# Strong nuclear force - change with distance

1. Homework Statement

a) The strong nuclear force cannot fall off as an inverse-square law. Why not?

b) Give a reason for thinking that the strong nuclear force must become a repulsion at very short range.

2. Homework Equations
Non identified.

3. The Attempt at a Solution
I have searched the Internet for phrases in the questions and found nothing satisfactory. The text book mentions quarks but nothing of colour, up/down, or mesons so I'm guessing the answers don't require that level of theory.

For a) the expression "fall off" must mean reducing from a maximum ...

For b) it seems simplistic to say the nucleons can't touch.

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dynamicsolo
Homework Helper
For a) the expression "fall off" must mean reducing from a maximum ...
It means that the "strong force" does not behave like gravity or the electrostatic force, which have r^2 [r being distance] in the denominators of their "force laws". Is there something you know about nuclei as systems or about nucleons that tells you the strong force doesn't behave like those other forces?

For b) it seems simplistic to say the nucleons can't touch.
Well, in fact, in a nuclei, the nucleons do "touch". The issue is this: if the "strong force" were only attractive, what would nucleons close to each other try to do? (And if they did, would nuclei exist?)

Andrew Mason
Homework Helper
1. Homework Statement

a) The strong nuclear force cannot fall off as an inverse-square law. Why not?

b) Give a reason for thinking that the strong nuclear force must become a repulsion at very short range.
The first question is easier than you think. If the strong nuclear force fell off as an inverse square law force, would it affect forces between atoms? Does it?

Well, in fact, in a nuclei, the nucleons do "touch". The issue is this: if the "strong force" were only attractive, what would nucleons close to each other try to do? (And if they did, would nuclei exist?)
What do you mean by "touch"? What defines the surface boundary of a nucleon?

Since nucleons are simply a composite of three quarks, and the quarks are attracted to each other by a strong force (mediated by gluon exchange), the attraction between nucleons can be seen simply as the result of the quarks in one nucleon being attracted to the quarks in the other nucleons - a sort of residual gluon force. The force between nucleons is analagous to the valence bonding force between atoms caused by two positive nuclei being attracted to the same electron.

AM

Thanks both The first question is easier than you think. If the strong nuclear force fell off as an inverse square law force, would it affect forces between atoms? Does it?
Thanks for the hint; I think I've got it now. If I were learning only by studying this text book, all I would know about the strong nuclear force is:

Rutherford's experiment was repeated some years later using accelerated alpha particles of considerably higher energy than those emitted from radioactive sources. At these energies the Rutherford scattering formula began to break down, that is, the distribution of scattered alpha particles no longer fit the prediction. This meant that one of the assumptions used to derive the equation was valid at low energy and not at high energy. At high energies the alpha particles get much closer to the target nucleus, so perhaps a new short-range force operates over these short distances and changes the interaction. This was the first evidence for the strong nuclear force, a short-range interaction that binds neutrons and protons together in the nucleus but has no long-range (i.e. greater than about 10^^-15 m) effects, and does not act on electrons.

and

Neutrons are unstable and protons repel one another, so the proton-neutron model of the nucleus needs some explaining! Protons and neutrons bind to one another by the strong nuclear force. This is a short range force which is strong enough, on the nuclear scale, to overcome the mutual electrostatic repulsion of protons and inhibit the decay of neutrons. We now know that both protons and neutrons are made of quarks, and the strong force is related to the force that binds quarks themselves together. The variation of strong force with distance is shown in the diagram. The strong force only acts on particles that are made of quarks; these are called hadrons. It has no effect on electrons.

The "no long range (i.e. greater than about 10^^-15 m) effects" implies that it falls off faster than an inverse square law (and any other law that asymptotically approaches zero) -- and thus that it does not affect forces between nucleii/atoms.

Since nucleons are simply a composite of three quarks, and the quarks are attracted to each other by a strong force (mediated by gluon exchange), the attraction between nucleons can be seen simply as the result of the quarks in one nucleon being attracted to the quarks in the other nucleons - a sort of residual gluon force. The force between nucleons is analagous to the valence bonding force between atoms caused by two positive nuclei being attracted to the same electron.
Sorry -- I'm still stuck on "Give a reason for thinking that the strong nuclear force must become a repulsion at very short range." One very good reason is that there is a diagram in the text book showing it so! Seriously, I can't find anything in the quotes from the text book, dynamicsolo or Andrew that allows "reasoning" for the strong nuclear force being/becoming repulsive at very short range.

dynamicsolo
Homework Helper
The "no long range (i.e. greater than about 10^^-15 m) effects" implies that it falls off faster than an inverse square law (and any other law that asymptotically approaches zero) -- and thus that it does not affect forces between nucleii/atoms.
There is another argument for the "strong force" being a "short range" force. Gravity follows an inverse square law and is always attractive. What is the largest structure that can be assembled by gravitational force? If the strong force acts this way, what would be the biggest nucleus that could be made?

Sorry -- I'm still stuck on "Give a reason for thinking that the strong nuclear force must become a repulsion at very short range."
Here's something to consider: if the strong force were only attractive, at what distance would the quarks stop drawing closer to each other? How big would nuclei be in that case? Would they even exist as such?

There is another argument for the "strong force" being a "short range" force. Gravity follows an inverse square law and is always attractive. What is the largest structure that can be assembled by gravitational force?
Mmm ... a galaxy? That's a "structure" in the sense of "Something made up of a number of parts that are held or put together in a particular way". The entire Universe fails that definition because a) it is a single "part" and b) it's not assembled in "a particular way" -- it's more diverse.

If the strong force acts this way, what would be the biggest nucleus that could be made?
No limit, if the only effective force is the strong force? (that feels like a big "if"!)

Here's something to consider: if the strong force were only attractive, at what distance would the quarks stop drawing closer to each other? How big would nuclei be in that case? Would they even exist as such?
I can't imagine how quarks would behave. If I imagine spheres of fixed radius then they touch but I do not know if "touch" is applicable. Perhaps they are more analogous to sponges, highly eleastic, in which case a mechanism analogous to "bulk modulous of elasticity" might apply. Or they might be more like atoms (closest in scale, at least) mostly empty space so, to a first approximation, a point.

Andrew Mason
Homework Helper
I can't imagine how quarks would behave. If I imagine spheres of fixed radius then they touch but I do not know if "touch" is applicable. Perhaps they are more analogous to sponges, highly eleastic, in which case a mechanism analogous to "bulk modulous of elasticity" might apply. Or they might be more like atoms (closest in scale, at least) mostly empty space so, to a first approximation, a point.
Gravitational force is always attractive so matter tends to collapse toward the centre of mass due to gravity. But this does not cause masses to collapse completely. At some point (the black hole being the exception) the gravitational force cannot overcome coulomb repulsion forces between atoms and the collapsing stops.

We know that a proton or neutron is mostly empty space containing three quarks. (A quark appears to be a point particle without internal structure). If the force between quarks is always attractive would the nucleus not simply collapse to nothing?

This is a trick question, however. The reason a nucleus does not collapse is the same reason an atom does not collapse despite the attraction between protons and electrons. It has to do with the uncertainty principle rather than the existence of a new kind of repulsive force.

AM

dynamicsolo
Homework Helper
Mmm ... a galaxy? That's a "structure" in the sense of "Something made up of a number of parts that are held or put together in a particular way". The entire Universe fails that definition because a) it is a single "part" and b) it's not assembled in "a particular way" -- it's more diverse.

No limit, if the only effective force is the strong force? (that feels like a big "if"!)
Just so. As AM points out, we don't observe any influence on electron orbitals due to the "strong force", so its "reach" does not appear to extend even 10^-10 m. from the nucleus (or, for inner orbitals of atoms far up in the periodic table, a fraction of 10^-11 m.).

But, also, if the "strong force" were long-range, a small nucleus would draw in any nucleons or other nuclei at all in its vicinity. There would, in principle, be no limit to the size a single nucleus could grow. Gravity can assemble single objects with masses billions of times the mass of the Sun (the supermassive black holes in some galactic centers), if the relative velocities of the original smaller objects is small enough.

The only reason we don't notice the long-range character of the electrostatic force is that it is far stronger than gravity, so particles of opposite charges are drawn together to form electrically neutral objects very quickly. (This is particularly evident at the temperature range we live in. It requires large average particle energies -- that is, high temperatures -- for an ionized gas to be sustained for long.)

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dynamicsolo
Homework Helper
We know that a proton or neutron is mostly empty space containing three quarks. (A quark appears to be a point particle without internal structure). If the force between quarks is always attractive would the nucleus not simply collapse to nothing?

This is a trick question, however. The reason a nucleus does not collapse is the same reason an atom does not collapse despite the attraction between protons and electrons. It has to do with the uncertainty principle rather than the existence of a new kind of repulsive force.
This is why I've been putting the term "strong force" in quotes. What is being described in the problem is actually a combination of the "color force" (chromodynamic force) between quarks (which behaves more like a "Hooke's Law" sort of force: weak at close range and stronger with increasing distance) and what is sometimes called the "exclusion force" which limits the number density of particles of the class to which quarks belong (those with spin of 1/2). [It isn't really a force as such, so much as a manifestation of a characteristic of such particles.]

Were there no "exclusion force", there would really be nothing to limit attractive forces and stop implosions of collections of particles. Matter as we know it would not exist...

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Thanks for your help (and sorry for the tardiness of the thanks).

The explanation goes well beyond my present knowledge or what could be expected of UK "A" level students.