A Semi-classical Interpretation of the Strong Nuclear Force

The effects of the Strong and Weak Forces can only be understood by use of quantum mechanics. Even so, it might still be helpful to try to obtain a more intuitive feel for how they work, even if the result is not a completely correct story.

The Strong Force is often, instead, called the Strong Interaction. At atomic scales, it is about 137 times stronger than the Electromagnetic Force, about 10¹³ times stronger than the Weak Force, and about 10³⁸ times stronger than Gravitation. They are all governed by a similar law, based on Proca's massive electromagnetism, which is a version of Maxwell's theory that includes the term for the Yukawa potential.

F = -K₁₂.q₁.q₂.(-1)^s₀.exp(-m₀.r/K₀)/(r.r)

where q₁ and q₂ are the charges (electrical, magnetic, weak isospin, weak hypercharge, strong colour, or mass, depending on which force is being considered) on the two bodies that are being attracted or repelled, with m₀ as the mass of the carrier of the given force and s₀ as its spin. In the case of electromagnetism, the force carrier is massless, so leading to the inverse-square law. Similarly for gravity, though, controversially, there is a suggestion that the graviton might have a small mass, perhaps around 10^-54 kg (NS, 11-Nov-2006, p36). Gravity is something of an exception for other reasons, too, including that it leads to like-charges attracting, where gravitational charge, namely mass, can only be positive; and the force carrier, if it is ever confirmed by experiment, is expected to have spin 2, instead of spin 1 (making gravity a tensor field, as opposed to vector fields for the others, and a scalar field for the Higgs boson, with spin 0).

At atomic scales, but distances greater than the diametre of one hadron, Strong Interaction approximates to being roughly constant, at about 100000 Newtons. It is, therefore, impossible for a single quark to exist by itself a property connected to QCD confinement and asymptotic freedom (NS, 04-Dec-1993, p25). Any attempt at removing a single quark an infinite distance from its partners would involve applying an infinite amount of energy, and even removing it a short distance would involve such large amounts of energy that new particles would be created instead (NS, 06-Jun-2015, p36). So, from outside the nucleus, we do not feel the effect of the Strong Interaction itself, but instead we just see the lumped effects from the whole nucleus (just as we see gravitation as if the mass were all concentrated at the centre of gravity of the object, and similarly for electrostatic or magnetic forces).

This lumped effect is known as the Residual Strong Force, and is to the Strong Interaction as van der Waals forces are to the Electromagnetic (coulomb) Force.

Fission

The fission event does not always produce the same products, but suppose that it typically produces one barium nucleus and one krypton nucleus (Z=92 breaks into Z=56 plus Z=36). That would mean that A=235 splits into A=137 plus A=84, plus or minus a few isotopes). So, according to this, there would be 235-221=14 surplus neutrons. Not all of these are useful for the chain reaction, being 'non-thermal'.

So, a certain proportion, α, of those neutrons will go on to hit other uranium nuclei, while the other proportion (1-α) will escape without doing anything further.

This is the essence of the chain reaction. The first neutron will lead to k neutrons (where k=αN) some small time interval later, and hence to k² electrons one interval later, k³ after that, and so on.

The question is, though, whether k<1 or k>1. It is only if you have enough uranium atoms around that you can achieve k>1. So, in a bomb, this is all you would need. In a nuclear power station, though, you must balance k at exactly 1. You do this by starting off with k slightly larger than one (so more than the critical mass is required again), and then moving the neutron-absorbing control rods around (or ultimately moving the fuel rods around) so that k is always hovering around the k=1 figure.

Inside the atomic nucleus, as a result of their strong attraction, quarks do not go into orbit round each other. That only works for the inverse-square law of gravitation and electromagnetism:

F_g = -G . M² / r²
F_e = q² / ( 4πε.r² )

With most of the forces, particles are more attractive (or less repulsive) when they are of opposite polarity. With the strong force, the polarity is three-way, and is referred to by metaphor with colour (with red+green+blue cancelling out in the same way as plus+minus, north+south or up+down).

Stable atomic nuclei

If three such quarks were to approach to less than 4 nucleon's width, the Residual Strong Force would be so strong that it would attract them even closer together, until some point (at very close range) at which the force becomes repulsive. Thus, the Residual Strong Force is repulsive for distances, r, much less than 1.7fm, but is strongly attractive at r=1.7fm, reducing roughly exponentially, as given by the Yukawa potential, after that:

F_r = - g² . exp( - m.r / K₀ ) / r

Since the electrostatic forces fall off with an inverse squared law, there must be a point where the two curves intersect. This seems to be at r=2.5fm, beyond which the Residual Strong Force ends up being weaker than the Electromagnetic Force. It turns out that this is somewhere between r=3.67 to 4 nucleon diameters. Putting this into the usual equation for the volume of a sphere, V=(4/3)π.r³, this means that the crossover is between V=207 and 268 nucleons. Since each nucleon weighs one atomic unit of mass, this means that it happens between A=207 (which is Pb) to 268 (which is Db). So this explains why it is so difficult to find chemical elements above Z=82, and to make them above Z=105: the electrostatic repulsion exceeds the binding forces in the nucleus, and the oversized nucleus ends up disintegrating (or, rather, the probability of it disintegrating increases, and so its half-life reduces).

In the equations above:

g = coupling constant between a fermion (a proton or neutron in this case) and a meson (a pion in this case)
m = mass of the meson (pion)
M = mass of the fermion
q = electrostatic charge on a proton
4πε = 10⁷/c² F/m, and
K₀ = a scaling constant, because I prefer to work in SI units, rather than in Planck units. (Since m.r needs a scaling of √(G.h/2πc³).√(c.h/2πG), K₀ is about 3.5x10^-43).
To get F_r = F_e at r=2.5x10^-15, it seems that g=7.1446x10^-7

According to this approximation (which does not model the repulsive effect of the force) F_r starts smaller than F_e, becomes equal at r=0.76fm, peaking at r=1.47fm, then reduces again until the two forces are equal at f=2.5fm, and then F_r is weaker than F_e for all r greater than that.

Making sense of the zoo of fundamental particles

By the 1960s, physicists had discovered an enormous zoo of sub-atomic particles. What has now emerged to become the Standard Model of quantum particles managed to explain it all, in terms of just a few fundamental particles. Even so, that few still amounts to 2x3x2 leptons, 2x3x2 quarks, 4 guage bosons, and a handful of other things. The following table is my attempt at understanding the taxonomy of these particles, starting from the map of the particles platted with electrical charge against weak field influence, for example as presented on page 246 of Goldberg (2013), The Universe in the Rearview Mirror: How hidden symmetries shape reality, Plume, Penguin Group, New York, USA, ISBN 978-0-14-218104-1.

int spin	snf	wnf	anti	properties	particle
0	0	0	0	-1, WIS, R	e^-/μ^-/τ^-
0	0	0	1	+1, WHC, L	e⁺/μ⁺/τ⁺
0	0	1	0	-0, WIS, L	eneutrino/μneutrino/τneutrino
0	0	1	1	+0, WHC, R	aeneutrino/aμneutrino/aτneutrino
0	1	0	0	-2/3, (WIS, L) or (WHC, R)	aup/acharm/atop
0	1	0	1	+2/3, (WHC, R) or (WIS, L)	up/charm/top
0	1	1	0	-1/3, (WIS, R) or (WHC, L)	down/strange/bottom
0	1	1	1	+1/3, (WHC, L) or (WIS, R)	adown/astrange/abottom
1	0	0	0	+0, WIS	photon
1	0	0	1	-0, WHC	Z⁰
1	0	1	0	+1, WIS	W⁺
1	0	1	1	-1, WHC	W^-
1	1				gluon

Each of the first eight rows show that they need a base-3 trit, to distinguish particle flavour. Each of the rows with snf=0 (not influenced by the strong nuclear force) can then be doubled, for spin-up and spin-down. Each of the rows with snf=1 (influenced by the strong nuclear force) can not only be doubled, for spin-up and spin-down, but then further tripled, for red/green/blue. Not mentioned explicitly in the cited book, each row with WIS should also be doubled further, for weak isospin up and weak isospin down, and so should the other rows with WHC, for weak hypercharge +1 and weak hypercharge -1.

Surprisingly, the above table suggests that anti-up, anti-charm and anti-top are the particles, and that up, charm and top are their anti-particles. Worse, it also suggests that the Z⁰ boson is the anti-particle of the photon, and that it has wnf=0 (not influenced by the weak nuclear force); there is clearly something wrong with my chosen representation, and will need correcting.

Using Karnaugh maps to simplify this, it seems that (not surprisingly) the choice between WHC and WIS is dependent on the anti=1. That is, unless there is an "or" alternative, in which case the inversee is true. So, WHC/notWIS is dependent on "anti XOR second-possibility". Meanwhile, positive electrical charge is dependent on "anti XOR integer spin". Lastly, left parity is dependent on "anti XOR (wnf XOR snf) XOR second-possibility".

Magnetic monopoles

It seems that magnetic monopoles do not exist in our universe, though there is nothing to prevent them from occurring in other configurations of universe. If we considered adding an extra parameter, em, for magnetic/not-electric, a zero value would lead to electrons, positrons, and all the other electrically-charged particles, while a one value would lead to northons, southons, and a similar set of magnetically-charged particles. Putting this in to the equation at the top of this page, the force between an electron and a northon would be that between unlike poles, in the em bit, but with a force-carrier has a spin of ½, so a force that is perpendicular to the line between the two particles.

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