Crystallography Class Notes 3 PDF

CRYSTALLOGRAPHY Programme: B.Sc. with Geology (Honours) Course: CRYSTALLOGRAPHY Paper Code: GLY102C CLASS NOTE - 3 Prof. Santanu Sarma, Department of Geology, Cotton University....

CRYSTALLOGRAPHY Programme: B.Sc. with Geology (Honours) Course: CRYSTALLOGRAPHY Paper Code: GLY102C CLASS NOTE - 3 Prof. Santanu Sarma, Department of Geology, Cotton University. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  Zone In many crystals a group of faces arranged with their intersecting edges parallel to each other. Collectively these faces form a zone. The general symbol for zone is [uvw]. All the faces belong to the same zone must have their mutual intersection parallel to a given direction, known as zone axis. The zone axis is passing through the center of the crystal. A zone is denoted by the zone axis. Zones may contain faces from more than one form. Relationship exist among the indices of the faces, that lie in the same zone  All the faces belong to the same zone must have their mutual intersection parallel to a given direction, known as axis of zone or zone axis. The position of the zone axis is expressed by what is known as zonal symbol.  The zonal symbol for a given zone may be obtained from the indices of any two faces lying in that zone.  Indices of every possible faces in a zone must have a definite relationship to the zonal symbol. For a given face with indices (x y z) in a zone having zonal symbol [u v w], the following equation known as Zonal Equation must hold true ux + vy + wz = 0 Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography   Zonal symbol of a given zone may be obtained from the indices of any two faces lying in that zone. Eg. If the faces (1 1 0) and (0 1 0) belong to zone, the zonal symbol [u v w] of the zone can be determined. Certain important mathematical relationship exist among the indices of the faces that lie in the same zone. All faces parallel to a common crystal axis are contained in a zone and have a common zero in their indices. (1 0 0), (0 1 0) (1 1 0), (2 1 0) are all in a zone which has c-axis as the zone axis and zonal symbol is [0 0 1] If the indices of two faces lying in the same zone are added to each other, the sum will be the indices of a face lying between them. The faces (0 0 1), (1 0 1) and (1 0 0) are lying in the same zone, and the face (1 0 1) is lying in between the other two faces. So the indices (1 0 1) can be obtained by adding the indices (0 0 1) and (1 0 0). [(001) + (100)] = (101)] The faces, which lie in the same zone have a common ratio between two corresponding numbers in the indices. The faces having the indices (1 1 0), (2 2 1) and (1 1 1) lie in the same zone and h : k = 1 : 1. Zonal symbol of a given zone may be obtained from the indices of any two faces lying in that zone. The faces (h1 k1 l1) and (h2 k2 l2) lie in the same zone. So the Zonal Symbol [u v w] can Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  be derived from the following equations – u = k1l2 – l1k2 v = l1 h2 – h1l2 w = h 1 k2 – k 1 h2 For a given face with indices (x y z) in a zone having the zonal symbol [u v w] the following equation known as Zonal Equation holds true. ux + vy + wz = 0 Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  Translational Symmetry Symmetry of the Internal Structure of the crystal The external shape of a well formed crystal reflects the presence or absence of the rotational or translational free symmetry. Translational free symmetry elements are axis of rotation, axis of roto-inversion, center of symmetry and mirror plane. In the definition of crystal the fundamental clause is an ordered atomic arrangement i.e. the crystal is a homogeneous solid possessing long range 3-D internal order. The regular external form of a crystal is the reflection of the ordered atomic arrangement of the crystal. Therefore 3D internal order underlay the euhedral form of crystal. So, here we shall discuss the ordered internal arrangement of crystal. Array and identipoints The regular arrangement of identical object each being identically oriented in space may be called an array of such object. If identical objects are regularly spaced along a line, it is called one dimensional array or linear array. If identical objects are regularly spaced in a plane, it is called two dimensional array or plane array. If identical objects are regularly spaced in space, it is called three dimensional array. These arrays represent a type of symmetry known as translational symmetry. In linear array if the identical object moved translationally along the vector b, it successively coincide with the successive identical objects. This vector ‘b’ is called the Unit of Translation along this line. A plane array may be considered to result from the successive translation of an Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography entire linear array along a second vector ‘a’. A 3-D array may be regarded as resulting from the repeated translation of a plane array along a vector ‘c’. So, arrays can be most easily described by specifying 1. Nature of the object being repeated in space 2. Scheme of repetition In order to visualize the array of identical object without referring to the shape of the object, each identical object can be replaced with a point called identipoints. This array of identipoints is called Point Lattice. Def. of Lattice: A lattice is a pattern of points in which every point has an environment that is equivalent to that of any other point in the pattern. A lattice has no specific origin as it can be shifted parallel to itself. In a lattice a regularly spaced point represent the locations of objects, which in chemical structure may be atoms, ions, molecules or ionic complexes. Depending on the scheme of repetition of a one, two and three dimensional array of point, the point lattice is called as 1. Line lattice or lattice raw 2. Plane lattice or net 3. Space lattice Types of line lattice or lattice raw Line lattice consists of identipoints equally spaced along a line. Only one type of line lattice is possible. Line lattice can differ only in respect of the distance between the points. This distance between the points is called the Unit of Translation and is represented by the length of a vector (t1). Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Types of plane lattice or net A two dimensional ordered pattern of identipoints can be created by considering a raw whose points are translationally repeated by a vector t 1 and the whole line lattice being bodily translated by a second vector t2 at an angle σ. So 2-D ordered pattern of identipoint can be created with two different spacing - Unit of translation along the raw = t1 - Unit of translation between the raw = t2 A plane lattice consists of a lattice raw which has been successively translated along a vector. Depending on the length of t1and t2 and the angle (σ) between t1 and t2 five different types of plane lattice or net may arise. 1. Oblique lattice or Clinonet t1 ≠ t2 and σ ≠ 90º Unit cell – Parallelogram 2. Rectangular lattice or Orthonet t1 ≠ t2 and σ = 90º Unit cell – Rectangle 3. Centered Rectangular lattice or Centered Orthonet There is choice of two possible differently shaped and sized unit cells. i) The unit cell is rectangular with an additional lattice point in its center. t1 ≠ t2 and σ = 90º Unit cell – Centered rectangle and non-primitive The resultant net is conventionally described as possessing two orthogonal vectors t1 and t2. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  ii) The unit cell is a non-rectangle with identipoints only at the corner. t1 = t2 and σ ≠ 60º, 90º, 120º Unit cell – Diamond Shaped and Primitive Either of the two unit cell choices will, when repeated indefinitely along two directions, produces the Centered Orthonet. 4. Hexagonal lattice or Hexanet t1 = t2 and σ = 60º, 120º Unit cell – Rhombus 5. Square lattice or Tetranet t1 = t2 and σ = 90º Unit cell – Square Unit cell in 2D Lattice or net The smallest building unit in a 2D pattern is known as unit cell. If unit cell is repeated indefinitely by translation along t1 and t2, the 2D lattice result. Different types of unit cell for five types of plane lattice are mentioned above Significance of space lattice in crystallography If a crystal could be magnified until its component atoms, ions or molecules become distinguishable, it would be seen to consist of 3-D array of these atoms or ions. If these identical atoms, ions or molecules are replaced with points (known as identipoints) the scheme of repetition of atoms/ions/molecules in forming a crystal’s internal structure can be summarized by the space lattice. Definition of space lattice A Space lattice consist of a net or plane lattice that has been successfully translated along a vector which does not lie in the 2-D plane lattice. The vector along which the net is translated successively called stacking vector. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  Or If identipoints are arranged in 3-D then the point lattice is called space lattice. Hence space lattice has three unit of translations in three dimensions. According to French crystallographer Auguste Bravais there are only 14 different types of 3-D array whereby atoms, ions or molecules can be packed together to form a crystal. So there could be only 14 different types of space lattice and they differ with respect to which of the five basic net being staked or successively translated length of the stacking vector as well as its angle to the net. 14 unique types of space lattice also known as Bravais lattice because of Bravais’ contribution to their study. 14 Bravais lattice can be represented by drawing the unit cell for each of them. A single unit cell represents the entire space lattice. Unit cell of a space lattice may be defined as the smallest possible portion of the space lattice with identipoints at its corners which still retains the same point group symmetry as the entire lattice. 14 types of space lattice can be distinguish on the basis of the Unit cell dimensions (i.e. The magnitude of the vectors that outline the edges of the unit cell) Angles made by the unit cell dimensions Type of the unit cell (Primitive or non-primitive) Unit cell dimension along X axis – a Unit cell dimension along Y axis – b Unit cell dimension along Z axis – c Angle between a and b is γ Angle between b and c is β Angle between c and a is α Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  Rotational – Translational Symmetry (Screw Axis and Glide Plane) The 10 symmetry elements are the basis of the 32 point groups represents the pure rotational symmetry. The 3D of the 14 types of space lattice represent the pure translational symmetry. In the internal structure of the crystal, the atoms are sometime so arranged that they can be conform to symmetry operations that combine both translation and rotation. These symmetry operations results the screw axis and glide plane. Screw Axis A rotational axis with a translation parallel to the axis of rotation is known as screw axis. Within a crystalline structure a set of equivalent atoms a1, a2, a3,..... May be related by an operation that combine o A rotation axis of 2, 3, 4 or 6 fold o With a translation parallel to the axis The screw axis symbol consists of a symbol for rotation axis followed by a subscript that represents the fraction of the translation involved in the operation. (Eg. 3 1, 62 etc.) Rotation is anticlockwise. Screw axis are said to be isogonal with the appropriate rotation axis. It means that 31 and 32 screw axis rotate the atoms through the same angle as the 3-fold rotational axis and aligned in the same direction. Rotational axis Screw axis Translational component 2 – fold 21 1/2 t 3 – fold 31 & 32 1/3 t & 2/3 t 4 – fold 4 1 , 42 & 4 3 1/4 t, 1/2 t & 3/4 t 6 – fold 61, 62, 63, 64 & 65 1/6 t, 1/3 t, 1/2 t, 2/3 t & 5/6 t Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Enantiomorphous Pair of Screw axis 31 and 32 41 and 43 61 and 65 62 and 63 An externally observed 2, 3, 4 and 6 fold axis of rotation may results from the presence in the crystal’s internal structure of a normal 2, 3, 4, or 6 fold axis of rotation or their screw axis Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Glide Plane In addition to the repetition of an object by a mirror reflection, a regular pattern can be generated by a combination of a mirror reflection and a translation. This operation is referred to as a Glide plane or Glide reflection. A mirror plane with a translational component parallel to the mirror plane (t/2 or t/4) is known as Glide plane. A glide plane operation involves – Reflection perpendicularly across a plane – Translation parallel to this plane for a vector equal to glide component. Specific glide direction can be identified in 3D patterns and expressed in terms of a set of axes such as a, b and c. Internal order as well as external morphology of a crystal refers to three axes a, b and c. There are three different types of Glide Plane – Axial Glide – Diagonal Glide – Diamond Glide Axial Glide: Axial Glide Planes whose glide component is parallel to a crystallographic axis and equal in length to one half the unit of translation along this axis. Axial glides are symbolised as a, b or c according to whether their glide component along a, b or c axis and equal to a/2, b/2 or c/2 respectively where a, b, c represent the unit cell edge. Diagonal Glide (n): Diagonal glides are planes whose glide component represents the vector sum of any of the following vectors: a/2, b/2, c/2. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography The Glide component could be a/2 + b/2, b/2 + c/2, c/2 + a/2, a/2 + b/2 + c/2 Diamond Glide (d): Diamond glides are planes whose glide component represents the vector sum of any of the following vectors: a/4, b/4, c/4. The Glide component could be a/4 + b/4, b/4 + c/4, c/4 + a/4, a/4 + b/4 + c/4 Diamond glides are so named because they occur in the structure of diamond. In the diamond glide (d) the translation components are one forth of the unit cell edge whereas for a diagonal glide (n) the translational components are equal to one half of the unit cell edge. Presence of periods of translation (as in lattice, glide plane or screw axis) cannot be detected morphologically because the translation involved are in the order of 1 to 10 Angstrom Unit. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography  Space Group When we combine the 14 possible space lattice (Bravais lattice) with the symmetry inherent in the 32 crystal classes (the translation free point group symmetry), as well as two symmetry elements that involves translation (screw axis and glide planes) the result is 230 possible types of arrangement in object in space, called space group. The translation free symmetry combinations are point groups whereas space group define the symmetry and translation in space. Space group represent the various ways in which atoms / ions / molecules in crystal can be arranged in space in a homogeneous array. If we ignore the translation component of 230 space group we would arrived at 32 point groups. Point Group is the translation free residue of a family of possible isogonal space group. Space group have the following characteristics  They are based on one of the 14 space lattice which is compatible with a specific point group.  They are isogonal with one of the 32 point group. Designation of Point group consist of a series of symmetry elements e.g. 2/m 2/m 2/m or 4/m 3 2/m. For each of the specific point group symmetry elements there are a possible space group elements (e.g. 2 -> 21, 3 -> 31, 32 or m -> a, b, c, n or d). The space group symbol is additionally preceded by a symbol that designate the general lattice type (P, A, B, C, I, F, R). So one of the possible symbol of space group that is Isogonal with point group 2/m 2/m 2/m could be P 21/b 2/c 21/a. Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography Distribution of 230 space group among the crystal systems Crystal System Space Group Point Group Bravais Lattice Triclinic 2 2 1 Monoclinic 13 3 2 Orthorhombic 59 3 4 Tetragonal 68 7 2 Hexagonal 27 7 1 Rhombohedral 25 5 1 Isometric 36 5 3 TOTAL 230 32 14 Prof. Santanu Sarma, Department of Geology, Cotton University. Class notes of Crystallography