Higgs boson

The Standard Model: Elementary Particles

Since the time of at least ancient Greece, human beings have tried to understand the fundamental constituents of all matter in the universe. Indeed, the idea that matter is comprised of discrete units dates back to many ancient cultures. For example, the etymology of the word “atom” can be traced back to the ancient Greek philosophers Leucippus and Democritus. References to the concept of atoms – the indivisible – has also been found in ancient Indian philosophical texts.

Today, we have a very good idea of the fundamental constituents of all matter in the universe. But we’re still missing some pieces of the puzzle.  That is to say, the picture is not yet complete. While the ancient Greeks speculated on the existence of atoms, primarily by way of philosophical and theological reasoning (as opposed to science-based methods), modern science has significantly advanced our understanding of the fundamental constituents of all matter. Whether we are close to discovering a “theory of everything” is debatable and uncertain. What we can say for certain, however, is that we already have an exceptionally successful theory which explains almost all experimental results and has precisely predicted a wide variety of phenomena, including new particles like the recently discovered Higgs boson. It is known as the Standard Model of Particle Physics.

In this post I want to focus primarily on introducing the three categories or groups of elementary particles that comprise the current version of the Standard Model. In the process, I also want to discuss some of them neat characteristics of these particles. In future posts I will then expand on the details and elaborate on the theoretical complexities and nuances. We’ll also dive deeper into the physics and explore the maths.

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Let’s start with some basic definition. What distinguishes an elementary particle from an atom or a molecule? Inasmuch that the ancient Greeks were concerned with atoms as fundamental indivisible objects, we can say today that atoms are not fundamental in the sense of elementary particles. An atom is the smallest constituent unit of ordinary matter; but it is also divisible, comprised of subatomic particles. On the other hand, an elementary or fundamental particle is indivisible – that is, an elementary particle is not composed of anything more basic.

One helpful way to think of the elementary particles of the Standard Model is by creating a sort of hierarchical picture. We take as our starting point the existence of matter. For example, many readers will already be familiar with how the stuff of everyday life is made of molecules which are then made of atoms. But as was already noted, atoms are divisible – they are composite objects made of smaller divisible parts: namely neutrons, protons and electrons.

Moreover, this is the basic picture one will learn in introductory science class: we have atoms, around which orbit electrons (electron cloud). The nucleus of an atom is then comprised of protons and neutrons. All three of these – electrons, protons and neutrons – are what we call fermions (they have odd half-integer spin). For the purposes of this post, it is assumed that one is already comfortable with this level of subatomic particles, such that one has sufficient knowledge of the electric force; the relative charges of electrons, protons and neutrons; and thus some sense of the physics of how equal charges repel. For example, one should be comfortable with the idea that inside the nucleus of an atom two or more protons would repel and fly apart if it were not for the strong nuclear force that holds protons and neutrons together. And the strong nuclear force is just that, very strong. So it takes a lot of energy to overcome it and to therefore break the nuclei apart.

It is worth noting that any reference to size in the hunt for ‘smaller’ and ‘smaller’ particles is a bit deceiving. When one thinks of particles, they might be inclined to think of something like a very tiny subatomic marble. But we’re not speaking of objects with such well-defined size and hardened surface, as one might relate in their mind with basic objects of everyday life. Particles indeed have particle-like properties in that they do carry discrete amounts of energy. But they also have wave-like properties, and when we observe and study elementary particles in great detail we find their wave-like properties.

So when we break the nucleus apart, and when we break open (in a manner of speaking) the separated protons and neutrons, we find even smaller constituent units. For instance, when we smash protons together in a particle collider, we find that they are made of even smaller particles known as quarks. More technically, they are comprised of up quarks and down quarks. More technically still, the recipe for a proton is two up quarks and one down quark. As for neutrons, they are made of two down quarks and one up quark!

What about electrons? Electrons are curious in that we still have no observable evidence that they are composite units (made up of smaller particles). The thing about discovering new particles is that it takes a lot of energy. (There is a terrific point of discussion that could be raised here in relation to Einstein’s famous equation, E=mc^2, which I’ve mentioned previously and will certainly discuss in more detail in a later post). Think of it this way: at the Large Electron-Positron Collider electrons are moving so fast that they can experience as much as 100 000 times an increase in mass. Although they are smashed at 99.999999999% the speed of light, we have seen no sign of anything smaller.

 

This means that quarks and electrons are two elementary particles. But in various experiments other elementary particles have also been found.

Of the matter particles, we have two categories: quarks which we already talked about and leptons. Together, these are also called fermions. There are 6 quarks: up, down, charm, strange, top, and bottom. There are also 6 leptons: the electron, electron neutrino, muon, muon neutrino, tau, and tau neutrino. Three of these leptons – electron, muon and tau – are also known as charged leptons. The other three types of leptons are, in fact, three types of neutrino. Neutrinos are neat in that, in many textbooks, they are considered massless. But there is good evidence to suggest neutrinos have mass, only that the mass is very tiny (smaller than electrons, for example). Another quirky feature of the neutrinos is how they tend to not interact with other particles. Moreover, one will occasionally read an account of neutrinos as being ‘shy’. Neutrinos are so abundant that some estimates suggest ~100 trillion of them pass through your body every second. But the tendency for neutrinos to not interact with other particles means that, as they collide with Earth for example, most of them will just pass directly through and emerge unscathed.

There are a few other interesting things about leptons. The etymology of the word, lepton, is derived from the Ancient Greek word ‘leptos’, meaning small. The first lepton discovered was the electron, the smallest of the three particles we typically associate with an atom. But what is curious is how, of the three charged leptons, the muon and tau are much heavier than the electron. Despite this difference, they each have the same basic properties. So a remaining question that remains is why are there three? What role or purpose does the muon and tau serve?

Putting this question aside,  you will have done well to notice that, as pictured above, the 6 quarks and the 6 leptons are also categorised by generation. The first generation, which is comprised of up quarks, down quarks, electron, and electron neutrino – these are the lightest and most stable particles.

There is a lot more to be said about all of these elementary particles, but it is important we leave room for the third category: bosons. The interesting thing about bosons is that there are two groups. First, we have gauge bosons. The cool thing about gauge bosons is that they are force-carriers. In other words, one can think of them as mediators for three of the four fundamental forces: strong, weak, and electromagnetic interactions. It turns out that there are 4 gauge bosons: gluon, photon, Z boson and W boson (the Z and W bosons transmit the weak force). These can also be described as vector bosons.

This leaves the famous Higgs boson. The Higgs is a scalar boson. We will talk more deeply about the Higgs in a future post (as with most other things presented here). Meanwhile, one of the truly fascinating properties about the Higgs boson has us return to a consideration of the Z and W bosons mentioned above. Before the predicted Higgs was observed at CERN, a problem existed with the gauge invariant theory for the weak force. The theory predicted that the photon – the gauge boson responsible for transmitting the electromagnetic force – and the W and Z bosons should have zero mass. While the photon is massless, the W and Z bosons are shown to have mass. So the following question for particle physicists emerged: what was giving these particles their mass? The answer is the Higgs boson. Indeed, it is the task of the Higgs to give the W and Z bosons their mass. But the Higgs field also helps explain why other fundamental constituents of matter, such as leptons and quarks, have mass.

With the inclusion of the Higgs, these particles are then the elementary particles of the Standard Model, which, as far as we know, have no substructure and are therefore pointlike. The up quark, down quark, electron and electron neutrino seem the most essential of the elementary matter particles, with the other 8 suspected to have played a key role in the early universe. Each one of the particles in the Standard Model also has an antiparticle. Antiparticles are something we’ll talk about more in latter post. Meanwhile, it is worth concluding with a brief introduction to the four fundamental forces.

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One of the great accomplishments represented by the Standard Model is how three of the four fundamental forces are included under the unifying principle known as the gauge principle. This is also why the Standard Model is described as a gauge theory. As a demonstrably proven and astonishingly successful theory, it brings together things like: a) Quantum Chromodynamics (QCD), which is the single theoretical framework of gauge theory in which three of the four fundamental forces are represented; b) Weinberg-Salam theory, which is a unified field theory that brings together the electromagnetic and weak nuclear interactions, or forces, in what we describe as “electroweak” interactions; c) finally, we also have Quantum Electrodynamics (QED). Behind all of these theories are many important theoretical concepts, such as gauge symmetry as well as other things like the concept of quantum fields. Symmetry plays an important role in particle physics, and the study of symmetry has even led to the production of new theories in the field. But, again, I’ll bracket these discussions for separate entries.

To close, the Standard Model is a theory that successfully describes fundamental particles and how they interact. More deeply, it describes a quark-lepton picture of matter and the quantum theory of the fundamental forces. These fundamental forces are the weak nuclear force, the strong nuclear force, electromagnetism and gravity. However, gravity is not included in the Standard Model. This exclusion refers to the issues relating quantum mechanics and the general theory of relativity. There are also other issues with the Standard Model, which suggest that our picture is not yet complete and that a physics beyond may be waiting to be discovered.

 

 

 

Mass and energy: From particle accelerators to thinking how much energy is stored in a mass of a glass of water

[latexpage]

The title of this article refers to a fun example that pertains to the relationship between mass and energy. But before we get to that, some definition is required. For the sake of general readership, I will try to keep things simple.

To start, we need to get into some physics. In particular, we need to look to special relativity and the principle of mass-energy equivalence. This principle tells us that energy has mass. To phrase it slightly differently, anything that has mass can be said to have an equivalent amount of energy. Thus, to the question of whether energy has mass, the answer is ‘yes’. This was a key lesson offered to us by Einstein’s famous equation,

[
E=mc^2
]

In that the fundamental quantities of mass and energy are directly related to one another, for the purposes of the present discussion we can re-write the above formula in a slightly more nuanced way [E_0=m_0c^2]. This means a particles rest mass has a rest energy. One will note how the rest energy is multiplied by [c^2] which is the speed of light.

This equation helps us explain, among other things,  a very important and certainly nuanced question in particle physics.

***

Let’s think of the Large Electron-Positron (LEP) collider at CERN. One of Einstein’s predictions concerned how mass increases without limit as the velocity of an object, or in this case a particle, approaches the speed of light. The implication is that, if we imagine a particle that has been projected, say, in the horizontal direction, travelling near the speed of light, whatever force we apply to this particle as it is travelling will have very little effect on its velocity.

Take a strip of paper and crumple it into a tiny ball. Pretend that this little ball of paper is our particle, moving in a horizontal direction nigh to the speed of light. Now take the tip of a pen, and as you move this particle in a horizontal direction, apply onto the particle – in other words, imagine the tip of the pen as some constant force being applied to the particle.

What happens is that, as our particle continues to approach the speed of light, as its velocity gets closer and closer to the speed of light, acceleration becomes more difficult. This is because there is an increase in inertia, and inertia relates to mass. And so, it can be said in simple terms that with this increase in inertia there is also an increase in mass.  The logical implication, if we keep things very handy wavy, is that an increase in mass also relates to an increase in velocity. Hence,

[m=gamma m_0]

Or we can say,

[m=frac{m_0}{sqrt{1-frac{v^2}{c^2}}}]

In the example of our imagined particle, it is travelling so fast that it has gained mass. Thus, the constant force that we have applied onto it struggles to induce any sort of acceleration  onto it.

Think of it this way, in the  Large Electron-Positron collider (LEP) electrons are moving so fast that they can experience as much as 100 000 times an increase in mass (Adams and Allday, 2013). This extra, or added mass, comes from the energy as a result of work done on the particle to achieve its acceleration.

Hence, if $E=mc^2$, what is a very cool thing to realise is not so much how energy has been converted to mass. In other words, one might be inclined to describe an energy-mass transfer (or vice versa), but they shouldn’t mistake this as a conversion of energy to mass. As noted above, the more accurate description is that energy has mass (there is no conversion).

For our particle, it has gained so much kinetic energy (KE) that its mass has increased, say for the sake of this example, almost 100 000 times its original value. This is a dramatic increase, to say the least. But it is not uncommon, at least when thought about in the context of today’s massive particle colliders.

This begs the question, why is there a need in our modern particle colliders to project particles at such speeds? The simple abbreviated answer is that, if we were to smash our particle into another, all of this energy has to go somewhere. This is just the law of the conservation of energy speaking. Thus, in the context of a collider, like LEP, much of that energy transfers to the creation of new particles. This is also why there is a tremendous push in particle physics to build even bigger and more powerful colliders that can project particles with even greater energy, with the aim of potentially discovering new and possibly even more fundamental particles.

***

In the future, we can discuss these points in a more rigourous way and also explore the maths.

But for the sake of keeping the present engagement short, let’s end with a wonderful thought. There is a very textbook, yet evocative example, that one will often read when it comes to learning how mass and energy relate with one another. Typically, the example consists of something very practical, like a glass of water. The question that follows is: how much energy is stored in this glass of water? The maths is quite simple.

Let’s assume we have poured ourselves a 500ml glass of water. We know that 1 milliliter of water (ml) equals 0.0010 kilograms (kg). We can therefore determine that our glass of water has 0.50 kg of mass. The question is: how much energy is stored in this glass? We think about this by simply substituting our values into Einstern’s equation:

 

energy and mass

That is an incredible amount of energy. To put it into perspective, $4.5×10^{16} J= 12.5 TWh$. To offer further context, in 2014 the demand for electricity in the UK was approximately 301.7TWh. That is a lot of energy in just 500ml of water.

***

There is potentially another way of arriving at this conclusion, if we were to consider constructing a simple theoretical model. It would look something like this:

We could start by noting that there are approximately $167.28×10^{23}$ H2O molecules in 500 ml of water. We know that each water molecule has two hydrogen atoms. Therefore, we can simply multiply the number of molecules in our glass by two: $167.28×10^{23} cdot 2 = 3.345×10^{25}$.

Now, let’s take as an assumption that in the process of nuclear fusion (it would also require nuclear fission, but I am leaving that out), 4 hydrogen atoms combine to make 1 helium atom.

There is also only ~0.7% conversion of initial mass to energy (that is a lot of potential fusion energy lost!).

So we can start by taking the number of molecules and divide by four, which gives the number of possible reactions.

[ R_{poss}=frac{3.345×10^{25}}{4}=8.364×10^{24}]

Now we must find, in our nuclear fusion reaction, how much energy would be released. We know that the energy released from one reaction would be approximately $4.3575 x 10^{-12}$ J per reaction. So, to get a sense of total energy potential, we could multiply R_poss by the amount of energy released in a single reaction,

[E = 8.364×10^{24} reactions cdot 4.3575×10^{-12}Jr^{-1}= 3.644×10^{13} Joules]

This number is much lower than previously estimated, due to the loss of energy in the fusion process.

Convert from joules to KWh, and we get, rounding our answer, $10100000$ kWh of potential fusion energy. Now if we convert from kilo-watt hours to mega-watt hours, we get $10100$ MWh. We can then finally convert this to tera-watt hours, which gives us $1.01×10^{-2}$.

In 2014, demand for electricity in the UK was approximately 301.7TWh.

So, on my estimating, a glass of water would contribute $frac{1.01×10^{-2}}{301.7} cdot 100% = 3.345×10^{-5}$ of the total need in the UK in 2014.

This was roughly calculated based on a 0.7% mass to energy conversion.  The earlier calculation assumed ~100% conversion, which would only be possible if we were able to cleanly annihilate matter with anti-matter. But if nuclear fusion/ fission were to improve over the coming decades, and this conversion were to increase from, say, 0.7% to 5%, then already we’re looking at a tremendous amount of energy captured from just a 500ml of water. Imagine, then, a 20% conversion of mass to energy or, dare I say, 50%!

References

Adams, S. and Allday, J. 2013. “Advanced Physics”. Oxford, Eng.: Oxford University Press.

Additional Sources

Molar mass of water: http://en.wikipedia.org/wiki/Properties_of_water

Energy released from single fusion reaction: https://en.wikipedia.org/wiki/Nuclear_fusion