Ядролық массаны есептеу формуласы

Formula to approximate nuclear mass based on nucleon counts

In nuclear physics, the semi empirical mass formula (SEMF) (sometimes also called the Weizsäcker formula, Bethe–Weizsäcker formula, or Bethe–Weizsäcker mass formula to distinguish it from the Bethe–Weizsäcker process) 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 formula gives a good approximation for atomic masses and thereby other effects. However, it fails to explain the existence of lines of greater binding energy at certain numbers of protons and neutrons. These numbers, known as magic numbers, are the foundation of the nuclear shell model.

The term is known as the volume term. The volume of the nucleus is proportional to A, so this term is proportional to the volume, hence the name. The basis for this term is the strong nuclear force. The strong force affects both protons and neutrons, and as expected, this term is independent of Z. Because the number of pairs that can be taken from A particles is , one might expect a term proportional to However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of pairs of particles that actually interact is roughly proportional to A, giving the volume term its form. The coefficient is smaller than the binding energy possessed by the nucleons with respect to their neighbors , which is of order of 40 MeV. This is because the larger the number of nucleons in the nucleus, the larger their kinetic energy is, due to the Pauli exclusion principle. If one treats the nucleus as a Fermi ball of nucleons, with equal numbers of protons and neutrons, then the total kinetic energy is , with the Fermi energy, which is estimated as 38 MeV. Thus the expected value of in this model is not far from the measured value.

The term is known as the surface term. This term, also based on the strong force, is a correction to the volume term. The volume term suggests that each nucleon interacts with a constant number of nucleons, independent of A. While this is very nearly true for nucleons deep within the nucleus, those nucleons on the surface of the nucleus have fewer nearest neighbors, justifying this correction. This can also be thought of as a surface tension term, and indeed a similar mechanism creates surface tension in liquids. If the volume of the nucleus is proportional to A, then the radius should be proportional to and the surface area to This explains why the surface term is proportional to It can also be deduced that should have a similar order of magnitude to .

By maximizing Eb(A, Z) with respect to Z, one would find the best neutron–proton ratio N/Z for a given atomic weight A.