Elements

The Periodic Table

Just about everyone is familiar with the image of the periodic table.

Even the name is familiar, you say "periodic table" to some one and chances are, providing they have had some education, they are likely to picture something that looks a little like the image to the left.

I have always loved the periodic table, when I was a kid and I first saw one in a class room I was instantly struck by just how beautiful it was. In those days there were 103 elements, the last being Lawrencium named after Ernest Lawrence the Nobel prize winning physicist who had quiet a lot to do with radioactivity. These days there are more than 112, the last being Copernicium.

Livermorium at 116 has been added as of May 2012, so far 35 atoms of Livermorium have been produced in the 12 years since it was first discovered!

The periodic table as it is now was first dreamt up by Dmitri Ivanovich Mendeleev and presented by him in 1871, it looks a little different to the modern day, but that is partly because there were only 56 known elements at the time.


What fascinated me about the periodic table was that it listed each and every element that makes up the known universe. All in one little table. Each element having one more proton than its predecessor. The elements are stacked into rows consisting of 2 elements, H and He, then 8, that's Li through to Ne, then 8, then 18, then another 18, then 32 and finally the remaining 25 to make a total of 112. If we get as far as element 118 then chances are we will have to start a new row.

Carbon, which gives us diamonds the hardest known material, sits right next to nitrogen which is a gas! Yet the differ by only a single proton. Potassium, the highly reactive metal that burns with a purple flame has only one more proton than Argon, a gas. Some of the elements become superconducting when cooled, others do not. Some are great at carrying electricity, others are semi conducting or have insulator properties.

The chemistry of the elements seems to be governed by the number of electrons each atom has around it, this is governed by the number of protons, the number of electrons being equal to the number of protons in electrically neutral atoms. While this makes sense in a way, it is still quiet staggering when you consider the differences between the different elements.

Some elements have radioactive isotopes which means that can undergo some form of particle emission from the nucleus and turn into another element. Some of the large atoms, Uranium for example can decay into two elements!

What I also find completely mind blowing is that apart from Hydrogen and Helium, which we currently believe have been here since the beginning of the universe, and excluding those that have been made in the lab, the rest were actually made inside stars.

Every atom that makes up our bodies is listed in the periodic table. We are mainly made from Hydrogen that has been here since the beginning of time. There may be atoms in your body that are 13 Billion years old and were there to witness the birth of the universe (that is if the Big Bang theory is correct!).

How cool is that, to think that we contains atoms of Hydrogen from the very beginning of the universe. Know wonder there are some days I feel old, it's because my constituent parts are really, really, really old.

Beta emissions and quarks, really?

Up becomes down and out comes a positron

All atomic nuclei are made up of different numbers of protons and neutrons, the different combinations of protons and neutrons give us the elements and their isotopes.

Radioactivity is the emission of particles from the nuclei of atoms.


Of all the different types of radiation, I think my favourite is probably beta emission and it is this I am going to talk about in this post.


Beta emission is where an electron, or a positron, is ejected from the nucleus of an atom. Now for those of you paying attention, I am sure you have just raised an eye brow and thought, "hold on a minute, you said that the nucleus is made up of protons and neutrons, no mention of electrons there!" and this is absolutely true. There are NO electrons in the nucleus of an atom. None whatsoever and yet, they can still be ejected from the nucleus!

I think this is absolutely fantastic.

Many believe that we can view a neutron as a proton and an electron bound together. By that logic, positron emission, the emission of a positively charge electron, implies that a proton is a positron and a neutron bound together. This is not right.

The current thinking is that protons and neutrons are actually made up of quarks. This is part of the Standard Model, which appears to have had a rather spectacular result recently with the discovery of the Higg's boson.

Quarks can change and as a result of this change electrons or positrons can be produced that come whizzing out of the atomic nucleus.

Take Fluorine 18, this has 9 protons and 9 neutrons. The stable form of fluorine is fluorine 19, 9 protons and 10 neutrons. The extra 1 neutron in fluorine 19 makes it a happy bunny. But fluorine 18, completely unhappy and it shows this by kicking out a positron, like so

¹⁸F -> ¹⁸O + e⁺

one of the protons has changed to a neutron, so only 8 protons remain, and the 9 neutrons become 10, making it an Oxygen atom. Oxygen 18 is also a happy bunny and does not decay any further.

So, this is the thing that I am going to be pondering for the next week or two, Fluorine 19 - happy, Fluorine 18 - unhappy, Oxygen 18 happy. One neutron difference between F19 and F18 is enough to make the second, F18 unstable. It seems that the neutron has a calming effect. Will ponder this in a later post.

Now, to finish, fluorine had 9 protons and 9 electrons to balance it out. Oxygen, the final product, only has 8 protons so only needs 8 electrons to balance it out and make it electrically neutral. So what happened to that extra electron?

Well, maybe, the positron, which is the anti particle of the electron, interacts with the extra electron to form something that we could call positronium. The two particles could then annihilate each other and decay into gamma rays.

Now that would be really cool.

Size of an atom

Helium nucleus surrounded its electron cloud

The size of the atomic nucleus is an absolute wonder to me. Atoms are incredibly small and yet they are massive compared with the size of the nucleus. When I was a kid I once read that if an atom of Helium was the size of the Albert Hall in London, then the nucleus would be about the size of an orange.

The size does vary according to the atom in question, Uranium atom nuclei are about 8 times larger than a hydrogen nuclei, which only contains a single proton. The largest come in at about 15 fm (Femto metres).

15 fm is tiny, it is about 10,000 times smaller than an atom, and this is why people often say that atoms are mostly made up of nothing. I think this is wrong. Atoms cannot be any smaller than they are, you cannot make them any smaller. You increase the pressure and eventually atoms will collapse into neutrons, this is what we believe happens in neutron stars. They do not collapse into smaller atoms.

So I think that they are actually as full as they possibly can be. They are full to capacity, which got me pondering to what they are actually full off.

Take the hydrogen atom, a single proton and a single electron. The proton is tiny and the electron is also very tiny. They carry equal but opposite charges. The proton is about 1830 times the mass of the electron. The interact to create the hydrogen atom. They both spin. This leads us to imaging the proton like a large planet with a small satellite, the electron in orbit around it, which is wrong^*, but looks nice.

The size of the hydrogen atom is about 0.03 nanometers, which is huge compared with either the electron or the proton, yet this is the smallest it can possibly be.

The best explanation we currently have for the behaviour of an atom comes from quantum mechanics (QM). I have been told that the Heisenberg Uncertainty Principle (HUP) explains why the electron, which is negatively charged, does not drop straight into the positively charged proton to form a neutron, after all opposites should attract.  I've never really understood or been convinced by the argument, seems a little odd to me. That said

Atoms do change size as we move through the period table, Hydrogen is the smallest and the largest has a radius almost 10 times the size of Hydrogen, which is comparable to the change in size of the atomic nucleus for the two extremes. Is this coincidental or is there something in this?

The reason we think the atoms change size is due to the electrons having to pack in around the atomic nucleus. Hydrogen has a single electron so it is small, Uranium has a whopping 92 electrons surrounding each atomic nucleus, so it is larger!

All atoms are small compared to the wavelength of visible light, which is about 1000 times bigger, which really baffles me, because visible light should not be able to see something as small as an atom, and yet interactions occur between visible light and atoms. Just take a look in the mirror! This is just visible light striking atoms and being reflected back.

The interaction of light and atoms is currently explained by Quantum electrodynamics or QED for short. A brilliant theory that has been tested over and over and has come through time and again. Something I will cover in another post.

So what is the point of this post?

Atomic nuclei are typically 10,000 time smaller than atoms, which are in turn 1,000 times smaller that visible light. So from visible light to atomic nuclei that is a change is scale of 10 million. Let's see if we can get that in perspective. Imagine a ball with a diameter of about 1 metre then the wavelength of light would be comparable to the width of the planet earth. Which is once again, too big for us to image.

I suppose the point is, than even on tiny scales, those of visible light, atoms and nuclei, the distance from the largest to the smallest is absolutely massive and is so big to be beyond our imagination.

What I still can't figure out is why atoms are the size they are and why nuclei are the size they are, why approximately 10,000 times, why not 50, or 5,000,000? will have to ponder, when I come up with something I'll let you know.

Pendulums

A wonder of the modern world

I was on the train from Slough to London pondering spin when I spotted a picture of a pendulum in a magazine.

I began to think that there are some similarities between spin and the behaviour of a pendulum and figured that it was definitely worth 20 minutes of my thinking time. Unfortunately the train to London was faster than my thought process and I arrived before I'd thought of anything interesting.

A couple of weeks later I am in a museum when I see an old antique grand father clock with a pendulum, so I gave it a little more thought, fortunately I had time to stop and think and this lead to this post.

The pendulum, discovered by Galileo apparently, is one of the most amazing inventions/discoveries ever. It completely changed the way we measured time.

You bung a weight on the end of a piece of string and you let it swing backwards and forwards. Provided that the swing angle is not too big then this thing will go on swinging in regular intervals for ages. Backwards and forwards, happy as Larry. This action is said to be isochronous, do like that word. It is this regular nature of the pendulum that allows its use in measuring time.

The equation of a pendulum is also rather straight forward, and is derived from Newton's second law of motion, F=ma, I got this from Wikipedia

L - length of the string
T - time period
g - local acceleration due to gravity
θ₀ - angle of the swing, small

What is amazing is that the actual period of the swing, the time it takes to swing from one side to the other and then back again, is directly proportional to the square root of the length of the string. Sorry, went into techno babble for a minute, what I meant to say was, the longer the string, the longer it takes to do one swing. The shorter the string, the shorter the time it takes to do a swing.

The time period of the pendulum is NOT dependant on the mass of the weight on the end of the string, 1 kg, 2 kg, provided it does not stretch the string, it does not matter what mass we use, the time period will stay the same. The only two factors we have to think about are the length of the string and gravity.

This is great, seriously, I genuinely can't help thinking there is something massively profound going on here, only we are not quite sure what that is yet.

Equation (1) above is very similar to another equation

C = 2π r  

which is just the equation for the circumference of a circle.

C - circumference of a circle,
r - radius of the circle

so if we take

r = √ L/g                                (2)

C = 2π √ L/g   

T =  2π √ L/g  

The circumference of the circle is the same as the time period, T. Which is cool in a way, as L gets smaller the circle will shrink and T will get smaller. L gets larger then the circle gets bigger and T increases.

This got me thinking about circles, area of circles, area of spheres and volume of a sphere, the equations for these are
area of circle, A =  π r² 

area of sphere, A_sphere =  4π r² 

volume of sphere, V =  4/3 π r³ 

using r in equation (2) above, we have


area of circle, A =  π L/g               (4)

area of sphere, A_sphere  =  4π L/g      (5)

volume of sphere, V =  4/3 π  √ L/g ³   (6)


If T from equation (1) is the time period, what are A, A_sphere  and V in equations (4), (5) and (6) physically? Do they actually represent anything at all? I am not sure at the moment. Will get the pencil and paper out and have a play. If I find anything interesting I'll put it in the next post.

Things you think about in Slough

Gravitons on a small scale

Finding myself in Slough (England) with very little to do I got thinking about spin again. I say again, because I did a little post on spin a while back. 

I can't help thinking that the great mysteries of the universe will be unvailed once we have a proper handle on spin. Anyway, here I am in Slough, won't bother explaining why I am here, other than to say tomorrow I get a train back to London. Oh happy days.

So spin.

It occured to me that the effect we call gravity is not actually dependent on mass, but rather on spin.

Mass certainly exists (at least I think it does for now) and this is one of the properties of matter that make it distinct from electromagnetic radiation. But here is the thing. Protons, electrons, photons etc all spin and so all have gravity.

Even though a photon has no mass it does have "gravity". Of course this means that the term gravity has a different meaning here from traditional gravity which is linked to mass.

So, here are some things I will be thinking about on my train back to London tomorrow.

If photons do have gravity is it related to the photon frequency? If so then won't xrays be deflected more that light rays when they pass near the sun?

Do neutrinos actually have no mass,  travel at the speed of light, but have enough spin to have gravity?

If we accelerate protons to almost the speed of light is the increase in mass actually an increase in spin?

Are neutrinos actually gravitons? don't know where that one came from! will give it some pondering though.

Looking forward to the journey.

E = m c squared?!

Small equation - big meaning

When I first started this blog I was planning on a post about Einsteins most famous equation. Then I got a bit side tracked. But I am back on track now, so lets see how we go.

E = m c² 

Probably the most famous equation in the world, certainly of the 20th century. This tiny equation that says such a lot. The equation showing mass-energy equivalence. While many people can quote, I suspect that a smaller number can tell you what each of the components of the equation are. So let's start there.

The equation has 3 components, E, m and c. We will tackle them in reverse order.

c - is the speed of light in a vacuum. By this we mean approximately 186,282 miles per second, or exactly 299,792,458 meters per second. We say "in a vacuum" to indicate that we are talking about the maximum speed of light. Light travels more slowly in materials such as water.

The closest we get to a vacuum is the space between galaxies, so strictly speaking we are saying the speed of light as it crosses these vast regions of inter galactic space. The speed of light in a vacuum is constant. This is the value of c that we are referring to in Einstein's E=mc² .

m - mass. Philosophically it can get a bit tricky here. Most of us get out mass from standing on scales. We weigh our selves. What we are doing here is getting our weight. Weight is our mass multiplied by the acceleration due to gravity. So what we are describing is a quantity of inertia! That said we know things have mass. Take a proton or the electron. From various experiments we have worked out that that have a well defined mass. So we'll go with that for now, I'll cover mass in detail in another post.

E - energy. Once again this is a bit of an oddity. We believe that energy can neither be created (produced) nor destroyed by itself. It can only be transformed from one state to another. Energy is an amount of something. If we have the same amount of something, then we have the same amount of energy.

So now we know what the 3 terms are that make up our wonder equation. The next thing to consider is how Einstein managed to show they were related. When you look at it, it is an amazingly simply equation, you would have thought that someone would have stumbled upon it way back when. The answer is that deriving the equation and understanding what it means takes a giant leap of the imagination. This is how Einstein made the leap...

Note: If your maths is a bit weak, take this slow and you'll get it, although it may take a couple of reads. Hang in there, its worth it.

Light, any electromagnetic radiation for that matter, has momentum. This can be measure and is found to be

P_photon = h / λ    --- (1)


P_photon - momentum
h - Planck's constant
λ - wave length of light

the shorter the wavelength the higher momentum. So gamma rays have the highest momentum. A photon also has energy and this is given by

E = P_photon c    ---(2) or P_photon = E / c

E - energy
P_photon - momentum of the photon
c - speed of light

Now image a train carriage that has a length of L (see diagram near the bottom). The carriage has a mass of M. The carriage is symmetrical in shape and mass. At the right hand end is a radioactive source. This source gives out a single photon. Hey, this is a thought experiment, so it can happen. The photon travels from the right end of the carriage to the opposite left end. It takes a finite time for the photon to reach the other end of the carriage. Now as soon as the photon takes off, the carriage recoils and takes off in the opposite direction.

The photon goes to the left, the carriage goes to the right.

When the photon reaches the other end of the carriage it is absorbed completely by the carriage wall and  the carriage stops. During the time of flight of the photon the the carriage has shifted, Δx.

Next bit is important - because the carriage has not been acted on by any external forces the center of mass of the carriage cannot have changed. But the carriage has actually moved to the side a distance, Δx. The only explanation is that the mass of the carriage has been redistributed slightly. The only thing that moved was a photon from one side to the other. The implication then is that the photon ray must have mass, m.

When the photon takes off with its momentum P_photon, the law of conservation of momentum tells us that the carriage goes in the opposite direction with momentum P_car ,and the two are the same, so

P_car = P_photon

now from standard mechanics we have

P_car = M v_car

P_car - momentum of the carriage
M - mass of the carriage
v_car - velocity of the carriage

P_car = M v_car  = P_photon = E / c   --- from equation 2 above, so

M v_car  = E / c    --- (3)  or  v_car  = E / (c M)


The next thing is to work out how long it takes the photon to travel from one end of the carriage. It travels at the speed of light, c, and has to travel the length of the carriage minus Δx the distance the carriage has traveled to meet the photon. Δx is small compared with L so we ignore it. So we have

t = L / c     --- (4)  , this is just the time it takes light to travel a distance L.

The velocity of the carriage is given by the distance travelled divided by the time it takes, so during the time of flight of the photon this is

v_car = Δx / t    ---(5) (for example, you go 60 miles in 2 hrs, v = 60/2 = 30 mph)

the t in (4) and (5) is the same so we can do some re-arranging to give

v_car = Δx c / L    ---(6), but take a look at (3) above, we have an equation for v_car so again re-arranging

E / (c M) = Δx c / L ---(7)

Now the center of the box is initially at a x=0, this is also the center of mass is x_m. After the photon has done its thing x_m is still the same, because the carriage has not been acted on by any external forces the center of mass of the carriage cannot have changed. The carriage has moved though by a distance Δx. This has happened because we have redistributed the mass of the carriage, a photon moved from one side to the other, taking a mass, m, with it.

This bit about center of mass is important, you have to get this part to crack it. You have to understand how we get from equation 8 to 9 below.

Look at the diagram below carefully. The top rectangle is the carriage before the photon is fired, the bottom of carriage after the photon is fired. The carriage has a mass of M and we imagine that it is distributed evenly at either end of the carriage. 

The carriage before and after the photon moves from right to left

It turns out that the center of mass is given by

Mass_left x Distance_left = Mass_right x Distance_right  . --- (8)

In the top carriage above (8) is just

M/2 x L/2 = M/2 x L/2   , which is the same on both sides and so is obviously true.

Now we have said that the center of mass is the same after the event, but after the photon has moved from right to left and the carriage has moved from left to right and we have

(M/2 + m) ( L/2 - Δx ) = (M/2 - m) ( L/2 + Δx )

expanding this gives a bit of a mess, take your time with this

(M/2)(L/2) + mL/2 - (M/2)Δx - mΔx = (M/2)(L/2) - mL/2 + (M/2)Δx - mΔx

The first term on each side is the same, so is the last, so we can simply remove them. Leaving

mL/2 -(M/2)Δx = - mL/2 + (M/2)Δx

doing a bit of swapping we get

m L = M Δx    --- (9) which becomes  m / M = Δx / L  --- (9a)

taking (7) we can replace Δx / L on the right with m / M, so we now have

E / (c M) = c m / M    and we are now on the home straight

E = c m ( c M ) / M = c m c  , we have just cancelled the Ms

and so with one last bit of re-arranging we get...

E = m c² 

Albert Einstein, that really is beautiful. Thank You.

Atoms

Atoms don't really look like this

Atoms. Just about everyone who has any form of education knows that the universe is made of atoms. Many also know that there are three sub-atomic particles that actually make up an atom. These are the proton, the neutron and the electron. A few will also know that the current best bet is that protons and neutrons are in turn made out of something called quarks.

Quarks I am not to sure about, despite the fact that they have been "seen". Some think that the quarks may in turn be made out of strings, this is were I draw the line. String theory for me is currently nonsense. The mathematicians have taken over the asylum.

Back to the atom. Before 1905 we didn't know for sure that atoms actually existed. So why didn't we know? We knew about gravity, electricity, light, radioactivity and loads of other things such as chemistry! Why was it that some people thought that the idea of atoms was little more than a mathematical abstraction, rather than something real? Well, for one thing they are remarkable difficult to see. They are really small. So small in fact that I don't believe it is possible for the human brain to comprehend just how small they are. You can talk about how many atoms make up a millimeter, but this number, about 6-10 million, is so large that you can't appreciate it.

So how did we finally crack this one. Well bring on Einstein, he had read about a paper written 80 years earlier by a bloke named Brown, a biologist (I think).

Now Brown had watched pollen particles in water and had noticed how they had bounced around in a random way. He hadn't been able to explain it and his idea and it could not be explained using classical thermodynamics.

Einstein was able to take this observation to determine the size of atoms. This was a brilliant paper (and that of the photoelectric effect) and won him a Nobel Prize. He did not win it for Special Relativity or General Relativity as many appear to think.

The paper also proved that classical thermodynamics was not valid on atomic scales. In fact Einstein opens the paper stating this.

What I think is great is that he actually comes up with an experiment (which he does not do himself, he leaves that to others) that will be able to calculate certain values that can then be used to determine the size of atoms.

The paper also derived an equation which showed that it would be possible to calculate the site of molecules and atoms.  The equation he derived was this

N = Avogadro's number = 6.0221415 x 10²³
R = Gas constant = 8.3144  (Aside: The Boltzmann Constant is just R divided by N)
T = temperature in Kelvin, so room temperature ~ 293 K
k = viscosity of the liquid  ~ 0.001 for water
λ_x = average distance moved in a given time during Brownian motion.

P = the size of the particle or molecule.


In 1908 Perrin began to study Brownian motion using the newly developed ultra-microscope. He carefully observed the Brownian motion of particles and provided experimental confirmation of  λ_x and P in  Einstein's equation. His experiments enabled him to estimate the size of water molecules and atoms as well as their quantity.

1908 was the first year that the size of atoms and molecules were reliably calculated from actual visual experiments. Perrin's work moved atoms from being hypothetical objects to observable entities. He was awarded the Nobel Prize in 1926 for his work.

It seems strange to me that just over 100 years ago atoms were still considered by many to be hypothetical and not really based on real objects. These days children are taught about atoms in primary school. 100 years from now will children accept "facts" from physics that we still consider just theoretical today?

Of course they will, I just wish that I was there to see what those facts will be.