Feeds:
Posts
Comments

Posts Tagged ‘Large Hadron Collider’

In this blog I derived, from first principles, the Lorentz transformations which are used in Einstein’s special theory of relativity to relate one frame of reference S to another frame of reference S^{\prime} which are moving relative to each other with a speed v.

\boxed {\begin{array}{lcl} x^{\prime} & = & \gamma (x - vt) \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & \gamma ( t - \frac{ v }{ c^{2} }x ) \end{array} }

So, these relate the length x and time t in two different reference frames which are moving relative to each other with a velocity v. One of the most intriguing and surprising consequences of Einstein’s special theory of relativity is that time is relative and not absolute. What this means in simple terms is that two observers in two reference frames S and S^{\prime} moving relative to each other with a velocity v will measure time to be passing at different rates.

Time dilation

This phenomenon is known as time dilation. Let us consider our two reference frames S and S^{\prime}. We will have a clock in frame S^{\prime}, which in that reference frame is stationary (e.g. a clock on a rocket ship, although the rocket ship is moving, the clock is stationary relative to the rocket ship).

Two successive events on the clock in S^{\prime} are separated by a time interval \Delta t^{\prime} which we are going to call the proper time T_{0}. The time interval in the other reference frame, S, is \Delta t = T. How does this compare to T_{0}?.

In the reference frame S^{\prime} the clock is stationary, so we can say that the location of the clock in the x-dimension, x^{\prime}, does not change. That is, \Delta x^{\prime} = 0.

Using our equation which relates t \; \text{and} \; t^{\prime} from above, we can write

\begin{array}{lcl} \Delta t & = & \gamma (\Delta t^{\prime} + \frac{v}{c^{2}} \Delta x^{\prime}) \\ \Delta t & = & \gamma \Delta t^{\prime} \; \; (\text{as} \; \Delta x^{\prime} = 0 ) \\ \end{array}

and so we can write

\boxed {T = \gamma T_{0}}

This means the time interval T in frame S will appear to be dilated by a factor of \gamma compared to the proper time interval T_{0}.


A clock travelling at close to the speed of light will run ore slowly compared to a stationary clock

A clock travelling at close to the speed of light will run more slowly compared to a stationary clock


Time dilation in Nature

We observe the effects of time dilation every day in Nature. Cosmic rays, high energy particles from space, strike molecules in our atmosphere and create particles from the high energy interactions (this is the same as happens in the Large Hadron Collider). One of the particles created in these reactions are muons, which decay very rapidly in about 2 microseconds second (2 millionths of a second). Given the distance between where they are created in the upper atmosphere and the Earth’s surface, they should not survive long enough to make it to the surface of the Earth. But they do. How? Because of time dilation, the muons are moving so quickly that \gamma is appreciable more than 1, meaning that 2 microseconds in the muon’s frame of reference is much longer in our frame of reference. So, in the muon’s frame of reference it is indeed decaying in let us say 2 microseconds, but in our frame or reference it could survive for maybe a millisecond (thousandth of a second) or more, long enough to reach the surface of the Earth.

The symmetry of relativity

One aspect of relativity which confuses a lot of people is that it is symmetrical. Although an observer in frame S will think that the clock in frame S^{\prime} is ticking more slowly, if an observer in S^{\prime} were to look at a clock which was at rest in frame S, that observer would think that the clock in frame S is moving more slowly. Each would think that their clock is behaving normally, and it is the clock in the other’s reference frame which is showing the effects of time dilation.

The twin paradox

If a twin sets off on a space trip where the rocket will travel close to the speed of light, then time dilation effects will come into play. This means that e.g. a 20-year old twin can set off on a space trip which for the twin who stays on Earth appears to last for 40 years, but because of time dilation effects maybe only 5 years will appear to pass for the twin on the rocket. Thus, the 60-year old twin who stayed on Earth will be greeted after 40 years by a 25-year old twin!!

In the example I have shown, 40 years for the twin who stays on Earth appears to pass as 5 years for the twin on the rocket. This means the time dilation factor is 40/5 = 8, and as the time dilation factor is just the Lorentz factor \gamma, this means the rocket will need to travel at a speed of 99.2\% of the speed of light.

HANG ON!!! you say, what about the symmetry of relativity? Surely the twin in the rocket will think that the twin on Earth is aging more slowly, so why doesn’t he return to find the twin on Earth is only 25 and he is 60? Or maybe, because of the symmetry, they will both be 60 when the travelling twin returns?

No, what one has to realise is that there is no symmetry in this trip. In order for the travelling twin to leave the Earth and travel at close to the speed of light he has to speed up considerably. Also, in order to come back he has to slow down and reverse his direction, speeding up again once he’s turned his rocket around to come back to Earth. And, as he approaches Earth, he will have to slow down again. These large accelerations (changes in speed) which the travelling twin experiences break the symmetry, and so it really is the case that the travelling twin will return younger than the twin who has stayed on Earth. How much younger depends on how close to the speed of light the travelling twin travels.

Back to the future

Although it is possible therefore to “travel to the future”, as our twin in the example above does, what is not possible is to travel to the past. In order to do this one would need to travel faster than the speed of light, which Einstein’s theory does not allow. The results of neutrinos travelling faster than the speed of light, announced back in the Autumn of 2011, proved to be incorrect. One of the reasons that story caused so much interest is that travelling back in time has all kinds of problems associated with it, the movie “Back to the future” illustrated some of them. I will discuss time travel more in another blog.

Time for a photon

I will finish this blog with a question about photons (particles of light). Remember that Einstein’s theory of special relativity is based on the premise that light always travels at the same speed in a vacuum. The nearest star system beyond our Solar System is the Proxima Centauri system, which is 4.2 light years away. That means it takes light 4.2 years to travel from this system to us, which in terms of kilometres is 40 trillion kilometres (4 \times 10^{13} kilometres!). Now you know why we use light years for such large distances.

So if light takes 4.2 years to travel the 40 trillion kilometres from Proxima Centauri to Earth, my question to you is


how long would it seem to take if you were a photon moving at the speed of light?


Answers on a postcard, or in the comment section below.

Read Full Post »

Types of particles

The Universe can be divided into three types of particles: matter, anti-matter and radiation (in the modern Quantum-mechanical view of Nature, radiation can also be treated as particles). Anti-matter is not just a science fiction idea, it was first proposed by Paul Dirac in the 1920s and is made every day in particle accelerators as well as in Nature. Today we can even make anti-hydrogen atoms. Clearly what we see in the Universe is composed of matter, not anti-matter. When matter and anti-matter come together they annihilate each other, producing lots of radiation in the form of high energy gamma rays. In a future blog I will discuss the ideas physicists have as to why our Universe seems to have more matter than anti-matter (if the amounts were exactly balanced all the matter and anti-matter would have mutually annihilated and there would be no matter left in the Universe, and hence no “us”).

 

The Universe is divided into matter, anti-matter and radiation.

The Universe is divided into matter, anti-matter and radiation.

 

The discovery of atoms

The word “atom” comes from the Greek word “atomos” which means “indivisible”. The idea of atoms thus dates back a couple of thousand years, but it was only in the 19th Century that evidence for their existence was really found. Through the work of John Dalton and others in the field of Chemistry, strong evidence was established that matter was composed of elementary building blocks, with each element being a different building block with different chemical properties. The Periodic Table of the elements was drawn up in the mid 1800s, and by the end of the 19th Century scientists had measurements of the masses of different elements, noting that e.g. Carbon was more massive than Hydrogen.

The first sub-atomic particle to be discovered was the electron, by J.J. Thomson in 1897. Then, in a series of experiments in 1909-10 the atomic nucleus was discovered by Ernest Rutherford and co-workes. Thus the modern picture of the atom emerged, negatively charged electrons in orbit around a positively charged nucleus. This is the so called “solar system model” because of its similarity to our Solar System. By the early 1930s it was known that the nucleus consisted of positively charged protons and of neutrons, which have no electrical charge.

 

The "solar system" model of the atom has the electrons orbiting the nucleus.

The “solar system” model of the atom has the electrons orbiting the nucleus.

 

The particle zoo

In the 1950s particle accelerators were used to probe the structure of matter. Initially electrons were accelerated to close to the speed of light, and smashed into stationary targets. As accelerators got more powerful physicists started accelerating protons, which are nearly 2,000 times more massive than electrons and hence much harder to accelerate. Physicists found a plethora of particles emerging from these particle accelerator collisions. Below is a picture of particle tracks in a typical bubble chamber, the device used for detecting these sub-atomic particles.

 

In the 1950s hundreds of new particles were being created in particle accelerators.

In the 1950s hundreds of new particles were being created in particle accelerators.

 

Physicists gave names to these new particles, sigma particles, pions, rho particles, D particles, kaons etc. So many new particles were being created in these experiments that physicists started running out of names for them. Some patterns started emerging. One was that particles could be divided into either hadrons (from the Greek word “hadrĂ²s” meaning “stout, thick”) or leptons (from the Greek word “lepton” meaning “fine, small, thin”).

 

Matter can be divided into hadrons (heavy particles) and leptons (light particles)

Matter can be divided into hadrons (heavy particles) and leptons (light particles)

 

Three quarks for Muster Mark

In the 1960s theoreticians tried to find a model which could be used to explain these hundreds of particles and the division into hadrons and leptons. It was Murrray Gell-Mann of Caltech who came up with the idea that the hadrons were composed of more fundamental particles which he called quarks. The word comes from a line in Finnegans Wake, a book written by James Joyce.

Three quarks for Muster Mark!
Sure he has not got much of a bark
And sure any he has it’s all beside the mark.

Initially Gell-Mann proposed three quarks as sufficient to explain all the observed hadrons, these three he called up, down and strange. However, we now believe we need an additional 3, making 6 quarks in all, to explain all hadrons. The names of the other 3 are charm, top and bottom.

 

The 6 quarks believed to constitute all hadrons
Name Generation Year proposed Year discovered
up 1st 1964 1968
down 1st 1964 1968
strange 2nd 1964 1968
charm 2nd 1970 1974
bottom 3rd 1973 1977
top 3rd 1973 1995

 

All hadrons are composed of quarks in this model. Protons and neutrons, the most well known examples of hadrons, are composed of 3 quarks. Any hadron which is composed of 3 quarks and which can decay into a proton is called a baryon. It may surprise you to know that a neutron, if it is not in a nucleus, will decay into a proton, with a half-life of about 14 minutes.

The other type of hadron is called a meson. Mesons are made up of just 2 quarks, and always in a quark-antiquark pair. Mesons cannot decay into a proton, as they have too few quarks.

 

Hadrons can be further divided into baryons and mesons.

Hadrons can be further divided into baryons and mesons.

 

The standard model

The standard model of particle physics is shown in the figure below.

 

The standard model of particle physics.

The standard model of particle physics.

 

You will notice in each box a number of figures. For example, for the up quark it has 2.4 MeV/c^{2} along the top, and 2/3 and 1/2 along the left hand side. The top figure refers to the rest mass of the particle expressed in energy (matter and energy are related via Einstein’s famous equation E=mc^{2}). This is the energy required to create this particle in an accelerator. The next figure, 2/3 in the case of the up quark, is the electric charge. For a proton, the 3 quarks which make it up are u,u and d, giving a charge of 2/3 + 2/3 – 1/3 = 1. For a neutron, the 3 quarks which make it up are u,d and d, giving a charge of 2/3 – 1/3 – 1/3 = 0.

The final figure, 1/2 for the up quark, is the quantum-mechanical spin of the particle. I will explain what this means in a separate blog. All quarks have a spin of 1/2, as do all leptons. Bosons have an integer spin.

The quarks and leptons fall into 3 generations. The first generation is normal matter. The 2nd and 3rd generations of matter seem to be heavier (more massive) versions of the 1st generation, and (apart from the 3 generations of neutrinos) will decay into particles in the 1st generation. We have no idea at the present time as to why Nature has 3 copies of matter, 3 generations. We currently believe that quarks are fundamental particles, and cannot be split up into anything simpler.

The best known example of a lepton is the electron, but another example many people have heard of is the neutrino. The electron and the neutrino are both 1st generation leptons, but there are 2nd and 3rd generation leptons just as there are 2nd and 3rd generation quarks making up the hadrons. We currently believe that leptons are, like quarks, fundamental particles.

The right hand column of the figure are bosons. In the modern quantum mechanical view of Nature, forces are carried (mediated) by particles called bosons. The photon is an example of a boson. It is a “particle of light”, but also the particle responsible for the electro-magnetic force. The weak nuclear force (responsible for radioctive decay) is mediated by the W and Z bosons, and the strong nuclear force (responsible for holding the nucleus together) is mediated by gluons.

You will notice that this figure does not include the famous Higgs boson. I will post a separate blog in the near future about the Higgs boson, why it was proposed, and whether CERN has actually discovered it with the Large Hadron Collider.

Read Full Post »

Today (Wednesday the 26th of September) is a big day for students of Coleg Morgannwg, where I have recently become head of Physics. It is the official opening of the brand new building that has been built in Nantgarw. To mark the occasion, the College has invited Dr. Lyn Evans, who is lead scientist of the Large Hadron Collider at CERN, to be part of the opening. Dr. Evans, known by the press as “Evans the atom“, grew up in Aberdare, which is up the valley from Nantgarw. He studied Physics at Swansea University, and then went on to also study for his PhD at Swansea. He has gone on to work at CERN and become leader of the Large Hadron Collider, the biggest and most expensive scientific experiment ever built.


Dr. Lyn Evans, from Aberdare in South Wales, is lead scientist at the Large Hadron Collider in CERN


The official opening of Coleg Morgannwg’s new campus will take place today, Wednesday the 26th of September. Dr. Lyn Evans will be the guest of honour


Coleg Morgannwg primarily serves the Rhondda Cynon Taf area of South Wales, an area which has seen huge social deprivation since the collapse of the mining industry in the 1980s. I only hope that the students who get to see and meet Dr. Evans today will be inspired to work hard and achieve their goals, as he is testimony that, with a good education, anything is possible for the young people of the South Wales valleys.

[Before you assume that Dr. Lyn Evans and this Dr. Evans are related, I should point out that Evans is an incredibly common last name in Wales. Why this is so will have to be explained in a future blog.]

Read Full Post »