The wave vector in 1st Brillouin Zone (B.Z) in Bloch theory refers to the vector that describes the periodicity of the crystal lattice in a solid material. It is used to describe the behavior of electrons in a crystal lattice and is an important concept in understanding the electronic properties of materials.
The wave vector is directly related to the energy of electrons in a crystal lattice through the energy-band structure. The energy of electrons is determined by the wave vector and the potential of the crystal lattice. As the wave vector changes, the energy of the electrons also changes, leading to different energy levels and energy bands.
The 1st Brillouin Zone is the first region in the reciprocal lattice of a crystal that contains all the information about the electronic properties of the material. It represents the smallest unit of the crystal lattice in reciprocal space and is used to study the electronic behavior of materials.
The wave vector is calculated using the Bloch wave function, which describes the behavior of electrons in a periodic potential. It is a combination of a plane wave and a periodic function, and the wave vector is determined by the periodicity of the crystal lattice and the energy of the electrons.
The wave vector and the crystal momentum are closely related in Bloch theory. The crystal momentum is defined as the product of the wave vector and Planck's constant, and it represents the momentum of the electron in the crystal lattice. As the wave vector changes, the crystal momentum also changes, leading to different energy levels and properties of the material.