The dd Cold Fusion-Transmutation Connection

LENR theory must explain dd fusion, alpha-addition transmutations, radiationless nuclear reactions, and 3-body nuclear particle reactions. Reaction without radiation requires many-body D+Bloch periodicity in both location and internal structure dependencies. Electron scattering leads to mixed quantum states. The radiationless dd fusion reaction is 2-D+Bloch → 4He++Bloch. Overlap between 4He++Bloch and surface Cs leads to alpha absorption. In the Iwamura et al. studies active deuterium is created by scattering at diffusion barriers. Constraints on theory Key characteristics of condensed matter cold fusion that have been challenging to explain are: 1) the overcoming of the Coulomb barrier, 2) the blocking of the decay modes that produce high energy particles, and 3) the conversion of nuclear energy into a heating of the environment. There is also an item 4. Experiments have shown that unexpected forms of "low energy nuclear reactions" (LENR) can occur in condensed matter, namely, the creation of various transmutation products. This paper speculatively assumes that these processes are part of the same physics scenario that is responsible for cold fusion. As Scott Chubb has stated [1], there is breaking of gauge symmetry, which means an ordering of deuteron wave-function phases. For dd fusion to occur, a subset of the deuterons in a metal must become self-organized into a coherent many-body deuteron system in which the deuterons act in concert like the atoms in a Bose-Einstein condensate. It may be that some of the teachings of BoseEinstein condensates, particularly of Bose-Einstein condensates in optical lattices, can provide insight into conditions that enable cold fusion reactions to occur. An intuitive fusion model This paper is concerned with groups of deuterons in deuterided multi-crystalline Pd metal in which there are small domains with relatively regular periodic order, which are called crystallites. The resulting metal deuteride is assumed to have a composition PdDx, where 0.5 < x < 1.[2] We assume that each of the crystallites contains two configurations of deuterons: a normal set of self-trapped interstitial D+, and a second population of highly delocalized D+ at a much smaller fraction y (y << 1). This second population consists of D+Bloch deuterons that are modeled in terms of a many-body D+Bloch wave function. The many-body D+Bloch wave function is assumed to have the translation symmetry of a Bravais lattice, i.e., the density associated with the wave function is invariant with respect to translation operations matching the displacements of a set of Ncell Bravais lattice vectors Rn.[3] The array specified by Rn may have either 2-dimensional or 3-dimensional array symmetry. If it has 2 dimensional symmetry, the occupied volume is at least one atom layer thick. In both geometries a scalar field ρ(r) is used to describe the D+Bloch charge density distribution. Array ressembles a dimpled metal sheet D molecule 2 D + Bloch Fig. 1. D2 molecule: Deuterons occupy side-by-side potential wells. Tunneling through a Coulomb barrier permits D+-D+ wave function overlap at the midpoint , which allows nuclear fusion. Reaction rate is too small to create detectable nuclear reaction. D+Bloch : The D+Bloch are in shallow potential wells within a metal lattice: High mobility spreads out each Bloch deuteron over a large set of communally occupied potential wells. A fraction of the charge of each Bloch-function deuteron is continuously present in each potential well all the time. Superposed wave functions overlap if the number of potential wells is sufficiently large. Defining terms By deuterons I mean a p-n nuclear pair, designated either by d or D+. The metal crystallite provides a periodic lattice potential Vlat(r) within which the deuterons reside. The Pd lattice provides 1 octahedral site potential well and 2 tetrahedral site potential wells per unit cell. The normal deuterons are known to occupy potential wells centered on random octahedral sites. It may be that the D+Bloch occupy the potential wells centered on the tetrahedral sites. The many-body wave function can be written in terms of products of single particle wave functions, each with the Bloch form φ(r+Rn) = eik⋅Rn φ(r).[3] A set of ND D+Bloch deuterons is assumed to be neutralized by a many-body electron system confined to the same volume and containing ND Bloch electrons. The ND-D+Bloch wave function system combined with the ND Bloch electron wave function system will be called a deuterium subsystem. The ND-D+Bloch wave function system by itself will be called a deuteron subsystem. In any experiment the Pd metal will contain many distinct deuterium subsystems. The term subsystem comes from Chernov et al., who attributed their observations to a set of deuteron subsystems in each of which the deuterons are more closely coupled to each other than to the lattice in which they are embedded[4]. I interpret this close d-d coupling as meaning that the deuterons are "merged" with each other as expressed by deuteron coordinate exchange in the many-body deuteron wave function. The palladium lattice with its neutralizing electrons and the normal deuterons with their neutralizing electrons will be called the environment. The conduction electron system of the PdDx lattice will be called the fermi sea. This definition of the fermi sea excludes the ND Bloch electron subsystem that neutralizes the deuteron subsystem, only because the electron subsystem is assumed to have a smaller coherence length. The environment, together with all the deuterium subsystems, is assumed to contain no deuterium molecules, i.e., no side-byside deuteron groupings. See Fig. 1. Also, the deuterium subsystems contain no D atoms, because there are no single electrons associated with a single D+. Defining terms in a wave equation, wave function model It is asserted that the cold fusion and transmutation processes can be adequately modeled by quantum mechanics. The Shrodinger wave equation, wave function approach is used. In this formalism Hψ=Eψ, where H is a differential operator, E is a constant, i.e., the system energy characterizing a stationary state, |ψ|2 is a number density field, and e|ψ|2 is a charge density field. As applied to modeling a deuteron subsystem containing ND deuterons, ψ is a many-body wave function. The Hamiltonian can be written in terms of configuration coordinates r1,r2, ....rND as H = Σ i=1 ND h /2mD ∇ri + eVlat(ri) + Σ i,j=1; i<j ND Unuc(|ri-rj|) + q2/|ri-rj| , where Vlat(ri) is the electrostatic potential provided by the environment plus the ND electrons of the deuterium subsystem, Unuc is the nuclear potential energy seeking to bind 2 deuterons at fm separation, and q is an effective point charge in a screened 2-D+ mutual Coulomb repulsion interaction. It will be shown that q is less than the electronic charge e when Double Bloch symmetry applies. Physical entities are identified by their wave functions. The nuclear interaction part of the process is treated as the decay of an excited state, rather than as a collision. A transition is a decay of a metastable state. A fluctuation is a reversing transition. The transfer of nuclear energy to the environment is treated as a scattering. A scattering changes a transition into a reaction, i.e., a reaction is a non-reversing transition. The potentially reactive state of the deuteron subsystem is viewed as a metastable state of a many-body D+Bloch. Each 2-D+Bloch pair in a nuclear Singlet spin configuration is potentially self-reactive and is metastable by 24 MeV relative to a 4He++Bloch configuration with the same deuterons. The non-Singlet spin pairings are not directly reactive and fail to support direct dd nuclear fusion. It is assumed that the coupling between the nuclear configuration and each Vlat(ri) interaction is sufficiently weak that the lattice interaction can be treated as independent of the nuclear configuration to a first approximation. This means that ρ(r) is almost unchanged by a fusion reaction. The decay transition can be modeled by the Fermi Golden Rule of time-dependent perturbation theory[5]. Transition rates are calculated using the Fermi protocol. Vlat(r) determines geometry The Hamiltonian H(ri) used to calculate the wave function of each single-particle D+Bloch uses the same lattice potential Vlat(ri) for each of the Bloch deuterons. For a deuteron subsystem containing ND D+Bloch, H(ri) uses a single Vlat(ri) for each of ND D+Bloch quasiparticles. The finite-lattice geometry of Vlat(ri) leads to stationary states in the form of "bands" made up of Ncell closely spaced energy levels. This band energy structure plays almost no role in the nuclear reaction process. Since the D+ are bosons, the ND D+Bloch quasiparticles of the deuteron subsystem can all occupy the same lowest energy level in the lowest energy band. Also, the nuclear reaction is not expected to change the occupation distribution of the ND-D+Bloch within the occupied band. For convenience, we model the deuteron subsystem with the assumption that all its D+Bloch occupy the lowest energy state of the lowest energy band, i.e., are in the electrostatic ground state. This ND-D+Bloch ground state is defined as the E=0 reference state. The D+Bloch concentration y is assumed to be small enough that no y-dependent mean charge density correction is required, as in Ref. 2. A 6-dimension problem Consider a set of 2-D+Bloch within the ND-D+Bloch. When viewed in the stationary state Bloch representation, the charge distribution ρ(r) of a 2-D+Bloch deuteron pair occupies the same set of Nwell potential wells as each single D+Bloch. To understand how dd fusion occurs one must recognize that a pair of Bloch-function deuterons in a manybody subsystem within a host metal, i.e., 2-D+Bloch, is a 6 degree-of-freedom system. A 3 degree-of-freedom wave function position factor φ(r) locates the 2-D+Bloch in/on the metal lattice. A second 3 degree-of-freedom factor g(r12) describes the pair's internal structure. (Similarly, a D2 molecule occupying a cavity in a metal is also a 6 degree-offreedom subsystem, with a 3 degree-of-freedom position factor describing the density distribution of the molecule within the cavity. A second 3 degree-of-freedom internal structure factor describes the molecule's rotation-vibration state). The nuclear interactions and the Coulomb repulsion interactions occur in the internal structure space rij, where the rij specify the vector separations characterizing each of the deuteron pairings. The H(rij) used to calculate wave function factors g(rij) include pair-interaction nuclear potential energies Unuc(rij) and pair interaction Coulomb repulsion potential energies Ucoul(rij) = q2/|rij|. H(rij) contains ND(ND-1)/2 nuclear potential energy terms, and an equal number of electrostatic Ucoul(rij) terms. The essential physics is contained in calculations for a single D+Bloch pair with designated vector separation coordinate r12.