Wavenumber Dependence of Multipolar Interactions in the Anderson Lattice

Multipolar interactions are derived in the orbitally degenerate Anderson lattice with a spherical Fermi surface and one conduction electron per unit cell of the simple cubic lattice. As the crystalline-electric-field (CEF) ground state of $f^1$ configuration, the four-fold degenerate $\Gamma_8$ is mainly studied. Intersite interactions up to a sufficiently distant pair are Fourier transformed to the wavenumber space. For the $\Gamma_8$ case, quadrupolar and octupolar interactions favor the staggered order with $\mib{q} = (1/2,1/2,1/2)$, while the dipolar interaction favors an incommensurate magnetic structure with a long modulation period. The latter is due to a Kohn anomaly which is found to be sharply peaked for interaction channels with large angular momenta. Implications of results are discussed for multipole orders in CeB$_6$, and the incommensurate magnetic structure in a quasi-cubic system CeB$_2$C$_2$.