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where I
0
is a constant defined by the conditions of the SAXS instrument. The intensity
distribution (SAXS pattern) as a function of~
q is uniquely determined by the structure
in terms of its electron density distribution. Therefore, in principle, the structure
associated with the electron density distribution can be uniquely determined from the
SAXS pattern. For instance, if the scattering is spherically symmetric, that is, Ið~
qÞ
depends only on q, then we have
1
sin qr
qr
IðqÞ¼4p
pðrÞ
dr
ð10
:
3Þ
0
where p(r) is the so-called pair-distance distribution function (PDDF) that gives the
number of different electron pairs with a mutual distance between r and r þ dr within
the particle. We can see that, like the electron spatial distribution functionrð~
rÞ, p(r)is
a function of the structure and can be given by the inverse transformation from the
scattering intensity
1
1
2p
pðrÞ¼
IðqÞqr sinðqrÞdq
ð10
:
4Þ
2
0
The equation gives the direct relationship between the measured scattering
intensity I(q) and the PDDF p(r). More theories and equations for SAXS can be
found in a number of textbooks and literature listed at the end of this chapter.
10.1.3 X-ray Source and Optics for SAXS
In a two-dimensional SAXS system, the scattered X-rays are measured in all 360
azimuthal angles simultaneously. Therefore, the X-ray source and optics have
different requirements in terms of the spectrum purity, beam divergence, beam size,
and beam cross-section profile. The X-ray source used for SAXS can be a sealed tube
or rotating anode generator (RAG). Small focal spots with high specific power loading
on the target is preferred since long beam path, low divergence, and small beam size
are typically required for SAXS. Any of the commonly used target material, such as
Cu, Co, Cr can be used for SAXS. The longer wavelength of Cr-K
a
radiationmay have
a gain in scattering angle resolution compared to Cu-K
a
, but suffers more from the
absorption and scattering of the beam path. Therefore, Cu-K
a
radiation is most
commonly used for SAXS. Another consideration in the target selection is to avoid
fluorescence. Due to the small scattering angles, the scattering patterns from the K
a1
line and K
a2
line are always merged together the monochromatization by graphite
crystals or multilayer mirrors can provide sufficient spectrum purity. Collimation is
the most critical part of a SAXS system. The collimation system defines the size,
shape, and divergence of the X-ray beam. Collimation also determines the resolution
of a SAXS system. Figure 10.1 shows the collimation of the SAXS system with a
pinhole collimator, sample, beam stop, and detector. The primary beam, consisting of
parallel and divergent components, is blocked by the beam stop. The maximum
