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[Josele]

I am following with great interest the thread Spinel Twins and also the thread Maclas in Spanish FMF.
Relating to tetrahedron [111] penetration twin, there is something I can not understand, perhaps you can help me to clarify this matter:

[Gerhard Niklasch]

Hello Josele,

no, a 90° rotation about any coordinate axis does not move the crystal structure of Tetrahedrite to a position that could also be obtained by a parallel translation without rotation. The structure really has tetrahedral symmetry.

Performing (say) a 90° rotation around the z axis followed by a reflection in the xy plane will map the structure into itself: this rotation-inversion is a symmetry of the tetrahedron inscribed into the obvious cube.

The space group is I-43m: You may contemplate the diagram which can be found e.g. here,
https://img.chem.ucl.ac.uk/sgp/large/217az1.htm
(link normalized by FMF)

You can find a crystal structure model on the Webmineral page linked from Mindat Tetrahedrite´s page

Enjoy,
Gerhard

[Josele]

Gerhard, thanks for your help and patience.
OK, now I understand (I think!), the matter is: tetrahedrite crystal lattice have not holohedral symmetry, crystallographic axes are just binary axes then a 90º rotation around the crystallographic axes do NOT lets the structure as it was in the original position, in oposition to that I said before.

Another question:
Why the previous twin is called Penetration twin on [111]?
If the twin operation is a 90º rotation around any of the crystallographic axes,
Would not be better to call it Penetration twin on [100]?

[Gerhard Niklasch]

Hello Josele,

This latter shape, as I understand it, is the most common form of twinning in Tetrahedrite: As the Handbook of Mineralogy puts it, "on {111} around [111] as twin axis; contact and penetration twins, commonly repeated." That is, the individuals have a plane equivalent to (111) in common, and are related by rotation around an axis orthogonal to this plane. This axis (a 3-fold rotation axis of the tetrahedron) is polar, unlike the situation for Spinel, and the polarity is preserved under this twinning operation. (The other three 3-fold axes are moved to new positions.) Twinning may be repeated, via a different plane and axis, resulting in rather complicated shapes.

Examples can be found via mindat's photo search by asking for the species and for the keyword "twin" in the description. A nice one is: https://www.mindat.org/photo-330485.html

Returning to your first star shape: Inversion through the coordinate origin would be the simplest way to describe it, without singling out any coordinate axis or rotation axis. But given how common twinning via the [111] rotation is in Tetrahedrite, the shape could also arise from repeated twinning, once via this rotation, and once via a reflection in the same (111) plane. In that case, one would expect the reflection twinning to occur on its own, too. I am not sure whether it does. A quick search found no examples and no mention of it.

In Spinel and other holohedral isometric crystals, the rotation and the reflection are indistinguishable, and the second operation would simply undo the first. Here they are not: The reflection reverses the polarity of the 3-fold axis orthogonal to the mirror plane. And thus the composition of the two operations also reverses it, and indeed it reverses all four 3-fold axes, resulting in something new that's at least logically possible as a twin of a tetrahedral crystal.

Cheers, Gerhard

P.S. Now I've found an example of the first star shape, but cheating a little: This is Helvine, not Tetrahedrite. It does confirm that such things happen in nature, though!
https://www.mindat.org/photo-186519.html

[Ru Smith]

Greetings,

Here's an example (I think) of the [111] twin in a Romanian tetrahedrite.

[Josele]

Gerhard, thanks for the detailed explanation. As you said, in the first case the twin operation is simply an inversion by the origin of crystallographic axes, which is not a symmetry center for a single tetrahedron but yes for the twin.
But still I think is confusing use the same name for these two twins.

Ru, thanks for your input, is heartwarming to confirm that theory corresponds to reality, ...although I have to admit that I'm unable to recognize the twin in your pictures.

[Ru Smith]

See if these two photos help, Josele. One looking roughly down the twin axis and the other orientated approximately as in your smorf drawing. As usual, not quite the ideal, but not bad either.

[Josele]

Ru, sorry in advance for abusing your patience, I know is a twin but I can't recognize it.

I suppose is due to crystal whim and photo deceiving...

[Ru Smith]

You have very sharp eyes, Josele!

Here attached is a view of the base of the twin together with an idealized drawing of what I see. What is clear is that the twin elements are rotated 180 degrees relative to each other about the twinning axis and that there's a shared basal face (though the element forming the "fins" is a tiny step off being flush with the face of the dominant element).

However, the angle you refer to in the dominant member of the twin is indeed a little larger than 120 degrees and seems to result from the presence of tristetrahedral faces ({211}). I've drawn the equilateral triangle of a simple tetrahedron face in the dotted line on the diagram. It seems that some sides of each tetrahedron have tristetrahedral faces expressed and others not. Turning the crystal around reveals that on some sides all three tristetrahedral faces (for one side of the tetrahedron) are expressed and on others only two.

The basal face, shared by both elements of the twin, is a single smooth plane with no sign of tristetrahedral faces.

I'd love to see some more examples.

[Josele]

Thanks Ru, after your clue of tristetrahedron modifications, now I begin to understand. Is a complex crystal, hard to see in photos.

[Ru Smith]

Basically correct, Josele. Here's a sketch looking down the twinning axis from the pointy end. I've drawn over the actual faces and edges and then extrapolated where the crystal contacts the matrix. One member of the twin is shaded blue, the other green. The edges of each tetrahedron are shown in bold lines.

Looking at non-twinned tetrahedrite crystals from this locality I see that they also show very variable development of faces on each side - some are simply tetrahedron faces with striations whereas adjacent sides of the tetrahedron reveal tristetrahedron and other faces.

[Mark Holtkamp]

Hi all,

Sorry for the confusion and the late reply. The drawings in the original post are from my website, after a bit of research I think the crystal in the first drawing is not a twin on [111] at all.

I based my drawing and the interpretation as a twin on [111] on a picture in Dana’s sytem of mineralogy, 7th edition, page 375. In my crystal drawing program I cannot recreate a twin on [111] that looks like the first drawing (when I made the original drawing a couple of years ago I had to create this twin another way for specific reasons).

I read a couple of old mineralogy handbooks, it seems there are 3 types of tetrahedrite twins:

1. penetration twins on [111] like in the second drawing. The two tetrahedrons share a common plane (when of equal size). 2. contact twins on [111] with {211} (or also {111}?) as composition plane. 3. penetration twins in which edges of both tetrahedrons make right angles. The twin element is {100}. I believe it could also be described as an inversion twin.

The twin in the first drawing clearly is of the third type. In my (recent) handbooks, and on the internet, I could not find a reference to this type of twin. Both Dana’s manual of mineralogy, 7th ed., and Klockmanns Lehrbuch der Mineralogie have a crystal drawing like this and refer to this as a twin om [111], which is wrong in my opinion.

Interestingly The System of Mineralogy of James Dwight Dana, 6th edition, has the following text, describing the tetrahedrite twin on {100}: “Twins (1) twinning plane o (o refers to the terahedron (MH)), contact twins with the composition face either parallel or perpendicular to the twinning plane, and penetration-twins, both common, twinning often repeated; also rarely twins (2) with axes parallel and symmetrical with reference to a cubic plane.”

The last twin described by Dana is the twin from the first drawing in the original post.
I wonder why in recent literature this twin law is no longer mentioned. Or did I miss something?

cheers, Mark.

[Mark Holtkamp]

After more reading I now believe that the tetrahedrite {100} twin doesn't occur in recent literature because it doesn't exist. The images in Strunz and Dana probably were copied from old references and falsly interpreted as twins on [111]. See the article by Sadebeck from 1872 which is online (A. Sadebeck, Über Fahlerz und seine regelmässigen Verwachsungen. Zeitschrift der Deutschen Geologische Gesellschaft 24 Heft 3, p. 427-464 (1872).). Apparently I cannot post a url here so I will attach a piece of text in an image. If you google the title of the article you'll find it.

Mark.

[Josele]

Mark, thank you doubly for your priceless web site (SMORF) and for your work as archaeologist of crystalography.
And again thanks for clarify this matter here.

[Mark Holtkamp]

Thank jou Josele also for spotting the error. You were right after all, it is a twin on [100] !

Mark.

[Rodney B Jackson]

Fantastic symmetry. The math is a bit deep for me though.

[chrisjacob]

Your discussion about the Tetrahedron penetration twin was very helpful for a newbie like me. I felt some parts very difficult to understand still the graph you shared help me in a better way to understand what you meant.






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CHris-

[Pete Richards]

The symmetry of tetrahedrite is a reduced isometric symmetry. The form {111}, for example, is a tetrahedron, not an octahedron. Thus a rotation of 90° about one of the axes introduces new symmetry elements and does not simply replicate the structure. I believe this is called twinning by merohedry.

Sorry, I now realize that I have missed a long and complex discussion....