Page 147 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Methods for Structural and Chemical Characterization of Nanomaterials 133

a single nanoparticle, we can rewrite Eq. 11 as:

Asqd 5 3 Vnp r e s2iqrd dr 1 3 Vsol sol s2iqrd dr (12)
r e
np
The scattering amplitude of this system can be rewritten in terms of
the scattering length density contrast between the nanoparticles and
solvent np sol :

r e
Asqd 5 3 Vnp np s2iqrd dr 1 3 Vsol sol s2iqrd dr (13)
r e
The last term of Eq. 12 is only present at q 0 and can be removed for
practical purposes. The scattering amplitude of a single nanoparticle is
then only dependent on the electronic density contrast between the
nanoparticles and solvent. The intensity can be written as:

2 s2iqsr2vdd
Asqd 5 AsqdArsqd 5 r e drdv (14)
Vnp
33
For dilute suspensions of nanoparticles, the intensity per unit volume
is obtained by summation of the scattering from each nanoparticle and
is often written as:
N 2
Isqd 5 V Psqd 5 fVnpPsqd (15)
np
V
where P(q) is the normalized form factor which is defined as:

1 2 s2iqsr2vdd
Psqd 5 r e drdv (16)
2 33
V np Vnp
The form factor P(q) is characteristic of the shape and scattering
length density contrast of the nanoparticle. For example, the form factor
of a spherical particle can be obtained as a function of the scattering
wave vector amplitude by:
3[sinsqrd 2 qr cossqrd] 2
2
Psqd 5 r a b (17)
sqrd 3
When particles are interacting through long-range forces in dilute sus-
pensions or simply by collision in a concentrated suspension, the scatter-
ing of the different nanoparticles is no longer independent and interferences
between nanoparticles must be accounted for. It can be shown that the scat-
tering intensity of the unstable suspension can be rewritten as:

Isqd 5 fV PsqdSsqd (18)
np