Spatial distribution functions 1590 1 2 3 4 5 6
r (Å)0
1
2
3
4
g(r)O O
O H
0 1 2 3 4 5 6
r (Å)0
1
2
3
4
5
6
N
(r)
(a) (b)Fig. 4.3(a) Oxygen–oxygen (O–O) and oxygen–hydrogen (O–H) radial distribution func-
tions for a particular model of water (Lee and Tuckerman, 2006; Marxet al., 2010). (b) The
corresponding running coordination numbers computed fromeqn. (4.6.23).
It is clear thatN 1 =N(rmin). As an illustration of the running coordination number,
we show a plot of the oxygen–oxygen and oxygen–hydrogen radialdistribution func-
tions for a particular model of water (Lee and Tuckerman, 2006; Marxet al., 2010)
in Fig. 4.3(a) and the corresponding running coordination numbers inFig. 4.3(b).
For the oxygen–oxygen running coordination number, the plot is nearly linear except
for a slight deviation in this trend aroundN(r) = 4. Thervalue of this deviation
corresponds to the first minimum in the oxygen–oxygen radial distribution function
of Fig. 4.3(a) and indicates a solvation shell with a coordination number close to 4
for this model. By contrast, the oxygen–hydrogen running coordination number shows
more clearly defined plateaus atN(r) = 2 andN(r) = 4. The first plateau corre-
sponds to the first minimum in the O–H radial distribution function andcounts the
two covalently bonded hydrogens to an oxygen. The second plateau counts two ad-
ditional hydrogens that are donated in hydrogen bonds to the oxygen in the first
solvation shell. The plateaus in the O–H running coordination number plot are more
pronounced than in the O–O plot because the peaks in the O–H radialdistribution
function are sharper with correspondingly deeper minima due to thedirectionality of
water’s hydrogen-bonding pattern.
4.6.2 Scattering intensities and the radial distribution function
An important property of the radial distribution function is that many useful observ-
ables can be expressed in terms ofg(r). These include neutron or X-ray scattering
intensities and various thermodynamic quantities. In this and the next subsections,
we will analyze this aspect of radial distribution functions.
Let us first review the simple Bragg scattering experiment from ordered planes in a
crystal, illustrated in Fig. 4.4. Recall that the condition for constructive interference is
that the total path difference between radiation scattered fromtwo different planes is
an integral number of wavelengths. Since the path difference (seeFig. 4.4) is 2dsinψ,