Aluminium is a monatomic fcc 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=4.05\,\mathrm{\AA}

. A triple-axis spectrometer sends in neutrons of energy

E_{\rm in}=20.0\,\mathrm{meV}

and, in one setting, detects neutrons scattered through an angle of

52.1^\circ

(the angle between

\mathbf{k}_{\rm in}

and

\mathbf{k}_{\rm out}

) with outgoing energy

E_{\rm out}=14.0\,\mathrm{meV}

. The sample is oriented so that the scattering vector

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

lies along

[111]

, close to the reciprocal lattice point

\mathbf{G}=(111)

. 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)) Using the law of cosines for the scattering triangle, find

|\mathbf{Q}|

. ((c)) State the one-phonon conservation condition that applies here (with the

\mathbf{G}

written explicitly), and use it with

|\mathbf{G}_{111}|=(2\pi/a)\sqrt3

to find the phonon wavevector

q

, and confirm the point

(q,\hbar\omega)

this measurement places on aluminium's dispersion curve by comparing

\hbar\omega

with

\hbar v_Lq

, taking the longitudinal sound speed along

[111]

v_L=6.5\times10^3\,\mathrm{m\,s^{-1}}

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