Introduction to Solid-State Physics – Lattices & Course Roadmap

Orientation and Scope of Solid-State Physics

• Solid-state physics focuses on atomistic origins of macroscopic properties; macroscopic extremes and sub-nuclear scales are largely excluded.
• Nuclear physics is treated as an external input: nuclei are considered rigid point charges.
• Core theoretical tools:
  • Quantum mechanics (electronic structure, bonding, Bloch states)
  • Classical mechanics & electromagnetism (vibrations, Maxwell response)
• Basic particles in the theory: nuclei (ions) + electrons.

Course Roadmap (as announced)

• Introduction (completed).
• Lattices & crystals (current topic).
• Reciprocal lattice & X-ray diffraction (derive reciprocal concept from diffraction data).
• Refresher on atomic physics & chemistry (valence electrons, shell structure).
• Chemical bonding: covalent, ionic, metallic, hydrogen, van-der-Waals; emphasis on continuous spectrum of bonding.
• Cohesive (bonding) energy & interatomic potentials.
• Vibrations:
  • Molecular normal modes (finite systems).
  • Phonons in infinite lattices.
• Electronic states in solids:
  • Tight-binding model (localized, atomic-like starting point).
  • Quasi-free (nearly-free) electron model (weak-potential perturbation of free electrons).
  • General principles: Bloch’s theorem, Brillouin zones, band structure.
• Consequences & applications:
  • Charge transport & conductivity.
  • Optical properties.
• If time permits: semiconductors, magnetism, other selected topics.

Lattices vs. Crystals: Conceptual Foundations

• Experimental fact: many solids exhibit long-range translational order (e.g.
  X-ray diffraction patterns).
• Theoretical idealization: crystal assumed infinite (extends from -\infty to +\infty); real materials finite → surfaces, grains break perfect translational symmetry but are treated later as perturbations.
• Goal of current lectures: develop precise mathematical language for periodic order.

Bravais Lattice: Formal Definition and Key Properties

• Definition (3-D):
  \mathcal{L}={\,\mathbf R=n1\mathbf a1+n2\mathbf a2+n3\mathbf a3\;|\;ni\in\mathbb Z\,} where \mathbf a1,\mathbf a2,\mathbf a3 are linearly independent primitive vectors.
• Generalization to d dimensions obvious (need d independent primitive vectors).
• Fundamental properties
  • Infinite set of points; finite clusters cannot satisfy definition.
  • Choice of primitive set is not unique – infinitely many triplets generate the same lattice.
  • Example: 2-D square lattice – green vs. pink primitive vectors both recreate identical point set.
  • Translational closure: if \mathbf R is in \mathcal L then m\mathbf R (any integer m) is also in \mathcal L.
  • Any choice of origin located at a lattice point is equivalent; shifting the origin merely re-labels n_i.
• Primitive cell: parallelepiped (2-D parallelogram) spanned by primitive vectors; contains exactly one lattice point when using the ‘empty-cell’ counting convention.

2-D Bravais Lattice Examples (with defining constraints)

• Square: |\mathbf a1|=|\mathbf a2|,\;\mathbf a1\perp\mathbf a2.
• Rectangular: \mathbf a1\perp\mathbf a2,\;|\mathbf a1|\neq|\mathbf a2|.
• Triangular (sometimes called ‘hexagonal’ in 2-D): |\mathbf a1|=|\mathbf a2|,\;\angle(\mathbf a1,\mathbf a2)=60^{\circ}\;(\text{or }120^{\circ}).
  • Yields densest packing of circles in 2-D.

Non-Uniqueness of Primitive Set illustrated

• For square lattice: \mathbf a1' = \mathbf a1,\; \mathbf a2' = \mathbf a1+\mathbf a_2 equally valid.
• For triangular lattice: choices at 60^{\circ} or 120^{\circ} equally ‘natural’; conventions differ across literature (~60 % vs. 40 % usage).

Non-Bravais Example: The Honeycomb Lattice

• Graphene’s 2-D pattern – visual honeycomb – not a Bravais lattice.
  • Proof via closure property: vector between nearest neighbours is a lattice vector; doubling it lands on an empty site → violates Bravais requirement.
• Necessitates extended concept: lattice plus basis.

Lattice with a Basis (Crystal Structure)

• Definition: provide
  • Bravais lattice \mathcal L generated by primitive \mathbf ai • Finite set of basis vectors {\mathbf b1,\dots,\mathbf b_N} (with N\ge 1).
• Full crystal structure:
  \mathcal C={\,\mathbf R+n1\mathbf a1+n2\mathbf a2+n3\mathbf a3\;|\;ni\in\mathbb Z,\;\mathbf R\in{\mathbf bj}\,}
  Equivalently:
  \mathbf r = n1\mathbf a1+n2\mathbf a2+n3\mathbf a3+\mathbf b_j.
• Special case: N=1,\;\mathbf b_1=\mathbf 0 → original Bravais lattice.

Simple Illustration

• Start from 2-D square Bravais lattice; choose basis \mathbf b1=(\tfrac12,0)a,\;\mathbf b2=(0,\tfrac12)a.
  • Generates checkerboard of points displaced from every Bravais site.
• Changing origin to a point in basis can set one \mathbf b_j=0, simplifying description.

Honeycomb via Basis

• Choose triangular Bravais lattice (primitive \mathbf a1,\mathbf a2).
• Basis: \mathbf b1=\mathbf 0,\;\mathbf b2=\tfrac23\mathbf a1+\tfrac13\mathbf a2 (vector to second sub-lattice).
• Resulting crystal structure reproduces graphene/honeycomb arrangement (two atoms per primitive cell → two-sublattice physics).

Real-World Relevance & Multi-Component Materials

• Materials with multiple atomic species must be described with N\ge 2 basis vectors to distinguish inequivalent sites.
• Basis size spans huge range:
  • Simple solids: N\le 8 typically.
  • Biomolecular crystals: proteins can yield N\sim10^5{-}10^6 (each atom in molecule appears in basis).
• Defects/impurities: single missing atom violates Bravais definition; treated later as perturbations atop ideal lattice.

3-D Bravais Lattices: Common Families (primitive versions only)

• Parameter counting for triclinic cell: 3 edge lengths + 3 inter-vector angles = 6 independent numbers (orientation is irrelevant).
• In literature: structural data tables list minimal independent parameters corresponding to lattice family (e.g., cubic → one lattice constant a; hexagonal → a,c).

Additional Remarks & Study Pointers

• Always specify:
  1. Choice of primitive vectors (lengths & relative angles).
  2. Basis vectors and associated atom types.
  3. Origin convention.
• Bloch’s theorem, Brillouin zones and band theory crucially rely on underlying Bravais periodicity; basis introduces sub-lattice degrees of freedom but not new reciprocal vectors.
• For problem-solving: verify Bravais vs. non-Bravais via “doubling rule” (if \mathbf R in lattice then 2\mathbf R must be) and ability to tile space with a single parallelotope.
• Understanding multiple valid primitive choices useful when switching between real-space and reciprocal-space formalisms.