Self-diffusion in -iron (body-centred cubic

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

Solid State Physics, Ch Point defects, dislocations and alloys

Problem

Self-diffusion in

\alpha

-iron (body-centred cubic, lattice constant

a=2.87\,\text{\AA}

) proceeds by single vacancy jumps: each atom has

z=8

nearest neighbours at a jump distance

s=\tfrac{\sqrt3}{2}a

, and jumps into a vacant neighbouring site with attempt frequency

\nu_0

and migration energy

E_m

. The equilibrium fraction of sites that are vacant is the single Boltzmann factor

c_v=e^{-E_v/k_BT}

for a vacancy formation energy

E_v

(a metal vacancy carries no compensating charge, so no factor of

\tfrac12

is needed here, unlike for a Schottky pair). ((a)) Starting from

D=\tfrac16 s^2\Gamma

for a jump rate

\Gamma

, and

\Gamma=zc_v\nu_0e^{-E_m/k_BT}

, show that

D=D_0e^{-Q/k_BT}

with

D_0=\tfrac16 s^2z\nu_0

and

Q=E_v+E_m

. ((b)) State, with one line of reasoning, what

D_0

and

Q

become for an interstitial solute (such as carbon) that already occupies interstitial sites without needing a thermally created vacancy. ((c)) For

\alpha

-iron self-diffusion take

E_v=1.6\,\mathrm{eV}

E_m=1.3\,\mathrm{eV}

and

\nu_0=5\times10^{12}\,\mathrm{s^{-1}}

. Compute

D_0

Q

and the self-diffusion coefficient

D_{\rm Fe}

T=1000\,\mathrm{K}

. ((d)) Carbon diffusing interstitially in

\alpha

-iron has measured

D_0=2.0\times10^{-6}\,\mathrm{m^2\,s^{-1}}

and

Q=0.87\,\mathrm{eV}

. Compute

D_{\rm C}

1000\,\mathrm{K}

and the ratio

D_{\rm C}/D_{\rm Fe}

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