Crystal morphology and Symmetry

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 ProjectionIntroductionCrystals have a set of 3D geometric relationships among their planar and linear featuresThese include the angle between crystal faces, normal (pole) to these faces, and the line of intersection of these facesPlanar features: crystal faces, mirror planesLinear features: pole to crystal faces, zone axis, crystal axes

Question: How can we accurately depict all of these planar and linear features on a 2D page and still maintain the correct angular relationships between them?Answer: With equal angle stereographic projection!Stereographic Projection Projection of 3D orientation data and symmetry of a crystal into 2D by preserving all the angular relationships

Projection lowers the Euclidian dimension of the object by 1, i.e., planes become lineslines become point

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, in contrast to structural geology, stereograms have no geographic significance, and cannot show shape of crystal faces!

Two Types of StereonetWulff net (Equal angle)Used in mineralogy & structural geology when angles are meant to be preservede.g., for crystallography and core analysisProjection is done onto both the upper and lower hemispheresSchmidt net (Equal area)Used in structural geology for orientation analysis when area is meant to be preserved for statistical analysisUses projection onto the lower hemisphere

Wulff net forMinerals -900+90-135+135The angle, is between the c crystal axis and the pole to the crystal face, measured downward from the North pole of the sphere6Wulf net

20406080 = 0 = 90 = -90 = -135 = 135+-Face Pole to face

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htmaxisaxisa axisThe angle is measured in the horizontal equatorial plane.The angle, is between the c crystal axis and the pole to the crystal face, measured downward from the North pole of the sphereWulff 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 along 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)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 a crystalFor example, the angle between crystal faces and rotation axes, or between axes and mirror planesTo understand these, we first give an introduction to stereographic projection!

Stereographic projection of a line

Each line (e.g., rotation axis, pole to a mirror plane) goes through the center of the stereonet (i.e., the thumb tack)

The line intersects the sphere along the spherical projection of the line, which is a point

A ray, originating from this point, to the eyes of a viewer located vertically above the center of the net (point O), intersect the primitive along one point

The point is the stereographic projection of the line. A vertical line plots at the center of the netA horizontal line plots on the primitiveSpecial cases of linesVertical 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 centerProjection of Planar ElementsCrystals 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

Stereographic projection of a plane:

Each plane (e.g., mirror plane) goes through the center of the stereonet (i.e., the thumbtack)

The plane intersects the sphere along the spherical projection of the plane, which is a series of points

Rays, originating from these points, to the eyes of a viewer located vertically above the center of the net (point O), intersect the primitive along a great circle

The great circle is the stereographic projection of the plane. The great circle for a:vertical plane goes through the centerhorizontal plane parallels the primitive

Special cases of planesStereographic projection of a horizontal face or mirror plane is along the primitive (perimeter) of the equatorial plane

Stereographic projection of a vertical face or mirror plane is along the straight diameters of the equatorial planeThey pass through the centerThey are straight great circles

Inclined faces and mirror planes plot along curved great circles that do not pass through the center

http://super.gsnu.ac.kr/lecture/wulff/wulff-1.htmlFace 1Face 2123Face 3pole to the crystal face

Face 1: verticalFace 2: inclinedFace 3: horizontalPreparing to plotMark N of the net as -90, E as 0, S as =+90 )(for ). Mount the stereonet on a cardboard. Laminate it. Pass a thumbtack through the center from behind the board

Secure the thumbtack with a masking tape from behind the cardboard

Place a sheet of tracing paper on the stereonet

Put a scotch tape at the center, from both sides of the tracing paper; pierce the paper through the pin

Tracing paper can now rotate around the thumbtack without enlarging the hole (because of scotch tape)

http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html-9000+90+135-135Symbols usedFor faces below the equator (when using lower hemisphere), place an open circle symbol () where the ray connecting the spherical projection of the pole to the plane intersects the equatorial plane This is the stereographic projection of the pole to the face

For faces above the equator (when using the upper hemisphere), place a solid circle symbol ( ) the ray connecting the spherical projection of the pole to the plane intersects the equatorial plane

Use a bulls-eye symbol () to show a point above the page that coincides with one directly below it (when using both hemispheres)

Reorient the stereogram such that lines of symmetry are north-south or east-westProjection of Linear ElementsWe can show all of the symmetry elements of a crystal and their relative positions stereographically

Edges, pole to crystal faces, and rotation or roto-inversion axes are linesWhen extended through the origin of the sphere, lines intersect the surface of the sphere as points

Each of these points, when connected to the upper or lower pole of the sphere (viewers eyes), is projected onto the equatorial plane, and depicted as a polygon symbol with the same number of sides as the fold of the axisMirror and Polygon SymbolsTo plot symmetry axes on the stereonet, use the following symbol conventions: Mirror plane: (solid line great circles) Crystal axes (lines):

Plotting the rotation axesVertical axes (normal to page) will only have one polygon symbol

Horizontal axes (in the plane of page) intersect the primitive twice, hence they have two polygons

Inclined axes will have one polygon symbol

An open circle in the middle of the polygon shows there is a center of symmetryMeasuring angle between facesThis is done using the poles to the faces!Three cases:On the primitive, the angle is read directly on the circumference of the net

On a straight diameter, the paper is rotated until the zone is coincident with the vertical diameter (i.e., N-S or E-W) and the angle measured on the diameter

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 circleGoing from one hemisphere to anotherDuring rotation of the pole to a face by a certain angle, we may reach the primitive before we are finished with the amount (angle) of 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 hemisphereHow to find reflection of a pointHaving 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, on the same small circle, to find point pIf the primitive is reached before p, then count inward along the same great circle Crystallographic AnglesInterfacial 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 poleUse a contact goniometer to measure the interfacial angles (also measures poles)ConventionBy 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 axisb-axis (normal of face 010) is east-westa-axis (normal of face 100) is north-south

See next slide!

http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htmaxisaxisa axisE-WUp and downN-SStereonet isthe equatorThe and anglesGenerally, it is the angles of the spherical projection, and ,that are given for each face of a crystalThese are measured with goniometer

If these are known, then the actual angles between any two faces can easily be obtained through trigonometry, or by the use of the stereonetThe angleThe 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 anglemeasured in the vertical plane containing the axis of the sphere and the face pole. Note: the (010) face has a angle of 90o(010) face is perpendicular to the b-axis

The angle is measured in the horizontal equatorial plane.Note: the (010) face has a angle of 0o!Plotting and

Suppose you measured = 60o and = 30o for a face with goniometer. Plot the pole to this face on the stereonet.

Procedure: Line up the N of the tracing paper with the N of the net. From E, count 30 clockwise, put an x (or a tick mark). Bring x to the E, and then count 60 from the center toward E, along the E-W line. Mark the point with . NOTE: The origin for the angle is at E (i.e., =0). - is counted counterclockwise, horizontally from E to the N on the primitive.+ is counted clockwise horizontally from E to S (i.e., clockwise) on the primitive.

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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 angle, except longitude is measured from the Greenwich Meridian, defined as = 0o

Latitude is measured in the vertical plane,up from the equator, shown as the angle . Thus, the angle is like what is called the colatitude (90o - latitude). Zone plottingZone: Two or more faces whose edges of intersection are parallel to a specific linear direction in a crystalThis direction is called the zone axis. A zone is indicated by a symbol similar to that for the 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; i.e., a zone is constructed by aligning the poles to these faces on a great circle

The zone axis (pole to the zone) is normal (i.e., 90o) to this great circle

On the stereogram, the lower hemisphere part of the zone great circle is dashed, while the upper great circle is solidLower-hemisphere faces are depicted by open circle symbol ()Upper hemisphere faces are depicted by filled circle symbol ()

Thus all poles in a zone are on the same great circle(111) (100) (111) (011) (100) all coplanar (= zone) 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 = 0o and = 90o

Positive angles will be measured clockwise on the stereonet, and negative angles will be measured counter-clockwise on the stereonet

Rules contdCrystal faces that are on the top of the crystal ( < 90o) will be plotted with the closed circles () symbol, and crystal faces on the bottom of the crystal ( > 90o) will be plotted with the "" symbol

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 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 stereonetRules contdMeasure the angle out from the center of the stereonet along the E-W axis of the stereonetNote 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 90o away from the intersection of the great circle along the E-W axis.That is, the zone axis is the pole to the great circle of the poles to the faces in a zone

As an example, the and angles for the (111) crystal face in a crystal model is shown here

Note again that the angle is measured in the vertical plane containing the c axis and the pole to the face, and the 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() as 90 D G as 0 D O

Fig 6.3We are looking along the primitive, and viewers eye is at the S pole for the two shown faces. The upper an lower hemisphere shown!

Example for an isometric crystalSee the isometric crystal axes in the next slide!

NOTE: This is a 3D view!a2 axisa1 axisa3 axisThe stereonet is the equatorial plane of the sphere!

Upper hemisphere stereographic projection of the poles to the upper crystal faces are shown by the () symbols

Viewers eyes are at the south poleStereographic projection of the isometric crystal in the previous slide

Symmetry elements of an isometric crystal.

Legend

Pole to the upper () and lower () crystal faces

NOTE: is measured as the distance (in o) from the center of the projection to the position where the pole to the crystal face plots

is measured around the circumference of the circle, in a clockwise direction away from the b crystallographic axis (010)

Stereographic projection of an isometric crystalAs 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.

a2 axisa1 axis (011) (1-11) (111) (001) (101) (11-1) (01-1) (1-1-1) (1-01) (1-10) (010) (1-00) (11-0) (1-1-0) (01-0) (110) (100) =45o =90o =45oa3 In the previous slide, only 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 angle of 45o

The angle for these faces is measured along a line from the center of the stereonet (where the (001) face plots) toward the primitive. For the (111) face the angle is 45o, and for the (110) face the angle is 90oExplanation of Previous SlideAs an example all of the faces, both upper and lower, are drawn 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.htmHow rotational axes are shownAxes that are parallel to the page are indicated by straight lines with proper polygons at the endSolid line if these are parallel to a mirror planeDashed otherwise

Oblique axes plot as polygons between center and primitiveThe distance between the polygon and center is proportional to the angle between the axis and pole to the face ( 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

3-D Symmetry Conventions

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