Introduction to Basic Terms of Band Structures

Harrison Model of Polyynic Carbyne Chains

“…43 On the other hand, we know that the contributions of the 2p y and 2p z orbitals in the linear carbon chain are the same due to symmetry, so that we only need to evaluate one of them. 44 Therefore, according to Fig. 1 and under the Harrison's model, the Hamiltonian of polyyne chain in the k-space with three orbitals per atom from the matrix form of the Schrödinger's Eq.…”

Lattice Entertain You: Paper Modeling of the 14 Bravais Lattices on YouTube

“…The structures adopted by solids are of considerable importance in chemical education, so much so that numerous articles on this topic have been published in this Journal . − Crystal systems, including the Bravais lattices (the 14 possible three-dimensional structures for which each single atom or ion is an identical environment), are discussed. − The relevance of these structures to wider chemical problems or space group symmetry has been noted. − Several systems for creating and utilizing models of crystal structures even beyond the Bravais lattices have already been developed. − …”

“…Kettle and Norrby emphasize the importance of the Brillouin zone, and highlight some features of the direct lattice that are shared by the reciprocal lattice. Meyer, Glaus, and Calzaferri discuss more advanced topics related to crystal structures, such as band structures and the density of states. These papers do not discuss either the construction of three-dimensional models, or their utility in pedagogy.…”

Teaching Reciprocal Space to Undergraduates via Theory and Code Components of an IPython Notebook

“…Components of the electronic structure include the band structure and density of states, topics covered in undergraduate physical chemistry and materials science courses. For example, the band structure diagram depicts the energy of bands as a function of k-points on a path connecting the high symmetry k-points in the irreducible Brillouin zone. , Understanding the meaning of these k-points requires knowledge of reciprocal space, the mathematical relationship between real and reciprocal space vectors, and the role that these reciprocal space vectors play in determining the physical properties of solids. Several studies have focused on these topics in the literature, but none have mathematically shown the stepwise transformation of these vectors along with a computational tool for automating the process. − Although the topics have been briefly discussed, the mathematical process and tools presented herein have yet been presented in the chemistry education literature.…”

“…For example, the band structure diagram depicts the energy of bands as a function of k-points on a path connecting the high symmetry kpoints in the irreducible Brillouin zone. 4,5 Understanding the meaning of these k-points requires knowledge of reciprocal space, the mathematical relationship between real and reciprocal space vectors, and the role that these reciprocal space vectors play in determining the physical properties of solids. Several studies have focused on these topics in the literature, but none have mathematically shown the stepwise transformation of these vectors along with a computational tool for automating the process.…”