1.1 Refinement and Analysis
For routine structure determinations, the independent atom model (IAM) is sufficient
to explain X-ray scattering with enough accuracy that atom positions and displacements can be confidently determined; this structure analysis paradigm has not
changed significantly since the early days of crystallography, nor have the underlying mathematical procedures. Nevertheless, increased computing power and optimized algorithms for linear algebra and Fourier transforms have increased the
practical limits of the complexity that can be handled within a reasonable timescale.
Interesting developments in the field have occurred at the frontiers of chemistry
and with increases in computational power, allowing excursions into new methods
and applications of crystallographic modelling, many of which have been incorporated into routine analyses. The use of crystal structure refinement to handle difficult
chemical and physical problems frequently leads to situations where extra information or assumptions are required to interpret or justify the results obtained. In
extreme cases the structure refinement fit can no longer independently confirm that
the structural model is correct, i.e. that it is a satisfactory approximation of the crystal
structure itself. These conditions can range from lack of good-quality data due to
data-limiting experimental conditions (e.g. such as high-pressure or hightemperature experiments), or intrinsic limitations of crystal quality (e.g. disorder,
solvent voids and stacking faults), to hard-to-model crystal structures which contain
multiple units related by non-crystallographic symmetry or modulation functions.
A key assumption for fitting a model by least squares minimization is that the
errors are random and normally distributed about their mean values. If scattered
X-rays are measured and processed carefully, the data will generally obey this
criterion; however the model itself can violate it by introducing systematic errors
into the fit.
1.2 Practical Elements of Refinement: The IAM Model
Using the independent atom model (IAM) to fit X-ray diffraction measurements
requires an atomic scattering factor for each element; for very precise work, it may
be necessary to use the scattering factor for the corresponding ion [1]. Scattering
factors are computed from the spherically averaged electron density arising from
relativistic Hartree-Fock atomic wave function calculations, or the analytical solution of the Schrödinger equation in the case of hydrogen atoms. The probability
density for an atom in real space and its Fourier transform, the scattering power in
reciprocal space, is shown in Fig. 1. The high densities close to the nucleus of the
atom account for the majority of the scattering interactions, and small deviations
away from the assumption of a spherically averaged X-ray scatterer, such as bonding
or lone pair electron density, are ignored within the IAM approximation.
Recent Developments in the Refinement and Analysis of Crystal Structures
45
For routine structure determinations, the independent atom model (IAM) is sufficient
to explain X-ray scattering with enough accuracy that atom positions and displacements can be confidently determined; this structure analysis paradigm has not
changed significantly since the early days of crystallography, nor have the underlying mathematical procedures. Nevertheless, increased computing power and optimized algorithms for linear algebra and Fourier transforms have increased the
practical limits of the complexity that can be handled within a reasonable timescale.
Interesting developments in the field have occurred at the frontiers of chemistry
and with increases in computational power, allowing excursions into new methods
and applications of crystallographic modelling, many of which have been incorporated into routine analyses. The use of crystal structure refinement to handle difficult
chemical and physical problems frequently leads to situations where extra information or assumptions are required to interpret or justify the results obtained. In
extreme cases the structure refinement fit can no longer independently confirm that
the structural model is correct, i.e. that it is a satisfactory approximation of the crystal
structure itself. These conditions can range from lack of good-quality data due to
data-limiting experimental conditions (e.g. such as high-pressure or hightemperature experiments), or intrinsic limitations of crystal quality (e.g. disorder,
solvent voids and stacking faults), to hard-to-model crystal structures which contain
multiple units related by non-crystallographic symmetry or modulation functions.
A key assumption for fitting a model by least squares minimization is that the
errors are random and normally distributed about their mean values. If scattered
X-rays are measured and processed carefully, the data will generally obey this
criterion; however the model itself can violate it by introducing systematic errors
into the fit.
1.2 Practical Elements of Refinement: The IAM Model
Using the independent atom model (IAM) to fit X-ray diffraction measurements
requires an atomic scattering factor for each element; for very precise work, it may
be necessary to use the scattering factor for the corresponding ion [1]. Scattering
factors are computed from the spherically averaged electron density arising from
relativistic Hartree-Fock atomic wave function calculations, or the analytical solution of the Schrödinger equation in the case of hydrogen atoms. The probability
density for an atom in real space and its Fourier transform, the scattering power in
reciprocal space, is shown in Fig. 1. The high densities close to the nucleus of the
atom account for the majority of the scattering interactions, and small deviations
away from the assumption of a spherically averaged X-ray scatterer, such as bonding
or lone pair electron density, are ignored within the IAM approximation.
Recent Developments in the Refinement and Analysis of Crystal Structures
45