what is the charge density and what are the units of it?

Answer 1

Charge density relates to the probability of an electron being in a certain location. Since we don;t know exactly where the electron is at a given moment, it is based on calculated probabilities. The higher the probabilitiy zone, the higher the "charge density". For example, in the innermost orbital, 1S, the charge density is higher closer to the nucleus and lower the further out you go. The units should be the same as you might measure the cahrge of an electron divided by the space being considered.

Answer 2

It is well known that all properties of a microscopic system of N particles, e.g. charge densities and chemical bonding, can be obtained from the solution of Schrödinger's equation, but assuming a macroscopic crystal (N approx. 1024) the equation cannot be solved. Using the density functional theory of Hohenberg, Kohn, and Sham, Schrödinger's equation can be transformed into a system of 1024 one-particle-equations, and each particle suffers from an effective potential Veff(r). With respect to the Hohenberg-Kohn-Theorem the effective potentials can be obtained from the electronic charge densities (Schlüter & Sham 1982). Hence, if the charge density is determined experimentally, all properties of the microscopic system can be calculated without having the exact solution of Schrödinger's equation.
The method

Using coherent elastic Bragg scattering theory of X-rays the Fourier transform of the charge density of the unit cell of a periodic crystal is referred to as the structure factor
Eq. (1)

where H is the scattering vector of the reflection, rho is the electronic charge density, and the integral runs over the volume Vc of the unit cell. Hence, the charge density can be obtained by the inverse Fourier transformation
Eq. (2)

The structure factors are only defined at the reciprocal lattice points, and (2) can be replaced by the sum
Eq. (3)

Assuming that the atomic structure has been solved a priori, a model of the crystal structure can be created and fitted to the experimental data. Since the number of observations, i.e. reflections, is large compared to the number of model parameters, the fit can be carried out using least-squares methods. Here, the sum of the squares of the discrepancies between observed and calculated structure factors is minimized, while the model parameters are adjusted.

A simple structure model is based on the assumption that all atoms are independently and that no chemical bonding exists (Independent Atom Model, IAM). Hence, the charge is represented by a sum of spherical charge distributions located at the atom positions
Eq. (4)

The fit of an IAM to the data, which is usually the first step of a structure refinement, is used to improve the model atom positions and to determine Debye-Waller factors, which describe the effect of thermal vibrations on the reflection intensities.
Figs. 1,2,3

Figs. 1 and 2 show charge density maps, which are obtained from observed structure factors and calculated structure factors, respectively. The data are taken from a measurement on Cuprite, Cu2O, and the maps show a copper atom, which is flanked by two oxygens in the upper right and lower left corners. The contour lines are at 0.1 e/Å3. The density in the region near the nucleus is much too large to be visualized properly by the contour program, but it is possible to show the differences of both densities by calculating a map from the differences of both structure factor sets. Fig. 3 shows such a residual density map. The existence of the differences, i.e. the features in Fig. 3, proves that the IAM is not the ideal model to describe aspheric charge densities, which result from the chemical bonds. Hence, an improvement is only possible, if aspherical models are applied.

Such multipole models divide the charge density of an atom into inner, core and outer, valence regions, which are both spherically symmetric, and a deformation part, which is mathematically described by real spherical harmonics. As an example, the Hansen-Coppens multipole model (Hansen & Coppens 1978) is described as follows
Eq. (5)

Pc, Pv, and Plm denote the electron population of the three parts, rhoc, rhov, and Rl are appropriate radial functions, and dlm are normalized spherical harmonic functions. The parameters kappa and kappa prime are used to model the radial distribution of the charge density. The effect of adjusting these parameters is comparable to expand or contract the valence shells. A similar model - the Rigid Pseudoatom Model - was proposed by Stewart (1969)
Figs. 4,5

Fig. 4 shows the aspherical deformation part of the copper atom shown in Figs. 1 - 3, and Fig. 5 presents the residual density after the multipole refinement. Compared to Fig. 3 a significant improvement of the model density in the vicinity of the atom position is obtained.
Why using High Energy Synchrotron Radiation

Structure factor measurements have recently considerably improved using new techniques at synchrotron radiation sources, e.g. CCD area detectors. While these techniques were mainly developped in order to speed up the data collection and not to improve the accuracy of the derived structure factors, our interests were mainly focussed on the latter topic. In this context it is essential to minimize systematic errors in the various corrections applied to the data in order to improve the reliability of structure refinements, which depends strongly on the procedures to calculate structure factors from the measured reflection intensities.

The integral intensity of a reflection H is corrected as follows
Eq. (6)

where L is Lorentz correction, P is the polarization correction, A is the absorption correction, T is the thermal diffuse scattering (TDS) correction and E is the extinction correction. The Lorentz correction is usually carried out by
Eq. (7)

The polarization correction is a function of both the beam properties and the scattering geometry, i.e. in the horizontal the intensity has to be corrected according to
Eq. (8)

where Q is defined as
Eq. (9)

and the horizontally and vertically scattered intensities Ih and Iv can be determined experimentally with high accuracy using polarization monitors.

The thermal diffuse scattering covers a broad region in reciprocal space and peaks at the reciprocal lattice points. It is corrected according to
Eq. (10)

and ITDS is usually calculated from elastic constants or from the sound velocity. TDS can be minimized by measuring at low temperatures.

The corrections mentioned above depend on the experimental conditions, but absorption and extinction terms are often less reliable, because they also depend on the sample shape and quality, and, hence, for proper corrections the crystal shape and composition have exactly to be known, which is rarely the case. Additionally, the a priori determination of extinction is not possible. Therefore, it is commonly described by an additional refinement parameters, which may be affected by other sample properties, and conclusively can introduce additional errors.

Assuming the possibiliy of reducing absorption and extinction on an absolute scale, major data quality improvement should be achieved. This can be for instance carried out using short wavelength radiation for the data collection. Such radiation is provided by insertion devices at modern synchrotron radiation facilities, so that high intense beams of 100 keV photons and above can be used to collect charge density data in a resonable time. Based on these considerations we first performed a feasibility study at HASYLAB beamline BW5. Next, data was collected on Cuprite, Cu2O and YBa2Cu3O6.98 .