Radius Ratio Rule radius ratio rule - University of Guelph · Radius Ratio Rule In ideal ionic crystals, coordination numbers are determined largely by electrostatic considerations

CHEM 2060 Lecture 15: Radius Ratio Rules L15-1

Radius Ratio Rule In ideal ionic crystals, coordination numbers are determined largely by electrostatic considerations. • Cations surround themselves with as many anions as possible and vice-versa. • This can be related to the relative sizes of the ions. ⇒ radius ratio rule Radius ratio rule states:

• As the size (ionic radius r) of a cation increases, more anions of a particular size can pack around it.

• Thus, knowing the size of the ions, we should be able to predict a priori

which type of crystal packing will be observed.

• We can account for the relative size of both ions by using the RATIO of the ionic radii:

€

ρ =r+

r−


Limiting Radius Ratios For a specific structure, we can calculate the limiting radius ratio, which is the minimum allowable value for the ratio of ionic radii (r+/r-) for the structure to be stable. Let’s start by looking at the CsCl structure.


CsCl (8:8) …Let’s calculate the limiting radius ratio for this structure.

Recall: we know the length of each side of the triangle: cube edge (1), face diagonal (√2), body diagonal (√3)

2r− + 2r+ = 2r− 3r+

r−= 3 −1= 0.732


NaCl (6:6) …Now we’ll calculate the limiting r+/r- for rock salt.

The plane is just the face of the unit cell.

Recall: we know the length of each side of the triangle: cube edge & face diagonal!

4r− = 2r− + 2r+( ) 2

r+

r−=22−1= 0.414


ZnS (4:4) …limiting radius ratio for zinc blende and wurtzite

The easiest plane to select is simply half of a tetrahedron!

r− = 14a 2

r−

2=14a

r− + r+ = r−

2"

#$

%

&'

2

+ r−( )2

r+

r−=

62−1= 0.225


So, there is a range to the radius ratio r+/r- for a given arrangement of ions. C.N. of cation r+/r- range possible structures 8 (cubic) ≥ 0.732 CsCl, CaF2 6 (octahedral) 0.414 – 0.732 NaCl, TiO2, CdCl2 4 (tetrahedral) 0.225 – 0.414 antifluorite, ZnS 3 (triangular) 0.155 – 0.225 Examples:

Beryllium sulfide, BeS

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rBe2+

rS2-

= 0.591.7 = 0.35 ∴ C.N. = 4

Sodium chloride, NaCl

€

rNa+

rCl-

= 1.161.67 = 0.69 ∴ C.N. = 6

Cesium chloride, CsCl

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rCs+

rCl-

= 1.811.67 =1.08 ∴ CN = 8


Radius Ratio rule doesn’t always work… Zinc sulfide, ZnS r+/r- = 0.52 ∴ C.N. = 6 is predicted…WRONG! Both forms of ZnS (zinc blende and wurtzite) have C.N.= 4. QUESTION: Why do you think the Radius Ratio rule breaks down in this example? (Hint: Remember what we said about assumptions!)


This graph compares actual structures with predictions by radius ratio rules from r+/r- (r-/r+ if cation is larger – in purple) NaCl structure is observed more than is predicted! Radius ratios are only correct ca. 50% of the time, not very good for a family of ionic solids! (Don’t spend too much time analyzing this graph…it is only meant to show you that the radius ratio rules are not very good at predicting structures…)


Close Packing

• An alternative way of looking at ionic solids.

• Anions are often larger than cations and therefore “touch”.

• Small cations then fit in the “holes” between anions.

• Think of co-packing softballs and golf balls in the most efficient way. 1926 Goldschmidt proposed ions could be considered as hard spheres when packing in solids. This reduces the problem of examining the packing of like atoms to that of examining the most efficient packing of any spherical object. e.g., how oranges are most effectively packed…


CLOSE-PACKING OF SPHERES A single layer of spheres is closest-packed with a HEXAGONAL coordination of each sphere!


A second layer of spheres is placed in the indentations left by the first layer. Space is trapped between the layers that is not filled by the spheres: TWO different types of HOLES (so-called INTERSTITIAL sites) are left:

• OCTAHEDRAL (O) holes with 6

nearest sphere neighbors • TETRAHEDRAL (T±) holes with

4 nearest sphere neighbors


When a third layer of spheres is placed in the indentations of the second layer there are TWO choices: Option #1 - The third layer lies in indentations directly in line (eclipsed) with the1st layer Layer ordering may be described as ABA (hexagonal close packed – hcp) Option #2 - The third layer lies in the alternative indentations leaving it staggered with respect to both previous layers Layer ordering may be described as ABC (cubic close packed – ccp)


Close-Packed Structures The most efficient way to fill space with spheres Is there another way of packing spheres that is more space-efficient? In 1611 Johannes Kepler asserted that there was no way of packing equivalent spheres at a greater density than that of a face-centered cubic arrangement. This is now known as the Kepler Conjecture. This assertion has long remained without rigorous proof, but in August 1998 Prof. Thomas Hales of the University of Michigan announced a computer-based solution. This proof is contained in over 250 manuscript pages and relies on over 3 gigabytes of computer files and so it will be some time before it has been checked rigorously by the scientific community to ensure that the Kepler Conjecture is indeed proven! An article by Dr. Simon Singh © Daily Telegraph, 13th August 1998 http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/oranges.html


Features of Close-Packing Coordination Number = 12 74% of space is occupied Simplest Close-Packing Structures ABABAB.... repeat gives Hexagonal Close-Packing (HCP) Unit cell showing the full symmetry of the arrangement is Hexagonal


ABCABC.... repeat gives Cubic Close-Packing (CCP) Unit cell showing the full symmetry of the arrangement is

Face-Centred Cubic


The most common close-packed structures are METALS. A NON-CLOSE-PACKED structure adopted by some metals is:-

(Like CsCl)


Copper

Tungsten

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Radius Ratio Rule radius ratio rule - University of Guelph · Radius Ratio Rule In ideal ionic crystals, coordination numbers are determined largely by electrostatic considerations