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## Aberfan – Vera Rich (poem)

50 years ago today, on 21 October 1966, a tragedy happened in a small mining village in Wales which horrified the world. At 9:15am, Pantglas school in a place called Aberfan was engulfed by a river of coal debris. 116 children (more than half of the school’s pupils) and 28 adults were killed. Dozens more were rescued from the horror, with people from Aberfan and surrounding villages digging with their hands in a desperate attempt to save some lives.

The tragedy was due to a tip of coal waste (“slag heap” as they were often called) which had been piled on the side of the mountain against which the village nestles, and was entirely preventable. For months the local council had been warning the National Coal Board (NCB) of the risk, but the NCB had taken no notice.

In a tribunal held after the tragedy, the NCB were found guilty of negligence and of corporate manslaughter. However, they never paid a penny of compensation to the families, nor did they pay to have the numerous slag heaps rendered safe. Local families had to raise the money to do this themselves. After years of campaigning, in 1997 the newly-formed Welsh Assembly government finally repaid the families the money that they had raised. Some 10 years later the Welsh Assembly government paid the families a much larger sum, to correct for the inflation in the intervening 40 years.

I have been to the cemetery and memorial park in Aberfan. It is a beautiful tribute and memory to the tragedy that happened that wet October day in 1966.

Here is a very moving poem simply called Aberfan by Vera Rich, an English-born poet.

I have seen their eyes, the terrible, empty eyes
Of women in a glimmerless dawn, and the hands
Of men who have wrestled through long years with the dark
Underpinning of the mountains, strong hands that fight

In dumb faith that what was once flesh born of their flesh
And is earth of the earth, should rest in the earth of God,
Not that of the devil’s making…

The Tip had crouched like a plague-god, with the town,
A victim in reversion, held beneath
A vast, invisible paw… Not a lion to toss
A proud, volcano-mane of destruction, crouched
Like a rat, it waited…

I have seen their eyes, and the empty hands of men,
And they walk like victims of a second Flood
In a world no longer home, where the void of sky
Between tall mountains looms as a cenotaph
For a generation of laughter…

I have seen them
Walking, near-ghosts, wraiths from a half-formed legend
Of this more-than-Hamelin, where, on an autumn Friday,
Between nine and ten of the clock, death raised his flute
And the children followed…

## Fortunate Son – Creedence Clearwater Revival (song)

Today I thought I would share this great anti-war song – “Fortunate Son” by Creedence Clearwater Revival. It was released in September 1969, and is specifically about the lucky men who were born into families which, somehow, meant that they were not called up for the draft to fight in the Vietnam war.

These were the senators’ sons, the millionaires’ sons, the fortunate sons. Sons like George W. Bush, who miraculously found himself in the National Guard, far away from any danger, rather than in Vietnam fighting. I wonder why? Oh, maybe because his father, George H. Bush, had the political clout and importance to make sure his precious son didn’t go and fight in the jungles of Vietnam, unlike the poor white and black men who were drafted there.

As the draft went on, it became more and more apparent how many fortunate sons were avoiding going to war, thanks to their family’s influence in bending the rules. And how many poor blacks and whites had no choice, they were forced to go and would be jailed should they refuse. The Vietnam war was wrong on so many levels, but the inequity of the draft was certainly one of its wrongs.

“Fortunate Son” was released in September 1969, and talks of the privileged few who, somehow, avoided the Vietnam war draft.

“Fortunate Son” is rated at 99 in Rolling Stone Magazine’s list of the 500 greatest songs of all time. It really is a great song, I am surprised that I haven’t blogged about it before.

Some folks are born, made to wave the flag
Ooo, they’re red, white and blue
And when the band plays “Hail to the Chief”
Ooo, they point the cannon at you, Lord

It ain’t me, it ain’t me, I ain’t no senator’s son, son
It ain’t me, it ain’t me, I ain’t no fortunate one, no

Some folks are born, silver spoon in hand
Lord, don’t they help themselves, y’all
But when the taxman comes to the door
Lord, the house looks like a rummage sale, yeah

It ain’t me, it ain’t me, I ain’t no millionaire’s son, no, no
It ain’t me, it ain’t me, I ain’t no fortunate one, no

Yeah, yeah
Some folks inherit star spangled eyes
Ooh, they send you down to war, Lord
And when you ask ’em, “How much should we give?”
Ooh, they only answer “More! More! More!”, y’all

It ain’t me, it ain’t me, I ain’t no military son, son
It ain’t me, it ain’t me, I ain’t no fortunate one, one
It ain’t me, it ain’t me, I ain’t no fortunate one, one

Here is a video of the song. Enjoy!

## Robert Kennedy – the mindless menace of violence (speech)

Just over 7 years ago, in early 2009, I bought a CD of some of Robert Kennedy’s greatest speeches. Whilst his brother John F. Kennedy gave some memorable speeches, for me Bobby Kennedy possibly surpassed JFK with his eloquence. One of his most moving and wonderful speeches has been passing through my mind these last two weeks or so; with the senseless shootings of innocent black people by police in the United States, the killing of the five policemen by an assassin in Houston, the horrific terrorist attack in Nice on Bastille Day which has killed at least 84 people, many of them children, and the failed coup in Turkey with over 100 dead. And, just as I was putting this blog together yesterday, the shooting of 3 more police officers in Baton Rouge.

Robert Kennedy (RFK) served as Attonery General under his brother’s Prisidency, but in 1965 he entered the Senate as one of the senators for New York. On 16 March 1968, RFK announced that he would run for the presidency, and set about touring the USA to garner support for his campaign. On the evening of 4 April, he was due to give a speech in Indianapolis when he learnt en-route of the assassination of Martin Luther King. He broke the news to the gathered crowd, many of whom had not heard the news until Bobby Kennedy told them. He gave a very moving and powerful speech on that evening, and I may blog about that particular speech another time.

But, today I am going to share the speech that he gave the day after MLK’s assassination, on 5 April 1968. The speech is entitled “The mindless menace of violence“, and it was delivered at the Cleveland Club in Ohio.

Kennedy toured the country as part of his campaign to become President of the United States, concentrating to a large part on some of the poorest communities in the country, where he met with dissaffected whites, blacks and latinos who had been left behind by the ‘American Dream’.

“this mindless menace of violence in America which again stains our land and every one of our lives.”

It is quite a long speech, nearly 10 minutes long, but bear with it and I think you will be struck by its eloquence. Bobby Kennedy wrote the speech himself, putting it together in the hours after the horror of MLK’s assassination had sunk into his mind.

The speech opens with these lines….

This is a time of shame and sorrow. It is not a day for politics. I have saved this one opportunity to speak briefly to you about this mindless menace of violence in America which again stains our land and every one of our lives.

It is not the concern of any one race. The victims of the violence are black and white, rich and poor, young and old, famous and unknown. They are, most important of all, human beings whom other human beings loved and needed. No one – no matter where he lives or what he does – can be certain who will suffer from some senseless act of bloodshed. And yet it goes on and on.

Why? What has violence ever accomplished? What has it ever created? No martyr’s cause has ever been stilled by his assassin’s bullet.

No wrongs have ever been righted by riots and civil disorders. A sniper is only a coward, not a hero; and an uncontrolled, uncontrollable mob is only the voice of madness, not the voice of the people……

But, Bobby Kennedy was also deeply concerned with the economic disparities in the United States, and with the sickening racism which had profoundly disturbed him. He later goes on to say…

……

For there is another kind of violence, slower but just as deadly, destructive as the shot or the bomb in the night. This is the violence of institutions; indifference and inaction and slow decay. This is the violence that afflicts the poor, that poisons relations between men because their skin has different colors. This is a slow destruction of a child by hunger, and schools without books and homes without heat in the winter.

This is the breaking of a man’s spirit by denying him the chance to stand as a father and as a man among other men. And this too afflicts us all. I have not come here to propose a set of specific remedies nor is there a single set. For a broad and adequate outline we know what must be done.

Followed immediately by these words…

When you teach a man to hate and fear his brother, when you teach that he is a lesser man because of his color or his beliefs or the policies he pursues, when you teach that those who differ from you threaten your freedom or your job or your family, then you also learn to confront others not as fellow citizens but as enemies – to be met not with cooperation but with conquest, to be subjugated and mastered.

The entire text can be found here at the John F. Kennedy presidential library website.

There are several versions of this mesmerising speech on YouTube, but many seem to have had an annoying soundtrack of some music added. I feel the added music detracts from hearing Bobby Kennedy’s words, which are powerful enough and do not need any music to make them more dramatic. So, the version I have included here is just RFK’s incredible words.

What strikes me most when I hear or read these words of Bobby Kennedy is how little progress we have made. One could argue that we have digressed; there are more mass shootings now in the USA than in the 1960s when these words were spoken. There is more terrorism and conflict than ever. And, in the presumptive Republican Party presidential nominee Donald Trump, we have a man who is the very antithesis of the wonderful ideals for which Bobby Kennedy stood.

I would say “enjoy” this video, but I am not sure that one can enjoy this speech. It is moving, harrowing, thought-provoking, upsetting, but also uplifting. To think that RFK was himself assassinated within a few months of giving this speech, it only adds poignancy to his words and highlights even more the truth and sadness of the mindless menace of violence

## The Bob Dylan “Judas” heckle

Fifty years ago yesterday (17 May 1966), one of the seminal moments in 20th Century popular culture took place in the Manchester Free Trade Hall. Bob Dylan, who had burst onto the folk scene a few years before, was playing to a packed crowd towards the end of his gruelling 1966 World tour. The first half of his set was vintage Dylan, just the man (poet) and his guitar. The crowd were enraptured.

But, it all turned sour in the second half, when Dylan was joined by his band, The Hawks, and proceeded to do an ‘electric’ set. The crowd became restless. Many left; others booed, stamped their feet or started chanting. When he came back on to do his encore, things came to a head.

“Judas!” a man shouted.

“I don’t believe you.” Dylan replied. Then he started getting ready for the encore song. A few seconds later Dylan added

“You’re a liar!”

Then, he turned to his band and said “Play it fucking loud”,  and they ripped into an angry version of Like a Rolling Stone. This is the moment as captured on film, it forms the closing scene of Martin Scorsese’s fascinating documentary No Direction Home.

There is also a very interesting in-depth audio documentary about this whole seminal incident, Ghosts of Electricity, made by Andy Kershaw for BBC Radio 1 and broadcast in 1999. It is available here on Andy Kershaw’s website.

Andy Kershaw’s fascinating documentary about the Bob Dylan “Judas” incident, which was originally broadcast in 1999 on BBC Radio 1.

The whole concert was recorded and circulated as a bootleg for many years. For some reason, it became known as the Royal Albert Hall Concert, even though it had happened at the Manchester Free Trade Hall; possibly because the 1966 World tour ended at the Royal Albert Hall on the 26 and 27 May. Dylan sanctioned an official release of the concert in 1998.

The cover for Bob Dylan’s “Royal Albert Hall Concert” CD, which includes the “Judas” heckle. In fact, the concert was recorded at the Manchester Free Trade Hall on 17 May 1966.

## Who invented calculus?

One of the physicists in our book Ten Physicists Who Transformed Our Understanding of Reality (follow this link for more information on the book) is, not surprisingly, Isaac Newton. In fact, he is number 1 in the list. One could argue that he practically invented the subject of physics. We decided to call him the ‘father of physics’, with Galileo (whose life preceded Newton’s) being given the title of ‘grandfather’.

Newton was, clearly, a man of genius. But he was also a nasty, vindictive bastard (not to mince my words!). He didn’t really have any close friends in his life; there were plenty of people who admired him and respected him, and of course he had colleagues. But, apart from a niece whom he seemed to dote on in later life, and two men with whom he probably had love affairs, he was not a man who sought company. He was probably autistic, but lived at a time before such conditions were diagnosed or talked about.

Isaac Newton (1643-1727), the ‘father of physics’. He relished in feuding with other scientists

One sort of interaction that he did seem to enjoy with other people though was feuds. In fact, he seemed to thrive on feuding with other scientists. He loved to argue with others, which is not uncommon amongst academics. He had strong opinions which he liked to defend; this is normal. But, Newton took these disputes to an extreme; if he fell out with someone he would do everything he could to destroy that person.

Although I am sure that he had many ‘minor’ arguments, he had three main feuds with fellow scientists. These three men were

• Robert Hooke – curator of experiments at the Royal Society
• Gottfried (von) Leibniz – the German mathematician
• John Flamsteed – the first Astronomer Royal

In each case, he did his level best to destroy the other man. Each of these feuds is discussed in more detail in our book, but in this blogpost I will give a brief summary of his feud with Leibniz.

The feud came about because Newton refused to believe that Leibniz had independently come up with the mathematical idea of calculus. It was a recurring theme throughout Newton’s life that he sincerely believed that he was special. He had deep religious views (some would say extreme religious views). As part of these views, he believed that he had been specially chosen by God to understand things that others would never be able to understand.

Thus, when he heard that Leibniz had developed a mathematics similar to his own ‘theory of fluxions’ (as Newton called it), he naturally assumed that the German had stolen it from him. There then ensued a 30-year dispute between the two men, with Newton very much the aggressor.

Gottfried (von) Leibniz (1646-1716), German mathematician and co-inventor of calculus

It escalated from a dispute to a feud, and culminated in the Royal Society commissioning an ‘official investigation’ to establish propriety for the invention of calculus. When the report came out in 1713 it came out in Newton’s favour. But, by this time Newton was not only President of the Royal Society, but he had secretly authored the entire report. It was anything but impartial. Leibniz died the following year, a broken man from Newton’s relentless attacks.

One should, of course, be able to to admire a person for their work but not admire them in the least for the person that they were. Newton, in my mind, falls very firmly into this category. His contribution to physics is unparalleled, but I don’t think he was the kind of person one would want to know or even come across if one could help it!

Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

As I mentioned in this blog here, a few months ago I contributed some articles to a book called 30-Second Einstein, which will be published by Ivy Press in the not too distant future. One of the articles I wrote for the book was on Indian mathematical physicist Satyendra Bose. It is after Bose that ‘bosons’ are named (as in ‘the Higgs boson’), and also terms like ‘Bose-Einstein statistics’ and ‘Bose-Einstein condensate’. So, who was Satyendra Bose, and why is his name attached to these things?

Satyendra Bose was an Indian mathematical physicist after whom the ‘boson’ and Bose-Einstein statistics are named

Satyendra Bose was born in Calcutta, India, in 1894. He studied applied mathematics at Presidency College, Calcutta, obtaining a BSc in 1913 and an MSc in 1915. On both occasions, he graduated top of his class. In 1919, he made the first English translation of Einstein’s general theory of relativity, and by 1921 he had moved to Dhaka (in present-day Bangladesh) to become Reader (one step below full professor) in in the department of Physics.

It was whilst in Dhaka, in 1924, that he came up with the theory of how to count indistinguishable particles, such as photons (light particles). He showed that such particles follow statistics which are different from particles which can be distinguished. All his attempts to get his paper published failed, so in an act of some desperation he sent it to Einstein. The great man recognised the importance of Bose’s work immediately, translated it into German and got it published in Zeitschrift für Physik, one of the premier  physics journals of the day.

Because of Einstein’s part in getting the theory published, we now know of this way of counting indistinguishable particles as Bose-Einstein statistics. We also name particles which obey this kind of statistics bosons; examples are the photon, the W and Z-particles (which mediate the weak nuclear force), and the most famous boson, the Higgs boson (responsible for mediating the property of mass via the Higgs field).

With the imminent partition of India when it was gaining independence from Britain, Bose returned to his native Calcutta where he spent the rest of his career. He died in 1974 at the age of 80.

You can read more about Satyendra Bose, Bose-Einstein statistics and Bose-Einstein condensates in 30-second Einstein, out soon from Ivy Press.

## Derivation of Planck’s radiation law – part 4 (final part)

In part 3 of this blog series I explained how Max Planck found a mathematical formula to fit the observed Blackbody spectrum, but that when he presented it to the German Physics Society on the 19th of October 1900 he had no physical explanation for his formula. Remember, the formula he found was

$E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} -1) } \; d\lambda$

if we express it in terms of wavelength intervals. If we express it in terms of frequency intervals it is

$E_{\nu} \; d \nu = A^{\prime} \nu^{3} \frac{ 1 }{ (e^{ a^{\prime} \nu / T } - 1) } \; d\nu$

Planck would spend six weeks trying to find a physical explanation for this equation. He struggled with the problem, and in the process was forced to abandon many aspects of 19th Century physics in both the fields of thermodynamics and electromagnetism which he had long cherished. I will recount his derivation – it is not the only one and maybe in coming blog posts I can show how his formula can be derived from other arguments, but this is the method Planck himself used.

As we saw in the derivation of the Rayleigh-Jeans law (see part 3 here, and links in that to parts 1 and 2), blackbody radiation can be modelled as an idealised cavity which radiates through a small hole. Importantly, the system is given enough time for the radiation and the material from which the cavity is made to come into thermal equilibrium with each other. This means that the walls of the cavity are giving energy to the radiation at the same rate that the radiation is giving energy to the walls.

Using classical physics, as we did in the derivation of the Rayleigh-Jeans law, we saw that the energy density (the energy per unit volume) is

$\frac{du}{d\nu} = \left( \frac{ 8 \pi kT }{ c^{3} } \right) \nu^{2}$

After trying to derive his equation based on standard thermodynamic arguments, which failed, Planck developed a model which, he found, was able to produce his equation. How did he do this?

### Harmonic Oscillators

First, he knew from classical electromagnetic theory that an oscillating electron radiates (as it is accelerating), and he reasoned that when the cavity was in thermal equilibrium with the radiation in the cavity, the electrons in the walls of the cavity would oscillate and it was they that produced the radiation.

After much trial and error, he decided upon a model where the electrons were attached to massless springs. He could model the radiation of the electrons by modelling them as a whole series of harmonic oscillators, but with different spring stiffnesses to produce the different frequencies observed in the spectrum.

As we have seen (I derived it here), in classical physics the energy of a harmonic oscillator depends on both its amplitude of oscillation squared ($E \propto A^{2}$); and it also depends on its frequency of oscillation squared ($E \propto \nu^{2}$). The act of heating the cavity to a particular temperature is what, in Planck’s model, set the electrons oscillating; but whether a particular frequency oscillator was set in motion or not would depend on the temperature.

If it were oscillating, it would emit radiation into the cavity and absorb it from the cavity. He knew from the shape of the blackbody curve (and, by now, his equation which fitted it), that the energy density $E d\nu$ at any particular frequency started off at zero for high frequencies (UV), then rose to a peak, and then dropped off again at low frequencies (in the infrared).

So, Planck imagined that the number of oscillators with a particular resonant frequency would determine how much energy came out in that frequency interval. He imagined that there were more oscillators with a frequency which corresponded to the maximum in the blackbody curve, and fewer oscillators at higher and lower frequencies. He then had to figure out how the total energy being radiated by the blackbody would be shared amongst all these oscillators, with different numbers oscillating at different frequencies.

He found that he could not derive his formula using the physics that he had long accepted as correct. If he assumed that the energy of each oscillator went as the square of the amplitude, as it does in classical physics, his formula was not reproduced. Instead, he could derive his formula for the blackbody radiation spectrum only if the oscillators absorbed and emitted packets of energy which were proportional to their frequency of oscillation, not to the square of the frequency as classical physics argued. In addition, he found that the energy could only come in certain sized chunks, so for an oscillator at frequency $\nu, \; E = nh\nu$, where $n$ is an integer, and $h$ is now known as Planck’s constant.

What does this mean? Well, in classical physics, an oscillator can have any energy, which for a particular oscillator vibrating at a particular frequency can be altered by changing the amplitude. Suppose we have an oscillator vibrating with an amplitude of 1 (in abitrary units), then because the energy goes as the square of the amplitude its energy is $E=1^{2} =1$. If we increase the amplitude to 2, the energy will now be $E=2^{2} = 4$. But, if we wanted an energy of 2, we would need an amplitude of $\sqrt{2} = 1.414$, and if we wanted an energy of $3$ we would need an amplitude of $\sqrt{3} = 1.73$.

In classical physics, there is nothing to stop us having an amplitude of 1.74, which would give us an energy of 3.0276 (not 3), or an amplitude of 1.72 whichg would give us an energy of 2.9584 (not 3). But, what Planck found is that this was not allowed for his oscillators, they did not seem to obey the classical laws of physics. The energy could only be integers of $h\nu$, so $E=0h\nu, 1h\nu, 2h\nu, 3h\nu, 4h\nu$ etc.

Then, as I said above, he further assumed that the total energy at a particular frequency was given by the energy of each oscillator at that frequency multiplied by the number of oscillators at that frequency. The frequency of a particular oscillator was, he imagined, determined by its stiffness (Hooke’s constant). The energy of a particular oscillator at a particular frequency could be varied by the amplitude of its oscillations.

Let us assume, just to illustrate the idea, that the value of h is 2. If the total energy in the blackbody at a particular frequency of, say, 10 (in arbitrary units) were 800 (also in arbitrary units), this would mean that the energy of each chunk ($E=h \nu$) was $E = 2 \times 10 = 20$. So, the number of chunks at that frequency would then be $800/20 = 40$. 40 oscillators, each with an energy of 20, would be oscillating to give us our total energy of 800 at that frequency.

Because of this quantised energy, we can write that $E_{n} = nh \nu$, where $n=0,1,2,3, \cdots$.

### The number of oscillators at each frequency

The next thing Planck needed to do was derive an expression for the number of oscillators at each frequency. Again, after much trial and error he found that he had to borrow an idea first proposed by Austrian physicist Ludwig Boltzmann to describe the most likely distribution of energies of atoms or molecules in a gas in thermal equilibrium. Boltzmann found that the number of atoms or molecules with a particular energy $E$ was given by

$N_{E} \propto e^{-E/kT}$

where $E$ is the energy of that state, $T$ is the temperature of the gas and $k$ is now known as Boltzmann’s constant. The equation is known as the Boltzmann distribution, and Planck used it to give the number of oscillators at each frequency. So, for example, if $N_{0}$ is the number of oscillators with zero energy (in the so-called ground-state), then the numbers in the 1st, 2nd, 3rd etc. levels ($N_{1}, N_{2}, N_{3},\cdots$) are given by

$N_{1} = N_{0} e^{ -E_{1}/kT }, \; N_{2} = N_{0} e^{ -E_{2}/kT }, \; N_{3} = N_{0} e^{ -E_{3}/kT }, \cdots$

But, as $E_{n} = nh \nu$, we can write

$N_{1} = N_{0} e^{ -h \nu /kT }, \; N_{2} = N_{0} e^{ -2h \nu /kT }, \; N_{3} = N_{0} e^{ -3h \nu /kT }, \cdots$

Planck modelled blackbody radiation as a series of harmonic oscillators with equally spaced energy levels

To make it easier to write, we are going to substitute $x = e^{ -h \nu / kT }$, so we have

$N_{1} = N_{0}x, \; N_{2} = N_{0} x^{2}, \; N_{3} = N_{0} x^{3}, \cdots$

The total number of oscillators $N_{tot}$ is given by

$N_{tot} = N_{0} + N_{1} + N_{2} + N_{3} + \cdots = N_{0} ( 1 + x + x^{2} + x^{3} + \cdots)$

Remember, this is the number of oscillators at each frequency, so the energy at each frequency is given by the number at each frequency multiplied by the energy of each oscillator at that frequency. So

$E_{1}=N_{1} h \nu , \; E_{2} = N_{2} 2h \nu , \; E_{3} = N_{3} 3h \nu, \cdots$

which we can now write as

$E_{1} = h \nu N_{0}x, \; E_{2} = 2h \nu N_{0}x^{2}, \; E_{3} = 3h \nu N_{0}x^{3}, \cdots$

The total energy $E_{tot}$ is given by

$E_{tot} = E_{0} + E_{1} + E_{2} + E_{3} + \cdots = N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots)$

The average energy $\langle E \rangle$ is given by

$\langle E \rangle = \frac{ E_{tot} }{ N_{tot} } = \frac{ N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots) }{ N_{0} ( 1 + x + x^{2} + x^{3} + \cdots ) }$

The two series inside the brackets can be summed. The sum of the series in the numerator, which we will call $S_{1}$ is given by

$S_{1} = \frac{ x - (n+1)x^{n+1} + nx^{n+2} }{ (1-x)^{2} }$

(for the proof of this, see for example here)

The series in the denominator, which we will call $S_{2}$, is just a geometric progression. The sum  of such a series is simply

$S_{2} = \frac{ 1 - x^{n} }{ (1-x) }$

Both series  are in $x$, but remember $x = e^{-h \nu / kT}$. Also, both series are from a frequency of $\nu = 0 \text{ to } \infty$, and $e^{-h \nu /kT} < 1$, which means the sums converge and can be simplified.

$S_{1} \rightarrow \frac{x}{ (1-x)^{2} } \text{ and } S_{2} \rightarrow \frac{ 1 }{(1-x)}$

which means that $\langle E \rangle = (h \nu S_{1})/S_{2}$ is given by

$\langle E \rangle = \frac{ h \nu x }{ (1-x)^{2} } \times \frac{ (1-x) }{1} = \frac{h \nu x}{ (1-x) }$

and so we can write that the average energy is

$\boxed{ \langle E \rangle = \frac{h \nu}{( 1/x - 1) } = \frac{h \nu}{ (e^{h \nu/kT} - 1) } }$

## The radiance per frequency interval

In our derivation of the Rayleigh-Jeans law (in this blog here), we showed that, using classical physics, the energy density $du$ per frequency interval was given by

$du = \frac{ 8 \pi }{ c^{3} } kT \nu^{2} \, d \nu$

where $kT$ was the energy of each mode of the electromagnetic radiation. We need to replace the $kT$ in this equation with the average energy for the harmonic oscillators that we have just derived above. So, we re-write the energy density as

$du = \frac{ 8 \pi }{ c^{3} } \frac{ h \nu }{ (e^{h\nu/kT} - 1) } \nu^{2} \; d\nu = \frac{ 8 \pi h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu$

$du$ is the energy density per frequency interval (usually measured in Joules per metre cubed per Hertz), and by replacing $kT$ with the average energy that we derived above the radiation curve does not go as $\nu^{2}$ as in the Rayleigh-Jeans law, but rather reaches a maximum and turns over, avoiding the ultraviolet catastrophe.

It is more common to express the Planck radiation law in terms of the radiance per unit frequency, or the radiance per unit wavelength, which are written $B_{\nu}$ and $B_{\lambda}$ respectively. Radiance is the power per unit solid angle per unit area. So, as a first step to go from energy density to radiance we will divide by $4 \pi$, the total solid angle. This gives

$\frac{ 2 h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu$

We want the power per unit area, not the energy per unit volume. To do this we first note that power is energy per unit time, and second that to go from unit volume to unit area we need to multiply by length. But, for EM radiation, length is just $ct$. So, we need to divide by $t$ and multiply by $ct$, giving us that the radiance per frequency interval is

$\boxed{ B_{\nu} = \frac{ 2h \nu^{3} }{ c^{2} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu }$

which is the way the Planck radiation law per frequency interval is usually written.

## Radiance per unit wavelength interval

If you would prefer the radiance per wavelength interval, we note that $\nu = c/\lambda$ and so $d\nu = -c/\lambda^{2} \; d\lambda$. Ignoring the minus sign (which is just telling us that as the frequency increases the wavelength decreases), and substituting for $\nu$ and $d\nu$ in terms of $\lambda$ and $d\lambda$, we can write

$B_{\lambda} = \frac{ 2h }{ c^{2} } \frac{ c^{3} }{ \lambda^{3} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \frac{ c }{ \lambda^{2} } \; d\lambda$

Tidying up, this gives

$\boxed{ B_{\lambda} = \frac{ 2hc^{2} }{ \lambda^{5} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \; d\lambda }$

which is the way the Planck radiation law per wavelength interval is usually written.

## Summary

To summarise, in order to reproduce the formula which he had empirically derived and presented in October 1900, Planck found that he he could only do so if he assumed that the radiation was produced by oscillating electrons, which he modelled as oscillating on a massless spring (so-called “harmonic oscillators”). The total energy at any given frequency would be given by the energy of a single oscillator at that frequency multiplied by the number of oscillators oscillating at that frequency.

However, he had to assume that

1. The energy of each oscillator was not related to either the square of the amplitude of oscillation or the square of the frequency of oscillation (as it would be in classical physics), but rather to the square of the amplitude and the frequency, $E \propto \nu$.
2. The energy of each oscillator could only be a multiple of some fundamental “chunk” of radiation, $h \nu$, so $E_{n} = nh\nu$ where $n=0,1,2,3,4$ etc.
3. The number of oscillators with each energy $E_{n}$ was given by the Boltzmann distribution, so $N_{n} = N_{0} e^{-nh\nu/kT}$ where $N_{0}$ is the number of oscillators in the lowest energy state.

In a way, we can imagine that the oscillators at higher frequencies (to the high-frequency side of the peak of the blackbody) are “frozen out”. The quantum of energy for a particular oscillator, given by $E_{n}=nh\nu$, is just too large to exist at the higher frequencies. This avoids the ultraviolet catastrophe which had stumped physicists up until this point.

By combining these assumptions, Planck was able in November 1900 to reproduce the exact equation which he had derived empirically in October 1900. In doing so he provided, for the first time, a physical explanation for the observed blackbody curve.

• Part 1 of this blogseries is here.
• Part 2 is here.
• Part 3 is here.