# Liquid Drop Model And Shell Model Pdf

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- Nuclear Physics
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- Nuclear Physics

*In nuclear physics and nuclear chemistry , the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. Gapon in The model was developed in following independent work by several physicists, most notably Eugene Paul Wigner , Maria Goeppert Mayer and J.*

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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 , [1] which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. 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. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force a residual effect of the strong force , there is a similarity to the structure of a spherical liquid drop. While a crude model, the liquid drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding energy.

The corresponding mass formula is defined purely in terms of the numbers of protons and neutrons it contains. The semi-empirical mass formula states the binding energy is:. Each of the terms in this formula has a theoretical basis.

While typically expressed by its basic five terms, further terms exist to explain additional phenomena. 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. 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.

This is because the larger the number of nucleons in the nucleus, the larger their kinetic energy is, due to the Pauli exclusion principle. 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. The basis for this term is the electrostatic repulsion between protons. To a very rough approximation, the nucleus can be considered a sphere of uniform charge density. The potential energy of such a charge distribution can be shown to be. Using the fine structure constant , we can rewrite the value of a C :. The theoretical justification for this term is more complex.

The Pauli exclusion principle states that no two identical fermions can occupy exactly the same quantum state in an atom. At a given energy level, there are only finitely many quantum states available for particles. What this means in the nucleus is that as more particles are "added", these particles must occupy higher energy levels, increasing the total energy of the nucleus and decreasing the binding energy. Note that this effect is not based on any of the fundamental forces gravitational , electromagnetic, etc.

Protons and neutrons, being distinct types of particles, occupy different quantum states. One can think of two different "pools" of states, one for protons and one for neutrons. Now, for example, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy than the available states in the proton pool.

If we could move some particles from the neutron pool to the proton pool, in other words change some neutrons into protons, we would significantly decrease the energy.

The imbalance between the number of protons and neutrons causes the energy to be higher than it needs to be, for a given number of nucleons. This is the basis for the asymmetry term. The actual form of the asymmetry term can again be derived by modelling the nucleus as a Fermi ball of protons and neutrons. Its total kinetic energy is. Thus we get. The first term contributes to the volume term in the semi-empirical mass formula, and the second term is minus the asymmetry term remember the kinetic energy contributes to the total binding energy with a negative sign.

The discrepancy is explained by our model not being accurate: nucleons in fact interact with each other, and are not spread evenly across the nucleus. For example, in the shell model , a proton and a neutron with overlapping wavefunctions will have a greater strong interaction between them and stronger binding energy. This makes it energetically favourable i.

One can also understand the asymmetry term intuitively, as follows. In addition, small differences between Z and N do not have a high energy cost.

This term captures the effect of spin -coupling. It is given by: [5]. The dependence on mass number is commonly parametrized as. The value of the exponent k P is determined from experimental binding energy data. Due to the Pauli exclusion principle the nucleus would have a lower energy if the number of protons with spin up were equal to the number of protons with spin down. This is also true for neutrons. Only if both Z and N are even can both protons and neutrons have equal numbers of spin up and spin down particles.

This is a similar effect to the asymmetry term. This means that the actual effect for large nuclei will be larger than expected by that model.

This should be explained by the interactions between nucleons; For example, in the shell model , two protons with the same quantum numbers other than spin will have completely overlapping wavefunctions and will thus have greater strong interaction between them and stronger binding energy. The same is true for neutrons. The coefficients are calculated by fitting to experimentally measured masses of nuclei. Their values can vary depending on how they are fitted to the data and which unit is used to express the mass.

Several examples are as shown below. The formula does not consider the internal shell structure of the nucleus. The semi-empirical mass formula therefore provides a good fit to heavier nuclei, and a poor fit to very light nuclei, especially 4 He. For light nuclei, it is usually better to use a model that takes this shell structure into account. This is roughly 1 for light nuclei, but for heavy nuclei the ratio grows in good agreement with experiment.

By substituting the above value of Z back into E b , one obtains the binding energy as a function of the atomic weight, E b A. The liquid drop model also allows the computation of fission barriers for nuclei, which determine the stability of a nucleus against spontaneous fission.

It was originally speculated that elements beyond atomic number could not exist, as they would undergo fission with very short half-lives, [10] though this formula did not consider stabilizing effects of closed nuclear shells. From Wikipedia, the free encyclopedia. Formula to approximate nuclear mass based on nucleon counts. Models of the nucleus. Nuclides ' classification. Nuclear stability. Radioactive decay. Nuclear fission. Capturing processes. High-energy processes. Nucleosynthesis and nuclear astrophysics.

High-energy nuclear physics. Proceedings of the Royal Society A. Bibcode : ZPhy Archived from the original PDF on 30 September Retrieved 30 September Nuclear Physics A. Bibcode : NuPhA. Introductory Nuclear Physics. Fundamental University Physics. Quantum and Statistical Physics.

Addison-Wesley Publishing Company. External Properties of Atomic Nuclei. Encyclopedia of Physics. Bibcode : HDP EPJ Web of Conferences. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Wikimedia Commons.

## Nuclear Physics

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 , [1] which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. 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. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force a residual effect of the strong force , there is a similarity to the structure of a spherical liquid drop.

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Collective model , also called unified model , description of atomic nuclei that incorporates aspects of both the shell nuclear model and the liquid-drop model to explain certain magnetic and electric properties that neither of the two separately can explain. In the shell model, nuclear energy levels are calculated on the basis of a single nucleon proton or neutron moving in a potential field produced by all the other nucleons. Nuclear structure and behaviour are then explained by considering single nucleons beyond a passive nuclear core composed of paired protons and paired neutrons that fill groups of energy levels, or shells.

This theory is based on the liquid drop model proposed by George Gamow. According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons protons and neutrons , which are held together by the strong nuclear force. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The interior nucleons are completely surrounded by other attracting nucleons.

Additional pages are accessible via the navigation bar on top.. The topics discussed in Nuclear Science and Technology I or II are selected from the following list which covers the materials for a more extensive, two-semester nuclear science course. The place holders in the list of topics will in due time be replaced by the actual presentation files. Electronic lecture presentations can be downloaded in various formats if marked by colored links.

### Nuclear Physics

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Sayyad Nagoorvali. Samit Kr. This project consists of a study of basic nuclear models, such as Liquid Drop Model, Shell Model and the Nilsson model for nuclei with different shapes at the ground state. The Liquid drop model of a nucleus is aid in the calculation of the nuclear binding energy. This model assumes that the ground state of nucleus is spherical.

Liquid-drop model , in nuclear physics , a description of atomic nuclei in which the nucleons neutrons and protons behave like the molecules in a drop of liquid. If given sufficient extra energy as by the absorption of a neutron , the spherical nucleus may be distorted into a dumbbell shape and then split at the neck into two nearly equal fragments, releasing energy. Although inadequate to explain all nuclear phenomena, the theory underlying the model provides excellent estimates of average properties of nuclei. Russian-born American physicist George Gamow formulated the model in , and Austrian physicists Lise Meitner and Otto Frisch used it in to explain nuclear fission. Liquid-drop model Article Additional Info.

It is shown that a consistent treatment of nuclear bulk and surface effects leads to an improved version of the liquid-drop mass formula with modified symmetry and Coulomb terms. If in addition shell effects are modelled through the counting of the number of valence nucleons, a very simple mass formula is obtained with an rms deviation from the database of atomic masses of about keV. A similar description of nuclear radii is suggested with specific reference to the neutron skin.

* Я д-думал, - заикаясь выговорил Бринкерхофф. - Я думал, что вы в Южной Америке. Лиланд Фонтейн окинул своего помощника убийственным взглядом.*

But in this nuclear scale, the fluid is made of nucleons protons and neutrons.

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Figure shows the energy levels predicted by the shell model using The Liquid Drop Model of the Nucleus. In the the Liquid Drop.