Understanding Solid State Physics Problems and Solutions

The correlation between the microscopic composition of solids and their macroscopic (electrical, optical, thermal) properties is the goal of solid state physics. This book is the deeply revised version of the French book Initiation à la physique du solide: exercices comméntes avec rappels de cours, written more than 20 years ago. It has five sections that start with a brief textbook introduction followed by exercises, problems with solutions, and comments and that are concluded with questions. It presents a quasi-systematic investigation of the influence of dimensionality changes, from 1D to 3D, via surfaces and 2D quantum wells, on the physical properties of solids. The aim of this book is to teach solid state physics through the use of problems and solutions giving orders of magnitude and answers to simple questions of this field. The numerous comments and problems in the book are inspired from some Nobel Prize–winning research in physics, such as neutron diffraction (1994), quantum Hall effect (1985), semiconducting heterostructures (1973), and tunnel microscope (1986), superconductivity (1987). The book will be helpful for undergraduate- and graduate-level students of solid state physics and chemistry and researchers in physics, chemistry, and materials science.

Preface Tables 1.Crystal Structure and Crystal Diffraction Course Summary A. Crystal Structure 1. Definitions 2. Simple and Multiple Lattices 3. Lattice Rows and Miller Indices 4. Point Symmetry 5. The 7 Crystallographic Systems and the 14 Bravais Lattices 6. Space Symmetry B. Diffraction and the Reciprocal Lattice 1. Bragg’s Law 2. X-Rays 3. Reciprocal Lattice (Exs. 12, 13, and 19) 4. More Detailed Analysis of Diffraction Exercises Exercise 1: Description of some crystal structures Exercise 2: Mass per unit volume of crystals Exercise 3: Construction of various crystal structures Exercise 4: Lattice rows Exercise 5a: Lattice rows and reticular planes Exercise 5b: Lattice rows and reticular planes (continued) Exercise 6: Intersection of two reticular planes Exercise 7: Lattice points, rows and planes Exercise 8: Atomic planes and Miller indices—Application to lithium Exercise 9: Packing Exercise 10a: Properties of the reciprocal lattice Exercise 10b: Distances between reticular planes Exercise 11: Angles between the reticular planes Exercise 12: Volume of reciprocal space Exercise 13: Reciprocal lattice of a face-centered cubic structure Exercise 14: Reciprocal lattice of body-centered and face-centered cubic structures Exercise 15: X-ray diffraction by a row of identical atoms Exercise 16: X-ray diffraction by a row of atoms with a finite length Exercise 17: Bravais lattices in 2D: Application to a graphite layer (graphene) Exercise 18a: Ewald construction and structure factor of a diatomic row Exercise 18b: Structure factor for a tri-atomic basis; Ewald construction at oblique incidence (variation of Ex. 18a) Exercise 19: Reciprocal lattice, BZs, and Ewald construction of a twodimensional crystal Exercise 20: X-ray diffraction patterns and the Ewald construction Exercise 21a: Resolution sphere Exercise 21b: Crystal diffraction with diverging beams (electron backscattered diffraction: EBSD) Exercise 22: Atomic form factor Exercise 23: X-ray diffusion by an electron (Thomson) Problems Problem 1: X-ray diffraction by cubic crystals Problem 2: Analysis of an X-ray diffraction diagram Problem 3: Low energy electron diffraction (LEED) by a crystalline surface: absorption of oxygen Problem 4: Reflection high energy electron diffraction (RHEED) applied to epitaxy and to surface reconstruction Problem 5: Identification of ordered and disordered alloys Problem 6: X-ray diffraction study of a AuCu alloy Problem 7: Neutron diffraction of diamond Problem 8: Diffraction of modulated structures: application to charge density waves Problem 9: Structure factor of GaxAl1–xAs Problem 10: Structure factor of superlattices Problem 11: Diffraction of X-rays and neutrons from vanadium Problem 12: X-ray diffraction of intercalated graphite Questions 2. Crystal Binding and Elastic Constants Course Summary A. Crystal Binding 1. Statement of the Problem 2. Rare Gas Crystals 3. Ionic Crystals 4. Metallic Bonds 5. Covalent Bonds B. Elastic Constants 1. Introduction 2. Stress 3. Strain 4. Hooke’s Law 5. Velocity of Elastic Waves Exercises Exercise 1: Compression of a ionic linear crystal Exercise 2a: Madelung constant for a row of divalent ions Exercise 2b: Madelung constant of a row of ions –2q and +q Exercise 3: Cohesive energy of an aggregate of ions Exercise 4: Madelung constant of a 2D ionic lattice Exercise 5: Madelung constant of ions on a surface, an edge, and a corner Exercise 6: Madelung constant of an ion on top of a crystal surface Exercise 7: Madelung constant of parallel ionic layers Exercise 8: Cohesive energy of a MgO crystal Exercise 9: Ionic radii and the stability of crystals Exercise 10: Lennard-Jones potential of rare gas crystals Exercise 11: Chemisorption on a metallic surface Exercise 12: Anisotropy of the thermal expansion of crystals Exercise 13: Tension and compression in an isotropic medium. Relations between Sij, Cij, E (Young’s modulus) and s (Poisson coefficient), l and m (Lamé coefficients) Exercise 14: Elastic anisotropy of hexagonal crystals Exercise 15: Shear modulus and anisotropy factor Exercise 16: Elastic waves in isotropic solids Problems Problem 1: Cohesion of sodium chloride Problem 2: Cohesion and elastic constants of CsCl Problem 3: Van der Waals–London interaction. Cohesive energy of rare gas crystals Problem 4: Velocity of elastic waves in a cubic crystal: Application to aluminum and diamond Problem 5: Strains in heteroepitaxy of semiconductors Questions 3. Atomic Vibrations and Lattice Specific Heat Course Summary 1. Vibrations in a Row of Identical Atoms 2. Lattices with More Than One Atom per Unit Cell 3. Boundary Conditions 4. Generalization to 3D 5. Phonons 6. Internal Energy and Specific Heat 7. Thermal Conductivity Exercises Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C–C=C–C= Exercise 2a: Vibrations of a 1D crystal with two types of atoms m and M. Exercise 2b: Vibrations of a 1D crystal with a tri-atomic basis Exercise 3: Vibrations of a row of identical atoms. Influence of second nearest neighbors Exercise 4: Vibrations of a row of identical atoms: Influence of the nth nearest neighbor Exercise 5: Soft Modes Exercise 6: Kohn Anomaly Exercise 7: Localized phonons on an impurity Exercise 8: Surface acoustic modes Exercise 9: Atomic vibrations in a 2D lattice Exercise 10: Optical absorption of ionic crystals in the infrared Exercise 11: Specific heat of a linear lattice Exercise 12a: Specific heat of a 1D ionic crystal Exercise 12b: Debye and Einstein temperatures of graphene, 2D, and diamond, 3D Exercise 13: Atomic vibrations in an alkaline metal: Einstein temperature of sodium Exercise 14: Wave vectors and Debye temperature of mono-atomic lattices in 1-, 2-, and 3D. Exercise 15: Specific heat at two different temperatures Exercise 16: Debye temperature of germanium Exercise 17: Density of states and specific heat of a monoatomic 1D lattice from the dispersion relation Exercise 18: Specific heat of a 2D lattice plane Exercise 19: Phonon density of states in 2D and 3D: evaluation from a general expression Exercise 20a: Zero point energy and evolution of the phonon population with temperature Exercise 20 b: Vibration energy at 0 K of 1, 2, and 3D lattices (variant of the previous exercise) Exercise 21: Average quadratic displacement of atoms as a function of temperature Problems Problem 1: Absorption in the infrared: Lyddane–Sachs–Teller relation Problem 2: Polaritons Problem 3: Longitudinal and transverse phonon dispersion in CsCl Problem 4: Improvement of the Debye model: Determination of qD from elastic constants application to lithium Problem 5: Specific heats at constant pressure Cp and constant volume Cv: (Cp – Cv) correction Problem 6: Anharmonic oscillations: thermal expansion and specific heat for a row of atoms Problem 7: Phonons in germanium and neutron diffusion Problem 8: Phonon dispersion in a film of CuO2 Problem 9: Phonons dispersion in graphene Questions 4. Free Electrons Theory: Simple Metals Course Summary 1. Hypothesis 2. Dispersion Relation and the Quantization of the Wave Vector 3. Electron distribution and density of states at 0°K: Fermi energy and Fermi surface in 3D 4. Influence of Temperature on the Electron Distribution: Electron-Specific Heat 5. Electronic Conductivity 6. Wiedemann–Franz Law 7. Other Successful Models Obtained From the Free Electron Formalism Exercises Exercise 1: Free electrons in a 1D system. Going from an atom to a molecule and to a crystal Exercise 2: 1D metal with periodic boundary conditions Exercise 3: Free electrons in a rectangular box (FBC) Exercise 4: Periodic boundary conditions, PBC, in a 3D metal Exercise 5: Electronic states in a metallic cluster: Influence of the cluster size Exercise 5b: Electronic states in metallic clusters: Influence of the shape Exercise 6 (Variation of Ex. 5 and 5b): F center in alkali halide crystals and Jahn–Teller effect Exercise 7: Fermi energy and Debye temperature from F and P boundary conditions for object

Jacques Cazaux (1934–2014) was emeritus professor at the University of Reims, France. He did his undergraduate work in physics at the University of Sorbonne, Paris, and obtained his PhD in 1970 from the College of France, Paris, by submitting the thesis titled Anisotropy of Plasmons in Graphite. He then joined the University of Reims as a professor of solid state physics, and there he initiated a research laboratory on surface analysis (XPS and Auger) and material characterization (electron probe microanalysis, electron and X-ray microscopies). His research focused on the physics of secondary electron emission, and he authored more than 150 articles published in scientific journals. In recognition of his special contribution to scientific knowledge, Prof. Cazaux was invited as a speaker at more than 50 international meetings and was on the board of several scientific committees.

The correlation between the microscopic composition of solids and their macroscopic (electrical, optical, thermal) properties is the goal of solid state physics. This book is the deeply revised version of the French book Initiation à la physique du solide: exercices comméntes avec rappels de cours, written more than 20 years ago. It has five sections that start with a brief textbook introduction followed by exercises, problems with solutions, and comments and that are concluded with questions. It presents a quasi-systematic investigation of the influence of dimensionality changes, from 1D to 3D, via surfaces and 2D quantum wells, on the physical properties of solids. The aim of this book is to teach solid state physics through the use of problems and solutions giving orders of magnitude and answers to simple questions of this field. The numerous comments and problems in the book are inspired from some Nobel Prize–winning research in physics, such as neutron diffraction (1994), quantum Hall effect (1985), semiconducting heterostructures (1973), and tunnel microscope (1986), superconductivity (1987). The book will be helpful for undergraduate- and graduate-level students of solid state physics and chemistry and researchers in physics, chemistry, and materials science.