Introduction to Mathematical Physics/N body problem and matter description/Crystalline solids

Crystalline solids ([#References|references]) are periodical arrangement of atoms or molecules. \index{crystal} Translation invariance symmetry allows to calculate approximations of quantum states of such systems (see section secsolidmq). Statistical physics allows to evaluate properties at equilibrium (see section secgasparfq). Continuous approximation allows to deal for instance with elasticity (see sections sepripuiva and  secmaterelast). Magnetic properties of solids are of great interest. Solids can be classified according to the orientation of the magnetic momentum carried by each elementary brick constituting the solid (for instance a small molecule) has a small magnetic moment \index{magnetic moment} or spin. If orientation of those spins is random, crystal is said paramagnetic \index{paramagnetic} (see figure ---figparamag---). Average magnetisation is then zero. If spins are oriented along a privileged direction, crystal is called ferromagnetic \index{ferromagnetic} (see figure ---figferromag---). There exists then a non zero magnetization. If spins have directions alternatively opposed (see Figure ---figantiferromag---), crystal is called anti ferromagnetic. Ising model, (see section secmodising) is a simple model that allows to describe paramagnetic -- ferromagnetic transition  that appears for certain materials (for example iron) when temperature decreases.