Electron Capture in a Fully Ionized Plasma
Properties of fully ionized water plasmas are discussed including plasma charge density oscillations and the screening of the Coulomb law especially in the dilute classical Debye regime. A kinetic model with two charged particle scattering events determines the transition rate per unit time for electron capture by a nucleus with the resulting nuclear transmutations. Two corrections to the recent Maiani et.al. calculations are made: (i) The Debye screening length is only employed within its proper domain of validity. (ii) The WKB approximation employed by Maiani in the long De Broglie wave length limit is evidently invalid. We replace this incorrect approximation with mathematically rigorous Calogero inequalities in order to discuss the scattering wave functions. Having made these corrections, we find a verification for our previous results based on condensed matter electro-weak quantum field theory for nuclear transmutations in chemical batteries.
pacs:23.40.-s, 31.15.V-, 94.05.Fg
In recent years we have been working on electro-weak interaction inverse beta decay by including including electro-magnetic interactions with collective plasma modes of motion[1, 2]. We have applied this theory to electron capture in a water plasma to explain observed nuclear transmutations on the cathode surface of a chemical cell. While the original theory was formulated in terms of electro-weak quantum field theory in a many body context, a reasonable alternative relies on physical kinetic plasma theory[5, 6, 7, 8] to describe a water plasma. The theoretical kinetic model gives rise to electron capture rates per unit time per unit cathode surface area in a water plasma in agreement with the quantum field theoretical model and is in agreement with experiments.
Objections based on the kinetic model in a cold plasma were raised by the Rome group Ciuchi et.al.. They find electron capture rates about two orders of magnitude lower than our previous work. The objection was answered by pointing out that the water plasma in a chemical cell can light up the laboratory and thereby represents a hot plasma. The hot fully ionized plasma gives rise to an increased electron capture rate in agreement with experiment and in agreement with our previous results. This has been previously and fully discussed[10, 11].
Most recently it has been predicted by the Rome group, Maiani et.al., that a cold plasma has a higher rate of electron capture than does a hot plasma. This has been predicted by Maiani on the basis of (i) the Debye screening of the attractive Coulomb interactions between the electron and the proton and (ii) by the quasi-classical WKB approximation to the s-wave electron-proton wave function. The prediction is in flagrant disagreement with experiments which exhibit a hot water plasma nuclear transmutations and do not exhibit such transmutations in a cold plasma. The Maiani computation fails theoretically because (i) The Debye screening length is applied in regimes wherein it is clearly invalid and (ii) the WKB approximation is applied in the long De Broglie wave length regime but in reality the WKB approximation is valid only in the short De Broglie wave length regime. One of our purposes is to correct the errors made by Maiani. When the properly rigorous mathematics is applied we recover our previous and correct re3sults.
In Sec.II, rigorous sum rules for the plasma oscillation frequency and the plasma screening length is reviewed. A general thermodynamic expression for the screening length is found in Sec.II.1. The Debye screening length is then derived in Sec.II.2 and the regime of the validity of the Debye theory is clearly specified.
In Sec.III the expression for an electron capture transition rate per unit time per nucleus is derived in terms of the electron nuclear correlation function
describing the density of electrons at positions sitting right on top of a nucleus at position . The effects of the plasma on electron capture transition rates is described by . For the case of a water plasma on the cathode surface of a chemical cell exhibiting nuclear transmutations, the length scales are discussed and estimated in Sec.III.1. In Sec.II.2, a hot Debye screened plasma result is derived for equivalent to our previous calculations[10, 11] but in dissagreement with Maiani et.al. for reasons discussed above.
The Rome group in reality calculates for some densities in the quantum degenerate zero temperature regime wherein Thomas-Fermi quantum screening plasma replaces classical Debye screening plasma. In practical terms, this regime requires the solution of the radial s-wave potential scattering equation
wherein is the two charged particle screened potential. In solving the problem for pure s-wave scattering one conventionally takes the limit . Maiani unconventionally employs the WKB method valid only for wherein phase shifts other than s-wave gain importance. These two energy limits are different. Maiani is in error in his calculation of of s-wave potential scattering wave functions.
For completeness of presentation, in Sec.IV we review the proper method of computing s-wave scattering wave functions employing the mathematically rigorous variable phase formalism of Calogero. In the limit , one may compute the scattering wave functions in terms of the variable scattering length as discussed in Sec.IV.1. The implications for electron capture rates is discussed in Sec.IV.2. In the concluding Sec.V a summary is given of the results of this work.
Ii Screening Theory
The dielectric response of a plasma to external charge distributions is described by a wave number and complex frequency dependent dielectric function in the upper half frequency plane . The static dielectric response function determines the one photon exchange screened Coulomb interaction between two charges and ,
with a Coulomb law screening function
as shown in FIG. 1.
The screening length may be defined as the equivalent limits
If we choose the limit first, then the plasma conductivity is defined and obeys a dispersion relation,
The frequency of plasma oscillations may be found from the sum rule
In Eq.(8) the plasma component with charge and mass exists with a density of per unit volume. The thermodynamic velocity may be defined as
The zero frequency conductivity defines a plasma relaxation time via or equivalently the charge diffusion coefficient which in virtue of Eq.(9) yields the Einstein relation
The above results are rigorously true for non-relativistic Coulomb plasma phases of matter. It is important to derive one further thermodynamic sum rule for the screening length .
ii.1 Thermodynamic Sum Rules
The thermodynamic pressure completely determines the equations of state of the plasma
Let us consider the charge in a macroscopic subvolume and the charge contained within that subvolume. Since the plasma is neutral, the mean charge is zero; i.e.
There are nevertheless charge fluctuations within the subvolume
wherein is a uniform electrostatic potential, the self capacitance of the volume is and statistical thermodynamic fluctuation theory has been invoked. Evidently,
The energy associated with a uniformly charge macroscopic subvolume obeys
Theorem: The screening length is determined by the thermodynamic identity
Depending on the equations of state implicit in Eq.(11), different screening lengths will appear in different regimes. For example, if the electrons are in a high temperature regime then the classical Debye screening length holds true. If the electrons are in a low temperature degenerate regime then the Thomas-Fermi screening length holds true. Let us consider the Debye screening regime.
ii.2 Debye Screening
If the charged particles in the plasma are dilutely distributed then the particle number fluctuations
We will consider below electron proton scattering wherein the electron has a heavy mass with an effective Bohr radius
wherein the last inequality on the right hand side of Eq.(21) requires that quantum corrections to Debye screening theory can be neglected. Debye screening itself is a purely classical effect. In the opposite regime that describes the quantum degeneracy, Thomas-Fermi screening is required. In all cases the central theorem Eq.(17) for screening lengths holds true. Finally, the Debye theory for the thermal velocity in Eq.(9) yields
that again indicates the classical nature of the Debye screening theory.
Iii Electron Correlations
Consider the following electron capture process in a nucleus in the vacuum,
One can employ a complex scattering length in the center of inertia reference frame to describe Eq.(23) in the low relative velocity limit; The elastic and total cross sections are thereby
When the nuclei is embedded in a condensed matter plasma, the transition rate per nucleus obeys
wherein is the density of electrons at positions that reside right on top of the nucleus at position ; It is the electron nuclear correlation function
For the case of a heavy electron dressed in a cloud of photons scattering off a proton producing a neutrino and a neutron
iii.1 Numerical Length Scales
(ii) The Landau length is
Let represents the number density of electrons which is the same as the number of protons in the neutral plasma. (iii) The mean particle spacing is given by
(iv) The Debye screening length is
(v) The thermal De Broglie wave length is
Equivalently, the thermal De Broglie wave length is determined by
A list of order of magnitude estimates for the above length scales in a chemical cell exhibiting nuclear transmutations is given in TABLE 1. Eq.(21) for the validity of Debye theory may be expressed as
iii.2 Application to Electron Capture
Iv s-State Wavefunctions
Employing the definitions
the s-wave radial wave function
is governed by the potential scattering Eq.(2),
Calogero defines a variable phase and variable amplitude defined by
The s-wave phase shift is computed in virtue of the limits
The second order differential Eq.(38) is thereby replaced by two first order differential equations
Of interest in what follows is the limit . This small energy regime is described by the variable scattering length.
iv.1 Variable Scattering Length
The scattering length is here defined in terms of the s-wave phase shift as
The variable scattering length is defined in terms of the variable phase according to
The s-wave cross section is thereby
Finally, the relative probability, i.e. relative amplitude squared, for the scattering particles to be on top of one another compared with being widely separated
iv.2 Degenerate Electron Capture
Suppose we consider an attractive screened Coulomb potential of the form
wherein the screening length is not required to be a Debye screening length. Then has the form
The scattering length thereby obeys
The cross section is bounded from below by
As the screening length grows ever larger the scattering cross section also diverges; i.e. the total cross section of an unscreened Coulomb potential is infinite.
Properties of a fully ionized water plasma have been discussed. The theory of the screening of the Coulomb law has been rigorously derived from a thermodynamic viewpoint. A kinetic model was reviewed determining the transition rate per unit time for electron capture by a nucleus and the resulting nuclear transmutations. Corrections to the recent Maiani et.al. calculations have been discussed. The regime of validity for Debye screening length has been derived. The WKB approximation erroneously employed by Maiani in the long De Broglie wave was corrected employing the mathematically rigorous Calogero formalism in potential scattering. We have stood by our previous results on the rates of electro electron capture processes for a water plasma in chemical cells.
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