 Hello everyone. In the previous lecture, we discussed about the cell model and in the cell model we discussed the nucleons orbiting in their individual orbitals. So, it is sort of a weak interaction among the nucleons, though the nuclear force is a strong force. The nucleons in their individual orbitals do not interact with the other nucleons in other orbitals and by solving the Schrodinger equation of a nucleon using the potential. The potential though it is non-central, we assume that it is central potential offered by other nucleons to the particular nucleon. We get the energy states of the nucleons and whereby you populate those levels to get the configuration of the nucleons in the nucleus and the cell model could explain the magic number. So, that will be the application of the cell model. What are the different applications? What were the limitations we discussed in terms of liquid top model? We can now try to see how this cell model scheme explains those things. So, the cell model explains the magic numbers. Just now we saw in the previous lecture, by introduction of the spin orbit coupling between the L and the S value. The coupling of L and S, we could regenerate the magic numbers. Essentially, because of the strong spin orbit coupling, the L plus half state is lowered and the L minus half is raised and the gap between L plus half and L minus half increases as the L value increases. Now, the high value of sufficient energy of protons and neutrons for magic number of nucleons, those nuclei which have magic number of nucleons. This can be explained by large energy gap above the closed cell. So, when you want to, so that is like a tightly bound nucleons in the closed cell are tightly bound to remove the any individual nucleon on that tightly bound cell, it requires extra energy and that is why the sufficient energies are higher. If you have a neutron more than or a proton more than the magic number, then again it is easy to remove that particular neutron or proton from the nucleus having magic number of nucleons. The absorption cross section for the such nuclei which have magic number of protons and neutrons again is very very low because again the large energy gap above the closed cell. The ground state and this is important now, the cell model can explain the ground state spin and parity of the nuclei. In fact, the liquid drop model could not explain the spin and parity of nuclei in their ground state. Also, the liquid drop model could not explain the nuclear isomers by using in cell model, we can explain the existence of isomeric states. The nuclear isomers are nothing but excited states of nuclei which are having metastability, T means having higher applied. So, they do not come down by gamma decay immediately. Normally, you know, gamma decay will take place in picoseconds time, but the isomers will take about maybe it could be milliseconds, seconds or minutes, hours and so. In addition to these properties, there are other properties of nuclei, the magnetic dipole moment of the nuclei and the electric quadruple nuclei also can be explained using the cell model, but I will not discuss these two points in this particular course. Now, let us see how we can explain the magic numbers. Just now, we discussed this that the magic numbers arise whenever there is a large gap between, in fact, this is called the cell, essentially the concept of cell has come because then we have a cell like after this 8 and 20, this 8 nucleon, there is a large gap. So, this is one cell, one cell, let us say cell 1, second cell, third cell like this, this is third cell and then you have the fourth cell. So, this is actually a cell and so, there is a gap between two cells, another cell we have. So, the different orbitals levels have merged together and in fact, there is a now, you can see the nucleons can occupy the states, the energy of these orbitals are very close and the result of that close energy between the different orbitals, we will discuss subsequently in explaining the properties of nuclei, but with the ground state spin parity. So, the concept of cell has come because after that close configuration, there is a large energy gap and so, the nucleon, the nucleus will try to attain that configuration. So, if there is a neutron extra over this cell configuration, close cell configuration, it will be easily able to give that nucleon from the, to form the close cell configuration and that is what is the concept behind the cell. So, the Lixpin orbit coupling we introduced to explain the magic numbers, the high sufficient energy of the nuclei having magic number of protons or neutrons can be explained by the large energy gap here. So, this is a very compact nucleus and similarly, the low sufficient energy for the nuclei one above the magic number again because of the magic number configuration above the magic number, if there is one nucleon, it can be easily removed. So, energy is required to remove a nucleon over the magic number is much less. Again, suppose you have a nucleus having 82 protons, then it has got a low neutron absorption cross-section because it would not like to take a neutron to become a one nucleon more than the magic number of. So, this is how properties of the, particularly with regard to the magic number can be explained. Now, let us see how can explain the ground state spin and parity of nuclei, the bulk of this lecture we will spend more on this one. So, when we say, so we will construct the cell model states for nuclei using the filling of the nucleons in the cell model states and we will use that parity, what is parity is nothing but the minus 1 to the power L, that is the L state. So, S, P, D, F, they have their parity because essentially parity is the symmetry of a function. For example, if you take X to minus X and this function changes sign, we say it is an odd parity and if by changing X from X to minus X, there is no change in the sign of the function, we say it is even parity, like cos theta is an even function, sin theta is an odd function. Similarly, any function, any orbital, S orbital, D orbital, G orbital, L values are even. So, this has got, this has got even parity because the L values are even. So, any L value 0, 2, 4 is or even parity and any L value 2, 1, 3 and 5 are called negative parity. So, depending upon the L value the orbital state populated by a nucleon, we can state away say what will be the parity of that nucleus. Now, let us see the what are the number rules for determining the spin and parity of the nuclei. So, first is the even even nuclei, it is very simple, even even nuclei, all the nucleons are paired up and therefore, this spin will be 0 and parity will be plus. So, the ground state spins and of the even even nuclei are always 0. So, any nucleus which is even even, you can straight away say that spin will be 0. When it comes to odd a nuclei, either the neutron will be broad or proton will be odd. So, the spin of that nucleus, the ground states of the nucleus will be the spin of that, the J value of the last occupied orbital. So, J of the, I will be equal to J of the last occupied orbital and we will see subsequently how to do that. There are, there are odd odd nuclei. In the case of odd odd nuclei, there is a odd proton and there is a odd neutron and so, the odd proton and odd neutron coupled together to give you the spin of the nucleus and it is little complicated. So, we will, there are rules which are called Nordheim's rules. By using Nordheim's rule, we will try predict the ground state spin of odd odd nuclei. So, now let us see how to build the cell model states. I will give you some examples like helium 4. Now, helium 4 has got two protons and two neutrons. So, you will say, so when I say pi, essentially I am being proton and when I say neutron, nu is the neutron state. In the single particle, this cell model is also called single particle model. Single particle means you will take individual nucleon in a potential and generate the label scheme. So, sometimes cell models are also called single particle model and for single particle cell, the states are like proton state as pi and neutron states are nu. So, what is the orbital occupied by the proton is say pi s half and nu s half. So, two neutrons are paired in the nu, this half half, two protons are paired up in this. So, I equal to 0 plus, very easy to consider. If you try to recollect the scheme. So, I can see here that you have first this s s orbital. So, it will be s half, then you have 1 s 1 p, 1 p will split into 1 p 3 by 2, 1 p half, then you have 1 d will split into 1 d 5 by 2, 1 d 3 by 2 and you will have 2 s will be 2 s half and like that you can build up the scheme. So, for oxygen 16 again, oxygen 16 you will have I think there is some mistake here, oxygen 16. So, it is 8, 8 it should be 0 plus or even even nucleus. I s half 2, 1 p 3 by 2 and 1 p half or 8 nucleons occupy and again neutrons nu 1 s half 2, 1 p 3 by 2, 4 and 1 p half. So, it is actually 0 not 1, 1 not half. So, any in fact any even even nucleus you will have the I equal to 0. Let us come to the ordered nuclei now. So, oxygen 17, 8 protons and 9 neutrons. So, it is what you have to see is the which is the orbital occupied by the ninth neutron, the odd neutron that is the ninth one and if you start filling from here. So, you will see s half, s half 2 p half. So, for the protons forget about protons about the neutrons. So, you will have 9 neutrons you populate s half 2, 1 p 3 by 2, 4, 1 p half 2 and you are left with one neutron that will go to 1 d phi by 2. So, the spin the j value of the occupied orbital will be the spin of that particular nucleus. So, I equal to phi by 2 since it is a d orbital it is phi by 2 plus. So, that is how we calculate the spin and parity of the ground state of a nucleus. For example, you have oxygen 15 and equal to 7, 7 neutrons. So, you will see again proton states we do not have to bother s the neutron states you have now s half 2, 1 p 3 by 2, 4 and 1 p half 1. So, 1 p half is the last occupied orbital by the odd neutron and accordingly the spin is j is half and parity is p is minus. So, this is how you can explain you can derive the spin and parity of the ground state of a nucleus by building up the cell model scheme and finding out what is the j value of the last occupied orbital. So, we discussed the even the nuclei ground state spin is 0, we discussed the odd a nuclei, odd a nuclei ground state spin is the j value of the last occupied nucleon, last occupied orbital and depending upon the orbital it is the parity can be defined. And now let us see for ordered nuclei how we can calculate the ground state spin. So, for ordered nuclei we have a odd proton spin JP we will call and we have a odd neutron spin Jn. This JP and Jn will couple together to give you the resultant spin of the nucleus. Now, in this case there are Nordheim's rules which apply to what we call as the smith groups. Now, what are the smith groups actually what happened now if I have not we are not discussing the magnetic dipole moment of the nuclei, but when you plot the magnetic dipole moment of the nuclei last occupied orbital either in J plus, L plus half or L minus half states, what was found that the magnetic dipole moments for these two groups of nuclei. What are the two groups means that whether the nucleus is the levels are the last occupied orbital is J plus, L plus half or L minus half they lie on two lines. And these two lines L plus half have been magnetic dipole moment in one way and the L minus half state magnetic dipole moment the other magnitude. So, there is a gap it is the two groups they are quite different magnetic dipole moment depending on the J value and there they are called the smith groups. So, a smith group is actually the classification based on the magnitude of the magnetic dipole moment that is L plus half or L minus half. So, that same analogy we derived here not he said that if the odd neutron and odd proton they belong to the different smith groups. So, when we say different smith group means one smith group is L plus half other smith group is L minus half that is what we mean by smith groups. Even we can suffice to say if they belong to different a value of L plus half or L minus half. So, if the odd proton and odd neutron belong to different smith groups that means one of them is L plus half other one is L minus half and vice versa. So, if J p is L plus half and J n is N minus half then J the I the spin of the nucleus is the the magnitude of J p minus J n different between the sum of different between the J p and J n absolute value that will be the because the spin is not negative sum the spin is always positive sum integral down the value the value it can be plus half integral integral but it is plus. So, J p minus J n will decide the spin of the nucleus. Just to give you an example chlorine 38 chlorine 38 chlorine atomic number 17 and you have so 17 proton and 21 neutron how you try to accommodate them. So, let us see how 17 17 neutron protons you can compile you will see that the last proton will go to D 3 by 2 state. Now, the 21st neutron will go to F 7 by 2 state. So, you have you can see here that D 3 by 2 is L minus half and F 7 by 2 is L plus because D 3 D 3 by 2 is L minus half and F 7 by 2 is L plus half. So, they belong to the proton and neutron to pi orbitals having different smith groups and so J p will be J p will be 3 by 2 which is L minus half J n is 7 by 2 is L plus half. So, the spin nuclear spin of this chlorine 38 it will be different between J p and J n and that is 3 by 2 minus 7 by 2 or 7 by 2 minus 3 by 2 it is the mod of that. So, that is equal to 2 and chlorine 38 ground state spin is indeed. So, the smith group this Nodding's rule very well explains the spin of the nucleus even having a odd odd configuration. The second aspect is that if the odd proton and odd neutron belong to the same smith group. So, there is a in fact it is not that easy to predict the spin of this nuclei having odd proton or neutron in the same smith group. But then we say that for such a nuclei the nuclear spin is more than J p minus J n. So, it is it is not easy to just predict it will be what value but it will be more than J p minus J. So, this is a very simple classification where then you you have to see what value it is. For example, the aluminum 26 the aluminum 26 having 13 proton and 13 neutrons. So, the 13 th proton will occupy d 5 by 2 and the 13th neutron will occupy d 5 by 2 because same value of proton and neutron numbers. So, J p is 5 by 2 which is L plus half state d 5 by 2 is L plus half state J n is 5 by 2 L plus half state. So, here is the case where both proton and neutron occupy the same smith group that is L plus half. For such a nucleus I is more than J p minus J n that is it will be more than now 5 by 2 minus 5 by 2 0 it is more than 5 5 0. But experimentally the value has to be found to be 5. So, you can see here it has a range it could be from 0 to 5 I by 2 plus 5 by 2 is 5. So, it could it could be anywhere and so that is the kind of guess it gives that it it is in that range. But you cannot exactly pinpoint what will be the value of the spin for such nuclei which are the odd proton and odd neutron belong to the same smith group. So, now in fact the cell model it is not completely successful in predicting the ground state already we have seen in the ordered nuclei. So, there are a lot of cases where ordered nuclei cell model cannot give exactly it will give a range it will be more than this or it will be equal to this depending upon the in which smith group they occupy. But for the ordered nuclei or even nuclei is very same it is 0 ordered nuclei by and large the cell model can explain the ground state spin and pet. But there are some discrepancies and that discrepancies we will discuss here. So, discrepancies in the cell model prediction and the observed ground state spin and pet. There are two important phenomena which in fact they will lead to these discrepancies. First is the paining energy. Now what happens you know in order nuclei so there are cases in a particular cell where there are different orbitals and which are close by in energy the energy gap is very small. So, if the thumb rule is that if a odd proton is there and it happens to be occupying a high spin state that is a G9 by 2 H11 by 2 then and if there is a nearby low spin state which is filled up then what happens there is a transition that the paired nucleon will right to remain in the high spin state because there will be a gain in the paining energy. So, the paining energy depends upon the J value. So, as you discussed know the higher the L value the gap between the two states L plus half L minus half increasing. So, if there is a one odd proton or odd neutron it will be jumped to a low spin state and so the pair will go to the high spin state and that would lead to discrepancies in the cell model prediction. So, whatever you predict based on this cell model the spins are different from the prediction and that I try to explain using this slide. So, arsenic 75 atomic number is 33. So, we have the last you have so you have 40 42 neutrons. So, this 33rd proton what is the spin state further proton you can see here last proton state F5 by 2. F5 by 2 is the occupied state and according to this one it should be I5 by 2 minus but actually the observed value is 3 by 2. So, here I try to explain the F7 by 2 will be 28 protons then we have after that 2 P3 by 2. So, 4 28 plus 4 32 and plus the last one goes to F5 by 2. So, according to cell model the proton should occupy F5 by 2 and hence its spin should be I by 2 minus but you see here even just before that there is a P3 by 2 level and that P3 by 2 level is. So, the instead of that 33rd proton occupying F5 by 2 the pair goes to F5 by 2 and the odd proton occupies the P3 by 2. So, this is what happens that the paired configuration is having higher energy if it higher the lower energy if it is in a high spin state and so the F5 by 2 gets stabilized by a paired configuration and the odd nucleon goes into the P3 by 2. Similarly, iodine 127 Z equal to 53. So, 53rd proton you can see last proton state is G7 by 2. So, it should be 7 by 2 plus and so you can see here the cell model states the 50 neutron 50 proton configuration 50 numbers here and then we have the G7 by 2 and the D5 by 2. So, the 53rd proton should have occupied 1 G7 by 2 and spin state should have been 7 by 2 plus but experimental value of this nucleus is high by 2 plus and again you can see here the next state to this is D5 by 2 which is vacant. So, it will try to remain it is the odd proton in the G7 by 2 is not preferred over that in the D5. So, that is the explanation that the odd proton will prefer to remain in a lower j value compared to that in the higher j paired configuration will remain in the higher j. So, this is the one of the explanations for discrepancies in the cell model predictions over the experimentally observed values. So, many cases you will find that this kind of changes will take place and then another explanation another there are some cases where you know even this cannot explain this pairing energy effect cannot explain the ground state spin and parity particularly in the case of mid-cell nuclei. So, this mid-cell nuclei what I mean by mid-cell nuclei. So, when we have 50, 80, 2, 1, 26 these are cell somewhere here. So, if around 100 around 100 if the proton number is 100 or neutron number is 100 then for those nuclei the nucleus is deformed in their ground state and for such a nuclei again you will find you can just to give an example to illustrate this point RBM169 has got n equal to 101 and so the 100 100 first neutron if you see the cell model state it should occupy the pi by 2 state and accordingly its spin should be pi by 2 minus but actually the observed spin is 1 by 2 minus and there is no half minus state there is no s orbital or p orbital p orbital it should be actually you can see it should be corresponding p half but there is no p orbital p half in the vicinity of this f5 by 2 and therefore it was becoming difficult to explain how you can get this kind of ground state spins. So, one of the explanations is that the collective motion, collective model means in the nucleus so other than the cell model there is one more model called collective model which I am not able to discuss because of the positive of time. The collective model considers that the all the nucleons inside the nucleus undergo a collective motion like the molecules have vibrations and rotations the nucleus also has its vibrational states and the nutritional states they are called the collective states of the nuclei and so apart from liquid drop and cell model there is one more state one more model called collective model and the collective model has been validated by observation of the spectra the rotational spectra rotational spectra you know the ground state then the low life states of a even even nucleus 0 plus 2 plus 4 plus and the energy gaps you know you get from the moment of inertia rotational energy of a rotational of a nucleus you can predict v b j j plus 1 where b just goes to add where 2 i and so people have even found out the moment of inertia there is a fixed ratio of the energy states for the nearby low lying states okay collective nucleus. So, for the collective motion people have seen that if you if the nuclei follow that kind of a relationship you can associate this with the rotational states of the nuclei and so there is a mixing of collective states and the single particle states or cell model states and that is one of the explanations for this observable spin values being quite different from cell model predictions. So, this is another area where the discrepancies can explain and lastly I will discuss the nuclear isomers the nuclear isomers are the long-lived states of nuclei excited states and it happens whenever there is a large difference between the excited state and the ground state of the nucleus. So, the gamma decay is not easy to happen and so the gamma decay is hindered. So, when the gamma decay is hindered the nucleus and the excited state having got a long lifetime. So, you can see here Kermia 113 48 protons and 65 Newton that this the ground excited state of this nucleus is 11 by 2 plus and ground state is half plus. So, the cell model states you can predict the ground state of this nuclei and excited states are there. So, there is a large delta spin change in the gamma decay and therefore, this isomeric state has a half life of 14.1 years. Similarly, the technician 99 M excited state half minus ground state 9 by 2 plus because of the large change in the spin during the gamma decay this decay is hindered and therefore, isomeric state has got a half life of six ups. So, again you can see here the states Z is equal to 43 for technician the 9 by 2 is the state it is the 43rd proton will go to 9 by 2 state and then there is a excited. So, if you populate the excited state half plus half minus its decay to ground state is hindered and so we can see this 99 M technician is the workhorse of nuclear medicine used in single proton emission computer tomography and this happens the isomeric states are found just below the magic number. So, you can see here the magic number is 82 and right from 70 to 82 or even right from 63. So, these are the all the tellurium isotopes have got isomeric states having these odd number of neutrons. So, the 71st to 82nd neutron enter H11 by state and nearby S half the state is there. So, again the nucleus decides where to occupy the pair will go to H11 by 2 and the odd neutron will go to S half. So, whenever there is a magic self-configuration just below that there is a large difference in the spin states of the levels and the excited state and ground state has difference and therefore, the gamma decay is hindered. So, this is what explains the energy the nuclear isomers and there are several nuclei having isomeric states many of them are having half lives seconds to hours to even years. So, cell model can therefore explain the properties of nuclei which are based on the wherever there is a fluctuation in the property of the nucleus in terms of the mass or in the ground state spin and parities binding energies and so on. This can be explained by cell model very well. Thank you very much.