A crystal, which has a lattice spacing of 0.50 nm, has many lattice vacancies wh

Question

A crystal, which has a lattice spacing of 0.50 nm, has many lattice vacancies which act as individual electron traps. Models of lattice vacancies can be used to predict behavior: Suppose the lattice traps can be approximated to a series of unconnected infinite sided boxes of width 0.50 nm. Sketch accurately the allowed wavefunctions and the associated probability densities of the trapped electron using the infinite side box model, for n = 1, 2, 3. What is the minimum energy a trapped electron can have (express your answer in eV)? UV laser is pointed at the crystal, and the light from the laser has a wavelength of 275 nm. Using the infinite side box model, would you expect the trapped electrons to absorb the laser photons via the photoelectric effect? Explain your answer. Would you expect, using the infinite side box model, the trapped electrons to scatter the laser photons via the Compton scattering effect? Explain your answer;

Explanation / Answer

Hi,

They are telling us to use the model of an infinite sided box. This means that the electrons trapped inside those boxes can only move in one dimension and along a certain distance (which in this case is equal to 0.5 nm).

(i) Unfortunately I cannot draw the wavefunctions, but at least I can show you the mathematical expression of said functions:

f(x) = (L/2)1/2 sin (nx/L) ; where L is the length of the box, n is an integer that represent the energy levels and x is the position of the particle (in this case electron) between 0 and L.

If n = 1 ::::::: f(x) = 2 sin (2x)

if n = 2 ::::::: f(x) = 2 sin (4x)

if n= 3 :::::::: f(x) = 2 sin (6x)

Note: in all the previous cases, L = 0.5 nm

(ii) The energy of an electron trapped in one of this boxes is equal to:

E(n) = (h2 / 8mL2) n2 ; where h is Planck Constant (which value is 6.626*10-34 J*s), m is the mass of the electron (which value is 9.109*10-31 kg), L is the distance of 0.5 nm and n is an integer referent to the energy level.

The minimum level an electron can have will be the energy of an electron when n is minimum, so n = 1:

E = [ 6.626*10-34 J*s ] 2 / [ 8*(0.5*10-9 m)2(9.109*10-31 kg) ] = 2.410*10-19 J = 1.50 eV

(iii) According to the fotoelectric effect, the electron can absorb all the energy of the photon if said energy is enough to free the electron from the metal; if the energy that the electron can give is not enough, then it is re-emitted.

In this case, the energy of any photon emitted by the UV laser is equal to 4.512 eV (was previously calculated). That energy will be hardly enough to promote an electron in the minimum level of energy to the second one:

E2 = 4E1 = 6.00 eV :::::::: (4.512 eV + 1.50 eV = 6.012 eV)

However, said amount of energy will not be able to free the electron from the metal so the energy will be re-emitted and no electron will scape the surface of the metal.

To sum up, I will not expect the crystal to absorb the laser photons via the fotoelectic effect.

(iv) The Compton scattering effect occurs when a photon colides with an electron and gives it part of energy, therefore the photon will have and incresed wavelength.

As in this effect the electrons can receive energy, even if they don't scape from the material, I would expect that part of the laser to be scattered via this effect.

I hope it helps.

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