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What holds the nucleus of an atom together? |
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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 stsbly 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 could be stable.
But even if a theory explains empirical facts that does hot mean that it is necessarily true. It only means the theory might be physically true. There might be an alternate true explanation of the empirical facts. And if a theory predicts somethings which do not occur then even it explains somethings 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 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.
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 an abbreviated version of an alternative of what holds a nucleus together. The full version is at Nucleus.
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 interactions may exceed those of the spin pair formations. But because the interaction force between like nucleons is repulsion there must be a proper proportion between the numbers of neutrons and protons for the net interaction to be an attraction. For heavier nuclei that requires there to be about fifty precent more neutrons than protons. That 150 percent ratio will be explained later.
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 the effects but the same pattern prevails throughout the dataset of nearly three thousand nuclides.
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 exactly 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.
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 increment binding energy at n=p=10 in 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 are16.9 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 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.
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..
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 the cumulative sum. 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 rotate as a vortex ring to keep the neutrons and protons (which are attracted to each other) separate. 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(I(I+1))^{½}, where
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 is then of the form
At rates of rotation of many billions of times per second a ll 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.
An alpha module 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 can be a source of stability. Here is how that works.
Without loss of generality the force between two nucleons can be represented as
where s is the separation distance between them, H is a constant, q_{1} and q_{2} a re 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 surprising yet obvious result see Repelling spheres.
When the question of what composition of neutrons and protons give the minimum energy for a nuclide with a fixed to tal number of nucleons (n+p) is analyzed the result is
That is to say, for q=−2/3 the minimum energy and most stable isotopes should be the ones for which n=(3/2)p−¼.
If the electrostatic repulsion between two protons is taken to a ratio d time the nucleonic force repulsion betwee them t he the number of neutrons which minimizes the energy of the nuclide is
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.
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 figures 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 c_{nn} and c_{pp} indicate that the force between two like nucleons is a repulsion. The positive value for c_{np} indicates the force between two unlike nucleons is an attraction.
The value of c_{pp} is not numerically less than that of c_{nn}; it is numerically larger. This cannot be, because the 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. An alternate model taking spin pairing into account explains 99.995 percent of the variation in the binding energies of the 2931 nuclides.
Let NN, NP and PP denote the numbers of neutron-neutron, neutron-proton and proton-proton spin pairs, repectively. Likewise nn, np and pp denote the numbers of neutron-neutron, neutron-proton and proton-proton interactions, repectively. According to the model, binding energy is a linear function of these variables with no constant term. Here are the results of the regression analysis.
The Regression Results | ||
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Pair | Coeff | t-Ratio |
NN | 13.83460453 | 76.96453664 |
NP | 10.31173861 | 34.68479304 |
PP | 4.310005984 | 8.327179424 |
nn | -0.192544389 | -37.34739731 |
np | 0.278037752 | 35.89755615 |
pp | -0.489577522 | -40.83013014 |
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 poitive coeficient for np confirms that the interaction of unlike nucleons is an attraction.
The regression coefficent for pp is more negative than the one for pp, as it should be, because the electrostatic repulsion between two proton is added to the repulsion between two likr nucleons.
The coefficent 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.
Let the nucleonic charge of a proton be designated as +1 and that of a neutron as q. Thus q is the nucleonic charge of the neutron relative to that of a proton. The value of q can be estimated from the values of the regression coefficients in a model in which the binding energies of nuclei are a function of the numbers of the various spin pairs and interactions. The coefficent for the number of neutron-neutron interactions should be proportion to q² whereas the coefficent for the number of neutron-proton interactions should be proportion to q. Thus the ratio of these two coefficient gives an estimate of q.
The true value is undoubtably a ratio of simple whole numbers. The best estimate of q is as −2/3. In any case it is of opposite sign from that of a proton and smaller in magnitude.
Thus in every way the regression results confirm the assertions of the Alpha Module Ring model of nuclear strucure. 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.
A nucleus is also held together by 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 nucleonic interactions in a nucleus to be a net attraction 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.
Thus a nucleus is held together partly 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 concentric spheres. So all of the nuclear forces, repulsions as well as attractions, are involved in holding a nucleus together.
For a 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 would
not have been written.
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