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
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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!
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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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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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