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Is there any information on the actual number of Nordheim theory? A paper I found says, the resistivity of a binary alloy follows, $$\rho=x(1-x)(V_a-V_b)^2 $$ ,where $V_a$ and $V_b$ are "potential", which I don't know exactly. And the paper shows the experimental plot of the resistivity of Cu-Au alloy.

I want to estimate the resistivity of Cu-In alloy, but I could not find the specific number of $V_a$ and $V_b$ in this Nordheim theory.

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The Nordheim rule is essentially empirical. Thus if you measure (or look up) the resistivity at one concentration you can estimate what it will be at different concentrations. In general this will be a function of the preparation of the alloy anyways (for example what sort of crystal structure the 'solvent' has or how thin the film is).

I didn't come across the $(V_b^2-V_a^2)$ term, but what I usually see is an equivalent which looks like $g \langle \psi | \Delta U | \psi \rangle$ where $g$ is some sort of density of states, $\psi$ is the (bloch) wavefunction for the (fully periodic) average lattice, and $\Delta U$ is the difference between the potentials of the two crystal lattices. The term I wrote then gives the scattering (in a perturbative sense) off the difference potential. A more detailed consideration 1 shows that the concentration shows up as $x(1-x)$ only when the difference potential is 'random'. If the alloy formed a perfect lattice (e.g. AuCu$_3$) then it would not hold.

References:

(1) G.Hall "Nordheim's theory of the resistivity of alloys" Phys. Rev. 1959

(2) textbook chapter (pdf)

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  • $\begingroup$ Thank you. So what I need is to get some resistivity values for certain proportions to estimate that of the other proportion. Do you know the resistivity of Cu-In alloy? Also I want to know the resistivity of Ag-In alloy, too. $\endgroup$ – Long Dec 10 '18 at 2:57

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