## The energy of nuclear reactions

Atomic masses are listed here. From this table we can see that most of the nuclear energy produced in the universe comes from the fusion of hydrogen into helium in the main sequences of stars.

The fact that deuterium is much less abundant than helium, means that it undergoes further reactions soon after it is produced. Thus the fusion of hydrogen into deuterium appears the slowest process in the chain, which thus determines its overall speed. But this is also the least energetical reaction: it gives away "only" about 0.077% of its mass into energy, and part of this energy (a half ?) is lost in the form of neutrinos. Thus the energy it gives to the star is equal to the energy E=mv²/2 the same quantity of matter undergoing this reaction would have if sent at the speed v = c 0.00077 = 0.0277 c = 8318 km/s.

Then, the fusion of deuterium into helium is the most energetical reaction. It converts 0,64% of the mass into energy, that is the energy it would have at the speed of c 2* 0.0064 = 34000 km/s.
The 3 next elements (Li, Be, B) have a higher mass per nucleus than He, and thus are not produced in significant quantities (they easily disappear by further reactions into other elements). Next is carbon : the fusion of helium into carbon, in the giant phase of stars, releases 0.065% of the mass into energy, corresponding to the energy of a speed of 11000 km/s.

The next important reaction is from carbon to oxygen. The rest of elements only have a small place in the composition of the universe.

See the binding energy curve for a global picture of the difference of mass per nucleus between the different elements.

The fact that chemical energies are much smaller per mass, is the reason why we don't speak about the difference of mass in a reaction : this difference would be too small to be measurable in practice.

### The radius of nuclei

The nuclei have a roughly constant density of protons and neutrons : their radius R as function of the atomic mass number A satisfies R=r0 A1/3 where  r0 = 1.25 fm (femtometre) = 1.25 × 10−15 m, so that each individual nuclus roughly takes the volume of a sphere with radius r0.

In fact the law that determines the volume taken is the Pauli exclusion principle, saying that no two particles can have the same quantum state. But in fact 4 nuclei can take the same space-time location : a pair of protons and a pair of neutrons (as they are spin 1/2 particles, each one has 2 possible spin states). The total number of states of movement taken by the nucleus is thus A/4.
Each pack of 4 nuclei takes the volume of a sphere with radius r0*41/3= 2 fm.

To be compared with the range L of the interaction between nuclei (such that the intensity of interaction depending on distance x undergoes a decrease by the factor exp(-x/L)), deduced from the mass 135 MeV/c2 of pions as

L= ℏc/135 MeV= 1.46 fm.

As quantum physics gives each state of movement a volume of h3 in the 6-dimensional phase space (3 coordinates of position, 3 coordinates of momentum), the volume it takes in this phase space is h3A/4=VV' where V is the space volume V=4πR3/3, and V' is the momenta volume V'=4πp3/3 where p is the maximum momentum taken by a nucleon, p=Mv where v is the maximum speed and M=1.67×1027 kg is the mass of a nucleon.

This equation thus says (R/r0)3h3/4=(4π/3)2(Rp)3

It simplifies into v=((3/π)-2/3/4)h/Mr0=85,000 km/s.

It is no coincidence that this result has the same order of magnitude as those calculated in the previous section.
Now if quantum mechanics determines the nuclei to be so small, it also determines, from the same Pauli exclusion principle, the atoms themselves to be quite bigger. There are 2 reasons for this : one is that electrons have smaller mass, the other is that they are kept in place by the electric force instead of the nuclear force. Let us now measure the electric force.

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