stp_chap5 - Chapter 5 Magnetic Systems c 2010 by Harvey...

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Chapter 5 Magnetic Systems c circlecopyrt 2010 by Harvey Gould and Jan Tobochnik 12 May 2010 We apply the general formalism of statistical mechanics developed in Chapter 4 to the Ising model, a model for which the interactions between the magnetic moments are important. We will find that these interactions lead to a wide range of interesting phenomena, including the existence of phase transitions. Computer simulations will be used extensively and a simple, but powerful approximation method known as mean-field theory will be introduced. 5.1 Paramagnetism The most familiar magnetic system in our everyday experience is probably the magnet on a refrig- erator door. This magnet likely consists of iron ions localized on sites of a lattice with conduction electrons that are free to move throughout the crystal. The iron ions each have a magnetic moment and, due to a complicated interaction with each other and with the conduction electrons, they tend to line up with each other. At sufficiently low temperatures, the moments can be aligned by an external magnetic field and produce a net magnetic moment or magnetization which remains even if the magnetic field is removed. Materials that retain a nonzero magnetization in zero magnetic field are called ferromagnetic . At high enough temperatures there is enough energy to destroy the magnetization, and the iron is said to be in the paramagnetic phase. One of the key goals of this chapter is to understand the transition between the ferromagnetic and paramagnetic phases. In the simplest model of magnetism the magnetic moment can be in one of two states as discussed in Section 4.3.1. The next level of complexity is to introduce an interaction between neighboring magnetic moments. A model that includes such an interaction is discussed in Sec- tion 5.4. 229
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CHAPTER 5. MAGNETIC SYSTEMS 230 5.2 Noninteracting Magnetic Moments We first review the behavior of a system of noninteracting magnetic moments with spin 1/2 in equilibrium with a heat bath at temperature T . We discussed this system in Section 4.3.1 and in Example 4.1 using the microcanonical ensemble. The energy of interaction of a magnetic moment μ in a magnetic field B is given by E = μ · B = μ z B, (5.1) where μ z is the component of the magnetic moment in the direction of the magnetic field B . Because the magnetic moment has spin 1/2, it has two possible orientations. We write μ z = , where s = ± 1. The association of the magnetic moment of a particle with its spin is an intrinsic quantum mechanical effect (see Section 5.10.1). We will refer to the magnetic moment or the spin of a particle interchangeably. What would we like to know about the properties of a system of noninteracting spins? In the absence of an external magnetic field, there is little of interest. The spins point randomly up or down because there is no preferred direction, and the mean internal energy is zero. In contrast, in the presence of an external magnetic field, the net magnetic moment and the energy of the system are nonzero.
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