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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 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. 

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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.

Radiation in a cavity

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

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.

 

 

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There has been quite a bit of mention in the media this last week or so that it is 100 years since Albert Einstein published his ground-breaking theory of gravity – the general theory of relativity. Yet, there seems to be some confusion as to when this theory was first published, in some places you will see 1915, in others 1916. So, I thought I would try and clear up this confusion by explaining why both dates appear.

Albert Einstein in Berlin circa 1915 when his General Theory of Relativity was first published

Albert Einstein in Berlin circa 1915/16 when his General Theory of Relativity was first published

From equivalence to the field equations

Everyone knew that Einstein was working on a new theory of gravity. As I blogged about here, he had his insight into the equivalence between acceleration and gravity in 1907, and ever since then he had been developing his ideas to create a new theory of gravity.

He had come up with his principle of equivalence when he was asked in the autumn of 1907 to write a review article of his special theory of relativity (his 1905 theory) for Jahrbuch der Radioaktivitätthe (the Yearbook of Electronics and Radioactivity). That paper appeared in 1908 as Relativitätsprinzip und die aus demselben gezogenen Folgerungen (On the Relativity Principle and the Conclusions Drawn from It) (Jahrbuch der Radioaktivität, 4, 411–462).

In 1908 he got his first academic appointment, and did not return to thinking about a generalisation of special relativity until 1911. In 1911 he published a paper Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravitation on the Propagation of Light) (Annalen der Physik (ser. 4), 35, 898–908), in which he calculated for the first time the deflection of light produced by massive bodies. But, he also realised that, to properly develop his ideas of a new theory of gravity, he would need to learn some mathematics which was new to him. In 1912, he moved to Zurich to work at the ETH, his alma mater. He asked his friend Marcel Grossmann to help him learn this new mathematics, saying “You’ve got to help me or I’ll go crazy.”

Grossmann gave Einstein a book on non-Euclidean geometry. Euclidean geometry, the geometry of flat surfaces, is the geometry we learn in school. The geometry of curved surfaces, so-called Riemann geometry, had first been developed in the 1820s by German mathematician Carl Friedrich Gauss. By the 1850s another German mathematician, Bernhard Riemann developed this geometry of curved surfaces even further, and this was the Riemann geometry textbook which Grossmann gave to Einstein in 1912. Mastering this new mathematics proved very difficult for Einstein, but he knew that he needed to master it to be able to develop the equations for general relativity.

These equations were not ready until late 1915. Everyone knew Einstein was working on them, and in fact he was offered and accepted a job in Berlin in 1914 as Berlin wanted him on their staff when the new theory was published. The equations of general relativity were first presented on the 25th of November 1915, to the Prussian Academy of Sciences. The lecture Feldgleichungen der Gravitation (The Field Equations of Gravitation) was the fourth and last lecture that Einstein gave to the Prussian Academy on his new theory (Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847), the previous three lectures, given on the 4th, 11th and 18th of November, had been leading up to this. But, in fact, Einstein did not have the field equations ready until the last few days before the fourth lecture!

The peer-reviewed paper of the theory (which also contains the field equations) did not appear until 1916 in volume 49 of Annalen der PhysikGrundlage der allgemeinen Relativitätstheorie (The Foundation of the General Theory of Relativity) Annalen der Physik (ser. 4), 49, 769–822. The paper was submitted by Einstein on the 20th of March 1916.

The beginning of Einstein's first paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916 and

The beginning of Einstein’s first peer-reviewed paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916


In a future blog, I will discuss Einstein’s field equations, but hopefully I have cleared up the confusion as to why some people refer to 1915 as the year of publication of the General Theory of Relativity, and some people choose 1916. Both are correct, which allows us to celebrate the centenary twice!

You can read more about Einstein’s development of the general theory of relativity in our book 10 Physicists Who Transformed Our Understanding of Reality. Order your copy here

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At number 28 in Rolling Stone Magazine’s list of the 100 greatest songwriters is Woody Guthrie. I wonder how many of you reading this, particularly non-Americans, have heard of Woody Guthrie? I first heard of him when I became interested in Bob Dylan; he was a hero of the young Dylan and, in fact, Dylan wrote a song of tribute to him on his first album, “Song for Woody”.

Guthrie was born in Oklahoma in 1912, and so lived through the depression of the 1930s in a state which was particularly blighted by the depression and the ‘dust-bowl’. This led to thousands of farmers and their families from Oklahoma leaving for California (this story is the basis of John Steinbeck’s wonderful novel The Grapes of Wrath). Living through these dismal years had a long-lasting effect on Guthrie, and he spent a career as a folk-singer and activist championing the rights of downtrodden people.



At number 28 in Rolling Stone Magazine's list of the 100 greatest songwriters of all time is Woody Guthrie.

At number 28 in Rolling Stone Magazine’s list of the 100 greatest songwriters of all time is Woody Guthrie.



Probably Guthrie’s best known song is “This Land is Our Land”, a song which has been translated into many languages, including into Welsh. Guthrie wrote this song in 1940, basing the lyrics to fit with an existing melody. He then altered the lyrics in 1944, which is when he first recorded it, and it was released in 1945. You can read more about the two different versions of the song here on its Wikipedia page.


This land is your land This land is my land
From California to the New York island;
From the red wood forest to the Gulf Stream waters
This land was made for you and Me.

As I was walking that ribbon of highway,
I saw above me that endless skyway:
I saw below me that golden valley:
This land was made for you and me.

I’ve roamed and rambled and I followed my footsteps
To the sparkling sands of her diamond deserts;
And all around me a voice was sounding:
This land was made for you and me.

When the sun came shining, and I was strolling,
And the wheat fields waving and the dust clouds rolling,
As the fog was lifting a voice was chanting:
This land was made for you and me.

As I went walking I saw a sign there
And on the sign it said “No Trespassing.”
But on the other side it didn’t say nothing,
That side was made for you and me.

In the shadow of the steeple I saw my people,
By the relief office I seen my people;
As they stood there hungry, I stood there asking
Is this land made for you and me?

Nobody living can ever stop me,
As I go walking that freedom highway;
Nobody living can ever make me turn back
This land was made for you and me.




Here is a video of Guthrie performing this song. Enjoy!




Which is your favourite Woody Guthrie song?

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As I have outlined in parts 1 and 2 of this series (see here and here), in the 1890s, mainly through the work of the Physikalisch-Technische Reichsanstalt (PTR) in Germany, the exact shape of the blackbody spectrum began to be well determined. By mid-1900, with the last remaining observations in the infrared being completed, its shape from the UV through the visible and into the infrared was well determined for blackbodies with a wide range of temperatures.

I also described in part 2 that in 1896 Wilhelm Wien came up with a law, based on a thermodynamical argument, which almost explained the blackbody spectrum. The form of his equation (which we now know as Wien’s distribution law) is
\boxed{ E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } e^{ -a / \lambda T } d \lambda }

Notice I said almost. Below I show two plots which I have done showing the Wien distribution law curve and the actual blackbody curve for a blackbody at a temperature of T=4000 \text{Kelvin}. As you can see, they are not an exact match, the Wien distribution law fails on the long-wavelength side of the peak of the blackbody curve.

Comparison of the Wien distribution law and the Planck law. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum

Comparison of the Wien distribution law and the actual blackbody curve for a blackbody at a temperature of T=4000 \text{Kelvin}. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum.

 

Comparison of the Wien distribution law and the Planck law. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum

A zoomed-in view to highlight the difference between the Wien distribution law and the actual blackbody curve for a blackbody at a temperature of T=4000 \text{Kelvin}. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum.

Planck’s “act of desperation”

By October 1900 Max Planck had heard of the latest experimental results from the PTR which showed, beyond any doubt, that Wien’s distribution law did not fit the blackbody spectrum at longer wavelengths. Planck, along with Wien, was hoping that the results from earlier in the year were in error, but when new measurements by a different team at the PTR showed that Wien’s distribution law failed to match the observed curve in the infrared, Planck decided he would try and find a curve that would fit the data, irrespective of what physical explanation may lie behind the mathematics of the curve. In essence, he was prepared to try anything to get a fit.

Planck would later say of this work

Briefly summarised, what I did can be described as simply an act of desperation

 

What was this “act of desperation”, and why did Planck resort to it? Planck was 42 when he unwittingly started what would become the quantum revolution, and his act of desperation to fit the blackbody curve came after all other options seemed to be exhausted. Before I show the equation that he found to be a perfect fit to the data, let me say a little bit about Planck’s background.

Who was Max Planck?

Max Karl Ernst Ludwig Planck was born in Kiel in 1858. At the time, Kiel was part of Danish Holstein. He was born into a religious family, both his paternal great-grandfather and grandfather had been distiguished theologians, and his father became professor of constitutional law at Munich University. So he came from a long line of men who venerated the laws of God and Man, and Planck himself very much followed in this tradition.

He attended the most renowned secondary school in Munich, the Maximilian Gymnasium, always finishing near the top of his class (but not quite top). He excelled through hard work and self discipline, although he may not have had quite the inherent natural ability of the few who finished above him. At 16 it was not the famous taverns of Munich which attracted him, but rather the opera houses and concert halls; he was always a serious person, even in his youth.

In 1874, aged 16, he enrolled at Munich University and decided to study physics. He spent three years studying at Munich, where he was told by one of his professors ‘it is hardly worth entering physics anymore’; at the time it was felt by many that there was nothing major left to discover in the subject.

In 1877 Planck moved from Munich to the top university in the German-speaking world – Berlin. The university enticed Germany’s best-known physicist, Herman von Helmholtz, from his position at Heidelberg to lead the creation of what would become the best physics department in the world. As part of creating this new utopia, Helmholtz demanded the building of a magnificient physics institute, and when Planck arrived in 1877 it was still being built. Gustav Kirchhoff, the first person to systematically study the nature of blackbody radiation in the 1850s, was also enticed from Heidelberg and made professor of theoretical physics.

Planck found both Helmholtz and Kirchhoff to be uninspring lecturers, and was on the verge of losing interest in physics when he came across the work of Rudolf Clausius, a professor of physics at Bonn University. Clausius’ main research was in thermodynamics, and it is he who first formulated the concept of entropy, the idea that things naturally go from order to disorder and which, possibly more than any other idea in physics, gives an arrow to the direction of time.

Planck spent only one year in Berlin, before he returned to Munich to work on his doctoral thesis, choosing to explore the concept of irreversibility, which was at the heart of Claussius’ idea of entropy. Planck found very little interest in his chosen topic from his professors in Berlin, and not even Claussius answered his letters. Planck would later say ‘The effect of my dissertation on the physicists of those days was nil.’

Undeterred, as he began his academic career, thermodynamics and, in particular, the second law (the law of entropy) became the focus of his research. In 1880 Planck became Privatdozent, an unpaid lecturer, at Munich University. He spent five years as a Privatdozent, and it looked like he was never going to get a paid academic position. But in 1885 Gottingen University announced that the subject of its prestigoius essay competition was ‘The Nature of Energy’, right up Planck’s alley. As he was working on his essay for this competition, he was offered an Extraordinary (assistant) professorship at the University of Kiel.

Gottingen took two years to come to a decision about their 1885 essay competition, even though they had only received three entries. They decided that no-one should receive first prize, but Planck was awarded second prize. It later transpired that he was denied first prize because he had supported Helmholtz in a scientific dispute with a member of the Gottingen faculty. This brought him to the attention of Helmholtz, and in November 1888 Planck was asked by Helmholtz to succeed Kirchhoff as professor of theoretical physics in Berlin (he was chosen after Ludwig Boltzmann turned the position down).

And so Planck returned to Berlin in the spring of 1889, eleven years after he had spent a year there, but this time not as a graduate student but as an Extraordinary Professor. In 1892 Planck was promoted to Ordinary (full) Professor. In 1894 both Helmholtz and August Kundt, the head of the department, died within months of each other; leaving Planck at just 36 as the most senior physicist in Germany’s foremost physics department.

Max Planck, who in 1900 found a mathematical equation which fitted the entire blackbody spectrum correctly.

Max Planck who, in 1900 at the age of 42, found a mathematical equation which fitted the entire blackbody spectrum correctly.

As part of his new position as the most senior physicist in the Berlin department, he took over the duties of being adviser for the foremost physics journal of the day – Annalen der Physik (the journal in which Einstein would publish in 1905). It was in this role of adviser that he became aware of the work being done at PTR on determining the true spectrum of a blackbody.

Planck regarded the search for a theoretical explanation of the blackbody spectrum as nothing less than the search for the absolute, and as he later stated

 

Since I had always regarded the search for the absolute as the loftiest goal of all scientific activity, I eagerly set to work

 

When Wien published his distribution law in 1896, Planck tried to put the law on a solid theoretical foundation by deriving it from first principles. By 1899 he thought he had succeeded, basing his argument on the second law of thermodynamics.

Planck finds a curve which fits

But, all of this fell apart when it was shown conclusively on the 2nd of February 1900, by Lummer and Pringsheim of the PTR, that Wien’s distribribution law was wrong. Wien’s law failed at high temperatures and long wavelengths (the infrared); a replacement which would fit the experimental curve needed to be found. So, on Sunday the 7th of October, Planck set about trying to find a formula which would reproduce the observed blackbody curve.

He was not quite shooting in the dark, he had three pieces of information to help him. Firstly, Wien’s law worked for the intensity of radiation at short wavelengths. Secondly, it was in the infrared that Wien’s law broke down, at these longer wavelengths it was found that the intensity was directly propotional to the temperature. Thirdly, Wien’s displacement law, which gave the relationship between the wavelength of the peak of the curve and the blackbody’s temperature worked for all observed blackbodies.

After working all night of the 7th of October 1900, Planck found an equation which fitted the observed data. He presented this work to the German Physical Society a few weeks later on Friday the 19th of October, and this was the first time others saw the equation which has now become known as Planck’s law.

The equation he found for the energy in the wavelength interval d \lambda had the form
\boxed{ E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} - 1) } \; d\lambda }

(compare this to the Wien distribution law above).

After presenting his equation he sat down; he had no explanation for why this equation worked, no physical understanding of what was going on. That understanding would dawn on him over the next few weeks, as he worked tirelessly to explain the equation on a physical basis. It took him six weeks, and in the process he had to abandon some of the ideas in physics which he held most dear. He found that he had to abandon accepted ideas in both thermodynamics and electromagnetism, two of the cornerstones of 19th Century physics. Next week, in the fourth and final part of this blog-series, I will explain what physical theory Planck used to explain his equation; the theory which would usher in the quantum age.

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In the first part of this blog (here), I described how experimenters at the Physikalisch-Technische Reichsanstalt (PTR) determined the true spectrum of blackbody radiation during the 1890s, By the year 1900, primarily by the work of Heinrich Rubens, Ferdinand Kurlkbaum, Ernst Pringsheim and Otto Lummer, the complete spectrum, from the ultraviolet through the visible and into the infrared, was known for the very first time. As the true shape of the blackbody spectrum started to emerge from this experimental work, theoreticians tried to find a theory to explain it.

The first to meet with any success was Wilhem Wien. As I mentioned in the first part of this blog, in 1893 he came up with his displacement law, which gave a very simple relationship between the wavelength of the peak of the spectrum and its temperature.



\lambda_{peak} = \frac{ 0.0029 }{ T }

where \lambda_{peak} is the wavelength of the peak in metres, and T is expressed in Kelvin.

By 1896 Wien had come up with a theory to explain the shape of the spectrum (even though the shape in the infrared was not fully known at that time). In what we now call ‘Wien’s distribution law’ or ‘Wien’s approximation’, he tried to explain the blackbody spectrum using thermodynamic arguments, and assuming that the gas molecules obeyed the Maxwell-Boltzmann speed distribution for molecules (or atoms) in a gas. I will not derive that explanation here, but if any readers wish me to derive it I can do so at a later date.



Wilhelm Wiens, who in 1893 came up with Wiens displacement law, and in 1896 with the Wien distribution.

Wilhelm Wiens, who in 1893 came up with Wiens displacement law, and in 1896 with the Wien distribution.



Wien’s distribution law (1896)

What Wien suggested was that the energy of a black body in the wavelength interval d \lambda was given by

E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } f( \lambda T) d \lambda

Wien found, using the Maxwell-Boltzmann distribution law for the speed of atoms (or molecules) in a gas, that the form of the function f( \lambda T) was

f( \lambda T ) = e^{ -a / \lambda T }

and so

\boxed{ E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } e^{ -a / \lambda T }  d \lambda }

where A \text{ and } a were constants to be determined.

If we wish to express this in terms of frequency \nu instead of wavelength \lambda then we need to remember that, from the wave equation, c = \nu \lambda and so \lambda = c/\nu. But, we also need to rewrite d\lambda in terms of d\nu and to do this we write


\nu = \frac{ c }{ \lambda } \rightarrow d \lambda = \frac{ -c }{ \nu^{2} }\; d \nu


We can ignore the minus sign as it is just telling us that as the frequency increases the wavelength decreases, and so substituting for \lambda \text{ and } d\lambda we can write
that the energy in the frequency interval d \nu is given by


E_{\nu} d \nu = \frac { A \nu^{5} } { c^{5} } e^{ -a \nu / cT } \frac{ c }{ \nu^{2} } d \nu


\boxed{ E_{\nu} d \nu = A^{\prime} \nu^{3} e^{ -a^{\prime} \nu / T } d \nu }


where A^{\prime} \text{ and } a^{\prime} are also just constants to be determined.

Wien’s ‘law’ breaks down

As I will show next week, Wien’s distribution law gave good (but not perfect) agreement with the blackbody curve on the short-wavelength side of the peak (what we now call the ‘Wien-side’ of the peak). But, as experimental results on the long-wavelength side started to emerge from the PTR, it became clear that his ‘law’ did not work on that side; it broke down on the long-wavelength side and showed very poor agreement with the actual observed curve.

Next week, in part 3 of this blogpost, I will also describe how and why Planck got involved in the problem, and what the solution he concocted was; the law which would correctly describe the blackbody spectrum and usher in the quantum age.

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For the final part of my series to commemorate the 50th anniversary of the discovery of the Cosmic Microwave Background (CMB), today I’m going to show the original papers announcing this momentous discovery to the scientific community. I should point out that I have taken these photographs to portray the historical context, even though it is not easy to read what they say. The papers have been scanned and are available online for free in both gif and pdf format, follow this link to get them.

The announcement of the CMB’s discovery came in two back-to-back papers in the July 1st edition of The Astrophysical Journal (see the front page below). On pages 414 to 419 Robert Dicke and his team from Princeton (Dicke, Peebles, Roll and Wilkinson, 1965, ApJ, 142, pp414-419) described the theoretical work they had been doing which predicted a relic radiation from a hotter denser early Universe.



The front page of the July 1st 1965 volume of Astrophysical Journal, in which the Penzias and Wilson CMB paper is to be found.

The front page of the July 1st 1965 volume of Astrophysical Journal, in which the Penzias and Wilson CMB paper is to be found.





Figure 1 from Dicke etal. in which they plot the "possible thermal history" of the Universe. It is due to the high temperatures in the early Universe that blackbody radiation would have been emitted when the Universe changed from being a plasma to being neutral ("re-combination" or "decoupling") - shown in this figure as happening when the Universe had a radius of   10^{-3} (one thousandth of its current size)

Figure 1 from Dicke etal. in which they plot the “possible thermal history” of the Universe. It is due to the high temperatures in the early Universe that blackbody radiation would have been emitted when the Universe changed from being a plasma to being neutral (“re-combination” or “decoupling”) – shown in this figure as happening when the Universe had a radius of 10^{-3} (one thousandth of its current size)





The part of Dicke etal's paper in which they refer to Penzias and Wilson's observations.

The part of Dicke etal’s paper in which they refer to Penzias and Wilson’s observations.



Then, immediately following on from this paper, on pages 419 to 421 is the paper by Penzias and Wilson (Penzias and Wilson, 1965, 142, pp419-421). For the announcement of one of the most important discoveries in the history of science, both the title and content are very understated.
The title is A Measurement Of Excess Antenna Temperature at 4080 Mc/s, hardly a title to grab the attention.



The beginning of Penzias and Wilson's paper. It possibly has the most understated title of any scientific paper of such importance.

The beginning of Penzias and Wilson’s paper. It possibly has the most understated title of any scientific paper of such importance.



The paper is nearly entirely technical, detailing their experiment and the steps they had taken to ensure that they accounted for the origin of every signal detected, apart from the “excess antenna temperature” of the title. At the end of the first paragraph of the paper is the following sentence – their only reference to its possible origin.



The only reference to the possible explanation for Penzias and Wilson's "excess antenna temperature" (i.e.. signal) is the line "a possible explanation for the observed excess noise temperature is the one given by Dicke, Peebles, Roll and Wilkinson (1965) in a companion letter in this issue."

The only reference to the possible explanation for Penzias and Wilson’s “excess antenna temperature” (i.e.. signal) is the line “a possible explanation for the observed excess noise temperature is the one given by Dicke, Peebles, Roll and Wilkinson (1965) in a companion letter in this issue.”



The paper was submitted on the 13th of May 1965, as can be seen below.



The end of Penzias and Wilson's paper, which was submitted on the xx of May 1965.

The end of Penzias and Wilson’s paper, which was submitted on the 13th of May 1965.



Although the paper appeared in the July 1st volume of Astrophysical Journal, the New York Times had picked up on the story and ran its discovery as headlines in their issue on the 21st of May 1965. Although press releases of major discoveries are now often made when the paper is submitted, I would imagine it was rather unusual in the 1960s for scientific discoveries to be published in the popular press before the journal article had appeared. Does anyone now of other examples from this time and before?



A rather fuzzy screen capture of the front page of the New York Times from the 21st of May 1965

A rather fuzzy screen capture of the front page of the New York Times from the 21st of May 1965



And this is the actual article, from the New York Times archives (one has to pay to get such articles, but it is not much).



The actual article as it appeared on the front page

The actual article as it appeared on the front page




The remainder of the article from the 21 May 1965 edition of the New York Times on the CMB's discovery

The remainder of the article from the 21 May 1965 edition of the New York Times on the CMB’s discovery



That concludes my series to mark the 50th anniversary of this most important of discoveries. If you want to read far more about the history of the CMB’s discovery, as well as its 1948 prediction and what we can learn from it, then check out my book by following this link.



My book "The Cosmic Microwave Background - how it changed our understanding of the Universe" is published by Springer and can be found by following this link.

My book “The Cosmic Microwave Background – how it changed our understanding of the Universe” is published by Springer and can be found by following this link.



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