Extension of the QHO Model to Periodic Crystals

These flashcards cover key concepts from the lecture on the quantum harmonic oscillator model and its extension to periodic crystals, specifically in the context of statistical thermodynamics, vibrational analysis, and wavevector representation in solid-state physics.

Boltzmann Distribution

The probability distribution of states in statistical mechanics, derived from statistical thermodynamics.

Statistical Thermodynamics

A branch of physics that employs the principles of statistics to explain thermodynamic properties.

Dynamical Matrix

A mathematical representation used to describe the vibrational modes of a system.

Hessian Matrix

A square matrix of second-order partial derivatives used in optimization and stability analysis.

Vibrations in Polyatomic Molecules

The oscillatory motion of atoms within a molecule, which can be characterized using normal modes.

Phonons

Quantum mechanical representations of vibrational energies within a crystal lattice.

Potential-Energy Surface (PES)

A multidimensional surface that describes the energy of a system as a function of molecular geometry.

Transition-State Theory

A theory that describes the rate of chemical reactions in terms of an activated complex that forms during the reaction.

Eyring Equation

An equation that relates the rate of a reaction to the temperature, enthalpy, and entropy of activation.

Supramolecular Materials

Materials composed of multiple molecules held together by non-covalent interactions.

Bloch Theorem

A theorem stating that wave functions in a periodic potential can be expressed as a product of a cell-periodic function and a plane wave.

Wavevector

A vector that relates to the direction and wavelength of waves in a periodic structure.

Brillouin Zone

The unique unit cell in reciprocal space that contains all distinct wavevectors for a periodic system.

Irreducible Brillouin Zone

The smallest portion of the Brillouin zone that contains all unique wavevector directions.

Diagonalization

The process of converting a matrix into a diagonal form, simplifying the analysis of its eigenvalues and eigenvectors.

Degrees of Freedom

The number of independent parameters that define the state of a physical system.

Computational Intractability

A term describing problems that are too complex to solve in a reasonable amount of time with given resources.