Sodium radial distributions

Sodium fluoride, NaF, is a favorable choice for X-ray analysis of the lattice energy of an ionic crystal. Both Na and F are relatively light atoms, and the Na 3s-radial distribution, though diffuse, is not quite as spread out as the Li 2s shell (single-C values are 0.8358 and 0.6396 au-1, respectively see appendix F), and therefore contributes to a larger number of reflections. 

Fig. 5.4 Radial distribution functions for sodium, showing the inner shells, and orbitals with n=3. Note the different degrees of penetration, 3s being most penetrating, and 3d least so.

The radial distribution functions of 3s, 3p, and 3d orbitals together with that of the sodium core (ls22s22p6). 

Radial Distribution Functions. What happens when X-ray diffraction occurs in liquids To understand this (see also Section 3.11), it is best at first to consider only a single-species liquid one could have in mind not a binary molten salt such as liquid sodium chloride, but, say, liquid sodium. 

If one takes g n gir) as the pair correlation function defined in Eq. (5.2), the radial distribution function represents the number of particles of 5 in a shell up to rj around A. If r, is then (see Fig. 5.7), one can regard Eq. (5.2) as giving the coordination number of A in the liquid. In the example chosen for simplicity, species A is the same as species B but this of course is only true for radial distributions of monatomics, e.g., sodium. It is found in practice that in a liquid, g B r) settles to unity by the third or fourth atom away from the reference atom A. 

Figure 8.15 Comparison of radial distributions of oxygen atoms conditional on the simplest metal ions in typical aqueous solutions obtained by ab initio molecular dynamics (AIMD). See Asthagiri etal. (2004c) for details. The potassium result was presented by itself in higher detail in Fig. 7.7, p. 157. Notice that the lithium result (displaced vertically by 2) and the sodium result (displaced vertically by 1) have inner shells clearly defined on the basis of the g r). For lithium, the occupancy of that inner shell is almost exclusively 4. For sodium, the principal occupancy is 4, but there is a statistical admixture of another oxygen that also serves to blur the primary minimum this occupancy is indicated by 4-1. For potassium, this statistical characterization is 4 - 2, as was also shown differently by Fig. 7.7 this leads to the occultation of the principal minimum in that case.

The radial distribution of electron probability density for the sodium atom. The shaded area represents the 10 core electrons. The radial distributions of the 3s, Ip, and 3d orbitals are also shown. Note the difference in the penetration effects of an electron in these thiee orbitals. 

The computed radial distribution functions (g(r)) are shown in Fig. 18. The g(r) between the Ca ions, the Na and the 02 type atoms (carboxylic PGA oxygen) exhibits a sharp, well defined peak at about 2.5 A The calcium plot is markedly more pronounced than the sodium one. This suggests calcium ions play a relevant bridging role between the chains while sodium mainly interacts with the peripheral carboxyl groups. The Ca-OW and Na-OW g(r) (OW= water oxygen) are very similar and shows two well defined hydration shells at about 2.5 and 4.5 A. The sodium one is a little more pronounced on the second peak, suggesting its water shell is more lasting and complete. 

Fig. 1 Diagrams depicting a a layer of a cubic sodium chloride crystal b a monoclinic 1,3-dimethylimidazolium chloride ionic-liquid crystal c two radial distribution functions (RDFs) in liquid l-dodecyl-3-methylimidazolium hexafluorophosphate. Anions and cations are depicted in red and blue. In the cases of b and c the blue circles represent the centroid of the imidazolium rings of the cations. The alternating sequences of red and blue circles in a and b as well as the two curves in phase opposition in c clearly indicate the existence and nature of the polar networks in ionic condensed phases...

Figure 27.31 Radial distribution curve in liquid sodium.

Figure 27.31 shows the radial distribution function, 4nr p, in liquid sodium. The upper drawing interprets the peaks in terms of shells of atoms around the central atom. At 400 pm from the central atom, the average number of atoms in the liquid is 10.6. This number is determined by the shaded area under the curve. The vertical lines show the number of atoms in successive shells in solid sodium. 

McKeown, D. A. (1987) Radial Distribution Analysis of a Series of Silica-rich Sodium Alumino-silicate Glasses Using Energy Dispersive X-ray Diffiaction, Phys. Chem. Glasses, 28, 156-163. 

Figure 1 presents a comparison between the CHARMM run and the AMBER run for the resulting radial distribution functions of the three pairs of charged species in the mlp system sodium-sodium, sodium-phosphorus (as the center of the phosphate... 

Figure 1. Radial distribution functions (rdf) for the ionic species in the 2 ns mlp simulations carried out with CHARMM23 (solid line) and AMBER 4.1 (broken line). A sodium - sodium rdf B phosphorus - sodium rdf C phosphorus -phosphorus rdf.

Sposito G, Park SH, Sutton R (1999) Monte Carlo simulation of the total radial distribution function for interlayer water in sodium and potassium montmoiillonites. Clays Clay Miner 47 192-200 Springborg M (1997) Density-Functional Methods in Chemistry and Materials Science. John Wiley and Sons, Chichester... 

Other ionic systems were also studied for example, solutions of soap micelles such as potassium oleate were found to give sharp small-angle X-ray diffraction bands [4]. Brady carried out a Fourier analysis of the X-ray diffraction of solutions of sodium dodecyl sulfate to calculate the radial distribution function g(r) and the number of nearest neighbors (N ) [5]. At about 30%, the N was about 12, indicating that the spherical micelles tended to assume a close-jmcked hexagonal arrangement. 

Fig. 2. Pair-wise radial distribution functions obtained from MD simulations (SPC/E water, 298 K, 1 atm) for halides of sodium and caesium.

Figure 12. Calculated cation-water radial distribution function os. center of mass separation R for the dilute aqueous solution of sodium at T = 25 C...

Schiesser and Lapidus (S3), in later studies, measured the liquid residencetime distribution for a column of 4-in. diameter and 4-ft height packed with spherical particles of varying porosity and nominal diameters of in. and in. The liquid medium was water, and as tracers sodium chloride or methyl orange were employed. The specific purposes of this study were to determine radial variations in liquid flow rate and to demonstrate how pore diffusivity and pore structure may be estimated and characterized on the basis of tracer experiments. Significant radial variations in flow rate were observed methods are discussed for separating the hydrodynamic and diffusional contributions to the residence-time curves. 

Fig. 6.4 The radial charge distribution of the screening clouds around sodium, potassium, magnesium, and aluminium ions in free-electron environments of the appropriate equilibrium metallic densities. The arrows mark the positions of the first nearest neighbours in hep Mg and fee Al, the first and second nearest neighbours in bcc Na and K. (After Rasolt and Taylor (1975) and Dagens et al. (1975).)...

It is of interest to consider the experimental radial electron density distribution in the ions Na+ and Cl- in sodium chloride in relation to corresponding results for the free ions calculated by the self-consistent field method. In Fig. 3 data from the experimental study of Schoknecht... 

---