 Hello everyone welcome to the second class on the advanced characterization techniques module comprising with X-ray diffraction. In the last class we had understood how X-ray is can be used to study the structure of materials and what exactly constitutes diffraction. We had ended up calculating the reflection condition for various crystalline structure and figured out that for BCC the condition for diffraction is that H plus K plus L is even and for FCC we need HKL all even or odd. Having understood the theoretical basis of X-ray diffraction in terms of Ewald sphere and reciprocal space in the present lecture we are going to focus on different experimental techniques for doing X-ray diffraction. We know that the entire technique all the techniques comprising of X-ray diffraction have one thing in common and that is essentially the Bragg's law which goes as so we know that for a given crystalline material the D is related to the lattice parameter. At the same time the wavelength lambda and the angle theta are the two variables. So all the diffraction techniques essentially deals with either varying lambda or theta. In order to have a look at those diffraction techniques let us look at this table. The first and foremost that was developed was the Laue method which is essentially used to characterize single crystalline material. As you can understand that a single crystalline material has a fixed reciprocal space and to get diffraction is much more difficult compared to a polycrystalline material wherein you would expect that all different grains will have a corresponding reciprocal space which is of all different orientations. Therefore we need to use a variable or white radiation. Now the white radiation essentially ensures that we have an evald sphere with varying radius. This essentially ensures that the chances of the evald sphere meeting some part of the reciprocal space or reciprocal point is much much higher and hence the probability of diffraction increases. Particularly this is very important for single crystalline materials wherein diffraction condition is very stringent. Having said that in an even more challenging case is wherein we do not have any idea about the existing crystal structure of the material. In that case not only the reciprocal space orientation is unknown the reciprocal space itself is unknown or rather the reciprocal space vectors are unknown. In order to map this reciprocal space generally we use what is known as a rotating crystal method wherein we use a fixed wavelength and the sample is angle or theta is varied in most of the to most of the values. This is essentially used to determine the unknown crystal structure. I would like to mention that though it is much easier for me to write it here that rotating crystal method can give us information about determination of unknown crystal fixed wavelength lambda and by varying angle theta it is quite complicated and a field in itself all together. What most of the time we end up using in a laboratory scale x-ray diffraction is essentially the device error or a normal powder diffractometer in which case the wavelength is fixed and the theta is varied. Now this kind of this technique we use routinely to characterize powders as well as polycrystalline materials. This kind of technique provides a wealth of information about the lattice parameter the phases present and various other things which I will demonstrate in the later few slides. So we know that a normal x-ray powder diffractometer that we use in our laboratories gives us output in the form of I or intensity versus to theta. Another question that arises and is very important is we talked about Bragg's law and we all have recollected by now that Bragg's law deals with n lambda is equal to 2 d sin theta. So the obvious question arises is from where does we get this value of 2 theta. So I would again like to draw back your attention to the first figure that we had seen in the last class. Herein we see that the detector makes an angle of 2 theta with the incident beam. Therefore if the sample is moving at an angle of theta the detector is essentially moving at an angle of 2 theta. Therefore whenever we do normal powder diffraction we essentially end up getting a I versus 2 theta plot. From this plot we have to calculate theta and use it in the Bragg's law. However in order to understand which peaks are present in the diffractogram which is shown over here we can do simple calculations. Then we go back to the Bragg's law and if we rewrite it considering first order reflection we do get lambda is equal to 2 d sin theta and then substituting d for cubic crystals it is a divided by h square plus k square plus l square and then substituting and rearranging the terms we do get that our h square plus k square plus l square is proportional to sin square theta. So once you obtain a diffractogram essentially we calculate the theta at which the peaks are present and then calculate the ratio of sin square theta. However I would like to caution you that this is applicable only for single phase materials. In case of multi phase materials one needs to use a software that can take care of all the peaks which are present in the diffractogram. Having said that I hope you have noticed that the diffractogram is characterized by a couple of things. We see a variation in intensity of the peaks as well as the peak broadening and all this gives us a lot of information about the structure of the material. So let us go back and concentrate on powder diffraction. So depending on the Bragg's lattice various diffraction conditions can easily be calculated and we find out that for simple cubic all diffractions are possible while for body centered h plus k plus l even are necessary present while h plus k plus l odd are necessarily absent while for phase center cubic h k l unmixed are always present while h k l mixed are always absent. I have just noted down different values of h square plus k square plus l square and the kind of peaks that we get in a simple cubic phase center cubic and a body center cubic crystal. Here in we see that the simple cubic material shows a peak rich diffractogram meaning that you have plenty of peaks in the diffractogram of a material that has simple cubic crystal structure. However when we go to the phase center cubic structure we see that the diffractogram is not that heavily populated. Another important thing that we notice is that we do have two peaks which are close and the third peak is far off. This is essentially because of the diffraction condition or the reflectors that are present in a FCC wherein we do get reflection from 348, 1112 and 16. However when we look at BCC we do see that the reflection condition is essentially 246810 which essentially indicates that the peaks are equidistant in the diffractogram for BCC materials. Having said that the point that I want to emphasize is that just looking at the diffractogram of a single phase material we can make a guess about the crystal structure of the material. Another important caveat that I would like to bring to your notice and which is generally missed in most of the textbooks is that we do not get a peak even in simple cubic case for h square plus k square plus l square is equal to 7 and this is essentially because the sum of no 3 squares adds up to 7. So there is nothing so special about this number. Having said that you have you must have noticed the intensity of the peaks which I had shown in the earlier slide here in after gaining knowledge about the kind of reflections and the crystals and the Bravais lattice I hope you can make a guess that this material showing two peaks very close separated by a third peak and again two peaks close separated by another peak corresponds to a FCC crystal structure with reflections at 3, 4, 8, 11, 12, 16 and corresponding to 111, 200, 220, 113 and so on. Having said that this is the peak of nano crystalline nickel which we will see in sometime. So as I had already mentioned if you look at the intensity of the diffracted beam for a powder sample it gets affected by the polarization factor which we had considered in the last class. At the same time we also looked at the structure factor which essentially determines what is going to be the intensity of your diffracted peak. Another important parameter which we had not touched upon by now is the multiplicity factor. In this regard I will like to point your attention to the 100 peak and the 111 peak in cubic crystals. So if you look at 100 peak the multiplicity of 100 peak is about is 6 while that of 111 peak is 8. Therefore we can believe that if we consider only the multiplicity factor the ratio of intensity of the 111 peak will be 8 by 6 times that of 100 peak. Another important parameter that affects the diffraction condition is the Lorentz factor which is essentially due to the fact that diffraction occurs not only exactly at the Bragg angle but at angle slightly away from the Bragg angle. The kind of polarization because of Lorentz contribution as well as the original polarization factor is generally combined together and when plotted together shows this kind of a variation. So whenever we compare different diffractograms or different peaks in the same diffractogram we have to account for the Lorentz factor. Another important parameter that generally affects diffraction and we tend to and which is generally ignored is the absorption factor. We all know that whenever x-rays are incident on a material part of it gets absorbed and therefore there is an additional path length that is to be considered. However in most materials the absorption of x-rays is not significant and it can be easily neglected. One important parameter that is not really valid for room temperature deformation rather room temperature x-ray diffraction but becomes very important at high temperature is the temperature factor. We all know that all the calculation for structure factor and x-ray intensity is based on atoms sitting at a particular point in the crystal structure. However we all know that the atoms are not actually sitting at a particular point in the lattice but they are constantly vibrating. Now this vibration of atoms is directly proportional to the temperature and hence with increasing temperature there is a decrease in the intensity of the diffraction pattern and this becomes particularly important for doing high temperature study involving phase transformations. Having understood the various factors which govern the intensity of the diffracted beam will now shift our focus and have a look at the actual way we go about indexing. So all the peaks in the diffractogram can be indexed manually and then we can go back and compare them with a lookup table. So the earlier lookup tables which comprised of all the database was maintained by the joint committee for powder diffraction standards. So these are like standard charts wherein for specific materials all the diffraction information is tabulated. This committee has now been renamed or re-christianed into the international centre for diffraction data. So this database contains all the data pertaining to all the available materials which have been identified till date. Having said that I will now give you a glimpse of how does one of the file looks in the database. Since we are discussing on nickel I have chosen the file for nickel and herein you can see all the information contained in the file. So in addition to the lattice parameter and the angles we also see that other important information like the density volume of the unit cell are also mentioned. Having said that the most important parameter that we get from this database is what is known as the stick pattern. So this essentially shows the theoretical variation of intensity with respect to the theta value for indeed diffractogram. Herein we can see that the different peaks show different intensity. Now the variation of intensity is directly proportional to the structure factor of the corresponding orientations. Since we are talking about FCC nickel we can see that the 111 peak which has the highest structure factor and multiplicity shows the highest peak and the other peaks namely the 200, 220, 113 and the remaining peaks also show intensity in accordance with the structure factor and multiplicity calculation. Now these are theoretical intensity variation shows a nice match with the theoretical calculations because they are obtained on random powder samples. However we know that most of the materials that we use are hardly anything but random and this is what is shown in our nickel slide. So this nickel which I was showing in the last few slides essentially is a bulk electro deposited nano pruseline nickel. So we see quite a few important things in the diffractogram. The first and foremost important thing that we seen the diffractogram is that the peaks are replaced the strong peaks that we saw in the earlier stick pattern are replaced with broadened peaks. Therefore we do get diffraction at not just the bragg angle but at an angle slightly away from the bragg angle. This essentially is due to the mosaic structure of the crystals. In this particular case this is essentially due to the nano pruseline structure of the nickel sample under consideration. Another important observation that we make is the change in intensity level of the different peaks. In the stick pattern we do see that the first peak namely the 111 peak is the strongest peak followed by the other peaks. However when we look at this diffractogram we do see that the 111 peak is no longer the strongest peak. However the 200 peak shows the highest intensity. This essentially indicates that the sample under consideration shows preferred crystallographic orientation which is also known as texture. Another important point that is to be mentioned is that the exact lattice parameter of the nano pruseline nickel obtained from this diffractogram is slightly different from the one that was expected in the stick pattern. In addition we see that the intensity ratio is also different. All these indicates towards the presence of texture, strain as well as broadening due to size and micro structure. In addition there are techniques which claim that they can get an estimate of operation of twinning, stacking faults and even dislocations using X-ray diffractograms. We are not going to discuss all these techniques in details in the present lecture but I do want to emphasize that X-ray diffraction which is generally taken as a normal mundane tool is a very powerful tool and when complemented with other microscopic tools can provide you a wealth of information regarding the micro structure of the material. Another misnomer that encounter is powder diffraction but we see that we end up using the material mostly all polycrystalline material bulk materials for doing X-ray diffraction and it gives us the same kind of information as we do get in fact much more than what we get in case of powder diffraction and hence I personally believe that powder X-ray diffraction is essentially a misnomer and it should be replaced by polycrystalline X-ray diffraction. Since polycrystalline X-ray diffraction encompasses all the information pertaining to the size, strain and crystallographic texture which is relevant to a given micro structure. Having seen and covered a broad spectrum of X-ray diffraction and what all capabilities that X-ray diffraction normal laboratory scale X-ray diffraction study can do we will now focus and try to understand what are the actual you know hardware or the instrumentation that makes it possible. So we will go very slowly in this part and try to understand how actually it works. So when we go to a normal X-ray diffraction facility we come across what is known as a diffractometer. A diffractometer comprises as we all know we need a source to generate X-rays having said that we just cannot use the X-rays we need to massage them you know kind of modify them to get them in a particular shape particular size and particular coherence so that we get proper Bragg's diffraction. Having said that once our X-rays interact with the matter we also need to collect the signal and then use this signal to analyze our results. This is what is shown here in a small schematic so we see we need a source to generate X-rays we need what are what is known as the incident beam optics to massage the X-rays and produce X-rays in the proper format in which we want then the X-ray is made to interact with the sample and then what all signal we get is essentially passed through the diffracted beam optics again to ensure that we are looking at a particular wavelength and then finally it goes to the detector. I would like to again and again mention that the entire exercise of going for incident and diffracted beam optics is to ensure one that we are looking at the wavelength of interest because most of the studies that we are looking at the wavelength has to be constant and second we are ensuring that the profile of the beam remains the same therefore the need for the optics in X-ray diffraction. Before we move ahead let us look at the diffractometer itself so I hope you have appreciated the diffractometer is nothing but an assembly which comprises of the source the detector the optics and the sample so diffractometer can be obtained essentially in what is known as a theta-theta geometry now what is theta-theta geometry we have gone and we have looked at the diffraction condition time and again so let me just go back and let us have a look at the diagram which we have seen for obtained times so let us focus on this diagram and here in we can see that as I had mentioned that our detector is moving at 2 theta and the sample is fixed over here so in there are two ways of doing it so one is you move your sample which is over here at theta and the detector also by theta this will essentially ensure that your detector is still maintaining 2 theta with the incident beam so this is known as the theta-theta geometry so essentially this can be done by you know keeping your sample fixed and moving the tube and the detector both at theta-theta so this is known as theta-theta geometry in the other case what we do is we fix the tube and move the sample at theta and the detector at 2 theta so this is essentially known as the theta 2 theta geometry which is shown over here now another configuration that we use on a routine scale is the vertical configuration and the horizontal configuration the vertical configuration as the name suggests includes all your source and detector on a vertical circle this essentially ensures that your sample is on a horizontal sample holder this essentially ensures that you can study powders very easily however if you are studying thin films or carrying out texture analysis you may like to use a horizontal configuration which essentially ensures that your sample is vertical having said that the diffractometer essentially has to ensure that the diffraction condition is satisfied not only that the diffraction condition has to be satisfied at not just the Bragg's angle but means at the Bragg angle as well as at different inclinations of the sample as well as the rotation of the sample to improve the statistics therefore a diffractometer consist of a sample which consist of a sample holder that can give different kind of rotations to the sample under consideration so we have what is known as a 3 circle and a 4 circle diffractometer in a 3 circle diffractometer the 2 theta and omega, omega corresponds to the theta which essentially ensures that the 2 angles are corresponding to the movement of the detector and the movement of the sample while the angle phi and chi correspond to the rotation of the sample about the normal axis and the rotation of the sample about an axis which is perpendicular which is going through the sample which is known as chi this is what is shown in the figure over here. So we see that in the case if 2 theta and omega are coupled it will be a 3 circle diffractometer while in case they are uncoupled it will be a 4 circle diffractometer having said that state of the at synchrotron sources which we are going to talk about in the next few slides and the neutron sources which we are again going to talk in the last class later can use as complicated as 6 circle diffractometers which comprise of theta phi and chi for the sample delta and gamma rotations for the detector as well as the mu rotation for the entire assembly. Let us not go into the details of the 6 circle diffractometer but the point that I want to pass on is that the diffraction condition is not just simple theta theta which may come theta theta or theta to theta 2 circle diffractometer but in order to do advance studies and do proper sample alignment we do need this flexibility of rotating the sample in different ways and therefore the need of 3 circle and 4 circle diffractometers. I am sure that you can find out a course on crystallographic texture in NPTEL courses wherein this will be covered in details at the same time in your in a course comprising of thin film diffraction we will also find the great use of using a 4 circle diffractometer. Having said that about we talked a lot about you know kind of rotation of the sample and all but the most important thing in my opinion or for that matter for diffraction to occur is the alignment of the sample at the same time the ability to look at the region of interest. Many a times we may find some particular region of interest in the given sample and therefore the XYZ translation is very important. The XY translation helps us to move and focus our X-rays to the point of interest while the Z translation essentially ensures that the sample follows the alignment exactly. So if the sample is exactly at the center of the diffractometer as seen over here we see that diffraction condition is satisfied. If it is not there at the center and slightly above or below it we may see that all the peaks in the diffractogram may be shifted and this may be just an artifact of sample misalignment. Now the sample misalignment can be taken care of by using a knife edge or a laser. State of the art diffractometers use a video microscope with laser to align the sample in a particular align the sample perfectly. Now let us go back and look at part by part what constitutes actually a diffractometer. So we know that X-ray diffractometer comprises of a source, the optics and the detector. So let us first look at the source for this we need to understand X-ray generation in details. So we know that for X-ray generation conventionally we use a X-ray tube at the same time if we need more intensity we use what is known as a rotating anode X-ray tube which comprises of a rotating anode. Recently there has been a development of liquid metal X-ray sources which provide high intensity X-ray at with a very small spatial size. Another important source of X-ray which has captured imagination of researchers over the last couple of years is the synchrotron which essentially provides wavelength ranging from infrared to X-rays. So X-ray tube essentially comprises of electrons which are accelerated towards a metal anode. All of us know that electrons which are emitted because of thermionic emission travel and hit the metal target and produce X-rays. I am not going to go in details of X-ray generation in details but I will touch upon it in a later slide. However I hope you can appreciate that if all the electrons are hitting the metal target there is going to be a temperature rise and therefore water cooling which is not shown over here is essential. However I hope you appreciate that the electrons are hitting the metal anode at the same particular point and hence it can the flux of electrons can only be to an extent which can sustain so that the metal can sustain this flux of electrons without getting evaporated or melting. Therefore in order to avoid this deficiency there has been a development of what is known as rotating anode. In case of rotating anode source we have a rotating anode comprising of a metal with high melting point like tungsten and moly. This rotating anode which is physically rotating at the same time which is cooled essentially ensures that the electrons which are hitting the anode are hitting in different regions ensuring that the flux of electrons that are heating the anode and producing X-rays can be much much higher. The rotation of the anode essentially ensures that the sample the flux of electrons and therefore the flux of X-rays can be higher without overheating the anode. This essentially ensures that the X-ray intensity obtained from a rotating anode is much higher than that of obtained from a normal X-ray tube. Another important observation is that the X-ray tube comprises of metal anode which can be of different elements like copper, chromium, moly etc. However for rotating anode we do need a high melting point element so that it can endure the flux that is incident on the anode and release X-rays of very high intensity. Just to show you the different kind of rotating anodes that are available we do have a small angle anode where essentially you see that this angle over here is essentially quite small and it gives a small focal spot while in case the angle is large we do see that we get a large focal spot. Now depending on our requirement we can go for a large or small focal spot. Having said that as I had mentioned rotating anode of tungsten or moly gives you high flux. This micro focus rotating anode is at best 10 times brighter than a conventional X-ray tube. Another recent development in the area of X-ray source has been the evolution or the development of what is known as liquid anode X-ray tubes. I would like to mention that the most important problem that is essentially the achilles heel in getting high X-ray intensity in X-ray tubes as well as rotating anodes is the heat or the removal of heat from the anode. Now this becomes the biggest problem and therefore we cannot heat the material or we cannot bombard the anode with electrons as the material tends to melt or evaporate if the current is increased. However with liquid anode we do have a constant flow of liquid and the electrons are always hitting a new interface comprising of liquid. Therefore there is no problem of overheating and we can obtain very high intensity as good as 100 times brighter than the X-ray tube. Another important advantage is the beam size that we can obtain with a liquid anode X-ray tube which essentially ensures that we get a very high intensity and small focus X-ray beam. The metals like gallium and gallium indium and tin alloys are used for generation of liquid anode X-ray tubes. A lot of literature is available recently and I request you to go through what all existing literature on the liquid anode technique. Another important thing that I had mentioned which can provide a very good source of X-rays is synchrotron which provides intense beam. A line says which I have borrowed from the Canadian light source essentially says that it is brighter than 1000 cents. However as we all know the access to synchrotron radiation is many a times limited. So I would just like to mention a few things about synchrotron won't go in much details but we know that synchrotron is a very high brilliance provides X-ray source with very high brilliance and coherence. It provides X-ray bulb emitting all radiations from IR to X-rays and it can be as high as 10 power 10 times brighter than a normal X-ray source. So here you see that essentially it comprises of a large ring comprising of straight and curved sections. A very good collection of synchrotron nodes is available on this Berkeley site and I request you to go through Professor Atwood lecture on synchrotron which and wherein he covers the details of synchrotron generation as well as various experimental techniques that can be used using synchrotron. So we know that the essence of synchrotron lies in the fact that moving charge essentially emits radiation and we have electrons which are traveling at velocity close to that of light in a synchrotron. These electrons which are traveling at the speed of light are focused using electromagnets which are used in different geometry mainly the bending magnet in the curved sections as well as the regular and the undulator in the straight sections. I will just show you how the profile of these things look like. So if you look at the bending magnet and if you look at the let us call it say H omega versus the fraction we see that this is how we get for the distribution of energy. This is how we get for what is known as a bending magnet when we go to the axis are the same when we go to a Wiggler we get something like this. So this is just slightly shifted so this is my Wiggler. However when I go to an undulator which again comprises the only difference between all these things is the way the electromagnets are arranged we do get high intensity. So we can understand that using a combination of these bending magnet Wigglers and undulators we can get something a spectrum which is similar to a very high intensity as shown in this case. So now coming back to our normal X-ray diffraction all of us know that as we increase the voltage initially we get no intensity at all which is essentially known as the short wavelength limit. So if you look at this curve this again I am sure must be covered in your course on X-ray diffraction but since this is advanced course I thought we will go through it again we see that for a particular wavelength there is no intensity. So this essentially is decided by what is known as the short wavelength limit and if this is given by 12400 divided by V where V is your voltage and you will see that if the voltage is lower than a particular for a given voltage the lambda short wavelength limit below this lambda short wavelength limit there is no radiation. I hope you remember that this formula is obtained by just doing the energy balance for the encompassing the kinetic energy of your electrons and the energy or rather the energy of the X-ray that is expected to be emitted. Another important thing is as we increase the voltage we see that we do get a continuous spectrum. Now the intensity of the continuous spectrum is given by aizv to the power m where m is generally equal to 2 and Z is your atomic number and A is a constant and I is the current. So we see there is a continuous spectrum however when the voltage is further increased we do see a characteristic peaks in the spectrum. Now this characteristic peak essentially occurs according to Mosley's law which essentially says the frequency the square root of the frequency is essentially equal to a constant into Z-sigma which is another constant. So depending on the atomic number we do see that different metals will give rise to different peaks. Now the intensity of the peak or the characteristic peak ik is given by biv-vk to the power n where n is a constant equal to generally 1.5 and vk is the characteristic voltage that is needed for getting the characteristic radiation. I hope all of you remember that a characteristic radiation is essentially obtained when a high energy electron knocks out an inertial electron from the atom and when an outer shell electron essentially replaces or jumps to this inertial electron the difference in the energy level is essentially emitted in terms of a characteristic x-ray. Having refreshed our knowledge of conventional x-ray diffraction let us now go back to the instrumentation part we have seen that how we can get a peak in the normal x-ray source which comprises of as we saw see here two peaks which are essentially obtained because of k alpha and k beta radiation. I would also like to mention that this characteristic peak k alpha essentially comprises of two peaks which are known as k alpha 1 and k alpha 2 and the k beta comprises of k beta 1 and k beta 2. I am not going in details of these things considering that you have gone through it in your basic course on x-ray diffraction. So having seen this characteristic peak the first and the foremost important job at our hand is to separate the two peaks and for this we use what is known as the filter. All of you must be aware of the absorption edge phenomena in which we know that a material can absorb x-rays of a particular wavelength and this is characteristic of the atomic number of that element. Having said that we know that generally for copper to remove copper k beta nickel filter is essentially used. However when we use a filter we see that there is an reduction in the intensity of k alpha 2. Therefore the choice of proper thickness of the filter is very essential to ensure that we do get a high intensity k alpha beam. So this filter comprises of nothing but a thin foil of the metal which is inserted in the path of the x-ray beam. Having said that the x-ray beam as we are all aware tend to diverge. In order to you know kind of limit the divergence of the beam we use what are known as slits. So the slits are used to limit the size of the beam. At the same time we can use the slits to alter the beam profile and the so called solar slits shown over here can essentially reduce the divergence of the x-ray beam and provide parallel or rather semi-parallel coherent x-rays to us. It can be appreciated that you know narrow slits can lead to lower intensity and can give you give us can lead to a narrow peak and a lower intensity. Another important hardware that we use is essentially the mirror. Now the mirror comprised of you know multi layers of heavy elements on silicon which essentially acts as a diffraction grating and can be used to separate k alpha 1 and k alpha 2 which are very close to each other. At the same time we can also use what is known as a monochromator for which consists of a particular crystal like the one shown over here namely the silicon and graphite crystal. The silicon being a much better monochromator than a graphite to separate the k alpha 2p. Another important thing that I would like to mention about the mirror which I missed out is that all of us are aware that x-rays cannot be focused. However having said that the mirrors can ensure that the divergence of the x-rays is limited and therefore we do get a nicely nice coherent x-ray beam which can be resolved or rather limited in terms of spatial dimensions. So now let us look at the different profiles that we can obtain. So we know that a normal diffractogram is going to or a normal source is going to give us x-rays which are tending to diverge. Using slits we have now limited them, passing through solar slits we have ensured that most of them are parallel. Now using a particular mirror we have ensured that the Bragg's law in the mirror is satisfied and all the planes or rather all the lines x-ray beams are diffracted only in one particular direction. So this essentially ensures that we are getting a parallel set of or rather a parallel x-ray beam and this can be used in what is known as essentially a parallel beam geometry. Now not only that what we can do is we can also put what is known as the solar slit which we had used in the earlier case at the receiving side and then a mirror so that we ensured that our detector is only seeing the parallel x-rays that are getting diffracted from the sample. So this is essentially known as the parallel beam geometry. However normally what we use in a normal x-ray diffractometer for powder and polycrystalline bulk materials is a para focusing geometry which comprises of filters and all but not much of mirrors and solar slits all we have is a few or rather are a few divergent slits in the source side and the detector side. Having said that there are certain applications like what I had mentioned like texture measurement or residual stress measurement where we need a point focus. Herein we can see that the x-ray is focused into a point a good observation or rather a good option is to use a rotating anode source which provides a very small but coherent source of x-rays. However those sources are very rare and pretty costly so a normal point source can be obtained using what is known as a poly capillary. A poly capillary essentially comprises of a lot of optical fibres and the x-ray beam which is incident essentially undergoes complete internal reflection ensuring that we do get a high intensity of x-ray beam with smaller spatial size. So this can be used essentially for characterizing texture and residual stress while the normal parallel beam which I had shown over here can be used mostly for characterizing thin film samples. So I would just like to give you a comparison of the parallel beam and para focusing geometry but before I do that I would like to mention that even in the point focus we do have a nice parallel beam geometry. Only thing is the spatial size of this particular geometry is reduced by an order of magnitude. So to compare the parallel beam and para focusing geometry I would say that the x-rays are aligned for the parallel beam while the x-rays are diverging for the para focusing geometry. The parallel beam geometry gives lower intensity for bulk samples and that is why we do not use it regularly while for normal bulk and powder samples we use the normal para focusing geometry which gives higher intensity. However whenever we come to smaller samples or thin film samples we go and use a parallel beam geometry because para focusing geometry gives us a lower intensity. Now another important thing is that the instrumental broadening is independent of orientation of diffraction vector with specimen normal. Now remember we had talked a lot about these angles which tend to rotate the sample as well as rotate your detector namely the phi the chi which essentially you know kind of rotate your sample along the axis which is perpendicular to the surface of the sample at the same time along axis which is passing through the sample. So this which is known as the chi rotation looking at the pen this is the chi rotation and this is the phi rotation. So you can imagine that in order to you know get good diffraction data in such a case the most important thing is to use a parallel source or a parallel beam geometry and therefore almost all thin film measurements crystallographic texture and residual stress measurement are carried out using parallel beam geometry and this is what is mentioned in the last slide. At the same time we know that the para focusing geometry is very suitable for normal brick brand and geometry this is the this is something that we go to the normal x ray diffractometer take in our sample put it in a diffractometer and that this is what we get. However the parallel beam geometry as I mentioned is useful for stress texture and also for grazing incidence x ray diffraction. We will cover this gi x already part in the next to next lecture but keep this in mind that the entire geometry the entire spectrum of diffraction can be covered just by playing with the optics and the source used in a normal diffractometer. So having said that now let us look at what exactly happens once the sample has interact once the x rays has interacted with the sample having after this interaction we do get a signal. Now once we get this signal we can again pass it through the optics part which I had shown earlier to ensure that we are getting a particular wavelength I would repeat that the entire thing that we are talking about the powder diffraction or poly crystal diffraction and its various sub domains essentially deals with only a single wavelength and therefore it is important to get information from a particular wavelength. Having said that this is probably the only reason why we need a lot of optics in the kind of source side which essentially ensures that you are producing only one particular wavelength to interact with your sample. However I hope you remember that you know the interaction of x rays with material can also lead to what is known as fluorescence and produce x rays with a different frequency or wavelength. If that happens you do not want that noise to come and interfere with your detector signal and therefore you have to again put the same optics or similar optics rather from your source side to your detector side to ensure that you are looking at one particular wavelength of your interest. Once the x ray beam passes through all these optics it then encounters the detector which essentially reads the intensity of the x rays and the angular dependence. So detector comprise can be classified depending on dimensions as 0d, 1d and 2d. So let me first talk about the concept of dimensionality of detectors. So all of us know that the Bragg's law is not simple as simple as it sounds and it is anything but just a reflection of x rays from so this is how generally we draw Bragg's law incident beam and this is the diffracted beam. But all of us know that this is not exactly what happens. In fact what gives us a much better picture is ensuring that all this is essentially we have a cone of diffraction. If you remember the more generic picture is the x rays coming over here and this getting diffracted at a particular angle forming a cone. So this is the actual this is a simplified version and is not this is just a schematic for you know understanding what is diffraction. A more realistic view is like this wherein we see not only one but a multiple number of multiple number of diffraction peaks occur. Now having appreciated that diffraction comprises of diffraction cones rather than diffraction lines I would like to mention that the detector is essentially has to see part of this thing. So we can imagine that if we take something like a normal film a photographic film which we do generally in the Debye-Scherrer method we know that we do see the intersection of these cones in terms of these arcs right. So this is probably the oldest detector that has been used or the oldest two dimensional detector that has been used to study x-ray diffraction. Now I will go back and show you the state of the detectors which can do the counting of x-ray signal and convert it to electronic signal. So we have what is known as a single photon detector which reads only one theta or one angle at a particular time. So this comprises of the normal scintillation detector comprising of sodium iodide which essentially on hitting the x-ray we do get the emission of a photo electron and this photo electron is then passed through a photo multiplier cube to generate a electrical signal and this electrical signal is proportional to the number of x-ray photons hitting the sodium iodide crystal. We can also use what are known as proportional counters and xenon gas which again lead to the interaction of x-ray with the gas leads to the formation of a photo electron and which can be amplified to obtain current. Another important thing is semiconductors. So in case of semiconductor detector what we have is once the x-ray interacts with the semiconductor we do get electron hole pair and therefore a current which is directly proportional to the number of x-ray photons you know hitting your detector. So the detector can give us information in single dimension or what is known as 1D detector. If we take the detector and replace it with a wire which can give us information not at a particular value of theta or 2 theta for that matter but over a range of 2 theta it is known as a position sensitive detector. The state of the detector the old fashion what I showed over here the device error ring of a photographic plate what we do get essentially is a 2 dimensional ccd detector. So the 2 dimensional ccd detector is not just giving me information. So this is the old Debye-Scherrer photographic film photographic film instead what we have is a nice area detector which covers a large area and shows me patterns like this which are nothing but the intersection of my diffraction cones. So these peaks essentially gives us information not only in 2 theta which is shown over here but also you see there is another angle over here which information you are getting. So this information corresponds to what is known as the gamma or chi angle which we had talked and corresponds to orientation of the Chris of the sample. So essentially we talked about only diffraction which comprises of this particular thing where you know the x-rays are getting diffracted but the sample can also be tilted accordingly and therefore you see this is what corresponds to this you know diffraction from not only the planes which are directly aligned this way but also the planes which are aligned at angle corresponding to chi. So we can I hope you appreciate that in a normal 2 dimensional detector you get a lot of information or a much better view of reciprocal space. Having said that we know that what all the entire concept of x-ray detectors is based on converting the x-ray photon into a photoelectron or electron hole pair then you use a photo multiplier tube or amplifier to increase the intensity of the signal and essentially get an electrical signal which can be correlated with the theta or rather 2 theta value and the intensity. Having said that what is important for the choice of a particular detector we need to focus on the resolution which is essentially the ability to distinguish between two energies at the same time what is more important is to look at the energy proportionality which is ability to produce signal which is proportional to the exact energy of the x-ray photon. Another important parameter is the sensitivity which ensures that we can detect low intensity levels. In the state of that x-ray diffractometer that we are talking about we talk about the speed is very important because we need to capture dynamic phenomena occurring in during the process like heating or phase transformation. At the same time we have to get a better view of the reciprocal space as I shown showed for the 2D detector which essentially ensures that we get a better view of the reciprocal space and get more information about the x-ray diffraction pattern. I hope by the end of this lecture you have got a rough idea about the instrumentation part of x-ray diffractometer. In the next couple of lectures we are going to talk about two of the most important diffraction techniques namely the small angle x-ray scattering and grazing incidence small angle x-ray scattering. So see you after the next class.