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ssss
Joined: 19 Sep 2012
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 Posted: Sep 20, 2012 00:16 Post subject: How are Miller Indices related stereographic projections? (am I on right track?) |
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I have to write about a page or two on this and I'm not sure which way to go with this.
Basically I have that each face has a pole that is positioned based on its Miller index value. This pole extends outward to touch a theoretical sphere that surrounds the unit cell.
A stereographic projection is a 2 dimensional representation of the sphere on which the point where the face pole meets the sphere can be plotted.
Any advice would be greatly appreciated.
Thanks.
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prcantos
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| Posted: Sep 20, 2012 03:34 Post subject: Re: How are Miller Indices related stereographic projections?(am i on right track?) |
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Hi, ssss. Could you please be a bit more specific? I am not sure about the question you ask.
Anyway, I remember that stereographic projection is an homeomorphism (topological equivalence) between these three elements:
1) The sphere minus one point
2) The whole plane
3) The circle minus his edge
This allows a 2-dimensional understanding of 3-dimensional concepts, which is easier at drawing on paper. The points of the sphere (leave one point out) can be plotted on the circle (leave the edge out). Miller indices are only a shorten notation for geometrical planes; planes can be represented by a unitary normal vector, whose terminal point belongs to the unitary sphere; therefore stereographic projection can be usen on it.
Please give me some minutes, I will try to write an example.
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prcantos
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| Posted: Sep 20, 2012 07:08 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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Here you have a mathematical approach. It is only a very simple example using a crystalline net which is isometric to euclidean space (e. g. a cubic crystal).
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Pablo Rodríguez Cantos
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 08:46 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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Thank your for taking the time to reply. The question is as follows:
" How and why do crystallographers label crystal faces? How are these labels related to stereographic projections that ultimately lead to point group classifications"
Instructions from prof: Use any source and site it. It is intentionally vague and open ended. Go in any direction that you wish to.
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Vinoterapia
Joined: 03 Feb 2009
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Location: Houston, Tx
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| Posted: Sep 20, 2012 09:58 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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Answering your question is not a quick thing, thus I prefer to give you some bibliography where you can find, at least, part of the answer and probably get other leads:
Dana-Hurlbut, Manual of Mineralogy. There are many editions of this classical book on mineralogy, you will probably find it in any library.
Donald B. Peck (2007). Mineral identification, a practical guide for the amateur mineralogist.
Donald F. Bloss (1971) Crystallography and crystal chemistry.
I hope you can get the answer in some of these books.
Regards.
José Luis.
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Pete Richards
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| Posted: Sep 20, 2012 10:04 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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| ssss wrote: | | ..." How and why do crystallographers label crystal faces? How are these labels related to stereographic projections that ultimately lead to point group classifications"... |
Faces can be labeled with Miller indices or with letters, with a table that gives the correspondence between the letters and the Miller indices. Labeling them in this way quantifies the orientations of the faces and their relationship to other faces, though you have to know the unit cell to completely "fix" the orientaitons of the faces relative to the crystal axes.
Stereographic projections are geometric (2-D) representations of the face positions in 3-D. They preserve the symmetry of the crystal, and so can be used to understand its symmetry. They convey somewhat the same information as face labels, but unless you have a stereo-net graduated in degrees the information is only useful in a semi-quantitave manner. Each point on the projection corresponds to a specific face, and can be assigned Miller indices, though this is not usually done. With a little practice, it is as easy, or even easier, to recognize the point-group symmetry of a crystal from a stereographic projection than it is to recognize the symmetry by inspecting the crystal itself. This is because crystals are rarely perfect in terms of the relative sizes of different faces, but they are nearly perfect in terms of the angles between them, which is what the stereographic projection displays.
I like to think of a stereographic projection by analogy to a globe. The crystal is placed at the center of a sphere (usually with a major symmetry axis oriented north-south. For each face, a line through the center of the sphere exends outward perpendicular to the face until it pierces the sphere. The spot where it intersects the sphere is analogous to a point on a globe described by latitude and longitude - on the sphere this point is located by two angles, rho and phi, rho being the analog of latitude except measured from the north pole downward rather than from the equator upward, and phi is analogous to longitude. The stereographic projection collapses the sphere and its set of points "straight down" onto a plane placed at the equator.
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Collecting and studying crystals with interesting habits, twinning, and epitaxy |
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 10:54 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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| Thank you. I appreciate the help.
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 17:32 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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Does anyone know of any online source of information that explains why crstallographers label crystal faces?
Thanks.
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 17:45 Post subject: Why do crystallographers label crystal faces? |
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Some online sources would be very helpful.
Thanks!
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gemlover
Joined: 31 Dec 2008
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Location: Easley, SC
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| Posted: Sep 20, 2012 18:12 Post subject: Re: why do crystallographers label crystal faces? |
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To separate and identify them in writing and discussion. Simplistic answer.
John
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John Atwell Rasmussen, Ph.D.. AJP
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 18:16 Post subject: Re: why do crystallographers label crystal faces? |
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| Thank you.
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Jim Robison
Joined: 17 Nov 2010
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| Posted: Sep 20, 2012 18:36 Post subject: Re: why do crystallographers label crystal faces? |
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Bob's Rockshop has a good, although detailed, writeup on crystals, terms, forms, and the way various crystal faces are numbered and why. Try the following. There is a tremendous amount of information, and it takes a good long while to begin to understand it. Don't be discouraged.
https://www.rockhounds.com/rockshop/xtal/index.shtml
(link normalized by FMF)
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ssss
Joined: 19 Sep 2012
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| Posted: Sep 20, 2012 18:56 Post subject: Re: why do crystallographers label crystal faces? |
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| Thank linked page is amazing. Thank you.
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prcantos
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| Posted: Sep 21, 2012 09:15 Post subject: Re: How are Miller Indices related stereographic projections? (am I on right track?) |
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| ssss wrote: | | ...How are these labels related to stereographic projections... |
I do remark: stereographic projection cannot be directly calculated on MIller indices. In our case stereographic projection is a mapping defined on points of the sphere, so you have to do identifications (justified in the euclidean space) between theses elements:
1) The Miller index of a face in a crystal
2) The associated geometric plane
3) The unitary normal vector of that plane
4) The terminal point of that vector, which belongs to the unit sphere
Stereographic projection can be calculated on this point. Assume the identifications to speak about the projection and Miller indices. Explicit calculation may be developed with formulae (as I did before) or using some different media such stereonets and software. I suppose that geologists do this in an optimal way.
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Pablo Rodríguez Cantos
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