[Encyclios]

Characteristics of crystalline solids

The main characteristics of crystalline solids are mentioned as below:

• Crystalline solids show a regular structure and have a definite geometrical shape.
• The sharp freezing point is found in crystalline solids. This is because the distance between the same atoms/molecules or ions is the same and remains constant, unlikely from amorphous solids.
• The heat of fusion is definite and fixed as the regularity in crystal lattice remains the same and is ideal.
• Crystalline solids are also known as true solids as they don’t tend to flow like pseudo-solids.
• When we cut a crystal solid with a knife, we obtain a flat and smooth surface.
• The nature of crystalline solid is anisotropic; that is, the properties turn out to be different in a different direction.
• Crystalline solids depict both long-range and short-range order.

A single macroscopic crystal is usually identifiable by their geometrical shape, consisting of flat faces with specific, particular orientations.

The scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification.

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Determination of the dynamic behavior of free electrons in the crystal lattice

Quantitatively, the dynamical behavior of free electrons in the crystal lattice is determined by a wave function satisfying the Schrödinger equation which, given the characteristics of the potential created by the lattice atoms, is very complex (W. Pauli, 1927; A. Sommerfeld, 1928; F. Bloch, 1928). Therefore, simplified models of the potential are generally introduced.

If a crystal lattice is considered one-dimensional, the solution of Schrödinger’s equation shows that electrons can have only energies of value within well-defined intervals, called allowed bands, separated by forbidden energy bands, called precisely forbidden bands. In a representation in which the electron energy is expressed as a function of the wave number, that is the inverse of the wavelength associated with the electron, these bands are called Brillouin zones.

Similar results are obtained for a three-dimensional crystal lattice if you consider the wave number as a vector quantity. Within each band, electrons can only take on a certain number of discrete energy values, each of which constitutes an energy level. Given a band with N levels, this, for the Pauli exclusion principle, can contain 2N electrons.

In complete bands, i.e. containing 2N electrons, the average velocity of these electrons is zero and therefore they do not contribute to the current circulating in the body; in bands occupied only partially the absence of an electron is equivalent to the presence of an identical charge, but of opposite sign, i.e. positive, called hole.

The distribution of electrons in a band is regulated by Fermi-Dirac statistic and in it is fundamental the Fermi level, which corresponds to the maximum value of energy that can be reached by electrons at a temperature of 0 K.

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Electron distribution in a band according to fermi-dirac statistics

When the Fermi level corresponds to a permitted band, this band is only partially occupied and therefore, by applying a potential difference, an electric current is produced and the solid is a conductor. In insulators, instead, the Fermi level is a forbidden band.

At 0 K the allowed bands below this level are completely occupied and the upper ones are completely empty; therefore, in no case, electric current is produced. At temperatures other than 0 K, due to thermal agitation, some electrons may jump into a permitted band, but the energy required is high and the probability of such a jump occurring is low; therefore, in each case, the corresponding electric current is very weak.

When the forbidden band containing the Fermi level is very narrow this probability is greater and the corresponding current can appreciate. The types of solids that fall into this pattern are called intrinsic semiconductors.