# Low energy neutron production by inverse *β* decay in metallic hydride surfaces

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## Abstract

It has recently been argued that inverse-*β* nuclear transmutations might occur at an impressively high rate in a thin layer at a metallic hydride surface under specific conditions. In this note we present a calculation of the transmutation rate, which shows that there is little room for such a remarkable effect.

## Keywords

Annihilation Cross Section Pion Decay Neutron Production Inclusive Rate Mass Renormalization## 1 Introduction and main results

*β*reaction:

*Q*-value (

*Q*=

*M*

_{ p }+

*m*

_{ e }−

*M*

_{ n }<0). The authors advance the hypothesis that mass renormalization from the interaction of electrons in atoms with some electromagnetic external field (e.g. due to a laser source) may increase the effective electron mass, so as to make the reaction (1) possible. The tilde sign on the electron symbol underscores that the electron bound to the proton is a ‘dressed’ one, with an effective mass denoted by

^{1}

*λ*=

*g*

_{ A }/

*g*

_{ V }≈1.25 is the axial to vector coupling in neutron

*β*decay.

*β*≈20, the authors find

*ψ*(0)|

^{2}, which gives the probability for finding the electron and the proton at the same point.

^{2}This factor is a mandatory consequence of the local Fermi lagrangian density: With the electron and proton fields computed at the same point, the factor |

*ψ*(0)|

^{2}indicates that it must be difficult for the lagrangian to induce transitions from an initial state with, say, the proton localized in our lab and the electron on Mars. We present a calculation of the rate of (1) done in two independent ways.

One is to consider the decay of the \(\tilde{e}p\) bound states with spin 0 and 1, relating the amplitude to the basic lagrangian (6) in a way analogous to the quark model calculation of pion decay made time ago by Van Royen and Weisskopf (VRW) [6].

*σ*is the unpolarized cross section for process (1). This is the basic formula in the theory of electron K-capture, also used [7] to derive the decay rate of positronium from the annihilation cross section of an electron–positron pair into photons.

*β*defined in (2),

*a*the Bohr radius,

*a*=(

*αm*

_{ e })

^{−1}≃0.54 Å and

*α*the fine structure constant, and find, finally:

*β*≈2

*β*

_{0}:

*β*=61 (i.e.

*m*

^{∗}≈30 MeV).

In the following, we give the details of the calculation of *Γ*. In the end, we comment on the high values of *β* considered in [1, 2, 3, 4, 5].

## 2 Decay rates of S-wave bound states

*H*

_{0}and

*H*

_{1}, with total spin equal to zero and one, respectively. We may describe this decay with two phenomenological parameters,

*f*

_{0,1}according to: The Fermi constant has been inserted for convenience,

*H*

_{0}and \(H_{1}^{i}\) are field operators describing the annihilation of either

*H*state and we describe the creation of the neutrino with the antineutrino field,

*ν*

_{ c }. The situation is entirely similar to the decay

*π*

^{−}→

*μ*

^{−}

*ν*

_{ c }, described by the phenomenological parameter

*f*

_{ π }.

^{3}

^{4}

*H*state according to

*r*and

*s*are spin indices,

*ϕ*(

*r*,

*s*) are the Clebsch–Gordan coefficients appropriate to the spin of

*H*, (

*a*

^{ p,e })

^{†}are the proton and electron creation operators and

*f*(

*) the momentum space wave function, with*

**p***f*

_{0}is obtained by comparing (15) with (16), e.g.:

*f*

_{0,1}are irrelevant): with the

*x*-space wave function given by Similarly:

## 3 Inclusive rate from the unpolarized cross section

## 4 The value of *β*

*β*of the order of or even larger than 20 are certainly unusual in condensed matter physics, especially for bound electrons. An estimate of

*β*is given in [1, 2, 3, 4, 5] where

*A*is given in terms of the plasma frequency

*Ω*

_{ p }of the protons:

*Ω*

_{ p }are estimated to be of order

*Ω*

_{ p }≈0.1 eV [10]. With this value of

*Ω*

_{ p }and Open image in new window [1, 2, 3, 4, 5]:

*Ω*

_{ p }=0.8 eV used in [1, 2, 3, 4, 5] leads to

*β*≈20 and in any case below threshold for nuclear transmutation to occur.

## 5 Conclusions

A correct calculation gives a neutron production rate from (1) about 300 times smaller than estimated in [1, 2, 3, 4, 5], for the value of the mass renormalization factor *β*≈20 considered there. In turn, it is questionable that values of *β* can be realized, in particular for bound electrons, so large as to give rise to useful nuclear transmutation rates. A more detailed analysis of the attainable values of *β* is needed to obtain more definite conclusions on this interesting phenomenon, should it exist at all.

## Footnotes

- 1.
See Eq. (107) of Ref. [5].

- 2.
A. Polosa in the seminar by Y. Srivastava in Roma, May 31, 2012.

- 3.
We understand that different flavors of neutrinos appear in

*π*and*H*decays. - 4.
When separating the term in

*γ*^{0}from the one in*γ*^{ i }we have used the relations*γ*^{0}*p*=*p*and \(\bar{n} \gamma^{0}=\bar{n}\), following from the non relativistic approximation.

## Notes

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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