1549380323-Statistical Mechanics Theory and Molecular Simulation

Spatial distribution functions 161

This simple analysis suggests that a similar scattering experiment performed in a
liquid could reveal the presence of ordered structures, i.e. significant probability that
two atoms will be found a particular distancerapart, leading to a peak in the radial
distribution function. If two atoms in a well-defined structure are at positionsr 1 and
r 2 , then the function exp[iq·(r 1 −r 2 )] will peak when the phase difference is an
integer multiple of 2π. Of course, we need to consider all possible pairs of atoms, and
we need to average over an ensemble because the atoms are constantly in motion. We,
therefore, introduce a scattering function

S(q)∝

1

N

〈

∑

i,j

exp (iq·(ri−rj))

〉

. (4.6.29)

Note that eqn. (4.6.29) also contains terms involving the interference of incident and
scattered radiation from the same atom. Moreover, the quantityinside the angle brack-
ets is purely real, which becomes evident by writing the double sum as the square of
a single sum:

S(q)∝

1

N

〈∣

∣

∣

∣

∣

∑

i

exp (iq·ri)

∣

∣

∣

∣

∣

2 〉

. (4.6.30)

The functionS(q) is called thestructure factor. Its precise shape will depend on certain
details of the apparatus and type of radiation used. Indeed,S(q) could also include
q-dependent form factors, which is why eqns. (4.6.29) and (4.6.30) are written as
proportionalities. For isotropic systems,S(q) should only depend on the magnitude|q|
ofq, since there are no preferred directions in space. In this case, it isstraightforward
to show (see Problem 4.11) thatS(q) is related to the radial distribution function by

S(q) = 4πρ

∫∞

0

dr r^2 (g(r)−1)

sinqr

qr

. (4.6.31)

If a system contains several chemical species, then radial distribution functionsgαβ(r)
among the different species can be introduced (see Fig. 4.3). Here,αandβrange over
the different species, withgαβ(r) =gβα(r). Eqn. (4.6.31) then generalizes to

Sαβ(q) = 4πρ

∫∞

0

dr r^2 (gαβ(r)−1)

sinqr

qr

. (4.6.32)

Sαβ(r) are called thepartial structure factors, andρis the full atomic number density.
Fig. 4.5(a) shows the structure factor,S(q), for the Lennard-Jones system studied in
Fig. 4.2. Fig. 4.5(b) shows a more realistic example of the N–N partial structure factor
for liquid ammonia measured via neutron scattering (Ricciet al., 1995). In both cases,
the peaks occur at wavelengths where constructive interference occurs. Although it
is not straightforward to read the structural features of a system off a plot ofS(q),
examination of eqn. (4.6.31) shows that at values ofrwhereg(r) peaks, there will be
corresponding peaks inS(q) for those values ofqfor which sin(qr)/qris maximal. The
similarity between the structure factors of Fig. 4.2(a) and 4.2(b) indicate that, in both
systems, London dispersion forces play an important role in their structural features.