Download Subtitles for Mod-01 Lec-2 Symmetry in Perfect Solids

from now on for the next few lectures we will be dealing with perfectly crystalline solids as prototypes of condensed matter as we have already seen such perfectly crystalline solids have a good deal of symmetry and the symmetry plays a central role in the understanding of the behavior of such solids we have also seen in lecture one that when there are phase transitions there are symmetry changes so we need a method for describing quantitatively in a standard sort of way the symmetry in perfect solids so we first talk about what is known as a symmetry operation what do we mean by a symmetry operation a symmetry operation is something we will come to various kinds of symmetry operations later the general definition is if you perform a symmetry operation on an object then the operation will bring it into a new configuration but this new configuration cannot be recognized from its original configuration so if you perform the symmetry operation and ask someone to observe the object before and after he would not be in a position to say that this operation has been performed so this object goes into a symmetry related new configuration examples of such con operations are translation translation of a row of regularly situated objects by an integral number of the repeat distance there is also another class of operations namely rotation for example rotation of a square if you take a square and rotate it about an axis perpendicular to its plane by an integral multiple of pi by 2 something like this so if i have a square like this and rotate it by pi by 2 pi by 2 it goes to a new symmetry related configuration but the square looks the same so there is no way of saying that there has been such a rotation by either pi by 2 or pi or 3 pi by 2 and so on the third class of symmetry operations are reflections if you have a symmetric object and if you have a plane of symmetry then if you you can reflect the object at this plane of symmetry so that the object gets comes here on the other side and doesn't look different the fourth category of symmetry operations or inversion of an object about a point so you have translations by regular repeat distances then rotations so these are also known as symmetry axis because a rotation is always performed about an axis of rotation so then a plane of symmetry which causes reflection symmetry is known as a mirror plane so and then inversion is always about a point which is known as the center of inversion so these are all examples of symmetry elements let us go to translation symmetry you can see from figure 2 1 where we have a regular arrangement of identical objects in three dimensional space and a b and c are the translation vectors along the three mutually orthogonal directions so you have each object can be a set of atoms or molecules arrange they are arranged regularly at various points along this three dimensional array so that obviously has translation symmetry so if you take this periodic distances periodicity along the three axis which we will call x y and z axis if they are a b and c respectively and if r is the position vector of a given lattice point then the translational periodicity requires that any other lattice point in this array is given has the position vector r prime which is n one a plus n two b plus n three c starting from the origin of course so n1 n2 n3 are integral numbers integers which can be positive you can go in the positive x direction or in the negative x direction so it will be plus r minus one plus r minus two you can go through two repeat distances you can go by three repeat distances and so on you can have an infinite number up to infinity for an infinite array of lattice points so that is what is being shown in figure two one next we go on to rotational symmetry this is translation now for example if you take a rectangular object and if you rotate this rectangle about an axis perpendicular to its plane and passing through its geometric center by an angle pi then it reaches an identical configuration so obviously for a rectangle a two fold rotation axis about this point is easily seen to be a symmetry operation since this rotation brings the rectangle to a new configuration which is identical to the initial configuration so if i have an equilateral triangle so it has a two fold rotation symmetry about this point a rectangle if you have an equilateral triangle this has a three-fold rotation symmetry about this axis so this is an axis passing through the centroid of this equilateral triangle so if you have a square so then we have a four fold rotation symmetry four full rotation axis about the geometric center of the square so this is two fold axis is a three fold axis this is a four fold rotation axis why do i say twofold because if i rotate by pi it comes to a different configuration and then you rotate again by pi it comes back to the original so you need two such rotations in order to bring it to the original configuration here you have to do the symmetry of rotation three times in order to bring the triangle into a self coincidence similarly you need four such rotations to bring the square into self coincidence and if you have a regular hexagon like this this has a six fold rotation symmetry about its geometrical center so its a six fold axis for a regular hexagon so you can see that a two-fold a three-fold four-fold and six-fold axis are the common rotation axis in solids which are perfectly periodic and have a translation symmetry we do not have a 5 fold rotation axis because i do not want to discuss this but i will simply state that this is not consistent with translation symmetry so the rotational symmetry a fifo rotation symmetry if it is present in a perfectly crystalline solid then it will violate the translation symmetry and so this is not present so what we have are only two four three four four four six four one fold of course is trivial it is just nothing is just a rotation by pi 2 pi so it obviously brings into self coincidence but for just completeness we will also include a one fold rotation axis so one two three four and six so here n fold axis where n equals 1 2 3 4 and 6 these are the rotation axis symmetry axis which are consistent with the translation symmetry of a perfectly periodic solid so figure 2 2 shows what is known as a stereographic projection by stereographic projection what we mean is we project on a sphere like this and the axis rotation axis a two fold axis for example is shown like this as in this case so you have an object which is repeated by the two fold rotation from here to this similarly a three-fold axis is shown like this and the object is repeated like this here it is a 4 fold rotation and here it is a 6 4 rotation so these 4 axis are shown by stereographic projection next we consider a plane of symmetry or a mirror plane a plane of ceremony or a mirror plane reflects the symmetric object as shown in the figure two three a so you can see that there is a mirror plane and that reflects an object about the mirror plane so takes the object into a new configuration which looks identical to the initial configuration however the lateral inversion present in the case of a mirror causes a change for the case of a right handed coordinate system will get inverted at the mirror plane into a left handed coordinate system so because of that the change in the handedness are so such a symmetry operation is known as an enantiomorphic symmetry operation this also true of a centre of inversion where the object which is symmetric with respect to an inversion about the geometric center so there is a center of inversion again an inversion operation also changes the right handed into your left handed system and vice versa so this is shown in the figure 2 3 b where there is a center of symmetry and this gets inverted by at the origin and there is an accompanying change in the handedness so in all these cases except translation in the case of the all these rotational axis there is an axis of rotation and that means that any point which lies on the axis of rotation is not changed so you have to have some other it should the point should be outside in order to for it to go to a new configuration so it leaves all this symmetry operation a rotation axis whether it is one fold two fold three four four fold or six volt leaves all the points which lie along this axis of rotation invariant they do not change similarly in the case of a mirror plane i will show this like this a mirror plane again all the objects lying in this mirror plane do not change the point has to lie outside the plane in order to get reflected and go to a new configuration any point which lies on this mirror plane does not undergo any change by the operation of the mirror symmetry therefore all these points are left invariant by the mirror plane similarly in the case of inversion the inversion is about the origin and nothing happens to the origin when you perform an inversion operation so at least this centre of symmetry remains invariant so all these three symmetry operations rotations reflections and inversion all these leave at least one point in variant in space therefore these rotation these rotation reflection and inversion operations are collectively known as defining the so called point group of the solid the solid is said to possess by virtue of the symmetry elements they belong they are said to form a point group the point group classification is very important for describing and understanding the effect of symmetry on the properties and characteristics of crystalline solids so we have it can be shown that there are 32 point groups in three dimensional space there are only thirty two combinations of the symmetry elements which leave a point in variant in three dimensional space we say that they are they form the so-called point group at this point we should understand what is meant by the term group here now this group is a mathematical group of symmetry elements now what does it mean when you can we say that a set of elements constitute a group this is important for us to understand so let us now go on to describe the so called group postulates group postulates what constitutes a group any collection of elements do not form a group so if you have a collection of elements let us write them as a b c etcetera so this is the collection of elements they can be objects there can be a collection of symmetry elements they are said to constitute a group if and only if the following requirements are satisfied what are the various satisfaction requirements to be satisfied in any such collection in order for them to form a group we must first specify what is the rule of combining these elements the rule of combination combination rule should be specified for example in the set of elements known as integers numbers 1 2 3 like that plus and minus positive and negative integers the rule of combination can be for example addition so that you can write 1 plus 1 or 1 plus 2 1 plus 3 like that you can combine the various elements what do you get if you say 1 plus 1 if you combine the element with itself you get 2 which is also an element of the group if you combine 1 and 2 you get 3 which is also an element so the general rule is that if you have a b by a b i mean the combination of a and b elements and this should result in a an element c which should also be a member of the group for any a and b so this is known as the closure property any two elements a and b if you specify the rule of combination and then combine a and b then they should lead to a third element c in general which should also belong to this collection then only this is one property which has to be satisfied for the elements a b c etcetera to form a group the next property which should be satisfied is the the following a b times c should be equal to a times b c this is associative the combination is associative so here for example you have 2 and 3 gives you 5 and then 1 that is 1 and 2 is 3 and then you combine it with 3 you get 6 and same is the result here 2 and 3 is 5 and then you combine it with 1 you get 6 again so you get the same result so this is very necessary for the collection to form a group but it is not necessary that a b should be equal to b a this is not necessary need not they can be but if they are then in that specific case if a b equal to b a then this group for all the elements then it is known as an abelian group then you must always have the third property which is that there should always be in this collection of elements a b c there should always be an identity element so the identity should be necessarily an element of this collection then in addition there is a fourth property which is known as the inverse of an element if you have an inverse that means suppose we have a general element x in this group of collection of element then you have also x to the power minus one so that the x x inverse equals x inverse x equals i that is the definition of an inverse so the inverse of an any element should also be a member of the group for example if you have the collection of positive integers then the corresponding negative integer is the inverse of a given positive integer this is also true these group postulates are satisfied by for example a 4 fold axis let us see let us consider a four fold rotation axis and see how the group requirements are satisfied in this case so taking a four fold axis i have the collection of elements in this is if i take a square like this i have a b c and d a rotation by pi by two takes it pi by two rotation anti-clockwise brings rotates the square and brings it to a configuration where a goes here b goes here c goes here and d goes here if we didn't have these letters then these two are indistinguishable so what happens let us see what happens to the group postulates so the various operations are associated symmetry operations associated with this rotation so let us say this rotation by pi by 2 let us call it r is pi by 2 rotation anticlockwise about geometric center of square we call it r then if i rotate it subsequently by another pi by 2 it will again go into a self coincidence so that is no shown as r square a second rotation we can go further and make a third rotation by pi by 2 which we call r cube and then r to the power 4 brings it back to the original configuration and i call it i so these are the various r r square r cube and r to the power 4 equal to i these are the symmetry elements are the elements of this group which corresponds to a square the symmetry operations of a square so we can easily check that all these group postulates are satisfied in the case of these four elements for example a rotation by pi by 2 in an anticlockwise direction its inverse is a rotation by the same angle pi by 2 in a clockwise direction so r inverse for equals r cube r square inverse equal to r square you can easily verify these things so this is in all these cases we have the situation a b equal to b a or r square is r cube which is also equal to r square r so it is an abelian group so we can see that this is the point group corresponding to the four fold rotation axis acting on a square we call it a cyclic group because these elements are forming a set of elements which perform a cyclic set of operations it brings his back into a cycle into the original configuration so this symbol c means cyclic cyclic point groups and to say it is a four fold rotation we call it c four we also can represent it simply as four point group four so in this way we can see that we have the point groups one two three four and six which can be written also as c one c two c three c four and c six these are two independent notations one is known as the international notation this is known as the shown flies notation the second notation shown flash notation is usually preferred by spectroscopies whereas the international notation is preferred by x-ray crystallographers for example so these are the five rotate pure rotation point groups there are no point groups with symmetry rotational symmetry higher than 6 and as we already saw a 5 fold rotation is not allowed because of translation cement so these are the five point groups which are the most basic point in addition we have a plane of symmetry a mirror plane which is shown as m for a mirror or it is also called it is also a cyclic group as you can easily check it is also called c s s being the letter for spiegel the word spiegel in german which means mirror so that is another point group which is cyclic which is also a sixth point group in addition to these five and then you have an inversion which is shown as simply i or it is also shown as r it is known as one bar so a one fold axis with an inversion center gives you this or this also reflect written as ci in schwannfels notation so the pure rotation axis mirror planes and symmetry have a symmetry center of symmetry constitute the cyclic point groups which are which may be represented as 1 2 3 4 6 m 1 bar or equivalently c 1 c 2 c 3 c 4 c six c s and c i these are all the cyclic point groups we said that there are 32 point groups of which now we have enumerated seven how do you get additional point groups additional point groups are got by combining these various point groups so you can combine any two point groups in a way as to satisfy group postulates then you can derive new find growth what do we mean by combining point groups two groups can be combined by forming the so called direct product of two groups of any two point groups leads to combinations of the different rotation axes among themselves or the rotational axis with mirror symmetry or the rotational axis with a center of inversion all this would give rise to new point groups let us see one example let us take a two fold rotation axis which gives rise to the point group known as c 2 or simply 2 let us take this how do you form the suppose we take a combine this with a mirror plane or c s suppose we combine these two points so what are the elements in this case c two i have the identity and then a c two axis c two square will be i so we do not have anything else and here in the case of a mirror we have the identity and a mirror if you repeat the mirror operation it comes back to identity so these are the two elements in a two-fold point group two and these are the two elements in the point group m so suppose i form the direct product direct product means combining the various elements together all the elements so i have the products like i into i which obviously is i identity product of identity with identity is identity itself similarly identity into c two is just c 2 identity with m is m and then you have c 2 and m if you combine a 2 fold rotation axis with a mirror plane then you can easily verify that this leads to an inversion center so these are the four elements derived by the direct combi direct product of these two so we have got a new group which has the elements i c two m i i means the center of inversion so these four elements give rise to a new point group which is represented as two by m a two fold rotation axis with a mirror plane lying perpendicular to it which this combination leads to an inversion center automatically so this is a new point group it is also known as c two h the two fold axis is taken to lie along the vertical direction so that the mirror plane perpendicular to it is a horizontal mirror plane so c two h means a c two axis which is combined with a horizontal mirror plane so that is the shown fly's notation for 2 by m this is the international notation similarly we can also combine a 2 fold axis with a vertical mirror a vertical axis with a parallel mirror lying parallel in the same plane so that is a vertical mirror plane then you can see that the combination of these you can verify that this will give rise to a second set of mirror planes which are like this so you have two mirror planes so two vertical mirror planes which are parallel to the rotation axis so you write the m next to the two fold axis this is known also as c two v a vertical mirror plane similarly we can have a three fold axis with a vertical mirror which is known as c three v a four fold which is known as c four v and a six fold which is known as c six so these are the additional combinations which you can obtain by combining the rotational and plane of symmetry now you can also combine the various rotation axis this kind of combination will give rise to a twofold axis combined with another two fold axis perpendicular to it will also lead to a third set of third two fold axis if you have one two fold axis like this another like this then that will generate a third two fold axis perpendicular to it so this is two two two similarly you can have three two four two two and six two two so you have dihedral axis so these are all known as the d n the dihedral point groups so this is known as d two d three d four d six additional point groups can now be obtained by combining these groups dihedral groups with mirrors or inversion center so we get in this way additional point groups which have the following nomenclature you can add mirror planes lying perpendicular and mirror planes lying perpendicular to the principle rotation axis two fold axis here so you will get d two h where the mirror is perpendicular to the z axis the vertical axis therefore it is a horizontal plane so this is also this is can also be written in this way similarly i can have h d four h and d six h we can also have d and v these are the d n h we can also have d and v and we can show that we can also add mirrors diagonal mirror planes diagonal mirror plane so it is called d and d in addition to all these we have the so called cubic point groups the cubic point groups are symmetries associated with a tetrahedron or an octahedron both of which can be inscribed in a cube so it is known as a tetrahedral point group or an octahedron t for tetrahedron and this is for octahedral tetrahedral and octahedral symmetries are implicit in a cube in cubic symmetry and you can also have additional symmetries like t h and o h by combining the elements of tetrahedral or octahedral symmetry with horizontal mirror planes or we can have other symmetries which we will see in greater detail now these symmetry elements all together this gives you the 32 point groups so all these are shown in stereographic projection you