CND-23

Bloch T⁵ resistivity

ρ = ρ₀ + A T⁵. Low-temperature phonon resistivity of a pure metal.

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Electrons in solidsBloch T⁵

Governing equation

\rho=\rho_0+A T^5

$T \in \left[1, 40\right]$

where

\rho_0: Residual (nΩ·m)
A: Bloch prefactor (aΩ·m/K⁵)
T: Temperature (K)
\rho: Resistivity (nΩ·m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\rho=\rho_0+A T^5

$T \in \left[1, 40\right]$

$T = 1$ $\rho = 0.202$

Along the curve

②

Domain of the variable — blue

T_{\min} = 1,\quad T_{\max} = 40

$T \in \left[1, 40\right]$

$T = 1$ $T = 1$

Along the curve

③

Response curve — green

\rho=\rho_0+A T^5

$T \in \left[1, 40\right]$

$T = 1$ $\rho = 0.202$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\rho}{\mathrm{d}T}

$T \in \left[1, 40\right]$

$T = 1.245$ $\rho^{\prime} = 0.026$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-23 — Bloch T⁵ resistivity) is the form associated with Bloch T⁵. Working symbols: $\rho_0$ , $A$ , $T$ $\rightarrow$ $\rho$ . Bloch–Grüneisen. At high T the same phonons give ρ ∝ T.

Purpose

Purpose: compute

\rho

from

\rho_0

A

T

in Condensed matter via

\rho=\rho_0+A T^5

ρ = ρ₀ + A T⁵. Low-temperature phonon resistivity of a pure metal. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho_0 = 0.200\,\mathrm{nΩ·m}

A = 2.000\,\mathrm{aΩ·m/K⁵}

T = 20.000\,\mathrm{K}

, the governing relation

\rho=\rho_0+A T^5

yields

\rho = 6400.200\,\mathrm{nΩ·m}

. A ρ(T) that starts as T⁵ then becomes linear. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Residual \rho_00.2 nΩ·mBloch prefactor A2 aΩ·m/K⁵Temperature T20 K

Outputs

Resistivity \rho6400.200 nΩ·m

Reading speed

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Narration of this film

A ρ(T) that starts as T⁵ then becomes linear.

Bloch–Grüneisen. At high T the same phonons give ρ ∝ T.

Reading speed

Watch on YouTube