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The Coulomb barrier, named after Coulomb's law, which is in turn named after physicist Charles-Augustin de Coulomb, is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo a nuclear reaction.
This energy barrier is given by the electric potential energy:
where
A positive value of U is due to a repulsive force, so interacting particles are at higher energy levels as they get closer. A negative potential energy indicates a bound state (due to an attractive force).
The Coulomb barrier increases with the atomic numbers (i.e. the number of protons) of the colliding nuclei:
where e is the elementary charge, and Zi the corresponding atomic numbers.
To overcome this barrier, nuclei have to collide at high velocities, so their kinetic energies drive them close enough for the strong interaction to take place and bind them together.
According to the kinetic theory of gases, the temperature of a gas is just a measure of the average kinetic energy of the particles in that gas. For classical ideal gases the velocity distribution of the gas particles is given by Maxwell–Boltzmann. From this distribution, the fraction of particles with a velocity high enough to overcome the Coulomb barrier can be determined.
In practice, temperatures needed to overcome the Coulomb barrier turned out to be smaller than expected due to quantum mechanical tunnelling, as established by Gamow. The consideration of barrier-penetration through tunnelling and the speed distribution gives rise to a limited range of conditions where fusion can take place, known as the Gamow window.
The absence of the Coulomb barrier enabled the discovery of the neutron by James Chadwick in 1932. [1] [2]
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle, transforming into an isobar of that nuclide. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in so-called positron emission. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band or valley of stability. For either electron or positron emission to be energetically possible, the energy release or Q value must be positive.
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
In physics, the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
Nuclear fusion is a reaction in which two or more atomic nuclei, usually deuterium and tritium, combine to form one or more different atomic nuclei and subatomic particles. The difference in mass between the reactants and products is manifested as either the release or absorption of energy. This difference in mass arises due to the difference in nuclear binding energy between the atomic nuclei before and after the reaction. Nuclear fusion is the process that powers active or main-sequence stars and other high-magnitude stars, where large amounts of energy are released.
Nuclear fission is a reaction in which the nucleus of an atom splits into two or more smaller nuclei. The fission process often produces gamma photons, and releases a very large amount of energy even by the energetic standards of radioactive decay.
In particle physics, Rutherford scattering is the elastic scattering of charged particles by the Coulomb interaction. It is a physical phenomenon explained by Ernest Rutherford in 1911 that led to the development of the planetary Rutherford model of the atom and eventually the Bohr model. Rutherford scattering was first referred to as Coulomb scattering because it relies only upon the static electric (Coulomb) potential, and the minimum distance between particles is set entirely by this potential. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of "elastic scattering" because neither the alpha particles nor the gold nuclei are internally excited. The Rutherford formula further neglects the recoil kinetic energy of the massive target nucleus.
The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.
A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.
In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as
In nuclear physics, the semi-empirical mass formula (SEMF) is used to approximate the mass of an atomic nucleus from its number of protons and neutrons. As the name suggests, it is based partly on theory and partly on empirical measurements. The formula represents the liquid-drop model proposed by George Gamow, which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. It was first formulated in 1935 by German physicist Carl Friedrich von Weizsäcker, and although refinements have been made to the coefficients over the years, the structure of the formula remains the same today.
The DLVO theory explains the aggregation and kinetic stability of aqueous dispersions quantitatively and describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so-called double layer of counterions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, . For two spheres of radius each having a charge separated by a center-to-center distance in a fluid of dielectric constant containing a concentration of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa potential,
A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit. This type of collision is common in plasmas where the typical kinetic energy of the particles is too large to produce a significant deviation from the initial trajectories of the colliding particles, and the cumulative effect of many collisions is considered instead. The importance of Coulomb collisions was first pointed out by Lev Landau in 1936, who also derived the corresponding kinetic equation which is known as the Landau kinetic equation.
The nuclear force is a force that acts between hadrons, most commonly observed between protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. Since protons have charge +1 e, they experience an electric force that tends to push them apart, but at short range the attractive nuclear force is strong enough to overcome the electrostatic force. The nuclear force binds nucleons into atomic nuclei.
The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, such as the Coulomb interaction. The equation was first suggested for the description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph.
Electron scattering occurs when electrons are displaced from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz force. This scattering typically happens with solids such as metals, semiconductors and insulators; and is a limiting factor in integrated circuits and transistors.
The plasma parameter is a dimensionless number, denoted by capital Lambda, Λ. The plasma parameter is usually interpreted to be the argument of the Coulomb logarithm, which is the ratio of the maximum impact parameter to the classical distance of closest approach in Coulomb scattering. In this case, the plasma parameter is given by:
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.
The Gamow factor, Sommerfeld factor or Gamow–Sommerfeld factor, named after its discoverer George Gamow or after Arnold Sommerfeld, is a probability factor for two nuclear particles' chance of overcoming the Coulomb barrier in order to undergo nuclear reactions, for example in nuclear fusion. By classical physics, there is almost no possibility for protons to fuse by crossing each other's Coulomb barrier at temperatures commonly observed to cause fusion, such as those found in the sun. When George Gamow instead applied quantum mechanics to the problem, he found that there was a significant chance for the fusion due to tunneling.
Weber electrodynamics is a historical alternative to Maxwell electrodynamics developed by Wilhelm Eduard Weber. In this theory, Coulomb's Law becomes velocity dependent. Weber electrodynamics does not describe electromagnetic waves and is incompatible with special relativity.
The Sommerfeld parameterη, named after Arnold Sommerfeld, is a dimensionless quantity used in nuclear astrophysics in the calculation of reaction rates between two nuclei and also appears in the definition of the astrophysical S-factor. It is defined as
