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A Semi-classical Understanding 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 100 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:

  • F = -K12.q1.q2.(-1)s0.exp(-m0.r/K0)/r2
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 1e-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.

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:

  • Fg = -G . M2 / r2
  • Fe = 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).

If three such quarks were to start orbiting at 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:

  • Fr = - 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 Fr = Fe 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) Fr starts smaller than Fe, 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 Fr is weaker than Fe for all r greater than that.

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