Hello I am wondering how to calculate XRR density using the common formula for the critical angle of
θc ≈ sqrt(δ)
where
δ=(electron _radius*electron_density*( λ)^2)/2π
where the electron density is = 0.007/Angstrom^3
electron radius = 2.817E-5 Angstrom
lambda = 1.544 Angstrom
The resulting value is much smaller than any reasonable critical angle?
What am I doing incorrectly?
Dear Muneeb Ahme ,
I do not know from where you fetched your formula.
Please see the relations to calculate the critical angle at page 8 of:
Presentation Attenuation Length and Penetration Depth of X-rays
They were picked up from the Parratt paper, the details of which are given on page 9.
I will send you this paper.
By the way: pay attention; yours and the 'Parratt' formula will give the angle in radians, but not in degree...
Please convert accordingly
θc[deg] = θc[rad]*180/pi
180/pi ~ 57,3; that might be the factor you are missing...
Additional remark:
from page 8 you see, that
θc ≈ sqrt(2*δ) and not θc ≈ sqrt(δ) as you wrote above....
So your calculated values of θc are always much smaller than any real ones.
Best regards
G.M.
Dear Muneeb Ahme ,
I hope this answers your question:
1) The measured intensity at the detector during an XRR measurement
I(Q) ∝ f(Q).f*(Q)
with f(Q) = f1(Q) + i.f2(Q) = TF[ρ_e] which means that the structure factor f(Q) is the Fourrier transform of the electron density ρ_e, and is composed of two correlation factors f1 and f2 responsible for dispersion and absorption respectively.
2) From the following expression of the refractive index:
n = 1 - δ( λ) + iβ( λ) ≃ 1 - δ( λ).
This is an approximation that is valid for surface reflectivity (but not really the case in transmission mode as absorption is much more significant). The optical theorem gives the following relation (see: S Brück et al 2011 New J. Phys. 13 033023):
δ( λ) = (N_a . f1( λ) . electron_radius . ρ_m . λ^2)/(2 . pi . M)
=> δ( λ)≃ (electron_radius . ρ_e . λ^2)/(2 . pi)
with N_a is the avogadro number, M the molar mass and ρ_m the mass density.
4) Snell's law at critical angle:
cos(θc) = 1 - n = 1 - δ( λ) with cos(θc) ≃ 1 - (θc^2 /2)
Consequently:
θc ≈ sqrt(2*δ)
We finally deduce the electron density:
ρ_e = (pi . θc^2)/(electron_radius . λ^2)
Best regards,
Elhoucine
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