# Advanced condensed matter physics by Leonard M. Sander By Leonard M. Sander

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Extra info for Advanced condensed matter physics

Example text

If we integrate Eq. 50) with respect to s we find: ds p(r, s) = (N − 1) n(r) , −1 = n dR(g(R) − 1). 54) The last line is for a translationally invariant system. 16 is big enough for a particle. We will use these ideas later in our discussion of interacting electrons. For magnets we can go through the same discussion. The relevant correlation function is of the magnetization, namely G(r, s) = m(r)m(s) . 55) If the magnet is ordered then if |r − s| → ∞ we expect that: G(r, s) → m(r) m(s) . 56) If this is so, the magnet is said to have long-range order.

19) i=j We need not consider the kinetic energy because, as is well known from statistical physics, it is the same for the solid and the gas if classical mechanics is valid. To see how to perform the sum, consider only the attractive terms. 20) where N is the total number of atoms. Note that in this case we need to go beyond nearest neighbors since the potential is a power law. In the magnetic case the dependence is exponential and nearest neighbors often suffice. 1. With a bit of work, more terms can be added.

K' q=G k Fig. 12 The Ewald construction. 3 Crystals with a basis If the crystal has a basis, we return to the above, and write the sum over positions as a sum over the Bravais lattice, Rj , and a sum over the positions of the atoms in the basis, sl , so that the lth atom in the ith cell is at Ri + sl . Then: f (q) = exp(iq · Ri ) i exp(iq · sl )fl (q). l Thus: eiq·[Ri −Rj ] | f (q)|2 = S(q) i, j = S(q)N 2 δk,k +G G 2 S(q) = eiq·sl fl (q) . 40) l The first factor is called the structure factor in this context (see footnote above).