Hundred years after the conjecture of the British astronomer
Eddington that the sun is powered by nuclear fusion of hydrogen, new physics theory may help make energy harvesting by nuclear fusion soon a reality. Researchers as well as investors funding fusion megaprojects are asked to deal with new relativistic corrections for mass and energy proposed by Suleiman in his Information Relativity Theory (
IRT). These corrections were calculated in this contribution. It will help to decide whether a venture will be successful and to save big investments when in doubt. The assumed optimal kinetic energy for controlled nuclear fusion must be corrected to a somewhat higher level. At very high kinetic energy in the upper GeV range, it remains not enough baryonic mass to be transformed in energy. The fusion probability faded out to zero already at the golden limit of the recession speed of
between target nucleon and projectile nucleon. Cold nuclear fusion, if ever possible, is recommended for protons rather than deuterons at highest experimental possible temperatures around 1000 (K) and needs fine-tuned kinetic nucleon energy. It would be also of interest whether a golden ratio based nuclear fuel confinement chamber could be beneficial. In this connection, also cold nuclear fusion setups should be discussed. Nature is governed by the golden ratio and criticality of physical systems influenced by it, and nuclear physics is not an exception. Computer simulations of the underlying controlled nuclear fusion processes should gain profit from
IRT corrected starting information and may tackle anew possible low energy nuclear transmutations considering the wave-like dark components of matter and energy.
Controlled Thermonuclear Fusion <i>IRT</i> Theory Relativistic Mass Correction Low Energy Nuclear Reaction Golden Ratio Golden Limit of Nuclear Fusion1. Introduction
Big ideas often suffer from teething problems caused by the indolence with which just great scientists insist on what they have learnt and worked out. This contribution should serve to support ideas of a new physics and recommends application of these ideas to big projects of the mankind such as thermonuclear fusion research and its technical realization. Success in application of the mentioned theory would also promote the theory itself.
Relativistic mass respectively energy corrections are very important when dealing with nuclear fusion and the dynamics of moving particles, where light atoms combine to larger ones accompanied by conversion of some mass into huge energy. A new theory of Ramzi Suleiman, named Information Relativity theory (IRT), has fixed an overlooked flaw in Newton’s theory that leads to changes in relativistic corrections [1]. Many formal explanations or physical constructs that bothered long time the world of physics are overcome by the new exciting theory. Recently, the present author has performed relativistic mass corrections on the gyromagnetic factor of the electron [2] that also would change the charge of the electron a little bit [3]. It is connected with the important question whether we need at all the construct of quantization [1] [4]. Ever the golden ratio as the dominate number of nature is involved in such considerations, and so the IRT theory has to do with this fundamental number as will be shown below. More than this, the genetic code of the DNA is based on a golden mean hierarchy [5]. In this way, it is stringent to deal with the new theory in every area of physics (and of life) knowing that even in fusion science and technology several disciplines of physics are combined. Especially attention must be paid to the wave-like dark component of mass energy density respectively energy density (see Figure 1). It is not the intention of the author to keep the reader away from studying the IRT theory in detail for himself. Therefore, only a sparse introduction was given. A concise overview on controlled nuclear fusion ideas and different technologies was recently summarized by Clynes [6] and should not be explained in detail. This isn’t a review article with an almost complete reference list, but a contribution motivating others to think outside the box. In the long run, we have to move away from the large energy projects towards more decentralized producers and distributors for the benefit of our environment.
During the preparation of this contribution, the successful ignition of a controlled thermonuclear fusion reaction maintaining the reaction for about 10 seconds was reported from the staff of the CFETR reactor in Hefei, China.
Furthermore, a recent contribution about the creation of elements from nuclear transmutation in Earth lower mantel is highly recommended [7].
2. New Relativistic Mass and Energy Corrections
Suleiman has set out to correct physical processes for time displacements between observer and moving bodies. Transformations for time duration, length, mass density as well as energy density were applied to a whole bunch of physical phenomena, which could be explained now in simple and beautiful clarity [1]. Indeed, this theory is an outstanding example for the beauty of the simple.
According to the IRT theory mass is transformed in dependence of the recession velocity of a moving body as [1]
m m 0 = 1 − β 1 + β . (1)
where β = v c is the recession velocity.
For the matter energy density eM of a moving body with velocity v and rest density ρo on yields
e M = 1 2 ρ v 2 = 1 2 ρ o c 2 1 − β 1 + β β 2 = e o 1 − β 1 + β β 2 , (2)
where e o = 1 2 ρ o c 2 .
The matter energy density reached its maximum at a recession velocity of β=
φ, where φ = 5 − 1 2 = 0.6180339887 is the golden mean. Replacement of this special value in Equation (2) gives
( e M ) max = e o 1 − φ 1 + φ φ 2 = e o φ 5 = e o ⋅ 0.09016994 ⋯ (3)
Remembering, φ5 represents Hardy’s maximum quantum entanglement probability [8]. This result was commented by the present author in a publication before mentioned [3].
Suleiman aptly characterized the behavior at the critical point βcr = φ as phase criticality at cosmic scale [1] [9]. However, φ5 scaling seems to be a more general quality of phase transitions [10].
The dark matter density transforms as
e D M e o = 2 β 3 1 + β (4)
If one calculates the energy density amounts (ratios) of matter and dark matter contributions at this point, one gets again a golden mean representation like Russian dolls nesting
The difference gives 2 φ 2 − φ 3 ≈ ( 2 − 1 ) 4 π , where 2 − 1 = 0.414213 ⋯ is the
silver mean. The case, where according to the Information Relativity theory of Suleiman [1] [9] just at the recession velocity of ß = 1/3 the matter and the dark matter density will be the same, delivers for the density amount the reciprocal of Lucas number L6 = 18 [2].
The relative energy density components are depicted in Figure 1 versus the recession velocity ß. Also the golden mean hierarchy was displayed.
Instead of the recession velocity it may be useful to choose another variable,
the redshift z = β 1 − β . Applied matter density transformations in terms of the redshift yield [1]
ρ M ρ 0 = 1 2 z + 1 (6)
ρ D M ρ 0 = 2 z 2 z + 1 (7)
As was demonstrated by Suleiman, an increase of the redshift z caused the matter density of the travelling corpuscular particle successively to diminish, while energy is transformed into the wave-like dark component and vice versa [1]. The complementary duality between matter density and dark matter density versus redshift was illustrated in Figure 2.
Both figures describe precisely and elegantly what needs to be considered when dealing with colliding particles of high speed during nuclear fusion, “where each moving particle will be permeated by the dark matter halo of the other one making the two physically entangled” [1].
However, many other scientist have studied the relativity aspect, for instance in the quantum world [11], or in space-time theories preferring the Selleri transformation [12].
3. Nuclear Fusion Reactions Reexamined
The first researcher who experimentally had demonstrated nuclear fusion was the Australian physicist Oliphant at Cambridge laboratory in 1934 [13]. The investigated nuclear fusion reactions are the basis of research and development up to now fusing nuclei of hydrogen isotopes deuterium and tritium respectively deuterium and helium-3 according to
D 1 2 + T 1 3 → H 2 4 e + n 0 1 + 17.6 MeV (8)
D 1 2 + H 2 3 e → H 2 4 e + H 1 1 + 18.4 MeV (9)
The stability of atomic nuclei can be found in textbooks of nuclear physics, where the average binding energy per nucleon is depicted versus the atomic mass number (Appendix FigureA1).
The energy gain according to the basic exothermic fusion reaction (8), for instance, can be calculated from the difference in the rest energy of the constituents given in MeV
whereas deuterium is abundantly available in seawater coming from comets that have reached the earth, for tritium there exists no sizable natural source due to its radioactive decomposition within a half-life time of about 12.3 years.
Tritium must therefore be produced by breeding from lithium [14], for instance by its reaction with neutrons produced during the fusion processes (tritium self-sufficiency).
In the near future, mankind will be able to extract by extraterrestrial mining H 2 3 e from the sunlit areas of the moon as a third-generation fusion fuel. This stable isotope stems from solar wind and has been stored in moon’s uppermost rock layers. A possible fusion reaction between these nucleons is
H 2 3 e + H 2 3 e → H 2 4 e + 2 p 1 1 + + 12.86 M e V (10)
An assumed cold nuclear fusion reaction, for instance, starting with two deuterons confined in an interstitial of the palladium metal lattice, would result in
D 1 2 + D 1 2 → H 2 4 e + 23.85 M e V (11)
The binding energy of the nucleons in the deuteron is relatively small and about 2.2 MeV. The nucleons (proton and neutron) are relatively far apart from each other by about 4 fm. Therefore, also a cascade of the following energetically more favorable reactions should be considered with three deuterons involved temporarily
D 1 2 + D 1 2 → H 2 3 e + n 0 1 + 3.27 M e V (12)
D 1 2 + H 2 3 e → H 2 4 e + p 1 1 + + 18.87 M e V (13)
n 0 1 + p 1 1 + = D 1 2 + 1.71 M e V (14)
Byproducts such as neutrons or sometimes tritium when excess of deuterium reacts with neutrons have been reported as observed experimentally [15]. A summing up of these reactions delivers again reaction (11). One can indeed place three neighboring deuterons in an octahedral interstitial of palladium besides nearby tetrahedral ones. In contrast to bulk PdDx, where only octahedral sites are occupied, in the subsurface region of nanometer-sized material of PdHx, also 30 percent of tetrahedral sites are occupied besides 70 percent of octahedral ones [16]. The same occupation scheme may be assumed for isomorphic PdDx. For cold fusion experiments, a high deuterium loading near the ratio Pd/D = 1 is of interest [17].
Now we want to set up the equation of motion for the colliding particles keeping a fuel of heavy water and lithium in the plasma state and accelerate deuterons towards the target. At very high particle speed we have to consider duality between both particles and waves (dark matter surrounding) using an elastic atomic collisions model as well as an elastic spring model associating the atomic collision model with particles and the spring model with waves. With respect to the speed probability distribution during a two-body collision between particles one can apply the Lattice Boltzmann Method (LBM) as a combination of the Boltzmann equation and the Maxwell-Boltzmann distribution function [18].
The tunneling probability PG of two nuclear particles to overcome the Coulomb barrier was derived by Gamov [19] being
P G ( E ) = exp ( − E G E ) (15)
The Gamov energy EG resulted in
E G = ( π α Z A Z B ) 2 2 μ c 2 (16)
and the kinetic energy is
E = 1 2 μ v 2 (17)
where α is Sommerfeld’s fine-structure constant, μ is the reduced mass, v is the speed between the two particles, c is the speed of light, and ZA respectively ZB are the proton numbers of the nucleons A respectively B. Importantly, the fine-structure constant α in relation (16) has to be changed marginally [2] [3].
Using the IRT transformation for mass according to relation (1), the reduced mass μ for a reaction involving two equal nuclear particles of rest mass m0 gets
μ = m 0 2 ⋅ 1 − β 1 + β (18)
Finally one gets together with Equation (2)
P G ( E ) = exp ( − 2 π α Z A Z B / β ) (19)
Following now Cooley [20] and considering that the particle’s speed v is distributed by a Maxwell-Boltzmann function one can finally verify the probability (reactivity) for a fusion reaction to occur with two opposing energy terms under the exponential function
〈 σ v 〉 = ( 8 π μ ) 1 / 2 ⋅ S 0 ( k T ) 3 2 ⋅ ∫ 0 ∞ exp [ − ( E k T + E G / E ) ] d E (20)
where σ is the nuclear cross section expressing the likelihood of interaction between projectile particle and target
σ A B ( E ) = S 0 E P G ( E ) (21)
The Maxwell averaged fusion cross section 〈 σ v 〉 or reactivity has the dimension of cm2·cm/s = cm3/s with a constant S0 (cm2keV). It should be noticed that the upper integration limit of ∞ in relation (20) is likely to be problematic because the speed of light is considered as limited. Therefore, Equation (23) in the following explanation represents a more concise form using β= v/c instead of v.
The maximum of the non-relativistic integrant of relation (20) can be found by setting the first derivative equal to zero [20]
E max = ( k T 2 ) 2 / 3 E G 1 / 3 (22)
giving
β max = 2 E max e D = 4 3 ⋅ ( k T ) 1 3 E G 1 6 e D = 4 3 ⋅ ( k T e D ) 1 / 3 ⋅ ( E G e D ) 1 / 6 (23)
Inserting relation (22) in Equation (20) results in
〈 σ v 〉 max = ( 8 π μ ) 1 / 2 ⋅ S 0 ( k T ) 3 / 2 ⋅ exp ( − 3 ( E G 4 k T ) 1 / 3 ) (24)
Approximating Equation (20) by a Gaussian, one can calculate the integral width wi yielding
w i = 2 π 3 k T ( E G 4 k T ) 1 / 6 (25)
and then the integral content
〈 σ v 〉 max ⋅ w i = 4 2 3 S 0 k T ⋅ μ ⋅ ( E G 4 k T ) 1 / 6 ⋅ exp ( − 3 ( E G 4 k T ) 1 / 3 ) (26)
The approximation by a Gaussian is compared in Figure 3 for the case of proton fusion applying kT = 1 keV. A better adaption between both curves would result when the maximum of the Gaussian would be shifted to somewhat higher values > 4.98 keV (for example see right side of Figure 3).
In TableA1 of the Appendix βmax values for selected kT have been compared for the non-relativistic case and the IRT corrected case. In Figure 4, the tunneling probabilities and Emax values at kT = 1 keV for proton and deuteron nuclei have been compared. Notice the significantly higher proton probability increased by a factor of about 68.5 compared to deuterium.
The representation of the fusion probability in relation to the kinetic energy can be done also in logarithmic form as performed in Figure 4 because β ∝ E k i n to compare both ß representation and kinetic energy one. If one would translate the given result in the parlance of cold fusion it seems to be evident that proton fusion should be happen much more likely than deuterium fusion, if possible at all.
The Information Relativity theoretical solution based on Equation (20) yielded the fusion probability in relation to the recession velocity β indirectly involving the golden mean, where 〈 σ β 〉 has the dimension of (cm2) and eD = 1875.61294257(57) (MeV) represents the rest energy of the deuteron [21] [22]. Integration is now taken within the changed limits from β = 0 to 1
〈 σ β 〉 = 1 c ( 8 ⋅ 2 ( 1 + β ) π m D ( 1 − β ) ) 1 / 2 ⋅ S 0 ( k T ) 3 2 ⋅ ∫ 0 1 exp ( − e D 4 k T 1 − β 1 + β β 2 − 2 π α Z A Z B β ) ⋅ e D ⋅ β ( 1 − β − β 2 ) 2 ( 1 + β ) 2 d β = 2 π ⋅ ( e D k T ) 1 / 2 S 0 k T ⋅ ∫ 0 1 β ( 1 − β − β 2 ) ( 1 + β ) 1 − β 2 ⋅ exp ( − e D 4 k T 1 − β 1 + β β 2 − 2 π α Z A Z B β ) d β (27)
The first integrand term being solely a function of ß has a maximum of f ( β ) = 0.147847 ≈ φ 4 = 0.145898 ⋯ at a recession speed of β max = 0.314026 ≈ 0.30901699 = φ / 2 (see Appendix). It means that the maximum relative particle speed should be about a third of that of light. When following the blue outlined curve in Figure 5, this parabola-like function has zeros at β = 0 respectively β = φ , but is not fully symmetrical, when mirrored at a vertical through β = φ / 2 . Because we are dealing with a polynomial of third degree, a third physically not meaningful zero is found at β = − φ − 1 . This pre-factor is responsible for fading out of the tunneling probability of nuclear fusion to zero at the golden mean limit β = φ , just at the recession speed, where the relative energy density of baryonic matter reaches its maximum. Once more this golden limit can be assigned as a phase critical point transforming matter and energy [1] [2].
The tunneling probability was calculated for the case of colliding deuterons depending of the recession speed as well as in dependence to the kinetic energy. The result was depicted in Figure 6 for the case of kT = 1 keV. The Emax = 6.61 keV value is slightly larger than that for the non-relativistic case with Emax = 6.27 keV. Also the half-width of the curve is pronounced greater using the IRT approach.
The β representation is better suited to be approximated by a Gaussian. One can further ask at which kT the maximal probability is realized. This can be done by
calculating d 〈 σ β 〉 d ( k T ) = 0 . Its solution yielded
( k T ) max = 1 6 e D 1 − β 1 + β β 2 (28)
for ß = ßmax. Inserting this result into Equation (27) leads to the following simplified expression for maximum kT
( k T ) max : 〈 σ β 〉 max = 12 ⋅ 6 π ⋅ exp ( − 3 2 ) S 0 e D ⋅ 1 − β − β 2 β 2 ( 1 − β ) 2 exp ( − 2 π α Z A Z B β ) (29)
Figure 7 depicts a set of curves for different kT values indicating the overall maximum for kT = 160.4 keV at ßmax = 0.023188 (Ekin= 240.7 keV) according to Equation (26). Covering all maxima with an envelope shows that the limiting value is ß = 1/3, where the probability at very high kT faded out to become completely zero. According to IRT at this special recession speed the amounts of matter density and dark matter density will be equal (Figure 1) [1] [2].
We will not write down the complex eighth degree polynomial resulting as first derivative of Equation (27) to determine the zeros and maxima of that function, but instead use an approximation by modifying the simpler non-relativistic result given in Equation (23). Whereas the low-kT data follow the non-relativistic result, the high-kT data can be approximated by a relation having the golden mean φ in its pre-factor and exponent
β max ( k T ) ≈ 4 3 φ 3 ⋅ ( k T e D ) φ / 2 ⋅ ( E G e D ) φ / 2 2 ≈ 4 3 2 φ ⋅ ( k T e D ) φ / 2 (30)
Another simple approximation valid over a larger kT range is given by the relation
β max ( k T ) ≈ exp ( 3 4 2 ⋅ ( k T e D ) 2 − 1 ) − 0.9993 (31)
The different relationships were double-logarithmically documented by Figure 8.
However, the reader may find a more appropriate approximation for relation (27) using Taylor series development [23] [24].
In addition, on can verify that at maximum kT it holds
E k i n k T max = 3 2 (32)
as well as [25]
β max ≈ π α Z A Z B = 0.022935 Z A Z B ≈ 6 α 5 φ 2 Z A Z B (33)
Again it should be pointed out that according to the IRT theory the charge of the electron respectively Sommerfeld’s fine-structure constant have to be marginally but in view of the given high accuracy still noticeably corrected [2].
In this contribution arbitrary unit for the fusion probability were used because some parameters such as the slightly energy-dependent (astrophysical) parameter S0 are not sufficiently correct known and should be verified by a modified model calculation (see Equation (21) and (23)).
A more quantitative prediction of the released power ε per unit mass and per second (power density) follows a non-relativistic approach already given by Cooley [20], but in an altered form and exemplified for protons
ϵ = 2 8 / 3 3 ⋅ c ⋅ ρ ⋅ X A X B m H 2 A A A B e P Q S 0 E G 1 / 6 ( k T ) 2 / 3 exp [ − 3 ( E G 4 k T ) 1 / 3 ] (34)
where XA respectively XB are the nucleon abundances, ρ is the mass density, Ai are the atomic numbers, c is the speed of light, and S 0 = 4 × 10 − 46 cm 2 ⋅ keV . The proton fusion reaction [18]
p + p → d + e + + υ e (35)
delivers an energy of
Q = 26.74 MeV − 0.52 MeV = 26.2 MeV (36)
when identical nucleons collide, X A = X B = 0.5 , the collision rate has in addition to be divided by 2. For conditions at the core of the sun (kT = 1 (keV), ρ = 150 g·cm−3) one can use the small energy unit of 1 (erg) = 10−7 (J) and calculate the non-relativistic power density to be about
ϵ suncore ≈ 11.8 erg ⋅ g − 1 ⋅ s − 1 (37)
Turning now to the relativistic relation (24) and approximate the Gaussian-like fusion probability integral by the area given by multiplication of the height 〈 σ β 〉 max with the integral width w i = π 2 ln ( 2 ) ⋅ w F W giving [26]
〈 σ β 〉 / S 0 = π 2 ln ( 2 ) 〈 σ β 〉 max ⋅ w F W (38)
Again, with proton abundances X A = X B = 0.5 and atomic number A A = A B = 1 one finally yields for the relativistic power density
ε * = 1 2 π 2 ln ( 2 ) c ⋅ ρ ⋅ X A X B m H 2 A A A B e P ⋅ Q ⋅ S 0 ⋅ 〈 σ β 〉 max ⋅ w F W (39)
For kT = 1 keV one can estimate 〈 σ β 〉 max = 0.001748 respectively w F W = 0.002039 giving
ε suncore * ≈ 13.2 erg ⋅ g − 1 ⋅ s − 1 (40)
Specialists in the field of controlled thermo-nuclear fusion can use own experimental values such as the chosen kT and mass density of the nuclear fuel to verify result obtained with relation (39) using the different concepts of low fuel density combined with extreme temperature contrary to high fuel density and less temperature.
If one briefly summarizes the main results, the following can be noted. There exists a natural golden limit, where nuclear fusion stops down to zero probability. This happened at the recession speed between nucleons of β = φ. The probability distribution curve as a function of the nucleon speed is pronounced broader than in the case of the non-relativistic approach. The maximum probability is found at markedly enhanced speed respectively kinetic energy (see right side of
Figure 5). In the GeV range of kT the maximum probability is found at β = 1 3 ,
straight at the recession speed at which baryonic matter energy and dark matter energy will become equal. It simply means the more moving matter is transformed to its wave-like dark component, the less is available to be transformed in released energy by fusion of nucleons.
In the range of high particle speed hot fusion takes place in the halo of surrounding dark matter energy emerging from moving particles. It should be considered in an adequate way by varying Equation (23). A related correction is in progress. Recently, such matter—dark matter coupling was excellently described for the case of disk galaxies by Suleiman [27]. Besides the action or involvement of dark matter in fusion physics also the validity of the Gamov approach [19] should be questioned critically.
Last but not least the probability for cold fusion remains an open question. Some IRT based calculations are summarized in the following chapter.
4. Cold Nuclear Fusion or Transmutation
For the cold fusion at low kT a very sharp energy distribution around the cross section probability maximum can be expected. A cold fusion experiment must optimally meet these requirements for maximum kinetic energy in order to be successful at all. In the following calculated fusion probability curves for deuterium respectively proton nuclei are shown. In Figure 8 the probability for deuteron fusion within the low-temperature range between 103 K and 105 K was presented depicting their small Gaussian-like distribution curves (T instead of kT was used). However, a further reduction of temperature failed to process the needed number of decimals with a home PC.
For the proton fusion the probability is markedly enhanced compared to deuterium fusion, and so the 500 K values could be calculated, too. If ever cold fusion is detectable, it should be realized using hydrogen at the experimentally highest possible temperature, for instance choosing about 1000 K = 727˚C, for which the result is depicted in Figure 9. For T = 1000 K one yields β max = 0.20375 corresponding to v max max ≈ 60 ( km / s ) as the optimum speed with which the nuclei should be moved towards each other. The full width at half maximum is about Δ β max = 1.84 × 10 − 5 (Figure 9). See also TableA1 in the Appendix.
The draft of a more quantitative picture is currently under progress. In advance some ideas will be presented in the next chapter (Figure 10).
5. Tesla’s Energetic Pyramid Conjecture and Beyond
In connection with the Great Pyramid, mention may be made of the vision of the great inventor Nicola Tesla to harvest free energy from what we now characterize as dark components of matter respectively energy. Intuitive knowledge of great thinkers shouldn’t be confused with obsession. The reader can find more information about this discussion in the references [28] and [29]. Be that as it may, the golden mean based architecture of the Great Pyramid can serve as a proposal for a quite asymmetric nuclear fuel confinement chamber. In this connection the reader may also study the in-ellipsoid approach for less-symmetric solids such as the Great Pyramid to stimulate new ideas [30]. Figure 11 shows a sketch of two opposed deuterium nucleons in an asymmetric-pyramidal micro-cage waiting for cold fusion to helium according to reaction (11), where the Great Pyramid served as a model.
Cold fusion experiments were described with atomic deuterium occluded in the cubic face-centered palladium lattice having 4 octahedral interstitials besides 8 tetrahedral ones in the unit-cell. First evidenced by Fleischmann and Pons [31], the reader may follow an explanation of lattice assisted low-energy nuclear fusion recently given by Garai [32], where two protons respectively deuterons are confined in the octahedral interstitials of the cubic Pd lattice. Theoretical considerations of Gurbich [33] indicate the problem of cold fusion.
Now we turn back to a pyramidal environment in contrast to the octahedral hole within the Pd lattice. If one would deposit Pd on the surface of a thin mica sheet and bend it cylindrically, the half-octahedral interstitial would be deformed towards the case described in Figure 9.
Possibly the occluded deuterons would be asymmetrically positioned at the
interface between mica and palladium as illustrated in Figure 11. It is suggested that a fusion reaction could be enhanced by such lattice deformation. However, we are still seeking for a more suitable host material. For micro-scale reactors one could deposit palladium on an atomic flat and highly reflecting (111) gold platelet [34] where the gold substrate serves also as a diffusion barrier for hydrogen or deuterium [35]. The PdHx respectively PdDx contact layer is strained due to mismatch between the lattice parameters of gold and PdHx (PdDx) and should be able to host even more hydrogen or deuterium in the widened lattice. The upload of deuterium into the Pd lattice was estimated to be about 1.2 × 1023 cm−3 [36]. Then one can calculate the mass density yielding ρ ≈ 0.8 g ⋅ cm − 3 .
The movement of the two nucleons against each other starting a fusion reaction can be supported by chaotic excessive resonance caused by focused ultrasonic energy loading of the confinement lattice bouncing back the nucleons from the walls with extremely high speed overcoming their Coulomb barriers [37] [38] [39] [40]. Fine-tuning of the applied cavitation energy according to the fusion probability maxima given in Chapter 4 is recommended. It is assumed that golden ratio governed geometry of the confinement can support chaotic bouncing of the nucleons against each other with a wanted distribution of collision angles. Because the particles moving with high speed are stretched in direction of the moving direction it makes a difference at which angle they collide. The relativistic longitudinal-transversal mass alteration already proposed in 1899 by Lorentz [41] should be recast in an IRT compatible version then applicable to the controlled nuclear fusion, too.
A Lattice-Boltzmann solution is currently in progress and should be published soon. Besides the lattice-assisted part of possible cold fusion the catalyzed fusion by neutral pions π0 should be considered. Fukuhara postulated that such bosons are responsible for all low-energy nuclear interactions [42].
If one speculates about the reality of a cold fusion of atomic nuclei it may appropriate remembering the history of diamond synthesis, where HPHT synthesis has been successfully supplemented by cost-efficient CVD synthesis under sub-ambient conditions. In this sense the same success is desirable for cold fusion currently under progress.
6. Keeping a Watch on Nuclear Fuel Resources on Earth and Moon
This remark takes a look into the future. What would extraterrestrial powers do on earth, fight against human beings or only harvest rough materials such as deuterium from the ocean or H 2 3 e from the sunlit areas of the moon to fuel their spaceships, as yet unobserved but friendly? May be they already manage it. If so, we should start an across area monitoring of the sea water deuterium content to detect significant local variation before the ocean flow equalizes any left local difference. By the way, we had the technique to do this immediately, gaining also profit for other scientific areas besides precaution for the future. With respect to the deuterium extraction from seawater we can assume that an extraterrestrial power surely knows processes like the archetype Girdler-sulfide process (GS process) [43], a two-temperature isotopic exchange process, or even seawater electrolysis, or a more sophisticated rapid gas effusion process.
7. Conclusion
There is a need of applying new physics to great technological challenges of our time such as providing the world’s sustainable energy needs through controlled nuclear fusion. Large investments in this technique could soon bear fruit if new relativistic corrections for mass and energy according to the IRT theory would be considered in computer models as well as in the equipment. Possible changes in the equipment are a task of experts, but the recalculation of energy balances of nuclear fusion reactions can also be up to cooperating readers. When exemplarily calculating the fusion probability for deuterons or protons with Suleiman’s IRT approach, one can confirm markedly different results in comparison to the non-relativistic case. The maxima of the probability shift to somewhat higher energy and the probability versus energy respectively recession speed curves are markedly broader than compared to non-relativistic calculations. Consequently, the IRT corrected probability results recommend to correct the existing assumptions to somewhat higher values. There exists a golden limit where already at a recession speed of β = φ the fusion probability is zero. At extreme temperatures (kT > 1 GeV), the maximum of the fusion probability is found at β = 1/3, where baryonic matter energy and dark matter energy becomes equal. A more quantitative calculation depends on the availability of some values that should be accessed by an elaborated model calculation. The new approach should inspire researchers and big investors in the fusion technology proceed in this direction of new physics, touching besides physics the philosophical foundations of our world. The presented ideas may be extended to more cosmological questions such as avoiding the “big bang” singularity hypothesis respectively the origin of the speed of light.
Conflicts of Interest
The author declares no conflict of interests regarding the publication of this paper.
Cite this paper
Otto, H.H. (2022) Nuclear Fusion Research and Development Need New Relativistic Mass and Energy Corrections Given by the Information Relativity Theory. Journal of Applied Mathematics and Physics, 10, 1813-1836. https://doi.org/10.4236/jamp.2022.105125
Appendix
Stability of Atomic Nuclei (FigureA1)
Maximum of the Function f(β)
The maximum of the polynomial equation as part of the integrand of Equation (23) can be found by setting the derivative equal to zero
Furthermore, the zeros of the quadratic equation as part of Equation (41)
β 2 − β + 1 = 0 (44)
deliver the golden mean β 1 = φ and its negative reciprocal β 2 = − φ − 1 = − ( φ + 1 ) [10].
A further examination of this function yielded the following results that were depicted in FigureA2, considering the decomposition into reciprocal terms according to Equation (43). The intersection point of the basic function with the
magenta curve at φ = 2 − 1 = 0.41421356 ⋯ is remarkable. The result representing the silver mean is given by solving the quadratic equation
This numerical similarity may be important when interpreting Suleiman’s result for rotating galaxies. When relating the galaxy’s scale radius rs, at which the rotation velocity equals half of its maximum value, to the galactic core radius rc,where the energy densities of matter and dark matter are predicted to be equal
[1], then one is faced exactly with the value r s r c = 1.70951 [27].
βmax respectively E values put in brackets were calculated for the non-relativistic case according to Equation (23). The maximum of 〈 σ β 〉 / S 0 is given in different color (TableA1).
IRT-corrected deuteron fusion data for selected values of kT
kT (keV)
βmax
Δβmax
Δβmax/βmax
E (keV)
〈 σ β 〉 max / S 0
10000
0.130213 (0.07878)
0.20244
1.5547
6118.5 (2910.1)
0.058405
8617
0.121788 (0.074967)
0.18575
1.5252
5444.8 (2635.2)
0.06656
2000
0.064038 (0.046070)
0.07773
1.2137
1691.4 (995.2)
0.2037
1000
0.047875 (0.036566)
0.05261
1.0989
976.5 (627.0)
0.3069
500
0.036135 (0.029022)
0.03593
0.9943
569.8 (395.0)
0.4179
200
0.025238 (0.021384)
0.02196
0.8701
284.0 (214.4)
0.51504
160.4
0.023188 (0.019868)
0.01953
0.8422
240.7 (185.1)
0.5197
100
0.019375 (0.016972)
0.01518
0.7832
168.8 (135.1)
0.4983
50
0.014985 (0.013471)
0.01054
0.7033
102.2 (85.1)
0.38616
20
0.010750 (0.009925)
0.00659
0.6133
53.0 (46.2)
0.18011
17.2347
0.01019 (0.009445)
0.00610
0.5988
47.7 (41.8)
0.15046
10
0.00840 (0.007878)
0.00465
0.5536
32.5 (29.1)
0.0664
8.6173
0.007965 (0.007497)
0.004276
0.5368
29.3 (26.4)
0.0507
1
0.003765 (0.003657)
0.001425
0.3785
6.63 (6.27)
4.05 × 10−5
0.1
0.001720 (0.001697)
0.000458
0.2660
1.39 (1.35)
2.198 × 10−13
0.01
0.0007925 (0.0007878)
0.000140
0.1767
0.294 (0.291)
1.552 × 10−32
Values of the relative width Δβmax/βmax that differ by a power of ten in temperature can be scaled by ln(2) division. In this way further values can be generated as given in TableA2 in brown color. In the same way one can scale down consecutive values of βmax by π − 1 = 2.16159 division respectively Δβmax by π = 3.14159
Full-width of fusion probability curves at “Low” temperature comparing deuterons with protons
T(K)
Deuterons
Protons
βmax
Δβmax
Δβmax/βmax
βmax
Δβmax
Δβmax/βmax
106
0.0016143
0.0004084
0.2489
105
0.0007538
0.0001300
0.1725
0.0009423
0.0001315
0.1912
104
0.0003485
0.0000414
0.1188
0.0004400
0.0000583
0.1325
103
0.0001617
0.0000131
0.0812
0.0002038
0.0000184
0.0903
5 × 102
0.0001617
0.0000129
0.0800
Fundamental Constants and Quantities (TableA3)
Used fundamental constants and quantities [<xref ref-type="bibr" rid="scirp.117548-ref3">3</xref>] [<xref ref-type="bibr" rid="scirp.117548-ref21">21</xref>] [<xref ref-type="bibr" rid="scirp.117548-ref22">22</xref>]
Notation
Symbol
Quantity
Unit
Speed of light
c
2.99792458 × 108
m/s
Boltzmann constant
k
8.617333262 × 10−11
MeV·K−1
Avogadro number
NA
6.02214076 × 1023
mol−1
Sommerfeld constant
Α
7.2973525693(11) × 10−3
-
Proton rest energy
H 1 1
938.27208816(29)
MeV
Deuteron rest energy
D 1 2
1875.61294257(57)
MeV
Golden mean
φ = 5 − 1 2
0.61803398874…
-
Silver mean
2 − 1
0.41421356237…
-
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between target nucleon and projectile nucleon. Cold nuclear fusion, if ever possible, is recommended for protons rather than deuterons at highest experimental possible temperatures around 1000 (K) and needs fine-tuned kinetic nucleon energy. It would be also of interest whether a golden ratio based nuclear fuel confinement chamber could be beneficial. In this connection, also cold nuclear fusion setups should be discussed. Nature is governed by the golden ratio and criticality of physical systems influenced by it, and nuclear physics is not an exception. Computer simulations of the underlying controlled nuclear fusion processes should gain profit from
IRT corrected starting information and may tackle anew possible low energy nuclear transmutations considering the wave-like dark components of matter and energy.