Category: Physics

The Colour of Music : Notes of Equal Temperament

The Colour of Music : Notes of Equal Temperament

Reading Time: 11 minutes

Equal temperament and just intonation are two of the methods used to divide an octave in western classical music. While just intonation is based on ratios of simple integers, the equal temperament system uses logarithmic intervals. In part II of this series, the Nerd Druid talks about equal temperament.

Music, sound, and frequency

Music is basically a set of well-ordered nice sounds. Sound is pressure waves in a medium–air, water, metal, the Earth. Not all pressure waves cause sound that we humans can hear. Earthquakes give rise to sounds of very long wavelength, too low for us to hear. Bats give off high frequency sounds, too sharp for us to hear. Our hearing range is $20Hz – 20kHz$; that is, we can[1] detect those pressure waves that oscillate at a rate of $20$ to $20000$ cycles per second.

Diagram showing hearing ranges of differnt animals. Humans are in orange, between 20Hz-20kHz. Elephants and moles (red) have a far lower range, while cats and dogs (green) have a wider hearing range (upto 40 kHz) than humans. Bats and dolphins are champions sound detectors, able to hear up to 160 kHz. Image Credit : Unknown. Image Source : Cochlea-dot-org.
Diagram showing hearing ranges of different animals. Humans are in orange, between 20Hz-20kHz. Elephants and moles (red) have a far lower range, while cats and dogs (green) have a wider hearing range (upto 40 kHz) than humans. Bats and dolphins are champions sound detectors, able to hear up to 160 kHz. Image Credit : Unknown. Image Source : Cochlea-dot-org.

Some animals are far better than us at this game. Cats and dogs can hear upto $40 kHz$, while bats and dolphins can hear sounds as sharp as $160 kHz$.

Notes and tunes

Since sound is music, it too is based on frequencies. Play a certain frequency and you get the fundamental musical species, a note. Play a series of frequencies, a set of notes, and you get a tune. Whether or not that is a nice tune depends very much on the notes you have chosen. Notes that have certain well-defined frequency relations among each other tend to produce nicer tunes.

In most well-developed music systems, such as western classical or Hindustani and Carnatic classical, the notes are labelled for ease of manipulation. Given a set of labelled notes, the problem is to assign frequencies to them so that they sound nice. In Part I of this series, The Colour of Music : Understanding Just Intonation, we discussed just intonation, a method of doing exactly this.

Image of keys in a piano. Remember that usually, pianos are tuned in equal temperament, not just intonation. Image Source : playpiano-dot-com.
Keys in a piano. Remember that usually, pianos are tuned in equal temperament, not just intonation. Image Source : playpiano-dot-com.

Just intonation : A quick recap

In just intonation, the frequencies you assign to various notes are in simple numeric ratios to the first note, the unison. Consider the standard western C-major scale, comprising notes

$$C-D-E-F-G-A-B-C^{\ast} \tag{1}$$

where $C$ (frequency $f$, say) is the unison and $C^{\ast}$ is an octave above it, with frequency $2f$. $G$ is a perfect fifth above $C$, and has a frequency of $\frac{3}{2} f$. $E$ is a major third above $C$, and has a frequency of $\frac{5}{4} f$. Using these three ratios $\displaystyle {}^2\!/\!_1, {}^3\!/\!_2, {}^5\!/\!_4$ you can generate every other note in the C-major scale in five-limit tuning. I like to call these three ratios the three generators of the just intonation series. You can either ascend by multiplying by the ratio, or descend by dividing with it, which is the same as multiplying with the inverse ratio. Thus, for instance, if you wish to go to $D$ from $C$, you need to ascend by two fifths and descend by an octave ($\uparrow$ fifth $\uparrow$ fifth $\downarrow$ octave), meaning you’d have to multiply by ${}^3\!/\!_2 \times {}^3\!/\!_2 \times {}^1\!/\!_2$.

The table below is a slightly modified form of the one from the previous article, and will hopefully explains what’s going on.

Note Up/Down Frequency
$C$ (unison) = $f$
$D$ (major second) $\uparrow$ fifth $\uparrow$ fifth $\downarrow$ octave $\frac{9}{8}f$
$E$ (major third) $\uparrow$ third $\frac{5}{4}f$
$F$ (perfect fourth) $\uparrow$ octave $\downarrow$ fifth $\frac{4}{3}f$
$G$ (perfect fifth) $\uparrow$ fifth $\frac{3}{2}f$
$A$ (major sixth) $\uparrow$ octave $\uparrow$ third $\downarrow$ fifth $\frac{5}{3}f$
$B$ (major seventh) $\uparrow$ fifth $\uparrow$ third $\frac{15}{8}f$
$C^{\ast}$ (octave) $\uparrow$ octave $2 f$

For further explanation, read Part I.

Pure intervals

Intervals that involve the ratios of simple integers are called pure or just intervals, since they correspond to sounds created by vibrations in physical objects.

Just intonation is a good way to assign frequencies, and since the intervals are pure, the sound produced by justly tuned scales tend to be quite pleasing to the ear. However, the ratios in the rightmost column are given with respect to the unison, and thus, if we want to move from a non-unison note to another such note, we’d be in a bit of a pickle.

How much of a pickle? Here’s a problem for you, see if you can solve it. I am at $A$, major sixth, ${}^5\!/\!_3 f$. I want to go to $D$, major second, ${}^9\!/\!_8 f$. What is the simplest way to get there using the three generators? Answer in the footnote[2]. Don’t peek till you’ve tried it yourself.

As you’ll find, simply going from $A$ to $D$ in just intonation involves quite a ride! Enough of it, then. Time for equal temperament.

Equal Temperament

Unlike just intonation, which has three generators, equal temperament has a single generator, and is thus mathematically simpler.

Well, sort of.

Black keys

So far I have discussed breaking up the octave into seven portions, seven notes in the C-major scale with frequencies assigned so that they sound nice. This corresponds to the seven white keys in a piano, as seen in the image above. Clearly, though, there is something incomplete about this picture. Where are the black keys? What frequencies do they represent? What are they called? Which scales include them?

Keys in a piano. The five black keys are in between the seven white keys. Image Source : playpiano-dot-com.
Keys in a piano. The five black keys are in between the seven white keys. Image Source : playpiano-dot-com.

Take a look at the table above. In the first column, I’ve written down the symbols designating each note. The names of the notes are in brackets. The adjective major appears quite a lot. In the C-major scale, where $C$ is the tonic or the root note, the white keys of the piano correspond to these major notes. The black keys sit in between the white keys, suggesting that their frequencies should be in-between those of the white keys.

Sharps $\sharp$ and flats $\flat$

Counting from $C$ on the left, you have 7 white keys, corresponding to $D-E-F-G-A-B$, and 5 black keys in between them, making up a total of 12 notes in an octave. The black notes are

$$C♯ – D♯ – F♯ – G♯ – A♯ \tag{2a}$$

where the note $C♯$ is pronounced C-sharp and corresponds to a frequency slightly higher than $C$. However, as you’ve no doubt noticed, the same note might well have been slightly lower than D, slightly flatter, if you will, and can thus be called $D♭$, D-flat. Thus the series might as well be called

$$D♭ – E♭ – G♭ – A♭ – B♭ \tag{2b}$$

or, if you are fastidious and have no eye for aesthetics, like this horribleness

$$C\sharp/D\flat – D\sharp/E\flat – F\sharp/G\flat – G\sharp/A\flat – A\sharp/B\flat \tag{2c}$$

The notes that can be both a $\sharp$ and a $\flat$ are called enharmonic equivalents.

Appending a $♯$ (sharp) to a note elevates the pitch (increases the frequency), while appending a $♭$ lowers the pitch (decreases the frequency).

But by how much?

Well, by a semitone.

Semitone

A semitone (s) is the unit interval, the smallest frequency increment. In a piano, the interval between successive keys is a semitone. For instance, the interval between the white $C$ key and the black $C♯/D♭$ key, or that between the black $F\sharp/G\flat$ and the white $G$ key, are both semitones. Thus, if you want to get to the fifth, $G$, from the root, $C$, you would have to cross 7 semitones. Similarly, to get to the third, $E$, you’d need to cross 4 semitones.

Image of keys in a piano, including the sharps and the flats. In order to go from A to D, you would need to take the following simple path : A - G♯ - G - F♯ - F - E - D♯ - D, a total of 7 steps. Image Source : johnpratt-dot-com.
Keys in a piano, including the sharps and the flats. In order to go from A to D, you would need to take the following simple path : A – G♯ – G – F♯ – F – E – D♯ – D, a total of 7 steps. Image Source : johnpratt-dot-com.

This massively simplifies our five-limit tuning table. Instead of dealing with three generators and going up and down, as you found out earlier, going from $A$ to $D$ in equal temperament simply involves knowing how many semitones separate the two notes. You don’t even need to involve the unison, you can simply count off 7 semitones to the left of $A$ and reach $D$. Voila!

Chopping up the octave

Time to get mathematical. Equal temperament tuning is based on logarithms, with the semitone as the unit. Clearly, if an octave has 12 semitone intervals, then we need to divide the range $f$ to $2f$ into twelve portions. A simple arithmetic method is to chop the interval $f$ up equally and assign $s = {}^f\!/\!_{12}$. If $f = 100 Hz$, then $s = 8.333…Hz$, and you are liable to be beaten up by people around you if use those notes.

Ok, not as simple as that, then. Remember that assigning frequencies to notes depends on whether those notes sound nice, not whether they are mathematically nice. Of course, in the case of just tuning, the ratios are mathematically nice. Clearly, we should attempt a method where the notes sound somewhat close to those in just intonation, but are also mathematically nicer.

The perfect(?) fifth

A good reference is the fifth, $G$. It is seven semitones away from the root, and its frequency ratio is $3:2$ with respect to unison. Therefore, while $G$‘s frequency is halfway between unison and the octave, $1.5 f$, there are 7 semitones on one side and 5 on the other. Clearly, then, we need a division method which gives smaller intervals early on and larger ones later on. Logarithms fit the bill perfectly.

Back to the semitone

I did get to logarithms in a roundabout way. Another simpler way to invoke logarithms is to realise that frequencies of the notes multiply , not add. Also, in equal temperament tuning, all semitones are equal. This immediately suggests that the best method of chopping up an octave into 12 intervals is to use, as the unit, a semitone interval of

$$s = \sqrt[ 12 ]{ 2 } = 2^{{}^1\!/\!_{12}} \tag{3}$$

Therefore, any note that is, say, seven notes away from another note, has a frequency that is higher by a factor of $7s = \sqrt[ 12 ]{ 2^7 } = 2^{{}^7\!/\!_{12}}$. This is the factor by which the frequency of a fifth $G$ in equal temperament is higher than the pitch[3] of the unison $C$. Similarly, the major third, $E$, is higher than the unison by a factor of $5s = \sqrt[ 12 ]{ 2^5 } = 2^{{}^5\!/\!_{12}}$, whereas the major sixth is higher by a factor of $10s = \sqrt[ 12 ]{ 2^{10} } = 2^{{}^5\!/\!_{6}}$.

Any note as root

Although I have used the unison as the root here, the same method can be used taking any key as root. For instance, I can proclaim that $A$ is root, and then descending to $D$ would simply involve lowering the pitch by a factor of $7s = \sqrt[ 12 ]{ 2^7 } = 2^{{}^7\!/\!_{12}}$. This also means that if I invert the system, and take $D$ as root, then $A$ is the fifth, a factor of $2^{{}^7\!/\!_{12}}$ higher.

Well, that’s all fine and dandy, but how does this fare with respect to that supreme condition of music, nicety? Do these logarithmic semitone intervals produce nice music? Is there any way to find out without tuning a piano and playing the notes?

Is Equal Temperament nice?

Well, yes, compare it to the just intervals. If the notes of equal temperament are close enough to just, then our job is done.

For that, first, how much is the twelfth root of two, exactly? How does one find its value? And why do I keep bandying the word logarithm about, when we haven’t really calculated the log of anything yet?

Well, it’s time for that. Problem #2 : calculate the value of $2^{{}^1\!/\!_{12}}$ using logarithms. The steps are in the footnote[4].

Have you tried it yourself? If not, do try, it’s fun.

Since we’re dealing with logarithms and exponentials here, the value of $s$ is not a rational number. However, for our purposes, we’ll take the value of $s = 1.059463$ to be sufficiently correct.

12-TET vs Just

Ok. Time to construct the twelve-tone equal temperament (12-TET) table. The frequency in equal temperament have been rounded off to the third place in decimal.

Name 12-TET Just
Unison (C) $2^{{}^0\!/\!_{12}} = 1.000$ $\frac{1}{1} = 1$
Minor Second (C♯ / D♭) $2^{{}^1\!/\!_{12}} = 1.059$ $\frac{ 16 }{ 15 } = 1.0666…$
Major Second (D) $2^{{}^2\!/\!_{12}} = 1.122$ $\frac{9}{8} = 1.125$
Minor Third (D♯ / E♭) $2^{{}^3\!/\!_{12}} = 1.189$ $\frac{6}{5} = 1.2$
Major Third (E) $2^{{}^4\!/\!_{12}} = 1.259$ $\frac{5}{4} = 1.25$
Perfect[5] Fourth (F) $2^{{}^5\!/\!_{12}} = 1.335$ $\frac{4}{3} = 1.333…$
Tritone (F♯ / G♭) $2^{{}^6\!/\!_{12}} = 1.414$ $\frac{7}{5} = 1.4$
Perfect[5:1] Fifth (G) $2^{{}^7\!/\!_{12}} = 1.498$ $\frac{3}{2} = 1.5$
Minor Sixth (G♯ / A♭) $2^{{}^8\!/\!_{12}} = 1.587$ $\frac{ 8 }{ 5 } = 1.6$
Major Sixth (A) $2^{{}^9\!/\!_{12}} = 1.682$ $\frac{5}{3} = 1.666…$
Minor Seventh (A♯ / B♭) $2^{{}^{10}\!/\!_{12}} = 1.782$ $\frac{ 16 }{ 9 } = 1.777…$
Major Seventh (B) $2^{{}^{11}\!/\!_{12}} = 1.888$ $\frac{15}{8} = 1.875$
Octave (C) $2^{{}^{12}\!/\!_{12}} = 2.000$ $\frac{2}{1} = 2$

The minor notes

This table also introduces additional five notes; the minor second, the minor third, the tritone, the minor sixth, and the minor seventh. Each of these notes is a semitone raised or lowered from one of the seven major notes. So, if you start playing at $C$ on a piano, these five notes correspond to the five black keys.

Equal temperament looks mathematically simple enough. All you need is to increment the power of the twelfth root of $2$ and you have a new note. However, as you can see, the frequencies of 12 tone equal temperament aren’t exactly equal to those of just intonation. This could potentially cause trouble, for as we know, the just intervals are pure, and intervals away from that might not sound nice.

Back to the perfect fifth

For instance, take a look at the fifth. In just, $G = 1.5f$, while in 12 tone equal temperament, $G = 1.498f$. That’s not equal, but that’s not massively far off too. However, it isn’t exactly $3:2$, and that is why I haven’t been calling it the perfect fifth anymore. However, the question still stands. Does an equal temperament $G$ sound more-or-less the same as the perfect fifth, or does it sound off?

Well, one way to verify this is to input the frequencies in some digital music software, have it play it back, and judge if the notes sound nice. However, clearly, not everyone is a good judge of good music. Besides, this is a very subjective (though perhaps more proper) way to analysing music.

The percentage difference

Instead, we science it out. In order for equal temperament to be nice enough, its frequencies needs to be within a certain percentage of the just frequencies. Agreed? All right, let’s take the normalised difference, or rather the percentage difference $\Delta$, given by

$$\Delta = 100 \times \frac{ E – J }{ J } \tag{4}$$

where $E$ and $J$ are the equal temperament and justly tuned frequencies, respectively.

I could list it out as another boring table, but diagrams, especially colourful diagrams, are far cooler. So, here’s a plot of the equal temperament and justly tuned frequencies, along with $\Delta$.

Plot of the frequencies of the twelve notes in an octave. The blue dots represent 12 tone equal temperament (12-TET) frequencies, while the red dots represent the justly tuned ones; they’re connected by straight line segments of the same colour for photogenic purposes. The yellow dots represent , the percentage difference between them. As is obvious, there is at most a 1% difference between the two frequencies, with the highest difference at F♯, Unison and octave are, of course, identical, while the fourth and the fifth are almost identical. Image Credit : <a href="http://nerdruid.com/">The Nerd Druid</a>.
Plot of the frequencies of the twelve notes in an octave. The blue dots represent 12 tone equal temperament (12-TET) frequencies, while the red dots represent the justly tuned ones; they’re connected by straight line segments of the same colour for photogenic purposes. The yellow dots represent , the percentage difference between them. As is obvious, there is at most a 1% difference between the two frequencies, with the highest difference at F♯, Unison and octave are, of course, identical, while the fourth and the fifth are almost identical. Image Credit : The Nerd Druid.

Blue dots (and line) are 12 tone equal temperament frequencies; red dots (and line) are justly tuned frequencies; yellow dots (and dashed line) are the percentage difference. $\Delta$ almost never goes beyond 1%. The highest difference is at $F\sharp$; the lowest are at $F$ and $G$. This means that, folks, equal temperament…or rather, 12-tone equal temperament should be quite good indeed.

And it is! You don’t really play it back to know that it is. Pianos are (almost always) tuned to 12-tone equal temperament. Just imagine a Chopin or a Mendelssohn or perhaps the Moonlight Sonata being played on an equal temperament piano.

Heaven!

The Reference Frequency

But how do you tune a piano? I mean, in the table above, we have normalised the frequencies; $C$ (unison) is $1$, while $C^{\ast}$ (octave) is 2. We need the frequencies in actual units, in hertz, to be able to actually tune a piano and have a good player play the Moonlight. Clearly, we need a reference frequency.

Well, more on that, and a bit of the history of equal temperament, in part III. In the meantime, here’s Captain A. F. Haddock, “playing the piano”.

Images of Captain (Retd.) Archibald Francis Haddock, "playing the piano". From Destination Moon (Objectif Lune). Image Credit : Hergé.
Captain (Retd.) Archibald Francis Haddock, “playing the piano”. From Destination Moon (Objectif Lune). Image Credit : Hergé.

 


Footnotes

  1. Well, most of us. As we grow older, the upper limit starts diminishing. People in their thirties can only detect sound of upto 17-18 kHz in frequency. ↩︎
  2. Beginning at unison $C$, I can get to $A$ using the following steps : $\uparrow$ octave $\downarrow$ fifth $\uparrow$ third. Writing this in the much more intuitive $+O-F+T$, we do the same for $D$, the major second: $-O+2F$. With a little bit of mental (or paper, if you wish), we see that we would need to add $-2O+3F-T$ to $A$ to get to $D$, since $(+O-F+T) + (-2O+3F-T)$ equals $(-O + 2F)$. This is the solution, then. Beginning at $A$, descend by two octaves, ascend by three fifths, and then descend by a third to get to $D$. Quite convoluted, as is obvious. Also, do note that we’d need to first know where the notes stand with respect to unison. There is no direct way to move from $A$ to $D$ in just, unlike in equal temperament, where you basically have to just…count. ↩︎
  3. Frequency is the technical, objective term, and refers the rate of oscillations of a wave. Pitch is a human subjective term, and is the sensation of frequency. For instance, one would generally use pitch in a comparative context, such as stating that the wail of a police siren is at a higher pitch than the notes of a cello. In contrast, frequency is mostly used in an absolute sense. ↩︎
  4. Let $s$ be the twelfth root of $2$, $s = 2^{{}^1\!/\!_{12}}$. Therefore, we can write $\log(s) = \frac{ \log(2) }{ 12 }$, which leads to
    $$s = e^{ \left( {}^{\log(2)}/_{12} \right) }$$

    which gives an approximate value of $s \approx 1.059463$. For our purposes we shall take this to be sufficiently equal. ↩︎

  5. Clearly, the values of the fourth and the fifth in 12 tone equal temperament are not exactly equal to $4:3$ and $3:2$, are they? Which is why you no longer call them perfect. ↩︎ ↩︎
The Colour of Music : Understanding Just Intonation

The Colour of Music : Understanding Just Intonation

Reading Time: 11 minutes

Just intonation and equal temperament are two of the methods used to divide an octave in western classical music. While just intonation is based on ratios of simple integers, the modern equal temperament system uses logarithmic intervals. In part I of this series, the Nerd Druid talks about just intonation.

Music, art or science?

What is your favourite piece of music?

Darned difficult question, I admit. In the modern world, where most urban and townspeople own a mobile phone, listening to music is pretty easy. If you have a smartphone, you can fill it up with all the music you want. If you have an ordinary mobile, you can still listen to radio[1]. And there are no dearth of musicians, and thus absolutely no lack of variety and choice when it comes to the type of music you’d want to hear. We live in a very musical world. Indeed, a most colourful world.

Photograph of a vintage bakelite radio. From the Bakelite Museum, Somerset, UK. Image Credit : Robneild.
A vintage bakelite radio. From the Bakelite Museum, Somerset, UK. Image Credit : Robneild.

Sound is a wave

But what is music? After all, it is but a series of sounds that are picked up by our ears and analysed by our brains. Pressure waves in air, like most sound, that can be transformed into electromagnetic waves for faster and longer transmission or for storage. And like all waves, sound waves have wavelengths and frequencies. A wavelength ($\lambda$) is the distance between two crests or two troughs, whereas the frequency ($\nu$) of a wave is the number of crests (or troughs) in a second. Multiply these and you have the velocity of the wave, $v = \lambda \nu$.

Diagram of sound as pressure waves in air. The regions of compression (labelled "condensation" in the image) are the "crests", those of rarefaction are the "troughs". The distance between two crests or two troughs is a wavelength.
Sound as pressure waves in air. The regions of compression (labelled “condensation” in the image) are the “crests”, those of rarefaction are the “troughs”. The distance between two crests or two troughs is a wavelength. Image Credit : Unknown
Diagram of sound frequencies. The shorter flute produces sound of a higher pitch, while the longer flute produces a low frequency sound. The same principle holds in a guitar; you hold a string to a fret to change the pitch. The closer to the guitar bridge the fret, shorter the effective length, higher the pitch. Image Credit : Unknown.
The shorter flute produces sound of a higher pitch, while the longer flute produces a low-frequency sound. The same principle holds in a guitar; you hold a string to a fret to change the pitch. The closer to the guitar bridge the fret, shorter the effective length, higher the pitch. Image Credit : Unknown.

Indian and western classical music

Consider Indian classical music. There are two primary branches : Hindustani classical, prevalent in the northern half, and Carnatic classical, popular in the southern half of the country. If you have had the fortune (or misfortune) of listening to a young music student practice his scales early in the morning[2], you’d probably have heard the following vocalisations repeated ad nauseam

Sa – Re – Ga – Ma – Pa – Dha – Ni – Sa*

If you’ve listened closely, you’ll know that each successive syllable here is at a higher pitch than the last; that is, it sounds more treble, sharper, while the ones previous sound more bass deeper.

This is true in Western music too. In the Italian genre, massively popularised by Julie Andrews in The Sound of Music, you have

Do – Re – Mi – Fa – Sol – La – Ti – Do*

while the German version, widely used nowadays, is

C – D – E – F – G – A – B – C*

You notice those asterisks attached to the notes at the end? We’ll come to that soon.

Photograph of Julie Andrews in the 1964 Hollywood movie The Sound of Music. The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.
Julie Andrews in the 1964 Hollywood movie The Sound of Music. The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.

Creating music

There are, of course, two ways for humans to produce music; either use one’s own body or use an external device. The first is usually called singing, while the second involves the use of musical instruments. Using either method effectively requires long periods of training. Untrained people, or those trained insufficiently, produce sound that is most unpleasant. A good example would be your neighbourhood musically enthusiastic but thoroughly ungifted early riser.

What does it mean to use either method effectively? Are there some criteria for good (or bad) music? Is it like modern art, entirely subjective, or can it be analysed, at least partially, in a scientific manner?

Well, of course it can! Most things in this world can be analysed scientifically. Even human stupidity can…but I digress.

Photograph of Einstein's Blackboard. A blackboard used by Albert Einstein in a 1931 lecture in Oxford. The last three lines give numerical values for the density (ρ), radius (P), and age of the universe. The blackboard is on permanent display in the Museum of the History of Science, Oxford. Photo Credit : decltype.
Einstein’s Blackboard. A blackboard used by Albert Einstein in a 1931 lecture in Oxford. The last three lines give numerical values for the density (ρ), radius (P), and age of the universe. The blackboard is on permanent display in the Museum of the History of Science, Oxford. Photo Credit : decltype.

Music and Frequency

It has, of course, to do with frequencies. When your voice skips from a Sa to a Re, or down from an A to a D, your vocal chords attempt to, with some help from your brain, find the correct frequency. That is, the frequency at which the lower note sounds right. If it is off, even by a bit, even non-practitioners realise that something is not quite how it should be.

If it sounds a tad subjective, it isn’t, really. Our brains are wired to respond favourably to music where the frequency relation between various notes are, in want of a better term, nice.

What is nice?

What is nice? Scores of people throughout history and geography have tried to find out. One of the first was Pythagoras, him of the right-angled triangle. The Greek polymath’s attempts culminated in the sixteenth century efforts of Zhu Zaiyu, a Chinese prince, who calculated the exact relation between the twelve notes in an octave that would make transposition simplest.

Illustration from 1913 showing Pythagoras teaching a class of women. Many prominent members of his school were women and some modern scholars think that he may have believed that women should be taught philosophy as well as men. Credit : Unknown.
Illustration from 1913 showing Pythagoras teaching a class of women. Many prominent members of his school were women and some modern scholars think that he may have believed that women should be taught philosophy as well as men. Credit : Unknown.

I might have gotten a little ahead of myself. Backing up now, slowly.

The keys of a piano

Consider a piano. There are white keys, and there are black keys. However, these are not arranged randomly. The black keys are always in between two white keys, and two successive black keys have, alternately, two or three white keys in between them. This is the patterns in most keyboard instruments nowadays.

Finding the C

If you have a piano at home, great! You’re rich! If you don’t, and instead have a keyboard, awesome. If not, don’t fret, use your imagination and trace your eyes to about the middle of the keyboard. The black keys are grouped in twos and threes, each black key always between two white keys . Pick a group of two black keys; any pair would do. Play the white key to the immediate right of the first of the two black keys[3]; that would be the cyan key in the second image below. The note you hear is a C. Find another such pair, play the first key. Again, the note you hear is a C. However, this C is the not the same as the first C. Depending on whether the second key you pressed is to the left or to the right of the first key you pressed, the note is lower or higher in pitch. For instance, in the first image below, if you play the C on the right, it will sound higher in pitch than the C on the left. However, somehow, these two still sound very similar.

Keys in a piano. Image Source : playpiano-dot-com.
Keys in a piano. Image Source : playpiano-dot-com.
Image of an 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry
An 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry

The same is true for any of the other keys. If you shift your focus to the second key in the pair, the one to the immediate right of the C, then you have a D. All such white keys are D. Their frequencies are also closely related. Hit one of the C keys in the middle, then find the next occurrence of the C, and play it. The second frequency will be, provided the piano is in tune, exactly double the first.

Frequency relations between the Cs

Which brings me back to the curious asterisks I had a while back, when I was writing out the DoReMi or the SaReGaMa or the CDE notations. Here’s the CDE again, for reference

C – D – E – F – G – A – B – C*

If you play this sequence on a piano[4] or a keyboard, you’ll find that the C* is twice the pitch of the C. So, if the first C has a frequency of 100 Hz, then the second C will have a frequency of 200 Hz. You could say

$$f_{C^{\star}} = 2 f_{C} \tag{1}$$
Animation of three blinking lights flashing at various frequencies. In musical parlance, the middle one is an octave higher than the top, while the bottom light is an octave higher than the middle, and two octaves higher than the top. Here f is the frequency in Hertz ("Hz"), meaning the number of flashes per second. T is the period in seconds ("s"), meaning the number of seconds per flash. T and f are always inverses. Image Credit : Sbyrnes321
Three blinking lights flashing at various frequencies. In musical parlance, the middle one is an octave higher than the top, while the bottom light is an octave higher than the middle, and two octaves higher than the top. Here f is the frequency in Hertz (“Hz”), meaning the number of flashes per second. T is the period in seconds (“s”), meaning the number of seconds per flash. T and f are always inverses. Image Credit : Sbyrnes321

Heinrich Hertz

A quick detour about the units. Hertz, abbreviated as Hz, is the unit of frequency. In plain English, it means cycles per second. In plainer English, it refers to the number of troughs (or crests) that the wave has per second. The unit is named after Heinrich Hertz, a German physicist who was the first to conclusively demonstrate the existence of electromagnetic waves, first postulated by James Clerk Maxwell in his electromagnetic theory of light. In the twentieth century, most common people would associate the word hertz with the radio; in the twenty-first, they associate it with clock and bus speeds in computers and smartphones.

Image of Heinrich Rudolf Hertz, German physicist. Image Credit : Robert Krewaldt.
Heinrich Rudolf Hertz, German physicist. Image Credit : Robert Krewaldt.
James Clerk Maxwell, Scottish physicist. Image Credit : Engraving of James Clerk Maxwell by G. J. Stodart from a photograph by Fergus of Greenock, sourced from the frontpiece in James Maxwell, The Scientific Papers of James Clerk Maxwell. Ed: W. D. Niven. New York: Dover, 1890.
James Clerk Maxwell, Scottish physicist. Image Credit : Engraving of James Clerk Maxwell by G. J. Stodart from a photograph by Fergus of Greenock, sourced from the frontispiece in James Maxwell, The Scientific Papers of James Clerk Maxwell. Ed: W. D. Niven. New York: Dover, 1890.

Octave

Anyway, back to music. If C* has twice the frequency of C, 200 Hz to its 100 Hz, then C* is said to be an octave higher than C. Similarly, if you go down (left on a piano) to the lower C, of frequency 50 Hz, then this C will be said to be one octave below C.

Scientific notation

All these C’s are a tad confusing. Which is why scientists use subscript indices to designate the various C notes. For our purposes, let out 100 Hz note be $C_1$, the 200 Hz be $C_2$, an octave higher at 400 Hz is $C_3$ etc. The 50 Hz note can be $C_0$, but hopefully we won’t need it.

On a piano, the keys begin from A, not C. There are a total of 88 keys, of which 52 are white keys, spanning a little more than seven octaves. The middle C is in the fourth octave, and is designated $C_4$.

Diagram of an 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Note that the piano is usually tuned in equal temperament, not just intonation. Image Credit : AlwaysAngry
An 88-key piano, with the octaves numbered and Middle C (cyan) and A4 (yellow) highlighted. Image Credit : AlwaysAngry

This octave structure in music makes labeling simple. You only need to think about one note, its double, and the notes in between. Once you have figured that out, it is a simple thing to reapply it to the other octaves.

Frequencies of intermediates

Right. We now know the frequencies of two notes, $C_1 = 100 Hz$ and $C_2 = 200 Hz$. We know $C_n$ too, where $n$ are integers, but they won’t be necessary here.

Our next job is to figure out what the frequency of the intermediate notes (D, E, F, G, A, B) could be. The problem is to chop up an interval of 100 Hz ($C_2 – C_1$) into seven pieces so that the notes don’t sound awful.

Do keep in mind that in actual music, the C notes are rarely at 100 Hz. What I am offering is just a simple pedagogical example. I will talk about actual musical frequencies later on.

Just intonation

One simple way is to use ratios of simple integers. Pythagoras used a version of this method way back in ancient Greece. Over the centuries, this method has been modified and improved, and by the time of renaissance and post-renaissance Europe, it had evolved into what is known as just intonation.

Do keep mind that a piano is not tuned in just intonation. Instead, a method called equal temperament is used. I’ll get to that in part II of this series, The Colour of Music : Notes of Equal Temperament.

Octave (C*) & the perfect fifth (G)

The octave is a perfect example of simple ratios–$C_2$ is $2:1$ times $C_1$. $C_1$ itself is a trivial example, for it has a ratio of $1:1$. The next ratio is, naturally, $3:2$. Multiply $C_1$ with $3/2$ and you get the note G or Ga or Sol, with a frequency value of $150 Hz$. This note is called the perfect fifth. This is one of the most important intermediate notes, and playing this together with the unison, C, makes for pleasing hearing. Or, as musicologists like to call it, consonance.

Perfect fourth (F), major third (E), major second (D)

The next simplest ratio is $4:3$. This is the perfect fourth, designated F, with a frequency value of $133.333… Hz$. This is followed by $5:4$ at $125 Hz$; this is E, also called the major third. We skip the next three ratios and move straight on to $9:8$, at $112.5 Hz$; this is D, the major second.

Do remember that all these ratios are with respect to the base note, C ($= 100Hz$), and not with respect to the previous note[5].

You do realise why these are called perfect fifths and fourths and major thirds and seconds, don’t you? That is the order in which they appear after C in the series CDEFGAB. That way, we can expect A and B to be the major sixth and seventh notes. Why major? Wait till Part II.

C-D-E-F-G

Here’s a quick table to organise matters (rounding off to one place after the decimal):

Name Symbol Ratio Value
Perfect Unison C 1:1 100.0 Hz
Major Second D 9:8 112.5 Hz
Major Third E 5:4 125.0 Hz
Perfect Fourth F 4:3 133.3 Hz
Perfect Fifth G 3:2 150.0 Hz

That takes care of the lower half of the octave in just intonation. Clearly, we can’t do much more with ratios of consecutive numbers anymore. Going higher than $9:8$ will only decrease the frequency, whereas we want to go beyond 150 Hz. Also, we don’t want to beyond $2:1$, so keep that in mind.

Major sixth (A)

Thus the next viable candidate is $5:3$, with a frequency value of $166.66… Hz$; this is A, the major sixth.

I’m certain that you’ve noticed by now that I’m not really talking about the ratios that have 5 in the denominator. Had I done so, $6:5 = 120 Hz$, $7:5 = 140 Hz$, and $8:5 = 160 Hz$ should have featured by now.

Fear not, they shall, in the next part of this series. They just don’t quite fall into the white major-ity.

That was a dreadful pun and I apologise.

Major seventh (B)

Moving on. We need one more note, the major seventh, note B. The next viable candidate is, naturally, $7:4$, giving a frequency value of $175Hz$. Which completes our octave comprising seven notes in ra…

Err, no. That’s not what happens.

For some strange reason, detectable only to people who actually understand music, $7:4$ is not used as ratio. Neither is $9:5$. In fact, the next note comes a lot further on, at $15:8$! So B has a frequency of $187.5Hz$, and we can update our table:

Name Symbol Ratio Value
Perfect Unison C 1:1 100.0 Hz
Major Second D 9:8 112.5 Hz
Major Third E 5:4 125.0 Hz
Perfect Fourth F 4:3 133.3 Hz
Perfect Fifth G 3:2 150.0 Hz
Major Sixth A 5:3 166.7 Hz
Major Seventh B 15:8 187.5 Hz
Octave C* 2:1 200.0 Hz

C major scale

In musical parlance, this comprises the C Major Scale in just intonation. This is the most basic scale[6] one uses, and has obvious pedagogic benefits. Musically, it’s pretty boring.

The following is an audio sample of C-major scale tuned using just intonation. Sound credit : Kyle Gann.

It is rather simple to generate these tunes using a computer nowadays. However, if you want to go old school, and get your hands dirty, here’s a nice way to find CDEFGAB while, hopefully, gaining better understanding of the system.

Finding the notes : octave, fifth, third

The tools at our disposal are ratios. Not just simple ratios, ratios of the first three prime numbers; $3:2$, $2:1$, and $5:2$. Since $5:2$ is greater than 2, we divide it by 2 and take it down to $5:4$. These are our principal ratios, the octave, the fifth, and the third, and it is possible to generate the C major scale from the C key by repeated combinations of these three ratios. So, for instance, to get to the perfect fourth, F, you need to start from C ($1:1$), go up an octave to C* ($2:1$) by multiplying with 2, then down a fifth to F ($4:3$) by dividing by 3/2.

The table will make things clearer:

Note Up/Down Multiply by… Final ratio
C Stay $\times \frac{1}{1}$ $1:1$
C* Up octave $\times \frac{2}{1}$ $2:1$
G Up fifth $\times \frac{3}{2}$ $3:2$
E Up third $\times \frac{5}{4}$ $5:4$
F Up octave, down fifth $\times \frac{2}{1} \times \frac{2}{3}$ $4:3$
B Up fifth, up third $\times \frac{3}{2} \times \frac{5}{4}$ $15:8$
D Up two fifths, down octave $\times \frac{3}{2} \times \frac{3}{2} \times \frac{1}{2}$ $9:8$
A Up octave, up third, down fifth $\times \frac{2}{1} \times \frac{5}{4} \times \frac{2}{3}$ $5:3$

Hmm. So that is why that weird 15:8 ratio. Makes sense now, doesn’t it?

Five limit tuning

I included the fourth column as an afterthought. This is why this system is called 5-limit tuning, because you only have powers of prime numbers below 5 in your toolbox. As you can see, it is enough for the C major scale.

This system of simple numerical ratios is called just intonation. In the next part in this series, The Colour of Music : Notes of Equal Temperament, I’ll take up equal temperament, a system involving logarithms. I’ll also look into scales more interesting than C major, and will discuss the minor notes.

Until then, here’s Signora Bianca Castafiore practising music. Poor Archibald. Poor poor Archibald.

Image of Signora Bianca Castafiore and Captain (Retd.) Archibald Francis Haddock. Image Credit : Hergé.
Signora Bianca Castafiore and Captain (Retd.) Archibald Francis Haddock. Image Credit : Hergé.

Footnotes

  1. Although, truth be told, most Kolkata FM channels are more ad than music these days. ↩︎
  2. At a time when you’d rather he shut up and let you sleep. ↩︎
  3. First means the one on the left ↩︎
  4. Easy enough. Find a C, then play all the white keys. ↩︎
  5. A mistake I had made while trying to wrap my head around this stuff. Things had gotten pretty interesting after a while. ↩︎
  6. The way I understand it, a scale is basically the set of notes you choose to play with during composition. As will be clear in Part II, the most natural way to divide up an octave is in twelve parts, not seven. You then choose 7 of these 12 notes available to you and form a sort of a scrabble rack. Then, you form musical words and phrases from this stack. Music is Scrabble. Hmm, never thought of it that way. ↩︎
Why Indigo? The Mystery behind Newton’s VIBGYOR

Why Indigo? The Mystery behind Newton’s VIBGYOR

Reading Time: 13 minutes

Four and a half centuries ago, Isaac Newton used a prism and a dark room to split white sunlight into its coloured components. He labelled seven primary colours, later known by the mnemonic VIBGYOR. Curiously, he chose indigo to be one of the seven, a colour that most people fail to pick out from the spectrum. Why did Newton choose this elusive colour to be one of his chosen seven? Why indigo? The Nerd Druid Investigates!

VIBGYOR and the rainbow colours

In my previous article, I spoke at length about the history of the colours in the rainbow, what VIBGYOR and its inverse ROYGBIV mean, and how non-anglophone cultures perceive the seven rainbow colours. Here’s a quick recap.

Rainbow in world cultures

Most cultures associate the rainbow with a god holding a bow. For the Estonians, it is the thunder god Erruk. For the Indians, it is either the thunder god Indra or the prince of Ayodhya, Rama. The Norse do not have a bow-wielding divine being. Instead, for them, the rainbow is the Bifröst bridge in divine Asgard.

Entry of the gods into Valhalla via the Bifröst bridge. Nordic mythology. Image Credit : Unknown.
Entry of the gods into Valhalla via the Bifröst bridge. Nordic mythology. Image Credit : Unknown.

Aristotle and the Greeks

The Greeks ascribed three primary colours to the rainbow : porphyra (dark purple), khloros (green), and erythros (red). Aristotle allowed for a fourth, yellow, but made it quite clear that this was a composite, being darker than white but lighter than red.

Newton and the Spectrum

In his 1671-72 paper New Theory about Light and Colours and then, in more detail, in the 1704 masterpiece Opticks, Newton describes how he placed a glass prism in front of a hole in his window, and, having darkened his room, observed on the opposite wall

the Spectrum … appear tinged with this Series of Colours, violet, indico, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying…

Newton's diagram of the spectrum of white light due to refraction from a glass prism. Image Credit : Isaac Newton, reproduced in "Memoirs of the life, writings and discoveries of Sir Isaac Newton", by David Brewster.
Newton’s diagram of the spectrum of white light due to refraction from a glass prism. Image Credit : Isaac Newton, reproduced in Memoirs of the life, writings and discoveries of Sir Isaac Newton, by David Brewster.

VIBGYOR and ROYGBIV in anglophone cultures

Post-Newton, anglophone cultures adopted ROYGBIV as a mnemonic for schoolchildren to learn and remember this seried of seven colours. Schoolchildren in the US were told that this refers to a man named Roy G. Biv, while those in the UK learnt a bit of history too, with Richard Of York Gave Battle In Vain.

The inverse, VIBGYOR, which agrees with Newton’s own colour order, is probably an Indian English thing.

BayNeeAaShoHoKawLa, the Bangla rainbow

Bengali people too had a nice mnemonic for the rainbow colours. BayNeeAaShoHoKawLa (বেনীআসহকলা) was, like VIBGYOR, made out of the first letters of the seven colours. However, there is a crucial difference. While VIBGYOR is

Violet > Indigo > Blue > Green > Yellow > Orange > Red

BayNeeAaShoHoKawLa, when translated to its equivalent English colour names, reads

Violet > Blue > Cyan > Green > Yellow > Orange > Red

Modern spectrum. Image credit : Unsure.
Modern spectrum. Image credit : Unsure.

A quick comparison with a good modern spectrum shows that the Bangla BayNeeAaShoHoKawLa is probably more accurate than VIBGYOR. So, the question is…

Why Indigo?

Most people were perfectly happy with indigo being the second (or the sixth, if you take ROYGBIV) for almost two and a half centuries after Newton’s spectrum. However, in the early part of the 20th century, some curious sould began to ponder the significance of seven primary colours. Some of these theories have long been discredited, others are perhaps more plausible. Analysis of Newton’s choice got a fresh impetus in the 1970s, and one would very much like to believe it was due a letter from a reader in a 1973 edition of the British popular science magazine New Scientist. The letter, by a Mrs H Davoll, of 10 Broadlands Avenue, Shepperton, Middlesex, UK, titled Why Indigo?, was published on 13 Dec 1973. In it, Mrs Davoll says that

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence…I feel that there is an element of the Emperor’s clothes situation in this matter, most people not daring to admit that they cannot distinguish another colour between blue and violet.

Let’s find out why, shall we?

Colour nomenclature : Sociopolitical Effects

Compare VIBGYOR with BayNeeAaShoHoKawLa again. I’ll lay them out one after the other, along with a modern spectrum :

Violet > Indigo > Blue > Green > Yellow > Orange > Red

Violet > Blue > Cyan > Green > Yellow > Orange > Red

Modern spectrum. Image credit : Unsure.
Modern spectrum. Image credit : Unsure.

Let’s imagine, for a moment, that both versions are accurate, and actually refer to the same thing. If that is so, then would we be amiss in thinking that, for some reason, what Newton thought was blue is what we know as cyan nowadays? There are precedents for this sort of linguisting colour labelling phenomenon. Homer, the blind Greek poet of the Epics, referred to

…the wine-dark sea…

while we, social media savvy modern humans, have great difficulties in figuring out the correct colour of an evening gown. Take the colour of the sky, for instance. Some would say it is cyan, some would prefer azure, while people with simpler vocabularies, such as me, would probably have to make do with sky-blue.

Image of the RGB colour star. The three primary colours--red, blue, green--at the three vertices of the upright triangle. The secondary colours--magenta (R+B), cyan (B+G), yellow (G+R)--at the three vertices of the inverted triangle. Tertiary colours rose (R+M), violet (M+B), azure/teal (B+C), spring green/sea blue (C+G), chartreuse green (G+Y), and orange (Y+R) form the intermediate vertices of the 12-pointed star. Image Credit : DanPMK
RGB colour star. The three primary colours–red, blue, green–at the three vertices of the upright triangle. The secondary colours–magenta (R+B), cyan (B+G), yellow (G+R)–at the three vertices of the inverted triangle. Tertiary colours rose (R+M), violet (M+B), azure/teal (B+C), spring green/sea blue (C+G), chartreuse green (G+Y), and orange (Y+R) form the intermediate vertices of the 12-pointed star. Image Credit : DanPMK

Newton’s indigo is today’s blue?

Take a look at the RGB colour star. Would you say that the colour of the sky is what is labelled as azure in the diagram? Or is it perhaps closer to cyan? If it were up to me, I would probably opt to lighten azure a little and call it sky-blue.

Therefore, isn’t it at all possible that, just as we posited that Newton’s blue might be today’s cyan, Newton’s indigo might also be today’s blue? After all, indigo had been adopted in Europe as a natural dye since about a hundred years before Newton’s experiment. It is entirely possible Newton himself owned clothing that had been coloured blue using the indigo dye. Would it be impossible for him to possess and wear a blue bowtie, dyed with indigo, and state, “I wear an indigo bowtie now. Indigo bowties are cool!”

Orange and Indigo : Fruit and Dye

Consider oranges. They are a fruit, but they are also a colour. Which came first?

Well, the fruit did. Portuguese merchants brought sweet Indian oranges to Europe in the late 15th century, displacing the bitter Persian oranges grown in southern Europe. The word orange itself has a circuitous route,

narangas (Sanskrit) > narang (Persian) > naranj (Arabic) > naranza > narancia > arancia (Italian) > orange (French, English)

Photograph of an orange and a half. Image Credit : Oasis Botanical Company, unknown photographer.
An orange and a half. Image Credit : Oasis Botanical Company, unknown photographer.

After arriving on English shores, the fruit lent its name to the colour; the first English usage was in 1512. By Newton’s time, a century and a half later, the colour was not just connected to delicious sweet oranges, but had entered daily use, supplanting yellow-red, saffron, and citrine as the most popular word describing that particular colour.

Similarly, indigo, though originally a dye, must have been closely connected with anything dark blue, prompting Newton to use it in his spectrum.

Photograph of a piece of indigo plant dye from India, c. 2.5 inches (6.35 cm) square. Image Credit : Evan Izer (Palladian)
Piece of indigo plant dye from India, c. 2.5 inches (6.35 cm) square. Image Credit : Evan Izer (Palladian)

All this was, of course, quite unnecessary when it came to the other five colours, Newton’s Originals. Red, Yellow, Green, Blue, and Violet were all in use for centuries by the time Newton landed in the scene, and had probably lost their charm as colour names. Perhaps orange and indigo were still exotic enough to be used fashionably.

Why Indigo?

If so, why wasn’t Mrs H Davoll equally bothered about orange? That is because of how current affairs and recent history shaped our perception off colour and usage of certain colour names. Unlike the fruit orange, a daily item in our lives, the dye indigo isn’t as ubiquitous.

The various names for dark blue

For instance, British naval uniforms were first introduced in 1748. They were coloured blue using indigo dye. Soon, though, that particular dark shade of blue came to be known as navy blue. Over the years, indigo has lost out to navy as a label for a dark blue in the fashion world. Among painters, the preferred term for a dark blue pigment is ultramarine.

Painting of Captain Edward Vernon (1723-1794), British Naval officer, wearing a dark blue uniform probably dyed with indigo. Painted in the mid-18th century, oil on canvas. Image Credit : Francis Hayman (painter), Royal Museums Greenwich UK.
Captain Edward Vernon (1723-1794), British Naval officer, wearing a dark blue uniform probably dyed with indigo. Painted in the mid-18th century, oil on canvas. Image Credit : Francis Hayman (painter), Royal Museums Greenwich UK.

Ever since the discovery of the first artificial dye in the mid-19th century, the use of the dye indigo has dropped off steadily, and so has the use of the colour term. While navy blue and ultramarine have survived into the 21st century, indigo has fallen by the wayside. So much so that nowadays, the word is used only when referring to the rainbow, or recollecting Newton’s work.

Colour perception : Psychological Effects

In 1920, Edridge-Green developed a theory of colour perception in which he separated colour vision into seven classes or psycho-physical units. Naturally, these units were based on Newton’s seven colours. According to Edridge-Green, a normal person is hexachromic; she should be able to see six colours–red, orange, yellow, green, blue, and violet. People with extraordinary vision, those who can all seven of Newton’s colours, are classified as heptachromic. These people have eyes and vision sensitive enough to distinguish and detect an indigo in between blue and violet.

It is difficult to believe Newton was a normal hexachromic like most other people; surely one of the greatest human geniuses must have extraordinary vision too, no?

Well, no. In fact, Newton’s vision was actually somewhat poor. So much so that, during his prism spectrum experiment, he had to ask

…an Assistant whose Eyes for distinguishing Colours were more critical than mine…

to draw the boundaries of the spectral colours. In his own words, this was

…because my owne eyes are not very criticall in distinguishing Colours…

Newton’s assistant

Image of Isaac Newton (1642-1727) English scientist and mathematician, using a prism to break white light into spectrum. With Cambridge room mate John Wickins. Engraving of 1874. Image credit : Getty Images.
Isaac Newton (1642-1727) English scientist and mathematician, using a prism to break white light into spectrum. With Cambridge room mate John Wickins. Engraving of 1874. Image credit : Getty Images.

Newton’s assistant in the prism experiment, the heptachromic, was possibly John Wilkins, his roommate at Cambridge. Wilkins and Newton were roommates for 20 years, and possibly enjoyed a sort of Holmes-Watson dynamic. Wilkins left Cambridge in the early 1680s and moved 240 km away to Stoke Edith. There he joined the parish, got married, and had a son. The departure of his Watson hit Newton hard; he buried himself even deeper into his research, and channeled his loneliness into work. In this period, a meeting with the astronomer Edmund Halley led him to channel his energies into what would ultimately become the Principia Mathematica, and with it the birth of calculus.

No, not the Tintin character.

Colour perception : Physical Effects

Raman and bright sunlight

Edridge-Green’s theory in no longer in favour, and has been discredited. Instead, C.V. Raman, in his book The Physiology of Vision (1968) had suggested that perhaps it was Newton’s use of sunlight that led to indigo. Raman’s hypothesis was based on the following fact; use a bright enough source of light for your spectrum experiment and you might well be able to make out a band of colour in between blue and violet, a colour that might as well be called indigo.

Photograph of Chandrasekhara Venkata Raman, Indian physicist. Image Credit : The Nobel Foundation.
Chandrasekhara Venkata Raman, Indian physicist. Image Credit : The Nobel Foundation.

Sounds plausible. However, if you do indeed perform the experiment with sunlight, then, under ideal viewing conditions, you should be able to detect as many as 200 separate hues of colour. Whether you choose to name all of them, or merely a certain subset, would quite likely depend on your personal preferences, shaped by your culture and background.

Thus, I’m afraid we’re still on Why indigo?.

The Glass Prism

Modern optical lenses and prisms are made out of two materials, crown glass and flint glass. The latter contains lead, is denser, and has a higher refractive index, which means light rays bend more while passing through it1 Newton probably used crown glass, although it seems his prism had a slightly higher refractive index than is usual. This was possibly because his prism had some lead in it too. Nevertheless, the spectrum it creates seems heavier on the blue-end.

 

Image showing comparison of Newton's prism spectrum diagram and the modern spectrum. Notice how the green, yellow, orange, and red bands in Newton's diagram are narrower than they should be, while the purple/violet, indigo/blue, and blue/cyan bands are wider. This is possibly because of the kind of glass Newton's prism was constructed out of. Image Credit : (Modern Spectrum) Unknown. (Newton's Spectrum) Isaac Newton. (Composite) The Nerd Druid.
Comparison of Newton’s prism spectrum diagram and the modern spectrum. Notice how the green, yellow, orange, and red bands in Newton’s diagram are narrower than they should be, while the purple/violet, indigo/blue, and blue/cyan bands are wider. This is possibly because of the kind of glass Newton’s prism was constructed out of. Image Credit : (Modern Spectrum) Unknown. (Newton’s Spectrum) Isaac Newton. (Composite) The Nerd Druid.

This bias towards the blue end might have prompted Newton and Wilkins to identify an extra colour in between violet and blue.

Colour number : Mystical Effects

In addition to mathematics and physics, Newton had considerable interest in alchemy, astrology, and theology. He wrote almost two million words on these subjects, a number that is almost twice the million words he wrote on science. His mystical interests have led him to be accused of being

…misled by a predilection for the number seven which during many ages has been regarded with a sort of mystical veneration…

It is entirely possible Newton was as fascinated with the number seven as the ancients were. There were enough references to that number in antiquity for Newton to have been influenced enough to have expanded his original five colours to seven, by inserting orange and indico.

Seven celestial bodies

The number of known celestial bodies in antiquity was seven : the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. Uranus and Neptune were discovered after Newton’s time, and Pluto is now a dwarf planet.

Seven days in a week

There are seven days in a week. Be it the Gregorian or Julian calendars of Europe, or the Bangla and neighbouring calendars of India, we always seem to find seven days. Also, these calendars attribute the days of the week to the seven celestial bodies–Tuesday to Mars, Wednesday to Mercury, Thursday to Jupiter, Friday to Venus, and Saturday to Saturn, the first two being rather obvious.

Image of seven planets, seven days, seven metals, seven gods. Image Credit : Unknown, possibly JoeDubs(?).
Seven planets, seven days, seven metals, seven gods. Image Credit : Unknown, possibly JoeDubs(?).

Seven metals of antiquity

The number of known metals in antiquity was seven–Gold, Silver, Iron, Mercury, Tin, Copper, and Lead. These were associated with the days of the week as well as the planets.

Planets, Days, Metals, Gods

The table below shows the ancient associations.

Metal Day Planet Greek/Roman god Norse god
Gold Sun Sunday Helios/Sol Sunna/Sól
Silver Moon Monday Selene/Luna Máni
Iron Mars Tuesday Ares/Mars Tyr
Mercury Mercury Wednesday Hermes/Mercury Odin
Tin Jupiter Thursday Zeus/Jupiter Thor
Copper Venus Friday Aphrodite/Venus Freya
Lead Saturn Saturday Cronus/Saturn
Animation of planets and days, again. Image Credit : Unknown, possibly JoeDubs(?).
Planets and days, again. Image Credit : Unknown, possibly JoeDubs(?).

Seven Deadly Sins

There are seven deadly sins in Christian theology

  1. Envy
  2. Greed
  3. Pride
  4. Lust
  5. Gluttony
  6. Sloth
  7. Wrath

Of course, one could counteract these evil evilnesses by embodying the seven virtues.

Seven heavens and worlds

Different ancient cultures and religions believed in seven heavens, or worlds divided in parts of seven. For instance, in Hinduism, there are fourteen worlds, and they are divided thus (I’m quoting directly from Wikipedia)

According to some Puranas, the Brahmanda is divided into fourteen worlds. Among these worlds, seven are upper worlds which constitute of Bhuloka (the Earth), Bhuvarloka, Svarloka, Maharloka, Janarloka, Tapoloka and Satyaloka, and seven are lower worlds which constitute of Atala, Vitala, Sutala, Talatala, Mahatala, Rasatala and Patala.

Seven day Creation myth

Creation myths in the Abrahamic religions, particularly Christianity, propound the belief that god built Creation in seven days.

Seven Wonders of the Ancient World

There were Seven Wonders of the Ancient World

  1. Great Pyramid of Giza
  2. Hanging Gardens of Babylon
  3. Temple of Artemis at Ephesus
  4. Statue of Zeus at Olympia
  5. Mausoleum at Halicarnassus
  6. Colossus of Rhodes
  7. Lighthouse of Alexandria.
A collage of The Seven Wonders of the (ancient) world, depicted by 16th-century Dutch artist Maarten van Heemskerck.
A collage of The Seven Wonders of the (ancient) world, depicted by 16th-century Dutch artist Maarten van Heemskerck.

Seven Liberal Arts

In ancient Greece, knowledge of the Seven Liberal Arts were considered essential for a free person. These were

  1. Grammar
  2. Logic
  3. Rhetoric
  4. Arithmetic
  5. Geometry
  6. Theory of music
  7. Astronomy.

Correspondingly, or perhaps not, the number of core subjects one has to write one’s tenth level exam in Bengal, the dreaded Madhyamik, is also seven. Bengali and English are the languages, History and Geography make up the humanities, while Mathematics, Physical Sciences (Phys+Chem) and Life Sciences (Biology) make up the sciences.

However inspired Newton might have been from these many examples to expand his colour roster to seven, one should remember that he was, after all, a consummate logical scientist. Would he really have no scientific reason for inserting indigo and orange? Shouldn’t one justifiably expect scientific answers to why indigo, as also why orange? Answers that are not only scientific, but also reveal a crucial insight about how the world works?

Colour pitch : Light and Music

DO – RE – MI – FA – SOL – LA – TI

Julie Andrews in the 1964 Hollywood movie</em> The Sound of Music. <em>The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.
Julie Andrews in the 1964 Hollywood movie The Sound of Music. The most famous song in this musical, Doe-a-deer, teaches the Do-Re-Mi scale. Image Credit : sound-of-music-dot-com.

If you have seen The Sound of Music, you will be familiar with these seven notes. Sung joyously by Julie Andrews, these seven are the names of the notes in the diatonic scale. These can also be represented as

C – D – E – F – G – A – B

or as

Sa – Re – Ga – Ma – Pa -Dha – Ni

in Hindustani classical music.

Musical notes are audio frequencies

Musical notes are, essentially, audio frequencies. In Western music, the A key in the fourth CDEFGAB series of the piano, denoted as A4, is set to exactly 440 Hz, and other notes are derived from it by changing frequency in units called semitones. Each octave has 12 semitone intervals. For instance, C4 is 9 semitones lower than A4. Under the popular equal temperament scheme, this makes C4‘s frequency 261.63 Hz.

A larger unit in music theory is the whole tone. It is double the semitone, and together, these two are instrumental in setting scales. For instance, in a diatonic scale of seven notes, there are 5 whole tones (T) and 2 semitones (S), always in groups of TTS and TTTS. Where you start from determines which mode you are following.

Ionian Mode

The CDEFGAB series that I began this section with is in the oft-used Ionian mode, with intervals TTSTTTS. Thus, in order to go from C to D, or from D to E, you need to increase by a whole tone T. E to F is a semitone S. F to G, G to A, and A to B are three whole tone (T) intervals TTT. Finally, to get to the higher C, you need to increase by a semitone (S).

Dorian Mode

In Newton’s time, however, the Dorian mode was much more in vogue, with notes DEFGABCD and intervals TSTTTST. The table below might help

Mode Ionian Dorian
Notes C–D–E–F–G–A–B–C D–E–F–G–A–B–C–D
Intervals T–T–S–T–T–T–S T–S–T–T–T–S–T

In the Dorian mode, the semitones occur at positions 2 and 6, exactly at the positions orange and indigo appear. Adding Newton to the table above (and removing Ionian mode), we get the answer to why indigo.

Mode Dorian Newtonian
Notes D–E–F–G–A–B–C–D R-O-Y-G-B-I-V
Intervals T–S–T–T–T–S–T P-S-P-P-P-S-P

where P are primary colours and S secondary.

 

Figure 4 (pg 91) from Newton's _Opticks_ (1704), slightly cropped. I have added the colours, the notes, and the Dorian intervals. Answers the question : Why indigo? Image Credit : Isaac Newton / The Nerd Druid.
Figure 4 (pg 91) from Newton’s _Opticks_ (1704), slightly cropped. I have added the colours, the notes, and the Dorian intervals. Answers the question : Why indigo? Image Credit : Isaac Newton / The Nerd Druid.

Newton’s Insight

As Newton himself writes (Opticks, pg 92)

…I delineated therefore in a Paper the perimeter of the Spectrum FAPGMT, and … I found that the … rectilinear sides MG and FA were by the said cross lines divided after the manner of a musical Chord…to be in proportion to one another, as the numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2, and so to represent the Chords of the Key, and of a Tone, a third Minor, a fourth, a fifth, a sixth Major, a seventh, and an eighth above that Key: And the intervals Mα, αγ, γε, εη, ηι, ιλ, and λG, will be the spaces which the several Colours (red, orange, yellow, green, blue, indico, violet) take up.

Appreciate Newton’s insight and genius. What are colours? They are simply frequencies of light; their pitch are seen and not heard, their instruments of detection are the eyes and not the ears. They are both, ultimately, waves. Very different types of waves, granted, but waves nevertheless, with frequencies and wavelengths.

Why indigo? Because Music!

Image of three centuries of colour scales, beginning with the pioneer, Isaac Newton, who connected DEFGABCD with ROYGBIVR. Image Credit : Unknown.
Three centuries of colour scales, beginning with the pioneer, Isaac Newton, who connected DEFGABCD to ROYGBIVR. Image Credit : Unknown.

 


References

Books

  1. Aristotle : Meterology, Greece (350 BCE).
  2. Newton, Isaac : Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light . Also two treatises of the species and magnitude of curvilinear figures, Sam Smith & Benj. Walford, for the Royal Society (MDCCIV, 1704).

Papers

  1. Newton, Isaac : New Theory about Light and Colours, Philosophical Transactions (1672).
  2. McLaren, K. : Newton’s Indigo and references therein, Color Research and Application (1985).

Articles

  1. Fisher, Len : Perceptual thresholds: Music inspired Newton’s rainbow, Nature (2015).
  2. Morr, Kelly : Why are there 7 colors in the rainbow?, 99designs, (2016).
VIBGYOR : Newton’s Rainbow and Indigo

VIBGYOR : Newton’s Rainbow and Indigo

Reading Time: 11 minutes

VIBGYOR is a popular mnemonic for the seven rainbow colours. 450 years ago, Newton split white light into its coloured components and labelled seven of them. Curiously, most people see only six. The Nerd Druid Investigates!

VIBGYOR

One of the very first English words I had learnt was VIBGYOR. Of course, it wasn’t really a word, but it was associated deeply with that thing that all childhood craves, colour.

Violet. Indigo. Blue. Green. Yellow. Orange. Red.

Red is the colour of passion, of love, of anger. Orange tastes sweet and sour, and reminds one of winter, and the Dutch; it is also the colour of greed. Yellow is the brightest of all colours, but is also associated with cowardice, and fear. Green is Life itself, of its great willpower to “…always find a way…“, as Ian Malcolm loves repeating. Blue, the colour of the skies and the seas, is calm, and instills hope. Indigo has a colourful history as a natural dye, and a confused one as regards its place in all this. And when a nor’wester, an April thunderstorm gathers clouds so deep that they look violet, you know the evening’s going to get more interesting.

Image of the Green Lantern emotion spectrum, representing VIBGYOR. Don't be alarmed if the emotions and colours don't quite match what you believe. I, as do most, associate Love with red. This is simply the Green Lantern emotion spectrum, an invention of DC comics. The white logos in the centre of each flag are the symbols of each of the coloured Lantern corps. The most well known are, of course, the Green Lantern Corps, sitting pretty right at the centre, analogues of the Jedi from Star Wars. Image Credit : Unsure, but probably DC comics.
The Green Lantern emotion spectrum, representing VIBGYOR. Don’t be alarmed if the emotions and colours don’t quite match what you believe. I, as do most, associate Love with red. This is simply the Green Lantern emotion spectrum, an invention of DC comics. The white logos in the centre of each flag are the symbols of each of the coloured Lantern corps. The most well known are, of course, the Green Lantern Corps, sitting pretty right at the centre, analogues of the Jedi from Star Wars. Image Credit : Unsure, but probably DC comics.

ROYGBIV

Richard Of York Gave Battle In Vain

Up until a few days ago, I used to think that VIBGYOR was the most common colour mnemonic in the English-speaking world. Turns out this is only true for India. For the US and the UK, the mnemonic of choice is ROYGBIV.

Now, that is of course simply VIBGYOR reversed; the Brits and the Yanks seem to prefer starting off with red. While ROYGBIV doesn’t quite pronounce as sweetly as VIBGYOR (vibjeeyohr), it does make up a (sort of) a name, Roy G. Biv. This is how pre-schoolers in the US learn their colours. The British are far more dextrous; they also have thicker history books. Thus,

Richard Of York Gave Battle In Vain

Richard of York was, of course, Richard III, the last king of England to die in battle. In 1485, he was defeated and killed at the Battle of Bosworth Field, an event that ended the War of the Roses1. The victor, Henry of the House of Tudor, ascended the throne as Henry VII.

Portrait of King Richard III. Artist unknown. Photograph Credit : National Portrait Gallery, London, UK.
Portrait of King Richard III. Artist unknown. Photograph Credit : National Portrait Gallery, London, UK.

Aristotle

Summer in Kolkata

Summers in India are hot. Summers in the Eastern metropolis of Kolkata are hot and very humid. Sweltering and suffocating are two English adjectives that attempt capture a Kolkata summer. They fall well short of the mark.

Curiously, the Kolkata summer this year, 2018, has been uncharacteristically…pleasant. Yes, it has been very hot and yes, it has been very humid. But, interspersed within these short bouts of I-want-to-run-away-to-the-hills, there has been rain and high wind and storms. Big, violent storms. And rain at times it usually does not do so in Kolkata. And the almost constant presence of clouds has made the sunsets absolutely gorgeous.

Rainbows

…the rainbow necessarily has three colours, and these three and no others.

It is not difficult to imagine that, during these spells of rain, there might come instances, short periods of time when, looking up towards the heavens, one would see that slate-grey rain clouds covering half the firmament, delivering their watery loads to the thirsty patches below, while on the other end, the sun, having peaked out tentatively from its nebular veil, would shine gloriously for an instant, showering its silver rays through the curtain of rain water.

Essentially, one expects rainbows.

Photograph of a double rainbow appearing over the 18th hole during the third round of the Utah Championship on July 13, 2013 in Sandy, Utah. Photo Credit : Stan Badz/PGA TOUR.
A double rainbow appears over the 18th hole during the third round of the Utah Championship on July 13, 2013 in Sandy, Utah. Photo Credit : Stan Badz/PGA TOUR.

Rainbows are a staple of cultures throughout the geography and history of the world. In Estonian, for instance, the rainbow is the bow wielded by the thunder-god Erruk. In the Nordic Eddas, and now in the Marvel films, the rainbow comes from the Asgardian Bifrost, the bridge of many colours. In Bangla, the word is Ramdhonu, or Ram’s Bow. In Hindi, this transforms to Indradhanu, Indra’s Bow. Perhaps unsurprisingly, Indra is the thunder-god in the Hindu pantheon.

The Greeks

Clearly, the ancients realised the correlation, if not the causation, behind rainbows and unsettled weather. Aristotle, the great Greek natural philosopher, was perhaps the first to peer closer to the rainbow in an attempt to classify the colours within, perhaps in a hope to divining its nature and purpose. Aristotle attempted to reconcile the colours of the rainbow with his theory that all colours came from white and black. In his book Meterologica, he says

…the rainbow necessarily has three colours, and these three and no others.

Bust of Aristotle. Marble, Roman copy after a Greek bronze original by Lysippos from 330 BC; the alabaster mantle is a modern addition. Caption text from Wikipedia. Image Credit : Ludovisi Collection. Image Photographer : Jastrow.
Bust of Aristotle. Marble, Roman copy after a Greek bronze original by Lysippos from 330 BC; the alabaster mantle is a modern addition. Caption text from Wikipedia. Image Credit : Ludovisi Collection. Image Photographer : Jastrow.

Aristotle’s triad of rainbow colours is the same as that suggested by his predecessor, Xenophanes of Colophon. These are porphyra (dark purple), khloros (green), and erythros (red). Aristotle allows for a fourth colour, yellow, a non-primary bright colour that is darker than white but lighter than red, and lives in between red and green.

RGB, rods and cones

From a modern perspective, Aristotle and his predecessor is surprisingly correct. Greek colour names could be a bit…confusing, and porphyra could well be blue. Which means, to them, the three primary colours are blue, green, and red. RGB. All the other colours stem from them. Mix R and G and you have Y, yellow. G + B = C(yan); R + B = M(agenta). White and black, light and dark, add extra dimensions to these colours.

The anatomy of the eye. The retina, that is the screen at the back of the eye, has two types of light-detecting cells. The cylindrical **_rod cells_** detect intensity of light, while the more conical **_cone cells_** detect colour. There are three classes of cone cells, each detecting one of three RGB channels. Image Credit : Unsure.
The anatomy of the eye. The retina, that is the screen at the back of the eye, has two types of light-detecting cells. The cylindrical rod cells detect intensity of light, while the more conical cone cells detect colour. There are three classes of cone cells, each detecting one of three RGB channels. Image Credit : Unsure.

We now know why that is so. Our eyes have three classes of colour detecting cells. Some of these cones are sensitive to red light, some to green light, and others to blue light. The retina also has rod cells; these detect brightness (or darkness). Together, the three cone types and the rods recreate a gamut of colours for human stimulation.

Why, then, do we talk about the seven primary colours? How exactly did VIBGYOR come about?

Newton

Blame Isaac.

Image of a portrait of Isaac Newton (1642-1727). The portrait of Newton is a copy of one painted in 1689 by Sir Godfrey Kneller, which is owned by the 10th Earl of Portsmouth. This copy was painted by Barrington Bramley and donated to the Isaac Newton Institute for Mathematical Sciences in 1992 by the Director of the Institute, Sir Michael Atiyah, who unveiled it at the opening in July of that year. It shows Newton at the height of his scientific acumen, before he went to London to take charge of the Mint.
Portrait of Isaac Newton (1642-1727). The portrait of Newton is a copy of one painted in 1689 by Sir Godfrey Kneller, which is owned by the 10th Earl of Portsmouth. This copy was painted by Barrington Bramley and donated to the Isaac Newton Institute for Mathematical Sciences in 1992 by the Director of the Institute, Sir Michael Atiyah, who unveiled it at the opening in July of that year. It shows Newton at the height of his scientific acumen, before he went to London to take charge of the Mint. Image Credit : Isaac Newton Institute of Mathematical Sciences, Cambridge, UK.

Before Albert, these two words would be oft-heard in the Halls of Physics. Newton was single-handedly responsible for kick-starting and rejuvenating several prime physics disciplines. While he is most well-known for the incident with the apple, his prism comes a close second.

The Original or primary colours are, Red, Yellow, Green, Blew, and a Violet-purple, together with Orange, Indico, and an indefinite variety of Intermediate gradations.

After he was done with gravity in the 1660’s, Newton turned his attention to light. He had read Aristotle and the other Greeks, and was rather motivated by them. Like Aristotle, he too wanted to figure rainbows out, to find out what light is. And he had an analytical tool Aristotle did not; the glass prism.

Newton’s Spectrum

On a bright and sunny day, Newton darkened his room, made a small hole in the window, and placed his prism in front of the hole. Sure enough, on the wall opposite, he saw a beautiful technicolour spectrum. In his own words;

…in order thereto having darkened my chamber, and made a small hole in my window-shuts, to let in a convenient quantity of the Suns light, I placed my Prisme at his entrance, that it might be thereby refracted to the opposite wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby…

Image of Isaac Newton using a prism to break white light into spectrum. With Cambridge room mate John Wickins. Engraving of 1874.Image credit : Getty Images.
Isaac Newton using a prism to break white light into spectrum. With Cambridge room-mate John Wickins. Engraving of 1874. Image credit : Getty Images.

Once the prism was set, and the spectrum ready, all Newton had to do was to walk over to the wall and mark out the colours of the rainbow. Instead, he asked his assistant (possibly his Cambridge roommate, John Wickins) to do it, remarking later that

because my owne eyes are not very criticall in distinguishing Colours

Not only does this reveal an interesting facet of Newton—that the intensely immensely competitive man did not consider himself perfect–it also shows how English spelling has evolved and changed over the years.

Newton’s Opticks

In 1671-72, Newton published “New Theory about Light and Colours” in the Philosophical Transactions of the Royal Society, in which he reported and explained his results. He wrote (emphasis mine)

There are therefore two sorts of Colours. The one original and simple, the other compounded of these. The Original or primary colours are, Red, Yellow, Green, Blew, and a Violet-purple, together with Orange, Indico, and an indefinite variety of Intermediate gradations.

Image of the cover of Opticks (1704), by Isaac Newton. Image Credit : Isaac Newton / Bibliotheque Nationale de France.
Cover of Opticks (1704), by Isaac Newton. Image Credit : Isaac Newton / Bibliotheque Nationale de France.

More than 30 years later, in his Opticks, he modified his earlier statement slightly

the Spectrum … appear tinged with this Series of Colours, violet, indico, blue, green, yellow, orange, red, together with all their intermediate degrees in a continual succession perpetually varying

including updating the spelling of blue.

Of course, he did more than just label the colours. Careful observations and accurate sketching mixed with a healthy dose of Newtonian analytical genius told him that, as white sunlight passes through a prism, red refracts, that is, changes direction the least, while violet refracts the most. Nowadays we know that light is composed of photons, and that refraction is merely photons decelerating as they pass through an optically denser medium such as glass2. Photons of lower energies, such as those that appear red to us, decelerate the least, while the higher energy violet photons decelerate the most.

Image of Fig. 12, from Newton's Opticks (1704), showing the prism ABC and how the rays refract. G/gamma is the stop, and the sunlight enters through the hole F/phi. Since the sun is so far away, the rays are parallel. Once through the prism, the parallel rays split into components, each parallel to each other. Newton has labelled five of these regions; P/pi (violet), Q/chi (indigo and blue), R/rho (green), S/sigma (yellow and orange), T/tau (red). These fall on the screen _mn_ and form the spectrum. Image Credit : Isaac Newton.
Fig. 12, Opticks (1704), showing the prism ABC and how the rays refract. G/gamma is the stop, and the sunlight enters through the hole F/phi. Since the sun is so far away, the rays are parallel. Once through the prism, the parallel rays split into components, each parallel to each other. Newton has labelled five of these regions; P/pi (violet), Q/chi (indigo and blue), R/rho (green), S/sigma (yellow and orange), T/tau (red). These fall on the screen _mn_ and form the spectrum. Image Credit : Isaac Newton.

Which is all fine and dandy, and proves once again what an absolute magician Isaac Newton was.

But why seven colours?

And we thus come to the rub of the matter. Or the hub of the matter, where lies the rub…

Never mind. This is the central question. Why seven? Why indico, or indigo?

Indigo

Well, first of all, a little bit about indigo. Indigo, as a dye, has ancient origins. According to Pliny the Elder, the Harappans extracted the dye from a certain plant (Indigofera tinctoria) that grew in the Indus valley. The Ancient Greek term for the dye was Ἰνδικὸν φάρμακον (Indikon farmakon). This later became indicum in Latin and later indigo in Portuguese. The Silk Route brought indigo to Europe, when Marco Polo reported about it in 1289. However, a further three centuries went by before the European textile landscape realised the potential of the dye, and started large-scale manufacture. A further three centuries went by before the first artificial dye was invented. Things have rather moved up nowadays, with quantum dots being the latest and best in the colour business.

Photograph of Indigo, historical dye collection of the Technical University of Dresden, Germany. Image Credit : Shisha-Tom.
Indigo, historical dye collection of the Technical University of Dresden, Germany. Image Credit : Shisha-Tom.

Around Newton’s time, clothing dyed in indigo was quite in vogue. The dye itself had an air of exotic orientalism about it, and it might have been quite fashionable to say, “Ah, is that there an indigo doublet that I spy?” instead of saying “I want that blue vest.” Newton was human, though perhaps less so than most, and he must not have been entirely out of step with the times. He must have felt, “I’ll use indico a lot. Indico is cool.”

Cyan

Which perhaps makes sense, except that to some people, it doesn’t. Most of the human race, when asked to identify the colours in a sunlight-prism spectrum, manage to name only six colours. A few sharp-eyed ones might manage seven, but there is a crucial difference. There is no indigo. The colour series is, for most,

Violet > Blue > Cyan > Green > Yellow > Orange > Red

What is cyan, then?

Image of the RGB colour star. The three primary colours--red, blue, green--at the three vertices of the upright triangle. The secondary colours--magenta (R+B), cyan (B+G), yellow (G+R)--at the three vertices of the inverted triangle. Tertiary colours rose (R+M), violet (M+B), azure/teal (B+C), spring green/sea blue (C+G), chartreuse green (G+Y), and orange (Y+R) form the intermediate vertices of the 12-pointed star. Image Credit : DanPMK
RGB colour star. The three primary colours–red, blue, green–at the three vertices of the upright triangle. The secondary colours–magenta (R+B), cyan (B+G), yellow (G+R)–at the three vertices of the inverted triangle. Tertiary colours rose (R+M), violet (M+B), azure/teal (B+C), spring green/sea blue (C+G), chartreuse green (G+Y), and orange (Y+R) form the intermediate vertices of the 12-pointed star. Image Credit : DanPMK

In the RGB colour star above, red, green, and blue are considered primary colours, in tune with human physiology. By mixing two of these primary colours in equal amounts, you get the secondary colours cyan (blue + green), yellow (green + red), and magenta (red + blue). Mix a primary with its adjoining secondary, and you have the tertiary colours. My favourite is obviously chartreuse green, the best colour there is.

Cyan (B+G) is basically what most people refer to as sky-blue. Although, truth be told, looking at the colour star, it kinda looks like azure might be closer to sky-blue. Either way, there is a pretty prominent colour between blue and green on this colour star, and it has non-English analogues as well.

VIBGYOR in Bengal : BayNeeAaShoHoKawLa

I am from Bengal, India. Although we do a lot of English stuff, Bangla is our language. Our first words are in Bangla, and our first connections with this wondrous world is via words in Bangla. And does Bangla have an analogue to VIBGYOR?

Well, of course it does. Although, sadly, urban Bengali kids born after the nineties might not quite have heard of

BayNeeAaShoHoKawLa

In Bangla, বেনিআসহকলা.

I’ll break it down. BayNeeAaShoHoKawLaa3. The colours, and their English equivalents, are

  1. Beguni (baygoonee, বেগুনী): Violet
  2. Neel (kneel, নীল) : Blue
  3. Aakashi (aakaashee, আকাশী)4 : Cyan/Sky-blue/Azure
  4. Shobuj (showbooj, সবুজ) : Green
  5. Holud (howlooð, হলুদ) : Yellow
  6. Komola (kawmohlaa, কমলা) : Orange
  7. Laal (laal, লাল) : Red

Notice any colour missing?

The Modern Spectrum

Image of the modern spectrum. Image credit : Unsure.
Modern spectrum. Image credit : Unsure.

Red photons have the least energy (among visible photons), and so have the longest wavelength. In the image, red appears at around 700 nm. Violet, at the other end of the visible spectrum, has the most energetic photons, and therefore the shortest wavelengths. In the image, violet appears around 400 nm. The human visual range is approximately that, roughly between 400nm to 700 nm. Any lower and we get high energy ultraviolet (UV) rays (< 380 nm); any higher and you have the low energy infrared rays (> 740 nm). Too much of UV is bad and can give you skin cancer; too much of infrared is also bad and can cook you.

Newton’s indico falls between 430 and 450 nm. Looking at the image, to me at least that looks rather blue, or at least blue-violet. I don’t know if you’ll do better. On the contrary, there is a clear cyan band between 475 nm and 500 nm.

Image of prism and spectrum. Image credit : Unsure.
Image of prism and spectrum. Image credit : Unsure.

Here we clearly find a violet, a blue, a cyan, a green, a yellow, and a red. No clear indigo, and, surprisingly, a somewhat muddled orange. Seems BayNeeAaShoHoKawLa works better than VIBGYOR.

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence.

Newton wasn’t aware of BayNeeAaShoHoKawLa though, and generations of people after him have sworn by VIBGYOR (or ROYGBIV). No questions were asked till the early part of the 20th century, when people began to analyse indigo and why Newton’s colour spectrum has seven colours. However, the greatest boost probably came in 1973, from a very unlikely source.

Helen Davoll and New Scientist

New Scientist is a popular science magazine5 based in London. Like other magazines, New Scientist too has a Letters section. In the issue dated 13 Dec 1973, among letters from readers providing valuable comments and input on issues as diverse as Jupiter’s Red Spot, methane power, and whether Uri Geller was a charlatan, there was one letter that was, perhaps, a little different. Titled Why Indigo?, the letter begins with these words :

Ever since I first learned the sequence of colours in the rainbow, I have been puzzled and annoyed by the inclusion of indigo in this sequence.

A sentiment, I’m sure, most of us share.

Image of the cover of New Scientist 13 December 1973. Image Credit : New Scientist.
Image of the cover of New Scientist 13 December 1973. Image Credit : New Scientist.

The letter was sent by a Mrs H Davoll, of 10 Broadlands Avenue, Shepperton, Middlesex, UK. She goes on to state that

I feel that there is an element of the Emperor’s clothes situation in this matter, most people not daring to admit that they cannot distinguish another colour between blue and violet.

Once again, I for one am completely in agreement with her.

This letter was followed by no less than five replies from various readers, and was preceded and succeeded by a number of printed peer-reviewed publications analysing the same question Mrs Davoll had asked, Why Indigo?

Next on VIBGYOR

In the next installment of this series on VIBGYOR, The Nerd Druid shall investigate these explanations. The Nerd Druid shall also attempt to uncover who Mrs Davoll was, and will trace the sequence of comments and letters in the New Scientist. Until then, we should remember that this June…

Pride

…is Pride month in the US. For the LGBTQ community, it is a time to come out in droves and celebrate life as normal human beings, to stand out from the stigma and oppression that accompanies them. It is a time for them to appreciate the full spectrum of which conspicuously omits indigo life. Which is why, appropriately, their symbols is the rainbow flag (six colours, no indigo):

Rainbow Pride Flag. The Pride Flag is the symbol of the LGBTQ community. It has six colours. Unsurprisingly, there is no indigo. Image Credit : [Ludovic Berton](https://www.flickr.com/people/23912576@N05)

References

Books

  1. Aristotle : Meterology, Greece (350 BCE).
  2. Newton, Isaac : Opticks or, a Treatise of the reflexions, refractions, inflexions and colours of light . Also two treatises of the species and magnitude of curvilinear figures, Sam Smith & Benj. Walford, for the Royal Society (MDCCIV, 1704).

Papers

  1. Newton, Isaac : New Theory about Light and Colours, Philosophical Transactions (1672).
  2. McLaren, K. : Newton’s Indigo, Color Research and Application (1985).
Quantum Dots of All Hues : A Modern Marvel

Quantum Dots of All Hues : A Modern Marvel

Reading Time: 8 minutes

Quantum dots are fluorescent nanocrystalline semiconductors that seem to magically change colour with size. The Nerd Druid investigates this 21st century marvel.


Alexei Ekimov discovered quantum dots in a glass matrix in 1981. Four years later, in 1985, Louis Brus discovered quantum dots in colloidal solutions. Their discovery is now at the heart of nanotechnology.

What are quantum dots?

Quantum dots are extremely small semiconducting crystallites. You could fit anywhere between 10-50 atoms along the diameter of a typical quantum dot. The entire structure contains hundred to thousands of atoms.

How big are quantum dots?

Image of size comparison of a standard football, an average human hair, and a typical quantum dot. Lengths are in nanometers. The football is about 20 cm in diameter, the human hair is about 80 μm in diameter, and the two quantum dots (indicated by the yellow circles in the inset of the rightmost image) are approximately 2.5 nm in diameter.
Size comparison of a standard football, an average human hair, and a typical quantum dot. Lengths are in nanometers. The football is about 20 cm in diameter, the human hair is about 80 μm in diameter, and the two quantum dots (indicated by the yellow circles in the inset of the rightmost image) are approximately 2.5 nm in diameter. Image credit : S Pitchaimuthu, Swansea University.

The image above is another excellent way of comparing the size of quantum dots with everyday objects. The one on the very left is a standard football, while the image at the center is that of a human hair. The football is about 20 cm in diameter; the hair is about 80 μm (= 0.08 mm) in diameter. The inset of the rightmost image shows two quantum dots cirlced in yellow; each of them is about 2.5 nm in diameter1. That is about the width of DNA. You could fit about 30000 quantum dots side by side along the width of a human hair, and 80 million quantum dots side by side along the diameter of a football.

What do quantum dots do?

Quantum dots fluoresce; shine light on them and they will glow. In addition to being photoluminescent, quantum dots are also electroluminescent. Pass electric current through them and they glow; put them in a high electric field and they glow.

Image of quantum dots with vivid colours with gradually stepping emission from violet to deep red. These have been manufactured on a large scale at PlasmaChem GmbH.
Quantum dots with vivid colours with gradually stepping emission from violet to deep red. These have been manufactured on a large scale at PlasmaChem GmbH. Image credit : Antipoff/PlasmaChem

Photoluminescence & electroluminescence in everyday life

Why, though, is that so awesome? Plenty of everyday items are photoluminescent and electroluminescent. For instance, your home fluorescent lamp is photoluminescent; electricity passing through the mercury vapour inside the tube causes it to fluoresce and emit ultraviolet radiation. Thankfully, the white layer of phosphor material coating the inner surface of the tube absorbs this UV light and fluoresces, emitting visible light2. Similarly, your car dashboard dials and your calculator backlight are electroluminescent.

Photograph of thirty Fluorescent lamps with reflections giving the impression of many more lamps, in Shinbashi, Tokyo, Japan.
Thirty Fluorescent lamps with reflections giving the impression of many more lamps, in Shinbashi, Tokyo, Japan. Image credit : Oimax (Flickr)
Photograph of 1966 Dodge Charger dashboard, showing electroluminescent dials.
1966 Dodge Charger dashboard, showing electroluminescent dials. Image credit : Jonathan Gibbs (FastbackJon)
Animation of LCD, both unlit and with electroluminescent backlight switched on.
Animation of LCD, both unlit and with electroluminescent backlight switched on. Image credit : Grimlock

Quantum dots produce colour

No, the reason why quantum dots are awesome is their colour-producing capability. Take a look at the image of those coloured bottles above; each contains colloidal quantum dots and have been irradiated with UV light. You can see the entire VIBGYOR there; from left, Violet, Indigo/Blue, Blue/Cyan, Green, Yellow, Orange, Red, and, er, Red again! VIBGYORR, very pirate-y, savvy3?

But LCDs and LEDs also produce colour

It is not impossible to have coloured fluorescent lamps. With the invention of the blue LED (light-emitting diode) in the 1990s, we now have LED lamps of all colours. LCD (liquid crystal display) screens on televisions, computers, and mobile phones are nowadays capable of creating millions of colours. Organic LED screens and their evolved cousins, the AMOLED screens, have made it possible to have ultraHD TV, computer and mobiles.

Why quantum dots, then?

Quantum dots can produce any colour

Take a magnifying glass and peer into the screen you are reading this on. If you don’t have one, use a drop of water. You’ll see that the pixels on the screens have three colours : red, blue, and green. Combinations of this RGB trio make up most of the colours we perceive. This is because our eyes also only have three types of colour-detecting cone cells, and these are sensitive in the red, green, and blue channels.

Red, Green, and Blue LEDs

LEDs are bulk semiconductors. The problem with them is that you can’t quite use the same material to make different colours. For instance, you can use AlGaAs (aluminium gallium arsenide) or GaAsP (gallium arsenide phosphide)4 to fabricate red LEDs. However, if you want a blue LED, you’d need to use InGaN (indium gallium nitride). You’ll not be able to make red LEDs out of InGaN, or green ones out of AlGaAs. That is how light-emitting semiconductors work.

Quantum dot colour depends on their size

Quantum dots are very different in that respect. You can tune their colour at will by simply changing their sizes! Larger quantum dots (5-6 nm) emit orange and red light. Medium sized ones (3 nm) emit green light, while smaller ones (2 nm) emit blue light. The ones in Swansea experiment, those circled in yellow in the earlier image, are about 2.5 nm in diameter and emit blue-green light. These technicolour quantum dots can all be fabricated from the same material.

Graph of quantum dots : size and colour. Different sized quantum dots made of the same material are being irradiated by blue light. As you increase their sizes, the quantum dots fluoresce with colours of longer wavelengths. The smaller ones emit The smaller ones emit blue, blue-green, and green light, while the larger ones emit orange and red light.
Quantum dots : size and colour. Different sized quantum dots made of the same material are being irradiated by blue light. As you increase their sizes, the quantum dots fluoresce with colours of longer wavelengths. The smaller ones emit The smaller ones emit blue, blue-green, and green light, while the larger ones emit orange and red light. Image credit : Nanosys Inc.

An Amazing Technicolour Dreamcoat

If you think about it, this is quite remarkable. For LEDs, the only true colours you get are red, green, and blue5; the colour step-size for them are the differences in frequency between red and green, and that between green and blue. With quantum dots, however, the colours they emit is dependent on their sizes, which in turn is dependent on how many atoms they have. Thus, the colour step size here is devastatingly smaller than LEDs–take out one atom, or put one in, and you get a different frequency. Thus, a different colour. Voila! Human eyes are quite incapable of distinguishing that fine a colour step-size. To us, then, quantum dots can indeed produce any colour.

How do quantum dots produce so many colours?

Why is that? Aren’t quantum dots semiconductors too?

Yes, they are. However, unlike bulk semiconductor diodes, quantum dots do not have Avogadro’s number level of atoms6. In fact, the smallness of quantum dots means that quantum effects begin to make their presence felt.

Atoms & semiconductors

Atomic energy levels

Electrons is an atom are arranged according to their energies. Those with lower energies inhabit lower energy levels–or shells–and stay closer to the central nucleus, while those with higher energies live in higher shells. Now, shine light on a group of atoms. Some of the photons you fire will interact with the atoms. If the energy is correct, electrons might absorb the photon, gain energy, and jump to a higher shell. Pretty soon, however, that energised electron will want to fall back to the energy level it came from. It will do so by emitting a photon of its own. The energy of the photon emitted will be exactly equal to the energy difference between the two energy levels. If this happens fast, within nanoseconds, then this is fluorescence.

Schematic of fluorescence. An electron absorbs a high-energy photon and jumps to an excited state. After a series of vibrational transitions where no photon is emitted, the electron fluoresces back down to its ground state, emitting a photon. The emitted photon has a lower energy and longer wavelength (green) than the incident photon (blue).
An electron absorbs a high-energy photon and jumps to an excited state. After a series of vibrational transitions where no photon is emitted, the electron fluoresces back down to its ground state, emitting a photon. The emitted photon has a lower energy and longer wavelength (green) than the incident photon (blue). Image credit : Jacobkhed.

Semiconductor energy bands

A similar thing happens with light emitting diodes. However, since LEDs are semiconductors, they don’t have the discrete energy levels that atoms have. Instead, their energy levels form two separate continuum bands. An electron from the lower valence band absorbs energy and jumps up to the higher conduction band. In LEDs, it soon jumps back down, emitting a photon. The energy of the photon emitted is the sum of the energy of the band gap and the excitation energy of the electron :

E = Eband-gap + Eexcitation

In the image below, the brown double arrow shows the band gap. LEDs made of a particular material have a fixed band gap value, and thus can only emit light of a certain colour.

Schematic of the splitting of energy levels in quantum dots due to the quantum confinement effect, semiconductor band gap increases with decrease in size of the nanocrystal.
Splitting of energy levels in quantum dots due to the quantum confinement effect, semiconductor band gap increases with decrease in size of the nanocrystal. Image credit : Sigma-Aldrich / Merck.

Quantum dot confinement energy

Particle in a box

Contrast this with quantum dots. Since they are so small, electrons are trapped within them. Quantum mechanical effects kick in, and the energy levels of the electron are like that a particle trapped in a box7. This confinement has its own energy, and adds to the total energy of the fluorescent photon.

Efluorescence = Eband-gap + Econfinement + Eexcitation

It is this Econfinement that makes the quantum dot into a technicolour marvel. You see, according to quantum mechanics, the energy of a particle in a box depends on how large the box is. More particularly, the energy goes as the inverse square of the dimension of the box; decrease the box size by 2, and your energy will multiply by 4.

Electron in a quantum dot box

For an electron in a quantum dot, the quantum dot itself is its box. The inverse square law implies that smaller dots have higher Econfinement, and vice verse. You can see that in the energy level schematic above. As the size of the quantum dot decreases (red > green > blue), the energy gap increases (blue double arrow > green double arrow > red double arrow).

You can increase or decrease this gap by changing the size of the box, that is, the quantum dot. Theoretically, you can change it in steps of a single atom, that is, by a few tenths of a nanometers. The corresponding change in Efluorescence will be extremely fine, and virtually undetectable by human eyes.

Practicality

Of course, technology limits the amount of fine-tuning you can do. For instance, you can’t manually pick up an atom and deposit it within a quantum dot crystal, or pick one out of it. Yet. Everything depends on your crystal deposition process, what materials you use, what conditions you are working in, and how good you are. Still, modern quantum dots are good enough to have the highest true colour resolution among all others.

That is the magic of a quantum dot.

Is that all?

Er, no. That is just the tip of the iceberg as far as quantum dots are concerned. They have tons of other uses other than being fancy screens. Solar panels made out of quantum dots are more efficient than their non-QD counterparts. Also, biomedical applications of quantum dots have picked up steam recently. Turns out quantum dots are brilliant at imaging inside the body, and are being used as such. Also, they make superb organic dyes, and are a 21st century extension of William Perkin’s dream.

Quantum dots aren’t universally fantastic, though. To manufacture them, you need to use compounds of heavy metals that are incredibly toxic to humans, other animals, and the environment. This severely limits their in vivo use. To counter that, a group of Indian-Welsh scientists have very recently teamed up to fabricate quantum dots from the extract of tea leaves. Not only does this decrease the overall toxicity, these tea-powered British Popeye quantum dots seem to be devastatingly effective at invading and killing lung cancer cells.

More on this in the next edition of The Nerd Druid Investigates!, Quantum Dots from Tea Leaves Kill Lung Cancer Cells! In the meantime, here’s a picture of Tintin and Snowy drinking tea.

Tintin and Snowy drinking tea in The Blue Lotus, while a shadowy figure looms outside the window.
Tintin and Snowy drinking tea in The Blue Lotus, while a shadowy figure looms outside the window. Image credit : Hergé.

 

References

Papers

  1. Bawendi, Moungi G. & Steigerwald, Michael L. & Brus, Louis E. : The Quantum Mechanics of Larger Semiconductor Clusters (“Quantum Dots”), Annual Reviews of Physical Chemistry (1990)
  2. Reimann, Stephanie M. & Manninen, Matti : Electronic structure of quantum dots, Reviews of Modern Physics (2002)
  3. Yoffe, A.D. : Semiconductor quantum dots and related systems: Electronic, optical, luminescence and related properties of low dimensional systems, Advances in Physics (2010)
  4. Dey, Samrat et al. : The confinement energy of quantum dots, arXiv (2012)

Other resources

  1. Quantum Dots | Sigma-Aldrich
  2. Quantum Dots | Brus group, Columbia University
  3. Nanotechnology Timeline, National Nanotechnology Initiative
E 0102 : A mysterious isolated neutron star

E 0102 : A mysterious isolated neutron star

Reading Time: 11 minutes

Neutron stars are exotic remnants of supernova explosions of massive stars. Most rotating neutron stars have extraordinarily high magnetic fields. These pulsars very often have a binary companion star from which they steal stellar matter. Recently, a neutron star in a neighbouring galaxy was found to have neither a binary companion nor a strong magnetic field. Curiously, the position of the supernova remnant with respect to the gases thrown off at the time of supernova does not quite tally. The Nerd Druid investigates E 0102, the lonely offset neutron star.

Neutron stars

Neutron stars are supernova remnants that are not quite massive enough to form black holes1. They are extremely dense objects and have phenomenally strong gravitational fields. The pressure in the interior of a neutron star one of the highest found naturally in the universe, second only to the pressure inside protons. Temperatures within neutron stars right after formation can be as high as hundreds of billions of degrees, but escaping neutrinos cool the star down.

Schematic of pulsar, showing the rotation axis, the magnetic axis, and the twin beams.
Schematic view of a pulsar. The sphere in the middle represents the neutron star, the curves indicate the magnetic field lines, the protruding cones represent the emission beams and the green line represents the axis on which the star rotates. Image and caption from Wikipedia.

Pulsars

Rotating neutrons stars that also have high magnetic field emit synchrotron radiation2. This radiation forms a beam along the magnetic axis of the neutron star; as the star rotates about its rotational axis–which may or may not coincide with the magnetic axis–these beams are observed at precise regular intervals. These are pulsars, celestial timekeepers that are highly accurate. Thanks to the beams, pulsars are easy to detect.

Optical mosaic image of the Crab nebula, thanks to the Hubble Space Telescope. The bluish glow from the central region is due to synchrotron radiation.
Hubble Space Telescope mosaic optical image of the Crab Nebula. The bluish glow from the central region of the nebula is due to synchrotron radiation.

Pulsars are almost always binary stars; their stellar companions are often main-sequence or giant stars that have not yet reached supernova, or, due to lack of mass, never will. Most pulsars are found in the Milky Way galaxy and its neighbourhood.

So, basically, pulsar neutron stars are social beings that have magnetic personalities, but you’re likely to get burnt if you venture too close. Clearly, they have issues.

Composite optical and X-ray image of the inner Crab nebula. The central part shows the pulsar wind nebula, with the red star in the centre being the Crab pulsar.
Composite optical/X-ray image of the Crab Nebula, showing synchrotron emission in the surrounding pulsar wind nebula, powered by injection of magnetic fields and particles from the central pulsar. Image and caption from Wikipedia.

Central Compact Objects (CCOs)

However, not all neutron stars are pulsars. There are some that have very low magnetic fields compared to their pulsar cousins. Called central compact objects (CCOs), these supernova remnants lack the active wind nebulae3 that are a telltale signature of pulsars. The pulsars PuppisA and CassiopeiaA (CasA), both located within the Milky Way, are excellent examples of CCOs.

Similarly, not all neutron stars have a binary companion. The Magnificent Seven, named after the spaghetti Western, are a group of seven isolated neutron stars. A decade ago, an eighth such star was found, and was promptly named Calvera, after the villain of the movie. Going by that logic, a ninth star might well be named Gabbar4.

It is quite rare to find neutron stars that both CCOs and isolated. Vogt et al have recently found one that is located outside the Milky Way. However, in doing so, they have uncovered a rather puzzling mystery.

E 0102

The object of interest is the supernova remnant 1E 0102.2-7219. E 0102 for short, it is one of the few oxygen-rich young supernova remnants (O-rich SNR) in the Magellanic clouds5, and the only low magnetic field isolated neutron star outside the Milky Way. E 0102 is a little more than 200,000 light years away from Earth, and is only about two thousand years old.

Chandra/ACIS X-ray image of E 0102. The blue and the purple are false-coloured fast-moving highly ionized gases. This forms a lightyears-wide shell around what is thought to be the origin of the supernova. Note the small blue dot south-south-west of the centre, and about midway to the outer ring.
Chandra/ACIS X-ray image of E 0102. The blue and the purple are false-coloured fast-moving highly ionized gases. This forms a lightyears-wide shell around what is thought to be the origin of the supernova. Note the small blue dot south-south-west of the centre, and about midway to the outer ring.

Observing E 0102 with Chandra and Hubble

Unlike pulsars, CCOs such as E 0102 cannot be detected by looking for synchrotron radiation. Instead, one has to rely on blackbody radiation from both the CCO and the supernova debris surrounding the CCO. Images from NASA’s Chandra X-ray Observatory (CXO, named after Indian astrophysicist S. Chandrasekhar) show that E 0102 is dominated by a fast-moving large ring-shaped structure (blue and purple in the X-ray image). This is, quite possibly, the forward envelope of the supernova blast wave. Ejected at supernova at speeds of millions of kilometres per hour, this shell is now, two thousand years later, approximately 12 light years wide. Older optical images from the Hubble Space Telescope show beautiful filaments of superheated highly ionized oxygen filaments (green in the optical image) within the shell.

Hubble Space Telescope (HST) and VLT/MUSE optical image of E 0102. The green filaments (HST) are highly-ionized oxygen-rich filaments of supernova ejecta. The bright red ring (MUSE) is slow-moving low-ionization gas surrounding an optical dark spot.
Hubble Space Telescope (HST) and VLT/MUSE optical image of E 0102. The green filaments (HST) are highly ionized oxygen-rich filaments of supernova ejecta. The bright red ring (MUSE) is slow-moving low-ionization gas surrounding an optical dark spot.

Position of E 0102 supernova remnant

Usually, one would expect, given the data available, that the supernova remnant should be right in the middle of the ring that Chandra observed. Unfortunately, as you can see from the X-ray image above, there is no corresponding X-ray source at the centre of the ring. There is, however, an X-ray point source slightly offset from the centre of this large ring, a bit towards the south-west. Given available data, that cannot be the remnant.

Vogt et al disagree. And they evidence, persuasive evidence. And to do that, they have moved beyond what NASA has had to offer.

ESO, VLT, & MUSE

The European Southern Observatory is an international collaborative effort at observing the southern skies. Established in 1962 with 5 members, ESO now boasts a roster of 15 member states that sponsor the installation, operation and maintenance of state-of-the-art telescopes, all located in Chile. Of them, the imaginatively named Very Large Telescope (VLT) is probably one of the most advanced optical telescopes available for research now.

The Multi Unit Spectroscopic Explorer (MUSE) is an advanced spectrograph attached to the VLT. Unlike older devices, MUSE provides high-end resolution in both spatial and spectral domains. Which basically means that it produces very clear pictures with sharp colours.

Observing E 0102 with MUSE

Instead of relying on old Hubble and Chandra data, Vogt et al pointed the VLT towards E 0102. New optical data from MUSE showed that, in addition to the highly ionized fast-moving large gaseous ring, E 0102 also shows a smaller low-ionization slow-moving gas ring (bright red in the optical image) that is about a sixth of the former’s size. The origin of this new ring cannot be explained from the optical picture alone.

Animation of the optical and X-ray channels of E 0102. Overlaying one on the other strongly indicates that the blue off-centre dot in the X-ray image is indeed the CCO.
Superimposing the Chandra X-ray image on top of the HST+MUSE optical one shows that the blue off-centre X-ray dot does indeed fall within the red optical gas ring, and that it is indeed the CCO.

The fun begins when you put the optical and X-ray images on top of each other. It is immediately obvious that the smaller red optical ring neatly encircles the small blue X-ray dot. This perfect coincidence convinced Vogt et al that the small blue dot was indeed the hitherto undiscovered supernova remnant they had been looking for.

Is that really the supernova remnant?

Apparent and absolute magnitudes

Well, sort of. Stargazing is a funny job. Since the stars are so far away, the night sky looks more or less like an upturned bowl, with the stars stuck to its underside. Take Venus and Sirius, for instance. Venus is the brightest point object in the night sky, while Sirius is the brightest star in the night sky. Naturally, the Sun and the Moon are disq-ualified.6 Now, going by apparent brightness, Venus outshines Sirius by a factor of 23.337. Actually, of course, Venus is a planet, while Sirius is a really bright star. This is evident in their actual luminosities : Sirius is 22.5 times as luminous as the Sun, which in turn is 550 million times brighter than Venus.

Constellations and distances

The same confusion holds for distances. Constellations are groupings of stars that are close together. For instance, Orion (Kalpurush), one of the most distinctive and conspicuous constellations in the night sky, has seven major stars. Of these, the two brightest are Rigel, a blue-white supergiant, and Betelgeuse, a red supergiant. The most distinctive feature of the constellation is the belt; Alnitak, Alnilam and Mintaka are the three stars that make up Orion’s belt. All these stars, being part of the same constellation, seem to lie close to each other on the same plane.

Photograph of Orion's belt. On the northeast (top right) is Mintaka, on the southwest is Alnitak, and at the centre is Alnilam, the brightest of the trio.
Orion’s belt. On the northeast (top right) is Mintaka, on the southwest is Alnitak, and at the centre is Alnilam, the brightest of the trio. Image credit : Davide De Martin (http://www.skyfactory.org); Digitized Sky Survey, ESA/ESO/NASA FITS Liberator.

The operative word is, naturally, seem. Here is how far the stars actually are from Earth (lyr = lightyear) :

  1. Betelgeuse = 640 lyr
  2. Rigel = 863 lyr
  3. Alnitak = 800 lyr
  4. Alnilam = 1340 lyr
  5. Mintaka = 915 lyr

Clearly then, if we move away from Earth, the constellations will not retain their familiar shapes.

So, is it the CCO or not?

The same issue is relevant when it comes to E 0102’s CCO. Wouldn’t it be possible that the blue X-ray point source simply lies in the background of the red low-ionization ring of gas. Is it impossible to imagine that the extraordinary evidence provided by MUSE might simply be chance alignment?

Well, after four paragraphs and 300 words, you would be entitled to a detailed explanation. Sadly, Vogt et al simply say that the area of the structure seen by MUSE is too small. They calculate that there might be at most 8 × 10-5 sources in the background that are as bright or brighter.

Rather compelling, that number, ain’t it.

What’s more, the MUSE data does not provide any evidence for active pulsar wind nebulae, or the presence of a binary companion to the supernova remnant. Vogt et al can very well claim to have found the first truly isolated CCO outside the Milky Way.

Which is when all hell breaks loose.

The E 0102 offset mystery

The problem, of course, is that the E 0102 CCO does not quite lie at the centre of the large gas ring.

A bit of firecrackers

Picture a firecracker going off in the night sky. Imagine one of those expensive ones that, um, look good and are made well. Soon after explosion, the expanding sparkles form a ball, while whatever part did not explode (there’s always some) stays more or less at the centre of this ball. Of course, gravity soon makes all of this come crashing down onto the Earth.

Now imagine you are in charge of firecrackers on a cricket night, and you are a bit short on funds. You buy perhaps one of those fancy firecrackers and scrounge the bottom of the barrel for the rest. Your idea is to impress people with that one good one, the one that looks very symmetric, and then quickly pass the other asymmetric ones under the radar. You succeed, for there are people in the audience that are clearly more interested in the badly made ones, the ones where the explosion remnant doesn’t quite remain in the middle of the ball.

Photograph of fireworks at the Sydney Harbour Bridge, taken on New Year's Day 2017. Image credit : Saeed Khan.
Fireworks at the Sydney Harbour Bridge, taken on New Year’s Day 2017. Image credit : Saeed Khan.

Why does that happen? Well, simply because the explosive mixture in the cracker wasn’t uniform or homogeneous enough. There might have been bumps or hollows, or some parts might have had more or less of one component than necessary for the correct amount of burn. Essentially, the firecracker must have had some asymmetry to begin with.

How a supernova works

A similar specific set of circumstances might have been applicable with E 0102. Here’s how a supernova works : at the end of its life, massive stars8 throw off most their stellar material in a titanic explosion. What remains behind is the superdense core; if massive enough, it forms a black hole; if not, neutron star. The stellar material is thrown off at the rate of thousands of kilometers a second. This translates into a speed of a few lightyears every thousand years. The gaseous material thrown off forms a spherical shell around the supernova remnant, very much like the fast-moving large ring of high-ionization gas in E 0102 as detected by Chandra’s Advanced CCD Imaging Spectrometer (ACIS).

Two scenarios

Had E 0102 behaved like that good high-end expensive firecracker, its supernova remnant should have been right at the centre of this big shell. However, given the large offset, clearly this isn’t the case. This leaves us with two scenarios :

  1. The supernova was indeed centred at the centre of the large X-ray ring, and that, somehow, the supernova remnant has migrated to where it is now.
  2. The supernova was centred at the centre of the smaller optical ring, more or less where the CCO is now, and somehow the larger gas envelope got distorted.

Both scenarios are plausible. Both scenarios are problematic. Which is of course what makes this such a delicious mystery.

Supernova remnant migrated

Optical image (VLT/MUSE) of the centre of large gas shell (white +) and CCO position (white crosshairs) in E 0102. Image taken from Vogt et al, arXiv:1803.01006
Optical image (VLT/MUSE) of the centre of large gas shell (white +) and CCO position (white crosshairs) in E 0102. The black bar in the key equals 3 parsec or 9.78 lightyear. Image taken from Vogt et al, arXiv:1803.01006

For

The greenish blob is optical light from the oxygen-rich gases floating around E 0102. The small white + is where the centre of this expanding mass of gas should lie. The white crosshair marker slightly southwest of the centre marks the position of the CCO. As you can see, that portion does not have any oxygen. The scale below converts between seconds of arc and parsecs (pc); the thick black line is equal to 3 pc or 9.78 lyr. If you take a ruler and measure it, you’ll find that the distance between the centre and the CCO is approximately 1.8 pc or about 6 lyr.

6 lightyear! That is a lot of distance to cover. If our first scenario is correct, and the stellar core was indeed at the white + when the progenitor star went supernova, then it must have given a mighty kick to the CCO for it to have travelled 6 lyr in 2000 yr. However, the radius of the green blob is almost 4.5 pc (= 14.5 lyr). Also, 6 lyr in 2000 yr equates to a speed of about 9000 km/s (about 3% lightspeed), which is about what a supernova is capable of. The idea of a massive kick seems legit.

Against

However, there is a but. Where did the low-ionization slow-moving gas ring come from? The smaller ring is centred on the current position9 of the CCO, which means that somehow, the CCO must have reached the crosshair position, stopped (which is impossible), ejected a whole lot of gas (also impossible), and then waited as that gas very quickly spread to a diameter of a lightyear.

The timeline doesn’t match. Nor do the evidences tally with one another. And the physics is quite impossible. Stellar cores post-supernova do not have any more gas to expend. Yet we have a new gas ring. Optical analysis proves that this gas is slow-moving. Yet the timeline demands that it move ultrafast. Also, how in the world would a stellar core moving at 0.03c brake?

Hmm, trouble. Shall we try the alternate scenario, then?

Large gas ring distorted

Optical image of the inner gas ring (pink) and CCO position (crosshairs) in E 0102. Image taken from Vogt et al, arXiv:1803.01006
Optical image of the inner gas ring (pink) and CCO position (black crosshairs) in E 0102. The black bar in the key equals 3 parsec or 9.78 lightyear. Image taken from Vogt et al, arXiv:1803.01006

For

In the second scenario, the supernova didn’t kick the remnant away, and the CCO has been drifting more or less at its present position, as indicated by the black crosshair. The pink shell of low-ionization gas almost perfectly, but not exactly, encircles the CCO. Vogt et al show that the CCO has drifted only about 0.36 pc (= 1.17 lyr) from the centre of the smaller ring, thereby setting an upper limit for its drift velocity to be 170 kms-1, a far cry from 9000 km-1 in the kick scenario.

The pink inner shell could well have been ejected at supernova. Alternatively, the progenitor star might have been ejected it a few thousand years before supernova. This indicates that E 0102 might have been unstable, a variable star that would shed stellar matter periodically.

Against

Which begs the question : why did the bow shock of the supernova not blow this inner ring away? Also, and this is a bigger problem, the explosion site is now located well away from the centre of the X-ray emissions. In the image below, the white crosshair again marks the position of the CCO, while the bright bands of white, red, blue and some green make up a false-coloured image of the Chandra ACIS X-ray data.

X-ray image (Chandra/ACIS) of the outer gas ring and CCO position (white crosshairs) in E 0102. Image taken from Vogt et al, arXiv:1803.01006
X-ray image (Chandra/ACIS) of the outer gas ring and CCO position (white crosshairs) in E 0102. The black bar in the key equals 3 parsec or 9.78 lightyear. Image taken from Vogt et al, arXiv:1803.01006

According to Vogt et al, this might have been possible due a very specific set of circumstances. For instance, before exploding, stellar wind from the progenitor might have skewed the direction of the shock wave so that it is now off-kilter–a bit like one of those cheap firecrackers. Alternatively, variations in density in interstellar dust and gas might have altered the flow of the ejecta, so that some parts would move faster than the others. Thinking in terms of electricity10, the interstellar medium, though a vacuum, would have enough dust and gas to be inhomogeneous enough to distort the ejecta.

A specific set of circumstances

Which begs the question : how is the outer ring so round and regular?

Vogt et al attribute this to a specific set of circumstances. Essentially, it still is a mystery. However, with increased precision analysis of MUSE data, Vogt et al are confident they will have the answer for us. Until that time, here is a composite image of the Crab Nebula, showing infrared (Spitzer), optical (Hubble), and X-ray (Chandra) bands. You can make out the remnant as a bright blue-white dot near the centre.

Chandra (X-ray), Hubble (optical), and Spitzer (infrared) composite image of the Crab Nebula. Credit : X-ray: NASA/CXC/SAO/F.Seward; Optical: NASA/ESA/ASU/J.Hester & A.Loll; Infrared: NASA/JPL-Caltech/Univ. Minn./R.Gehrz.
Chandra (X-ray), Hubble (optical), and Spitzer (infrared) composite image of the Crab Nebula. Credit : X-ray: NASA/CXC/SAO/F.Seward; Optical: NASA/ESA/ASU/J.Hester & A.Loll; Infrared: NASA/JPL-Caltech/Univ. Minn./R.Gehrz.

 

References

Papers

  1. Vogt et al. : Identification of the Central Compact Object in the young supernova remnant 1E0102.2-7219, arXiv (2018)

Articles

  1. Press Release : Astronomers Spot a Distant and Lonely Neutron Star, Chandra X-ray Observatory (2018)
  2. PTI/Firstpost : NASA discovers rare neutron star outside of Milky Way and releases a stunning image of it (2018)

The Great Telescopes and their instruments

  1. Hubble Space Telescope (NASA)
  2. Chandra X-ray Observatory (NASA)
  3. Very Large Telescope (ESO)
  4.  Spitzer Space Telescope (NASA)

Image sources

  1. Composite, Optical, and X-ray images of E 0102 : Chandra X-ray Observatory
  2. Other images linked directly
Semiconductors : Junction Diodes and Transistors

Semiconductors : Junction Diodes and Transistors

Reading Time: 8 minutes

Semiconductors are at the heart of modern human civilization. Every word I type using this laptop is an implicit thank you to the incredible versatility and utility of semiconductors, as well as to the wonderful ingenuity of the scientists and engineers who have harnessed the potential of these magical materials to their fullest.

The following is part I of a brief overview of the physics of semiconductors. This is meant to be a starter pack, so to say. For deeper understanding, I suggest that you read a good textbook.

In this article, The Nerd Druid will talk about intrinsic and extrinsic semiconductors, p-n junctions, p-n junction diodes, and bipolar junction transistors and their applications.

Insulators and conductors

Let’s begin with sodium. Sodium is a metal. This means that it is a good conductor of heat and electricity. However, since sodium is very reactive, it isn’t very smart to use it to conduct electricity. Household wires are usually made of copper, which does a fine job of conducting current.

Image of sodium metal
Sodium metal

In comparison, nonmetallic elements and most compounds are poor conductors. Diamond, an allotrope of the nonmetal carbon, is a very good insulator. Ironically, graphite, another carbon allotrope, is a very good conductor. Graphene–atomic monolayers of graphite–is even better at this conduction business. 1 Air is good insulator, and it takes a huge voltage to pass electrons through air. Happens regularly around thunderstorms. Water is a decent conductor, since it has a lot of free protons (H+ cations) that carry the current.

Image of forked lightning. Extremely high potential differences between the Earth and thunderstorm clouds breaks down the insulation of air, forming forked lightning.
Forked lightning. Extremely high potential differences between the Earth and thunderstorm clouds breaks down the insulation of air, forming forked lightning.

Most natural materials found on Earth are either insulators or conductors.

Semiconductors

Intrinsic semiconductors

Some elements and compounds, however, do not identify themselves as either. Elements such as silicon and germanium, and binary compounds such as gallium arsenide and silicon carbide are semiconductors. When heated, semiconductors begin to conduct electricity. This behaviour is quite the opposite of conductors, where resistance increases with increasing temperature. These are intrinsic semiconductors.

Atomic orbitals

Before we proceed, a quick review of atomic orbitals might be helpful. A more detailed overview can be found here.

Electrons are arranged within atoms in orbitals. The energy of the electrons residing in them groups orbitals together in shells. Shells further away from the nucleus have higher energy than those closer to the nucleus. The most commonly encountered orbitals are s, p, and d. Each orbital can have at max 2 electrons.

Consider silicon. A neutral silicon atom has 14 electrons. These are arranged thus : 1s22s22p63s23p23d0. The first shell of silicon has 2 electrons (lowest energy) residing in the filled s-type orbital (1s). The second shell has 8 electrons (higher energy), 2 in the 2s and six in the 2p. The third shell has the remaining 4 electrons (highest energy); 2 in 3s and 2 is 3p. The 3d orbitals are completely vacant.

Image of silicon electronic configuration
Silicon electronic configuration

Extrinsic semiconductors

The behaviour of intrinsic semiconductors can be altered by adding small amounts of another element. For instance, silicon semiconductors can be thus doped using boron or phosphorus. Boron is trivalent, meaning it has three electrons in its outer shell (1s22s22p63s23p1); phosphorus is pentavalent, meaning it has five electrons in its outer shell (1s22s22p63s23p3). Since silicon has only four, phosphorus doped silicon has extra electrons which are free to move about. Similarly, boron doped silicon has fewer electrons, the absent electrons leaving behind holes, which are also free to hop about. The former is a n-type semiconductor, where electrons are the charge carriers 2. The latter is a p-type semiconductor, where the holes are the charge carriers 3 These extrinsic semiconductors are far more conductive than their undoped counterparts.

p-n junction

A single wafer of semiconductor can also be doped in two different ways; the boundary between the p-type and n-type regions form a p-n junction. When electrodes are attached to the two sides, this becomes the basic building block of all semiconducting electronic devices.

Depletion region

When a p-n junction is fabricated, there are more holes on the p-side and more electrons on the n-side. Immediately, holes and electrons begin migrating across the junction 4. This sets up a diffusion current pointing from the p-side to the n-side.

As holes diffuse over to the n-side, they leave behind negative bound ions. Similarly, diffusing electrons leave behind positive bound ions. Once on the n-side, the holes recombine with the electrons there and disappear. The electrons emulate this behaviour on the p-side. Soon enough, a region forms around the junction which has very few charge carriers. This is the depletion region.

Image showing formation of the depletion region in a p-n junction semiconductor
Formation of the depletion region in a p-n junction semiconductor

The cations and anions left behind set up an electric field pointing from the n-side to the p-side, thus opposing the diffusion current. Once a sufficient number of electrons and holes have diffused, this opposing field is strong enough to stop the diffusion current altogether, and set up its own drift current pointing from the n-side to the p-side.

p-n junction diode

Connect the p-n semiconductor to electrodes. This makes it a p-n junction diode. There are two ways this can be done.

Forward bias

In the first scenario, connect the anode (+ve electrode) to the p-side, and the cathode (-ve electrode) to the n-side. This is the forward bias condition.

The anode will inject holes into the p-side, while the cathode will inject electrons into the n-side. This will push and squeeze the depletion region so that it is narrow enough for electrons and holes to jump across. Once this happens, a current flows across the p-n semiconductor.

Illustration of forward bias in a diode. As the forward bias voltage increases, the deplection region gets narrower and finally collapses
As the forward bias voltage increases, the depletion region gets narrower and finally collapses. This allows current to flow through the diode.

Energetically speaking, the depletion region creates an energy barrier. In forward bias, if the voltage is not high enough, electrons and holes will not have sufficient energy to surmount the depletion barrier. Which is why the current-voltage graph of a p-n junction diode shows a knee in forward bias. The voltage above which the diode turns on is called the cut-in or the forward voltage (VF).

Reverse bias

In the second scenario, reverse the polarity 5. Connect the anode to the n-side and the cathode to the p-side. They will inject holes and electrons as before, though this time this will lead to greater recombinations and will serve to further widen the depletion region.

Illustration of reverse bias in a diode. Reverse bias widens the depletion region, making it almost impossible for current to flow through the diode.
Reverse bias widens the depletion region, making it almost impossible for current to flow through the diode.

Energetically, the reverse voltage will raise the height of the depletion region potential barrier. In this reverse bias state, the diode does not let current pass. Well, not unless the voltage is high enough so that the current flows in the opposite direction as before. That is breakdown. But we don’t want that to happen.

Transistors

Diodes are electronic valves; they allow current to pass in one direction. As such, they are very useful in circuits. However, they can be made much more useful if two p-n junction semiconductors are attached end-to-end in series, creating three separate zones. Attach electrodes to these three zones and you have a bipolar junction transistor (BJT).

There are two ways to do this; you either get a p-n-p transistor or an n-p-n transistor. The middle zone is called the base, and the ones on the two ends are called the collector and the emitter. Now, connect electrodes to each of these regions and apply potentials.

Image of an n-p-n transistor, with n-type collector and emitter, and p-type base.
An n-p-n transistor, with n-type collector and emitter, and p-type base.

Of course, BJTs are not the only type of transistor around. Field effect transistors work in a slightly different fashion. I’ll not be discussing FETs here.

Modes of operation : Switch and amplifier

The transistor can operate in four modes. One can control this by tuning the base voltage VB, the emitter voltage VE and the collector voltage VC. In what follows, I shall use an n-p-n transistor to illustrate the mechanism. You can convert this into a p-n-p by merely flipping the “>” signs and reversing the current.

Saturation mode : VB > VC , VB > VE

Recall that the transistor is n-p-n, so that the base is p and the emitter and collector are n. Thus, VBE > 0 and VBC > 0 imply that both the base-emitter (BE) and base-collector (BC) junctions are in forward bias mode. This setup behaves like a closed circuit; current flows freely between the collector and the emitter. For n-p-n, the collector current IC points from the collector to the emitter. Electrons, the majority carriers in n-p-n, are emitted by the emitter and collected by the collector 6.

The transistor is now ON.

Cut-off mode : VB < VC , VB < VE

The signs are reversed; VBE < 0 and VBC < 0. Both the BE and BC junctions are reverse biased. No current flows in the transistor, and it behaves like an open circuit.

The transistor is now OFF.

By reversing the voltages, one can thus use the transistor as a switch.

Forward active mode : VC > VB > VE

Here, VBE > 0 but VBC < 0; the BE junction is forward biased, while the BC junction is reverse biased. In this mode, the transistor acts as an amplifier–a small input base current IB gives rise to a very large output collector current IC.

IC = β IE

where β is large, usually 100.

Image of a transistor in forward active mode, acting as an amplifier. The n-type emitter emits electrons, a majority of which reach the collector due to the large potential difference between C and E.
A transistor in forward active mode, acting as an amplifier. The n-type emitter emits electrons, a majority of which reach the collector due to the large potential difference between C and E.
Reverse active mode : VC < VB < VE

This is basically the same as forward active, except β is much smaller. This mode is seldom used in practice.

Schematic of the four modes of a transistor; saturation (ON), cutoff (OFF), forward active (amplifier), and reverse active.
The four modes of a transistor; saturation (ON), cutoff (OFF), forward active (amplifier), and reverse active.

Applications

Transistors are the basic building blocks of computers and modern electronics, and have wide-ranging applications.

As a switch

Keep the transistor in switch mode and assign labels to the states of the switch; 0 for OFF, 1 for ON. The transistor now functions as a bit, the fundamental unit of classical computing.

You can string a few transistor switches together and form logic gates. These follow Boolean algebra, and allow you to perform OR, AND and NOT operations. More complex combinations of transistors give NOR, NAND, and XOR gates.

Bits and logic gates can be used together to implement a series of instructions. This implementation of an algorithm is a computer program.

Finally, by connecting transistors together in certain ways, it is possible to make it stay either in the saturation or the cut-off states until a new input has been applied. Such flip-flops can either be in 0 or 1 states, and thus can store bits. These are the building blocks of computer memory.

As an amplifier

Hearing aids are wonderful applications of transistor amplifiers. The microphone converts sounds into electrical current. It then feeds this into the base of a transistor which is operating in the active mode. The collector current, greatly amplified, feeds into a loudspeaker. Prof Calculus (in moon rocket mode) would greatly approve.

A panel from the tintin book Destination Moon, showing Professor Calculus and Captain Haddock
Professor Calculus and Captain Haddock, from Destination Moon.

References and Further Reading

  1. All About Circuits
  2. ExplainThatStuff
  3. SparkFun
Exploring the Subatomic World : Electrons and Nuclei

Exploring the Subatomic World : Electrons and Nuclei

Reading Time: 6 minutes

Atoms were once thought to be indivisible and fundamental. Nineteenth century chemistry was based around atoms, culminating in Dmitry Mendeleev’s extraordinary Periodic Table. The discovery of the electron in 1897 led to a subatomic revolution, and the twentieth century has revealed the rich internal structure of the atom, culminating in the Standard Model, the subatomic counterpart of Mendeleev’s Table. In this series of brief articles, The Nerd Druid traces the wonderful history of the subatomic.

Subatomic particles

Protons, neutrons and electrons are the three most well-known subatomic particles. Earth has an abundance of free protons and electrons. All you need for the former is to somehow make electricity happen. An easy way to do that is to rub amber with fur 1. For the latter, water works pretty well, since it has plenty of free protons roaming about. Of course, it helps to keep in mind that protons are simply ionised hydrogen atoms.

The Electron

The ancient Greeks noticed that if you rub amber with fur, the amber tended to attract small light objects. This, along with lightning strikes, was the only connection humans had with electricity until modern times.

Image of Garfield (the cartoon cat character) rubbing his fur against Jon's pants. This creates static electricity. The charged fur repels each other.
Garfield rubbing his fur against Jon’s pants creates static electricity. The charged fur repels each other.

Cathode Rays

In the latter half of the nineteenth century, German and English physicists found that if you pull air out of sealed container and insert a cathode 2 into it, you see a glowing discharge. Pump out more air, lower the air pressure further, and this cathode discharge glows brighter. The pioneers of cathode ray physics were Johann Hittorf (Germany) and Eugen Goldstein (Germany); they did their work in 1869 and 1876 respectively.

William Crookes (England) created the first cathode ray tube in the 1870s by creating a high vacuum. He observed that the glow had now become a sort of a ray, moving from the cathode to the anode 3. Reckon we’d have to thank Crookes for all the televisions and older computer monitors.

Crookes also applied a magnetic field to the cathode rays and made them deflect. This showed that the rays were charged. Arthur Schuster (Germany-England) modified Crookes’ setup. He sandwiched the cathode ray between two parallel plates, one positive and one negative. The electric potential between the plates made the rays bend and strike the positive plate. This proved that the rays were negatively charged.

Discovery of the electron

Schuster didn’t stop there. By varying the current fed into his parallel capacitor, Schuster was able to vary the degree by which the cathode rays were deflected. He measured and tabulated these results, and was thus able to calculate the charge-to-mass (Q/m) ratio of the cathode rays. His results were astonishing–the Q/m ratio of the cathode rays seemed to be more than a thousand times what was expected!

Unfortunately, Schuster’s results were ignored, probably because they didn’t quite conform to what people knew about atoms. The prevalent idea at that time was that cathode rays were some sort of atoms or molecules, and thus they were expected to have a mass at least equal to the hydrogen ion.

Image of J.J. Thomson, English physicist. Experimentally obtained evidence for the existence of electrons. Thus showed that cathode rays are independent particles. Showed that these negatively charged particles are the same as the ones produced by radioactivity, by heated materials, and by illuminated materials. Suggested the plum pudding model for the distribution of positive charges and electrons within the atom. Discovery of the electron threw open the subatomic realm.
J.J. Thomson, English physicist

In 1897, J.J. Thomson (England), working with colleagues John Townsend (Ireland) and H.A. Wilson (England), showed that cathode rays were, indeed, individual charged particles. A few years later, Millikan and Fletcher performed the famous oil-drop experiment and accurately measured the charge of this new particle, the electron 4. Since the charge-to-mass ratio of the electron had already been measured, it was now simple to calculate the mass of the electron. Simply put, the electron has the same magnitude but opposite sign of the charge of a hydrogen ion. However, it had almost two thousandth its mass.

The Atomic Nucleus

Thanks to Thomson and his fellow cathode ray physicists, the atom was no longer a black box. Negatively charged cathode rays were actually particles called electrons, and they lived inside an atom. Sometimes some of these electrons would get knocked off the atom. The charged atom would then be a positive ion, a cation. At all other times, the atom would be strictly neutral. Clearly, the atom within contained an amount of positive charge equal to the negative charge it contained due to the electrons. The question was, where did the positive charge reside, and where were the electrons in relation to this?

Thomson had an answer to this. He envisaged electrons embedded within a uniform diffuse distribution of positive charge within the atom, much like plums in a pudding. The Geiger-Marsden experiment (1909) put paid to this plum pudding model soon enough.

Image of Thomson's plum pudding model (above) and Rutherford's Gold Foil experiment (below, aka the Geiger-Marsden expt.).
Thomson’s plum pudding model (above) and Rutherford’s Gold Foil experiment (below, aka the Geiger-Marsden experiment)

Hans Geiger 5 (England) and Ernest Marsden (England-New Zealand), working under the direction of Ernest Rutherford (New Zealand-England), fired positively charged alpha particles 6 at a metal foil. While most of the alpha particles whizzed through without any change in momentum, a tiny fraction (1-in-20000) deflected by almost 90 degrees. Rutherford concluded (in 1911) that this must be because all the positive charge in the atom is tightly packed inside a tiny volume at its centre. This, of course, is the atomic nucleus.

There were quite a few experiments performed by Geiger and Marsden. The most famous one is also referred to as Rutherford’s Gold foil experiment.

Nationalities

A quick aside about the nationalities of the people involved. A majority seem to be from England and Germany. The only two scientists from the US are Millikan and Fletcher, and they did their work in the 20th century. This seems to suggest that the nerve-centre of cutting edge physics in the nineteenth century was very much a few nations in Europe.

Nuclear density and pressure

The atom is mostly empty space 7 Atomic diameters are usually of ångström order (1 Å = 10-10 m). Nuclear diameters are a hundred thousand times smaller, usually of femtometer order (1 fm = 10-15 m). However, nuclei carry most of the atom’s mass. Due to their small sizes, they are incredibly dense objects.

A quick back-of-the-envelope calculation for carbon (C-12 isotope) shows that its nuclear density is approximately 1.25×1015 gm/cc. To put that into perspective, the density of ordinary water, under normal atmospheric pressure and standard room temperature, is approximately 1 gm/cc. Atomic nuclei are hundreds of trillion times denser than tap water.

Before we proceed, I’m going to presume that you are aware that atomic nuclei contain protons and neutrons. I’ll get to them in the next part of this series.

You’d think that such high densities would mean that the pressure inside the nucleus would be immense. You’d be correct, but there are places in this universe which make nuclear pressures seem like cotton candy. I’m talking about the interior of neutron stars, where pressures reach absurd values of 1034 pascal. In comparison, standard atmospheric pressure is about a hundred thousand pascal, 105. This of course makes neutron star cores very very hot. The story of how they cool down is quite interesting.

Up until a few days ago, this was the highest pressure found in the universe. Recently however, scientists at Jefferson Lab (the USA) have found that proton pressure is ten times that inside neutron stars. That is a mind-boggling million trillion trillion times that of standard Earth sea-level pressure.

Yes. A million trillion trillion times. 10 followed by 30 zeros.

In the next article, we encounter nucleons.


References

List of People : Who Did What

  1. Dmitry Mendeleev : Russian chemist
    • Designed the Periodic Table of Elements
  2. Johann Wilhelm Hittorf : German physicist
    • Discovered that a cathode within an evacuated chamber emitted a glow
    • Found that the intensity of the glow increased as pressure is lowered
  3. Eugen Goldstein : German physicist
    • Showed that the rays from Hittorf’s glow cast a shadow
    • Named the rays cathode rays
  4. Sir William Crookes : English chemist and physicist
    • Invented the extreme low pressure cathode ray tube
    • Showed that cathode rays travel as a straight beam between a cathode and anode
    • Showed that cathode rays deviate in a magnetic field, proving that they are charged
  5. Arthur Schuster : German-British physicist
    • Placed the cathode ray between the plates of a parallel plate capacitor
    • Showed that the beam deviated towards the positive plate
    • Thus showed that cathode rays are negatively charged
  6. J.J. Thomson : English physicist
    • Experimentally obtained evidence for the existence of electrons
    • Thus showed that cathode rays are independent particles
    • Showed that these negatively charged particles are the same as the ones produced by radioactivity, by heated materials, and by illuminated materials
    • Suggested the plum pudding model for the distribution of positive charges and electrons within the atom
    • Discovery of the electron enabled the subatomic world to be probed
  7. George Johnstone Stoney : Irish physicist
    • Introduced the term electron for units of electricity
  8. Robert Millikan : American physicist
    • Performed the famous oil-drop experiment and measured the charge of the electron
  9. Harvey Fletcher : American physicist
    • Performed the famous oil-drop experiment and measured the charge of the electron
  10. Ernest Rutherford : New Zealand-British physicist
    • The father of subatomic and nuclear physics
    • Called the greatest experimentalist since Faraday
    • Supervised the Geiger-Marsden experiment or the Gold foil experiment that disproved the plum-pudding model and discovered the atomic nucleus
    • Discovered the proton
    • Suggested that protons within the nucleus have a neutral partner
    • Named them neutrons
  11. Hans Geiger : German physicist
    • Performed the Geiger-Marsden experiment that disproved the plum-pudding model and discovered the atomic nucleus
    • Invented the Geiger counter, a detector of radioactivity
  12. Ernest Marsden : New Zealand-English physicist
    • Performed the Geiger-Marsden experiment that disproved the plum-pudding model and discovered the atomic nucleus

Original papers

  1. Thomson, J.J. : Cathode Rays, Philosophical Magazine (1897)
  2. Rutherford, Ernest : The scattering of α and β particles by matter and the structure of the atom, Philosophical Magazine (1911)
  3. Millikan, R.A. : On the Elementary Electrical Charge and the Avogadro Constant
Proton Pressure : How do Quarks Tackle Stress?

Proton Pressure : How do Quarks Tackle Stress?

Reading Time: 9 minutes

Proton pressure, theoretically estimated to be higher than even neutron stars, has been notoriously difficult to measure experimentally. Recently, however, particle physicists have used a nice trick to measure proton pressure. The Nerd Druid delves deeper.


Subnucleonic particles

Quarks

Particle physics is the study of extremely small particles. In the nineteenth century, atoms were thought to be indivisible and fundamental. Then, in 1897, J.J. Thompson discovered the electron. Following the Geiger-Marsden experiment in 1909, Ernest Rutherford in 1911 postulated the existence of the proton. The now-familiar subatomic trio was completed when James Chadwick discovered the neutron in 1932.

Image of a comparison of various physical properties of the three most familiar subatomic particles.
Comparison of various physical properties of the three most familiar subatomic particles.

In the three decades following 1932, the use of higher and higher energies in particle accelerators led to the discovery of a veritable particle zoo. In 1964, Murray Gell-Mann and George Zweig attempted to simplify matters. They proposed that protons, neutrons and all the other hadrons were not as fundamental as once thought. Instead, they suggested that smaller particles called quarks are the true fundamental particles that make up hadrons.

Image of Murray Gell-Mann (left) and George Zweig (right) were the first to propose the existence of quarks. 10 years after their proposal, the discovery of the charm quark in 1974 definitively proved Gell-Mann and Zweig to be correct. Gell-Mann received the Nobel Prize in physics in 1969.
Murray Gell-Mann (left) and George Zweig (right) were the first to propose the existence of quarks. 10 years after their proposal, the discovery of the charm quark in 1974 definitively proved Gell-Mann and Zweig to be correct. Gell-Mann received the Nobel Prize in physics in 1969.

The naming of quarks

Quarks are possibly one of the first particles named without reference to a Greek or Latin dictionary. Murray Gell-Mann, the man responsible for the name, had borrowed the word from the novel Finnegan’s Wake by James Joyce. At one point in the novel, a character, presumably drunk, exclaims

Three quarks for Muster Mark!

the words quarks and Muster possibly drunk modifications of quarts and Mister.

The Nerd Druid has once attempted to read Finnegan’s Wake. In his opinion, had Joyce not written Ulysses, Finnegan’s Wake would have been, without doubt, one of the most difficult books to read in the English language.

The discovery of the quarks

According to Gell-Mann and Zweig, there were three types or flavours of quarks. These were the up, down and the strange quarks. In 1970, Glashow, Iliopoulus and Maiani (GIM) proposed that a fourth flavour, the charm quark, should also exist. Three years later, Kobayashi and Maskawa’s proposal of top and bottom quarks increased the number of quark flavours to six.

Despite such stunning theoretical success, particle physicists were initially unwilling to accept the quark model. Richard Feynman proposed an alternative and called them partons. However, increasingly complex and beautiful experiments in the decade since Gell-Mann and Zweig seemed to weigh evidence in favour of quarks. Deep inelastic scattering experiments at the Stanford Linear Accelerator Centre (SLAC) found particles that would later be identified with the up and the down quark. The SLAC experiment also indirectly suggested that strange quarks should exist.

The November Revolution

The final breakthrough came in the November of 1974. Unlike its more illustrious and political Russian cousin, the particle physics November Revolution took place at SLAC and at Brookhaven National Laboratory, both in the USA. The two teams almost simultaneously discovered the charm quark and its antiparticle, bound together to form a meson. Burton Richter, who led the SLAC team, named it the ψ (psi) meson. Samuel Ting, who led the Brookhaven team, preferred to call it the J meson. Due to the joint announcement of their discovery, the J/ψ meson is the only particle with a two letter name.

Images of Samuel Ting (left) and Burton Richter (right), discoverers of the J/psi meson. They received the 1976 Nobel Prize in Physics for their work.
Samuel Ting (left) and Burton Richter (right), discoverers of the J/ψ meson. They received the 1976 Nobel Prize in Physics for their work.

The J/ψ discovery validated the quark model and led to rapid changes in high-energy particle physics. One of the major innovations to come out of this incredible time is what is known as the Standard Model of particle physics. To put it in a nutshell, the Standard Model puts the entire workings of the universe–sans gravity–in a nutshell. Thus, the name November Revolution.

The Standard Model

The standard model did to high-energy particle physics what Dmitry Mendeleev’s periodic table had done for chemistry almost exactly a hundred years earlier. It cut through all the noise and gave a beautiful tabular representation of all the fundamental particles that interact among each other via the four fundamental forces 1.

Image of a schematic depiction of the Standard Model of elementary particles. Shown are the six quark flavours, the three leptons and their corresponding neutrinos, the four force-carrying vector gauge bosons, and the solitary mass-carrying scalar gauge boson, the Higgs.
A schematic depiction of the Standard Model of elementary particles. Shown are the six quark flavours, the three leptons and their corresponding neutrinos, the four force-carrying vector gauge bosons, and the solitary mass-carrying scalar gauge boson, the Higgs. (Image Credit : Wikipedia).

The Fermions

In the image above, the six purple boxes represent the six flavours of quarks, while the six green boxes represents the three leptons and their neutrino counterparts. These twelve particles are fermions : particles that follow Fermi-Dirac statistics and have half-integer spins.

The Bosons

The particles on the right half of the image are bosons. These follow Bose-Einstein statistics and have integer spins. The four red boxes represent vector gauge bosons. These carry and mediate three of the four fundamental forces of nature; the fourth, gravity and its mediating gravitons, are still out of the standard model’s grasp. The yellow box on the very right represents the solitary scalar gauge boson, the Higgs, which carries and mediates mass.

Thus, all in all, 17 particles (and their antiparticles) are involved in almost everything that happens in the universe. The Nerd Druid will, hopefully, expand on the standard model in a later post.

Proton Constituents

Quarks are the building blocks of protons, neutrons, and a host of other particles. Together, these quark-based particles are called hadrons. For instance, the J/ψ meson is a hadron. There are two types of hadrons:

  1. Baryons : Composed of three quarks. These can be any flavour.
  2. Mesons : Composed of a quark and its antiquark.

The J/ψ is a meson because it comprises the charm (c) quark and its antiquark. Protons are composed to two up quarks and a down quark (uud), while neutrons have two down and one up quark.

Image of the quark composition of protons and neutrons. On the left, two up and one down quark make up a proton. On the right, two down and one up quark make up a neutron. The quarks have different "colour charges" that add up to "white", and are joined by gluon flux tubes.
On the left, two up and one down quark make up a proton. On the right, two down and one up quark make up a neutron. The quarks have different “colour charges” that add up to “white”, and are joined by gluon flux tubes. Do note that the flux tubes as shown here are wrong. They do not act along the sides of the triangle making up the quarks, but rather form a Y-shape, much like the flux capacitor in the movie Back to the Future.

Why does a proton not break up?

An up quark has charge +2/3, a down quark has charge -1/3. This makes the interior of nucleons a very interesting place. Take the proton, for instance. It has two positively charged up quarks within it, each of which is attracted towards, and attracts, the third negatively charged down quark. However, since they are both positive, they repel each other. How in the world does the proton not break up?

{Atomic nuclei have a similar problem. Going by common sense, the positive protons should never be confined to such a small region.}

If conditions are energetically favourable, bound protons transmute into neutrons. Free protons, one of the stablest objects in the universe, only do so if bombarded with high energy particles. The quarks within them never fly apart on their own, although, electromagnetically speaking, they should be flying apart. What the proton needs is an attractive force that keeps the various quarks together. One candidate is gravitation. However, that is approximately 1040 times weaker than the EM force. So that’s no help.

The Strong Force

Luckily, there is a force stronger than even the EM force. This is the imaginatively named strong force. While the EM force is mediated by photons, the strong force is mediated via gluons.

Take a quick peek at the standard model diagram. The topmost red box on the right half is the gluon. It is a vector gauge boson, and functions as a force-carrier. The strong force holding the quarks together manifests as constant exchanges of gluons between the quarks. From a quantum field theoretic viewpoint, the three quarks within the proton live in a sea of frothing bubbling gluon field.

{The strong force is, of course, also responsible for holding the nucleus together, and manifests itself via constant exchange of gluons between protons and neutrons.}

Proton Mass

The mass of an atom is approximately the same as the mass of its nucleus, since electrons are 1839 times lighter than neutrons, and 1836 times lighter than protons. Adding the masses (the rest masses) of the nucleons is the mass of the atom minus the binding energy needed to keep the nucleus together. However, this is still a small percentage of the nucleonic rest masses.

Inside a proton, the situation is rather different. The total rest mass of the three quarks (2u+1d) is only about 10% of the total mass of the proton. The rest 90% of the mass is provided by the highly energetic gluon field. Since it has energy, and by Einstein’s Mass-Energy equivalence, E = mc2, the gluon field has relativistic mass.

Image of a schematic of the masses of the six different quark flavours. The larger the balls, the higher the mass. The mass of the proton (grey) and electron (red) are shown in the southwest corner. Note that the total mass of 2u+1d would be quite a bit lesser than that of the proton.
A schematic of the masses of the six different quark flavours. The larger the balls, the higher the mass. The mass of the proton (grey) and electron (red) are shown in the southwest corner. Note that the total mass of 2u+1d would be quite a bit lesser than that of the proton.

Proton Pressure

Probing the inside of a proton is not easy. While it is possible to build up a three-dimensional image of the proton using the EM force, probing the mechanical properties of the proton is another matter entirely.

The Energy-Momentum Tensor

The energy-momentum tensor provides a complete classical dynamical description of a particle. It encodes information such as its mass and its momentum, and the shear stresses and pressures on it. Einstein’s Field Equations, which form the backbone of general relativity, relates this tensor with the curvature of spacetime. In the image, the E-M tensor is represented as a 4-by-4 matrix. The energy density, shown in the red box on the top left corner, is equivalent to the mass. The rest of the diagonal, the green band, represents pressure.

Image of the energy-momentum tensor, represented as a 4-by-4 matrix. The northwest corner (red) is mass. The northern and western walls (yellow-orange) are momentum. The elements along the diagonal (green) are pressure acting along the three space axes.
The energy-momentum tensor, represented as a 4-by-4 matrix. The northwest corner (red) is mass. The northern and western walls (yellow-orange) are momentum. The elements along the diagonal (green) are pressure acting along the three space axes.

Gravitational Form Factors

To obtain information about the mechanical structure of a proton, you’d need to prod it with a gravity probe. However, a gravity probe will not directly measure the matrix elements of the E-M tensor. Instead, what it can do it measure what are called gravitational form factors. In 1966, Heinz Pagels developed the concept of these GFFs and also analytically calculated them. Pagels, however, was not very confident that it would ever be possible to measure the detailed structure of a particle using the GFFs, since there was

…very little hope of learning anything about the detailed mechanical structure of a particle, because of the extreme weakness of the gravitational interaction.

And therein lies the rub. Gravity is just too weak a force at the scale of subnucleonic particles.

Image of Heinz Pagels, who constructed the gravitation form factors for nucleons in 1966, at the age of 27. He was extremely capable of explaining science in a simple way. Pagel's work in chaos theory provided the inspiration for the character of Ian Malcolm in Michael Crichton's Jurassic Park.
Heinz Pagels constructed the gravitation form factors for nucleons in 1966, at the age of 27. He was extremely capable of explaining science in a simple way. Pagel’s work in chaos theory provided the inspiration for the character of Ian Malcolm in Michael Crichton’s Jurassic Park.

Generalised Parton Distributions

If restricted to EM probes, then, using something called generalised parton distributions, researches can build up a 3D model of the proton. The word parton here is not quite used in the sense Feynman envisioned it; the term parton nowadays collectively refer to quarks, antiquarks, and gluons. GPDs were introduced around the year 1994. In the quarter century till then, they have emerged as an extremely valuable tool for probing hadrons.

However, the puzzle was still incomplete. GPDs are electromagnetic objects, while GFFs are gravitational. Proton investigations required a bridge between these two.

Measuring Proton Pressure

Deeply Virtual Compton Scattering

The bridge was finally proposed in 1997, when the Xiandong Ji suggested that a method called deep virtual compton scattering (DVCS) could be used to map GFFs onto GPDs. In this process, a beam of electrons is fired into atomic nuclei. Some of these electrons shoot through protons. In doing so, they transfer some of their energy to one of the quarks inside the proton. This is mediated by a virtual photon. The proton, thus energised, doesn’t hold on to this excess energy for long. It soon emits another photon and drops down to its original state.

Magnitude of Proton Pressure

By analysing these photons, Jefferson Lab scientists Volker Burkert, Latifa Elouadrhiri, and Francois-Xavier Girod (BEG) have built up a mechanical picture of the interior of the proton. In particular, they have calculated proton pressure for the very first time. BEG have found that quarks near the centre of the proton feel pressures of the order of 1035 pascal.

To put that into perspective, standard atmospheric pressure at sea level on Earth is slightly more than a million pascal. That is 106.

The centres of neutron stars used to be the undisputed champions of the universe when it came to ultrahigh pressures. They’ve been dethroned, for proton pressure is about 10 times the pressure found within neutron stars.

Distribution of Proton Pressure

Image of the proton pressure distribution. The y-axis is in units of force (positive outward), while the x-axis shows distance from proton centre (in femtometer, 10^-15 m). The repulsive proton pressure near the centre, directed outwards, is balanced by the confining proton pressure from the outer edges of the proton, directed inwards.
Pressure distribution within a proton. The y-axis is in units of force (positive outward), while the x-axis shows the distance from proton centre (in femtometer, 10^-15 m). The repulsive pressure near the centre, directed outwards, is balanced by the confining pressure from the outer edges of the proton, directed inwards.

According the BEG analysis, the proton pressure at the centre of the proton is highest. It is also repulsive, threatening to push the three quarks apart. However, proton pressure near its edge, though less in magnitude, is compressive in nature. This confines the quarks to within the interior of the proton, within a diameter of about 2 fm (1 fm = 10-15 m).

BEG are not done yet, though. They are waiting for better and precise data that they hope to obtain soon. This will help them polish some of their current results on the distribution of proton pressure. BEG are confident that their method of DVCS-GPD-GFF will reveal more about the interior of the proton, such as internal shear forces and the its mechanical radius.


References

Original Paper

Burkert, V. and Elouadrhiri, L. and Girod, F-X. : The pressure distribution inside the proton, Nature (2018).

Reference papers

  1. Pagels, Heinz : Energy-Momentum Structure Form Factors of Particles, Physical Review (1966)
  2. Maxim V. Polyakov, Maxim V. and Schweitzer, Peter : Forces inside hadrons: pressure, surface tension, mechanical radius, and all that, arXiv (2018)
  3. Ji, Xiangdong : Gauge-Invariant Decomposition of Nucleon Spin, Physical Review Letters (1997)
  4. Ji, Xiangdong : Deeply virtual Compton scattering, Physical Review D (1997)
  5. Rutherford, Ernest : The scattering of α and β particles by matter and the structure of the atom (2011)
  6. Chadwick, James : Possible Existence of a Neutron, Nature (1932)
  7. Belitsky, A.V. and Radyushkin, A.V. : Unraveling hadron structure with generalized parton distributions, Physics Reports (2005)

Other resources

  1. Perdrisat, Charles and Punjabi, Vina : Nucleon Form Factors, Scholarpedia (2010)
  2. Wikipedia
    1. Atoms#Subatomic Particles
    2. Proton#History
    3. Neutron#Discovery

Articles

  1. Press Release : First measurement of subatomic particle’s mechanical property reveals distribution of pressure inside proton
  2. Press Release : Quarks feel the pressure in the proton
    1. EurekaAlert (2018)
    2. Jefferson Lab (2018)
  3. Mandelbaum, Ryan F. : Scientists Calculate the Pressure Inside a Proton and It’s Higher Than in a Neutron Star, Gizmodo.com (2018)
  4. Micu, Alexandru : Pressure in protons’ cores is over ten times greater that that in neutron stars
  5. News Staff : Physicists Measure Pressure Distribution inside Proton, Sci-News.com (2018)
  6. Yirka, Bob : Best of Last Week – Measuring pressure inside a proton, flying wireless robot insects and yogurt may treat inflammation, ScienceX.com (2018)
  7. McRae, Mike : Protons Contain 10 Times More Pressure Than a Neutron Star, According to First-Ever Measurement, Science Alert (2018)
  8. Grossman, David : Pressures Inside a Proton Are More Extreme Than Inside a Neutron Star, Popular Mechanics (2018).
  9. Williams, Matt : The Pressure Inside Every Photon is 10x That Inside Neutron Stars, Universe Today (2018)
Direct Urca : How Thieving Neutrinos Cool Neutron Stars

Direct Urca : How Thieving Neutrinos Cool Neutron Stars

Reading Time: 11 minutes

Neutron stars are one of the most interesting and exotic objects in the universe. Formed as remnants of stellar cores post-supernova, core temperatures of these ultradense stars at formation can be as high as hundreds of billions of kelvin. Neutron stars cool off by shedding neutrinos–either via the superfast direct Urca process, or by the much more sedate modified Urca process. While the latter mechanism has been observationally confirmed, there has been no compelling evidence for the direct Urca process. Until now. The Nerd Druid investigates how neutrinos rob neutron stars of their heat, and how quick are their ghostly pickpockety hands.

Stars

Stars are fascinating objects. The sun is one. It is middle-aged, having burnt through about half of its ten billion year lifespan. The sun is a yellow-white main sequence star of average luminosity and mass. There are other types of stars; some are far more massive and luminous than the sun, some are far less massive and much dimmer. Some stars shine blue-hot, some are cool and very red. Just like living beings, stars have a beginning and an end. Near the end of their lives, stars shed excess mass, either via spectacular explosions called supernovae or in far more docile fashion. Near the end of their lives, stars shed excess mass, either via spectacular explosions called supernovae or in far more docile fashion. In the latter, small superdense stellar core remnants called white dwarfs are formed. In the former, tiny ultradense stellar core remnants called neutron stars may be formed, unless the mass of the remnant is high enough to trigger an unstoppable collapse, resulting in a black hole.

Image of the white dwarf companion to Sirius, the brightest star in the sky.
Image of Sirius A and Sirius B taken by the Hubble Space Telescope. Sirius B, which is a white dwarf, can be seen as a faint point of light to the lower left of the much brighter Sirius A.

The Chandrasekhar limit

Stars are massive objects, and thus generate high gravitational fields. Logic dictates that this gravity field should act in on itself, squeezing it strongly enough for it to collapse. However, since the star is constantly fusing hydrogen to form helium and, in the process, releasing a lot of energy, the thermal pressure outward balances the gravitational pull inward. When stellar fusion fuel runs out, however, the outward thermal pressure is no longer sufficient, and the star collapses.

But not forever. As the star shrinks, density increases, atoms get compressed, and, as one point of time, electrons try to cram into each other’s orbits. This is when Pauli’s exclusion principle comes in; simply put, it states that identical electrons cannot cohabit. This creates an outward electron degeneracy pressure, and this now balances the inward gravitational pull. This is how white dwarfs are formed.

Image of Indian theoretician and astrophysicist Subrahmanyan Chandrasekhar. He calculated the mass limit for stellar core remnants to collapse to neutrons stars.
Indian theoretical physicist and astrophysicist Subrahmanyan Chandrasekhar

In the 1920s, Subrahmanyan Chandrasekhar, the Indian astrophysicist, was trying to figure out what happens near the end of a star’s life. He calculated that, if after shedding excess material, the stellar remnant had a mass equal or less than 1.4 times that of the sun’s mass (1.4 M), then it will form a white dwarf. This is the Chandrasekhar limit. Stars that originally had a mass of about 8 M or less would typically form white dwarfs.

The TOV limit

For stars with remnant masses higher than 1.4 M, even the electron degeneracy pressure isn’t enough to counteract the grav collapse. Atoms get squished together until their nuclei stand shoulder-to-shoulder, and densities and pressures go through the roof. Between remnant masses 1.4 M to 3 M, however, the gravitational contraction isn’t quite strong enough to breach the final barrier : the neutron degeneracy pressure. For, you see, Pauli’s exclusion principle applies not just to electrons, but to all fermions. Fermions obey something called Fermi-Dirac statistics and have half-integer spins. Electrons, protons and neutrons are some of the familiar fermions; all of them are spin-½ particles.

Compact stars of remnant masses between 1.4 M and 3 M are neutron stars. The extreme pressure and temperature within the star causes nucleonic transmutations–neutrons get converted into protons and vice versa. The extreme pressure and temperature within the star causes nucleonic transmutations–neutrons get converted into protons and vice versa, though the latter happens on a far higher scale. The stellar core mass limit of 3 M is called the TOV limit (Tolman-Oppenheimer-Volkoff), and is the counterpart of the Chandrasekhar limit for neutron degeneracy pressure.

Stars whose remnant masses are greater than TOV limit have no further defences. For them, gravitational collapse in unstoppable, eventually leading to black holes and singularities.

The Chandrasekhar limit has been theoretically and observationally confirmed to be 1.4 M. The TOV limit is theoretically anywhere between 1.5M to 3M, though most estimates have put at around 3 M. Recent observational evidence from the LIGO and Virgo gravitational wave detectors, obtained during the merging of two neutron stars, sets a lower bound on the TOV limit at 2.17M.

What are Neutron stars?

Black holes

Black holes are regions where the curvature of spacetime is so high that the escape velocity from within this region is greater than c, the speed of light. Basically, nothing escapes a black hole. Black holes enclose singularities, which are regions of spacetime where pressure and density are infinite. Black holes and singularities are formed as a result of stellar collapse : the singularity is the supernova remnant of the massive star, while the black hole is the spacetime envelope that separates the singularity from the rest of the universe. This separation is necessary since the laws of physics, especially those of general relativity, cease to work at the singularity.

Characteristics of a neutron star

A teaspoonful of neutron star (volume about 5 ml) would have a mass of almost 900 times the Great Pyramid of Giza, and a weight about 15 times the weight of the moon.

Unlike spacetime singularities, pressure, density and gravity within neutron stars aren’t infinite. But they are massively high, higher than most macroscopic regions in this universe. A teaspoonful of neutron star (volume about 5 ml) would have a mass of almost 900 times the Great Pyramid of Giza, and a weight about 15 times the weight of the moon. The gravity of a neutron star is about 200 billion times that of the Earth, resulting in escape velocities that are a third to a half of lightspeed. Magnetic fields of neutron stars can be anywhere from a 100 million to 1 quadrillion (1 million billion, 1015) times as strong as Earth’s.

And all of that is a diameter of about 10 km. That’s about the size of a decent-to-smallish city.

The small size of a neutron star also makes it spin very very fast. Most stars, if not all, spin about their axis. After supernova, the stellar neutron cores retain this angular momentum. However, these are much smaller than the stars themselves. Thus, by conservation of angular momentum, neutron stars spin very fast. Typical rotational periods are of the order of seconds. However, some neutron stars spin much faster. One, in particular, spins at 716 Hz. That is 716 times a second, giving it a surface speed of about 0.24 c.

How hot are neutron stars?

The high pressure, density, and gravity ensure that, at birth or post-supernova, a neutron star is immensely hot. Initial temperatures can be as high as 100 billion to 1 trillion kelvin (1011 – 1012 K). As a comparison, the temperature at the sun’s core is a paltry 15 million kelvin (15*106 K). However, unlike the sun, a neutron star has no nuclear furnace within it to keep generating energy. Thus, by the laws of thermodynamics, neutron stars must cool down.

How do neutron stars cool down?

High school physics has taught us that objects cool via three primary mechanisms:

  • Conduction : Heat is transferred from the hot body to the cooler body via direct contact
  • Convection : Heat is transferred from a hot part of a fluid to a cooler part via actual movement of mass
  • Radiation : Heat is given off by a hot body in waves of electromagnetic radiation
What neutron stars need are phantom particles that are capable of smuggling energy away. Neutrinos fill this role rather nicely.

The first two are not applicable here, obviously. Conduction is irrelevant because there isn’t a cold body that the neutron star can lean up against and give up its heat. Also, convection will simply redistribute heat within the neutron star. So that’s no good either.

Curiously, cooling via typical electromagnetic radiation is also not an option during the earlier turbulent times of a neutron star’s life. Its interior is too opaque for photons of any wavelength to breach–any photons that do attempt to escape quickly encounter something to interact with, and are either reabsorbed or scattered away.

What neutron stars need are phantom particles that are capable of ferrying energy away while passing under the radar of most electromagnetic interference effects that the roiling innards of the neutron star can cough up. Neutrinos fill this role rather nicely.

George Gamow

George Gamow was a Russian-American astrophysicist and cosmologist. Apart from his many contributions towards physics, especially the physics of the big bang, Gamow was also a well-known science educator. His book One Two Three…Infinity, aimed at school-level readers, explains concepts of science and mathematics in fascinatingly simple ways. In his series of books Mr Tompkins…, the protagonist, CGH Tompkins, dreams of various worlds where the values of the physical constants are different from that of this universe. Tompkins’ initials stand for, arguably, the three most important physical constants:
* c : lightspeed; special relativity
* G : Newtonian gravitation constant; classical gravity
* h : Planck’s constant; quantum mechanics.

Image of Russian theoretician and cosmologist George Gamow. In the 1940s, he, along with fellow astrophysicist Mario Schenberg, coined the term "Urca process" (that is, the direct Urca process) and coined the term after the casino they were visiting at the time.
Russian theoretical physicist, cosmologist, and popular science author George Gamow.

Sometime in the 1940s, much before One Two Three…Infinity or Mr Tompkins…, George Gamow was visiting a casino in Rio de Janeiro with friend and fellow astrophysicist Mário Schenberg. They were both working on supernova remnants at that time and, seeing their money disappear on the roulette table, one of them said to the other that

“the energy disappears in the nucleus of the supernova as quickly as the money disappeared at that roulette table”

Now I’m not entirely sure who said this to whom; some sources attribute this to Schenberg, while others claim that it was Gamow who said it. Nevertheless, what is true is that this was said, and it is a wonderful description of how these supernova remnants cool.

Beta decay

Neutron stars are so-called because most of the protons within them have, via a process called beta decay, transformed into neutrons. Clearly, it isn’t possible to this to occur : p → n, since protons are positively charged and neutrons are chargeless. Thankfully, the proton-to-neutron transformation also yields a positron, which is basically an antielectron and has positive charge, balancing out the proton charge. A cascade of these p-to-n processes mean that neutron stars are left with far more neutrons than protons.

Image of the two types of beta decay. In the neutron-to-proton decay, Carbon-14 transmuted to Nitrogen-14, having gained a proton and lost a neutron. In the p-to-n decay, Carbon-10 transmutes to Boron-10, having lost a proton and gained a neutron. In the first, an electron and an antineutrino are also emitted. In the second, a neutrino and a positron are emitted. These reactions are similar to the direct Urca process.
The two types of beta decay. (Top) A neutron decays to form a proton, an electron, and an antineutrino. (Bottom) A proton decays to form a neutron, a positron, and a neutrino.

The quark picture

A quick digression here. Protons and neutrons are not fundamental particles; they can be further subdivided into quarks. There are six types (or flavours, as they are curiously called) of quarks. Of these, the up and down quarks are the most common. A proton consists of two up quarks and a single down quark, while a neutron has one up and two down quarks; p = uud, n = udd. An up quark has a charge of +2/3, while a down quark has a charge of -1/3, where the unit and sign of charge is fixed by that of the electron, which, of course, has charge -1. Clearly, therefore, a proton’s charge is (+2/3 + 2/3 – 1/3 =) +1 and a neutron’s is (+2/3 – 1/3 – 1/3 =) 0, as expected.

Protons and neutrons are baryons, particles that contain a triplet of quarks. These are distinct from electrons and neutrinos, which are leptons. The former are composite particles, while the latter are fundamental particles.

Neutrinos and antineutrinos

Apart from positrons, the decay of a proton also creates a neutrino:

(1) p → n + e+ + νe

Neutrinos are phantom particles that are nearly massless, carry no charge, and almost never interact with normal baryonic matter. Trillions of neutrinos pass through the Earth every second; only a few are either absorbed or scattered. Here the symbol stands for an electron neutrino. There are two other types of neutrinos, though they won’t feature here.

The proton-to-neutron reaction is not the only one going on inside a neutron star. The reverse also happens, when a neutron transmutes into a proton while releasing an electron (to balance the electric charge) and an antineutrino (νe). The reaction is

(2) n → p + e + νe

Neutrinos and antineutrinos are similar in most respects. They differ in two aspects : lepton number and chirality. The first is easy enough to explain, the second not as much.

Consider the reactions (1) and (2). They involve the baryons p and n, and the leptons e and νe. A fundamental principle of particle physics reactions states that the baryon number and the lepton number in a reaction must be conserved. This is a bit like atomic and molecular chemistry, where oxidation numbers need to be conserved. In both the reactions, there is one baryon on either side of the reaction, thus conserving baryon number. As for lepton number, there is one anti electron and one neutrino on the RHS, giving a total lepton number of (-1) + (+1) = 0, which is fine. The same is true in Eq 2, where the electron has lepton number +1 and the antineutrino has lepton number -1.

The p ↔ n transmutation reactions are called beta decay because of the emitted electrons/positrons that form beta rays.

The Urca processes

The direct Urca process

Post-supernova, something similar to beta decay happens within a neutron star. Neutrons are converted into protons and vice versa, while neutrinos and antineutrinos are emitted. The reactions are:

(1) n → p + ee

(2) p + e → n + νe

This is the direct Urca process, and is the simplest and fastest method by which neutron stars can cool down. A neutron star with an initial temperature of 100 billion to even 1 trillion kelvin can, by utilising the direct Urca, cool down to 1 billion kelvin in the order of minutes. That is seriously fast. Almost as fast as coins disappearing down the roulette wheel.

A neutron star with an initial temperature of 100 billion to even 1 trillion kelvin can, by utilising the direct Urca, cool down to 1 billion kelvin in the order of minutes.

However, such fast cooling drastically reduces the number of nucleons that are thermally excited enough to agree to direct Urca. If the proton fraction falls below 1/9, then it is no longer possible to simultaneously conserve energy and momentum via the direct Urca process. If the temperature falls below 1 billion K, then, at standard neutron star densities, calculations indicate that the proton fraction should be 1/25, thus stopping the direct Urca process.

The modified Urca process

At this point, the modified Urca process then takes over:

(1) N +n → N + p + ee

(2) N +p + e → N + n + νe

where N is any nucleon, n or p. Notice that the only difference between the modified and the direct processes is in the number of baryonic reactants : the direct process has a single baryon on the LHS, the modified process has two. At lower proton concentrations and lower temperatures, this modification helps conserve energy and momentum at the same time. However, it is far less efficient than the direct process, with a rate of cooling that is almost a million times slower. After a while, the interior cools sufficiently for it to be transparent to X-ray photons, and, for the next million years or so, neutron stars remain visible in the X-ray EM band.

Evidence for the direct Urca process

There is sufficient evidence for neutron star cooling via modified Urca and X-ray emission. However, it is far more difficult to observe cooling via direct Urca. Also, it is quite possible that the initial proton concentration might be too low, and direct Urca never actually takes place. Until now, this has been a purely theoretical question. Recently, however, scientists have been able to measure the X-ray output of a quiescent neutron star and have concluded that the direct Urca process must have taken place.

MXB 1659-29

The neutron star in question is labelled MXB 1659-29. Given that there are almost 2000 known neutron stars in the Milky Way and the neighbouring Magellanic Clouds, we are perhaps lucky that the label is as simple as that. MXB 1659-29 is actually a binary star, one of whose members is a neutron star. Neutron stars as one member of a binary are actually quite common. The neutron star pulls in matter from its companion star, the accretion showing up as X-ray emissions of intensity far higher than the usual.

Illustration of the accretion disk around a neutron star, created when matter is pulled in from its binary companion star. After accretion, neutron stars need to cool down. Analysis of X-ray emissions from such cooldowns provide insights about the direct Urca process of neutrino cooling.
An artists’ view of accreting neutron star (Credit: Tony Piro)

This accretion is not always a continuous process. In recent times, MXB 1659-29 has accreted matter twice, once in 2001 and then, fifteen years later, in 2016. In between these two accretion events, MXB 1659-29 was in a quiescent phase. It is during this quiet phase that researchers observed and analysed the X-ray emission spectra of MXB 1659-29. And came to the conclusion that MXB 1659-29 has indeed, during its formative years, gone through a phase of enhanced cooling via the direct Urca process.

Proton fraction

The findings also set a lower bound on the proton fraction. If direct Urca has indeed taken place, then the proton fraction must have, during that time, been at least 1/9. Theories that predict a far lower proton fraction than 1/9 would have to be discarded or modified. Thus, theories that predict a far lower proton fraction would have to be discarded, or at least given a stringent dressing-down.

Besides, further analysis of these observational finding should tell scientist more about the inner workings of the ultradense high-temperature regions within neutron stars. Particularly, it should shine further light on the superconductivity and superfluidity of the interiors of neutron stars. However, that is a topic for another day.

The name URCA

The name URCA,, or rather, Urca, is not an acronym. The casino that Gamow and Schenberg visited was the Cassino de Urca. In Gamow’s southern Russian dialect, urca meant robber or gangster. This gels well with astrophysicist and neutron star expert Madappa Prakash’s statement, where he states that

“The neutrino is a thief; it robs energy from the star…”

I prefer calling the neutrinos the magpies of the neutron star Marlinspike Hall. Naturally, Castafiore’s emerald plays the role of heat, to be stolen away by La gazza ladra.

A panel from "The Castafiore Emerald" by Herge, showing Tintin having recovered the eponymous and presumed missing emerald. The emerald was actually stolen by a magpie, in much the same way that neutrinos steal a neutron star's heat away via the direct Urca process.
From “The Castafiore Emerald”, by Herge

Sources

Original papers

  1. Brown, Cumming, Fattoyev, Horowitz, Page, Reddy : Rapid Neutrino Cooling in the Neutron Star MXB 1659-29, Physical Review Letters (2018).
  2. Lattimer, Pethick, Prakash, Haensel : Direct URCA process in neutron stars, Physical Review Letters (1991).

Articles

  1. Lattimer, James M. : A Rapidly Cooling Neutron Star, APS Physics Viewpoint (2018).
  2. Conover, Emily : Neutron stars shed neutrinos to cool down quickly, ScienceNews (2018).

Other resources

  1. Redd, Noah T. : Neutron Stars: Definition & Facts, Space.com (2018).
  2. Miller, M. Coleman : Introduction to Neutron stars
  3. Cartlidge, Edwin : Neutron star has superfluid core, PhysicsWorld (2011).
  4. Conover, Emily : Collision illuminates the mysterious makeup of neutron stars, ScienceNews (2017)
  5. Wikipedia : The Urca Process