Band gaps - fused silica glass

[accountkiller]

Homework Statement

Glass is transparent to visibile light under normal conditions; however, at extremely high intensities, glass will absorb most of the light incident upon it. This works through a process known as multiphoton absorption. In this process, several photons are absorbed at the same time. If very intense light whose photons carry 2 eV of energy is shined onto a material with a band gap of 4 eV, that light can be absorbed through two-photon absorption, because two photons have the right amount of energy to bridge the band gap. What is the minimum number of photons of 800 nm light that are needed to equal or exceed the band gap of fused silica glass?

Homework Equations

E = hf
c = Yf ... (I mean the Y here to be delta)

The Attempt at a Solution

The energy of a photon with a wavelength of 800 nm is:
f = c/Y = c/(800E-9) = 3.7375E14 Hz
f = 3.7375E14 Hz

E = hf = h(3.37375E14) = 2.476E-19 Joules
E = 2.476E-19 J

The energy of a 800-nm photon is 2.476E-19 Joules per photon.

... The band gap is 4 eV, which = 6.408E-19 J

So I do the band gap energy divided by the energy of one photon to give me the number of photons, right? ...
(6.408E-19 Joules)/ [2.476E-19 Joules/photon] = 2.588 photons.

But MasteringPhysics says no. Where did I go wrong?

 

[fzero]

What is 0.588 of a photon?

 

[accountkiller]

Well, yes, that's a bit crazy. But I didn't know any other way to do it, so I just tried putting in whatever I was getting just to try it. Sometimes, MasteringPhysics gives tips on how to go from our wrong answer to the right one.

So then, my method is wrong. How do I do it?

 

[fzero]

A single photon is the base object with energy $$hc/\lambda$$. If 2 photons with this wavelength are not enough to bridge the band gap, what is the minimum number of photons that do?

 

[accountkiller]

Ohh, I see now. The 2.4831E-19 J is per TWO photons, not per one photon. So then my answer would be 6. Thank you for the extra help!

 

[fzero]

No the band gap is equal to 2.588 times the energy of a single photon. What is the smallest number of photons such that their total energy is equal or greater than the band gap?