### Do nuclear reaction rates depend on environment?

Claus Rolfs and his group have found experimental evidence for the dependence of the rates of nuclear reactions on the condensed matter environment (see this and this). See also the comments of Lubos Motl about these findings whose tone is not too difficult to guess. For instance, the rates for the reactions

^{50}V(p,n)

^{50}Cr and

^{176}Lu(p,n) are fastest in conductors. The model explaining the findings has been tested for elements covering a large portion of the periodic table.

** 1. Debye screening of nuclear charge by electrons as an explanation for the findings**

The proposedd theoretical explanation is that conduction electrons screen the nuclear charge or equivalently that incoming proton gets additional acceleration in the attractive Coulomb field of electrons so that the effective collision energy increases so that reaction rates below Coulomb wall increase since the thickness of the Coulomb barrier is reduced.

The resulting Debye radius

R_{D}= 69(T/n_{eff}ρ_{a})^{1/2} ,

where ρ_{a} is the number of atoms per cubic meter and T is measured in Kelvins. R_{D} is of order .01 Angstroms for T=373 K for n_{eff}=1, a=1 Angstrom. The theoretical model predicts that the cross section below Coulomb barrier for X(p,n) collisions is enhanced by the factor

f(E)=(E/E+U_{e})exp(πη U_{e}/E).

E is center of mass energy and η so called Sommerfeld parameter and

U_{e}= 2.09×10^{-11}(Z(Z+1))^{1/2}×(n_{eff}ρ_{a}/T)^{1/2} eV

is the screening energy defined as the Coulomb interaction energy of electron cloud responsible for Debye screening and projectile nucleus. The idea is that at R_{D} nuclear charge is nearly completely screened so that the energy of projectile is E+U_{e} at this radius which means effectively higher collision energy.

The experimental findings from the study of 52 metals support the expression for the screening factor across the periodic table.

- The linear dependence of U
_{e}on Z and T^{-1/2}dependence on temperature conforms with the prediction. Also the predicted dependence on energy has been tested. - The value of the effective number n
_{eff}of screening electrons deduced from the experimental data is consistent with n_{eff}(Hall) deduced from quantum Hall effect.

^{22}Na β decay rate in Pd (see this), a metal which is utilized also in cold fusion experiments. This might have quite far reaching technological implications. For instance, the artificial reduction of half-lives of the radioactive nuclei could allow an effective treatment of radio-active wastes. An interesting question is whether screening effect could explain cold fusion and sono-fusion: I have proposed a different model for cold fusion based on large hbar here.

** 2. Could quantization of Planck constant explain why Debye model works?**

The basic objection against the Debye model is that the thermodynamical treatment of electrons as classical particles below the atomic radius is in conflict with the basic assumptions of atomic physics. On the other hand, it is not trivial to invent models reproducing the predictions of the Debye model so that it makes sense to ask whether the quantization of Planck constant predicted by TGD could explain why Debye model works.

TGD predicts that Planck constant is quantized in integer multiples: hbar=nhbar_{0}, where hbar_{0} is the minimal value of Planck constant identified tentatively as the ordinary Planck constant. The preferred values for the scaling factors n of hbar correspond to n-polygons constructible using ruler and compass. The values of n in question are given by

n_{F}= 2^{k} ∏_{i} F_{si},

where the Fermat primes F_{s}=2^{2s}+1 appearing in the product are distinct. The lowest Fermat primes are 3,5,17,257,2^{16}+1. In the TGD based model of living matter the especially favored values of hbar come as powers 2^{k11}.

It is not at all obvious that ordinary nuclear physics and atomic physics should correspond to the minimum value hbar_{0} of Planck constant. The predictions for the favored values of n are not affected if one has hbar(stand)= 2^{k}hbar_{0}, k≥ 0. The non-perturbative character of strong force suggests that the Planck constant for nuclear physics is not actually the minimal one. As a matter fact, TGD based model for nucleus implies that its "color magnetic body" has size of order electron Compton length. Also valence quarks inside hadrons have been proposed to correspond to non-minimal value of Planck constant since color confinement is definitely a non-perturbative effect. Since the lowest order classical predictions for the scattering cross sections in perturbative phase do not depend on the value of the Planck constant one can consider the testing of this issue is not trivial in the case of nuclear physics where perturbative approach does not really work.

Suppose that one has n=n_{0}=2^{k0} > 1 for nuclei so that their quantum sizes are of order electron Compton length or perhaps even larger. One could even consider the possibility that both nuclei and atomic electrons correspond to n=n_{0}, and that conduction electrons can make a transition to a state with n_{1}<n_{0}. This transition could actually explain how the electron conductivity is reduced to a finite value. In this state electrons would have Compton length scaled down by a factor n_{0}/n_{1}.

For instance, suppose that one has n_{0}=2^{11k0} as suggested by the model for quantum biology and by the TGD based explanation of the optical rotation of a laser beam in a magnetic field (hep-exp/0507107). The Compton length L_{e}=2.4×10^{-12} m for electron would reduce in the transition k_{0}→ k_{0}-1 to L_{e} =2^{-11}L_{e}= 1.17 fm, which is rather near to the proton Compton length since one has m_{p}/m_{e}≈ .94×2^{11}. It is not too difficult to believe that electrons in this state could behave like classical particles with respect to their interaction with nuclei and atoms so that Debye model would work.

The basic objection against this model is that anyonic atoms should allow more states that ordinary atoms since very space-time sheet can carry up to n electrons with identical quantum numbers in conventional sense. This should have been seen.

** 3. Electron screening and Trojan horse mechanism**

An alternative mechanism is based on Trojan horse mechanism suggested as a basic mechanism of cold fusion. The idea is that projectile nucleus enters the region of the target nucleus along a larger space-time sheet and in this manner avoids the Coulomb wall. The nuclear reaction itself occurs conventionally. In conductors the space-time sheet of conduction electrons is a natural candidate for the larger space-time sheet.

At conduction electron space-time sheet there is a constant charged density consisting of n_{eff} electrons in the atomic volume V= 1/n_{a}. This creates harmonic oscillator potential in which incoming proton accelerates towards origin. The interaction energy at radius r is given by

V(r)= α n_{eff}r^{2}/2a^{3} ,

where a is atomic radius.

The proton ends up to this space-time sheet by a thermal kick compensating the harmonic oscillator energy. This occurs below with a high probability below radius R for which the thermal energy E=T/2 of electron corresponds to the energy in the harmonic oscillator potential. This gives the condition

R= (Ta/n_{eff}α)^{1/2} a .

This condition is exactly of the same form as the condition given by Debye model for electron screening but has a completely different physical interpretation.

Since the proton need not travel through the nuclear Coulomb potential, it effectively gains the energy

E_{e}= Zα/R= (Zα^{3/2}/a)×(n_{eff}/Ta)^{1/2},

which would be otherwise lost in the repulsive nuclear Coulomb potential. Note that the contribution of the thermal energy to E_{e} is neglected. The dependence on the parameters involved is exactly the same as in the case of Debye model. For T=373 K in the ^{176}Lu experiment and n_{eff}(Lu)=2.2+/-1.2, and a=a_{0}=.52 Angstrom (Bohr radius of hydrogen as estimate for atomic radius), one has E_{e}=28.0 keV to be compared with U_{e}=21+/- 6 keV of Rolfs et al (a=1 Angstrom corresponds to 1.24× 10^{4} eV and 1 K to 10^{-4} eV). A slightly larger atomic radius allows to achieve consistency. The value of hbar does not play any role in this model since the considerations are purely classical.

An interesting question is what the model says about the decay rates of nuclei in conductors. For instance, if the proton from the decaying nucleus can enter directly to the space-time sheet of the conduction electrons, the Coulomb wall corresponds to the Coulomb interaction energy of proton with conduction electrons at atomic radius and is equal to α n_{eff}/a so that the decay rate should be enhanced.

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-Matrix" contains this piece of text too. See also the chapter TGD and Nuclear Physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy"

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