A New Approach to Extracting Radial Pair Correlation
Functions from X-Ray Scattering Experiments
By Tony Link
Iowa State University REU Program
July 19, 2002
radial distribution function for oxygen
The study of liquid water has
atoms in the molecule and applying
gone on for many years and with the
least-squares non-linear fitting
increases in the development of
computations to best fit the data.
condensed matter theories and advances
in x-ray and neutron diffraction, the
search for a better understanding of
Background:
The study of structure of water is
earth's most prevalent liquid is taken up
done in two methods, x-ray and neutron
and furthered by researchers each and
scattering. In the x-ray scattering, the x-
every year. However, the structure and
ray photons are bounced off the electrons
dynamics that give the water molecule its
in the molecule. Most of the electrons
properties are still not well understood
are located around the oxygen molecule
today. My contribution to this search has
and some theorists suggest that the role
been the analysis of the x-ray diffraction
of the chemical binding of the hydrogen
data in a recent experiment done at the
to the oxygen places a greater amount of
Advanced Photon Source on both bulk
charge in the space located around the
water and water at the surface. The
oxygen molecule, which gives
analysis consisted of constructing a
approximately 7:1 ratio for the weight of
the O-O terms vs. the O-H terms in the
and molecular structure of the liquid
scattering1. Therefore most of the
water.
scattering data for x-ray diffraction is
The intensity equation for the
related to the distribution of oxygen
scattering in momentum transfer space
atoms in the molecule.
for a simple liquid of is2:
Neutron diffraction experiments
sin(Qr )
are similar in nature although the
I (Q ) =< F 2 > + < F > 2 ∫ 4πρ r 2 ( g ( r ) − 1)
experiments are done on D20 due to the
This equation has three main
large amount of incoherent scattering
components to it, the molecular form
involved with hydrogen during neutron
factors and the major component of the
diffraction experiments. These
static structure factor. The two
experiments generally are used to find
molecular form factors, <F>2 and <F2>,
the other correlation functions, the radial
are terms for describing the scattering
functions associated with deuterium or
based upon the constituents of the
hydrogen with respect to a deuterium or
molecule which makes up the liquid.
hydrogen located at the origin, and
The first form factor, <F2>, is a sum of
deuterium or hydrogen with respect to an
the scattering intensities of the atoms that
oxygen atom located at the origin, the
make up the molecule, namely one
gOH(r) and gHH(r) functions. These three
oxygen atom and two hydrogen atoms
radial distribution functions, the oxygen-
and describes the scattering from a single
oxygen, oxygen-hydrogen, and the
molecule. The other form factor, <F>2,
hydrogen-hydrogen, describe the atomic
describes the intermolecular scattering
Qr
dr
occurring within the liquid. These terms
between pair of atoms in radial space. In
are calculated using the Debye
x-ray scattering experiments upon water,
approximation which assumes a
the function describes the oxygen-
spherical electron density for each atom
oxygen pair correlation and is denoted,
within the molecule. 2 While this
gOO(r). The function contains
approximation is flawed in the lack of
information pertaining to how an oxygen
perfect spherical electron densities in
atom’s neighboring oxygen atoms
nature, it, however, is still an excellent
position themselves with respect to itself.
approximation for calculating these
The function tells the probability of
values. The final component is the
locating another oxygen atom a radius r
Fourier transform of the radial
away from an oxygen atom located at the
distribution function. This integral forms
origin.
most of the static structure factor S(Q)
which is generally derived from the
intensity by removing the form factors.
S (Q ) = 1 + h(Q )
h(Q) = ∫ 4πρ r 2 ( g (r ) − 1)
sin(Qr )
dr
Qr
The radial distribution function is
the object of desire in most scattering
experiments. The purpose of the
function is to describe the correlations
Figure 1: Intensity vs. Momentum Transfer
with experimental data in blue and best fit
achieve in red. Inset: Depictions of the
molecular form factors.
Experiment:
The following experiment was
performed by D. Vaknin at the Advanced
Analysis on the data was
Photon Source (APS) located in Argonne
conducted through the development of
National Laboratory. This experiment
custom functions which were ran
used a different geometry than the
through the non-linear squares fitting
conventional approach to water studies
process provided by C-Plot. The
by imploring a reflective geometry
algorithm was developed with a different
instead of a transmissive geometry. This
approach to the gOO(r) extraction from
geometry setup was used to probe both
the experimental data. The conventional
the surface of the water and the bulk of
approach for extracting gOO(r) from the
the water by using an angle of incidence
intensity data is to remove the form
below or above the critical angle for total
factors, and doing a Fourier Transform
scattering. The data collected from the
of the following form upon that result to
experiment was intensity vs. momentum
acquire gOO(r) 3.
transfer, I(Q).
g oo (r ) = 1 +
1
sin(Qr )
Q 2 hoo (Q)
dQ
2
∫
Qr
2π ρ
The new approach attacks the
problem from the other side by modeling
a gOO(r) with scalable parameters and
inserting this function into a modified
Figure 2: Diagram of the experiment showing
the incident and final vectors and angles as
photons were reflected off of the water.
Analysis:
version of the intensity equation. The
modeling process was based upon
creating a block representation of the
This equation has necessary fitting
oscillating extrema with a series of step
functions of varying heights. Each of
these step functions is convoluted with
its own complementary error function of
a varying width to form a smooth
continuous function containing all of the
structural information of the radial
function.
parameters: Ai, ri, and σi. These 3
parameters describe the features of all
the blocks which represent the various
extrema and are the parameters that are
fitted with the least squares non-linear
fitting. The Ai is the parameter
associated with the height of the
convoluted peak, ri is the position of the
convoluted peak, and σi is the width of
the convoluting error function.
This model g(r) is plugged
directly into the intensity equation with
some additional factors to account for
various corrections. The form of this
Figure 3: Comparison of the convoluted block
structure of step functions and the smooth
function after the convolution with the
complementary error functions.
equation is:
The process is easily defined as the
C
I(Q) =[ 2 +(< F2 >+< F >2
Q
following mathematical function:
g (r ) =
r − ri
1
( Ai − Ai −1 )erfc(
)
∑
σi
2
∫ 4πρ r
2
( gOO (r ) − 1)
sin(Qr )
dr )e−Qt ]I 0
Qr
This equation is very similar to the
equation for scattering for simple liquids
with a few noticeable additions. The
C
Q2
term is a correction for capillary waves,
microscopic vibrations occurring at the
surface. These waves become a
dominating when analyzing results from
near the surface of a liquid. The
e − Qt term is a correction to account for
photon absorption within the liquid.
Finally the I0 term is a scaling factor for
the fitting process.
It can be noted that there is exists both
similarities and difference in comparison
to a g(r) produced by the conventional
method using some recent experimental
data. An important similarity that exists
is the basic structure of both radial
functions is the same with the differences
occurring in the details of the peaks and
troughs. The most noticeable difference
occurs in the height and width of the first
peak. However, an interesting result
from the difference is that the area
Results:
underneath each peak is approximately
The process of fitting produced the
equal. The integral value associated with
following gOO(r):
those peaks is representative of the
number of nearest atomic neighbors in
any sphere of radius r, and therefore both
peaks are describing a similar number of
nearest neighbors. The differences
between the two results are related to the
Figure 4: Comparison of the gOO(r) produced
by this method with the gOO(r) from a
recently publish article.1
distribution of these atoms. Our result
suggests that there may exist a greater
uncertainty in the positions of these
Q, however, there is a noticeable
atoms.
similarity in structure between the two
sets of data. The two data sets have
Future:
The process described is a general
purpose method to calculated radial
distribution functions for any simple
bulk liquid; however, the process was
developed to compare the distribution
differences between the details of that
structure suggesting that there may be a
different g(r) for the water surface.
Future work and analysis will be done to
see how the inhomogeneity affects the
surface water.
difference between water in bulk and
water at an interface. In addition, further
study needs to be completed in exploring
the uncertainties in the parameters which
give rise to many possible radial
distribution functions which equally fit
the data.
In the future additional work will
be done to study water at the interface.
Preliminary work comparing the bulk
water and the surface water has begun.
As can be seen the capillary waves
dominate the intensity at low values of
Figure 5:Intensity vs. Momentum Transfer
comparing the bulk water data(purple) with
the surface water data(blue).
Conclusion:
The development of a new
approach based upon the modeling of the
radial distribution function gives another
tool in analyzing the data from x-ray
scattering experiments upon liquids.
This approach can be applied to study
much more than the structure of simple
1
Sorenson, Hura, Glaseser, Head-Gordon.
"What can x-ray scattering tell us about the
radial distribution functions of water?".
Journal of Chemical Physics, Vol. 133 N.
20. 22 November 2000
2
bulk liquids but also can be applied to
the study of liquid interfaces. Results
which have came from this approach
have so far suggested that there may be
more uncertainty within the actual
positions of near atomic neighbors in the
structure of a liquid than previously
observed. However, the dynamics of
molecular liquids still hold information
for future discoveries and this approach
may provide another means of studying
these interactions.
Acknowledgements:
This research and procedure was
developed under the supervision of D.
Vaknin and J. Morris at Ames
Laboratory, a U.S. Department of Energy
funded research site.
A. Guiner, X-ray Diffraction in Crystals,
Imperfect Crystals, and Amorphous Bodies
(Dover, New York, 1994)
3
A. Isihara, Condensed Matter Physics
(Oxford Press, New York, 1991)