GISAXS Conversion Formulae


Important numbers in brief
Quantum 1 pixel size : 70.7 micron x 70.7 micron
Quantum 1 pixel number : 1152 x 1152
Quantum 1 active area : 81.45mm x 81.45mm
Quantum 1 distance fluorescent screen to housing : 3 mm
Silver behenate lattice constant : 58.38 Angstroem
 

For quantitative information, you need to convert from pixels to q-space. First step is to convert from pixels to mm using the pixel size of the camera (called "conversion" in the following). For the Quantum 1 detector often used at G1 the pixel size is

conversion = 70.7 microns.
Then you can convert to angles knowing the distance L of sample to CCD and the position (in pixels) of the direct beam. When you measure L_measured from the outside frame of the CCD to the sample, you have to add 3mm to account for the distance of the fluorescent screen from the outside frame of the Quantum 1 detector:
L = L_measured + 3 mm
Next thing you need to know is the x-ray wavelength lambda. If you use a silver behenate standard, the wavelength can be derived from the silver behenate lattics constant and Bragg's law:
lambda=2*d*sin(0.5*ring_radius*conversion/L)
where ring_radius is the radius (in pixels) of the first behenate ring. d is the behenate lattice constant of 58.38 Angstroems.
If several rings are visible (depending on your flightpath length), this value can be further refined. If you use a second standard, which provides a second Bragg condition, you can even determine lambda and L simultaneously.

Finally you are ready to determine the q-values. Now for GISAXS the perpendicular component q_perpendicular has a different formula than the q-component parallel to the sample surface q_parallel:

q_perpendicular = (2*PI/lambda) * {sin(alpha) + sin(beta)}
lambda is the x-ray wave length. alpha is the incident angle, which can be calibrated with the help of the position of the specular beam:
alpha = 0.5 * atan({ypixel#(specular beam) - ypixel#(direct beam)}*conversion / L)
beta is the exit angle from the surface:
beta   = atan({ypixel# - ypixel#(direct beam)}*conversion / L) - alpha


q_parallel is determined by the in-plane scattering angle psi:

q_parallel = (2*PI/lambda) * {sin(psi)}
psi = atan({xpixel# - xpixel#(direct beam)} * conversion / L)
which corresponds to the usual SAXS conversion formula.

In the above I assumed that the pixel number in the x direction (xpixel#) is in the direction parallel to the sample surface and the pixel number in the y direction (ypixel#) is perpendicular to the surface. Otherwise I am using the conventions from my paper.
 

Small-Angle Limit

Now, if you are in the small-angle limit, i.e. if the maximum scattering angles are smaller than 5 deg, the following holds:

alpha (in rad) = sin(alpha) = tan (alpha)
For the Quantum 1 detector this will be the case if  L is larger than
L > {0.5 * 70mm * root(2)} / tan(5deg) = 570 mm                         (direct beam in the center of the CCD)

L > {70mm} / tan(5deg) = 800 mm                                                 (direct beam on the bottom of the CCD)

L > {70mm * root(2) } / tan(5deg) = 1450 mm                               (direct beam in the corner of the CCD)


In this case the above formulae simplify: With
 

alpha + beta = {ypixel# - ypixel#(direct beam)} * conversion / L


we get:
 

q_perpendicular = (2*PI / lambda) * ({ypixel# - ypixel(direct beam)} * conversion / L)
p_parallel           = (2*PI / lambda) * ({xpixel# - xpixel(direct beam)} * conversion / L)


i.e. in the small-angle approximation we regain the usual SAXS conversion formulae. You will have to check your maximum scattering angle to see whether you can apply the approximation. Furthermore you should determine the alpha value for each GISAXS image, which is important for the modelling. The thus determined angle is more reliable than the one we found in the reflectivity scan, because the absolute zero of the theta motor is usually only known to within 0.1 deg.