Solid State Physics. Philip Hofmann. Читать онлайн. Newlib. NEWLIB.NET

Автор: Philip Hofmann
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Физика
Год издания: 0
isbn: 9783527837267
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long distances between the nuclei, Eqs. (2.12) and (2.13) can be rewritten in the form

      where the plus and minus signs apply to the singlet and triplet state, respectively. In this representation, the energy change upon bonding contains two contributions, of which one depends on the relative spin orientations of the electrons (plus-or-minus upper X) and the other does not (upper C). The energy difference between the two states is then given by 2 upper X, where upper X is called the exchange energy. In the case of the hydrogen molecule, the exchange energy is always negative. Equation (2.14) is a remarkable result, because it means that the energy of the system depends on the relative orientation of the spins, even though these spins are not explicitly mentioned in the Schrödinger equation.

      We will encounter similar concepts in the chapter about magnetism, where the underlying principle for magnetic ordering is very similar to what we see here: Through the exchange energy, the total energy of a system of electrons depends on their relative spin orientations, and therefore a particular ordered spin configuration is energetically favored. For two electrons, the “magnetic” character is purely given by the sign of upper X. For negative upper X, the coupling with two opposite spins is favorable (the “antiferromagnetic” case), whereas a positive upper X would lead to a situation where two parallel spins give the lowest energy (the “ferromagnetic” case).

      In metals, the valence electrons of the atoms are removed from the ion cores, but in contrast to ionic solids, there are no electronegative ions to bind them. Therefore, they are free to migrate between the ionic cores. These delocalized valence electrons are involved in the conduction of electricity and are therefore often called conduction electrons. One can expect metals to form from elements for which the energy necessary to remove outer electrons is not too big. Nevertheless, this removal always costs some energy that has to be more than compensated by the resulting bonding. Explaining the energy gain from the bonding in an intuitive manner is difficult, but we will at least try to make it plausible. Obviously, the ultimate reason must be some sort of energy lowering.

      One energy contribution that is lowered is the kinetic energy of the conduction electrons. Consider the kinetic energy contribution in a Hamiltonian, upper T equals minus italic h over two pi squared nabla squared slash 2 m Subscript normal e. A matrix element left pointing angle upper Psi StartAbsoluteValue upper T EndAbsoluteValue upper Psi right pointing angle measures the kinetic energy of a particle, and upper T upper Psi is proportional to the second spatial derivative of the wave function, that is, its curvature. For an electron localized at an atom, the curvature of the wave function is much larger than for a nearly free electron in a metal, and this is the origin of the energy gain. A simple implementation of this idea is found in Problem 4, which inspects the lowering of the kinetic energy resulting from delocalization of an electron over the size of a typical crystal unit cell rather than just over the volume of an atom.

      We can now also understand why metals prefer close‐packed structures. First of all, metallic bonding does not have any directional preference. Second, close‐packed structures secure the highest possible overlap between the valence orbitals of the atoms, maximizing the delocalization of the electrons and thereby the kinetic energy gain. The resulting structures also maximize the number of nearest neighbors for any given atom, again giving rise to strongly delocalized states.