56
3 Molecular Liquids
Each partial form factor (l = C, H or O for typical organics) has the form
s l (q, ω) = f l (q) p l (q, ω)
(3.15)
with the partial molecular structure factor
p l (q, ω) =
i
exp[i q · (r i − R 0 )]
(3.16)
with R 0 being the center of molecule. Then, we have
l
s l (q, ω)
2
=
l
m
f l (q) f m (q) p l (q, ω) p m (q, ω)
∗
.
(3.17)
Since f l (q) does not depend on the molecular orientation ω, the orientational averaging is necessary for products of p l (q, ω). Using these quantities, the averaged
scattering intensity from a single molecule is given by
| f av (q)|
2
=
l
m
f l (q) f m (q) p l (q, ω) p m (q, ω) ∗ .
(3.18)
The formulation given above is rigorous under the assumption, no correlation in
orientation between neighboring molecules. The validity relies on the choice of the
“molecules” as a constituting particle of the sample. Once we chose the molecule,
the other structural information is included in the structure factor of liquid, S(q).
Note that we only measure the net intensity of scattering.
A naïve way to define the “molecule” is to assume that atoms within the orientational correlation length constitute a molecule. This strategy will only be the
way for the molecular association to extend over a long distance, as in the case of
chain and/or network. On the other hand, if the association mode is well defined and
not-extending, molecular clusters with a well-defined structure may be regarded as
a molecule. If dividing the experimental I (q) by | f av (q)|
2 calculated under such an
assumption gives S(q) resembling that for a simple liquid such as rare gas elements
and hard spheres, the assumption seems plausible. In this context (or the way of
proceeding analysis), the calculation of | f av (q)|
2 based on available structural data
is crucial.
If we have the cartesian coordinates of atoms in a molecule (or a cluster), we
redefine the origin of the coordinate by subtracting the coordinates of the center of
electron density,
R i = r i − R 0
(3.19)
within a molecule with
3 Molecular Liquids
Each partial form factor (l = C, H or O for typical organics) has the form
s l (q, ω) = f l (q) p l (q, ω)
(3.15)
with the partial molecular structure factor
p l (q, ω) =
i
exp[i q · (r i − R 0 )]
(3.16)
with R 0 being the center of molecule. Then, we have
l
s l (q, ω)
2
=
l
m
f l (q) f m (q) p l (q, ω) p m (q, ω)
∗
.
(3.17)
Since f l (q) does not depend on the molecular orientation ω, the orientational averaging is necessary for products of p l (q, ω). Using these quantities, the averaged
scattering intensity from a single molecule is given by
| f av (q)|
2
=
l
m
f l (q) f m (q) p l (q, ω) p m (q, ω) ∗ .
(3.18)
The formulation given above is rigorous under the assumption, no correlation in
orientation between neighboring molecules. The validity relies on the choice of the
“molecules” as a constituting particle of the sample. Once we chose the molecule,
the other structural information is included in the structure factor of liquid, S(q).
Note that we only measure the net intensity of scattering.
A naïve way to define the “molecule” is to assume that atoms within the orientational correlation length constitute a molecule. This strategy will only be the
way for the molecular association to extend over a long distance, as in the case of
chain and/or network. On the other hand, if the association mode is well defined and
not-extending, molecular clusters with a well-defined structure may be regarded as
a molecule. If dividing the experimental I (q) by | f av (q)|
2 calculated under such an
assumption gives S(q) resembling that for a simple liquid such as rare gas elements
and hard spheres, the assumption seems plausible. In this context (or the way of
proceeding analysis), the calculation of | f av (q)|
2 based on available structural data
is crucial.
If we have the cartesian coordinates of atoms in a molecule (or a cluster), we
redefine the origin of the coordinate by subtracting the coordinates of the center of
electron density,
R i = r i − R 0
(3.19)
within a molecule with