Guide Modern Theory of Magnetism in Metals and Alloys: 175 (Springer Series in Solid-State Sciences)

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The book gives an introduction to the metallic magnetism, and treats effects of electron correlations on magnetism, spin fluctuations in metallic magnetism, formation of complex magnetic structures, a variety of magnetism due to configurational disorder in alloys as well as a new magnetism caused by the structural disorder in amorphous alloys, especially the itinerant-electron spin glasses. Physics of the Interstellar and Intergalactic Medium. Bruce T. College Chemistry. Steven Boone. The Thermodynamics of Phase and Reaction Equilibria. Ismail Tosun.

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Dirk Dubbers. Note that the two particle operator has been decoupled so that 1. This is called the Hartree—Fock approximation.


In the Hartree—Fock approximation, the Hamiltonian 1. The third term at the l. The last term also originates in the Coulomb interaction, but the wave functions have been exchanged due to the anti-symmetric property of the Slater determinant. It is referred as the exchange po- tential.

The Hartree—Fock wave function is the best Slater determinant at the ground state according to the variational principle [4]. Taking the Hartree—Fock wavefunctions as the basis functions, one can express the Hartree—Fock Hamiltonian 1. The second term at the r. The exchange potential in the Hartree—Fock equation 1. Slater proposed an approximate local exchange potential, using the free-electron wave functions. It is given as follows [4]. The effective potential in the Hartree—Fock self-consistent equation is spherical in the atomic system when we apply the potential 1.

We can express the atomic 8. The nl shells are therefore written as ns, np, nd, nf, etc. With the use of the Hartree—Fock atomic orbitals, a many electron state of an atom is expressed in the form of the Slater determinant. The ground-state electronic structure of an atom is constructed according to the Pauli principle. For example, in the case of Ar, we have the ground state 1s22s22p63s23p6. Since the outermost electron shell of Ar is closed, the magnetic moment of Ar vanishes. The ground state of electrons in an atom should be obtained by minimizing the Coulomb energy.

The intra-orbital Coulomb interaction i. The third term in 1. These effects suggest that the magnitude of the total spin S is maximized at the ground state. On the other hand, the 9. One might expect that such an interaction energy is reduced when electrons move around the nucleus in the same direction avoiding each other. Thus we expect that the magnitude of the total angular momentum L should be maximized at the ground state under the maximum magnitude of total spin. These properties are summarized in Fig.

The value of J at the ground state is determined by the spin—orbit interaction. The spin—orbit interaction 1. More- over the spin—orbit interaction energy is much smaller than JH in 3d transition metals. We can then verify the following relation in the subspace. N and mz denote the d electron number and the orbital magnetic quantum number, respectively. J denotes the total angular momentum. Examples of the 3d transition metal ions are also given in the bottom row Thus we have an effective Hamiltonian for the spin—orbit interaction in the Hund- rule subspace as follows.

The observation of the Curie law in the susceptibility measurement is regarded as an indication of the existence of local magnetic moments. The question then is whether electrons tend to remain in each atom or to itinerate in the solid. In the case of the latter, magnetic properties may be quite different from those expected from atomic ones and one has to start from the itinerant limit in order to explain their magnetism. Whether electrons in solids are movable or not is governed by the detailed bal- ance between the energy gain due to electron hopping and the loss of Coulomb in- teraction.

The simplest example may be the case of the hydrogen molecule H2. The bonding orbital is occupied by two electrons according to the Pauli principle. It leads to the total energy gain 2 t. This is the covalent bond in the hydrogen molecule. In this state, electrons hop from one atom to another, and thus they are movable. The covalent bond is stabilized by the kinetic energy gain of independent electrons.

It consists of the polarized state in which the 1s orbital of an atom is doubly occupied and the orbital of another atom is unoccupied, and the neutral state in which each atomic orbital is occupied by an electron. When the Coulomb interaction between electrons are taken into account, the covalent bonding state is not necessarily stable because it contains the polariza- tion state with double occupancy on an atomic orbital.

Assume that the loss of the intraatomic Coulomb interaction energy in the covalent bonding state is given by U, and consider the neutral-atom state as the state in which each electron is localized on an atom. In solids, we expect the same behavior as found in the hydrogen molecule.

Solid State Magnetism (Lecture 20): Quantum mechanical description of Magnons

Let us consider the behavior of electrons when atoms form a solid. The latter is expressed in It is not easy to treat the electrons in solids described by the Hamiltonian 1. In order to discuss both the itinerant and localized behaviors of electrons in solids, we can consider a simpler Hamiltonian consisting of one orbital per site as follows. The Hamilto- nian 1. The Hubbard model 1.

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Nevertheless it describes the localiza- tion of electrons as well as their itinerant behavior in solids. Let us consider the atomic limit of the model. The ground-state energy E0 of the atomic limit is obtained by minimizing the energy with respect to the number of double occupancy in solid. Assume that the number of lattice points is given by L. Magnetic moments on the sites with an electron are active in this case as shown in Fig.

Because there are L! Thus the system is a metal. We have then an insulator with local magnetic moments at each site in this case. The electrons in such systems are mobile. Note that spins of itinerant electrons are also mobile. Assume that there is a band for a non-interacting system whose band width is W. When the Coulomb interaction U is increased, each atom tends to be occupied by one electron, and electron hopping to neighboring sites tends to be suppressed in order to reduce the on-site Coulomb interaction energy. Zakharov, E. Kondratenko, L. Pitaevskii, I. Khalatnikov, G.

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