How Does Band Gap Influence Lattice Spacing in Materials?

[Sr1]

Homework Statement

(a) (10 Points) Diamond has a band gap Eg equal to 5.5eV (at standard temperature and pressure). Use this number to derive a rough estimate of the lattice spacing, a, of the diamond lattice. Do you expect the true lattice spacing to be larger or smaller than your estimate?

(b) (5 Points) What is the minimum wavelength at which a diamond in a jewelry store is opaque? How does this wavelength depend on the size of the diamond?

Relevant Equations

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How can we link the band gap to lattice spacing?
For (a), if we purely do dimension analysis, then I would guess $$a=\frac{\hbar c}{E_g}$$. But what's the reason behind this answer, and will the true lattice spacing be larger or smaller?
For (b), I guess $$\lambda=\frac{\hbar c}{E_g}$$ due to band gap = photon energy. But I have no idea on the second question.
Also, dose it make sense to have $\lambda=a$?

 

[Sr1]

ok from dirac comb model, $$\frac{\sqrt{2mE}}{\hbar}a\sim \pi,$$ then $a=0.26nm$. The remaining questions are: Do you expect the true lattice spacing to be larger or smaller than your estimate? How does this wavelength(band gap) depend on the size of the diamond?