Professor Asghar,

Many thanks for commenting in such detail (here and elsewhere). Despite obvious differences in perspective, I am not sure how mutually-exclusive our views are. Specifically, I would agree with you that the shell model’s description of “independent” nucleon states is “unassailable”. But the theoretical contortions that are needed to get to that description in a...

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Professor Asghar,

Many thanks for commenting in such detail (here and elsewhere). Despite obvious differences in perspective, I am not sure how mutually-exclusive our views are. Specifically, I would agree with you that the shell model’s description of “independent” nucleon states is “unassailable”. But the theoretical contortions that are needed to get to that description in a “nuclear gas” are, I believe, dubious. Moreover, the fact that the final description of nuclear states is precisely replicated in the lattice model suggests an alternative. As most nuclear theorists would admit, the other (liquid-drop, cluster, etc.) models of nuclear structure theory are fundamentally not quantum mechanical (QM) and are, for that reason, widely held to be limited analogies between macroscopic objects and a small set of nuclear properties, but are far from being comprehensive theories of the nuclear world. In that sense, I think the choice for the “one truly comprehensive theory” of the nucleus will necessarily be either the QM lattice or the QM shell model.

There are lots of specialist topics where the two models differ and where a decisive experiment might yet distinguish between them, but let me ask you about the shell model’s invocation of the exclusion principle to justify the “orbiting” of nucleons in the dense nuclear interior. Weisskopf (1951) introduced that idea soon after the first publications on the shell model (1949), and stated it frankly as a hypothesis that needed examination: “It remains to be proved whether this [exclusion] effect is sufficient to establish independent orbits in low-lying states of nuclei in spite of the existence of strong interactions” (Blatt and Weisskopf, 1953, p. 778).

But the hypothesis was soon taken by others to be a fundamental “truth,” and has since been generalized to the form you mention, where the exclusion “principle” is said to be a “force” of nature! Gravity, EM, weak and strong forces….. and the fifth “force” of one particle excluding another particle from entering its QM space by virtue of “exclusion”. I am not the only one who finds that logic a little bit strange. Herzberg (1937, p. 123), Condon and Shortley (1935, p. 167), Yang and Hamilton (“its physical basis remains an open puzzle”, 1996, p. 193) and a handful of others over the years have commented that, as true as the exclusion principle is in describing certain fermion phenomena in both atomic and nuclear physics (and even QCD), it still needs to be explained on the basis of underlying physical forces.

The invocation of the exclusion principle in nuclear theory is an example of the kind of domino effect that faulty assumptions produce: If you assume an experimentally unknown central nuclear potential around which nucleons orbit, you then need to explain how the nucleons can squeeze past each other in a substance as dense as nuclei. If the nuclear volume were much larger or the nucleons much smaller, then nucleon collisions might be sufficiently infrequent and their “orbiting” might be justified. But a typical heavy nucleus like Lead has more than 200 nucleons (rms radius, 0.9 fm) inside of a rather small nuclear volume (rms radius, 5.5 fm), making the nuclear interior about one-half filled with nucleons and making the nucleon “mean-free path” extremely short. So, having made the faulty assumption of a central nuclear potential, a nuclear gas is implied, but in order to justify nucleon orbiting, the exclusion principle must then elevated to the level of a “force of nature”.

So, I see what the shell model theorists are up to, but a less tortured view of nuclear structure is to accept the short-range nuclear force inherent to the liquid-drop model (and experimentally known!) – and simply forget about the novel use of the exclusion principle as a force of nature. All of the shell model QM states of nucleons are retained in a frozen liquid-drop (the nucleon lattice) – and the lattice shares most of the macroscopic properties already described in the liquid-drop model.

In the end, does the lattice model today accomplish as much as the shell model, independent-particle model, the liquid-drop model and the cluster models have achieved?! The answer is: Not yet. But, if you would be so kind as to send me 10,000 graduate students and a few years of supercomputer time, I think we will leave the shell model in the trash bin of history where Ptolemy hangs out.

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Many thanks for your comments again, Professor Asghar!

The various models of the nucleus have a long history, going back to the 1930s, and I often refer to the Fermi-gas model, the shell model and the independent-particle model (IPM) collectively as the “gaseous-phase” models. As you note, in fact, their theoretical foundations are quite different. The Fermi-gas model was little more...

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Many thanks for your comments again, Professor Asghar!

The various models of the nucleus have a long history, going back to the 1930s, and I often refer to the Fermi-gas model, the shell model and the independent-particle model (IPM) collectively as the “gaseous-phase” models. As you note, in fact, their theoretical foundations are quite different. The Fermi-gas model was little more than an analogy with the gas contained within a fictitious boundary. The early shell model was constructed around the idea of a central potential well – in analogy with atomic structure, but of course without any central attracting object that could act as the central potential. The “central potential well” was assumed to be the net result of the many local, nucleon-nucleon interactions... an interesting idea, maybe, but purely hypothetical as a mechanism for binding nucleons together into a stable nucleus. Everyone understood and most textbooks acknowledged that the central potential well was a dubious ploy, but the shell model nonetheless had many theoretical successes.

As you have commented, the modern IPM utilizes a nuclear force. From the perspective of theoretical coherency, the use of a realistic short-range nuclear force is huge progress. Unfortunately, the modern IPM also requires a theoretical “trick” to make the model work. That is,the Pauli exclusion principle must be “enhanced” to maintain the “gaseous” phase that the short-range force, on its own, would not allow.

As connoisseurs of nuclear structure theory well know, the exclusion principle was first suggested by Weisskopf in 1950 as a possible mechanism to allow nucleons to move like the particles within a gas under the influence of the shell model’s (fictitious) central potential well. He argued that “exclusion” might justify the shell model’s underlying theoretical assumptions, but the exclusion principle has been, over the decades, gradually “enhanced”. Instead of a simple statement that “no two fermions can have the same set of quantum numbers” (which was Pauli’s original formulation of “exclusion”), the exclusion effects have been referred to as “Pauli blocking” and even as the “Pauli force”. It is not clear, dynamically, what that “force” is (and others have complained about this reification of Pauli exclusion into a force of nature), but without the “blocking” of the powerful short-range nuclear force effects, the nucleus would condense into a high-density liquid or solid.

That is of course precisely what the fcc lattice version of the IPM predicts. If it is assumed that, without a Pauli “force”, nucleons condense to a solid of nucleons (a close-packed, antiferromagnetic face-centered-cubic lattice with alternating proton and neutron layers), then it is easily shown that the properties of that particular lattice neatly reproduce all of the quantum number properties of the nucleus – for which the IPM is justly famous.

So, I return again to the seemingly paradoxical position that I have stated before: I fully agree with Prof. Asghar that the “shell model” (IPM) has been wildly successful over many decades. Those successes cannot be ignored, but I maintain that the gaseousness of the shell model (and the implied nucleon orbiting and the Pauli blocking of the inevitable collisions of orbiting nucleons) is not necessary to explain the IPM quantal regularities, insofar as the lattice contains the same regularities. From a conventional point of view, it may seem unfair for the “new kid in town” – the fcc lattice model – to stake a claim for the shell model’s theoretical successes, but I maintain that it is simply uncertain which model is mimicking the other!

Without the quantum mechanical properties (the independent-particle states) of the IPM, the extensive and rigorous data on thousands upon thousands of nuclear isotopes become incoherent, so that the known shells and subshells of the IPM – and the occupancies of nucleons in those shells and subshells under the influence of the exclusion principle – appear to be essential. But I think that factors of serendipity and chance, more than logical necessity, led to the overwhelming predominance of the gaseous-phase shell model and it becoming the central paradigm of nuclear structure physics, while Wigner’s original proposal of the lattice structure of the nucleus in 1936 was seen simply as a mathematical analog (and Everling’s suggestion of the lattice as a coherent nuclear model in 1958, and Lezuo’s solid-phase formulation in 1974, and subsequent developments in the same direction have been ignored).

There are undoubtedly other criteria that can distinguish between the solid-phase and the gaseous-phase models, but it appears that neither model can assert that the well-established independent-particle systematics of the nucleus unambiguously supports one model over the other.

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Dear Norman,

Thank you very much for your supporting comments. I would love to finally start quantitative consideration, but it is a really tough job: finding the exact Lagrangian combined with performing really difficult numerical simulations. Unfortunately I cannot find cooperation to work on such nonstandard approach, prof. Faber shares the belief of "complete soliton model" existence,...

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Dear Norman,

Thank you very much for your supporting comments. I would love to finally start quantitative consideration, but it is a really tough job: finding the exact Lagrangian combined with performing really difficult numerical simulations. Unfortunately I cannot find cooperation to work on such nonstandard approach, prof. Faber shares the belief of "complete soliton model" existence, but he focuses on much more complex trial of expanding his basic model. Maybe you could think of somebody who could be interested in cooperation to move such approach forward?

About "building up quantum phenomena" - while I was on Emergent Quantum Mechanics congress a year ago (proceedings), most of the speakers were talking about double-slit experiment ... I think there are already good intuitions where quantum phenomenas come from (e.g. by great Couder's experiments) - they are the consequences of two factors: the wave conjugated with particle and that QM is kind of thermodynamical level theory - averaging complex dynamics, what is greatly seen e.g. in the shell nuclear model. I believe it is finally the time to search for concrete models below this effective quantum description, but looking at the congress, there are very few people interested in finally making the next step ...

About combining "complete soliton approach" with different models, it doesn't leave freedom for that: we just take a simple field (e.g. real symmetric tensor field) and Lagrangian: kinetic term (like Faber's interpreting field curvature as electromagnetic field) and potential term (Higgs-like - with topologically nontrivial energy minimum) ... and ask for consequences. Of course there is some freedom of choosing Lagrangian within a single approach, but most of consequences are very stiff - topological. So basically it is all or nothing approach - a single essential qualitative discrepancy and it probably goes to trash.

So practically the only way to combine it with other models is to see them as its effect - what can be very valuable while choosing the Lagrangian or to disqualify given approach. Thank you for the Palazzi's phenomenological models - I will try to see them through my model.

Cheers

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