Partial wave expansion of the RKKY indirect exchange interaction in metals

Abstract

Using the full partiall wave series for the exchange coupling constant J(k,k′)=∑iL=0 (2L+1) PL(k̂⋅k̂′ JL(k,k′), in standard notation, for an s‐d type exchange interaction between a local spin (electronic or nuclear) and free conduction electrons in metals, the corresponding full partial wave series for the long range, indirect, exchange interaction (Ruderman–Kittel–Kusuya–Yosida, RKKY) between two such local spins is derived. In particular, the asymptotic long range form of the exchange coupling constant is F (R)∼−(9πn2/EF) JRKKY2 cos(2kFR)/(2kFR)3, where JRKKY=∑∞L=0 (2L+1)(−1)L JL(kF, kF). For constant JL(k,k′)=JL the exchange coupling F(R) is exactly calculated for all distances R in terms of standard functions. To obtain this result nonstandard indefinite integrals of spherical Bessel functions have been caLculated and the results are given in terms of recursion relations. Also, a closed form is obtained for the Cauchy principal value of the singular integral F∞0 dy y2 JL(y)/(y2−x2). Results are given for the electron spin polarization due to a local spin.