 Namaste. Myself, Dr. Mrs. Preeti Sunil Joshi working as assistant professor in Valkan Institute of Technology, Solapur. In this session of crystallography, we are going to study the different symmetry operations in crystal systems. Learning outcomes are by the end of this session, students will be able to define symmetry and reveal different symmetry operations. The contents include introduction, different symmetry operations. Students, let us first discuss about symmetry. Now here you can see some images which are very famous and familiar to you. Please pause the video and think that which images are symmetrical and which are not symmetrical. Yes, students you are correct. Images of Motorola, Audi, Mercedes Benz, McDonald's are symmetrical and the images Facebook and Nike are not symmetrical. Very good. Now students, see some other pictures. Image A is a wallpaper, image B and C are the flowers and image D is of butterfly. Now look at these images and tell, are these images symmetrical? Please pause the video and think. Now look at this one, image of day, night, day. Is this also symmetrical? Yes, all these images are symmetrical. Often we don't realize symmetry but we continuously live with the symmetry. So what is symmetry? Symmetry is the consistency, the repetition of something in space or in time. As shown in the examples, we can see in the wall drawing, the petals of flowers, the two sides of a butterfly, the succession of night and day. So to find out the symmetry in these objects, what do you have done? Yes, you have done some operations. Isn't it? We have done some operations. Now let us again look at these images. We found symmetry by repetition of patterns in a wall drawing or in flowers. The wall drawing shows repetition by translation. The flowers show repetition by rotations. The flower on the left shows repetition around an axis of rotation of order 8. The second flower shows an axis of rotation of order 5. In addition, each petal in both flowers shows a plane of symmetry which divides it into two identical parts. The same as it occurs with the butterfly shown on the right. Then the last one, symmetry by repeating events, day, night, day. So students, now how you will define symmetry or symmetrical object? An object is described as symmetric with respect to a transformation if the object appears to be in a state that is identical to its initial state after the transformation. Now next question comes is that why study of symmetry is required in crystallography? Or why we find symmetry in crystals? In crystals, symmetry is used to characterize crystals, identify repeating parts of molecules and simplify both data collection and nearly all calculations. Also, the symmetry of physical properties of a crystal such as the thermal conductivity and optical activity must include the symmetry of the crystal. Thus, a thorough knowledge of symmetry is essential for a crystallographer. A clear brief description of crystallographic symmetry was prepared by Robert Woundery. So our discussion of symmetry in crystallography should begin with a description of crystals. A crystal is a solid composed of a periodic array of atoms that is a representative unit is repeated at regular intervals along any and all directions in a crystal. In crystals, the atoms are arranged in a periodic fashion or rather periodicity of the structure is an essential feature of a crystal. So these regularly repeating blocks are known as unit cells. The dimensions of the unit cells are described by the lengths of the three axes A, B and C and the three axial angles alpha, beta and gamma. There are many choices of repeating blocks in any given lattice. The main principles defining the lattice is that each lattice point must be in an identical environment as any other lattice point and that is and that the individual blocks in the lattice must have the smallest volume possible. And this is the reason why we find symmetry in crystals. The internal symmetry is mirrored in the external shape of the perfect crystals. So we therefore study the external symmetry. In crystallography most types of symmetry can be described in terms of an apparent movement of the object such as some types of rotation or translation. The apparent movement is called the symmetry operation. The locations where the symmetry operations occur such as a rotation axis, a mirror plane and inversion center or a translation vector are described as symmetry elements. Generally speaking we can say that finite objects can contain themselves or may be repeated by the following symmetry operations i.e. inversion, rotation and reflection which is also known as mirror symmetry. Now let us study these operations in detail. The inverse operation occurs through a single point called the inversion center. See this image each part of the object is moved along a straight line through the inversion center to a point at an equal distance from the inversion center. Look at this another example which shows hands left and right related through a center of symmetry. Next is symmetry by rotation. A crystal is said to possess an axis of symmetry if when the crystal is rotated about the axis the atomic arrangement looks exactly the same more than once during one complete revolution. In a rotation operation represents a counter clockwise movement of 360 by n degree around an axis through the object. If an n fold rotation operation is repeated n times then the object returns to its original position. So crystals with a periodic lattice can also have axis with 1, 2, 3, 4 and 6 fold symmetry axis. The other types of axis are not consistent with the condition of periodicity. A 1 fold rotation operation implies either a 0 degree rotation or a 360 degree rotation and is referred to as an identity operation. A 2 fold rotation operation moves the object by 360 by 2 that is equal to 180 degree it is called a 2 fold axis or diode axis that is arrangement looks the same twice in one revolution. A 3 fold rotation operation moves the object by 120 degree and the axis is known to be triad axis. Similarly for tetrad the object is moved through 90 degree and for hexad the object is moved through 60 degree. Although objects themselves may appear to have 5 fold, 7 fold, 8 fold or higher fold rotation axis these are not possible in crystals. Crystals can only show 2 fold, 3 fold, 4 fold or 6 fold rotation axis. The reason is that the external shape of the crystal is based on a geometric arrangement of items. In fact if we try to combine objects with 5 fold and 8 fold apparent symmetry we cannot combine them in such a way that they completely fill the space as shown in the figure here. Now the next symmetry is by deflection. A mirror symmetry operation is an imaginary operation that can be performed to reproduce the object. It is assumed that the object possessing mirror symmetry is cut into 2 halves and placing one of the halves on a mirror the reflection in the mirror reproduces the other half of the object. The plane of the mirror is an element of symmetry referred to as a mirror plane and is symbolized with the letter M. The rectangle on the left has a mirror plane that runs vertically on the page and is perpendicular to the page. The rectangle on the right has a mirror plane that runs horizontally and is perpendicular to the page. The dashed parts of the rectangles below show the part of the rectangles that would be seen as a reflection in the mirror. The rectangles shown above have 2 planes of mirror symmetry. Students now tell can we consider this line of symmetry for this given rectangle? Please pause the video and think for a minute. So, let us check your answer. If we cut the rectangle along a diagonal such as the labeled M as shown in the left diagram reflected the lower half in the mirror then we could see what is shown by the dashed lines in the right diagram. Since this does not reproduce the original rectangle the line M does not represent a mirror plane. So, students note that a rectangle does not have mirror symmetry along the diagonal lines. Students mark the point that reflection symmetry must satisfy certain conditions. Reflection symmetry occurs when a line is used to split an object or shape in halves so that each half reflects the other half. Sometimes objects or shapes have more than one line of symmetry. In addition to these there exist four hybrid operations. They are rotoreflection, roto inversion, screw translation and glide reflection. Rotoreflection is a combination of an n-fold rotation followed by a reflection in a plane perpendicular to the rotation axis. Roto inversion is a combination of an n-fold rotation followed by an inversion. In screw translation n-fold rotation axis is coupled with the translation parallel to the rotation axis and in glide reflection a mirror plane is coupled with a translation parallel to the reflecting plane. So, before discussing these operations let us first understand the difference between reflection and inversion. On students now consider this two-dimensional object and imagine its reflected and inverted image. Pause the video and think for a while. Yes, students now look at these images. Which one is reflected and which one is inverted? Yes, students you are correct. Image to the left is reflected image and to the right is inverted image. Reflection occurs along a plane whereas inversion produces an inverted image through an inversion center. If we draw lines from every point on the object through this inversion center and out an equal distance on the other side. So, in both the cases we find symmetry, congruent pairs. Now look at these illustrations of four hybrid operations. The first one, roto reflection in this object is rotated through certain angle and then reflected through a plane perpendicular to the rotational axis. Second symmetry operation is roto inversion in this object is rotated through certain angle and then inverted through a point on the rotational axis. When the third symmetry operation screw translation it is an operation in which a pattern is generated at regular identical intervals and look at the footprints and the last one is the glide reflection. It is a two-step process reflection followed by translation and now look at this footprint pattern which shows translation combined with a reflection. Thank you.