Minerals
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A mineral is a naturally occurring solid chemical substance formed through biogeochemical
processes, having characteristic chemical composition, highly ordered atomic structure, and
specific physical properties.
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By comparison, a rock is an aggregate of minerals and does not have a specific chemical
composition.
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International Mineralogical Association (IMA, http://www.ima-mineralogy.org/) approved the
following definition in 1995:
"A mineral is an element or chemical compound that is normally crystalline and that has been formed as a result
of geological processes."
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According to this definition and classification scheme, biogenic materials were excluded from the
mineral kingdom:
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"Biogenic substances are chemical compounds produced entirely by biological processes without a
geological component (e.g., urinary calculi, oxalate crystals in plant tissues, shells of marine molluscs, etc.)
and are not regarded as minerals. However, if geological processes were involved in the genesis of the
compound, then the product can be accepted as a mineral."
There are currently more than 5,000 known minerals, according to the IMA, which is responsible
for the approval of and naming of new mineral species found in nature. Of these, perhaps 100 can
be called "common", 50 are "occasional", and the rest are "rare" to "extremely rare".
RRUFF database: http://rruff.info/
American Mineralogist Database: http://rruff.geo.arizona.edu/AMS/amcsd.php
Webmineral database: http://webmineral.com/
Types of chemical bonds and their general characteristics
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Because of the nature of ionic and covalent bonds, the materials produced by those bonds tend
to have quite different macroscopic properties. The atoms of covalent materials are bound tightly to
each other in stable molecules, but those molecules are generally not very strongly attracted to other
molecules in the material. On the other hand, the atoms (ions) in ionic materials show strong attractions
to other ions in their vicinity. This generally leads to low melting points for covalent solids, and high
melting points for ionic solids.
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Ionic Compounds
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–
–
–
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Crystalline solids (made of ions)
High melting and boiling points
Conduct electricity when melted
Many soluble in water but not in nonpolar liquid
Why is diamond an exception?
Covalent Compounds
–
–
–
–
Gases, liquids, or solids (made of molecules)
Low melting and boiling points
Poor electrical conductors in all phases
Many soluble in nonpolar liquids but not in water
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Elements from opposite ends of the periodic table will generally form ionic bonds. They will have large
differences in electronegativity and will usually form positive and negative ions. The elements with the
largest electronegativities are in the upper right of the periodic table, and the elements with the smallest
electronegativities are on the bottom left. If these extremes are combined, such as in RbF, the
dissociation energy is large. Hydrogen is an exception to that rule, forming covalent bonds.
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Elements which are close together in electronegativity tend to form covalent bonds and can exist as
stable free molecules. Carbon dioxide is a common example.
Oxidation state of atoms
Oxidation state is an indicator of the degree of oxidation of an atom in a chemical compound. The formal
oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different
elements were 100% ionic.
Oxidation states are typically represented by integers, which can be positive, negative, or zero. In some
cases the average oxidation state of an element is a fraction, such as 8/3 for iron in magnetite.
The increase in oxidation state of an atom through a chemical reaction is known as an oxidation; a
decrease in oxidation state is known as a reduction. Such reactions involve the formal transfer of
electrons, a net gain in electrons being a reduction and a net loss of electrons being an oxidation. For pure
elements, the oxidation state is zero.
Oxidation state
General rules for simple compounds without structural formulae:
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Any pure element (even if it forms diatomic molecules like chlorine, Cl2) has an oxidation state
(OS) of zero. Examples of this would be Cu or O2.
For monatomic ions, the OS is the same as the charge of the ion. For example S2- has an OS of 2, whereas Li+ has an OS of +1.
The sum of OSs for all atoms in a molecule or polyatomic ion is equal to the charge of the
molecule or ion, so that the OS of one element can be calculated from the OS of the other
elements. For example, in (SO3)2- (sulfite ion), the total charge of the ion is -2, and each oxygen
is assumed to have its usual oxidation state of -2. The sum of OSs is then OS(S) + 3(-2) = -2, so
that OS(S) = +4.
Do not confuse the formal charge on an atom with its formal oxidation state, as these may be
different, and often are different, in polyatomic ions. For example, the charge on the nitrogen atom
in ammonium ion NH4+ is +1, but the formal oxidation state is -3, the same as it is for nitrogen in
ammonia. In this case, the charge on the atom changed, but its oxidation state did not.
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Ionic radius, \is the radius ascribed to an atom's ion. Although
neither atoms nor ions have sharp boundaries, it is useful to treat
them as if they are hard spheres with radii such that the sum of
ionic radii of the cation and anion gives the distance between the
ions in a crystal lattice.
Ions may be larger or smaller than the neutral atom, depending on
the ion's charge. When an atom loses an electron to form a cation,
the lost electron no longer contributes to shielding the other
electrons from the charge of the nucleus; consequently, the other
electrons are more strongly attracted to the nucleus, and the radius
of the atom gets smaller. Similarly, when an electron is added to an
atom, forming an anion, the added electron shields the other
electrons from the nucleus, with the result that the size of the atom
increases.
The ionic radius is not a fixed property of a given ion, but varies
with coordination number, spin state and other parameters.
Nevertheless, ionic radius values are sufficiently transferable to
allow periodic trends to be recognized. As with other types of
atomic radius, ionic radii increase on descending a group. Ionic
size (for the same ion) also increases with increasing coordination
number, and an ion in a high-spin state will be larger than the
same ion in a low-spin state. In general, ionic radius decreases
with increasing positive charge and increases with increasing
negative charge.
An "anomalous" ionic radius in a crystal is often a sign of
significant covalent character in the bonding. No bond is
completely ionic, and some supposedly "ionic" compounds,
especially of the transition metals, are particularly covalent in
character.
Ionic radius
Bond valence method
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The bond valence method (or bond valence sum) is a popular method in
coordination chemistry to estimate the oxidation states of atoms and expected bond
lengths.
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The basic idea is that the valence V of an atom is the sum of the individual bond
valences vi surrounding the atom:
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The individual bond valences in turn are calculated from the observed bond lengths.
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Ri is the observed bond length, R0 is a tabulated parameter expressing the (ideal)
bond length when the element i has exactly valence 1, and b is an empirical constant,
typically 0.37 Å.
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Bond Valence wizard:
http://orlov.ch/bondval/index.html#4
Characteristics of crystalline state
• Great majority of minerals are crystals
• The two most important characteristics of a crystal are:
– Periodicity
– Symmetry
Description of crystal structure and direct space
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Unit cell parameters
Crystal system
Metric symmetry of the unit cell does not determine
the crystal system – the symmetry does
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Space group
Wyckoff positions and fractional atomic
coordinates
Site occupancy factors
Atomic displacement parameters (anisotropic
or isotropic)
Crystal data
Formula sum
Z
Crystal system
Space group
Unit cell dimensions
SiO2
3
trigonal
P 31 2 1 (no. 152)
a = 4.8815(7) Å
c = 5.3816(15) Å
111.06(4) Å3
2.695 g/cm3
Cell volume
Density, calculated
Atomic coordinates
Atom
SI1
O1
Wyck.
3a
6c
Anisotropic displacement parameters (in Å 2)
Atom
SI1
O1
U11
0.00618
0.01193
U22
0.00551
0.01158
U33
0.01209
0.02161
U12
0.00309
0.00913
U13
0.00051
-0.00280
U23
0.00026
-0.00316
x
0
0.27246
y
0.46621
0.41287
z
2/3
0.78002
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Unit cell is the simplest element of the crystal that is repeated by translations
to form infinite crystal lattice.
Geometrically, the unit cell is fully characterized by 6 parameters describing
the lengths of the edges and the inter-edge angles.
Alternatively, a set of principal translation vectors a1, a2, a3 can be described,
to form a basis of direct space reference system.
The content of the unit cell (atomic positions) are usually described using
fractional atomic coordinates (x,y,z)f, which are coordinates in the direct
space reference system.
Alternatively, atoms can be described in Cartesian coordinates of laboratory
reference system.
A conventional choice of the Cartesian reference system to describe unit cell
contents is given by matrix A:
a b cos
A  0 b sin 
0
0
c cos 
c(cos  cos  cos ) / sin 
c sin 2   cos2   cos2   2 cos cos  cos



1/ 2



/ sin  

Unit cell and
direct space
xyzC  Axyzf
Metric tensor
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Metric tensor (symmetric, positive definite) of direct space can be calculated as
follows:
T
GA A
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Matrix A is a Cholesky decomposition of the metric tensor.
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Determinant of the Metric tensor is equal to the square of the unit cell
volume:
detG   V 2
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Unit cell volume can also be calculated from vector equation:
V  a1  a2  a3 
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Inverse of the metric tensor, Gr=G-1 is the reciprocal space metric tensor
Calculation of density
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Unit cell of SiO2 quartz has a volume of 111.06 A3
The Z-number is 3
Atomic mass unit: 1u = 1.660 10-24 g
Avogadro’s number NA=6.022 1023 1/mol
What is the density of quartz?
What is the molar volume of quartz?
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All properties of a crystal remain invariant under lattice
translation, therefore only angular dependence of the
properties is of importance.
From the point of view of diffraction it is important to consider
families of parallel planes of lattice points.
These families of planes can be characterized by their vectors
normal. It is convenient to introduce another reference frame
basis, built on vectors “reciprocal” to the direct space basis
vectors:
This space is called reciprocal space. Reciprocal space
coordinates of vectors are always integer and are called Miller
indices.
The length of the reciprocal space vectors is defined as inverse
of the interlayer spacing in direct space.
The Brillouin zone is a primitive unit cell of the reciprocal
lattice.
By analogy to Cholesky decomposition of the direct
space metric tensor, a reciprocal space Cartesian basis
matrix B can be defined. B=A-1
Reciprocal space
Reciprocal space
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z
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hkl
X-ray
x
y
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Direct space relates to atoms in the unit cell, while reciprocal
space relates to peaks in diffraction experiment.
Vectors in reciprocal space correspond to families of planes
in the crystal (the direction of the rec. vector is normal to the
family of planes).
In a conventional crystal (as opposed to incommensurately
modulated crystal or quasi-crystal) vectors in reciprocal
space can occupy only points on a 3-dimensional grid.
Grid coordinates of reciprocal vectors are known as Miller
indices hkl
Geometry of the reciprocal space grid is determined by the
unit cell of the crystal. And can be directly measured in
diffraction experiment.
D-spacing is equal to inverse of the reciprocal vector length.
Coordinates of vectors in reciprocal space are described in
laboratory (instrument-related) reference coordinate system.
Seven crystal systems
Crystal System
Characteristic
Symmetry
Triclinic
1× 1-fold
-1
Monoclinic
1× 2-fold
2/m
Orthorhombic
3× 2-fold
Tetragonal
Syngony
Unit-Cell Parameters
Indep.
Parameters
a ≠ b ≠ c; α ≠ β ≠ γ
6
a ≠ b ≠ c; α = γ = 90°; β ≠ 90°
4
mmm
a ≠ b ≠ c; α = β = γ = 90°
3
1× 4-fold
4/mmm
a = b ≠ c; α = β = γ = 90°
2
Trigonal (see note)
1× 3-fold
6/mmm
-3m(R)
a = b ≠ c; α = β = 90°; γ = 120°
2
Hexagonal
1× 6-fold
6/mmm
a = b ≠ c; α = β = 90°; γ = 120°
2
Cubic
4× 3-fold
m-3m
a = b = c; α = β = γ = 90°
1
Scalar and vector equations to
calculate d-spacing
d
1
Bhkl
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UB matrix relates the Miller indices of reciprocal vector (hkl) with
its Cartesian coordinates in lab reference system (xyz) at zero
goniometer position.
Orientation matrix
xyz=UB hkl
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Columns in the orientation matrix are coordinates of the principal
vectors in reciprocal space 1 0 0 , 0 1 0 and 0 0 1.
UB matrix is composed of two sub-matrices, U and B
U describes the orientation of the crystal axes with respect to
the laboratory reference system, while B stores information
about the unit cell parameters.
By inverting the above equation one can calculate what are the
Miller indices of a measured reciprocal vector xyz
hkl=UB-1 xyz
z
hkl
X-ray
x
y
Symmetry operations in crystals
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Simple symmetry operations is crystals include:
Identity 1
Inversion -1
Proper rotation axes 2, 3, 4, 6
Improper rotation (rotoinversions) -3, -4, -6
Mirror plane m
Only symmetry operation that allow infinite tiling of space are allowed
Simple symmetry operations
inversion center
2-fold axis
Axis is always described by direction parallel (e.g. along a), whereas
plane is described by direction normal (e.g. perpendicular to b)
Symmetry element can be described by its type and general equation of
its invariant points:
mirror plane normal to b, located at y=1/2 is (x,1/2, z)
3-fold axis along to c, located at x=1/2, y=0 is (1/2,0, z)
Transforming atom coordinates with symmetry
operations
Mirror plane
All symmetry operations in crystals can be
expressed by combination of rotation,
translation and inversion.
For example, a matrix
describing a reflection in mirror
plane perpendicular to a,
located at x=0 (0,y,z) is:
 1 0 0
 0 1 0


 0 0 1 
(x,y,z) → (-x,y,z)
Write transformation matrices and transformation formulae
for a 2-fold and 4-fold axis, parallel to c, located at 1/2, 0, 0
(1/2, 0, z)?
1/2 +X,1/2 -Y, -Z
Point groups (describe external
morphology of crystals)
Crystal System
32 Crystallographic Point Groups
Triclinic
1
-1
Monoclinic
2
m
2/m
Orthorhombic
222
mm2
mmm
Tetragonal
4
-4
4/m
422
4mm
Trigonal
3
-3
32
3m
-3m
Hexagonal
6
-6
6/m
622
6mm
Cubic
23
m-3
432
-43m
m-3m
-42m
4/mmm
-62m
6/mmm
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A lattice is a regular array of points. Each
point must have the same number of
neighbors as every other point and the
neighbors must always be found at the
same distances and directions.
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All points are in the same environment.
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A Bravais Lattice is a three dimensional
lattice, characterized by translation
symmetry, which tiles space without any
gaps or holes.
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There are 14 ways in which this can be
accomplished. Bravais Lattices contain
seven crystal systems and four lattice
centering types.
Bravais lattice types
Symmetry operations involving translation
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Screw axes always have translation along the direction parallel. Subscript denotes
fraction of the translation period (e.g. 42 has a translation period of ½, 41 and 43 have
a period of ¼, 62 has a period of 2/6). Translation period is never more than ½.
Glide planes can have one (a, b, c) or two (n, d) translation directions. Translation is
never along the direction perpendicular. The letter describing the plane type denotes
the directions of translation (e.g. a has translation along x, b along y, c along z, n
along two directions). All glide translations have a period of ¼, except for d, for which
the period is ¼.
Glide plane
Screw axis
What symmetry operation is described by the
following transformation?
(x,y,z)→(1/2 +x,1/2 -y, -z )
21
(X, ¼, 0)
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All simple symmetry operations have invariant points
(points that transform into themselves as a result of the
operation).
Examples:
• mirror plane - all points on the plane
• Rotation axis – all points along the axis
• Inversion – the point where the inversion
center is located
An atom located on a symmetry element invariant point
has special properties. Such position is referred to as
“special position”.
There are less independent parameters needed to
describe an atom located on a special position.
A position within unit cell which is not located on any
symmetry element invariant point is called general
position.
Multiplicity of a position is the number of equivalent
repetitions of this position within the unit cell.
General positions have higher multiplicity than special
positions.
Symmetry elements involving translations do not have
invariant points and do not generate special positions.
General and special
atomic positions
Understanding space group symbol
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Thre are 230 space groups in 3 dimensions.
They were first enumerated by
Fyodorov (1891), Barlow (1894) and
Schönflies (1891).
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Space group symbol consists of
Capital letter describing the Bravais
lattice type, and three groups of lower
case letters/numbers describing
symmetry elements in different
symmetry independent directions
Explain space group symbols:
I-43m
P-421c
Pmc21
P3112
Crystal
System
Symmetry Direction
Primary
Secondary
Tertiary
Triclinic
None
Monoclinic
[010]
Orthorhombi
c
[100]
[010]
[001]
Tetragonal
[001]
[100]/[010]
[110]
Hexagonal/
Trigonal
[001]
[100]/[010]
[120]/[1 `1 0]
[100]/[010]/
[001]
[111]
[110]
Cubic
Centrosymmetric, polar and chiral space groups
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Crystals with centrosymmetric space groups (including inversion centers) exhibit some common
special properties: do not have piezoelectric effect and do not cause second harmonic generation
(frequency doubling)
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Symmetry element that include inversion symmetry are:
–
–
–
-1
2k/m rotation axis+perpendicular plane
-3
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Acentric space groups are also called polar.
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The point groups that possess no improper (-3,-4,-6) rotations and no inversion centers are called
enantiomorphic. Enantiomorphic molecules have right-handed and left-handed forms, although
in nature usually one form strongly predominates over the other. An example of enantiomorphic
mineral is quartz.
Location of symmetry elements in the unit cell
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Space group symbol equivocally defines the relative location of the
symmetry elements.
Symmetry elements repeat every half unit cell.
There are certain conventions regarding the choice of origin in the unit
cell (e.g. usually on an inversion center, if present).
Most space groups can be represented in multiple settings (e.g.
P21/c=P21/n).
Only primary symmetry element are described in the space group
symbol. There usually are additional symmetry elements that are
consequences of the presence of the primary elements.
Find coordinates of all equivalent general positions in the unit cell
Thermal vibrations and atomic displacement parameters
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Atoms in crystals vibrate about their equilibrium positions.
These vibrations are thermally activated and their amplitude increases with temperature.
To account for “blurring” of electron density around atoms resulting from thermal vibrations
crystallographers use atomic displacement parameters (ADP) describing radius of a sphere or
shape and size of ellipsoid of highest probability of finding given atom.
Because of anisotropy of interatomic interactions (e.g. bonding) the vibration amplitudes might be
anisotropic as well.
Isotropic ADP require one parameter per atom, anisotropic ADP require up to 6 parameters per
atom (3 axial lengths and 3 parameter describing orientation of the ellipsoid). Number of
independent anisotropic ADPs is modified by crystal and site symmetry.
Atomic vibrations in the presence of rigid bonds can cause errors in determination of bond lengths
using crystallographic methods. To account for this effect a TSL modeling (accounting for libraton)
is used.
Chemical substitution and site occupancy factors
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Atoms and ions of similar size (radius) and similar charge often substitute for each other in crystal
structures, without changing crystal symmetry. This effect is know as solid solutions.
It is also possible for atoms in the crystal lattice to be missing from their crystallographic sites,
leading to formation of vacancy defects.
If defects or substitutions show long-range order, the crystal symmetry changes.
In the description of the crystal structure site occupancy factor account for the degree of filling of
the given crystallographic site by the given atom type (1=full, 0=completely empty) averaged over
the whole crystal.
Software for visualization and analysis of crystal structures
XtalDraw (free)
http://www.geo.arizona.edu/xtal/group/software.htm
PowderCell (free)
http://www.ccp14.ac.uk/ccp/web-mirrors/powdcell/a_v/v_1/powder/e_cell.html
Endeavour (commercial, 60 day free fully functional demo)
http://www.crystalimpact.com/endeavour/download.htm