How Can I Calculate the Dielectric Susceptibility of Silicon Correctly?

[jpapa]

Hi everybody,

i'm trying to calculate the dielectric susceptibility of Silicon (Si) using the formula

[tex] \chi_{\vec g}=S(\vec g)(-\frac{\omega_p^2}{\omega^2})\frac{F(\vec g)}{Z}e^{-M} [/tex]

where [itex]S(\vec g)[/itex] is the structure factor, [itex]\omega_p[/itex] is the plasma frequency, [itex]\omega[/itex] is the frequency, [itex]F(\vec g)[/itex] is the atomic form factor, Z is the atomic number of Silicon (14) and M is the Debye-Waller factor (i take it equal to zero).

I'm interested in (111) planes, so [itex]S(\vec g)=4+4i[/itex].

The problem is that I'm not getting the correct result in my calculations. I'd like to know if i can find somewhere values of the dielectric susceptibility with respect to the energy [itex]\omega[/itex]. I work with physical units. Thus [itex]\omega[/itex] is given in eV and I'm in the X-ray region (keV).

Additionally i'd like to know if the values from http://physics.nist.gov/PhysRefData/FFast/html/form.html" that i use for the atomic form factor [itex]F(\vec g)[/itex] are correct or whether i need to make some changes.

What confuses me is that only the outer 2 electrons contribute to the flux in the crystal and since the crystal system of Si is fcc with 2 atoms base that means than in every unit cell exist [itex](8\frac{1}{8}+6\frac{1}{2})\times 2 \times 2[/itex] free electrons. Do i need to include that somewhere in the formula above?

Thank you all in advance,
John

 

[eljose79]

Hello John,

Thank you for your question. Calculating the dielectric susceptibility of a material can be a complex task, and it is important to make sure all the variables and factors are taken into account. Let's break down the formula you have mentioned and address some of your concerns.

Firstly, the structure factor S(\vec g) represents the arrangement of atoms in the crystal lattice and takes into account the interference of x-rays with the crystal structure. In your case, you mentioned that you are interested in (111) planes, which have a face-centered cubic lattice. However, the value you have given for S(\vec g) seems incorrect. The structure factor for (111) planes in a face-centered cubic lattice is actually 8+8i, not 4+4i. This difference in structure factor can significantly affect your calculation.

Next, the atomic form factor F(\vec g) represents the scattering amplitude of an atom at a particular momentum transfer \vec g. The values from the NIST database are correct for a neutral atom, but it is important to note that in a crystal, the atom is no longer neutral due to the presence of neighboring atoms. This can cause the atomic form factor to deviate from the values in the database. You may need to use a more advanced form factor calculation that takes into account the crystal environment.

You also mentioned that only the outer 2 electrons contribute to the flux in the crystal. This is true, but it is already taken into account in the atomic form factor, which is a function of the atomic number Z. So there is no need to include it separately in the formula.

Lastly, it is important to note that the Debye-Waller factor M is not always zero. It takes into account the thermal vibrations of the atoms in the crystal and can have a significant effect on the final result. It is usually experimentally determined, so you may need to find literature values for M for silicon in the X-ray region.

I hope this helps clarify some of your concerns. It is always a good idea to double-check all the variables and factors in your calculations to ensure accuracy. Good luck with your research!