section_path: Quantum-mechanical origin of magnetism
section_name: Quantum-mechanical origin of magnetism
paragraph_text: Quantum-mechanical origin of magnetism
  
  While heuristic explanations based on classical physics can be formulated, diamagnetism, paramagnetism and ferromagnetism can be fully explained only using quantum theory. A successful model was developed already in 1927, by Walter Heitler and Fritz London, who derived, quantum-mechanically, how hydrogen molecules are formed from hydrogen atoms, i.e. from the atomic hydrogen orbitals u A {\displaystyle u_{A}} and u B {\displaystyle u_{B}} centered at the nuclei A and B, see below. That this leads to magnetism is not at all obvious, but will be explained in the following.
  
  According to the Heitler–London theory, so-called two-body molecular σ {\displaystyle \sigma } -orbitals are formed, namely the resulting orbital is:
  
  Here the last product means that a first electron, r, is in an atomic hydrogen-orbital centered at the second nucleus, whereas the second electron runs around the first nucleus. This "exchange" phenomenon is an expression for the quantum-mechanical property that particles with identical properties cannot be distinguished. It is specific not only for the formation of chemical bonds, but also for magnetism. That is, in this connection the term exchange interaction arises, a term which is essential for the origin of magnetism, and which is stronger, roughly by factors 100 and even by 1000, than the energies arising from the electrodynamic dipole-dipole interaction.
  
  As for the spin function χ (s 1, s 2) {\displaystyle \chi (s_{1},s_{2})}, which is responsible for the magnetism, we have the already mentioned Pauli's principle, namely that a symmetric orbital (i.e. with the + sign as above) must be multiplied with an antisymmetric spin function (i.e. with a − sign), and vice versa. Thus:
  
  I.e., not only u A {\displaystyle u_{A}} and u B {\displaystyle u_{B}} must be substituted by α and β, respectively (the first entity means "spin up", the second one "spin down"), but also the sign + by the − sign, and finally r by the discrete values s (= ± 1 ⁄ 2); thereby we have α (+ 1 / 2) = β (− 1 / 2) = 1 {\displaystyle \alpha (+1/2)=\beta (-1/2)=1} and α (− 1 / 2) = β (+ 1 / 2) = 0 {\displaystyle \alpha (-1/2)=\beta (+1/2)=0}. The " singlet state ", i.e. the − sign, means: the spins are antiparallel, i.e. for the solid we have antiferromagnetism, and for two-atomic molecules one has diamagnetism. The tendency to form a (homoeopolar) chemical bond (this means: the formation of a symmetric molecular orbital, i.e. with the + sign) results through the Pauli principle automatically in an antisymmetric spin state (i.e. with the − sign). In contrast, the Coulomb repulsion of the electrons, i.e. the tendency that they try to avoid each other by this repulsion, would lead to an antisymmetric orbital function (i.e. with the − sign) of these two particles, and complementary to a symmetric spin function (i.e. with the + sign, one of the so-called " triplet functions "). Thus, now the spins would be parallel (ferromagnetism in a solid, paramagnetism in two-atomic gases).
  
  The last-mentioned tendency dominates in the metals iron, cobalt and nickel, and in some rare earths, which are ferromagnetic. Most of the other metals, where the first-mentioned tendency dominates, are nonmagnetic (e.g. sodium, aluminium, and magnesium) or antiferromagnetic (e.g. manganese). Diatomic gases are also almost exclusively diamagnetic, and not paramagnetic. However, the oxygen molecule, because of the involvement of π-orbitals, is an exception important for the life-sciences.
  
  The Heitler-London considerations can be generalized to the Heisenberg model of magnetism (Heisenberg 1928).
  
  The explanation of the phenomena is thus essentially based on all subtleties of quantum mechanics, whereas the electrodynamics covers mainly the phenomenology.