Four measurements on a hypothetical ionic crystal

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Electronic transport in solids

Solid State Physics, Ch Point defects, dislocations and alloys

Problem

Four measurements on a hypothetical ionic crystal, a tracer-diffusion couple, a polycrystalline steel and a metal wire are each summarised below. Translate each into the physical quantity asked for. ((a)) An Arrhenius plot of

\sigma T

against

1/T

for an ionic crystal has an intrinsic-region slope corresponding to an activation energy of

1.90\,\mathrm{eV}

and an extrinsic-region slope corresponding to

0.85\,\mathrm{eV}

. Find the Schottky formation energy

E_S

and the migration energy

E_m

. ((b)) A tracer-diffusion experiment gives

D = 1.0\times10^{-16}\,\mathrm{m^2\,s^{-1}}

T=1200\,\mathrm{K}

and

D = 5.0\times10^{-15}\,\mathrm{m^2\,s^{-1}}

T=1400\,\mathrm{K}

. Find the activation energy

Q

and the prefactor

D_0

D = D_0 e^{-Q/k_BT}

. ((c)) A Hall-Petch plot of yield stress against grain size for a steel gives

\sigma_y = 220\,\mathrm{MPa}

at grain size

d=100\,\mu\mathrm{m}

and

\sigma_y = 310\,\mathrm{MPa}

d=25\,\mu\mathrm{m}

. Find

k_y

and

\sigma_0

\sigma_y = \sigma_0 + k_yd^{-1/2}

. ((d)) A metal wire has

\rho(300\,\mathrm{K}) = 1.7\,\mu\Omega\,\mathrm{cm}

and

\rho(4.2\,\mathrm{K}) = 8.5\times10^{-4}\,\mu\Omega\,\mathrm{cm}

. Using an increment of

1\,\mu\Omega\,\mathrm{cm}

per atomic per cent of impurity, find its impurity concentration in ppm.

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