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The nuclear U bend

OK you might think from the title that it is some sort of nuclear toilet or drain, as the U bend is normally found under sinks or toilets. But it is something different. I am going to try to explain to you about the driving force behind beta decay.

Now for many of you the things behind nuclear processes might be a complex mystery which you feel that you can not understand, I will admit that there are some things in nuclear physics which I can not understand but there are some simple things which go a long long way. One of the most important simple things is the fact that when protons and neutrons bond to each other and make a nucleus there is binding energy.

What you have to understand is that down quarks can turn into up quarks

Now the binding energy does vary from nuclide to nuclide, a optimum ratio of down quarks to up quarks exists for every isobar. It is important to understand that the quarks from the neutrons and protons are mixed up together to make a bigger particle. The nucleus in an atom is a group of quarks so the protons and neutrons no longer exist in reality, but for ease of counting many people treat the nucleus as if these neutrons and protons still exist.

Now some combinations of quarks have lower energies than others, such as two neutrons and two protons makes a very stable nucleus. This can be thought of as being like the special chemical stability of the noble gases.

This translates into the following quarks

2 protons : two down quarks and four up quarks

 2 neutrons : four down quarks and two up quarks

helium-4 nucleus : six down quarks and six up quarks

This special stability of the helium-4 nucleus explains alpha decay but we will leave that for another day.

By the way, isobar might be a new word it means a line along the chart of the nuclides where all the nuclides (atoms) have the same ‘mass’.

For very light elements such as helium and carbon the best down quark to up quark ratio is 1:1 while for heavy elements like lead and uranium when the energy is lowest the number of down quarks is greater than the number of up quarks. As a result when uranium is broken up in nuclear fission the new nuclides are too rich in neutrons (oops I mean down quarks). What will happen next overall is the down quarks will be converted into up quarks (beta decay).

Like many things the system will try to get to the lowest energy, like chemical thermodynamics these energy differences do not tell you how fast the system will get to the new lower energy state.

Because mass can be converted into energy (or the other way around) it is possible by very careful measurements of the mass of atoms to see the basis for the energy differences behind beta decay. I started my quest by looking at the exact masses for all isotopes where the sum of the number of quarks is 309 (sum of the protons and neutrons is 103). If the exact masses of the atoms are measured then the differences in energy can be estimated.

As I expected the graph of energy is a U shape, at the bottom is Rh-103 which is the only stable nuclide with that mass. As the up quark to down quark ratio is changed more away from the ideal for this mass then the energy gets higher and higher. Due to the imperfections in the measurement of atomic masses it is not a perfect U shape.

Energy as a function of proton number for isotopes with mass number of 103. This is based on the exact atomic masses.

While if data based on radioactive decay energies is used then the curve looks better, note for this new graph that the zero point is at a different point. For the above graph the zero point is for the 103 protons/neutrons as 103 particles (plus the electrons). In the new graph the zero point is the energy of the nuclide with the lowest energy.

Energies of nuclides with mass number of 103, this is based on radioactive decay energies.

What happens is that atoms which are on the top left end of the curve tend to roll down the curve by the emission of electrons (common or garden beta decay) while those on the top right end roll down the curve by either electron capture or positron emission (also beta decay).

Now if we consider the energy of every isotope with a mass of 104, if we use the gold standard of using radioactive decay energies we get a U shaped curve but this one is not a nice U. It has a wiggle on it. Below is my graph which I calculated.

The graph showing the energies of different atoms with mass numbers of 104, this is based on radioactive decay energies.

Oh, does not look so good now does it ! But there is some hope which will explain a little more about what is going on. If we consider which atoms have even numbers of both protons and neutrons and which have odd numbers of both then we can see something special.

The graph of energy as a function of proton number for nuclides with mass numbers of 104, even even and odd odd nuclides have different symbols

What you should notice is that the nuclides with even numbers of both protons and neutrons have lower energies than you would expect while those which have odd numbers of both particles have higher energies. This is because of the way that the nucleus fills up groups of slots for quarks.

Ru-104 has 44 protons and 60 neutrons, so as a result it has 148 up quarks and 164 down quarks.

Rh-105 has 45 protons and 59 neutrons, so as a result it has 149 up quarks and 163 down quarks.

It is very likely that the up and down quarks are paired in the more stable atoms.

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2 Responses

  1. […] but are not able to or willing to explain how it happens. Now some time ago I explained the driving force behind beta decay, today we are going to start to deal with alpha […]

  2. […] In the s process the nuclei of atoms are subjected to a slow neutron bombardment process which forms new neutron rich nuclides. These then undergo beta decay to increase their atomic number. What happens is that a down quark is converted into a up quark, an antineutrino and an electron. This quark converting process moves the nucleous closer to the bottom of a U shaped energy graph. […]

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