Methodology of the Cold Fusion Research +

Transcription

1 Reports of CFRL (Cold Fusion Research Laboratory), 8-5, pp (December, 2008) Methodology of the Cold Fusion Research + Hideo Kozima* Low Energy Nuclear Laboratory, Portland State University, Portland, OR 97207, USA ; pdx00210@pdx.edu, Website; *Permanent address; Cold Fusion Research Laboratory, Yatsu, Aoi, Shizuoka , Japan. ; hjrfq930@ybb.ne.jp, Website; + This paper is an extended version of the paper with the same title in the Low-Energy Nuclear Reactions Sourcebook (Vol. 2) eds. J. Marwan and S. Krivit, to be published by the American Chemical Society from Oxford University Press in The man of science must work with method. Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house. Most important of all, the man of science must exhibit foresight. Henri Poincaré, Science and Hypothesis, p (Translated by W.J.G.). Dover Publications, Inc Library of Congress Catalog Card Number Contents Abstract 1. Introduction 2. Brief Survey of Researches of the Cold Fusion Phenomenon (CFP) 3. Difficulty to Explain Nuclear Reactions in Solids at around Room Temperature without Acceleration Mechanisms 4. Phenomenological Approach using Concepts which do not contradict the Knowledge of Physics 5. Complexity in the Cold Fusion Phenomenon revealed by Experimental Facts and Their Explanation ` 6. Conclusion Acknowledgement References 1

2 Abstract On the brief summary of researches on the cold fusion phenomenon (CFP) performed in these twenty years, an appropriate methodology of the research for this curious phenomenon is discussed and the phenomenological approach is recommended for it at present stage of investigation. There have been found several quantitative relations and regularities or laws between observables in this phenomenon; ratios of the number N a of nuclear reactions producing a definite observable a to another N b, the stability effect of nuclear transmutation, the inverse-power law for the frequency-intensity relation of an effect, and bifurcation of effects in temporal progress of events (excess power, neutron emission). As an illustration of the phenomenological approach appropriate for the CFP, a full introduction of a phenomenological model proposed already is given with successful applications and with a trial to verify it from a microscopic point of view. The above mentioned laws are tentatively explained by the model. We could anticipate various application of this phenomenon in the course of and after establishment of the science of the CFP. 1. Introduction The science is a conversation between a human mind and its environment and also is its result. The human mind may have an initiative to start a conversation using an idea induced by the environment in it. However, the conversation always needs opponents in its process to develop the idea. Through the conversation, the idea originally born in the mind transforms into a new form absorbing new elements fed by the environment through the conversation. Therefore, the most important aspect in the science is the process of the evolution of the idea to reach a new level of recognition of the environment even if the resulting science is too brilliant to attract the attention of the successive people. In these almost twenty years of researches of the cold fusion phenomenon (CFP) in various materials (CF materials), mainly transition-metal hydrides and deuterides, and hydrocarbons with periodic array of carbon atoms and also some biological systems, we have obtained a number of various experimental data sets and elaborated on a few theoretical trials to explain the curious facts revealed by these data sets. Generally speaking, the experimental facts, especially those of excess energy and several cumulative observables such as transmuted nuclides, have unquestionably revealed abnormal behavior of components in CF-materials difficult to explain by the known knowledge developed in the 20 th century in nuclear physics and in solid-state physics. 2

3 We may specify at the present stage of the researches in the CFP that plenty of experimental facts are lacking satisfactory theory and we need such an effort as specified by H. Poincare s words cited in the head of this paper. Likewise, it is necessary to have an appropriate design of the house and an effective cement to combine blocks together to construct a building. It is better to define terminology at first to make our logic more clear. The CFP (Cold Fusion Phenomenon) stands for nuclear reactions and accompanying events occurring in solids with high densities of hydrogen isotopes (H and/or D) in ambient radiation including environmental thermal neutrons belonging to Solid-State Nuclear Physics (SSNP) or Condensed Matter Nuclear Science (CMNS), an interdisciplinary field between nuclear physics and solid-state physics [1, 2]. CF materials stand for materials in which the CFP occurs. There are three kinds of CF materials; (1) transition-metal hydrides/deuterides and some proton conductors, (2) hydrocarbons with a periodic array of host nuclei (carbon C) and (3) some biological bodies [1]. To perform a theoretical trial for a consistent explanation of experimental facts with wide variety, we have to have a strategy for the investigation. The methodology recommended in this paper is one of such trials based on historical experience in modern physics. There has been proposed a scheme of stages of the development of science as three steps called 1) phenomenological, 2) substantial, and 3) essential steps (three step hypothesis of science evolution). According to this scheme, a theoretical approach to a new problem starts with a phenomenology based on experimental facts assuming a hypothesis to explain them using concepts of existing science. In the next step it uses a model, a system of hypotheses (or premises) with some adjustable parameters based on experimental facts. Finally this involves a theory (logical system) based on established general principles to explain new facts. It should be noticed that a hypothesis turns into a principle in the course of development as these in the special theory of relativity by A. Einstein show a good example. Therefore, theoretical trials to attack an inexplicable experimental data obtained in the CFP may be classified into three categories; 1) hypothesis, 2) model, and 3) theory as explained in the previous work [1, Appendix B]. In this paper, we give a brief explanation of the trials to construct a science of the CFP using whole experimental facts such as its elements especially putting a weight on undeveloped fields in nuclear physics and solid-state physics. In conclusion, we recommend a phenomenological model for the CFP at the present stage of its development. 3

4 2. Survey of Experimental Facts of the Cold Fusion Phenomenon (CFP) When they started their research of the cold fusion phenomenon (CFP) published in 1989, Fleischmann, Pons and Hawkins (F-P-H) presumed fusions of two deuterons d = 2 1 H (d d fusion reactions) in a solid palladium deuteride PdD x (x = ) simply as a result of a two-body reaction assisted by the environment and sought results coincident with knowledge of nuclear physics; production of a helium-3 3 2He and a neutron n, or a triton t ( 3 1H) and a proton p ( 1 1H), or a helium-4 4 2He and a gamma photon γ in a rare (10 7 times less) rate compared to the former two. They already realized at this stage discrepancies between their presumption and their experimental data and suggested a possibility of new reaction schemes. The mainstream researches done in following years in this field have traced along the same line of F-P-H s looking for the answer yes or no for the coincidence between the presumptions and the selected experimental data sets. By the way, there have been obtained extraordinary data sets not only in deuterium systems but also in protium ones exceeding the range encompassed by the originally presupposed d d fusion reactions. One of these data is the nuclear transmutations (NT) generating almost all nuclides on the periodic table of elements in both deuterium and protium systems (cf. [1] Section 2.5 and Table 11.5; [2] Chapter 2 and Tables 11.2 and 11.3). Besides the researches looking for the possibility of the d d fusion reactions in solids, there have been found several statistical laws between observables as a natural result of scientific endeavors, i.e. a conversation with nature; the inverse-power law for the frequency-intensity relation in excess power production [1], the stability effect for generation of new nuclides by the NT [1], the bifurcation of effects (excess power production), and recursion and chaotic behaviors of effects in the course of experiments (cf. Chapter 5 for full explanation). Looking over the history of modern physics, we notice the existence of many examples of statistical laws and the qualitative reproducibility in complex systems besides the rigorous quantitative reproducibility for phenomena occurring in simple systems in idealized conditions, e.g. laws of a falling body. A most familiar example of the statistical laws in nuclear physics is the decay of radioactive nuclides. Statistically, there is a law describing the temporal change of the number N(t) at a time t of a parent nuclide with a decay constant τ if t is not too large compared with τ; N(t) = N 0 exp (- t /τ). (2.1) It is not possible to know the behavior of an individual parent nuclide even if we know 4

5 the average number N(t) of the nuclide. However, it is enough to know only the statistical law for construction of atomic piles because its output is determined by an average over whole reactions in the pile at the time. The same logic in these cases does not apply to some cases where the effect of each nuclide is discriminated individually. A characteristic of the limitation of a statistical law is shown by the example of the unstable nuclide. Let us consider the decay law (2.1), again. If we measure the product of a decay process, e.g. alpha particles emitted by a radium nucleus A 88Ra (A = 223, 224, or 226), the number of α particles observed by a counter in a definite time interval decreases with a decrease of N 0 reaching a result not expressed by the equation (1) for a short time interval is not appropriate for an average. The scale of the appropriate time interval for averaging depends on the nature of the phenomenon. Returning to the CFP, we notice many factors that make the effects in the CFP statistical. Let us survey the circumstances where CFP occurs. We confine our investigation to cases where there are no initial or artificial high-energy particles more than 10 kev, which makes the situation interdisciplinary between nuclear physics and the CFP different from that where the CFP occurs. 2.1 Necessary conditions for the CFP. First of all, we list the necessary conditions for the CFP which we know from our experience in these twenty years. All systems for the CFP are composite; there are several components (agents) in the CF materials in non-equilibrium, dynamical conditions interacting nonlinearly. The composite systems are therefore complex. It is emphasized that all systems should be dynamical to produce the CFP which has been observed by now. Dynamical means not static, non-equilibrium or far from equilibrium and is realized by electrolysis, discharge, or temperature variation, etc. In the dynamical system, the following characteristic composition of materials is necessary to induce the CFP; existence of special host nuclei on the periodic lattice and interlaced protons or deuterons. Host nuclei known effective for the CFP by now are carbon C, titanium Ti, nickel Ni, and palladium Pd. Furthermore, there are some positive data with iron Fe, tungsten W, platinum Pt, and gold Au. Also, the existence of thermal neutrons seems necessary for occurrence of the CFP; the CFP does not take place where there are no background thermal and epithermal neutrons, and the CFP is intensified by artificial irradiation of thermal neutrons ([1] Section , [2] Chapter 8). These facts are often neglected or ignored but should be seriously considered as the conventional approaches have failed to properly treat the 5

6 situation of CF materials and we need a new approach as shown in Section Characteristic Cause-Effect Relations of the CFP Secondly, we consider the characteristic cause-effect relations of the CFP. The causes of the CFP are atomic with scale length of about 10 1 nm = m (= 1A ) and energy range of about a few tens mev (1 mev = 10 3 ev). The effects are nuclear, conventionally realized by nuclear reactions that occur with scale length of about 1 fm = m and with energy more than a few MeV (1 MeV = 10 6 ev). This means that the cause and the effect relate to each other by differences of a factor 10 5 in length and of more than 10 8 in energy. These large differences in scale and energy influence characteristics of the CFP. If the necessary conditions for the CFP are atomic structure sensitive, an occurrence of an event (nuclear) easily destroys the condition (atomic) for the CFP arranged carefully or by emergence in the complex system. This may be a cause of the irreproducibility (rather the qualitative reproducibility) and the sporadic occurrence of events in the CFP. 2.3 Characteristics of Events in the CFP in terms of Research Fields. Looking into our experience obtained in these twenty years, we notice several characteristics of the CFP in atomic and nuclear levels. They are classified into several branches depending on the viewpoints. Chemical There are several chemical characteristics such as favorable combinations of host metals, hydrogen isotopes, and electrolytes. i) Favorable combinations of host transition-metals, hydrogen isotopes, (and electrolytes in electrolytic experiments); Pd-D (-Li), Ni-H (-K), Ti-D, H (-Li), and C-H. In the transition-metal systems, a high ratio of concentrations (H, D)/(C, Ni, Pd, Ti) is favorable or necessary for the CFP. This may show that the good periodicity of (H, D) and/or (C, Ni, Pd, Ti) in CF materials is a necessary condition for the CFP. ii) Metallic catalysts which do not occlude hydrogen (Pt, Au, W, Fe) also show the CFP in some situations. iii) Some hydrocarbons (XLPE, Phenanthrene) show nuclear transmutations in the CFP. It is noticed that the CF materials listed above in i) and ii) are all metallic catalysts and complex chemical reactions occur on their surface. Unfortunately, we do not know the relationship between the catalytic power of these metals and the CFP in them yet 6

7 and that remains as our future problem. Nucleochemical There are several regularities or laws between observables which should be investigated in nuclear chemistry. i) Stability effect; stable nuclides are generated more often in which the isotopic ratio differs from the natural one reflecting the characteristic of the system [1]; the latter example is Ag from Pd (cf. [1] Section 2.11). ii) Inverse-power law for the frequency-intensity relation of effects (especially for excess power) (cf. [1] Section 2.12) iii) Recursion, bifurcation, and chaotic behaviors in temporal variation of effects (neutron emission and excess power) (cf. Chapter 5). iv) Localization of the trace of events at regions of a few μm diameters within surface/boundary layers of a few μm thickness; Examples are in many data sets. Examples of boundary layers are the Patterson power cell (beads), the Au/Pd/CaO multi-layer by Yamaguchi, and MnO/Pd structure by Iwamura (cf. [1] Section (c); [2] especially Sections 6.5a, 9.1e). v) There are a few numerical relations between observables. Especially important is the ratio of numbers of nuclear reactions N a and N b supposed to give observables a and b, respectively. Experimental data shows that N Q N NT N t 10 7 N n, N He m N Q, N γ 0, N He3 0 (m = 2 4), where N NT means the number of reactions participating in the nuclear transmutation and N Q means that producing the excess energy Q defined by a relation N Q = Q (MeV)/5 (MeV) (cf. [1] Section 2.3; [2] Section 11.1). Statistical characteristics of the effects; Several characteristics related to statistical quantities of observables have been discovered and they show that the CFP is a phenomenon regulated by many-body effects in the compound system. 1. Qualitative reproducibility or irreproducibility 2. Unpredictability or Sporadicity; unpredictable or sporadic occurrence of events in the CFP 3. Favorable non-equilibrium, dynamical conditions; almost all positive effects have been observed in systems in dynamical or non-equilibrium situation. 2.4 Success of a Phenomenological Approach; TNCF and ND models 7

8 As fully explained in Section 4, we could give qualitative explanations of numerical relations between numbers of events N a of a and N b of b. The theoretical ratios (N a /N b ) th were in accord with the experimental ones (N a /N b ) ex by a factor of about 3 [1, 2]. Explanations of statistical laws discovered by now have been given by our model on an assumption that the parameter n n of the model is identified with the parameter of the l.d.e., explained in Section 4. The statistical laws include the stability effect, the inverse-power law or the 1/f law, the bifurcation and the chaos in excess energy generation, and the recursion behavior of neutron emission. 3. Difficulty to Explain Nuclear Reactions in Solids at around Room Temperature without Acceleration Mechanisms In this section, we discuss briefly difficulties to explain the CFP summarized in previous sections by theories based on principles and knowledge established in modern physics. These difficulties are the origin of disbelief of many scientists in the CFP and should be considered seriously by researchers in this field. In nuclear physics, low energy nuclear reactions in free space are a well investigated subject and detailed knowledge has been accumulated in almost 80 years after the discovery of the neutron in 1932 and the substantial birth of nuclear physics. A nucleus (or nuclide to specify its constituent characterized by the number of protons Z and the number of nucleons A, and the energy state) has a volume V proportional to A and therefore the radius of the nucleus R N is proportional to a cube of A with a constant C, a value of which is about 1 fm = m; R N =CA 1/3. (3.1) This means that nucleons (protons and neutrons) are interacting with an attractive force called the nuclear force with an action range of about 1 fm (femtometer or fermi) except the inter-nucleon distance r is not too small ( r 0.1 fm). Let us consider an interaction of two nuclei with proton numbers Z and Z. When the mutual distance R between two nuclides is larger than a distance R N fm, the force exerted between two nuclei is the repulsive Coulomb force F c proportional to the inverse square of the distance R due to the charges on the nuclides; F c = c ZZ e 2 / R 2, with a constant c. When the distance R diminishes to the magnitude of an order of R N, the nuclear force starts to work to attract each other and the two nuclei fuse. Thermonuclear fusion reactors investigated for more than fifty years to realize a sun on the earth are machines to realize d-d or d-t fusion reactions in plasmas with high density (10 15 cm 3 ) and temperature (10 8 degree Kelvin). 8

9 When the two nuclei approach in the range of the nuclear force each other overcoming the repulsive Coulomb force by some means, the two nuclei coalesce forming a compound, intermediate nucleus A ZX *. The compound nucleus A ZX * stabilizes through several branches as established in nuclear physics; emitting an electromagnetic radiation (γ ray) to become A ZX, emitting a light particle as n, p, heliuim-4, or exerting a fission. In CF materials, if entirely different nuclear reactions from those in the free space occur there, we have different events somewhere in the nuclear process from the initial to the final through the intermediate stage. The conceivable stage is one of the followings; (1) when two nuclei are approaching, (2) stage of forming a compound nucleus, (3) when a compound nucleus is formed, and (4) after a compound nucleus is formed. Let us investigate possible effects of CF materials on these nuclear processes. As an example of illustrative investigation, we use the most popular case of PdD crystal out of CF materials. The specific situation of deuterons in this crystal is listed up as follows; (a) existence of itinerant electrons, (b) existence of nuclei A 46Pd (A = ) on the lattice points (lattice nuclei), (c) existence of a possible electric force on deuterons at interstitials. It is not impossible that these factors cooperate to produce new effects, which might be stronger than individual effects. We consider, however, only these factors individually in this paper due to a common sense that a cooperative effect is not stronger than effects induced by individual causes even if it has different qualitative characteristics from them. It is advisable to refer to a report [3] of DOE in 2004 on a proposal which tried to show a possibility of the d d fusion reactions in the CF material. The member of a panel asked to check the proposal denied the trial to verify the d d fusion reactions due to the inappropriate assumptions made in the logical course of the proof. We check the factors (a) (c) listed above individually. 3.1 Electrons To consider the effect of electrons to the CFP, we consider the fundamental nature of elementary particles. In quantum mechanics, a particle can take restricted values of momentum p and position r by the uncertainty principle different from in classical mechanics where they can take arbitrary exact values. Let us consider this situation in one-dimensional case, for simplicity [1, Section 3.4.2]. In one dimensional system, the uncertainties of the momentum Δp and the position 9

10 Δx of a particle with a mass m have to satisfy the uncertainty relation; Δx Δp ћ/2. (3.2) The constant ћ is defined as the Planck constant h divided by 2π and has a value; ћ = J s. (3.3) According to the uncertainty relation (3.2), an uncertainty Δx of position and an uncertainty Δ p of momentum relate to each other and they cannot decrease independently. The momentum p is related to the energy E of the particle by a relation; E = p 2 /2m. (3.4) This also means an uncertainty of energy ΔE of the particle is restricted by the uncertainty of position Δx through the uncertainty of momentum Δp. If we confine a particle in a small space thus making Δx small, then we have a larger value of the momentum due to the increased Δp, resulting in an increased value of the energy. It is a commonplace written in every textbook of quantum mechanics using the uncertainty relation (3.2) why there is a minimum energy state with an interatomic distance R min. This reasoning by the energy balance forbid to make the interatomic distance decrease further to an order of magnitude of r N cm when there are no strong force exerted to enforce two protons approach. When the inter-proton distance becomes close tor N 10 5 a H where the nuclear force works by a screening effect of electrons confined in the small space to decrease the Coulomb repulsion between two protons, we can calculate how high the energy of electrons E e should be to be confined in the small space between two protons to give a following value [1]; E e Δp 2 /2m ћ 2 /m a E H ev (= 10 5 MeV). This value shows it is impossible to screen the inter-proton Coulomb repulsion by electrons in ordinary conditions of CF materials. This estimation also applies to D-D system instead of H-H system without any change of factors. Thus, in a system without a special acceleration mechanism, it is impossible to expect d-d fusion reactions to be realized by any screening effect of electrons in the system. There are several trials to overcome the above explained shortage of the screening effect by electrons without respectable positive results [1]. 3.2 Phonons The effects of the lattice nuclei are taken into consideration by phonons. Let us consider PdD crystal, one of the typical CF materials of fcc transition-metal deuterides and hydrides, for example. The lattice structure of PdD is an fcc lattice of Pd interlaced 10

11 with that of D ([1] Fig ). Ions of Pd and deuterons are oscillating with thermal energy around their equilibrium points, lattice points and interstitial sites, respectively. The thermal motion of ions and deuterons are equivalently described as oscillations of lattices of Pd ions and deuterons as usually done in solid-state physics. The oscillation of a lattice is quantized to be described by phonons, quasi-particles with quasi-momenta and quantized energy. The phonon is treated in parallel to the photon, the quantized state of the electromagnetic field ([1] Section 3.4.3). Physics of a system composed of charged particles and electromagnetic fields is called Quantum Electromagnetic Dynamics (QED), which inspired M. Fleischmann to expect Fleischmann s hypothesis [4]. There are several researchers, including M. Fleischmann, who considered that phonons will help realization of d-d fusion reactions in CF materials as photons worked to explain such quantum electrodynamic effects as Lamb shift and abnormal magnetic moment of an electron. The different characteristics of the photon and the phonon make it difficult to show possibility of d-d (or p-d) fusion reaction in CF material as was shown before [1]. In short, the difference of continuous and discontinuous media results in too different characteristics of oscillations in both systems as explained as follows. In the case of a discontinuous medium like a crystal lattice, there is a minimum wavelength, twice the distance between adjacent mass points. This is a strong restriction of the oscillations to accelerate charged particles for fusion overcoming the Coulomb repulsion between them. The idea of phonons to accelerate two particles to approach (or to keep away from) each other is easily imagined by the phase of their motions. The acceleration becomes a maximum when the phase difference is 180 degrees where the wavelength is a minimum. Thus, the force has a maximum determined by the number of oscillations restricted by the lattice structure and cannot be increased at will to realize the d-d fusion reactions. In the continuous medium, there is no finite minimum wavelength and therefore no limitation for the degree of freedom of photons to add up to become infinity. This is the case of QED where photons played decisive roles in several electromagnetic interactions between charged particles. Thus, we understand why phonons do not play a spectacular role in assisting nuclear fusion reactions of d-d or p-d pairs. 3.3 Electric Field There are other trials which use the electromagnetic force possibly existing in CF 11

12 materials to make d-d fusion reactions feasible without any exterior acceleration mechanism. In this case, the difference of masses of the electron and nuclei, or typically the proton, is an obstacle to them ([1] Section 3.4.4). The CF materials are mainly metals in which itinerant electrons are easy to move under an electric field. If there is an electric field, a charged particle receives a force proportional to its charge. A force to a particle means there is an acceleration that is inversely proportional to its mass. The velocity gained in a unit time is proportional to the acceleration exerted. Therefore, an electron with mass m e moves under an electric field about 1800 times faster than a proton with a mass about 1800 times m e. The effect of the electric field therefore absorbed by electrons before it works on deuterons or other charged particles. Magnetic field has usually weak effects on charged particles with small velocity and we need not care much about it in the CFP. Thus, several factors in CF materials that are not in the free space seem to have no respectable effects on the CFP. If these factors do not have fantastic effects on d-d fusion reactions, it is far from expectation to have tremendous effects on three or more particle fusion reactions as some researchers dream about. There are several standard treatments of d-d fusion reactions in CF materials by physicists in nuclear physics and plasma physics denying increases of fusion probability by huge orders of magnitude to realize the CFP in CF materials at around room temperature [5, 6, 7]. 3.4 Neutrons As we explained in section 2.1, existence of thermal neutrons should be considered to be one of the necessary conditions. As is well known, the neutron is a particle unstable in a free state and decays by beta disintegration with the decay constant of 886.7±1.9 s (or the half-life of 616 s) into a proton p, electron e and neutrino ν e liberating energy of MeV (Caso 1998). n p + e +ν e MeV. (3.5) Mass of the neutron m n is kg or m e, where m e is the electron mass; kg, Wave Nature of Neutron It is well established now that any microscopic particle has wave nature and it is true 12

13 for the neutron, too. This nature of the neutron is widely used in many applications as in the neutron diffraction and neutron optics (cf. [2] Section 12.17). A neutron with a momentum p has a characteristic wavelength (called de Broglie wave length) λ = ħ/ p = ħ/(2m n E) 1/2 (3.6) which takes a value cm for a kinetic energy of 88 mev and cm for 25 mev (the thermal energy at 300 K). The wave nature of the neutron is used widely in neutron diffraction for structural analyses of materials and also in neutron traps in multi-layer structure of crystals ([2] Section 12.17). The latter may have a close relation with our trial of quantal verification of the premises of the TNCF model. Recommendation of Phenomenological Approach The experimental results briefly surveyed in Chapter 3 and difficulties summarized in this Chapter clearly show that the phenomenological approach to the CFP introduced above in Section 3.4 and fully explained in the next Chapter is the appropriate method to develop the science of the cold fusion phenomenon at the present stage of investigation. The phenomenological approach to the CFP is strongly supported by the difficulty to explain various events in the CFP as a whole by a theory which uses partial improvements of the conventional techniques as shown in the next section. 4. Phenomenological Approach using Concepts which do not contradict the Knowledge of Physics As was shown in previous chapters, the cold fusion phenomenon contains complicated events occurring in various materials appealing to a new approach which is different from existing sciences. As the history of science shows, it is necessary to use a phenomenological approach to such a phenomenon for which known knowledge contradicts the facts obtained in it. In the phenomenological approach, we use premises (assumptions) based on facts even if they contradict with knowledge of established sciences. 4.1 Characteristics of Experimental Facts used in the Phenomenological Approach As surveyed in Section 2, there are observed several characteristics of the CFP which are difficult to explain by using knowledge of nuclear physics and solid-state physics. In Section 3, we investigated the difficulty to explain these experimental facts using knowledge of modern physics. The standard approach to such a problem far 13

14 distant from the common sense of present day science should be a phenomenological one as the history of science tells us. The key factor we have used in the model for the CFP is the role of neutrons to participate in nuclear reactions in room-temperature solids. We summarize here experimental facts related to neutrons with an addition of theoretical success obtained by the phenomenological model. The background neutron is composed of neutrons with various energies ubiquitous around us. Their energy ranges from thermal (about 25 mev) to epithermal (up to about 1 ev) and higher. They are mainly remains of neutrons generated at higher atmosphere by collisions of high-energy cosmic ray with gases of atmosphere. At first, the generated neutrons have energies about several million-electron-volt (MeV) and after many collisions with nuclei of gases in atmosphere reach to surface of the earth. The number of their flow density at middle latitude zone is about 10 2 /cm 2 s. Tritium on the surface of the earth is mainly generated by the reaction of the background neutrons with a deuteron (nucleus of deuterium) contained in water about 0.02 % (1/5000) ([1] Topic 8). 1. Positive evidence of neutron effect There are several evidences of neutron effect on the CFP. (Shani, Celani, Lipson, [2] Section 8.2) 2. Null result without thermal neutrons There are several definite evidences showing no CFP when there is no background neutrons (Ishida, Jones, [2] Section 8.1) 3. Success of models with neutrons The TNCF model assuming existence of thermal neutrons in CF materials has shown its ability to give qualitative and semi-quantitative explanations for experimental results. (Section 4.3 and [2] Section 11) Thus, it is natural to follow a phenomenological approach as far as we can go. 4.2 Underdeveloping Area of Nuclear Physics and Solid-State Physics To develop a new model based on experimental facts, it is necessary to assume several presumptions not contradictory to the knowledge of modern science. In relation with presumptions of the model explained in later Subsections, we point out here several developing fields in nuclear physics and in solid-state physics. The under-developing field introduced in this section will have a close relation to our quantal verification of our model given in Section 4.6. Exotic nuclei. Exotic nuclei with a large excess number of neutrons have been observed recently 14

15 which have extended wavefunctions of neutrons out of the periphery of the nucleus (especially for nuclides with medium mass numbers). It is also in the process of investigation to know the nature of the wavefunction of neutrons at excited states around zero energy level. Delocalization of proton/deuteron wavefunctions in Ti, Ni, Pd It is not well known why hydrogen isotopes in Ti, Ni, Pd, - - where the CFP occurs have different character from those in Mo, Ta, V, where no CFP is observed. One difference known is the localized wavefunctions of protons in the latter and perhaps non-localized ones in the former. 4.3 Phenomenological TNCF Model assuming Thermal Neutrons in CF Materials From the experimental facts obtained in the CFP, we can establish a phenomenological model to investigate events in the CFP as a whole discarding other approaches such as a theoretical explanation of some specific events starting from the first principles of physics Premises of the TNCF Model The TNCF model is a phenomenological one and the basic premises (assumptions) extracted from experimental data sets are summarized in the previous works [1, 2]. We give here explanations of some important premises in terms of experimental facts which the premises rest on. To consider the fact that the CFP occurs only when there are thermal neutrons, we assumed the Premise 1. Premise 1. We assume a priori existence of the quasi-stable trapped thermal neutrons with a density n n (which is an adjustable parameter). The density n n is determined by an experimental data set assuming the same interaction cross-section σ of the neutron and another nuclide to that in a free space. The quasi-stability of the trapped neutron means that the neutron trapped in the crystal does not decay until a strong perturbation destroys the stability while a neutron in the free space decays with a time constant of 887.4±0.7 s (the half-life of 615 s). To take into the fact that the nuclear reactions in the CFP occur at surface/boundary region, we assumed a multiplicative parameter ξ in the Premise 2. Premise 2. The trapped thermal neutron in a solid reacts with another nucleus in the surface/ boundary regions of the solid, where it suffers a strong perturbation. The reaction of the trapped neutron with another nucleus in these regions occurs as if they are in the free space. We express this property by taking the parameter with values ξ = 1 15

16 at the surface/boundary regions and ξ = 0.01 otherwise as suggested by experimental results. To determine the value of the parameter n n in terms of experimental results, we used following Premises 4 7. Premise 4. Product nuclei of a reaction lose all their kinetic energy in the sample except they go out without energy loss. Premise 5. A nuclear product observed outside of the sample has the same energy as its initial (or original) one. Premise 6. The amount of the excess heat is the total liberated energy in nuclear reactions dissipated in the sample except that brought out by nuclear products observed outside. Premise 7. Tritium and helium measured in a system are accepted as all of them generated in the sample. To convert the amount of the excess energy Q to the number of nuclear reactions N Q, we assumed Premise 11 rather arbitrary considering experimental result as a whole. Premise 11. In the calculation of the number of an event (a nuclear reaction) N Q producing excess heat Q, the average energy liberated in a reaction is assumed as 5 MeV unless the reaction is identified: N Q = Excess heat Q (MeV)/ 5 (MeV). 4.4 Neutron Drop Model Extension of the TNCF Model When there are neutron drops in cf-matter formed around surface/boundary regions by the mechanism discussed above, we can use the neutron drop A ZΔ and a small neutron-proton cluster A Zδ in the nuclear reactions as a simultaneous feeder of several nucleons to nuclides; A Z Δ + A Z X A a Z zδ+ A + a Z + zx *, A a Z zδ + A + a a Z + z z X + a z X, (4.1) A Z Δ + A+ 1 ZX * A Z Δ * + A + 1 ZX A Z Δ+ A + 1 ZX + x, (4.2) A Z δ + A ZX A + A Z + Z X * A + A Z + Z X + x. (4.3) The neutron-proton cluster A Zδ is supposed to be a unit of nucleons absorbed simultaneously by a nuclide to form a new nuclide as in Eq. (4.3). In the reactions (4.2) and (4.3), the symbol x means not a photon in the free space but another particle (a neutron or a neutron-proton cluster) in cf-matter [8]. 4.5 Experimental Data explained by the Models Using the models with an adjustable parameter, we could explain experimental data 16

17 sets consistently. The most fascinating explanation is given for several observables measured simultaneously adjusting a single parameter n n as explained in the end of Subsection 2.5; (N a /N b ) th m (N a /N b ) ex, (m = 3 5). (4.4) Concrete results have been given in tables in the previous works [1, 2]. 4.6 Quantum Mechanical Explanation of Premises assumed in the Model Neutrons in solids are not fully investigated until now perhaps because of their short lifetime of about 887 seconds in the free space [9]. However, the wave nature of the low energy neutron has been used more widely in technology (the neutron guide and others) and science (the neutron trap to study nature of neutrons). From our point of view, however, the research is in an infantile stage and we have much work to do in studying neutron physics in solids, especially fcc transition-metal hydrides and deuterides. In addition to the behavior of neutrons in the solids there are undeveloped fields of neutrons in exotic nuclei as described in Section 4.2. Thus, we can expect new states of neutrons in transition-metal hydrides and deuterides when there is an optimum situation where several conditions are fulfilled to realize the neutron valence bands below zero as discussed in previous papers [8, 10] and also expect new phenomena related to the neutron valence band. Several words should be added about the states of neutrons in the boundary regions in solids. At boundaries of a crystal, there is aperiodicity of the crystal lattice and disturbance to the neutron Bloch waves. There appear new states due to the disturbance such as surface states different from the Bloch states with different energies. As was noticed in Chapter 2, there is much evidence of nuclear reactions in CFP that is difficult to explain without participation of neutrons, including those called decay-time shortening and NT in surface layers of electrodes in electrolytic systems and in surface regions of cathodes in discharge systems. In the TNCF model, this surface nature of CFP is taken into the model by the instability factor ξ of the trapped neutron assuming a value 1 (ξ = 1) in the surface layer and 0.01 (ξ = 0.01) in volume [1, Section 3.2]. 5. Complexity in the Cold Fusion Phenomenon revealed by Experimental Facts In addition to the individual facts of excess power generation and the nuclear transmutation, it is useful to have such new materials to construct the science of the CFP as statistical regularities or laws for observables in the CFP. In this case, the approach is 17

18 fundamentally qualitative as many cases should be in complexity. So it is somewhat different from where we have an individual fact supposedly the result of a nuclear reaction. There are several experimental data sets suggesting that the CFP is a phenomenon characterized by complexity. The composition of the materials where the CFP occurs suggests to us an idea that the CFP should have a nature of complexity and these data have given an evidence supporting the suggestion. The inverse-power law for the frequency-intensity relation, bifurcation, chaotic, and recursion-like evolution of effects have been discovered in several extensive data sets. It is interesting to notice that these statistical laws seem to be qualitatively explained by the TNCF model using the adjustable parameter n n as the parameter of the l.d.e. explained below. 5.1 Theoretical Foundation of Complexity in Cold Fusion Materials There are theoretical works which revealed interesting features of the recursion relations f(p) = p b eff (p) (5.1) where the recursion function f(p) satisfies a condition figuratively expressed in Fig. 1 (the Feigenbaum condition). Fig. 1. Dependence of f(p) = p b eff (p) on p after Feigenbaum [11]. When the recursion function which satisfies the condition depicted in Fig. 1 18

19 figuratively, it has been investigated its dependence on the parameter using the logistic difference equation, or l.d.e. (a simplified form for the function f(p) in Eq. (5.1)); x n+1 = λ x n (1 x n ) (0 < x 0 < 1). (5.2) The dependence of the behavior of the system depicted by the l.d.e. (Eq. (5.2)) on the parameter is shown in Fig. 2. There appears period-doubling and chaos in terms of the increase of the parameter [12]. The main figure in Fig. 2 depicts x on the ordinate (x is x n at n = ) vs. the parameter λ on the abscissa of the logistic difference equation, i.e. l.d.e., Eq. (5.2). The inserted figures in Fig. 2, a) Steady state, b) Period two, c) period four, and d) chaos, depict variations of x n with increase of suffix n (temporal variation if n increases with time) for four values of the parameter λ; a) 1 < λ < 3, b) 3 < λ < 3.4, c) λ 3.7, d) 4 < λ. The region a), b) and d) correspond to Steady state, Period two and Chaotic region in the main figure, respectively. The features of the recursion relations illustrated in Figs. 1 and 2 have been found in experimental data sets obtained in the CFP as shown below. Fig. 2. Bifurcation diagram to show period- doubling and chaos (From Chaos by J. Gleick [12]. p.71). 19

20 5.2. Data set by De Ninno et al. The experimental data set obtained by De Ninno et al. [13] shows neutron emission from TiD x samples in dynamical processes of absorption/desorption of deuterium when the sample temperature T varied between 77 K and the room temperature. As was explained in the previous paper [14], the variations of the neutron emission shown in Fig. 3 simulate the characteristics of the l.d.e. depicted in Fig. 1 [12]. In this figure, the abscissa represents time elapsed and also the temperature of the CF material TiD x. As the data by Dash et al. have shown, the temperature of the sample works as a parameter which measures feasibility of the occurrence of excess heat generation (one of the main events in the CFP). If we can consider the neutron emission is also described similarly to the excess power, the abscissa of Fig. 3 corresponds to that of Fig. 1. And then the envelope of the data illustrated in Fig. 3 is compared to the curve in Fig. 1 showing an event (neutron emission in this case) of the CFP as described by a recursion relation satisfying the Feigenbaum condition shown by the curve. Fig. 3. Diagram showing the temporal evolution of the neutron emission from TiDx sample during the run A (April 15-16, 1989). The values indicated on the ordinate are integral counts over periods of ten minutes (Fig.3 of [13]). Fig. 4. Diagram showing the temporal evolution of the neutron emission counts (ordinate) during the run B (7-10 April, 1989) by De Ninno et al. [13]. The values indicated on the ordinate are integral counts over periods of 10 minutes. 20

21 We notice a characteristic of the time pattern of neutron emission appeared in Fig. 4 that there are two levels of emission as if they are quantized. We can give a possible explanation for this behavior by the bifurcation as appeared in Fig. 2.The inserted diagram period two shows an appearance of two stable states by bifurcation according to the increase of the parameter λ in Eq. (5.2). We may identify two levels that appeared in Fig.4 as the two states shown in the period two diagram in Fig Data set by McKubre et al. The experimental data set of excess energy generation in Pd/D 2 O + LiOD/Pt electrolytic systems by McKubre et al. [15] is one of the most extensive calorimetric results obtained in this field. The excess power P ex (W/s) has been observed as functions of the loading ratio η (= D/Pd) and electrolytic current density i (A/cm 2 ) in addition to temporal variation of P ex. Based on our quantum mechanical investigation, we may assume that the number density n n of trapped neutron increases with η which makes the super-nuclear interaction more effective to build up the neutron band below zero [1, 8] when other conditions are kept constant. We investigate the data by McKubre et al. [15] from this point of view. Then, Fig. 5 shows the temporal evolution of their data ([15] Fig. 5) which is to be compared with the inserted figures of Fig. 2 of [12]. Fig. 5. Variation of Excess Power, Uncertainty and Loading ratio with time [15]. In the initial part from 400 to 550 h of Fig. 5, the loading ratio η increases from 0.8 to 0.92 and the excess power P ex increases up to about 0.8 W while with a burst at around h. This increase of P ex with η (or n n in our interpretation) reminds us of the recursion function f(x) shown in Fig. 1; the excess power P ex, which is proportional to n n, increase with n n until a point where f(n) becomes a maximum. The bursts of P ex appear while η does not change much as we see in Fig. 5. We may interpret these variations of P ex as a presentation of bifurcation as appeared in inserted figures of Fig. 2 in the case of the l.d.e., Eq. (5.2). 21

22 Fig. 6 Variation of Excess Power with Loading ratio [15] Figure 6 shows the variation of the excess power P ex as a function of the loading ratio ([15] Fig. 7). The excess power depicted in this figure shows such a chaotic behavior for η > 0.89 as appeared in Chaotic region of Fig. 2 of [12]. Furthermore, there appear several chaotic behavior at I 0.1, 0.4, 3.9, 5.1 and 7.1 in Fig. 7 where are plotted P ex vs. cell current I. As far as we know at present, there are no explicit relations between the cell current I and the parameter n n (density of the trapped neutrons) we used to explain Fig. 5 and 6. The occurrence of the chaotic behavior in Fig. 7 may be accidental depending on other uncontrollable factors in the system. Fig. 7 Variation of Excess Power with Cell Current 22

23 Fig. 8. Excess power pulses during a 14 hour period of an experiment (070108) of Kozima et al. [16] which lasted 12 days as a whole. 5.3 Data set by Kozima et al. Experimental data set of the excess energy in Pd/D 2 O + H 2 SO 4 /Pd electrolytic systems by Kozima et al. [16] gives another illustration of complexity in the CFP. They observed temporal variations of P ex with bursts up to 20 W as shown in Fig. 8 similar to those observed by McKubre as shown in Fig. 5. These features of the excess power P ex as a function of time remind us of the behavior of the l.d.e. as shown in inserted figures of Fig. 2 of [12]. 5.4 Inverse-power dependence of frequency of events on their intensity With a statistical treatment of the data sets obtained by Dash et al., we could show the inverse-power dependence of the effect in the lower parts and the bifurcation of the excess power generation in the upper parts of x (= P ex ) of Fig. 9 [16]. To plot this figure, we assumed that the amount of excess power increases with temperature of the sample generalizing property of the system that P ex is finite only if the temperature of the system is higher than a critical temperature T c 90 degc. Fig. 9. Distribution of the frequency N P (= y) producing excess power P ex (= x). To depict log-log curve, values of N P and P ex were arbitrarily multiplied by 10 n. (x = 100 in this figure corresponds to P ex = 1 W). The data points in the range from x = 20 to 60 are considered to be on a straight line with a gradient 2. 23

24 We recognize the linear dependence of y (= N P ) on x (= P ex ) in Fig. 9 at lower parts of x. The behavior of P ex (= x) depicted in the upper parts of x illustrates explicitly the bifurcation of the state in the l.d.e. appeared at 3 < b < 3.4 in Fig. 2. The same regularity, the inverse-power dependence, of the excess power is also shown by the data sets obtained by McKubre et al. and by those collected by E. Storms as depicted in Figs. 10 and 11, respectively. In Fig. 11, we see a deviation from the straight line at large values of the excess power similar to that we see in Fig Inverse-Power Law of Excess Heat Generation log N i logp with P in W, log P = -1 (0.1) +0.3 Fig. 10. log N Pe vs. log P with a gradient 1 ([1] Fig. 2.13). Fig. 11. Excess Heat results from 16 years of Cold Fusion experiments with a power output equal to, or larger than, 0.5 W (N = 125): The frequency distribution is characterized by an inverse-power law with an exponent of (By Haiko Lietz [18] with data collected by Storms [17]). 5.5 Positive Feedback of Nuclear Reactions in the CFP We have shown several examples of the bifurcation (or bifurcation-like behavior) of 24