= Finite-momentum thresholds of an itinerant spin-flip continuum
The edges of an <itinerant spin-flip particle-hole continuum> follow from both excitation <energy> and occupation constraints. Consider a zero-<temperature> three-dimensional parabolic-band model with majority states occupied for $|\mathbf k|\leq k_F$ and an entirely empty minority band. For exchange splitting $2M$ and $\epsilon_k=\hbar^2k^2/(2m)$, a majority-to-minority flip carrying <wavevector> $\mathbf q$ has
$$
E(\mathbf k,\mathbf q)=2M+\frac{\hbar^2q^2}{2m}+\frac{\hbar^2\mathbf k\cdot\mathbf q}{m}.
$$
For $q>0$, the scalar product is bounded by $\pm k_Fq$, giving continuum edges
$$
E_\pm(q)=2M+\frac{\hbar^2q^2}{2m}\pm\frac{\hbar^2k_Fq}{m}.
$$
The extremes occur at opposite points of the occupied <Fermi surface> along $\mathbf q$. At $q=0$ the interval collapses to the single exchange energy $2M$.
The empty-minority assumption requires sufficient splitting; otherwise <Pauli exclusion principle> blocks some final states and the edges must be recomputed over the allowed region. These are independent particle-hole thresholds, not a calculation of the collective <ferromagnetic magnon> dispersion. A collective mode below the continuum can therefore have a different low-<momentum> energy scale.
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