A single crystal of aluminium (fcc

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

Solid State Physics, Ch Diffraction as measurement: form factors, thermal motion and refinement

Problem

A single crystal of aluminium (fcc, conventional cubic lattice constant

a=4.05\,\mathrm{\AA}

, Debye temperature

\Theta_D=428\,\mathrm{K}

, molar mass

26.98\,\mathrm{g\,mol^{-1}}

) is heated from

300\,\mathrm{K}

900\,\mathrm{K}

. Treat the atomic displacements as harmonic and Gaussian, so that

2W=\langle u_x^2\rangle Q^2

for this cubic crystal, and use the Debye estimate

\langle u_x^2\rangle=3\hbar^2T/(Mk_B\Theta_D^2)

per axis at each temperature. ((a)) Estimate the percentage of the

(220)

Bragg intensity that has moved into the diffuse background between

300\,\mathrm{K}

and

900\,\mathrm{K}

. ((b)) A laboratory source delivers

\mathrm{Cu\,K}\alpha

X-rays, for which the linear attenuation coefficient of aluminium is

\mu_X=130\,\mathrm{cm^{-1}}

. Thermal neutrons have an attenuation length of

100\,\mathrm{mm}

in aluminium. Estimate the factor by which a neutron beam penetrates further into the crystal than the X-ray beam, and say briefly why this matters for measuring the weak diffuse background.

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