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Subject
Geology
Paper No and Title
Crystallography and Mineralogy
Module No and Title
Module Tag
Min. Ib
Principal Investigator
Co-Principal Investigator
Co-Principal Investigator
Prof. Talat Ahmad
Vice-Chancellor
Jamia Millia Islamia
Delhi
Prof. Devesh K Sinha
Department of Geology
University of Delhi
Delhi
Prof. P. P. Chakraborty
Department of Geology
University of Delhi
Delhi
Paper Coordinator
Content Writer
Reviewer
Prof. Naresh C. Pant
Department of Geology
University of Delhi
Delhi
Prof. Naresh C. Pant
Department of Geology
University of Delhi
Delhi
Prof. Santosh Kumar
Department of Geology
Kumaun University
Nainital
Paper: Crystallography and Mineralogy
GEOLOGY
Module:
Table of Content
1. Learning Outcomes
2. Introduction
3.
3.1 Rule 1
3.2 Rule 2
3.3 Rule 3
3.4 Rule 4
3.5 Rule 5
4. Summary
Paper: Crystallography and Mineralogy
GEOLOGY
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1. Learning outcomes
After studying this module, you shall be able to:
Learn about Hexagonal and Cubic Close packing
Learn about the role of coordination number in ionic arrangement
Learn relationship of bonding and ionic radii
Understand the role of electrovalence in the ionic bonding
2. Introduction
The major anion in large number of minerals is oxygen and since it has a large
electronegativity, the bonds involving oxygen has a significant ionic character.
Linus Pauling conceived of cations and anions as hard spheres so tightly packed that
the spheres are touching each other and devised rules which help in many aspects of
crystal chemistry of mi
Rule deals with the analysis of bond length and bond strength in a crystal. The
charge and electrovalency of an atom plays an important role. The size of an atom or
ion depends on the size of the nucleus and the number of electrons.
3.
3.1. Rule 1: Around every cation, a polyhedral of anions exists in direct contact
with the cation. The cation-anion distance is determined by the radius sums
and the coordination number is determined by the radius ratio. Coordination
Number: It is the number of anions surrounding or in the immediate vicinity
of the cation. It depends on the relative sizes of the cations and anions. If all
the atoms in a crystal are of the same size then maximum of 12 anions can
surround a cation, leading to the 12-fold coordination. There are two ways in
which atoms can be packed in in 12-fold coordination. If we consider an
original layer of atoms as spheres of equal size touching each other, we can
call this as layer A. The voids between the spheres are of triangular shapes.
There exist two types of voids of triangular shape in a single layer of atoms
Paper: Crystallography and Mineralogy
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of equal size. Those triangular shapes with the triangles pointing up can be
called as B voids and those with the triangles pointing downwards can be
named C voids. If the next layer of atoms is added in such a way that they
occupy the space above the B-voids, and then add the third layer above the A
layer lies over the base A layer such that it lies over B-voids and the third, A
layer lies over the atoms of original A layer at the base, with a B layer
between them. This type of stacking is termed as hexagonal closest packing,
where the c-axis is oriented parallel to the AB AB layer.
Fig. 1 (a) A group of similar sized atoms stacked to form a layer A. (b) Layer
B is placed over the B-voids of layer A, resulting into the AB AB .stacking
pattern in the hexagonal closest packing, (c) Layer C is placed over the Cvoids of the original layer A, resulting in ABC ABCABC. Stacking pattern,
also known as cubic closed packing.
If after adding the layer of B atoms we place the next layer so that the atoms
occupy positions over the C voids in the layer A, and continue process
upward, we get a stacking pattern as ABC ABCABC
This type of
stacking pattern is known as the cubic closest packing. It results in a cubic or
isometric lattice with the
axis perpendicular to the layers. This is a very
simple case of stacking where equal sized atoms has been considered. If we
systematically decrease the size of the cation in such a way that it still
touches the surrounding anions, and the anions touch each other, it will result
Paper: Crystallography and Mineralogy
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in an 8-fold coordination, also known as the cubic coordination, closed
packing. Vertical cross section cutting through anions A and B. because the
shape of the object constructed by drawing lines through the centers of the
larger ions of a cube. If we keep on reducing the size of the cation, then a
scenario will come when the cation will become too small to touch the
surrounding anions. Hence, there is a limiting radius ratio that will occur
when the Rx/Rz becomes too small. A vertical section running between the
anions A and B and simultaneously running through the cation is shown in
Figure 2.
Fig. 2 8-fold coordination in cubic closed packing. Vertical cross section
cutting through anions A and B.
Using the Pythagoras theorem, we can write:
(2Rz+2Rx) 2 = (2Rz) 2 + (2O2Rz) 2
2Rz + 2Rx = (4Rz2 + 8Rz2)1/2
2Rz + 2Rx = (12Rz2)1/2
2Rz + 2 Rx = 3.464 Rz
2Rx = 1.464 Rz
Giving Rx/Rz = 0.732
Thus, for Rx/Rz< 0.732 the cation will be too small or will rattle in its site
and the structure will have to change to 6-fold coordination.
Paper: Crystallography and Mineralogy
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Six-fold coordination is also called octahedral coordination because the
shape defined by drawing planes through the centre of the larger ions is an
octahedron. Octahedral coordination is stable at the Rx/Rz = 0.732, but as
the size of cation gets smaller, eventually a limiting size is reached where the
ion will rattle in its site. The no rattle limit can be determined by looking at
the horizontal plane running through the ions labeled C and D.
Fig. 3 6-fold coordination.
In this case, we can write:
For Rx/Rz<0.414 the structure goes into 4-fold coordination. Planes through
the centres of the larger atoms in this case will form a tetrahedron, so 4-fold
coordination is also called tetrahedral coordination. The no rattle limit is
reached when Rx/Rz = 0.225. Beyond this triangular coordination results.
Paper: Crystallography and Mineralogy
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Figure 4: 4-fold coordination.
Table1: Summary of Rx/Rz ration and resulting coordination number.
Rx/Rz
C.N.
Type
1.0
12
Hexagonal or Cubic Closed Packing
8
Cubic
1.0
0.732
0.732
0.414
6
Octahedral
0.414
0.225
4
Tetrahedral
0.225
0.155
3
Triangular
2
Linear
<0.155
3.2. Rule 2: In a crystal structure, the sum of the strength of all the charges
produced by cations and anions should be null. It means that strength of
cation should be negated by the strength of anions. Electrovalency is
defined as:
e.v = Charge on the ion / C.N.
For example, in case of NaCl, each positively charged Na ions are
surrounded by 6 Cl-1 ions in the octahedral coordination, having coordination
number = 6. Thus e.v. = 1/6. Therefore, the strength of the negative charge,
which actually effects Na ion, is 1/6. Hence one positive charge is balanced
by the 1/6 negative charge of 6 Cl ions, 6*1/6 = 1. Similarly, each Cl ion is
Paper: Crystallography and Mineralogy
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surrounded by 6 Na atoms in octahedral coordination. So, again the 1/6 of a
positive charge from each Na reaches the Cl ion and thus the Cl ion sees
6*1/6 = 1 positive charge.
Figure 5: Schematic representation of the charge balance in NaCl in 6-fold
coordination and in CaF2 in 8-fold coordination.
Similarly, in the CaF2 structure, each Ca2+ ion is surrounded by 8 F- ions in
8-fold coordination. The e.v. reaching the Ca ion from each of the F ions is
thus ¼. Since there are 8 F ions, the total charge reaching the Ca ion is 8*1/4
or 2, thus balancing the charges.
In case of NaCl, where the charge is exactly balanced on both the anions and
cations and bonds are of equal strength from all directions, the bonds are
known as isodesmic. While in case of carbonate structural group, CO32-, the
C4+ ion is in triangular coordination with O2-. Here e.v. = 4/3 (C has a charge
of 4 and coordination number of 3). Thus, 3 oxygens contributes 4/3 charge
to the Carbon ion, and the charge on the carbon is balanced. However, each
O still has 2/3 of a charge that is unused, which leads to formation of bonds
with other elements, with the carbonate structural group. In cases like this,
where the electrostatic valency is greater than ½ the charge on the anion, the
anion will be more strongly bonded to the central coordinating cation than it
can be bonded to other structural groups. This type of bonding results in
bonds of unequal strengths, hence it is called anisodesmic. In cases where the
e.v. is equal to 1/2, as in case of SiO4, where each Si has e.v. of 1 and the
Paper: Crystallography and Mineralogy
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oxygen atoms are left with a -1 charge which is not shared. Since this -1
charge is exactly equal to ½ of the total charge of oxygen, the bonding with
other ions may be equally strong as with Si. In this case, the bonding is said
to be mesodesmic.
3.3. Rule 3: The sharing of edges and faces by two anion polyhedral decreases
the stability of an anionic structure. The decrease in stability is due to the
repulsion faced by the two cations when they are brought in proximity by
shared edges and faces. The effect is greater for cations with high charge and
low C.N.
Figure 6: For tetrahedral coordination, if the distance between the cations in
the polyhedrons that share corners is taken as 1, then sharing edges reduces
the distance to 0.58, and sharing of faces reduces the distance to 0.38.
3.4. Rule 4: In a crystal composed of different cations, those of high valency and
small coordination number tend not to share polyhedron elements with one
another. This tends to increase the distance between highly charged cations,
to reduce the electrostatic repulsion between them. For example, in Olivine,
FeMgSiO4, the structure contains distinct SiO4 tetrahedra, which do not
share any oxygen with each other. The lower valence Fe and Mg cations are
surrounded by polyhedral, which share oxygens.
3.5. Rule 5: There are few different types of cation and anion sites in a crystal.
Even though a crystal may have tetrahedral sites, octahedral sites and cubic
Paper: Crystallography and Mineralogy
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sites, most crystals will be limited to this small number of sites, although
different elements may occupy similar sites.
Well this makes understanding of crystal structure possible and easy.
Let us now consider application of these rules to a structure for example that
of mineral periclase which has formula MgO. The radius of Mg+2 is 0.66 Å
while that of the O-2 is 1.40 Å. The ration of 0.47 suggests that Mg should
be in six-fold coordination (octahedral coordination- range 0.41-0.73). With
this the bond strength for the Mg-O bonds is +2/6 = 1 /3. Each O-2 is bonded
to six Mg+2 (i.e. each O-2 is at a corner of six different octahedra) which
implies exact local charge balance. The structure of periclase can be views
as layers of Mg-filled oxygen octahedral, all of which share all of their
edges (12) with adjacent octahedral
4. Summary
We gained an understanding of the HCP ad CCP type of close packings. The radius
ratios of cations and anions not only control the environment of cations but also the
nature of bond. Polyhedrons that share corners have the maximum distance between
cations followed by those polyhedral which share edges. Polyhedra sharing faces
have cations closest to each other. Therefore, we have learnt a set of rules, which
consider the bonding cations and anions as hard spheres and help us in
understanding the crystal chemistry of minerals.
Paper: Crystallography and Mineralogy
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Frequently Asked QuestionsQ1. Derive the radius ratio for two-fold coordination?
Q2. Differentiate between the ansiodesmic and isodesmic?
Q3. How does hexagonal closed packing and Cubic closed packing form?
Q4. In which case in polyhedral arrangement the cations will be closest and in which
the farthest?
Multiple Choice Questions1. Electrovalence of Na ion in NaCl is
(a) 1/2
(b) 1/6
(c) 2/3
(d) 2/6
Ans: b
2. Sharing of edges of anionic polyhedra
(a) Increases the stability of the polyhedra
(b) Have no effect on the polyhedra
(c) Decreases the stability of polyhedra
(d) None of the above
Ans: c
3. Electrovalence of Ca in CaF2 is
(a) 1/4
(b) 1/2
(c) 2/3
(d) 3/4
Ans: a
4. NaCl is
(a) None of the below
(b) Mesodesmic
(c) Anisodesmic
(d) Isodesmic
Ans: d
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5. SiO4 is
(a) Anisodesmic
(b) Mesodesmic
(c) Isodesmic
(d) None of the above
Ans: b
Suggested Readings:
1. Klien, Cornelis and Hurlbut, Cornelius S., (1985). Manual of Mineralogy
(after James D. Dana), 20th Edn. John Wiley & Sons, New York. ISBN:
0471805807, 978-0471805809.
2. Putnis Andrew (1992), An Introduction to Mineral Sciences, 1st Edn.,
Cambridge University Press, UK. ISBN: 0521429471, 978-0521429474.
Paper: Crystallography and Mineralogy
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