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A SOLUTION TO THE 80 YEARS OLD PROBLEM OF THE NUCLEAR FORCE

Authors:
  • University of Padova (Italy) & Primordial Dynamic Space Research (Verona - Italy)

Abstract

Nuclear structure theory has recently undergone a renaissance, attributed to isotopic anomalies in chemical systems at energies well below the expected ~10 MeV nuclear level and surprising ab initio super-computer calculations of nuclear properties, under the assumption that nucleons have well-defined intranuclear positions (x≦2 fm). Considering a magnetic structure of nucleons consistent with classical physics, using the Biot-Savart law with carriers "in phase", we have made connected lattice calculations of nuclear binding energies and magnetic moments, obtaining results comparable with other Copenhagen-style nuclear models.
International Journal of Applied and Advanced Scientific Research (IJAASR)
Impact Factor: 5.655, ISSN (Online): 2456 - 3080
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34
A SOLUTION TO THE 80 YEARS OLD PROBLEM OF THE
NUCLEAR FORCE
Paolo Di Sia
University of Padova, School of Engineering and Neuroscience Department, Via
Giustiniani 2, 35128 Padova, Italy
Cite This Article: Paolo Di Sia, “A Solution to the 80 Years Old Problem of the
Nuclear Force”, International Journal of Applied and Advanced Scientific Research,
Volume 3, Issue 2, Page Number 34-37, 2018.
Abstract:
Nuclear structure theory has recently undergone a renaissance, attributed to isotopic anomalies in
chemical systems at energies well below the expected ~10 MeV nuclear level and surprising ab initio super-
computer calculations of nuclear properties, under the assumption that nucleons have well-defined intranuclear
positions (x2 fm). Considering a magnetic structure of nucleons consistent with classical physics, using the
Biot-Savart law with carriers “in phase”, we have made connected lattice calculations of nuclear binding
energies and magnetic moments, obtaining results comparable with other Copenhagen-style nuclear models.
Key Words: Classical Physics, Low-Energy Nuclear Reactions, Transmutations, Biot-Savart Law, Magnetic
Attraction & Nuclear Binding Energies
1. Introduction:
Since 1989, many experimental findings have indicated isotopic anomalies in “chemical” systems at
energies well below the expected ~10 MeV nuclear level. In addition to this, since 2007 remarkable ab initio
super-computer calculations of nuclear properties have been made under the assumption that nucleons have
well-defined intranuclear positions in a nucleon lattice (x2 fm).
Such theoretical work is the so-called “Nuclear Lattice Effective Field Theory” (NLEFT). Assuming
the localization of nucleons to rather small intranuclear volumes (x < 2.0 fm), the Copenhagen interpretation
implies a very low uncertainty in position associated with high uncertainty in angular momentum, with
restrictions on what properties the particles themselves might have. As a consequence, non-Copenhagen
theoretical “unconventional” assumptions are now routinely made as a computational necessity in NLEFT.
We made unconventional lattice calculations of nuclear binding energies and magnetic moments,
finding good results that compare favorably with more complex theoretical results from nuclear models that are
consistent with the Copenhagen interpretation of quantum mechanics.
We have calculated the nuclear binding energies of all stable/near-stable isotopes, and the magnetic
moments of all stable odd-even, even-odd, and odd-odd isotopes whose magnetic moments have been
experimentally measured. By specifying the positions of nucleons within a close-packed nucleon lattice, every
nucleon is assigned a set of quantum numbers (n, l, j, m, i, s, and parity) based solely on its Cartesian
coordinates [1-4]. This description of nucleons in the lattice is isomorphic with the symmetries known from the
independent-particle model (IPM, ~shell model) of conventional nuclear structure theory.
Then we showed that LENR transmutation data on Lithium, Nickel, and Palladium isotopes can be
simulated using the nuclear lattice and the magnetic nuclear force, because of the identity between the gaseous-
phase IPM and the fcc lattice [5,6].
We and others have, in fact, developed the fcc lattice of nucleons as a model of nuclear structure and
shown that its numerical results concerning nuclear size, shape, density, etc. well compare with the 30+ other
models of nuclear structure developed throughout the 20th century [5].
2. Technical Details:
The Biot-Savart law allows one to calculate the magnetic field generated by electric currents, obtaining
the mutual force between coils as due to the contribution of infinitesimal length elements, and ignoring any
phase relation between the currents. If phase relations are brought into consideration, the situation radically
changes, bringing to a modified potential energy.
We considered two circular coils (1) and (2), in which circulate the currents i1 and i2, respectively. Let R
be the common radius of the coils, placed within a canonical orthogonal Cartesian coordinate system (xyz). The
simplest configurations are: (i) the coils in two parallel planes; (ii) the coils in the same plane.
(i) Coil 1 lies in the plane (xz), while coil 2 is in a parallel plane at a distance y (Figure 1).
Using Biot-Savart and Laplace laws, the force perceived at coil 2 under the action of the field generated
from coil 1 is given by:
1
3
12
121
2
2
210
12 4CC r
rld
ld
π
iiμ
=F
. (1)
Let P1=(x1, 0, z1) be a generic point of the coil 1 and P2=(x2, y, z2) a generic point of the coil 2, Eq. (1) can be
rewritten in the form:
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12
3
12
12
1
1 2
3
12
122
210
12 4r
r
ldld
ld
r
rld
π
iiμ
=F
C C
 
(2)
Figure 1: The case of coils in parallel planes
and its first part is zero, being the integral of a gradient extended to a closed line. With further algebra,
considering cylindrical coordinates and the binomial series of the denominator, up to the first order, Eq. (2)
becomes:
j
y
Rπ
π
iiμ
=F
4
42
210
12 6
4
(3)
with intensity:
 
4
4
210
12 2
3
y
Rπiiμ
=F
. (4)
In the hypothesis that the two currents are in phase, Eq. (2) gives:
j
y
Rπiiμ
=F ph
2
2
210
12
, (5)
 
2
2
210
12 y
Rπiiμ
=F ph
, (6)
implying:
 
12
2
12 3
2F
R
y
=F ph
. (7)
(ii) The two coils are placed in the same plane, for example (xz) (Figure 2).
Figure 2: The case of coils in the same plane
If P1 is the generic point of coil 1, P2 that of coil 2, and d is the distance between the two centers, using the same
procedure as the previous one, we get:
 
4
4
210
12 3
d
Rπiiμ
=F
, (8)
 
2
2
210
12 d
Rπiiμ
=F ph
, (9)
International Journal of Applied and Advanced Scientific Research (IJAASR)
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. (10)
The corresponding energies are respectively:
(i)
3210
2yπ
μμμ
=E
, (11)
yRπ
μμμ
=Eph 2210
, (12)
(ii)
3210 xπ
μμμ
=E
, (13)
xRπ
μμμ
=Eph 2210
, (14)
with in evidence the magnetic moments [4].
3. Results:
As an example, considering the case of two nucleons placed at a distance y = 2 fm and the nucleon
radius R = 0.5 fm, energies (11, 12) are respectively: E = 3.97 KeV; Eph = 0.127 MeV.
It is possible also to introduce an appropriate exponential phase factor, of the form “exp(-λr)”; Eqs (12,14) can
be rewritten as:
rλ
ph e
yrπ
μμμ
=(r)E
2210
, (15)
rλ
ph e
xrπ
μμμ
=(r)E
2210
. (16)
In Table 1 we have summarized the values of Eph for different combinations of values of x, y and R.
Case (i)
Case (ii)
y (fm)
R (fm)
Eph (MeV)
x (fm)
R (fm)
Eph (MeV)
0.5
0.2
3.18
2
0.1
3.19
1
0.2
1.59
2
0.2
0.80
1.5
0.2
1.06
2
0.3
0.35
Table 1: Values of Eph corresponding to different R of coils and distances x and y.
We have calculated the tuning factor values for fixed R, binding energies and distance among coils in the case
(i) (Table 2).
Case (i)
y (fm)
R (fm)
Binding energy (MeV)
Tuning factor (m-1)
1.0
0.2
3.0
3.18 · 1015
1.0
0.4
2.0
4.04 · 1015
1.0
0.6
1.0
2.89 · 1015
1.0
0.8
0.5
2.02 · 1015
Table 2: Tuning factor values for fixed R, binding energies and distance among coils.
An attractive magnetic energy obtained from the classical Biot-Savart interaction without consideration
of the phase relationship (4 keV) would be only a small contribution to nuclear binding energies, but 0.136 MeV
between nearest neighbors is already a significant percentage of the mean nuclear binding in either the context
of the LDM or the fcc lattice. Specifically, default structures for 90Zr, 200Hg, and 238U lattice nuclei have,
respectively, 370, 865, and 1036 nearest-neighbor interactions, corresponding to mean energies of 2.1, 1.8, and
1.7 MeV (per nearest-neighbor “bond”).
Considering now the case (i) (analogous calculations can be made for case (ii)), making the assumption
about the nucleon dipole: R = y = 0.2336 fm,
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It follows that the magnetic interaction between two protons, two neutrons, or one proton and one
neutron is 5 MeV; the short-range nuclear force with various spin and isospin combinations can explain the
basic trend in nuclear binding energies.
With the adjustement above, we can conclude that the magnetic nuclear force is sufficient to explain
nuclear binding. Binding effects (5MeV) among neighboring nucleons are enough to achieve binding for 273
isotopes built in the fcc software, implying that a magnetic nuclear force is consistent with the lattice model [6].
4. Conclusions:
Funding of LENR research should focus on the basic experimental science of isotopic transmutation
effects, regardless of their possible technological utility. Once the empirical data are unambiguous, ab initio
computational simulations should become possible.
We have developed the fcc lattice of nucleons as a model of nuclear structure, showing that its
numerical results concerning nuclear size, shape, density, etc. compare well with the 30+ other models of
nuclear structure developed throughout the 20th century [5].
To date, nuclear “modeling” contributes little or nothing to the fundamental unresolved issue of the
nature of the nuclear force holding nuclei together. In the present work, we addressed the question of the nuclear
force acting between nucleons in a close-packed nuclear lattice.
The validity of results depends crucially on the three variables R, x, y. A center-to-center internucleon
distance of approximately 2.0 fm gives a core nuclear density of 0.17 nucleons/fm3 [5], nuclear core density
normally cited in the textbooks since the electron-scattering experiments of Hofstadter in the 1950s (somewhat
larger values (0.13~0.16) for the “mean” density (core plus skin region) are also cited in the literature).
Similarly, the nucleon RMS radius for both protons and neutrons is known experimentally to be ~0.88
fm [7,8]. Nevertheless, the nuclear dipole that results in the magnetic moments of +2.79 and -1.91μ, respectively,
might have dimensions somewhat different from the matter distribution within the nucleon, so that calculations
of magnetic force effects over a broad range of dipole sizes are relevant.
The followed way, with the novelty of the particular use of the Biot-Savart law, is therefore a
possible solution to the 80 years old problem of the nuclear force.
References:
1. Di Sia P. and Dallacasa V., Quantum percolation and transport properties in high-Tc superconductors”,
International Journal of Modern Physics B, Volume 14, Page Number 3012-3019, 2000.
2. Di Sia P. and Dallacasa V., Origine Magnetica Della Forza Nucleare: Un Calcolo Interessante”,
Periodico di Matematiche, Volume VIII, Issue 1(2), Page number 1-8, 2001.
3. Di Sia P., Magnetic force as source of electron attraction: A classical model and applications”, Journal
of Mechatronics, Volume 1, Issue 1, Page Number 48-50, 2013.
4. Dallacasa V., Di Sia P., and Cook N.D., The magnetic force between nucleons. In: Models of the
Atomic Nucleus”, Springer, Heidelberg, Page Number 217-220, 2010.
5. Cook N.D., and Dallacasa V., Face-centered solid-phase theory of the nucleus”, Physical Review C,
Volume 35, Page Number 1883-1890, 1987.
6. Cook N.D., Models of the Atomic Nucleus, Springer, Heidelberg, 2010.
7. Hofstadter R., “Electron Scattering and Nuclear Structure”, Reviews of Modern Physics, Volume 28,
Page Number 214-253, 1956.
8. Sick I., “Nucleon radii”, Progress in Particle and Nuclear Physics, Volume 55, Page Number 440-450,
2005.
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Origine Magnetica Della Forza Nucleare: Un Calcolo Interessante
  • Di Sia
  • P Dallacasa
Di Sia P. and Dallacasa V., "Origine Magnetica Della Forza Nucleare: Un Calcolo Interessante", Periodico di Matematiche, Volume VIII, Issue 1(2), Page number 1-8, 2001.
The magnetic force between nucleons
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  • Di Sia
  • P Cook
Dallacasa V., Di Sia P., and Cook N.D., "The magnetic force between nucleons. In: Models of the Atomic Nucleus", Springer, Heidelberg, Page Number 217-220, 2010.