Elastic and inelastic neutron scattering studies of Li(Ni$_{1-x}$Fe$_{x}$)PO$_4$ single crystals reveal anomalous spin-wave dispersions along the crystallographic direction parallel to the characteristic wave vector of the magnetic incommensurate pha
se. The anomalous spin-wave dispersion ({it magnetic soft mode}) indicates the instability of the Ising-like ground state that eventually evolves into the incommensurate phase as the temperature is raised. The pure LiNiPO$_4$ system ($x=0$), undergoes a first-order magnetic phase transition from a long-range incommensurate phase to an antiferromagnetic ground state at {it T}$_N$ = 20.8 K. At 20% Fe concentrations, although the AFM ground state is to a large extent preserved as that of the pure system, the phase transition is second-order, and the incommensurate phase is completely suppressed. Analysis of the dispersion curves using a Heisenberg spin Hamiltonian that includes inter- and in-plane nearest and next-nearest neighbor couplings reveals frustration due to strong competing interactions between nearest- and a next-nearest neighbor site, consistent with the observed incommensurate structure. The Fe substitution only slightly lowers the extent of the frustration, sufficient to suppress the IC phase. An energy gap in the dispersion curves gradually decreases with the increase of Fe content from $sim$2 meV for the pure system ($x=0$) to $sim$0.9 meV for $x=0.2$.
Inelastic neutron scattering (INS) experiments were performed to investigate the spin dynamics in magnetoelectric effect (ME) LiCoPO$_4$ single crystals. Weak dispersion was detected in the magnetic excitation spectra along the three principal crysta
llographic axes measured around the (0 1 0) magnetic reflection. Analysis of the data using linear spin-wave theory indicate that single-ion anisotropy in LiCoPO$_4$ is as important as the strongest nearest-neighbor exchange coupling. Our results suggest that Co$^{2+}$ single-ion anisotropy plays an important role in the spin dynamics of LiCoPO$_4$ and must be taken into account in understanding its physical properties. High resolution INS measurements reveal an anomalous low energy excitation that we hypothesize may be related to the magnetoelectric effect of LiCoPO$_4$.
