Crystallography & Mineral Chemistry GLY3632 PDF 2nd Semester 2024

Department of Physics, Chemistry & Material Science 2nd semester, 2024 Crystallography & Mineral Chemistry GLY3632 Formatted and revised by Dr. P....

Department of Physics, Chemistry & Material Science 2nd semester, 2024 Crystallography & Mineral Chemistry GLY3632 Formatted and revised by Dr. P. N. Hishimone Introduction Lecturers Dr. P. N. Hishimone; office – H135, Ext. 3712, [email protected] Prof. H. Sommer (Geology Dept., Southern campus) Lab Instructor Ms. Hileni Thomas ([email protected]) Lectures Mondays 11:30 – 12:25 Tuesdays 11:30 – 12:25 Wednesdays 11:30 – 12:25 Thursdays 13:30 – 14:25 Labs Mondays 14:30 – 17:25 (Venue: W169) Class-representative: PAHEJA NOVENGI 2 Recommended textbooks ① ② ③ For extra readings ① Sands, D.E. Introduction to Crystallography; Dover Books on Chemistry Series; Dover Publications, 1993; ISBN 9780486678399. ② Klein, C. Manual of Mineral Science; John Wiley & Sons, Incorporated, 2003; ISBN 9780471427674. ③ West, A.R. Solid State Chemistry and its Applications; Wiley, 2014; ISBN 9781119942948. 3 Course content ① Crystallography Crystals, lattices, and crystal symmetry; crystal morphology: and Crystal projections; Space groups, internal order, and translational symmetry; Crystal structures and crystal chemistry, X-ray crystallography, and X-ray diffraction. ② Mineral Chemistry Crystals minerals in the earth’s crust: Chemical analytical techniques (X-ray diffraction, X-ray Fluorescence, electron microbe analysis); mineral compositions and variations; exsolutions; calculation of mineral analyses; Graphic representative of mineral composition. 4 Crystallography: ② Crystal Symmetry Outline Definition of symmetry The 32 crystal classes Crystal planes and indices Crystal forms Symmetry – An object or figure is said to have symmetry if some movement of the figure or operation on the figure leaves it in a position indistinguishable from its original position. 5 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements Axis of rotation Symmetry of a H2O molecule After rotation the O-atom and the H- Before 180° rotation atoms look exactly the same as before rotation After 180° rotation Inspection of the object and its surroundings after some movement or operations will not reveal whether or not the operation has been carried out. 6 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… (I) Rotation axis – An axis about which a similar crystal face appears more than once in a revolution of 360° is a rotation axis. (II) Mirror plane - A plane diving a three dimensional body into two similar halves, each being the mirror image of the other. (III) Centre of symmetry - A point at the centre of a crystal through which an imaginary straight line can be joined to an equivalent point an equal distance beyond the centre. (IV) Rotoinversion axis - A combination of rotation by 360/n followed by an inversion across a centre to a new position, the process being repeated until the original starting position is obtained. 7 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… (I) Rotation axis – An axis about which a similar crystal face appears more than once in a revolution of 360° is a rotation axis. A symmetry element for which the operation is a rotation of 360°/n is given the Hermann-Mauguin symbol n. Fig. A six-fold axis: Hermann-Mauguin Notation = 6. A similar crystal face appears every 360°/n of rotation 360°/6 = 60° The axis may be: two-fold (a diad axis); n = 2, three-fold (a triad axis); n = 3, four-fold (a tetrad axis); n = 4, or six-fold (a hexad axis); n = 6 8 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… (II) Mirror plane - A plane diving a three dimensional body into two similar halves, each being the mirror image of the other. The Hermann-Mauguin symbol for a mirror plane is m. A mirror plane (m) 9 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… (III) Centre of symmetry - A point at the centre of a crystal through which an imaginary straight line can be joined to an equivalent point an equal distance beyond the centre. The centre of inversion is denoted by ഥ. the Hermann-Mauguin symbol 𝟏 Fig. A crystal with a centre of symmetry (c). The symmetry element is a point, and the operation consists of inversion through the point. 10 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… (IV) Rotoinversion axis - A combination of rotation by 360/n followed by an inversion across a centre to a new position, the process being repeated until the original starting position is obtained Inversion or rotoinversion axes may be 1-fold, 2-fold, 3-fold, 4-fold, or 6- fold. ഥ. The Hermann-Mauguin symbol is 𝒏 A four-fold rotoinversion axis 11 ②-a: Crystal Symmetry… Symmetry operations and symmetry elements… Identity - Any direction in any object is a one-fold axis, since a 360° rotation merely restores the original position. This symmetry element is called the identity and is symbolized by 1. 12 ②-a: Crystal Symmetry… The thirty-two crystal classes The number of possible symmetry elements and their combinations is only 32. The 32 possible elements and combinations of elements are identical to the 32 possible crystal classes to which crystals can be assigned on the basis of their morphology or their internal atomic arrangement. ➯ The 32 possible non-identical crystal classes are also known as point groups. Point groups: Point ⇒ symmetry operation leaves at least one particular point unmoved during the operation. Group ⇒ relates to the mathematical theory of groups that allows for a systematic derivation of all possible & nonidentical symmetry combinations. 13 ②-a: Crystal Symmetry… The thirty-two crystal classes… (Hermann-Mauguin notation) ✔ Crystal classes (point groups) are designated by combinations of symbols for symmetry elements. ✔ Only the minimum symmetry required to define a class is stated. Order of notation: Principal axis → symmetry plane normal to P-axis → secondary axes Please check: https://youtu.be/a0q9nX2Zbl4 Examples: 2/m means 1 principal diad axis normal to a mirror plane. 4 2 2 4 The short symbol for 4/m 2/m 2/m (𝑚 𝑚 𝑚) is 4/mmm (𝑚mm) but can not be mmm. 14 ②-a: Crystal Symmetry… The thirty-two crystal classes… Table. Characteristic symmetry and relationship between crystal axes and symmetry notation of crystal systems (from Klein & Hurlbut 1993). 15 ②-a: Crystal Symmetry… The thirty-two crystal classes… Table. Characteristic symmetry and relationship between crystal axes and symmetry notation of crystal systems (from Klein & Hurlbut 1993)… 16 ②-a: Crystal Symmetry… Hermann-Mauguin notation Triclinic system ഥ Two crystal classes: 𝟏, 𝟏 Characteristic symmetry: 𝟏 fold (inversion or identity) only Space lattice: a≠b≠c;α≠γ≠β 17 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Monoclinic system Three crystal classes: 𝟐, 𝐦, 𝟐/𝐦 Characteristic symmetry: one 𝟐- fold rotation axis and/or one mirror plane (m) Space lattice: a ≠ b ≠ c ; α = γ = 90° ≠ β 18 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Orthorhombic system Three crystal classes: 𝟐𝟐𝟐, 𝐦𝐦𝟐, 𝟐/𝐦𝟐/𝐦𝟐/𝟐 Characteristic symmetry: Three mutually perpendicular directions about which there is binary symmetry. Space lattice: Orthorhombic 2/𝑚2/𝑚2/2 a ≠ b ≠ c ; α = γ = β = 90° 19 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Tetragonal system ഥ, 𝟒/𝐦, Seven crystal classes: 𝟒, 𝟒 𝟒𝟒𝟐, 𝟒𝐦𝐦, 𝟒𝟐𝐦, 𝟒/𝐦𝟐/𝐦𝟐/𝟒 Characteristic symmetry: One 𝟒- fold rotation axis. Space lattice: a = b (≠ c) ; α = γ = β = 90° Tetragonal 4/𝑚2/𝑚2/𝑚 20 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Hexagonal system ഥ, 𝟔/𝐦, Seven crystal classes: 𝟔, 𝟔 ഥ𝐦𝟐, 𝟔/𝐦𝟐/𝐦𝟐/𝐦 𝟔𝟐𝟐, 𝟔𝐦𝐦, 𝟔 Characteristic symmetry: One 𝟔- fold rotation axis. Space lattice: a = b (≠ c) ; α = β = 90° ; γ = 120° Hexagonal 6/𝑚2/𝑚2/𝑚 21 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Trigonal system ഥ , 𝟑𝟐 , Five crystal classes: 𝟑 , 𝟑 ഥ𝟐/𝐦 𝟑𝐦, 𝟑 Characteristic symmetry: One 𝟑- fold rotation axis. Space lattice: a = b = c ; α = γ = β ≠ 90° Trigonal 3ത 2/𝑚 22 ②-a: Crystal Symmetry… Hermann-Mauguin notation… Isometric system ഥ, Five crystal classes: 𝟐𝟑 , 𝟐𝐦𝟑 ഥ𝟐/𝐦 𝟒𝟑𝟐, 𝟒𝟑𝐦, 𝟒/𝐦 𝟑 Characteristic symmetry: Four 𝟑- fold rotation axis. Space lattice: a = b = c; α = β = γ 23 ②-a: Summary and examples Rotation axes, mirror planes and the notations 24 ②-a: Summary and examples… Rotation axes, mirror planes and the notations… 25 ②-a: Summary and examples… Rotation axes, mirror planes and the notations… 4 fold axis 3-fold axis 2-fold axis 26 ②-a: Summary and examples… Rotation axes, mirror planes and the notations… 0° 120° 240° 360° 27 Crystallography: ② Crystal Symmetry Outline Definition of symmetry The 32 crystal classes Crystal planes and indices Crystal forms ➯ Crystals frequently have polyhedral shapes bounded by flat faces. ➯ Crystal faces correspond to sets of parallel planes, each set representing a stacking of layers of molecules. ➯ A growing crystal adds molecules more easily on some planes than on others, and the corresponding crystal faces experience greater development. 28 ②-b: Crystal planes and indices Constancy of interfacial angles ✔ In any two crystals of the same substance the angles between corresponding faces, measured in a plane normal to the edge between them, is always the same. Regular and irregular crystals of same mineral NB: The above is the law of constancy of interfacial angles 29 ②-b: Crystal planes and indices… Crystallographic reference axes ✔ Crystal faces are described with reference to a +z set of coordinates termed crystallographic reference axes chosen by convention to -x -y coincide with important symmetry axes and/or important directions of edge in the +y crystal. +x ✔ The axes should intersect at a common point -z in the centre of the crystal and need not be all in the same plane 30 ②-b: Crystal planes and indices… Parameters, intercepts, and indices ✔ In order to indicate the relative positions of +z crystal faces it is necessary to have a system of -x a b notation -y c ✔ In this Figure the shaded face makes intercepts of a, b and c on the axes x, y, and z, respectively +y ✔ Such a plane is termed a parametral plane +x and is taken to define the unit length along -z each reference axis 31 ②-b: Crystal planes and indices… Parameters, intercepts, and indices… +z -x a b -y c +y +x -z ➯ The relative lengths are called parameters and the ratios a:b:c are termed axial ratios ➯ The parameters and axial angles constitute the crystal elements which are specific for a particular crystal. 32 ②-b: Crystal planes and indices… Miller system of indices ✔ Provides a notation for describing +z the slopes of other crystal faces in -x a b terms of unit lengths. -y c ✔ Intercepts are written in the order a, b, c, and the letters are +y conventionally omitted. +x ✔ Reciprocals of the intercepts are multiplied through to clear fractions, -z to give the Miller indices. 33 ②-b: Crystal planes and indices… Miller system of indices… +z Plane Intercepts Reciprocals MIs -x Parametral 1 1 1 1/1 1/1 1/1 (111) a b -y c Light blue 231 1/2 1/3 1/1 (326) +y ✔ The 3 Miller indices together constitute the +x Miller Symbol of the face and define its relative slope. -z ✔ Crystal faces parallel to one or more of the crystallographic axes make intercepts at infinity. 34 Crystal planes and indices… Important notes To refer to a general face that intersects all three crystallographic axes where the parameters are not known, we use the notation (hkl). 1. For a face that intersects the b and c axes with general or unknown intercepts the notation would be (0kl), 2. for a face intersecting the a and c axis, but parallel to b the notation would be (h0l), and similarly 3. for a face intersecting the a and b axes, but parallel to c we would use the notation (hk0) 35 ②-b: Crystal planes and indices… Bravais-Miller indices ✔ In the hexagonal system, 4 indices, the Bravais-Miller indices, are required because there are 4 axes (three horizontal at 120° and one vertical). What are the B-M indices in this case? 36 Crystal planes and indices… Important notes… Since the hexagonal system has three "a" axes perpendicular to the "c" axis, both the parameters of a face and the Miller Index notation must be modified. The modified parameters and Miller Indices must reflect the presence of an additional axis. This modified notation is referred to as Miller- Bravais Indices, with the general notation (hkil) ✓ An important rule to remember in applying this notation in the hexagonal system, is that whatever indices are determined for h, k, and i, h+k+i=0 37 Crystal planes and indices… Important notes… For a similar hexagonal crystal, this time with the shaded face cutting all three axes, we would find for the shaded face in the diagram that the parameters Note how the "h + k + i = 0" rule applies here 38 ②-b: Crystal planes and indices… Axial ratios ✔ The parametral plane defines the unit lengths of the reference axes and establishes the axial ratio, which is characteristic for each substance. ✔ The unit lengths in directions x, y, and z are called a, b and c, respectively. ✔ The axial ratios express the relative, not the absolute lengths of the cell edges that correspond to the crystallographic axes. z b a c y x 39 ②-b: Crystal planes and indices… Axial ratios… ✔ For orthorhombic sulphur the unit cell dimensions are given as: a = 10.47 Å, b = 12.87 Å and c = 24.49 Å ✔ Using b as a unit of measure the lengths of a and c relative to b are a/b:b/b:c/b =X:1:Y ✔ In the case of sulphur the axial ratios Fig. The orthorhombic unit cell of a:b:c = 0.813 : 1 : 1.903 sulphur with axial ratios a:b:c = 0.813 : 1 : 1.903 40 Crystallography: ② Crystal Symmetry Outline Definition of symmetry The 32 crystal classes Crystal planes and indices Crystal forms Crystal form implies an assemblage of similar faces, which together may wholly or partly constitute the exterior of a crystal. ✔ Forms are described with reference to reference axes. ✔ The terms unique, special and general forms describe the relationship of faces to symmetry elements. 41 ②-b: Crystal planes and indices… Unique form One developed from a plane normal to an axis of symmetry. ✔ Such a plane has only one slope in relation to crystallographic reference axes. ✔ It has a unique Miller symbol (always same digits, though the order in which they are written changes). 42 ②-b: Crystal planes and indices… Special form One that may have varying slopes relative to the crystallographic axes, but is normal or parallel to a symmetry axis or plane, resulting in the same pattern of numbers in the Miller symbol For example, {h0l} in the orthorhombic system Special form: {h0l} 43 ②-b: Crystal planes and indices… General form A face that is not specially related to any element of symmetry may have any combination of numbers as its Miller symbol (i.e. it is not parallel or normal to a symmetry axis or plane). ➯ Such a form will have a symbol of the type {hkl} 44 ②-b: Crystal planes and indices… Closed and open forms Describe the arrangement of faces developed as a form by the operation of a particular complement of symmetry on a given plane. ➯ Closed forms are those forms which, when developed alone, will enclose space. ➯ Open forms are those that do not by themselves enclose space. In a crystal they must be combined with another form or forms. 45 ②-b: Crystal planes and indices… Closed forms: cubic forms ➯ Crystal forms of the isometric crystal system Note that they are all closed forms 46 ②-b: Crystal planes and indices… Closed forms: cubic forms Bipyramid Sphenoid Trapezohedron Rhombohedron Scalenohedron ➯ Crystal forms of the isometric crystal system Note that they are all closed forms 47 ②-b: Crystal planes and indices… Open forms Pedion: a single plane not repeated by any Dome: A pair of intersecting planes, symmetry repeated by a mirror plane or by a diad Pinacoid: a pair of parallel planes axis produced by the symmetry operating on Prism: A set of planes forming an open- an initial plane ended tube 48 ②-b: Crystal planes and indices… Crystal habit The term habit refers to the characteristic shape a crystal commonly assumes. ➯ It may be a single form or a combination of forms Fig. Cube (a), octahedral (o) and Fig. Cube and distorted cube dodecahedral forms 49 ②-b: Crystal planes and indices… Crystal habit & forms Growth stages: 1. Nucleus with only octahedral faces. 2. Crystal with octahedral and cube faces. 3. Development of cube faces with growth. 4. Final external form of crystal. 50 ②-b: Crystal planes and indices… Zones A zone is the arrangement of a group of faces with parallel intersection edges. ➯ Considered collectively, these faces form a zone. ➯ A line through the centre of the crystal that is parallel to the lines of face intersections is called the zone axis. In this figure the faces m’, a, m, and b are in one zone 51 ②-b: Crystal planes and indices… Zones axis The faces m’, a, m, and b are in one zone. The line designated is the zone axis. Similarly b, r, c, and r’ are in another zone. The zone axis in this case is the line designated 52 ②-b: Crystal planes and indices… Zones axis… To determine the zone axis of two faces (hkl) and (pqr): 1. Write twice the symbol of one face and directly beneath, twice the symbol of the other face. 2. The first and last digits of each line are disregarded and the remaining numbers, joined by sloping arrows, multiplied. 3. In each set the product of 2 is Mnemonic method of zone determination subtracted from the product of 1 as: kr – lq, lp – hr, hq - kp 53 This section ends here 54