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4.3 Solid-State Chemistry: Crystal Systems, Unit Cells, and Band Theory
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1 Chapter 1: Exam Overview, Chemistry Law (RA 10657), and Laboratory Safety
1.1 PRC Chemist Licensure Examination: Format, Schedule, Grading, and Passing Standards 1.2 Republic Act 10657 (Chemistry Profession Act) and Professional Ethics 1.3 Chemical Laboratory Safety, GHS Classification, Hazardous Waste Management, and Quality Assurance
2 Chapter 2: Inorganic Chemistry: Atomic Structure, Periodicity, and Chemical Bonding
2.1 Atoms, Ions, Compounds, Inorganic Nomenclature, and Stoichiometry 2.2 Quantum Numbers, Electronic Configurations, and Periodic Trends 2.3 Chemical Bonding Models: Lewis Structures, VSEPR Theory, and Molecular Polarity 2.4 Molecular Orbital Theory and Magnetochemistry of Diatomic Molecules
3 Chapter 3: Inorganic Chemistry: Coordination Compounds and Transition Metals
3.1 Coordination Chemistry: Nomenclature, Formulas, and Isomerism 3.2 Crystal Field Theory, Ligand Field Theory, d-Orbital Splitting, and Reaction Mechanisms 3.3 Electronic Absorption Spectra, Tanabe-Sugano Diagrams, and Magnetic Properties
4 Chapter 4: Inorganic Chemistry: Main Group, Reactivity, Solid State, Nuclear, and Applied Topics
4.1 Systematic Descriptive Chemistry of Main Group Elements (s- and p-Blocks) 4.2 Reactivity of Inorganic Compounds: Acid-Base Concepts, HSAB, and Redox Diagrams 4.3 Solid-State Chemistry: Crystal Systems, Unit Cells, and Band Theory 4.4 Bioinorganic Chemistry: Metalloproteins, Oxygen Carriers, and Metal Ion Toxicity 4.5 Organometallic Chemistry: 18-Electron Rule, Carbonyl Complexes, and Catalysis 4.6 Nuclear Chemistry: Stability, Decay, Transmutation, and Radioanalytical Methods 4.7 Industrial, Environmental, and Atmospheric Inorganic Chemistry and Advanced Materials
5 Chapter 5: Analytical Chemistry: Measurements, Statistics, Classical Wet Analysis, and Chemical Equilibria
5.1 Analytical Measurements: Apparatus, Units, Concentration, Sampling, and Sample Preparation 5.2 Analytical Data Evaluation: Errors, Uncertainty, Statistics, and Calibration 5.3 Gravimetric Analysis and Precipitation Equilibria ($K_{sp}$) 5.4 Neutralization Titrimetry, Polyprotic Systems, Buffers, and Alpha Fractions 5.5 Complexometric (EDTA), Redox, and Precipitation (Argentometric) Titrimetry
6 Chapter 6: Analytical Chemistry: Electroanalytical Methods and Separations
6.1 Potentiometry, Ion-Selective Electrodes, and Voltammetric Techniques 6.2 Analytical Separations: Solvent Extraction, SPE, Ion Exchange, Precipitation, Masking, and Electrodeposition 6.3 Fundamentals of Chromatography: Theoretical Plates, Resolution, and van Deemter Equation 6.4 Gas Chromatography (GC): Columns, Stationary Phases, Detectors, and Applications 6.5 High-Performance Liquid Chromatography (HPLC): Normal vs. Reversed Phase and Method Development
7 Chapter 7: Analytical Chemistry: Spectroscopic, Mass Spectrometric, and Other Instrumental Methods
7.1 Molecular UV-Vis Spectroscopy and Quantitative Beer-Lambert Law 7.2 Infrared (FTIR) and Raman Spectroscopy: Vibrational Modes and Instrumentation 7.3 Atomic Spectroscopy: Atomic Absorption (AAS) and Inductively Coupled Plasma (ICP-OES/MS) 7.4 Mass Spectrometry: Ionization Sources, Mass Analyzers, and Fragmentation Analysis 7.5 Other Instrumental Methods (XRF, Flame Emission, CE, SFC, Thermal Analysis, Electronics) and Method Selection
8 Chapter 8: Physical Chemistry: Gases, Thermodynamics, and Thermochemistry
8.1 Ideal and Real Gases and the Kinetic Molecular Theory 8.2 First Law of Thermodynamics: Heat, Work, Internal Energy, and Enthalpy 8.3 Second and Third Laws of Thermodynamics: Entropy, Carnot Engine, and Absolute Entropies 8.4 Gibbs and Helmholtz Free Energy, Spontaneity, and Fundamental Equations 8.5 Chemical Potential, Partial Molar Quantities, Fugacity, and Activity
9 Chapter 9: Physical Chemistry: Equilibrium, Phase Equilibria, Solutions, and Electrochemistry
9.1 Chemical Equilibrium: Equilibrium Constants, Le Chatelier's Principle, and the van 't Hoff Equation 9.2 Phase Rule, Phase Diagrams of One- and Two-Component Systems, and Clausius-Clapeyron Equation 9.3 Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes 9.4 Electrochemistry: Galvanic Cells, Standard Potentials, and the Nernst Equation 9.5 Electrolyte Conductance, Ionic Mobility, Transport Numbers, and Concentration Cells
10 Chapter 10: Physical Chemistry: Kinetics, Surface Chemistry, and Quantum Mechanics
10.1 Reaction Rates, Rate Laws, and Integrated Rate Equations 10.2 Temperature Dependence of Rates: Arrhenius Equation, Collision Theory, and Transition State Theory 10.3 Reaction Mechanisms: Steady-State Approximation, Chain Reactions, and Catalysis 10.4 Surface Chemistry, Adsorption Isotherms, Colloids, and Macromolecular Solutions 10.5 Postulates of Quantum Mechanics, Operators, Particle in a Box, and Harmonic Oscillator 10.6 Atomic Spectra, the Hydrogen Atom, Many-Electron Atoms, and Term Symbols
11 Chapter 11: Organic Chemistry: Structure, Stereochemistry, and Hydrocarbons
11.1 Systematic IUPAC Nomenclature and Conformational Analysis of Alkanes and Cycloalkanes 11.2 Stereochemistry: Chirality, Enantiomers, Diastereomers, (R)/(S) Configuration, and Optical Activity 11.3 Alkenes and Alkynes: Addition Reactions, Markovnikov Rule, and Stereospecificity 11.4 Aromaticity: Hückel's Rule, Electrophilic Aromatic Substitution, and Directing Effects
12 Chapter 12: Organic Chemistry: Functional Groups, Mechanisms, Synthesis, and Structure Elucidation
12.1 Nucleophilic Aliphatic Substitution ($S_N1$, $S_N2$) and Elimination ($E_1$, $E_2$) Mechanisms 12.2 Alcohols, Phenols, Ethers, and Epoxides: Structure, Synthesis, and Reactivity 12.3 Carbonyl Compounds I: Aldehydes and Ketones, Nucleophilic Addition, and Oxidation/Reduction 12.4 Carbonyl Compounds II: Carboxylic Acids and Derivatives, Enolate Chemistry, and Condensations 12.5 Amines: Basicity, Synthesis, Reactions, and Arenediazonium Chemistry 12.6 Structure Elucidation: NMR Spectroscopy, Combined Spectral Problems, and Qualitative Organic Tests 12.7 Organic Synthesis Strategy, Protecting Groups, Rearrangements, and Pericyclic Reactions
13 Chapter 13: Biochemistry: Cells, Biomolecules, Enzymes, and Membranes
13.1 Cells, Organelles, Water, and Biological Acid-Base Buffers 13.2 Amino Acids, Peptides, and Hierarchical Protein Structure 13.3 Protein Purification and Biochemical Analysis of Proteins, Carbohydrates, and Lipids 13.4 Enzyme Nomenclature and Classification, Cofactors, Coenzymes, and Vitamin Precursors 13.5 Enzyme Kinetics: Michaelis-Menten Model, Lineweaver-Burk Analysis, and Enzyme Inhibition 13.6 Carbohydrates, Lipids, Biological Membranes, and Membrane Transport
14 Chapter 14: Biochemistry: Metabolism, Bioenergetics, Photosynthesis, and Molecular Genetics
14.1 Carbohydrate Catabolism and Anabolism: Glycolysis, Gluconeogenesis, Glycogen, and Pentose Phosphate Pathway 14.2 Citric Acid Cycle, Mitochondrial Electron Transport, Oxidative Phosphorylation, and $\beta$-Oxidation 14.3 Photosynthesis: Light Reactions, Photophosphorylation, and the Calvin Cycle 14.4 Lipolysis and Lipid Biosynthesis: Fatty Acids, Triacylglycerols, and Cholesterol 14.5 Nitrogen Metabolism: Nitrogen Fixation, Amino Acid Metabolism, and the Urea Cycle 14.6 Nucleic Acids, DNA Replication, Transcription, Translation, and Recombinant DNA Techniques 14.7 Regulation of Gene Expression, Mutation and Mutagens, DNA Repair, Carcinogenesis, and cDNA Cloning
4.3 Solid-State Chemistry: Crystal Systems, Unit Cells, and Band Theory
Key Takeaways Explain This Topic AI
The 7 crystal systems partition into 14 distinct Bravais lattices based on translational symmetry, with the orthorhombic system possessing all four centering types (Primitive P, Body-centered I, Face-centered F, Base-centered C).
Cubic unit cells exhibit distinct geometrical relationships between cell edge a and atomic radius r: Simple Cubic (Z=1, CN=6, a=2r, APF=0.524), Body-Centered Cubic (Z=2, CN=8, a=4r/√3, APF=0.680), and Face-Centered Cubic (Z=4, CN=12, a=2√2r, APF=0.740).
Close-packed sphere lattices achieve maximum theoretical packing efficiency (74.05%) via hexagonal close packing (hcp, ABABAB stacking) or cubic close packing (ccp/fcc, ABCABC stacking); a lattice of N spheres contains N octahedral holes (radius ratio 0.414) and 2N tetrahedral holes (radius ratio 0.225).
Ionic crystal prototypes are determined by stoichiometry and cation-to-anion radius ratios (r+/r-): ZnS zinc blende (CN 4:4, 0.225-0.414), NaCl rock salt (CN 6:6, 0.414-0.732), CsCl (CN 8:8, >0.732), and CaF2 fluorite (CN 8:4).
Band theory distinguishes conductors (partially filled bands or overlapping bands), semiconductors (band gaps Eg between 0.5 and 3.0 eV), and insulators (Eg > 3.0 eV); silicon conductivity is selectively engineered via n-type doping with Group 15 donors (P, As) or p-type doping with Group 13 acceptors (B, Ga).
Last updated: October 2026
4.3 Solid-State Chemistry: Crystal Systems, Unit Cells, and Band Theory
Quick Answer: Solid-state chemistry investigates the three-dimensional crystalline arrangements of atoms, ions, and molecules and their resulting electronic band structures. Key board exam competencies include classifying crystals among the 7 crystal systems and 14 Bravais lattices ; deriving unit cell relationships for simple cubic (sc) , body-centered cubic (bcc) , and face-centered cubic (fcc) lattices; calculating theoretical crystallographic densities ( ρ = Z M / N A V c \rho = ZM / N_A V_c ρ = ZM / N A V c ); determining hole distributions ( N N N octahedral holes and 2 N 2N 2 N tetrahedral holes per close-packed array of N N N spheres); applying radius ratio rules ( r + / r − r_+/r_- r + / r − ) to predict structural prototypes ( N a C l , C s C l , Z n S , C a F 2 NaCl, CsCl, ZnS, CaF_2 N a Cl , C s Cl , Z n S , C a F 2 ); solving Bragg's Law ( n λ = 2 d sin θ n\lambda = 2d\sin\theta nλ = 2 d sin θ ); and distinguishing conductors, insulators, and n n n -type vs. p p p -type semiconductors using electronic band theory.
The 7 Crystal Systems and 14 Bravais Lattices
A crystal lattice is an infinite periodic array of points in space generated by repeating a fundamental parallelepiped termed the unit cell . The geometry of a unit cell is defined by six lattice parameters: three edge lengths ( a , b , c a, b, c a , b , c ) and three interaxial angles ( α \alpha α between b b b and c c c , β \beta β between a a a and c c c , and γ \gamma γ between a a a and b b b ).
All crystalline solids belong to one of 7 crystal systems , which in turn give rise to exactly 14 Bravais lattices when combined with the four centering operations (Primitive P P P , Body-centered I I I , Face-centered F F F , and Base-centered C C C ):
Crystal System Axial Constraints Interaxial Angles Bravais Lattices Centering Types Present
Cubic a = b = c a = b = c a = b = c α = β = γ = 90 ∘ \alpha = \beta = \gamma = 90^\circ α = β = γ = 9 0 ∘ 3 P , I , F P, I, F P , I , F
Tetragonal a = b ≠ c a = b \neq c a = b = c α = β = γ = 90 ∘ \alpha = \beta = \gamma = 90^\circ α = β = γ = 9 0 ∘ 2 P , I P, I P , I
Orthorhombic a ≠ b ≠ c a \neq b \neq c a = b = c α = β = γ = 90 ∘ \alpha = \beta = \gamma = 90^\circ α = β = γ = 9 0 ∘ 4 P , I , F , C P, I, F, C P , I , F , C (all four centering types)
Rhombohedral (Trigonal) a = b = c a = b = c a = b = c α = β = γ ≠ 90 ∘ < 120 ∘ \alpha = \beta = \gamma \neq 90^\circ < 120^\circ α = β = γ = 9 0 ∘ < 12 0 ∘ 1 R R R (or P P P )
Hexagonal a = b ≠ c a = b \neq c a = b = c α = β = 90 ∘ , γ = 120 ∘ \alpha = \beta = 90^\circ, \gamma = 120^\circ α = β = 9 0 ∘ , γ = 12 0 ∘ 1 P P P
Monoclinic a ≠ b ≠ c a \neq b \neq c a = b = c α = γ = 90 ∘ , β ≠ 90 ∘ \alpha = \gamma = 90^\circ, \beta \neq 90^\circ α = γ = 9 0 ∘ , β = 9 0 ∘ 2 P , C P, C P , C
Triclinic a ≠ b ≠ c a \neq b \neq c a = b = c α ≠ β ≠ γ ≠ 90 ∘ \alpha \neq \beta \neq \gamma \neq 90^\circ α = β = γ = 9 0 ∘ 1 P P P (lowest symmetry, no rotational axes)
Exam Note: The orthorhombic system is uniquely versatile because it is the only crystal system that accommodates all four Bravais centering types ( P , I , F , C P, I, F, C P , I , F , C ). Conversely, the triclinic system has the lowest symmetry ( C 1 C_1 C 1 or C i C_i C i ).
Cubic Unit Cell Geometries, Packing Factors, and Density Calculations
In cubic crystals, atoms are modeled as hard, touching spheres of radius r r r . The number of net equivalent atoms per unit cell ( Z Z Z ) is calculated by accounting for fractional site sharing:
Corner atom: shared by 8 adjacent unit cells → 1 / 8 \rightarrow 1/8 → 1/8 atom per corner.
Edge-center atom: shared by 4 unit cells → 1 / 4 \rightarrow 1/4 → 1/4 atom per edge.
Face-center atom: shared by 2 unit cells → 1 / 2 \rightarrow 1/2 → 1/2 atom per face.
Body-center atom: unshared, residing entirely within the cell → 1 \rightarrow 1 → 1 atom.
Lattice Type Net Atoms / Cell ( Z Z Z ) Coordination Number ( C N CN CN ) Sphere Touch Line & Edge Relation Unit Cell Volume ( V c V_c V c ) Atomic Packing Factor (APF) Typical Metal Examples
Simple Cubic (sc) 8 × 1 8 = 1 8 \times \frac{1}{8} = 1 8 × 8 1 = 1 6 Touches along cube edge: a = 2 r a = 2r a = 2 r V c = 8 r 3 V_c = 8r^3 V c = 8 r 3 π 6 ≈ 0.5236 \frac{\pi}{6} \approx 0.5236 6 π ≈ 0.5236 ( 52.4 % 52.4\% 52.4% ) α \alpha α -Polonium (rare)
Body-Centered Cubic (bcc) 8 × 1 8 + 1 = 2 8 \times \frac{1}{8} + 1 = 2 8 × 8 1 + 1 = 2 8 Touches along body diagonal: 3 a = 4 r ⟹ a = 4 r 3 \sqrt{3}a = 4r \implies a = \frac{4r}{\sqrt{3}} 3 a = 4 r ⟹ a = 3 4 r V c = 64 r 3 3 3 V_c = \frac{64r^3}{3\sqrt{3}} V c = 3 3 64 r 3 π 3 8 ≈ 0.6802 \frac{\pi\sqrt{3}}{8} \approx 0.6802 8 π 3 ≈ 0.6802 ( 68.0 % 68.0\% 68.0% ) α − F e , C r , W , M o , N a , K \alpha-Fe, Cr, W, Mo, Na, K α − F e , C r , W , M o , N a , K
Face-Centered Cubic (fcc / ccp) 8 × 1 8 + 6 × 1 2 = 4 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4 8 × 8 1 + 6 × 2 1 = 4 12 Touches along face diagonal: 2 a = 4 r ⟹ a = 2 2 r \sqrt{2}a = 4r \implies a = 2\sqrt{2}r 2 a = 4 r ⟹ a = 2 2 r V c = 16 2 r 3 V_c = 16\sqrt{2}r^3 V c = 16 2 r 3 π 3 2 ≈ 0.7405 \frac{\pi}{3\sqrt{2}} \approx 0.7405 3 2 π ≈ 0.7405 ( 74.0 % 74.0\% 74.0% ) C u , A g , A u , A l , N i , P t , γ − F e Cu, Ag, Au, Al, Ni, Pt, \gamma-Fe C u , A g , A u , A l , N i , Pt , γ − F e
The Atomic Packing Factor (APF) Derivation for FCC
APF = V atoms V cell = Z × 4 3 π r 3 a 3 = 4 × 4 3 π r 3 ( 2 2 r ) 3 = 16 3 π r 3 16 2 r 3 = π 3 2 ≈ 0.7405 ( 74.05 % ) \text{APF} = \frac{V_\text{atoms}}{V_\text{cell}} = \frac{Z \times \frac{4}{3}\pi r^3}{a^3} = \frac{4 \times \frac{4}{3}\pi r^3}{(2\sqrt{2}r)^3} = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2}r^3} = \frac{\pi}{3\sqrt{2}} \approx 0.7405\, (74.05\%) APF = V cell V atoms = a 3 Z × 3 4 π r 3 = ( 2 2 r ) 3 4 × 3 4 π r 3 = 16 2 r 3 3 16 π r 3 = 3 2 π ≈ 0.7405 ( 74.05% )
Worked Crystallographic Density Calculation
Problem: Metallic copper crystallizes in an fcc lattice with an atomic radius r = 1.278 A ˚ = 1.278 × 10 − 8 cm r = 1.278\text{ \AA} = 1.278 \times 10^{-8}\text{ cm} r = 1.278 A ˚ = 1.278 × 1 0 − 8 cm . The molar mass of copper is M = 63.546 g/mol M = 63.546\text{ g/mol} M = 63.546 g/mol . Calculate the theoretical crystallographic density of copper in g/cm 3 \text{g/cm}^3 g/cm 3 .
Solution:
For an fcc unit cell, Z = 4 Z = 4 Z = 4 .
Compute the unit cell edge length a a a : a = 2 2 r = 2 × 1.4142 × ( 1.278 × 10 − 8 cm ) = 3.615 × 10 − 8 cm a = 2\sqrt{2}r = 2 \times 1.4142 \times (1.278 \times 10^{-8}\text{ cm}) = 3.615 \times 10^{-8}\text{ cm} a = 2 2 r = 2 × 1.4142 × ( 1.278 × 1 0 − 8 cm ) = 3.615 × 1 0 − 8 cm
Calculate the unit cell volume V c V_c V c : V c = a 3 = ( 3.615 × 10 − 8 cm ) 3 = 4.724 × 10 − 23 cm 3 V_c = a^3 = (3.615 \times 10^{-8}\text{ cm})^3 = 4.724 \times 10^{-23}\text{ cm}^3 V c = a 3 = ( 3.615 × 1 0 − 8 cm ) 3 = 4.724 × 1 0 − 23 cm 3
Apply the crystallographic density formula: ρ = Z ⋅ M N A ⋅ V c = 4 × 63.546 g/mol ( 6.022 × 10 23 mol − 1 ) × ( 4.724 × 10 − 23 cm 3 ) = 254.184 28.448 = 8.935 g/cm 3 \rho = \frac{Z \cdot M}{N_A \cdot V_c} = \frac{4 \times 63.546\text{ g/mol}}{(6.022 \times 10^{23}\text{ mol}^{-1}) \times (4.724 \times 10^{-23}\text{ cm}^3)} = \frac{254.184}{28.448} = 8.935\text{ g/cm}^3 ρ = N A ⋅ V c Z ⋅ M = ( 6.022 × 1 0 23 mol − 1 ) × ( 4.724 × 1 0 − 23 cm 3 ) 4 × 63.546 g/mol = 28.448 254.184 = 8.935 g/cm 3 This value perfectly matches the experimentally measured density of copper ( 8.94 g/cm 3 8.94\text{ g/cm}^3 8.94 g/cm 3 ).
Close Packing Systems and Interstitial Sites
Close packing of equal-sized spheres in two dimensions produces a hexagonal close-packed layer where each sphere touches 6 neighbors. Stacking these layers produces two distinct close-packed crystal systems with an identical, maximum theoretical packing efficiency of 74.05 % 74.05\% 74.05% and a coordination number of 12 :
Hexagonal Close-Packed (hcp): Stacking sequence follows an alternating A B A B A B … ABABAB\dots A B A B A B … pattern. The third layer spheres lie directly above the first layer spheres. Space group P 6 3 / m m c P6_3/mmc P 6 3 / mm c ; unit cell contains 2 atoms (primitive) or 6 atoms (hexagonal prism). Typical metals: M g , Z n , T i , C o Mg, Zn, Ti, Co M g , Z n , T i , C o .
Cubic Close-Packed (ccp / fcc): Stacking sequence follows a three-layer A B C A B C … ABCABC\dots A BC A BC … pattern. The third layer occupies the alternate set of triangular interstitial depressions, and spheres repeat every fourth layer. Space group F m 3 ˉ m Fm\bar{3}m F m 3 ˉ m . Typical metals: C u , A g , A u , A l Cu, Ag, Au, Al C u , A g , A u , A l .
Interstitial Holes in Close-Packed Lattices
In any close-packed lattice containing N N N spheres , there are exactly:
N N N Octahedral Holes: Formed by six spheres arranged at the vertices of a regular octahedron. In an fcc cell ( N = 4 N = 4 N = 4 ), octahedral holes are located at the center of the unit cell (1 hole) and at the centers of the 12 edges ( 12 × 1 / 4 = 3 12 \times 1/4 = 3 12 × 1/4 = 3 holes), yielding a total of 1 + 3 = 4 1 + 3 = 4 1 + 3 = 4 octahedral holes. Critical radius ratio: r h r s = 2 − 1 ≈ 0.414 \frac{r_h}{r_s} = \sqrt{2} - 1 \approx 0.414 r s r h = 2 − 1 ≈ 0.414 .
2 N 2N 2 N Tetrahedral Holes: Formed by four spheres in a tetrahedral cluster. In an fcc cell ( N = 4 N = 4 N = 4 ), there are 2 × 4 = 8 2 \times 4 = 8 2 × 4 = 8 tetrahedral holes, located entirely inside the unit cell, one within each of the 8 smaller octants along the body diagonals. Critical radius ratio: r h r s = 3 2 − 1 ≈ 0.225 \frac{r_h}{r_s} = \sqrt{\frac{3}{2}} - 1 \approx 0.225 r s r h = 2 3 − 1 ≈ 0.225 .
Ionic Crystal Prototypes and Radius Ratio Rules
In binary ionic crystals ( M x X y M_x X_y M x X y ), structures are idealized as a close-packed or primitive lattice of larger anions ( X n − X^{n-} X n − ) with smaller cations ( M m + M^{m+} M m + ) occupying interstitial holes. The coordination number of the cation is governed by the radius ratio ( r + / r − r_+ / r_- r + / r − ):
Limiting Radius Ratio ( r + / r − r_+/r_- r + / r − ) Cation Coordination Number Preferred Geometry Structural Prototype Structure Description
0.225 − 0.414 0.225 - 0.414 0.225 − 0.414 4 Tetrahedral Zinc Blende ( β − Z n S \beta-ZnS β − Z n S ) FCC array of S 2 − S^{2-} S 2 − ; Z n 2 + Zn^{2+} Z n 2 + occupies half of the tetrahedral holes (4:4 coordination, Z = 4 Z=4 Z = 4 )
0.225 − 0.414 0.225 - 0.414 0.225 − 0.414 4 Tetrahedral Wurtzite ( α − Z n S \alpha-ZnS α − Z n S ) HCP array of S 2 − S^{2-} S 2 − ; Z n 2 + Zn^{2+} Z n 2 + occupies half of the tetrahedral holes (4:4 coordination, hexagonal)
0.414 − 0.732 0.414 - 0.732 0.414 − 0.732 6 Octahedral Rock Salt ( N a C l NaCl N a Cl ) FCC array of C l − Cl^- C l − ; N a + Na^+ N a + occupies all octahedral holes (6:6 coordination, Z = 4 Z=4 Z = 4 )
0.414 − 0.732 0.414 - 0.732 0.414 − 0.732 6 : 3 Octahedral / Trigonal Planar Rutile ( T i O 2 TiO_2 T i O 2 ) Tetragonal cell; T i 4 + Ti^{4+} T i 4 + coordinated by 6 O 2 − O^{2-} O 2 − , O 2 − O^{2-} O 2 − coordinated by 3 T i 4 + Ti^{4+} T i 4 + (6:3 coordination)
> 0.732 > 0.732 > 0.732 8 Cubic Cesium Chloride ( C s C l CsCl C s Cl ) Primitive cubic array of C l − Cl^- C l − ; C s + Cs^+ C s + at body center (8:8 coordination, Z = 1 Z=1 Z = 1 , NOT bcc!)
> 0.732 > 0.732 > 0.732 8 : 4 Cubic / Tetrahedral Fluorite ( C a F 2 CaF_2 C a F 2 ) FCC array of C a 2 + Ca^{2+} C a 2 + ; F − F^- F − occupies all 8 tetrahedral holes (8:4 coordination, Z = 4 Z=4 Z = 4 )
< 0.414 < 0.414 < 0.414 (inverted) 4 : 8 Tetrahedral / Cubic Antifluorite ( L i 2 O , N a 2 O Li_2O, Na_2O L i 2 O , N a 2 O ) FCC array of O 2 − O^{2-} O 2 − ; alkali cations occupy all 8 tetrahedral holes (4:8 coordination, Z = 4 Z=4 Z = 4 )
Perovskite ( A B X 3 ABX_3 A B X 3 ) A = 12 , B = 6 A = 12, B = 6 A = 12 , B = 6 Cuboctahedral / Octahedral Perovskite ( C a T i O 3 CaTiO_3 C a T i O 3 ) Cubic cell; C a 2 + Ca^{2+} C a 2 + at corners ( C N = 12 CN=12 CN = 12 ), T i 4 + Ti^{4+} T i 4 + at center ( C N = 6 CN=6 CN = 6 ), O 2 − O^{2-} O 2 − at face centers
Crucial Exam Trap: C s C l CsCl C s Cl is not body-centered cubic (bcc)! A true bcc lattice requires identical atoms at both the corner and body-center positions. Because C s + Cs^+ C s + and C l − Cl^- C l − are distinct chemical species with different scattering factors, C s C l CsCl C s Cl consists of two interpenetrating primitive cubic lattices, possessing simple cubic Bravais symmetry.
X-Ray Diffraction and Bragg's Law
X-ray diffraction (XRD) probes long-range periodic order in crystals because X-ray wavelengths ( λ ∼ 0.5 – 2.5 A ˚ \lambda \sim 0.5\text{--}2.5\text{ \AA} λ ∼ 0.5 – 2.5 A ˚ ) match interplanar lattice spacings ( d h k l d_{hkl} d hk l ).
- Bragg's Law Formulation
- n λ = 2 d h k l sin θ n\lambda = 2d_{hkl}\sin\theta nλ = 2 d hk l sin θ
Where:
n n n is the diffraction order (an integer, typically 1).
λ \lambda λ is the X-ray wavelength (e.g., C u K α = 1.5418 A ˚ Cu\text{ } K_\alpha = 1.5418\text{ \AA} C u K α = 1.5418 A ˚ ).
d h k l d_{hkl} d hk l is the perpendicular distance between adjacent lattice planes characterized by Miller indices ( h k l ) (hkl) ( hk l ) .
θ \theta θ is the glancing Bragg angle between the incident beam and the crystal plane.
For a cubic crystal with edge length a a a , the interplanar spacing is:
- d h k l = a h 2 + k 2 + l 2 d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} d hk l = h 2 + k 2 + l 2 a
- Systematic Absences (Selection Rules)
Destructive interference among centered lattice planes systematically eliminates certain reflections, allowing experimental determination of the Bravais lattice type from diffraction patterns:
Primitive Cubic ( P P P ): All reflections allowed (all combinations of h , k , l h, k, l h , k , l observed: 100 , 110 , 111 , 200 , 210 , 211 , … 100, 110, 111, 200, 210, 211, \dots 100 , 110 , 111 , 200 , 210 , 211 , … ).
Body-Centered Cubic ( I I I ): Reflections observed only when the sum of indices is even: h + k + l = 2 n h + k + l = 2n h + k + l = 2 n (e.g., 110 , 200 , 211 , 220 , … 110, 200, 211, 220, \dots 110 , 200 , 211 , 220 , … ). Reflections like 100 , 111 100, 111 100 , 111 are absent.
Face-Centered Cubic ( F F F ): Reflections observed only when h , k , l h, k, l h , k , l are unmixed (either all odd or all even): 111 , 200 , 220 , 311 , 222 , … 111, 200, 220, 311, 222, \dots 111 , 200 , 220 , 311 , 222 , … . Reflections like 110 , 210 110, 210 110 , 210 are systematically absent.
Band Theory of Solids: Conductors, Semiconductors, and Insulators
When N N N atomic orbitals overlap across a crystal lattice, they split into N N N discrete molecular orbitals. Because N ∼ 10 23 N \sim 10^{23} N ∼ 1 0 23 , the energy separation between adjacent levels becomes infinitesimally small ( ∼ 10 − 22 eV \sim 10^{-22}\text{ eV} ∼ 1 0 − 22 eV ), coalescing into continuous energy bands separated by forbidden band gaps ( E g E_g E g ) :
Valence Band (VB): The highest energy occupied band of electrons at absolute zero ( 0 K 0\text{ K} 0 K ).
Conduction Band (CB): The lowest unoccupied or partially filled energy band through which delocalized electrons travel.
Fermi Level ( E F E_F E F ): The chemical potential of electrons; the highest occupied energy level at 0 K 0\text{ K} 0 K , where the Fermi-Dirac occupation probability f ( E ) = 0.5 f(E) = 0.5 f ( E ) = 0.5 .
Electronic Class Band Gap Energy ( E g E_g E g ) Electron Distribution at 0 K 0\text{ K} 0 K Temperature Dependence of Conductivity Mechanism
Conductor (Metal) E g = 0 eV E_g = 0\text{ eV} E g = 0 eV Partially filled band (e.g., N a Na N a 3 s 1 3s^1 3 s 1 ) or overlapping VB and CB (e.g., M g Mg M g 3 s 3s 3 s and 3 p 3p 3 p ) Conductivity decreases as T T T rises ( d σ d T < 0 \frac{d\sigma}{dT} < 0 d T d σ < 0 ) Thermal lattice vibrations (phonons) increase electron-phonon scattering
Semiconductor 0.5 eV ≤ E g ≤ 3.0 eV 0.5\text{ eV} \le E_g \le 3.0\text{ eV} 0.5 eV ≤ E g ≤ 3.0 eV Filled VB, empty CB separated by small gap ( S i = 1.12 eV , G e = 0.67 eV , G a A s = 1.42 eV Si = 1.12\text{ eV}, Ge = 0.67\text{ eV}, GaAs = 1.42\text{ eV} S i = 1.12 eV , G e = 0.67 eV , G a A s = 1.42 eV ) Conductivity increases exponentially as T T T rises ( d σ d T > 0 \frac{d\sigma}{dT} > 0 d T d σ > 0 ) Thermal excitation promotes electrons across E g E_g E g : σ ∝ e − E g / 2 k B T \sigma \propto e^{-E_g / 2k_B T} σ ∝ e − E g /2 k B T
Insulator E g > 3.0 eV E_g > 3.0\text{ eV} E g > 3.0 eV Filled VB, completely empty CB separated by wide gap (Diamond E g ≈ 5.5 eV E_g \approx 5.5\text{ eV} E g ≈ 5.5 eV ) Extremely low conductivity at all practical temperatures Thermal energy k B T k_B T k B T ( 0.026 eV 0.026\text{ eV} 0.026 eV at 298 K 298\text{ K} 298 K ) is far too small to bridge E g E_g E g
Extrinsic Semiconductors: n n n -Type vs. p p p -Type Doping
Pure silicon ( S i Si S i , Group 14) is an intrinsic semiconductor with modest room-temperature conductivity. Introducing trace impurities (parts-per-million) through doping dramatically alters its carrier concentration:
n n n -Type Semiconductors (Negative Carriers):
Doped with Group 15 pentavalent donor atoms ( P , A s , S b P, As, Sb P , A s , S b ).
Four of the five valence electrons form covalent bonds with the host silicon lattice, leaving one extra unbonded electron.
This extra electron occupies a discrete donor energy level ( E d E_d E d ) located just below the conduction band edge ( Δ E d ≈ 0.045 eV \Delta E_d \approx 0.045\text{ eV} Δ E d ≈ 0.045 eV for P P P in S i Si S i ).
At room temperature, thermal energy fully ionizes donor impurities, promoting electrons into the conduction band.
Electrons are the majority charge carriers ; positive holes in the valence band are minority carriers. The Fermi level shifts upward toward the conduction band.
p p p -Type Semiconductors (Positive Carriers):
Doped with Group 13 trivalent acceptor atoms ( B , A l , G a , I n B, Al, Ga, In B , A l , G a , I n ).
The dopant has only three valence electrons, creating an electron deficiency (a "hole") in the covalent silicon network.
This creates a discrete, unoccupied acceptor energy level ( E a E_a E a ) located just above the valence band edge ( Δ E a ≈ 0.045 eV \Delta E_a \approx 0.045\text{ eV} Δ E a ≈ 0.045 eV for B B B in S i Si S i ).
Valence electrons are easily excited thermally into these acceptor levels, generating mobile positive holes in the valence band.
Holes are the majority charge carriers ; electrons are minority carriers. The Fermi level shifts downward toward the valence band.