Pycnonuclear fusion

Pycnonuclear reactions (Ancient Greek: πυκνός, romanized: pyknós, lit. 'dense, compact, thick'; Not to be confused with cold fusion) — nuclear fusion reactions which occur between zero-point energy nuclei bound in their crystal lattice. The primary difference between thermonuclear reactions and pycnonuclear reactions is that in thermonuclear reactions the Coulomb barrier is overcome due to the thermal energy of the nuclei, while in pycnonuclear reactions it is overcome due to the zero-point oscillations of nuclei in their crystal lattice. In quantum physics, the phenomena can be interpreted as overlap of the wave functions of neighboring ions. Under the conditions of above-threshold ionization, the reactions of neutronization and pycnonuclear fusion can lead to the creation of absolutely stable environments in superdense substances.

The current consensus on the rate of pycnonuclear reactions is not coherent. This is because every approximation is related to the extrapolation of present data on nuclear interactions towards zero-point energy nuclei, and because pycnonuclear are very sensitive towards the defects within the crystal lattice of the high-density material. Thus, current efforts have concentrated on modelling the deformations in the crystal lattices, in order to arrive at more appropriate approximations of the rate of pycnonuclear reactions. Closest approximations of pycnonuclear reaction models remain to be works by Salpeter-Van Horn (1969) and Afanasjev et al., and were also the pioneers in the description of thermal effects on the rate of pycnonuclear reactions.

The term "pycnonuclear" was coined by A.G.W. Cameron in 1959, but research showing the possibility of nuclear fusion in extremely dense & cold compositions was published by W. A. Wildhack in 1940.

Astrophysics

Pycnonuclear reactions can occur anywhere and in any matter, but under standard conditions, the speed of the reaction is exceedingly low, and thus, have no significant role outside of extremely dense systems and neutron-rich and free electron-rich environments, such as the inner crust of a Neutron star. A feature of pycnonuclear reactions is that the rate of the reaction is directly proportional to the density of the space that the reaction is occurring in, but is almost fully independent of the temperature of the environment.

The pycnonuclear reactions occurred most violently in the initial phases of the universe, as the baryonic matter was a billion times denser than today. Pycnonuclear reactions are still observed today in neutron stars or white dwarfs, with evidence present of them occurring in lab-generated deuterium-tritium plasma. Some speculations also relate the fact that Jupiter emits more radiation than it receives from the Sun with pycnonuclear reactions or cold fusion.

White Dwarves & Neutron Stars

In white dwarves, the core of the star is cold, which results in the hydrogen protons arranging themselves into a crystal lattice. At higher densities, the crystal lattices of neutron-rich nuclei are forced closer together due to gravitational collapse of accreting material, and at a critical point, fusion occurs. The zero-point oscillations of nuclei in the crystal lattice with energy equal to $E_{0}\thicksim \hbar w$ can overcome the Coulomb barrier, actuating pycnonuclear reactions. Because upon the contact of the ions, the probability for further interaction practically does not depend on the ions' initial quantum state. The increasing mass of the white dwarfs as a result of their accretion also increases the density of the core of the star, allowing for the pycnonuclear reactions to accelerate further. Following the pycnonuclear reactions, the core of the star heats up, and the nuclear fusion reactions turn into thermonuclear reactions, which serves as a catalyst for the formation of type Ia supernovas.

Since the core of neutron stars was approximated to be $3*10^{14}$ g/cm³, at such extreme densities, pycnonuclear reactions play a large role in the fusion processes of the inner crust. In neutron stars, pycnonuclear reactions are possible energy sources for gamma-ray bursts, as the burning of neutron-rich nuclei (${\ce {^{34}Ne + ^{34}Ne}}$) serves as a major heat source.

In white dwarves and neutron stars, the nuclear reaction rates can not only be affected by pycnonuclear reactions, but also by the plasma screening of the Coulomb interaction. A Ukrainian Electrodynamic Research Laboratory "Proton-21", established that by forming a thin electron plasma layer on the surface of the target material, and, thus, forcing the self-compression of the target material at low temperatures, they could stimulate the process of pycnonuclear fusion. The startup of the process was due to the self-contracting plasma "scanning" the entire volume of the target material, screening the Coulumb field.

Mathematical model

It can be generally accredited that the transition from pycnonuclear to thermonuclear reactions occurs are Debye temperature $\theta \approx \hbar w/k$, where $w$ is the typical vibrational frequency of the nuclei within the crystal lattice. It is impossible to differentiate between thermonuclear and pycnonuclear reactions at extremely low temperatures, as in those conditions even the reacting nuclei is bound to a crystalline lattice. A study published by Y. B. Zeldovich showed a simplified expression for the rate of pycnonuclear reactions:

${\textstyle Q\thicksim nvr^{-3}\sigma P,P\thicksim exp(-\gamma ),\gamma \thicksim R^{2}/r^{2}}$, where

• n - the concentration of nuclei
• r - oscillation amplitude
• $R\thicksim n^{-1/3}$ - distance between the nuclei in the crystal lattice
• $\sigma =S(\delta )/\delta$ nuclear reaction cross section, where $S(\delta )$ is the astrophysical factor, where $\delta ,v=rw$
• P - coefficient of transmission through the Coulomb barrier

In Earthlike conditions, the $\hbar w$ is less than 1 eV, $r\leq 10^{-9}$ and $R\thicksim 10^{-8}$, which makes the coefficient of transmission through the Coulomb barrier P extremely low, and, thus, pycnonuclear reactions play a very small role in conditions of low density.

However, a study published in 1990 showed that at intermediate temperatures and high densities, it is possible for proton–proton chain reactions to occur even in the core of the Earth. This was done by building a mass–luminosity relation graph of giant planets and the Earth, which showed a direct correlation. The presence of Earth on the correlation line indicated that the core of the Earth also had fusion reactions going on in it. Factoring in the high density of the Earth's core would indicate that pycnonuclear reactions play a role.

A contender for the best approximation of the rate of pycnonuclear reactions belongs to the study published by Salpeter & van-Horn. In their approximations, they consider the effects of "weak" and "strong" electron "screening", as well as anisotropy and a more accurate model of crystal lattice potential, and provided an expression showing the correlation between the rate of pycnonuclear reactions and the surrounding temperature (see the role of screening in § Astrophysics). At intermediate points of lower temperatures or higher densities, where $E_{coul}\gg k_{B}T$, but $E_{pk}$ of the Gamow peak is greater than $E_{coul}$, the chemical composition starts forming condensed phases. Salpeter concluded that in intermediate cases, the rate of pycnonuclear reactions is simply equal to the rate of thermonuclear reactions, but with factored in parameters of weak-screening correction factor $e^{U_{w}}$, where $U_{w}={\sqrt {(}}3){\frac {Z_{1}}{Z_{2}}}\left({\frac {\mu _{e}}{Z_{1}}}\right)^{1/2}\zeta \Gamma _{Z1}^{3/2}$ multiplied by the rate. Salpeter et al. arrived at the following expression for the rate of pycnonuclear reactions at extreme temperatures, such as absolute zero. In this case, the energy required to bypass the Coulomb barrier is lower than the product of the Boltzmann constant and the temperature, while the Gamow peak is approx. equal to the Coulomb barrier energy:

$P_{0}={\rho \over A}A^{2}Z^{4}S\gamma \lambda ^{7/4}exp(-\varepsilon \lambda ^{-1/2})$ reactions cm^-3 sec^-1,

where

$\gamma =3.90*10^{46},\varepsilon =2.638$

or

$\gamma =4.76*10^{46},\varepsilon =2.516$

Salpeter established general guidelines on the role of pycnonuclear reactions, how to classify them depending on the temperature & density, and how to simply differentiate between the types of nuclear fusion depending on temperature and Gamow peak energy.

However, according to approximations from Shapiro, the rate of the pycnonuclear reaction can be heavily affected by imperfections/defects in the crystal lattice of the interaction nuclei, which were not considered in the study published by Salpeter.

Afanasjev et al. introduced an analytic model which revolved around astrophysical S-factor - $S(E)$ and the parameter of effective potential $U(r)$, using the barrier penetration formalism and Sao Paulo potential. Initially, an analytic method that required 4 parameters was built, but was elaborated in 2012 to be dependent only on 3 parameters.

Afanasjef et al. recognize that the proposed analytic $S(E)$ model has a lot of uncertainties. For example, the presence of free neutrons between the reacting nuclei may change the effective potential parameter $U(r)$, which would make approximations of the value of $U(r)$ unreliable.

See also

• Cold fusion
• Thermonuclear reaction
• Accretion (astrophysics)