# The Theory of Ferromagnetism By W. Heisenberg, Leipzig. With 1 illustration. (Received on May 20, 1928.) Weiss's molecular forces are reduced to a quantum mechanical exchange phenomenon; specifically, these are the exchange processes recently successfully used by Heitler and London to explain homogeneous valence forces. Introduction. Ferromagnetic phenomena have been formally well explained by the known Weiss theory. This theory is based on the assumption that each atom in the crystal experiences a directing force from the other atoms of the lattice, proportional to the number of already directed atoms. The origin of this atomic field, however, was completely unknown. The interpretation of Weiss's forces based on classical theory was hindered by the following difficulties: Magnetic interaction forces between atoms are always several orders of magnitude smaller than the atomic fields derived from ferromagnetic experiments. Electrical interactions, although leading to the correct order of magnitude, would rather be expected to be proportional to the square of the cosine of the angle between two atoms, contrary to the assumptions of the Weiss theory. Other difficulties were discussed in detail by Lenz, and Ising showed that even the assumption of directing, sufficiently strong forces between neighboring atoms in a chain is not sufficient to produce ferromagnetism. The ferromagnetic question complex has entered a new stage through the Uhlenbeck-Goudsmits theory of the spin electron. In particular, it follows from the known factor g = 2 in the Einstein-de Haas effect (which was measured in ferromagnetic substances) that only the magnetic moments of the electrons, not the atoms, are oriented in a ferromagnetic crystal. This eliminates the possibility of interpreting Weiss's forces as electrical interactions depending on the relative spin direction of the electrons, since we know such forces do not exist. Furthermore, Pauli showed that neglecting the electron interactions in a metal using the Pauli-Fermi-Dirac statistics always results in paramagnetism or diamagnetism. § 1. Model-based Foundations of the Theory. The basic idea of this theory is that the empirical results present a situation similar to that encountered earlier with the spectrum of the helium atom. From the terms of the helium atom, it seemed that a strong interaction existed between the spin directions of two electrons, leading to the splitting of the term scheme into singlet and triplet systems. This difficulty was resolved by the discovery that the apparent strong interaction was indirectly caused by a resonance or exchange phenomenon, characteristic of all quantum mechanical systems of identical particles. It is therefore natural to use this exchange phenomenon to explain ferromagnetic phenomena. We will show that Coulomb interactions together with the Pauli principle are sufficient to produce the same effects as the molecular field postulated by Weiss. Only recently have the mathematical methods for treating such a complex problem been developed in the important studies of Wigner, Hund, Heitler, and London. Before proceeding to the actual calculation,# The Theory of Ferromagnetism By W. Heisenberg, Leipzig. With 1 illustration. (Received on May 20, 1928.) Weiss's molecular forces are reduced to a quantum mechanical exchange phenomenon; specifically, these are the exchange processes recently successfully used by Heitler and London to explain homogeneous valence forces. Introduction. Ferromagnetic phenomena have been formally well explained by the known Weiss theory. This theory is based on the assumption that each atom in the crystal experiences a directing force from the other atoms of the lattice, proportional to the number of already directed atoms. The origin of this atomic field, however, was completely unknown. The interpretation of Weiss's forces based on classical theory was hindered by the following difficulties: Magnetic interaction forces between atoms are always several orders of magnitude smaller than the atomic fields derived from ferromagnetic experiments. Electrical interactions, although leading to the correct order of magnitude, would rather be expected to be proportional to the square of the cosine of the angle between two atoms, contrary to the assumptions of the Weiss theory. Other difficulties were discussed in detail by Lenz, and Ising showed that even the assumption of directing, sufficiently strong forces between neighboring atoms in a chain is not sufficient to produce ferromagnetism. The ferromagnetic question complex has entered a new stage through the Uhlenbeck-Goudsmits theory of the spin electron. In particular, it follows from the known factor g = 2 in the Einstein-de Haas effect (which was measured in ferromagnetic substances) that only the magnetic moments of the electrons, not the atoms, are oriented in a ferromagnetic crystal. This eliminates the possibility of interpreting Weiss's forces as electrical interactions depending on the relative spin direction of the electrons, since we know such forces do not exist. Furthermore, Pauli showed that neglecting the electron interactions in a metal using the Pauli-Fermi-Dirac statistics always results in paramagnetism or diamagnetism. § 1. Model-based Foundations of the Theory. The basic idea of this theory is that the empirical results present a situation similar to that encountered earlier with the spectrum of the helium atom. From the terms of the helium atom, it seemed that a strong interaction existed between the spin directions of two electrons, leading to the splitting of the term scheme into singlet and triplet systems. This difficulty was resolved by the discovery that the apparent strong interaction was indirectly caused by a resonance or exchange phenomenon, characteristic of all quantum mechanical systems of identical particles. It is therefore natural to use this exchange phenomenon to explain ferromagnetic phenomena. We will show that Coulomb interactions together with the Pauli principle are sufficient to produce the same effects as the molecular field postulated by Weiss. Only recently have the mathematical methods for treating such a complex problem been developed in the important studies of Wigner, Hund, Heitler, and London. Before proceeding to the actual calculation,