Yukawa potential

In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the particle, r is the radial distance to the particle, and α is another scaling constant, so that 1 / α m {displaystyle 1/alpha m} is the range. The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is (1/meters). The Coulomb potential of electromagnetism is an example of a Yukawa potential with e − α m r {displaystyle e^{-alpha mr}} equal to 1 everywhere. This can be interpreted as saying that the photon mass m is equal to 0. In interactions between a meson field and a fermion field, the constant g is equal to the gauge coupling constant between those fields. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron. Prior to Hideki Yukawa's 1935 paper, physicists struggled to explain the results of James Chadwick's atomic model, which consisted of positively charged protons and neutrons packed inside of a small nucleus, with a radius on the order of 10−14 meters. Physicists knew that electromagnetic forces at these lengths would cause these protons to repel each other and for the nucleus to fall apart. Thus came the motivation for further explaining the interactions between elementary particles. In 1932, Werner Heisenberg proposed a 'Platzwechsel' (migration) interaction between the neutrons and protons inside the nucleus, in which neutrons were composite particles of protons and electrons. These composite neutrons would emit electrons, creating an attractive force with the protons, and then turn into protons themselves. When, in 1933 at the Solvay Conference, Heisenberg proposed his interaction, physicists suspected it to be of either two forms: on account of its short-range. However, there were many issues with his theory. Namely, it is impossible for an electron of spin 1/2 and a proton of spin 1/2 to add up to the neutron spin of 1/2. The way Heisenberg treated this issue would go on to form the ideas of isospin. Heisenberg's idea of an exchange interaction (rather than a Coulombic force) between particles inside the nucleus led Fermi to formulate his ideas on beta-decay in 1934. Fermi's neutron-proton interaction was not based on the 'migration' of neutron and protons between each other. Instead, Fermi proposed the emission and absorption of two light particles: the neutrino and electron, rather than just the electron (as in Heisenberg's theory). While Fermi's interaction solved the issue of the conservation of linear and angular momentum, Soviet physicists Igor Tamm and Dmitri Ivaneko demonstrated that the force associated with the neutrino and electron emission was not strong enough to bind the protons and neutrons in the nucleus. In his February 1935 paper, Hideki Yukawa combines both the idea of Heisenberg's short-range force interaction and Fermi's idea of an exchange particle y in order to fix the issue of the neutron-proton interaction. He deduced a potential which includes an exponential decay term ( e − α m r {displaystyle e^{-alpha mr}} ) and an electromagnetic term ( 1 / r {displaystyle 1/r} ). In analogy to quantum field theory, Yukawa knew that the potential and its corresponding field must be a result of an exchange particle. In the case of QFT, this exchange particle was a photon of 0 mass. In Yukawa's case, the exchange particle had some mass, which was related to the range of interaction (given by 1 / α m {displaystyle 1/alpha m} ). Since the range of the nuclear force was known, Yukawa used his equation to predict the mass of the mediating particle as about 200 times the mass of the electron. Physicists called this particle the 'meson,' as its mass was in the middle of the proton and electron. Yukawa's meson was found in 1947, and came to be known as the pion.