Let's say I irradiate a sample that has an absorbance β with light of a wavelenght λ1. Is there a way to relate the initial λ1 and the diffracted/scattered λ2 using β?
Most materials will have λ2=λ1. β (which is a function of the wavelength) just tells you how much light gets reflected/scattered.
Absorbance is wavelength dependent You can find the absorbance as a function of wavelength for many materials in palik's handbook of optical propeties. Most materials have peak absorbance where there is resonance between the energy levels of the various energy carriers and the incident photon energy (wavelength)
My sample has parabolic bands instead of quadratic, they touch inside the Brillouin zone, so I'm not sure as to how to find the reason of the ressonance with this kind of e-band.
What kind of sample do you have? Parabolic band and quadratic band are the same thing. There are different band corresponding to the different levels and it is the transition between these levels that I was talking about. You can have two different types of transitions, interband and intraband. Interband transition is the transition between the conduction and valence bands (electrons and holes) Intraband transitions are the transitions between the quantized levels within the conduction or valence band.
I am not familiar with this material. Is it similar to graphene dispersion relation, where you have a discontinuity at k=0 (or infinite effective mass). To know the absorption, you must do the following. Using Fermigolden rule. calculate the transition rate for an electron from one band to another due to the absorption of the photon. This is done as follows. You have to find the matrix element, <i|H'|f>, where, H' is the perturbing potential and in this case can be treated as a dipole moment caused by a classical electromagnetic wave, i.e H' = eEr*exp(ik*r). The initial and final states of the electrons are given by plane waves modulated by block functions. Finally, you need the density of states g(E). This is where your dispersion relation comes into play. In the case of silicene, assuming that it is similar to graphene, i would assume a 2D density of states.
There's literature about the use of FGR to calculate the absorption in graphene, it yields the result A= Nπα, N is the number of layers (in my case, I use single layer silicene, or single layer graphene in the case you are proposing), and α is the fine structure constant. So for N=1, A= πα. But this doesnt help me understand this high intensity i'm seeing in dark field, i cannot relate the two facts