Crystallography: Symmetry at submicroscopic to macroscopic scales
So far we have talked about arrangements of just a few atoms and ions at the atomic level.
In a crystal, an atom or a group of atoms can be REPEATED by an OPERATION to generate a three-dimensional structure
The operations include ROTATION AXES, INVERSION, and MIRROR PLANES (Sen Fig. 2.7)
1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotation axes are possible
Rotation and inversion can be combined into ROTOINVERSION AXES (Sen Fig. 2.8)
A mirror plane is the same as 2-fold rotoinversion.
The five rotation axes and five rotoinversion axes can act individually or in combination in 32 ways.
NOW-- so far weve been talking about the atomic level.
But the repetitions weve talked about can propagate from atomic scale to the scale of mm or cm that we call MACROSCOPIC.
CRYSTAL FACES
The symmetry at an atomic level shows up at the macroscopic level in crystal faces.
Crystal faces develop on planes that contain high densities of atoms or groups of atoms.
Symmetry of faces is described by the same five rotation axes and five rotoinversion axes. As at the atomic level, these combine into 32 combinations, called CRYSTAL CLASSES.
Some of those crystal classes have symmetry in common, allowing further division into 6 (or 7) CRYSTAL SYSTEMS.
CRYSTAL SYSTEMS (Sen Fig. 2.12)
The systems are isometric, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic
In addition to symmetry elements, each crystal system is assigned a set of CRYSTALLOGRAPHIC AXES
Assignment rules are in Sen pp. 34-35.
Hexagonal can be subdivided into hexagonal & rhombohedral.
In monoclinic system the b-axis is the 2-fold or the perpendicular to the mirror plane
HOW MANY MINERALS IN EACH SYSTEM?
You might think, Nature being the chaos that it is, that most minerals would be triclinic.
But Nature actually prefers symmetry -- a brick wall with orderly bricks is more stable than a wall with randomly-oriented bricks
Only ~2% of minerals are triclinic.
41% are monoclinic or orthorhombic.
26% are isometric (cubic).
UNIT CELLS (Sen p. 40)
In 1784 Hauy recognized that crystals were built of small repeated units. We now call these UNIT CELLS.
At the atomic level, it is the smallest part of the 3-D structure that, by repetition, can generate the entire structure.
The crystallographic axes are also the axes of the unit cell.
AN ASIDE ABOUT SYMMETRY AT THE ATOMIC SCALE
p p p p p p p p p p
---------------------------------------------------------
b b b b b b b b b b
At the atomic scale, operations like reflection can be combined with a TRANSLATION operation.
The additional operation allows 230 different symmetries at the atomic scale, called the 230 SPACE GROUPS.
MILLER INDICES (Sen p. 42-43)
At the atomic scale, we can name planes by seeing where the plane intersects the crystallographic axes
Find the intercepts in terms of unit cell dimensions for x, y, and z
Invert each intercept
Clear fractions to obtain integers
If the number is < 0, put a bar above it
We do the same operations at the macroscopic level (Sen Fig. 1.16)
o Note that (113), (111), and (110) do not intersect a and b axes at the same place -- we reduce integers to the lowest possible numbers
o Thus (111) really denotes the ORIENTATION of that face
o (110) does not intersect c, so the intercept is infinity and the Miller index 0
CRYSTAL FORMS (Sen p. 45-46)
In an octahedron the (111) face can be rotated by 4-fold symmetry to generate the other 3 faces on top of the octahedron, and then those faces can generate the 4 faces on the bottom by a mirror plane.
All those faces, related by symmetry, are called a FORM. The octahedron is denoted {111}
Some other cubic system forms are shown.
Forms that completely enclose a crystal are called CLOSED FORMS (Sen Fig. 2.18), and those that do not are OPEN FORMS.
A prism is an open form, so another form must also be present to completely enclose the crystal.
PIEZOELECTRICITY
Many physical, magnetic, and electrical properties of minerals are related to crystallography.
Here we mention one, piezoelectricity.
Piezoelectricity results from a nonuniform charge distribution within a crystals unit cells. When such a crystal is mechanically deformed, the positive and negative charge centers displace by differing amounts. So while the overall crystal remains electrically neutral, the difference in charge center displacements results in an electric polarization within the crystal.
That nonuniform charge distribution can only occur if the crystal lacks a CENTER OF SYMMETRY, as the class 622 hexagonal crystal.
21 of the 32 crystal classes lack a center of symmetry.
Quartz was discovered to be piezoelectric in the 1880s by Jacques & Pierre Curie.
When strained, an electrical current is produced.
In CONVERSE-piezoelectricity, passage of an electrical current will strain the crystal. A quartz wafer in digital watches vibrates at a fixed frequency due to an AC current. That vibration tunes the watch to count time.
There are many applications of piezoelectricity today, including loudspeakers, microphones, sonar wave generation, and car bumper sensors.
In class I will demonstrate a piezoelectric sensor and a converse-piezoelectric ceramic vibration damper in my ski.