[3] **viXra:0902.0005 [pdf]**
*submitted on 14 Feb 2009*

**Authors:** Akito Takahashi

**Comments:** recovered from sciprint.org

This paper gives the follow-up study on the basics for our previous papers, Study on
4D/Tetrahedral Symmetric Condensate Condensation Motion by Non-Linear Langevin Equation,
Symposium 998-Low Energy Nuclear Reactions Source Book, ACS, published on August 2008
from Oxford University Press.
Pertaining to the quantum mechanics, the basics of new approach using the stochastic
differential equation (Langevin equation) is written for quantifying dynamic motion of
known molecules as D_{2}^{+}, D_{2} and D_{3}^{+}
as well as D-atom state. The role of the Platonic
symmetry in these known molecules are discussed for deducing simple one-dimensional
(R_{dd} dependent; here R_{dd} is distance between nearest d-d pair) Langevin equation and
making quantum-mechanical ensemble averaging to obtain equation for expectation value.
The methodology is applied for more complicated D-clusters as 4D/TSC and 6D/OSC which
would keep the Platonic symmetry, by introducing the force fluctuation deviating from
the ideal Platonic symmetry. Time-dependent TSC and OSC trapping potentials which take
balance to getting back to the Platonic symmetry from the distorted states were defined
and used for numerical solution of Langevin equation. Finally, time-dependent fusion
rate formula for simultaneous 4D interaction was obtained based on the Fermi's golden
rule and one-pion exchange potential of strong interaction. The 4D fusion is regarded to
cause radiation-less excess heat and ^{4}He ash in metal-deuterium systems under dynamic conditions

**Category:** Condensed Matter

[2] **viXra:0902.0004 [pdf]**
*submitted on 14 Feb 2009*

**Authors:** Akito Takahashi, Norio Yabuuchi

**Comments:** recovered from sciprint.org

Tetrahedral symmetric condensate (TSC) with 4 deuterons and 4 electrons has been proposed as
a seed of clean 4D fusion with 4He product in condensed matter. To solve molecular dynamics
motion of 4D/TSC condensation, a nonlinear Langevin equation was formulated with a Coulombic
main condensation force term under Platonic symmetry, 6 balancing forces by quantum mechanical
electron clouds of dde*(2,2) EQPET molecules on 6 faces of TSC cube and a random quantum
mechanical fluctuation term f(t) for d-d distance. Gaussian wave functions for d-d pairs and
their ground state energies were first obtained by variational method, for D2 and EQPET molecules.
Then same sigma-value was used for time-dependent Gaussian wave functions of d-d pairs of TSC
system to calculate the ensemble-averaged <f(t)> for changes of Coulomb energy and force of
distorted TSC system deviated from the ideal double Platonic symmetry. Molecular dynamics
calculation with TSC Langevin equation by the Verlet time-step method was then done. We
obtained mean relative final-stage d-d kinetic energy 13.68 keV with -130.4 keV deep trapping
TSC potential at Rdd-minimum = 25 fm and time-to-TSC-minimum =1.4007 fs. Mean kinetic energy
of electron of a "d-e-d-e" EQPET molecule of TSC system was estimated as 57.6 keV at Rdd =25
fm. These time-dependent trapping potential for d-d pair of TSC can be approximated by HMEQPET
potentials with the empirical relation of m=4.36x10^{4}/R_{dd}, (Rdd in fm unit), continuously as a
change of condensation time or Rdd(t). Barrier factors for fusion reactions as a function of
Rdd(t) and 4D fusion rate per TSC generation were calculated using these HMEQPET potentials
and Fermi's golden rule. We found that 4D/TSC got to the TSC-minimum state with 10 fm-20 fm
radius in 1.4007 fs and 4D fusion rate was 100 % per 4D/TSC generation-condensation. Thus we
concluded that 4He production rate by 4D/TSC was equal to two times of 4D/TSC generation rate
in condensed matter (e.g., PdDx).

**Category:** Condensed Matter

[1] **viXra:0902.0003 [pdf]**
*submitted on 14 Feb 2009*

**Authors:** Akito Takahashi

**Comments:** recovered from sciprint.org

This paper gives further discussions and explanations on the timedependent
quantum-mechanical behaviors of electron-clouds in 4D/TSC
condensation motion by Langevin equation, in comparison with steady
ground state electron orbits and their de Broglie wave lengths for D-atom
and D_{2} molecule.

**Category:** Condensed Matter