 Let us consider a few more points about clusters and importantly about what we call the magic numbers associated with some of these clusters. Clusters can be synthesized by various techniques like sputtering, supersonic jet or gas condensation, laser ablation, vaporization etcetera. Some of these clusters are very suitable for ceramic materials while others may be good for carbon and other specific kind of compounds which may be whose clusters we want to construct. Characterization of clusters requires important techniques like mass spectrometry which not only tells us the mass of the various species which are produced during a certain process, but also helps us separate out the various clusters which may be produced as a mixture in the original process. Transmission electron microscopy can give us the local structure of these clusters and also the defect structure within these clusters X-ray photo emission spectroscopy etcetera are other techniques which can add to our information that we gain about the clusters. We had been mentioning about the stability of certain number of atoms ions or molecules in a cluster. So, let us see what is the origin of the stability and what kind of numbers we are going to encounter when we are talking about these magic numbers. We typically encounter magic numbers though it is not often stated in that language that magic numbers are usually associated with nucleons in the nucleus of an atom and the electrons in the orbitals. We already know that for instance suppose I have a s orbital then I know that only two electrons can sit there p orbital can accommodate six of them and I also have the formula something like 2 n square which go into a particular shell. Therefore, though we do not often say this it is obvious that given Pauli's exclusion principle and the Hoon's rule we know that only a certain specific number of electrons can be accommodated in a particular electronic configuration and these can constitute what we call magic numbers at the level of the atom when we are talking about electronic configurations. Similarly, nucleons also have their own magic numbers that means not that any set of nucleons are allowed within the nucleus. But the term magic number is not often used to refer to these kind of numbers which come out of such an what you might call in filling rule or such a kind of an electronic configuration. But in the case of these clusters this is of more often than not the term which is used analogous to these terms which we are using for electronic or nucleonic configuration are atomic clusters having their own magic numbers. Mass spectroscopy shows peaks corresponding to certain number of atoms in a cluster. The magic numbers for stability of clusters these are the magic numbers we are talking about these reflect the bonding characteristics between the atoms. Therefore, what we are saying here is that when I make a have a process in hand where in clusters of various sizes can be produced simultaneously. Then I would find that not all clusters are equally abundant in other words there are peaks corresponding in the mass spectroscopy to certain number of atoms in a cluster. And the reason behind this is obvious that these clusters are more stable than the others. And that is why they are produced in a higher abundance in corresponding to other clusters which may not be that stable couple of examples we see here first. Where we can talk about xenon which is a noble gas and sodium clusters the subscript here n refers to number of atoms in a cluster. So, suppose I am talking about the stability of xenon clusters then we find that those clusters with 13, 19, 25, 55, 71, 87, 147 etcetera atoms are more stable. Then say for instance a cluster with something in between like a 15 atom, a 16 atom or a 17 atom cluster. Now, these numbers why are these the automatic question arises is that why are certain number of cluster atoms in a cluster more stable than the others. And there are two reasons as we shall see in the next slide, but one of the reasons is that these numbers correspond to the mechaicosahedral kind of a cluster. And xenon is of course, not the only example couple of more common examples are calcium and magnesium and they also show these exact precise kind of a what we might call magic numbers. This kind of an ordered arrangement is incompatible with crystalline symmetry. This arrangement is different from multiply twin crystals which may be icosahedral packed. So, in the case of xenon we noticed that certain magic number of clusters are more frequent than the others. And this is similar to the case for calcium and magnesium also. And when I am talking about a 19 atom cluster or a 25 atom cluster this is not a crystalline packing. It is a packing with icosahedral symmetry which is different from what you might call a crystalline kind of an order which is a long range order. And it is also different from multiply twin crystals which may also mimic some of this symmetry. And in a few slides back we had all in fact talked about some such example of a multiply twin crystal. And we had pointed out that often this kind of a system would mimic a symmetry higher than that present actually in any one of the crystalline variants when this is a crystalline variant. And we had also pointed out that in the case of nano crystals or a nano particle this kind of a twinning may be more feasible. Because of the lack of constraints of bulk as compared to a as compared to the bulk crystal where in such kind of a twinning may not be that easy. Therefore, these stable xenon clusters have certain numbers which stabilize them and these are the magic numbers. And these clusters can have what you might call an they they are coming from geometrical origin of packing around with a symmetry which is like the icosahedral symmetry. On the other hand if you look at a sodium n cluster a different set of magic numbers emerge. In other words the first thing we note is that there is no universal set of magic numbers. And depending on the kind of effect we are taking into account and the kind of system we are talking about there are a set of magic numbers which evolve. In the case of sodium we note that not again not all numbers are equally probable numbers like 8, 20, 40, 58 and 92 atom clusters are more stable than the other ones for instance a 12 atom cluster or a 14 atom cluster. So, this stability can be understood as arising from valence electrons of sodium moving in a spherical potential. We will have a little more to say about this in the coming slides and also lot more to say perhaps later. But essentially we see that to summarize this slide two important aspects come out. Number one that not all clusters are equally stable. Number two there are magic numbers associated with the stability of clusters. These number three these magic numbers are system dependent and the reason behind these magic numbers is also not the same. So, suppose I want to understand these magic numbers I can think of that in two ways one is a geometrical reason another is an electronic kind of a reason. Now, these filled shells of electron shells leading to a spherically symmetric charge density lead to a van der Waals kind of interaction between atoms which is what is stabilizing the kind of xenon clusters which is in no different in behavior from say the other inert glasses like neon argon krypton because these filled electron shells themselves do not have a propensity for bonding like for instance an unfilled electron shell. And therefore, the bonding between two atoms in a cluster is more of van der Waals type and even when larger congregations of these atoms give rise to in crystalline array even in that case the bonding at the fundamental level between the atoms remains of the van der Waals type. That means that these obviously will have a very low melting point and also a very low perhaps a sublimation or a vaporization point vaporization point as compared to something which is covalently or ionically bonded. In the case of the Makai aqueous aedle clusters we know that the aqueous aedle symmetry is incompatible with translational symmetry because this is the symmetry typically found in quasi crystals. Microns size aqueous aedle clusters also have been synthesized not only do we have these small clusters which have aqueous aedle symmetry which is what we saw in the case of these numbers which are coming for instance 30, 19, 25, 55, etcetera these are can be considered as small clusters because we have very small finite number of atoms. But much larger clusters and now we are talking about micron size crystals not crystals, micron size aqueous aedle clusters have been synthesized and this can be thought of as a massive achievement. That means that even in this micron size cluster we have managed to avoid crystallization and we have maintained this what you might call and growing order from a point outward to get these kind of micron size cluster. So, this is a very interesting kind of a aqueous aedle cluster we are talking about whose scale is not in the few atom level whose scale is not even in the nanometer level, but whose scale is in the micron level. Under high pressures and we are talking about 5 gigapascals B 6 O clusters with aqueous aedle shape have also been synthesized. So, these are couple of beautiful examples of not only of inert gas kind of an element which interacts with van der Waals type, but also even compounds which have been synthesized in aqueous aedle shape. So, these are very interesting examples. Another important point to note in the case of these clusters and again we are talking about small clusters here. Bond lengths of the atoms inside the cluster are smaller than the bond lengths of the surface atoms. That means now we have already seen this that surface is an important part of an nano crystal or a cluster may be small or big and this is clearly seen in this example. Where in the bond lengths are not uniform the atoms inside the cluster are better bonded. I have a smaller lattice parameter or we cannot call it lattice parameter inter atom distance as compared to the atoms on the surface which faces surface relaxation and hence the interior of the cluster is under higher pressure. So, these are some important points we need to note here. Number one that though there is a tendency for larger clusters to not exist clusters, but become actually crystalline, but there have been examples specific examples where in larger size clusters have been synthesized. Second thing that the inner cluster not all atoms are equivalent not all bond lengths are equivalent and as expected the surface atoms have a certain bond length which is different from the atoms in the interior of the cluster. So, we had said that there are two reasons why we can have a magic number of atoms in a cluster and we took two typical examples for this the xenon n which is predominantly position ordering kind of a system and the sodium n system which is predominantly an electronic shell kind of a structure. In other words here the stability is coming from geometrical reasons and here it is coming from the electronic structure reasons. So, there are two different reasons and we also seen the numbers like for instance the numbers for the electronically stabilized structure is like we have seen is 8, 20, 40, 58 etcetera while in the case of this there are icosahedral what you might call close pack configuration because icosahedral kind of a configuration gives us a close packing, but only in the short range because the long range we typically prefer have a crystalline kind of an order. So, they have the different kind of a magic numbers. Now, and we had already seen that this stability in the case of sodium can be thought of as arising from the shell structure of valence electrons of sodium moving in a spherical potential. The important and the crux of the whole this slide is that this whole cluster can be thought of as a super atom. In other words now we have the concept of an atom with whom we know that whose stability depends on a certain kind of an electronic configuration. That means, a filled shell would actually given atom a stability an electronic stability. Similarly, here we are not talking about individual atoms constituting the basic building block, but the entire cluster behaving like a super atom. In other words now that is the reason that when I add a single atom to for instance an 8 atom cluster or an atom to a 13 atom cluster I do not have a stable configuration. You require to add say from 13 6 more atoms to go to 19 atoms before you have a stable cluster because now this is akin or very similar to our case of electronic configuration where in you have a filled or a half filled orbital which gives us a certain stability. And the origin of the stability to some to reiterate is coming can coming from purely geometrical or structural factors or it can come from electronic reasons in other words electronic shell structures. And this whole cluster now can be thought of as a super atom and if they have these magic numbers then they are stable and this super atom can itself function as a building block in some kind of an hierarchical or a higher order assembly. And we will take up a few of these higher order assemblies as we go along in what we may call an hierarchical construction. So, I will summarize these two important slides in summary there are some stable magic numbers available these stable magic numbers of clusters have a structure which is very different from that of a long range crystalline order or some kind of a defected crystal like a twin crystal. These magic numbers can arise from geometrical or electronic stabilization geometrical stabilization implies that I am getting a close pack structure. And in this kind of sometimes these clusters can be grown to larger sizes and this whole cluster can behave like what you may call a super atom which itself can be the fundamental building block in a larger assembly. So, having made a cluster we may want to ask some of the questions right here and we need to address some issues the answers for some of these questions will be perhaps I am not at fully mean understood some of these questions may be answered later. These are important questions before we go from the level of an atom to a cluster which we already achieved to a much larger unit like a crystal. And we have already asked ourselves some of these questions like what is bulk what is a bulk like property etcetera. So, the issues are when are there sufficient atoms to call a cluster a cluster. So, and we will precisely address these issues in some of the coming slides, but is it enough that I have a few unit cells or 10 unit cells or 3 unit cells or a 3 by 3 by 3 unit cell can I call it a crystal. When does a collection of metal atoms show metallic character. That means now I am talking about metal atoms I am talking about the bulk structure being a metal rather than individual atom and there is no point talking about individual atom to be a metallic or non-metallic. Therefore, when does my now atoms which would be metallic in the bulk start to behave like a bulk metal. So, this is the second question I can ask myself when does the energy levels of a semiconducting cluster become continuous. That means, that we know that semiconductors have bands and these bands do not overlap for instance the valence and conduction bands do not overlap. And if you have only a few atoms in the cluster then obviously, you have discrete energy levels there is not enough sufficient atoms to give us a semi continuous levels which you observe in the case of a complete large crystal consisting of say a mole of atoms. If you notice then you notice that energy levels remain discrete up to 7 nanometer clusters in the case of cadmium sulphide and even up to 14 nanometers for gallium arsenide. Of course, the exact transition in many of these cases as you would have appreciate is going to be blurred there may be states intermediate to what you might call a bulk metal and those which you might have to call individual clusters. But, nevertheless these are interesting questions which we need to keep in mind when we are trying to study the properties of clusters and growing larger and larger clusters with some point of time merge in merge into the bulk level. Now, let us take up some individual examples of clusters and we will consider clusters which are like alkali halide clusters. We will also take up some semiconducting clusters we will also consider metallic clusters and we will scale ourselves to consider semiconductor nanoparticles. So, we will consider a few type of clusters to understand again the aspect of what you might call the magic number and also the to understand how a cluster even though locally might resemble what you might call the bulk kind of a structure it is definitely different from the bulk structure. So, these couple of things have to kept in mind of course, we already seen those other examples where the cluster level structure is totally different from the bulk level structure. So, this is have to be has to be kept in mind. So, examples of alkali halide clusters are and we know that alkali halide clusters can be ionic in nature, lithium fluoride, sodium chloride, copper bromide, cesium iodide etcetera. And again like in the case of the other clusters we saw that there is an irregularity in cluster abundance when we look at the mass spectrograph. In other words certain number of atoms in the cluster are more stable than certain other number n and when we are talking about n in these cases we are talking about for instance a CS CSI n kind of a cluster which it has a net positive charge. And when we look at the cluster like this it will become obvious and here we are seeing for instance a sodium chloride kind of a cluster and here an important point has to be kept in mind that we are actually though this diagram resembles or is showing something we might actually see for the unit cell of a sodium chloride kind of a structure we should know this is not a unit cell. So, this should not be constitute as a unit cell what we are talking about here is a cluster. Of course, a real cluster would not look like this the actual structure of a cluster the actual structure will be relaxed distorted with respect to this ideal structure as expected. Because this of course, is just a geometrical representation in ideality this can function as a unit cell in a much larger what you may call structure where in you have few maybe a few million atoms, but when you are talking about small size clusters and this obviously, is not going to be there what you call the ideal geometric configuration is not the stable structure. Now, we have seen that this formula for instance CS CSI n plus let us see what is the origin of this kind of a formula. Now, suppose I take a cluster like this and now I calculate the number of atoms in a if this were a unit cell in an infinite or a large crystal. Then each one of these corner ions would actually be contributing one fourth one eighth to this unit cell. Similarly, and blue ion which is chlorine in this example considered would actually be contributing one fourth to this unit cell, but in a cluster we make a calculation based on net number of atoms. That means, there is a complete contribution to this structure and therefore, there are phi of these here and 4. So, that makes it 14 sodium ions in this cluster, but then similarly, when you count the number of chlorine here it is 1 2 3 4 4 in the mid plane plus 1 5 plus 4. So, that makes it 12 plus 1 13 chlorine and that leaves me with a net charge of plus 1 for the entire cluster. Now, if I look at the stability of such kind of clusters I would notice that there are some nice geometrical patterns which evolve here like a 3 by 3 by 3 cluster is found to be more abundant. Similarly, a 3 by 3 by 5 a 3 by 5 by 5 etcetera clusters are found and these clusters have number of atoms going as 13, 22, 37, 62. So, this is not number of atoms of course, this is n which would go into this formula here n wherein the number of chlorine atoms is what is n and the number of sodium atoms is 1 more than the number of chlorine atoms giving rise to a net positive charge to the entire complex or the cluster. If you talk about larger clusters with n equal to 171, 364, 665 etcetera and we are specifically taking an example of sodium iodide here are very symmetric and start to resemble the bulk structure with an FCC lattice. That means that again there is a transition from cluster like behavior wherein in some sense it resembles the larger unit to some extent unlike the previous cases where it started taking an icosahedral form like the example we saw here. Here, there was no resemblance between the short range cluster shown in the right bottom and the real crystal structure which the system would adopt, but here there is some resemblance though not a complete resemblance between the unit cell of a larger cluster or a larger crystal and a cluster which is observed for small number n. Therefore, at some point of time for instance with the number of number n starts to touch 665 or 1000 or more then you try to observe that it has a bulk kind of a behavior and this unit which we are seeing tends to go to a more what you might call geometrically perfect configuration. So, when we consider the alkali halide clusters we see that there is a market difference from the clusters we saw for the sake of xenon etcetera. And here there are other kind of magic numbers coming, but these magic numbers can be easily understood in terms of the geometry of the cluster which we are seeing here as for the example of sodium chloride. Now, if you go to semi conducting clusters we again see a richness of possibilities at this point of time of course, we will not go into the details of all these possibilities, but it is important to note that there is a lot of richness there is lot of variety there is a lot of physics which can come out of studying these kind of semi conducting clusters. And when I am using the word semi conducting clusters what it is typically implied here that these are semi conducting in bulk form. So, we are referring to something like silicon germanium etcetera which are semi conductors in the bulk form. And we are not implying that the clusters themselves are semi conducting. So, this is not the implication. So, we have been using terms for instance later on like metallic clusters again here we would actually imply that the bulk form is actually metallic or the bulk form is semi conducting and not the cluster itself which is actually comprising of a small number of atoms. And we already seen there are market differences in the electronic structure and the physical structure between the clusters and that of the bulk form. Silicon and germanium prefer maximum coordination number is in cluster which is unlike carbon. So, there is a difference between for instance carbon germanium silicon which all adopt diamond cubic structure. And which are all we might call between semi conducting and insulating because carbon has a larger band gap as compared to silicon and germanium. And but they also have this diamond cubic structure and with a 4 coordination around each one of those silicon or germanium atom. Very small clusters when you are talking about very small we are talking about n less than 10 of silicon and germanium are difficult to produce and stabilize. So, while we are seen for the case of some of these earlier what you might call noble gases we could easily synthesize some of the larger clusters which was stable. But here even though there is a possibility of synthesis of smaller clusters they are usually difficult to produce especially for silicon and germanium. Cluster beams with n greater than 60 for silicon and n greater than 50 for germanium have easily been produced. So, it is in the small scale regime or the small size regime for clusters that is difficult to produce, but larger clusters have been easily synthesized for silicon and germanium. When silicon 12 plus is fragmented silicon n plus with n is equal to 10, 8, 6, 4 are produced. So, there are techniques by which actually you can produce these smaller clusters that is by actually fragmentation of larger clusters and hence you can produce some small clusters as well. Small clusters of silicon can have metallic or covalent character that means that they can be different from what the bulk form is. Small clusters with higher coordination number than the crystalline state because crystalline state we know has a coordination number 4 imply a metallic character. For instance Si 7 has some kind of a metallic character as compared to say something like Si 4. Among small clusters it has been observed experimentally that n is equal to 7 has pentagonal bipyramidal structure. That means that again we are noting that it is not the bulk structure which is stable for the small clusters and there are certain specific geometries which come out when you are talking about the structure of these small clusters. And again we have already noted that in the case of small clusters that 2 neighboring n's for instance n equal to 4 or n equal to 5 need not have a either similarity in structure or similarity in property. So, it is now case by case that I need to take up these small clusters n equal to 7, n equal to 8 etcetera are very different in their structure and properties. Medium size clusters when I am referring to medium sized I am meaning 20 to 25 atoms exhibit cage like structure essentially with sp3 kind of a character. Minimization of dangling bonds on the surface lead to a compact spherical shapes. So, for larger clusters you get these cage like structures and the bonding characteristic we cannot call it exactly sp3 but gets close to sp3. For instance we know in the C 60 cluster we know that which is a which is much larger than of course, this n equal to 20 to 25 we know that the bonding characteristic is somewhere between sp2 and sp3. For large clusters non greater than n equal to 500 the bulk structure which is the diamond cubic structure is retried. So, to summarize this slide about semiconductor clusters there is a richness of possibilities number 1 of course, we note that it is actually difficult to produce some of these small clusters as compared to some of the larger clusters and of course, the very large clusters we as we expect take up the bulk structure which is the diamond cubic structures. When I am talking about the small clusters each one of these clusters could have a different property. For instance the n equal to 7 cluster has a pentagonal bipyramidal structure and also shows metallic character which is unlike the large silicon crystal which we know is actually semiconducting. Having produced some of these clusters their stability could be very different and therefore, stabilizing some of these clusters is more easy as compared to some of the other clusters and it is specially. So, for the smaller size clusters like the 4 atom cluster or the 6 atom cluster or the 8 atom or 10 atom cluster. Therefore, when I am talking about semiconducting clusters which is now we are talking about semiconducting in the bulk form then there is actually a vast richness of possibilities and this richness of possibility helps us actually in tailoring clusters to a given kind of a property which then as we noted before can be used in a kind of a construction at a larger scale. Next we take up metallic clusters and here we are talking we already seen one class of metallic clusters in our example or a few class of metallic clusters in our examples before which was the sodium in kind of a cluster and we noted that we also have calcium magnesium kind of a clusters and we also seen that some of these clusters like the sodium in shows a certain magic numbers which we are again noting down here. So, the magic numbers which sodium clusters exhibit we have noted is 8, 20, 40, 58 etcetera and similarly lithium potassium cesium etcetera also seem to show a similar kind of a behavior and again to reiterate we are talking about metallic clusters implying that the bulk form is metallic. The stability of metallic clusters is determined by the quantization of electronic orbitals these clusters are referred to as quasi atoms or giant atoms or super atoms as the case we have seen before. So, this entire cluster is now starting to behave like a single quantum mechanical system and therefore, I have to talk about not the electronic configuration of the individual atoms alone, but of the entire cluster and now this cluster is like a quasi atom and it can be treated like this. Agelium model is used for the calculation of the energy of the neutral sodium clusters energy shows minimum for n equal to 2, 5, 8, 13, 18, 19, 20 etcetera. So, there are been theoretical advances in this area and people have tried to calculate that where are the stability regimes for a cluster consisting of sodium atoms. Now, we already seen that by doing experiments we come up these numbers which are shown on the right hand side which is numbers like 8, 20 etcetera. So, when you do a theoretical calculation it is seen that this there is a larger set of numbers which emerge and we can understand this kind of a larger set of numbers by considering that now we are talking about two kinds of electronic or shell kind of electronic configuration. One those corresponding to a complete closed shell configuration and the other corresponding to a half filled configuration. So, obviously, the closed shell configuration will have a higher stability as compared to the half filled configuration and these numbers as you can see here 8, 20 etcetera which correspond to experimental results are coming from a closed shell configuration while others which may be included in this list like a 5, 13 etcetera are coming from a half filled configuration. Therefore, a game similar to what we normally play for a single atom is being played out at a larger scale consisting of a larger number of atoms, but it is basically the same principles of electronic filling of shells or and we note already that a full filled shell or which is called a closed shell or a half filled shell can give us certain kind of a stability. The surprising thing of course, is that you would expect that only that very small clusters which are about say 10 atoms, 20 atoms etcetera would having this dominance of this what you might call long range electronic interactions, but electronic shell structures seem to play a dominant role even when the cluster is consisting of a few hundreds of atoms. So, this is something very interesting and therefore, we had noted previously in the case of the semiconducting clusters that we actually had to go to something like 500 atoms or more before we actually retrieve what you might call the total bulk kind of a structure. So, if you want to summarize small clusters are stabilized by electronic shell structures where large clusters are stabilized by atomic shells and in atomic shells we know that the general rules of you know electronic configuration and also the pair wise interaction leading to the crystal aerobid. The shell structure is not compatible with the bulk structure which is retrieved for about n equal to 20,000 atoms. That means that these are also theoretical calculation which tell us that this the two rules one a local rule which is giving us these clusters and one is the rule for bulk wherein you are worried about a different set of parameters controlling the structure and, but there is a transition which we can see typically occurring at about say for instance more than 1000 atoms. And when you go to say something a cluster of 20,000 atoms then we see that it is come clearly the bulk structure we are talking about. 20 plus and cesium plus have magic numbers which are different from that observed for for instance our case of sodium and these magic numbers at 10, 20, 28, 35, 46, 54 and this is again due to electronic shell structures. Though the numbers are different the basic physics giving rise to these numbers is no different. Clusters of calcium magnesium barium have magic numbers with n equal to 13, 19, 26, which arise from compact patting of atoms. That means that again we have the two example two kinds of rules coming in one we may say this is coming from geometry or structure. And here we are talking about physical structure the other coming from what you might call the electronic structure. So, even within the case of metals we have both the examples which we had cited before which we had when we classified we said the reason for these magic numbers can come from geometrical aspects or from electronic aspects. And in the case of metallic clusters we see that both these kind of stabilities come into play and some of them are dominated by in the case of calcium magnesium or dominated by the geometrical aspects while others are dominated by electronic shell structure aspect. The next scale is the scale of the what may call semiconducting nanoparticles. So, these are and many of these nanoparticles we are going we are talking about here are actually compound nanoparticles. These include the cadmium sulphide, cadmium selenide, cadmium telluride, zinc sulphide and zinc sulphide which is doped with manganese etcetera. And these are obviously very different then in terms of the synthesis difficulty in terms of maintaining what you may call stoichiometry etcetera very different from these pure elemental clusters which we talked about like the case of xenon or sodium. So, and we know that some of these compound semiconductors actually are technologically very very important and they have been studied in diverse context intensely and nanoparticles of many of these have been synthesized and they show very very interesting properties. Here we are of course briefly introducing the concept of these semiconducting nanoparticles and we are not really taking up it in any detail, but we will return to some of these important particles like cadmium telluride, cadmium sulphide and cadmium selenide which are important from even the optical property aspects and many other semiconducting properties. For basic studies and applications size, shape and surface characteristics of these particles need to be controlled. So, this is obvious that now when I am talking about nanoparticles there are additional parameters that I need to control which include not only the size of these particles, but what you may call the shape and also the surface characteristics which I want to perhaps maintain pure. 3-5 semiconducting nanoparticles like gallium nitride and gallium phosphide are more covalent as compared to 3-5 2-5 semiconducting nanoparticles like zinc sulphide. Therefore, when you go to nanoparticle size then the what you may call even the bonding characteristics can change depending on the kind of compound you are talking about and typically we make want to characterize some of these properties and these bonding characteristics and we use various kind of techniques which includes various spectroscopic techniques like UV visible, fluorescence, Raman, X-ray photoelectron etcetera. We also use of course, microscopy techniques to determine the particle shapes etcetera and local defect structure which includes transmission electron microscopy, atomic force microscopy, scanning tunneling microscopy etcetera. And of course, we use various kind of diffraction studies which can give us not only the particle size, but the strain in this particles which include X-ray diffraction and electron diffraction. So, after this what you may call a very brief introduction to semiconducted nanoparticles we will return to some of these other these compounds when we actually talk about electronic properties of these particles. But nevertheless here we have stress to emphasize that we have the semiconducting clusters and also which is now made of pure elemental things like silicon and germanium, but we can also have semiconducting compound clusters and also semiconducting nanoparticles which are made of compounds. So, this makes large gamut of materials which we can study and use them for our benefits. When we had next we move on to the topic of what is known as self assembled ordered nano structures. When we dealt with the topic of nano manufacturing we came across a concept called nano manipulation. We said that suppose I want to make an useful assembled structure which is coming using these clusters and particles. Then I may want to take and position these what you may call clusters and nano crystals in a very precise fashion may be on a substrate may be in a three dimensional array. I had pointed out that though this is a beautiful technique and beautiful techniques are available for this kind of a manipulation using nano tweezers, but this is not only cumbersome, but definitely not amenable to mass manufacturing or even scaling up. So, but there are certain other classes which we had pointed out that there are systems which assemble themselves and these are very beautiful systems. Because here we do not have to actually put the system into a given order for instance I may want to assemble a two dimensional crystal out of nano particles or I may want to even assemble it in a three dimensional form. I do not have to do it myself the system actually organizes given a certain kind of a process conditions which I would use for to get a certain kind of an assembly. Self assembling process is a method that organizes and orders units and I am talking about units there could be very small units like molecules, they could be nano crystals or they could be nano particles and of course, they can also be nano structures of various kinds and the interactions leading to this kind of a self assembly is non covalent interactions like hydrogen bonding, van der Waals bonding forces, electrostatic forces etcetera. So, in this process we control only the process parameter and the system organizes or orders its units and this units could be of course, starting from the very small like the molecules to somewhat larger unit like nano crystals and even larger particles and the interactions leading to this kind of an ordering is the weak interactions typically like hydrogen bonding or van der Waals bonding. And if any of these particles are charged in the charge itself electrostatic charge you would help it in the ordering process. Under suitable processing conditions ordering can take place of these units which we are talking about without external intervention. And I am meaning external intervention I mean what you may call a control at the cluster or at the level of the nano crystal. Of course, I am going to intervene in this in the sense of controlling under suitable processing conditions ordering can take place and the ordering we are talking about is the ordering of the units without external intervention. When I mean external intervention is I am referring to intervention at the level of the nano crystal or the individual nano particle, but I will definitely be controlling the process parameters which include temperature, pressure, the amount of solvent etcetera. And this optimization of process conditions being is an important in actually achieving this process. The kind of order I obtain the kind of units the size to which they pack etcetera is depend on the processing conditions which I am going to optimize. Various kind of self assemblies I can visualize are ordered self assembled nano crystals, ordered mesoporous materials and hierarchically ordered materials. So, in all these cases I have of course, the basic structural unit and this structural unit is somehow organized in a fashion which we call a self assembled way of organization. And of course, we already seen the example of the nano crystals being self assembled in materials where pores is the main focus of the material. Then we can actually have ordered mesoporous materials and we will see even examples of this. And finally, we can also hierarchically order materials that means, ordering is not just at one lens scale, but at multiple lens scales. So, we can to summarize this slide self assembled or those kind of a nano structures which assemble themselves are in some sense a good solution for the making what you might call a mass production of these kind of a crystals or nano structures. Because if we are going to manipulate it unit by unit then actually this kind of a process is definitely not amenable to scale up or not amenable to large scale production. And when we are talking about self assembled nano structures the process parameters are very important. And there are classes of these self assembled nano structures which include self assembled nano crystals, mesoporous materials and hierarchically ordered materials. So, let us take up some examples of these ordered self assembled nano structures. And in this each nano crystal serves as a basic building block or a super atom. So, now I am not building a crystal starting from atoms. I am building a crystal starting from a nano crystal itself. And the end product is referred to as a nano crystalline solid. And we will also note there are alternate terms used in literature sometimes for this nano crystalline solid like one such example one such term is often used is what is known as a super lattice. And we will point out that it is better to avoid the term super lattice in this context because it can actually be confusing. So, in this set of lectures we will typically use a term nano crystalline solid or nano crystalline solid. And this nano crystalline solid can have translational orientational or both translational and orientational kind of an order. The starting point for my building up of this nano crystalline solid could be a semiconductor crystal like cadmium selenite, cadmium telluride, indium phosphide, cadmium sulphide etcetera could be a metallic nano particle or a nano crystal like gold, silver, nickel, platinum, cobalt etcetera could even be an oxide like titanium oxide, cobalt oxide or Fe 2 O 3. So, I am starting with some kind of a crystalline material, a crystalline nano particle and I am making a larger unit crystal. And if I am talking about a metallic for instance a system and I am say starting with a gold as a nice typical example because many kind of nano crystalline solids have been synthesized using gold. Then the bare nano crystal surface are very reactive and they may be they have to be kept before the self assembly takes place. So, in some sense when I am dealing with metallic nano particles I am not truly talking about what we call a monolithic kind of a nano crystalline solid, but it is a composite now of a gold nano particle which is been coated with some second layer which is now helping me to avoid the coalescence to form larger which is usually a highly twin crystal and hence the individual entity of individual nano crystal is lost. Suppose I do not do this coating then what would happen then these because now these nano particles have a tendency to melt at lower temperature the surface diffusivity high and they can get into those kind of configurations which are not allowed for bulk which include this twin configuration. So, they would coalesce they would get twin and therefore, my individual nano particle will lose its identity. Therefore, if I want to preserve the identity of my individual nano particle or a nano crystal to actually synthesize a nano crystalline solid then I typically have to cap it with a capping layer. In other words now my unit in the assembly is not a what you might call a bare metallic gold nano particle, but is actually a gold nano particle coated with some kind of a surfactant molecule as in the schematic shown below and this surfactant molecule is now the region by which one unit is going to interact with an other unit. If surfactant molecules are applied to the surface of gold nano particle they can retain their size and shape. Now, I do not have the fear of these gold nano particles you know coalescing or you know absorbing certain other species which may actually spoil my structure, but nevertheless this system is definitely not a pure system it is actually now a composite of gold and a surfactant molecule which is now helping me isolate these molecules. Hence, with coating of surfactant molecules which sometime are called passivation molecules the surface of nano crystals becomes hydrophobic and can be dissolved in a non-polar solvent forming a stable colloid. So, one such synthesis method to form this organized self assembled nano crystal is actually coating it with a hydrophobic passivation molecule on the surface and which can be dissolved to form a stable colloid. If the solvent is evaporated the nano crystals do not coalesce, but arrange themselves in the form of assemblies and that is why this is called a self assembled system. Because here all you are doing is controlling the process parameter which includes the you know the kind of solvent which you added initially the evaporation or rate of the solvent and if the evaporation rate is slow enough then this we have these molecules or these units which are the nano crystals coated with the surfactant molecules of sufficient you know time to slowly arrange themselves into an ordered structure which can be a nano crystalline solid. So, let us look at this schematic where we have this say for instance a gold kind of a nano particle and these gold nano particles now have been coated with these molecules which are schematically shown by this wiggly arrows and typically in some synthetic methods these wiggly arrows represent what you might call a non-polar represent a passivation molecule which is hydrophobic and when therefore, helps us this can be dissolved in a non-polar solvent and later on when the evaporation this gives rise to an structure where the fundamental unit is of course, a gold nano particle with this cover layer, but now these have been arranged in a crystalline array which we call a nano crystalline solid. So, this is a crude schematic wherein we are representing a two dimensional crystal here which is now a nano crystalline solid. The kind of arrangement which I can get from a single what you might call a passivation molecule a single solvent etcetera is dependent on the process parameters that means I may use different conditions to obtain different kind of ordering, but nevertheless all of these would be come with all this come under the what you might call the self assembled structures and specifically here we are referring to a crystalline solid obtain which is a nano crystalline solid and this should not be confused with a nano crystal which is each one of these. So, each one of this is a nano crystal, but this whole assembly here is a nano crystalline solid. So, this distinction has to be kept in mind and now we have what you might say ordering a two length scales one length scale being the lattice parameter of this gold. For instance suppose this is my gold and I am talking about a length scale which is the lattice parameter of gold which is ACU, but the second length scale is the lattice parameter of this crystal itself which can be thought of as this. So, there are two distinct length scales in this problem of course, instead of starting with a gold which is a crystalline material I could have started with an amorphous glassy kind of a bead and done this kind of an assembly in which case of course, I may or may not obtain of course, the same kind of structure, but for simplicity assuming that we are getting the same kind of a structure then we do not have this underline crystalline order which is in the case of gold. Therefore, this is what you might call a very very interesting beautiful example of a two level hierarchical crystal. In other words there is ordering a two length scale one length scale which is of the order of angstroms and one which is of the order of nanometers or tens of nanometers or actually could be even larger. Therefore, this what we are talking about here is a new kind of a crystal all together which is to be very clearly has to be distinguished from the kind of crystals which we typically talk about when you are talking about atomic or molecular crystals. A few more points in this regard are to be noted single sized mono dispersive nano crystals can be crystallized more easily as compared to nano crystals with a range of sizes. This is in some sense is obvious. Suppose I did not start with a single size for this unit I started with a multiple size for this unit then it is unlikely or it is difficult to crystallize that structure because obviously, there will be mismatch in the way the packing takes place. This is in some sense the opposite of what you might call the confusion principle for glasses where in it is stated suppose I want to make a glassy or amorphous material then I would put in very many different kind of elements with very many with different sizes. Now, this implies that now when the crystal is trying to grow from say a melt then atoms have to precisely arrange in without actually leading to much strain in the lattice and this is not possible because now the atoms are all of different type and they in the end end up arranging themselves typically in an amorphous fashion. Of course, this is depend on the processing parameters suppose I have a multi element alloy and I cool it faster then the chance of actually making a amorphous is much larger then come say for instance suppose I took a pure element like aluminum and I cooled it to produce to try to produce a glass. Therefore, a single size for these units is preferred if I am trying to produce a crystal and if I am trying to produce a crystal by for instance a slow evaporation rate. The forces responsible for holding the cap layers together are the weak interactions typically Van der Waals forces. So, the bonding at the level of the individual nano crystal which is the gold nano crystal is obviously metallic type, but the bonding leading to the formation of the nano crystal in solid is not any one of the strong forces, but is actually of the weak kind which could typically be the Van der Waals kind of forces. Because the interaction is now mediated by these what you might call surfactant molecules. So, these surfactant molecules are actually talking to each other whenever there are two units in contact and therefore, this interaction between these molecules is not of the strong type and it is a non covalent interaction which is responsible for it and therefore, even though gold might melt at higher temperature, but this solid itself may melt or can be dissolved at a lower temperature. Therefore, the favorable conditions for the formation of a self ordered assembly of nano crystals is are single size for the nano crystals along with of course, now we are talking not only about the nano crystal itself we are talking about it along with the passivation layer, the passivation layer itself which is actually giving rise to the interaction between these units and slow evaporation rate of the solvent. So, the slow evaporation rate as I pointed out is very similar to that the rules which are applicable for even normal crystals growing from the melt. If I cool faster then the chance of producing an amorphous material is larger as compared to cooling slowly wherein there is sufficient time for the atoms to arrange in a precise crystalline order. Silver nano crystals have been assembled in an FCC lattice and the resulting self assembled structure has orientational and positional order. So, one beautiful example of this kind of what you might call self organized nano crystal or nano crystalline solid is the example of gold nano crystals and these gold nano crystals have a beautiful shape which is the shape of the tetrachydecahedral. So, we know that there are two semi regular space filling solids one of them is the tetrachydecahedron other is of the rhombic dodecahedron. And this tetrachydecahedron shown here has is a semi regular solid because it has got square and hexagonal phases all vertices of course, are identical because they consist of a four phase joining with two six phases. So, these vertices are antical and this beautiful space filling solid has been often used as a model for grain structure and this now this polyhedral crystal of silver itself is an unit in the assembly leading to a larger crystal which itself has an FCC lattice. So, now again we have a two levels of ordering we are talking about here one within the silver particle which is now has a very specific shape which is the shape of the tetrachydecahedron. And the next level is this silver particle or silver crystal organizing itself into an FCC lattice. And now we have a larger lattice parameter for this FCC lattice and at each one of these lattice points I am going to put one of these silver tetrachydecahedron. And therefore, this is a nice example of what you might call a nano crystalline solid having two lens scales. And an important point to note is that this kind of an ordered crystal has both positional and orientational order. In other words when I am putting this nano crystal as a motif in this FCC crystal its orientation is maintained precisely with respect to now my basis vectors which are now my three basis vectors for this nano crystalline solid. So, I maintain my orientation with respect to my the x y z coordinate axis and the typical orientation relationship which has been synthesized. So, far is that the 1 1 0 of the lattice is parallel to the 1 1 0 direction of the silver. And the 0 0 1 direction of the lattice is parallel to the 1 1 bar 0 direction of the silver crystal. Therefore, I have now synthesized as crystal which is both positionally and orientationally order having two lens scales. So, this is a very what you might call a different kind of a crystal from what we encounter which are made of just of atoms. Gold nano particle synthesized by inverse micelle root have a diameter about 4.5 nanometers organized into an FCC lattice other synthetic methods have led to the formation of a b a b kind of packing. In other words as I pointed out that actually I can change my processing parameters to obtain for instance an a b c a b c kind of a packing of these units. And in the case we have a specific example being considered we are talking about a gold nano particle. Then I can get both an FCC kind of a lattice or an h c p kind of a structure where in we have an a b a b kind of a packing. Therefore, multiple options are possible here and this definitely is a new class of what you might call crystals and a new class of an ordered solid which is very very different from any other class which we you normally might talk about. A few important points again have to be stressed in this context in spite of the analogy used it should be remembered that 3 d packing of atoms is different from 3 d packing of nano crystals. So, there are very distinct reasons for us to differentiate these crystals compared to the atomic crystal which could be a building block in this kind of an structure. Atoms of a single species have the same size, but even what we call a mono dispersed crystal have a small variation in size. So, when I am talking about a mono dispersed size I typically mean and here we are not in the cluster regime where very certain only certain number of atoms and certain geometries are very very stable, but we are talking about nano crystals. That means that here even the term mono dispersed does not imply exactly the same size and exactly the same number of atoms for or exactly the same shape for each one of those building crystals. Therefore, there is a small variation in size and therefore, what I might say is that we have to differentiate these from what you might call the 3 d packing of nano crystals. Nano crystals can be faced it while we know that atoms are not and this capping we talked about that means we are capping this or putting an outer shell of these molecules. This capping layer can actually alter the shape this passivation layer itself contributes to the shape and therefore, even though the starting crystal may be polyhedral the nano crystal which or the unit which we are starting of to make this nano crystalline solid may actually may not be polyhedral and in any case this assembly is definitely different from the case of the 3 d packing of what you may call atoms. Interatomic spacing is fixed by the type of bonding inter particle bonding is variable based on the characteristics of the passivation molecule and can be varied. In the case of a normal crystal there is only typically fixed bonding of course, it could be mixture of covalent and ionic it could just be pure metallic etcetera. But whatever the case may be for a given set of atoms or compounds I am talking about the interatomic spacing is fixed and that cannot be changed. But in the case of this nano crystalline solid I am putting in a passivation layer and that implies that based on the type of passivation molecule I am using based on the thickness of this passivation molecule or if I am putting it in a configuration like this the length of the passivation molecule my bonding characteristics may change and its interaction with the neighboring units in the assembly also can be varied. That means that I have a certain degree of freedom when it comes to when it comes to the kind of assembly I want the kind of crystal I can manufacture using these kind of what you may call nano crystals and passivated nano crystals. Interparticle bonding is not of covalent or metallic type or ionic type. In the case of normal crystals we note that typically and we are talking about normal crystals I am talking about non-molecular crystals we know that typically the bonding is covalent metallic or ionic. But in this case it is usually the weak interaction. So there are reasons which make these 3D crystalline nano crystalline solids very interesting and very similar to our usual atomic crystals or molecular crystals. But we have to differentiate them based on these parameters which I have just now stated. The particle size also may play an important role in the self-assembling process. For example the CDSE dispersions the interaction become repulsively stabilizing from attractive as the particle size is increased. That means at one size regime of the CDSE particles I have an attractive kind of interaction and other size regime I may have a repulsive kind of a stabilization. That means that the bonding or the interparticle interactions may change character as I am wearing my particle size which is not the case as you know for the simple atomic or molecular crystals wherein you do expect that because now the size of the atom is fixed and therefore the interaction is not going to change with the you know the kind of crystalline making. CDSE nano crystals have been self-organized into 3D quantum dot nano crystalline solids and many a times in this context the term super lattices used which in my view should be avoided because the term super lattice refers as we know to the case wherein a structure is composed of two sub lattices. So when I am using as term super lattice I imply as in these cases it is clear that the system is not actually comprising of any sub lattices the system actually has an order at the lower length scale and also an order at the higher length scale. Therefore, I cannot call this a super lattice but whenever somebody is reading literature and this kind of a term is called a super lattice. You should understand that what one is referring to is not the usual super lattice but a different kind of a super lattice or what you may call a nano crystalline solid. Clusters of TiO2 have been agglomerated into spheres which then form a motif for the super lattice in a two level ordering. So, there are other examples like the case of TiO2 wherein you first assemble these clusters into a sphere and the sphere themselves becomes a motif for a crystalline ordering. Therefore, there is a two level of ordering I am talking about one level wherein you have the individual clusters of TiO2 forming a sphere and the sphere itself forming a crystalline order. Of course, the first level is not a crystalline packing but put together the first and second level consists of a two level ordering which is again a very interesting.