 Now, we are back to diffraction. After discussing the difference in diffraction experiments between reactor and the pulse neutron source, we are back to the diffraction or structure determination that means no energy analysis using neutrons. Now, most of us are aware that we actually as soon as we make a sample, we take it through a extra powder diffraction and we try to see whether we can identify the peaks that are listed in the international table form diffraction from powder. So, we can do powder diffraction for phase identification which is the simplest thing to do for strain analysis, preferred orientation and crystallographic and magnetic structure. So, all these are identified, all these experiments can be clubbed under powder diffraction I should say, which is the most commonly used technique with neutrons also with x-rays and I will discuss some of the techniques. Most importantly, I will introduce you to crystallography and magnetic structure determination together with extra crystallography is required for various samples. Then if you can make a single crystal, a good single crystal because in neutron diffraction not only you need a single crystal if you want to do ab initio structure determination, but you also need to make a slightly larger single crystal because neutron intensity is poorer, but still in case of hydrogen bonded crystals which is not possible to study with x-rays single crystal diffraction is done, I will very briefly mention this part. Next, another important thing is local structure in liquid and amorphous systems. This is extremely important for neutrons because for determination of local structure you need to go to a very high Q and neutrons have an advantage that it can penetrate deep. Since it can penetrate deep, we can take a reasonable dimension of a sample which will give the bulk liquid or a bulk amorphous system and you can find out the local order in the system. I will introduce you to this technique with large Q range experiments to find out local structure. Another very important technique which has become extremely popular not only with physics people, but chemists, metallurgists and many others it is small angle neutron scattering. Small angle you can also call it small Q often we are talking about angle angle and Q we will use interchangeably. So, small angle neutron scattering is nothing but a small Q neutron scattering and since it is small Q we can always say that if Q in an experiment Q max is small then the length resolution is twice pi by Q max. If Q max equal to 0.1 angstrom inverse then this comes to around 60 angstrom inverse 60 angstrom that means inherently if I do an experiment in the small Q range say up to 0.1 angstrom inverse then that experiment will tell me about structures with a resolution inherent resolution of 60 angstroms. So, with 60 angstrom resolution I cannot see the crystal structure or the crystallographic structure in a system what I will see is the in average over this length scale. So, that is a mesoscopic length scale to compare a typical diffraction experiment will be around going up to 10 angstrom inverse. This might go to 120 degrees 130 degrees in an experiment whereas 1 angstrom inverse experiments are almost a near direct beam experiments. I will discuss it when I discuss the small angle instruments. So, small angle is used heavily for determining structure at mesoscopic length scale and often many researchers are not keen to understand to get information at near crystallographic or near atomic level resolution, but at mesoscopic level resolution. For example, we may talk about micelles forming in liquids or as I told earlier in my transparency pores in solids or precipitates in metallurgical samples this can be used by this can be understood by using small angle neutrons. Another technique which has come up recently is neutron reflectometry for thin film structure. So, today especially for applications thin films are very important and the neutron reflectometry deals with films deposited on various substrates and the reflection I mean optical reflection, optical reflection not Bragg reflection, optical reflection, reflection following Snail's law, but from the reflected beam reflected intensity we can understand the thickness of these layers from a heterostructure that is reflecting neutrons, their interface roughnesses they can be determined using neutrons and x-rays and neutrons can also give you magnetization magnetization as a function of depth which is unique again for neutrons and all this we will be discussing when we discuss neutron reflectometry which will be coming under the heading of mesoscopic length scales. So, this will be my flow that I will start with powder diffraction various techniques briefly single crystal deuteron diffraction then liquid and amorphous systems and local structure small angle neutron scattering and neutron reflectometry for thin film structure under the heading of diffraction or structure at various length scale. Now for diffraction experiments we I told you earlier that the for Bragg diffraction from a lattice K minus K prime should be equal to G a reciprocal lattice vector and for a for an elastic experiment the magnitude of K and K prime are same. So, this is a requirement that K minus K prime is equal to G now I can show you that this requirement of K minus sorry K minus K prime equal to G this translates to K minus G or K plus G both ways equal to K prime plus or minus G equivalent this comes to 2 K dot G plus G square equal to 0 this is I am quoting from Kitten the most basic solid state physics book and K dot G is K is nothing but twice pi by lambda and G is giving by 2 pi by d h k l where d h k l is the d spacing of h k l set of planes and G is equal to twice per h k l basically this 2 K dot G plus G square I can write 2 K dot G if I consider plus G and minus G equivalent is equal to G square or 2 K G sin theta because K this is K prime this is G so K dot G will be K G sin theta is equal to G square or 2 sin theta sorry 2 K sin theta K sin theta equal to G substituting K is equal to twice pi by lambda and G is equal to twice pi by d h k l I will get back 2 d h k l sin theta equal to lambda which is my Bragg's law. So, many of us have started with Bragg's law but I started K minus K prime equal to G because now I want to and they are equivalent as I showed you just now I want to get back to something called Ewald construction. Now if K minus K prime is equal to G this construction was done by Ewald so if I consider an incident beam K i and the reflected beam or the scattered beam ending there is a incident beam ending on a reciprocal lattice so this is a reciprocal lattice then when K minus K prime equal to G I get a reflected beam that means if I consider the three-dimensional reciprocal lattice and with this construction I can say that whenever the equation K minus K prime equal to G whenever the scattered beam hits a reciprocal lattice vector then I have got a diffracted beam is this direction so this is known as Ewald construction. This is the most fundamental construction in case of diffraction atomic diffraction that the I have superimposed the incident beam and this is the origin of the reciprocal space and with that I just show that K minus K prime equal to G so in this thing that means I can have a reciprocal scattered beam in this direction because it is hits similarly I can have a reciprocal lattice vector being equal to K minus K prime equal to G so all these points which are falling on this sphere intercepting this sphere from the reciprocal lattice point they signify that there is a diffracted beam in this direction and that's what our Laue pattern that we see from single crystals are all about so Laue pattern is nothing but it comes out straight away from the Ewald construction and in case of a polychromatic beam I must add here that in case of pulse neutron sources since you have polychromatic beam so your K you have got because K dictates the size of the sphere that you were plotting you can see that this sphere or the circle in two dimension is the radius of K but K for a range of wavelength is polychromatic beam there is a lambda minimum there is a lambda maximum lambda maximum means K is minimum and lambda minimum means K is maximum so I have a range of lattice point wavelength and you can see I have superimposed these two spheres this is the origin of the circle this is a so they are superimposed on the reciprocal lattice vector and all these reciprocal lattice vectors will satisfy 2d sin theta equal to lambda for one of the wavelengths in this beam of lambda max to lambda mean and then again you will have a reflection Bragg reflection in that direction so I take you back to again to Kittel actually here I hope some of you have done this experiment we have done this during master's day so this shows how a powder crystal now looks like so this is called Debye camera this is known as Debye camera where a photographic plate was wrapped inside this circle this is the incoming beam this is the outgoing beam this is the outgoing beam this is the incoming beam and the powder crystal is kept at the center but I myself did was a copper crystal if I remember it correct 40 years back now here instead of a single crystal you all construction now imagine I have got a powder crystal if I have got a powder crystal then imagine these are incident beam so if one crystal light satisfied it in this direction there will be other crystal lights so in having if this and this is the incident beam I can have crystal lights equally oriented with the same theta but at different angle so that means I will be rotating the crystal around the incident beam and the reflected beam will describe a circle in the reciprocal space so now these are perfect powder crystal which will have a Debye Sharer cone this is known as a Debye Sharer cone this is a Debye Sharer cone and what I showed you in this photograph basically a strip of photographic plate that is intercepting this Debye Sharer cone now this is the incident beam hole there's a hole in the strip and there's outgoing beam hole and you can see that this is the largest smaller de-spacing and I can calculate the de-spacing from the radius of this because this gives me the angle from the radius of the Debye camera and from lambda I can find out from this radius of this cone what is the de-spacing so what is the de-spacing so this experiment was done or is done to find out various de-spacing you can see various de-spacings so we'll satisfy different angles because the angle 2d sin theta equal to lambda in this case it was monochromatic beam of copper k alpha so it was 1.54 angstrom it can be something else but usually in laboratory sources we use 1.54 angstrom so that lambda is known knowing the radius of this beam around the incident point we know the theta and once we know the theta 2d sin theta equal to lambda lambda is known we can find out de-spacing so this experiment that detects the circles or the Debye Sharer cones which gives us the d this if I reduce to a strip detector or a one-dimensional position sensitive detector this gives one particular cube or theta and what I showed you earlier this is x-ray Debye camera but what you see here each one is one part of that Debye Sharer cone from a powder crystal from a powder crystal so this is the very beginning of our knowledge about x-ray diffraction but basically this is a monochromatic x-ray beam as I showed this is a specimen and they break up into the this is around this is the low angle and the specimen this is the outgoing beam this is the high angle so this is sorry I'm sorry the back reflected beam is high angle and the forward reflected beam in the low angle so this is around the this hole this particular radius this is the at the center of this circle and this is near the back scattered beam and you can see the back scattered x-rays because of the atomic form factor they are of much lesser intensity than the forward scattered beam because the form factor for almost all atoms in case of x-rays is fall like that form factor so with theta or with q so at large angle we have lesser intensity because of the form factor structure remaining same so we see the same circles same Debye Sharer cones but with lower intensity when you do x-ray diffraction we do exactly the same thing with neutrons in neutron sensitivity I mean in neutron crystallography a position sensitive detector is equivalent to the photographic plate only the size dictates how much of the cone that you are intercepting but they have better intensity resolution the instrument that I showed you it can identify the de-spacing from the radius of the cone around this incoming and outgoing paths and but in case of in case of neutron there is better intensity resolution and we can carry out phase determination or more detailed experiments to read well fitting so an ab initio crystallography is possible with single crystals and powders a single crystal not with powders there are various applications of neutron crystallography apart from physics I will also tell you how strains can be determined using neutron diffraction specially neutron diffraction because neutrons can penetrate deep in industrial pieces of materials which is not possible by any other using any other radiation so also neutrons are unique for magnetic structure so we will discuss all those in the next lecture