Born-Landé equation

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The Born-Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918 Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.

${\displaystyle E=-{\frac {N_{A}Mz^{+}z^{-}e^{2}}{4\pi \epsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right)}$

where:

• N_A = Avogadro constant;
• M = Madelung constant, relating to the geometry of the crystal;
• z⁺ = numeric charge number of cation
• z^- = numeric charge number of anion
• e = elementary charge, 1.6022×10^-19 C
• ?₀ = permittivity of free space

  4??₀ = 1.112×10^-10 C²/(J·m)

• r₀ = distance to closest ion
• n = Born exponent, typically a number between 5 and 12, determined experimentally by measuring the compressibility of the solid, or derived theoretically.

Video Born-Landé equation

Derivation

The ionic lattice is modeled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.

Electrostatic potential

The electrostatic potential energy, ${\displaystyle E_{\text{pair}}}$, between a pair of ions of equal and opposite charge is:

${\displaystyle E_{\text{pair}}=-{\frac {z^{2}e^{2}}{4\pi \epsilon _{0}r}}}$

where

${\displaystyle z}$ = magnitude of charge on one ion

${\displaystyle e}$ = elementary charge, 1.6022×10^-19 C

${\displaystyle \epsilon _{0}}$ = permittivity of free space

${\displaystyle 4\pi \epsilon _{0}}$ = 1.112×10^-10 C²/(J m)

${\displaystyle r}$ = distance separating the ion centers

For a simple lattice consisting ions with equal and opposite charge in a 1:1 ratio, interactions between one ion and all other lattice ions need to be summed to calculate ${\displaystyle E_{M}}$, sometimes called the Madelung or lattice energy:

${\displaystyle E_{M}=-{\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r}}}$

where

${\displaystyle M}$ = Madelung constant, which is related to the geometry of the crystal

${\displaystyle r}$ = closest distance between two ions of opposite charge

Repulsive term

Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to ${\displaystyle 1/r^{n}}$ so that the repulsive energy term, ${\displaystyle E_{R}}$, would be expressed:

${\displaystyle \,E_{R}={\frac {B}{r^{n}}}}$

where

${\displaystyle B}$ = constant scaling the strength of the repulsive interaction

${\displaystyle r}$ = closest distance between two ions of opposite charge

${\displaystyle n}$ = Born exponent, a number between 5 and 12 expressing the steepness of the repulsive barrier

Total energy

The total intensive potential energy of an ion in the lattice can therefore be expressed as the sum of the Madelung and repulsive potentials:

${\displaystyle E(r)=-{\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r}}+{\frac {B}{r^{n}}}}$

Minimizing this energy with respect to ${\displaystyle r}$ yields the equilibrium separation ${\displaystyle r_{0}}$ in terms of the unknown constant ${\displaystyle B}$:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} E}{\mathrm {d} r}}&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r^{2}}}-{\frac {nB}{r^{n+1}}}\\0&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r_{0}^{2}}}-{\frac {nB}{r_{0}^{n+1}}}\\r_{0}&=\left({\frac {4\pi \epsilon _{0}nB}{z^{2}e^{2}M}}\right)^{\frac {1}{n-1}}\\B&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}n}}r_{0}^{n-1}\end{aligned}}}$

Evaluating the minimum intensive potential energy and substituting the expression for ${\displaystyle B}$ in terms of ${\displaystyle r_{0}}$ yields the Born-Landé equation:

${\displaystyle E(r_{0})=-{\frac {Mz^{2}e^{2}}{4\pi \epsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right)}$

Maps Born-Landé equation

Calculated lattice energies

The Born-Landé equation gives a reasonable fit to the lattice energy

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Born Exponent

The Born exponent is typically between 5 and 12. Approximate experimental values are listed below:

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See also

• Kapustinskii equation
• Born-Mayer equation

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References

Source of the article : Wikipedia