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The Coulomb Barrier Not Static in QED," a Correction to the Theory by Preparata on the Phenomenon of Cold Fusion and Theoretical Hypothesis

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The Coulomb Barrier Not Static in QED, Frisone, F. "The Coulomb Barrier not Static in QED," A correction to the Theory by Preparata on the Phenomenon of Cold Fusion and Theoretical hypothesis. in ICCF-14 International Conference on Condensed Matter Nuclear Science. 2008. Washington, DC. “The Coulomb Barrier not Static in QED” A correction to the Theory by Preparata on the Phenomenon of Cold Fusion and Theoretical hypothesis Fulvio Frisone Department of Physics of the University of Catania Catania, Italy, 95125, Via Santa Sofia 64 Abstract In the last two decades, irrefutable experimental evidence has shown that Low Energy Nuclear Reactions (LENR) occur in specialized heavy hydrogen systems [1-4]. Nevertheless, we are still confronted with a problem: the theoretical basis of LENR are not known and, as a matter of fact, little research has been carried out on this subject. In this work we seek to analyse the deuteron-deuteron reactions within palladium lattice by means of Preparata’s model of the palladium lattice [5,15]. We will also show the occurrence probability of fusion phenomena according to more accurate experiments [6]. We are not going to use any of the research models which have been previously followed in this field. Our aim is to demonstrate the theoretical possibility of cold fusion. Moreover, we will focus on tunneling the existent Coulomb barrier between two deuterons. Analysing the possible contributions of the lattice to the improvement of the tunneling probability, we find that there is a real mechanism through which this probability could be increased: this mechanism is the screening effect due to d-shell electrons of palladium lattice. The accordance between theoretical and experimental results will prove the reality of cold fusion phenomena and show the reliability of our model. 1. Introduction Our research has shown that cold fusion phenomena were verified by the Coherence Theory of Condensed Matter. In this theory [5] it is assumed that the electromagnetic (e.m.) field plays a very important role on dynamic system, due to elementary constituents of matter (i.e. ions and electrons). Due to charged matter, considering the coupling between e.m. equations and the Schrödinger equation of field matter operator, it is possible to demonstrate that the matter system shows a dynamic coherence in proximity of e.m. frequency 0.. Hence, it is possible to define the coherence domains, whose length is about CD =2/0 . The simplest model of matter with coherence domain is obviously the plasma system. In classical plasma theory we have to consider the plasma frequency p and the Debey length that measures the Coulomb force extension, i.e. the coherence domain length. In order to address the crucial issue of the nuclear fusion reaction involving the deuterons that pack the Pd-lattice, we 1 must have a rather detailed understanding of the environment in which such nuclear process will eventually take place. For a system with N charge Q of m mass within a V volume the plasma frequency can be written as: Qe N (1) k p m V Introducing the dimensional variable pt we can rewrite the above equations as follows: 1 * n ( ) n kr kr a kr kr a n n ( ) (2) 2 k p n r 1 i * * kr ikr mkr kr n a n n (t)n ( ) (3) 2 2 nn Defining the state as: (4) n ( ) n n and the e.m. field amplitude as: 3 A d (5) 8 k kr kr r we can rewrite as follows: 2 Aa A a (6) t 3 i 2 A A imA a (7) 2 3 If we only concentrate on a short time dynamics, we can write: (0) 0 (8) then, creating a difference in the (6) and using the (7) with 0 , we can easily obtain: i 2 (9) Aj Aj 0 a j ,al 0 2 3 2 (10) Al Aj 3 This has the same form of the equations of the coherence domains as the case in which: 0 0 2 g 2 3 16 2 g 2 c 27 3 2 So, a solution for the ideal plasma exists. More in detail, we can define: k ak (11) 1 / 2 2 g0 3 In this case the coupling critical constant is: k g0 Ak (12) i A A g (13) k 2 k 0 k so to admit the following holding quantity: * i * * * (14) Q Ak Ak Ak Ak Ak Ak kk 2 while for the Hamiltonian’s it is easy to compute: E 1 * * * (15) H Q Ak Ak igAkk Akk N p 2 With the aim of seeing if there are any externals solutions, we write i k uke (16) i Ak Auke (17) where and A are positive constants and uk is a complex vector. Changing these ones in (12) and (13) we have: (18) 2 A g (19) 0 1 3 2 g 2 0 (20) 2 0 and by the condition Q=0 (we cannot have a net charge flow in a plasma), we have: g 2 1 0 0 (21) 2 1 / 2 2 In this case it is easy to observe that for g0 it is unlikely that both the equations (18) 3 and (19) are satisfied. This result means that the energy of an ideal quantum plasma does not have a minimum; in other words, an ideal quantum plasma does not exist. That must not be surprising, because the Hamiltonian’ describes this plasma as a system whose amplitude oscillations are arbitrary, whereas the limit over a certain amplitude does not exist in a real plasma. 3 But by: 1 (a a ) 2m p it is easy to obtain: 1 2 2 (22) m p Also, in the plasma approximation as an homogeneous fluid, we suppose that our Hamiltonian’s stops have to be valid for the oscillations bigger than the following: 1/ 2 1/ 2 V 2 a (23) max N that is when plasma oscillations are of the same order as the inter-particle distance a. In order to create some more realistic models of plasma, we want to compute the breaking amplitude max obtained by the combination of the equations (22) and (23) for a gas of electrons. Using the definition of p , we have: 1 / 3 1 1 / 4 1 / 2 max m p ma e (24) 3 taking a 2.5Å the result is max 2.7 Å This simple calculation shows how it is possible to change our quantum ideal plasma in a real plasma. As the oscillations remain very low in a plasma, a two level model can be a good approximation (the dynamics only includes the first excited state). A consequence of this approximation consisting in the reduction of the plasma in a homogeneous fluid is the changing of the plasma frequency p as follows: Q N (25) p m V Moreover, we study the “nuclear environment” that is supposed existent within the palladium lattice D2-loaded and at room temperature as predicted by the Coherence Theory. As a matter of fact, some physicists declared that it is possible to observe traces of nuclear reactions [1,2,3] when the palladium lattice is loaded with deuterium gas. For this reason many of these physicists define that as a Low Energy Nuclear Reaction (LENR). 4 One of the most interesting experiments shows that in the D2-loaded palladium case the most frequent nuclear reactions are: 1) D D3H p 4.03MeV 2) D D4H p 23.85MeV The aim of our work is to propose a “coherence” model, by means of which we can explain the occurrence of reactions 1) and 2) and their probability according to more reliable experiments. First of all, we will start from the analysis of the environment, i.e. of plasmas present within palladium (d-electron, s-electron, Pd-ions and D-ions); we will do so using the coherence theory of matter. Then, we will use the effective potential reported in ref. [7,8] and add the role of lattice perturbations, by means of which we will compute the D-D tunneling probability. 2. The plasmas within non loaded palladium According to the Coherence Theory of Condensed Matter, in a palladium crystal at room temperature the electron shells are in a coherent regime within a coherent domain. In point of fact, they oscillate in tune with a coherent e.m. field trapped in coherent domains. So, in order to describe the lattice environment we must study the plasma of s-electron and d- electron. a) The plasma of d-electrons Similar arguments were proposed by Preparata, but starting from a new formulation of condensed matter theory known as Coherence Theory. In this theory we can visualize the plasma formed by d-shell electrons as consisting of shells charged nd e (for palladium nd =10) radius rd =1 Å and thickness a fraction of one Angstrom. The classical plasma e n N d (26) d m V is d-electrons plasma frequency. But according to the coherence theory of matter we must adjust this plasma frequency of a factor 1.38. We can understand this correction by observing that the formula (26) is obtained assuming a uniform d-electron charge distribution. But of course the d-electron plasma is localized in a shell of radius R (that is about 1Å), so the geometrical contribution is: 6 1.38 (27) If we rewrite the renormalized d-electron plasma frequency with d , we have d 41.5eV / (28) and the maximum oscillation amplitude d is about 0.5 Å. b) The plasma of delocalized s-electrons The s-electrons are those neutralizing the absorbed deuterons ions in the lattice.
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