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## Derivation of Planck’s radiation law – part 2

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.

## 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.

## Chelsea well beaten at The Etihad

Defending champions Chelsea continude their poor start to the season by going down 3-0 to a rampant Manchester City yesterday (Sunday the 16th).  Afterwards Jose Mourinho described it as a ‘fake result’. I have no idea what that means, except that he felt that Chelsea were the better team in the second half and so, somehow, the two late goals scored by City didn’t count. What a load of rubbish. Chelsea were thoroughly outplayed by a more hungry City, and were far from being the better team in either half.

Chelsea’s poor start must be quite a worry to Mourinho. They lost the FA Charity Shield match against Arsenal, they had a draw at home against Swansea (as a Welshman, the only time I have divided loyalties is when Chelsea are playing a team from Wales); and then a thorough beating yesterday at The Etihad. Mourinho claims, rather bizarrely, that he has deliberately held off on the pre-season preparation for his team, so that they start the season fresh. Again, what a load of rubbish.

The fact that he replaced club captain John Terry (of whom I am no fan, the man was (is?) an abusive racist) at half time  to me suggests not all is well at Stamford Bridge. In addition, there has been a very public spat over one of the team doctors being from coming onto the pitch in the match against Swansea; Mourinho is usually very good at controlling such stories from leaking out to the wider world.

Of course it is still very early days, with only two matches into a 38 match season it is far too early to rule Chelsea out of defending their title. But, whereas Man City have come out and started with two 3-0 wins, Chelsea look to be lacking cohesion or hunger. Now it is up to Mourinho to show why many consider him to be the best manager in football, and get Chelsea’s season kick-started.

## Sympathy for the Devil – Rolling Stones (song)

Today I thought I would share this amazing song – “Sympathy for the Devil”, by the Rolling Stones. This is probably my favourite Stones song, although there are many that I like a great deal. I’m therefore quite surprised that I have never blogged about this song before, but there we go. Good things are worth waiting for. It is listed as number 32 in Rolling Stone Magazine’s 500 greatest songs of all time.

“Sympathy for the Devil” is possibly my favourite Rolling Stones song.

Like with Lennon and McCartney of The Beatles, Mick Jagger and Keith Richards have their songs credited to both of them, irrespective of whether one or the other didn’t contribute to it. In the case of “Sympathy for the Devil”, it would seem that Jagger wrote the song, with little (or no?) input from Richards. The song recounts some of the atrocities committed throughout human history in the name of the devil, and is a nice history lesson for those who, like me, stopped studying the subject formally at 14!

Please allow me to introduce myself
I’m a man of wealth and taste
I’ve been around for a long, long year
Stole many a man’s soul to waste

And I was ’round when Jesus Christ
Had his moment of doubt and pain
Washed his hands and sealed his fate

Hope you guess my name
But what’s puzzling you
Is the nature of my game

I stuck around St. Petersburg
When I saw it was a time for a change
Killed the Tsar and his ministers
Anastasia screamed in vain

I rode a tank
Held a general’s rank
When the blitzkrieg raged
And the bodies stank

Hope you guess my name, oh yeah
Ah, what’s puzzling you
Is the nature of my game, oh yeah
(Woo woo, woo woo)

I watched with glee
(Woo woo, woo woo)

I shouted out,
Who killed the Kennedys?
When after all
It was you and me
(Who who, who who)

I’m a man of wealth and taste
And I laid traps for troubadours
Who get killed before they reached Bombay
(Woo woo, who who)

Hope you guessed my name, oh yeah
(Who who)
But what’s puzzling you
Is the nature of my game, oh yeah, get down, baby
(Who who, who who)

Hope you guessed my name, oh yeah
But what’s confusing you
Is just the nature of my game
(Woo woo, who who)

Just as every cop is a criminal
And all the sinners saints
Just call me Lucifer
‘Cause I’m in need of some restraint
(Who who, who who)

So if you meet me
Have some courtesy
Have some sympathy, and some taste
(Woo woo)
Or I’ll lay your soul to waste, mm yeah
(Woo woo, woo woo)

Hope you guessed my name, mm yeah
(Who who)
But what’s puzzling you
Is the nature of my game, mm mean it, get down
(Woo woo, woo woo)

Woo, who
Oh yeah, get on down
Oh yeah
Oh yeah!
(Woo woo)

Tell me baby, what’s my name
Tell me honey, can ya guess my name
Tell me baby, what’s my name
I tell you one time, you’re to blame

Oh, who
Woo, woo
Woo, who
Woo, woo
Woo, who, who
Woo, who, who
Oh, yeah

What’s my name
Tell me, baby, what’s my name
Tell me, sweetie, what’s my name

Woo, who, who
Woo, who, who rick
Woo, who, who
Woo, who, who
Woo, who, who
Woo, who, who
Oh, yeah
Woo woo
Woo woo

Here is a video of this amazing song. Enjoy!

## The 100 best Beatles songs – number 47 – Things We Said Today

At number 47 in Rolling Stone Magazine’s 100 best Beatles songs is “Things We Said Today”, which is from their fourth album “A Hard Day’s Night”, released in 1964 to coincide with the film of the same name.

At number 47 in Rolling Stone Magazine’s list of the 100 greatest Beatles songs is “Things We Said Today”.

McCartney wrote this song about Jane Asher, to whom he was engaged for a while (and whose brother Tony Asher co-wrote the Beach Boys’ song “God Only Knows”, which I blogged about here, with Brian Wilson). Apparently it was Asher and her family who got McCartney listening to classical music, and so e.g. the strings which became quite prominent in his music about 1965 came about because of this influence.

As usual the song is attributed to Lennon and McCartney, but as far as I am aware John Lennon played no part in the writing of this song. It certainly strikes me as being pure McCartney, but not as soppy as some of his efforts. Like most songs in this top 100 list, it was never released as a single by The Beatles.

You say you will love me
If I have to go
You’ll be thinking of me
Somehow I will know
Someday when I’m lonely
Wishing you weren’t so far away
Then I will remember
Things we said today

You say you’ll be mine, girl
Till the end of time
These days such a kind girl
Seems so hard to find
Someday when we’re dreaming
Deep in love, not a lot to say
Then we will remember
Things we said today

Me, I’m just the lucky kind
Love to hear you say that love is luck
And though we may be blind
Love is here to stay and that’s enough

To make you mine, girl
Be the only one
Love me all the time, girl
We’ll go on and on
Someday when we’re dreaming
Deep in love, not a lot to say
Then we will remember
Things we said today

Me, I’m just the lucky kind
Love to hear you say that love is luck
Though we may be blind
Love is here to stay and that’s enough

To make you mine, girl
Be the only one
Love me all the time, girl
We’ll go on and on
Someday when we’re dreaming
Deep in love, not a lot to say
Then we will remember
Things we said today

Here is a video of this wonderful song. Enjoy!

Which is your favourite song from “A Hard Day’s Night”?

## The Penzias & Wilson CMB discovery paper

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.

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.

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 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 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 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

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 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.

## Need You Tonight – INXS (song)

At number 65 in BBC Radio 2’s list of the 100 greatest guitar riffs is “Need You Tonight” by INXS. It has an infectious guitar riff and driving rhythm, and was the break-through song from this Australian band, catapulting them and their lead singer Michael Hutchence to super-stardom. Hutchence, who became the partner of Bob Geldof’s estranged wife Paula Yates, died in 1997 under tragic circumstances having apparently hung himself. There is some controversy whether this was suicide, or an “autoerotic asphyxiation” event gone wrong (as later claimed by Yates).

At number 65 in BBC Radio 2’s list of the 100 best guitar riffs is “Need You Tonight” by INXS.

“Need You Tonight” was released as a single in September 1987 and got to number 2 in the Disunited Kingdom singles charts, to number 1 in the US, and to number 3 in their native Australia.

All you got is this moment
Twenty-first century’s yesterday
You can care all you want
Everybody does yeah that’s okay

So slide over here
And give me a moment
I’ve got to let you know
I’ve got to let you know
You’re one of my kind

I need you tonight
‘Cause I’m not sleeping
That makes me sweat

How do you feel
I’m lonely
What do you think
Can’t think at all
Whatcha gonna do
Gonna live my life

So slide over here
And give me a moment
I’ve got to let you know
I’ve got to let you know
You’re one of my kind

I need you tonight
‘Cause I’m not sleeping
That makes me sweat

So how do you feel
I’m lonely
What do you think
Can’t think at all
Whatcha gonna do
Gonna live my life

So how do you feel
I’m lonely
What do you think
Can’t think at all
Whatcha gonna do
Gonna live my life

So slide over here
And give me a moment
I’ve got to let you know
I’ve got to let you know

So slide over here
And give me a moment
I’ve got to let you know
I’ve got to let you know

You’re one of my kind

Here is a video of this great song. Enjoy!

## Derivation of Planck’s radiation law – part 1

One of my most popular blogposts is the series I did on the derivation of the Rayleigh-Jeans law, which I posted in three parts (part 1 here, part 2 here and part 3 here). I have had many thousands of hits on this series, but several people have asked me if I can do a similar derivation of the Planck radiation law, which after all is the correct formula/law for blackbody radiation. And so, never one to turn down a reasonable request, here is my go at doing that. I am going to split this up into 2 or 3 parts (we shall see how it goes!), but today in part 1 I am going to give a little bit of historical background to the whole question of deriving a formula/law to explain the shape of the blackbody radiation curve.

## ‘Blackbody’ does not mean black!

When I first came across the term blackbody I assumed that it meant the object had to be black. In fact, nothing could be further from the truth. As Kirchhoff’s radiation laws state

A hot opaque solid, liquid, or gas will produce a continuum spectrum

(which is the spectrum of a blackbody). The key word in this sentence is opaque. The opaqueness of an object is due to the interaction of the photons (particles of light) with the matter in the object, and it is only if they are interacting a great deal (actually in thermal equilibrium) that you will get blackbody radiation. So, examples of objects which radiate like blackbodies are stars, the Cosmic Microwave Background, (which is two reasons why astronomers are so interested in blackbody radiation), a heated canon ball, or even a canon ball at room temperature. Or you and me.

Kirchhoff’s 3 radiation laws, which he derived in the mid-1800s

Stars are hot, and so radiate in the visible part of the spectrum, as would a heated canon ball if it gets up to a few thousand degrees. But, a canon ball at room temperature or you and me (at body temperature) do not emit visible light. But, we are radiating like blackbodies, but in the infrared part of the spectrum. If you’ve ever seen what people look like through a thermal imaging camera you will know that we are aglow with infrared radiation, and it is this which is used by Police for example to find criminals in the dark as the run across fields thinking that they cannot be seen.

The thermal radiation (near infrared) from a person. The differences in temperature are due to the surface of the body having different temperatures in different parts (e.g. the nose is usually the coldest part).

Kirchhoff came up with his radiation laws in the mid-1800s, he began his investigations of continuum radiation in 1859, long before we fully knew the shape (spectrum) of a blackbody.

## Germans derive the complete blackbody spectrum

We actually did not know the complete shape of a blackbody spectrum until the 1890s. And the motivation for experimentally determining it is quite surprising. In the 1880s German industry decided they wanted to develop more efficient lighting than their British and American rivals. And so they set about deriving the complete spectrum of heated objects. In 1887 the German government established a research centre, the Physikalisch-Technische Reichsandstalt (PTR) – the Imperial Institute of Physics and Technology, one of whose aims was to fully determine the spectrum of a blackbody.

PTR was set up on the outskirts of Berlin, on land donated by Werner von Siemens, and it took over a decade to build the entire facility. Its research into the spectrum of blackbodies began in the 1890s, and in 1893 Wilhelm Wien found a simple relationship between the wavelength of the peak of a blackbody and its temperature – a relationship which we now call Wien’s displacement law.

Wien’s displacement law states that the wavelength of the peak, which we will call $\lambda_{peak}$ is simply given by

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

if the temperature $T$ is expressed in Kelvin. This will give the wavelength in metres of the peak of the curve. That is why, in the diagram below, the peak of the blackbody shifts to shorter wavelengths as we go to higher temperatures. Wien’s displacement law explains why, for example, an iron poker changes colour as it gets hotter. When it first starts glowing it is a dull red, but as the temperature increases it becomes more yellow, then white. If we could make it hot enough it would look blue.

The blackbody spectra for three different temperatures, and the Rayleigh-Jeans law, which was behind the term “the UV catastrophe”

By 1898, after a decade of experimental development, the PTR had developed a blackbody which reached temperatures of 1500 Celsius, and two experimentalists working there Enrst Pringsheim and Otto Lummer (an appropriate name for someone working on luminosity!!) were able to show that the blackbody curve reached a peak and then dropped back down again in intensity, as shown in the curves above. However, this pair and others working at the PTR were pushing the limits of technology of the time, particularly in trying to measure the intensity of the radiation in the infrared part of the spectrum. By 1900 Lummer and Pringsheim had shown beyond reasonable doubt that Wien’s ad-hoc law for blackbody radiation did not work in the infrared. Heinrich Rubens and Ferdinand Kurlbaum built a blackbody that could range in temperature from 200 to 1500 Celsius, and were able to accurately measure for the first time the intensity of the radiation into the infrared. This showed that the spectrum was as shown above, so now Max Planck knew what shape curve he had to find a formula (and hopefully a theory) to fit.

In part 2 next week, I will explain how he went about doing that.