GY 302: Crystallography & MineralogyGY 302: Crystallography & MineralogyUNIVERSITY OF SOUTH ALABAMALecture 5: Space Groups, Crystal Growth and TwinningLecture 5: Space Groups, Crystal Growth and TwinningLast Time1. Stereo Projections and the Wulff Net2. Stereo projections of the Point GroupsStereo projectionsEnvision the cube suspended in a sphereStereo projectionsNow pass a line that is perpendicular to each crystal face (poles) outward to intersect the sphere001001010010001Stereo projectionsNow envision the equatorial cross section of the spherical projection.There are 2 possible stereonet configurations:1) Wulff Net (equal angles)2) Smith Net (equal area)Projection:Equal AngleRadius:3.50 inchesWulff NetNSWE10170350190201603402003015033021040140320220501303102306012030024070110290250801002802600 DATASource: D. AllisonStereo projectionsNow envision the equatorial cross section of the spherical projection.There are 2 possible stereonet configurations:1) Wulff Net (equal angles)2) Smith Net (equal area)Projection:Equal AreaRadius:3.50 inchesSchmidt NetNSWE10170350190201603402003015033021040140320220501303102306012030024070110290250801002802600 DATASource: D. AllisonStereonet comparisonsProjection:Equal AngleRadius:3.50 inchesWulff NetNSWE10170350190201603402003015033021040140320220501303102306012030024070110290250801002802600 DATAProjection:Equal AreaRadius:3.50 inchesSchmidt NetNSWE10170350190201603402003015033021040140320220501303102306012030024070110290250801002802600 DATAStereonets courtesy of D. AllisonPreferred for crystallographyStereo projections010001100100010 and 001 Faces010010Stereo projections001111/111111/111111/111111/111The stereonet allows for precise measurements of angular relationships between crystal facesStereo projections001All faces are 90° apart from one another…-45°45°90°0°Stereo projections001All faces are 90° apart from one another…90°90°90°Stereo projections010001…and rotated 45° from the faces of a cubic form45°45°Chalk BoardThe Point Groups•Orthorhombic Point Groups; 2-fold rotational axes or 2 fold-rotational axes and mirror planesThe Point Groups•Tetragonal Point Groups; a single 4-fold rotational axis or a 4 fold-rotoinversion axisToday’s Agenda1. Space Groups (230!)2. Controls on crystal growth3. Twins and twinningSpace GroupsThis is yet another means of addressing elements of symmetry in crystals.Space GroupsThis is yet another means of addressing elements of symmetry in crystals.1. Bravais Lattices: simple translational symmetry in 3-dimensions (14)Space GroupsThis is yet another means of addressing elements of symmetry in crystals.1. Bravais Lattices: simple translational symmetry in 3-dimensions (14)2. Point Groups: combination of reflectional, rotational and inversion symmetry in 3-dimension (32)Space GroupsThis is yet another means of addressing elements of symmetry in crystals.1. Bravais Lattices: simple translational symmetry in 3-dimensions (14)2. Point Groups: combination of reflectional, rotational and inversion symmetry in 3-dimension (32) 3. Space groups: essentially a combination of Bravais Lattices and Point Groups (point symmetry+ translational symmetry; 73+157 = 230)Space GroupsHow it works:• Groups of atoms or ions within a crystalline substance (e.g., CO32-in calcite) at the nodes in a Bravais Lattice possess the symmetry of one of the 32 point groups.Space GroupsHow it works:• Groups of atoms or ions within a crystalline substance (e.g., CO32-in calcite) at the nodes in a Bravais Lattice possess the symmetry of one of the 32 point groups. • Translational operations required to reproduce the lattice in 3 dimensions then repeat the atomic/ionic groups throughout 3-dimensional space (and the point group symmetry it possessed). This generates 73 space groupsSpace GroupsHow it works:• Groups of atoms or ions within a crystalline substance (e.g., CO32-in calcite) at the nodes in a Bravais Lattice possess the symmetry of one of the 32 point groups. • Translational operations required to reproduce the lattice in 3 dimensions then repeat the atomic/ionic groups throughout 3-dimensional space (and the point group symmetry it possessed).• Point group symmetry must be consistent with the symmetry of the Bravais Lattice (isometric point groups for isometric lattices)Space GroupsBut there are two other combination symmetry operations that must be considered:Space GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureGlide planeSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureaStep 1: translationSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureStep 2: reflection in the glide planeSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureStep 1: translationaSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureStep 2: reflection in the glide planeSpace GroupsBut there are two other combination symmetry operations that must be considered:Glides: a combination of translation and reflectionPyroxene structureEtc., etc., etc…..The Point GroupsBut there are two other combination symmetry operations that must be considered:Screw Axes: a combination of translation and rotationPossible combinations:2-fold, 3-fold, 4-fold 6-foldaAn example of a 4 fold screw axisThe Point GroupsBut there are two other combination symmetry operations that must be considered:Screw Axes: a combination of translation and rotationPossible combinations:2-fold, 3-fold, 4-fold 6-fold4-fold screw axis (4 translations of distance “a” in 360° of rotation)a1234Space GroupsThe combination of reflection, rotation, inversion, glide and screw axes with Bravais Lattices leads to an additional 157 space groups (total 230).Space GroupsThe combination of reflection, rotation, inversion, glide and screw axes with Bravais Lattices leads to an additional 157