References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry:

References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry: Holt, Reinhardt, and Winston, New York, 545 p. Klein, C., and Hurlbut, C.S.Jr., 1993, Manual of Mineralogy (after James Dana), 21st edition: John Wiley & Sons, New York, 681 p. Stereographic Projection

A convenient way of looking at the symmetry of a crystal Projection of 3D orientation data and symmetry of a crystal into 2D using spherical projection Projection lowers the Euclidian dimension of the object by 1, i.e., planes become lines, and lines become point! The poles (normals) of crystal faces are projected onto the inside surface of a sphere as points, and then reprojected onto its equatorial plane

In mineralogy, it involves projection of faces, edges, mirror planes, and rotation axes onto a flat equatorial plane of a sphere, in correct angular relationships In mineralogy, stereograms have no geographic significance, and cannot show shape of crystal faces!

Why need projection? Projection of all crystal faces of a crystal leads to many great circles or poles to these great circles These great circles and poles allow one to determine the exact angular relationship, and symmetry relationships for example, between crystal faces and rotation axes, or between axes and mirror planes

Stereonets Wulff net (Equal angle) Used in mineralogy & structural geology when angles are meant to be preserved e.g., for crystallography and core analysis Schmidt net (Equal area) Used in structural geology for orientation analysis when area is meant to be preserved for statistical analysis

Wulff net

Wulff Stereonet (Equal Angle net) Shows the projection of great circles and small circles Great circle: line of intersection of a plane, that passes through the center of the sphere, with the surface of the sphere (like lines of longitude on Earth) NOTE: Angular relationships between points can only be measured on great circles (not small circles)! Small circle: loci of all positions of a point on the surface of the sphere when rotated about an axis such as the North pole (like lines of latitude on Earth) In mineralogy the upper and lower hemispheres are used In structural geology only the lower hemisphere is used

Great & Small circles View From a Pole

http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html

Preparing to Plot Mount your stereonet on a cardboard. Laminate it. Pass a thumbtack through the center. Secure the thumbtack with a masking tape Place a sheet of tracing paper on the stereonet Put a scotch tape at the center of the tracing paper, and pierce the paper and tape through the pin The tracing paper can now rotate around the thumb tack without enlarging the hole (because of scotch tape)

Projection of the pole to a face! Visualize that one of the faces of a crystal, positioned at the center of a sphere, dips down to the right (from the horizontal) Imagine that when the pole of this face is extended up to inner surface of the upper hemisphere, a bulb turns on at point P! Also imagine that you are watching the bulb from the South pole of the sphere (i.e., your eyes are at the S pole) The ray of light, coming off point ‘P‘ to reach your eyes at point ‘S’, intersects the equatorial plane at point p ’ Point p ’ is the stereographic projection of point ‘P’, i.e., of the pole of the crystal face

How Projection is done (Figure 9.29) The pole (i.e., normal) to each crystal face is extended until it intersects the inner surface of the hemisphere (upper or lower) say at point ‘P’ If this point is in the upper hemisphere, connect it to the S-pole (no geographic significance!) If this point is in the lower hemisphere, connect it to the N-pole

Figure 9.29 cont’d For faces below the equator (when using lower hemisphere), place an open circle symbol ( ◯ ) where the line intersects the equatorial plane (this is the stereographic projection of the face) For faces above the equator, using the upper hemisphere, place a solid circle symbol (●) where the line intersects the equator Use a bull’s-eye ( ◉ ) to show a point above the page that coincides with one directly below it (using both hemispheres) Reorient the stereogram such that lines of symmetry are north- south or east-west

Projection of Planar Elements Crystals have faces and mirror planes which are planes, so they intersect the surface of the sphere along lines These elements can be represented either as: Planes, which become great circle after projection Poles (normals) to the planes, which become points after projection

Projection of Linear Elements We can show all of the symmetry elements of a crystal and their relative positions stereographically Edges, pole to crystal faces, and rotation or rotoinversion axes are lines When extended through the origin of the sphere, lines intersect the surface of the sphere as points Each of these points, when connected to the North or South pole of the sphere, is projected onto the equatorial plane, and depicted as a polygon symbol with the same number of sides as the ‘fold’ of the axis

Mirror and Polygon Symbols To plot symmetry axes on the stereonet, use the following conventions: Mirror plane: ― (solid line)

Special cases of planes Stereographic projection of a horizontal face or mirror plane is along the perimeter of the equatorial plane This plane is called the ‘primitive’ Stereographic projection of a vertical face or mirror plane is along the diameters of the equatorial plane, i.e., They pass through the center They are straight ‘great circles’ Inclined faces and mirror planes plot along curved great circles that do not pass through the center

Special cases of lines Vertical lines (e.g., rotation axes, edges) plot at the center of the equatorial plane Horizontal lines plot on the primitive Inclined lines plot between the primitive and the center

Plotting the rotation axes Vertical axes (normal to page) will only have one polygon Horizontal axes (in the plane of page) intersect the primitive twice, hence they have two polygons Inclined axes will have one polygon An open circle in the middle of the polygon shows there is a center of symmetry

Mirror and Polygon Symbols To plot symmetry axes on the stereonet, use the following conventions: Mirror plane: ― (solid line)

Measuring angle between faces Three cases: 1. On the primitive, the angle is read directly on the circumference of the net 2. On a diameter, the paper is rotated until the zone is coincident with the vertical diameter and the angle measured by the diameter 3. On a great circle (an inclined zone), rotate the paper until the zone coincides with a great circle on the net; read the angle along the great circle

Going from one hemisphere to another During rotation of the pole to a face by a certain angle, we may reach the primitive before we are finished with the rotation In this case we are moving from one hemisphere to another Move the pole back away from the primitive along that same small circle you followed out to the primitive, until it has been moved the correct total number of degrees Then note its new position with the point symbol for the new hemisphere

How to find reflection of a point Having a symmetry (mirror) plane and a point p, find the reflection of point p (i.e., p ’) across the mirror: Align the mirror along a great circle Rotate point p along a small circle to the mirror plane Count an equal angle beyond the mirror plane to find point p’ If the primitive is reached before p ’, then count inward along the same great circle

Convention By convention (Klein and Hurlbut, p.62), we place the crystal at the center of the sphere such that the: c-axis (normal of face 001) is the vertical axis b-axis (normal of face 010) is east-west a-axis (normal of face 100) is north-south

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm

3-D Symmetry Conventions http://www.kean.edu/~csmart/Mineralogy/Lectures

Crystallographic Angles Interfacial angle: between two crystal faces is the angle between poles to the two faces. The interfacial angle can be measured with a contact goniometer These angles are plotted on the stereonet

Making a stereographic projection of a crystal face pole Use a contact goniometer to measure the interfacial angles (also measures poles)

The ρ and φ angles Generally, it is the angles of the spherical projection, ρ and φ, that are given for each face of a crystal (measured with goniometer) If these are known, then the actual angles between any two faces can easily be obtained through trigonometry, or by use of the stereonet

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm

The ρ angle The ρ angle, is between the c axis and the pole to the crystal face, measured downward from the North pole of the sphere A crystal face has a ρ angle measured in the vertical plane containing the axis of the sphere and the face pole, and a φ angle measured in the horizontal equatorial plane Note that the (010) face has a ρ angle of 90 o

Plotting ρ and φ Suppose you measured  = 60 o and  = 30 o

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm These angular measurements are similar to those we use for latitude and longitude to plot positions of points on the Earth's surface For the Earth, longitude is similar to the φ angle, except longitude is measured from the Greenwich Meridian, defined as φ = 0 o Latitude is measured in the vertical plane, up from the equator, shown as the angle θ. Thus, the ρ angle is what is called the colatitude (90 o - latitude).

Zone plotting Zone: two or more faces whose edges of intersection are parallel to a specific linear direction in a crystal. This direction is called the zone axis. A zone is indicated by a symbol similar to that for Miller Indices of faces, the generalized expression for a zone is [uvw], e.g., all faces parallel to the c axis in an orthorhombic crystal are said to lie in the [001] zone All faces in a zone lie on a great circle Zone is constructed by aligning the poles to these faces on a great circle The zone pole is normal (i.e., 90 o ) to this great circle On the stereogram, the lower hemisphere part of the zone circle is dashed, while the upper is solid Lower-hemisphere faces are represented by open circles

The following rules are applied: All crystal faces are plotted as poles (lines perpendicular to the crystal face. Thus, angles between crystal faces are really angles between poles to crystal faces The b crystallographic axis is taken as the starting point. Such an axis will be perpendicular to the (010) crystal face in any crystal system. The [010] axis (note the zone symbol) or (010) crystal face will therefore plot at φ = 0 o and ρ = 90 o Positive φ angles will be measured clockwise on the stereonet, and negative φ angles will be measured counter-clockwise on the stereonet

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm The pole to a (010) face will coincide with the b crystallographic axis, and will impinge on the inside of the sphere at the equator, i.e., on the primitive We define this face (010) as having a φ angle of 0 o For any other face, the φ angle will be measured from the b axis in a clockwise sense in the plane of the equator

Thus all poles in a zone are on the same great circle (111) (100) (111) (011) (100) all coplanar (111) (100) (111) (011) (100) all coplanar (= zone)

Rules cont’d Crystal faces that are on the top of the crystal (ρ 90 o ) will be plotted as " ◯ " signs. Place a sheet of tracing paper on the stereonet and trace the outermost great circle. Make a reference mark on the right side of the circle (East) To plot a face, first measure the φ angle along the outermost great circle, and make a mark on your tracing paper. Next rotate the tracing paper so that the mark lies at the end of the E-W axis of the stereonet

Rules cont’d Measure the ρ angle out from the center of the stereonet along the E-W axis of the stereonet Note that angles can only be measured along great circles. These include the primitive circle, and the E-W and N-S axis of the stereonet. Any two faces on the same great circle are in the same zone. Zones can be shown as lines running through the great circle containing faces in that zone The zone axis can be found by setting two faces in the zone on the same great circle, and counting 90 o away from the intersection of the great circle along the E-W axis

As an example, the ρ and φ angles for the (111) crystal face in a crystal model is shown here. Note again that the ρ angle is measured in the vertical plane containing the c axis and the pole to the face, and the φ angle is measured in the horizontal plane, clockwise from the b axis. http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm

D and E are spherical projections, i.e., where the pole to the faces intersect the inside of the sphere D' and E' are stereographic projections, when DS and ES intersect the equator (when projected to the south pole) Distance GD' = f(ρ) as ρ  90 D’  G as ρ  90 D’  G as ρ  0 D’  O as ρ  0 D’  O Fig 6.3

Example: Stereogram in next slide

Projection of the upper faces to the south pole

Stereographic projection for faces of an isometric crystal Note how the ρ angle is measured as the distance from the center of the projection to the position where the crystal face plots The φ angle is measured around the circumference of the circle, in a clockwise direction away from the b crystallographic axis, or the plotting position of the (010) crystal face.

As defined in our projection, the N and S poles would plot directly above and below the center of the stereonet. On the stereonet, we see several different components that we define here.

In the previous slide, the upper faces of an isometric crystal are plotted. These faces belong to forms {100}, {110}, and {111} Form: set of identical faces related by the rotational symmetry (shown by poles/dots in stereograms) Faces (111) and (110) both have a φ angle of 45 o The ρ angle for these faces is measured along a line from the center of the stereonet (where the (001) face plots) toward the edge. For the (111) face the ρ angle is 45 o, and for the (110) face the ρ angle is 90 o Explanation of Previous Slide

As an example all of the faces, both upper and lower, for a crystal in the class 4/m 2/m in the forms {100}(hexahedron - 6 faces), {110} (dodecahedron, 12 faces), and {111} (octahedron, 8 faces) in the stereogram to the right. Rotation axes are indicated by the symbols as discussed above Mirror planes are shown as solid lines and curves, and the primitive circle represents a mirror plane. Note how the symmetry of the crystal can easily be observed in the stereogram. http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm

How rotational axes are shown Axes that are parallel to the page are indicated by straight lines with proper polygons at the end Solid line if these are parallel to a mirror plane Dashed otherwise Oblique axes plot as polygons between center and primitive The distance between the polygon and center is proportional to the angle between the axis and pole to the face (ρ angle )

Mirror planes (see Figure 9.20 of Perkins) Horizontal mirror planes (in the plane of the page) plot as solid primitive Vertical mirror planes (i.e., normal to the page) plot as solid straight line through the center Inclined mirror planes (inclined to the page) plot as solid curved great circle

Axes normal to Mirror planes See Figure 9.21 Perkins For example: 1/m, 2/m, 3/m, 4/m, and 6/m

Depiction of Rotoinversion Involve a combination of rotation and inversion in one operation See Figure 9.22 Perkins, p. 191 for the rotoinversions on stereonet i.e., 1 -, 2 -, 3 -, 4 -, and 6 - AxisRotation axisWe use these equivalents 1 -360 I (inversion center) 2 -180m 3 -120 4 -90 6 -603/m

Guide for the handed out exercise Page 1: Use the interfacial angles given for the top crystal (the bottom crystal is given as a guide!), and determine the  and  angles for each labeled face (use the Powerpoint slides as a guide for how to do it). The table gives the interfacial angles for all the faces. Do the following: First decide on the system of the crystal. Then orient the crystallographic axes. For example, if the top crystal is tetragonal, then the pole to faces ‘y’ and ‘z’ (i.e., top and bottom faces) is the c axis, and the poles to faces ‘b’ and ‘a’ are the ‘a 2 ’ and ‘a 1 ’ axes, respectively, such that  and  for face ‘b’ are 0 o and 90 o, respectively, and  and  for face ‘a’ are 90 o and 90 o, respectively, and  for face y is 0 o (has no  ). So, faces ‘a’ and ‘b’ plot on the primitive, and faces y and z plot at the center of the stereonet. Plot and label the faces on the Wulff net. Add the symmetry elements. Use solid symbol for the faces whose poles intersect the upper hemisphere, and open symbols for those with poles intersecting the lower hemisphere.

Guide for the handed out exercise Page 2: Use the interfacial angles given for the top crystal, and determine the  and  angles for each labeled face. Do the following: First decide on the system of the crystal. Then orient the crystallographic axes. For example, if the crystal is monoclinic, the ‘b’ axis must be selected at the intersection of the principal faces, perpendicular to the mirror, and parallel to the 2-fold axis. The zone axis is the ‘c’ axis. See slide #32. Also read the note at the bottom of page 2! Having the ‘b’ axis (  = 0 o and  = 90 o ) and the ‘a’ and ‘c’ axes, plot all the faces on the stereonet. Plot and label all the faces on the Wulff net. Add the symmetry elements. Use solid symbol for the faces whose poles intersect the upper hemisphere, and open symbols for those with poles intersecting the lower hemisphere.

Guide for the handed out exercise Page 3: The  and  angles are given for a crystal faces ‘a’ through ‘r’ of a crystal (not shown). Plot all the faces on the Wulff net using these angles Label all the points on the net to correspond with the faces Put the elements of symmetry (e.g., 2-fold axes, m planes) on the net

References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry:

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References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry:

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Presentation on theme: "References: Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New Jersey, 483 p. Bloss, F.D., 1971, Crystallography and Crystal Chemistry:"— Presentation transcript:

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