Let's start with illustrational one-dimensional crystalline potential as a periodic function of coordinate against the distance from a chosen starting atom V (x) = V (x+a) ( see fig. ). In solid state physics, an ideal, pure, or perfect crystal is defined as a crystalline material in which the constituent atoms, ions, or molecules are arranged in a perfectly ordered, repeating three-dimensional lattice structure. This means that every unit cell of the crystal is identical, and there are no defects or impurities within the structure. The lattice extends infinitely in all directions, and the periodic arrangement results in well-defined physical properties, such as symmetry, electronic band structure, and phonon dispersion, that can be described using crystallography principles and quantum mechanics. In practice, no real crystal is perfect, but the concept of an ideal crystal serves as a useful theoretical model for understanding the fundamental properties of crystalline solids. Some discussion exists about the validity of the definition of covalent bonding and the relation of the electron delocalization to covalent bonding . If we assume translational symmetry of the crystal -meaning that the Green's function will be dependent only on the relative position ⃗ R -⃗ R ′ ) -we get 114