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Methods for Structural and Chemical Characterization of Nanomaterials 131

2
where d /d
(u) is the differential scattering cross-sectional area (m ).
This cross section is characteristic of the interaction between the mate-
rial and the incident beam. The scattering intensity by a sample is the
differential scattering cross section per unit volume and is expressed as
1
the inverse of a length (m ).

Te
d (9)
I 5 1/V ds/d
sud 5 N/sN 0 S
This quantity is experimentally accessible, provided that the thick-
ness e s is known and the transmission T is measured with the scatter-
ing properties N/(N 0 ). If the exact composition of the sample is
known, e s can be deduced from the transmission measurement accord-
ing to the following relationship: T N(0)/N 0 e
e , where µ∗ is the
absorption coefficient, which depends only on the scattering volume
composition.
The experimental setup directly measures a differential scattering
cross section per unit volume if the detector geometry is well defined
(
) and if the same detector is used to measure the direct beam (N 0 )
and the scattered beam (∆N). Some experimental configurations have
become standard, as for example the Bonse-Hart Ultra Small Angle
X-ray Scattering apparatus, where the same punctual detector is
scanned from the direct beam to the scattered beam and because
is
precisely defined by the optics before the detector. Generally, however,
a standard must be used to calibrate the instrument. For X-ray scat-
tering, a practical standard is pure water. Indeed, the scattering of pure
water is related to its isothermal compressibility T and is well defined
(I 0.016 cm 1 ). The transmission of pure water also gives the thick-
ness as the mass attenuation coefficient for this medium, which is known
∗ 2
(µ / (water) 9,91 cm /g at 8 keV). Using water as a standard, however,
requires highly sensitive instruments. Other standards with known
d /d
values may also be used. For light scattering, pure liquids can
be employed (benzene is often used for these purposes) or for less sen-
sitive low angle light scattering instruments, calibrated pinholes can be
used as a standard [Thill et al., 2002].

General expression of the scattering intensity. For an incident radiation source
of wavelength the incident beam has a wave vector k i of amplitude
|k i | k i 22 / . The scattered wave vector k d making an angle u with
k i has the same amplitude for an elastic scattering process and defines
a scattering wave vector q k d k i. Thus, the amplitude of the scat-
tering wave vector is q 4 / sin( /2). A static scattering experiment
yields structural information on the dispersed phase at a typical spa-
tial scale of 1/q. For nanoparticles, the interesting range of scattering
1
wave vectors is 10 2 to 1 nm . This range is observable for reasonable