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Physics / Chapter 41: Conduction of Electricity in Solids / Section 41-4

Metals — Question 7

Section 41-4 · Physics pages 6864–6865

Question 7 Residual resistivity separates impurities from phonon scattering In a temperature range where the model is valid, a metal has ρ(T) = ρ0 + bT. Measurements give ρ(100 K) = 1.20 × 10−8 Ωm and ρ(300 K) = 3.20 × 10−8 Ωm. T ρ ρ0 Schematic; labels, not drawing lengths, specify the data. (a) Infer residual resistivity and b. (b) If purification halves only ρ0, find the new resistivity at 300 K and the ideal residual-resistivity ratio ρ(300)/ρ0.
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Question 7 - Solution Residual resistivity separates impurities from phonon scattering T ρ ρ0 (a) Method and calculation b = 3.20 × 10−8 −1.20 × 10−8 300 −100 = 1.00 × 10−10 Ωm/K. ρ0 = 1.20 × 10−8 −100b = 2.00 × 10−9 Ωm . (b) Interpretation and further calculation ρnew(300) = 1.00 × 10−9 + 300b = 3.10 × 10−8 Ωm . RRRnew = 31.0 (RRRold = 16.0). Physical check The residual term models temperature-independent elastic scattering. The linear phonon term is an assumed fit over the measurement range, not a universal low-temperature law.

Question 7

Residual resistivity separates impurities from phonon scattering

In a temperature range where the model is valid, a metal has ρ(T)=ρ0+bT\rho(T)=\rho_0+bT. Measurements give ρ(100K)=1.20×10−8Ωm\rho(100\,\mathrm K)=1.20\times10^{-8}\,\Omega\,\mathrm m and ρ(300K)=3.20×10−8Ωm\rho(300\,\mathrm K)=3.20\times10^{-8}\,\Omega\,\mathrm m.

Diagram shown in the original SVG worksheet.

  1. Infer residual resistivity and b.

  2. If purification halves only ρ0\rho_0, find the new resistivity at 300 K and the ideal residual-resistivity ratio ρ(300)/ρ0\rho(300)/\rho_0.

Question 7 - Solution

Residual resistivity separates impurities from phonon scattering

Diagram shown in the original SVG worksheet.

(a) Method and calculation

b=3.20×10−8−1.20×10−8300−100=1.00×10−10Ωm/K.b=\frac{3.20\times10^{-8}-1.20\times10^{-8}}{300-100}=1.00\times10^{-10}\,\Omega\,\mathrm{m/K}.ρ0=1.20×10−8−100b=2.00×10−9Ωm.\boxed{\rho_0=1.20\times10^{-8}-100b=2.00\times10^{-9}\,\Omega\,\mathrm m}.

(b) Interpretation and further calculation

ρnew(300)=1.00×10−9+300b=3.10×10−8Ωm.\boxed{\rho_{\mathrm{new}}(300)=1.00\times10^{-9}+300b=3.10\times10^{-8}\,\Omega\,\mathrm m}.RRRnew=31.0(RRRold=16.0).\boxed{{\mathrm{RRR}}_{\mathrm{new}}=31.0}\quad({\mathrm{RRR}}_{\mathrm{old}}=16.0).

Physical check

The residual term models temperature-independent elastic scattering. The linear phonon term is an assumed fit over the measurement range, not a universal low-temperature law.

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