 Today we will be discussing application of neutron crystallography and in this lecture I will presume that almost all the samples that I will be discussing are powder samples and I will start with some of the practical applications of neutron crystallography and when I say crystallography I mean crystallography structure using neutrons and their applications and I will towards the end I will come to magnetic diffraction in this lecture but before I start with the lecture proper I think I should repeat a little bit of what I said in the previous lecture because I think it is important for us to understand various constructions and the various techniques that we use to determine the position of the Bragg peaks. In the last lecture I discussed something called Ewald's construction. Possibly many of you are familiar with this technique but we always go with the formula Bragg diffraction Bragg's law 2D sin theta equal to n lambda. This is the most commonly used formula and where we assume diffracting planes and the diffracting planes and the way it is done actually the path difference between these two rays are 2D sin theta. If theta is the angle of incidence and it can be shown that the path difference between these two rays basically are 2D sin theta and that should be equal to integer number of wavelength lambda for constructive interference. But there is another way we can visualize diffraction which I said last time that if I have a reciprocal lattice here a reciprocal lattice here a reciprocal lattice here a reciprocal lattice and then if I have my initial wave vector key which is comprising a value of twice pi by lambda and a direction if I end it on one of the reciprocal lattice points and around that I draw a circle or sphere in three dimension then whenever this circle touches one of the reciprocal lattice points I will have a k prime vector which will or rather the outgoing vector that will giving a Bragg diffraction. So whenever there is this circle so if I have a shorter lambda then this circle twice pi by lambda will be larger if I have a longer lambda it will be smaller and the same circle will be a smaller dimension if my please excuse my drawing. So depending on lambda the diameter of the circle matters as radius is k and the diameter is 2k and whenever this evolved circle is sphere we call it evolved sphere touches one of the reciprocal lattice points we will have a diffracted beam. So that's how we wrote the selection rule that k minus k prime should be equal to g a reciprocal lattice vector. So k minus k prime should be equal to g a reciprocal lattice vector where the lattice reciprocal lattice vector the value is twice pi by dhkl there is a lattice reciprocal lattice vector for a d spacing d please understand that this expression and this expression that equivalent if I start going from here I know that I can write k dot g plus g plus g squared equal to zero and from there I can show you if I say twice pi by lambda and twice pi by dhkl here and g dot and this is sin theta and I can also take g negative or positive so then it becomes k dot g is equal to equal to g square or this is equal to 2 pi by dhkl square so this cancels out and we get 1 here so then should be equal to cool and then from there I get 2d sorry this g on the other side I'm sorry sorry sin theta should be equal to 2 pi by dhkl and then 1g square as there so into 2 pi into 2 pi ultimately what I did is one second d 2 pi by lambda sin theta equal to 2 pi by lambda dhkl this gives you 2d sin theta equal to lambda for this transfer to this so with this so starting from k minus k prime equal to g I can also get back to 2d sin theta equal to lambda and which allows us this eval construction that k minus k prime equal to g now coming back to powder crystals now the thing is that I talk about 2d sin theta equal to lambda with respect to one set of crystal planes but and then the reflected direction is this but if I rotate the crystal around the incident beam that means I have a powder crystal in which I have got crystallized of all orientation then I will work out a cone and for a powder crystal this cone is the signature of one particular dhkl and then I discuss with you the Debye-Shearer photography in which there is a cylindrical body in which at the center I have a powder sample I have got an entrance slit and then exists exit slit and then there is a photographic plate wrapped inside the Debye-Shearer camera then the diffracted this cone is arrested on this photographic plate this is at large angle this is at small angle so when I open the strip and develop the photographic plate this is what we start there when we did powder diffraction then I then I have goods I have got these circles corresponding to various displacings on the photographic plate and the one which is near the exit point slit that means here they are the low Q values and they are more intense and the one which is near the entrance slit they are less intense because of the form factor in x-rays going down so this photographic plate basically intercepts a part of this a part of this Debye-Shearer cone corresponding to every d spacing and knowing this radius and knowing this radius of this circle this circle and this radius of this Debye-Shearer camera we can find out easily what is the d spacing corresponding because we can easily calculate the theta for that plane but the given lambda of x-rays so this is the same thing we do with powder crystals with neutrons so powder crystals powder crystals plus neutron diffraction we do the same thing exactly what we do with x-rays that means with the powder crystal we try to get a part or a slice of the Debye-Shearer cone and knowing the position of the detector or in case of position sensitive detector knowing the angle at which it has hit the detector we can find out the intensity and we can find out not just g-spacing but we can do a ribbed fitting for detailed structure analysis in case of neutrons so I repeated a part of what I did earlier so now I will try to introduce you to various techniques that can give you structural information that can give you strain and micro strain and particle size with neutrons it can give preferred orientation and most importantly the powder diffraction and from a magnetic sample and then obtaining the magnetic structure using the diffraction data and lastly I will mention how single crystal diffraction is done because most of the time it is easy to make or usually we get a powder sample and single crystals are rare but if you need to solve an ab initio structure or structure in case of hydrogenous crystals if you can make typically a sample which is 2 millimeters across at least 5 millimeters to 10 millimeters across then we can do single crystal diffraction for ab initio structure now with this let me get on to the studies including screen now neutron has a very big advantage because they can penetrate very deep if we compare the penetration depth of neutron with respect to any other radiation x-rays of course tens of pichrons and then x-rays are absorbed very strongly in case of electrons it is not even 100 angstroms before it is absorbed in case of neutron this value goes to 10s of centimeter so because of that it is very suitable to study strain in case of industrial samples or samples of large size and this strain in industrial samples is a very important factor when we talk about various welding joints and other kind of samples or material bodies which are under strain continuously or under repeated strain so the basic theory is very simple if we can measure the Bragg peak from a sample then the Bragg peak can give you compressive or tensile stress strain in a sample simply from the Bragg relation 2d sin theta equal to lambda because if the Bragg peak shifts if it shifts to lower lambda sorry I am sorry to lower theta for a given lambda of neutrons then that means that d has increased because 2d sin theta equal to lambda or if it happens other way then d has decreased and the macro strain for a sample under stress is given by d minus d0 by t where d0 is the lattice spacing for strain free sample so in this case what I mean is that if I have a lattice under strain under some field so either the d spacing can shorten when it is a compressive strain or it can elongate but this we can find out from the position of the Bragg peak position of the Bragg peak if the lattice has elongated then we can see that d will increase if the d increases then theta decreases and the Bragg peak shifts to the lower value and if the lattice is compressed that means d has reduced so for the same peak the lattice will move to the higher theta value so in principle it is very easy to understand but there are technical challenges which I will discuss with you so this is the experiment comprises in measuring the position of a Bragg peak that's all a determined strain but the fact remains that when I am trying to determine the strain in a sample then the sample is not uniformly strained and it's a large let us consider it's a large industrial sample so like a welding joint or a piece of a railway track I will show you the photographs and the experimental arrangement should be such that I can select a certain volume in the sample in which I am attempting to measure the strain so that is to be done by using what is known as nose cones that means we have the same arrangement earlier the way we had earlier but here it is not a PSD based system because we are not trying to find out the structure our target is to find the strain in a sample so knowing the sample for example it can be stainless steel knowing the d and lambda of neutrons we can choose the angle of diffraction and often we use a end on detector not a position sensitive detector to get the diffracted intensity so we have an incident beam as you can see here and we have a sample here the large sample here and we have a detector so this is a monochromatic this is the detector but what is unique here I can show you these two green collimeters which are known as nose cones why these nose cones are required because you see I have a large industrial sample but I want to determine the strain in certain volumes so if I make a tight cone like this of course this will also reduce the resolution to some extent but if I have this code then the beam is focused and then if I have a nose cone on the detector side then the outgoing beam is also inside the inside this has to go pass through this nose cone and there is a detector here I have scaled up everything so to make it clear so now these nose cones dictate that what is this volume from which the diffraction is taking place so now this volume can be as small as 3 by 3 by 3 millimeters to maybe 1 centimeter by 1 centimeter by 1 centimeter depending on how tight you make these nose cones and since the volume is defined so now let us consider that the incoming monochromatic beam and the bragg reflected beam for a certain wavelength in a certain direction going to a detector positioning the sample in front of the beam we can choose the volume that we are planning to study so that means now this industrial sample can be moved up and down left and right and if you look at the three dimension then there may be also rotations and with this we can figure out which volume of sample which part of the sample is getting irradiated for a certain setting of the sample and then we can see the diffracted beam get the d experimental and also can compare it with the d0 which is strain free now this is a question that how do we decide what is the d0 value with our strain so now that is done either if we know that the part of the sample is strain free for example if you are trying to detect the strain around a welding joint now we know that where the welding joint is I might go 5 centimeters or 10 centimeters or 20 centimeters away from the welding joint and measure the d spacing and consider that as my d0 that means d spacing without strain and compare it with the d spacing that you obtain from the sample volume around the welding joint and then I can compare these two and I can figure out what is the strain in a given volume and I can keep on sliding my slide a slide rotate translate my sample in the beam path so that I can find out for example for a welding joint across the welding joint away from the welding joint what are the strains and make a map of the strain values in a welded joint and this is a very lucrative or rather sort after technique today but the fact is that they often the samples can be to the tune of tons that means they are very heavy samples so we should have a very heavy x y z translation stages rotation stages which can handle such heavy loads maybe from kilograms to tons so for example I have chosen some examples from the in from the literature so this is a work done to understand the structure connecting ferritic carbon steel and austenitic stainless steel through a welding so you can see that here in this figure a welding joint is made between these two these two pipes and this is a part of a reactor so it is important to know because often such welding joints go through thermal cycling so how good is the welding joint and what is the sort of strain distribution around a welding joint this is more of an engineering problem than a basic science problem but neutron does excellent job of mapping the strain for example here you can see this is these are the sample volumes as I explained to you just now which have been irradiated by the nose cone and a small volume here gives gave the strain in these locations so a photograph of the thing you can see that this is the x y omega stage this experiment was done at two locations for the same welding joints you can see the sample here here the welded pipe joints and you can see the detect incident slit and the detector slits typically for steel the diffraction angles are close to 90 degree so so we can play with rather we can move this sample and get various various volumes as I said so you can see for this welded joints we can do a scanning of along the pipe around the what should I say the body around the body of the welded joint in the goings in circles what is known as a hoop and also we can do a radial direction that means across the thickness of the tube so all these studies can be done map and then decision can be taken about the quality the procedure of the thing here I show that they had plotted in axial means along the axis of the cylindrical tube radial means as you go out in a cylindrical geometry and hoop is along the body of the sample and they have tried to plot the strain overlaying the axial hoop and radials together over here across the welding joint and the they have mentioned the lattice strain in some unit over here so this is a practical example which has been done by using a nickel based alloy welding material also this is this was done at firm to garking this is a piece of the railway line which had seen a lot of passage of trains over it and then a piece of it was taken and the stress mapping strain mapping was done using neutron diffraction so you can see the map so these are the red is the more stress part and there are you can see this is a section of this pipe of this railway line and they have done a mapping of the various parts of this railway line to determine the map this map was obtained from the strain measurements using as I told showed you earlier the diffraction techniques with tight nose cones for the incoming and the outgoing beam so this is macroscopic strain on large samples so this macroscopic strain means the entire sample is strained I mean at a macroscopic level means at least the strain part is more than strain part are larger than microns and you can see that the volumes can be 10 centimeter cube to maybe 100 centimeter cube for such studies that depends on the opening in the nose cones for such experiments there is another way of doing particle size and microstrain measurements using neutron this technique is well known with respect to x-rays and in the next part of the lecture I will attempt to explain to you how the same thing is done using neutrons