Revision:Edexcel Physics Unit 3C - Nuclear and Particle Physics

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Edexcel Physics Unit 3C - Nuclear and Particle Physics

Topic 3C – Nuclear and particle physics

C1 Stable and unstable nuclei

Nuclear matter & The Four Fundamental Interactions

Fundamental Forces

4 forces exist in the universe

Gravitational Force

• The force one mass exerts on another. It is attractive only and has an infinite range. Is mediated by the graviton. Affects all particles with mass.

Electromagnetic force

• Force due to object an object having a charge. This also has an infinite range but is much strong than the gravitational force. It can be both attractive and repulsive and so is rarely notice over large noticed other large distances as the forces usually cancel each other out. Is mediated by the photon. Affects all particles with charge.

Weak Nuclear force

• Force responsible for beta decay and has the shortest range of the forces, 1 x 10^-18m. Is mediated by the W+, W- and Z. Affects all particles.
• N.B. Salam and Weinberg showed that this force could be unified with the electromagnetic force. The mediator’s only act in a different manor to each other at low energies but at high energies behave similarly. At high energies it is combined with Electromagnetic force to become the electroweak force.

Strong Nuclear force or colour force

• Holds quarks together, there this force holds a proton together and cause attraction between a proton and a neutron. Has the property of confinement, it will only bind together combinations that have no colour e.g. red + blue + green or red + antired.
• Has a very small range, 1 x 10^-15 and so it only acts between nucleons that are next to each other in the nucleus. Is mediated by the gluon. Affects only Hadrons (Particles made from Quarks, see classification of particles)

N.B. All exchange particles or mediators are virtual, ie not real and so cannot be measured.

The positively charged protons in a nucleus repel due to the electromagnetic force. The force between two protons, of equal charge, +e can be calculated by the formula:

Where

• k = 9 x 10⁹ N m²C^–2
• e = charge on one proton
• r is a typical distance 10^-14

This would mean that the force between 2 protons, as in the helium nucleus, is 2.3N.

On a nuclear scale is the very large and will cause all nuclei to fly apart at high speeds, but as we exists this does not happen.

Because this does not happen another force must exist that is stronger than the electromagnetic force but acting over a shorter distance. This strong nuclear force is about 100 times stronger than the electrostatic force but only reaches as far as one nucleon and it’s neighbour.

In large nuclei there comes a size when the longer ranged electrical force from a large number of protons will overcome the shorter ranged strong nuclear force. When this happens 2 things may occur:

1. An alpha particle may be emitted.
2. The nuclei break apart and nuclear fission occurs.

Size of a nucleus

• Alpha scattering experiment show that the diameter of a nucleus is approximately 1 x 10^-4
• The volume of the nucleus is about 1x10^-12th of total volume of the atom
• As almost all atomic mass is in the nucleus it must be very dense
• Electron diffraction experiments allow more precise estimates of the size of the nuclei to be made. This shows:
• All nuclei have approximately the same density, about 1 x 10^17 Kgm^-3
• The more nucleons the larger the radius of the nucleus
• The nucleon number, A, is proportional to the cube of the nuclear radius, r, or

, therefore

This means that if the nuclei of ⁶⁴Cu and ³²S are compared as copper has twice the amount of nucleons as sulphur the radius is 2^1/3 = 1.26 times as large. Similarly, the copper nucleus has a radius which is 4^1/3 = 1.59 times that of ¹⁶O.

Also volume of a nucleus is

Substituting the first equation we get:

[Unparseable or potentially dangerous latex formula. Error 6 ]

N – Z curve for nuclides

See Graph for region of stability etc.

From the graph it can be seen that:

1. If z < 20, a stable nucleus N = Z
2. If z > 20 a stable nucleus N > Z

This is so as for large amounts of protons a greater than proportional number of neutrons are needed to provide the strong force needed to prevent the atom form flying apart. Nuclei above the stability line are called neutron – rich. They can become more stable by decreasing the number of neutrons. The nucleus decays by β – emission, leading to 1 less neutron and one less proton, bringing it closer to the stability curve and the N, Z ratio closer to, or equal to 1. E.g.

Unstable nuclei below the stability decay by β+ emission. This leads to 1 more neutron and 1 less proton, again bringing the N, Z ratio closer to 1. E.g.

The emission of an alpha particle has little effect on the neutron – proton ratio for isotopes close to line N = Z. It occurs many in larger nuclei. In these it shifts the N: Z ratio slightly towards the neutron, e.g. The decay of thorium – 228, N:Z ratio from 1.53 to 1.55

Notes on Decay Chains and Radioactivity

Decay Chain or Radioactive Series: - When the decay of an unstable nucleus results in another unstable nucleus which in turn may result in another unstable nucleus.

See Graph of thorium decay chain.

• Begins with thorium – 232 and ends with lead – 208
• Graph of A against Z.
• On this graph alpha decay is shown by a movement 4 down and 2 to the left.
• Beta – minus decay involves a movement of one unit to the right
• Half life (t1/2) – the average time taken for the number of undecayed nuclei of the isotope to halve.
• Becquerel: The rate of decay of a sample of radioactive material is measured in becquerels (Bq). An activity of 1Bq represents a rate of decay of 1 s-1
• The rate of decay is affected by only 2 things, the isotope involved and the number of undecayed nuclei
• The graph of activity against time shows a curve of exponential decay. This means that it takes the same amount of time for the activity to half no matter where you start on the graph. :

N.B. Due to the uncertainty principle radioactive decay is an apparently random process so there will be some variation from the curve with experimental data.

If you have N number of a particular number of an isotope, the number remaining after 1 half life is N/2, 2 half lives, N/4 and n half lives N/2^n

The Decay Constant

The relationship between the rate of decay, or activity, of a radioactive isotope is:

Activity = λN

Where the activity is measures in Bqs, N is the number of undecayed nuclei present and λ is the decay constant of the substance. λ unit is s-1

The half life of a radio-active isotope and its decay constant are related by the equation:

Notes on Radioactive Dating

Radioactive dating uses the fact that we know the half-life of a radioactive element. We can find the ratio of amount of this isotope to the amount of it’s daughter product present in a sample. Eg.1 Carbon Dating:

The carbon in the atmosphere contains 98.89% Carbon – 12, 1.11% Carbon – 13 and traces of Carbon 14. This has a half-life of 5730 years and is formed continuously as high energy cosmic collide with nuclei in the atmosphere causing to break up, producing some daughter nuclei and some neutrons. This causes them to lose a proton and form carbon – 14:

This then combines with Oxygen to form Carbon dioxide. This is taken in by plants along with Carbon Dioxide made from Carbon – 12. This means that carbohydrates in all plants contain radioactive carbon in the same ratio as it exists in the atmosphere. This is so for all other organism and plant or animal based materials.

When the plant dies it does not take in any more carbon dioxide and therefore as the carbon 14 decays the ratio of carbon 14 to carbon 12 falls, this means that the age of a piece of wood can be found by measuring the ratio of carbon – 14 to carbon – 12, assuming that the same amount of carbon 14 was in the atmosphere as it is today.

A Modern piece of wood has an activity per Kg (its specific activity) due to carbon – 14 of 250 s^-1.

If the specific activity of an old piece of wood is 125 s^-1Kg^-1 than it one half life old, 5730 years.

If the zero age specimen has a specific A₀a specimen t years old has a specific activity A than:

Where k is the decay constant of carbon – 14

A similar principle can be used to date rocks.

In a decay chain of uranium – 238 ending with lead – 206 the uranium – 238 has a half-life of 4.5 x 10⁶ years, the isotopes in between having relatively short half-lives. This means that the age of the rock can be estimated by comparing the numbers of uranium – 238 isotopes to lead – 206.

Energy and the Nucleus

Binding energy

Particles in a nucleus have binding energy. It is the net sum of the attractive strong nuclear force and the repulsive electromagnetic force. This binding energy can thought of as nuclear potential energy. If a nucleus that has a positive potential energy it will release energy when it breaks apart, its particles are mainly repelling and is unstable. A nucleus with a negative potential energy is stable; it requires a net input of energy to break apart.

However you cannot calculate the binding energy of a nuclear system by adding up the force on the individual nucleons. This is because mass and energy are equivalent. If an object has potential energy this causes an increase in the objects mass. In every day life this is not noticeable, but on the quantum scale it is. For example the mass of a helium nucleus < the mass of the 4 individual nucleons. This is because the nuclease has a negative potential energy, so the mass of the 4 nucleons is less. As we know the change in mass we can calculate the binding energy using as mass/energy must be conserved.

Often a more useful way of looking at this is the Binding Energy Per Nucleon. (BEPN). This is the energy needed to take one nucleon out of a nucleus, doing work against the attracting forces.

Graph of BEPN against nucleon number:

The graph shows that iron has the highest BEPN, and so it therefore it is the most nuclearly stable element.

NB BEPN can also be shown as an inversion of this graph, showing the potential energy, which is negative p 394 in textbook. This shows that as iron has the least potential energy and therefore by moving nuclei closer to iron the potential energy in the atom deceases. This means that energy must be given out.

When a nucleus that is heavier than iron decays by alpha emission it loses potential energy. The difference in energy is called the decay energy. It is transformed to the kinetic energy of the decay products and the energy of any waves emitted.

Units:

1 unified atoms mass unit (u) is defined to be 1 twelfth of the mass of a carbon atom.

A proton with 0 potential energy has a mass of 1.0073 u and a neutron 1.0087u

As mass and energy are equivalent 1u = 930 MeV

An electron volt is the energy transferred when an electron moves through a potential difference of 1 volt.

Below is a description of the different forms of decay that occur and the forms of emission.

Nuclear Decays

Alpha decay occurs when in a nucleus Strong Force >> Electromagnetic Force This means that it squeezes the nucleus, giving all the particles in it greater velocity. This makes it more probable that an alpha particle, the most stable of configuration of small numbest of protons and neutrons, to escape, being emitted along with a gamma ray. This is because the nucleus now has less energy, being closer to a ground state, so the extra energy is emitted in the form of a gamma wave. Nb. this ray is emitted after a random interval of time after the alpha particle.

However an average alpha particle in a nucleus has an energy of 5 MeV. An energy of 30 MeV is needed to overcome the strong force. One theory to explain this is tunnelling. This says that, as the position of the alpha particle is not certain it can be represented by a probability wave. The higher the amplitude of this wave the greater the chance that that is where the alpha particle is. This wave exits all over the atom but does not end inside the nuclear potential energy barrier. Why? There is a small but finite chance that the alpha particle can exist outside this barrier and therefore decay. The chance that this occurs is the decay constant.

As there is a known amount of energy to be lost the alpha particles will have a definite energy. Therefore sources emit alpha particles with one or a few finite energies.

This occurs when down quark changes into an up quark, i.e. when a neutron changes into a proton:

n = udd = 2/3 - 1/3 –1/3 = 0

p = uud = 2/3 +2/3 – 1/3 = 1

1 d changes to u , causing the partial to become positive rather than neutral. The weak force is responsible for this decay. A W- is emitted for this to take place. The process is random for the same reason as for the decay of an alpha particle. For the decay to take place the W- must be outside the 30MeV barrier. The W- causes an electron and an antielectron neutrino ( υe ) to be produced.

NB strike through denotes line above the top throughout, ie anything with a strikethrough is an antiparticle.

Two particles are produced however neutrinos are virtually undetectable. This means that although a known amount of energy is released from the atom some is given to the neutron and some to the electron. This means that beta particles have a range of possible energies.

This range of energies for an element is shown in the following graph.

This occurs when an up quark changes into a down quark.

p = uud = 2/3 +2/3 – 1/3 = 1

n = udd = 2/3 - 1/3 –1/3 = 0

This produces a W+ particle, which creates a positron (beta plus or antielectron) and an electron neutrino (e+ + υe )

The distribution of energy is the same as that as beta minus.

Gamma Radiation

This radiation is emitted after the decay of a nucleon due to the exited state of the atom. As it moves to a lower energy state the energy lost is emitted as a gamma ray. As a set amount of energy needs to be lost there are 1 or a few finite frequencies at which gamma rays can be emitted.

For Feynman diagrams on these decays see the final section.

Classification of Particles

All particles have what is called spin. This is the number of times the particle looks the same when rotated through 360 degrees; a measure of rotational symmetry. All particles can have spin up or down. For example a particle with spin 2 will look the same after being turned around 180 degrees. All force carrying particles, called Bosons have an integer particle spin. All mater however has spin, ½, so matter particles have to be rotated twice before they look the same. These are divided up into two groups. The Hadrons are made of quarks. These in turn are divided into 2 groups, the Mesons, made up of two quarks and baryons, made of 3. Only Hadrons are affected by the strong nuclear force. The other group are leptons

Leptons

Name	Symbol	Mass	Lifetime	Charge	Spin
Electron		0.5511 MeV	Stable	-1	½
Muon		105.6 MeV	2 x 10-6	-1	½
Tau		0 < 50 eV ?	Stable?	0	½
Electron Neutrino		≈0	Stable?	0	½
Muon Neutrino		0 <0.50eV?	Stable?	0	½
Tau neutrino		0<70MeV?	Stable?	0	½

N.B. All Leptons have a lepton number of 1 and strangeness of 0.

The Existence of antimatter

Every particle has an antiparticle. This particle is the exact opposite of the particle. For example a positron, antielectron, has a charge of +1. Antiparticles however do not have negative mass. This is probably because mass is attractive only. Also they do not have negative spin. As spin is rotational symmetry a negative spin is exactly the same as a positive one.

When a particle meets its antiparticle they annihilate producing gamma rays. The energy of the gamma rays can found using e = mc^2

Quarks and anti quarks

These come in different flavours:

Quarks also have colour, though this is independent of what flavour they are. See Strong Nuclear Force for more information.

Mesons are made of a quark and an anti quark, in a combination that makes white, e.g. Blue and anti blue. Mesons are often unstable and decay very quickly. They only occur naturally in cosmic radiation. Most of them are name after Greek letters, and many have 3 varieties, + - and 0, e.g. Pi-zero, pi-plus, pi-minus. Some other mesons are Kaon, J/PSI, D and Upsilon.

Baryons are made of 3 quarks with colour red, green and blue. The most common and stable are the proton and the neutron, however there are others, and many also have a + - and 0 variety. Some examples of other hadrons are the Lambda, Sigma, Xi, omega , charmed lambda.

Conservation Laws in Particle Interactions

The Fundamental particles obey the basic conservation laws:

• Conservation of Momentum
• Conservation of Mass/Energy
• Conservation of Charge

It has been observed that there are also a set of quantum numbers that are conserved:

• Baryon number: - The number of baryons must be conserved
• Lepton number: – The number of leptons must be conserved
• Spin: - The total spin must be conserved
• Strangeness: - Some particles are observed to decay much slower than would be expected, this is due to a property called strangeness that must be conserved, however sometimes in the short term as far as we know this rule is broken, but at the end of the decay strangeness is conserved

E.g. Beta minus decay

For the 4 fundamental interactions see 1st section.

Forces described in terms of exchange particles

As is mentioned in the 1st section forces can describe by the exchanging of an exchange particle, or mediator. This is part of a theory called Quantum Electrodynamics (QED). These mediator particles are called Bosons:

N.B. Graviton is undetected at time of writing

N.B.2 Mesons such as the Pion carry the Strong Force over larger distances, such as across a nucleus. This is called Residual Strong Nuclear Force.

One way of looking at exchange of bosons is this is done by change momentum, e.g. 2 electrons get close so they “shoot” photons at each other, the closer they get the more photons the shoot. As photons have momentum the shooting of them will change the electrons direction

Feynman Diagrams

These diagrams developed by Richard Feynman, one of the main architects of QED are a way of visually representing force interactions.

The arrows show movement in time, which moves from left to right along the diagrams. Antiparticles are shown with an arrow that goes backwards in time; this makes sense as they can be thought of as matter travelling backwards in time.

Some examples of Feynman diagrams all shown bellow:

As can be seen from the above diagrams a W+ transfers positive charge, a W- negative charge and a Z0 no charge.

This shows a positive muon (an anti lepton) decays to becoming an anti muon neutrino. It emits a W+ which forms a positron and an electron neutrino.

Although the book says time goes left to right more correctly it goes along the y axis, and space on the x axis. An angle of 45 degrees is the speed of light.

Comments

• Suitable for: A Level physics, especially for Edexcel Unit 3B - Nuclear and Particle Physics.
• Written by: RichyP.
• From this post.