Page 146 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

132 Principles and Methods

angles for very small wavelengths that are typical of X-rays and neu-
trons ( ~ 0.1 nm). Thus, the following notations, also very general, are
.
more adapted to X-ray scattering.
The scattering amplitude, A(q), from a scattering volume, V is:

Asqd 5 rsrde 2iqr dr (10)
3
v
where qr gives the phase shift between two scatterers separated by
the vector r and (r) is the density of scattering length. The density of
scattering length is (r) ∑ i (r)b i i (r) i (r) being the local density of scat-
i
terers of type i and b i is the scattering length. In the case of x-rays, the
photons interact with all the electrons in the sample. Thus, the scat-
tering length is the Thomson scattering length of a single electron b e
2
2
e /(4 0 mc ) 0.282 10 14 m and i (r) is simply the local density of elec-
trons in the scatterers e (r). The scattering intensity per unit volume is
given by the following expression:
AsqdArsqd
Isqd 5 (11)
V

Sample preparation. No particular limitations exist for the type of media
that can be analyzed by SAXS. The sample can be in a solid, liquid, or
gaseous phase. For liquids or gases, the sample must be put in a
container that has windows to allow for a beam path. It is further nec-
essary that the container walls that compose the window do not inter-
act with the X-rays. Thin plastic sheets, thin mica sheets, or beryllium
are some typical materials used as windows. The thickness of the sample
is the most critical parameter. It has to be large enough so that the
X-rays can interact with the matter and thin enough so that both
the incident beam and scattered beam can cross the sample. A good
criterion for choosing the sample thickness is when it is possible to
detect a transmitted beam. Indeed, from Eq. 9, it can be shown that the
quantity of scattered photons is proportional to eT, where e is the sample
thickness and T is the transmission. As e ~ ln(1/T ), the optimal trans-
∗
mission is obtained for the maximum value of T ln(1/T ) which is
obtained for T 0.36. For example, with an 8 keV incident X-ray beam,
∗ 2 3
in the case of water (µ r 10 cm /g, r 1 g/cm ), the maximum scat-
tered signal will be measured for a thickness of e ln(1/0.36) / (10 ∗ 1.) ~
∗ 2 3
0.1 cm; for cerium oxides (µ r 290 cm /g, r 6.5 g/cm ), optimum thick-
∗ ∗
ness falls down to e ln(1/0.36)/(290 6.5) 5 µm.
Scattering by a nanoparticle dispersion. We now consider calculation of scatter-
ing for a solvent of homogeneous scattering length density rsol containing
N nanoparticles with a homogeneous scattering length density rnp. For