1 X-Ray Birefringence Imaging (XBI): A New Technique …
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X-ray beam. This rotation changes the orientation of the sample reference axis (z s -
axis) relative to the direction of linear polarization (X-axis) of the incident X-ray
beam. Clearly, χ-rotation is analogous to the sample rotation commonly carried out
in the polarizing optical microscope (Fig. 1.1). Normally, χ = 0° is defined as the
orientation in which the sample reference axis is horizontal (i.e., with the z s -axis
parallel to the X-axis).
Rotation of the sample around the reference axis is called φ-rotation. Clearly,
φ-rotation does not change the orientation of the reference z s -axis relative to the
direction of linear polarization (X-axis) of the incident X-ray beam, but it does
change the orientation of the material (x s y s -plane) relative to the direction of linear
polarization of the incident X-ray beam.
In XBI studies, it is common to carry out a complete two-dimensional mapping
by recording XBI images as a function of both χ and φ. Due to practical limitations
in moving the goniometer, the range of values of χ and φ that can be accessed is
typically about 180° in each case.
1.3.3 The Polarization Analyzer
The role of the polarization analyzer in the XBI experiment (analogous to the function of the analyzer in the polarizing optical microscope shown schematically in
Fig. 1.1) is to select the vertical component of linear polarization of the X-ray beam
transmitted through the sample. However, unlike the transmission-based polarization analyzer (e.g., a polaroid sheet) used in the polarizing optical microscope, the
experimental setup for XBI uses a diffraction-based polarization analyzer. The polarization analyzer is a large single crystal (typically silicon or germanium) positioned
and oriented such that the X-ray beam transmitted through the sample is diffracted
at the analyzer, with the diffracted beam directed towards the detector. Ideally, the
angle of diffraction at the analyzer (in the setup shown in Fig. 1.2) should be exactly
2θ = 90° so that the X-ray beam diffracted from the analyzer comprises only the
vertical component of linear polarization. However, as the X-ray wavelength used in
the XBI experiment is dictated by selecting a suitable X-ray absorption edge for an
element in the material, and as only a relatively restricted set of analyzer crystals are
available, it is unlikely that the XBI experiment can be set up such that the diffraction angle at the analyzer is exactly 2θ = 90°. Nevertheless, once the wavelength is
selected according to the X-ray absorption edge of interest, the analyzer crystal is
chosen as the one that gives a diffraction angle as close as possible to 2θ = 90°. In
practice, provided the diffraction angle is within a few degrees of 90°, the analyzer
operates effectively (although not perfectly), selecting predominantly the vertical
component of the X-ray beam transmitted through the sample. For XBI experiments
in which the X-ray energy corresponds to the Br K-edge (E ≈ 13.474 keV), suitable
analyzer crystals are Si(111) and Ge(111), which gives diffraction angles for the
(555) reflection of 2θ = 94.4° and 2θ = 89.5°, respectively.
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X-ray beam. This rotation changes the orientation of the sample reference axis (z s -
axis) relative to the direction of linear polarization (X-axis) of the incident X-ray
beam. Clearly, χ-rotation is analogous to the sample rotation commonly carried out
in the polarizing optical microscope (Fig. 1.1). Normally, χ = 0° is defined as the
orientation in which the sample reference axis is horizontal (i.e., with the z s -axis
parallel to the X-axis).
Rotation of the sample around the reference axis is called φ-rotation. Clearly,
φ-rotation does not change the orientation of the reference z s -axis relative to the
direction of linear polarization (X-axis) of the incident X-ray beam, but it does
change the orientation of the material (x s y s -plane) relative to the direction of linear
polarization of the incident X-ray beam.
In XBI studies, it is common to carry out a complete two-dimensional mapping
by recording XBI images as a function of both χ and φ. Due to practical limitations
in moving the goniometer, the range of values of χ and φ that can be accessed is
typically about 180° in each case.
1.3.3 The Polarization Analyzer
The role of the polarization analyzer in the XBI experiment (analogous to the function of the analyzer in the polarizing optical microscope shown schematically in
Fig. 1.1) is to select the vertical component of linear polarization of the X-ray beam
transmitted through the sample. However, unlike the transmission-based polarization analyzer (e.g., a polaroid sheet) used in the polarizing optical microscope, the
experimental setup for XBI uses a diffraction-based polarization analyzer. The polarization analyzer is a large single crystal (typically silicon or germanium) positioned
and oriented such that the X-ray beam transmitted through the sample is diffracted
at the analyzer, with the diffracted beam directed towards the detector. Ideally, the
angle of diffraction at the analyzer (in the setup shown in Fig. 1.2) should be exactly
2θ = 90° so that the X-ray beam diffracted from the analyzer comprises only the
vertical component of linear polarization. However, as the X-ray wavelength used in
the XBI experiment is dictated by selecting a suitable X-ray absorption edge for an
element in the material, and as only a relatively restricted set of analyzer crystals are
available, it is unlikely that the XBI experiment can be set up such that the diffraction angle at the analyzer is exactly 2θ = 90°. Nevertheless, once the wavelength is
selected according to the X-ray absorption edge of interest, the analyzer crystal is
chosen as the one that gives a diffraction angle as close as possible to 2θ = 90°. In
practice, provided the diffraction angle is within a few degrees of 90°, the analyzer
operates effectively (although not perfectly), selecting predominantly the vertical
component of the X-ray beam transmitted through the sample. For XBI experiments
in which the X-ray energy corresponds to the Br K-edge (E ≈ 13.474 keV), suitable
analyzer crystals are Si(111) and Ge(111), which gives diffraction angles for the
(555) reflection of 2θ = 94.4° and 2θ = 89.5°, respectively.