80 INSTRUMENTAL METHODS
the crystal and thed spacing of the crystal according to Bragg ’ s law as shown
in equation 3.2 :
(^) ndλθ=2sin (3.2)
Detectors for quantitative measurement of X - ray absorption spectra must
measure the fl ux (photons s − 1 ) of the X - ray beam. Ionization chambers consist-
ing of X - ray transparent windows on each end of a chamber holding an inert
gas work well as transmission detectors for concentrated samples. For trans-
mission detectors, ln( I 0 / I ) is proportional to the absorption coeffi cient of the
absorbing atom, μ ( I 0 = incident X - ray photon intensity, I = transmitted inten-
sity), according to Beer ’ s Law:
(^) IIe= 0 −μt (3.3)
and the absorption coeffi cient varies according to the following:
μ≈
pZ
AE
4
3 (3.4)
where
μ = absorption coeffi cient
Z = atomic number
A = atomic mass
E = X - ray energy
Fluorescence excitation techniques provide a more sensitive detection system
in which fl uorescent X - ray photons (a fraction of the ionized absorbing atoms
relax by emission of a fl uorescent X - ray photon) are counted as the photon
energy is scanned. The signal generated is proportional to the absorption coef-
fi cient, μ , of the absorbing atom.
The data collected are subjected to Fourier transformation yielding a peak
at the frequency of each sine wave component in the EXAFS. The sine wave
frequencies are proportional to the absorber – scatterer (a – s) distance Ras. Each
peak in the display represents a particular shell of atoms. To answer the ques-
tion of how many of what kind of atom, one must do curve fi tting. This requires
a reliance on chemical intuition, experience, and adherence to reasonable
chemical bond distances expected for the molecule under study. In practice,
two methods are used to determine what the back - scattered EXAFS data for
a given system should look like. The fi rst, an empirical method, compares the
unknown system to known models; the second, a theoretical method, calculates
the expected behavior of the a – s pair. The empirical method depends on having
information on a suitable model, whereas the theoretical method is dependent
on having good wave function descriptions of both absorber and scatterer.