Abstract
Recently we showed that the nuclear transmutation rates are largely overestimated in the Widom–Larsen theory of the so-called ‘Low Energy Nuclear Reactions’. Here we show that unbound plasma electrons are even less likely to initiate nuclear transmutations.
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1 Introduction
Claims of electron–proton conversion into a neutron and a neutrino by inverse beta decay in metallic hydrides have recently been raised [1, 2], in the context of the so-called Low Energy Nuclear Reactions (LENR). The condition for the reaction to occur is a considerable mass renormalization of the electrons, to overcome the negative Q-value that, otherwise, would forbid the reaction to occur. Defining a dimensionless parameter, \(\beta \), in terms of the electron effective mass, \(m^\star \),Footnote 1 one needs
a reference value, \(\beta =20\) was estimated in [1].
It is not clear at all if such spectacularly large values of \(\beta \) can be obtained in metallic hydrides and under which conditions. Nonetheless, assuming a given value of \(\beta \), a calculation of the neutron rate can be obtained in a straightforward fashion from known electroweak physics. A calculation along these lines has been presented in Ref. [3] for the case of an electron bound to a proton, superseding the order-of-magnitude estimate presented in [1].
More recently, the authors of Ref. [2] have argued that nuclear transmutations should most likely be started by unbound plasma electrons. Assuming a fully ionized plasma and completely unscreened electrons, they find a rate which is enhanced, with respect to the value obtained for bound electrons, by the so-called Sommerfeld factor, \(S_0\) (\(c=1\)):
where \(\alpha \) is the fine structure constant and \(v\) is the average thermal velocity of the electrons defined byFootnote 2
with the numerical value in correspondence to \(\beta =20\) and to the temperature \(T \approx 5 \times 10^{3}~\)K, estimated in [2] as the temperature that can be reached by hydride cathodes. However, the assumption of completely unscreened electrons may be unrealistic. We consider here the situation in the presence of Debye screening, which, in a different context, has been recently analyzed in Refs. [4, 5]. We find that at large densities, the plasma enhancement saturates to a value determined by the Debye length, \(a_D\):
with
and \(a_B\) the Bohr radius.
2 Debye length
Static charges are screened in a plasma. The potential of the electric field of a test charge at rest in a plasma is (in Gaussian units)
where \(a_D\) is the Debye length defined by
The two lengths \(a_{e,i}\) are associated to electrons and ions, respectively, and are given by [6]
and
The difference in temperature between electrons and ions is expected to occur naturally because of the large difference of mass which impedes the exchange of energy in electron–ion collisions. Here we will make the approximation \(a_D = a_e\), which leads to the numerical value
or a Debye mass \(m_D\):
We therefore get a Debye length of about nine atoms (compared to \(a_B = 0.5 \) Å) in correspondence to the reference temperature \(T \approx 5 \times 10^{3}~\)K and a reference density \(n_e=10^{20}\) cm\(^{-3}\). When considering the \(n\) dependence, we shall restrict ourselves to the range
Values between \(10^8\) and \(10^{14}\) cm\(^{-3}\) are typical of glow discharges and arcs, whereas a value of about \(10^{22}\) cm\(^{-3}\) is the free electron density in copper [7]. Around \(2.5\times 10^{21}~\text {cm}^{-3}\) the Debye length equals the Bohr radius.Footnote 3
3 Critical velocity
The Sommerfeld factors in a plasma, Eqs. (40) and (43), can be obtained from an intuitive argument as follows (see the appendix for a derivation from the Schrödinger equation following [4, 5]).
We consider the critical value of the velocity as defined by
Under this condition, the de Broglie wavelength of the particle is equal to the Debye length:Footnote 4
For larger velocities, the wavelength is smaller and the particle probes a region of space smaller than \(a_D\), where it sees an essentially unscreened Coulomb potential. Under these conditions, we have to use \(S_0\), Eq. (2).
For smaller velocities, as \(v \rightarrow 0\), the wavelength gets larger than \(a_D\). The Sommerfeld factor saturates to the value on the r.h.s. of (13) since the particle explores increasingly large portions of the neutral plasma, and the screened Sommerfeld factor in Eq. (4) has to be considered.
The critical velocity defined by (13) is
We consider our electrons to be at \(v_\mathrm{th}\), Eq. (3). At the reference point, this is larger than \(v_\mathrm{crit}\); hence we should apply the unscreened result, \(S_0\). With increasing density, however, \(v_\mathrm{crit}\) grows above \(v_\mathrm{th}\) (at \(n\sim 2\times 10^{20}\) cm\(^{-3}\)) and one should apply the screened result, \(S\).
4 Transmutation rates
To translate the previous discussion into the expected rates for transmutation from electrons in a plasma, we first recall the rate for the transmutation from bound electrons [3]:
The total rate is obtained by multiplying the result \( \Gamma _\mathrm{bound}\) by the volume and by the ion density, which we take equal to the electron density, \(n\), because of global neutrality:
In the case of plasma electrons, screened and unscreened rates are obtained by the substitution
and the rate is proportional to \(n^2\):
\(S\) and \(S_0\) corresponding, respectively, to the screened Debye plasma and to the unscreened Coulomb case.
For convenience, we normalize the rates in plasma to the rate in Eq. (17), computed for \(\beta =20\), already a considerably large rate, although a factor of \(\sim 300\) smaller than claimed in [1], and we shall see if we can get anywhere close to unity or higher.
Ratios corresponding to the screened plasma (Sommerfeld factor \(S\)) and to the unscreened one (Sommerfeld factor \(S_0\)), for the case \(\beta =20\). The previous discussion indicates that we must use \(S_0\) for \(v_\mathrm{crit}\le v_\mathrm{th}\) and \(S\) for \(v_\mathrm{crit}\ge v_\mathrm{th}\). The result is represented by the thick line
The formulas are
and
for the two cases.
In Fig. 1 we display the ratios corresponding to the screened plasma (Sommerfeld factor \(S\)) and to the unscreened one (Sommerfeld factor \(S_0\)), for the case \(\beta =20\). The previous discussion indicates that we must use \(S_0\) for \(v_\mathrm{crit}\le v_\mathrm{th}\) and \(S\) for \(v_\mathrm{crit}\ge v_\mathrm{th}\). The result is represented by the thick line.
The rate for electron capture from plasma never comes anywhere close to the capture rate for bound electrons derived in [3] for the same value of \(\beta \), let alone to the larger rate quoted in [1]. Our results are in line with the lack of observation of neutrons in plasma discharge experiments recently reported in [9].
Notes
To avoid confusion, we underscore that the mass renormalization in (1) has nothing to do with the velocity dependent relativistic mass. We consider extremely non-relativistic electrons. The situation is closely analogous to muon capture in muonic atoms; in that case \(m^\star \) being replaced by the muon mass.
We shall use the numerical values \(k=8.617 \times 10^{-5}\)eV/K, \(e^2 / \hbar c=\alpha =1/137.043\) and set \(c=\hbar c=1\).
Electron capture occurs spontaneously during the formation of neutron stars, when the Fermi energy of the electrons increases above the threshold value, due to the gravitational pressure. This occurs at electron densities \(\gtrsim 10^{31}\) cm\(^{-3}\).
We use \(\hbar =1\), so that \(h=2\pi \).
In the conventions of [8], \(A=2\).
References
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Acknowledgments
We thank Giancarlo Ruocco and Massimo Testa for interesting discussions.
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Appendix: Sommerfeld factor for electrons in screened and unscreened plasma
Appendix: Sommerfeld factor for electrons in screened and unscreened plasma
Let us consider an attractive screened potential in the plasma in the form
The radial Schrödinger equation for the two body (\(e^-\)–ion) wavefunction, \(\chi (r)\), reads
Changing \(r\) into the nondimensional variable \(x\):
we get
In the limit of small or vanishing \(v\) we write the equation as
and in terms of the effective momentum:
We solve it by the WKB method, which gives
We can use the WKB approximation as long as
that is,
At the value where the exponential bends, namely \(\epsilon x =1\), we have
and the condition that this region is within the range of validity of WKB is then
with \(\beta \) defined as in Eq. (1).
For \(\beta =20\) and \(a_D\) from Eq. (10), we find
On the other hand, the smallest velocity we consider is the thermal velocity, Eq. (3), which is safely within the region of validity of the WKB approximation. Note that \(v_\mathrm{WKB}\) is simply proportional to the critical velocity \(v_\mathrm{crit}\) defined in (13):
We are interested in the square modulus of the wavefunction at the origin relative to its unperturbed value (transmutation is taking place at the origin), the ratio being the Sommerfeld enhancement:
where we have used the fact that \(R_{k\ell }(x)\sim x^\ell \) as \(x\rightarrow 0\). The constant \(A\) depends on the normalization of the radial function at large distances.Footnote 5 Since \(R_{k,\ell =0}\) goes to a constant as \(x\rightarrow 0\), we need that \(\chi _k(x)\rightarrow 0\) as \(x\rightarrow 0\) or
thus giving
Within the region of validity of the WKB approximation, \(k\gtrsim \varepsilon \), we have
where \(A\) is chosen to be the same constant as appears in (35). Therefore
the last term in parentheses being much smaller than 1. The maximum value attainable by \(S_k\) is at the border of the WKB approximation limit, i.e. for \(k\sim \epsilon \), Eq. (32), and we have
In the limit \(\epsilon \rightarrow 0\), the Schrödinger equation (23) is solved analytically. The \(^\prime \)in\(^\prime \) wavefunction in the continuous spectrum of the attractive Coulomb field is given by
where \(F=~_1F_1 \) is the Kummer function (hypergeometric confluent). Here \(\mathbf{k\cdot r}\) corresponds to \(mv\times r\), measured in units \(1/m\). Thus it is the nondimensional quantity \(v/\alpha \). The same would hold writing \(kr = (k/\alpha m)(\alpha m r)\).
In these respects \(k/\alpha m\rightarrow k\) is dimensionless, \(k = v/\alpha \), and we understand the factor \(e^{\pi k/2}\), or the term \(\Gamma =(1- i/k)\). The \(k = v/\alpha \) appears in the Schrödinger equation (23).
The action of the attractive Coulomb field on the motion of the particle near the origin can be characterized by the ratio of the square modulus of \(\psi ^{(+)}(0)\) to the square modulus of the wavefunction for free motion, \(\psi _ k(r) = e^{i{\varvec{k}}\cdot {\varvec{r}}}\). Using \(\Gamma ^*(z) =\Gamma (z^*)\), \(F(i/k, 1, 0) = 1\) and
we get the result
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Maiani, L., Polosa, A.D. & Riquer, V. Neutron production rates by inverse-beta decay in fully ionized plasmas. Eur. Phys. J. C 74, 2843 (2014). https://doi.org/10.1140/epjc/s10052-014-2843-1
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DOI: https://doi.org/10.1140/epjc/s10052-014-2843-1