CND-12

Madelung energy

U = − N α k e² / r. Ionic lattice energy with Madelung constant α.

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Electrons in solidsMadelung

Governing equation

U=-N\alpha\dfrac{ke^2}{r}

$r \in \left[1.5, 5\right]$

where

N: Amount (mol)
\alpha: Madelung α (—)
r: Nearest neighbour (Å)
U: Lattice energy (kJ)

The figure, part by part

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①

Governing law — red

U=-N\alpha\dfrac{ke^2}{r}

$r \in \left[1.5, 5\right]$

$r = 1.5$ $U = -161.9$

Along the curve

②

Domain of the variable — blue

r_{\min} = 1.5,\quad r_{\max} = 5

$r \in \left[1.5, 5\right]$

$r = 1.5$ $r = 1.5$

Along the curve

③

Response curve — green

U=-N\alpha\dfrac{ke^2}{r}

$r \in \left[1.5, 5\right]$

$r = 1.5$ $U = -161.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}U}{\mathrm{d}r}

$r \in \left[1.5, 5\right]$

$r = 1.522$ $U^{\prime} = 104.9$

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-12 — Madelung energy) is the form associated with Madelung. Working symbols: $N$ , $\alpha$ , $r$ $\rightarrow$ $U$ . NaCl has α = 1.748. The Born–Landé repulsive wall is omitted on this sheet.

Purpose

Purpose: compute

U

from

N

\alpha

r

in Condensed matter via

U=-N\alpha\dfrac{ke^2}{r}

U = − N α k e² / r. Ionic lattice energy with Madelung constant α. 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

N = 1.000\,\mathrm{mol}

\alpha = 1.748\,\mathrm{—}

r = 2.820\,\mathrm{Å}

, the governing relation

U=-N\alpha\dfrac{ke^2}{r}

yields

U = -86.1\,\mathrm{kJ}

. A rock-salt lattice, an attractive well. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Amount N1 molMadelung α \alpha1.748 —Nearest neighbour r2.82 Å

Outputs

Lattice energy U-86.1 kJ

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Narration of this film

A rock-salt lattice, an attractive well.

NaCl has α = 1.748. The Born–Landé repulsive wall is omitted on this sheet.

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