Question

(a) What maximum light wavelength will excite an electron in the valence band of diamond to the conduction band? The energy gap is 5.50 eV. (b) In what part of the electromagnetic spectrum does this wavelength lie?

Short answer

a) The maximum wavelength of the light that will excite electrons in the valence band of diamond to the conduction band is 226 nm.

b) The wavelength lies ultraviolet region in the electromagnetic spectrum.

Step-by-step solution

The given data

Energy gap of diamond,$E_g=5.50\hspace{0.33em}eV$

Understanding the concept of energy gap

The valence and the conduction band are separated by an energy gap. When the electron jumps from the conduction band to the valence band, a photon is released and the energy of the photon is equal to the energy gap between those two bands.

Formula for the wavelength of the released photon is given as-

$\mathit{E}\mathbf{=}\frac{\mathbf{h}\mathbf{c}}{\mathbf{\lambda }}$

Here, c is the speed of light in vacuum, E is the energy of the photon, $\mathit{\lambda }$ is the wavelength of photon and his the plank’s constant.

a) Calculation of the maximum wavelength of the light

Using the concept and the given energy gap in equation (i), we can get the value of the maximum wavelength required for the excitation as follows:

$$\begin{gathered} {\lambda }_{max}=\frac{hc}{E_g} \\ =\frac{\left(6.626\times {10}^{-34}J.s\right)\left(3\times {10}^{-8}m/s\right)}{\left(5.5eV\right)\left(1.6\times {10}^{-19}J/eV\right)} \\ =2.26\times {10}^{-7}m \\ =226\mathrm{nm} \end{gathered}$$

Hence, the wavelength of the photo is 226 nm.

b) The wavelength region in the electromagnetic spectrum

Comparing with the electromagnetic spectrum, we can see that the wavelength $\lambda =226\mathrm{nm}$lies in the ultraviolet region.

Hence, the region of the spectrum is ultraviolet region.