1. geometric smearing in GIWAXS / RWAXSh l
side view a
CCD screen f L
top view w y
beam height h
sample length l
incident angle a
sample-detector distance L
scattering angle y
Dx: geometric smearing of diffraction spot
Dy: apparent angular smearing
(note: this analysis assumes
a sharp Bragg reflection) Dy Dx

2. formulae and numbers
if l is the length of sample along the beam and h the beam height,
the footprint of the beam is given by: f = min{l, h/sin(a)}
geometric smearing of the diffracted beam on the detector
Dx = (L+f/2) tan(y) – (L-f/2) tan(y) = f tan(y)
apparent angular broadening due to smearing
Dy = {Dx cos(y)} / {L / sin(y)} = (f/L) tan(y) sin(y) cos(y)
conclusion:
no problem for GISAXS (y<5°)
problem for GIWAXS past 20°
solutions: small samples or larger angle of incidence at cost of intensity
vertically focused beam (requires focusing mirror)

3. ring doubling at very low incidence
lower regions get “shadowed”
as x-rays cannot penetrate bumps
any more due to total external reflection
Conclusions:
Depth profiling below ac is extremely problematic
for wavy surfaces.

4. bandwidth smearing total smearing: Dytot = {Dygeo2 + Dybw2}1/2
the D1 multilayer mono has a bandwidth Dl/l of 1.5%
Bragg’s law yields:
l = 2 d sin(q) and Dl = 2 d cos(q) Dq or
Dq = tan(q) Dl/l
or in terms of scattering angle y = 2q
Dy‘ = 2 tan(y/2) Dl/l
total smearing: Dytot = {Dygeo2 + Dybw2}1/2