GEOMETRY CONVENTIONS - Two-Dimensional X-Ray Diffraction - page 44

is not fixed on the sample but on the instrument center. In practice, the f axis may be
built above the Z axis. Since both axes are overlapping, Z translation will not move f
axis away from the instrument center.
To analyze the diffraction results relative to the sample orientation, it is necessary
to define the sample coordinates S 1 , S 2 , and S 3 . The sample coordinates S 1 , S 2 , and
S 3 have the same directions as the sample translation coordinates X, Y, and Z,
respectively. However, the origin of the sample coordinates is considered fixed on
the sample, or more precisely, on the sample spot that is measured by the diffraction.
The S 1 -S 2 plane is the sample surface plane and S 3 is the sample surface normal. In
principle, it is important to notice carefully the distinction between the two sets of
coordinates. The XYZ translation coordinates indicate the relative sample position.
For example, samples are moved to multiple locations by an XYZ translation stage
to collect diffraction mapping data sets, while the S 1 S 2 S 3 coordinates are fixed on
the sample with origin at the measurement spot. The S 1 S 2 S 3 coordinates are mainly
used to establish the relationship between the sample orientation and diffraction
pattern. Practically, there is no need to distinguish the two coordinate systems most
of the time because the measurement spot on the sample should also be on the
instrument center.
The f rotation in the goniometer is intentionally chosen as a left-hand rotation so
that the diffraction vectors will make a right-hand rotation observed in the sample
coordinates S 1 S 2 S 3 . Figure 2.13(a) shows a diffraction vector H hkl and its projection
on S 1 axis. In Figure 2.13(b), the sample made a left-hand rotation of f.The
projection of the same diffraction vector on the sample rotates away from the S 1 axis
in a right-hand rotation of f. The corresponding rotations in the two-dimensional
polar coordinate system, three-dimensional cylindrical coordinate system, and
spherical coordinate system are all right-hand rotations. Therefore, it is imperative
to use a right-hand f rotation when describing diffraction vectors in the sample
space so that orientation-sensitive diffraction results can be properly mapped into
the sample space by mathematical operations. In summary, there are a total of six
parameters in the sample space, including three independent rotations (v, c, f)and
three orthogonal translations (X, Y, Z). The position of a sample in the laboratory
coordinates can be uniquely determined by these six parameters.
2.4.2 Variation of Goniometer Geometry
Many variations of goniometer geometry exist owing to historical reasons and
preferences for various applications. Not all diffractometers have the complete six
axes. That means some of the axes should be assigned as constants. The diffractometer
may be set in a horizontal orientation or vertical orientation relative to the ground
level. The v anglemay be achieved by rotating the sample against a stationary primary
X-ray beam or by rotating the primary X-ray beam against the sample. The
diffractometer may also be categorized with a left-hand goniometer or right-hand
goniometer depending on the relative position of the primary beam, goniometer, and
operator. Figure 2.14 shows two typical diffractometer configurations. Figure 2.14(a)
is the configuration of a horizontal diffractometer with the left-hand goniometer