TRENDS AND FUTURE PROSPECTS - Total-Reflection X-Ray Fluorescence Analysis and Related Methods - page 482

of the crystalline sample. The photon energy or wavelength of the incident
beam is known, and the glancing angle is varied by means of a goniometer.
Consequently, the interplanar spacing of the corresponding crystal lattice
plane can be calculated by the reverse Bragg equation for the m th order of
diffraction:

d m λ

1
sin α 2

(7.3)

Usually, a crystalline sample provides several reflections with different sets
of Miller indices, hkl , even for one single crystal and moreover for different
crystals of a sample. For a rhombic lattice with lattice constants a , b ,and c ,
several interplanar spacings are possible for different combinations of the
Miller indices:

h 2
a 2

k 2
b 2

l 2
c 2

1
d 2 hkl

(7.4)

For a cubic lattice, the three constants a , b , and c are equal. Interplanar
spacings are listed for about 50 000 substances in an extensive data file of
JCPDS (Joint Committee on Powder Diffraction Standards). As an example,
Figure 7.24 shows a spectrum of a car finish with white pigments of TiO 2 , ZnO,
and Ba 2 SO 4 . XRD can be applied for the determination of the crystal structures
of known samples but also for the identification of chemical compounds
possibly in addition to element-specific X-ray fluorescence analysis. Conse-
quently, XRD can be used for phase analyses and the study of phase transitions,
for their dependence of temperature and pressure, for the investigation of the
texture of samples, and of mechanical stress and strain.
X-ray diffraction and total reflection were already combined by Yoneda and
Horiuchi in 1971 as described by Horiuchi 22 years later [142]. First, they
determined different elements in seawater, in blood, and in spring water.
Moreover, an ultrathin organic film of n -C 33 H 68 (an alkane) was investigated.
Such a Langmuir-Blodgett film has a molecular structure with vertical axes
perpendicular to the substrate. Diffraction occurred at the crystal lattice planes
(110) and (200) of the film. Background noise from scattering within the
substrate was removed or significantly reduced by total reflection. In accord
with Bragg's law, weak diffraction peaks were found at two particular angles
corresponding to the d values. The evaporated film with a thickness up to
100 nm was studied during a heating process between 30 and 60 ° C invacuo .
Between 1993 and 1995, the combination of total reflection and diffraction
of X-rays was developed mainly by Horiuchi and coworkers [143-148]. It was
called TR-XRD (total reflection X-ray diffraction) or GI-XRD (grazing inci-
dence X-ray diffraction). Because of the low penetration depth at total
reflection, it is a powerful tool for evaluating thin near-surface layers, such