Crystal Structure, Reciprocal Lattice, and Lattice Dynamics
This document details the fundamental physics of crystalline solids, from their periodic atomic arrangement to the dynamic vibrations that govern their thermal properties. It covers the mathematical framework of the reciprocal lattice and derives the essential equations for diffraction (Bragg's Law) and phonon dispersion.
1. Periodic Arrangement in Solids
A crystalline solid is characterized by a periodic arrangement of atoms in space. Let \(\rho(\mathbf{r})\) denote the atomic (or electron) density. Periodicity implies:
where \(\mathbf{R}\) is a lattice translation vector: \[ \mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, \quad n_i \in \mathbb{Z}. \] This translational symmetry is the fundamental origin of diffraction phenomena in crystals.
2. Why Scattering Experiments Reveal Crystal Structure
Crystal structure is probed using wave scattering:
- X-ray scattering: Sensitive to electron density.
- Neutron scattering: Sensitive to nuclei and magnetic moments.
- Electron scattering: Strong interaction with charge.
All these techniques rely on the interference of waves scattered from periodically arranged atoms.
3. Scattering Geometry and Wave Description
Incident and Scattered Waves
Incident plane wave: \[ \psi_{\text{in}} = e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} \] Scattered wave observed far from the crystal: \[ \psi_{\text{out}} = \frac{e^{i(\mathbf{k}'\cdot\mathbf{r} - \omega t)}}{R} \]
Define the scattering vector: \[ \mathbf{q} = \mathbf{k}' - \mathbf{k} \]
4. Scattering Amplitude and Intensity
The scattering amplitude is proportional to the Fourier transform of the density: \[ A(\mathbf{q}) \propto \int \rho(\mathbf{r}) e^{i\mathbf{q}\cdot\mathbf{r}} \, d^3r \]
The measured intensity is: \[ I(\mathbf{q}) \propto |A(\mathbf{q})|^2 = \left| \int \rho(\mathbf{r}) e^{i\mathbf{q}\cdot\mathbf{r}} \, d^3r \right|^2 \]
Physical meaning: diffraction directly measures the squared modulus of the Fourier transform of the crystal density.
5. Fourier Transform of a Perfect Crystal
1D Lattice
Atomic density: \[ \rho(x) = \sum_{n} \delta(x - na) \] Fourier transform: \[ \tilde{\rho}(k) = \sum_n e^{ikna} \] This sum is non-zero only when: \[ k = \frac{2\pi m}{a}, \quad m \in \mathbb{Z} \] Thus, scattering occurs only at discrete reciprocal lattice points.
3D Crystal
Density: \[ \rho(\mathbf{r}) = \sum_{\mathbf{R}} \delta(\mathbf{r} - \mathbf{R}) \] Fourier transform: \[ \tilde{\rho}(\mathbf{q}) = \sum_{\mathbf{R}} e^{i\mathbf{q}\cdot\mathbf{R}} \] This is non-zero only if: \[ e^{i\mathbf{q}\cdot\mathbf{R}} = 1 \quad \forall \, \mathbf{R} \]
6. Reciprocal Lattice
Definition
Vectors \(\mathbf{G}\) satisfying: \[ \mathbf{G}\cdot\mathbf{R} = 2\pi m \] form the reciprocal lattice.
Reciprocal Basis Vectors
Given direct lattice vectors \((\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3)\): \[ \mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \] (with cyclic permutations).
Any reciprocal lattice vector: \[ \mathbf{G} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3 \] Note: Reciprocal basis vectors are not unique; physical diffraction conditions are invariant.
7. Intensity and Laue Condition
The scattered intensity becomes: \[ I(\mathbf{q}) \propto \left| \sum_n e^{i(\mathbf{q}-\mathbf{G})\cdot\mathbf{R}_n} \right|^2 \] Hence: \[ I \neq 0 \quad \text{only if} \quad \mathbf{q} = \mathbf{G} \] This gives the Laue condition:
8. Miller Indices
If a plane cuts the crystallographic axes at: \[ \left(\frac{a}{h}, \frac{a}{k}, \frac{a}{l}\right) \] it is labelled as the \((hkl)\) plane.
The plane normal is parallel to \(\mathbf{G}_{hkl}\), and the interplanar spacing is: \[ d_{hkl} = \frac{2\pi}{|\mathbf{G}_{hkl}|} \]
9. Bragg’s Law from Reciprocal Space
From the Laue condition: \[ |\mathbf{k}' - \mathbf{k}| = |\mathbf{G}| \] Since \(|\mathbf{k}| = |\mathbf{k}'| = 2\pi/\lambda\): \[ 2k \sin\theta = \frac{2\pi}{d} \] Therefore:
10. Atomic Motion and Interatomic Potential
Atoms are not rigidly fixed; they vibrate about equilibrium positions. The interatomic potential is anharmonic, leading to:
- Thermal expansion
- Finite heat capacity
- Phonon interactions
11. One-Dimensional Monoatomic Chain
Consider atoms of mass \(m\), spacing \(a\), spring constant \(K\).
Equation of Motion
\[ m \ddot{u}_n = K(u_{n+1} - 2u_n + u_{n-1}) \]
Trial Solution
\[ u_n = A e^{i(\omega t - k n a)} \] Substitution gives: \[ m \omega^2 = K(e^{ika} + e^{-ika} - 2) \]
Dispersion Relation
12. Physical Limits
Long Wavelength Limit (\(ka \ll 1\))
\[ \omega \approx v_s k, \quad v_s = a \sqrt{\frac{K}{m}} \] (Sound waves)
Zone Boundary (\(k = \pi/a\))
Adjacent atoms oscillate out of phase.
13. Periodicity in Reciprocal Space
\[ e^{ikna} = e^{i(k + 2\pi/a)na} \] Thus:
This explains Brillouin zones and band folding.