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Wikiversity/Cold fusion/Theory/Estimates

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This discussion, apparently by User:Mattvers, was moved from the higher level page Cold fusion/Theory.[1]


Despite the original rejection of the Fleischmann-Pons experiment the concept of cold fusion in Deuterium saturated solid in not entirely out of sense. Only the parameters could have been estimated to optimistically like for a cheap and simple electrolytic chemical lab. The mole volume of Palladium is about 10 cm3. The mole of Deuterium perfectly saturating this volume in ratio one to one atom of both i.e. 2 g and having density 0.180 kg/m3 at standard conditions will occupy the volume of 10 000 cm3. Using the equation of state of the perfect gas it means that Deuterium in Palladium is effectively under pressure at least 1000 times of the atmospheric value. Also the standard Hartree theory of uniform charged Bose gas having this density predicts a measurable fusion rate and the fusion rate could be further greatly enhanced if Deuterium was confined in two-dimensional Palladium layers due to the geometric centrifugal attraction [1]. No experiments were made yet for example with more complicated structures like one atom thin layer of Deuterium saturated Palladium multiply altered with one atom layer of neutron reflector like Graphene to enhance confinement to two dimensions and all in very low temperature except Palladium powder mixtures with ionic near-supercoductors[2]. The positive outcome of Palladium-Zirconia-based-powder experiments appears however to be really a hot fusion and moderated few body fusion-scattering-fusion chain reaction in cold fusion environment due to the anty-moderating action if superionic conductors between Palladium sponges.

The simplest theory qualitatively predicting the cold fusion in palladium is easy for tritium and immediately suggests also the electrolysis of the super-heavy tritiated water T2O in order to saturate the palladium with tritium and produce it at the same time. Similarly to electrons in metal the energy levels of deuterium nuclei in the crystal lattice of palladium have the band structure.[3] Because the total spin of the tritium nucleus is fractional similarly to the electron it is a fermion and it is subjected to Fermi-Dirac statistics. In order to achieve fusion it is enough to concentrate tritium atoms in the crystal lattice of palladium to the level when the Fermi energy on the Fermi surface will reach 4 keV i.e. the energy reachable in Farnsworth–Hirsch devices working like TV vacuum tube. Electrons in semiconductors of certain kind because of small effective mass may reach relativistic energies. Because the Deuterium nuclei strongly repel with the Coulomb interaction which can be approximated by the short-range Dirac-Delta function and then the fermionization can occur the theory extends also on Deuterium. The Fermi energy for the uniform fermionic gas depends on its density <math>N/V</math> according to the formula

<math>E_F = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3}</math>

Substituting <math>m</math> as the mass of the tritium nucleus for the required value of 4 keV because of the small value of the Planck constant one gets however a giant and impossible density of required Tritium nuclei 2 x 1010 per one unit cell of palladium which under normal conditions is around 1. This is even 10 times more than for the close packing which corresponds to the distance between Deuterium nuclei in muonic Hydrogen (1.2 x 109) or if muonic Deuterium was put there instead so at that density they even would not have to have any kinetic energy at all.

In the tight-binding scenario on the other hand one can assume that two Deuterium ions are in neighbouring interstitial positions in harmonic oscillator Trojan packet-like ground state when the frequency of the harmonic oscillator is calculated by expanding the Coulomb potential of closest Pd+ neighbours around the equilibrium position. The rates estimated here are purely from geometric and quantum mechanical considerations treating that the fusion event is a quantum relative position measurement subjected to uncertainty of the conjugate but not from the experimental value of A, <math>\Gamma_f=A |\Psi(x_1-x_2=0)|^2</math> and the fusion happens with probability 1 if only two particles are close enough. The quantum fusion rate is calculated as the probability to find both Deuterium nuclei within the proton electromagnetic volume <math>4 \pi/3 \times a_{Pt}^3</math> per the quantum measurement time from the uncertainty principle. The fusion is then from overlapping of the wave functions of two closest minima. It turns out however this frequency (and the fussion rate) from the symmetry is exactly 0 for both the 6 nearest neighbours and from next 8 members of the face centered Palladium lattice. The dynamics is therefore intrinsically nonlinear and characteristic to the quartic oscillators which is the higher non-vanishing order. In mean-coordinate approximation this however can be renormalized back to be effectively harmonic with the frequency dependent on the temperature. Also correction from the Coulomb repulsion between Deuterium breaks the symmetry and leads to non-zero frequency. This also suggests that the normal state of Deuterium sublattice inside the Palladium lattice at zero temperature is always superradiant due to the mutual repulsion between nuclei and only temperature can make the normal state. Furthermore the fusion probability per time unit has a strong maximum as a Planck-like <math>x^3 e^{-x^2/2}</math>function of the ratio <math>x</math> of the Palladium lattice constant and the harmonic oscillator ground state spatial spread (the frequency). At the maximum however tuned for example by the temperature the highest fusion rate calculated in this approximation is of the order 10-6s-1 which is 3 fusion events per year per D-D pair and the temperature for that needs to be below 10-4K. It suggests that 1010 fusing pairs formed in the originally reported experiment if the numer 4 x 104s-1 of fusion events (estimated neutron count) was right and they were somehow by unknown reasons at the maximum at room temperature. This theory neglects however D-D interaction at all at interstitial positions as it was non-interracting bosonic gas and therefore also tunneling Gamow factors far from the harmonic approximation. The relevant oscillator frequencies in better approximation must be therefore much above the characteristic frequency <math>\hbar/m_{D}a_{Pd}^2</math> constructed from the mass of the Deuterium nucleus and the Palladium lattice constant leading to much lower rates and the maximum is really never reachable. The same value at the maximum (and F-P claimed rate) i.e. of the order of <math>(a_{Pt}/a_{Pd})^3 \times \hbar/m_{D}a_{Pd}^2</math> is also immediately obtained from the Hartree approximation to Deuterium two nuclei gas with constant density in the volume of one lattice cell <math>a_{P}^3</math> as it was the infinite potential well since the potential is quartic. However this volume appears to be too small for this approximation while in the condensed matter theory it is the volume of whole solid[4]. This level of mutual confinement (Pycnodeuterium) however was claimed by Yoshiaki Arata to explain positive outcome of his fusion experiments with Palladium nanoparticles. The ground state of quartic oscillator is pretty much like <math>e^{-{x^6}}</math> and is quite rectangular so the density is quite uniform for one Deuterium however the Coulomb interaction will brake that while two nuclei are in the well. Far from the maximum because of the exponential factor from Gaussian overlap it can be as low as 10-200s-1 which is not measurable at all.

The standart Hartree theory applied to Deuterium nuclei in Palladium as the uniform Bose or Fermi gas at 0 temperature in the uniformly negative background of charge [5] and with the density one nucleus per unit cell which includes the interactions appears to fail even if it gives from purely quantum considerations unmeasurably low fusion rate <math>N^{-2/3}(a_{Pt}/a_{Pd})^3 \times \hbar/m_{D}a_{Pd}^2</math>, 10-21Hz for the number of nuclei N of the order of the Avogadro number 1024 (macroscopic mole size of the electrode) and the assumption that they all stay as self-consistent and undistorted by the Coulomb scattering plain waves is equivalent to assumption of hyper-tunneling with the negligence of Gamow factor. Putting however the higher experimental value of A from the muonic catalysis[6] <math>A=2.01 \times 10^{-16}\approx 2 \times (4 \pi/3)a_{Pt} \hbar /m_{D} </math> cm3/s corresponding to much higher characteristic frequency <math>\hbar/m_{D}a_{Pt}^2</math> (which is also the frequency of the harmonic oscillator with the ground state with the spread of the size of the proton) and the volume of two nucleons the rate predicted from Hartree theory is measurable and higher i.e. 107Hz (even 1000 times higher then in F-P original experiment) per one cubic centimeter of the electrode volume[7]. This ground state for bosons is also Bose-Einstein condensate and this is quite impossible that Bose-Einstein conditions for Deuterium nuclei in Palladium were reached at the room temperature and even if the experiment was repeated near 0 temperature putting Deuterium saturated Palladium into the fridge the Gross-Pitaevskii ground state predicting measurable fusion rate seems to be too rough and approximate to take care of sharp Coulomb repulsion for this purpose and simply predicts hyper-tunneling. The Hartree-Fock theory (for Tritium and Coulomb-fermionized Deuterium) however gives the fusion rate exactly zero and only from the fact that both the antysymmetric wave function and the two-particle density vanishes when the pair coordinates equal and the only way tritium could fuse in Palladium is that spin anty-parallel Cooper-like pairs would be in symmetric spatial state.

Cold fusion in analogy to the recombination control of the positronium[8] at rates equivalent to those obtained with ultra-tight muonic Deuterium molecule would be possible if the Palladium lattice in fluctuations together with the Deuterium diffusion provided the conditions to generate binary star Trojan wave packets from its nuclei. [9] This however to achieve the confinement and the fusion rate equivalent to that of muonic catalysis would require that they would move on circles in binary star configuration with cyclotron frequency and the speed of light in magnetic field of the order od 106 Tesla. Since the cyclotron orbit must be so tight and the mass ratio between the electron and the muon is slightly above the half of the fine structure constant this cyclotron frequency corresponds to the frequency of the gamma radiation slightly below the necessary to cause the electron-positron pair production. This is 0.1% of that what is inside the neutron stars magnetars and 10 000 times more then those in Tokamak which does not seem to be possible. The current experiments with exploding electromagnets can barely provide 1000 T for milliseconds. Pairing of this kind would require also highly improbable counter-channeling with simultaneous turn-on of the magnetic field during the close approach event since the generation of such probability focused states is a non-trivial problem of quantum control itself. The channeling nuclei would also have to have subluminal speeds and therefore relativistic energies and the fusion would have to occur inside the unit cell of Palladium. Even if the need of magnetic field could be weaken to the laboratory values or even removed and the compression and fusion occurred in Langmuir configurations consisting of one negative ion from the Palladium cell and two nuclei of Deuterium the rotation frequency of the pair to provide sufficient compression should be of the order of gamma radiation and the velocity also near the speed of light i.e. with the kinetic energy sufficient to cause the fusion on the direct scattering. The rate enhancement would be only due to prolong exposure to each other. No energy pumping during the electrolysis appears to provide such activation energy except the accelerator beam injection of at least 4 keV ions readily achievable in TV-like Farnsworth–Hirsch fusor. Since the negative ion so point-like is not really possible if the nuclei were squeezed by the circularly polarized electromagnetic field this field strength should be as giant as on the muonic hydrogen Bohr orbit which is about 2x1016 V/m but this is so strong that can accelerate the ion to 4 keV at the same Bohr radius distance. The three-body mechanism mentioned here assuming that first someone would somehow overcome the problem of no-go values of the parameters implied by the fusion conditions would however explain very low reproducibility of the experiment without the prior knowledge of the theory since the fussing three body configurations in magnetic and electric fields consisting of two ions of Deuterium and one crystal lattice ion of Palladium exist always within very complicates stability (and therefore fusion) diagrams that depend on the frequencies and strengths of the fields that could be generated by the build-up charges and currents inside the solid lattice and by the applied voltage potential. Those could also depend on the level of Palladium doping and the electrode geometry.


  1. Bialynicki-Birula, I (2002). "In- and Outbound Spreading of a Free-Particle s-Wave". Physical Review Letters 89 (6): 060404-1-4. doi:10.1103/PhysRevLett.95.103001. 
  2. Physicist Claims First Real Demonstration of Cold, Fusion May 27, 2008, phys.org
  3. Alefeld, Goerge; Völkl, Johann (1978), Hydrogen in Metals, Springer-Verlag, ISBN 3540087052. 
  4. Engvild, K. G.; Kowalski, L. (2003), "Triple Deuterium Fusion By Three Body Recombination Between Deuteron And The Nuclei Of Lattice Trapped Deuterium Molecules", Proceedings of Tenth International Conference on Cold Fusion. 2003. Cambridge, MA: LENR-CANR.org (World Scientific, Inc): 75-98. 
  5. Kittel, Charles (1963), Quantum Theory of Solids, John Wiley & Sons, pp. 75-98. 
  6. Jackson, J. D. (1957). "Catalysis of Nuclear Reactions between Hydrogen Isotopes by <math>\mu^-</math> Mesons". Physical Review 106 (2): 330-339. 
  7. Gou, Qing-Quan (2010). "Full theory of cold fussion". J. At. Mol. Sci.(Chines) 1 (1): 87-92. doi:10.4208/jams.091009.100309a. 
  8. Apparatus and method for long-term storage of antimatter, US Patent 7709819
  9. Kalinski, Matt; Hansen, Loren; David, Farrelly (2005). "Nondispersive Two-Electron Wave Packets in a Helium Atom". Physical Review Letters 95 (10): 103001. doi:10.1103/PhysRevLett.95.103001. PMID 16196925.