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What holds a nucleus together? |
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These are effectively forces of attraction. The binding energy associated with the formation of a spin pair consists of that due to formation itself and the change in potential energy due to any rearrangement or adjustment of the structure of the rest of the nucleus.
The forces associated with spin pair formation are exclusive, as will be explained later, and this precludes them from being identified as the nuclear strong force.
This name is inappropriate because it is not all that strong at relevant distances compared with the forces involved in spin pair formation. A more appropriate name would be nucleonic force, the force between nucleons.
Under this force like nucleons are repelled from each other and unlike ones attracted.
This force only affects interactions between protons. Neutrons have no net electrostatic charge but have a radial distribution of electrostatic involving an internal positive charge and a negative outer charge.
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.
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 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.
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 and protons that like nucleons are repelled from each other and unlike attracted..
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 best estimate of q is as −2/3 although there is some possibility that it is −3/4. In any case it is of opposite sign from that of a proton and smaller in magnitude.
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 a better visual depiction of an alpha particle.
The sharp drop off in the incremental binding energy of neutrons after 41 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). Those 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 a quantum. 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 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 rates of rotation were found to be
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 British science journal Nature in its online version of August 24, 2010 reported the following:
In 2002, Oak Ridge physicist Paul Koehler and his colleagues used the neutron beam to measure 'neutron resonances' in each of four different isotopes of platinum. The resonances are particular energies at which the neutrons are especially likely to be absorbed by the platinum nuclei. The motion of protons and neutrons inside the platinum nuclei affects the pattern of resonances. And according to random matrix theory, a mathematical theory that for decades has been crucial for calculating the behaviour of large nuclei, those motions should be chaotic.Yet, as Koehler and his colleagues report this month in Physical Review Letters (P. E. Koehler et al. Phys. Rev. Lett. 105, 072502; 2010), their analysis of the ORELA data found no sign that the nucleons in platinum were moving chaotically. By looking at the strength of the resonances, rather than just their spacing, the group rejects the applicability of random matrix theory with a 99.997% probability. Instead, the nucleons seem to move in a coordinated fashion. "There's no viable model of nuclear structure that could explain this," says Koehler.
The description of the results is compatible with the Alpha Module Model of nucler structure.
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} 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 surprising yet obvious result see Repelling spheres.
The explanation of why the Larmor proposition does not apply is that it is for point particles and the radiated energy is proportional to the square of the charge. If a charge of Q is divided into M pieces which are effectively point particles the result is M pieces each having the effect of (Q/M)², which reduces to Q²/M. If M→∞, as it would for a spatially distributed charge, the effect goes to zero. Thus the Larmor proposition is irrelevant for real charged particles because they are spatially distributed rather than being point particles.
Regression equations for the binding energies almost three thousand nuclides based upon the model presented above have coefficients of determination (R²) ranging from 0.999 to 0.99995. See Statistical Performance for the details.
In a nucleus wherever possible the nucleons are linked together through 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 sphere shell.
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 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 a 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 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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