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A chart of the cluster structure of nuclides. This is a two-dimentional chart in which the x-axis gives the triton cluster number and the y-axis gives the alpha cluster number. No nucleon has more than one deuteron cluster.

The Peter Fimmel extended cluster model (ECM) of the atomic nucleus is a model of nuclear structure based upon the distribution of nucleons into two-, three- and four-body clusters. It assigns each proton (p) and neutron (n) of any atomic nucleus to from one to three of four naturally occurring clusters. The clusters are the deuteron (p,n), the triton (p,n,n), 3-He (p,p,n) and the alpha (p,p,n,n). A single procedure for the assignment of protons and neutrons among the clusters of a nucleus accommodates all the nuclides of the nuclear landscape.

The choice of the four clusters and the nucleon distribution procedure derive from a set of principles that arises from the process of neutron capture in the low-energy regime.

Structure and interaction models compliment one another and are each important tools for improving the understanding of the atomic nucleus and the interpretation of nuclear data. The ECM differs from earlier models in three main respects:

1. It assigns all the nucleons of every nucleus to a cluster
2. The addition or removal of a proton or neutron causes a discrete change of nuclear structure. Unlike shell models, every structure is complete; changes in nucleon number do not improve or complete substructure. The cluster concept means that nuclear collectivity is discrete and dynamic
3. The relationship between structural collectivity and nucleon number assigns a definite number of potential nuclear structures to each of the natural elements. The total number of isotope structures available for the 92 natural elements is 8,556, which far exceeds the actual number of naturally occurring nuclides.

The ECM is congruent with several light nuclear phenomena and provides structural explanations for a number of observations,which include:

• nuclear β-decay is given a structural, necessary initial condition.
• The tetraneutron and the islands of particle stability beyond the neutron drip line of fluorine, neon and sodium are structurally unified.
• The onset of neutron emission in the neutron-rich isotopes of the light elements is closely correlated to structure.
• Weakly bound halo nucleons show a correlation with structure.

The model makes testable predictions.

History

The bound atomic nuclei of the naturally occurring elements are composed of from two to over 230 nucleons. Their stability is almost as varied as their composition. Intuition suggests that the proton and neutron constituents of individual nuclei are arranged in an orderly manner and that the nuclear force is closely related to the nucleon arrangement; nuclear structure the nuclear interaction are related. Models of the atom have been the subject of discussion and enquiry for over two millennia. Following the discovery of the neutron in 1932, modern efforts to better understand the nucleus began and prompted the development of several models of nuclear structure. The liquid drop and [[]]shell models are examples of models devised to explain particular observations.

The liquid drop model was developed by Neils Bohr as a means of explaining the fission of heavy nuclei and the associated energy emission. The splitting of uranium into smaller nuclei was likened to dividing a large liquid drop into two smaller 'daughter' drops. The emission of radiation by an excited nucleus was likened to the evaporation of surface molecules from a liquid.

The shell model was created to explain the nuclear magic numbers. By the 1940s it was realised that some atomic nuclei were particularly strongly bound and that the strength of their bond was related to the numbers of their protons and neutrons. These were known as the nuclear magic numbers, viz: 2, 8, 20, 28, 50, 82, and 126. The nuclear magic numbers bore a resemblance to the periodic relationship of electron numbers to the differential chemical reactivity observed among the chemical elements. Chemical periodicity was explained by the theory of electron shells. The electron shell concept was adopted as the basis for the shell theory of atomic nuclei by Maria Goeppert-Mayer and Hans Jensen; who independently developed the nuclear shell model, as a means of explaining the nuclear magic numbers. An important feature of the model is that single protons and neutrons are assumed to move independently - thus the concept of nuclear single-particle motion. By contrast with the empirical motives for the development of the liquid drop and shell models, the ECM arose from an enquiry into what principles might be instantiated by neutron capture among the isotopes of hydrogen and helium. It was not developed in order to explain neutron capture or any particular nuclear behaviour or observations, it simply arose from deeper underlying principles.

The ECM rationale

The structure that arises from the ECM is not superimposed on nuclear phenomena; there is no template to be fitted to nuclear data. Neither does it have a geometric aspect—there are no shapes, spheres, shells or paths traced out by individual or grouped particles. The model is non-geometrical.

Neutron capture by the isotopes of hydrogen and helium

The serial addition of three single neutrons to one proton, in a suitable time frame, forms first the deuteron and then the triton. The triton does not bind further neutrons, but with a T1/2 of ~12.5 yrsit decays to 3He, which is stable and able to bind the third neutron to form 4He. The process is then at an end since 4He is stable and does not bind a single neutron (or a single proton). The conclusions that are drawn from the process of serial single-neutron capture include:

• The formation of nuclei by individual nucleons tends toward the production of alphas.
• Nucleon excess is structurally preserved by the three-body mirror clusters (3H, 3He).
• Because the addition or removal of a nucleon annihilates one cluster and creates another, clusters are reactive to nucleon number.
• Nuclear cluster collectivity is therefore discrete and dynamic—there are no in-between states.

Table 1: Carbon nuclear structures

nucleons			structures
A	Z	N	α	T	d
7	6	1		h+4p
8	6	2	1+4p
9	6	3	1	h+2p
10	6	4	2+2p
11	6	5	2	h
12	6	6	3
13	6	7	2	1	1
14	6	8	2	2
15	6	9	1	3	1
16	6	10	1	4
17	6	11		5	1
18	6	12		6
19	6	13		6+n
20	6	14		6+2n
21	6	15		6+3n
22	6	16		6+4n
Table 2: Light nuclear high-mass limit isotopes and neutron emission Nuclide Tritons Decay Mode 3-H 1 β 4-H 1+n n 6-He 2 β 7-He 2+n n 9-Li 3 β, β+n* 10-Li 3+n n 12-Be 4 β, β+n 13-Be 4+n n 15-B 5 β 16-B 5+n n 18-C 6 β, β+n 19-C 6+n β+n, β 21-N 7 β, β+n 22-N 7+n β, β+n, β+2n 24-O 8 β, β+n 25-O 8+n n 27-F 9 β, β+n 28-F 9+n n 30-Ne 10 β, β+n, β+2n 31-Ne 10+n β, β+n 33-Na 11 β, β+n, β+2n 34-Na 11+n β+2n, β, β+n 36-Mg 12 β 37-Mg 12+n β, β+n Table 3: Structures of the generally accepted light neutron halo nuclei Nuclide Tritons Halo 6-H 2 2n halo 7-He 2+n 8-He 2+2n 2n halo 9-Li 3 10-Li 3+n 11-Li 3+2n 2n halo 11-Be 3+d 1n halo 12Be 4 13-Be 4+n 14-Be 4+2n 2n halo 15-Be 5 16-Be 5+n 17-B 5+2n 2n halo 18-C 6 19-C 6+1n 1n halo Table 4: The structures of the islands of particle stability beyond the neutron drip-lines of F, Ne and Na Nuclide Tritons Decay Mode[11] 27-F 9t β,β+n 28-F 9t+n n 29-F 9t+2n β+n,β 30-F 9t+3n n 31-F 9t+4n Particle stable 30-Ne 10t β,β+n,β+2n 31-Ne 10t+n β,β+n 32-Ne 10t+2n β,β+n 33-Ne 10t+3n n 34-Ne 10t+4n Particle stable 33-Na 11t β,β+n,β+2n 34-Na 11t+n β+2n,β,β+n 35-Na 11t+2n β,β+n 36-Na 11t+3n n 37-Na 11t+4n Particle stable Tetra-n 4n Particle stable

The nucleon assignment procedure

The principle that underlies the model derives from two imperatives, viz. (a) the tendency to alpha formation and (b) the conservation of nucleon excess. Those imperatives distribute the nucleons of each A > 4 nuclide among from one to three of the four clusters according to the following rules:

• Each excess neutron forms a triton, if there are sufficient protons for the cluster. (excess means additional to N = Z.)
• Remaining pairs of neutrons form alphas.
• If there is a residual neutron following alpha formation it forms a deuteron if one proton remains and a 3-helium if two or more remain.

Any nucleons which this procedure does not assign to a cluster are passengers because they neither change nor contribute to the structure of the nucleus. The structure of each naturally occurring isotope of the natural elements is therefore unique.

Discrete dynamic collectivity

Nuclear structure displays discrete dynamic collectivity; it reacts to changes in nucleon complement at the level of its constituent clusters. Structure changes discretely, it is not built up gradually by nucleon addition or removal. A one-nucleon change that characterises contiguous pairs of isotopes or isotones causes the annihilation of some clusters and the creation of others. A nucleon whose addition or removal does not change nuclear structure, while obeying the rules of nucleon assignment to clusters, is not part of that structure. Such a nucleon is a passenger. Consistent with the absence of nucleon shells, there is no motion of individual particles. Each proton and neutron of the nuclides of an isotope series is a member of a two-, three- or four-body cluster; none, save passengers, are free of a nuclear bond.

Model consequences

The double imperative of the model gives each M > 4 with N ≥ Z nuclide a unique structure. For any isotope series, the serial increase in neutron number is accompanied first by increasing alpha numbers among proton-rich nuclides, then increasing triton numbers until T = Z. The T = Z isotope marks the exhaustion of the potential for cluster reactivity, and additional neutrons are referred to as passengers. Table 1 shows the isotope structures of carbon.

Key to Tables: A = mass number; Z = proton (nuclear charge); N = neutron; α = alpha; T = 3-body cluster (h = 3He, numeral = triton); t = triton; d = deuteron; p = passenger proton; n= passenger neutron. Decay modes in Tables 2 and 4 are given in descending order of branching ratios.

Because discrete dynamic collectivity of an isotope series does not extend beyond the T = Z nuclide, the model gives a high mass limit for the isotope series of every natural element, viz. the A = 3Z nuclide. The isotope low mass limit is the Z + n structure.

Model correlations

Many of the model correlations follow from the concept of the high mass limit to the series of isotopes of any element.

The neutron drip line

The decay modes about the neutron drip line among the light elements correlate with model structure. It can be seen in Table 2, that β-decay is the dominant decay mode of the high mass nuclide of the twelve lightest elements. By contrast, neutron emission is the sole decay mode of the one-neutron passenger nuclide of seven elements, and with β-decay is the dominant decay mode of a further two elements,

Halo nuclei

Light halo nuclei exhibit two main properties: * a low binding energy of the halo nucleons to the core of the nucleus * an extended size of the nucleus. The most prominent feature of halo nucleons is their low separation energies, which are usually <1 MeV. The concept of passenger neutrons is consistent with the structures for many of the generally accepted neutron halo nuclides of the light elements, shown in Table 3. Consistent with the concept that the nuclear interaction and structure are related, passenger neutrons, that do not contribute to nuclear structure, are expected to be weakly bound. Measured separation energies and large rms radii are characteristic of the A=3Z+2n nuclides of helium, lithium, beryllium and boron, which are two-neutron halo nuclides. These data on the generally accepted halo nuclides represent support for the ECM.

The islands of particle stability

The model reveals a remarkable structural feature common to the single-nuclide islands of particle stability beyond the neutron drip lines of fluorine, neon and sodium. Each particle-stable island nuclide (31F, 34Ne and 37Na) consists of the high-mass limit isotope of each element plus four passenger neutrons. Four neutrons have also been observed to bind in the formation of the isolated particle-stable tetraneutron,as shown in Table 4.

Model predictions

• Nuclear break-up interactions, at certain resonance energies, are predicted to produce cluster signatures consistent with the model. *The model predicts the observation of further bound neutron matter.

Model-consistent observations

Examples of model-consistent signatures that have been observed include:

• the triple alpha structure of 12C
• the 16O + 3He structure of 19Ne
• the double triton decay mode of the 18 MeV resonance in 6He, with a branching ratio of 90±10%.

See also

• "Table of ECM Nuclear Structures of the Naturally Occurring Isotope Zones" (PDF).
• Peter J. Fimmel (2009). "On the W-boson NN Interaction and the Extended Cluster Model of the Nucleus". Physics Essays. 22: 104–111.

References

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Catagories: Atom | Atomic nucleus | Nuclear physics | Nuclear structure | Hadrons