I did XRR analysis to calculate the thickness of thin film. can any body explain what is critical angle in XRR measured graph. and how to measure this critical angle???
Dear Ali,
the critical angle in XRR is the critical angle of total reflection thetac.
This angle is defined via cos(thetac) = realpart(n) (eq. 1)
with n being the complex index of refraction:
n= 1-delta - i*ß (eq. 2).
The real part (1-delta) is a little bit smaller than 1 and this parameter governs the refraction.
ß bears the attenuation properties of the material.
From eq. 1 and eq. 2 we conclude:
cos(thetac) = 1-delta and from that (thetac)2 = 2*delta (eq. 3).
However determination of thetac from the measured graph is only easily done when ß is neglectable with repect to delta; i.e. in the case of very weak x-ray absorbing material.
In this case thetac is equal to the angular position of the steep cut off of the reflection curve going from total reflection regime around 100% rapidly down to a few percent and even much lower. Here the angular region of refraction shows up and the Kiessig fringes are produced in the case of a thin film.
In the general case the x-ray attenuation is not small enough. In these cases there is only a gradual decrease of reflectivity observed. So the thetac does not up as a sharp signature.
Here you have to fit the reflectance with respect to delta and ß via the appropriate Fresnel equations. When knowing delta then apply eq. 3 to calculate your thetac.
The Fresnel formalism is decribed in
"Surface studies of solids by total reflection of x-rays";
L.G. Parrat; Phys. Rev. Vol. 95, No. 2; pp. 359-369 (1954)
and
"Ultra-soft x-ray reflection, refraction and production of photo-electrons (100-1000eV region)";
B. L. Henke; Phys. Rev. A, Vol. 6 No. 1; pp. 94 - 104 (1972)
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