= Solution
For this calculation define the net <wavevector> gained by the <neutron> as $\mathbf Q=\mathbf k_f-\mathbf k_i$, the opposite of a common sample-transfer convention. Since $\mathbf Q\perp\mathbf k_i$,
$$
E_f-E_i=\frac{\hbar^2}{2m_n}\bigl[(\mathbf k_i+\mathbf Q)^2-k_i^2\bigr]=\frac{\hbar^2Q^2}{2m_n}.
$$
The <perpendicular neutron recoil and phonon absorption> condition therefore requires \b[$\omega(Q)=\hbar Q^2/(2m_n)$]. In the supplied periodic dispersion, interpret its $q$ as this transfer $Q$ in the <extended-zone representation of a phonon dispersion>. Put $x=Qa/2$ and $C=m_nav/\hbar\simeq15.84$. Nonzero intersections obey
$$
x^2=C|\sin x|.
$$
There is no root in $0<x\leq\pi/2$: $\sin x\geq2x/\pi$ there and $C>\pi^2/4$. On $(\pi/2,\pi)$ the ratio $x^2/\sin x$ increases strictly from $\pi^2/4$ to infinity, so exactly one root occurs. Nor can there be a higher root: $x\geq\sqrt C$ gives $x^2\geq C\geq C|\sin x|$, with strict inequality at the possible boundary. For $\pi\leq x\leq\sqrt C$, use $|\sin x|\leq x-\pi$; $x^2-C(x-\pi)>0$ throughout this short interval, since its minimum there is $C(1+\pi-\sqrt C)>0$. Thus the only nonzero magnitude satisfies
$$
\boxed{\pi/a<Q<2\pi/a.}
$$
Solving the exact dispersion, which is permitted instead of the linear approximation, gives
$$
\boxed{Q=1.069\times10^{10}\,\mathrm{m^{-1}},\quad \omega=3.610\times10^{12}\,\mathrm{rad\,s^{-1}},\quad f=0.575\,\mathrm{THz}.}
$$
A straight chord between the zone edge and the next zone centre gives $\omega\simeq4v/a-2vQ/\pi$. Its intersection solves $(\hbar/2m_n)Q^2+(2v/\pi)Q-4v/a=0$, yielding $Q\simeq1.006\times10^{10}\,\mathrm{m^{-1}}$ and $f\simeq0.508\,\mathrm{THz}$ as a less accurate estimate.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2011/paper-1-phonon-transfer.png]
{title=Neutron recoil intersects a periodic acoustic branch}
{description=Original dispersion and recoil curves using the supplied parameters. The nonzero exact intersection lies in the second extended-zone interval; the dashed chord is the permitted linear approximation.}
The reduced-zone <phonon> <wavevector> is $q_{\rm red}=Q-2\pi/a=-1.873\times10^9\,\mathrm{m^{-1}}$ along the chosen positive transfer direction. The difference is a reciprocal-lattice vector, not an additional independent <phonon> outside the first zone. At low <temperature> the absorption probability contains the <Bose-Einstein distribution> factor $n_B(\omega)=[e^{\hbar\omega/k_BT}-1]^{-1}$, which tends to zero. Here $\hbar\omega/k_B\simeq27.6\,\mathrm K$, so this absorption channel is strongly suppressed well below that scale. <elastic scattering> and kinematically allowed spontaneous <phonon> creation need not disappear.
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