Download Mineralogy - Carleton College

Mineralogy
Carleton College
Winter 2003
Lattice and its properties
• Lattice: An imaginary 3-D framework, that
can be referenced to a network of regularly
spaced points each of which represents the
position of a motif.
Lattice and its properties
• line lattice
• plane lattice
• space lattice
– unit cell
– primitive and non-primitive cells
Lattice and its properties
• I can generate a lattice
line from a lattice
point by translating
my lattice point with a
vector (a)
Lattice and its properties
• I can generate a lattice
line from a lattice
point by translating
my lattice point with a
vector (a)
vector a
Lattice and its properties
• I can generate a lattice
line from a lattice
point by translating
my lattice point with a
vector (a)
vector a
Lattice and its properties
• I can generate a lattice
line from a lattice
point by translating
my lattice point with a
vector (a)
vector a
Lattice and its properties
• Plane lattice: by
introducing another
vector b, that is not in
the same direction as
a, I can produce a
plane lattice
vector a
vector b
Lattice and its properties
• Space lattice, by
introducing another
vector c, which is not
in the same plane as a
and b, I can generate a
space lattice
c
b
Unit Cell
• The smallest representative unit of structure
which when repeated in 3-D gives the
whole crystal.
Structure:
• Nearly all minerals are crystalline solids composed of
atoms or ions held in an orderly, 3-D array by inter atomic
forces. Such array of atoms are called crystal structure and
are characterized by periodic duplication of any grouping
of atoms along any line through the structure.
• In other wards the ordered arrangement of atoms or group
of atoms within crystalline substance.
Unit Cell
• How to choose a Unit cell from plane
lattice?
Choice of a Unit Cell
Choice of a Unit Cell
• Look at this pattern, it
is produced by simple
translations.
• There are several
possible choices for
the Unit Cell.
Choice of a Unit Cell
Choice of a Unit Cell
Choice of a Unit Cell
C
Choice of a Unit Cell
• A lattice point occurs
where the corners of
four cells meet, and
therefore, 1/4 point per
corner lies in a give
cell (1/4 * 4=1)
C
Choice of a Unit Cell
• Unit Cells that include
one lattice point, such
as A, and B are called
primitive Cells.
• Unit Cell C is Nonprimitive.
C
Choice of a Unit Cell
• Many different cells
containing a single
lattice point may be
chosen.
C
Choice of a Unit Cell
• How do you chose the
Unit Cell?
C
– To keep the
translations short
– To provide as highly
specialized a lattice
geometry as possible
– To have the cell shape
comparable with the
shape of the crystal
Symmetry of a Lattice:
• Lets see what symmetry exist in a lattice for
a moment and we will come back to Unit
Cell
Elements of symmetry operations:
• Symmetry operations: Movements
performed on an object such that when
completed, the object looks the same as
when you started.
– These include:
Elements of symmetry:
–
–
–
–
–
–
–
–
Translation
Reflection
Rotation
Inversion
Roto-inversion
Roto-reflection
Glide
screw axis
Elements of symmetry:
• What elements of repetition exist?
– Translation
vector a
vector b
Elements of symmetry:
• What elements of repetition exist?
– Reflection/Mirror
• Mirror plane: plane passed through object such that
the images on opposite sides of the plane are mirror
images of one another
Elements of symmetry:
• What elements of repetition exist?
– Reflection
Elements of symmetry:
• What elements of repetition exist?
– Rotation
• Rotation Axis - An axis through the object, around
which the object is rotated such that the original
"motif" (or appearance) is repeated a specific
number of times during 360 degrees
Elements of symmetry:
• What elements of repetition exist?
– Rotation
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 90 degrees will give me..
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 90 degrees will give me..
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 90 degrees will give me..
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 90 degrees will give me..
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 90 degrees will give me..
Elements of symmetry:
• What elements of repetition exist?
– Here is a different unit cell
Elements of symmetry:
• What elements of repetition exist?
– Here is a different unit cell
Elements of symmetry:
• What elements of repetition exist?
– Rotation of 60 degrees gives me another motif
Elements of symmetry:
• What elements of repetition exist?
– Rotation
• 1 axis
• 2 axes
• 3 axes
• 4 axes
• 6 axes
360 degrees
180 degrees
120 degrees
90 degrees
60 degrees
Elements of symmetry:
• What elements of repetition exist?
– Inversion
Elements of symmetry:
• What elements of repetition exist?
– Roto-inversion
• first a rotation, then an inversion of 180 degrees
Elements of symmetry:
• What elements of repetition exist?
– Roto-reflection
Elements of symmetry:
• What elements of repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of
repetition exist?
– Glide
Elements of symmetry:
• What elements of repetition exist?
– screw axis
• This include translation and rotation together
Screw Axis
,
,
,
,
,
Rotation of 90 degrees
t1
Unit Cell
• Unit Cell parameters
– a, b, c – (
c
sides)
– a, b, g – angles
b
Unit Cell
• Unit Cell parameters
c
b a
a
– a, b, c – sides
– a, b, g –
b
angles
Translation Symmetry
• A translation is simply moving an object in
some direction (a, b, c) without a rotation.
Hence a point (x, y, z) is translated to the
point (x+a, y+b, z+c).
Translation Symmetry
• Crystalline materials have
structures with translational
symmetry. The unit cell of the
crystal contains the smallest
atomic group that is needed to
define the structure under
repetition.
Translational Nets in 2-D
• There are five different ways to
translate a point in twodimensions. Here is the first
simple net.
Translational Nets in 2-D
• There are five different ways to
translate a point in twodimensions. Here is the second
simple net.
Translational Nets in 2-D
• There are five different ways to
translate a point in twodimensions. Here is the third
simple net.
Translational Nets in 2-D
• There are five different ways to
translate a point in twodimensions. Here is the fourth
simple net.
Translational Nets in 2-D
• There are five different ways to
translate a point in twodimensions. Here is the fifth
simple net.
Translational Nets in 2-D (cont.)
• The diamond net can also be
defined in terms of a “centered
rectangular net” with a1 = a2
and g = 90degrees.