1. Radius Determination GLY 4200 Fall, 2018
Before the radius ratio concept can be used, the ratio must be determined.
Although it is easy to get a radius value which is nearly correct, it is rather difficult to determine a very precise value.
It is also important to know that there are different types of radius, depending on the type of bonding.

2. X-ray Diffraction Image
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X-ray crystallography can be used to determine the exact atomic positions within a crystal.
X-rays are similar to light, because both are forms of electromagnetic radiation. Like light, X-rays can undergo diffraction if the diffracting bodies are the proper distance apart.
X-rays have more energy and hence a shorter wavelength than light.
It was found that atoms in a crystal are spaced properly to diffract X-rays.
Images like these can be used to determine crystal structures

3. Comparison of Radii
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Before examining the types of radii, we need to know some general trends.
1. Radii decrease on going from left to right across the periodic table.
2. Radii increase on going down a column of the periodic table.
Examination of the diagram shows one exception. The radii of the inert or Nobel gases are larger than the elements to their left.
Why? Unlike a ball, an atom doesn't have a fixed radius. The radius of an atom can only be found by measuring the distance between the nuclei of two touching atoms, and then halving that distance. Since Nobel gas elements are only very weakly bonded together, they do not compress like other elements due, and their radii are larger.

4. Squashed versus Unsquashed
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In the left hand diagram, bonded atoms are pulled closely together. The measured radius is less than if they are just touching. The bonding may be either metallic or covalent, and the type of atomic radius being measured is called the metallic radius or the covalent radius, depending on the bonding.
In the right hand diagram, the atoms are just touching. The attractive forces are much less, and the atoms are essentially "unsquashed". This measure of atomic radius is called the van der Waals radius after the weak attractions present in this situation.
Left, metallic or covalent radius
Right, Van der Waals radius

5. Radii Across a Row
Diagram uses metallic radii for metallic elements, covalent radii for elements that form covalent bonds, and van der Waals radii for those (like the noble gases) which don't form bonds
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We can now begin to explain the trend of radii across the periodic table. Elements on the left side of the periodic table are metals, and we can use metallic radii. Elements toward the right side have covalent bonding, and we use covalent radii. But the inert gases are held together only by van der Waals forces, and so have larger radii. But why do the metallic or covalent radii shrink as you cross a row, adding electrons?
Metallic and covalent radii are a measure of the distance from the nucleus to the electrons which make up the bond. From lithium to fluorine, those electrons are all in the 2-level, being screened by the 1s2 electrons. The increasing number of protons in the nucleus as you go across the period pulls the electrons in more tightly. The amount of screening is constant for all of these elements. Why don’t the 2s orbitals screen the 2p orbitals? Prior to bonding, existing s and p orbitals within a shell (in this case, n = 2) are reorganized or hybridized into new orbitals of equal energy. When these atoms are bonded, there aren't any 2s electrons as such.
Covalent radius are generally much easier to determine – they are simply the arithmetic man of the distance between atoms attractive or repulsive electrostatic force (Coulomb’s Law).

6. Coulomb’s Law F ≃ (Z1Z2)/r2 where Zi = charge on each ion
r = distance between ions
This procedure generally works well – even for prediction.

7. Spin State High-spin versus low-spin
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HS refers to a high-spin state.
Some metals have two possible electronic configurations - High Spin and Low Spin. Low spin involves pairing of electrons in orbitals.
The two states have different magnetic, optical and structural properties. One of the differences is the difference in ionic radius, with low-spin radii being smaller than high-spin radii. .
High-spin versus low-spin

8. Covalent Radius Example
C-C spacing in diamond is 0.154nm
Metallic Si-Si spacing is 0.234nm
C radius = 0.077nm
Si radius = 0.117nm
Si – C = nm predicted
0.193 nm observed

9. Electron Density Contour Map of LiF
Electron density (electrons/volume) is the preferred method of determining ionic radius
Electron density contour map of LiF: a section through part of the unit cell face
The electron density (electrons A-3) is constant along each of the contour lines
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The hardest type of radii to determine are ionic radii. Ionic bonding does not involve equal sharing, so we have to figure out a method of splitting the distance between ions into two unequal parts.

10. Electron Density Variation
Electron density variation between Li+ and F-
Note the variation has a very flat bottom.
M, G and P indicate the true minimum, and the Goldschmidt and Pauling ionic radii
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X-ray mappings of electron density can be used to determine what the splitting should be.
But the distance between atoms is not enough to determine the radius of each atom.
If a series of measurements are made between different compounds, such as NaCl, KCl, KF, NaF, RbF, RbCl, etc., and if one radius is assumed, or determined by other means, a consistent series of values can be worked out.
This has been done by a number of people – each set is internally consistent but the sets do differ on the exact values. Both Shannon and Prewitt and Whittaker and Muntus have published tables of ionic radii. It is very important not to mix a radius from one set with another from a different set.
Several trends are apparent when studying ionic radii.
Ionic radii increase going down a group
Ionic radii decrease with increasing charge for any isoelectronic series of ions - Na+, Mg2+, Al3+, Si4+

11. Ionic Radii versus CN
From Shannon and Prewitt, Acta Cryt. B25, 725 (1969) and B26, 1046 (1970).
Ionic radii increase with increasing coordination number
Ionic radii decrease with increasing oxidation state - Fe3+ is smaller than Fe2+
What forces are responsible for holding ions together, but at a certain distance? The electrostatic attraction of the nuclei for the other ions electrons holds the ions together, while the electron – electron and especially, the nuclei – nuclei repulsion keep the ions apart – the observed radii are the best values that can be determined for the average ionic radius.
However if the resonance effect is present to a large extent the radius will not be the ionic radius. The handout sheet gives a series of ionic radius for various ions in different coordination polyhedron.
If we knew whether a crystal was held together by ionic or covalent bonds it would be easier to predict the bond lengths. There is one method used to predict the degree of ionic character a bond has. This is the electronegativity, a concept that was developed by Linus Pauling. Electronegativity is based on the idea of polarizability.
Polarizability is the amount of deformation that an ion undergoes in a given environment. Klein and Dutrow, p60 figure 3.20, gives a table of electronegativities. The greater the difference in electronegativities, the greater the percent ionic character:

12. Ionic Character Example: CaF Difference = 3.0 Close to 90% ionic
Example: SiN
Si = N = 3.1
Difference = % ionic