Gamow's Theory of Alpha Decay AND Geiger Nuttal Law
Summary
TLDRThis 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
🔬 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.
🧲 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.
🌪️ 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.
📊 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.
Mindmap
Keywords
💡Alpha Decay
💡Alpha Particle
💡Nuclear Force
💡Coulomb Force
💡Quantum Tunneling
💡Nuclear Potential Well
💡Half-Life
💡Geiger-Nuttall Law
💡Gamow's Theory
💡Kinetic Energy
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
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