Overview

Vibrations of Crystals with Monatomic BasisThis book provides a detailed, concept-oriented introduction to the vibrational behavior of crystals that contain one atom per primitive cell. It explains how atoms arranged in a regular lattice can oscillate collectively, forming waves that propagate through the crystal. These vibrations play a fundamental role in determining the mechanical, thermal, and electronic properties of solids. The analysis begins by defining the displacement usu_sus of the sss-th atom from its equilibrium position and describing how entire atomic planes move in phase when a vibrational wave travels through the lattice. For any wave vector kkk, the system supports three vibrational modes-one longitudinal (parallel to the wave direction) and two transverse (perpendicular to the wave direction). To determine how these vibrations behave, the book introduces the elastic restoring forces acting between neighboring atoms. Assuming a linear response and nearest-neighbour interactions, the force on any atom depends on the difference in displacement between that atom and its adjacent planes. Applying Newton's second law leads to the fundamental equation of motion for the lattice. Using a harmonic trial solution us=Ae-iωtu_s = A e^{-i\omega t}us=Ae-iωt and expressing atomic displacements in terms of a wave vector, the book derives the dispersion relation: ω2=2Cm(1-coska), \omega^2 = \frac{2C}{m}(1 - \cos ka),ω2=m2C(1-coska), where CCC is the force constant, mmm is the atomic mass, aaa is the lattice spacing, and kkk is the wave vector. This relation describes how the vibrational frequency varies with kkk, showing that the frequency becomes zero at k=0k = 0k=0 (all atoms move together) and reaches a maximum at the boundary of the first Brillouin zone k=±π/ak = \pm \pi/ak=±π/a. Using trigonometric identities, this relation is rewritten as: ω=4Cm∣sin(ka/2)∣, \omega = \sqrt{\frac{4C}{m}} \sin(ka/2),ω=m4C∣sin(ka/2)∣, revealing the periodic nature of lattice vibrations. The book explains how this dispersion curve governs the propagation of vibrational energy, the group velocity, and the behaviour of low- and high-frequency modes. Overall, this book gives a complete understanding of lattice dynamics in simple monatomic crystals. It builds the foundation for advanced topics such as phonons, thermal conductivity, heat capacity, scattering processes, and vibrations in more complex crystal structures.

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