Unrealistic radioactive decay

today I am going to write about one of the problems that I had to solve for a physics seminar. The submissions for this series are already closed and you can look up the solutions so I know that I answered correctly to this very interesting problem.

Imagine you have two types of particles, A and B. They are radioactive, meaning they keep falling apart but in a very peculiar way, A decays into B and B decays into A. This doesn’t happen in a real world because the particles decay into something smaller, they break up, but this is a hypothetical scenario very simplified, we do not care about what is happening on the inside. The question is, what is the ratio of the particles at any point in time?

Since the answer itself and the calculation are not trivial I will mainly try to make some facts about this problem obvious and then show the answer.

There is one part that I didn’t mention in the setup. Radioactive particles do decay but there is a very important value that characterizes how fast. It is either (half-life) which tells you in what time will half of particles decay (if there is enough of them it will give the right results) or in other words when the time passes one particle has 50% chance to decay. It seems that also decay constant, which I like better, is used which is basically half-life except the larger the value is, the faster the particles will decay.

The problematic part of this exercise is that when part of the A particles decay they will increase the pile of B particles which means that more particles will decay into A and so on, this is a cycle. To get to an important point it is good to try some simple case of such decay.

Let’s say we have 200 of A particles and 100 of B and half of both will decay in 1 hour. In 1 hour:


Next hour:


It is obvious that from now on the amount won’t change. This little experiment revealed something obvious, there, first of all, no particles get lost, there is always the same amount present: A+B=constant and with a bit more experimenting it would become more apparent that there is an equilibrium between A and B meaning there is always some amount of A that when it decays it will equalize the amount that decayed from B, this equilibrium will shift depending on the length of half-life or the decay constant. From these thought experiments that reveal the behavior of this problem, we need to use some math that I will not get into here to get the result that you can try to play around with in Desmos.




Various fields for dummies!

fields? Well yes, this is probably the last post before I will write about Higgs boson. Today you will learn that everything is made up of fields and that particle is just a special vibration.

Take for example magnet, it is something totally special if you think about it. Not that it sticks to stuff but that you can feel how it wants to stick without actually touching anything, just through EMPTY SPACE!

Magnet has his magnetic field all around, that is why we can feel it from some distance and the strength decreases with distance increasing.

On the picture above you can see lines of force around magnet.
This is electromagnetism and now we know that there is electromagnetic field around all the stuff that have charge. We know that we can magnetize needle using electric current.

This field is not only around magnets but also around all things that have mass, this is called gravitational field and even that it does not sound very exciting to us, in Newtonian age it was very cool when Newton found out that there is similarity between apple falling on his head and Moon orbiting Earth!

What happens when you start to rapidly move with source of electromagnetic field? Well you create waves. If you do it well enough such waves will create light! This light then comes to you in quantum! If you do it more rapidly you will get gamma rays and if you do it less rapidly you will get radio waves. This is just a vibration of field.

Or when you take radioactive decay. There is neutron which somehow changes itself into proton, electron and antineutrino. This is just a change of vibration in various fields.

Now there are 4 fields if I do not count Higgs field. What physicists want to achieve is the prove that all those fields are just one field and everything is just a various vibration of it, creating the effect of gravity, electromagnetism and so on. This would be the Theory of Everything.


Conservation laws of physics

today I want to talk about conservation laws of physics, at least some which I know about since there are just lot of them. Probably during your middle school/high school you have already encountered them. In chemistry when you have some reaction, for example:

Li + H2SO4—> Li(SO4)­ + H2

This could seem good but WordPress does not let me write upper index and (SO4) is -II which is by rule transfered to lithium —> Li2(SO4)

If you ever saw this in school you may remember that now you need to add something since if we count there are two lithium atoms coming from the reaction while only one was there before, we have violated the law of conservation of matter, mass and probably many other laws.
It should go like this:

2Li + H2SO4—> Li2(SO4)­ + H2

Now there is equal amount of stuff on both sides of equation, only thing that happened was that those atoms changed their place.

Neutron decayimg_lrg/virtual_w.jpg not found

To see some laws of conservation we can write up the neutron decay during radioactive decay:

Neutron(0) —> Proton(+) + Electron (-) + anti-neutrino(0)

There is conservation of electric charge, we can test it: neutron has charge exactly 0. Proton has positive charge and electron has exactly the same charge but it is negative, when you put this together it is exactly zero. There is also anti-neutrino but it has charge zero too, so as you can see it does not violate the conservation of electric charge (I am not saying it is right, I am just saying that one law was not violated).

So as you can see from this rule, it may seem that this anti-neutrino thing was not necessary at all, but there are other laws of conservation which would be violated without this tiny particle.

Also here the conservation of quarks is not violated, there are three coming in and three coming out.

Leptons are particles that are not affected by strong nuclear force, those are electrons, mions, tauons and all types of neutrinos.

There is conservation of leptons, there is none coming in the reaction and two are coming out.. hmm
Electron is one lepton while anti-neutrino is anti-lepton which means that they will cancel out and there is zero of leptons in whole reaction, kind of.

So those are some basic laws of conservation but I read that there could be some special circumstances under which the amount of leptons or quarks could change.. look at the amount of matter in the Universe, it is much greater than the amount of anti-matter maybe this is such a case.

Still the most important law is the conservation of energy. Energy or mass since as Einstein’s equation says: E=mc2  —> mass is just different form of energy.

Neutron has little higher mass than proton, this little mass left is the mass which is than transferred to create electron and anti-neutrino.

At last I was very curious how does it work with photon.
Everyone knows that photon has 0 mass and that is why he is able to travel at the speed of light.


What? So photon does not have energy or what?  (c2=E/m … c2=E/0 (Universe just exploded))

The problem here is that E=mc2 works only for objects that are not moving, that are on one place which photon is not.

The full equation goes like this:
E^{2}=p^{2} c^{2} + m^{2} c^{4}.P is the magnitude of the momentum of vector p.

\boldsymbol{p}=\hbar\boldsymbol{k},Where ħ is the reduced Planck constant and k is the wave vector which is:

Where λ is the wavelength of photon!! Finally we got to something which is understandable for me.


PS: I tried to get to some normal value using wavelength of orange light but I was not able to get something normal so I will update with new post when I will know what I did wrong.

Pictures from Planck constant page and neutron decay page