Solid crystals

jd12345

Just wondering - What factor decides whether a solid takes up a crystalline structure or an amorphous structure? And if it takes a crystalline structure , what decides if it will be cubic, tetragonal, hexagonal , trigonal, monoclinic or triclinic?

DrDu

Amorphous structures are generally not equilibrium structures.
The crystal structure depends on the substance making up the crystal just like the molecular structure of gasses depends on the atoms forming the molecules.

jd12345

Do we have a mathematical way of determining that which solid will take what crystalline structure yet?

DrDu

That is still very difficult in the general case. For a molecule, there are usually a limited number of possibilities for the possible structures (isomers) while there are infinitely many possible crystal structures. However, in many cases the structure can be determined when there are only a limited number of reasonable structures using e.g. using ab initio programs, density functional theory.

DrDu

Mathematical proofs that a given structure is really a global minimum are notoriously hard. I think even the proof that a regular dense packing of spheres has maximal density was finished only some years ago.

M Quack

Even relatively simple compositions can give rise to several different crystal structures, for example TiO2.

http://ruby.colorado.edu/~smyth/min/tio2.html

Small tweaks of temperature or pressure can lead to changes in the crystal structure. Water ice has a particularly rich phase diagram, or elemental Sulfur.

Predicting ab-initio what crystal structure a certain chemical composition will form is impossible - today - except maybe in some particularly simple systems.