 We already seen the importance of surface and it is role in altering the properties of materials, especially at the nano scale. Now, we will take up a formal definition of what is surface energy, what is surface stress, how should we differentiate between these concepts when it comes to liquids, vis-a-vis the concepts when solid, crystalline solids and further how can I go about classifying what are known as clusters which are small, which are large and then nano crystals and how the shape and other aspects come out naturally when we are talking about clusters and nano crystals. Now, I may want to assemble a solid or a liquid starting with the gaseous state. In the gaseous state the atoms are far apart of course, the need not be atoms these entities could also be molecules, but for simplicity here we consider atoms and we assume that in the gaseous state since the atoms are far apart there is essentially no interaction. Of course, there will be collisions as these atoms move about, but the kind of interaction we are talking about is an attractive or repulsive interaction which continuously exist between a set of atoms. And when you want to assemble a solid or liquid what we do is that we take the gaseous state and put it together in the form of a solid or liquid which has been scorned schematically here. If it is of course, a solid as you can see on the example here it would be an ordered state where you can see that there is a certain periodicity and order. Of course, I could always assemble an amorphous solid also, but for simplicity we will consider an crystalline solid to start with. In the solid or liquid state the atoms are typically very close to each other and this formation of the liquid and assuming now that I am below my melting point or below the freezing point respectively for the liquid and the solid then the system would like to lower its Gibbs free energy by being in a state which is the state of a liquid or a solid. Therefore, below the melting or boiling points I have an increasing energy and therefore, the gaseous state represents a state of high energy and the solid or liquid state represents a state of low energy. Now, when we assemble a solid for instance from the gaseous state and now this is first what you might call a conceptual experiment we are doing. Then we see that of course, the atoms have come close to each other in the solid state and more importantly this central atom for instance is bonded to its neighbors and in this process of bonding the energy of the system is lower and I am specifically referring to the Gibbs free energy assuming we are working at constant temperature and pressure. Now, but the atoms on the surface as you can clearly see for suppose I take an atom here is only bonded partially compared to its full potential and therefore, these represents a state of higher energy. Now, when we do a calculation for lowering of energy of the system on the formation of a condensed state we assume that all the atoms are fully bonded and you can see that in this process we have done an over counting because now we assumed all the atoms are fully bonded while the atoms on the surface are only partially bonded. Therefore, atoms in the surface represent an higher energy with respect to the bulk and more importantly to keep in mind the reference state now is the bulk for instance in this example the crystalline state and this higher energy state is not with respect to the gaseous or the non interacting state. Therefore, when I say my surface is an higher energy what I really imply it is got an higher energy with respect to the bulk atom and not with respect to the gaseous state from which of course, I started my thought experiment. This is a very important point often not emphasized on, but it is important to keep it in mind because now when I say the surface is higher energy state I should not think that it is an higher energy state with respect to the gaseous state and I am assuming that I am below the freezing point now and it is a purely a coincidence of the fact that I did not over counting that I have even have to talk about something known as the surface energy. Suppose I were to do an accurate calculation in other words I only count those atoms in the bulk when I try to lower find find out how much energy they are lowering and separately treat my surface atoms and find out how many how much energy they were lowered then I do not even have to consider a concept like a surface energy. But typically this is not how it is done it is done by assuming that all atoms are bonded well and then I do a correction with respect to the fully bonded state which I call the surface energy term hence the reference state for the surface energy is the bulk and not the gaseous state. Now, this automatically implies that it cause energy to put an atom on the surface and when I say put an atom I put I take a bulk atom and move it to the surface then it is going to cost me energy as compared to the bulk this is the origin of the surface energy which is typically given a symbol gamma. Now, the surface since the atoms on the surface have an higher energy this is to wants to minimize the surface area the atoms want to go closer to each other or in fact into the solid and this minimum tendency for minimization of area or the tendency to shrink is origin of what is known as surface tension. We shall see now there is an important caution will be kept in mind that often surface energy and surface tension are sometimes interchangeably used but this is not to be done. Now, the origin or the what you call the seed for this kind of a misconception or a equivalence comes on the fact that suppose I track my dimension the surface stress or surface tension it is force per unit length then it is f by l and the units turn out to be Newton per meter. The units of gamma my surface energy is energy per unit area which is e by a energy being in joules area being meter square and joules is Newton meter by meter square. Therefore, units of gamma is Newton per meter which is exactly identical to the units of surface tension just because they are dimensionally identical we should not think they are identical quantities physically these represent very different type of quantities. Gamma is a scalar that means surface energy as we know is a scalar while surface tension is a second order tensor. Further we will see very soon that how these two quantities are different or how we should differentiate liquids and solids these are with respect to these quantities what additional quantities and parameters do I need to consider when I am talking about solids and liquids and more importantly what is the true origin of this surface tension. So, these are things we will consider next just to summarize this slide we have two important quantities when we deal with surfaces one is surface energy other is surface tension dimensionally they are very similar or they are identical. But, we should note that they are physically very different kind of quantities gamma is a scalar and surface tension is a second order tensor liquids in liquids when I am talking surface energy and surface tension are equivalent concepts and sometimes they can be interchangeably used. But, in the case of solids it is important to note that surface energy is not equivalent to surface tension of course, in very under very special circumstances they may become equivalent. But, for now we will assume that surface energy and surface tension represent different concepts when it comes to solids additionally surface energy is could be anisotropic when it comes to solids. So, this is something which is not present in liquids liquids we assume that the surface energy is isotropic that means it is not direction dependent. Therefore, I need to differentiate between liquid and solid surfaces when it comes to these quantities like surface energy and surface tension and let us do that next. So, we have a comparison of the solid and liquid surfaces now when I want to characterize a liquid surface I have a few parameters which I need to specify I need to specify the two quantities we just now dealt with the surface energy which we have given a symbol gamma the surface tension which had given a symbol sigma. And essentially you are characterizing the surface structure itself the liquid surface structure it characterizes by one number which is the surface density. In other words number of atoms per unit area which sits on the surface. So, essentially it is very easy for me to characterize a liquid surface because surface energy and surface tension represent equivalent concepts. And I just need one parameter which is my surface density to characterize my surface. And this is coming from the fact that liquid surfaces cannot support shear stresses. Suppose, I apply shear on a surface then the material will flow and therefore, liquids cannot support shear stresses. So, therefore, we have a term like surface tension now I want to compare this liquid surface with the surface of a solid. Then I would note that I could talk about something known as a surface energy. And instead of talking about the term surface tension I may have to talk about the true quantity which is surface stress which is a second order tensor. And in two dimensions it has got four numbers or four values which need to be specified additionally a surface is also associated with surface torque. The origin of surface torque is related to the fact that the surface energy could be anisotropic that means that it depends on my crystallographic plane. Suppose, I am talking about a crystalline surface the energy is going to depend on that. And therefore, suppose I am talking about two kinds of surfaces in two dimensions let me call this the one one kind of a surface vis-a-vis this zero one kind of a surface. The energy of this is expected to be different from the energy of this surface. Therefore, if an additionally these are of course, two important surfaces which are called low index plane surfaces. But I could also have a surface which is close to the gamma zero one surface or this is normally zero one surface which I can call a viscinal surface. Now, suppose this is my zero one surface I could visualize as surface which is inclined to this zero one surface at a small angle. So, I am considering an angle which is small. So, let me redraw this. So, this is my zero one surface and I am talking about a small angle with respect to this one surface which I call a viscinal surface. Now, since these low index planes are expected to have higher density and they are also typically associated with lower energy while these kind of high index planes would have higher energy the tendency for this plane to rotate into this orientation. Of course, it really does not rotate and typically you would find actually break up into steps and terraces, but that is a detail we will not consider now. For now I will note that this tendency for rotation to a low index direction or a low energy direction is the origin of what we may call the surface torque. Therefore, when I am trying to characterize a solid surface I need to specify more quantities as compared and I have to go into little more detail as compared to liquid surfaces. Now, the first of which we noted was that the surface energy and surface stress are not equivalent concepts anymore in a solid. The solid surface also associated with the surface torque and we clearly know that the surface the crystalline the surface of a crystalline material has structure and has more numbers need to be specified to characterize a solid surface. Macroscopically suppose I want to construct a surface from an infinite material then I need to perform a cut this cut itself will require 5 numbers to specify. So, for now I will simplify the problem and say suppose I want to make a surface then I need a certain orientation which I need to make this cut and this orientation can be given by 2 numbers or 2 independent miller indices in 3 dimensions and additionally I need to note where is this cut inside a crystallographic cell. Therefore, when I make a cut like this for instance in a sodium chloride crystal then I would notice that I would actually produce a polar surface and depending on where I make the cut I may have a set of negative charges sitting on the surface or negatively charged ions sitting on the surface or a set of positively ions sitting on the surface. Therefore, when I am making a surface in a solid I need to specify more numbers than I need to specify for the case of a liquid surface and in a crystalline surface I need to know all the lattice constants. In amorphous if I am not talking about a crystalline substance I am talking about amorphous surface then I need to know the density and a short range order parameter. So, in a sense a solid surface is very much unlike a liquid surface and that is when I am just even trying to create the surface for instance that is what the parameter I was mentioning when you try to make a cut. But additionally after the surface has been created it will be associated with quantities like surface stress and surface torque and as we had noted before a surface in a crystalline solid may undergo relaxation and reconstruction. Therefore, there are possibilities which do not exist in the case of liquid surfaces there in the case of solids the term surface tension which is actually should be avoided refers to surface stresses. Now, we will see what is the connection between the term surface stress and the term surface tension. The surface energy can be understood gamma which we have been dealing with can be understood as the reversible work required to create a unit area of surface. So, this can be understood for instance by using a soap film for instance between two sliders. So, I can take two sliders and create a soap film and assuming this one of these sliders is fixed and this is a soap film and I can pull this surface pull this slider here. So, this is my another slider which I can pull and then see how much work I need to do to create an unit additional area. Suppose, I am starting with this area A and after pulling I may create an area which is now this new area is A plus delta A then I would need to track how much work is required to create an additional unit area of surface and of course, I will be working at constant volume temperature and chemical potential with respect to all the species. So, this is an alternate way of looking at what is meant by surface energy. Now, I would like to connect my term surface tension in the case of solids with the term surface stress. Surface tension is the average of the surface stresses in two perpendicular directions that means that I take my sigma x x component and I take my sigma y y component and average them that is the term sigma which I have been referring to as surface tension. So, in some sense you can think of surface tension as the 2 D analogous term to the definition of hydrostatic pressure in three dimensions. So, in three dimensions we have the concept of hydrostatic pressure and we know that if you apply hydrostatic stresses on any volume of body then this body cannot undergo plastic deformation it can only undergo volume changes. So, we have a two dimensional analog of that which is now my surface tension which is the average of the two orthogonal stresses. An additional factor comes into play when I am talking about a surface which is the concept of surface entropy. Now, I am talking about surface energy I could just be talking about the just the internal surface energy or I could be talking about the enthalpy of the surface or I could be talking about the Gibbs free energy of the surface. Now, when I am talking about the Gibbs free energy of the surface I need to involve the concept of surface entropy and as the surface atoms have higher degree of freedom to move and hence they have a higher entropy. Though they may have a higher energy which is tending to I mean called destabilize the surface or put a surface at a lower stability level than the bulk. But, entropically a surface is more stable as compared to the bulk atom and the competition of the two is going to decide at what temperature I am going to have a stable surface with respect to a bulk. Surface stress at any point on the surface is the force acting across any line on the surface which passes through this point and in the limit that the length length of the line goes to 0. So, this is a way of determining surface stresses, but as we have seen that surface stress can be thought of as the two dimensional analog of the hydrostatic pressure which is a term used in three dimensions. So, let me summarize some of the concepts so far. Surface energy is not an energy in itself it is an energy which comes because of over counting and when I am talking about surface energy I am actually talking about the reference state being the fully bonded state or the bulk crystalline state. Of course, it may not be crystalline state it can be an amorphous state or liquid, but for simplicity we consider a bulk crystalline state. When we are talking about quantities associated with the surface I need to talk about quantities like surface energy, surface tension which are dimensionally identical and additionally I need to involve concepts like surface torque and surface stress. Surface tension itself can be thought of as an average of the two orthogonal stresses the sigma x plus sigma y by 2 and therefore, it is a 2 d analog of the hydrostatic pressure. When I am talking about solid surfaces I can no longer call the surface energy and surface stress to be representing equivalent concepts and additionally solid surfaces have a surface torque and when I am talking about solid surfaces I may note that the surface might have undergone relaxation or reconstruction and therefore, the crystallography of the surface could be very different from the crystallography of the bulk. So, some of these points I will just summarize which you already noted that liquid surfaces are characterized by a single number which is surface density. The short range order in liquids is spatio temporally varying hence no structure can be assigned to the surface. So, if at all we are going to assign any kind of structure to the surface in a liquid it is going to be a frozen snapshot of what is happening over a period of time. Crystalline solids have a definite 3 d and hence structure in 3 d and hence additional parameters are required to characterize them. The order of this at the surface of a crystal can be very different from the bulk and this is arising from surface relaxation and surface reconstruction and even in absence of these quantities you would expect that the surface is going to be somewhat different from just the bulk termination. Amorphous solids have short range order, but no long range order under low temperature conditions in short times. That means, we are talking about low atomic mobility regimes the atomic entity positions are temporarily fixed. So, I may have short range order on a amorphous surface, but no long range order and under typically at low temperatures I will assume that this short range order persist over long times. Now, we are already noted that crystals tend to have a certain phase stud shape and how can we obtain this shape of a crystal is the next question and this question is very very closely related to the fact that we are considering surface energy. So, we are already pointed out that surface energy depends on the direction that means it is an anisotropic quantity. For instance suppose I am talking about a 3D crystal then my 0 1 0 surface may have a different energy as compared to a 1 1 1 surface which is expected to have a different energy compared to a 1 1 2 surface which will again be different from every other surface which I can think of in a crystal. Depending on the crystallography or the symmetry obviously a suppose I am talking about a cubic crystal then the 0 1 0 surface energy is will be identical to the 1 0 0 surface energy which will be identical to the 0 0 1 plane surface energy. So, this is coming from the crystal symmetry. Now, if I grow a crystal then typically it ends up in a certain polyhedral shape which is called the growth shape of a crystal. And suppose I now anneal this crystal for long times then the system will tend to go it is to into its low lowest energy shape state which is an equilibrium shaped state in which case the crystal will put out those phases which gives the system overall low energy state. And how do we determine this low energy state of the crystal in other words how do I determine the equilibrium shape of a crystal this is done by a construction known as the wolf construction. And to a reiterate the wolf construction is going to give us the long time stable state of the crystal and not the growth shape of the crystal. Because crystals may grow in shapes which are slightly bit depend on the kinetics and kinetics of attachment etcetera. And those phases which grow very fast will exhaust themselves unlike that when you talk about equilibrium shape it is actually in unique shape which gives a system the low energy state. The way it is done is that we start with the variation of the surface energy with respect to theta. So, we have a polar plot of the surface energy with respect to theta. And you can see that suppose I am considering a 4 fold direction in a cubic crystal then this gamma plot will also have a 4 fold symmetry. Another important point to notice that this gamma plot would have certain cusps here as you can see which represent low energy surface orientations. Then we draw a radius vector from the origin to intersect the wolf plot like for instance O A in the plot I take my O origin and I draw a tangent and allow it to intersect. Now the figure formed by the inner envelope of all the perpendiculars is the equilibrium shape. So, you can see if I keep on drawing inner perpendiculars then I get a shape which is now for instance given by this blue box and this is my equilibrium shape of the crystal. Of course, this is schematic here and various crystals would have different kind of equilibrium shapes which may have 111 surfaces 100 surfaces. But typically these are low index surfaces which are thrown out by a crystal. A point to be noted is that those low energy orientations obviously or the cusp orientations are the more preferred ones and typically will show up in the what you might call the wolf construction. So, essential message from this is that because of anisotropine surface energy crystals will tend to have polyhedral shapes. And in the case of nano crystals because it is surface dominant system the system would or the crystal will tend to go towards its equilibrium polyhedral shape in much shorter time scales as compared to a bulk crystal. And of course, we know that most of the bulk crystals we deal with typically could be a polycrystalline material which means that we are not actually going to get a single kind of a wolf construction which we are talking about here. So, wolf plot gives us the equilibrium shape from the equilibrium shape it is of course, not uniquely possible to go in the reverse direction make on wolf plot. Wolf plots with sharp cusp represent the equilibrium shape which is typically a polyhedral. If of course, the anisotropy is small then you may not get a nicely faceted crystal as you can see here with very low number of planes representing the outer surface. The width of the crystal facets is was proportional as one by the surface energy. That means that the largest facets are the ones with the lowest energy over the crystal. Obviously, we put out those surfaces which have the lowest energy. Now, we move on to a equivalent, but in some sense distinct concepts that how do I classify my nano crystals or clusters based on certain kind of a property. Now, we have seen that we can go to the nano scale from two possibilities. One I can make a bottom up approach. That means I start with a few atoms or a few molecules and try to make larger and larger clusters and at some point of time if these cluster of atoms or molecules arrange themselves in what you may call crystalline order then I am I can be actually talking about a nano crystal. So, we will introduce three terms here with respect to properties. One I will call the small cluster regime. One will be the large cluster regime and of course, I will go into a region which is called the nano particle regime. And in as we shall see now that one of the special class of interesting nano particles are the nano crystals or the crystalline nano particles. And for now, we will assume that this nano particle is a single crystal. That means it is does not have further grains into it. In reality of course, when I am making a top down approach you might notice that some of the particles could actually be polycrystalline. Now, why do I want to differentiate these three classes and how is it how is it meaningful in understanding their properties is what we will address next. Later on we will also take up an alternate way of classifying some of these because in literature often you would find not only classifications of the kind just now I mentioned, but there are other size regimes which are defined like people define molecules, people define micro clusters, small particles and micro crystals. Some of the regimes and what you may call the property regimes and the definitions would tend to overlap, but depending on the literature you are reading and with respect to certain specific properties one of these two may be used by an author to actually describe what you may call a cluster of atoms. Now, how can I differentiate my three regimes the small cluster regime, the large cluster regime in the nano particle regime and why do I want to do that is what we are considering. In the small cluster regime which could now be in the sub nanometer regime the structure and properties do not vary in a monotonic way and we shall soon see that in this regime we will have certain numbers called the magic numbers when you are talking about clusters. The important point note suppose I am changing a property here which could be a property related like magnetization it could be some other property I am talking about and there is the dotted line here which talks about the bulk value of this property. That means if I take a bulk material and I measure its magnetization it will have a certain value. Now, the three regimes which are approximate of course here drawn here are these three regimes the regimes of the small clusters the regime of the large clusters and the regime of the nano crystals. In the small cluster regime if you notice that just by a mere addition of one atom a property could actually change very very drastically. Now, this could be for instance a stable configuration this could be an unstable configuration and hence the property could change very drastically by mere addition of one atom or the removal of an atom. So, if you track the property in the small cluster regime you will see that actually it is going undergoing large fluctuations. The nature of these fluctuations and the range of these fluctuations obviously depend on the property and when we get down to some of the properties you will be able to see that how actually these properties could vary. But the essential thing here is that I cannot talk about an average property in this case I have to take each cluster for instance a cluster of 12 atoms a cluster of 9 atoms separately and study its properties. And of course this as you can visualize would be a challenging task I need to first of all separate out these clusters of the same size then make a measurement on the properties. And any presence of any other size clusters would contaminate my results. In the case of the large clusters which is basically intermediate between the cluster and the particle or the nano crystal there is still fluctuation in the properties. And here we are talking about a size regime for instance in the nanometer size regime there are still fluctuations. But these fluctuations are much more gradual and typically if you add for instance a few atoms or one or two atoms you do not expect that these properties is going to change in a very drastic way as it would happen in the case of the small cluster regime. So, here also there is some fluctuations in the properties here also you would notice that there are some possibilities of certain sharp minima and maxima. But the overall trend line is not fluctuating that large when you actually add one or a few of the atoms or molecules to the cluster. On the other hand we expect that somewhere all these fluctuations have to die down and I have to approach the bulk value of my property. Where in now I can I do not have to no longer deal with individual clusters like a 30 atom cluster or 32 atom cluster separately. But I can talk about an average property which of course does not change with size and that is my bulk limit. But somewhere in the large nano crystal regime I would expect that this limit is obtained. And this is of course easy to understand because this is now my what you may call the more monotonic kind of a variation in property. So, here I am not observing fluctuations the system is slowly tending towards the bulk limit in a monotonic fashion. How many atoms actually constitute a small cluster? How many atoms would I put together to form a large cluster or nano crystal? Of course depends on the kind of property I am studying and the kind of system I am considering. So, for instance suppose I am talking about as we can study the magnetic property ferromagnetic property and I am talking about nickel atoms this variation could be different vis-a-vis for instance suppose I am talking about some totally different property like for instance the corrosion resistance or oxidation or adsorption of as foreign molecule vis-a-vis some other particle like a ion particle. The intermediate regime is a regime where I would presume that the crystalline order has not yet been established. In small clusters they are not even sufficient number of atoms to even talk about for instance a long range crystalline order. Because when I am talking about a crystal I am assuming that at least there are few tens of unit cells or a few hundreds of unit cells in each dimension and this long range order which I call a crystalline order has been established. But in the case of small clusters it is meaningless to talk about it. In the case of nano particles which is now also as we can consider an example of nano particles the crystalline state there it is clear that I have enough number of unit cells to classify that as the nano crystal. In the large cluster regime this is a some sort of a gray area where I perhaps do not have enough number of atoms or unit cells to classify that as a nano crystal. These large clusters may be crystalline in some sense that they may not represent just a polyhedral shell but in this toward this end they may actually have a few unit cells. But along the way you would notice that the structure might change from that of some kind of a cluster behavior where I have no kind of unit cell formation to a regime where you actually have a crystalline structure. Here though we are not explicitly talking about the structure we are actually talking about the property. But it is very clear that this property would be a sensitive function of the structure and therefore it is meaningful for us to correlate the structure with the property we are talking specifically. So, to summarize this slide there are three regimes which we can talk about which is a small cluster regime, the large cluster regime and the nano particle regime. And the specific examples of nano particles we are talking about here are nano crystals and took for something to qualify as a crystal we should have sufficient number of unit cells. So, that a long range order has been established in the small cluster regime which is in some sense a very interesting regime. A mere addition of one particle or one atom or one molecule can actually lead to a drastic change in the properties. And these fluctuations with respect to size would die down in the large cluster regime where in the oscillations become smaller and of course in the bulk in the nano crystal regime it becomes monotonic. So, let us now talk about going from clusters to nano crystals a little further. Clusters are aggregates of atoms of course they could be aggregates of molecules or ions also usually less than a few thousand atoms. So, when I am talking about clusters typically I am dealing with a less than about few thousand atoms. Typically when the term cluster is applied to a collection of small number of atoms. So, we do not use the term cluster for large collection for instance in the case of a nano crystal. The structure and properties of clusters are often very different from the constitutive atoms and their bulk counterparts. So, this is something to be kept in mind for instance suppose I could be talking about a material which is anti ferromagnetic in the bulk, but which could turn ferromagnetic in the small cluster regime. Therefore, there is a drastic change in properties which are possible when I go to the cluster regime as compared to the bulk counterpart. If the cluster is made up of molecules the individual molecules themselves could have a certain kind of a property, but the property of the cluster could be very different from the property of the individual molecule or the constituent atoms. At this level of the cluster there is not sufficient atoms to classify as a crystal as we saw because which there are insufficient atoms to establish this long range periodic order. If an atom is added to a small cluster of atoms reconstruction usually takes place I mean reconstruction with respect to the structure which is the rearrangement of the atoms. And we have seen that this implies that the properties of two clusters which are close to each other in terms of number of atoms could be very different. And this is what makes the study of these clusters very exciting. At a certain larger size the critical size the bulk structure will be established when I am when we mean a bulk structure we already seen we are referring to the bulk structure not only in terms of the tangible kind of a definition, but here we are actually referring to it in terms of the property based definition. Small metal clusters for example, on substrate assume icosahedral bipyramidal etcetera kind of shapes and obviously these icosahedral bipyramidal shapes do not represent crystals. In many cases the size it is about 150 nanometer the crystalline structure is obtained. That means that now in the small cluster regime there are certain polyhedral shapes which are not crystalline or we just they are just clusters. When you reach about 150 nanometers you obtain the large bulk crystalline limit and in between the two you could actually be stabilizing for instance different kind of crystal structures which are not the bulk crystal structure. So, let us summarize some of the aspects we have seen with respect to clusters and nano crystals. The shape of nano clusters and also nano crystals can be very different from their bulk counterparts. And now we are talking about the shape of the external polyhedral which forms in the equilibrated shape which we have seen that how we can obtain from a wolf construction, but wolf construction typically applies to large crystalline macroscopic tangible bulk kind of a crystal to rather than to these small clusters. And therefore, I may have wolf construction for the nano crystal or the bulk crystal and I may have these clusters and these two shapes typically are very different. Large crystals may adopt large clusters and of course, small clusters may adopt a crystal structure. Now, we are talking about large clusters because in the case of small clusters we cannot even talk about the crystal structure. The large clusters may adopt a different crystal structure as compared to the stable bulk crystal structure. So, this we have noted already and we will also see examples of phase transformations later on wherein when we reduce the crystal size we see that new phases are stabilized. We already seen couple of those examples in the case of Fe 2 or 3 etcetera. Given the large surface to volume ratio the equilibrated shape can be attained quickly in nano crystals and also in large clusters. So, if I have a shape which is in when I am synthesizing a solid or it is produced by certain method and it is not the equilibrium shape it can reach the equilibrium shape very very quickly. Now, we will see next that a certain magic number of atoms may be stabilized in clusters and we will also see what is the origin of these magic number of atoms. In other words when I am talking about a cluster the cluster instead of behaving like a collection of individual atoms tends to start to behave like something like a super atom. That means, the entire cluster itself can be visualized as an entity in itself which can be thought of as a super atom and not only that these nano crystals themselves may serve as motifs in higher order hierarchical construction of crystals. And we will take up some examples wherein of course, in many cases you have what is called self assembly wherein I start with a nano crystal and then make use this as a motif in making a larger unit crystal wherein of course, I am not worried about the ordering of atoms within this nano crystal. But, I am talking about ordering of the nano crystal themselves in a periodic fashion to give rise to a what you might call a crystal of an higher order. So, I can have an hierarchical construction of crystals and this is a very interesting concept which we will explore further. One interesting example we talked about shapes of crystals and when you are talking about a shape of a crystal we are typically assuming that is arising naturally from some kind of a polyhedral arrangement. But, if you look at some of the examples like for instance I am here showing a multiply twin gold nano particle a schematic of that which is about 35 nanometers. And we have talked about this kind of a structure with respect to what we may call a rotational twin. So, we had given this as an example of a rotational twin. Suppose, I have twinning occurring in the enable crystal. So, typically some of these twin boundaries would exist go in a for instance in a copper crystal would extend from one green boundary to another. So, let me draw that schematically here suppose I am talking about a copper poly crystal. So, I have a green structure like this and I may notice that within a single green you may have one twin boundary and of course, another twin boundary extending from one green boundary to another. So, suppose I have crystallographic planes here these crystallographic planes would be reflected here by this and this other twin would actually restore my orientation. So, this is of course, twinning in a polycrystalline material, but many a times this kind of twinning would lead to internal stresses in the material. But, in the case of a nano particle because of the way these arise and the fact that they are not embedded in a matrix they actually these stresses could not be large and you could actually stabilize an nano particle which is having for instance in this case a 5 fold rotational twin. Now, of course, we had noted in that context of this rotational twins that if I had to put a selected area diffraction aperture here I would mimic a system of higher symmetry. In fact, I would think as if there is some 5 fold symmetry arising from this kind of a twin particle. The two variants in an ideal circumstance in such a crystal would could be related to a 72 degree rotation. A 72 degree of course, is an ideal kind of a rotation I am talking about and if you refer to some of the publications you would notice that the actual angles involved could be close to 72 degree, but they are may not be exactly 72 degree for instance they could be 71.2 etcetera. So, they are close to 72 degrees giving this some sort of a close to a 5 fold rotation. In twinning of a diamond film for instance the angles have been found to be again close to about 72 degrees, but the interesting thing here is that in systems wherein I am talking about twin as a defect in a perfect crystalline order they could be much more easily stabilized in the case of nano particles. In fact, in many nano particles we will see that the deformation mode will switch from that of what you may call a dislocation plasticity to a twinning kind of a plastic much more easily. And this is some good feature that we have in nano particles and nano crystals that certain defects are much more easily formed and some of these defects may actually may not be found in their bulk counter parts. And additionally of course, you may have gold nano particles about 20 nanometer which can form within icosahedral shape. So, we know that icosahedral symmetry is not compatible with translation, but still you find nano particles which can have these kind of a shape which is otherwise not seen for bulk crystals. So, with respect to crystal structure with respect to properties with respect to defect structures within the material we may note that nano particles and nano crystals and nano clusters could be very very different from their bulk counter parts and this is something which is very very important for us to note. We had stated initially when we consider the classification of nano clusters visa or small clusters visa we large cluster visa we nano particles or nano crystals that there are alternate classifications used in literature. And we had even noted when we try to make the assembly called a cluster that we may not need not start with an atom we could actually start with an ion or we could even start with the molecule. And therefore, it is for us good to revise certain other classifications which are typically found in literature one of them is shown here, where in a slightly different terminology as compared to what we have just considered is introduced. In some sense the micro crystal which is mentioned here is very very similar to our nano crystal in some sense, but the size regimes could be overlapping with these, but may not be identical with what we are classifying here. So, we could be starting with molecules and here we are talking about individual molecules like CO 2 it could be O 2 etcetera. Then we go to the regime known as the micro clusters regime, then we have the small particle regime and you may notice that the micro cluster regime is similar to our what you may call the small cluster regime. And the small particle regime can again be somewhat coincidence with the large cluster regime and micro crystals as you can see are much larger, where in you have about 10 power 6 number of entities. So, you said atoms here, but we should again note once again that if the constituent entities need not be atoms they could be ions or molecules. So, for molecules it is about 1 for micro clusters it is about 10 power 2 to 10 power 3 entities for small particles it is about 10 power 4 to 10 power 5 and larger than 10 power 6 can be thought as micro crystals. The radius in nanometers for micro clusters is about 1 nanometer or it could be sub 1 nanometer and small particles are about 10 nanometers and micro crystals are larger than 15 nanometers. When you are talking about bonding characteristics or the atomic arrangement molecules of course, you have the usual rule of bonding. In other words they could be covalently bonded or the bonding could be polar covalent etcetera. In micro crystals micro clusters the surface and interior atoms have behavior very different from bulk atoms. Now, I am comparing the surface atoms sitting in a micro cluster and the interior atom in a micro cluster with what you might call a bulk atom in a bulk crystal. And I am talking about a bulk crystal I am considering something having more than about 10 power 6 may be 10 power 8, 10 power 10 kind of an atoms or even larger number of atoms. So, here the surface is different from the surface which we can think of in a bulk material. The interior atoms are also going to be different compared to that what we see in the case of a bulk crystal or a bulk particle. In small particle regime the interior atoms are similar to bulk that means they have already grown so large that their overall what you might call environment and configuration is somewhat similar to the bulk interior atom, but definitely not identical. But the surface atoms still behave very differently because still suppose I am considering a small particle still I cannot ignore the curvature effects. And sooner during these course of these lectures we will see that how surface effects can alter the equilibrium between a matrix and a small particle. And this is the origin of what you might call the Gibbs Thomson effect, but essentially the surface atoms still are very very different from the surface atoms in a bulk crystal where in of course curvature effects can be ignored. In micro crystal surface is still an active site, but this can be approximated to the surface of a bulk crystal. When I am talking about electronic properties for instance and of course I can go on in the list of properties which I am going to consider, but for just now for now I am going to just talk about this as an example electronic properties. The energy levels are discrete of course for a molecule we know how a molecular energy levels are they are similar to an atomic energy level. In the case of micro clusters the valence electrons exhibit shell structure with magic numbers. So, we will return to this concept very soon where in I am talking about certain magic numbers and certain kind of a stable electronic configuration, but nevertheless here too I do not expect any kind of a continuous band formation and still the energy levels are discrete. In small particles still quantum size effects are going to dominate and in the micro crystal level you would expect that you can now talk about more like continuous bands which you talk for a bulk crystal. And therefore, you could talk about conduction properties for instance which is very similar to that of a bulk crystals, but here too you would expect the surface plasmon are playing a dominant role and similarly other surface effects coming from electrons on the surface is expected to play a dominant role in determining the properties of the material.