 Hello everyone, in the previous part of this lecture we discussed the radioactive decay chain and also derived the expression for the activity of a daughter isotope which is also radioactive and based on that equation in fact we generalize that equation for multiple decays to find out the activity of granddaughters. We also discussed that if the parent is shortly compared to the daughter isotope then this is a case of no equilibrium after the parent activity has decayed down the daughter isotope decayes with its own half-life. Now I will discuss a different situation where the parent is long-lived than the daughter isotope and that is where we will call it as a radioactive equilibrium. So this is the case of a parent being long-lived than daughter isotope and there are two cases of this type here. In the first case the parent is roughly 10 times half-life I mean 10 times more half-life than the daughter isotope and there is another case where the parent is much longer lived maybe 100 times or even more than that. So we will discuss these two cases separately and they have slightly different implications that's all. So let us take the case of this 10 times more parent half-life than daughter and I have given an example here 140 barium having a half-life of 12.8 days decays to length of 140 having half-life of let us say 1.6 days so roughly 10 times the parent it is not very hard and faster it has to be 10 times it could be 9 times 8 times or 12 times no but just order of that and why it is so I will explain with this one and it is decaying to 140 cerium which is stable. So let us see mathematically what happens and then see graphically what happens. So when we have a large time elapsed the line elapsed so again here I will put the equation here activity of daughter equal to activity of parent lambda 2 upon lambda 2 minus lambda 1 e raised to minus lambda 1 t minus e raised to minus lambda 2 t this is the general equation for the activity of the daughter product here length of 140 as a function of time. So what we are discussing here that when the time so here lambda 1 is smaller than lambda 2 so when the time elapsed is much more compared to the half-life of the daughter the daughter is short lived so after several half-lives of the daughter this term e raised to minus lambda 2 lambda 2 is very high so e raised to lambda 2 term tends to 0 when this term tends to 0 you can see here activity of daughter a2 equal to a10 lambda 2 upon lambda 2 minus lambda 1 e raised to minus lambda 1. You see here what is happening now this is the decay of parent whereas we are seeing the change in the activity of daughter so daughter isotope is decaying with the half-life of parent after a sufficiently long time that means several half-lives of daughter so after several half-lives of daughter the daughter activity starts decaying with the half-life of parent so even after several half-lives because that daughter activity being fed from the parent the daughter duty is there but it is decaying with the half-life of parent. So as a result of that the daughter activity reaches a maximum and then decays with the half-life of parent so that means this is like a transient that is why it is called transient equilibrium for a moment the daughter activity reaches the maximum and then starts decaying with the half-life of parent so I have tried to explain this in this graph which is again in the linear scale we are plotting a not log a as a first of time and so the activity of parent the parent is longing the parent is decaying in this way and the daughter activity starts going from 0 because initially there is no daughter activity reaches more than that of parent now you see the implication of this term in this equation this term lambda 2 upon lambda 2 minus lambda 1 will be more than 1 so at that case you will see a2 will become so you can write this as a1 lambda 2 upon lambda 2 minus lambda 1 a1 means a10 raised to lambda 1 t and so a2 will be more than a2 a2 will be more than a1 and this is the condition which is meaning by this equation so the activity of daughter becomes more than that of parent and again after some time starts decaying with the half-life of parent so that is what we call as the transient equilibrium okay so now let us do an exercise of resolving the total activity into that of the parent and daughter in the case of a transient equilibrium so when we have a freshly purified parent isotope then the total activity of this sample if we have a freshly purified the total activity will increase and then subsequently decrease in this fashion here I am plotting total activity in the logarithmic scale as a function of time and from this data let us try to find out the half-life of parent and the that of the daughter and also their initial activity we can find out so as you know the total activity can be given as the parent activity that is a10 e raised to lambda 1 t plus the daughter activity a10 lambda factor e raised to lambda 1 t minus e raised to minus lambda 2 t so now what happens that when that lambda 2 is become quite large because that lambda 2 means the decay constant of daughter that is higher than lambda 1 then this exponential term this term tends to 0 and therefore the total activity becomes parent activity and both of them following the half-life of parent so you can see here the parent activity at a later time is in a logarithmic scale it is state times that means it is exponentially decaying and it should be a decay constant of so this is the data which represents the half-life of parent so you can extrapolate this to zero time and draw a line parallel to this line this is the decay of the parent in the total activity and so from here you can find out the half-life of parent now the excess activity from here to here is due to the growth of the daughter so if you subtract point by point from here then you will get the growth of the daughter which will grow become more than that parent activity and then again decay with the half-life of parent so this is the growth of daughter activity and if the if we had not done the separation of daughter from parent then if you can extrapolate this to zero time that gives you activity of daughter which was present in the sample before the separation was done so this was the activity of daughter which was in the sample and this is the activity which is growing in the sample for the subtraction of this to this this data will give you the daughter decay the daughter which had been separated from the parent as a result of ridiculous separation which is now kept separately will be decaying with its own half-life and so from the decay data you can find out the lambda and hence the half-life of daughter so this is how you can resolve the parent total activity into that of the parent and the daughter you can find out their half-lives you can find out their initial activities this was the exercise that we I thought you should better to do it for the transcendent equilibrium similarly you can do for the secluded. Now we discussed the situation that the daughter activity grows reaches a maximum and then starts decaying with the half-life of parent so what is that time to reach the maximum daughter activity let us discuss this in this slide the activity of daughter changes with time using this expression a 1 0 lambda factor into a raised to minus lambda 1 t minus a raised to minus lambda 2 t so the time at which the activity of daughter activity atoms become maximum then the derivative of a 2 will be 0 so let us differentiate this a 2 as a function of time and equate this equal to 0 to get that time so when this expression is equal to 0 t becomes t m this may maximum time so this you can say if you if you put this at the constant terms at k like a 1 0 lambda factors will be constant so we have differential of this minus lambda 1 here is minus lambda 1 t m plus lambda 2 so minus minus will become plus here is minus lambda 2 t m equal to 0 so this equation boils down to lambda 2 here is minus lambda 2 t m equal to lambda 1 here is to minus lambda 1 t m so you can now try to separate the time factor so you bring the lambda 1 lambda 2 this side it becomes here is to lambda 2 t m minus lambda 1 t m equal to lambda 2 by lambda 1 and now you can arrange them l n lambda 2 by lambda 1 equal to t m take out lambda 2 minus lambda 1 so t m that time to reach maximum daughter activity equal to l n of lambda 2 by lambda 1 upon lambda 2 minus lambda 1 so essentially you have here t m equal to l n lambda 2 by lambda 1 upon lambda 2 minus lambda 1 using this expression you can try to find out and if you want to put this l n in terms of the logarithm you can put 2.303 log base 10 lambda 2 by lambda 1 upon lambda 2 minus lambda 1 so using this formula you can calculate the time when the daughter activity will reach maximum in a case of a transient equilibrium for that matter even even in the case of a no equilibrium case the daughter activity will reach a maximum and you can calculate at what point of time it will happen okay now I will come to the second case of equilibrium and that is called as the secular equilibrium where the parent is much much longer lived than daughter roughly of the order of 100 so it is not a very hard and fast it has to be exactly 100 times but of the order of it could be even 200s or even 1000s also so but it is more than let us typically more than 100 or more than and one example I have given here is tonsium 90 undergoes beta minus decay to yttrium 90 which also undergoes beta 1 degree discondium 90 the half lives 28 years 64 hours so let us say typically less than 3 days 2.64 days and decondium 90 is stable so again use the expression assumptions that at very large time compared to the half life of the daughter e raised to minus lambda 2 term become 10 to 0 because lambda 2 into t will become very very large positive number so exponential of a very large negative of a large negative positive number will become 0 so a 2 equal to a 1 0 lambda 2 upon lambda 2 minus lambda 1 e raised to minus lambda 1 t now compare this transient equilibrium case here lambda 2 is much much larger than lambda 1 because t 1 is much larger than t 2 and so in such a case you can neglect lambda 1 with respect to lambda 2 so in the denominator this term lambda 2 by minus lambda 1 will become lambda 2 and so this will get cancelled out so because of this approximation this term vanishes it becomes 1 and so what happens in this term a 2 equal to a 1 0 e raised to minus lambda 1 t and this term is nothing but activity of dot parent so you can see here the very term meaning of secular means they are same in the case of transient equilibrium the daughter activity is decaying with the half life of parent but it is not same in fact it is more than the parent activity because of the lambda factors in the case of secular equilibrium once the equilibrium is established the activity of daughter and parent become same a 2 equal to a 1 this i have tried to illustrate using this graph here the activity now see though it is linear scale i have put the graph as a linear the activity that means during the period of our observation the activity is not changing the parent activity is not changing and so a 1 is becoming flat but the daughter activity will start growing and finally will become equal to that of the parent activity so this is the difference between the case of the secular and transient equilibrium in transient equilibrium the parent is not that long lived that it will not decay with time parent is also decaying but the daughter grows and becomes more than that of parent activity and then starts decaying with the half life of parent now i will give you an example of this secular equilibrium here and here now i have taken this as a long scale because the growth of the daughter is like a linear that means the activity is plotted in the linear scale along the scale so again this is the experiment one can do in the laboratory to dissolve the total activity into parent and daughter activity and then find out their half lives so suppose you take a parent isotope and you do a chemistry to separate it purify it you have pure freshly purified parent activity then it will total activity will grow like this okay and then you extrapolate to zero time so let there will be time here so extrapolate zero time this is the zero time now in secular equilibrium so the parent is very long lived so you can draw this is the decay of parent you can draw a parallel line to this this will represent from this point this will represent the parent decay so these two these two lines you know this is the decay of a freshly purified parent isotope and this is that this is the decay of activity total activity if you did not purify the parent so when you did not purify the parent activity was here that is the total activity of parent plus daughter and they are in equilibrium once we separate that there is no daughter activity daughter would become zero and it starts growing in the parent totalized so this you can find out from again this separation these points you separate you subtract this data from this data this is the total activity this is the without separation and with separation if you separate if you subtract then you will get this data that is representing the daughter activity is it like you know if you do a separation of daughter activity put it in a separate flask then there it is not growing from parent it will decay in its own half life and that is what is represented by this graph so from this graph you can find out the half life of daughter and the total activity is the total activity the activity of daughter is nothing but the same as that of parent because they are in the secular equilibrium so that is why the parent activity is here it will become double after equilibrium is damaged now you can see here this is the total activity and this is the total activity at equilibrium and this is the parent activity so this because of the logarithmic scale this is equal to this that is what I wanted to convey that in logarithmic scale it is such that like one to two and two to four are there's a different scale in the log scale and the typical example I have seen these are shown here the experiment in fact is done in the laboratories to take 137 cgm having half life of 30 years and it goes beta minus 2 the isomeric state of 137 barium which has a half life of 2.54 minutes and which is limits the gamma ray of 661 kv to 137 barium ground state which is stable so in the laboratory experiments the the cgm 137 is held in a column which is selectively taking up cgm and then when you elute the barium you take some solution so to provide solution barium 137 will be eluted or cgm will not be eluted and so whatever barium is eluted which will decay with the half life of 2.54 minutes because it is not growing now it is in a separate test tube and so you can get this data only from the separated barium 137 and after again you know sometime it will again grow like that time to reach the maximum activity will be typically four times that dotted half life so you can do this experiment multiple times from a a column which is containing the 137 cgm and that like for years together you can do the same call so this is a experiment in the laboratory for radiochemical separations now I will just discuss the why I put took so much time to discuss this equilibrium cases because they have a lot of applications so the applications of radioactive equilibrium in particularly you know nuclear medicine or even in if you want to use the isotope frequently then if you have a parent which is long lived you can make a generator so radio isotope generators the radio isotope generators are based on the radioactive equilibrium so I will give you two examples one of transient equilibrium and one of circular equilibrium this is the case of technetium 99 m generator technetium 99 m is the called as the workhorse of nuclear medicine it is used in diagnosis of the diseases in the body so it is actually having a half life of six hours so you cannot know you produce this isotope take to the hospitals and then you have to keep on supplying every day at the technetium 99 m but if you have a generator system you have a parent which is long lived like 66 99 momentum 66 hours half life and it is decaying by beta minus 2 99 m technetium which undergoes internal transition to gamma decay to 99 g technetium having half life of 10 to 5 years so now let us see you calculate the time when the activity of technetium 99 will reach a maximum using this formula just now we derived this formula the time taken for 99 m technetium to reach its maximum is 2.8 hours so the technetium negative activity will grow in the sample reach a maximum and you can then separate technetium 99 m use it for the investigation diagnosis like a spec analysis and then after one one day 22.8 hours again you have same amount of technetium in the column so this is what is the concept of a radio isotope generator so here on the left hand side I try to plot the activity of molybdenum so this is the activity of molybdenum which is decaying exponentially it is a linear scale and now the technetium activity grows in the 2.8 let us say typically one day it reaches this value and then you elude this technetium 99 from that column molybdenum remains in the column and this activity now you can use in the hospitals for diagnostic purposes next day again you elude technetium 99 now the molybdenum is decaying so this activity also will be less you again elude technetium 99 next day morning carry out the tests that day third day you again do the separation fourth day and fifth day and so so normally you know Monday the hospitals are supplied with molybdenum 99 and for that week because this half life is 66 hours 2.6 days or so so you can carry on this this test for the whole week and next week again you have fresh lot of molybdenum 99 now I will put a question mark here why does the technetium 99 m activity do not exceed that of 99 molybdenum I explained that in that transient equilibrium and daughter activity will be more than that of the parent so it should have actually gone up and more than molybdenum but in this particular case what happens you are measuring the gamma ray from the technetium 99 m sample and the gamma ray have certain abundance 400 decay this gamma is written only 87 times so they this is called abundance percentage 87 percent so this is not the complete decay of the 99 m takes them only the 87 percent of the time technetium 99 m decaying this gamma ray is imitated and that is why the gamma ray activity becomes less than the parent activity instead of going more than per molybdenum it is becoming less and so this is a typical case of a transient equilibrium where you can use generators based on this concept for the applications in healthcare now another application of this generator is the 99 strontium 90 strontium and this is the case of a secular equilibrium where the parent is much longer than dot so you see here that strontium 90 is having half life of 28 years undergoing beta minus decay to 90 strontium 90 having half life 64 years and this is undergoing beta minus decay to 90 strontium which is stable so now this is a case of secular equilibrium because the parent is much much longer than that of the dot the ratio of half life is more than 100 it is quite high so what I have plotted here is the activity of a case where activity of this strontium 90 will not change with time so the half life is 28 years you are measuring for a day or two or few days it will there will no change but the activity of itram 90 will grow so it was let us say it was 2.6 days for about 10 days 3 4 half lives the activity of itram 90 will grow and then it will become flat it will become equal to that of the parent so at this point of time itram 90 can be separated and again it will start growing and you can again you reach the maximum value so here both strontium 90 and itram 90 are two beta meters that means the activity of the dot will be equal to parent there is nothing like a gamma ray abundance factor coming into picture here and also the every time over a period of even say few months the activity of dot will again become equal to that of parent so you can use this generator to separate 90 itram which is also another important isotope useful in therapeutic applications itram 90 used in many therapeutic applications in nuclear medicine so this is a case of circular diffibrium where you have itram 90 generators now lastly I come to the case of a new rate of reaction in production of radioisotopes we have just seen multinerm 99 or strontium 90 so they are also produced in some process like fission or in a nuclear reaction so the decay growth of activity produced in a nuclear reaction also follows similar pattern as we discussed in the radioactive decay chain previous so here I will give you an example of a nuclear reaction sodium 23 captures a neutron becomes sodium 24 and which means say gamma ray the prompt gamma is emitted and we have the 924 sodium ground state undergoing beta minus decay to magnesium 24 which is stable so what we are essentially seeing the how the activity of sodium 24 will change with time when it is produced by n gamma reaction on sodium 23 so this is analogous to a going to b by nuclear reaction and b going to c the c is magnesium 24 so instead of a going to b by beta minus decay or alpha decay here we have a going to b by n gamma reaction so the profile will be similar so here we can set up the equation for the formation of b dnb by dt equal to r this is a rate of reaction minus dnb lambda b and you can again solve the same way that is dnb by dt plus nb lambda b equal to r and if you recall the integration factor e raised to lambda bt e raised to lambda bt if you integrate what we will be getting is nb lambda b equal to r 1 minus e raised to minus lambda bt so the activity of a daughter though those steps I have not shown because they are similar to what we showed earlier for the case of a going to b going to c and so here this activity this so this is called a saturation factor that means the activity of b will grow this by this factor 1 minus e raised to minus lambda t and what I have shown here is the same thing as a function of time this factor saturation factor this is the rate of the reaction so saturation factor will grow in this fashion and become equal to 1 factor some 4 5 half lives so this in fact this graph we used to decide for how much time we need to irradiate this particular target in the reactor or accelerator so suppose you have one half life then 50 percent is produced to get 100 percent you get about four half lives and more than that there is no gain so this is an idea to fix the time of irradiation and the same profile you get for the activity of daughter activity when you are producing it in the irradiation so that's all I have to say and in the next lecture now I will talk about you get structure and stability thank you very much