 In this lecture we will continue with phonon measurements and we will be discussing single crystals. So primarily we are talking about doing experimental measurements of phonon dispersion curves using neutrons. Here before I discuss further it is important then that I discuss the way the experiments are being done. I have told you several times and showed you these spectrometers known as triple axis spectrometers. It was discovered by B.N. Brockhouse for studying inelastic dynamics or inelastic neutron scattering to reveal dynamics in materials, condensed matter in general. So as I told you that this is the monochromatic drum, this is the first axis, the sample is the second axis and then the analyzer around the sample which finds some of the final energy is the third axis and then there is the detector. But I will go little bit more details of it. The thing is that first in the beam path as I showed you inside this drum at the center of the drum there is a monochromatic. The role of monochromator is to get a monochromatic beam of neutrons. Now if you consider the reactor flux as Maxwellian then the role of this monochromator is that we have a known d spacing so 2d monochromator sin theta equal to n lambda. So we choose one specific monochromatic beam by choosing the plane of the monochromator and the angle with respect to the beam and finally the total deviation which is 2 theta monochromator. This is the deviation of the beam with the direct beam and this fixes the lambda. But now I must mention a few things. For a monochromator as I mentioned earlier I cannot use a perfect single crystal because a perfect single crystal will give a very narrow beam of reflected neutrons so the intensity will be a problem. So what we use actually are known as mosaic crystals. What does a mosaic crystal do? So instead of usually in textbooks you find when you do the Bragg's law like this planes of the Bragg's law and you will write 2d sin theta equal to n lambda when you assume that this is a perfect single crystal. But in reality what we do actually this crystal if it is a single crystal it is traced or strained in a way that this crystal becomes consists of small crystallites like this. So the crystal is broken into crystallites which have slight angle with respect to each other. And then the beam of 2d sin theta equal to n lambda 2d sin theta equal to n lambda if this angle theta monochromator varies a little then you will also have a spread on lambda and that's a desirable thing. So that means instead of choosing a very narrow beam from the incident beam neutron beam we try to choose something which is broader but now the question comes by doing this I have improved the intensity but compromised resolution yes because now our wavelength lambda has got a spread delta lambda. In this I can also mention another another phenomena that is second order contamination we write 2d sin theta equal to n lambda. So for example suppose I am taking 1 1 reflection from a copper monochromator monochromator. Now for almost the same angle the 2 2 2 reflection from lambda by 2 if this gives me lambda from lambda by neutrons will contaminate the reflected beam. Now there are various ways that I can get rid of this monochromator this contamination because that we can put a filter in the beam often we can put a filter in the beam which will cut down the lambda by 2 or the more energetic neutrons and allow the primary neutrons to go in. So we want a broader beam but away from contamination due to second order reflection and the mosaic spray will give me intensity and of course I make a little bit of compromise. Now the analyzer this analyzer also analyzes the energy of the outgoing beam by choosing a bright plane often it is pyrolytic graphite let us say sometimes pyrolytic graphite and this plane is chosen at an angle you can say theta analyzer then this lambda analyzer is known because of the angle everywhere there are contributions from instrumental resolution which is mosaic spray and ultimately these 2 mosaic sprays will sum up in square but the fact is that this will give me better intensity to do the experiments. So now what I have is 2 changes one is Ki minus Kf equal to Q this Q is not equal to 4 pi by lambda sin theta I must mention it specifically because now it is an inelastic process so length of Ki is not equal to length of Kf so this is not 4 pi by lambda sin theta but this is Ki by Kf of different lengths and h square Ki square by 2 m minus h square Kf square by 2 m is the energy difference between the 2 this is the energy transfer in the inelastic experiment and Q square equal to Ki square plus Kf square minus 2 Ki Kf cos theta these are magnitude if theta is the angle between Ki and Kf so now these are the equations which will be valid for an inelastic neutroscattering experiment and then we have to arrange our sample analyzer and the detector in a way that either we can perform the scan along a path is Q e or Q omega space Q is the momentum transfer e or omega h cross omega is the energy transfer we can follow a path in a constant Q or in a constant e path so let me just quickly show you if you remember I was showing you the phonon dispersion relations somewhat like this so this is omega versus Q so I can do this scan along a constant Q path or I can do the scan along a constant energy path but now again I must caution you that I have got three values for momentum one is Q this is the phonon momentum phonon momentum when I'm plotting omega versus Q this is a phonon wave vector and this is limited between 0 to pi by a and on the negative side minus pi by a for a linear lattice of length l so if I go to twice pi by a twice pi by a that gives me one reciprocal lattice wave vector g but I can always reflect back anything I have got beyond pi by a from here to here just by subtracting just by subtracting twice pi by a from this value I can bring it back to the first brillouin which is from here to here often you will find and I will show you the experimental results later that we do the plot in first brillouin second brillouin for another symmetry direction third brillouin for another symmetry direction because the phonon dispersion relations in a crystalline lattice depends the direction of propagation or the direction of the wave vector Q and there are specific symmetry directions in a crystal along which the phonon dispersion relations will change depending on the direction that you are following and all of them needs to be found out experimentally so now quickly so I have got a Q which is phonon vector for the as I said phonon momentum vector Q is the momentum transfer momentum transfer in the experiment and G most importantly the reciprocal lattice vector reciprocal that is vector so before we start or you plan an experiment for phonon dispersion relation we must know the crystallographic structure crystallographic structure and we need a single crystal single crystal for experiments for experiments because phonon dispersion relations are along symmetry directions in the crystal and for that we need a single crystal which we can orient for our experimental purpose so with this let me show you two more I tried to show you triple axis spectrometers from reactors so the structure is similar this is the monochromator this is the sample and there is a analyzer on three axis and the detector comes so this is an iron 22 so it is the basic structure is same but what are added advantage are you can see here I have taken it up from the online description of the instrument so you can see that it can have incident energies in the range of 5 to 100 mab reasonably large energy range can be covered it has got a vertical magnetic field up to 15 tesla horizontal magnetic field up to 4 tesla and we can also do high pressure experiments because phonons are the collective oscillation of the atomic lattice sides with pressure there are lots of changes because inter atomic distances change and the phonons have different changes so you can do pressure at 1 gpa or high pressure experiments are possible here similarly iron 12 is another three axis spectrometer for cold neutrons so we can do because it's a cold neutron spectrometer we have can do lattice dynamics at low frequencies or low energies when I have cold neutrons that means typically the energies are in the range of less than or in the below 5 millilektron volt and wavelength is more than 4 angstrom such an instrument needs a cold neutron source installed at the reactor and that's why this is an added advantage here that you can use the cold neutrons in ILA Grenoble for studying slower dynamics lattice dynamics at low frequencies slower dynamics critical scattering and it has got a weak static magnetic field that they can study magnetic multi layers and thin films and because of slower dynamics most of the biological model membranes their dynamics are extremely slow or magnetic excitations can be used like magnomes using this spectrometer but basic structure of the instrument that I described here the triple axis axis spectrometer at druba the ion 20 to thermal triple axis at ILL and the spectrometer at the other spectrometer ion 12 at ILL the basic structure is same here you can see because I am working with at low energies the monochromotor is curved monochromotor is curved means you have a curvature like this so this curved monochromotor will tend to focus the beam so beam dimensions will be small dimension will be small but divergence will increase so at the cost of again divergence or resolution I can get better intensity but this is a curved monochromotor you can also have vertically curved monochromotor but experiment done in the horizontal plane in that case you can gain intensity but you will not lose resolution in the horizontal plane so mostly you can see if you see this photograph I couldn't get photograph for those these are very heavy equipments like detector or rotation stages so always or almost always I don't know of any instrument doing using a vertical geometry the experiment is done in horizontal geometry horizontal geometry so all the experiment that I the instrument that I showed everywhere it is horizontal geometry so that means q which in general should be having three components q q x q y q z if I consider the z is the vertical axis then generally q is q x and q y the plane with the angle between the monochromotor and the angle at which you put the analyzer we can decide what q values I'm choosing and I can do a constant q scan so here if you see then let's see this is the incident energy and wave vector this comes on the sample now this angle this angle decides what is the wave vector transfer q now after it is complicated so now I have got a analyzer here and the detector rotates around it so at this angle when I rotate the rotate the analyzer and the detector in I rotate this in theta I rotate the detector to theta so they're coupled once I once I couple them for a drag peak they remain coupled but this kf changes so kf goes to kf prime because this angle has changed so now you are taking from this beam you are taking another neutron whose energy is slightly different because this angle is different because 2d sin theta analyzer equal to n lambda when I increase the theta I increase the lambda so that means kf that is kf prime 2 pi by lambda prime is longer is shorter because lambda is longer so now I have another ki minus kf at this angle and now I can adjust this angle and this angle to keep ki minus kf at a fixed q and this is the momentum vector transfer in this experiment earlier when we did structure work it was much simpler I had a sample then either I had a detector directly here or I had position sensitive detector which covered the whole q range and at that time what we were doing actually we were doing the structure scattering law which I explained to you earlier we're integrating over energies and we are not bothered about the outgoing energy now we are trying to find out s of q for energy transfer omega and that's why all this travel with an analyzer and its angle