 Hello everyone. So, in the last lectures, we discussed the beta decay and prior to that we had discussed the alpha decay and in all these discussions, our effort helped me to explain the energetics of the decay as well as the decay probability that is somehow by a model we try to calculate the uplives for the decay of this variable processes like alpha, beta and so in the gamma decay also we will try to discuss how to calculate the uplives and if you can predict the uplives for the gamma decay and also what are the selection rules. So, that in fact dictates the uplives depending upon the different spins and priorities involved in this one, the uplives are changing. So, we discussed that this lecture goes for gamma decay. So, gamma decay essentially is not from the ground state of a nucleus but whenever we populate a nucleus in its excited state by alpha decay or beta decay or it could be electron capture or so. So, if the nucleus is left in excited state, then the excited nuclear state emits gamma ray to come to the ground state. For example, the Koval 60 having a half life of 5.27 years undergoes beta minus decay to excited states of nickel 60 and these states are actually 4 plus, 2 plus and 0 plus. So, this excited nuclear states of nickel 60 undergo gamma decay. So, you have this, this is the you know very famous radio isotope Koval 60 used in many applications. So, this 4 plus state emits 1.173 and may be gamma ray to come to 2 plus state and then 2 plus state emits a 1.32 gamma may be gamma ray to come to ground state. So, this these are the you know. So, gamma decay essentially takes place from the excited states of a nucleus and these gamma rays they are photons. So, they carry integral units of angular momentum unlike the beta decay which carry half spin. Now, the range of half lives, what are the different values of half lives? Range of the half life for gamma decay spans. So, gamma decay can take place in like picoseconds also or it can be years also. So, the ones which are emitting at more than you know picoseconds nanosecond microsecond they will be called as the metastable states or isomeric states. So, for example, 178 half anium has got in fact 2 isomeric states M1 and M2 and the half lives are 31 years and let us say 4 seconds. So, the half life essentially depends upon the change in the angular momentum involved associated with the gamma decay. So, half lives of gamma rays gamma decay can be from 5 picoseconds to years. The energy of the gamma ray essentially the energy of the gamma ray is nothing but the spacing between the levels of the nuclei. So, it could be from QKEV to few MEV. So, there are gamma rays of 50KEV also there are gamma rays of 1 to 2 MEV also. And then from the probability for decay has been in fact calculated by Weisskopf using Weisskopf theory that is called the single particle model and according to which the half life if the energy is high then half life is short. So, that means the higher energy gamma decay are more allowed in a way you can say that say that allowedness depends upon energy also and we will see more of it in subsequent part of this lecture. Now, let us first discuss the selection rules for the gamma ray. So, how what happens actually when a gamma ray in emission takes place the gamma ray like you know in atoms or molecule in a molecule when there is a excitation absorption or emission of a photon then we say there is a change in the dipole moment of the molecule. So, essentially it is the change in the charge density associated with the transitions. So, in the case of gamma rays we will have two types of transitions called electric multiple EL, electric multiple and magnetic multiple ML. So, the electric multiple transitions are associated with the change in the charge density. So, if there is a change in charge density. So, like for example, if a nucleus vibrates the charge density changes and so, we will have the associated electric transition. When we have a change in the current density now how you produce a current positive electron is moving in a circular orbit then it is generating a charge moving in a circular orbit generating a magnetic field. So, there are nucleons moving in the orbits of the nucleus and there is a change in the current density in the nucleus we say it is associated with the magnetic multiple transition. So, the gamma transition is taking place a single gamma ray can have either as electric character or the magnetic character. So, we will discuss this more details as we go along. And so, the whatever the angular momentum that the gamma ray photon is carrying will call it capital L and it will carry the integral value of the angular momentum. So, L could be 1, 2, 3 and so on and that we call as the multipolarity of the gamma ray. So, the multipolarity depending upon the multipolarity 1, 2, 3, 4 we can have if L equal to 1 then 2 to the power 1 to the power 2 raise to L is the multipolarity. So, L equal to 1 we will call as dipole gamma ray, L equal to 2 we will call quadrupole, L equal to 3 octopole, L equal to 4 hexadecable. So, as we go to higher and higher L value the transition becomes more and more hindered because gamma ray it is difficult for the gamma ray to carry large angular momentum. And in fact, when we when there is a excited nucleus it has to emit gamma ray to by changing a large angular momentum and if it is possible then it will find that the particle emissions, neutron, proton, emissions are more flavored from the excited nuclei if the energies are more than the binding energy of nuclei. So, the gamma ray essentially do not carry much angular momentum. So, if it has to carry more angular momentum that gamma ray decay is hindered. So, we now we will try to see how to classify this the gamma ray decay selection rules in terms of the change in the angular momentum delta i and the change in the particle delta pi. And depending upon the value of the delta i and delta pi we can categorize a transition as EL or ML we will see the selection rules very shortly. So, the selection rules for the gamma decay essentially are similar to the selection rules for optical transitions in molecules or atoms. And taking the analogy from the molecular transitions like ultra-invisible transitions then the matrix element for a decay an optical transitions can be written in terms of the wave function for the final state which is the complex conjugate of that the dipole moment operator for the electric transitions e into r is the charge and the r is the distance and psi is the wave function of the initial state of the molecule or nucleus and then this is the volume of the g tau dx dy dz over the volume of the system. So, now this particular matrix element we will say psi f star e r psi i it should be positive parity for a allowed transition. So, the parity of the transition matrix if it is positive then we say the transition is allowed and that will dictate the parity of the individual states psi i and psi m. For example, electric dipole operator e r so, how do you define the parity is a minus 1 to the power n. Now, if it is a function like psi the function is changing psi when we go from x to minus x then we say this function has got odd parity if it does not change the sign we say even parity. So, e r essentially r is the distance and so in x or r will be having negative parity and so it is odd parity. So, pi will be negative for electric dipole operator. Now, so one of the elements of this transition matrix is odd or negative parity. So, if this function has to be allowed then psi i and psi f have to be of opposite parity. So, that suppose suppose this is odd, odd into odd even even into even even. If it is even even into odd odd then odd into odd odd even. So, if one of them is odd other has to be even for a electric dipole transition that is how we decide what would be the parity of the two states. So, if it is an even transition l equal to 1 then minus 1 to the power l is negative. So, parity for even transition is negative. So, then psi i and psi f have to be of opposite parity that means there has to be a change in the parity when we go from psi i to psi f that means delta pi B s means change in there should be a change in parity. Yes, delta pi s means there should be change in parity and the magnetic dipole operator has opposite parity with respect to electric dipole transition and therefore, for M1 transition delta pi is known. That means if a gamma ray has to be of M1 type then there should not be change in parity from ii to if. So, that is how we decide the parity of the transitions. Just to give a then also to elaborate it further suppose you have an initial state of a nucleus excited state ii the spin of the excited state and pi i is the parity of that and by emission of a transition of molecular tl it is taking to a state and then state if and parity pi. Then so the angular momentum carried by the gamma ray will be in the range of ii minus if to ii plus if it can take that many values. Now, the lowest one is the most probable. So, when ii equal to if then l can not take 0 because 0 is not possible gamma ray has to be associated with the spin. So, l will take the value 1, 1, 2 to so to maximum value ii to ii plus if plus l equal to 0 is not possible and also it is important to note that if ii equal to 0 and if equal to 0 0 to 0 transition is forbidden for if ii minus if is 0 but then there could be ii plus if will be non-zero. So, like for example here so l can take anything more than 0 so 1, 2, 2 ii plus if. So, that is how we will decide the the multiplicity of the gamma rays depending upon delta i and delta pi. So, let us see here. So, l can be decided based on this ii minus if to ii plus f and delta pi we will know from the functions. So, if l equal to 1 dipole and delta pi is s then that then this will call e1 transition. So, electric dipole transition and if l equal to 1 and delta pi is no there is no change in parity we will call it m1 transition. So, that is how we classify the different transitions into el and ml where l can take value 1, 2, 3, 4 and so on. Similarly, when l equal to 2 and there is change in parity then we say it is m2 transition. When l equal to 2 there is no change in parity then we say e2 transition because as we go to high minus 1 to the power l when minus 1 power 2 then it becomes positive it becomes positive. So, accordingly the roles will change and you can we can just put the value for different transitions in terms of l and pi and you can see whether it will be el or ml. Now, in the case of el and ml the ml transitions are weaker than compared to el transitions that we will discuss very soon. So, if there is a ml and el plus 1 for example, m1 and e2 then what will happen this e2 starts competing with m1 and that then you will find that some of the gamma ray transitions have a mixture of m1 plus e2. But if there is a m2 there is a e2 then e2 and m3 m3 is a weaker. So, you will not have the el mixture of e2 and m3. So, similarly e1 and m1 e1 is much stronger than m1 so you will not have a mixture of e1 plus m1. So, only in case of m1 plus e2 or m3 plus e4 means the higher e can compete with the lower m. So, you can have admixture of and in fact there are experimental techniques to determine the admixture of the ml. Now, we will just see the how we will just elaborate this illustrate by an example of cobalt-60. In fact, cobalt-60 has got a very detailed decay scheme from the excited states of cobalt-60 to plus state which is having half gamma ray limiting gamma ray 59 kV to phi plus state. So, let us see we will have the we will see based on selection rules discussed previously we will see the multiplicity of this transition this transition and this transition in this particular discussion. So, for the first transition let us say cobalt-60 m to cobalt-60 was 60 metastable state to cobalt-60 ground state. So, it is 5 and 2 ii is 2 and if is 5. So, it can take the value l can take the value from 5 minus 2 to 5 plus 2. So, you can have 3, 4, 5, 6, 7 and here the parity of both the states is positive. So, there is no change in parity delta pi is no. Now, you can see here and l equal to 3 and delta pi is no you see here l equal to 3 delta pi is no you will have m3 type of transition. So, predicted values of m, l are m3, e4, m4, m5, e6 and m7 and as I discussed the higher multiplicities are having lower probability. So, ideally it should be a m3 transition, but because e3 can compete with m3. So, e4 can compete with m3. So, we will have the observed value is m3 plus e4. There is a admixture of e4 with m3. So, this is the type this is how we can equally predict the multiplicity of a particular gamma detection. So, for this it is ml m3 plus e4. Now, let us come to 1172 transition kv transition here this one. So, that is the de-excitation of the excited states of nickel 60 the 4 plus state. So, we have now 4 plus to 2 plus. So, the gamma ray can take place 4 minus 2, 2, 4 plus 2 that means 2, 3, 4, 5, 6. So, it the gamma ray so, it is 4 plus 2 transition the l value can take 2, 3, 4, 5, 6 and the delta pi is known. So, when the delta there is no change in the parity then you can see again here delta pi is known and it is 2. So, it is e2 transition. So, we will have this as e2 and m3 cannot compete with this because m3 will be much, much weaker because it is it is the other way around m2 can with m2 e3 can compete with the e2 m3 cannot compete. So, this will be e2 type. So, the predictor is e2 m3 e4 m5 predictor is e2 and observed is also e2. Now, let us come to this transition 2 plus to 0 plus in 2 plus to 0 plus we have 2 minus 0, 2 plus 0 is 2 only and no change in parity. So, again this also will be e2 times. So, predicted value l equal to 2 delta pi no is e2, observed value is also e2. So, that is how you can predict and then experimentally there are experiments to measure the multiplicity of a gamma ray transition. So, these are the selection rules for gamma decay. You can find out the multiplicity of a particular gamma ray. Now, let us come second part how to predict the decay constant for a gamma decay. So, gamma decay constant, see the gamma ray probabilities essentially we will just discuss in terms of the what is actually happening when there is a gamma decay. Say a nucleon is essentially getting transition from one state to other state. So, for the gamma ray the wavelength of the gamma ray can be given by this hc by lemma e hc by e. So, we are actually defining here the reduced wavelength called lambda upon 2 pi because this is called angular wavelength because this is a situation of the angular momentum. The gamma ray photon is carrying angular momentum. So, we define in terms like angular momentum we say h cross 1 h cross h by 2 pi. Similarly, associated wavelength we say lambda cross. So, the wavelength of a gamma ray we can calculate based on hc by e. So, you can see here this is the h value h h 6.62 minus 27 this is the c value h h cross minus h upon 2 pi and you convert the MeV into joules joules into MeV. So, it will become 1.974 minus 13 meters where e is in MeV. So, now let us compare this value with the radius of the nucleus. Typically, for a nucleus of mass 100 you can calculate the radius in terms of r0 e raise to one third that becomes 6.4910 power minus 15 meters. So, what essentially we want to highlight is that wavelength of the gamma ray for example, for a 1 MeV gamma ray then the lambda cross is much larger than the radius of the nucleus or the r by lambda cross is a much much less than 1. So, the probability of emission or absorption of a gamma ray photon decreases with L as r by lambda cross for the power 2 L. So, why this has come this actually come from these Weisskopf's theory of statistical decay of gamma rays. Essentially, it is that the wavelength the wavelength of the nucleon the wavelength of the nucleon inside the nucleus can be compared with the nuclear dimensions. So, the radius of the nucleus is similar to the wavelength of the nucleon and the wavelength of gamma ray is of the order of lambda the lambda you can calculate. So, r by lambda cross essentially dictates ratio wavelength of the nucleon to the photon and the probability of emission or absorption essentially depends upon this ratio of the two wavelengths. Secondly, the ML transitions are much weaker than EL transitions because the ML transitions are essentially associated with the velocity of the nucleon and whereas the EL transitions are essentially the velocity of the light. So, this is another factor we can discuss this in terms of the moment the magnetic moment and the electric moment. So, we can derive in fact why how why does this ML to EL transition depends upon the p by c square we can take the ratio of square of magnetic and electric moments. So, the magnetic dipole moment of a nucleus can be written in terms of e h cross upon 2 m c or e h upon 4 pi m c this is the nuclear magneton and electric dipole moment can be written as e into r. So, the ratio becomes h cross upon m c r square the square of the ratio. Now, for the nucleons the reduced wavelength is equal to h cross by m v and the nucleon wavelength is close to the r. So, we can replace this r by h cross by m v in this formula. So, the ratio of the magnetic dipole moment to electric dipole moment can be written as h cross by m c upon h cross by m v square. So, it becomes p by c. So, essentially the magnetic transitions are associated with the change in the current that means the nucleonic motion which is associated with the velocity of nucleon and the electric transition associated with the change in the charge density. So, that is associated with the photonic velocity. So, c that is why the magnetic transitions are much weaker than the electric times. Now, let us see how to calculate the decay constant in the Weiskopf's theory of single particle model. So, Weiskopf's formula for the decay constant for electric transition is lambda e, this is the decay constant for the not to be confused with the wavelength, decay constant for the electric transition is 2 pi v e square upon h cross c in the statistical factor upon into r by lambda cross to the power 2l, where s is the statistical factor, the series expands series is evolving in the l values and the lambda cross is the reduced the reduced wavelength of the gamma ray. So, that is 197 by e, the 1.974 minus 13 meter to write in Fermi's, then 197 Fermi's and m v v. So, now, we can substitute this value r by this lambda cross in terms of 197 by e. So, the decay constant becomes this formula e 2 pi v e square upon h cross c s r is 1 e upon 197 to the power n and from this lambda value we can calculate the half life or the lifetime, 1 by lambda is lifetime because for the excited states normally instead of half life we say lifetime. So, for different electric transitions e 1, e 2, e 3, e 4 you can see the tau values are this and for the magnetic transitions there. So, you can see the tau values are higher for magnetic transition because that means the lambda values are shorter. So, the lambda value for magnetic transitions are they are weaker than electric transitions and one more thing is observed that this these calculated values they are in fact much higher than the observed values. The observed values for electric transitions are observed minus 13, minus 14 seconds, but you will see some of the observed calculated values are minus 9, minus 3. So, this is Weisskopf's single particle theory can give sort of order of estimates, but the difference the large differences between calculated and experimented values essentially explain can be explained by other types of motions in them. It is like all the collective models collective motion like the nucleus can vibrate, nucleus can rotate. So, nucleus in fact these gave an idea the fact that the experimental values of the half lives of the excited states are much lower than the single particle estimates by Weisskopf gave to the speculation that apart from the single particle model that means the nucleons moving in their orbit specifies by orbitals the nucleus as a whole is undergoing collective motion and that it has also got the collective states like biviscent states and rotational states and then one can calculate the moment of inertia of the nucleus using the excited states of the gamma ray of the excited. So, there is an evidence for the collective model of the nucleus also. Now, as I discussed that the higher L values are hindered. So, there are alternative routes to excitation by gamma ray and that is one of these internal conversion. This is an alternative de-excitation process in the gamma ray or you can say it is a radius in less transition non-radiative transition instead of the gamma ray emission from excited state to ground state that energy is used up in emitting an electron from the orbitals K shell L cell L Hamser. So, an electron from the atomic cells is ejected with energy E gamma minus the binding energy of that electron. So, the energy of the electron where does it like to suppose it is a beta decay for example, mercury 203 emitting beta minus 203 and you have a continuous beta spectrum. These conversion electrons K electron L electron M electron they will have sharp energy because they their energies are well defined. Energy of gamma ray electron K electron gamma ray energy minus the binding energy electron is well defined and so, over a continuous spectrum you will see this sharp peaks due to electrons K electron L electron and gamma M. So, the energy of the K shell electron that is limited will be lower because this binding energy is higher. So, higher the binding energy of the shell lower with the energy of the electron. So, different conversion electrons appear in the over the continuous spectrum, but in the case of electron capture where there is no gamma ray beta emission then the gamma ray conversion will be giving rise to sharp peaks in the electron spectrum and you see the electrons because of the K conversion you will have the electrons at lower energy. So, you can see here put K conversion 3.67 KV L conversion 30.55 MVV and M conversion 34.48 KV and their percentages are also given in this. So, conversion electron whenever there is a large spin difference between the spin states of excited state and ground state lower state then you will find the gamma ray is converted and it is a partly conversion not that gamma ray is not limited at all let us say you can say some percentage of gamma is limited some percentage of conversion electrons are limited. So, there is something called in the conversion coefficient. That means what percentage of the gamma emission is converted that will be called as the conversion coefficient. So, the conversion coefficient is denoted by alpha that is it is like you know the branching decay you recollect know the branching decay of a nucleus into two modes in that decay constant add up. So, here the conversion coefficient is the ratio of decay constant for conversion electron and internal conversion and for the gamma ray decay. So, or you can also say number of conversion electrons upon number of gamma protons imitate the ratio is called as the conversion coefficient. So, you can measure the gamma ray and you can measure the conversion electron and find out the ratio to obtain the conversion coefficient. So, the net decay constant lambda will be lambda E plus lambda gamma because that is the like the decay constant add up. And so, if you take lambda gamma outside it becomes 1 plus alpha that means lambda E by lambda gamma is alpha. And so, depending upon whether you have the k conversion l conversion and conversion we have k alpha k alpha l alpha m. So, alpha the conversion coefficient is some of the individual conversion coefficients. So, higher the multiple order the higher the alpha k. So, as for lower multiple value gamma ray decay is more probable for higher multiple value conversion is more. And this is true for alpha l alpha m also. Similarly, with regard to jet if the jet of the nucleus is high then conversion coefficient is high if the energy of the gamma ray is high conversion coefficient is low. So, internal conversion is more favored with gamma ray of low energy with and the higher atomic number nuclei higher multiplicity. So, there are a lot of experiment people do to determine the conversion coefficient k by l ratio also and that tells you about the multiplicity of the gamma ray. Lastly, I have already discussed the nuclear isomers. So, don't have to discuss much these excited states of nuclei lifetime more than the ecosystem that living beyond the single particle estimates will be as wherever there is a gamma ray decay is hindered. And you can see typical example where m 137 m 2.5 minutes you can see large spin difference technician 99 m half minus 2 9 by 2 plus you know 125 m 9 by 2 minus 2 half plus they are having gamma decay as well as conversion also. But where m 2 10 m c for 6 years and this gamma decay is totally hindered which is emitting alpha because there is a large difference 9 minus 2 1 minus. So, nuclear isomers are here particularly near the mid-cell nuclei or the near the close cell celestial closures gamma the nuclear isomers are very common and essentially their existence is due to the large spin difference between the two accepted statistics and the ground state. So, we will discuss this more in the how to detect this nuclear isomers one can develop methods for determining their lifetimes there are methods to determine lifetimes. So, we will discuss more now on detection of this different act of reducing subsequent lectures as of now I will close here. Thank you very much.