Using numerical calculations of equi-energy contour plot of the Fourier transform of the spin Hamiltonian, we study the magnetic phase diagram ($J_{ 2}$ and $J_{ 3}$) of the rutile type $\beta$-MnO$_{ 2}$, where $J_{ 1}$ ($<0$) is fixed and is the antiferromagnetic interaction along the diagonal direction, $J_{ 2}$ is the interaction along the c axis, and $J_{ 3}$ is the interaction along the a axis. The magnetic phase diagram consists of the multricritical point (the intersection $J_{ 2}J_{ 3}=J_{ 1}^{ 2}$ and $J_{ 2}+J_{ 3}=2J_{ 1}$), the helical order along the c axis, the $(h=1/2,k=0,l=1/2)$ phase, the helical order along the a axis, and the phase $(h=0,k=0,l=1)$. The shift of the location of the magnetic Bragg peak in the $(h,0,l)$ reciprocal lattice plane is examined with the change of $J_{ 2}$ and $J_{ 3}$ in the phase diagram. The shift is discontinuous on the first-order phase transition, and is continuous on the second-order phase transition. The detail of our magnetic phase diagram is rather different from that reported by Yoshimori.
Categories: cond-mat.str-el
