 So, I pardon my not so good diagrams, so if I take a plane of atoms because in solids for example, if it is a simple cubic lattice, let's see simple cubic lattice with atoms at the corner, I can talk about planes of atoms or I can talk about diagonal planes of atoms. So, I can talk about diagonal planes of atoms or 1 0 0 plane of atoms, so I can talk about various planes of atoms. So, here one way of showing this displacement, these are longitudinal displacement that means this is one plane I have shown a line in this plane of the board. Now, this plane, the next plane is displaced by some amount and the next plane is displayed little more, the next plane little more, so the wave is going to the highest temperature then it starts reducing and so on. So, this is a displacement, the displacement is along the along the movement of the along the propagation of the wave vector, propagation vector, so if this is k, the particle movement is parallel to k, particle movement is parallel to k, it is known as longitudinal movement. On the other hand, this displacement from one plane to another can be in a direction normal to the propagation vector. Here I just take one example of the plane which is of the movement, if I call it xz in the xz plane, the atoms from this plane to this plane it is displaced up, next plane it is further displaced up, then in the next plane it comes down and possibly in the next plane it comes up to normal, so you can see a sinusoidal chain. But here the propagation vector is perpendicular to the direction of displacement, so this is a normal reflection, normal displacement, similarly I can think of another plane, let me just draw it here which is perpendicular to the xz plane, let me call it y direction and the same displacement of the atomic planes I can conceive in the y, yz, y direction and the propagation is direction in z, so then also either kx perpendicular or ky, so there can be two directions or two planes of movements which are normal to the propagation direction of the vector. So I have got longitudinal waves and I have got transverse waves and they are the phonons in this case which can be called as acoustic phonons, so let me just quickly tell you, if there is only one atom per unit cell, unit cell then there are three degrees of freedom, now for three degrees of freedom, so there will be, now there will be acoustic phonons where two of them will be transverse as I told you earlier, if the propagation direction was, the q was parallel to the z direction, one can be in the x direction, other movement can be in the y direction, so I have three acoustic modes, two transverse and one longitudinal, that means in this case the displacement is along the k direction, k direction here it is I called z in the z direction, so I have got three acoustic modes which are possible when I have a one atom per unit cell. Another interesting aspect is that from the formula that we used the us plus 1 by us is given by, that means the neighboring atom displacement are given by e to the power i k a, now e to the power i k a it means us plus 1 by us, if k a is equal to pi or let us take when k a equal to pi, then I have e to the power i pi, the us plus 1 by us e to the power i pi means that exactly pi out of phase with respect to each other, now if I go to let us say pi plus 0.2 or pi plus 0.2 pi, so this part becomes 1 and what I have got is e to the power 0.2 pi, now this is something which I will try to explain to you, in case of Brillo zone, the zone boundary comes at pi by a and minus pi by a, what is the Brillo zone? I am just showing that in the reciprocal lattice space when the nearest neighbor is at a distance 1 by a, then you go to pi by a, it is pi by a twice pi by a because that is the reciprocal lattice vector, when the real lattice vector is a twice pi by a, then you do a perpendicular bisector at pi by a, same thing is true in the negative direction minus pi by a, so minus pi by a to plus pi by a form the first Brillo zone and when k is equal to pi or k is equal to pi by a, then from 0 I have reached the Brillo zone boundary, now when I go k greater than pi or k or greater than pi by a a point here as I said 0.2 pi, I can reflect it by 2 pi by a to minus 0.8 pi by a, so I can bring it back by reflection to the first Brillo zone, so anything outside the first Brillo zone, so if I go to the second Brillo zone here, draw it will look clearly, so this is pi by a and this is one of the acoustic modes, when I go to the second Brillo zone, then it is just a mirror image of this and anything here I can reflect back to the first Brillo zone, same thing is true, it will look like this, so that means my dispersion relation of omega versus q, omega versus q is limited to the first Brillo zone and positive side of the first Brillo zone, I can also draw the negative side but it will be just the mirror image of this one and anything beyond first Brillo zone can be reflected back, so k a greater than pi can be taken to k a less than pi just by reflection, so this is what is done in phonon diffraction, phonon studies, so I have just shown you the relative displacements for an acoustic mode, now if there are more than one atom, suppose there are two atoms, suppose there are two atoms per unit cell and ideal example is a bcc lattice, bcc lattice, there is one at the body center and there are eight atoms at the corners coming to one atom over the two atoms per unit cell, then you have got six degrees of freedom, when you have six degrees of freedom, we have got three acoustic modes, two transverse plus one longitudinal but then they are with three what are known as optic modes, so now my dispersion relation will also have a branch known as optic branch, I will come to this later in little more detail, so I have got optic branch and I have got acoustic branch, acoustic branch, why this is called acoustic branch because I will show you in this acoustic branch when you go to q equal to 0, when q tends to 0, q is twice pi by lambda proportional to 4 pi by lambda, so that means lambda becomes long or you go into sound limit, then this becomes clearly linear, so we know that the sound wave propagates with 330 meters per second velocity and here also I can calculate d omega by dq at q equal to 0, we will go to some wave limit and that is why but further down when we have got shorter wavelengths or larger q values then they are actually the relative oscillations they propagate with different velocities, the group velocities are different and there but because of this q going to 0 limit this is known as acoustic branch, acoustic branch I will come to the optical branch in the next lecture, so but before that let me just tell you that I talked about three acoustic modes, two transverse and one longitudinal for any number of atoms, others are optical branch, optical branches basically it show us that this is an example taken from here, in the acoustic mode you can see the movement of the points they can atoms like atom position in a crystalline lattice like aluminum, so if I am talking about aluminum phonons then these are the positions of the aluminum lattice points and their movements are like this they are part of a wave and they move in the same direction whereas when I am talking optical modes then they move against each other, this positive and negative are not charges rather it shows that if this atom moves in the positive direction the negative or the nearest neighbor moves in the opposite direction and its neighbor moves in the opposite direction, so atoms are moving against each other with their center of mass fixed, so these are the optical modes, how to define optical modes I will come to it later but this comes when you have got either two equivalent atoms in the unit cell or in equivalent atom if they are in equivalent atoms and if this positive and negative are the ionic charges for the lattice sides then you can see that this movement these movements of the atoms at the lattice sides look like a dipole oscillating and it can absorb energy from an optical wave or it can give energy an optical wave and that's why this when the atoms move against each other they are known as optical modes and their dispersion relations are also different compared to the acoustic modes I will come to this later, so we have acoustic modes and optical modes and we need to do measurements now for measurements typical I just draw one mode like this one mode like this and one mode like this omega versus q, so I have to measure in q omega space these points on the dispersion curve so for this we need to have instruments which can measure q and which can measure energy transfer and we should also be able to probe either along the constant q or along the constant omega and every time we touch one of these dispersion curves we will get a peak because that is an allowed phonon and that energy transfer is allowed so we have to design instruments to measure these changes and I just show you the triple axis spectrometer at Dhruva you can see this is the monochromator drum at the center of which we have the monochromator then we have a sample here followed by another rotation stage on which we have an analyzer and then a detector so the geometry of the experiment is like this there is a monochromator which can rotate around its axis to change the incident energy we have a sample which can rotate around its axis if it's a single crystal then you can find out various planes atomic planes and various two in reflection and then you have an analyzer crystal here which analyzes the energy of the outgoing beam by Bragg so this gives the energy of the incident beam by Bragg reflection and this gives energy of the reflected beam or the scattered beam again using Bragg reflection and then this rotates along with the detector in theta to theta mode and that's why this is called triple axis triple axis spectrometer spectrometer because there is one axis for monochromator second axis for the sample third axis for the energy analysis of the analyzer so here we can measure the energy transfer and we can also measure the momentum transfer and I will come to this in a little greater detail so monochromator sample and analyzer the three axis actually the first triple axis spectrometer was designed by Bartram Baukhaus Bragghaus at Canada and he got the Nobel Prize for studies on dynamics of material there are other instruments like filter detector spectrometers spin-nico spectrometers I will touch upon all of these spectrometers that are used for inelastic neutrons scattering and I stop here