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Physics Procedia 20 (2011) 404–419

Space, Propulsion & Energy Sciences International Forum - 2011

Conventional physics can explain cold fusion excess heat
S. R. Chubb*
Infinite Energy Magazine, 9822 Pebble Weigh Ct, Burke, VA 22015-3378

Abstract In 1989, when Fleischmann, Pons and Hawkins (FP), claimed they had created room temperature, nuclear fusion in a solid, a firestorm of controversy erupted. Beginning in 1991, the Office of Naval Research began a decade-long study of the FP excess heat effect. This effort documented the fact that the excess heat that FP observed is the result of a form of nuclear fusion that can occur in solids at reduced temperature, dynamically, through a deuteron (d)+d 4He reaction, without high-energy particles or rays. A key reason this fact has not been accepted is the lack of a cogent argument, based on fundamental physical ideas, justifying it. In the paper, this question is re-examined, based on a generalization of conventional energy band theory that applies to finite, periodic solids, in which d’s are allowed to occupy wave-like, ion band states, similar to the kinds of states that electrons occupy in ordinary metals. Prior to being experimentally observed, the Ion Band State Theory (IBST) of cold fusion predicted a potential d+d 4He reaction, without high energy particles, would explain the excess heat, the 4He would be found in an unexpected place (outside heat-producing electrodes), and high-loading, x 1, in PdDx, would be required.

© 2011 Published Elsevier B.V. Selection and/or peer-review under under responsibility of Institute for © 2011 Published byby Elsevier B.V. Selection and/or peer-reviewresponsibility of Institute for Advanced Advanced Studies in the Space, Propulsion and studies in Space, Propulsion and Energy Sciences Energy Sciences
PACS: 03.65.XP, 03.75.-b, 03.75Kk, 03.75.LM, 03.75.Nt Keywords: Cold Fusion; Ion Band States; Nuclear Fusion in Solids; Energy Band States

1. Introduction In 1989, when Fleischmann and Pons (FP) [1] made their initial cold fusion announcement, considerable confusion occurred. A key reason for this involved a misunderstanding about what they actually found in their experiments. Initially, everyone believed that Jones et al. [2] and Fleischmann, Pons, and Hawkins discovered the same thing: “cold fusion.” Jones did observe neutrons, but they probably came from either a very low-level, conventional hot fusion reaction, or some other nuclear

* Corresponding author. Tel.: 703-309-0493; fax: +0-000-000-0000 . E-mail address: scott.r.chubb@alumni.princeton.edu .

1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Institute for Advanced studies in Space, Propulsion and Energy Sciences doi:10.1016/j.phpro.2011.08.036

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process that does not involve fusion; while although FP initially claimed they found neutrons, they found them at a level that was a billion times too low to explain their most important discovery: excess heat. In fact, it is known now that the large amounts of excess heat that FP observed occurred from a form of fusion, but not hot fusion. FP discovered their excess heat from a nuclear reaction that occurs when a deuteron (d) fuses with a second d, to form 4He, without the “usual” gamma ray that occurs in one of the conventional (but infrequent) hot fusion reactions. Because FP’s discovery is fusion, but not hot fusion, the name cold fusion actually fits, and it is more appropriate to think of the neutrons that Jones and FP found as occurring either from hot fusion or through Low Energy Nuclear Reactions (LENR) that do not involve fusion; while the reaction that created excess heat that FP found is cold fusion, and it has no neutrons. Although considerable controversy followed the initial announcements, research at a small number of laboratories continued. And with time, this number has grown significantly. Even at an early stage in the associated research, the amounts of excess heat that were observed, which were so large that they could not be accounted for by any possible chemical energy process, were confirmed by a number of experienced electrochemists and material scientists. Some of the individuals who were involved in the more important, early successful experiments were McKubre, Tanzella, Crouch-Baker, Huggins, Oriani, Bockris, Miles, and Arata and Zhang, as documented in the book by Beaudette [3]. The Office of Naval Research conducted a decade long study that also documented the heat is real and that it occurs from the d+d 4He reaction [4]. The most precise and quantitative relationship between total amount of excess heat that accumulates over time and the comparable accumulation of helium atoms was reported by McKubre et al. [5]. Here, it was found necessary to chemically thermally cycle the palladium cathodes to drive much of the helium out of the metal into the gas phase. An important, more recent development is the fact that excess heat can be created through the same kind of d+d 4He reaction (without high energy particles) when deuterium gas is loaded into nanometer scale crystals of Pd and other composite materials involving Pd [6]. Talbot Chubb and the author suggested in 1989 that deuterons (d’s), in fully-loaded palladium deuteride (PdD), could behave very differently than in free space, by occupying the kinds of states (energy band states) that electrons occupy in periodically ordered solids. Based on this conjecture, it was suggested [7, 8, 9] that the normal rules about fusion might not apply to the cold fusion (CF) claims by Fleischmann and Pons (FP) and that it might be possible to account for what FP suggested through alternative forms of fusion that could result from the very different physical situation that might be possible, as a consequence. Because evidence exists [10, 11, 12] that in certain situations, hydrogen and deuterium nuclei occupy these kinds of “wave-like,” ion band states (IBS’s) in Casella [13] and on the surfaces [10, 11, 12] of transition metals, a physically plausible argument can be made that the starting point of the associated argument makes sense. Because d’s are bosons, on the length scales associated with conventional electromagnetism, implicit in this hypothesis is the idea that by occupying the lowest energy (band) state, the d’s could form a Bose Einstein Condensate (BEC). At the time this suggestion was made, confusion about its relevance resulted [14] because it was widely believed that BEC’s could form only at very low temperatures, as opposed to a situation in which they are induced (as in the case of laser-cooling of alkali vapors) dynamically through the presence of externally applied forces. In addition, the suggestion that IBS’s could be involved in cold fusion was not widely publicized for two reasons: 1) The experiments were not widely believed to be valid; and 2) Perceptions about the limitations of conventional energy band theory. Over the years, the associated IBST has evolved considerably. It has, in fact, been used to provide a number of predictions about excess heat in cold fusion that were subsequently observed. In responding to the critics of the theory, the author has shown that the basis of the criticism, associated with limitations of energy band theory, reflect biases that are the result of the approximate justification for the theory and the way this justification is usually presented. In addressing this problem, the author has developed a generalization of the conventional theory, based on a procedure (referred to as Generalized Multiple Scattering Theory), that provides a framework for both generalizing conventional energy band theory and

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for understanding how, in the context of conventional physics, excess heat in cold fusion, through coherent forms of nuclear fusion, in which the wave-like features of “particles” associated with conventional energy band theory can be formally re-expressed in a manner that quantifies how the associated effects can occur without high energy particle emission. The paper is organized in the following way. In the next section, an overview of IBST is presented that provides important background information about the new physics that is assumed in the associated description, as well as effects that the theory has predicted. In the following section, the key features of Generalized Multiple Scattering Theory are reviewed. In the final section, an explanation is provided of how nuclear reactions in PdD can take place, again, based on the conventional laws of physics. 2. Overview Of The Ion Band State (IBST) Through the associated ion band state theory (IBST) [6, 7, 8, 9], a number of effects were predicted that were subsequently observed experimentally, including the following:
1. That 4He is created when excess heat is produced at appropriate levels that account for the heat through a nuclear reaction [15, 16, 17, 18]; 2. That the 4He is found in an unanticipated location, outside heat-producing electrodes [15, 16, 17, 18]; 3. That there is an apparent requirement that for the effect to occur, “high-loading” (defined by the limit x 1 in PdDx) is necessary [5]; and 4. That the effect occurs without the emission of any high energy particles or radiation [4, 15, 16, 17, 18].

The “high-loading” limit, which is emphasized in this paper, applies to electrolysis experiments. In gas-loading experiments [6] involving nm powders of Pd, “high-loading” might not be required as a result of interfacial chemistry effects. In responding to the critics of the IBST, the author identified important effects, involving particle indistinguishability, degeneracy, and band theory that can be applied not only to the PdD fusion problem but in more general terms. In particular, although energy band theory is usually formulated in terms of the semi-classical limit of a single-particle, eigenvalue problem, as it applies to infinitely-repeating periodic systems; it was actually derived by Bloch through a formalism (multiple scattering theory) that has more general applicability. In fact, although conventional multiple scattering theory is defined in terms of the scattering properties of single particles with an array of scattering centers, it can be generalized [19],” to situations involving arbitrary collisions between arbitrary numbers of particles in many-body systems. Although the resulting equations are usually so general that they are intractable, in a limited set of circumstances, involving finite, periodic lattices, they can be used to establish a hierarchy of processes, involving different rates of reaction, as a result of approximate translational invariance. Within this context, conventional energy band theory can be derived as a limiting situation associated with the lowest energy excitations of these kinds of lattices. These occur through perfectly resonant forms of interaction, involving rigid translations of the lattice in situations in which fluxes of particles into and away from the region of space occupied by the lattice vanish. Implicit in this limit is the importance of approximate translation symmetry. In fact, because it is never possible to determine the precise location of the boundaries of a lattice, it is never possible to tell precisely where it begins and ends and whether or not its center-of-mass (CM) is in motion or at rest. Implicitly, this result is associated with a form of Galilean relativity: it is never possible to tell if an outside observer is in motion or at rest, relative to a periodic (or approximately periodic), finite array of atomic centers and electrons. As a result of this symmetry, a huge degeneracy exists, associated with rigid translations, in which the separations between all of the particles remain fixed. In fact, this symmetry, which also is a (somewhat trivial) form of gauge symmetry, provides a mechanism for the relative momentum between one particle and a second particle to change arbitrarily (as a result of generalized forms of Umklapp processes, which can also be viewed as forms of broken gauge symmetry) at a point through implicit forms of coupling involving the electromagnetic field during

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collision processes. (This is the basis of the Mossbauer effect.) In more general terms, the associated many-body configurations, associated with finite, ordered lattices provide a environment for particleparticle collisions that are dominated by the effects of particle indistinguishability and degeneracy. This fact provides justification for a number of important, new results, associated with collisions in solids that have not been widely appreciated. In particular, at an early stage, our suggesting that if d’s occupy IBS’s (and form what we referred to as Bose Bloch Condensate) it might be possible for new forms of fusion to take place was widely criticized [14], based on the observation that conventional band theory only applies in situations in which details about particle-particle interactions can be ignored. In fact, although in conventional band theory, this appears to be the case, implicitly, an important reason for this is that a semi-classical limit of the associated physics has been widely used, and details about the underlying many-body physics have been ignored. In this context, degeneracy (or near-degeneracy), resulting from particle indistinguishability and periodic order has important implications. In particular, as in the Mossbauer effect, through a real effect, implicit in the symmetry associated with rigid lattice translations that preserve periodic order, it is possible for a lattice to “recoil” elastically, as a whole, in response to a collision at a point. In the generalization of band theory [19] to many-body, finite systems, the same symmetry is invoked and leads to a huge degeneracy. Because indistinguishable particles are involved in these systems, implicitly, additional degeneracies are also present. The combined effects provide a means for particles to have appreciable overlap at many, periodically displaced “points” (as discussed below), simultaneously, for finite periods of time, in a manner that can result in new forms of collisions in which momentum is transferred from the locations where overlap can occur, rigidly to the lattice as a whole. When these idealized forms of motion are initiated by collisions resulting from the overlap between d’s in IBS’s, they can result in forms of coupling that can cause nuclear fusion to take place in which small amounts of momentum and energy from many different locations are transferred coherently to the solid as a whole and subsequently transferred to many different particles in a cooperative fashion. As a consequence, in agreement with experiment, the associated nuclear energy is predicted to be released without high-energy particles. Because the theory assumes the underlying physical situation involves collisions that can arise from very similar configurations that are degenerate or nearly degenerate, the resulting process, if it occurs, is very different from processes that involve collisions in free space. Furthermore, as opposed to the situation in free space, where, in order for fusion to occur, d’s are “forced” to occupy a common volume (by over-coming a Coulomb Barrier), in the situation in the solid, effectively, a form of decay can occur, in which d’s not only are allowed to have appreciable overlap (in order to minimize energy) by occupying IBS’s, but 4He can be created (also in an IBS form). Thus, as opposed to being forced to occupy a common volume externally (as in free space) by requiring the d’s to have high velocity, through quantum mechanical effects the solid can cause the d’s to do this, without requiring that any of them have appreciable velocity. An important point is that the Bose exchange symmetry of d’s, as well as the possible processes associated with the underlying periodic order (and the potential, coherent forms of interaction that can occur when d’s occupy IBS’s) are associated with a very particular limit, involving low energy (ion band state-like) fluctuations in charge, in which only a very small concentration of d’s (~10-4 per unit cell) is involved. It is also important that this limit, which is consistent with the electronic structure of the relevant structure [5, 20], which involves highly loaded PdD (defined by PdD x , x 1), is predicted to involve a dynamical process that actually minimizes the underlying Coulombic repulsion associated with d-d fusion [8]. In the lowest energy configurations, the associated collisions release momentum and energy at the boundaries of the lattice. An important point is that the associated degeneracy (and near degeneracy) and resulting overlap and interaction are coupled to each other through interaction with the lattice (and, implicitly, its electronic structure). In this context, as opposed to over-coming the conventional “Coulomb Barrier” through processes that are initiated when localized particles possessing a well-defined

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S.R. Chubb / Physics Procedia 20 (2011) 404–419

(nearly constant) relative momentum at high velocity collide at a point, a more sophisticated (implicitly time-dependent) “Barrier” is involved. Because the underlying dynamics is associated with rigid lattice recoil, the associated problem of overcoming the “Barrier” (as outlined below) is related to crystal size and a number of additional environmental factors. In the case of PdD, a key result (that is also observed [5, 20]) involves the prediction that a near full-loading ( PdD , 0) condition is required for excess heat to be created. (A 1 new result is the prediction that this near full-loading condition is required because approximate periodic order is necessary for the d’s to occupy band states and, because of the anti-bonding characteristics of the electronic structure in PdD, which is responsible for the d’s behaving as ions.) Because the reaction is initiated from a state in which interacting d’s possess Bose exchange symmetry, details about the electromagnetic interaction are important far from the location of the reaction, and (as opposed to the situation in the more common ( d d 3 H p and d+d 3He+n) fusion reactions, the electromagnetic and nuclear interactions are not separable. (In the case of the “rarelyoccurring,” d+d 4He+ , reaction a similar lack of separability between electromagnetic and nuclear interactions also takes place.) Because of this fact and because of the large degeneracy and overlap with many-body configurations in which the final state product possesses 4He, through the IBST, it was predicted [7, 8, 9, 21] (prior to observation) that (in order to minimize energy), the fusion process that creates this product in a solid (as opposed to 3H and protons or 3He and neutrons) would be dominant in the heat-producing reactions. Because of the requirement that periodic order be present, through the theory, it was suggested that this product would be initiated through the occupation of transient 4He in IBS’s that would be neutralized at the boundaries of the solid, where it would be released in atomic form (again prior to and in agreement with experiment [15, 16, 17, 18]). In the next two sections, it is demonstrated that appreciable reaction rates for fusion of the form, d+d 4He + 23.8 MeV ( in which the 23.8 MeV is dispersed throughout the solid), can occur, near fullloading in PdD. Because the associated rates result from the lowest energy excitations of the lattice, they can be estimated using the generalization of multiple scattering theory [19, 22]. In the resulting limit, overlap between d’s in IBS’s is allowed to take place inside the lattice near nuclear dimension (defined when the separations between the CM’s of d’s become vanishing-ly small on the length scales associated with electromagnetic interaction in the lattice) as a result of discontinuous changes in momentum (wave function cusps) at the locations where this takes place. (These kinds of abrupt changes in momentum are allowed to take place, as a consequence of broken gauge symmetry, in which the gauge associated with the vector potential is allowed to shift by a constant amount in one region—the nuclear region, where nuclear overlap is allowed to take place—relative to the rest of the solid) through rigid translations of the lattice. Nuclear reaction (which, technically occurs outside the lattice) involves a transition at these locations through processes associated with d+d fusion that can be modeled (in a manner that is consistent with multiple scattering theory and the constraints associated with a lower bound on reaction rate) through a shift in the zero of energy of the lattice and the coherent transfer of momentum from the reaction to the solid as a whole. 3. Generalized Multiple Scattering Theory And Its Relationship To Band Theory The wave-like behavior of quasi-particles that occurs in the conventional energy band theory formalism can be rigorously understood from wave-particle duality effects that result from a form of Galilean Relativity. In particular, for a solid possessing total mass MT, each possible velocity ( V j ) associated with the relative motion of an observer and a solid that is stationary involves a different centerof-mass momentum Pcm,i M T V j , which in turn can be used to define a particular wave-vector,

S.R. Chubb / Physics Procedia 20 (2011) 404–419

409

ki

m

Pcm ,i MT

,

(1)

for each particle possessing mass m. Perfect resonance [23] occurs when the possible values of the wavevector ki are used to define the energies and 2) That each value of ki i

ki of possible excitations of the system, and through

two requirements [19]: 1) That the set of allowable wave-vectors be restricted to the first Brillouin zone; possess a discrete (invertible) Fourier transform, associated with the (finite) set of Bravais vectors Rn that is used to define the lattice:

ki n n

eiki

Rn

, ,

(2) (3)

n

1 N cell

ki e i iki ¥Rn

where Ncell is the number of unit cells in the lattice. The many-body wave function of each excitation can be expressed through a generalized form of Bloch’s theorem [19, 22], associated with the band energies and wave-vectors used in equations (2) and (3). As a consequence of these equations and the relationship between the wave-vectors and the large number of possible Galilean transformations, associated with rigid translations, it follows that an implicit degeneracy is present in which each value of ki ki ki

satisfies
Gn

(4)

and is associated with a situation in which the lattice as a whole is allowed to move rigidly with a CM Gn , defined by one of the reciprocal lattice vectors Gn . momentum Pcm ,n Although equation (4) holds only in the absence of perturbations in the interior of the solid and at its boundaries, in fact, on some time scale, it actually applies for all of the particles and low-lying excitations (which may involve quasi-particles, including phonons) in the system. As a result, a huge degeneracy exists. In practice, perturbations, which break this symmetry, always exist, and for this reason, usually, the only ones that have an impact on the dynamics appear to be the ones that involve electrons that occupy energy band states or phonons. An important point is that implicitly the degeneracy associated with equation (4) potentially can result in a situation in which the energy of a perturbation results in a coupling between many different particles and/or states, each involving a different CM momentum, that preserves periodic order in the interior of the solid. In situations in which d’s occupy IBS’s, provided the crystal lattice is sufficiently large, values of Pcm ,n Gn can be significantly larger than the momentum that is required to initiate a nuclear reaction. For this reason, provided overlap between d’s can take place over length scales associated with nuclear reaction (and we shall see that this can occur when the relative momentum between interacting d’s changes sufficiently rapidly over short distances), fusion can occur. When d’s interact and undergo fusion by occupying IBS’s, the degeneracy associated with equation (4), which is the basis of Bloch’s theorem, leads to forms of overlap and interaction which have no counterpart in conventional fusion in free space, where a semi-classical picture applies in which momentum is transferred only between two particles and is constrained to change slowly. Because in the situation involving energy band states, the associated change in momentum involves a form of rigid lattice recoil, it can provide a mechanism for transferring energy and momentum, non-locally, in an

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elastic fashion that implicitly leads to a coupling between many particles. For this reason, potential fusion reactions that can be initiated from this kind of process need not involve particles that have high energy and momentum. A lower bound for potential nuclear reactions that can result from the associated IBS picture can be estimated using the logic that was used, involving generalized multiple scattering [19, 22], to extend conventional energy band theory to situations involving finite lattices. The associated argument involves an analysis of the time evolution of the overlap between the ground state (GS) many-body wave function with the comparable wave functions of the lowest lying excitations. The first step in the GS r ,.....,rn ,t 1 analysis is the observation that as a function of time t, because
GS

r1 ,.....,rn ,t describes the GS, it is

required to have minimal coupling with outside processes, and its overlap with any other many-body state ' r1 ,.....,rn ,t must be minimized and remain constant; while, for a sufficiently large solid, the GS and the lowest lying excitations of the solid, are required to possess the symmetry associated with rigid lattice translations, defined by equation (1). A requirement for this to occur is:

'| t

Gs

d 3 r1 ...d 3 rn d r
3

(

'* t

GS

)
(5)

' | v( r )|

GS

V V' '| | i

GS

0,

where terms in the second equality are defined by the many-body Schroedinger equations of ' and GS . In general, the associated integrations are unrestricted. To extend energy band theory to finite lattices, it is appropriate to consider situations in which minimal overlap in “bulk regions” (associated with equations (1)-(4), takes place. In this situation, the unrestricted integrations over all of the coordinates in the multi-dimensional integral, term by term, can be restricted to regions in the bulk, based on the criteria that to find a possible GS, the associated overlap between this state and other states in the bulk region is required to be minimized. In this limited context, by restricting states to have minimal overlap with GS , additional restrictions are imposed on GS (subject to the implicit assumption that, in general, at the boundaries of the bulk region, possible discontinuities in the gradient and vector potential, are allowed to take place). Then, the analysis associated with extending band theory to finite lattices proceeds by restricting the multi-dimensional integrations in equation (5), exclusively to the bulk region. In the more general situation, considered here, rate expressions can be derived for different regions (bulk and non-bulk), associated with the requirements that charge be conserved (in bulk regions) or not conserved (in non-bulk regions), subject to the constraint that in bulk regions states are required to have minimal overlap with GS . Also, in equation (5), is the matrix element associated with ' v r GS the (off-diagonal) contribution to the (many-body) particle velocity operator v, defined by its overlap with the states ' and GS :
' |v r |
GS j

d 3 r1 ..d 3 rn

3

r rj

1 m j 2i

'*

rj

GS

rj

'*

ej
GS

c

'* Aeff rj

GS

,

(6)

where Aef((r)=(A(r)+A’(r))/2 is the arithmetic mean between the vector potential A’(r) associated with the state ' and the comparable vector potential A(r), associated with the state GS , and the final term in equation (5) is defined by the difference between the many-body potential energies associated with states ' and GS . . In particular, this last term, is given by

S.R. Chubb / Physics Procedia 20 (2011) 404–419

411

'|

V V' | i

GS

'|

Vem V 'em | i

GS

'|

Vs V ' s | i

GS

,

(7)

' |Vem V 'em | GS is the difference in electromagnetic potentials associated with coupling where between the vector potentials A’(r) and A(r),
' |Vem V 'em |
GS

d 3r

A r c

A' r

J r ,

(8)

defined by the associated current J(r),
'|J r |
GS j

d 3 r1 ..d 3 rn

3

r rj

ej m j 2i

'*

rj

GS

rj

'*

ej
GS

c

'* Aeff rj

GS

,

and [in equation (7)], the remaining contribution to the difference in potential energy is defined by any change in electrostatic and other (for example, nuclear) contributions to the energy, associated with the transition from ' (where the non-electro-dynamic portion of the potential energy is Vs’) to GS (which has a corresponding non-electro-dynamic potential energy Vs). When GS has minimal coupling to the bulk, equation (5) holds identically, outside the bulk, provided the total internal flux of all particles into and away from the bulk region also vanishes. Thus, if the flux of particles, across all boundaries in the bulk vanishes, and the energies of the different states are the same within the bulk region, it follows from equation (2) that, d 3r | v( r ) |
GS V

dS n

| v(r ) |

GS

(i )

1 V

d 3 r1....d 3 rn

*

(V V )

GS

(9)

where the integration in the final term can either extend only over the bulk region or regions outside the bulk, and the surface integral (associated with v(r) ) extends over the boundary of the region where the contribution to the matrix element involving V-V’ is evaluated. When the integration only includes the bulk region, and its boundary, when particle flux vanishes,
V

dS n

| v( r )|

GS

0 , and the bulk

contribution to the matrix element involving V-V’ vanishes. This fact is used to establish the extension of band theory to finite lattices [19, 22]. In general, this surface integral may include separate contributions from regions where v may become discontinuous (which are allowed to occur whenever V-V’ becomes singular). (As discussed below, an idealized limiting case, associated with locations where nuclear reaction can occur in the fusion problem, involves allowing for such discontinuous behavior—through wave function cusps.) equation (5) vanishes identically whenever the energies associated with GS and ' are the same. This means equation (9) can also be used to evaluate the rate R for any reaction that is initiated from an initial state possessing a particular E. In particular, from the Lippman-Schwinger equation, it follows [19] that each term in the expression for R conserves energy and involves the square of a matrix element involving V-V’ that (as a result of equation (7) and Gauss’ law) can be represented in terms of a surface integral

d 2r
S

o

|v r |

exact

CF exact involving a different (exact) representation of a possible final state the velocity operator v r :

CF , the initial state

o

, and

412

S.R. Chubb / Physics Procedia 20 (2011) 404–419

R where 1

2
F

E

Eexact C F

|
S

d 2r

o

|v r |

exact

CF

|2 ,
(10)

is the lifetime of the state.

4. Nuclear Reactions in PDD As noted in the last section, in the situation associated with fusion, we are suggesting that the reaction occurs when a small concentration (

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