PHYS01200704032 Debes Ray - Homi Bhabha National Institute

Chapter 3: Multi-Technique Approach for Characterization of Gold Nanoparticles
for a system could be different because of differences in their interactions with matter and
therefore provide complementary information.
3.5.1. Theory of Small-Angle Scattering
The scattering of radiation (X-ray/neutron) by a scatterer is characterized by a single
parameter b referred as the scattering length. If the wave vectors of incident and scattered
waves are k i and k f respectively, a wave scattered by a scatterer at a point r in the sample will
thus be phase shifted with respect to that scattered at the origin by a phase factor e -Q.r , where
Q = k f - k i is the wave vector transferred in the scattering process. The integral scattering
cross-section for a nucleus is given by σ = 4πb 2 and it can be looked upon as an effective area
presented by the scatterer to the incident radiation. The scattering cross-section describing the
flux scattered into the solid angle dΩ and normalized to the irradiated sample (V T ) volume is
called macroscopic differential scattering cross section and is expressed for an assembly of
scatterers in a macroscopic sample as [176-179]
d 1
( Q) bj
exp( iQr
.
j
) (3.6)
d V
T
j
2
where b j is the bound scattering length and r j is the position vector of j th scatterer in a sample,
and the bracket represents an average over all possible orientations.
Since SANS deals with the study of large scale heterogeneities rather than locating the
individual scattering centers, the summation over b j can be replaced by a volume integral
over scattering length density ρ(r) as defined as
bj
r r
j V
( ) d
(3.7)
T
The summation in the above equation extends over all the nuclei in the volume V T .
For the two-component system, particles dispersed in a medium having scattering
length densities ρ p and ρ m , respectively, equation simplified to
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