Principles of Crystal
Our discussion of bonding led us to conclude that the bond character
is determined primarily by the electronic configurations and electronegativity
values of the combining elements, whereas the crystal structure (which controls
the shape and system of the crystal) is a function of the plane lattice and the
coordination number(s) (C.N.).
I- Coordination Number
1-
Is the number of nearest neighbors
in a crystal structure. It depends on the radius
ratio + type of hybridization in covalent ones (Table 1) and the radius ratio in ionic structures (Table
2).
2-
Whereas the C.N. is strongly
controlled by the radius ratio, the radius of an ion (or atom) may (and usually
does) change as a function of the C.N. (Fig. 1; Table 3). Other factors
influencing the size of an ion are its charge, the atomic number of its
element, and the “shielding” effect of some electrons, especially those of the
penultimate shell. Of particular interest is the “lanthanide contraction”.
3-
Coordination numbers are usually
2, 3, 4, 6, 8, or 12. C.N.’s of 5, 7, 9, 10, or 11 are much less common, but
can occur in some cases!
Some Examples of
C.N. (Fig. 2)
Halite: NaCl
Na : Cl = 6 : 6
Sphalerite: ZnS
Zn : S = 4 : 4
Fluorite: CaF2
Ca : F = 8 : 4
II-
Structure Controls for minerals with metallic bonding:
·
Bonding takes place between
electropositive elements of similar electronegativities.
·
Radii of the combining
elements are also similar; radius ratio (rA/rB) @ 1.
·
Three types of packing
(Fig. 3):
1.
Hexagonal close packing “HCP”:
layers arranged in the pattern: ABAB. C.N. = 12.
2.
Cubic close packing “CCP”:
layers arranged in the pattern: ABCABC. C.N. = 12.
3.
Body centered cubic packing:
lower density structure; C.N. is always 8.
·
Close packing results in
the high density observed for metals and alloys.
III-
Structure controls for minerals with covalent bonding:
·
Hybridization of the
orbitals forces this type of bonding to have a strong directional character.
·
Because of hybridization,
the covalent radius of an element will be different from its ionic or metallic
radii.
·
The structure of the
mineral will be controlled by (i) the covalent radii of the elements; (ii) the
type of hybrid resulting (Table 1); same factors that affect the C.N.!
IV-
Structure controls for minerals with ionic bonding:
Pauling’s Rules for ionic compounds:
Rule 1:-
Cation – anion distance in a structure determined by radius
sums, C.N.’s determined from r+/r- (Table 1).
For this rule, the following points are very important:
1- The r+/r- limiting values for the different
coordination numbers are derived by simple rules of geometry and trigonometry
(Figs 4 & 5).
2- If the r+/r- falls very close to one of
the limiting values (e.g. 0.414), then the same compound may have two different
structures (in this case, one with a C.N. of 4; the other with 6). This may
result in polymorphism.
3- Because Oxygen is the most abundant element in the Earth’s
crust, and because it is strongly electronegative, most of the common
rock-forming minerals have a significant ionic character. Accordingly, relative
to Oxygen, elements will have a fairly predictable size and coordination
number. Common coordination numbers for some cations are listed in Table 3. Please make a strong effort to remember
these!
Rule 2:
Electrostatic valency
principle: The total strength of the valency bonds that reach an anion from
all the neighboring cations is equal to the charge of the anion (Fig. 6).
Electrostatic valency =
Z/C.N. (Where Z is
the charge of the cation).
Strength of bond = sum of
its electrostatic valencies.
This rule is useful for constraining the types of polyhedra
in a crystal structure.
Example:
Find the # of Ca2+ ions bonded to O in the structure of calcite,
knowing that the C.N. of Ca2+ (to O) = 6. Use Pauling’s second rule
knowing that the carbonate radical has a trigonal planar structure.
e.s.v. for CO3-2 = Z (for C)/ C.N. =
4/3
e.s.v. for Ca 2+ = 2/6 = 1/3
Let x be the # of Ca2+ linked to O in Calcite,
\x . 1/3 + 4/3 = 2
Þ
x = 2.
Example 2:
e.s.v for Si – O = 4/4 = 1
= ½ of 2 Þ (2
Si can link to one O)
\2
or more tetrahedra can become linked at their apex by sharing one of their
oxygens.
This process, by which different tetrahedra are linked to
each other, results in the formation of rings, chains, sheets or frameworks,
and is known as polymerization.
Types of ionic
compounds according to electrostatic bond strength:
1- Isodesmic compounds: Crystals
or compounds in which all bonds are of equal strength are known as isodesmic
(Example: the Spinel group (MgAl2O4)). In the case of the
spinels, there are 2 polyhedra. The contribution of oxygen to each of these two
polyhedra is 2, which makes the mineral isodesmic. Isodesmic compounds are
characterized by high symmetries, crystallizing in the cubic, tetragonal or
hexagonal systems. Oxides, halides, and hydroxides are isodesmic.
2- Anisodesmic compounds: Are ones that have different bonds with
different bond strengths. Sulfates and carbonates are anisodesmic. Carbonates,
phosphates and sulfates fall into this group.
3- Mesodesmic compounds: Are ones in which one particular cation
always accounts for half of the total bond strengths in the compound. Silicates
are mesodesmic.
Rule 3:
Polyhedra are usually joined at their corners; joining at
their edges is much less common, whereas joining by sharing faces is rare (Fig.
7). Sharing of faces among polyhedra generally decreases the stability of the
structure, especially if the centers of these polyhedra are occupied by cations
with high valency and small coordination numbers. This effect is especially large
when the radius ratio approaches the lower stability limit of the polyhedron.
Rule 4:
In a crystal with different cations, those cations with high
valency and small C.N. tend not to share polyhedral elements with each other.
However, if they do, the polyhedra become distorted. This is necessary to
minimize cation – cation repulsion.
Rule 5:
The number of essential sites in a crystal structure that
are substantially different (i.e. with differences in size and coordination)
tends to be small. On the other hand, many of these sites may be occupied by
different ions, leading to the phenomenon of “isomorphism”. This is known as
the “principle of parsimony”.
Example:
Amphiboles are minerals that often contain 13 different elements (or more!),
and are characterized by the formula:
A0-1B2C5T8O22(OH,F,Cl)2
Although this group of minerals has a large # of different
sites (relative to most other minerals), there are only 5 such “different”
sites:
A: occupied by K or Na in 8 or 12 coordination
B sites: filled with Na, Li, Ca, Fe2+, Mg and Mn2+
in six or 8 coordination
C sites: filled with Mg, Fe2+, Mn, Al, Fe3+,
or Ti all in six coordination
T sites: filled with Si or Al, in 4 coordination; and
finally,
Hydroxyl sites: where F and Cl may substitute.
IV- Calculation of
the number of formula units per unit cell “f”:
Density of mineral = weight of unit cell/volume of unit cell
(V)
Weight of the unit cell = molecular weight . # of formula
units in the cell (f)/ # of formula
units in 1 mole.
The # of formula units in 1 mole is Avogadro’s # (A = 6.023
. 1023).
D = Mol. Wt . f/A.
V
f = D . A . V/Mol.
Wt.
V- Isostructuralism
and Isotypism:
Isotypes:
Two minerals are considered isotypes if they have the same structures (arising
from similar C.N.’s and r+/r- values), but otherwise have
no chemical relations between them. Examples: Galena and Halite, NaNO3 and CaCO3,
Stishovite and Rutile.
Isostructural
groups: Minerals are considered isostructural if they have the same
structures (same C.N.’s), and same anionic group (but obviously different
cations, and hence different r+/r- values. Examples are
members of the aragonite group (aragonite, strontianite, witherite, and
cerussite).
VI- Some important
crystal structures
1- Diamond (covalent): F 41/d 
2/m; C.N = 4 (Fig. 8)
2/m; C.N = 4 (Fig. 8)
2- Graphite (covalent): P 63/m m c; C.N. = 3 (Fig. 9)
3- Sphalerite: F 
3 m, C.N. = 4 (Fig.
10)
3 m, C.N. = 4 (Fig.
10)
4- Halite (ionic): F 4/m 2/m, C.N. = 6 (Fig.
11)
2/m, C.N. = 6 (Fig.
11)
5- Pyrite (ionic/metallic): P 2/m , C.N. = 12:6 (Fig. 12)
, C.N. = 12:6 (Fig. 12)
6- CsCl (ionic): P 4/m 2/m, C.N. = 8 (Fig.
13)
2/m, C.N. = 8 (Fig.
13)
7- Fluorite (ionic): F 4/m 2/m, C.N. = 8:4 (Fig.
14)
2/m, C.N. = 8:4 (Fig.
14)
8- Calcite: P 2 m, C.N. = 6:6:2
(Fig. 15).
2 m, C.N. = 6:6:2
(Fig. 15).
9- Spinel: F 41/d 2/m, C.N.: Mg:O = 4:1,
Al:O = 6:3, Mg:Al = 4:2 (Fig. 16)
2/m, C.N.: Mg:O = 4:1,
Al:O = 6:3, Mg:Al = 4:2 (Fig. 16)
10- Perovskite: P 21/c 21/m 21/n,
C.N.: Ca:O = 12:4, Ti:O = 6:2, Ca:Ti = 8:8 (Fig. 17)
VII-
Structure of Silicates
·
The bond between Si and O
has a mixed character, and is approximately 50% ionic and 50% covalent.
·
Remember, silicate
structures are mesodesmic.
·
Si is tetravalent, whereas
O is divalent. Therefore, if both elements are to form an electrically neutral
compound, it would be SiO2 (equivalent mineral
would be quartz or one of its polymorphs).
·
A Si atom may also combine
with two more oxygens to form a negatively charged group (SiO4)-4. This group
will always be tetrahedral as dictated by the radius ratio of Si to O. This is
the basic building block of all silicate structures.
·
For this group to form an
electrically neutral mineral, it needs to combine with two divalent cations,
one monovalent and one trivalent cation, or four monovalent cations, of a
specific size or size range. The most common minerals with this structure are
olivines: (Fe,Mg)2SiO4 formed of single tetrahedra. Separate
tetrahedra that are adjacent to one another may polymerize by sharing Oxygens, giving rise to more complex
structures or networks of tetrahedra with different charges.
·
The reason for
polymerization of tetrahedra can be understood from applying Pauling’s second
rule; the e.s.v. between Si and O = 4/4 =1, which is equivalent to ½ the
bonding strength of O-2. Two tetrahedra may therefore be linked
through a bridging or shared oxygen.
·
The degree of
polymerization will depend on the conditions of formation or crystallization of
the silicate mineral, and may be used to subdivide silicates into six groups
(Fig. 18; Table 4).
Types of silicate structures:
(1) Nesosilicates
(orthosilicates): with isolated tetrahedra linked by bonds sharing oxygen
through cations. Example: olivine.
(2) Sorosilicates:
consisting of two tetrahedra sharing one oxygen (Si2O7)-6. Example: hemimorphite Zn4Si2O7(OH)2.H2O
(3) Cyclosilicates: more
polymerized, consisting of closed rings of 3, 4 or 6 tetrahedra each sharing 2
oxygens (Si3O9)-6, (Si4O12)-8, (Si6O18)-12. Example:
tourmaline
(4) Inosilicates:
consisting of single chains of tetrahedra, each sharing two oxygens (Si2O6)-4 (e.g. pyroxenes), or double chains of
tetrahedra in which each tetrahedron shares three oxygens (Si4O11)-6 (e.g. amphiboles).
(5) Phyllosilicates:
Continuous sheets of hexagonal networks of tetrahedra formed by sharing three
oxygens (Si4O10)-4. Individual
sheets are bonded to each other by the "interlayer cations",
resulting in the minerals developing excellent cleavage which separates it into
individual sheets. Example: micas.
(6) Tectosilicates:
Three-dimensional networks of tetrahedra, each sharing all four of its oxygens
(e.g. Feldspars).
It should be noted that Al+3 may substitute for Si+4 in many of the above structures, as both
cations are of comparable sizes. This substitution increases the negative
charge of the overall structural group, and must be compensated for in any
mineral by another substitution that will increase the total number of positive
charges to maintain electrical neutrality.
