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Among the common states of matter (gas, liquid and solid), the liquid and solid states have higher density. It is often useful to classify the states of matter according to their relative kinetic and potential energy scales (KE vs PE):

Gas: KE >> PE;

Liquid: KE ~ PE;

Solid: KE < PE.

Therefore, the solid state is characterized by very localized constituent units (atoms, molecules or ions). At all temperatures, atoms in a solid vibrates in the vicinity of their equilibrium positions. Another significant difference between the solids and fluid phases (liquid and gas) is that a solid has a non-zero shear modulus; that is, a finite deformation of a solid can only be caused by a finite force.

The solid state materials can be categorized based on the presence/absence and the type of order. In a crystalline material, the crystalline order is characterized by the translational symmetry, which are defined by the unit cell vectors and the sums of their integer multiples. That is, if you start from one point in a crystal, and travel along a vector that is the sum of integer multiples of the unit cell vectors, at the end of your trip, you will be sitting at a point in an environment identical to your starting point. A result of the three-dimensional periodicity is that a crystal shows sharp peaks in the diffraction patterns when irradiated with x-rays, electron or neutrons. In an amorphous material, such translational symmetry is not present, but local order may still be observed because of local bonding. The diffraction patterns of amorphous materials do not have sharp peaks, a signature of the lack of long-range order. In a quasicrystal, three-dimensional translational symmetry is absent. However, the structure of a quasicrystal may be viewed as a projection of periodic structure in higher dimensions, and as a consequence, quasicrystals exhibit sharp diffraction peaks -- hence the name "quasicrystal".

There are other ways to categorize solid state materials. Based the electrical conductivity, there are metals, insulators and superconductors. Based on magnetic properties, a solid can be said to be diamagnetic, paramagnetic, ferromagnetic, ferrimagnetic or antiferromagnetic. Based on the bonding and constituent units, there are covalent, metallic, and molecular solids.

Even though real solids have finite sizes and definite shapes, typically we start understanding the structure and properties of a solid by first looking at the thermodynamic limit, that is, infinite size without a boundary (or often, with a periodic boundary condition). Quantum theory, in conjunction with statistical mechanics, is at the heart of modern understanding of the solid state.It will always keep it's shape

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