 Good morning everyone, initially I will start with the questions which are on crystallography. Then there are few questions on quantum mechanics which was on the earlier on the Saturday session. So then I will talk about them thing, those things but let us start with the questions on the crystallography portion. So the first question is from MGM College of Engineering, Dandere. This is a question which was not very clear to me but I thought I made some sense out of the question. So let me just repeat. 3D Bravais lattice should have 3 directions and 3 primitive vectors but in FCC Bravais lattice primitive vectors i, vector j, vector k are missing while representing a, b, c respectively. In slide 12 I saw this particular thing. See the things that it should have 3 directions and 3 primitive vectors, it does have 3 primitive vectors. So what I had written there was a, b, c. So what I had written was that your, the vector a is a by 2 j plus k then I have written b as a by 2 k plus i and c as a by 2 i plus j. So these are indeed 3 vectors, it involves all the 3 directions i, j, k there is absolute no ambiguity. My feeling is that this question has been asked that why i is missing here, why j is missing here, why k is missing here. Well that is natural because you know eventually you have to look at 3 different directions. These 3 directions could be mixture of these 3 directions, you know any of these vectors. For example if we take simple cubic, I write a is equal to a i, there is no j and k appearing here. Similarly we write b is equal to a j and c we write as a k. So it is not necessary that in every vector all the 3 directions should appear, it so happens that in b, c, c it does appear but in simple cubic it does not appear and in f, c, c it does not appear. So I do not see any contradiction as far as this question is concerned. In fact you can always visualize this particular thing because if you are going half the distance along the y directions and z direction it means you are looking at that particular point which is centering the y z plane. This is a particular point which is centering x and z plane. This is the point which is connecting, which is the center of x y plane. So I do not see any contradiction, I hope I have understood the question. The second question is from Kavi Kuluguru Institute of Technology in Nagpur. The Miller index of a plane is never infinity but can be 0, is there, can it be negative? Yeah that is actually a good question because I have unfortunately could not discuss that particular aspect of negative coefficient. You can always have a situation in which you know a particular plane intersects not just x, maybe intersecting let us say x direction and y direction or a direction and b direction in the positive side but in the c direction it has an intercept along the negative direction. So if it has a negative direction the intercept in that particular direction will have negative value. For example you can have intercept which is let us say minus a, a and a. So this intercept is minus a it means along the x direction or along, let us just take cubic to make life simple. Let us assume that this is simple, this is a simple cubic. In that case you know the intercept of this particular plane is along in the x direction in the is the minus a while along the y direction and the z direction it is along the plus a. So this is perfectly possible you can always imagine a particular plane which is intersecting x direction in the negative a direction and not in the positive a direction. In the intercepts will be minus 1, 1, 1 you take reciprocal this becomes minus 1, 1, 1. When you write this particular mirror indices you put actually not negative sign here but on the top of you write bar 1, 1 so when you write a plane like this or for that matter in general I write a plane hkl or I write a plane h let us say kl it means in this plane has intercept along the a direction in the negative direction here it has intercept or negative intercepts along b and c direction and a positive intercept along a direction. So having a negative intercept is perfectly possible and when we represent the mirror indices of these planes I put a negative sign or a rather bar on top of that number indicating that that particular intercept has happened in the negative direction. So there is no ambiguity as far as this particular aspect is concerned. Now this next question is from Vulture Institute of Technology, Solaapur the iron has BCC and has one atom basis means it gives number of atoms per unit cell or anything else but the BCC number of atoms per unit cell is 2 that is perfectly correct. See remember if it is a primitive unit cell primitive unit cell will always have one particular point per cell but when I am saying 2 points per unit cell actually I am not using primitive unit cell but I am using a conventional unit cell and remember I have always been mentioning that in the case of cubic lattices normally we always take conventional cell as the simple cubic cell unit cell it means essentially it is in the form of a cube when it is in the form of a cube they are no longer primitive vectors. See when I write a is equal to ai b is equal to aj and c is equal to ak they are the primitive vectors of simple cubic lattice they are not the primitive vectors of BCC lattice okay but on the other hand because of my convenience I want to talk of a cubic unit cell therefore I am treating this particular lattice as sort of simple cubic lattice and I am using the same unit cell which is cubical in the shape. In that case if I do I will get 2 points per unit cell but then we should realize that this unit cell is not really a primitive unit cell but is a conventional unit cell. So I hope I have answered this particular question then there is a next question about random stacking and poly typism this is also again interesting question in fact again this I have avoided talking about it but let us just understand it. See when we are talking about close packing structures we talked only of two type of structures which are a b a b a b c a b c what it means that a b a b a b keeps on repeating every time it is a b c a b c a b c which keeps on repeating every time like that this we said is HCP this we said FCC. Now in principle these are not the only possible type of structures you could in fact have structures which I mean all that we want that the two layers that we put one on the top of other should not have the same symbol. So I will not have possibility of let us say a b c when I put let us say a b c after c I could put a or b after c I cannot put c again because then it is no longer close packing. So you can put let us say c b then you can put again c then you can put again a you can put again let us say b see all that you want that the subsequent two numbers cannot be identical. So if I put b on the top of b then that does not form a close packing all of them will be close packing structure but if you look at this particular case this appears that along the third direction we have essentially a random stacking because you have a b c then you have b c a then you know you can have a very stable random stacking and then many times we say that this particular material is crystalline or regularly arranged in two dimension but a sort of random in the third direction. So this is what we meant by random stacking that you could always have random stacking in the third direction whether a really a crystal system exists like that or not that is a different question okay but theoretically speaking it is always possible to have these type of behaviors. Now what is polytypism in fact in polytypism you know what happens you know some of such type of random I mean not I will call random periodic structures are found where periodicity is obtained after a very very large number of such stacking for example you could have this type of thing then it can repeat again exactly in the same fashion a b c b c a b. Now what you will call because repetition here is of this particular type of thing so essentially in the c direction your periodicity range has become much larger now they 1 2 3 4 5 6 7 so that 7 layers after that the periodicity is maintained. Now it appears that certain materials have been observed to have such type of stacking in which the repetition in the c direction where this particular structure gets repeated happens at a very very large you know sort of distance okay many times you have something like 100 layers and after that it starts stacking. Of course there is always a question why a particular material you know sort of slices in a particular structure which has to do with the inter atomic forces and minimization of energy that is a different subject all together. Now in I mean these type of things where you have a structure which is repeating after a long time is generally called polytypism so there is always a question why this polytypism I mean what are the type of forces which can give rise to polytypism. Now people feel that is not really I mean I am again I am not expert in this polytypism I can give only from whatever my sort of general information is concerned. These polytypism is generally believed to be not because of really a forces because normally these forces will not may not be that long range okay so generally we believe that there are certain type of defects or certain other type of things which actually give rise to this type of periodicity. So basically the moral of the story whatever is I am saying is that we have considered only two type of stacking when we are talking of close packing structures one is A B A B A B type here repetition occurs after two cells A B then repeats A B A B A B then we have a second type where repetition occurs after three so you have A B C A B C A B C A B C but then in principle it is possible that you have various other type of stackings at least theoretically possible the only thing that you want the two symbols may not be should not be exactly same one after another okay so you can have for example A B C B C A A B and for example this seven layers could repeat after that or it may not repeat at all if it does not repeat at all then we will call it random stacking if they repeat after a very large number of iteration we will call these things as a polytypism so that is what I would like to say. Now there is another question this is from E wrote the Kongu engineering college E wrote it says clarify crystallite grain and particles this also I think I will spend some time because I generally realize that many people are sort of confused even our students are confused see normally an ideal structure is a crystalline structure okay but as we have said that many times you do not have a material I mean we do not use the material which is actually single crystal because it is I mean difficult or it is more expensive to make single stand materials in many cases okay if you do not care about making it perfectly crystalline material you know you make what we call as a polycrystal in that case you know these are comparatively simpler to make. So when we say a polycrystal essentially it means that it consists of a large number of single crystal crystalline zone which amongst themselves are ran oriented maybe randomly maybe in some preferential way but it does not consist of a single single crystal it is a combination of a large number of single crystals. So let me draw a just a let us say top view of let us say a particular material okay the material could be let us say going deeper into inside it now if you look at the top okay for example here you could have a particular type of single crystal material and let us suppose the x axis of that let us just take cubic case to make it life or simple to understand it. So maybe this is x direction this is y direction this is z direction y direction and this will be z direction like this alright. Now you have another neighboring particular region which is also single crystal in its own way but x axis is not in this particular fashion but x axis let us suppose in this particular direction and then y direction x like this and z is like this okay the third one could have another type of orientation with respect to so this could consist of a very large number of these single crystals I mean I am just giving some arbitrary structure okay in which all these individual particles are single crystals but they are rotated it could be randomly it could be in a particular artificial fashion arbitrarily with respect to the original let us say single crystal like. So now such type of materials we generally call as a polycrystalline material because this particular material consists of very large number of single crystals so that is why word poly it is a polycrystalline material and generally where you look at one of the single crystal within this polycrystalline this we generally use the word grain a grain means this is a single crystalline region within a polycrystalline material these grain sizes the size of this particular grain is very very important at least in this my area of recording okay one would like to control the grain size so that you obtain a particular type of magnetic properties so these grain sizes are extremely important as far as application this particular material is concerned so this is what we mean by a grain in a polycrystalline material a small single crystalline region could would which is alike a single crystal okay would be called a grain and of course the grain sizes could be in general different when we talk of the grain size we have to talk about let us say the average grain size we could talk about their distribution of the grain sizes there are so many things that we talk about the grain sizes when we say particles particle essentially means that you are actually for example these days we talk of a lot of nanoparticles essentially it means that your material is really very very tiny here the material is big this is of let us say macroscopic dimension but these grains could be exceedingly small for example they could be nano crystalline where the typical sizes are of the order of let us say 10 nanometers or 15 nanometers or they could be much bigger grains where they could be of the order of microns okay but depends again on the type of material that you are using okay but when we are talking of particles particles actually the shape the size itself is a very very small so if I take let us say a nanometer particle it means I mean typically nanopowders that typically the size of the particle itself would be of the order of 10 nanometers or 20 nanometers okay so particle is really a very very tiny thing is like you know one of the particle of a powder let us say for example okay of course this particle size could be bigger they may not be nanopowders okay it could be let us say micron size particles powders where you can have the typical you know particle size of the order of microns. Now again I mean chances if you are making a very fine powder which is the nanopowder chances that they are grains within this powder may not be there it may be there so whether these particles are polycrystalline or a single crystalline that is a different issue but basically when we are saying powder powder essentially means that they are really a tiny material just like I mean a normal talcum powder that you normally use at home so you see that it consists of a very very fine particles so these are just the particles. Now I just want to I think somebody has asked this particular question about the orientation see what happens or texture what could happen that as I said that these grains here in principle could be randomly oriented with respect to each other but many times it happens that these are not randomly oriented with respect to each other for example this particular material could be what we call as a oriented material it could happen in such a fashion that your C axis of all these grains is always normal to the film plane I am talking of thin film or you can talk of any other material okay that they could be all in this particular direction if that happens so their x y axis could be randomly with respect to each other but C axis is same then we call that this is an oriented material or oriented film okay it is also possible that there is a some preferential alignment but not a complete alignment okay it is you know if you look at the number of grains for which C axis is oriented like this maybe much larger in comparison to grains for which C axis is oriented like this or like this or like this or like this so we have some texture it means the and the the grains are not really random with respect to each other they are in orientation is not random with respect to each other okay so what we call is that these materials are textured. Okay now let us go to the next question next question is differentiate crystals and pristlites I mean when you say crystallite generally we always mean small particles or crystallite could also mean a grain it depends on the reference to the context the next question is from I think Sheikhao Ganan engineering college how the Hkl values of polycrystalline sample material is to be found out see whenever we are talking of Hkl value we always assume the material is to be single crystal okay when we are talking of polycrystalline material okay it essentially means that the grains are oriented with respect to each other in fact you still use the same Miller indices to define what is the orientation of one particular grain with respect to another. So I mean when we define Miller indices it is always with respect to a single crystal material if it is polycrystalline material individual planes of these particular grains will be different like the example which I had given you just now. What is the physical meaning of Miller indices physical meaning I have tried to explain this is basically to identify various planes. Next question is for an FCC lattice is there a easy method to find out the coordination number of course I mean coordination number can generally I mean I do not think there is a I mean a clear shortcut way of you know you just look at this one particular atom and find out how many nearest neighbors it could happen it could have that is what is called coordination number. So as you can see in the FCC material if you take any particular point you will find 12 point which are nearest neighbor to it so in fact in the case of FCC let me just draw this particular picture. So if you take this particular point then you will find the nearest neighbors are the face centers because this is at a distance of a by root 2 while this is a distance of a so this is the nearest neighbor. So you will find that this particular point there are four faces 1 2 3 4 to which this is common so all these four faces will have centers so there will be four nearest neighbor four will be in this direction and four will be in this direction so it has 12 nearest neighbor so the coordination number is 12. So this generally you know when I think only by visualization one can find it out I am not sure whether there is any other method of this. The next question is from Raigarh MES Pilliz Institute of Information Technology for representing a plane direction is it correct to take the reciprocal of the given indices rather than dividing by the highest integer yes that is the standard way of taking the reciprocal okay this is the way we do it I mean that is the convention we follow so I mean we cannot say why we do this way this is the way it is done. The next question is from JP Co Institute of Information Technology Noida in general for cubic lattice Hkl is perpendicular to Hkl plane how to prove it of course it is a little longer longish proof but you know it can you know it is true that you know Hkl direction is perpendicular to Hkl planes or rather normal to Hkl planes normal to Hkl planes only for cubic lattices. In fact in general if you take any arbitrary lattice you will find its G vector which is normal to the plane Hkl if you write G as HA plus KB plus LC where Hkl are integers so this is a translation vector in the reciprocal lattice then you will find that this turns out to be normal to Hkl plane. So this is the most general way of representing looking at the angles between these things only for cubic lattices it can be shown that Hkl direction turns out to be normal okay this what you have to do in fact you know I do not remember the exact derivation you take this particular Hkl plane okay then this particular intercept here is generally believed to be A by H if this A direction this is B by K and this is C by L then you write take you write this particular vectors A by H B by K and C by L then find out what is this direction this direction can be written in terms of this direction then you look at the direction HA let us say HA plus KB plus LC see remember this is the direct vector this this was reciprocal lattice vector okay then show that this is perpendicular to this then you write take this particular direction show again that this is perpendicular to this then take this particular direction show again that this is perpendicular to this so you can show that this eventually turns out to be normal so there is a standard way of deriving it okay I mean this you can try it yourself if you have any difficulty you can probably contact me later we will try to sort it out. The next question is that how to calculate grain and particle size from XR which is also a very interesting question which again something which I have not talked about in the XRD but because this may be of general interest so I thought I will anyway mention about this particular thing see generally if you take a really infinite crystal and infinite solid the X-ray piece that you will be getting should be sort of delta function okay normally this is never a delta function for example by uncertainty principle the wavelength that you are getting there is always a small distribution so in principle this particular line will have what we call as a natural line width okay because new self or lambda itself has a small amount of width because of uncertainty principle so you cannot when you are talking monochromatic beam it cannot be 100 percent monochromatic there is always going to be you know certain other wavelengths mixed up so therefore there will be always what we call as a natural line width of this particular line okay but generally when you are taking a finite sized material okay then you find that there is a further broadening whether many other reasons of broadening but let me put it like that if we assume that this particular line is further broadened it becomes like this okay and this is mainly because of the finite in fact you can show that the finite size of a crystal will broaden this particular line then you look at the full width at the half maximum it means you go half the way here and look at what is the width okay there is a well known formula which is called Sherer formula okay which relates this particular width to the average green size or average particle size depending on which material that you are trying to take the x-ray I mean whether you are taking a let us say a thin film for example or for example a powder it can give you the average green size or average particle size okay this is a very I mean this particular thing gives a sort of I will say approximately correct value only when the green sizes are smaller than 100 nanometers if it is more than 100 nanometers then you know this particular expression does not give you a very correct size green size but this is sort of shortest and quickest method to measure green sizes in fact that is the reason that lot of students including my own students when they publish the paper for the first estimate of the green size they actually always use the x-ray Sherer formula taking the width of these particular lines and taking from that particular thing what should be the approximate value of the average green size that is what they use okay so this is very very commonly used let me put it like that if you have multiple materials okay the relative values relative changes in the green size can be fairly accurately predicted by Sherer formula but if you want to take really the correct green sizes okay it is not I mean one cannot be 100% sure or one cannot be I mean even reasonably sure that these green sizes are alright and one of the other very very important thing which sort of changes the green sizes changes the which increases the width of the x-ray line is because of the stresses which are always present inside the material so I say under quotes you know what there is one way of finding this gives you one of the ways of finding out the green sizes in fact as probably some of you are aware the best way of finding out the green sizes most accurate way is to do a dark field imaging in the transmission electron microscope but you know to make samples for transmission electron microscope is not very easy okay then you must have this particular facilities facility people also use AFM or SEM to get an approximate idea of green size and its distribution so remember using these methods you can even get distribution while the Sherer formula using x-rays will only give you the average green size there is no way that you can calculate the the distribution of the green sizes but as I said this being quickest at any way when the students are working for their PhD okay they will do XRD to find out whether they have made the material XRD has to be done for every of their sample and when they do the sample they will anyway get this widths from that they can always get an idea of the green sizes so as a very large number of publications which come these days probably use only Sherer formula to calculate the green size but whatever might be the accuracy of it and then there is one more question which is from Dehradon University of petroleum and energy studies which is what is the physical significance of the atomic radius since atom has no physical boundary is the distance between nucleus and valence electrons okay it is again a very very interesting question I mean it is true that I mean physicists used to long time back used to avoid calling a atomic radius okay because atomic radius see I mean I mean though we have been using a hard sphere model but you know we know that atoms are not really hard spheres in fact if they were really hard spheres they do not have any physical vibrations they do not have any lattice vibrations okay but as we always say that we always try to approximate the I mean we try to take some idealized picture and then only we start talking about these things otherwise you know it is not the physics is going to too tough if we want to have all the realistic things okay let me put it like that when we say about approximately the atomic radius which I have not used by the way this particular word atomic sphere it does mean that I take this particular atom approximately as a sphere okay and in fact there are some books which always write which is probably a correct word they do not call it atomic radius they will call as the shortest I mean let us start the shortest half of the shortest distance of approach so if you put two atoms together of the exactly identical way what is the shortest distance between their nuclei that you can normally possess half of that we will call as atomic radius now as you see yourself say which is correct that the shape cannot really be put into spherical form properly if you go into it is all regressness okay but this does give you an idea of how the packing stacking takes place that is all I would like to say now let us go to the other set of questions which is for the yesterday's lecture okay then first question is from Shegaw again is it okay for it is okay for single crystals but how the polycrystalline sample is confirmed by XRD pattern again a very interesting question so I will just mention see unfortunately all these things I just gave a bird's eye view on these things I did not discuss the complete XRD diffraction and other things in its in more detail that would have taken probably another two three lectures if I have to do it a proper thorough job but let me just tell you that in the case of the standard XRD machines that we are using these days okay the material has to be polycrystalline in order that you see all the possible peaks so because if you remember what we have discussed that the condition of Bragg reflection which is specular reflection as well as 2D sin theta is equal to L lambda is satisfied for only these grains only for these particular planes essentially you have a situation when you have theta corresponding to this and this and this theta is continuously being varied okay you are varying this theta as well as you are varying this particular theta so you are looking only at the top of the plane you are not looking at any of these planes for example if I give you an example there can be multiple ways of drawing the planes but the way the modern XRD machine works I mean at least in theta to theta geometry of course there are other geometries also when this is not correct so in theta to theta geometry which is the most standard geometry which we use in XRD okay essentially you are scanning on the top only the top of the planes okay now if this material is actually a single crystalline then this particular plane is let us suppose is let us say 1 0 0 so this if these planes are 1 0 0 and if this material is single crystalline material okay then D of this will correspond to 1 0 0 direction so you can get only diffraction from 1 0 0 planes you will not get diffraction from any other plane because 1 1 1 plane does not exist here okay or 2 0 I mean let us say 3 1 1 plane for example does not exist there all right so if you are using a single stand material then you will in this particular thing and if this is oriented along 1 0 0 direction you will get only 1 0 0 peak or you will also get 2 0 0 peaks or 3 0 0 peaks provided it is allowed by the space group you know that that is what I have said that not all the reflection is allowed for every type of lattice okay let us assume this is simple cubic in fact there is no material which actually slices into simple cubic with a single atom basis but just for the sake of argument let us take a simple cubic material then you have 1 0 0 2 0 0 3 0 0 because this second order reflection from 1 0 0 this third order reflection from 1 0 0 okay only these reflections will be present all others will be absent because those planes do not exist here okay now only when I am using a polycrystalline material then what will happen that for some particular material the top will contain this will be 1 0 0 for some other grain it may divide me 1 1 1 for some other grain it may be let us say 3 1 1 and if I have the very fine mesh of the grain sizes okay in principle there is a equal probability of finding every type of orientation on the top plane then only you will start getting all other peaks like 1 1 0 1 1 1 you know 2 1 1 3 3 1 1 then you start getting all the peaks so in fact this particular material this particular thing is most suited only for polycrystalline material and not really for single-strand material because of course you can do single-strand material but then you will get the peaks only corresponding to what is the plane on the top okay the second question was from Bulldana which okay it is actually the same college same question how you we use Bragg's law derived from reflection of x-rays from atoms but the technique used Bragg's spectrometer is called x-ray diffraction in fact as I have told you that the technique is only x-ray diffraction it so happens that because in x-ray diffraction condition the condition of a specular at diffraction is satisfied so many times we call it reflection we should strictly speaking should not use the word reflection we should call only x-ray diffraction okay but as I say people you know by loose talking or what we call as a misnomer people call it x-ray reflection okay it is not reflection in fact I have told you had it been a reflection when there is x-ray beam which is coming you always get reflected beam okay there cannot be any further condition imposed a normal reflection for example a light reflection is that whenever you are putting you know your light you always get reflected light okay you do not have any further condition satisfied but while here you have to have not only angle of incidence equal to angle of reflection but you must also have 2D sin theta is equal to a lambda so it is not necessary that will always get reflected beam so it is actually not reflection it is diffraction but somehow I mean I mean people many times use words which are sort of loose words so people call it x-ray reflection how do we measure a plane for a particular peak you know this actually has done to be done by fitting with that is not a very always for cubic light is comparatively simple because you can look by the ratios of no h square I mean sin square theta values and get the idea of what are the type of planes in fact this exercise we normally give to our students in our MSc class to work it out okay but if you are taking it really an arbitrary lattice these are not really very easy things and you these days of course they are support from the software's where you can give all these data and then they can tell you which is likely to be the structure of the particular material the next question is that in case of x-ray diffraction when x-ray as an electromagnetic wave is entering inside the crystal then why we are not taking under the consideration the refractive index of the material see thing is that you know where we are talking of the refractive index of the material when we are treating that material as a continuum material okay here the wavelengths are of the order of interatomic spacing see for example when we are using light okay light has a typical wavelength of 4000 5000 angstrom okay while you know the interatomic spacing is of the order of 1 angstrom so when this particular light is actually being incident on a let us say a glass slab okay we do not consider this glass slab to be made of atoms okay which are placed at 1 angstrom distance apart because the wavelength is too large so this particular material can be treated as a continuum material first of all we will take a vibration of a string okay and vibration of strings of course this string is consisting of atoms okay I never consider the vibration of the atoms in this particular case because my wavelength that I am going to excite is much much longer than this particular interatomic spacing so for all practical purposes I can treat this particular material as a continuum material as if it's perfectly continuous material then we talk in terms of tensions and other things which you are talking about the macroscopic property similarly when you are talking about refractive index we are talking about its macroscopic property treating this glass slab as a continuum medium but on the other hand when I am talking about x-ray diffraction x-rays have to in fact in case we are seeing x-ray diffraction their wavelengths has to be of the order of the order of interatomic spacing so typical x-ray wavelength that we are using are also of the order of 1 angstrom so then you cannot treat this particular material as a continuum material so these properties like refractive indexes etc have no meaning which are defined for a continuum medium I have to treat this particular material as a set of individual atoms okay then only I can talk about it and that is what we have done NaCl has interpreting 2 FCC lattice can we consider this NaCl lattice as a new lattice in reality apart from the 14 Bravais lattice of course you can always call it a lattice but not a Bravais lattice okay that is what I have been talking I have given you example of many structures which are periodic structures okay which will not be a Bravais lattice if we just replace all these particular points by their own points for example diamond structure okay if you replace all these carbon atoms by one particular point okay this will not form a Bravais lattice but on the other hand okay if you want to insist in terms of Bravais lattice and why we insist on a Bravais lattice because it is only a Bravais lattice which has that translational symmetry and this translational symmetry is very very important when we are talking of let us say band structure calculation okay when we are talking of let us say band properties or any other important properties it is very very important to talk about this the fact that there is a translational symmetry it means the environment of every particular particle every particular point is exactly identical okay this will not happen if I am going and I treat this particular thing as a non Bravais lattice because non Bravais lattice means you do not have this periodic symmetry periodic translation symmetry okay so you could call it but you know that will not be a Bravais lattice okay as I have discussed last time Bravais lattice can only be the can only be 14 why it is necessary to conceive the idea of a reciprocal lattice okay this is always a question of why we want to do it okay a lot of students do ask this type of questions and I start always arguing opposite fashion which is probably not a good way of doing but nevertheless I do to convince people for example why I say 2 multiplied by 2 is equal to 4 why we will not say 2 multiplied by 2 is 5 okay it is the way because that is the way we have defined it why we have defined it that way not 2 multiplied by 2 is equal to 5 why we have defined 2 multiplied by 2 is equal to 4 okay it is because there we are likely to face certain type of problems later okay in which it would be easy if I remember 2 multiplied by 2 is equal to 4 okay that was probably the simplest example but I remember I mean I remember as a high school student where first time the vector products were introduced to us dot product and cross product okay I mean we are sort of amazed when you talk of multiplication of two things there can be only one type of multiplication 2 multiplied by 2 can only result 4 it cannot result 2 multiplied by 2 we cannot get into 5 but when you are defining vectors like say dot products sometimes you can get a b cos theta sometimes you get a b sin theta okay sometimes it has direction sometimes it does not have direction so why do we define that way okay then we later realize that there is certain amount of easiness when we go to the later part of physics if I define the direction okay for example if I say the force is v cross b okay I have to just write v cross b so long you understand the cross product you understand what will be the direction what will be the magnitude of the force alright I remember you know some of the earlier textbooks you know which you are there I mean when I was essentially a child study you have so many right handed rule left handed rule you have to remember if this is the direction of velocity this is the direction I mean I do not remember those rules now anymore okay okay then this is the direction of the force so you have to for every law you have to remember what will be the direction of the thing now you have once defined using cross product v cross b things become much easier so all you have to remember is what is the definition of cross product once you remember that definition it makes easiness in understanding the later part of the physics we just say force is v cross b alright so that is all where we say torque torque is r cross f or when we say angular momentum angular momentum is r cross b okay so then direction magnitude everything is inherent I have to only explain you what is the cross product once then all those things can be expressed in terms of r cross b so that is the way that is the reason we define reciprocal lattice once we define reciprocal lattice okay you find it strange but then you do solid state physics you realize that things become so simple if I introduce the concept of reciprocal lattice if I would not have introduced reciprocal lattice to write those conservation of you know crystal momentum would have been much much more difficult I do not know how how do I will explain to it the next question is how are the planes for example 1 0 0 1 1 0 0 1 1 identified in a ground crystal okay in a ground crystal to identify is I mean see classic traditionally people have been doing by looking at you know how can you cleave and things like that okay I mean the only way I can think about is that if you do not do that you know you do an x-ray diffraction I mean I have told you how could how to find out you know using x-ray diffraction at least I have given you some idea okay how to how one can get some idea of orientation of the crystals in which particular plane and of course if it is a cubic direction 1 0 0 and 0 1 0 and 0 0 1 will all the 3 mill identically there is no way that you can distinguish between them for a particular glancing angle why we do not get same plane diffraction as shown in your table for example 1 0 0 and 2 0 0 that is what I tell you told you 1 0 0 and 2 0 0 1 0 0 is n is equal to 1 order diffraction from 1 0 0 plane 2 0 0 n is equal to 2 diffraction from 2 0 0 from 1 0 0 plane it does happen that it may be possible that 1 0 0 plane may not be present but 2 0 0 plane may be present because it again depends on the condition of constructive interference when you are talking of 2 times lambda the path difference may lead you to with 1 lambda this may give you destructive interference with 2 lambda it may give you constructive interference so that is possible this is nothing very surprising about it why we take the powder x-ray of thin films and single style x-ray for crystal I mean whatever is your material that you are making you are going to take x-ray only of that particular material can we use Wilma zones for real-time application of crystal or simple theoretical way again I say I will answer the same way Wilma zone is a very very important concept of solid state physics you will not be able to understand what are the realistic values of wave vectors which are allowed when you have a complete translational symmetry and only those k values which are lying in the Wilma zone are of importance I mean I do not think we can understand solid state physics concept without introducing Wilma zone so I mean though this looks somewhat theoretical but I mean I mean that way 2 multiplied by 2 is equal to 4 is also a theory okay but you know on the other hand you know we can see what is used the usefulness similarly once you go little bit advance in condensed metaphysics you realize that the concept of Wilma zone is very very interesting because I have told you that for example if you are taking the wave vectors let us say of electron in a band structure okay the wave vectors which are allowed for this particular electron to be present are only those which are in the first Wilma zone because the second Wilma zone whatever are the larger values of k they represent they can always be represented by an equivalent value of k in the first Wilma zone so maybe I am not sure whether you have understood this particular fact but only thing I would suggest that you know we will you can take it that you know the concept of Wilma zone is a very very important and interesting concept so I think I have exhausted all the questions that I have received so far on the crystallography what are remaining questions are the questions of quantum mechanics because there has already been some delay so I thought I will start with my lecture because I have left only with about half an hour so let me start with my lecture on relativity that we at least introduce special theory of relativity I think one theory of physics which almost everyone knows I mean even the person is not a physicist is the special theory of relativity and that is because of most brilliant work by Stein and people will definitely know about E is equal to mc square if they do not know anything else but this this relationship has become so what we call as a folklore almost everyone knows E is equal to mc square and everyone associates A to the brilliance of Einstein okay so this particular theory is a I mean I always enjoy this particular thing peaching to first year students okay and what is interesting that is more than 100 years that this particular theory was brought out and even today when I teach this to the first year students in IIT they have a shock they say no this is not possible I mean this is such a brilliant theory it is such a brilliant piece of work okay and as probably all of you know that they Einstein got Nobel prize much later after the discovery of special theory of relativity but you know that Nobel prize did not cite its work on special theory of relativity it only cited the explanation of photoelectric effect okay special theory of relativity was not sort of accepted by the people it is only at much later time when people have started having lot of experiments and then they found that many of the explanations many of the predictions which I said special theory relativity gave really turned out to be correct so today we believe in special theory of relativity but you know you realize that some of the postulates which are there are fairly shocking okay so let us start with special theory of relativity I will not be able because we have only four lectures and out of which also have spent some time here I will also be spending some time in answering questions so I do not think I will be able to do a small even the mildest justice to special theory of relativity okay but you know let us just go sort of at least introduce give you a feeling or what normally people call give you a flavor of special theory of relativity just to get you an idea of what is this all about okay so now let us go to there so what is my plan in special theory of relativity that I will identify first the issue with classical physics why special theory of relativity was needed and how this particular theory helped in overcoming those issues then give a flavor of some of the new definitions I am only talking about the flavor okay not talking too much of how do we obtain partly because of lack of time partly because they cannot strictly be speaking derived like we cannot derive Schrodinger equation will not be able to derive these things we give only a series of arguments but again even those arguments I will not be able to give these limited time and as I said I will focus more on concepts and not so much on mathematics okay so just to as I said I mean I am becoming purely defensive give just a flavor of the special theory of relativity now when I started the series of lectures you know I have talked about the two dark areas or what we call many times clouds on the classical physics which was in the late 19th century about the cloud one I had talked in detail about the failure of Maxwell Boltzmann doctrine regarding the equipartition of energy and specifically talked about the blackboard radiation and how it led to quantization of energy and how it led to eventually the quantum mechanics now at that time we had mentioned about the second cloud about which I did not talk and I said I will talk about it when we discuss special theory of relativity so this is the cloud about the motion of earth in ether so let us first understand this particular cloud in detail before we actually start special theory of relativity before we start let us first introduce the concept of a frame of reference probably all of you are aware but I mean I know in first year students in IIT everyone knows very well concept of frame of reference inertial non-inertial frame of reference but nevertheless I like to introduce it again just because in fact I said as a as my experience in teacher as a teacher I always like to launch a course from a particular point where the students are fairly comfortable okay so that they feel that they know the subject well then only go to more and more difficult topic in fact I have found the strategy to be helping very well because if there is a gap between what they know and what I am going to teach probably they were they are never going to appreciate what you know what I am going to talk about it so I introduce a concept which I know that at least the students in the first year level in IIT are very well familiar with which is the concept of an inertial frame of reference okay let us first say what is the frame of reference frame of reference is basically you tell where my observer is located okay observer could be located on earth observer could be located on let us say sun okay if it all is possible or a moon okay or it could be in a moving car or a moving train or a center of some galaxy or whatever it is all right so there is one particular observer which is sitting somewhere and there where that particular observer is able to draw his or her x y and z axis and has methods of measurement of distances and times etc so that particular observer can calculate what is the speed of a particle can find out what is the location of the particle can calculate what is the acceleration of the particle can know what is the force on a particle all the dynamical things what can know in that product sitting at a particular place okay so whatever is that particular place you know for example if it is earth then earth becomes my frame of reference if the observer is sitting on train that by train is my frame of reference in fact in normal classical mechanics especially the mechanics that we do it in high school level one of the very common mistake people make is to confuse the frames of reference I always say that choose a particular frame of reference and then only you can apply mechanics properly so it is very very important to realize where my observer is sitting where I am saying velocity which is the observer which is missing this velocity okay so now generally frame of references are considered of two type one is what is inertial frame of reference another is what is called as non-inertial frame of reference and inertial frame of reference is defined is that a isolated object moves with constant velocity so essentially it means that if we take one particular object very far away from all other objects then that particular particle is expected to move with a constant velocity in the classical mechanics in macroscopic world okay generally the forces I mean the forces are always occur as a pair I mean this is Newton's third law of motion that they always occur in pair so if there is a force on this particular body okay it must be if it is a real force it must be experienced by there must be another body which must apply this particular force on this particular body so there are two bodies which must interact then only this particular force from this particular this body can be applied on this particular body and when a force is seen on this particular body the body which is applying the force that will experience an equal and opposite force that is what is Newton's third law of motion remember the force and its reaction are on two different bodies so if this body is applying a force on this body then this body will also apply a force on this particular body and these forces are equal and opposite so all the forces occur in pair okay so there cannot be an isolated force on this body without looking at any body other body nearby which can apply a force on this particular body and also for most of the common forces that we come across we know that if I increase the distance between these forces generally the forces go down take gravitational force take electrostatic forces remember I am talking about macroscopic words okay not really talking about the particle I mean the nuclei etc of course so if I am able to remove the distance if I take this particular distance of these two bodies very far from each other then the force between these two bodies with respect to each other is definitely going to go down so if I am taking one particular object and move it to infinity or to a very large distance where it does not see any particle nearby any particular object nearby then there cannot be a real force on that particular particle and if my observer finds that particular object to be moving with a constant velocity then I will call that this observer is sitting and in an inertial frame of the frames now the word special in special theory of relativity means that we are dealing only with inertial frames so many times discussing these things I may not always call inertial inertial inertial but you should realize that when we are talking special theory of relativity we are talking only of inertial frame of reference if because some reason I have to use the word non-inertial I will specify at that time that is a non-inertial frame of reference otherwise we take this just as an inertial frame of reference whenever I am talking of frame of reference so I said unless specifically mentioned we shall assume that the observations are being made only in an inertial frame of reference so that is why this is special theory of relativity it talks only of inertial frames now we talk about Newtonian mechanics I am making two statements let us see whether all of you agree with that then I will show that this is actually correct first statement which I make that velocity is a frame dependent quantity if I go to a different frame I may find the velocity of the same particle to be different which is very normal very natural if I am standing on earth I may find a particular tree to be stationary but if I observe the same tree sitting in a train I will find that this tree moves backwards okay so this velocity is actually a frame dependent quantity so from where you are observing this particular particle okay that actually determines its velocity but what is important to realize that even though velocity is frame dependent quantity acceleration will not be same acceleration does not depend on the frame so whether I am sitting on the earth assuming earth to be an inertial frame which is strictly speaking it is not but assuming that earth is an approximately inertial frame of reference and there is a train which is moving with constant velocity which is also an inertial frame of reference okay in that particular case the two particular particles if a body is accelerating okay you will find that the acceleration turns out to be same both of them will calculate the same value of acceleration let us just show this by a particular diagram and convince yourself that the statement that I am making is true so let us suppose we have two frames of reference normally frames of reference always represented by a set of axes so when I frame I call it S frame okay this S frame I am denoting by a red axis so this is my x direction okay and this is my y direction and this is my z direction and I have another frame S frame where the axis I have written by blue color I am not sure whether you are able to see the color very clearly in your centers so this is your x frame this is your y frame and this is your z frame and because of some convenience I am choosing a particular set of axes one can show that this is not will not lose any generality I am assuming that x and x frame axes are always constant with respect to each other y and y frame axes are parallel and z and z frame axes are also parallel physics are not going to be dependent on what we take as x y and z axes okay we choose appropriately x y z axis so that my mathematics become simple not only to suppose that there is an object here of course let us assume that S frame with respect to S moves with respect to velocity okay and because S and S frame both are inertial frame of reference therefore if they have to be inertial frame of reference let us say S has to be is a inertial frame of reference S frame can be inertial frame of reference only if the relative velocity is constant as a function of time okay it means the velocity of S frame which I am again assuming it is on in this particular direction for this derivation will not depend on that particular thing you can see that derivation will not take I mean will be same in spite of the fact that you know we may not we may choose a different set of axes yeah all right so what we are trying to do is that we are looking at one particular point both observers as S and S frame are looking at this particular object S frame let us assume is moving relative to S with a velocity constant velocity V okay now the position vector of this particular point with respect to these x y z axes is this which I am writing as RT so there is no prime here because I am talking a frame of reference which is S also not primed so this is RT it means this R of course would be a function of time because in general this observer may find this particular point to be moving in any arbitrary direction okay so this R position vector will be a function of time at different time this R may be different now same point at the same time was being observed by another frame of reference which I am calling as S frame of reference okay and that particular person finds out its position vector to be R prime t okay so here I am using prime indicating that this is being observed by an observer S prime all right so as you can see very clearly through the identity that this RT is some of these two vectors this vector is capital RT which gives the relative position of O prime or position of O prime with respect to O so this is the vector which gives you the position of the origin of S frame of reference with respect to S so you can very easily see that this RT is R plus this R prime t so if I have to go from here to here okay this is the same as going from here and then going here so this RT is some of these two vectors capital RT plus R prime t okay now let us differentiate these two vectors with respect to time this is these two sides with respect to time okay this was my equation if I differentiate this with respect to time I will get velocity of this particular object dr dt is the object velocity dr prime dt will give me the velocity of the particle as will be measured by an observer sitting in S frame of reference and if I differentiate R with respect to t I will get v naught where v naught will be the velocity of S frame with respect to S frame all right so now you can see that velocity as will be measured by S frame will be different from the velocity which is being measured by S frame of reference and the equation is that v t is equal to v naught plus v prime t all right now I differentiate once more with respect to time if I differentiate velocity with respect to time I will get acceleration okay when I differentiate on the right hand side because this v naught has to be constant because we are talking only only if inertial frames okay so this will be zero time derivative so this will be equal to d v prime dt which is the acceleration of the same object or same particle being observed by S frame of reference so we can see that the acceleration of that particular particle measured by S and S prime will turn out to be identical so this is what is the statement that I made I had made earlier that in Newtonian mechanics you will find that velocity is a frame dependent quantity this velocity is a frame dependent quantity the two objects two frames of reference do not agree with the value of velocity that they have obtained according to one the velocity is something another according to another you have to add this velocity v naught then only you will get that particular velocity of the object but on the other hand both the observers will agree that the acceleration is same now I make another statement about Newton's law of motion okay I mean as I said earlier Newton's law normally we need a second law okay this is the Newton's law of motion f is equal to ma we just now see if we go from one inertial frame reference to another inertial frame of reference acceleration would be identical okay now if we assume if this is a very important if we assume that the forces seen on the particle is same in both the frame of reference which generally we thought of that way in Newtonian mechanics then this f is equal to ma can be equally well applied in all the inertial frames so if the forces are same in all the frame of reference f is equal to ma can be equally well applied into all inertial frame of reference okay so all inertial frames appear to be equivalent however probably once I have made this statement and I have emphasized all this respect provided force is same in all the frame of reference probably this would have clicked to you that their forces which also are velocity dependent forces I mean I have been giving example just now in the morning okay by taking the example of a cross product there is a Lorentz force v cross b okay it depends on the velocity of the charge so there is a particular charge which is moving with the velocity v then v cross b will be the force on that particular object and we have just now said that v is actually a frame dependent quantity does it mean that this force is also a frame dependent quantity okay so you will realize that so long we have not been talking about electromagnetic theory if you talk of purely mechanical motions okay you will find that probably Newton's law of motion can be equally well applied in all inertial frame of reference we start landing into problem when we try to mix up the electromagnetic theory with respect to the Newtonian mechanics and that is what I am going to emphasize and that is what I am going to call as the second cloud or second black spot okay on the classical mechanics. Let us understand this particular thing okay of course we say from Newton's law velocity can be evaluated in any frame we must specify the initial condition in a frame to know the velocity see equation is same f is equal to ma okay if you go from one frame to another frame the frame s will put its own initial condition and can find out what is the velocity at a given time of the particle frame s observer will put its own initial condition solve these equations and then calculate what will be the velocity at time t is equal to 0 and find out what is the velocity at time at particular time t knowing the force etc so all those things can be done. The problem is that you have a velocity dependent force which is q e plus v cross v we had just now said that this particular v depends on different frames if it depends on different frames does it mean that this force also depends on different force frames of course you can always say that this magnetic field is also represented by a motion of charge carriers okay and v will be frame dependent quantity okay in fact strictly speaking that is a correct in fact our professor who asked me to specify specifically mention about this particular point that in principle what is pure electric field or what is pure magnetic field in relativity okay can be mixture of the two. Unfortunately we will not be able to do in this particular series of lectures the transformation of electric field and magnetic field because that requires not more working okay but I am sure you can look into the books and look about the transformation of these forces okay but let us not go into those details at this particular moment but let us just understand the problem so we have first problem is that if v is frame dependent quantity does it mean that this magnetic force will also be frame dependent alright then if it is so then f is equal to ma cannot be equally applied in all the inertial frames okay so we have some issue but let us list the second issue which appears to be much more interesting and much more thought provoking in that sense see I think professor Ghosh has done electromagnetic theory and he must have derived this particular condition that C the speed of light turns out to be equal to 1 upon under root epsilon naught mu naught you realize that epsilon naught mu naught are fundamental constants this epsilon naught determines the force between two charges mu naught determines the forces between two current carrying wires alright both of them are believed to be fundamental constants. So therefore C is supposed to be fundamental constant okay let us look how does this second problem manifest itself when we are talking about you know different frames. So let us first look at this particular problem which I am sure all of us have noticed it let us suppose there is the observer ground okay let us assume that this ground is an inertial frame of reference though you know that ground is not strictly spring not inertial frame of reference but for many over discussion let us start assuming that this is an inertial frame of reference so this particular person notices one particular train or particular car going to the right hand side with the direction v let us assume that this are all unidirectional motion and v is along the x axis and he also notices that there is another car which is moving to the left hand side with a speed u which is an opposite direction alright now my question is that if there would have any other observer sitting in A this particular frame of reference or this particular car or this particular train what is the speed of this particular object or train v which he or she would have observed as all of you know that this would have been v plus u okay if you are sitting on a train and a train passes against you and from opposite direction it appears to be moving very fast okay this is because the relative velocity is v plus u on the other hand if the situation was somewhat different in the sense that this train was also moving in this direction the other train was also moving in the same direction then what will be the velocity of the of this particular train as seen by this okay this will be u minus v alright so this is also very clear that if the two I mean if you are driving on road and there is a car which is going by its side okay you find that the velocity of that particular car with respect to you is comparatively much smaller if the car was coming opposite to you you will find the speed to be much larger this is very very natural things that we always see in our daily life in mechanics now let us do the same experiment with light everything same other than that this carriage which was there this carriage I have replaced by a light source so this there is a light source of course light source is not moving okay it is only throwing light so light is moving so you have a source okay the light is coming and let us assume that this particular speed of light is c as being measured by this observer so it is exactly same condition so this observer find a carriage or car moving to the right hand side with the speed u he also finds that light is coming towards this particular car and as measured by this particular observer this particular light is c the speed of this particular light is c then putting the same arguments if an observer was sitting here he would notice that speed of light will be u plus c now if you have a different situation that the car is moving this particular way and a light is somebody shines a light from this particular side then a person sitting here remember this c and u both are being measured by this observer sitting on the ground then an observer sitting here would notice that the speed of light will be c minus u so just like the case here where these this observer notices two different velocities of this train b here also if we apply the same classical mechanics idea this observer would notice this particular observer will notice different speeds of light in this situation and in this situation okay it means as I said this should also apply to light if it applies to light then speed of light should also turn out to be frame dependent quantity but that is where we have a problem we have just now said that c depends on fundamental constants okay now if c is frame dependent quantity does it mean to say that these fundamental constants are also frame dependent okay does it mean that they are no longer fundamental that is where they should is speed of light frame dependent if yes is the following expression is also valid in all the frame of reference if it is so then epsilon naught and mu naught also are becoming frame dependent implying fundamental constants epsilon naught and mu naught are also frame dependent and if epsilon naught and mu naught are frame dependent quantities then electric forces and magnetic forces will also become frame dependent quantity because you know electric forces charge between two forces okay if you take the Coulomb's law one upon four epsilon naught qq prime upon r square okay if epsilon naught has changed okay force between two charges will change okay if I go to a different frame in which epsilon naught is different force between these same two charges will be different similarly if mu naught has become a different in a different frame then forces between two current carrying current will become different in different frames okay so this looks a little surprising that in mechanics we had always the situation where we appeared the forces are to be if you are talking about it is a friction force or we push a body okay you always feel that the force is same in respect to whether you are observing here or a person sitting in the train is observing but here it appears that if I change my frame of reference the electric force and magnetic forces will become frame dependent quantity it is not a situation which about which you are really having of course there is another possibility that whatever I am trying to say it is not really corrupt though velocity of course light could be frame dependent okay possible that light is frame dependent but somehow I do not want to have epsilon or mu naught to be frame dependent okay so let us assume another situation that this expression C is equal to one upon under root epsilon is not frame dependent okay it is specifically valid only in one spatial frame so this is what I am mentioning it or is it that the following expression is valid only some specific frame so let us assume that this frame is this particular equation that I have derived on the basis of electromagnetic theory it is not a universal equation okay this equation is valid only in certain spatial inertial frames okay are these two types are there two types of inertial frame okay see at the moment we have said I mean in Newton's law of mechanics if you remember we have always been telling all inertial frames are identical inertial when if you look at very standard definition of inertial frames people always say that these are the frames in which Newton's law of motion is valid when force is only the real force all right but now we have come to a situation that we have one expression which is this fantastic expression C is equal to one upon under root epsilon remember this is the expression which really led to the success of Maxwell's equation because this could predict the velocity of light which turned out to be equal to the experimental values that we had measured but really showed marvelously that the lights are electromagnetic waves all right now it is so appears that this particular expression is valid only in one type of inertial frames okay it is one of the possibilities okay and in other frames it is not valid this implies all the inertial frames to equivalent from the mechanical point of view are not equivalent from the point of view of electromagnetic theory whenever we are talking of charges and magnetic field then you will end up into problem probably all inertial frames are not equivalent there are certain special inertial frames okay in which this particular expression will be valid and there are certain other frames in which it will not be valid okay a special frame may then exist okay if you want we can call this as an absolute rest and that is what people thought that is something like absolute rest okay so only that particular inertial frame this particular expression which I have given C is equal to one upon under root epsilon or not may not is valid in other frames it is not valid so though they are all inertial frames okay I am not talking of non-inertial frames at all they are all inertial frames but out of those inertial frames they are certain specific frames okay only with which this particular expression is valid now what is your pick imagine yourself in the late 19th century when these things were being discussed okay you have two possibilities okay that you allow fundamental constants to become frame dependent there is one possibility second is that you allow only a special frame of reference that you say that this fundamental constants are not frame dependent frame dependent but on the other hand the expression is a frame dependent quantity means the C is equal to one upon under root epsilon or not may not is valid only in a special frame I mean ideally if I am sitting today if you can imagine yourself at that time probably you are not happy with any of the situation okay ideally you would not like any of these situations to happen but if there are no other choice given to you probably you will prefer what we call as a lesser evil okay out of these two possibilities the possibility which appears to be more favorable that you assume that there is a special inertial frame of reference to treat epsilon or mu not which we have always thought of as fundamental constants to a frame dependent does not I mean is bigger evil than treating that there is a special inertial frame of reference in which this particular you know expressions are valid this is what I have written do you allow fundamental constants to become frame dependent or you allow a special inertial frame ideally none but if there is no other choice given one is likely to prefer lesser evil which is the second choice and that is what people in the classical physicist area in the 19th century preferred that there is one special inertial frame of reference only in that particular frame of reference this expression is valid in no other frame of reference it was valid and this particular frame of reference was called ether so earlier ideas favored the concept of a special frame of reference and this was also supported by some of the other things about which were sort of what we call as uncomfort zone see all the waves that we knew required a medium to travel sound waves sound waves you know always requires a medium to travel okay like ocean waves they require a medium to travel on the other hand light comes from distant stars you know they reach earth where there does not seem to be any medium so people thought probably there is some medium but which we are not able to see okay there has to be some medium okay when we are talking of a wave to travel it requires a medium so that let there be a medium okay and this medium could be the same medium that we are talking ether so it was let me just read earlier ideas favored the concept of a special frame light also does not seem to require a medium to travel unlike other waves like sound waves so it was imagined that the universe is filled with ocean of ether light needs this medium to travel so you assume which was very very simple way of thinking it's a very natural way of thinking a classical way of thinking okay which I mean as I said even first year students who come here that's the way they always think that let's assume that the entire universe is filled with something okay it's like ocean okay ocean you know like you know you know different fishes different type of you know sea animals move okay so it's exactly the same thing you have the entire universe is filled with a medium called ether okay and in this ether medium all the planets galaxies etc are all within this particular medium this particular ether we can consider is an absolute rest and let's assume that this expression sees you go to one upon under root epsilon or mu naught is valid only in this particular ether medium it's not present in any other medium okay and let's assume that this particular medium signifies absolute rest it's implications all planets stars galaxies float in ether the speed of light is C given by the following expression only in this ether medium in no other medium this particular expression is correct okay if you calculate the velocity C and find it to be equal to one upon under root epsilon not mean you know that will be valid only in this ether frame in any other frame the velocity of light will be turning out to be different and in that particular case this particular expression will not be valid in other frames the speed of light would be different from C one can determine the speed of frame by measuring the speed of light in that frame and comparing with one in ether frame all right so you can always see you calculate the velocity of light in your frame of reference calculate what is its value compare with C is equal to what you get from C is equal to one upon under root epsilon or mu naught okay whatever is now then you can always calculate by standard relative velocity formula what will be the velocity of your frame of reference it has a larger implication okay probably one can see there is slight amount of an uncomfort in these arguments okay but nevertheless that's the way people are arguing because they had no other choice ether can be thought as signifying absolute rest hence we can ask a question this is very very important we can ask a question what is the absolute speed of sun or what is the absolute speed of earth without specifying the frame okay then what I have said in the classical mechanics if you want to talk of velocity okay you have to specify frame if I say okay in earth's frame okay this particular table which is in front of me is at rest but in sun's frame it is not at rest okay so when I am or in train's frame of reference this is not at rest if I am moving table is not rest table is moving backwards okay so if I am calculating the speed of any object if I am saying the speed the train is moving with the speed of let us say 100 kilometers per hour okay this is with respect to earth okay though I may not always say this thing but this is sort of implied say when I say that particular train you know the for example super fast trains can move with a speed of 350 kilometers per hour this is with respect to earth okay though we may not say okay but it is sort of understood okay with respect to sun the speed of the same train will be different okay if there was another train moving on a track parallel to it with respect to that train the speed will be different so in Newtonian mechanics we said that you cannot talk of speed unless you specify the frame but here I am lending into a different situation it says okay you can always say what is the speed of the earth without specifying with respect to what okay the idea would be that okay the speed has to be mentioned with respect to ether because ether is supposed to be absolute rest it is supposed to be filling everywhere okay so if I say what is the speed of the sun okay the sun speed has to be measured with respect to ether ether becomes your absolute rest you can give a unique value of velocity or speed to any object okay with reference to ether it is a very very important far reaching conclusion broader implication though all the frames are equivalent from the point of view mechanical point of view they need not be from the electromagnetic point of view see pure mechanics does not lead to this particular fact see so long we are not talking about electromagnetic theory so long we are not talking about electric forces magnetic forces we are not introducing the concept of light as electromagnetic wave we had no problem we thought that all inertial frames are equivalent okay but as soon as I introduce electromagnetic theory I started having problem okay it means is it so that inertial frames are equivalent only from mechanical point of view so long there is no electromagnetic theory involved so long there is no electromagnetic force involved okay but when we talk about electromagnetic theory there okay there is one inertial special frame of reference ether which exists but strictly speaking there is no real process which is purely mechanical okay there are always I mean there I mean if you know about friction forces they are actually because of the sort of electromagnetic forces between them at least if you want to observe something the light has to go there and it has to come back and reflect it to it I mean there is no a mechanical 100% mechanical force is it a chance that nature has made mechanical processes equivalent in inertial frames but not the electromagnetic processes okay are we really happy with the situation okay I am pretty sure that all of us will realize that we are not happy with the situation so let us just look and test whether we can look at the experiments I mean we always fall back upon experiments to tell whatever we are saying is correct or not so let us fall back on experiments okay and seriously that this particular statement of the presence of ether is correct or not so I think I will stop here I will talk about the famous Michael's and Morley experiment one of the most interesting experiment which tried to look for the ether and was unsuccessful okay there we will take a few questions today Don Bosco Don Bosco Guwahati you have some questions yes sir yes good morning good morning so my question is is the inertial frame an ideal concept can we really find inertial frames in nature because as far as I know the earth is not exactly an inertial frame definitely not the fact is that I mean there is no I mean a perfection doesn't exist in nature let's let's go to this particular thing I'm probably talking of philosophy there is nothing which is perfect okay so I don't think a perfectly inertial frame of reference exists but let me put it like that that for any given problem a frame of reference which is as close to inertial frame of reference as possible can always be imagined for example when I am trying to describe the motion of let's say I mean a car or a train or a plane okay earth is a reasonably good amount reasonably good type of inertial frame of reference of course I have to realize that then I have to introduce something which are called fictitious forces to understand every part of physics because we realize that earth is not strictly speaking inertial frame of reference but it depends the see what happens when we go from in mechanics when we go from inertial to non-inertial frame of reference we have to give rise to some other forces which are not real forces which we call as fictitious forces or pseudo forces okay if these forces are small enough so as not to disturb the motion of the object that I am interested in okay I can treat this particular frame of reference as inertial frame of reference now forces like centrifugal forces Coriolis forces though they are real forces which will be present on earth because earth is not really a inertial frame of reference so long these forces are completely much smaller than the force that I apply to a train to make it accelerate okay I can ignore the fact that earth is a non-inertial frame of reference so I can always find an approximately inertial frame of reference for a given situation thank you sir welcome Keva College of Engineering Nashik hello sir yes hello yeah yeah please I can hear you sir there are many microscopic events where we apply Newton's laws of motion yes can we apply the concept of inertial frame of reference or non-inertial frame of reference in that case see thing is that as I told you that you know see when you're talking of really microscopic you know see let me very clear see there is a limit of application of classical mechanics see classical I mean I am putting in somewhat subjective manner so I mean please excuse me for that as a scientific person one should not do it but just to give you a feeling see the traditional classical mechanics is valid when we are talking of bigger particles if you are talking of let's say particles like electrons or neutrons I mean neutrons or something like that we have to go to quantum mechanics which is a totally different mechanics okay similarly the classical mechanics is valid only when our speeds of the objects that we are observing is much smaller than the speed of light okay if the object speed turns out to be closer to speed of light the relativistic effects will start coming in so classical mechanics has its own limitations okay of course you can also thought of quantum particles which are moving very large speed in comparison to speed of light okay of that order of speed of light in that case we use what we call as a relativistic quantum mechanics which is much more advanced topic okay so so long you are within the limitation of classical mechanics you can always apply Newton's law of motion the only penalty you have to pay if you are not in a proper inertial frame of reference but in a non-inertial frame of reference that you have to give rise to certain forces which are we call as fictitious or pseudo forces the difference between real forces and fictitious forces that in real force they are always resulted they are always resulted by interaction between a pair of particles while pseudo forces for example centrifugal force Coriolis force okay they are not a outcome of an you know interaction between two bodies they are present just because you are in a non-inertial frame of reference now in a given non-inertial frame of reference if these influence of pseudo forces are much smaller in comparison to the forces which are real forces then you can ignore the fact that you are in a non-inertial frame of reference and talk only about the treat this as a inertial frame of reference that is what we do when we are sitting on earth as I said if I want to describe the motion of a car okay I always treat earth to be a inertial frame of reference which I know it is not because in principle this car will also be subjected to what we call as a Coriolis force will also be sort of experiencing a centrifugal force okay and it so happens that the amount of the this pseudo force okay is so small in comparison to the force that you will apply in order a car to accelerate okay is negligible so I treat this earth as a inertial frame of reference so depending upon the problem okay you may have to consider but on the other hand if you are sitting on a merry-go-round and want to walk along the merry-go-round okay then you have to apply centrifugal force you have to apply Coriolis force which are pseudo forces thank you yes Rangaswamy college hello good afternoon sir I have two questions yeah please go ahead it's from crystallography yeah can we have miller indices beyond using the three hkl indices in some solid state books I have seen the crystal planes of hexagonal lattices are shown like one one bar zero zero that's right you are right how will you say it is a four-dimensional okay that that's again a very interesting question see what happens strictly speaking in hexagonal crystals also you can use only three miller indices okay fourth index is superfluous in fact you can find the fourth index okay if you know the first three index the thing is that you take I mean as I said in hexagonal thing okay you take this as A and this as B and C as this now there is also possibility if I do not remember I think you take one and you take another or you take this and this so you take one more axis and take intercepts of one particular plane in three axes along this xy plane and fourth along this there this direction and then in principle you introduce a fourth miller index also okay which as I say is superfluous because it's not really necessary but why we introduce is because when we are talking of equivalent planes in hexagonal crystals you may find that this plane is equivalent to this particular plane is also equivalent to this particular plane so in order to and hexagonal is a very very important type of crystal structure so in order to introduce those equivalent planes sometimes prefer prefer prefer to write the fourth index also so it's only done in the case of hexagonal as far as I know okay and this is not really necessary it's just for the sake of convenience here another question another one more is yeah why why we use reciprocal space whether our samples and elements are evaluated in the direct lattice space only then why we use reciprocal space to illustrate the structure see what is the significance of using reciprocal space I mean reciprocal lattice space and I had tried to answer this question in the morning when I said that you know we do it because it makes certain lives certain things easier if you want to visualize a particular Bragg reflection people use for example the concept of Belouin zone the people use the concept of what is called evolved spheres about which I have not talked about it they are all drawn in the reciprocal lattice space to visualize what type of reflections that you are going to get okay so these days we have computer programs you know 50 years back we did not have computer programs almost everything has to be done manually you found people in the reciprocal lattice space to observe and to see which reflections you will get or which diffraction peaks you will be getting much simpler so it is only from the point of view of convenience that we introduce the concept of reciprocal lattice space and as I said if you are looking at the conservation of what we call as a crystal momentum which is one of the fundamental equations conservation equations in crystals okay you have to use a concept of reciprocal lattice it is basically for making our life easier Techno India Salt Lake Calcutta good afternoon sir good afternoon yes my question is actually not related with the topic you have covered in the session but I want to ask you that how we can explain the motion of proton why we can explain the motion of more proton through the help of relativity only why could we explain it with the help of Newtonian mechanics is there any way to explain it that if we apply Newtonian mechanics to the photon particle then it will fail and if we apply relativity to the photon particle it you can only explain the motion of proton see as I told you that you know Newtonian mechanics doesn't work it's no longer valid when you are talking with speeds closer to the speed of light and photons photon is a particle which travels with actually speed of light so obviously you are not your standard classical mechanics will not work there in fact in classical mechanics I mean you cannot you know you don't even have a concept that the particle can have zero rest mass because I mean there is nothing like a velocity dependent mass so obviously those ideas will not be valid in Newtonian mechanics so I can never imagine to apply Newton's law of motion to a conceptual photon okay because see as we will be seeing when we are coming I mean we are going little deeper into special theory of relativity we will feel that you will see strong departures of course I again I will be only only be able to talk about Lorentz transformation I will not be able to talk about let's say force transformation but you will realize that we get strong departures from Newton's law of motion as we our the speed of the particle approaches the speed of light can we apply some law of classical mechanics to photon and show that it fails as I said its rest mass is zero so if you apply any force on this particular thing it should not move but we know for example the photons does get affected by gravity for example you know you have a concept of things like black holes etc so when it's very very clear that you know this is a purely classical mechanics you cannot imagine to apply that thing okay we are stopping if you have any questions please post these questions so we will try to answer most of them yes