Sodium is a monatomic bcc metal with conventional cubic lattice constant

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Solid State Physics, Ch Phonons: quantisation, occupation and inelastic scattering

Problem

a=4.225\,\mathrm{\AA}

. Neutrons of energy

E_{\rm in}=10.0\,\mathrm{meV}

are scattered through an angle of

55.6^\circ

(the angle between

\mathbf{k}_{\rm in}

and

\mathbf{k}_{\rm out}

) and detected with a larger energy,

E_{\rm out}=16.0\,\mathrm{meV}

, so the neutron has gained energy from the crystal. The sample is oriented so that

\mathbf{Q}=\mathbf{k}_{\rm out}-\mathbf{k}_{\rm in}

lies along

[110]

, close to the reciprocal lattice point

\mathbf{G}=(110)

. A neutron's kinetic energy and wavevector are related by

E(\mathrm{meV})=2.072\,k^2(\mathrm{\AA^{-2}})

. ((a)) Find

k_{\rm in}

and

k_{\rm out}

\mathrm{\AA^{-1}}

. ((b)) Find

|\mathbf{Q}|

from the scattering triangle. ((c)) Identify which one-phonon process this is (creation or destruction), write its conservation condition with

\mathbf{G}

explicit, and find the phonon wavevector

q

and the point

(q,\hbar\omega)

this places on sodium's dispersion curve.

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