 Hello, everyone. In the previous lecture, I introduced a topic called Positon Annihilation Sproposcopy, which is a nuclear probe technique to study electron momentum distribution and also the defects or pore size in different types of materials. Now, I will discuss another technique, which is another nuclear probe technique. In fact, this is more like Mosbauer Sproposcopy, wherein we study the angular correlation between two gamma rays and determine about the electric field radiates, electronic environment in different types of materials. So, in this particular lecture, I will introduce the topic of PSE, then the theory of behind the angular correlation, what is the instrumentation for particular angular correlation, what are the requirements for a particular radioisotope to be used as a probe for PSE and some of the applications of PSE. So, just let me introduce what is the technique of particular angular correlation. Part of the angular correlation essentially let us first understand the angular correlation. Angular correlation between two gamma rays, suppose you have an isotope undergoing beta minus or beta plus decay. So, A x goes to A y by beta minus rather A plus 1 x by beta minus, A y only z plus 1 z. So, this particular decay scheme is for a beta minus emitting radioisotope and if it has got a cascade of gamma rays. So, you have gamma 1, you have gamma 2. The gamma ray will have their own multipolarities like magnetic dipole, electric quadpole and so on. These are the spin states of the three levels, the ground state and the excited state. Then these two gamma rays will have a definite angular correlation between them. What I mean by angular correlation? We have a source here and you put two detectors at a particular angle and you vary this detector and measure the coincidence counts. So, this is what we are going to measure called angular correlation. So, as a function of theta, you will find the coincidence counts between these two detectors are not same. Depending upon the spin states of the three levels, you may have low counts at 90 and higher counts at 180 or vice versa. It depends upon the angular momentum of the spin states and the multiplicity of the gamma rays. So, this is called the angular correlation and this angular correlation is used in finding out the spin states and multipolarities of the gamma rays. The expression for this angular correlation function double theta, it depends upon the this AKK are called the Directional Correlation Coefficient. They are the 3J symbols, the Clutch Gordon coefficients between I, I, L1, I for the one first radiation, I, L2, I for the second transition. So, that one can actually calculate using angular momentum coupling. And pK cos theta is the Legendre polynomial of K algebra. So, this is the general angular correlation between two gamma rays. But if you have certain conditions in a probe, that is the intermediate level has got a lifetime more than about 1 nanosecond and it has got a spin more than half. Then it will have certain quadruple moment electric quadruple moment. So, these are the two conditions. The intermediate level should have spin more than half so that it has got a quadruple moment and it has got a tau lifetime of the arrow nanoseconds. Normally lifetime will be picoseconds, then it is all no huge. With these two conditions, you will find that the electric field gradient around this, suppose you have a metal atom and you have it is coordinated to different ligands and you have this line matrix. So, the ligands offer a certain electric field gradient around the metal ion. If it is non-cubic, then this electric field gradient will interact with the quadruple moment to perturb this angular correlation. And this perturbation of angular correlation leads to perturbed angular correlation and this perturbation factor is called GK. So, the angular correlation is perturbed because of interaction of the quadruple moment of the intermediate nucleus with the surrounding electric field gradient. It can also change by interaction of the magnetic dipole moment of the nucleus with the surrounding magnetic field. There are magnetic materials. So, there the magnetic dipole moment can come to picture. Now, you can expand this legendary polynomial. So, you have A0, A22, G22 cos theta, P2 cos theta, A44, G44, E4 cos theta. So, normally we will take up only the second order term. We will this fourth order term also is important, but we require a little more elaborate arrangement. So, let me go into little more details about the theory of perturbed angular correlation. And try to give you an example of a probe atom which is a beautiful nucleus for perturbed angular correlation 181 hafnium. 181 hafnium you can produce by 180 hafnium, 180 hafnium and gamma gives you 181 hafnium. And this 181 hafnium undergoes beta minus decay in the half life of 42 days to the excited states of 181 tantalum. So, half plus state goes by 133 kV gamma re it goes to 5 by 2 states and by 482 kV gamma re go to 7 by 2 state of 181 tantalum. Now, you can see here this intermediate state has a tau of 10.8 nanosecond and has got a spin of i by 2 plus. So, it has got all the parameters, nuclear parameters which is making conducive for perturbed angular correlation. So, what happens? When this hafnium is put in an asymmetric electrical gradient like non-cubic geometry you have orthogonal, orthorhambic monoclinic type of geometry, then in that that like this positions know they will offer a non-cubic environment around the metal ion. And so, this non-cubic environment will have certain electric field gradient. So, this electric field gradient I will explain here this will interact with the electric quadrupole moment of the nucleus in its intermediate state and split this intermediate level into its magnetic sub-levels. So, you have the splitting in an electric field gradient is plus minus half plus minus 3 by 2 plus 5 by 2 unlike in NMR. In NMR each magnetic state splits separately. So, you have half splitting to plus half and minus half. So, magnetic splitting of this level will give you 6 levels plus minus half separate levels plus minus 3 by 2 plus 5 by 2 but electrical interaction splitting will give you only 3 levels. Now, so this so when you have first gamma imitate to this they are populated in a certain way. So, when we say perturbation of angular correlation essentially the population of this magnetic sub-levels had changed because of that perturbation. So, essentially what we have to do? You have to solve the Schrodinger equation for the this quadrupole interaction or magnetic interaction. You have to have the Hamiltonian containing the quadrupole moment and the electric field gradient or like you say nu dot h for NMR. Similarly, for q dot vz q not dot product this quadrupole moment and electric field gradient are tensor quantities. So, you have to solve the Schrodinger equation to get the eigenvalues and their amplitudes. So, you have the transitions among this magnetic sub-levels w1, w2, w3 and their amplitudes are this SKN coefficient. So, you have here the angular correlation function a0 plus a22 g2 p2 cos theta where g2 to the perturbation factor is a sum of coefficients, SKN coefficients are the amplitudes of the frequencies and cos omega nt omega n equal to 1, 2, 3 or omega 1, omega 2, omega 3. If the electric field gradient is axially symmetric, axially symmetric means this eta value axially asymmetry parameter if it is 0 we say electric field gradient is axially symmetric that is v xx equal to vy i. In that case this omega values are integral multiples of omega 1 that is omega 2 is on 2 omega 1, omega 3 is 2 omega 3 omega. In that case the frequency can be given in terms of eq vzz vzz is the electric field gradient 1 4 i 2 i minus 1 where vzz is the second derivative of the electric field gradient that is delta 2 v upon del z square. So, you essentially what you are getting is the electric field gradient vzz if it is axially symmetric and if it is axially asymmetric then you get this parameter asymmetry parameter which tells you what is the asymmetry in the electric field gradient. So, finally, the parameters you get out of TDPSE are asymmetry parameter and quadruple interaction frequency. If its asymmetry parameter is not 0 that means if it is not axially symmetric then the omega n's are not integral multiples of omega 1 they are dependent upon the eta. So, this was the theory of PSE you can go through the book on particular correlation and investigate understand more of it. Now, the instrumentation for PSE is similar to the lifetime spectroscopy or positive spectroscopy we have you have to have actually three detectors I have shown two detectors the sodium iodide thalium detectors can be used. So, one and a half inch by one and a half inch sodium iodide thalium coupled to PM2 PM2 and the preampifier then you take the anode output for timing and through a set of concentration, displacement and delay that becomes the stop signal of the time to the converter and same is from the other side. The energy you take the dinote output for amplifier get the gamma ray in here and you generate the fast coincidence signal from here this fast coincidence will stroke the TSE what you get in the MCA you get the time spectrum of MCA and you will have if you have one more detector here then you will get two time spectra 1 490 degree or 1 4 180 degree. So, basically you will require to get this W theta as a function of time since you are doing time differential as a function of time you record the time spectra and they will find if it is if there is a anode electrical gradient then you will see on the normal exponential there will be oscillations because of that perturbation. So, this you fit into the function W theta T into in this function and you get then G2 to T which is shown here. So, G2 to T is given by SKL cos omega and T and the Fourier transform of G2 to T will give you the frequency difference. So, these are all there are standard packages now software is available if you feed the data to data of 180 and 90 degree coincidence time spectra you can get the Fourier transforms directly. So, you essentially get W1, W2 like in FTNMR you get directly the frequency similarly here also you can get the FTNMR you get the data by Fourier transform Fourier transform of time domain spectra will give you the frequency domain. So, what essentially you do you have that counts at 180 degree here the counts at 90 degree and if you recall the this P2 cos theta is nothing but 3 by 2 cos theta minus 1. So, you can substitute for two angles the value of P2 cos theta you can get G2 to. So, you require two angle data to find out G2 to T because you have A0 and A2 to G2. A2 value is known A2 to 2 is minus 0.29 for half number 81 or 5 by 2 stator. So, from this experimental data you can get this frequency data. So, you need to get F1 F2 F3 omega 1 omega 2 omega 3 and you will also get the eta parameter from the fitting of this G2 to data. In fact, the ratio of omega 2 by omega 1 omega 3 by omega 1 will also give you the eta. So, let us see all the radioisotopes that you have they are not available for PSE there are few radioisotopes which can work as a PSE pro. Let us see what are the characteristics first is the of course the lifetime should be long enough maybe few hours to few days so that you produce it in the reactor or accelerators bring it to your laboratory put it in the system that pass slow coordinate setup and acquire the data. So, your data acquisition may take few hours or a few days depending upon the statistics that you want to acquire. Then secondly the intermediate level spin should be more than half for quadruple interaction. Intermediate level lifetime should be more than nanosecond why this nanosecond lifetime because the resolving time of the instrument when the two gammas are coming then the time resolution of the setup is observed of 500 picosecond. So, your lifetime should be more than the resolving time of the equipment so that you can see anything any exponential decay of the level can be seen on the lightning time spectrum. Then the energy of the gamma rays in cascades should be more than or about 100 kV or more because as less than 100 kV the window of detector will start invading the gamma ray and of course the the probe should be compatible to the host matrix. So, like half-nium if you are studying as a half-nium as a probe all these you know dyraths or even titanium zirconium compounds you can study. So, you could see the radii or the chemistry valency should match with the one of the elements in the sample. So, you can see here this is the list of radioisotopes the parent isotopes which you have to produce in the reactor the half-lives what is the mode of decay what is the daughter product the half-life of intermediate level lifetime and the two gamma rays which are in cascade. So, you can see jigs 62, Muldron 99, cadmium 111, indium 111, silver 111, berium 133, terbium 160, half-nium 181, mercury 199 m, lead 204 m, nitrogen 147, european 152. So, there are these are the kind of radioisotopes which one can use. So, chemistry of elements containing these isotopes these are the elements can be studied and sometimes you know like indium it will go very well in the rare earth and you travel in metal ion you can use indium as a probe. So, that kind of analogy you can use to study different types of matrices. Okay. So, let us see what are the areas in which you can apply particular angular correlation technique. So, you can use PSE in studying the phase transitions like you know if you start with the orthogonal or the orthorhombic matrix and you want to go it goes through monoclinic or tri-clinic this there will be going to be change in the elliptical gradient. So, any phase transition where there is a structural change or there is a magnetic transition you can study by TD-PSE and the sample remains intact except that you are introducing some radioactive isotope. You can study radiation effects in solids, radiations you can introduce maybe they may introduce defects or if you take an amorphous material it may generate crystallinity or if you have a spline matrix it may generate amorphous amorphosity or it can be even you know polymerization it can lead to change in the molecular weight. So, wherever there is a change in the chemical environment by radiation we can study by TD-PSE. The binding site of metal ion in bio molecules impact in bio biochemistry people are interested in knowing that like you have a big molecule protein molecule where is that this metal ion is going and trying to attach to this big molecule. So, what is the geometry around that you can study by using TD-PSE. Another is diffusion in solids. So, for example, hydrogen diffusion in different materials. So, when the when the before you have a material called palladium or they are they absorb hydrogen even zirconium absorbs hydrogen. So, when the hydrogen is going in the material it may be diffusing out and during the diffusion there is a change in the electronic environment around the probe atom and that change in the electronic environment can be proved by using TD-PSE. Complexity of the metal ion the ligand or polymerization of the metal ions can be also studied using TD-PSE. So, I will just give you some examples of particular correlation. One of our colleagues has studied what we call as the phase transitions in hafnium oxide. A hafnium oxide has a function of pressure, but at high pressure it was a hafnium oxide undergoes changes. So, that kind of study you can do. So, if you have hafnium oxide radiated in the reactor and at atmospheric pressure you study the ion differential perturbation correlation, you can determine the parameters like water pole interaction frequency. In fact, I forgot to tell you about this delta part actually this G22 equal to skn or omega nt. So, this is for a system which is perfect. There is no no fluctuation in the electric field gradient, but if the omega is changing. So, electric field gradient is changing. What is omega? Omega n can be n omega 1 and this omega 1 is eq Vzz upon 4i to i minus 1. So, this electric field gradient if it is changing, suppose the metal ion is not sitting in the same environment everywhere, then there will be a distribution of frequencies and then distribution of frequencies will give minus half delta square omega n square p square. So, this is like you can use a Gaussian distribution of frequencies. So, this delta is the distribution in the frequencies at percentage, what is that like 0.054 into 1 eta. So, then this is the asymmetry of the electric field gradient. So, what my student did in the part which work. So, at atmospheric pressure, this is the kind of data you get and then you apply, you put this half num oxide in a diamond and you will sell and then study. So, at 45 gigapascal, you can see that the water pole interaction parameters, the frequency, the delta value and the asymmetry parameter have been changed significantly. So, you can carry out this study as a function of pressure and see what type of geometry that it is changing when you apply pressure. So, pressure induced phase transitions can be studied by means of this technique. Similarly, the radiation effects in solids can be studied by TTPSE. In radiation effects in solids, one of the studies was recently carried out is perovskites, stonsome titanate and calcium titanate, they are being investigated as a post matrix for the waste that is generated in the reprocessing of the spent nuclear fuel. And therefore, a lot of materials are being investigated whether they can be used to accommodate the efficient products and minor actinides and therefore, unable to dispose of nuclear air pollution. So, this by gel combustion method, they were synthesized and so the first you verify them by x-ray diffraction and dope half num 181 as a probe. So, your doping concentration is of the row 0.1 percent or 2.5 percent. So, that you do not alter the crystalline structure very much and of course, titanium in the half num you can go and sit in the place of titanium. So, at their ionic radii are very close. Then after the you dope maybe you can anneal also or you can dope while preparing the compound and then study the time differential part of the angular correlation. So, PSE and TTPSE the difference is that in TTPSE you are recording the time spectrum. So, TTPSE is essentially PSE only and you can see as a first of time in nanosecond the 322 is in the case of Stonseff titanate is a cubic lattice. So, it is supposed to have no electric gradient, but even then you get some static interaction and in calcium titanate you get well defined electric gradient. So, like after this now you can study the electric gradient and then once you irradiate with the different radiations like gamma rays or electrons or alpha particles, then one can study effect of radiation. Essentially, you irradiate at a particular dose and determine the water core interaction parameters, the frequency W, the distribution of frequency delta and the asymmetry of the electric gradient. Similarly, the biological applications very interesting studies can be done by in biology where the particular metal ion is going to sit by the micromodic. So, one of the study I have taken from literature is the binding site coordination geometry of carboxypeptidase and what essentially they have done that they have used 111 cadmium as a probe nucleus and this 111 cadmium was doped into this material. So, it is essentially not a doping it is going to bind. So, this cadmium will bind the carboxypeptidase in a particular geometry and what is the coordination environment around cadmium one can study using TDPS. So, what they found that in the when the this carboxypeptidase is in the coded in the crystalline phase, in the crystalline phase they found that there is only one what we call as the NQI means a nuclear quadrupole interaction. A nuclear quadrupole interaction is nothing but the interaction between the quadrupole moment and the electrical gradient. So, this is also called as the NQR, nuclear quadrupole resonance like Bosbach's photoscope is like nuclear quadrupole resonance. So, here it is nuclear quadrupole interaction or we can also call it hyperfine interaction because the magnetic the intermediate level is split into it is a magnetic substrates. So, what they found that in the crystalline form you get distinct frequencies omega 1, omega 2, omega 3 and one can find out the electrical gradient and match with the X-ray data. But what they found is that when we you try it in the solution form, in the solution form they found like particularly sucrose solution at different chloride concentrations one molar sodium chloride and 8 millimolar chloride ion then they found that these peaks have come broad because see the peaks have become broad here also. So, why this peaks get broadened because they if there is a if there are two sides. So, cadmium is sitting in sides which are there is a fluctuation in the electrical gradient around cadmium and that can happen if there are two sides. So, some cadmium ions are sitting in one geometry other cadmium ions are in other geometry. So, there is a dynamic exchange between two binding sites like tautomerism. Now, the time scale the time scale of the that interaction exchange has to be at a time scale if it is a single peak then it is a much shorter time scale. If the if the there are two peaks you see then it is happening at a longer time scale than the TDPs. TDPs see time scales are of the row and that is how tico circuits that is the kind of difference because the resolving time of TDPs set up is of the row 500 tico circuits. So, any event which happens at of that time then you may not resolve it, but that may lead to broadening of the TDP system. So, the net result of this two exchange to two sides no exchanging. So, it can lead to like hydrogen bonding between two species to two functional groups in the molecule can lead to two different geometry around the metal ion. So, this kind of studies have been seen in TDPs and then we can explain the behavior why these structures are changing what is what is happening in the structure at those under those chemical conditions. Another very interesting in the example in the literature in the phase transitions is on what we call as the spin frustrated copper ferrites. So, these copper ferrites actually they have lot of applications and the powder neutron diffraction of this material showed two magnetic transitions at very low temperature 11 k and 16 k. So, you can see what they have done that you they want to know investigate what are the kind of geometries. So, neutron diffraction will give you that there are two transition taking place. Now, you want to go further investigation. So, using 111 indium as a pro they wanted to know what kind of structures what kind of interactions they are. So, it is a magnetic material and if there is a non cubic geometry then there will be electric interaction as well as magnetic interaction. So, there is a combination of electric and magnetic interaction. So, by electric field gradient interaction with quadrupole moment you have this like splitting of i by 2 into plus minus half plus minus 3 by 2 plus 5 by 2 and if you have a magnetic field it will split each one of them into its magnetic solid. So, you have six levels now. So, the combined electric and magnetic interaction can be studied by means of PDPS. So, what they found that that suppose the for example, you have the room temperature data and somewhere at 4.2 there is a broadening in the. So, this 11 k Kelvin is somewhere in middle of the 2 and so, they essentially try to study the as a when there is a transition. So, there is a phase transition between the in this materials. So, what is happening to the structure? The structure could change or the there could be changing them because of the magnetic interaction. So, this interplay between the electric and magnetic fields can lead to this phase action. So, what they actually found that the weak weak component between the two electric and magnetic was going to broaden their frequencies of the stronger company. So, these are the kind of studies people have studied using PDPS. So, there are innumerable examples. So, one can take up I will I just wanted to illustrate that if you know, if you know that the fundamentals of PDPSC or Poisson photoscopy and if you are working in a area where you know special information you want, then by setting up these instrumentations, you can go for studying the processes chemical environment around the metal ions and you it requires the middle of nuclear instrumentation. So, you require to have the proper knowledge of detectors, the energy resolution, time resolution. If you set up the system and have it moved that acquisition system, then you can study different areas in physics and chemistry. So, that is what was the purpose of these lectures that it induces you to take up the nuclear chemistry what are the areas in which one can take up research problems for investigation and they are not done in isolation. These days we have PDPSC setup or a Poisson setup. These are already you know, you need to have the complementary information from other techniques. So, you combine the information from other techniques like XRD or other techniques and use this data to obtain a complete picture around the system. So, depending upon the application you have or depending upon the problem you want to understand, one can choose different type of techniques and Poisson and PSE provide such avenues for you. So, I will stop here. Thank you very much.