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Auditioning Techniques

by: Elliott Yundt

Auditioning Techniques THE 230

Marketplace > Marshall University > Theatre > THE 230 > Auditioning Techniques
Elliott Yundt
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This 0 page Class Notes was uploaded by Elliott Yundt on Sunday November 1, 2015. The Class Notes belongs to THE 230 at Marshall University taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/233241/the-230-marshall-university in Theatre at Marshall University.


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Date Created: 11/01/15
Crystallography A crystal is a solid object with a geometric shape that re ects a longrange regular internal structure commonly with some element of symmetry 1 Space lattices Definition The regular internal structure of a mineral is manifested by the existence of a space lattice which is quotan array of points in space that can be repeated inde nitelyquot All quotpointsquot in a lattice have identical environments Fig 1 In the case of a mineral or crystal these quotpointsquot also known as motifs may be considered atoms ions or groups of atoms ions Note that a lattice has no origin A unit cell Fig 1 is the smallest number of quotpointsquot which completely de ne the space lattice The repetition of those points or unit cells in a space lattice is performed by certain operations which build the space lattice Criteria used for the selection of unit cells 1 The smallest sized unit that retains the characteristics of the space lattice Fig 1b 2 Edges of the cell should coincide with symmetry axes see below 3 Edges of the cell related to each other by the symmetry of the lattice Building a s ace lattice from motifs to lattices Motif gt Line lattice gt Plane lattice gt Space Lattice Operations a elements of symmetry b translations c glide planes d screw axes A Elements of symmetry Types i Axes of rotation 11 2 3 4 or Q If during the rotation of a crystal around an axis one of the faces repeats itself two or more times the crystal is said to have an axis of symmetry Symmetry axes may be two fold digonal if a face is repeated twice every 360 three fold trigonal if it is repeated three times four fold tetragonal if it is repeated four times or six fold hexagonal if that face is repeated 6 times Figure 2a shows these relations ii Center 1n or 11 If two similar faces lie at equal distances from a central point the crystal is said to have a centre of symmetry Fig 2e iii Planes gm When one or more faces are the mirror images of each other the crystal is said to have a plane of symmetry Fig 2f Motifs related to each other by mirror planes are known as enantiomorphs Fig 4 iv Axes of rota inversion 1 2 3 4 or 6 When two similar faces are repeated 2 3 4 or 6 times when the crystal is rotated 360O around an axis but in such a way that these faces appear inverted Therefore if the face is repeated 2 times during a full rotation the axis is known as a 2fold rotary inversion axis 3 times gt 3fold rotary inversion etc Figure 3 shows the types of rotary inversion axes Note that axes of rotary inversion can also produce enantiomorphs Fig 4 Equivalence 0fs0me symmetry elements NB Only rotary inversion axes of odd order imply the presence of symmetry centers Rnlesfor combinations of rotations and mirrors Table l 1 Rotary axes at 900 or 54 44 2 All operators must intersect at a point 3 Rotary axes amp mirrors are either perpendicular or parallel 4 Intersection of 2 vertical mirror planes produces a digonal rotary axis 5 Intersection of 3 vertical mirror planes produces a trigonal rotary axis 6 Intersection of 4 vertical mirror planes at 450 produces a tetragonal rotary axis 7 Intersection of 6 vertical mirror planes at 300 produces a hexagonal rotary axis It was found that there are only possible combinations of the above listed symmet elements These are known as point groups since the symmetry elements simply reproduce or repeat groups of points B Translations i Fig 5 Think of a translation as an operation by which new motifs are generated in a new location ie a means of reproduction for these motifs Translations that reproduce a motif in that direction towards the reader in the plane of the paper are labeled by the vector a those moving to the right of the original motif b and those in a direction perpendicular to the plane of the paper 5 The lengths of these translation vectors are designated a0 b0 and co respectively The magnitude of these translations is normally on the order of l 10 C Glide plane g Is a compound symmetry operation that results from the combination of a mirror plane and a translation Fig 6 see Table 2 for compound symmetry D Screw axes xy A screw axis forms by the combination of rotation and a translation Depending on the type of rotation there will be several types of screw axes eg 2 3 4 amp 6 Any of these screw axes will also be either righthanded or lefthanded and are given symbols as xy eg 31 where ifthe ratio yx is lt 05 the axis is right handed and if yx gt 05 the axis is lefthanded Fig 7 It is essential that the rotational and screw axes have the same rotational angle a phenomenon described as is0g0nal You will notice that there are no 5 fold rotary rotary inversion or screw axes in crystals This is simply because points or motifs related by fivefold symmetry will produce objects that are incapable of fitting together to produce closed space Fig 8 The combination of the elements of symmetry with translations glides and screw axes is what ultimately produces all possible crystal lattices known as the 230 space groups Breaking down this process into steps 39 39 39 w gl39d 139 39 motlfs wgt 5 0r 6 plane lattlces me In a anegt10 plane grou S 1 e mes mirror planes lide planes mirror planes screw axes elements of symmetrVgt 230 gt 17 twodimensional point groups space groups 11 Crystal Morphology A Crystal Faces The regular internal structure of a mineral is manifested by the development of surfaces that define the shape of the crystal and which may be related to one another by certain elements of symmetry These surfaces are known as crystal faces 0 Bravais Law Crystal faces are most likely to develop fully abundantly along those planes with the highest density of lattice points Bravais Law therefore states that the frequency by which a face is observed in a crystal is directly proportional to the number of points or nodes it intersects in a lattice Fig 9 Factors affecting the morphology of a crystal conditions of growth P solutions available direction of solution ow availability of open space 0 Steno s Law The angles between these faces known as the interfacial angles are always constant for the same mineral at the same temperature Interfacial angles are measured using a goniometer actually the goniometer measures that angle that is complimentary to the interfacial angle Fig 10 UIbUJNt I IIIII B Crystallographic Axes amp Interaxial angles In addition to quotcrystal facesquot and quotelements of symmetryquot crystals can be described and classified on the basis of their quotcrystallographic axesquot and quotinteraxial anglesquot Crystallographic axes are imaginary lines of reference inside a crystal that intersect at a common point the crystal centre and to which crystal faces can be referenced Any crystal has either 3 or 4 crystallographic axes The angles between these axes are known as the interaxial angles Figure 11 shows the three standard crystallographic axes a b and c and the nomenclature of the interaxial angles at i and y Note that a b and c are the names given to the positive sides of the crystallographic axes a7 b and E to the negative sides Similarly 0c 5 and 7 apply only to those interaxial angles between the a b andc not a band E 0 Criteria used for the selection of the crystallographic axes 7 In crystals that lack sufficient symmetry axes are selected to coincide with the lines of intersection between the major largest crystal faces Fig 12 Ideally they should be parallel to the edges of the unit cell and their lengths should be proportional to the edges of the unit cell Either 3 or 4 in number Perpendicular wherever possible As much as possible they should coincide with the symmetry axes and be perpendicular to the symmetry planes For monoclinic systems see below a is selected so that i gt 90 For triclinic systems at and i are each gt 90 In practice crystallographic axes are determined from Xray measurements C Relationship between crystal faces and crystallographic axes Indexing the crystal faces V V V V VV V Intercepts Points of intersection of the face with the crystallographic axes Coordinates Actual distances of the intercepts of a face with the crystallographic axes expressed in some unit of measurement eg mm cm inches etc Figl3 Parameters Distances ie coordinates of one face diVided by those of another to which it is parallel Expressed as 2a4blc They are intercepts referenced to axes a b and c The unit face Is a face that intersects all three crystallographic axes and is arbitrarily assigned the parameters la lb lc In the event that several faces intersect all three axes in the crystal the face with the largest area is selected as the unit face Indices Miller Indices Are whole numbers that represent the reciprocals of parameters after clearing the fractions by multiplying by one or more factors Law of rational indices Law of Haily The position of one face on a crystal can always be referred to that of another face on the same crystal by ratios which may be expressed in small whole numbers that seldom exceed 3 Zones All faces whose lines of intersection are parallel are considered a zone The zone axis is a line through the center of a crystal to which the edges of intersection of faces in a zone are parallel Fig 14 D Crystal Forms Two or more faces haVing the same geometric relations to the crystallographic axes and the same shape and which are related to each other by some element of symmetry in a crystal constitute a crystal form de ned as an assemblage of one or more faces which may partially or completely constitute a crystal exterior A crystal may thus consist of one or more forms Fig 15 III Towards a classi cation of crystals Systematic Crystallography As mentioned above there are only possible combinations of the symmet elements excluding the screw axes and glide planes that can be referred to sets of axes intersecting at a common point ie crystallographic axes Accordingly the external geometry of any crystal must correspond to one of these 32 sets known as quotclassesquot Many of these classes share some fundamental element or elements of symmetry Therefore these classes that have common symmetry elements are grouped together into a quotcrystal systemquot so that all 32 crystal classes fall into seven crystal systems o The Crystal Systems The crystal classes are grouped into seven crystal systems based on the following criteria a relative lengths of the crystallographic axes b number of crystallographic axes c values of the interaxial angles d some essential element of symmetry The seven crystal systems are Fig 16 Table 3 1 The Cubic system Three crystallographic axes a b c or l y 90 Essential element of symmetry is a threefold rotary or rotary inversion axis 2 The Tetragonal system Three crystallographic axes a b at c or l y 90 Essential element of symmetry is a fourfold rotary or rotary inversion axis 3 The Trigonal system Four crystallographic axes three of which are equal and coplanar and at angles of 120 fourth axis quotcquot is perpendicular to the other three axes and is characterized by commonly being a three fold axis of symmetry 4 The Hexagonal system Four crystallographic axes three of which are equal and coplanar a1 a2 a3 and at angles of 120 fourth axis c is perpendicular to the other three axes and is characterized by being a six fold axis of symmetry 5 The Orthorhombic system a at b at c or l y 90 Essential element of symmetry is atwofold rotary axis Some debate over choice of axes Generally c lt a lt b 6 The Monoclinic system a at b at c or y 90 l gt 90 Essential element of symmetry is a two fold rotary axis or a plane The 2fold rotational axis or the direction perpendicular to the mirror plane is usually taken as the b axis the a axis is inclined to the front 5 gt 90 and c is vertical 7 The Triclinic system a at b at c or at l at y No essential element of symmetry Additional criteria As much as possible i and or should be both gt 90 The most pronounced zone should be vertical The axis of this zone is then taken as c 001 should slope forward and to the right In general cltaltb o The Hermann Mauguin Symbols Each of the elements of symmetry is designated by a unique alphanumeric symbol If the same symmetry element occurs more than once in a crystal then it is designated by the symbol raised to the power of number of occurrences A string of Hermann Mauguin symbols for a crystal therefore completely summarizes its symmetry Note that a shorthand form of these symbols largely ignores the number of times an element is repeated as this can be guessed with little trouble and some experience We will stick to the longform in our class 0 Crystal classes amp crystal systems To clarify the relationship between the crystal systems and classes let us consider the cubic system This system de ned by having three equal crystallographic axes and interaxial angles of 90 is also characterized by having a threefold rotary axis of symmetry The cubic system has ve distinct classes each characterized by a unique set of symmetry elements Among this set of symmetry elements is the threefold rotary axis the essential element of symmetry for the cubic system The tetragonal system has seven classes all with a four fold or four fold rotary inversion axis of symmetry etc Table 4 lists the fundamental features of each one of the seven crystal systems as well as the classes belonging to each system Crystal forms Each crystal system is characterized by a number of speci c forms Some of these forms are unique to a particular system whereas others can occur in several different systems Within a system forms characteristic of a higher symmetry class can exist in crystals belonging to a lower symmetry class but not the reverse An example is the Octahedron characteristic of the hexoctahedral holohedral class 4m32m can actually exist in the Diploidal 2m3 class but the pyritohedron characteristic of the Diploidal class cannot exist in the holohedral class Forms that are common to most crystal systems are Pedion Pinacoid Dome Sphenoid Prism and Pyramid Figure 15a shows some common forms in the seven systems whereas Fig 15b shows two examples of complicated combinations of some forms in the cubic system These two gures show that in any system a crystal may be made of one or more forms For a full list and illustration of all 48 possible forms please refer to your textbook 0 The Bravais Lattices Investigating the shapes and types of unit cells for all crystal structures belonging to the different crystal systems reveals that there are only 14 possible different unit cells which when repeated through symmetry operations will de ne all possible lattices These different unit cells are known as Bravais Lattices and are really four basic types of unit cells applied to the 7 crystal systems to yield the 14 Bravais lattices Fig 17 The four basic types are Primitive P Body centered B or 1 Face centered F and End 7 Centered E or C Knowing that Halite is cubic can you gure out its type of Bravais Lattice Refer to Fig lb to help you out 0 Relationship between the 230 Space groups and the 32 Crystal classes Point groups gt A point group implies that one point within the lattice remains xed gt Point groups are de ned on the basis of all possible combinations of the elements of symmetry 3 there are 32 possible combinations gt Space groups are de ned by the combination of elements of symmetry glide planes and screw axes along with a de nition of the space Bravais lattice P A B C I F etc gt Space groups and point groups of the same system have to be isogonal gt The point group can therefore be derived from the space group by eliminating the lattice type screw axes and glide planes gt You can gure out the crystal system easily from its point group or space group Simply eliminate the space lattice symbol replace every screw axis by its isogonal regular axis of symmetry and replace every glide plane by the symbol m for the mirror Now you have the point group crystal class 0 The Stereographic projection The stereographic projection is a 2D graphical representation of the symmetry elements of a crystal or a crystal class as well as the relative locations of all its faces As such it is much easier to construct and read compared to a 3D drawing of a crystal the most common type of 3D view is known as the clinographic view Steps and rules for constructin a stereo raphz39c projection N E 4 V39 0 gt1 9 Imagine that the crystal is located in a large sphere From the center of each face of the crystal draw a straight line perpendicular to this face until it reaches the surface of the enclosing sphere Each one of these lines is known as a M and each face will now be represented by a point on the surface of the sphere This is known as a spherical projection Fig 18 Imagine cutting the sphere into two halves along the equator of this sphere You now have 2 hemispheres a northern hemisphere and a southern one The equatorial surface of this sphere a large circle with the radius of the sphere is now going to be used as the projection plane This circle is known as the Qrimitive circle All other circles through the sphere that maintain its same radius are known as great circles These may represent mirror planes Every point in the northern hemisphere representing a crystal face can now be projected onto the stereographic projection by drawing a line from each of these points to the south pole of the original sphere The point of intersection of this line with the projection plane marks the actual location or projection of this face and is marked with an x Fig 19 Points in the southern hemisphere representing crystal faces can be projected onto the same plane by drawing lines from these points to the north pole of the original sphere The points of intersection of these lines with the projection plane mark the location or projection ofthese faces and are marked with an o The location of the crystallographic axes is marked by lines with arrow feathering at their ends Fig 20 The c axis will be located at the center of the projection b to the right and a at the bottom in front Fig 20 Axes of rotation or rotary inversion represented by points are designated with their usual symbols at these points see above Fig 20 Symmetry planes are indicated by solid lines for planes coinciding with or containing the caxis a solid circle coinciding with the primitive circle if the plane is perpendicular to the c axis or solid arcs representing great circles or planes oriented at an angle of 450 to the caxis gt0 The size and shape of a crystal face projected are unimportant and not represented on a stereographic projection but its position relative to other faces is maintained All faces of the same zone will be represented by points lying on the same great circle A face common to 2 zones plots as a point at the intersection of two great circles Dashed arcs and circles on a stereographic projection represent great circles which do not coincide with symmetry planes O Figures 20 and 21 show stereographic projections and elements of symmetry for the heXtetrahedral class and some of the forms for the hexoctahedral class For a detailed explanation of how interfacial angles are used to plot crystal faces on the stereographic projection with the aid of a Wulf net Fig 22 refer to pages 241 7 248 of Klein 2002 Twinning Twin Rules 1 an aXis of even fold symmetry can never become a twin axis Only a 3 7 fold symmetry aXis can serve as a twin aXis 2 Twinning planes never coincide with symmetry planes in either part of the compound crystal 3 Twins that develop in classes with a center of symmetry have both a twin aXis and a twinning plane perpendicular to it 4 Twinning results in increased symmetry Common Twin 1 aws System Type Name Twin law Example Figure Cubic Contact Spinel plane Spinel 23 11 1 axis 11 1 Cubic Penetration Iron cross axis 001 Pyrite 24 Tetragonal Contact Geniculated plane Cassiterite 25 elbow 011 Rutile amp Zircon Trigonal Contact Calcite plane Calcite 26 10i1 Trigonal Cyclic Polysynthetic plane Calcite 27 1012 Trigonal Contact Brazil plane Quartz 28 1 1 2 0 Trigonal Contact Dauphine axis 0001 Quartz 29 Orthorhombic Cyclic Cyclic plane Aragonite 30 110 Orthorhombic Penetration Cross twins plane Staurolite 31a 031 Orthorhombic Penetration Oblique cross plane Staurolite 3 lb 231 Monoclinic Contact Simple axis 001 Orthoclase 32a quot 39 P quot Carlsbad axis 001 Orthoclase 32b Monoclinic Contact Manebach plane Orthoclase 33 001 Monoclinic Contact Baveno plane Orthoclase 34 021 Monoclinic Contact Swallow tail plane Gypsum 3 5 100 Triclinic Cyclic Albite or plane Plagioclase 36 polysynthetic 010 feldspars Triclinic Cyclic Pericline axis 010 Plag felds 37


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