Why Symmetry?

# Why Symmetry?

Symmetry constrains the chemistry, thermodynamic properties (energy) and physical properties of a crystalline solid. Magnets wouldn't retain their magnetism and lasers wouldn't work without symmetry. Any physical property that has directionality is effected by symmetry. Electrical, optical, magnetic and elastic properties all have directionality.

Q.1. Which of the physical properties we use in lab for hand samples have directionality? e.g. hardness (check kyanite), cleavage, density, luster. Explain...

Mineralogists and mineralogy students use symmetry to classify and characterize minerals but symmetry is widely studied outside of mineralogy especially by chemists and physicists because of its importance for understanding the properties of crystals.

# What is Symmetry?

Symmetry (as we learnt earlier in this lab) consists of these operations.
• Translation along a direction for a unit length
• Rotation about an axis through an angle
• Reflection across a plane (mirror plane)
• Inversion through a point
• Rotation-Inversion (combination of rotation and inversion)
• Glide (combination of reflection and translation)
• Screw Axis (combination of rotation and translation)

Symmetry consists of elements
• Rotation through 90o is a four-fold rotation. A four-fold rotation axis is a symmetry element.
• A mirror plane or symmetry plane is a symmetry element.
• A center of symmetry (Inversion Point) is a symmetry element.
Q.2. List the possible rotation axes used in crystallography, including rotation-inversion.

A collection of symmetry elements is known as a group. Because mathematicians like to show symmetry elements operating on points, the collection of symmetry elements is called a point group.

We have learned that minerals have

• a well constrained chemical composition
• crystal structure.

The chemical composition consists of different types of atoms. These atoms are arranged in a crystal structure that consists of atoms or groups of atoms that are repeated regularly in three dimensions.

This regular repeat is a type of symmetry known as translational symmetry.

One way of representing translational symmetry is by using an array of points known as a lattice. A lattice is an infinite array of points in space, in which each point has identical surroundings to all others. Each point is called a lattice point. The lattice represents the filling of space by the groups of atoms. It is important not to confuse lattice points (a mathematical construction) with atoms (physical objects). Sometimes atoms are at the lattice points and sometimes not. A unit cell in 2D is defined by two vectors. It's easy to work with this on paper (the first part of this lab).

Three vectors define a three-dimensional solid which is a mineral. Imagine three vectors connecting the points in the 3D movie.

Primitive. If the cell contains one lattice point it is primitive. This is the same as having lattice points only at the corners of the cell. All primitive cells in the same lattice have the identical volume. A primitive unit cell is the smallest unit cell that will define the crystal structure.

Look at the 6 lattices below. They are all primitive cells and are given the label P. (These lattice moovies were made by Nord Consultants using Crystalmaker on a Power Macintosh and with the kind permission of David Palmer.) Play with these lattices and make them move with the hand.

These lattices are all refered to using a specific notation. All are logical except for triclinic. Cubic is c, monoclinic is m, tetragonal is t, hexagonal is h, orthorhombic is o but triclinic is a where a stands for anorthitic. For example a triclinic primitive lattice in International Notation would be aP

Q.3.Label each lattice with it's crystal system, use the correct International Notation. (#4 is a little tricky, move it around a lot.) While you're at it - write down a, b, c, alpha, beta, gamma notations for each crystal system.

These primitive lattices are only 6 of the 14 known space lattices or more commonly called Bravis Lattices. The other 8 lattices are formed by adding additional lattice points and are called Non-Primitive. The cell is non-primitive if it contains more than one lattice point.

• are we making a new arrangement that still conforms to the crystal system?
• is it a new lattice?
The ways we can add lattice points is called centering:
• body centered - a point at the center of the primitive cell. This makes a new lattice denoted by the symbol I.

• all face centered - a point in the center of every face. This makes a new lattice denoted by the symbol F

• one-face centered - a point in the center of one face - and the corresponding face - the symbol is A, B or C depending on the face chosen.

Some crystal systems have four space lattices (P, I, F, C/A/B) some only P. Do you known which ones they are? Check it out on p. 230 of your text

Why are there only 14 Bravais Lattices? Mathematicians found in the 19th century that there were only 14 types of lattices that would fill space. That's it, only 14.

The choice of primitive or non-primitive unit cells is arbitrary. All the non-primitive lattices, I, F, A, B, C can be described using a primitive unit cell with one lattice point. BUT this smaller unit cell "by itself does not neatly, in a clear manner, display the full rotational symmetry of the crystal system". Burns and Glazer, 1990. Crystallographers and mineralogists usually chose the unit cell that best displays the symmetry while solid state physicists usually chose a primitive cell because it fully describes the electronic states in band theory.

Q.4.Look at the remaining Bravis lattices shown below and ascribe each to a crystal system.

Q.5. Why does the cubic system not have a A/B/C space lattice?

With all this useful information stored in our brains - lets now go to an on-line site in Switzerland called Escherweb.

WE will be working with space groups, you may want to refer to p. 237 in Perkins.

Hold the mouse button down and select new window with this link. Then Escherweb will be in one window and the lab in another so that you can read the questions and do the experiments without going back and forth. N.B. It takes a little while to load.

Q.6. Click on p1 at the top and the red ball at the bottom. Choose the arrow on the pull-down menu on the lower right. Choose a color for your arrow and click on the upper left-hand corner of the polygon. Draw what happens and explain in terms of the symmetry of p1. Discuss both the translational and rotational, etc. operations. Explain the symmetry notation p1.

ERASE

Q.7. Click on pm and place the arrow in the upper left-hand corner of the rectangle. Draw what happens and explain. make sure you find the mirror. Add a second arrow of a different color just below the first. Is there a horizontal mirror? Why / why not? Explain the symmetry notation.

ERASE

Q.8. Click on p6 and place the arrow (or a different symbol) in the upper left-hand corner. What happens? You may want to repeat this several times at different distances from the corner. Explain the symmetry notation.

ERASE

Q.9. Click on p4mm and place a diamond in the red and black triangle. How many diamonds appear in the cell. Draw what you see and explain the symmetry notation.

ERASE

Q.10. Click on cm and place the symbol at a corner. How many symbols appear in the cell. Compare this to pm (you may want to go back and forth and use a few symbols at a time. Explain the differences (This is also fun to try with p6 and p6mm.)

Hope you enjoyed your visit. A great inactive site. You can revisit this site, it is one of the links in the hyperlinks page for Geol 302.

Q.11.DEFINE SPACE GROUP

Enough of the geometry for a while, lets look at some simple crystal structures. We shall start with cubic as that is the system we are currently studying.

Here is the structure of Native Copper (Cu).

Q.12. What symbol would you use? P,F,C,I? Why?

Crystallographers also use the notation 4/m32m for this point group. Rotate the copper structure about so that you can see a 4-fold axis along <100> a 3-fold axis along <111> and mirror planes perpendicular to <100> and <100>.

Q.13. What symmetry do you see in this crystal of Halite (NaCl)?

Q.14.What symmetry in Fluorite (CaF2)?

Copper, halite and fluorite should all be F4/m32/m - check yourself.

Q.15.What about CsCl? and iron (Fe)?

Q.16.What would you call sphalerite (CuZn)?

As you notice, all these crystals have been non-silicates. This is because the silica tetrahedra is a better building block for describing silicates. Silicates often have much larger unit cells and are much more complex. The details are beyond an introductory course.

BUT using CrystalMaker's VR QuickTime movies, let's have a look at some symmetry in common rock-forming minerals.

Q.17.Go back to my home page and look at the rotating chain of tetrahedra. What is the symmetry axis? Remember to click the back arrow to return to the lab page.

Q.18.Here is part of a structure with a prominant symmetry element. The mineral is Indialite. What is the distinctive symmetry element?

Q.19.Here is part of another structure with a prominant symmetry element. The mineral is Ruby. What is the distinctive rotational symmetry element?

final Q Let's look at the silica tetrahedron. There are three movies. One shows a ball & stick model of an atom of silicon surrounded by 4 oxygens. One shows a space filling model of the same atomic structure. And one shows a polygon where the oxygens would be at the apices of the polygon. This four sided polygon is a tetrahedra. You will be hearing a lot about the tetrahedra, so look at each carefully and compare.

Q.20 Is this tetrahedral structure P, C, F or I? What is the crystal system and what symmetry operations do you see?

The End