Exchange Interactions
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(this chapter deals with fundamenta1s, and may be skimmed over during a first reading)
In the preceding chapters, we have shown that it is the exchange interactions between the spins of two atomic moments that are responsible for magnetic order. These interactions play a fundamental role, as the type of magnetic order that occurs depends on the sign of these exchange interactions, and the range over which they operate. The ordering temperature itself is generally of the same order of magnitude as the energy of the interactions. The aim of this chapter is to demonstrate the dif.ferent microscopic mechanisms that are the origins of these exchange interactions.
1. MANY-ELECTRON WA VE FUNCTIONS The existence of ordered states at high temperatures cannot be exp1ained by classica1 physics using the dipo1ar magnetic interaction, as this interaction is at 1east 100 times weak:er than required. Quantum mechanics has to be invoked to understand the origin of the exchange interactions that give rise to them. The exchange interaction in a solid has the same quantum origin as the interaction between electrons within one atom (chap. 7): the correlations between two e1ectrons 1ead to a difference in energy between configurations in which the spins are paralle1 and antiparallel. This originates in the Pauli princip1e, which stipu1ates that two e1ectrons cannot occupy the same quantum state. One of the consequences of this princip1e is that many-e1ectron wavefunctions must be antisymmetric with respect to the exchange of two e1ectrons. We will treat here the case of a two-e1ectron wave function 'P(l, 2), where 1 and 2 describe both the spatial coordinates and the spins of the electrons. Application of the Pauli principle 1eads to the re1ation: 'P(l, 2)
= -'1'(2, 1)
(9.1)
The wavefunction 'P(l, 2) can be written as a product of spatial q>(l, 2), and spin components x(l, 2). This 1eads to two types of wavefunctions depending on the symmetry of the functions q>(l, 2) and x(1, 2): '1'1(1,2)
= q>A(1,2)Xs(1,2)
(9.2)
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MAGNETISM - FUNDAMENTALS
Here S and A designate respectively symmetric and antisymmetric functions. In the case of two electrons, there is one antisymmetric wavefunction XA (1,2) that describes a singlet spin state: XA0,2) = (11.J2)(11î,2J..>-11J..,2î>), and three symmetric functions xsm (1, 2)(m =O, ±1) that correspond to a state with total spin S = 1 (the triplet state): Xs 1 (1, 2) = Il î, 2Î >, xs-l (1, 2) = Il J.., 2J.. >, and Xs0(1,2)=(11.J2)(11Î,2J..>-11J..,2î>). The energies ofthe two states can be calculated from the two-electron hamiltonian 'K,: EI(II) =
fJ t (r) 2 * (r' )j 2
Jd3rd3r' H (r, r') t (r) 1 * (r') 2 (r') 2 * (r) L = Jd3rC~>t (r)CI>2 * (r)
V =
(9.6)
H(r, r') being the Coulomb repulsion between electrons located at r and r'. It can be seen that the exchange interaction arises as a direct consequence of the antisymmetry of the wave functions, and corresponds to the difference in energy between the symmetric and antisymmetric spatial wave functions; in the simple case presented here, its
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