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[ssss]

Thanks in advance. It would be a big help. This is my first upper leval geology class and I'm a little overwhelmed.

This is my first draft. I still have to get my citations right and polish everything up. The paper is not supposed to be over 3 pages(not including figures) double spaced, so I still have room to add a page. Thanks for any tips you can provide.
Here was the question I was responding to:
" How and why do crystallographers label crystal faces? How are these labels related to stereographic projections that ultimately lead to point group classifications"

Crystallographers study the properties and symmetry of crystals. One of the fundamental aspects of the field is the labeling of crystal faces. The systematic separation of crystals based on the angles between each crystal face is helpful for identifying crystal samples. It is also useful for studying different aspects of the crystal such as the conditions present when it was formed or various physical properties. Although there are several systems for classification for crystal faces Miller Indices is the most common. (Haywick) William H. Miller developed the Miller Indices in the 1800’s. His system addressed the problem of a lack of a unified system of viewing and orientating crystals. (BOBS). There are a finite number of ways that a crystal face can exist relative to the adjacent faces and the Miller Indices is used to identify the different variations. (Haywick)

To describe the way this system of classification works it is useful to invasion a cube sitting in front of you (see fig. 4). The a,b, and c intercepts of each planar crystal form are given specific mathematical notations.

The side of the cube facing directly at you is composed of a series of planes that take steps in the direction of the x-axis. The distance between planes is labeled “a” The edges of the plane go off in the y, and z, direction in infinity. The labeling of these planes comes from the reciprocal of the intercept value. Fractions are not used so a lowest common denominator is utilized when necessary to get a whole number. For this plane, the values are 1 0 0 for the x, y, and z axes. For the opposite side of the cube, facing away from you, the same numbering system is used. The only difference being the values are negative which are written by overstriking the integer.

The side of the our imagined cube facing up comprised of a series of planes taking steps in the Y direction while extending into infinity in the x and z directions. This side of the cube is represented as 0 0 1 and the opposite face is designated by the negative of those values.

The face of the cube facing to the right has a Miller Index of 0 0 1. The negative of those values designates the side facing to our left.
The Miller Indices works the same even when there are planes that are not parallel to two sides of an axis (see fig. 1). Here, the Miller Index is 1 1 1. The values come from the points that the plane intersects each axis in the same way that occurs when labeling the faces on a cube.
Assigning each crystal face with a Miller Index is a useful tool in labeling each face but it is not enough by itself to label them. To convey the faces of a crystal and the faces between each, however, is best achieved through the use of stereographic projections (see Fig. 2)

[ssss]

A stereographic projection, like the Miller Indices, is a way to label and keep track of the faces of a crystal. In this system, a pole extends from the center of each crystal going outward to a imaginary sphere. The point of contact is marked. Once all the points have been fixed, a form of graph paper can be used to show these points on a 2 dimensional surface. The area and shape of the sphere is distorted so that the exact angles can be preserved.

There are a finite number of ways in which the angels of crystal faces can be oriented relative to the others. For this reason, it is possible to group crystals with similar geometry. Each of the point groups therefore, have a stereographic projection, specific to that point group.

[Jordi Fabre]

We can try to help you in your crystallographic doubts (How are Miller Indices related stereographic projections? (am I on right track?) but as you can figure, not do your studies for you.

Please submit just concrete requests, not drafts like this one and please note that FMF is a Forum not a chat, so in case you get not answer immediately ( https://www.mineral-forum.com/message-board/viewtopic.php?p=25759#25759 ) please don't push. Thank you.

[Matt_Zukowski]

You should know that I often abbreviate crystal as xtal. You should not do this in your paper, but be aware that I make this abbreviation.

Honestly, your paper seems confused and unorganized to me. I do think you have some understanding of the subject matter. But if you organized it better, you might see better where your holes in understanding are and then proceed to fill them.

I would suggest that you outline your paper before writing it. For example, the first question you are to answer related to how and why crystal faces are labeled by crystallographers. I’d start with how xtalographers label faces. This way you can use the “how” to go on to describe the “why.” After that you can go on to a) describe how stereographic projections work and b) how miller indicies relate to dots on a stereogram. Finally, you can describe point group classifications and present some examples of faces in different point groups and how, using the symmetry rules of the point group, you can plot other equivalent faces on the stereogram or other faces in the same form.

A beginning of such an outline might look something like this:

A) How xtal faces are labeled by xtalographers.
-miller indices – I’d show a figure of a couple of xtals with the miller indices labeled and then describe how the indices were determined.
-differentiate between particular faces, as in (111) and all the faces collectively known as a form {111}

B) Why xtal faces are labeled by xtalographers.
-it is impossible to describe xtals without some sort of terminology labeling a particular face.
-macro descriptions
-other applications like xray xtalography.

c) Stereographic projections
-Here I would use a figure to show a stereogram and plot all the faces that you showed in figures under item A above. I’d use text to describe how each dot on the stereogram relates to a face as defined by a particular miller indice.

D) Point group classifications
-Define what point groups are a give a couple of examples
-Describe how the dots that you have shown on your stereogram can be rotated or reflected depending upon the rules that define each point group.
-Note that all the faces that follow particular rules are of the same form.

Good luck

[Tracy]

Sorry, but I agree with Jordi. FMF should not be used for helping students with their homework.

- Tracy