Gradient effects on dust lattice waves in paramagnetic dusty plasma crystals

Dust lattice modes are studied in a hexagonal two-dimensional lattice in plasma crystal, including paramagnetic dust particles. the gradients of magnetic fields, electric fields, and dust charge and also the interaction of dipole-dipole take into account. these gradients modify the levitation condition and affect the frequencies of dust lattice waves. the coupling between in-plane and out-of-plane modes gives rise to the hybrid mode, which is always an unstable mode. however, intersection of the in-plane mode with other modes does not result in mode-coupling instability.