Representation of a Fiber Diffraction / Grazing Incidence pattern

• FiberIntegrator class

• New parameters:

  • sample_orientation

  • incident_angle

  • tilt_angle

• integrate2d_grazing_incidence: method to get qIP - qOOP patterns

• integrate2d_exitangles: method to get exit angles maps

• integrate2d_polar: method to get polar_angles maps

• FiberIntegrator do not use pixel splitting scheme

• The origin os the missing wedge

How to use pyFAI to display or reduce data from grazing-incidence/fiber scattering experiments

from pyFAI.test.utilstest import UtilsTest
from pyFAI.gui.jupyter import subplots, display, plot1d, plot2d
from pyFAI import load
from pyFAI.integrator.fiber import FiberIntegrator
from pyFAI.units import get_unit_fiber
from pyFAI.integrator.azimuthal import AzimuthalIntegrator
from pyFAI.io.ponifile import PoniFile
import fabio
import numpy
import time
import matplotlib.pyplot as plt
y6 = fabio.open(UtilsTest.getimage("Y6.edf")).data
air = fabio.open(UtilsTest.getimage("air.edf")).data
data = y6 - 0.03 * air
def update_style(ax):
    for img in ax.get_images():
        img.set_cmap("viridis")
        img.set_clim(10,500)
t0 = time.perf_counter()

Since version 2025.01, pyFAI uses a specific Integrator for this case: FiberIntegrator

The FiberIntegrator can be created by three ways:

• Using pyFAI.load declaring the pyFAI.integrator.fiber.FiberIntegrator as type_

• Importing and instantiating directly a pyFAI.integrator.fiber.FiberIntegrator

• Promoting an AzimuthalIntegrator instance using promote

poni_filename = UtilsTest.getimage("LaB6_3.poni")

# 1st way
fi = load(filename=poni_filename, type_="pyFAI.integrator.fiber.FiberIntegrator")

# 2nd way
poni = PoniFile(data=poni_filename)
fi = FiberIntegrator(dist=poni.dist, poni1=poni.poni1, poni2=poni.poni2,
                     wavelength=poni.wavelength,
                     rot1=poni.rot1, rot2=poni.rot2, rot3=poni.rot3,
                     detector=poni.detector,
                    )

# 3rd way
ai = load(filename=poni_filename)
fi = ai.promote(type_="pyFAI.integrator.fiber.FiberIntegrator")

FiberIntegrator

• integrate2d_grazing_incidence == integrate2d_fiber

• integrate1d_grazing_incidence == integrate1d_fiber

res2d_default = fi.integrate2d_grazing_incidence(data=data)
fig, ax = subplots()
plot2d(result=res2d_default, ax=ax)
update_style(ax=ax)
pass

• There are 8 possible orientations (1 -> 8). Default value = 1

fig, axes = subplots(ncols=4, nrows=2, figsize=(12,6))
results_so = []
for sample_orientation in range(1,9):
    res = fi.integrate2d_grazing_incidence(data=data, sample_orientation=sample_orientation)
    ax = axes.ravel()[sample_orientation - 1]
    plot2d(result=res, ax=ax)
    update_style(ax=ax)
    ax.set_title(f"SO: {sample_orientation}")
    ax.set_xlabel("")
    ax.set_ylabel("")

FiberIntegrator introduces three new parameters:

• sample_orientation (int) : default = 1

• incident_angle (η) (float) : default=0.0 (in radians). Tilt of the sample stage towards the beam (key parameter in grazing-incidence)

• tilt_angle (χ) (float) : default=0.0 (in radians). Horizon tilt of the sample stage (normally as close as 0.0 in grazing-incidence)

res_0 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.0, tilt_angle=0.0)
res_1 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.1, tilt_angle=0.0)
res_2 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.0, tilt_angle=0.5)
res_3 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.1, tilt_angle=0.5)
iangles = (0.0, 0.1, 0.0, 0.1)
tangles = (0.0, 0.0, 0.5, 0.5)
fig, axes = subplots(ncols=4, figsize=(15,5))
for iangle, tangle, ax, res in zip(iangles, tangles, axes.ravel(), [res_0, res_1, res_2, res_3]):
    plot2d(result=res, ax=ax)
    update_style(ax=ax)
    if ax != axes[0]:
        ax.set_ylabel("")
    ax.set_title(f"η={iangle} rad, χ={tangle} rad")

• incident_angle changes the accessible region along the out-of-plane axis (just like in a reflectivity experiment)

• tilt_angle rotates the pattern around the direct beam point (q=0nm-1)

pyFAI provides methods for other fiber units

• integrate_2d_grazing_incidence uses by default qIP_nm^-1 and qOOP_nm^-1 (switchable to A^-1 units)

• integrate2d_exitangles for horizontal and vertical exit angles in which the origin is the sample horizon (dependent of incident_angle)

• integrate2d_polar to represent the polar angles (as the arctan of qOOP / qIP) versus q modulus

iangle = 0.1
res_0 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=iangle, unit_ip="qip_A^-1", unit_oop="qoop_A^-1")
res_1 = fi.integrate2d_exitangles(data=data, sample_orientation=6, incident_angle=iangle)
res_2 = fi.integrate2d_exitangles(data=data, sample_orientation=6, incident_angle=iangle, angle_degrees=False)
res_3 = fi.integrate2d_polar(data=data, sample_orientation=6, incident_angle=iangle)
res_4 = fi.integrate2d_polar(data=data, sample_orientation=6, incident_angle=iangle, polar_degrees=False, radial_unit="A^-1", rotate=True)

fig, axes = subplots(ncols=5, figsize=(20,4))
for ax, res in zip(axes.ravel(), [res_0, res_1, res_2, res_3, res_4]):
    plot2d(result=res, ax=ax)
    update_style(ax=ax)

integrate1d_grazing_incidence

• This method slice regions of integrate2d_grazing_incidence maps (essentially, integrate1d using the new Fiber Units)

• Additional parameter to consider: vertical_integration : default = True

res2d = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12))
res2d_patch = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                               npt_ip=500, npt_oop=500,
                                               ip_range=(-2,2), oop_range=(0,20))

res1d_vertical = fi.integrate1d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                                  npt_ip=500, npt_oop=500,
                                                  ip_range=(-2,2), oop_range=(0,20))

fig, axes = subplots(ncols=3, figsize=(15,5))
plot2d(result=res2d, ax=axes[0])
plot2d(result=res2d_patch, ax=axes[1])
plot2d(result=res2d, ax=axes[1])
update_style(ax=axes[0])
update_style(ax=axes[1])
axes[1].get_images()[1].set_alpha(0.2)
plot1d(result=res1d_vertical, ax=axes[2])
axes[1].set_title("Integration patch")
axes[2].set_title("qOOP-integration")
pass

/gpfs/jazzy/data/scisoft/edgar/miniforge3/envs/edgar/lib/python3.12/site-packages/pyFAI/integrator/fiber.py:269: RuntimeWarning: invalid value encountered in divide
  intensity = sum_signal / sum_normalization

res2d = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12))
res2d_patch = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                               npt_ip=500, npt_oop=500,
                                               ip_range=(-8,8), oop_range=(5,7))

res1d_horizontal = fi.integrate1d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                                  npt_ip=500, npt_oop=500,
                                                  ip_range=(-8,8), oop_range=(5,7),
                                                    vertical_integration=False,
                                                   )

fig, axes = subplots(ncols=3, figsize=(15,5))
plot2d(result=res2d, ax=axes[0])
plot2d(result=res2d_patch, ax=axes[1])
plot2d(result=res2d, ax=axes[1])
update_style(ax=axes[0])
update_style(ax=axes[1])
axes[1].get_images()[1].set_alpha(0.2)
plot1d(result=res1d_horizontal, ax=axes[2])
axes[1].set_title("Integration patch")
axes[2].set_title("qIP-integration")
pass

APIs for integrate1d using the exit angles and polar angles

• integrate1d_polar: intensity profile as a function of the polar angle (or inverse, integrate a polar angle sector as a function of q)

• integrate1d_exitangles: intensity profile along a specific exit angle (horizontal or vertical)

sample_orientation=6
res2d_polar_1 = fi.integrate2d_polar(data=data, sample_orientation=sample_orientation)
res2d_polar_2 = fi.integrate2d_polar(data=data, sample_orientation=sample_orientation,
                                     polar_degrees=False, radial_unit="A^-1", radial_integration=True,
                                     )
res2d_exitangle_1 = fi.integrate2d_exitangles(data=data, sample_orientation=sample_orientation, incident_angle=0.0,
                                              npt_oop=1000, npt_ip=1000,
                                              )
res2d_exitangle_2 = fi.integrate2d_exitangles(data=data, sample_orientation=sample_orientation, incident_angle=0.0,
                                              npt_oop=1000, npt_ip=1000, angle_degrees=False,
                                              )


res1d_polar_1 = fi.integrate1d_polar(data=data, sample_orientation=sample_orientation,
                                     npt_oop=500, npt_ip=500,
                                     )
res1d_polar_2 = fi.integrate1d_polar(data=data, sample_orientation=sample_orientation,
                                     npt_oop=500, npt_ip=500,
                                     polar_degrees=False, radial_unit="A^-1", radial_integration=True,
                                     )
res1d_exitangle_1 = fi.integrate1d_exitangles(data=data,
                                              npt_oop=1000, npt_ip=1000,
                                              sample_orientation=sample_orientation, incident_angle=0.0)

res1d_exitangle_2 = fi.integrate1d_exitangles(data=data, npt_oop=1000, npt_ip=1000,
                                              angle_degrees=False,
                                              sample_orientation=sample_orientation, incident_angle=0.0)


fig, axes = subplots(ncols=4, nrows=2, figsize=(20,10))
plot2d(result=res2d_polar_1, ax=axes[0,0])
plot2d(result=res2d_polar_2, ax=axes[0,1])
plot2d(result=res2d_exitangle_1, ax=axes[0,2])
plot2d(result=res2d_exitangle_2, ax=axes[0,3])

plot1d(result=res1d_polar_1, ax=axes[1,0])
plot1d(result=res1d_polar_2, ax=axes[1,1])
plot1d(result=res1d_exitangle_1, ax=axes[1,2])
plot1d(result=res1d_exitangle_2, ax=axes[1,3])
for ax in axes.ravel():
    update_style(ax=ax)
pass

FiberIntegrator methods use non-pixel-splitting by default

• For AzimuthalIntegrator, integration methods use bbox pixel splitting scheme by default

• For integrate2d/integrate1d_grazing_incidence methods, the default pixel splitting scheme is set to no

res2d_no = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12))
res2d_bbox = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                              method=("bbox", "csr", "cython")
                                             )


fig, axes = subplots(ncols=2, figsize=(10,5))
plot2d(result=res2d_no, ax=axes[0])
plot2d(result=res2d_bbox, ax=axes[1])
update_style(ax=axes[0])
update_style(ax=axes[1])
axes[0].set_title("Pixel-splitting=no")
axes[1].set_title("Pixel-splitting=bbox")
pass

WARNING:pyFAI.integrator.fiber:Method ('bbox', 'csr', 'cython') is using a pixel-splitting scheme. GI integration should be use WITHOUT PIXEL-SPLITTING! The results could be wrong!
WARNING:pyFAI.geometry.core:No fast path for space: qip
WARNING:pyFAI.geometry.core:No fast path for space: qoop

res1d_no = fi.integrate1d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                                  npt_ip=500, npt_oop=500,
                                                  ip_range=(-2,2), oop_range=(0,20))
res1d_bbox = fi.integrate1d_grazing_incidence(data=data, sample_orientation=6, incident_angle=numpy.deg2rad(0.12),
                                              npt_ip=500, npt_oop=500,
                                              ip_range=(-2,2), oop_range=(0,20),
                                              method=("bbox", "csr", "cython")
                                             )

fig, axes = subplots(ncols=2, figsize=(10,5))
plot1d(result=res1d_no, ax=axes[0])
plot1d(result=res1d_bbox, ax=axes[1])
update_style(ax=axes[0])
update_style(ax=axes[1])
axes[0].set_title("Pixel-splitting=no")
axes[1].set_title("Pixel-splitting=bbox")
pass

WARNING:pyFAI.integrator.fiber:Method ('bbox', 'csr', 'cython') is using a pixel-splitting scheme. GI integration should be use WITHOUT PIXEL-SPLITTING! The results could be wrong!

Why the missing wedge?

The missing wedge appears when we represent the qz component versus the in-plane q component.

With a static geometry, we can only prove a spherical shell of the whole reciprocal space of the sample.

Every point of that shell is in “Bragg condition”; we call that shell the Ewald sphere.

What we see in the detector is the projection of the Ewald sphere into the surface of the detector.

We can represent that data in terms of the exit angles (horizontal and vertical)

But if we represent it with the units qoop-qip, we will see how a sphere can only intersect a rod in 1 or 2 points.

• If incident_angle=0.0, the only point of qoop in Bragg conditions is qoop=0

• If incident_angle!=0.0, there are two points in Bragg conditions: qoop=0 and qoop=incident_angle, which is the reflected beam.

res2d_0 = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.0)
res2d_01rad = fi.integrate2d_grazing_incidence(data=data, sample_orientation=6, incident_angle=0.1)

fig, axes = subplots(ncols=3, figsize=(15,5))
plot2d(result=res2d_0, ax=axes[0])
plot2d(result=res2d_01rad, ax=axes[1])
plot2d(result=res2d_01rad, ax=axes[2])
update_style(ax=axes[0])
update_style(ax=axes[1])
update_style(ax=axes[2])
axes[2].set_xlim(-2,2)
axes[2].set_ylim(10,15)
axes[0].set_title("0.0 rad")
axes[1].set_title("0.1 rad")
axes[2].set_title("0.1 rad_zoom")
pass

incident_angle = 0.1
q_bragg = 4 * numpy.pi / fi.wavelength * numpy.sin(2 * incident_angle / 2) / 1e9
print(f"Along qoop, we only prove 0.0nm^-1 and {q_bragg:.2f} nm^-1")

Along qoop, we only prove 0.0nm^-1 and 12.55 nm^-1

The same way, we get a missing wedge in the horizontal axis if we plot data as a function of (Qxz - Qy)

fig, axes = plt.subplots(ncols=3, figsize=(15,5))
arr_qx = fi.array_from_unit(unit=get_unit_fiber("qxgi_nm^-1"))
arr_qy = fi.array_from_unit(unit=get_unit_fiber("qygi_nm^-1"))
arr_qz = fi.array_from_unit(unit=get_unit_fiber("qzgi_nm^-1"))
display(arr_qx, ax=axes[0], label="q_horizontal")
display(arr_qy, ax=axes[1], label="q_vertical (OOP)")
display(arr_qz, ax=axes[2], label="q_beam")
pass

fig, ax = plt.subplots(ncols=2)
arr_qx = fi.array_from_unit(unit=get_unit_fiber("qxgi_nm^-1"))
arr_qy = fi.array_from_unit(unit=get_unit_fiber("qygi_nm^-1"))
arr_qz = fi.array_from_unit(unit=get_unit_fiber("qzgi_nm^-1"))
arr_qxy = numpy.sqrt(arr_qx ** 2 + arr_qy ** 2)
arr_qxz = numpy.sqrt(arr_qx ** 2 + arr_qz ** 2)
arr_qyz = numpy.sqrt(arr_qy ** 2 + arr_qz ** 2)
ax[0].pcolormesh(arr_qxz, arr_qy, data, vmin=0, vmax=500)
ax[1].pcolormesh(arr_qx, arr_qyz, data, vmin=0, vmax=500)
pass

/tmp/ipykernel_830751/1134296278.py:8: UserWarning: The input coordinates to pcolormesh are interpreted as cell centers, but are not monotonically increasing or decreasing. This may lead to incorrectly calculated cell edges, in which case, please supply explicit cell edges to pcolormesh.
  ax[0].pcolormesh(arr_qxz, arr_qy, data, vmin=0, vmax=500)
/tmp/ipykernel_830751/1134296278.py:9: UserWarning: The input coordinates to pcolormesh are interpreted as cell centers, but are not monotonically increasing or decreasing. This may lead to incorrectly calculated cell edges, in which case, please supply explicit cell edges to pcolormesh.
  ax[1].pcolormesh(arr_qx, arr_qyz, data, vmin=0, vmax=500)

print(f"Total run-time: {time.perf_counter()-t0:.3f}s")

Total run-time: 41.899s

Conclusions

• PyFAI provides a simple API to represent a data array as a function of in-plane and out-of-plane components of vector q.

• This is a standard way to display GIWAXS/GISAXS or fiber diffraction patterns.

• New parameters to take into account:

  • sample_orientation

  • incident_angle

  • tilt_angle

• Other possible new coordinates:

  • exit_angle_vert vs exit_angle_horz

  • polar_angle vs qtot