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Crystal Structures & Properties, 5. Single & Polycrystalline…
Crystal Structures & Properties
Crystallographic Points
Coordinates: \((q,r,s)\)
Position along axes: \(x=qa,\ y=rb,\ z=sc\)
\(a,b,c\) \= lattice parameters
\(q,r,s\) \= fractional coordinates
Crystallographic Directions：
Direction indices: \([uvw]\)
Parallel directions → same direction indices
Equivalent directions → same atomic spacing along the direction
Family of equivalent directions: \(\langle100\rangle\)
Method：
Identify starting and ending points.
Calculate endpoint − starting point.
Convert to lattice units.
Clear fractions; reduce to smallest integers.
Write as \([uvw]\).
Important notation: \([uvw]\) \= one direction; \(\langle uvw\rangle\) \= family of equivalent directions.
3. Crystallographic Planes — Miller Indices (p. 63–70)
Plane notation: \((hkl)\)
Hexagonal notation: \((hkil)\)
Method
Find the origin.
Determine axis intercepts in lattice units.
Take reciprocals.
Clear fractions to obtain smallest integers.
Write indices in parentheses.
Rules
Parallel to an axis → intercept \(=\infty\), reciprocal \(=0\).
If plane passes through origin → choose a parallel plane or new origin.
Never confuse \((hkl)\) (plane) with \([uvw]\) (direction).
Examples
Intercepts \((1,\infty,1)\) → \((101)\)
Intercepts \((1/2,1/2,1)\) → \((221)\)
Intercepts \((1,1,\infty)\) → \((110)\)
Linear & Planar Density (p. 74–76)
Linear Density (LD)
\[ LD=\frac{N_L}{L} \]
\(N_L\) \= atoms centred on the direction
\(L\) \= direction length
Unit: atoms per unit length
Example from lecture:
\[ LD_{110}=\frac{2}{4R}=\frac{1}{2R} \]
Planar Density (PD)
\[ PD=\frac{N_P}{A_P} \]
\(N_P\) \= atoms centred on the plane
\(A_P\) \= planar area
Unit: atoms per unit area, e.g. \(\mathrm{nm}^{-2}\)
Example from lecture:
\[ PD_{110}=\frac{2}{8R^2\sqrt2} =\frac{1}{4R^2\sqrt2} \]
Importance: LD and PD affect the slip process and deformation behaviour.
Principle
X-rays \= high-energy electromagnetic radiation.
Wavelength comparable to atomic spacing.
Constructive interference → diffraction peaks.
Destructive interference → reduced intensity.
textXRD identifies crystal structures, phases and composition.
Bragg's Law
\[ n\lambda=2d_{hkl}\sin\theta \]
\[ d_{hkl}=\frac{n\lambda}{2\sin\theta} \]
Rearranged formulas:
\[ \theta=\sin^{-1}\left(\frac{n\lambda}{2d_{hkl}}\right) \]
\[ \text{Diffraction angle}=2\theta \]
\(n\) \= order of reflection
\(\lambda\) \= X-ray wavelength
\(d_{hkl}\) \= interplanar spacing
\(\theta\) \= Bragg angle
Cubic Crystal: Interplanar Spacing
\[ d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}} \]
\(a\) \= lattice parameter
\((hkl)\) \= Miller indices
BCC & FCC Reflection Rules
BCC: diffraction occurs when \(h+k+l\) is even; odd sum → no diffraction.
FCC: diffraction occurs when \(h,k,l\) are all even or all odd; mixed parity → no diffraction.
Identify Cubic Structure Using XRD
\[ \sin^2\theta\propto h^2+k^2+l^2 \]
BCC allowed sequence: \(2,4,6,8,\ldots\)
FCC allowed sequence: \(3,4,8,11,\ldots\)
Compare ratios of \(\sin^2\theta\) for successive peaks with the allowed sequences.
Lattice Parameter from XRD
Starting from Bragg's law and the cubic spacing formula:
\[ \sin^2\theta=\frac{n^2\lambda^2}{4a^2} (h^2+k^2+l^2) \]
For first-order diffraction, \(n=1\):
\[ a=\frac{\lambda\sqrt{h^2+k^2+l^2}} {2\sin\theta} \]
Powder Diffraction
Randomly oriented powder/polycrystalline sample.
Correctly oriented planes produce diffraction peaks.
Peak positions and intensities help identify phases.
X-ray Diffraction (XRD)
5. Single & Polycrystalline Materials
-Single crystal: Continuous, uninterrupted crystal orientation.
-Polycrystalline: Many crystals (grains) separated by grain boundaries.
-Solidification: nucleation → grain growth → grain boundaries form.
Isotropy vs Anisotropy
-Isotropic: properties independent of direction.
-Anisotropic: properties vary with direction.
-Single crystals → generally anisotropic.
-Randomly oriented polycrystals → often isotropic.
-Textured polycrystals → may be anisotropic.
Application: Crystal orientation influences mechanical and electrical properties.