methods for characterizing...
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Chapter. 2 Physical Methods for Characterizing SolidsChapter. 2 Physical Methods for Characterizing Solids
2.1 Introduction
▪ Single Crystal X-ray diffraction: precise atomic positions, bond angles, bond lengths
long range ordered str.
: weakness → less suited on the str. positions of defects
→ difficult to grow single crystal
▪ powder X-ray diffraction - most commonly used
- phase purity analysis & assessing
- str. determination
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2. 2 X2. 2 X--ray Diffractionray Diffraction
2.2.1 Generation of X-rays
▪ Discovery: 1895, German Physist, William Rötgen → 1901, Noble Prize in Physics
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Wilhelm Conrad Röntgen (1845-1923)
A famous photo of his wife’s hand appeared in Nature and Science
“Eine Neue Art von Strahlen” (“On a New Kind of Rays”)
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▪ Nature of X-ray: X-rays are electromagnetic waves (EW)
: Every time a charge accelerate or decelerates, EWs are generated.
2. 2 X2. 2 X--ray Diffractionray Diffraction
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▪ white X-rays (origin):cathode (-) anode (+)
accelerating voltage
e-
2. 2 X2. 2 X--ray Diffractionray Diffraction
Ek = 1/2mv2 ≡ EE = eV
(kinetic and electro static E)
Eph = hcv = hc/λ
(E of a photon)
accelerating voltage
hc/λ = eV → λ = hc/eV
m = e- mass
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λ = (12.4 x 10-7)m/V
v = e- velocitye = e- charge (1.6 x 10-19 C)V = accelerating voltageh = Planck’s constant (6.625 x 10-34 J/sec) c = velocity of light in vacuum (2.998 x 108 m/sec)ν = frequencyλ = wavelength (λ = ν-1)
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▪ an angström: 1Å = 10-10 m
▪ λ = (12.4 x 10-7)m/v establishes the short wavelength limit (SWL)
- e.g.) V = 1000V; λSWL = 12400/1000 = 12.4 Å
2. 2 X2. 2 X--ray Diffractionray Diffraction
V = 10000V; λSWL = 12400/1000 = 1.24 Å
(L)
(M)
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▪ X-ray generation: ‘white-radiation’ + Kα, Kβ
: Kα, Kβ’s wavelength – characteristic of the anode metal (e.g. Cu, Mo)
: bombarding → knock-out e- @ K shell (n=1) → vacancy → filled by descending
e- from L shell (n=2): Kα or from M shell (n=3): Kβ
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2. 2 X2. 2 X--ray Diffractionray Diffraction
▪ X-ray generation: Kα, Kβ’s wavelength – characteristic of the anode metal (e.g. Cu, Mo)
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2. 2 X2. 2 X--ray Diffractionray Diffraction
▪ monochromatic radiation (single wavelength or a very narrow range of wavelengths) is required.
Kα, is used, but Kβ is not.
screened by a filter made of thin metal foil of the element adjacent (Z-1)
e.g.) Cu : uses Ni
Mo : uses Nb
by graphite (single crystal)
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2. 2 X2. 2 X--ray Diffractionray Diffraction
▪ sealed X-ray tube:
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2. 2 X2. 2 X--ray Diffractionray Diffraction
▪ X-ray generation:
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2. 2 X2. 2 X--ray Diffractionray Diffraction
2.2.2 Diffraction of X-rays
▪ crystalline solids: regular arrays of atoms, ions or molecules w/ interatomic spacing of the order
of 100 pm.
f d ff h l h f h d l h h f hfor diffraction, the wavelengths of the incident light ≈ the spacing of the
grating
▪ X-ray diffraction str. determination: W.H Bragg, W.L. Bragg (father and son)
str. of NaCl, KCl, ZnS, CaF2, CaCO3
: X ray diffraction acts like “reflection” from the planes of : X-ray diffraction acts like reflection from the planes of
atoms within the crystal at specific orientation
reflection only occurs when the constructive
interference conditions are fulfilled!!
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2. 2 X2. 2 X--ray Diffractionray Diffraction
▪ Bragg‘s Law: relationship among, 1) the diffraction angle (Bragg angle)
2) wavelength
3) interplanar spacing
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∴ nλ = 2dhklsinθhkl
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2. 3 Powder Diffraction2. 3 Powder Diffraction
2.3.1 Powder Diffraction Patterns
▪ crystallites: a finely ground crystalline powder
: randomly oriented to one another in powder samples
diff i h h lli i d h l f lfill h B - diffraction occurs when the crystallite are oriented at the correct angle to fulfill the Bragg
calculation
- 2θ: angle b/w incident & diffracted beam
- reflections lie on the surface of core whose semi-apex angles are equal to the deflection angle
2θ (Fig. 2.4(a))
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▪ Debye-Scherrer methods: a strip of film wrapped around the inside of a X-ray camera(Fig.2.4(b))
w/ a hole)
: a sample is rotated for more reflections into the diffracting condition
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2. 3 Powder Diffraction2. 3 Powder Diffraction
cont’d
▪ Debye-Scherrer methods: cores were recorded as arcs on the film
using radius of the camera
distance along the film from the center
2θ, dhkl can be calculated.
▪ recent method: automated diffractometer (Fig.2.5(a))
using CCD detector or scintillation
angle and intensities are recorded (Fig. 2.5(b))
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▪ ‘indexing the reflection’: assigning the corrected hkl index to each reflection
detecting which planes are responsible for each reflection
▪ possible for simple compound w/ a high symmetry, but extremely difficult for larger & less
symmetrical system
2.3.12.3.1--2. Reciprocal Lattice2. Reciprocal Lattice
▪ Introduced by P. Ewald in 1921
▪ Let a, b, c be the elementary translations (vectors) of a lattice (i.e. a directional lattice)
▪ A second lattice, reciprocal to the first one, is defined by introducing the translations a*, b*,
c*, which satisfy the following conditions:
a = 1/V(b x c)
b = 1/V(a x c)
a*·a = 1
b*·b = 1
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b 1/V(a x c)
c = 1/V(a x b)
b b = 1
c*·c = 1
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2.3.12.3.1--3.3. Properties of Reciprocal LatticeProperties of Reciprocal Lattice
▪ the family of planes (hkl) w/ interplanar distance d becomes a vector in reciprocal space
▪ d* = 1/d, and it is perpendicular to the corresponding (hkl) planes
▪ in an orthogonal lattice,,
a* = 1/a and it is perpendicular to bcb* = 1/b and it is perpendicular to acc* = 1/c and it is perpendicular to ab
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2.3.12.3.1--4.4. Reciprocal Lattice and Braggs’ LawReciprocal Lattice and Braggs’ Law
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2.3.12.3.1--5.5. Reciprocal lattice and Ewald’s SphereReciprocal lattice and Ewald’s Sphere
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2.3.12.3.1--6.6. Stationary vs. Rotating Crystal TechniqueStationary vs. Rotating Crystal Technique
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2.3.12.3.1--6.6. Rotating Crystal TechniqueRotating Crystal Technique
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2.3.12.3.1--7.7. Origin of Powder Diffraction PatternOrigin of Powder Diffraction Pattern
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2.3.12.3.1--8.8. Powder Diffraction Pattern of CuPowder Diffraction Pattern of Cu
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2.3.12.3.1--9.9. Powder Diffraction Pattern of LaBPowder Diffraction Pattern of LaB66
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2. 3. 2 Absences due to lattice centering2. 3. 2 Absences due to lattice centering
▪ For a primitive cubic system: reflection w/ the smallest Bragg angle → the largest dhkl spacing
e.g.) 100 plane → largest separation → reflection
010, 001 at the same position (∵ a = b = c)
dhkl = a/(√h2 + k2 + l2)
w/ Bragg equation,
λ = 2asinθhkl/(√h2 + k2 + l2)
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all integer #
h2 + k2 + l2 ↑ → 2θ ↑
2. 3. 2 Absences due to lattice centering2. 3. 2 Absences due to lattice centering
▪ Table 2.1
▪ body-centered, face-centered cubic: different line from the primitive system
∵ centering → destructive interference → extra missing reflection
known as systematic absences
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▪ 200 plane in the F-centered cubic unit cell
cell dimension: a → spacing: a/2
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