Instability threshold of a one-dimensional Bloch wall

Possible nucleation modes for a one-dimensional Bloch wall in a field antiparallel to the magnetization at the wall center are analyzed and the corresponding threshold instability fields are calculated. For the case of a mode uniform in the plane of the wall the exact result $H_b\mathrm{}\left(0\right)=\frac{4\pi M}{3}$ is obtained, but it is then shown that modes exhibiting buckling in this plane will have a lower threshold. These modes are characterized by the constraint that the wall azimuthal angle remains at its equilibrium value until an instability in the polar angle is reached. Rigorous upper- and lower-bound calculations show that the buckling-mode threshold instability field will be in the range $0.034\le \frac{H_b\mathrm{}\left(q\right)\mathrm{}}{4\pi M}\le 0.149$. An alternate nucleation mode, characterized by zero magnetostatic self-energy, is also analyzed. For this corrugating mode we find a rigorous upper bound to the threshold instability field of $\frac{H_c(q\to 0)}{H_k}=0.543\mathrm{}$. The implications of these results are discussed.