In 2005, A. Widom of Northeastern University and L. Larsen of Lattice Energy LLC have proposed a theoretical explanation for low energy nuclear reactions (LENR) in the framework of electroweak interactions of ultra-low momentum neutrons. Electroweak interactions are relatively well understood in the Standard Model as the electromagnetic and weak forces are unified in a single field theory. However, their appearance in condensed matter or solid state system is novel, and hence unfamiliar. Most condensed matter interactions are purely electromagnetic: electrons or ions usually have energies in the keV range, and nuclear reactions are absent. Weak interactions, however, such as the capture of an electron by a proton to create a neutron (reverse beta decay), require MeV's of energy to overcome the mass-energy difference, and are usually not observed. However, certain circumstances exist that can accelerate electrons to MeV energies within a condensed matter system, allowing weak interactions (and hence nuclear reactions) to occur.

The standard reaction for electron capture is \[\ell^-+p^+\to n+\nu_{\ell}\] whereby any negatively charged lepton can be captured to create a neutron and a neutrino. Note that there is no Coulomb barrier to this interaction; instead the Coulomb attraction actually encourages it, and we can often see this with muons. We do not readily observe the reaction with electrons, however, because of the mass-energy barrier. For the reaction to spontaneously occur, the mass of the lepton (let's deal just with electrons, since they are common and of practical interest) must satisfy a threshold criterion: \[m_e^*c^2>(m_n-m_p)c^2=\beta m_ec^2\approx 2.531m_ec^2.\] We see that the electron's mass must be augmented by a factor of 2.531 to give it enough energy to be captured by a proton. How can we "dress up" the mass of the electron in a condensed matter system? Local electromagnetic field fluctuations!

If we denote the four-momentum of an electron as \(p^{\mu}\) in an electromagnetic field \(F_{\mu\nu}=\partial_{\nu}A_{\nu}-\partial_{\nu}A_{\mu},\) we have the Hamilton-Jacobi equation \[\left(p_{\mu}-\frac{e}{c}A_{\mu}\right)\left(p^{\mu}-\frac{e}{c}A^{\mu}\right)=m_e^2c^2.\] Even if the field fluctuations average to zero, the remaining mean square fluctuations can add mass to the electron such that \[p_{\mu}p^{\mu}=m_e^{*2}c^2=m_e^2c^2+\frac{e^2}{c^2}\langle A_{\mu}A^{\mu}\rangle.\] Clearly, we must have that \[\beta=\frac{m_e^*}{m_e}=\sqrt{1+\left(\frac{e}{m_ec^2}\right)^2\langle A\cdot A\rangle}>2.531\] for neutron production, ignoring the mass of the electron neutrino. As a simplification, we can assume that the rms field fluctuations as a function of position satisfy \(\langle A\cdot A\rangle=c^2\langle|\vec{E}(\vec{r})|^2\rangle/\omega^2\) so that the local mass enhancement factor is \[\beta(\vec{r})=\sqrt{1+\frac{\langle|\vec{E}(\vec{r})|^2\rangle}{\mathcal{E}^2}},\] where \(\omega\) is the frequency scale of the fluctuations, and \(\mathcal{E}=m_ec\omega/e\) is a constant.

To calculate the field strength and proton oscillation frequency, we imagine a small displacement \(\vec{u}\) from equilibrium in a uniform cloud of negative charge, so that \[e\vec{E}=-\left(\frac{e^2n}{3}\right)\vec{u}=-m_p\omega^2\vec{u}\] where \(n\) is the electron density, which we approximate as \[n=|\psi(0)|^2=\frac{1}{\pi a_0^3}\] for the density at the origin of a hydrogen atom, with \(a_0\) the Bohr radius. With the distinctions \[\mathcal{E}_A=\frac{e}{4\pi a_0^2}\approx 5.142\times 10^{11}\text{ V/m}\] for the typical electric field in an atom, and \[\mathcal{E}_p=\frac{m_ec\omega}{e}\approx 1.55\times 10^{11}\text{ V/m}\] for the electric field strength due to the proton oscillations in the condensed matter system (value determined experimentally), we can see that \[\frac{\langle|\vec{E}(\vec{r})|^2\rangle}{\mathcal{E}_A^2}=\frac{16}{9}\frac{\langle|\vec{u}|^2\rangle}{a_0^2},\] giving us a mass enhancement factor of \[\beta=\sqrt{1+\frac{16}{9}\frac{\mathcal{E}_A^2}{\mathcal{E}_p^2}\frac{\langle|\vec{u}|^2\rangle}{a_0^2}}.\] Using the values above, we find that \[\frac{16}{9}\frac{\mathcal{E}_A^2}{\mathcal{E}_p^2}\approx 19.6,\] so that the electron's mass enhancement is a direct function of the mean square displacement of the protons in their collective mode oscillations. To obtain the threshold value of \(\beta\), this requires \(u_{rms}\approx 0.53a_0\), which is not an inconceivable displacement for the protons.

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