Crystal morphology I Free crystallizing solids will exhibit flat, smooth faces: euhedral all faces perfectly crystallized subhedralpartially crystallized faces anhedralno faces expressed macrocrystalline crystals can be distinguished by unaided eye > 0.2 mm Finegrained crystals: microcrystalline crystals can be distinguished by light microscopy >1  m cryptocrystallinecrystalline nature can only be revealed nanocrystalline through x-ray or electron diffraction/microscopy The crystal symmetry are characteristic for a certain phase and will correspond to one of the 32 point groups. The true symmetry of a crystal can, however, be hidden through distortions.

Crystal morphology II Euhedral arsenolite (As 2 O 3 ) crystals (White Caps mine Nye County, Nevada.) Anhedral silver specimen Hemihedral rhodonite (MnSiO 3 ) with small (euhedral) quartz crystals, Chiurucu mine, Dos de Mayo, Peru. 3 x 3.5 cm.

Crystal axes I The external form of crystals is related to the crystal structure, which has the same point group symmetry. Both external form and internal atomic arrangements of crystals are conveniently described using a set of three reference axes. The reference axes are, with the exception of the cubic system, non-cartesian. The coordinate system is oriented relative to the symmetry elements and the unit length along the axes is related to the arrangement of the atoms e.g. the periodicity. Interface angles measured on macroscopic crystals can be used to determine axial ratios. Two dimensional example: symmetry elements: Coordinate axes orientation: two mirrors two-fold axis perpendicular to the mirrors unit length: given by the periodicity along the axis a b External crystal form

triclinic system + c - c + b - b + a - a       ( ≠ 90°) a ≠ b ≠ c + c - c + b - b + a - a   a ≠ b ≠ c monoclinic system c: longest axis a: intermediate axis b: shortest axis b:  to the 2-fold axis or   to the mirror plane. c < a Crystal axes II

Crystal axes III orthorhombic system  a ≠ b ≠ c + c - c + b - b + a - a tetragonal system a,b,c: parallel to the 2-fold axes or perpen- dicular to the mirror planes. c < a < b  a 1 = a 2 ≠ c + c - c + a 2 - a 2 + a 1 - a 1 c: parallel to the 4-fold axes, a 1 and a 2 parallel to 2-fold axes or perpendicular to the mirror planes.

hexagonal/ trigonal system + c - c + a 2 - a 2 + a 1 - a 1 cubic system  a 1 = a 2 = a 3 Crystal axes IV   a 1 = a 2 ≠ c c: parallel to the 6 or 3-fold axis, a 1 and a 2 perpendicular to c and, if present, parallel to 2-fold axes or perpen- dicular to the mirror planes.  a 1, a 2,a 3 : parallel to the 4-fold or 2-fold axes, which are 54.44° from the 3-fold axes. + a 2 - a 2 + a 1 - a 1  a3 a3  a3 a3

Choice of axes and unit cell Criteria for the choice of axes and unit cell 1. Coincidence with symmetry elements of the structure 2. Axes should be related through symmetry elements of the structure 3. Smallest possible unit cell respecting 1. and 2. Standard settings for the 7 crystal classes Triclinic system c: longest unit cell edge a: intermediate unit cell edge b: shortest unit cell edge Monoclinic system b:   to the 2-fold axis or   to m a,c:  to b-axis and arranged to form a right-handed system, c > a. Orthorhombic system: a,b,c:  to the 2-fold axes or  to the mirror planes arranged to form a right-handed system, c is usually the longest, b the shortest axes. Tetragonal, trigonal or hexagonal systems: c:  to the 4-,3- or 6-fold axis a1,a2: in the plane  to c and, if present,  to the 2-fold axes or   to the mirror planes. Cubic system: a1,a2,a3:  to the 4- or 2-fold axes, which are 54.44° from the three 3-fold axes.

Face intercepts I Crystal faces are defined by indicating their intercepts on the crystallographic axes. The units along the axes is determined by the periodicity along theses axes: - c - b - a Intercepts: 5a : 3b : 2c= 5 : 3 : 2 + c + b + a 2c 3b 5a

Face intercepts II Faces parallel to an axis have an intercept with that axis at infinity + c - c + b- b + a - a aa Intercepts: 3a :  b :  c= 3 :  :   c  b

Face intercepts III Intercepts are always given as relative values, e.g. they are divided until they have no common factors. Parallel faces in the same quadrant have, therefore, the same indices + c + b- b + a - a aa 4c 2b 1b 2a 2c Intercepts: 4a : 2b : 4c= 4 : 2 : 4  div. by 2  2 : 1 : 2 - c Intercepts: 2a :  1b : 2c= 2 : 1: 2 Intercept ratios are called Weiss indices

Miller indices I The Miller indices of a face are derived from the Weiss indices by inverting the latter and, if necessary, eliminating the fractions. Reason for using Miller indices:- avoiding the index  - simplifies crystallographic calculations - simplifies the interpretation of x-ray diffraction Example: Weiss indices Miller indices  x Miller indices are placed in round brackets, e.g. (1 0 0). Commas are only used, if two digit indices appear, e.g. (1,14,2) Negative intercepts are indicated by a bar above the number, e.g. (1 0 0). Indices, which are not precisely known, are replaced by the letters h, k, l. This system is also used to indcate indices of faces with common orientation properties e.g. (0, k, 0)all faces parallel to the a- and c-axis (0, k, l)all faces parallel to the a- axis

Miller indices II Examples: - a 2 + a 1 - a 1  a 3  a 3 - a 2 (1 0 0) (0 1 0) (0 0 1) (1 1 1)

Crystal forms I All crystal faces which are related through one or several symmetry elements are called a crystal form. Example: cubic system, point group 4/m32/m, starting face (1 1 1) e.g cutting all the axes in the positive quadrant at the same distance (1 1 1) Applying the four-fold rotation to (1 1 1) creates 3 new faces Applying the mirror perpendicular to the 4-fold axis creates 4 new faces Applying all the other symmetry elements to the already created faces recreates the already existing faces. The eigth faces represent a crystal form, called an octaedron.

Crystal forms II (1 1 1) In the above example, all eight faces belonging to   enclose space, the form is said to be closed. All faces belonging to a crystal form are indicated by the Miller indices of the starting face inclosed in braces: faces belonging to the octahedron: (111), (111), (111), (111), (111), (111)  

Crystal forms III Fluorite cube from Hardin County, Illinois. All pictures: Galena cubes from Reynolds Co., Mo.

Crystal forms IV All pictures: Fluorite octahedrons from Grant County, New Mexico Magnetite octahedrons from Diamantina, Minas Gerias, Brazil

Example: tetragonal system, point group 4/m, starting face (1 0 0) e.g cutting only the a 1 axis Applying the four-fold rotation to (1 0 0) creates 3 new faces. Applying the mirror operation to (1 0 0) does not create new faces The form with the faces (1 0 0), (0 1 0) (1 0 0) and (0 1 0) =   does not enclose space, it is an open form, called a tetragonal prisma. Crystal forms V Applying the mirror operation to (0 0 1) creates one new face: (0 0 1). Applying the four-fold rotation to the same face does not create new faces The form consisting of the faces (001) and (0 0 1), e.g. two parallel quadrangles, is called a tetragonal pinacoid.

 11 ∞ -1∞ ∞ The faces correspond often to densly packed atom planes. The intercepts of macroscopic crystals faces have values in the billions. The intercepts of densly packed planes, however, will have a common multiplicator, which allows to reduce these numbers to small integers. Example: - densly packed planes - intercepts = Weiss indices Atomic structure and macroscopic crystal faces I

High resolution transmission electron microscopy image of a Mn 3 O 4 crystal inbedded in a matrix of metallic silver. The black dots within the inclusion are manganese atoms. (The oxygen atoms cannot be seen. The faces of the crystal follow dense packed manganes planes. Atomic structure and macroscopic crystal faces II

The repetitive unit has the same symmetry as before but not so the external shape of the crystal The growth velocity of symmetrically related faces are also “symmetric” e.g. they are equal - but only if the external conditions (pressure, temperature, concentration of crystal components in the solution etc.) are equal. That is rarely the case and differences in growth velocities leads to “unsymmetric”, distorted crystal shapes. Perfect, symmetric crystals are, therefore, rare in Nature. concentration high = high growth velocity of face A’ concentration low =low growth velocity of face A flow direction A A’ Atomic structure and macroscopic crystal faces III

Zones I Faces with parllel (common) intersection edges form a zone. a m m ee d d c k ee b Example: zone with faces c, k, and b zone with faces m, a, and b zone with faces c, d, and a The zone axis, e.g. the orientation of the edge common to all faces is indicated by a symbol in square brackets: [ u v w ]. zone axis 2 zone axis 1 zone axis 3

Zones II The indices of the zone axis can be derived from the Miller indices of any two faces belonging to the zone. The procedure is the same as employed to calculate the determinant of a matrix: Indices face 1 h 1 k 1 l 1 h 1 k 1 l 1 Indices face 2 h 2 k 2 l 2 h 2 k 2 l 2 1. product 2.product Index = 1. product - 2.product k 1 l 2 - l 1 k 2, l 1 h 2 - h 1 l 2,h 1 k 2 - k 1 h 2 = = = U VW Example: Indices face k: Indices face b: x0 - 1x1 1x0 - 0x0 0x1-3x0 = = = zone axis

Zones III A face (h k l) belongs to a zone [u v w] when the following zone axis condition is fulfilled: hu + kv + lw = 0 The Miller index of a face parallel to two given lines (u 1 v 1 w 1 ) and (u 2 v 2 w 2 ) is: h = v 1 w 2 - v 2 w 1 k = w 1 u 2 - w 2 u 1 l = u 1 v 2 - u 2 v 1 If three faces belong to the same zone (tautozonality),the following determinant must be zero: The same condition must be fulfilled for three coplanar directions. Adding or subtracting Miller indices of faces belonging to the same zone gives additional faces belonging to that zone.

Hexagonal Indices a1a1 a3a3 a2a2 c The hexagonal coordinate system is traditionally represented with four axis. The a 3 - axis, however, is superfluous. Nevertheless the four axes system is widespread and hexagonal face symbols are given with 4 Miller indices. For the third index i the following relation always holds: h + k = - i. For the above matrix calculations the third index is omitted and the results are always three membered indices, e.g. ( ) -> (2 1 0) Attention: The above relationships do not hold for directions! [ ] ≠ [2 1 0] The transformation from the three membered [u v w] to the four membered direction symbol [U V T W] is given by The reverse conversion by: a1a1 a2a2 a3a3 -a 3 Weiss: Miller:11-20