01-GLY3632_Introduction-220724-Portal.pdf

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 Wednesdays 14:30 – 17:25 (Venue: TBC) 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 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. 5 Crystallography: ① Introduction Outline Crystals Crystallization Crystal growth Crystals and lattices Lattice points Unit cells Bravais lattices Crystallography – The study of crystalline solids and the principles that govern their growth, external shape and internal structure. 6 ① Introduction… Minerals Naturally occurring solids generally formed by inorganic processes and they have an ordered internal arrangement of atoms and a definite chemical composition and physical properties that may vary within a definite range. Fig. 1 Image of fluorite, CaF2 (google.com) Minerals are known to possess the internal, ordered arrangement characteristic of crystalline solids and may be bound by smooth surfaces and assume regular geometric forms known as crystals. 7 ① Introduction… Crystal A crystal is a geometric form consisting of atoms arranged in a pattern that repeats periodically in three dimensions. Pattern: single atom, a group of atoms, a molecule or a group of Fig. 2 NaCl structure based on a face-centered molecules. cubic lattice with each ion surrounded by 6 neighboring ions of opposite charges outlining octahedral polyhedral. NB!! Periodicity or regularity of the arrangement of the patterns. Crystal structure (of NaCl): represents an arrangement of the Na+ and Cl– ions that leads to a potential energy minimum. 8 ① Introduction… Crystals vs minerals Crystal Mineral Solid material whose Naturally occurring solid formed constituent atoms, molecules, by an inorganic process and Definition or ions are arranged in an represented by a chemical ordered pattern extending in all formula with an ordered atomic three spatial dimensions. arrangement. Atoms, ions and molecules are Definite chemical composition Composition fixed in regular order, which (Minerals are inorganic extends in 3D. compounds). Structure and habit, hardness, Shape, atom composition, luster, diaphaneity, color streak, Characteristics bonds and defects. tenacity, cleavage, fracture, parting and specific gravity. 9 ① Introduction… Crystals vs minerals… Crystal Mineral Hexagonal, cubic, orthorhombic, tetragonal, Classification Silicates and non-silicates. rhombohedral, and monoclinic. Study Crystallography Mineralogy Quiz Please answer the following questions. 1. What is crystallography? 2. Define: (a) a mineral (b) a crystal. 10 ① Introduction… Crystallization Crystals are formed from solutions, melts and vapours. Solutions, melts and vapors are disordered states. The atoms are randomly distributed. With changing concentration, temperature and pressure, atoms may join in an ordered arrangement ⇒ forming a crystalline state. 11 ① Introduction… Crystallization from solutions ① By changing the concentration The concentration of a solution can be increased through evaporation. Fig. 3 A nucleus of NaCl in an evaporating lake (from Klein & Hurlbut, 1993). Example: if a solution containing NaCl is allowed to evaporate, the solution will contain more and more ions of Na and Cl per unit volume. When the remaining water can no longer retain the ions ⇒ Solid salt begins precipitate 12 ① Introduction… Crystallization from solutions… ① By changing the concentration… Very slow evaporation ⇒ ions will form one or a few crystals with characteristic shapes and often with common orientation. Rapid evaporation ⇒ Many centers of crystallization Fig. 3 A nucleus of NaCl in an evaporating lake (from Klein & will be set up, usually resulting in Hurlbut, 1993). many small, randomly oriented crystals. 13 ① Introduction… Crystallization from solutions… ② By lowering the temperature Hot water will dissolve slightly more salt than cold water. If the hot solution is allowed to cool, a point is reached where the solution becomes sufficiently concentrated that salt will crystallize. ③ By lowering the pressure Similarly, the higher the pressure, the more salt water can hold in solution, and lowering the pressure of a saturated solution will result in supersaturation and crystals will form. 14 ① Introduction… Crystallization from melts The formation of Igneous rocks from molten magmas is similar to the freezing of water. In a magma many ions are in an uncombined state, although there is considerable cross-linking of the ions and ionic groups. Crystal formation in a magma is a result of two competing tendencies: ① Thermal vibrations – tend to destroy the nuclei of potential minerals, and ② Attractive forces – tend to aggregate atoms into crystal structures. As temp. drops, effect of ① ⇓ and ② ⇑ 15 ① Introduction… Crystallization from melts… Cooling & crystallizing 1 2 3 Silicate liquid Growing crystals (magma) Fig. Generalized crystallization 4 5 6 from a magma (Columbus.edu). Network of intergrown crystals (all magma used up) ✓ If the magma is slowly cooled, crystallization is also slow ➱ Coarse-grained texture & mineral crystals are visible to the unaided eye. ✘ Lavas are cooled quickly on the earth’s surface, crystallization is very fast ➱ Fine-grained texture & mineral crystals are too small to be seen by the unaided eye. 16 ① Introduction… Crystallization from vapours If a vapour is cooled, the dissociated atoms or molecules are brought closer together, eventually locking themselves into a crystalline solid. Fig. Crystals of sulphur in basalt 17 ① Introduction… Crystal growth Nucleation is the first stage in the birth of If a crystal. In most cases nuclei are the initial products of precipitation or of crystallization. For a nucleus to survive it must grow rapidly enough to reduce its surface energy and thus its solubility. Fig. Portion of a sodium chloride crystal structure showing the size of the ions, magnified about 108 times ( from Sands 1993) The Nucleus is the result of the coming together of various ions to form the initial regular structural pattern of a crystalline solid. If a nucleus reaches a critical size through rapid deposition of further layers of ions, it will have a chance of surviving as a larger crystal. 18 ① Introduction… Crystal growth… The outer solid of a nucleus (or crystal) in contact with a saturated solution represents a surface of unsatisfied chemical bonds. Fig. Well bonded ions in the internal part of a NaCl crystal and unsatisfied chemical bonds at the outer surface of the crystal. The energy of a surface with unsatisfied chemical bonds is lowered when an atom attaches itself to it, and the amount of energy released by such an attachment depends on where such attachment occurs. 19 ① Introduction… Crystal growth… (klein & Hurlbut 1993) (klein & Hurlbut 1993) Fig. Attachment of ions at a submicroscopic step Fig. Submicroscopic blocks create steps for the lowers the energy of the crystal surface. attachment of new layers of ions on outer surface of a crystal. Example: In Ionically bonded crystals, such as NaCl, the energy of attachment is greatest at corners, intermediate at the edges and least in the middle faces. 20 ① Introduction… Crystal growth… (klein & Hurlbut 1993) Fig. Attachment of ions at a submicroscopic step lowers the energy of the crystal surface. The growth of a crystal to its optimum size depends on: 1. The availability of ions, and 2. Space. 21 Outline Crystals Crystallization Crystal growth Crystals and lattices Lattice points Unit cells Bravais lattices 22 ① Introduction… Crystals and lattices Recap: definition of a crystal and the important feature of a crystal? (a) (b) Na+ Cl– Na+ Cl– Fig. (a) NaCl structure with atomic centers shown. (b) Six-fold coordination of a single atom (from Battey 1981) Example: a crystal of NaCl contains many +ve and –ve ions held together by electrostatic attractions. Arrangement details depend upon the balancing of attractive and repulsive forces (electrostatic and ionic effects). 23 ① Introduction… Lattice points A wallpaper pattern is a useful two- dimensional analog of crystal. It can be of any complexity and the entire pattern repeats periodically in two-dimensions. Fig. A 2-D periodic structure with lattice points connected to form parallelograms ( from sands 1993) A system of reference points may be obtained by choosing one point at random – all points identical with this point constitute a lattice point. [exactly the same surroundings, identical in position relative to the repeating pattern or motif]. 24 ① Introduction… Lattice points… ✓ This set of identical points in two- dimensions constitutes a net. ✓ The three-dimensional distribution of points is referred to as a lattice or Space lattice. ✓ Row refers to the one-dimensional distribution of these points. Fig. A 2-D periodic structure with lattice points connected to form parallelograms ( from sands 1993) 25 ① Introduction… Unit cells Note! Connecting the lattice points with straight lines: ✓ In 2-D space ⇨ parallelograms (Fig.). ✓ In 3-D space ⇨ parallelepipeds. ✓ Repetition of these parallelepipeds by translation from one lattice point to another generates the space Fig. A 2-D periodic structure with lattice lattice. points connected to form parallelograms ( from sands 1993) This generating parallelepiped is called a Unit cell 26 ① Introduction… Unit cells… The smallest unit or parallelepiped, which when repeated in three-dimensions, will build the crystal, is called the Unit cell of the structure. Fig. Representation of a Unit cell in a crystal structure. The indicated unit cell has a simple cubic structure (only 84 Po). A unit cell is always parallelepiped, and it is sort of a template for the whole crystal. Knowledge of exact arrangement of atoms within one unit cell ⇛ knowledge of atomic arrangement for the whole crystal (in effect). 27 ① Introduction… Unit cells… The internal structural pattern of a crystal consists of the unit cells , each of which is a group of linked atoms in a fixed spatial relationship to one another. Fig. Representation of a Unit cell in a NaCl Fig. Representation of a Unit cell in a crystal crystal structure (Na+ ions are at the corners). structure. (Cl– ions are at the corners) Each corner of a unit cell is a point with identical surroundings in the same direction and each unit cell constitutes a space lattice in 3-dimensions. 28 ① Introduction… Unit cells… (size and shape of the unit cell) May be specified by means of the axial length of the three independent edges and the three axial angles between these edges. Fig. Six numbers specify the size and shape of a unit cell (i.e. a, b, c, β, γ and α ). a, b and c are the lengths of the three independent edges. β is the angle between a and c, γ is between a and b, and α is between b and c. 29 ① Introduction… Unit cells… (size and shape of the unit cell) May be specified by means of the axial length of the three independent edges and the three axial angles between these edges. V= abc(1 – cos2 α – cos2 β – cos2 γ + 2cos α cos β cos γ)1/2 For isometric structures, Since cos90° = 0, the formula reduces to V=a·b·c 30 Outline Crystals Crystallization Crystal growth Crystals and lattices Lattice points Unit cells Bravais lattices 31 ① Introduction… Bravais lattices Space lattices which represent possible ways in which points can be arranged periodically in 3D. ➯ Named after Auguste Bravais – identified 14 basic types of space lattices in 1848 The Bravais lattices are categorized into three groups: 1. Primitive (P) lattices 2. Body-centered (I) lattices 3. Face-centered (F, A, B, or C) lattices 32 ① Introduction… 1st things first: the seven (7) crystal systems Crystals are classified according to their atomic lattice or structure (i.e. the symmetries of their unit cells). In total, there are 7 groups ➯ Crystal systems 33 ① Introduction… Now the Bravais lattices The Bravais lattices are categorized into three groups: 1. Primitive (P) lattices – the unit cell has a lattice point at each corner only. 2. Face-centered (F, A, B, or C) lattices – the are lattice points either at centers of all faces (F), or at the center of one pair of faces (A, B or C in different cases), in addition to the lattice points at each corner. 3. Body-centered (I) lattices – there is a lattice point at the center of the cell in addition to the lattice points at each corner. a = b = c; α = β = γ Fig. three (3) Bravais lattices of the cubic (isometric) crystal system. 34 ① Introduction… The Bravais lattices… (i) (ii) (iii) Two (2) Bravais lattices One (1) Bravais lattice One (1) Bravais lattice a = b (≠ c) ; α = γ = β = 90° a = b (≠ c) ; α = β = 90° γ = 120° a=b=c; α = γ = β ≠ 90° (iv) Four (4) Bravais lattices a ≠ b ≠ c ; α = γ = β = 90° 35 ① Introduction… The Bravais lattices… (i) (ii) One (1) Bravais lattices Two (2) Bravais lattices a≠b≠c;α≠γ≠β a ≠ b ≠ c ; α = γ = 90° ≠ β 7 crystal systems ➯ 14 Bravais lattices (Cubic (P, F & I), Tetragonal (P & I), Hexagonal (P or C), Rhombohedral (R), Orthorhombic (P, C, F & I), Monoclinic (P & I), and Triclinic (P)) 36 ① Introduction… The Bravais lattices… Two (2) Bravais lattices a ≠ b ≠ c ; α = γ = 90° ≠ β Body centered monoclinic = Base centered monoclinic 37 Summary and examples… Unit cell calculations (number of atoms in a unit cell) In a crystal, a lattice point is shared by a number of unit cell. The contribution of each atom to a specific unit cell can be determined by the share rate (Sr in this course) The Sr are fractional numbers from 1 to 1Τ8 and are the same for any unit cell and only depend on the atom’s relative position within the unit cell. Atom’s position Sr Inside 1 On the plane 1Τ 2 On the edge 1Τ 4 On the vertex 1Τ 8 In order to adjust for the real numbers of atoms in a unit cell (Ncell), one has to multiply the number of observed atoms (Nobs) by their share rates (Sr) 38 Summary and examples… Unit cell calculations (number of atoms in a unit cell)… On the vertex On the plane Inside On the edge 39 Summary and examples… Unit cell calculations Let’s calculate the number of atoms in a unit cell (e.g. unit cell of NaCl) Na+ Atom’s position Sr Nobs Ncell Inside 1 0 0 On the plane 1Τ 6 3 2 On the edge 1Τ 0 0 4 On the vertex 1Τ 8 1 8 Total 4 Exercise – Use table above to calculate the number of Cl– ions in the unit cell of NaCl. 40 Summary and examples… Unit cell calculations… Let’s calculate the number of Cl– atoms in a NaCl unit cell Atom’s position Sr Nobs Ncell Inside 1 On the plane 1Τ 2 On the edge 1Τ 4 On the vertex 1Τ 8 Total 41 Summary and examples… Unit cell calculations (volume and axial lengths) The density of a NaCl crystal is 2.16 g/ cm3. What is the axial length of a NaCl unit cell? Answer: hints – Calculate the number of atoms per cm3 (via mol.) – From there we can get the number of unit cells per cm3. – Calculate the volume of one unit cell. – Calculate the axial length of one unit cell (Ans. 5.64 Å). Exercise – Show your calculations on the above exercise. 42 Summary: Exercises Exercise 10 August 2020 (Due 08 August 2022) Instructions: Please answer the following questions and send your answers to my email address. In your own words, please answer the following questions. 1. What is crystallography? 2. Define: (a) a mineral (b) a crystal. Calculate the number of Cl– ions in the unit cell of NaCl (shown in the figure). (show all your calculations) Unit cell calculations (volume and axial lengths) The density of a NaCl crystal is 2.16 g/ cm3. What is the axial length of a NaCl unit cell? (show all your calculations) (hints – 1. Calculate the number of atoms per cm3 (via mol.). 2. From there we can get the number of unit cells per cm3. 3. Calculate the volume of one unit cell. 4. Calculate the axial length of one unit cell (Ans. 5.64 Å). 43 Summary and examples… Unit cell calculations: Solutions Let’s calculate the number of Cl– atoms in a NaCl unit cell Atom’s position Sr Nobs Ncell Inside 1 0 0 On the plane 1Τ 6 3 2 On the edge 1Τ 0 0 4 On the vertex 1Τ 8 1 8 Total 4 44 Summary and examples… Unit cell calculations (volume and axial lengths) The density of a NaCl crystal is 2.16 g/ cm3. What is the axial length of a NaCl unit cell? (show all your calculations) 1. Calculate the number of atoms per cm3 (via mol.). 45 Summary and examples… Unit cell calculations (volume and axial lengths) The density of a NaCl crystal is 2.16 g/ cm3. What is the axial length of a NaCl unit cell? (show all your calculations) 2. From there we can get the number of unit cells per cm3. 46 Summary and examples… Unit cell calculations (volume and axial lengths) The density of a NaCl crystal is 2.16 g/ cm3. What is the axial length of a NaCl unit cell? (show all your calculations) 3. Calculate the volume of one unit cell. 4. Calculate the axial length of one unit cell (Ans. 5.64 Å). 47 1 st section ends here 48