|San José State University|
& Tornado Alley
What holds the nucleus|
of an atom together?
This is not the conventional explanation of what holds a nucleus together. The conventional explanation is merely a naming of what holds nuclei together; i.e., the nuclear strong force. This naming has no more empirical content than if physicists said something holds a nucleus together. The physicists at the time needed an explanation for how a nucleus composed of positively charged protons could stably hold together. They hypothesized a force which at shorter distances between protons is more attractive than the electrostatic force is repulsive, but at longer distances is weaker. The only evidence for this hypothetical nuclear strong force is that there is a multitude of stable nuclei containing multiple protons. According to the theory nuclear stability was aided by the neutrons of a nucleus being attracted to each other as well as to the protons. So the conventional theory is merely an explanation of how a nucleus containing multiple positive charges can be stable.
But even if a theory explains empirical facts that does not mean that it is necessarily true. It only means the theory might be physically true. There might be an alternate true explanation of those empirical facts. And if a theory predicts somethings which do not occur then even if it explains other things it cannot be physically correct.
According to the strong force theory of nuclear structure there should be no limit on the number of neutrons in stable nuclides. There should be ones composed entirely of neutrons. There should even be ones composed entirely of a few protons. These things do not occur physically. In fact there has to be a proper proportion between the numbers of neutrons and protons. In heavier nuclides there are fifty percent more neutrons than protons. Thus there are serious flaws with the conventional theory of nuclear structure; i.e., the nuclear strong force.
When the conventional theory of nuclear structure was formulated physicists thought that they could not be wrong, but, as will be be shown below, they were wrong, because their concept of nuclear strong force conflates two disparate phenomena: spin pairing, attractive but exclusive, and non-exclusive interaction of nucleons in which like-nucleons repel each other and unlike attract. The proof of this assertion is given below. This is an abbreviated version of an alternative of what holds a nucleus together. The full version is at Nucleus.
It should be noted that neutron-neutron and proton-proton can only exist within a nucleus; i.e., in conjunction with other spin pairs.
Under this force like nucleons are repelled from each other and unlike ones attracted. This astounding proposition will be proved later.
This force only affects interactions between protons. Neutrons have no net electrostatic charge but do have a radial distribution of electrostatic charge involving an inner positive charge and a negative outer charge.
However, in a nucleus having many nucleons the magnitude of the energy of the many small energy interactions might possibly exceed those of the few spin pair formations. But because the interaction force between like nucleons is repulsion there would have to be a proper proportion between the numbers of neutrons and protons for the net interaction to be an attraction or involve a significant reduction in the repulsion between like nucleons. For heavier nuclei that requires there to be 50 percent more neutrons than protons. That 150 percent ratio will be explained later.
The mass of a nucleus made up of many neutrons and protons is less than the masses of its constituent nucleons. This mass deficit when expressed in energy units through the Einstein formula E=mc² is called the binding energy of the nucleus. Binding energy is described as the energy required to break a nucleus apart into its constituent nucleons. The total binding energy of a nucleus also includes the loss in potential energy involved in its formation as a nucleus. When a nucleus is formed from its constituent nucleons there is a loss of potential energy but a gain in kinetic energy for a net energy loss that is manifested in the form of the emission of a gamma ray. Unfortunately the total binding energies are not known for the various nuclides except for the deuteron. However there is reason to believe that the loss of potential energy is proportional to the mass deficit binding energy. Nevertheless the analysis of the mass deficit binding energies reveal a great deal about the structure of nuclei. Much of this comes from an examination of incremental binding energies.
If n and p are the numbers of neutrons and protons, respectively, in a nucleus and BE(n, p) is their binding energy then the incremental binding energies with respect to the number of neutrons and the numbers of protons are given by:
The sawtooth pattern is a result of the enhancement of incremental binding energy due to the formation of neutron-neutron spin pairs. The regularity of the sawtooth pattern demonstrates that one and only one neutron-neutron spin pair is formed when a neutron is added to a nuclide. The above graphs are just illustrations of the effect but the same pattern prevails throughout the dataset of nearly three thousand nuclides.
The pattern of spin pairing described above prevails throughout the more than 2800 cases of the incremental binding energies of protons in addition to the more than 2750 cases of the incremental binding energies of neutrons. .
Here is the graph for the case of the isotopes of Krypton (proton number 36).
As shown above, there is a sharp drop in incremental binding energy when the number of neutrons exceeds the proton number of 36. This illustrates that when a neutron is added there is a neutron-proton spin pair formed as long as there is an unpaired proton available and none after that. This illustrates the exclusivity of neutron-proton spin pair formation. It also shows that a neutron-proton spin pair is formed at the same time that a neutron-neutron spin pair is formed.
The case of an odd number of protons is of interest. Here is the graph for the isotopes of Rubidium (proton number 37).
The addition of the 38th neutron brings the effect of the formation of a neutron-neutron pair but a neutron-proton pair is not formed, as was the case up to and including the 37th neutron. The effects almost but not quite cancel each other out. It is notable that the binding energies involved in the formation of the two types of nucleonic pairs are almost exactly the same, but the binding energy for the neutron-neutron spin pair is slightly larger than the one for a neutron-proton spin pair.
This same pattern is seen in the case for the isotopes of Bromine.
The components of the incremental binding energy of neutrons can be approximated as follows. For an even proton number look at the values of IBEn at and near n=p. Project forward the values of IBEn from n=p-3 and n=p-1 to get a value of ICEn for n=p; i.e.,
Likewise the values for IBEn can be projected back from n=p+1 and n=p+3 to get a value of IBEn for n=p without the effect of either an nn spin pairing or an np spin pairing. This procedure is shown below for the isotopes of Neon (10).
When this procedure is carried out numerically the results indicate that 42.7 percent of the incremental binding energy at n=p=10 are due to the nn spin pairing, 17.1 percent is due the np spin pair and the other 40.0 is due to the net interactive binding energy.
This domination of IBEn by spin pairing can only occur for small nuclides. For iron (p=26) the figures are 16.9 percent for the nn spin pairing, 12.8 percent for np spin pairing, and 70.3 percent due to the net effect of the interactive binding energy of the nucleons.
It is not just that effects of the spin pairings goes down for the heavier nuclei; it is that those of the interactions goes up. For more on the components of IBEn see Components of IBEn.
It is found that the increments in the incremental binding energies are related to the interactions of the nucleons. There are theorems (second difference theorem and cross difference theorem) that relate the second differences in binding energy to the interaction binding energy of the last two nucleons added to the nuclide. That binding energy corresponds to the slope of the relationship shown below.
Thus if the incremental binding energy of neutrons increases as the number of protons in the nuclide increases then that is evidence that a neutron and a proton are attracted to each other through the nucleonic force.
If the incremental binding energy of neutrons decreases as the number of neutrons in the nuclide increases then it is evidence that the interaction of a neutron and another neutron is due to repulsion. That is to say, neutrons are repelled by each other.
The above two graphs are just illustrations but exhaustive displays are available at neutrons, protons and neutron-proton pairs that like nucleons are repelled from each other and unlike attracted. The theoretical analysis for the proposition is given in Interactions.
The character of the interaction of two nucleons can be represented by their possessing a nucleonic charge. If the nucleonic charges of two particles are Ω1 and Ω2 then their interaction is proportional to the product Ω1Ω2. Thus if the charges are of the same sign then they repel each other. If their charges are of opposite sign then they are attracted to each other.
The electrostatic repulsion between protons simply adds to the effective charge of protons. The amount of the addition depends upon the distance separating the protons. There is no qualitative change in the characteristics of a nucleus due to this force.
The data on incremental binding energies establishes that whenever possible nucleons form spin pairs. Having established this principle it then follows that nucleons in nuclei form chains of nucleons linked together by spin pairing. Let N stand for a neutron and P for a proton. These chains involve sequences of the sort -N-P-P-N- or equivalently -P-N-N-P-. The simplest chain of this sort is the alpha particle in which the two ends link together. These sequences of two neutrons and two protons can be called alpha modules. They combine to form rings. A schematic of such a ring is shown below with the red dots representing protons and the black ones neutrons. The lines between the dots represent spin pair bonds.
It is to be emphasized that the above depiction is only a schematic. The actual spatial arrangement is quite different. For illustration consider the corresponding schematic for an alpha particle and its spatial arrangement.
The depiction of an alpha particle in the style of the above would be the figure shown on the left below, whereas a more proper representation would be the tetrahedral arrangement shown on the right.
Here is an even better visual depiction of an alpha particle.
The graph below demonstrates the existence of nucleon shells.
The sharp drop off in the incremental binding energy of neutrons after 41 neutron pairs indicates that a shell was filled and the 42nd neutron pair had to go into a higher shell.
Maria Goeppert Mayer and Hans Jensen established a set of numbers of nucleons corresponding to filled shells of (2, 8, 20, 28, 50, 82, 126) nucleons. Those values were based on the relative numbers of stable isotopes. The physicist, Eugene Wigner, dubbed them magic numbers and the name stuck. For more on this topic see Magic Numbers.
In the above graph the sharp drop off in incremental binding energy after 41 neutron pairs corresponds to 82 neutrons, a magic number
Analysis in terms of incremental binding energies reveal that 6 and 14 are also magic numbers. If 8 and 20 are considered the values for filled subshells then a simple algorithm explains the sequence (2, 6, 14, 28, 50, 82, 126).
First consider the explanation of the magic numbers for electron shells of (2, 8, 18, …). One quantum number can range from −k to +k, where k is an integer quantum number. This means the number in a subshell is 2k+1, an odd number. If the sequence of odd numbers (1, 3, 5, 7 …) is cumulatively summed the result is the sequence (1, 4, 9, 16, …), the squared integers. These are doubled because of the two spin orientations of an electron to give (2, 8, 18 …).
For a derivation of the magic numbers for nucleons take the sequence of integers (0, 1, 2, 3, …) and cumulatively sum them. The result is (0, 1, 3, 6, 10, 15, 21 …). Add one to each member of this sequence to get (1, 2, 4, 7, 11, 16, 22, …). Double these to get (2, 4, 8, 14, 22, 32, 44 …) and then take their cumulative sums. The result is (2, 6, 14, 28, 50, 82, 126), the nuclear magic numbers with 6 and 14 replacing 8 and 20. Note that 8 is 6+2 and 20 is 14+6. There is evidence that the occupancies of the filled subshells replicate the occupancy numbers for the filled shells.
These alpha module rings rotate in four modes. They must rotate as a vortex ring to keep separate the neutrons and protons which are attracted to each other. The vortex ring rotates like a wheel about an axis through its center and perpendicular to its plane. The vortex ring also rotates like a flipped coin about two different diameters perpendicular to each other.
The above animation shows the different modes of rotation occurring sequentially but physically they occur simultaneously. (The pattern on the torus ring is just to allow the wheel-like rotation to be observed.)
Aage Bohr and Dan Mottleson found that the angular momentum of a nucleus (moment
of inertia times the rate of rotation) is quantized to
h is Planck's constant divided by 2π and I is a positive integer. Using this result
the nuclear rates of rotation are found to be many
billions of times per second. Because of the complexity of the four modes of rotation each nucleon
is effectively smeared throughout a spherical shell. So, although the static structure of a nuclear shell is that
of a ring, its dynamic structure is that of a spherical shell.
The overall structure of a nucleus of filled shells is then of the form
At rates of rotation of many billions of times per second all that can ever be observed concerning the structure of nuclei is their dynamic appearances. This accounts for all the empirical evidence concerning the shape of nuclei being spherical or near-spherical.
For a nucleus consisting of filled shells plus extra neutrons (called halo neutrons) the dynamic appearance is a spherical core of filled shells with pairs of halo neutrons in orbits about the core.
For the 2929 nuclides the following variables were computed which represent the formation of substructures.
To represent the interactions between nucleons the following variables were computed.
The model indicates that nuclear binding energy of nuclides is a linear function of these variables. Here are the regression equation coefficients and their t-ratios (the ratios of the coefficients to their standard deviations).
The Results of Regression Analysis
Testing the Alpha Module Ring
Model of Nuclear Structure
|Number of |
|Number of |
Proton-Proton Spin Pairs
Not in an Alpha Module
|Number of |
Neutron-Proton Spin Pairs
Not in an Alpha Module
|Number of |
Neutron-Neutron Spin Pairs
Not in an Alpha Module
It should be noted that there is a great difference among the frequencies of the extra spin pairs. There are 2919 with an alpha module and only 10 without. There are 2668 nuclides with extra neutron-neutron spin pairs, but only 164 out of the 2929 nuclides which have one or more extra proton-proton spin pairs. There are 1466 with an extra neutron-proton spin pair.
The coefficient of determination (R²) for this equation is 0.9998825 and the standard error of the estimate is 5.47 MeV. The average binding energy for the nuclides included in the analysis is 1072.6 MeV so the coefficient of variation for the regression equation is 5.47/1072.6=0.0051. Most impressive are the t-ratios. A t-ratio of about 2 is considered statistically significant at the 95 percent level of confidence. The level of confidence for a t-ratio of 923 is beyond imagining.
It is notable that the coefficients for all three of the spin pair formations are roughly equal. They all are larger from what one would expect from the binding energies of small nuclides.
The regression coefficients for the nucleonic force interactions have some especially interesting implications. Without loss of generality the force between two nucleons with charges of Ω1 and Ω2 can be represented as
where H is a constant, s is the separation distance and f(s) could be a constant or a declining function of s, possibly exp(−s/s0).
Let the nucleonic force charge of a proton be taken as 1 and that of a neutron as q, where q might be a negative number. The nucleonic force interactions between neutrons is proportional to q², and those between neutrons and protons would be proportional to q.
Thus the ratio of the coefficient for neutron-neutron interactions to that for neutron-proton interaction would be equal to q. The value of that ratio is
This is confirmation of the value of −2/3 found in previous studies. Thus the nucleonic force between like nucleons is repulsion and attraction between unlike nucleons.
The values involving proton-proton interactions are most likely affected by the influence of the electrostatic repulsion between protons. That force would be as if the charge of the proton were (1+d) where d is the ratio of the electrostatic force to the nucleonic force. More on this later.
An alpha module thus has a nucleonic charge of +2/3=(1+1-2/3-2/3). Therefore two spherical shells composed of alpha modules would be repelled from each other if the spherical shells are separated from each other. This would be a source of instability. But if the spherical shells are concentric the repulsion is a source of stability. Here is how that works. As noted before without loss of generality the force between two nucleons with charges of Ω1 and Ω2 can be represented as
where s is the separation distance between them, H is a constant, q1 and q2 are the nucleonic charges and f(s) is a function of distance. For the nucleonic force it is presumed that f(s) is a positive but declining function of distance. This means that the nucleonic force drops off more rapidly than the electrostatic force between protons.
When one spherical shell is located interior to another of the same charge the equilibrium is where the centers of the two shells coincide. If there is a deviation from this arrangement the increased repulsion from the areas of spheres which are closer together is greater than the decrease in repulsion from the areas which are farther apart. This only occurs for the case in which f(s) is a declining function. If f(s) is constant there is no net force when one sphere is entirely enclosed within the other. For more on this surpris
The regression of the number of neutrons on the number of protons gives the equation
The coefficient 1.57054 corresonds to |q|=2/3 and d=0.078.
Regression equations for the binding energies of almost three thousand nuclides based upon the model presented above have coefficients of determination (R²) ranging from 0.9999 to 0.99995 with all of the regression coefficients being of the right sign and relative magnitude. See Statistical Performance for the details.
Let n and p be the numbers of neutrons and protons, respectively, in a nuclide. The number of neutron-neutron interactions is equal to n(n-1)/2. This will be denoted as nn. Likewise the number of proton-proton interactions is p(p-1)/2 and this will be denoted as pp. The number of neutron-proton interactions is np.
The binding energy due to these interactions is a function of the separation distances of the nucleons. Here no distinction is made for separation distances so the results will be for the average separation distance of the nucleon.
The regression equation expressing the attempt to predict the binding energy of a nuclide from the numbers of the interactions of its nucleons is
There is no constant term because if nn=np=pp=0 the BE must be zero.
The conventional model of nuclear structure is then expressed as
According to the Conventional Model the coefficient for proton-proton interactions should be less than that for neutron-neutron interaction because of the electrostatic repulsion between protons.
Here are the results of the regression analysis for the 2931 nuclides.
The figure in the square brackets below a coefficient is its t-ratio, the ratio of the coefficient to its standard deviation. The t-ratios indicate that the coefficients are statistically significantly different from zero.
The assertions of the Conventional Model of nuclear structure are not born out. Two of the three coefficients are negative. The negative values for cnn and cpp indicate that the force between two like nucleons is a repulsion. The positive value for cnp indicates the force between two unlike nucleons is an attraction.
The value of cpp is not numerically less than that of cnn; it is numerically larger. This cannot be, because the electrostatic force between two protons is known to be a repulsion.
The coefficient of determination (R²) for the above regression equation is 0.924. But given that almost all of the regression coefficients are wrong in terms of sign or relative magnitude a higher value of R² is evidence against the conventional model rather than for it.
The above regression coefficient values can be explained by allowing for a neutron to have a different nucleonic charge than a proton. But more importantly the Conventional Model leaves out the effects of the spin pairing of nucleons. The alternate Alpha Module Ring model of nuclear structure presented above which takes spin pairing into account explains 99.99 percent of the variation in the binding energies of the 2931 nuclides.
It had already been established that the interaction of like nucleons is a repulsion and the negative coefficients for nn and pp confirm that. The positive coefficient for np confirms that the interaction of unlike nucleons is an attraction.
The regression coefficent for pp is more negative than the one for nn, as it should be, because the electrostatic repulsion between two protons is added to the repulsion between two like nucleons.
The coefficient of determination (R²) for the regression equation is 0.9999. Modifications of the model, such as taking into account the shell structures of the nucleons, raises that value to 0.99995.
Thus in every way the regression results confirm the assertions of the Alpha Module Ring model of nuclear structure. This is in contrast to the Conventional Model in which almost all of its assertions are denied by empirical analysis.
In a nucleus wherever possible the nucleons are linked together through exclusive spin pair formation into rings of alpha modules which rotate in four different modes at rapid rates. This rapid rotation results in each nucleon being effectively smeared uniformly throughout a spherical shell.
The binding energy of a nucleus is also affected by the nonexclusive interactions of nucleons due to their having a nucleonic charge. If the nucleonic charge of a proton is taken to be 1 then statistical analysis of binding energies indicate that the nucleonic charge of a neutron is −2/3. This results in like nucleons being repelled from each other through nucleonic interaction and unlike nucleons being attracted. For the interactions of neutrons with protons in a nucleus to reduce the effect of the repulsion between like nucleons there must be a proper balance between the numbers of neutrons and protons. This balance in heavier nuclei requires about fifty percent more neutrons than protons.
The nucleons are organized in spherical shells containing at most certain numbers of nucleons. These nuclear magic numbers are explained by a simple algorithm.
Dynamically a nucleus is basically composed of concentric spherical shells which repel each other. This mutual repulsion results in a stable arrangement in which the centers of the concentric spherical shells coincide. This only occurs for repulsion forces that drop off faster than inverse distance squared.
The dynamic concentric spherical shells of the nuclear core are in most cases surrounded by halo neutrons in orbits.
Thus a nucleus is held together largely by the linkages created by the formation of spin pairs. The rings of alpha modules rotate to create the dynamic appearance of concentric spherical shells which are held together through the repulsion of the nucleonic forces. Neutron spin pairs outside of the concentric spheres are held by their attraction to the core. So all of the nuclear forces, repulsions as well as attractions, are involved in holding a nucleus together.
For a further review and critique of the conventional theory of nuclei see A statistical test of the conventional theory of the nucleus
For more on the physics of nuclei and other things see New pages.
Dedicated to K. Serventi
without whose medical and
people skills this article would
not have been written.
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