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## 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.

## Tân yn Llŷn (A Fire in Llŷn)

Last week, in this blog here, I shared a song “Dros Gymru’n Gwlad”, performed by Dafydd Iwan but written by the Reverend Lewis Valentine. I mentioned in that blog that Lewis Valentine held a special place in 20th century Welsh history, so today I am giving that history.

Lewis Valentine (1893-1986), together with Saunders Lewis (1893-1985) and D.J. (David John) Williams (1885-1970) were the three men who were involved in this particular event. Valentine was a Baptist minister in North Wales. Saunders Lewis was born and brought up in Liverpool in a Welsh-speaking family (his father was a minister in a Welsh-speaking chapel in Liverpool). He became a celebrated playwright and lecturer in English at Swansea University, and the founder in 1925 of Plaid Cymru, the ‘Party of Wales’. D.J. Williams (never known as David John!) was born in Rhydycymerau in rural Carmarthenshire, and in addition to writing short stories he was an English teacher at the Grammar School in Fishguard, West Wales (I went to that school in the 1970s but by that time it was a comprehensive school). In 1936, in protest to the

• ‘English’ preparations for war
• English imperialism in Wales (some 500,000 people had protested against the construction of the bombing school)
• the destroying of an historical Welsh landmark (Penyberth had been used for centuries as a stopping point for pilgrims going to Ynys Enlli (Bardsey Island), which is at the end of the Llŷn peninsula)

the three of them set fire to an RAF bombing school on the Llŷn Peninsula, at a place called Penyberth. At the time the men were in their early forties, and deliberately chosen by Plaid Cymru as the three were all middle-aged and respectable pillars of their communities.

DJ Williams (left), Lewis Valentine (centre) and Saunders Lewis (right); taken in 1936, the year they set fire to the bombing school in Penyberth on the Llŷn peninsula. In Welsh, they are often known as “y tri” (the three).

Penyberth is often seen as the first act of Welsh nationalism (patriotism) of the 20th Century. After setting fire to the bombing school, the three men made their way to the local police station where they gave themselves up and told the confused police officer what they had done and why. In the subsequent court case in Caernarfon a largely sympathetic jury of their peers failed to find them guilty, and so the trial was sent to the Old Bailey in London, where the three were found guilty and sent to jail. They each served 9 months in prison in Wormwood Scrubs. Saunders Lewis was, controversially, dismissed from his job at Swansea University before he had been found guilty of the crime. He was subsequently hired as a lecturer of English at Cardiff University (strictly speaking “University of Wales, Swansea” and “University of Wales, Cardiff”, as they were known at the time).

A plaque at the site of the arson of the bombing school in Penyberth.

An interesting historical quirk of their trial in Caernarfon is that, at that time (and up until the “Welsh Language Act” of 1967), a Welsh person had no right to give their testimony in Welsh in a court in Wales. Ever since the “Laws in Wales” acts of 1535-1542, English had been made the only language of legal proceedings in Wales. The only exception allowed to this rule was if one could prove that one’s English was inadequate. All three wished to give their testimonies in Welsh, but Lewis Valentine was the only one allowed to do so, as no evidence could be provided that he was anything like fluent enough in English.

As for the other two, Saunders Lewis had a degree in English from Liverpool University (the city where he was born and brought up); and D.J. Williams also had a degree in English from Aberystwyth (University of Wales, Aberystwyth), and had done post-graduate studies at Jesus College, Oxford! Additionally, at the time of the trial, Saunders Lewis was lecturing in English, and D.J. Williams teaching English at Fishguard Grammar School. Not surprisingly, their English was deemed to be good enough, and they were not allowed to testify in their own language.

If you want to read more about this episode of Welsh history, I can recommend the excellent book by Dafydd Jenkins, my copy is shown below.

My copy of the book “Tân yn Llŷn” by Dafydd Jenkins, which I bought in 1986.

Had you ever heard of Penyberth, or any of “y tri” before?

## Dros Gymru’n Gwlad – Dafydd Iwan (song)

Today I thought I would blog about this beautiful song (hymn), “Dros Gymru’n Gwlad” (For Wales, our country), written by the Reverend Lewis Valentine, and here performed by Dafydd Iwan. It is sometimes referred to as Wales’ second national anthem. This version of the song is, in fact, entitled “Gweddi Dros Gymru” (A Prayer for Wales) by Dafydd Iwan; but it is the same song, just with a different (and maybe more apt) title.

“Gweddi dros Gymru” (Prayer for Wales) appeared on Dafydd Iwan’s 1986 album “Gwinllan a Roddwyd” (A vineyard was given).

I will blog next week about who Lewis Valentine was, because he holds a particular place in Welsh history for an act of defiance he committed in 1936 along with DJ Williams and Saunders Lewis. But, today I will just concentrate on this song/hymn.
“Dros Gymru’n Gwlad” is usually set to the tune of Sibelius’ Finlandia, as it is in the video I include below.

Here are the words (in Welsh)

Y winllan wen a roed i’n gofal ni;
D’amddiffyn cryf a’i cadwo’n ffyddlon byth,
A boed i’r gwir a’r glân gae1 ynddi nyth;
Er mwyn dy Fab a’i prynodd iddo’i hun,
O! crea hi yn Gymru ar dy lun.

O! deued dydd pan fo awelon Duw
Yn chwythu eto dros ein herwau gwyw,
A’r crindir cras dan ras cawodydd nef
Yn erddi Crist, yn ffrwythlon iddo Ef;
A’n heniaith fwyn â gorfoleddus hoen
Yn seinio fry haeddiannau’r Addfwyn Oen.

And now for my translation. As always, I am not going to attempt to retain any rhyme or meter, just translate the words as best I can; so that you get the meaning of what Lewis Valentine was trying to say in his song/hymn.

For Wales our country, O Father I raise a wail,
This pure vineyard which was given to us to care for;
May You protect it vigorously and keep it forever faithful,
And let the true and the pure find in her a nest;
For your Son who bought it for himself,
Oh! create a Wales in Your image.

Oh! Let there come a day when the breezes of God
Are once again blowing over our wilted acres,
And the awful wasteland under the grace of showers from heaven
Gardens of Christ, fruitful to Him;
And her old sweet language with a cheerful vigour
Ringing out on high, the deserves of the Gentle Lamb.

Here is an alternative translation which I found. It is far more poetic and less clumsy than mine, but less true to what Lewis Valentine was actually saying in his lyrics.

For Wales our land O Father hear our prayer,
This blessed vineyard granted to our care;
May you protect her always faithfully,
And prosper here all truth and purity;
For your Son’s sake who bought us with His blood,
O make our Wales in your own image Lord.

O come the day when o’er our barren land
Reviving winds blow sent from God’s own hand,
As grace pours down on parched and arid sand
We will bear fruit for Christ by his command,
Come with one voice and gentle vigour sing
The virtues of our gentle Lamb and King

Here is a video I have created on YouTube of Dafydd Iwan’s version of this song/hymn.

Had you heard of this song before?

## Why do we have leap seconds?

At midnight on the night of Monday the 30th of June, an extra second was added to our clocks. A so-called leap second. Did you enjoy it? Me too 🙂 I got so much more done….. But, why do we have leap seconds?

In this blog here, I explained the difference between how long the Earth takes to rotate $360^{\circ}$ (the sidereal day) and how long it takes for the Sun to appear to go once around the Earth (the mean solar day). We set the length of our day, 24 hours, by the solar day. If there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, then there should be $24 \times 60 \times 60 = 86,400 \text{ seconds}$ in a solar day. But, there aren’t! The Earth’s rotation is not consistent, that is if we measure the length of a mean solar day, it is not consistently 86,400 seconds. This difference is why we need leap seconds.

A leap second was added at midnight on the 30th of June. It was the first leap second to be added since 2012.

But, how do we accurately measure the mean solar day (the average time the Sun appears to take to go once around the sky) , and what is causing the length of the mean solar day to change?

## How do we define a second?

When the second was first defined, it was defined so that there were 86,400 seconds in a mean solar day. But, since the 1950s, we have a very accurate method qof measuring time, atomic clocks. Using these incredibly accurate time pieces (the most accurate atomic clocks will be correct to 1 second over some tens of thousands of years) we have been able to see that the mean solar day varies. It varies in two ways, there is a gradual lengthening, but there are also random changes which can be either the Earth speeding up or slowing down its rotation.

## How do we measure the Earth’s rotation so accurately

In order to measure the Earth’s rotation accurately we use the sidereal day, which is roughly four minutes shorter than the mean solar day. By definition, the sidereal day is the time it takes for a star to cross through a local meridian a second time. But, actually, stars in our Galaxy are not good for this as they are moving relative to our Sun. So, in fact, we use quasars, which are active galactic nuclei in the very distant Universe; and use radio telescopes to pinpoint their position.

## The gradual slowing down of the Earth’s rotation

There is a gradual and unrelenting slowing down of the Earth’s rotation, which may or may not be greater than the random changes I am going to discuss below. This gradual slowing down is due to the Moon, or more specifically to the Moon’s tidal effects on the Earth. As you know, the Moon produces two high tides a day, and this bulge rotates as the Earth rotates. But, the Moon moves around the Earth much more slowly (a month), so the Moon pulls back on the bulge of the Earth, slowing it down. To conserve angular momentum, the Earth slowing down means the Moon moves further away from the Earth, about 3cm further away each year.

## The random fluctuations in the Earth’s rotation

In addition to the unrelenting slowing down of the Earth’s rotation due to the Moon, there are also random changes in the Earth’s rotation. These can be due to all manner of things, including volcanoes and atmospheric pressure. These random fluctuations can either speed up or slow down the Earth’s rotation.

We have been having leap seconds since the 1970s when atomic clocks became accurate enough to measure the tiny changes in our planet’s rotation. Since them we have added a leap second when it is decided that we need it, typically but not quite once a year. However, having that extra second at the end of June can cause glitches with computers, and so there are discussions to remove the leap second and replace it with something larger on a less frequent basis.

## The Prediction of the Cosmic Microwave Background – the original paper

Last week I reposted my blog about the prediction of the cosmic microwave background (CMB), which I had originally written in April 2013. This month, July, marks the 50th anniversary of the first detection of the CMB, and I will blog about that historic discovery next week. But, in this blog, I wanted to show the original 1948 paper by Alpher and Hermann that predicted the CMB’s existence.

I learnt far more about the history of the CMB’s prediction whilst researching for my book on the CMB, which was published at the end of 2014 (follow this link to order a copy). In doing my research, I found out that many of the things I had been been told or had read about the prediction were wrong, so here I wanted to say a little bit more about what led up to the prediction.

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.

## Gamow did not predict the CMB

Many people either do not know of the 1940s prediction of the CMB, or they attribute its prediction to George Gamow. In fact, it was his research assistants Ralph Alpher and Robert Hermann who made the prediction, but as head of the group it is often Gamow who gets the credit.

Ralph Alpher had just finished his PhD on the origin of the elements, and after the publication of the famous Alpher, Bethe, Gamow paper (see my blog here about that), Gamow started writing a series of papers on the nature of the early Universe. One of these papers was entitled “The Evolution of the Universe”, and it appeared in Nature magazine on the 30th of October 1948 (Nature 1948, volume 162, pages 680-682) – here is a link to the paper.

Gamow’s October 1948 paper in Nature was entitled “The Evolution of the Elements”.

Although a man of huge intellect and inventiveness, Gamow was often sloppy on mathematical detail. Alpher and Hermann spotted an error in some of Gamow’s calculations on the matter-density, and so wrote a short letter to Nature magazine to correct these mistakes. The letter is entitled “Evolution of the Universe”, nearly the same title as Gamow’s paper, but with no “The” at the start. The letter is dated 25 October 1948. It appeared in Nature magazine on the 13th of November 1948 (Nature 1948, volume 162, pages 774-775) – here is a link to the paper.

Here is the paper in its entirety (it is short!), and I have highlighted the part which refers to a relic radiation from the early Universe, what would become known as the cosmic microwave background.

The original paper (letter) by Alpher and Hermann which makes the first prediction of the cosmic microwave background (CMB). It was published in Nature magazine on the 13th of November 1948.

As you can see, the prediction is not the main part of the paper, it just forms two sentences!

Next week, I will blog about the accidental discovery of the CMB by Penzias and Wilson, which was published 50 years ago to this month (July).

## How long is a day?

How long is a day? It seems like a stupid question. As everyone knows, there are 24 hours in a day. The Earth rotates on its axis once every 24 hours. Or does it?

## The difference between a ‘solar day’ and a ‘sidereal day’

In fact, there is a slight difference between how long the Earth actually takes to rotate $360^{\circ}$ (the ‘sidereal day’, the day as measured by the motion of stars in the sky) and how long it takes for the Sun to appear to go once around the Earth (the ‘solar day’). This is because, during the course of a day, the Earth has moved a little bit in its orbit about the Sun, and so the Earth has to rotate a little bit more than $360^{\circ}$ to bring the Sun back over the local meridian. We measure our day by the solar day, as otherwise the time of local noon would drift away from midday more and more, which we clearly do not want. (You may notice that this is related to the difference between a sidereal month and a synodic month, which I discussed here.)

The difference between a solar day and a sidereal day, which comes about because of the Earth’s motion about the Sun.

## Kepler’s 2nd law

This difference is easy to measure, with a sidereal day being, on average, 4 minutes shorter than a solar day. This means that stars rise about 4 minutes earlier from day to day, or over the course of a month about 2 hours earlier. But, this 4 minute difference is not constant. It changes because the Earth is orbiting the Sun in an ellipse, not a circle. This means that the Earth’s speed in orbit changes, it travels faster when it is closer to the Sun (in January), and slower when it is further from the Sun (in July). This fact, which was first noticed by Kepler, is now known as Kepler’s 2nd law of planetary motion. It is illustrated below.

Kepler’s 2nd law of planetary motion states that a planet will sweep out an equal area in equal time, so that in the same period of e.g. 1 month, the three areas A will be equal. This means that a planet travels quicker when it is near the Sun, and slower when it is further away.

When the Earth is travelling quicker it has to rotate a little bit more to complete a solar day, and when it is travelling slower it has to rotate a little bit less. So, the length of the actual solar day changes in the course of a year, but in a cyclical fashion (this is known as the equation of time, something I will explain more in a future blog). The equation of time is the reason for the East-West motion of the Sun as shown in the analemma, which I discuss here.

Because of these changes in the difference between a sidereal day and a solar day at any given time of the year, we define something called the mean solar day, and it is the mean solar day which should be 24 hours, or 86,400 seconds long. But, it isn’t! In a blog next week, I will explain how the Earth’s period of rotation is not consistent, and this is why we had a leap second at midnight on the 30th of June this year.

## When did Newton die?

In the last week I have been putting together the final edits for the book that I have been writing with Brian Clegg – Ten Physicists who transformed our understanding of reality, which will be published later this year. One of the issues which we needed to clarify in this editing process were the dates of Newton’s birth and death. The reason this is an issue is that the calendar system was changed in the period between the 1500s and 1700s, which spans the years that Newton was alive. In 1582 the Gregorian calendar was introduced by Pope Gregory XIII, because the Julian calendar’s system of having a leap year every four years is not exactly correct (I will blog separately about the details of why having a leap year every four years is not correct).

‘Ten Physicists who transformed our understanding of reality” will be out December

Different countries adopted the Gregorian calendar at different times, with Catholic countries adopting it before Protestant ones. Newton was born in England in the mid-1600s, and when he was born England was still using the Julian calendar. Under the Julian calendar, he was born in the early hours of the 25th of December 1642. But, by that time, many European countries were using the Gregorian calendar, and so had someone in e.g. France heard of his birth on that day (imagine radio existed!), their calendar would have said that the date was the 4th of January! But, which year, 1642 or 1643? This is where another subtlety of calendars arrises, because starting the year on the 1st of January is something else that changed during this period.

In England, the year traditionally began on the 25th of March, and so the 4th of January (the one 10 days after the 25th of December) was still in 1642! The 4th of January 1642 actually came after the 25th of December 1642, because the year did not change to 1643 until the 25th of March! (The year starting on the 25th of March is also why the financial year (Tax year) in Britain still starts on the 6th of April, the date to which that the 25th of March was adjusted when the Gregorian calendar was finally adopted in Britain.)

However, in France (for example) they switched to starting their year on the 1st of January in 1564 (prior to this France started their year at Easter), so again this hypothetical person in France who heard of Newton’s birth would have said the date was the 4th of January 1643 (for more about this see here).

The first page of the Papal bull announcing the introduction of the Gregorian calendar, which was published on the 24th of February 1582

A similar confusion arises over Newton’s death. Under the Julian calendar, he died on the 20th of March 1726. At the time of his death, England was still using the Julian calendar, and was also still starting its year on the 25th of March (it switched to the Gregorian calendar and to starting its year on the 1st of January in 1752). So, had Newton died just 5 days later his date of death would have been the 25th of March 1727, which to any casual reader would imply he was a year older than he actually was. To someone in France, the date of Newton’s death would have been the 30th of March 1727.

Confused? 😉