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2-4: The Liquid Drop Model

Needless to say, a nucleus is a quantum mechanical many-body system consisting of many nucleons. We can not directly watch it with our eyes. In such case, how can we see the structure of a nucleus? Of course, the fundamental theory to govern the nuclear structure is Quantum Mechnics, and the basic equation to describe it must be the Schroedinger equation. It is however very difficult and almost impossible to precisely solve the many-body Schroedinger equation for such a complicated system as nuclei.
For such a complicated system, we sometimes assume a simple model which is very easy to treat and understand. If this model can reproduce a lot of experimental data on nuclei in a unified way, then we could understand the structure of the nuclei.
Here, let us discuss the liquid drop model, which was the oldest one introduced for the nuclear structure.

[The Liquid Drop Model]
Just after the discovery of the neutron by Chadwick, Ivanenko and Heisenberg proposed an idea that nuclei are composed of protons and neutrons (1932).
Then it becomes the next question what structure do nuclei have.
C. Fv. Weizsaecker (Germany, 1912- ) imaged that the nucleus might be something like a raindrop. This is called a liquid drop model.

[Weizsaecker-Bethe's Mass Formula]
On the basis of the liquid drop model, Weizsaecker and H. A. Bethe (Germany, USA, 1906-2005) proposed a simple formula for the nuclear binding energy.
Let the proton and neutron masses be and , respectively. And let the mass of nucleus of the proton number Z and the neutron number N be M (Z, N ). As learned on the the last page, the binding energy of a nucleus, B (Z, N ), is given by

Weizsaecker and Bethe found that the above binding energy may approximately be obtained by a formula,

This is the famous Weizsaecker-Bethe's Mass Formula, which is sometimes called the semiempirical mass formula.
Meaning of each term in the above mass formula can be understood according to the idea of the liquid drop model.
The first term is proportional to the nuclear volume and called the volume energy, which is the main part of the nuclear binding energy.
The second term is proportional to the area of the nuclear surface, and it is considered as the surface energy due to the surface tension.
The third term is the Coulomb energy which is the energy loss due to the Coulomb repulsive forces between protons. Among the nuclei with the same mass number A , the energy loss due to the Coulomb forces becomes larger in nuclei with larger proton number Z. Because of this Coulomb energy, nuclei with larger N compared to Z have less energy loss, so that, in the larger mass number region, the neutron number N is, in general, greater than the proton number Z .
The next term, the fourth term, shows an opposite effect; namely, there is a property that, the less the difference between the proton number Z and the neutron number N is, the greater the energy gain becomes. This effect is represented by the fourth term, which is called the symmetry energy. This effect comes from the symmetry property in the nuclear force.
In the nucleus, a pair of equivalent nucleons (proton-proton or neutron-neutron) couples with each other more strongly than a pair of non-equivalent nucleons (proton-neutron). This is also due to the property of the nuclear force. The last term, the fifth term, represents this effect, which is called the even-odd mass difference.
The more detailed discussions about the last two terms, the symmetry energy and the even-odd mass difference, are omitted here, because they would be very specialized and complicated.
The parameters in the above Weizsaecker-Bethe mass formula should be adjusted to the experimental nuclear masses. The results are


[Comparison with Experimetal Data]
In order to confirm how well the Weizsaecker-Bethe mass formula can reproduce the experimental data of nuclear binding energies, the comparison with experiment is shown in the following figure, in which the red curve shows the values of the binding energy per nucleon obtained by the Weizsaecker-Bethe formula, and the black points indicate the experimental data, which are equivalent to those in the figure at the bottom of the preceding page. You should be careful because the scales of ordinates in these two figures are somewhat varied for the sake of convenience.

[Comparison between Mass formula and Experiment]
The black points indicate the experimental data of the binding energy per nucleon. The red curve shows the values obtained by the Weizsaecker-Bethe mass formula. Be careful because the scale of the ordinate is largely expanded.
As seen in the above figure, the Weizsaecker-Bethe mass formula can reproduce well the experimental data for a wide range of nuclei. We can therefore conclude that the liquid drop model is enough valid in nuclei.
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