Ionic Radius and Effective Atomic Radius

Atoms and ions in crystalline minerals pack together at regular distances and in specific geometric arrangements, and to understand and predict those arrangements it is useful to treat each ion as a small hard sphere with a definite radius. ^[1] This approximation is not strictly accurate - the actual volume of an atom is defined by electron probability clouds that are not spherical (except for electrons in s shells) and have no sharp boundary - but it is justified by the results it produces: treating ions as spheres leads to conclusions about mineral structures that are consistent with what detailed structure studies actually observe. ^[1]

Defining Effective Radius

The effective radius of an atom or ion is defined as the radius it would have if it behaved as a hard sphere, and it is derived from the measured distance between the centers of adjacent atoms or ions in a crystal structure. ^[1] For an uncharged atom, this is the atomic radius; for a charged ion, it is the ionic radius. The techniques of X-ray diffraction allow measurement of bond lengths, and from many such measurements across many different structures it is possible to derive reliable estimates of individual ionic radii even though X-ray methods alone cannot apportion the bond length between the cation and anion without additional analysis. ^[1]

Bond Length Equations

For a covalent or metallic solid made of a single element, the bond length L between two identical atoms is simply twice the effective atomic radius R: ^[1]

L = 2R

For an ionic solid, the bond length L between cation and anion equals the sum of their individual effective ionic radii R_c (cation) and R_a (anion): ^[1]

L = R_c + R_a

The Oxygen Reference: Why 1.26 Å, Not 1.40 Å

All ionic radii tabulations require an anchor value - a radius assigned to one ion from which all other radii are derived. The value used is that of O^2-: effective ionic radii in the standard tables are based on an O^2- radius of 1.26 Å when O^2- is in contact with six cations (i.e., in 6-fold coordination). ^[1] An older set of tables used 1.40 Å for O^2-, and those values were widely distributed, but the 1.26 Å system is now preferred because the 1.40 Å values lead to the absurd result that some cations appear to have negative radii. ^[1] To convert from the 1.26 Å system to the 1.40 Å system, subtract 0.14 Å from all cation values. ^[1]

It must also be noted that the radius of oxygen is not truly fixed: it varies depending on the electronegativity of the cations with which it bonds, being larger when bonding with low-electronegativity cations and smaller with higher-electronegativity ones. ^[1] However, for the common cations in most rock-forming minerals, the assumption of a fixed ionic radius works well enough to be useful for understanding silicate and oxide structures. ^[1]

Controls on Effective Ionic Radius

Two principal variables influence how large an ion appears to be in a crystal structure: its oxidation state and the number of anions surrounding it.

Oxidation State

For any element that can occur in more than one oxidation state, ion size decreases as positive charge increases. ^[1] Cations are always smaller than the equivalent neutral atom, and anions are always larger. The reason is straightforward: increasing positive charge means the nucleus holds the remaining electrons more tightly and pulls them closer, shrinking the ion; adding electrons to form an anion means the nucleus has a weaker hold on the outer electrons, allowing them to spread outward. ^[1]

Coordination Number

The number of anions in contact with a cation - the coordination number (CN) - also influences effective ionic radius. As the coordination number increases, so does the effective ionic radius of the cation. ^[1] In physical terms, a cation in a large hole surrounded by many anions has more room to spread out; a cation in a tight four-fold tetrahedral site is squeezed smaller by the surrounding anions. In a sense, cations expand or shrink to fill the space available to them. ^[1]

Spin State in Transition Metals

An additional complexity arises for transition metals that have d orbitals containing four, five, six, or seven electrons. ^[1] These elements can adopt either a high-spin or a low-spin configuration. In the high-spin state, electrons minimize pairing within the d orbitals; in the low-spin state, electrons are maximally paired, producing a smaller and more compact ion. ^[1] The high-spin state is generally preferred in nature because unpaired electrons represent a lower energy configuration than forced pairing within the same orbital. ^[1] However, certain bonding geometries require hybridization schemes that favor the low-spin arrangement instead. Iron is the most geologically important element that exhibits different spin states, and under crustal conditions the high-spin configuration dominates in the great majority of minerals. ^[1]

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