 This material characterization course in the last 2 class we are discussing about the diffraction phenomenon in the transmission electron microscopy. And especially in the last class we have seen that the diffraction phenomenon what we have seen in an x-ray diffraction is all applicable to this TEM also and the only difference we have seen in terms of the wavelength difference between the in an accelerated electron beam versus in an x-ray beam. And then we also discussed all these aspects in terms of a walled sphere concept as well as reciprocal lattice concept and then today we will continue in this discussion to complete the I mean all the aspects of diffraction involving the walled sphere as well as reciprocal lattice. So if you look at the walled sphere and its relation to reciprocal lattice another important aspect can be visualized what I am trying to describe in this slide a parameter called a deviation parameter s. In an XRD also we have discussed that there is something called you have the set of planes which diffracts with exact Bragg condition and which is not exact Bragg conditions. So we have discussed those aspects in much more detailed manner and similarly the use of a walled sphere concept and reciprocal lattice also clearly demonstrate this aspect through this deviation parameter s. So look at the schematic here is this is the deviation I mean this is the walled sphere and then you have the intersection points which is designated as I mean q and s and I can play as a schematic. What you are seeing here is basically the vector form of a Bragg's law that is g is equal to k minus k naught that we have seen the deviation parameter is actually defined as to what extent the diffraction is occurring with from the exact Bragg angle so that is the idea. So how far it is deviating from the exact Bragg angle so that is what it is depicted here. So you see that you look at this q, o, s this kind of a point where your diffraction happens exactly at the point s but then suppose if your theta is slightly different or deviating by the amount delta theta then the point is slightly away from this walled sphere surface. So you can see that the intensity profile also is drawn accordingly so you have two aspects where you have the maximum intensity you have a negative s as well as a positive s over the I mean the thickness of the diffracting volume and accordingly you can write that you know the new formula for this vector form of Bragg law g prime is equal to g plus s. So since it is a positive s then it is g plus s so the figure introduces a vector parameter s which is a measure of deviation of the reciprocal lattice point from the exact Bragg position. So you have some quantitative data here to measure this how far the diffraction takes place away from the exact Bragg angle and so on. So this particular concept we will be using this even we can demonstrate in terms of in a diffraction experiment when I discuss the kikuchi line I will be able to demonstrate particular peak positions where you can see exactly whether it is a positive s or a negative s and also where exactly you can see s is equal to 0 all these situations you can demonstrate easily with the kikuchi pattern. This is the actually the allowed condition for the I mean conditions for the allowed reflections in the most of the crystal systems and where we talk about a different I mean unit cells where you have the selection rules which is present. We will use this table in an appropriate time when we do the I mean CBED and then other indexing procedures for time being we will skip this we will come back to this table once we go to this CBED and so on. So now I would like to discuss something little more detail on a kikuchi pattern in a transmission electron microscopy. So we have some background about this kikuchi pattern in even in a scanning electron microscopy lectures and one important point we have to understand is the diffraction whatever we have discussed so far dealt with elastically scattered electrons. Now we are going to talk about the inelastically scattered electron so that is a primary difference then we will look at the initial remarks. The inelastically scattered electrons can subsequently be elastically scattered that is Bragg diffracted by a lattice planes to produce a phenomenon known as kikuchi lines. Please remember this is inelastically scattered electrons that means the electrons has much I mean lost their energy and you they will form in a form of a diffuse spot in an electron diffraction when such an electron further subjected to a elastically scattered that means without losing further energy if it undergoes a Bragg diffraction it produces a kikuchi line. So that that point you have to remember kikuchi lines will be best seen in a diffraction patterns from areas of specimen that have low density of defects and are of about half the thickness that the beam can penetrate or thicker. So what it means is you should in order to obtain or in order to visualize a kikuchi pattern you need to have a sample where the defect density is minimal as well as it should be sufficiently thick your sample should be sufficiently thick enough to visualize this effect otherwise you will normally see a single crystal diffraction pattern rather than a kikuchi pattern. So that is a another reference one can have if the specimen is thinner only spots will be seen if it is very thick only kikuchi line will be seen. So look at this electron diffraction pattern where we are going to discuss about the exact Bragg condition as well as the deviation from the exact Bragg condition and so on. So what we are now going to discuss is finding your way around the reciprocal lattice. So we are looking around the reciprocal lattice and we will see what is the effect of this thickness specimen thickness and then tilting also we can be discussed. So the change in the spot pattern by tilting by small angle the intensities will change and not the position what is the advantage. So you can just do this kind of a simple tilting experiments to in order to obtain a two beam conditions or if you are interested in particular exciting a particular diffraction positions and so on. So look at this four diffraction pattern carefully you have both a spot pattern as well as a kikuchi line what you have to remember in a kikuchi pattern as we discussed in the previous diffraction lectures it is the 2D projection of a parabola. So you have one pair of a bright line and a dark line so you can see that this is a bright line and a dark line we have it has been marked here A A dash B B dash. So this is the pair line that is you can always call a pair kikuchi pair of lines or you can say kikuchi band and so on you can this can be described in many ways and what you are seeing here is you see that the 2 2 2 set of plane I mean the diffraction spot designated as 2 2 2 has been excited more and then you have after a small tilt this band has moved into the center that means you have the 2 2 2 and this the corresponding the negative indices 2 bar 2 bar 2 bar indices have the equal intensity here and this is this condition belong to your deviation parameter s is equal to 0 and you have you see that the the 2 2 2 spot is now excess in intensity in this tilting exercise. So that means it has got or we can say that the bright line as further away from this 2 2 2 spot and here you see that the excess intense line it is further the other side just opposite sides to 2 2 2 spot. So these 2 conditions belong to a positive s as well as a negative s. So you have positive deviation from the bright condition this is a negative deviation from the bright conditions. So this is what I just said a kikuchi line diffraction can clearly demonstrate the deviation parameter and so on and we can just see what is the origin of this kikuchi line from how the kikuchi line form. So look at this schematic where you suppose if you are looking at one inelastically scattered beam and then we are now talking about a 2 rays ray 1 and then the ray 2. Please remember the ray 1 intersects at this point which is close to the I max and the ray 2 intersects the profile at this point which is slightly away from this ray where the ray 1 intersects here. So it is far further from the I max this is closer to the I max the intersection point so keep that point in mind then we will continue the discussion. So to explain the kikuchi line this is a schematic which is shown assume that this is a specimen thick specimen and this is an incident beam and you have the diffracted beam which represented as G and D and you can see that this is the ray 1 and then this is a ray 2 with reference to the previous schematic on the intensity profile of inelastically scattered beam you have ray 1 and ray 2 suppose if you assume this then we have the remarks which is pertaining to these 2 rays are as follows as ray 1 is closer to the forward direction than the ray 2 it is more intense and an excess number of electrons over the background will arrive in the back focal plane at B and there will be a deficiency of electrons at D thus there is a bright line at B and a dark line at D in the diffraction pattern and these are all kikuchi lines. So there is a very simple not complicated explanation just with respect to this inelastically scattered intensity profile namely ray 1 and ray 2 how it reaches the screen and then the bright line and dark line is simply I mean explained on the basis of the intensity of each of this line that is the excess of electron and then less I mean I would say that excess line or a dark line you can say that a bright line or a dark line you can say that. So if you tilt this specimen in a small angle you can bring back this G and D coincide with your transmitted beam and the diffracted beam originally. So this is one simple explanation how the kikuchi line will form the diffracted rays actually form cones of semi angle 90-theta called causal cones what we see in the diffraction pattern is pair of parabolas where the cones intersect the evolved sphere the parabolas appear as straight lines in the diffraction pattern because the angle involved are very small the plus or minus G pair of lines and the region between them is known as kikuchi band the angular separation of the pair of line is 2 theta their spatial separation in the diffraction pattern in the back focal plane is G and the lines are perpendicular to the G vector each reflection has associated with associated pair of kikuchi lines attached to it and this is demonstrated in this schematic diagram. This schematic we have already seen in the X-ray diffraction as well. So just for a recap so how this suppose if you have the HKL plane and this is your a diffracted beam from the both the sides since we are talking about a thin sample I mean you have both sides the pattern comes and then when this parabola intersects a 2D plane here you see and these two parabola I mean cones are called a causal cone and which intersects the evolved sphere appearing like a kikuchi line here. So you have a positive I mean positive HKL and as well as negative HKL we can say that and similarly you can visualize these line formation through this simple ray diagram in the specimen where the electron diffracts in this form of excess line as well as deficient line what we have shown in the diffraction pattern. If the specimen is tilted by a small angle alpha the lines will move a distance r across the pattern on the screen and from the simple geometry we can say r is equal to L alpha where L is the camera length. To characterize certain defects such as dislocations and stacking faults it is important to set up a two beam conditions in which only one set of lanes is strongly diffracting that is the relevant reciprocal lattice point lies exactly on the reflecting sphere that is s is equal to 0. It is also important that s is positive if you want to form images of defects with good contrast. So now you appreciate from these two points why we talk about kikuchi lines I mean why it is important how it can be used and also the importance of the deviation parameter s. So I have just shown how the deviation parameter can be shown whether it is a positive s or a negative s with respect to the intense line whether it exactly intersects the for example the 222 set of I mean plane diffraction spot where you have the excess line going further or the closer to the transmitted beam the condition of s is decided or it can be visualized practically and this condition is important to obtain the images of some of the important defects with a good contrast. So that is how these parameters are used and at least these are all the preliminary or very basic use of usage of this techniques. So you may wonder why we do all this so we are not discussing the complete details of all this techniques and the phenomenon just for a basic idea you should have how these kikuchi line forms and what is the use of it to begin with. So we can now look at the some of the schematic where it clearly demonstrates the deviation parameters with respect to reciprocal lattice point and then exact exact black conditions. So what you are now seeing is this is a black condition and you have a positive s that means your actual reciprocal lattice point is completely inside the sphere I mean and then you see that the corresponding effect on the kikuchi line on the diffraction pattern and this is a negative s then this part will come out of the evolved sphere and then this is a point where it is in the equidistance I have shown this kind of a situation in an actual diffraction pattern in the pattern B where I have shown that it is equidistance from the both positive as well as negative. So this is different from the exact black condition do not confuse these two so this is you have a positive and negative effect s parameters are all equal magnitude at this orientation here it is a completely oriented towards exact black condition and so on. So other some of the uses of kikuchi maps is to just to find out the symmetry you can just by looking at the kikuchi maps you will be able to tell what kind of symmetry the your crystal has for example if you have the 111 zone of germanium showing a 3 fold symmetry this is a 3 fold symmetry what you are seeing here is this is 3 fold symmetry and the 001 zone of magnesium showing a 6 fold symmetry so these are all some of the immediate application just by looking at a diffraction pattern you are able to obtain a basic information about the system crystal system and so on they are very powerful in that manner. So you can also look at the zone axis when the electron beam is close to the beam axis the position of the zone axis relative to the undivided beam can readily be identified. So look at the zone axis identification of zone axis also will be easier once you have some details about this kikuchi maps and this is a schematic of a kikuchi map of a for a diamond cubic crystals which I have already shown this for the completing this section I am just giving this example again. So the kikuchi lines and the kikuchi maps are one of the most important aids we have when orienting or determining the orientation of the crystalline materials identification of orientation of the specimen is essential for any form of quantitative microscopy. So the bottom line is if you are interested in carrying out quantitative microscopy in electron transmission electron microscopy the knowledge of kikuchi lines and their you know identification or operating this for a exact condition to generate this is very essential. Whether you are analyzing the dislocation merger vector by diffraction contrast imaging grain boundaries with lattice resolution or measuring chemistry variation by electron energy loss spectroscopy or x-ray energy dispersive spectroscopy they are especially useful when combined with the map of zones and poles that is direction of plane normals on the stereographic projections. So using the stereographic projections as I told you yesterday you will be able to identify the plane normals which are very parallel to the zone axis. If you recall in the yesterday's lecture where I showed how the planes which are connected to the I mean the plane which are parallel to the zone how the diffraction pattern exactly appear the diffraction pattern appear at the 90 degree to the plane orientation. So that concept is exploited simply using a stereographic projection if you have a 001 zone then the plane normal which all intersects the periphery of the stereographic projection will be normal to the zone axis that means all the planes will be parallel to the zone axis. So that also can be exploited in combination with kikuchi maps to do quantitative analysis. So that is the information you should have at this point of time. So it is a summary the kikuchi lines consist of an excess line and an deficient line. In the diffraction pattern the excess line is further from the direct beam than the deficient line. The kikuchi lines are fixed to the crystal so we can use them to determine the orientation accurately. The trace of the diffracting planes is midway between the excess and the deficient lines. So these are all some of the key points one can remember. Now we will move on to another important diffraction phenomenon called convergent beam electron diffraction in TEM. So we have some background for this topic when I introduced the transmission electron microscopy we talked about some of the instrumentation operations like parallel beam as well as convergent beam and then the name itself clearly says that it will give you very a small probe whether it is a micro probe or nano probe and then and with lot of you have the tilting possibility in a TEM then you have the advantage of looking around a large volume of reciprocal space. Please watch my words a large volume of reciprocal space not the real space. So you will get lot more information using a tilting experiments and a convergent beam electron diffraction we will see what it is. Need good understanding of crystallography and space groups to follow this technique. So you have to have some basic knowledge on crystallography and space group point groups to exploit this technique that is a prerequisite otherwise this is very difficult. So in principle it is a straightforward matter to decide whether or not the crystal is cubic hexagonal or has a lower symmetry from the observation of the geometries. So that is so powerful by looking at the pattern you will be able to decide all these things. If for example node diffraction patterns with the 4 or 6 fold symmetry are found the crystal cannot be a cubic and hexagonal with the large tilt angles that are available in modern TEMs it is possible to examine large volumes of reciprocal space. So you will be able to extract lot of information from the very small volume that is a geometrical information or crystallography information from the very small volume that is what it means. So what are the typical things one can get from this technique specimen thickness unit cell and precise lattice parameter and crystal system and a true 3D crystal symmetry and inertia morphism if present. So all these things can be analyzed using this technique we are not going to demonstrate all these things because of the that is not the scope of this course but you should know what is this technique and what is the meaning of it what it does and what is the use at least to that extent you will be able to have some idea. So when a convergent electron beam is used to form a diffraction pattern a range of incident angle leads to a significant excitation of the reflections from the higher order law way zones or holes. If the electron beam is exactly along the zone axis of a crystal it is clear that the spacing H star why H star we are now talking about a reciprocal space of the reciprocal lattice layers per particular to the electron beam. The zone axis repeat in the crystal can be derived from the radius G of the higher order law way zones. So look at this schematic again we are talking we are exploiting the evolved sphere concept as well as the reciprocal lattice here just what we have just seen in the Kikuchi lines also. For the first order law way zones assuming the angle alpha is small if you if you assume this alpha is small from this geometry you look at this this is the you know this is an evolved sphere then whatever the point which are intersecting will form a diffraction pattern and you just look at the q fp q fp this triangle and if you assume this alpha is too small then we can show that 2 alpha is equal to lambda g1. So what is g1 g1 is this distance and lambda is 1 by lambda here so it is 2 alpha is equal to lambda g1 radiance and from the triangle fop you can say that fop this small triangle you can write alpha is equal to H star by g1. So from these 2 relation you can write H star is equal to g1 square lambda by 2 for the Foles line that is first order law way zones you can write like this and similarly H star is equal to g2 square lambda by 4 for souls that is second order law way zones. So you have you can do this for a second order law way zones which interacts from this this point so please understand what we see normally the diffraction pattern is 0 order law way zone and then you have first order law way zone second order law way zone and so on as you claim this you know the later like this wherever the you have the intersection point here you will see the higher order law way zone pattern so we will see some practical example. So the spacing of the reciprocal layers is given by H star is equal to n divided by RUVW where RUVW is the zonal repeat and the value of n takes account of the systematic absence due to the lattice type. See this is the again a selection rule if you remember we have just shown some table where all the crystal systems where the you know systematic absence of reflections will be tabulated so that table can be referred to I mean to arrive at this conclusions what is that absence systematic absence due to lattice type whether it is a cubic or a BCC or FCC and so on. So for a primitive or a rhombohedral lattice n is equal to 1 for any axis for an phase centered lattice n is equal to 2 if u plus v plus w is even otherwise n is equal to 1 for a body centered lattice n is equal to 2 if u v w are all odd otherwise n is equal to 1 for a C lattice n is equal to 2 if u and v are both or odd otherwise n is equal to 1 the same information which is tabulated in the previous slide I have brought it back for the reference for all lattice where the hexagonal indices have been used n is equal to 3 if u minus v plus w not equal to 3 and this is a selection rule where n is an integer otherwise n is equal to 1. From the first order lobby zone ring the zonal repeat is therefore given by RUVW is equal to 2n divided by lambda g1 square which is equal to 2n divided by lambda into lambda l which is a camera constant divided by R1 square. So where R1 is the radius of the folds measured on the film and lambda is the camera constant. So you have the general relations to obtain a different laway zones basically. Similarly you can write for the souls ring a 4n R is equal to 4n divided by lambda g2 square is equal to 4n by lambda into lambda l by R2 whole square in terms of unit cell parameters the values of the zonal repeat you can also represent the zonal repeat in terms of unit cell parameters then you can put it in this formation R square is equal to u square alpha square plus v square b square plus w square c square plus 2vwbc cos alpha plus 2wuc alpha cos beta plus 2uvab cos gamma. So this is a general information about the zonal repeat please remember this is with respect to the walled sphere intersecting the reciprocal point. So and it is done with the convergent beam electron diffraction. So now we will see some typical example again the before we go to the example this is the schematic how whatever we have just discussed this is another form of putting it so this is a reciprocal lattice rods and then you have the souls and then folds and souls and so on you have the all this you know sphere which with the different tilting conditions you will be able to see the different laway zones and this is the typical CBED pattern you see the zero order laway zone the first order laway zone you can see the second order laway zone very faintly which is coming here. So look at this slide you have a 001 CBED pattern from the spinel and what you are seeing here is it is a pattern obtained with the micro probe with a 30 micrometer condenser aperture B is taken with a coastal conditions with a nano probe size of 100 micron condenser aperture. So the pattern is for the spinel which is having a cubic crystal with the space group Fd3 bar m so the reflections in the folds have the spacing along x star and y star that is half of souls. So what is that stated here so this is the first order laway zone pattern here what is given the the orientation suppose you consider this and x direction and this is an y direction the spacing along x star and y star that is half of souls that means the distance here what you see here it is only half in the first order laway zones in both x axis as well as the y axis so that indicates a presence of a glide plane perpendicular to the z axis that is parallel to the plane of projection. So you have the glide plane in parallel to the plane of projection that is this projection so you have you can identify a glide plane with this kind of an information so this is one typical example we can we can just I just want to give you by looking at a simple CBED pattern you are able to get a very important information about the space group and space group information from the of the crystal so that is the powerfulness of the CBED and we can see one more example a 010 CBED pattern from the asymptotic magnesium germanium oxide the reciprocal lattice is monoclinic in shape the folds is displaced by half a repeat in the x direction. So the schematic of this is here because the pattern is so weak here so you see that even the zero order laway zone is slightly in oblique nature and then the repeat distance in the x direction as well as the other direction is half of the zero order reflection so that again gives the information of C glide plane parallel to 010 in the space group. You see this is the usefulness of a CBED just by looking at a pattern and then you have you will be able to some of the space group information of the particular given crystal so we will not get into the details of this technique I just want to tell you what is CBED how it is formed and what is the usefulness of this and in an advanced characterization techniques as I said when you do a quantitative microscopy here again you can you can use this CBED to measure the precise thickness of the sample and precise lattice parameter and orientation and then in addition to this space group information so these are all the typical applications of CBED and again the table I have just brought back for the reference so this is the diffraction group symmetry seen in CBED pattern from the different zone axis you will see that this is a point group and this is the different UVW directions where you will see all this information this is been tabled here you can find this also in most of the transmission electron microscopy standard textbooks and similar table regarding the relationship between I mean the diffraction groups and the full symmetries of CBED patterns are also tabled in most of the textbooks just for a reference I have given so with that I want to finish this diffraction discussion in a TEM so what to summarize the diffraction in TEM what you have to appreciate is diffraction is very powerful tool in transmission electron microscopy so it is I would say it has been completely exploited in this technique that is why at TEM itself it become most powerful technique because of this aspect of the microscopy so if you have a thorough knowledge on a crystallography and symmetry that is a group theory you will be able to exploit this technique otherwise it is not been used to the fullest extent possible so now we will move on to the other topic in the TEM that is an imaging we will just look at what are the kinds of our types of imaging possible we just mentioned very briefly what does mean by a bright field and a dark field in during the instrumentation details of the microscope and we will also look at the types of techniques which gives a different images and what are the other details we will see it in the next class thank you.