Neutrino oscillations and Nobel Prizes

This year’s Nobel Prize in Physics was awarded to Takaaki Kajita and Arthur McDonald for the discovery of neutrino oscillations. This is some really awesome physics which required some careful experimentation and some interesting new theory; it is also one of the things that got me interested in astrophysics.

Neutrinos

Neutrinos are an elusive type of subatomic particle. They are sometimes represented by the Greek letter nu \nu, and their antiparticle equivalents (antineutrinos) are denoted by \bar{\nu}. We’ll not worry about the difference between the two. Neutrinos are rather shy. They are quite happy doing their own thing, and don’t interact much with other particles. They don’t have an electric charge (they are neutral), so they don’t play with the electromagnetic force (and photons), they also don’t do anything with the strong force (and gluons). They only get involved with the weak force (W and Z bosons). As you might expect from the name, the weak force doesn’t do much (it only operates over short distances), so spotting a neutrino is a rare occurrence.

Particle Zoo

The charming bestiary of subatomic particles made by Particle Zoo.

There is a large family of subatomic particles. The electron is one of the most familiar, being a component of atoms (and hence you, me, cake and even marshmallows). The electron has two cousins: the muon (not to be confused with the moo-on) and the tau particle. All three have similar characteristics, with the only real difference being their mass. Electrons are the lightest, muons are about 207 times heavier, and tau are about 17 times heavier still (3477 times the mass of the electron). Each member of the electron family has a neutrino counterpart: there’s the electron-neutrino \nu_e, the muon-neutrino \nu_\mu (\mu is the Greek letter mu) and the tau-neutrino \nu_\tau (\tau is the Greek letter tau).

Neutrinos are created and destroyed in in certain types of nuclear reactions. Each flavour of neutrino is only involved in reactions that involve their partner from the electron family. If an electron-neutrino is destroyed in a reaction, an electron is created; if a muon is destroyed, a muon-neutrino is created, and so on.

Solar neutrinos

Every second, around sixty billion neutrinos pass through every square centimetre on the Earth. Since neutrinos so rarely interact, you would never notice them. The source of these neutrinos is the Sun. The Sun is powered by nuclear fusion. Hydrogen is squeezed into helium through a series of nuclear reactions. As well as producing the energy that keeps the Sun going, these create lots of neutrinos.

The pp chain

The nuclear reactions that power the Sun. Protons (p), which are the nuclei of hydrogen, are converted to Helium nuclei after a sequence of steps. Electron neutrinos \nu_e are produced along the way. This diagram is adapted from Giunti & Kim. The traditional names of the produced neutrinos are given in bold and the branch names are given in parentheses and percentages indicate branching fractions.

The neutrinos produced in the Sun are all electron-neutrinos. Once made in the core of the Sun, they are free to travel the 700,000 km to the surface of the Sun and then out into space (including to us on Earth). Detecting these neutrinos therefore lets you see into the very heart of the Sun!

Solar neutrinos were first detected by the Homestake experiment. This looked for the end results of nuclear reactions caused when an electron-neutrino is absorbed. Basically, it was a giant tub of dry-cleaning fluid. This contains chlorine, which turns to argon when a neutrino is absorbed. The experiment had to count how many atoms of argon where produced. In 1968, the detection was announced. However, we could only say that there were neutrinos around, not that they were coming from the Sun…

To pin down where the neutrinos were coming from required a new experiment. Deep in the Kamioka Mine, Kamiokande looked for interactions between neutrinos and electrons. Very rarely a neutrino will bump into an electron. This can give the electron a big kick (since the neutrino has a lot of momentum). Kamiokande had a large tank of water (and so lots of electrons to hit). If one got a big enough kick, it could travel faster than the speed of light in water (about 2/3 of the speed of light in vacuum). It then emits a flash of light called Cherenkov radiation, which is the equivalent of the sonic boom created when a plane travels faster than the speed of sound. Looking where the light comes from tells you where the electron was coming from and so where the neutrino came from. Tracing things back, it was confirmed that the neutrinos were coming from the Sun!

This discovery confirmed that the Sun was powered by fusion. I find it remarkable that it was only in the late 1980s that we had hard evidence for what was powering the Sun (that’s within my own lifetime). This was a big achievement, and Raymond Davies Jr., the leader of the Homestake experiment, and Masatoshi Koshiba, the leader of the Kamiokande experiment, were awarded the 2002 Nobel Prize in Physics for pioneering neutrino astrophysics. This also led to one of my all-time favourite pictures: the Sun at night.

The Sun at night!

The Sun at night. Solar neutrinos as detected by Super-Kamioknade looking through the Earth. I think this is the astronomical equivalent of checking if the fridge light does go off when you close the door. Credit: Marcus Chown & Super-Kamiokande.

The mystery of the missing neutrinos

Detecting solar neutrinos was a big success, but there was a problem. There were only a fraction of the predicted number. This became known as the solar neutrino problem. There were two possibilities, either solar physicists had got their models wrong, or particle physicists were missing a part of the Standard Model.

The solar models were recalculated and tweaked, with much work done by John Bahcall and collaborators. More sophisticated calculations were performed, even folding in new data from helioseismology, the study of waves in the Sun, but the difference could not be resolved.

However, there was an idea in particle physics by Bruno Pontecorvo and Vladimir Gribov: that neutrinos could change flavour, a phenomena known as neutrino oscillations. This was actually first suggested before the first Homestake results were announced, perhaps it deserved further attention?

The first evidence in favour of neutrino oscillations comes from Super-Kamiokande, the successor to the original Kamiokande. This evidence came from looking at neutrinos produced by cosmic rays. Cosmic rays are highly energetic particles that come from space. As they slam into the atmosphere, and collide with molecules in the air, they produce a shower of particles. These include muons and muon-neutrinos. Super-Kamiokande could detect muon-neutrinos from cosmic rays. Cosmic rays come from all directions, so Super-Kamiokande should see muon-neutrinos from all directions too. Just like we can see the solar neutrinos through the Earth, we should see muon-neutrinos both from above and below. However, more were detected from above than below.

Something must happen to muon-neutrinos during their journey through the Earth. Super-Kamiokande could detect them as electron-neutrinos or muon-neutrinos, but is not sensitive to tau-neutrinos. This is evidence that muon-neutrinos were changing flavour to tau-neutrinos.

Sudbury Neutrino Observatory

The Sudbury Neutrino Observatory detector, a 12-metre sphere containing 1000 tonnes of heavy water which is two kilometres underground. Credit: SNOLAB.

The solar neutrino problem was finally solved in 2001 through measurements of the Sudbury Neutrino Observatory (SNO). SNO is another Cherenkov detector like (Super-)Kamiokande, but it used heavy water instead of regular water. (High-purity heavy water is extremely expensive, it would have cost hundreds of millions of dollars for SNO to buy the 1000 tonnes it used, so it managed to secure it on loan from Atomic Energy of Canada Limited). Using heavy water meant that SNO was sensitive to all flavours of neutrinos. Like previous experiments, SNO found that there were not as many electron-neutrinos from the Sun as expected. However, there were also muon-neutrinos and tau-neutrinos, and when these were added, the total worked out!

The solar astrophysicists had been right all along, what was missing was that neutrinos oscillate between flavours. Studying the Sun had led to a discovery about some of the smallest particles in Nature.

Neutrino oscillations

Experiments have shown that neutrino oscillations occur, but how does this work? We need to delve into quantum mechanics.

The theory of neutrino oscillations say that each of the neutrino flavours corresponds to a different combination of neutrino mass states. This is weird, it means that if you were to somehow weight an electron-, muon- or tau-neutrino, you would get one of three values, but which one is random (although on average, each flavour would have a particular mass). By rearranging the mass states into a different combination you can get a different neutrino flavour. While neutrinos are created as a particular flavour, when they travel, the mass states rearrange relative to each other, so when they arrive at their destination, they could have changed flavour (or even changed flavour and then changed back again).

To get a more detailed idea of what’s going on, we’ll imagine the simpler case of there being only two neutrino flavours (and two neutrino mass states). We can picture a neutrino as a clock face with an hour hand and a minute hand. These represent the two mass states. Which neutrino flavour we have depends upon their relative positions. If they point in the same direction, we have one flavour (let’s say mint) and if they point in opposite directions, we have the other (orange). We’ll create a mint neutrino at 12 noon and watch it evolve. The hands more at different speeds, so at ~12:30 pm, they are pointing opposite ways, and our neutrino has oscillated into an orange neutrino. At ~1:05 pm, the hands are aligned again, and we’re back to mint. Which neutrino you have depends when you look. At 3:30 am, you’ll have a roughly even chance of finding either flavour and at 6:01 pm, you’ll be almost certain to have orange neutrino, but there’s still a tiny chance of finding an mint one. As time goes on, the neutrino oscillates back and forth.

With three neutrinos flavours, things are more complicated, but the idea is similar. You can imagine throwing in a second hand and making different flavours based upon the relative positions of all three hands.

We can now explain why Super-Kamiokande saw different numbers of muon-neutrinos from different sides of the Earth. Those coming from above only travel a short distance, there’s little time between when they were created and when they are detected, so there’s not much chance they’ll change flavour. Those coming through the Earth have had enough time to switch flavour.

A similar thing happens as neutrinos travel from the core of the Sun out to the surface. (There’s some interesting extra physics that happens here too. A side effect of there being so much matter at the centre of the Sun, the combination of mass states that makes up the different flavours is different than at the outside. This means that even without the hands on the clock going round, we can get a change in flavour).

Neutrino oscillations happen because neutrino mass states are not the same as the flavour states. This requires that neutrinos have mass. In the Standard Model, neutrinos are massless, so the Standard Model had to be extended.

2015 Physics Nobel laureates

2015 Physics Nobel laureates, Takaaki Kajita and Arthur B. McDonald. Credit: Nobel Foundation.

Happy ending

For confirming that neutrinos have mass, Takaaki Kajita of Super-Kamiokande and Arthur McDonald of SNO won this year’s Nobel Prize. It is amazing how much physics has been discovered from trying to answer as simple a question as how does the Sun shine?

Even though neutrinos are shy, they are really interesting characters when you get to know them.

Now that the mystery of the missing neutrinos is solved, what is next? Takaaki Kajita is now involved in another project in the Kamioka Mine, the construction of KAGRA, a gravitational-wave detector.

KAGRA control room

The control room of KAGRA, the gravitational-wave detector in the Hida Mountains, Japan. I visited June 2015. Could a third Nobel come out of the Kamioka Mine?

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Tips for scientific writing

Second year physics undergraduates at the University of Birmingham have to write an essay as part of their course. As a tutor, it is my job to give them advice on how to write in a scientific style (and then mark the results). I have assembled these tips to try to aid them (and make my marking less painful).

Writing well is difficult. It requires practice. It is an important skill, yet it is something that I do not believe is frequently formally taught (at least in the sciences). However, there are many resources online, and many universities offer support for learning key skills, like the University of Birmingham Library. Scientific and other technical writing can be especially hard, as it has its own rules that can be at odds with what we learn at school (when studying literature or creative writing). Reading the work of others is a good way for figuring out what works well and what does not.

In this post, I include some tips that I hope are useful (not everyone will agree). I begin by considering how to plan and structure a piece of writing (section 1), from the largest scale (section 1.2) progressing down to the smallest (section 1.4); then I discuss various aspects of technical writing (section 2), both in terms of content and style, including referencing (section 2.5), which is often problematic, and I conclude with some general editing advice (section 3) before summarising (section 4). If you have anything extra to add, please do so in the comments.

1 Structure and planning

The structure of your writing is important as it reflects the logical flow of your arguments. It is worth spending some time before you start writing considering what you want to say, and what is the best order for your ideas. (This is also true in exams: I have found when trying to answer essay questions it is worth the time to spend a couple of minutes planning, otherwise I am liable to miss out an important point). I frequently get frustrated that I must write linearly, one idea after another, and cannot introduce multiple strands at a time, with arguments intertwining with each other. However, putting in the effort to construct a clear progression does help your reader.

1.1 Title and audience

The first thing to consider is what you want to write and who is going to read it. Always write for your audience, and remember that professional scientists and the general public look for different things (this blog may be a poor example of this, as different posts are targeted towards different audiences).

Having thought about what you want to say, pick a title that reflects this. Don’t have a title “The life and works of Albert Einstein” if you are only going to cover special relativity, and don’t have a title “Equilibrium thermodynamics of non-oxide perovskite superconductors” if you are writing for a general audience. If your title is a question, make sure you answer it. It might be a good idea to write your title after you have finished your main text so that you can match it to what you have actually written.

1.2 Beginning, middle and end

To help your audience understand what you are telling them, begin with an introduction, and end with a summary. This is also true when giving a talk. Start by explaining what you will tell them, then tell them, then tell them what you told them. Repetition of key ideas makes them more memorable and help to emphasise what your audience should take away.

At the beginning, introduce the key ideas you will talk about. If you are writing an essay titled “The Solar Neutrino Problem“, you should explain what a solar neutrino is and why there is a problem. You might also like to explain why the reader should care. Sketching out the contents of the rest of the work is useful as it prepares the reader for what will follow: it’s like warm-up stretches for the mind. The introduction sets the scene for the arguments to follow.

The main body of your text contains most of the information, this is where you introduce your ideas and explain them. It is the burger between the buns of the introduction and conclusion. For longer documents, or subjects with many aspects, you might consider breaking this up into sections (and subsections). Using headings (perhaps numbered for reference) is good: skimming section headings should give an outline of the contents. Some sections within the main body might be sufficiently involved to merit their own introduction and summary. There should be a clear progression of ideas: if you find there is a big jump, try writing some text to cover the transition (“Having explained how neutrinos are produced in the Sun, we now consider how they are detected on the Earth”).

After presenting your arguments, it is good to summarise. As an example, a summary on the solar neutrino problem could be:

“Experiments measuring neutrinos from the Sun only detected about a third as many as expected. This could indicate either a problem with our understanding of solar physics or of particle physics. It is not possible to modify solar models to match both the measured neutrino flux and observations of luminosity and composition; however, the reduced flux could be explained by introducing neutrino oscillations. These were subsequently observed in several experiments. The solar neutrino problem has therefore been resolved by introducing new particle physics.”

Don’t introduce new arguments at this stage, this is just as unsatisfying as reading a murder mystery and discovering the murderer was someone never mentioned before. In my solar neutrino example, both the solar models and neutrino oscillations should have been discussed. Distilling your argument down to few lines also helps you to double-check your logic.

Either as part of your summary, of following on from it, end your writing with a conclusion. This is what you want your audience to have learnt (it should be the answer to your question). It is OK if you cannot produce a concrete answer, there are many cases where there is no clear-cut solution, perhaps more data is needed: in these cases, your conclusion is that there is no simple answer. To check that you have successfully wrapped things up, try reading just your introduction and conclusion; these should pair up to form a delicious (but bite-sized) sandwich.

1.3 Paragraphs

On a smaller scale, your writing is organised using paragraphing. Paragraphs are the building blocks of your arguments; each paragraph should address a single point or idea. Big blocks of text are hard to read (and look intimidating), so it is good to break them up. You can think of each paragraph as a micro-essay: the first sentence (usually) introduces the subject, you then go on to elaborate, before reaching a conclusion at the end (see section 1.2). To check that your paragraph sticks to a single point (and doesn’t need to be broken up), try reading the first and last sentences, usually they should make sense together.

1.4 Sentences

Paragraphs are constructed from sentences. Ensure your sentences make sense, that they are grammatically correct and that their subject is clear.

Vary your sentence length. In technical writing there is often the temptation, even amongst the best writers, to include long, convoluted sentences in order to fully describe a complicated idea and include all the relevant details, but these can be hard to read, both because of the complexity of their structure, which may require significant mental effort to unpack, and because by the time they finally conclude, the reader has forgotten the initial topic of the over-long, rambling sentence. Brevity gives impact. Shorter sentences are easier to understand. Breaking up your ideas helps the reader. Short sentences also get boring. They seem repetitive. They are tiring to read. They can send your reader to sleep. It is, therefore, better to have a range of sentence lengths. Include some short. In addition to these, have some longer sentences, as these allow you to join up your ideas. If you are unsure where to break up long sentences, look for commas (or semi-colons, etc.); if you are unsure where to put commas, read the sentence and see where you would pause.

2 Writing style and referencing

Having discussed how to structure your writing, we now move on to what to write. Technical writing has some specific requirements with regards to content, these might seem peculiar when first encountered. I’ll try to explain why we do certain things in technical writing, and give some ideas on how to incorporate these ideas to improve your own writing.

2.1 Be specific

The most common mistake I come across in my students’ work is the failure to be specific. The following two points (sections 2.2 and 2.3) are closely related to this. As an example, consider making a comparison:

  • Poor — “Nuclear power provides more energy than fossil fuel.”
  • Better — “Per unit mass of fuel, nuclear fission releases more energy than the burning of fossil fuel.”
  • Even better — “Nuclear fission can produce ~8000 times as much energy per unit mass of fuel as burning fossil fuels: the same amount of energy is produced from 16 kg of fossil fuels as by using 2 g of uranium in a standard reactor (MacKay, 2008).”

Here, we have specified exactly what we are comparing, given figures to allow a quantitative comparison, and provided references for those figures (see section 2.5). If possible, give numbers; don’t say “many ” or “lots” or “some”, but say “70%”, “9 billion” or “six Olympic swimming pools”.

Weak modifiers like “very”, “quite”, “somewhat” or “highly” are another example where it is better to be specific. What is the difference between being “hot” and being “very hot”? I might say that my bowl of soup is very hot, but does that tell you any more than if I just said it was hot? It is tempting to use these words for emphasis, surely if I were talking about the surface of the Sun we can agree that’s very hot? Not if you were to compare it to the centre of the Sun! Often, what is hot or cold, big or small, fast or slow depends upon the context. What is hot for soup is cold for the Sun, and what is cold for soup is hot for superconductors. It is much better to make distinctions by using figures: “The surface of the Sun is about 6000 K”.

It is OK to use “very” if you define the range where this is applicable, for example “High frequency radio waves are between 3 MHz and 30 MHz, very high frequency radio waves are between 30 MHz and 300 MHz ultra high frequency radio waves are between 300 MHz and 3 GHz.”

2.2 Provide justification

When putting forward an argument, it is necessary to include some evidence or justification to back it up. It is not sufficient merely to assert your opinion because you need the reader to follow your reasoning. If you are using someone else’s argument, you should provide a citation (section 2.5); the reader can then check there to find the reasoning. However, if it is an important point you might like to add some exposition. If you are being good about providing quantitative statements (section 2.1), you are already part way there as you can use those figures a back-up. For example, if discussing global warming, it is easy to argue it is important if you have already included figures on how many people would lose their homes to rising sea-levels, or if comparing materials, it is straightforward to argue that aluminium is better for making aeroplanes than steel if you have already included their densities. Sometimes, all that is required is an explanation of your reasoning, for example, “It is a good idea to build nuclear power plants because this reduces reliance on fossil fuels” or “It is not advisable to lick the surface of the Sun because it doesn’t taste of golden syrup.” Here, the reader might disagree that it is a good idea to build nuclear power plants, but they understand that you are using dependence on fossil fuels as an argument instead of, say, environmental issues, or the reader might agree that it is a bad idea to lick the Sun, but might have been thinking more about its temperature than its flavour. Even if the reader does not agree with your conclusions, they should understand how you reached them.

A similar idea is to show rather than tell. Don’t tell me that something is a fascinating topic or an exciting concept, get on with explaining it! Similarly, don’t just say something is important, but explain why it is important. This allows the reader to decide upon things themselves, if you have justified your arguments then they should follow your logic.

2.3 Use the correct word

In technical writing there is often a specific word that should be used in a particular context. In common usage we might use weight and mass interchangeably, in physics they have different meanings. This sometimes trips people up as they naturally try to find synonyms to reduce the monotony of their work. Always use the correct term.

Technical language can be full of jargon. This makes things difficult to understand for an outsider. It is important to define unfamiliar terms to help the reader. In particular, acronyms must be defined the first time they are used. As an example, “When talking about online materials, the uniform resource locator (URL), otherwise known as the web address, is a string of characters that identifies a resource.” Avoid jargon as much as possible; try to always use the simplest word for the job. It will be necessary to use technical terms to describe things accurately, but if they are introduced carefully, these need not confuse the reader.

A particular pet-peeve of mine is the use of scare quotes, which I always read as if the author is making air quotes. If quoting someone else’s choice of phrase then quotation marks are appropriate, and a reference must be provided (section 2.5). Most of the time, these quotation marks are used to indicate that the author thinks the terminology isn’t quite right. If the terminology is incorrect, use a different word (the correct one); if the terminology is correct (if that is what is used in the field), then the quotation marks aren’t needed!

2.4 Use equations and diagrams

Most physics problems involve solving an equation or two. For these mathematical questions, I am always encouraging my students to explain their work, to use words. When writing essays, I find they have the opposite problem: they only use prose and don’t include equations (or diagrams). Equations are useful for concisely and precisely explaining relationships, it is good to include them in writing.

Equations may put off general readers, but they improve the readability of technical work. Consider describing the kinetic energy of a (non-relativistic) particle:

  • With only words — “The kinetic energy of a particle depends upon its mass and speed: it is directly proportional to the mass and increases with the square of the speed.”
  • Using an equation — “The kinetic energy of a particle E is given by E = (1/2)mv^2, where m is its mass and v is its velocity.”

The second method is more straightforward, there is no ambiguity in our description, and we also get the factor of a half so the reader can go away at calculate things for themselves. This was just a simple equation; if we were considering something more complicated, such as the kinetic energy of a relativistic particle

\displaystyle E = \left(\frac{1}{\sqrt{1- v^2/c^2}} - 1\right)mc^2,

where c is the speed of light, it is much harder to produce a comprehensive description using only words. In this case, it is tempting to miss out reference to the equation. Sometimes this is justified: if the equation is too complicated a reader will not understand its meaning, but, in many cases, an equation allows you to show exactly how a system changes, and this is extremely valuable.

When including an equation, always define the symbols that you are using. Some common constants, such as \pi, might be understood, but it is better safe than sorry.

Equations should be correctly punctuated. They are read as part of the surrounding text, with the equals sign read as the verb “equals”, etc.

Using diagrams is another way of providing information in a clear, concise format. Like equations, diagrams can replace long and potentially confusing sections of text. Diagrams can be pictures of experimental set-up, schematics of the system under discussion, or show more abstract information, such as illustrating processes (perhaps as a flow chart). The cliché is that a picture is worth a thousand words; as diagrams are so awesome for conveying information, I’m not even going to attempt to give an example where I try to use only words. Below is as example figure, which I have chosen as it also includes equations.

“Figure 1 shows the proton–proton (pp) chain, the series of thermonuclear reactions that provides most (~99%) of Sun’s energy (Bahcall, Serenelli & Basu, 2005). There are several neutrino-producing reactions.”

The pp chain

Figure 1: The thermonuclear reactions of the pp chain. The traditional names of the produced neutrinos are given in bold and the branch names are given in parentheses. Percentages indicate branching fractions. Adapted from Giunti & Kin (2007).

Graphs can be used to show relationships between quantities, or collections of data. They can be used for theoretical models or experimental results. In the example below I show both. Graphs might be useful for plotting especially complicated functions, where the equation isn’t easy to understand. There are many types of graph (scatter plots, histograms, pie charts), and picking the best way to show your data can be as challenging as obtaining it in the first place!

“In figure 2 we plot the orbital decay of the Hulse–Taylor binary pulsar, indicated by the shift in periastron time (the point in the orbit where the stars are closest together). The data are in excellent agreement with the prediction assuming that the orbit evolves because of the emission of gravitational waves.”

Periastron shift of binary pulsar

Figure 2: The cumulative shift of periastron time as a function of time of the Hulse–Taylor binary pulsar (PSR B1913+16). The points are measured values, while the curve is the theoretical prediction assuming gravitational-wave emission. Taken from Weisberg & Taylor (2005).

All diagrams should have a descriptive caption. It is usually good to number these for ease of reference. If you are using someone else’s figure, make sure to cite them in the caption (see section 2.5).

Tables can also be used to present data. Tables might be better than plots for when there are only a few numbers to present. Like figures, tables should have a caption (which includes relevant references if the data is taken from another source), they should be numbered, and they should be referred to explicitly in the text.

When writing, it is useful to remember that different people learn better through different means: some prefer words, some love equations, and other like visual representations. Including equations and figures can help you communicate effectively with a wider audience.

There are conventions for how to present equations, graphs and tables. I shall return to this in future posts. The rules may seem arcane, but they are designed to make communication clear.

2.5 Referencing

At the end of any good piece of technical writing there should be a list of references, hence I have tackled referencing last in the section. (Sometimes this is done in footnotes rather than the end, but I’m ignoring that). However, referencing should not be considered something that is just done at the end, or something that is tacked on at the end as an after-thought; it is one of the most important components of academic writing.

We include references for several reasons:

  1. To show the source of facts, figures and ideas. This allows readers to verify things that we quote, to double-check we’ve not made an error or misinterpreted things. It also shows distinguishes what is our own from what we have taken from elsewhere. This is important in avoiding plagiarism, as we acknowledge when we use someone else’s work.
  2. To provide the reader with a further source of information. It is not possible to explain everything, and a reader might be interested in finding out more about a topic, how a particular quantity was measured or how a particular calculation was done. By providing a reference we give the reader something further they can read if they want to (that doesn’t mean our work shouldn’t make sense on it’s own: you should be able to watch The Avengers without having seen Iron Man, but it’s still useful to know what to watch to find out the back-story). By following references readers can see how ideas have developed and changed, and gain a fuller understanding of a topic.
  3. To give credit for useful work. This is linked to the idea of not claiming the ideas as your own (avoiding plagiarism), but in addition to that, by referencing something you are publicising it, by using it you are claiming that it is of good-enough quality to be trusted. If you are to look at an academic article you will often see a link to citing articles. The number of citations is used as a crude measure of the value of that paper. Furthermore, this linking can allow a reader to work forwards, finding new ideas built upon those in that paper, just as they can work backwards by following references.
  4. To show you know your stuff. This might sound rather cynical, but it is important to do your research. To understand a topic you need to know what work has been done in that area (you can’t always derive everything from first principles yourself), and you demonstrate your familiarity with a field by include references.

You must always include citations in the text at the relevant point: if you use an idea include your source, if you introduce a concept say where it came from. It is not acceptable just to have a list of references at the end: does the reader have to go through all of these to figure out what came from where?

There are multiple styles for putting citations in text. The two most common are the following:

  • Numeric (or Vancouver) — using a number, e.g., [1], where the references at the end form an ordered list. This has the advantage of not taking up much space, especially when including citations to multiple papers, e.g. [1–5].
  • Author–year (or Havard) — using the authors and year of publication to identify the paper, e.g., (Einstein, 1905). This has the advantage of making it easier to identify a paper: I’ve no idea what [13] is until I flick to the end, but I know what (Hulse & Taylor, 1975) is about.

Which style you use might be specified for you or it might be a free choice. Whichever style you use, the important thing is to include relevant references at the appropriate place in the text.

Having figured out why we should reference, where we should put references and how to include citations in the text, the last piece is how to assemble the bibliographic information to include at the end (or in footnotes). Exactly what information is included and how it is formatted depends on the particular style: there are endless combinations. Again, this might be specified for you or might be a free choice, just make sure you are consistent. Basic information that is always included are an author (this may be an organisation rather than a person), so we know who to attribute the work to, and a date so we know how up-to-date it is. Other information that is included depends upon the source we are referencing: a journal article will need the name of the journal, the volume and page number; a book will need a title, edition and publisher; a website will need a title and URL, etc. We need to include all the necessary information for the reader to find the exact source we used (hence we need to include the edition of a book, the date updated or written for a website, and so on).

There are numerous guides online for how to format references correctly. Some software does it automatically (I use Mendeley to produce BibTeX, but that’s not for everyone). The University of Birmingham has a guide to using Havard-style referencing that is comprehensive.

A final issue remains of which sources to reference: how do you know that a source is reliable? This is an in-depth question, so I shall return to it is a dedicated post.

3 Editing

Writing isn’t finished as soon as you have all your ideas on the page, things often take some polishing up. Some people like to perfect things as they go along, others prefer to get everything down in whatever form and go back through after. Here, I conclude with some tips for editing.

3.1 Be merciless

Keep your writing short. Don’t waste your readers’ time or overcomplicate things. Cut unnecessary words.

There are some phrases that are typically superfluous:

  • “Obviously…” — If it is obvious, then the reader will realise it; if it’s not, you are patronising them.
  • “It should be noted that…” — That would be why it’s written down! (I hope you are not writing things that shouldn’t be noted).
  • “Remember that…” — You’re reminding the reader by writing it.
  • Any of the modifiers like “very”, “quite” or “extremely” mentioned in the section 2.3.

3.2 Proof-read

The single best method to improve a piece of writing is to proof-read it. Reread what you have written to check that it says what you think it should. I find I have to wait for a while after writing something to read it properly, otherwise I read what I intended to write rather than what I actually did. Having others read it is an excellent way to check it makes sense (especially if you are not a native English speaker); this is best if they are representative of your target audience.

I hate it when others find a mistake in my writing. It’s like rubbing a cat the wrong way. However, each mistake you find and correct makes your writing a little better, and that’s really the important thing.

4 Summary

In conclusion, my main tips for good scientific writing are:

  • Plan what you want to tell your audience and how they will take your message away.
  • Say what you’re going to say (introduction), then say it (main text), then say what you said (conclusion).
  • Have a clear, logical flow, with one point per paragraph.
  • Be specific and back up with your points with quantitative data and references.
  • Use equations and diagrams to help explain.
  • Be concise.
  • Proof-read (and get a second opinion).

If you have any further ideas for improving essay writing, please leave a comment.