Crystal momentum k in 2D band structure is a mathematical concept that describes the momentum of an electron in a crystal lattice. It is a vector quantity that represents the momentum of the electron at a specific point in the Brillouin zone of the crystal.
The definition of crystal momentum k on a torus is a result of the periodic nature of the crystal lattice. The Brillouin zone, which represents all possible values of k, is shaped like a torus due to the periodic repetition of the crystal lattice in all directions. Therefore, defining k on a torus allows for a complete representation of the crystal's electronic structure.
Crystal momentum k is directly related to the energy of electrons in a crystal as described by the Bloch theorem. The energy of an electron in a crystal is a function of both its crystal momentum and the crystal potential. This relationship is represented by the energy dispersion curve in the band structure, where the energy is plotted against k.
No, crystal momentum k cannot be measured directly. It is a mathematical concept that is used to describe the behavior of electrons in a crystal lattice. However, its effects can be observed through experiments such as angle-resolved photoemission spectroscopy (ARPES) which measures the energy and momentum of electrons in a crystal.
The value of crystal momentum k affects the electronic properties of a material in several ways. It determines the energy levels and electronic states that are allowed in a crystal, which in turn affects its conductivity, optical properties, and other electronic characteristics. It also plays a crucial role in determining the behavior of electrons in response to external forces or fields.