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Question 5.25: Find the energy at the bottom of the first allowed band, for...

Find the energy at the bottom of the first allowed band, for the case β = 10, , correct to three significant digits. For the sake of argument, assume α/a = 1 eV.

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We’re looking for a solution to Eq. 5.73 with β = 10 and z \lesssim \pi: f(z)=\cos z+10 \frac{\sin z}{z}=1 .

f(z) \equiv \cos (z)+\beta \frac{\sin (z)}{z}              (5.73).

Mathematica gives z = 2:62768. So E=\frac{\hbar^{2} k^{2}}{2 m}=\frac{\hbar^{2} z^{2}}{2 m a^{2}}=\frac{z^{2}}{2 \beta} \frac{\alpha}{a}=\frac{(2.62768)^{2}}{20} eV =0.345 eV .

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