math sym 2 web - UMass Amherst

1 Mathematical Symmetry Symmetry operationsand Indicesand Indices311113311131111111111111111111111 111111111 Face InterceptsEach face of a crystal may be defined as a functionofitsinterceptwithoneormoreoffun ction of its intercept with one or more of the crystallographic in the orthorhombic system looking down the c-axisb1. 1a, 1b, c12. a, 1b, c,,41a41b3. 1a, 2b, c3a4. 1a, b, c2cc2a, 1b, 1c1a, 1b, 3c1a, 1b, 2cb2a, 2b, 2c,,aThe universally employed system for notation of crystal face intercepts was proposed by ceptsaspoposedbye Miller indices of a face consist of a series of whole numbers that have been derived from the intercepts of their inversions, and if necessary, the clearing of f bfractions.

2 Each of these Miller indices bintersect the ve are similarly defined as ltftithaelements of symmetry with SymmetryTheexternalshapeofacrystalreflec tstheThe external shape of a crystal reflects the presence or absence of translation free symmetryelementsinitsunitcellsymmetry elements in its unit not always immediately obvious, in most well formed crystal shapes, axis of rotation, ifii faxis of rotoinversion, center s of symmetry, and mirror planescan be symmetry operationsUsing linear algebra, it is possible to represent locations of points, lines or planes and operations performed onof points, lines or planes and operations performed on those planes by a matrixhi lil i fh ddIn the mineralogical version of the standard x,y,zCartesian system (a,b,c)

3 A 3x3 matrix will represent all the necessary points for any locationypyThis method allows for a semi-quantitative approach to s mmetr operations sing the same terms as ithsymmetry operations using the same terms as with Miller Indices180 [100]22 Plotstereo[001]2-1000-10[001]2ab010001cb [100]m-100010[100]mab010001cb[100]2 Plot-1000-10001[100]2110x=-1-10 PlotstereoBack to discussed operations may be combined, but the numberof( )combinationsislimited,toyynumber of ( unique) combinations is limited, to 32. Each of these is known as a point group, or crystal use of point derives from the observation that throughout each of the operations at least one point gppin the pattern is unmoved.

4 Group, comes from mathematical group crystal classes may be sub divided into one of 6 crystal of the 32 crystal classes is unique to one of h6(7)lthe 6(7) crystal systems:Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric (cubic)hhlfhhfbfhhlThe 7th system, trigonal, is often thought of as a sub set of the hexagonal , while all mirror planes and poles of rotation must intersect at one point, this point it lftbtft(i)itself may not be a centre of symmetry (i).Of the 32 classes, 21 are without, and 11 with AxesThe identification of specific symmetry pyyoperations enables one to orientate a crystal according to an imaginary set of reference ggylines known as the crystallographic are distinct and different from the classic CartesianAxesxyandzusedinotherCartesian Axes, x, yand z, used in other common day usage, such as plotting the exception of the hexagonal system, the axes aredesignatedabandcare designated a, b, and +orThisisThe ends of each axes are designated +or.

5 This is important for the derivation of Miller angles between the positive ends of the axes are designated , , and . lies between band c. lies between aand c. lies between aand of the 6 cystals systems has a fllhunique set of crystallographic axesTriclinic:Threeunequalaxeswithobliqu eanglesTriclinic: Three unequal axes with oblique : Three unequal axes, two are inclined to one another, the third is : Three mutually perpendicular axes of different :ThreemutuallyperpendicularaxestwoareTet ragonal: Three mutually perpendicular axes, two are equal, the third (vertical) is : Three equal horizontal axes (a1, a2, a3) and a h34thperpendicular vertical axis of different : Three perpendicular axes of equal.

6 Three equal axes and three equal angles that are not 90 .Triclinic: Three unequal axes with oblique angles. To orientate a triclinic crystalthemostcrystal the most pronounced zone be vertical. aand bare determined bytheintersectionsof bby the intersections of (010) and (100) with (001) a(001). The b axis should be longerthantheaaxislonger than the a unique symmetry operation in a triclinic system is a onefold axis of rotoinversion ( equivalent to a centre of symmetry or qyyinversion, i).All forms are pinacoids therefore must consist of two identical and parallel formingmineralsincludeCommon triclinic rockforming minerals include microcline, some plagioclases, and : Three unequal axes, two are inclined with obliqueanglesthethirdisperpendicularobli que angles, the third is perpendicular.

7 Orientation of a crystal has few constraints bis the only axis fixed by symmetry. cistypicallychosenontheccis typically chosen on the basis of habit and b and = 90 . There are some very rare cases where b equals 90 giving a pseudoorthorhombic unique symmetry operation in a monoclinic system /fldfhis 2/m a twofold axis of rotation with a mirror is the rotation, while a and clie in the mirror crystals have two forms: pinacoids and iprisms. lkflldCommon monoclinic rock forming minerals include clinopyroxene, mica, orthoclase and : Three mutually perpendicularaxesofdifferentlengthsperpe ndicular axes of different lengths.

8 Convention has it that a crystal is oriented such that c> b> Crystals are oriented so that cis parallel to crystal elongation. yg In this case the length of the b axis is a taken as unity and ratios are calculated unique symmetry operation in an orthorhombic system is 2/m 2/m 2/m Three twofold axis of rotation coinciding with2/m 2/m 2/m Three twofold axis of rotation coinciding with the three crystallographic to each of the axes is a mirror general class for the orthorhombic system arerhombicThe general class for the orthorhombic system are rhombic dipyramid{hkl}.Thhf fihliididThere are three types of form in the class: pinacoids, prisms, and orthorhombic rock-forming minerals include andalusite and sillimanite, orthopyroxene, olivine and : Three mutually perpendicular axes, two are equal the third (vertical) is shorterare equal, the third (vertical) is shorter.

9 The two horizontal axis in a tetragronal mineral are oriented in the plane of the horizontal Therefore ifacplane of the horizontal. Therefore, if a= b, cmust be in the There is no rule as to whether cis greater or less than = = The unique symmetry operation in a tetragonal system is 4/m 2/m 2/m Theverticalaxis(c)isalwaysafourfoldaxiso frotation2/m The vertical axis (c) is always a fourfold axis of are 4 two fold axis of rotation: 2 parallel to the llhdbhh crystallographic axes aand b, the others at 45 . are 5 mirror general class for the orthorhombic system is known as the dildiid llditetragonal dipyramidal :basalpinacoids,tetragonalThere are four types of form in the class: basal pinacoids, tetragonal prisms, tetragonal dipyramids, and ditetragonal tetragonal rock forming minerals include zircon, rutile and anatase, and : Three equal horizontal axes (a1, a2, a3) and a vertical axis of different length.

10 The three horizontal axis of a hlilitdithchexagonal mineral are oriented in the plane of the horizontal, with cin the vertical. Unlike the other systems the Bravais Millltftlfa3 Miller nomenclature for crystal faces is given by 4 numbers ( {0001}) The first three numbers are listed in a2order of a1, a2, 90 = = 90 = 120 The unique symmetry operation in the hexagonal system is a six fold axis of rotation, and the most common space group is 6/m ,pgp/2/m 2 foldrotationaloperation,whilethereThere vertical axis is the sixfold rotational operation, while there are a further 6 two fold axis of rotation in the horizontal plane (3 coincide with the anaxes).