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## Can Wales finally defeat Australia?

The waiting is nearly over. At 5pm today, Wales will take on Australia in the long anticipated showdown between the two countries. As the table below shows, including the match at the World Cup in October 2011, Wales have played Australia a total of 6 times in the last two years, and have lost on each occasion, usually narrowly. Of the three Southern Hemisphere “superpowers”, Australia are the country Wales have beaten the most, and the country Wales have beaten the most in the professional era.

Having won the 6 Nations in March by annihilating England 30-3 in the final match, and a successful Lions’ tour where Wales contributed 10 of the starting lineup in the 3rd and decisive Test, beating Australia today could cap a very successful 2013 for the Welsh team and for Warren Gatland and his coaching team. I also believe that, if we win, it will really boost our confidence ahead of what I expect to be a very competitive 2014 6 Nations. Thankfully we will have Alex Cuthbert back in the team, and despite our injury problems in the centre I believe Wales are fielding a very strong team.

I saw yesterday morning that the Wallabies coach Ewen McKenzie has ratcheted things up nicely by saying that Wales are “worried” about falling to their ninth straight loss to Australia. Certainly it’s not a good record.

Wales have a woeful record against the three Southern Hemisphere “giants”.

England look resurgent, and France and Ireland also look vastly improved from earlier this year. The 2014 6 Nations sees Wales play England in Twickenham, which is a place we have only won 3 times in the last 25-odd years. Beating Australia today will boost Wales’ players’ confidence hugely, and Wales are very much a team whose style of play works so much better when they are playing with self belief. So DERE ‘MLAEN CYMRU, let’s beat Australia today and get that particular monkey off our back!

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## She Drives me Crazy – Fine Young Cannibals (song)

Today I thought I would post this great song, “She Drives me Crazy” by Fine Young Cannibals. This was a hit in 1989, reaching number 5 in the Disunited Kingdom and number 1 in the USA. It is from their 1988 album “The Raw and the Cooked”.

The way I feel,
Things you do,
Don’t seem real.

Tell you what,
I got in mind,
‘Cause we’re runnin’,
Out of time,
Won’t you ever,
Set me free,
This waiting’ ’round’s,
Killeen’ me.

[Chorus]
She drives me crazy,
Uh uh,
Like no-one else,
Uh uh,
She drives me crazy,
And I can’t help myself,
Uh uh.

I can’t get,
Any rest,
People say,
I’m obsessed,
Every-thing,
That’s serious lasts,
But to me,
There’s no surprize,
What I have,
Fatal last truth,
Things go wrong,
They always do.

[Chorus]
She drives me crazy,
Uh uh,
Like no-one else,
Uh uh,
She drives me crazy,
And I can’t help myself,
Uh uh.

[Instrumental Section]

I won’t make it,
On my own,
No-one likes,
To be alone.

[Chorus]
She drives me crazy,
Uh uh,
Like no-one else,
Uh uh,
She drives me crazy,
And I can’t help myself,
Uh uh.

Chorus x2 to fade

Enjoy!

Which is your favourite Fine Young Cannibals song?

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## Learning from failure

Each week, in the free Metro newspaper is an excellent “science and discovery” feature called “MetroCosm”. It is by Ben Gilliland, whom from what I can remember reading is not a trained scientist at all, but he certainly has a gift for explaining complex scientific ideas clearly and succinctly. He has a web page, which can be found here.

Last week he had a very good article about Dark Matter, the elusive component of the Universe that we think comprises some 85% of all the matter there is, but which only announces itself through its gravitational effects. Finding out what Dark Matter is is one of the many challenges facing physicists and astrophysicists, and Ben talks in this article about the negative results obtained by a recent experiment to detect dark matter particles (the particles called WIMPs, I blogged about them here).

The part of the article I enjoyed the most is the part I have illustrated below, which talks of how science advances through “failure” as much as through success. Really an experiment is never a failure, unless it is done incorrectly. Even results which do not go the way scientists expect means the scientists can learn from the experiment. Thus, the negative results obtained in searching for WIMPs will help scientists refine their future searches, it is not a wasted effort.

As Ben Gilliland so correctly says, scientists can learn just as much (and sometimes more) from a negative result as they can from a positive one.

As Ben so rightly says, this is like the negative results one obtains in playing hide and seek. If one goes into a room and determines that no-one is there, this may be a “negative result”, but it is not a wasted effort as it tells you that you have already explored that part of the house. In science we explore in experiments what is technically called “parameter space”, which may in this case be the energy of the WIMPs, their mass (which is related to their energy), or the method we are using to detect them (which will depend on how they interact with normal matter).

And, as Ben also says, negative results can sometimes be greeted with even more enthusiasm by the science community than a positive result, as it can point towards a new theory that needs to be developed, possibly uncovering a deeper understanding of the underlying science. So, scientists are amongst the few who can, sometimes, welcome “failure” (negative results). And, without exception, we always learn from our negative results.

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## The 500 greatest albums – number 485 – “Vitalogy” (Pearl Jam)

At number 485 in Rolling Stone Magazine’s 500 greatest albums of all time is “Vitalogy” by Pearl Jam. Here is the list of albums from 490 to 481.

• 490 – “Tres Hombres” by ZZ Top (1973)
• 489 – “Destroyer” by Kiss (1976)
• 488 – “New Day Rising” by Hüsker Dü (1985)
• 487 – “She’s So Unusual” by Cyndi Lauper (1983)
• 486 – “That’s the Way of the World” by Earth, Wind and Fire (1975)
• 485 – “Vitalogy” by Pearl Jam (1994)
• 484 – “All the Young Dudes” by Mott the Hoople (1972)
• 483 – “Entertainment!” by Gang of Four (1979)
• 482 – “Guitar Town” by Steve Earle (1986)
• 481 – “Voodoo” by D’Angelo (2000)

“Vitalogy” is the only album in this list which I own. I have heard of ZZ Top, Kiss, Cyndi Lauper, Earth, Wind and Fire, Mott the Hoople and Steve Earle, but do not own any albums by any of them. I do know some of the hit songs that ZZ Top, Cyndi Lauper, Earth, Wind and Fire, Mott the Hoople and Steve Earle have had, and I even have a few of those songs on my iPod. Although I have heard of Kiss, they are very definitely not my kind of music; synonymous with the worst of American rock in my opinion. I have never even heard of Hüsker Dü, Gang of Four or D’Angelo, let alone heard their music.

At number 485 in Rolling Stone Magazine’s 500 greatest albums is “Vitalogy” by Pearl Jam.

“Vitalogy” was released by Pearl Jam in 1994, and I bought the album the same year, on the strength of the single “Better Man”, which was one of my favourite songs of 1994. The album reached number 1 in the USA and quite a few other countries. In the Disunited Kingdom its highest chart position was number 4.

The song “Better Man” reached number 1 in the US “Billboard mainstream rock” chart, spending a total of 8 weeks at number 1. As I was living in the US at the time, I have no idea if it was released as a single in the Disunited Kingdom. The words and music to “Better Man”were written by Peal Jam’s lead singer, Eddie Vedder. I will retain the original American (i.e. incorrect :P) spelling of words such as “colour”. I really like this song, and before blogging about this album I had not listened to either the album or this song in quite a few years. That is one of the great things about blogging about albums in this Rolling Stone Magazine 500 greatest albums list, I am either discovering new music or rediscovering music that I haven’t listened to in quite a while. In this song, Vedder’s voice is powerful and full of texture and depth, and the lyrics are great. I really like the way the song builds from a quiet, mournful song to a powerful “rock” ballad.

Waitin’, watchin’ the clock, it’s four o’clock, it’s got to stop
Tell him, take no more, she practices her speech
As he opens the door, she rolls over…
Pretends to sleep as he looks her over

She lies and says she’s in love with him, can’t find a better man…
She dreams in color, she dreams in red, can’t find a better man…
Can’t find a better man
Can’t find a better man
Ohh…

Talkin’ to herself, there’s no one else who needs to know…
She tells herself, oh…
Memories back when she was bold and strong
And waiting for the world to come along…
Swears she knew it, now she swears he’s gone

She lies and says she’s in love with him, can’t find a better man…
She dreams in color, she dreams in red, can’t find a better man…
She lies and says she still loves him, can’t find a better man…
She dreams in color, she dreams in red, can’t find a better man…
Can’t find a better man
Can’t find a better man
Yeah…

She loved him, yeah… she don’t want to leave this way
She feeds him, yeah… that’s why she’ll be back again

Can’t find a better man
Can’t find a better man
Can’t find a better man
Can’t find a better… man…

Here is a video with lyrics of this great song. Enjoy!

Which is your favourite Pearl Jam song? Do you own any of the albums from 490 to 481?

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## Red stars, white stars, blue stars

Anyone who has looked at the night-time sky for any length of time will have noticed that stars have different colours. Some stars are red, some are white, and some are blue. A nice example of this can be seen in the Orion constellation. Betelgeuse, the star in the top left hand corner, has a distinct red colour, Rigel, the star in the bottom right hand corner, has a distinct blue colour, and Saiph, the star in the bottom left hand corner is white.

The Orion constellation. Betelgeuse in the top left is a distinct red colour, Rigel in the bottom right is a distinct blue colour, and Saiph in the bottom left is white.

## Stars have different temperatures

These differences in colour are due to stars having different surface temperatures. As the figure below shows, the blackbody curve peaks at different wavelengths depending on a star’s temperature, so a red star is cooler than a white star, which in turn is cooler than a blue star. This is because of Wien’s displacement law, which I discussed in this blog.

Stars with different surface temperatures will appear to have different colours. This is due to their blackbody spectra peaking at different wavelengths.

## Fraunhofer’s spectrum of the Sun

In 1817 the German scientist Joseph von Fraunhofer published a spectrum of the Sun showing the continuum spectrum which had long been familiar, but superimposed on this were a series of dark lines. These dark lines were shown by Kirchhoff and Bunsen in 1859 to be due to absorption lines being produced by different elements. I discussed absorption spectra in this blog. We now know that the gases which produce the absorption lines are in the atmosphere of the Sun. The visible surface of the Sun is called the photosphere, and it is the photosphere of the Sun which produces its blackbody spectrum. The overlying, thinner gases in the Sun’s atmosphere produce the numerous absorption lines which Fraunhofer saw.

The spectrum of the Sun sketched by Fraunhofer in 1814/15, showing the dark lines he observed, superimposed on the Sun’s continuum spectrum.

## The Harvard stellar classification scheme

In the 1880s the Harvard College Observatory, under the Directorship of Edward C. Pickering, set about gathering the spectra of thousands of stars. The initial catalogue was published as the Draper Catalogue of Stellar Spectra in 1890. Williamina Fleming classified the spectrum using the letters of the alphabet, with with stars with the strongest Hydrogen absorption lines having the designation A, then the next strongest B, then C etc.

The Harvard stellar classification scheme was originally in alphabetical order based on the strength of the hydrogen absorption lines of a star You can see in this figure how the strength of the Hydrogen absorption lines (e.g. the $H\alpha, H\beta \text{ and } H\gamma$ lines) are strongest for the A, B and F-type stars.

In 1901 Annie Jump Cannon revised the system. First of all she dropped most of the letters, leaving A,B,F,G,K,M and O. Secondly she subdivided each of these into 10 divisions, so for example A0, A1, A2…. A9. Thirdly, she re-ordered the letters based on the stars’ surface temperatures, not the strength of the Hydrogen line, with the hottest stars first. This is what has led to the O,B,A,F,G,K,M (Oh Be A Fine Girl/Guy Kiss Me) system we have today. The O-class stars are the hottest, the B-class stars the next hottest, all the way down to the M-class stars which are the coolest.

Annie Jump Cannon

## The strength of the lines of different elements and their ions

The figure below shows the variation of the strength of the absorption lines of different elements and their ions as a function of temperature. Stars which have the strongest Hydrogen lines (A-class stars) have a surface temperature of around 10,000 K. The bright star Vega is an A0-class star, and is white in appearance. The Sun is a G2-type star, and its strongest lines are the Calcium II lines (singly ionised Calcium).

The strengths of the absorption lines of different elements (and their ions) as a function of temperature. The nomenclature e.g. Ca II means “singly ionised Calcium”, He I means “neutral Helium”, He II means “singly ionised Helium”, etc.

The reason the strength of the Hydrogen absorption lines peak at around 10,000 K whereas the singly ionised Calcium lines (Ca II) peak at around 6,000 K is something I will explain in a future blog.

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## When Irish eyes were crying

It was an agonising way to lose a match. With 80 minutes up on the clock, Ireland were 22-17 ahead and looking at their first victory over current World Champions New Zealand in 109 years of matches between them. Ireland had the ball, and were picking and driving to run down the clock, just as they should. Then, for some bizarre reason, one of the Ireland players (I think it was the scrum half) kicked possession away, and the nightmare scenario unfolded. New Zealand scored a try in the 82nd minute, to level the scores at 22-22. Then, stupidly, the Irish players charged the conversion before the kicker had started his run up, so having missed the kick the kicker got a second chance and slotted it through the uprights, to win the game 24-22. Agony for Ireland, and as a fellow Celt I felt their pain.

Ireland could have beaten New Zealand for the first time in their history, but lost to a try scored with 82 minutes on the clock

This really is a game that Ireland could have won. Not only could have, they should have. They were 19-0 up in the first half, with three tries to their credit with less than 20 minutes played. Even in the second half, when New Zealand started dominating, Ireland’s defence held firm. Ireland missed a very kickable penalty in the last 10 minutes which would have taken them into a 25-17 lead. But to be ahead with the 80 minutes up, to be in possession and to still lose the game is almost unforgivable. And, what I don’t understand is that the team who are most adept at running down the clock is the Irish region Munster. What on earth possessed Ireland to kick possession away when all they had to do was keep picking and driving and the game would have been theirs?

They say that teams learn more from defeat than from victory, but this will be a very very harsh lesson for Ireland. Not only could they have recorded their first ever victory over New Zealand, but they could have also denied New Zealand becoming the first country in the professional era to go a whole calendar year winning every game (14 of them).

Meanwhile, in Cardiff on Friday evening Wales put in a very mediocre display in beating Tonga 17-7. There were no scores in the second half, and I nearly fell asleep through boredom. Even the match commentators Eddie Butler and Jonathan Davies (both Welsh), referred to it as “dire”. Of course Wales’ big game is next Saturday against Australia, which I have said before could determine not only how Wales reflect on 2013 but also could determine how well we perform in the 2014 6 Nations. So I understand why Warren Gatland didn’t put out a full-strength team, he wanted to rest some players and also to give some players their first international experience.

But, even a half-strength Welsh team should play better than that. We should be putting 30-40 points on teams like Tonga if we have pretensions to challenge for the World Cup in 1015. And, a convincing victory by that kind of margin would have done so much to boost our confidence ahead of next Saturday’s game against Australia. Instead, after a clinical and dominant display against Argentina last week, we were dragged down to the level of a mediocre team and failed to excel. I just hope we can come out next Saturday and end a so-far very good season by finally claiming a long-overdue victory over Australia!

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## In Your Eyes – Peter Gabriel (song)

Today I thought I would share this great song by Peter Gabriel – “In Your Eyes”. It was released in 1986 and is from his 5th studio album “So”. “In Your Eyes” was not released as a single in the Disunited Kingdom. In the US it reached no 26 in the Billboard charts. One of the more unusual features of the song is some singing towards the end by Youssou N’Dour, a singer from Senegal, who sings some words translated into his native Wolof, a language of Senegal, The Gambia and Mauritania. When you consider that this song was recorded in 1985, it illustrates how Gabriel was one of the early champions of what became known as “World Music”

Peter Gabriel wrote both the words and the music. The lyrics are

Love I get so lost, sometimes
Days pass and this emptiness fills my heart.
When I want to run away
I drive off in my car
But whichever way I go
I come back to the place you are.

All my instincts, they return
And the grand facade, so soon will burn
Without a noise, without my pride
I reach out from the inside

In your eyes
The light the heat
In your eyes
I am complete
In your eyes
I see the doorway to a thousand churches
In your eyes
The resolution of all the fruitless searches
In your eyes
I see the light and the heat
In your eyes
Oh, I want to be that complete
I want to touch the light
The heat I see in your eyes

Love, I don’t like to see so much pain
So much wasted and this moment keeps slipping away.
I get so tired of working so hard for our survival
I look to the time with you to keep me awake and alive.

And all my instincts, they return
And the grand facade, so soon will burn
Without a noise, without my pride
I reach out from the inside

In your eyes
The light the heat
In your eyes
I am complete
In your eyes
I see the doorway to a thousand churches
In your eyes
The resolution of all the fruitless searches
In your eyes
I see the light and the heat
In your eyes
Oh, I want to be that complete
I want to touch the light,
The heat I see in your eyes
In your eyes in your eyes
In your eyes in your eyes
In your eyes in your eyes

The first Peter Gabriel song I heard was his hit single “Solsbury Hill”, which got to number 20 in the DUK in 1977. I then discovered that he had been in the band Genesis, but I must say that I am not a fan of progressive rock and have never taken to the early Genesis albums. I only like their stuff after he left, but ironically I like a lot of Gabriel’s solo work.

Here is the song. Enjoy!

Which is your favourite Peter Gabriel song? Did you prefer him with Genesis, or his solo work?

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## Comet ISON

On Tuesday (19th of November) I was on the BBC just before 7am talking about Comet ISON. I am going to be on TV this next Tuesday evening (26th of November) talking about the same topic, so I thought it would be a good idea to do a blog about it.

## What is a comet?

Back in this blog, I talked about Comet PANSTARRS. I briefly mentioned that comets come from either the Kuiper belt or from the Oort Cloud. But I did not really say what a comet is. A comet is a ball of ice, rock and dust, sometimes referred to as “a dirty ball of ice”. The ice includes water-ice (astronomers can also refer to carbon dioxide and ammonia and methane as “ice”, as they can freeze at the low temperatures found in parts of the Solar System). It is the boiling away of the water-ice as a comet approaches the Sun which gives it its distinctive appearance.

In fact, a comet has two tails. This picture of the comet Hale-Bopp, taken in 1997, shows the two tails quite distinctly.

A photograph of the 1997 comet Hale-Bopp, showing the “dust tail” (white) and the “ion tail” (blue).

The more distinctive white tail is also called the “dust tail”, and is due to dust and ice boiling away from the solid body of the comet as it gets heated up in the inner parts of the Solar System. This material forms a coma around the solid body of the comet, and as the comet hurtles through space it leaves some of this material behind in its wake. This tail points in the opposite direction to the comet’s direction of travel, directly behind where it is heading.

The less distinctive blue tail is called the “ion tail”. It is also due to ice and dust, but rather than being left behind by the comet’s motion through space, the material in the ion tail has been driven away from the comet by the Solar wind, a stream of charged particles coming from the Sun. So, the ion tail always points away from the Sun, which may not be in the same direction as the direction in which the comet is travelling.

A cartoon of the components of a comet

## What is Comet ISON?

Comet ISON was discovered by Valili Nevski and Artyom Novichonok on the 21st of September last year (2012) using the 0.4m International Scientific Optical Network (ISON), from which it gets its name. From studies of its path it is felt that it has an open, hyperbolic orbit, which suggests it does not come from the Kuiper belt, but rather from the Oort Cloud. It was presumably nudged out of the Oort Cloud by a passing star in our Galaxy. Because it has come from the Oort Cloud, it is a new comet in the sense that it has not been to the inner parts of our Solar System before, not will it return if it really is in an open (hyperbolic) orbit.

A photograph of Comet ISON

Comet ISON is rapidly approaching its closest approach to the Sun, something called perihelion. This will be on 28th of November, when it will be only 0.0124 AUs (Astronomical Units) from the Sun, which translates to 1.86 million kms. By Solar System standards, this is pretty close, and Comet ISON has been referred to by some as a “Sun-grazing comet”, so close to the Sun is its closest approach. About a month later, on the 26th of December (Boxing Day to those of us in the Disunited Kingdom) it will pass at its closest to the Earth, 0.43 AUs (64 million kms) away.

## Where should I look to see Comet ISON?

As comet ISON approaches the Sun it will be moving pretty quickly through the sky. That is to say, its position from night to night will be quite different. Here is a chart showing its position in the early morning sky at the beginning of December. Currently, as of today the 21st of November, it has just passed the bright star Spica in the constellation Virgo and has entered the constellation Libra. You should look for it in the sky just before Sunrise. It is not visible in the evening or middle of the night at present, only in the pre-dawn sky.

Comet ISON’s position in the sky in late November.

Where to find Comet ISON in early December

It has been said that Comet ISON could be one of the best comets for many many years. We do not really know yet, but given how close it is approaching to the Sun, and the observations of it so far, it could indeed put on a spectacular display for us. We will find out over the next few weeks, so keep your fingers crossed!

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## The 500 greatest albums – number 492 – “Touch” (Eurythmics)

Last week I reached number 6 in my countdown of the top 30 Greatest Albums of all Time, as listed by Rolling Stone Magazine. Before I continue with the top 5, I thought I would take a digression and talk about some of the albums in the top 500 which are not in the top 30, but which I personally own. The Rolling Stone Magazine list of the albums between 500 and 491 is as follows:

• 500 – “Aquemini” by OutKast (1998)
• 499 – “Live in Cook County Jail” by B.B. King (1971)
• 498 – “The Stone Roses” by The Stone Roses (1989)
• 497 – “White Blood Cells” by The White Stripes (2001)
• 496 – “Boz Scaggs” by Boz Scaggs (1969)
• 495 – “Give It Up” by Bonnie Raitt (1972)
• 494 – “Oracular Spectacular” by MGMT (2008)
• 493 – “Yankee Hotel Foxtrot” by Wilco (2002)
• 492 – “Touch” by Eurythmics (1984)
• 491 – “Born Under a Bad Sign” by Albert King (1967)

The first one I will blog about is at number 492 in the list, the album “Touch” by the 1980s band Eurythmics. This is actually the only album of this list from 500 to 491 that I do own, and I don’t own any albums by any of the other artists. In fact, I have only ever heard of B.B. King, The Stone Roses, Boz Scaggs, Bonnie Raitt and Wilco. I have no clue who the others are 😛

At number 492 in Rolling Stone Magazine’s 500 greatest albums is “Touch” by Eurythmics.

I was a fan of Eurythmics in their day, I used to love Annie Lennox’ voice. I was never much of a fan of synthesiser pop, but her voice and the good lyrics made up for it. I even saw the Eurythmics in concert in Adelaide Australia in early 1987. “Touch” was the Eurythmics’ 3rd studio album, released in 1983. It reached number 1 in the Disunited Kingdom and was a top 10 album in the USA too. Three of the tracks on the album were released as singles, namely

• Here Comes the Rain Again
• Right By Your Side
• Who’s That Girl?

I like all three songs, and I do feel they are the strongest songs on the album. Here is “Here Comes the Rain Again”. Enjoy!

Which is your favourite Eurythmics song? Do you prefer Annie Lennox’ solo stuff, or her work with Dave Stewart?

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## The Bohr model of the atom

In my blog on Niels Bohr, number 2 in The Guardian’s list of the 10 best physicists, I mentioned the Bohr model of the atom. In this blog I will go into more detail about this model, and how it agreed with the experimental results (the Rydberg formula) for the hydrogen atom.

## Quantised orbits

In 1911, Rutherford had proposed that atoms have positively charged nuclei, with the negatively charged electrons orbiting the nuclei. One of the problems with this idea was that an orbiting electron would be accelerating, by virtue of moving in a circle. The acceleration is directed towards the centre of the circle. It was well known that when an electron is accelerated it radiates electromagnetic waves. Calculations showed that the orbiting electrons on Rutherford’s model should radiate away their energy in a few microseconds (millionths of a second), and spiral towards the nucleus. They clearly were not doing this, but why?

Bohr suggested in a paper in 1913 that electrons would somehow not radiate away their energy if they were orbiting in certain “allowed orbits”. If they were in these special orbits, the normal laws of EM radiation would not apply. He suggested that these allowed orbits were when the orbital angular momentum $L$ could be written as

$L = \frac{ n h }{ 2 \pi } = n \hbar \text{ (Equ. 1) }$

($\hbar = \frac{ h }{ 2 \pi } \text{ where } h$ is Planck’s constant, and is given its own symbol in Physics at it crops up so often). What is orbital angular momentum? Well, it is the rotational equivalent of linear momentum. Linear momentum is defined as $\vec{p} = m\vec{v} \text{ where } m \text{ is the mass and } \vec{v} \text{ is the velocity}$. Notice, momentum is a vector quantity, this is important in doing calculations involving collisions, such as the ones I did in this blog.

By analogy, orbital angular momenutm is defined as

$\vec{L} = \vec{r} \times m \vec{v}$

where $\vec{r}$ is the radius vector of the orbit, which is defined as pointing from the centre of the orbit along the radius. For a circular orbit, where the radius vector is at right angles to the velocity vector, we can just write $L=mvr$ where $L$ is the magnitude (size) of the vector $\vec{L}$.

## The force keeping the electron in orbit

From Classical Physics, Bohr argued that the force which was keeping the electron in orbit about the positively charged nucleus was the well known Coulomb force, given by

$F = - \frac{ Z k_{e} e^{2} }{ r^{2} }$

where $k_{e}$ is the Coulomb constant (which determines the force between two 1 Coulomb charges separated by 1 metre), $Z$ is the atomic number of the atom, $e$ is the charge on an electron and $r$ is the radius of the orbit. The minus sign is telling us that the force is directed towards the centre, whereas our definition of the radius vector is that it is away from the centre, so they are in opposite directions.

We can equate this to the formula for the centripetal force on any object moving in a circular orbit, so we can write

$\frac{ m_{e} v^{2} }{ r } = \frac{ Z k_{e} e^{2} }{ r^{2} } \text{ (Equ. 2) }$

where $m_{e}$ is the mass of the electron and $v$ is the speed of its orbit.

## The Bohr radius

Re-arranging Equation 2 we can write

$v = \sqrt{ \frac{ Z k_{e} e^{2} m_{e} r }{ m_{e}^2 r^{2} } }$

which then allows us to write the angular momentum as

$m_{e} v r = \sqrt{ Z k_{e} e^{2} m_{e} r } \text{ which (from Equ. 1) } = n \hbar$

This allows us to write an expression for the “radius” of an electron’s orbit as

$r_{n} = \frac{ n^{2} \hbar^{2} }{ Z k_{e} e^{2} m_{e} }$

where $n$ is the energy level of the electron. The so-called “Bohr radius” is the radius of an electron in the $n=1$ energy level for hydrogen ($Z=1$) and can be written

$\boxed{ r_{1} = \frac{ \hbar^{2} }{ k_{e} e^{2} m_{e} } \approx 5.29 \times 10^{-11} \text{ metres } }$

This is, indeed, about the size of a hydrogen atom.

## The total energy of the electron

The total energy of the electron in its orbit is given by the sum of its kinetic energy and its potential energy. The kinetic energy is just given by $1/2 \; (mv^{2})$. What about the potential energy? The potential energy can be found by using the relationship between work and force; back in this blog I said that work was defined as the force multiplied by the distance moved. Energy is the capacity to do work, and is measured in the same units, Joules. So we can derive the potential energy of an electron in orbit due to the Coulomb force as

$P.E. = \int_{r}^{\infty} { F} dr = - \int_{r}^{\infty} \frac{ Z k_{e} e^{2} }{ r^{2} } dr$

where $dr$ is an incremental change in the radius. If we do this integration we get

$P.E. = - \frac{ Z k_{e} e^{2} }{ r }$

where the negative sign is telling us that we have to do work on the electron to increase its radius, or to put it another way that the force acts towards the centre but the radius vector acts away from the centre of the electron’s orbit. This means that the total energy $E$ is given by

$E = \frac{ 1 }{ 2 } m_{e} v^{2} - \frac{ Z k_{e} e^{2} }{ r }$

But, from Equ. 2 we can write the kinetic energy as

$\frac{ 1 }{ 2 } m_{e} v^{2} = \frac{ Z k_{e} e^{2} }{ 2r }$

So then the total energy $E$ can be written

$E = \frac{ Z k_{e} e^{2} }{ 2r } - \frac{ Z k_{e} e^{2} }{ r } = - \frac{ Z k_{e} e^{2} }{ 2 r }$
So, in the Bohr model, the energy of the $n^{th}$ energy level is given by

$\boxed{ E_{n} = - \frac{ Z k_{e} e^{2} }{ 2 r_{n} } \text{ or } -\frac{ Z^{2} (k_{e} e^{2})^{2} m_{e} }{2 \hbar^{2} n^{2} } }$

In the case of hydrogen, where $Z=1$ we can write

$\boxed{ E_{n} = -\frac{ (k_{e} e^{2})^{2} m_{e} }{ 2 \hbar^{2} n^{2} } \approx -\frac{ 13.6 }{ n^{2} } \text{ eV} }$

This was in perfect agreement with the Rydberg formula for the energy levels of hydrogen, which had been experimentally derived by the Swedish physicist Johannes Rydberg in 1888. As I will show in a future blog, Bohr’s model was a “semi-empirical” model, in that it was a step along the way to the correct model. It was produced by using a mixture of classical physics and quantum mechanics, and Bohr did not understand why his condition that only orbits whose angular momentum were equal to $n \hbar$ was true. The explanation was produced with the full theory of Quantum Mechanics in 1926 as a solution to Schrödinger’s wave equation for hydrogen.

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