Understanding Alpha Decay and GAMOS Theory

Hi, welcome back to my video once again. In this video, I want to give a brief introduction to the GAMOS theory of alpha decay and how it relates to the Giger-Nuttle law. So the alpha decay is a kind of a spontaneous radioactive decay process in which a large size nucleus, usually nucleus having mass number greater than 210, spontaneously undergoes a decay process which leads to the emission of an alpha particle. What is the alpha particle? An alpha particle is nothing but a helium nuclei. It has two protons and two neutrons. So it has a mass number of four. Now before diving into the GAMOS theory, let's spend a moment discussing why does an alpha decay happen in the first place? Why is it that only large sized nucleus undergo radioactive decay, which is the alpha decay process and not small size nucleus or medium size nucleus? The answer to this question lies in the nature of the nuclear force. So basically the nucleus is held together because of two kinds of forces. One is the nuclear force which is an attractive force and it acts between neutrons as well as protons and the other is coulombic force which is a repulsive force and it acts only between protons and it is trying to break apart the nucleus. Now it just so happens that at short distances of distances of around one femtometers to three femtometers the nuclear force which is attractive in nature is very much dominant. compared to the coulombic repulsion. So when we look at small size nucleus as well as medium size nucleus whatever nuclear forces exist easily dominates over the coulombic repulsion and we end up getting stable nuclear configurations. However, as the size of the nucleus becomes bigger and bigger and the distances between the nucleons inside the nucleus increases and it becomes larger compared to the distances in which the nuclear forces act, a very interesting thing happens. The coulombic force is now suddenly starts dominating over the nuclear forces because the distances between nucleons starts increasing and when we look at large size nucleus these local nuclear forces which only acts at short distances is not easily able to dominate over the coulombic repulsion so as we reach a particular size so for nucleus having mass number usually greater than somewhere around 200 the size is so big that nuclear forces are not able to dominate over the coulombic repulsion and therefore the nuclear structure becomes unstable. And the only way these kind of large-sized unstable nuclear configurations become stable is by losing some of the number of protons and neutrons and decreasing its size, which is what happens in the alpha decay process. Now a very interesting thing happens in this particular process. If we look at the different kinds of alpha decay processes happening for different kinds of nuclei, then it is seen that The kinetic energy of the alpha particle, the maximum kinetic energy of the alpha particle usually ranges from 4 to 9 mega electron volt. Now there's a very interesting puzzle associated with this amount of energy. To understand that puzzle, let's first look at the nuclear potential diagram of any given nuclear configuration. So all the particles which are stuck inside the nucleus basically experience some kind of a nuclear potential as a result of all its interactions and we can approximate the nuclear interactions by this kind of a potential well. So as the alpha particle is trying to come out of the nucleus it experiences this kind of a nuclear potential well. So inside the nuclear radius it experiences somewhat and approximately for our purposes of discussion resemble a square well potential and as it comes out of the nucleus it experiences a Coulombic repulsion which can be which is basically a function of 1 upon r where r is the radial distance away from the center of the nucleus. Now before the alpha particle comes out of the nucleus it experiences this kind of a potential as it is stuck inside the nucleus itself. What is interesting about our discussion is that the alpha particle is seen to have maximum energy of around 4 to 9 mega electron volts. However, if we make a calculation of different kinds of nuclear configurations and we look at their potential well, it is found that different kinds of nuclear configurations have a maximum height of around 25 to 30 mega electron volts. Now this is very puzzling. The alpha particle which comes out of the nuclear potential well is seen to have energies up to 9 mega electron volt while the potential itself has a height of around 25 mega electron volt. How can a particle having kinetic energy almost 15 to 20 mega electron volt less than the height of the potential barrier still escape the potential barrier. To understand this problem Let's think of it in a very simple example Let's say that I have this chalk and I throw the chalk vertically upwards then it basically goes goes to a particular height and it comes back. Why? Because this chalk is experiencing gravitational force. Now this chalk theoretically can escape the gravitational potential of this earth if I throw this chalk vertically upwards with a velocity greater than the escape velocity of the earth's gravitational potential. If I throw the shock upwards with a velocity greater than the escape velocity, then it has sufficient kinetic energy to overcome the gravitational potential of the Earth. So the shock will finally escape the gravitational potential and go to space. However, for all the cases in which I throw the shock with a velocity less than the escape velocity, it is always going to come back towards the Earth because it does not have sufficient kinetic energy to overcome the gravitational potential of the Earth. Now, let me propose another situation. If I throw the shock upwards... with a velocity less than the escape velocity, but this chalk still escapes to space, it still becomes free from the gravitational potential of the earth, then that is going to be puzzling, right? Because this does not have sufficient kinetic energy to escape the gravitational potential of earth. So how is it possible that this can penetrate the potential which is greater than the kinetic energy that it has? A same situation is happening here. The alpha particle which is stuck inside the nucleus has a kinetic energy much less. than the potential height itself. So how can the alpha particle escape the nucleus if its kinetic energy is less than the potential barrier itself? This is quite puzzling. The only explanation to this comes from what is known as quantum tunneling. So classically we cannot explain this kind of a behavior as I just now told you from this example of throwing a particle in the gravitational field of Earth. Classically we cannot explain it but there is an explanation from quantum physics which is known as quantum tunneling. Now according to quantum tunneling what happens is that there is a certain possibility in quantum mechanics for a particle to penetrate a barrier whose height is greater than the kinetic energy that it has. So if let's suppose there is a particle which has an energy E it faces a barrier has a height which has a height of v and length of l then this particle has a certain probability of penetrating through this particular barrier now why does this happen i will not go too much in detail in short we can say that the particle in quantum mechanics has a wave behavior associated with them so a particles motion can be understood by studying the wave mechanical behavior associated with the particle so a particles trajectory can in certain situations be replicated by wave motion so if I replicate a particles behavior using a certain wave mechanical equation then that equation basically tells us that this wave has a certain probability of penetrating through the barrier even though it has a kinetic energy which is less than the height of the barrier itself and this kind of a standard problem has a solution which basically tells us that the transmission probability of this kind of a particle having energy less than the potential height itself is somewhere around e to the power minus 2 k to l where L is the width of the barrier and k2 basically gives us the differences in the energy which is nothing but twice m v minus e upon h cross square whole square over right. So this is a standard sort of a solution that comes from quantum physics and what George Gamow did was that he borrowed this kind of an idea of quantum tunneling to the concept of alpha decay. So the puzzle that we had in the case of alpha decay, he borrowed the idea of quantum tunneling here. He said that in the same way that quantum tunneling is predicted in quantum physics, we can apply it in the case of alpha decay process. That means, let's suppose the alpha particle is a particle which is stuck in a potential well like this and the potential well has a height of around 25 mega electron volt but the alpha particle has an energy much less compared to the height. Let's suppose around 5 to 10 mega electron volt but that alpha particle still has a particular probability of escaping the potential well because we can replicate that alpha particle with some kind of a wave mechanical solution and this particular wave has a probability of escaping through this barrier only and the nature of the escape is given by probabilistic mechanics so this is in a sense the Gamos theory of alpha decay in which he borrowed the idea of quantum tunneling to explain the puzzling behavior of how an alpha particle can escape a potential having height greater than its kinetic energy. Now how does it relate to the Giger-Nuttle law? The relationship can be obtained if we make a comparison between two different alpha particles having two different kinetic energies. To understand, let's go back to our diagram. Let's suppose that we are looking at two different nuclear species undergoing radioactive decay but have they have comparable potentials and and they emit alpha particles having different kind of kinetic energies. Let's suppose one of the alpha particle comes out with energy, let's suppose E1 and there is another alpha particle which comes out of this potential having energy, let's suppose. E2 right. So what I'm saying here is that we are basically making a comparison between two alpha particles Having energies of E1 and E2 such that E2 is greater than E1 now based upon what I just now told you what kind of a prediction can we make about the nature of this kind of an alpha decay. So I just told you that the transmission probability of a particle escaping through a potential or a tunneling through a potential is given by this right. Wherever L here basically represents the width of the barrier. A simple statement that I can make from an expression like this is that if the barrier width is increased, then the probability of the particle tunneling through the barrier will decrease. So you can see that if we compare two different alpha particles having two different energies, in those cases, they basically experience different kind of effective potentials. So for the low energy alpha particle, the low energy alpha particle experiences a potential potential width of around this much right but the high energy alpha particle experiences a potential width of around this much basically right so as you can see the low energy alpha particle experiences a barrier width effectively which is greater than the width experienced by the high energy alpha particle what conclusion can we make from here we can make that the high energy alpha particle has a greater transmission probability compared to the low energy alpha particle So, the high energy alpha particle has a greater transmission probability compared to the transmission probability of the low energy alpha particle. What does this mean? This means that the alpha particle which is continuously striking the barrier of the nuclear potential well over and over and over again, in those cases the high energy alpha particle will have a greater probability of escaping through the potential. This means that the half-life, the half-life in the the case of the high energy alpha particle will be less compared to the half-life of the low energetic alpha particle. So if we look at the half-life of the high energetic alpha particle let's suppose E2 then that half-life will be less than the half-life of the low energetic alpha particle. That means the alpha particle having less kinetic energy will take a longer amount of time period to escape the potential barrier. So its half-life is going to be greater compared to the alpha particle having high kinetic energy. This is the essence of the Giger-Nuttle law. In short, what we can say is that those nuclear decay reactions which has higher half-life lead to low energetic alpha particle. compared to those nuclear reactions which have lower half-life or we can also make the statement that short-lived alpha particles have greater kinetic energy or longer lived alpha particles have lesser kinetic energy now this is experimentally proven by the Giger-Nuttel law So what Giger and Nuttall basically did is they looked at the kinetic energies and the half-life of large number of nuclear species undergoing alpha decay process and they plotted a particular graph between the half-life and the kinetic energy when they took the log of the half-life in the y-axis and the atomic number versus the square root of the kinetic energy in the x-axis they basically found that there is a kind of a straight line proportionality of both these two terms the achievements of the Giga Nuttel log can be replicated by the equation which is log 10 of the half-life is basically equal to z upon root over e multiplied by some constant a1 plus a2. Without going too much in details into this particular equation what Giger and Nuttall basically did is they looked at different kinds of nuclear species undergoing alpha decay and then they plotted a relationship between the half-life and their kinetic energies and from this graph what we can conclude is that if the half-life of some alpha particle is greater then its kinetic energy is going to be less. and if the half-life of some alpha particle is less then its kinetic energy is going to be high. So this is an experimental observation and the theoretical explanation for this kind of experimental observation came from the GAMOS theory of alpha decay thereby successfully explaining an experimental observation. So this is how the GAMOS theory borrowed the idea of quantum tunneling and explained the puzzling aspect of why alpha particles can penetrate through potential barriers which are greater than its kinetic energies and we can use this kind of idea to predict an experimental observation which is known as the Giga-Nuttle law. That way the Gamos theory of alpha particle also successfully gave an experimental validation to the idea of quantum tunneling. So this is sort of a brief introduction to the concept of Gamos theory of alpha decay and the Giga-Nuttle law. In my next video what I'm going to do is I'm going to take this particular expression, this standard expression of transformation probability coming from quantum mechanics and I'm going to derive this particular expression of Giga Nuttel law which is an experimental observation but theoretically I'm going to derive this expression from here. So if you're interested in the derivation of the Giga Nuttel law from the quantum tunneling expression using the GAMOS theory then you can follow my next video and I'll put a link of that video in the description. In that video I'm going to do a derivation from here to here. So that's it for today. Thank you very much.

What is the alpha particle? An alpha particle is nothing but a helium nuclei. It has two protons and two neutrons.

So it has a mass number of four. Now before diving into the GAMOS theory, let's spend a moment discussing why does an alpha decay happen in the first place? Why is it that only large sized nucleus undergo radioactive decay, which is the alpha decay process and not small size nucleus or medium size nucleus?

The answer to this question lies in the nature of the nuclear force. So basically the nucleus is held together because of two kinds of forces. One is the nuclear force which is an attractive force and it acts between neutrons as well as protons and the other is coulombic force which is a repulsive force and it acts only between protons and it is trying to break apart the nucleus.

Now it just so happens that at short distances of distances of around one femtometers to three femtometers the nuclear force which is attractive in nature is very much dominant. compared to the coulombic repulsion. So when we look at small size nucleus as well as medium size nucleus whatever nuclear forces exist easily dominates over the coulombic repulsion and we end up getting stable nuclear configurations. However, as the size of the nucleus becomes bigger and bigger and the distances between the nucleons inside the nucleus increases and it becomes larger compared to the distances in which the nuclear forces act, a very interesting thing happens. The coulombic force is now suddenly starts dominating over the nuclear forces because the distances between nucleons starts increasing and when we look at large size nucleus these local nuclear forces which only acts at short distances is not easily able to dominate over the coulombic repulsion so as we reach a particular size so for nucleus having mass number usually greater than somewhere around 200 the size is so big that nuclear forces are not able to dominate over the coulombic repulsion and therefore the nuclear structure becomes unstable.

And the only way these kind of large-sized unstable nuclear configurations become stable is by losing some of the number of protons and neutrons and decreasing its size, which is what happens in the alpha decay process. Now a very interesting thing happens in this particular process. If we look at the different kinds of alpha decay processes happening for different kinds of nuclei, then it is seen that The kinetic energy of the alpha particle, the maximum kinetic energy of the alpha particle usually ranges from 4 to 9 mega electron volt. Now there's a very interesting puzzle associated with this amount of energy. To understand that puzzle, let's first look at the nuclear potential diagram of any given nuclear configuration.

So all the particles which are stuck inside the nucleus basically experience some kind of a nuclear potential as a result of all its interactions and we can approximate the nuclear interactions by this kind of a potential well. So as the alpha particle is trying to come out of the nucleus it experiences this kind of a nuclear potential well. So inside the nuclear radius it experiences somewhat and approximately for our purposes of discussion resemble a square well potential and as it comes out of the nucleus it experiences a Coulombic repulsion which can be which is basically a function of 1 upon r where r is the radial distance away from the center of the nucleus.

Now before the alpha particle comes out of the nucleus it experiences this kind of a potential as it is stuck inside the nucleus itself. What is interesting about our discussion is that the alpha particle is seen to have maximum energy of around 4 to 9 mega electron volts. However, if we make a calculation of different kinds of nuclear configurations and we look at their potential well, it is found that different kinds of nuclear configurations have a maximum height of around 25 to 30 mega electron volts.

Now this is very puzzling. The alpha particle which comes out of the nuclear potential well is seen to have energies up to 9 mega electron volt while the potential itself has a height of around 25 mega electron volt. How can a particle having kinetic energy almost 15 to 20 mega electron volt less than the height of the potential barrier still escape the potential barrier. To understand this problem Let's think of it in a very simple example Let's say that I have this chalk and I throw the chalk vertically upwards then it basically goes goes to a particular height and it comes back. Why?

Because this chalk is experiencing gravitational force. Now this chalk theoretically can escape the gravitational potential of this earth if I throw this chalk vertically upwards with a velocity greater than the escape velocity of the earth's gravitational potential. If I throw the shock upwards with a velocity greater than the escape velocity, then it has sufficient kinetic energy to overcome the gravitational potential of the Earth.

So the shock will finally escape the gravitational potential and go to space. However, for all the cases in which I throw the shock with a velocity less than the escape velocity, it is always going to come back towards the Earth because it does not have sufficient kinetic energy to overcome the gravitational potential of the Earth. Now, let me propose another situation.

If I throw the shock upwards... with a velocity less than the escape velocity, but this chalk still escapes to space, it still becomes free from the gravitational potential of the earth, then that is going to be puzzling, right? Because this does not have sufficient kinetic energy to escape the gravitational potential of earth. So how is it possible that this can penetrate the potential which is greater than the kinetic energy that it has?

A same situation is happening here. The alpha particle which is stuck inside the nucleus has a kinetic energy much less. than the potential height itself. So how can the alpha particle escape the nucleus if its kinetic energy is less than the potential barrier itself?

This is quite puzzling. The only explanation to this comes from what is known as quantum tunneling. So classically we cannot explain this kind of a behavior as I just now told you from this example of throwing a particle in the gravitational field of Earth.

Classically we cannot explain it but there is an explanation from quantum physics which is known as quantum tunneling. Now according to quantum tunneling what happens is that there is a certain possibility in quantum mechanics for a particle to penetrate a barrier whose height is greater than the kinetic energy that it has. So if let's suppose there is a particle which has an energy E it faces a barrier has a height which has a height of v and length of l then this particle has a certain probability of penetrating through this particular barrier now why does this happen i will not go too much in detail in short we can say that the particle in quantum mechanics has a wave behavior associated with them so a particles motion can be understood by studying the wave mechanical behavior associated with the particle so a particles trajectory can in certain situations be replicated by wave motion so if I replicate a particles behavior using a certain wave mechanical equation then that equation basically tells us that this wave has a certain probability of penetrating through the barrier even though it has a kinetic energy which is less than the height of the barrier itself and this kind of a standard problem has a solution which basically tells us that the transmission probability of this kind of a particle having energy less than the potential height itself is somewhere around e to the power minus 2 k to l where L is the width of the barrier and k2 basically gives us the differences in the energy which is nothing but twice m v minus e upon h cross square whole square over right.

So this is a standard sort of a solution that comes from quantum physics and what George Gamow did was that he borrowed this kind of an idea of quantum tunneling to the concept of alpha decay. So the puzzle that we had in the case of alpha decay, he borrowed the idea of quantum tunneling here. He said that in the same way that quantum tunneling is predicted in quantum physics, we can apply it in the case of alpha decay process. That means, let's suppose the alpha particle is a particle which is stuck in a potential well like this and the potential well has a height of around 25 mega electron volt but the alpha particle has an energy much less compared to the height.

Let's suppose around 5 to 10 mega electron volt but that alpha particle still has a particular probability of escaping the potential well because we can replicate that alpha particle with some kind of a wave mechanical solution and this particular wave has a probability of escaping through this barrier only and the nature of the escape is given by probabilistic mechanics so this is in a sense the Gamos theory of alpha decay in which he borrowed the idea of quantum tunneling to explain the puzzling behavior of how an alpha particle can escape a potential having height greater than its kinetic energy. Now how does it relate to the Giger-Nuttle law? The relationship can be obtained if we make a comparison between two different alpha particles having two different kinetic energies.

To understand, let's go back to our diagram. Let's suppose that we are looking at two different nuclear species undergoing radioactive decay but have they have comparable potentials and and they emit alpha particles having different kind of kinetic energies. Let's suppose one of the alpha particle comes out with energy, let's suppose E1 and there is another alpha particle which comes out of this potential having energy, let's suppose. E2 right. So what I'm saying here is that we are basically making a comparison between two alpha particles Having energies of E1 and E2 such that E2 is greater than E1 now based upon what I just now told you what kind of a prediction can we make about the nature of this kind of an alpha decay.

So I just told you that the transmission probability of a particle escaping through a potential or a tunneling through a potential is given by this right. Wherever L here basically represents the width of the barrier. A simple statement that I can make from an expression like this is that if the barrier width is increased, then the probability of the particle tunneling through the barrier will decrease.

So you can see that if we compare two different alpha particles having two different energies, in those cases, they basically experience different kind of effective potentials. So for the low energy alpha particle, the low energy alpha particle experiences a potential potential width of around this much right but the high energy alpha particle experiences a potential width of around this much basically right so as you can see the low energy alpha particle experiences a barrier width effectively which is greater than the width experienced by the high energy alpha particle what conclusion can we make from here we can make that the high energy alpha particle has a greater transmission probability compared to the low energy alpha particle So, the high energy alpha particle has a greater transmission probability compared to the transmission probability of the low energy alpha particle. What does this mean?

This means that the alpha particle which is continuously striking the barrier of the nuclear potential well over and over and over again, in those cases the high energy alpha particle will have a greater probability of escaping through the potential. This means that the half-life, the half-life in the the case of the high energy alpha particle will be less compared to the half-life of the low energetic alpha particle. So if we look at the half-life of the high energetic alpha particle let's suppose E2 then that half-life will be less than the half-life of the low energetic alpha particle. That means the alpha particle having less kinetic energy will take a longer amount of time period to escape the potential barrier. So its half-life is going to be greater compared to the alpha particle having high kinetic energy.

This is the essence of the Giger-Nuttle law. In short, what we can say is that those nuclear decay reactions which has higher half-life lead to low energetic alpha particle. compared to those nuclear reactions which have lower half-life or we can also make the statement that short-lived alpha particles have greater kinetic energy or longer lived alpha particles have lesser kinetic energy now this is experimentally proven by the Giger-Nuttel law So what Giger and Nuttall basically did is they looked at the kinetic energies and the half-life of large number of nuclear species undergoing alpha decay process and they plotted a particular graph between the half-life and the kinetic energy when they took the log of the half-life in the y-axis and the atomic number versus the square root of the kinetic energy in the x-axis they basically found that there is a kind of a straight line proportionality of both these two terms the achievements of the Giga Nuttel log can be replicated by the equation which is log 10 of the half-life is basically equal to z upon root over e multiplied by some constant a1 plus a2. Without going too much in details into this particular equation what Giger and Nuttall basically did is they looked at different kinds of nuclear species undergoing alpha decay and then they plotted a relationship between the half-life and their kinetic energies and from this graph what we can conclude is that if the half-life of some alpha particle is greater then its kinetic energy is going to be less.

and if the half-life of some alpha particle is less then its kinetic energy is going to be high. So this is an experimental observation and the theoretical explanation for this kind of experimental observation came from the GAMOS theory of alpha decay thereby successfully explaining an experimental observation. So this is how the GAMOS theory borrowed the idea of quantum tunneling and explained the puzzling aspect of why alpha particles can penetrate through potential barriers which are greater than its kinetic energies and we can use this kind of idea to predict an experimental observation which is known as the Giga-Nuttle law.

That way the Gamos theory of alpha particle also successfully gave an experimental validation to the idea of quantum tunneling. So this is sort of a brief introduction to the concept of Gamos theory of alpha decay and the Giga-Nuttle law. In my next video what I'm going to do is I'm going to take this particular expression, this standard expression of transformation probability coming from quantum mechanics and I'm going to derive this particular expression of Giga Nuttel law which is an experimental observation but theoretically I'm going to derive this expression from here.

So if you're interested in the derivation of the Giga Nuttel law from the quantum tunneling expression using the GAMOS theory then you can follow my next video and I'll put a link of that video in the description. In that video I'm going to do a derivation from here to here. So that's it for today.

Thank you very much.

Transcript for:Understanding Alpha Decay and GAMOS Theory

Transcript for:
Understanding Alpha Decay and GAMOS Theory