- WAXS - Conversions
- WAXS in reflection geometry
- GIWAXS - Conventions
- Refraction correction
- Indexation
- Calibrations
- Questions & Comments

In the WAXS regime with scattering angles larger than 5 deg the usual small angles approximation alpha=sin(alpha)=tan(alpha) does not hold any more. Now we need to perform the full transformation from a plane (the detector) to a sphere (the Ewald sphere). As it turns out, this transformation is not very difficult.

If direct beam hits the area detector at (xpixelD, zpixelD) and an arbitrary point (xpixel, zpixel), then the scattering angle tth is given by

tan(tth)
= conversion sqrt{ (xpixel-xpixelD)^{2}
+ (zpixel-zpixelD)^{2}
} / LSD

The conversion from pixels to microns is 46.9 microns per pixel for the MedOptics CCD or 172 microns/pixel for Dectris Pilatus detectors, and LSD is the sample-detector distance. From this we determine the length of the scattering vector q the usual way:

q
= 4 PI sin(tth/2) / lambda

where lambda denotes the x-ray wavelength. For an isotropic sample we would be done now. In view of the further treatment we have already introduced the CHESS default hutch coordinate system with x towards the storage ring, y along the beam and the vertical direction z.

For anisotropic samples we introduce the detector azimuth angle azi around the direct beam with respect to the x axis.

Using the atan2 function which is used in transformations between 2D Cartesian and 2D polar coordinates, we get:

azi
= atan2(xpixel, zpixel)

(The atan2 unction is implemented in many programs and programming languages for the conversion from 2D cartesian to polar coordinates.)

Now we can determine the components of the incident and final wave vectors for elastic scattering in the lab system by

kilabx
= kilabz = 0

kilaby = k = 2 PI / lambda

kflabx = k sin(tth) cos(azi)

kflabz = k sin(tth) sin(azi)

kflaby = k cos(tth)

The scattering vector q is then obtained as usual as kilaby = k = 2 PI / lambda

kflabx = k sin(tth) cos(azi)

kflabz = k sin(tth) sin(azi)

kflaby = k cos(tth)

q
= k_{f,lab} - k_{i,lab}

The qy coordinate is very small and thus is often ignored for diffuse scatterers. Aternatively the parallel momentum transfer qpar can be introduced by

qpar
= sqrt(qx^{2}
+ qy^{2})

This notation makes use of the CHESS hutch coordinate system with qy along the beam and qz pointing up in the vertical direction; other choices of the coordinates are also often used in transmission geometry (e.g. qx, qy in the detector plane).

Radial integrations of such patterns can be obtained by converting to polar coordinates and integrating over azi. Such integration can also be obtained using fit2d. Textured samples, i.e. when the WAXS powder ring are inhomogenous, can also be analyzed conveniently with fit2d in polar coordinates (tth, azi) using the CAKE feature (see my fit2d primer).

alpha
incident angle
ki
incident wave
vector
del meridional angle (latitude)

psi in-plane scattering angle kf exit wave vector nu equatorial angle (longitude)

beta exit angle q=kf-ki scattering vector

phi sample aximuth - only important for anisotropic samples

psi in-plane scattering angle kf exit wave vector nu equatorial angle (longitude)

beta exit angle q=kf-ki scattering vector

phi sample aximuth - only important for anisotropic samples

Figure 1. Definition of surface scattering angles and lab scattering angles (following psi-circle convention)

reference: Smilgies & Blasini, J. Appl. Cryst. 40, 716-718 (2007).

For the surface coordinates we need to determine the incident angle alpha which is given by the surface alignment. Usually the samth turntable gets calibrated during line-up by measuring a reflectivity curve. If the reflectivity curve is too limited, the maximum of the reflectivity scan can be set to the critical angle of the sample which can be obtained from the CXRO website (http://henke.lbl.gov/optical_constants/).

alphaR
= alpha_c

For alpha=0, there are convenient formulae for psi and beta (see Handbook of International Crystallography under "Fiber Diffraction"):

tan(psi0)
= conversion (xpixel-xpixelD)/
LSD

tan(beta0) = conversion (zpixel-zpixelD)^{ }/
sqrt{ conversion^{2} (xpixel-xpixelD)^{2}
+ LSD^{2}}

tan(beta0) = conversion (zpixel-zpixelD)

psi0 and beta0 correspond to the longitude and latitude of the exit wave vector, respectively, when the incident beam points to the intersection of the Greenwich meridian and the equator. In fiber diffraction the angles psi0 and beta0 sometimes are called mu and nu, respectively; on a psi-circle diffractometer they are called nu and delta, respectively. We will call this frame the lab frame.

The scattering angle tth is related to psi0 and beta0 via the relation:

cos(tth)
= cos(psi0) cos(beta0)

For closer examination of the scattering features we need to introduce the surface frame relative to the sample surface.

If alpha is non-zero, but small, i.e. less than 1 deg as typically used in GIWAXS, a small correction applies to beta:

beta
= beta0 - alpha cos(psi0)

psi = psi0

psi = psi0

The scattering vector relative to the surface frame (qz in direction of the surface normal, qx along the projection of the beam onto the surface) is then given by (see for instance Smilgies, Rev. Sci. Instrum. 73, 1706 (2002) ):

qx
= k { cos(beta) cos(psi) - cos(alpha) }

qy = k { cos(beta) sin(psi) }

qz = k { sin(beta) + sin(alpha) }

qy = k { cos(beta) sin(psi) }

qz = k { sin(beta) + sin(alpha) }

So any point in the detector plane has a qy and qz component associated with it, but also a small qx compont. In surface diffraction from single crystals and 2D powders we would want to plot the scattering intensity as the perpendicular scattering vectot component q_perp versus the parallel component q_par, with

q_perp
= qz

q_par = sqrt(qx^{2}+qy^{2}) = k sqrt{ cos(alpha)^{2} -
2 cos(alpha) cos(psi) cos (beta) + cos(beta)^{2}}

q_par = sqrt(qx

Bragg scattering in GIWAXS is closely related to the various forms Grazing-Incidence Diffraction, as discussed in the book by Als-Nielsen and McMorrow.

Close to the incident plane given by psi=0, i.e. the scattering signals right above the beam stop, the Bragg condition cannot be fulfilled. Hence any scattering features in this region should be treated as diffuse. qx describes the offset of the cut of the Ewald sphere through the diffuse Bragg sheet from the actual Bragg reflection (at qx=0). Diffuse Bragg sheets are a means to characterize roughness correlations in thin films and multilayers/lamellar films. The theoretical description is in the framework of Distorted Wave Born Approximation (see Sinha et al., Phys Rev B 38, 2297-2311, 1988 and Gutmann et al, Physica B 283, 40-44, 2000).

This situation is similar to the meridional "reflections" in fiber diffraction, and the "reflections" close to the oscillation axis in protein crystallography, which are actually due to diffuse scattering. The limiting curve dividing the Bragg region from the diffuse scattering region has the form

q_perp
< q_perp_lim = sqrt{ 2 k q_par
- q_par^{2} }

for small incident angles alpha.
Thus, in the Bragg region (for q_perp
<
q_perp_lim), q_perp should be plotted versus q_par. Otherwise q_perp versus qy may be a better choice for
diffuse Bragg sheets.Figure 2. Limiting q_perp value versus q_par for a typical range of q values (in Å

For single crystalline films the sample will have to be oscillated during exposure of the detector (Smilgies et al., J. Synchrotron Rad. 12, 807–811 (2005)), so that the Bragg condition can be met for all points along the scattering rod covered by the detector. Finally the surface frame is mandatory to describe dynamic effects that occur when either incident and exit angle are located between the critical angles of film and substrate. See for instance Busch et al., J. Appl. Cryst. 39, 433-442 (2006).

If the scattering is dominated by the mosaicity of the thin film, or if determination of the mosaic distribution is the primary goal, the coordinate system that was introduced for texture studies (see in the WAXS section above) may be more suitable.

kpari' = kpari kparf' = kparf qpar' = (kparf' - kpari')

The perpendicular components can by obtained from Snell's law which reads in the x-ray case

|kzi'| = Re{ sqrt(kzi

kzf' = Re{ sqrt(kzf

kc is the critical wavevector as given by the critical angle of the film alpha_cF

kc = k sin(alpha_cF)

qz' = kzf' + |kzi'|

- Lee et al., Macromolecules 38, 3395 (2005); Macromolecules 38, 4311-4323 (2005).
- Busch et al., J. Appl. Cryst. 39, 433-442 (2006).
- Breiby et al., J. Appl. Cryst. 41, 262-271 (2008).

The refraction correction is essential for reflections with small exit angles {see Busch et al., J. Appl. Cryst. 39, 433-442 (2006) and Breiby et al., J. Appl. Cryst. 41, 262-271 (2008)}. The diffraction corrected q-values (qpar',qz') can then be used for indexing (see for instance Smilgies & Blasini, J. Appl. Cryst. 40, 716-718 (2007)). While the technicalities are well understood, indexation of surface scattering data from unknown lattices remains still a bit a matter of trial & error to get the six lattice parameters right. Lattices with high symmetry, such as the cubic lattices and the hexagonal close-packed lattice, can often be indexed in a straightforward way. Molecular lattices which are often monoclinic or triclinic are much harder to index. If the bulk structure is known, this can be a good starting point. An automated indexation routine based on Monte-Carlo sampling of a trial range of lattice structure parameters has been described by Hailey et al., J. Appl. Cryst. 47, 2090-2099 (2014))

The other important calibration is the direct beam position which should be measured regularly, and in particular, whenever the detector is moved. The GIWAXS macro set at D1 includes a convenient automatic macro for this measurement.

If this page was useful for your work I would appreciate very much, if you referenced it.