 I myself am Dr. Mrs. Preeti Sunil Joshi working as assistant professor in Valchan Institute of Technology, Sallapur. In this session of crystallography we are going to see the symmetry elements in cubic system. The learning outcomes are by the end of this session students will be able to define symmetry and reveal the symmetry elements in a cubic crystal system. The contents include introduction, center of symmetry, axis of symmetry and plane of symmetry. Now we take the example of a cubic crystal and illustrate the elements of symmetry it exhibits. Crystal symmetry refers to the balanced pattern of the atomic structure which is reflected in the external shape. Different species vary in the symmetrical arrangement of faces. These arrangements have certain planes and axis of symmetry. So accordingly the main symmetry elements of a crystalline solid are center of symmetry, axis of symmetry and plane of symmetry. Now we will discuss these elements in detail. Let's consider a cube as shown in the figure here. If the body centered point is considered and the body diagonals are drawn through it, each diagonal connects identical lattice points located at equal distances and in opposite directions from this point. This point acts as a point mirror which generates the second lattice at an equal distance in opposite direction also. Therefore this point is called center of symmetry or inversion point. So center of symmetry can be defined as it is an imaginary point in the crystal that any line drawn through it intersects the surface of the crystal at equal distance on either side. It is equivalent to the reflection through a point. Thus for a cubic crystal inversion point is located at the center. If a cube is rotated about a vertical axis then to complete a rotation of 360 degree we have to rotate the cube through 90 degree. We can see here in every rotation of 90 degree it is not possible to distinguish the cube from its congruent position. See the position of the front face is changing students. So in one complete revolution of 360 degree there are found to be four positions of the cube which are coincident with the original position or in another words we can say that each rotation of 90 degree brings the cube into self-coincidence or in congruent position. So this rotation axis is an axis of symmetry. So axis of symmetry can be defined as a line about which the crystal may be rotated such that after a definite angular rotation about this axis the crystal comes into a congruent position. So now in general if a rotation through an angle of 2 pi by n radians or 360 by n degrees about an axis brings a figure into a congruent position the axis is called an n fold axis of symmetry. Now let us consider different number of folds. So students for n is equal to 1 the crystal is rotated through 360 degree to achieve the self-coincidence and in that case the axis is known to be identity axis and each crystal possesses an infinite number of such axis. Now consider n is equal to 2. So for this the crystal has to be rotated through 180 degree to achieve self-coincidence and in that case the axis is known to be dyad axis. If n is equal to 3 the axis is called triad axis and the crystal has to be rotated through 120 degree. For n is equal to 4 the angle of rotation has to be 90 degree and the axis is termed as tetrad axis. Similarly if number of folds is equal to 6 the corresponding angle of rotation is 60 degree and in that case the axis is called hexad axis. And students as we have discussed earlier it is found that the crystalline solids show only 1, 2, 3, 4 and 6 fold symmetry. They do not show 5 fold or any symmetry axis higher than 6. Yes, can you remember the reason for that? Yes, the reason for this fact is a crystal is not just a solid body but it is one in which the atoms and molecules are internally arranged in a very regular and periodic fashion in 3 dimensional pattern. Consequently identical repetition of a unit can take place only when we consider 1, 2, 3, 4 and 6 fold axis. In view with this discussion the cubic crystal possesses 3 axis of symmetry. So for 2 fold symmetry the crystal is rotated through 180 degree. Here we can see the rotation of 180 degree shows the position of crystal as shown in the right figure and the second rotation of 180 degree achieves the self-coincidence position of the crystal. Two rotations we have to take therefore 2 fold symmetry and axis is called as diodexes. A line which is joining the middle points or center points of a pair of opposite parallel edges provides a diodexes. So students can you guess how many diodexes will be there in a cubic system? Yes, as there are 12 edges in a cube the number of diodexes will be 6. Now consider 3 fold symmetry. The crystal is rotated through 120 degree. Here we can see the change in the position of the cube with rotation of 120 degree. So by third rotation of 120 degree the cube achieves the self-coincidence position hence it is called as 3 fold symmetry and the axis is known as triad axis. So a line which joins the corner points of a pair of opposite parallel edges is known as triad axis. Students can you now guess how many triad axis will be there in a cubic system? Yes, as there are 4 diagonals in a cube the number of triad axis will be 4. Now for 4 fold symmetry the crystal is rotated through 90 degree. Here we can see the change in the position of the cube with every rotation of 90 degree. So by fourth rotation of 90 degree the cube achieves the self-coincidence position hence it is called as 4 fold symmetry and the axis is called as tetrad axis. A line which joins the centers of pairs of opposite parallel faces provides the tetrad axis. So students how many tetrad axis will be there? Yes, one normal to each pair of parallel faces hence the number of tetrad axis is 3. So the axis of symmetry in a cube are thus 6 triad, 4 triad and 3 tetrad. So there are total 13 axis of symmetry in a cube. Now let us see the third symmetry element that is plane of symmetry. Consider a cube and now consider a plane in the middle of the cube and which is parallel to one of the pair of opposite faces. If it is a plane mirror and one half of the crystal is cut and removed it the plane forms image of that half of the crystal in it. That means if we reflect one half of the crystal in the plane the image will coincide with the other half. Yes, therefore the plane is called the plane of symmetry or a mirror plane. So now we can define that when an imaginary plane can divide the crystal into two equal parts such that each part is the exact mirror image of the other then the crystal is said to have a plane of symmetry. So there are three such planes of symmetry which are parallel to the faces of the cube. Further consider a diagonal plane in the cube as shown in the figure here. If the plane is imagined to be a mirror it is readily seen that the prism behind it is the reflection of the prism in contact with it in its front thus it is also a mirror plane. Thus there are six such diagonal mirror planes in a cube as shown in the figure which is formed by a pair of opposite parallel edges. So in the end it may be noted that the full crystallographic symmetry of a cubic crystal comprises of the following 23 elements of symmetry one or one center of symmetry then 13 axis of symmetry which consists of six diode axis, four priad axis and three tetradaxis and nine planes of symmetry of which three are parallel planes of symmetry and six are diagonal planes of symmetry. Thank you.