Crystallography Lecture Notes PDF

For second year chemistry branch  Crystallography It is the science that deals with the study of crystals and crystallized materials that exist in nature, either in a single form or in crystal groups, in terms of the shape of these full-faceted crystals and the relationship...

For second year chemistry branch  Crystallography It is the science that deals with the study of crystals and crystallized materials that exist in nature, either in a single form or in crystal groups, in terms of the shape of these full-faceted crystals and the relationship of the crystal faces to each other, and the study of various natural properties. Crystal It is a part of a solid medium of homogeneous chemical composition, defined by smooth flat surfaces between which are symmetric relations, formed by natural factors and under suitable conditions of pressure and temperature, and has a certain atomic structure. 1- Crystal faces: They are the outer surfaces of the crystal, which determine the shape of the crystal and determine its regular geometric shape as a result of the internal atomic arrangement. It is usually flat surfaces. 2- Edges Characters: They are those lines that result from the meeting of two adjacent crystal planes or faces. 3- Interfacial angles: It is produced from the convergence of any two adjacent crystalline faces. 4- Solid angles: They are the angles resulting from the convergence of more than two crystal faces.  Crystal symmetry Crystal symmetry is the process that results in a certain group of crystal facets taking the same place as one of them. All repetition operations are called symmetry operations. Symmetry consists of the repetition of a pattern by the application of specific rules. Elements of symmetry: 1- Planes of symmetry 2-Axis of symmetry a- Rotation axis of symmetry b- Rotation inversion symmetry axis 3- Crystal symmetry center 1- Planes of symmetry=The Mirror Plane It is the plane that divides the crystal into two halves. It is noted that every face, letter or corner is similar on the other side of it. (Denoted by the symbol m). symmetry operation is reflection  2-Axis of symmetry a- Rotation axis of symmetry It is an imaginary line that passes through the center of the crystal if it is rotated around it, provided that a face, a letter or a corner angle is repeated twice or more during a full 360-degree rotation without displacement. -Diagonal - every 180 degrees. It is symbolized by the number 2. 2-Fold Rotation Axis -Trigonal _ every 120 degrees. It is symbolized by the number 3. 3-Fold Rotation Axis -Tetragonal _ every 90 degrees. It is symbolized by the number 4. 4-Fold Rotation Axis -Hexagonal _ every 60 degrees. It is symbolized by the number 6. 6-Fold Rotation Axis The standard international symbols for 2, 3, 4 and 6 are respectively  1-Fold Rotation Axis - An object that requires rotation of a full 360o to repeat itself has no rotational symmetry. Why are there no five or seven axes of symmetry? A structural unit with crystal symmetry must be able to repeat in space without leaving any voids or spaces. Thus, the two-, three-, four-, and six- symmetrical shapes are repeated to fill the space without leaving any gaps, while the pentagonal, heptagonal and octagonal shapes leave gaps between them when they are repeated, and this does not apply to the regular arrangement in the space of the building units in the three dimensions. B-Rotation inversion symmetry axis It is an axis of rotational crystal symmetry with the inversion through the center of the crystal, meaning that rotation alone is not enough for the crystal to restore a similar position to that of its previous shape, but it requires an inversion process for every facet, letter or corner angle across the center of the crystal. Similarly, we find that there are two-, three-, four-, and hexagonal inversion axes of symmetry. 3-Crystal symmetry center: It is an imaginary middle point in the center of the crystal, around which the crystal faces, letters and corner angles are arranged in pairs and in similar positions in opposite directions and at equal distances from this central point. Meaning that every crystal face, letter, or corner angle located on one side of the center of the crystal and away from it by a certain distance, must be opposite and corresponding to it on the opposite side and at the same distance a similar crystal face, letter or corner angle. The symmetry operation called inversion relates pairs of points or objects which are equidistant from and on opposite sides of a central point (called an inversion center or center of symmetry Crystal form : A group of similar faces in shape, area, physical and chemical properties, and position. Resulting from a group of symmetry elements that have a unified guide in a single crystal. crystal elements 1- Crystallographic axis: (a, b and c vectors). They are imaginary lines that intersect at the center point of the crystal and extend from the center of the crystal faces, edges, or corners in the crystal. The total shape is called Axial cross 1- Crystal systems with three crystal axes, two of which are horizontal and the other vertical: (such as the cubic, tetragonal, orthorhombic, monoclinic and triclinic system groups). 2- Crystal systems with four crystal axes, three of which are horizontal and the other vertical: (such as the hexagonal and trigonal groups). 2- Axial angles: α=β=γ -In three-axis crystal systems, there are three axial angles between the three axes: Between the axes c, b (alpha)., Between the two axes c, a (beta). Between the axes a, b (gamma). -In four-axis crystal systems, there are also three axial angles between the three axes: Between the horizontal axes a1,a2,a3 (gamma = 120°). Between each of the three horizontal axes and the vertical c axis (alpha = beta = 90°).  ‫‪+c‬‬ ‫األنظمة البلورية ذات‬ ‫الثالثة المحاور البلورية‬ ‫‪-a‬‬ ‫‪-b‬‬ ‫‪+b‬‬ ‫‪+a‬‬ ‫‪-c‬‬  ‫األنظمة البلورية ذات‬ ‫األربعة المحاور البلورية‬ ‫‪+c‬‬ ‫‪+ a3‬‬ ‫‪- a1‬‬ ‫‪- a2‬‬ ‫‪+ a2‬‬ ‫‪+ a1‬‬ ‫‪- a3‬‬ ‫‪-c‬‬ Crystal systems :‫األنظمة البلورية‬ a) Cubic system ( a = b = c ),( α = β = δ = 90° ) b) Tetragonal system ( a = b ≠ c ),( α = β = δ = 90 ° ). c) Orthorhombic system ( a ≠ b ≠ c ),( α = β = δ = 90 ° ). d) Monoclinic system ( a ≠ b ≠ c ) ,( α = δ = 90 ≠ β > 90 ° ). e) Triclinic ( a ≠ b ≠ c ) ,( α ≠ β ≠ δ ≠ 90 ° ). f) Trigonal system ( a1 = a2 = a3 ≠ c ) ,( α = β = 90 ≠ δ ° = 120 ° ). g) Hexagonal system ( a1 = a2 = a3 ≠ c ) ,( α = β = 90 ≠ δ ° = 120 °). All lattices, all crystal structures and all crystal morphologies which can be defined by the same axial system belong to the same crystal system. ` C - axis 1- If the c-axis of the crystal has a degree of hexagonal symmetry, then the crystal follows the hexagonal. 2- If the c-axis of the crystal has a degree of trigonal symmetry, then the crystal follows the trigonal. 3- If the c-axis of the crystal has a degree of quadrilateral symmetry, then the crystal follows: The tetragonal if a = b ≠ c The cubic if a = b = c 4- If the c-axis of the crystal has a degree of binary symmetry, then the crystal follows the orthorhombic. 4- If the c-axis does not have any degree of symmetry, then the crystal follows: monoclinic if the b axis is bisymmetric and there is m with (a,c) or one of these two elements. The triclinic if there is a single reversal axis of symmetry. Symmetry formula: The symmetry elements in the crystal can be written in terms of a law known as the full symmetry law using the symmetry symbols, 2, 3, 4, 6 for the rotational axes, m for the symmetry plane, n for the center of representation. 32/3m+n The Stereographic Projection Since crystals are three-dimensional objects, it is necessary to use projections in order to work with them on a flat surface. One such projection is the parallel projection onto a plane. The principle of this projection is A crystal, is placed at the center of a sphere. The normals to each face, if drawn from the center of the sphere, will then cut the surface of the sphere in the indicated points, the poles of the faces. The angle between two poles is taken to mean the angle between the normals n, not the dihedral angle f between the faces. These two angles are simply related as: n (angle of normals) = 180◦− f (dihedral angle). The poles are not randomly distributed over the surface of the sphere. In general they will lie on a few great circles, i.e. circles whose radius is that of the sphere. Those faces whose poles lie on a single great circle will belong to a single zone. The zone axis will lie perpendicular to the plane of the great circle.