Gamow's Theory of Alpha Decay AND Geiger Nuttal Law

Summary

This video script delves into the theory of alpha decay, explaining how large atomic nuclei undergo spontaneous decay by emitting alpha particles, which are helium nuclei. It discusses the balance between nuclear and Coulomb forces and how quantum tunneling allows alpha particles to escape despite lower kinetic energy compared to the potential barrier's height. The script also explores the Geiger-Nuttall law, which relates the half-life of an alpha decay to its kinetic energy, and promises a future derivation of this law from quantum mechanics principles.

Takeaways

• 🔬 Alpha decay is a type of radioactive decay where a large nucleus emits an alpha particle, which is a helium nucleus with two protons and two neutrons.
• 💥 The occurrence of alpha decay is due to the interplay between the attractive nuclear force and the repulsive Coulomb force, with the latter becoming dominant in larger nuclei.
• 🌌 In smaller and medium-sized nuclei, the nuclear force overcomes the Coulomb repulsion, leading to stable configurations, but in larger nuclei, the Coulomb force dominates, causing instability and decay.
• 🚀 The maximum kinetic energy of alpha particles typically ranges from 4 to 9 MeV, which is puzzling given the potential barrier's height of around 25 to 30 MeV.
• 🤔 The escape of alpha particles from the nucleus, despite having less kinetic energy than the potential barrier, is explained by quantum tunneling.
• 📉 Quantum tunneling allows particles to penetrate barriers that are higher than their kinetic energy due to their wave-like behavior in quantum mechanics.
• 📚 George Gamow applied the concept of quantum tunneling to explain alpha decay, suggesting that alpha particles can escape the nucleus through probabilistic mechanics.
• ⚖️ The Geiger-Nuttall law relates the half-life of an alpha decay to the kinetic energy of the emitted alpha particles, stating that shorter half-life decays produce higher kinetic energy particles.
• 📈 The Geiger-Nuttall law was empirically derived from plotting the relationship between half-life and kinetic energy, showing a straight-line proportionality.
• 🔑 Gamow's theory of alpha decay not only explains the puzzling behavior of alpha particles but also provides a theoretical foundation for the Geiger-Nuttall law.
• 📚 The next video will delve into the derivation of the Geiger-Nuttall law from the quantum tunneling expression, providing a deeper understanding of the theoretical underpinnings of alpha decay.

Q & A

What is alpha decay?

Alpha decay is a type of spontaneous radioactive decay process where a large-sized nucleus, typically with a mass number greater than 210, emits an alpha particle, which consists of two protons and two neutrons, essentially a helium nucleus with a mass number of four.

Why do large-sized nuclei undergo alpha decay?

Large-sized nuclei undergo alpha decay due to the imbalance between the attractive nuclear force and the repulsive Coulomb force. As the nucleus size increases, the nuclear force, which acts over short distances, is less effective in overcoming the Coulomb repulsion between protons, leading to an unstable configuration that seeks stability by reducing its size through alpha decay.

What is the relationship between the nuclear force and Coulomb force in a nucleus?

The nuclear force is an attractive force that acts between both neutrons and protons, holding the nucleus together. The Coulomb force is a repulsive force that acts only between protons, trying to break the nucleus apart. At short distances, the nuclear force is dominant, but as the nucleus grows larger, the Coulomb force becomes more significant due to increased distances between nucleons.

What is the typical range of kinetic energy for an alpha particle emitted during alpha decay?

The maximum kinetic energy of an alpha particle emitted during alpha decay usually ranges from 4 to 9 mega electron volts (MeV).

Why is there a discrepancy between the potential barrier height and the kinetic energy of the alpha particle?

The discrepancy arises because the alpha particle can escape the nucleus with less kinetic energy than the potential barrier height due to a quantum mechanical phenomenon known as quantum tunneling.

What is quantum tunneling and how does it apply to alpha decay?

Quantum tunneling is a quantum mechanical effect where a particle can penetrate a potential barrier that is higher than its kinetic energy. In the context of alpha decay, quantum tunneling provides a probabilistic mechanism for the alpha particle to escape the nuclear potential well despite having insufficient classical kinetic energy.

What is the Geiger-Nuttall law and how does it relate to alpha decay?

The Geiger-Nuttall law is an empirical observation that relates the half-life of an alpha decay process to the kinetic energy of the emitted alpha particle. It states that shorter-lived alpha decays result in higher kinetic energies of the alpha particles, and vice versa.

How did Gamow's theory of alpha decay provide an explanation for the Geiger-Nuttall law?

Gamow's theory of alpha decay used the concept of quantum tunneling to explain how alpha particles with less kinetic energy than the potential barrier height could escape the nucleus. This probabilistic approach to alpha decay successfully explained the observed relationship between half-life and kinetic energy as described by the Geiger-Nuttall law.

What is the significance of the Geiger-Nuttall law in understanding nuclear decay processes?

The Geiger-Nuttall law is significant as it provides a predictive tool for understanding the relationship between the half-life and kinetic energy of alpha particles in nuclear decay processes. It also validates the quantum mechanical concept of quantum tunneling in the context of nuclear physics.

What experimental observations led to the formulation of the Geiger-Nuttall law?

Geiger and Nuttall conducted experiments observing a large number of nuclear species undergoing alpha decay. They plotted the relationship between the half-life and the kinetic energy of the alpha particles and found a straight-line proportionality, which led to the formulation of the Geiger-Nuttall law.

Outlines

00:00

🔬 Introduction to Alpha Decay and Nuclear Forces

The video introduces the concept of alpha decay, a type of radioactive decay where a large nucleus emits an alpha particle, which is a helium nucleus consisting of two protons and two neutrons. It explains why only large nuclei undergo alpha decay, attributing it to the balance between nuclear force and Coulomb repulsion. The nuclear force is dominant in small nuclei, but as the nucleus grows larger, the Coulomb force becomes more influential, leading to instability and decay. The video sets the stage for discussing the gamma theory by explaining the fundamental forces at play within atomic nuclei.

05:02

🧲 Nuclear Potential and the Puzzle of Alpha Decay Energy

This paragraph delves into the nuclear potential energy barrier that alpha particles must overcome to be emitted from a nucleus. It presents a paradox where alpha particles have kinetic energies of 4 to 9 MeV, yet the potential barrier is typically 25 to 30 MeV high. The video uses an analogy of a chalk thrown upwards to illustrate the concept of potential energy and introduces the concept of quantum tunneling as the key to understanding how alpha particles can escape despite having less kinetic energy than the barrier height.

10:03

🌪️ Quantum Tunneling and the Geiger-Nuttall Law

The video explains quantum tunneling, a quantum mechanical phenomenon that allows particles to penetrate barriers that classically they shouldn't be able to. It applies this concept to alpha decay, showing how alpha particles can escape from the nucleus despite the high potential barrier. The paragraph then connects this theory to the Geiger-Nuttall law, which relates the half-life of a nucleus to the energy of the emitted alpha particle. It suggests that higher energy alpha particles have shorter half-lives and vice versa, which is an experimental observation supported by the gamma theory of alpha decay.

15:04

📊 Derivation of the Geiger-Nuttall Law from Quantum Mechanics

The final paragraph discusses the experimental validation of the gamma theory through the Geiger-Nuttall law. It mentions an upcoming video where the presenter will derive the Geiger-Nuttall law from the quantum tunneling expression, providing a theoretical foundation for the experimental observations. The video concludes by emphasizing the importance of quantum tunneling in explaining the behavior of alpha particles and its successful application in predicting the relationship between half-life and kinetic energy in alpha decay processes.

Keywords

💡Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus emits an alpha particle. In the context of the video, it is a spontaneous process that occurs in large-sized nuclei with mass numbers typically greater than 210, leading to the emission of a helium nucleus consisting of two protons and two neutrons. The script discusses alpha decay as the main theme, explaining why it happens and how it relates to the nuclear forces and quantum tunneling.

💡Alpha Particle

An alpha particle, as defined in the script, is essentially a helium nucleus composed of two protons and two neutrons, giving it a mass number of four. It is the particle emitted during alpha decay, which is a key concept in the video's exploration of nuclear decay processes.

💡Nuclear Force

The nuclear force is a strong, attractive force that acts between nucleons, such as protons and neutrons, within an atomic nucleus. The script explains that this force is crucial for holding the nucleus together and that its dominance over the Coulomb force is what stabilizes smaller nuclei. However, in larger nuclei, the nuclear force is less effective at overcoming the repulsive Coulomb force, leading to instability and alpha decay.

💡Coulomb Force

The Coulomb force is a repulsive force that acts between positively charged protons in the nucleus. The script discusses how this force becomes dominant in larger nuclei, counteracting the attractive nuclear force and leading to the instability that results in alpha decay.

💡Quantum Tunneling

Quantum tunneling is a quantum mechanical phenomenon where a particle can penetrate a potential barrier even if its energy is less than the barrier's height. The script uses this concept to explain how alpha particles can escape the nuclear potential well despite having less kinetic energy than the barrier's height, which is a central part of the gamma theory of alpha decay.

💡Nuclear Potential Well

The nuclear potential well is a conceptual model used to describe the potential energy landscape that particles within a nucleus experience. The script describes how alpha particles, when attempting to leave the nucleus, must overcome this potential well, which is higher than their kinetic energy, through quantum tunneling.

💡Half-Life

Half-life is the time required for half of the atomic nuclei in a sample to undergo radioactive decay. The script relates the concept of half-life to the kinetic energy of alpha particles, explaining that higher kinetic energy alpha particles have shorter half-lives and vice versa, which is a key aspect of the Geiger-Nuttall law.

💡Geiger-Nuttall Law

The Geiger-Nuttall law is an empirical relationship in nuclear physics that connects the half-life of a radioactive decay process to the energy of the emitted alpha particles. The script explains that this law was derived from experimental observations and later theoretically explained by the gamma theory of alpha decay, showing a direct relationship between half-life and alpha particle energy.

💡Gamow's Theory

Gamow's theory, as discussed in the script, is a theoretical framework that applies the concept of quantum tunneling to explain alpha decay. It was developed by George Gamow and provides a quantum mechanical explanation for how alpha particles can escape the nucleus despite the potential barrier being higher than their kinetic energy.

💡Kinetic Energy

Kinetic energy is the energy that a particle possesses due to its motion. In the script, it is mentioned that the maximum kinetic energy of alpha particles emitted during decay typically ranges from 4 to 9 mega electron volts. The kinetic energy of the alpha particle is a key factor in understanding the decay process and its relation to the Geiger-Nuttall law.

Highlights

Introduction to the gamma theory of alpha decay and its relation to the Geiger-Nuttall law.

Alpha decay is a spontaneous radioactive decay process for large-sized nuclei with mass numbers greater than 210.

An alpha particle is a helium nucleus with two protons and two neutrons, having a mass number of four.

Nuclei stability is determined by the balance between the attractive nuclear force and the repulsive Coulomb force.

Large nuclei become unstable due to the dominance of Coulomb repulsion over nuclear force at increased distances.

Alpha decay is a method for large unstable nuclei to achieve stability by reducing size through the loss of protons and neutrons.

The maximum kinetic energy of alpha particles ranges from 4 to 9 mega electron volts across different decay processes.

The nuclear potential well height is typically 25 to 30 mega electron volts, posing a puzzle for alpha particle escape.

Quantum tunneling explains how alpha particles can escape the nucleus with less kinetic energy than the potential barrier.

Gamow applied the concept of quantum tunneling to explain alpha decay, introducing a probabilistic approach to particle escape.

The Geiger-Nuttall law relates the half-life of an alpha particle to its kinetic energy, with shorter half-lives corresponding to higher kinetic energies.

The transmission probability of an alpha particle is influenced by the width of the potential barrier it must tunnel through.

The Geiger-Nuttall law was empirically derived from plotting the relationship between half-life and kinetic energy of alpha particles.

Gamow's theory provides a theoretical explanation for the Geiger-Nuttall law, connecting quantum mechanics to nuclear decay.

The video promises a future derivation of the Geiger-Nuttall law from the quantum tunneling expression in a subsequent video.

The gamma theory of alpha decay successfully explains the puzzling behavior of alpha particles and validates quantum tunneling.

Transcripts

hi welcome back to my video once again

in this video I want to give a brief

introduction to the gammas theory of

alpha decay and how it relates to the

giggle nut law so the alpha decay is a

kind of a spontaneous radioactive decay

process in which a large sized nucleus

usually nucleus having mass number

greater than 210 spontaneously undergoes

a decay process which leads to the

emission of a 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 gamma 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-sized

nucleus or medium-sized 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 Coulomb big

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 1 fantome meters

to 3 femtometers the nuclear force which

is attractive in nature is very much

dominant compared to the Coulomb

repulsion so when we look at small sized

nucleus as well as medium-sized nucleus

whatever nuclear forces exist easily

dominates over the Coulomb bit 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 Coulomb

big force now suddenly starts dominating

over the nuclear forces because the

distances between nucleons starts

increasing and when we look at large

sized nucleus these local nuclear forces

which only acts as short distances is

not easily a

to dominate over the coulomb big

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

Coulomb big repulsion and therefore the

nucleus 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 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 is 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 a resemble square

well potential and as it comes out of

the nucleus it experience as a columbic

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

the their potential well it is found

that different kinds of nuclear

configurations have a maximum height of

around 25 to 30 mega electron volts 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 shock

vertically upwards then it basically

goes to a particular height and it comes

back why because this chalk is

experiencing gravitational force now it

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 gravitation

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 this chalk 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

gravitation potential of the earth now

let me propose another situation if I

throw this chalk 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 it's 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 not only from this example of

showing 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 is 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 the 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 particles 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 hide itself is

somewhere around e to the power minus 2

K - L where L is the width of the

barrier and K - basically gives us the

differences in the energy which is

nothing but twice M V - 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 whale has a height of

around 35 mega electron volt but the

alpha particle has an energy

much less compared to the height let's

suppose around 5 to 10 mega electronvolt

but that alpha-particle still has a

particular probability of escaping the

potential well because we can replicate

that as a particle with some kind of a

wave mechanical solution and this

particular wave has a probability of

escaping through this barrier all right

and the nature of the escape is given by

probabilistic mechanics so this is in

essence the gammas 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 high greater

than its kinetic energy now how does it

relate to the gigger Nuttall 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 put potentials 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 V one and

there is another alpha particle which

comes out of this potential having

energy let's suppose E 2 right so what

I'm saying here is that we are basically

making a comparison between two alpha

particles having energies of e 1 and e 2

such that e 2 is greater then e 1 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 an a

particle escaping through a potential or

a tunneling potential is given by it

this right where L here basically

represents the width of the barrier a

simple statement that I can make from a

expression like this is that if the

barrier width is increased then the

probability of the particle tunneling

through that 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

kind of effective potentials so for the

low energy alpha particle the low-energy

alpha particle experiences a potential

width of around this much right but the

high energy of a particle experiences a

potential width of our own this much

basically right so as you can see the

low-energy alpha particle experiences a

barrier with 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 will 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 case of

the high-energy ray 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 it's half-life

is going to be greater compared to the

alpha particle having high kinetic

energy this is the sense of the Geiger

Nuttall law in short what we can say is

that those nuclear decay reactions which

has higher half-life lead to low

energetic alpha particles 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

one by the Giga not a lot so what gear

and not only basically did is they

looked at the kinetic energies and the

half-life of large number of nuclear

species undergoing an 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 what

these two terms the achievements of the

giggle Nuttall law can be replicated by

the equation which is now ten of the

half-life is basically equal to Z upon

root over e multiplied by some constant

k1 plus e2 without going too much

details into this particular equation

what Giga 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

gammas theory of alpha decay thereby

successfully explaining an experimental

observation so this is how the gammas

theory had 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

geeking at a loss that way the gammas

fear 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 you are not alone

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

gegen at a law which is an experimental

observation but theoretically I am going

to derive this expression from here so

if you are interested in the derivation

of the gegen at law from the quantum

tunneling expression using the gammas

theory then you can follow my next video

and I'll put a link of that video in the

description in that video I am going to

do a derivation from here to here so

that's it for today thank you very much