nucleus, atomic The core of the atom consisting of positively charged protons and neutral neutrons. Its size is 10-15m. The protons and neutrons are collectively called nucleons.
The strong forces that bind the nucleus are short ranged and charge independent. The nature of these forces are still not clear ( see under fundamental interactions ). The force between nucleons has spin dependent component, and a non central component besides the central component. The force turns repulsive at very short distances ( ~ fm).
A species of nucleus characterized by atomic mass number A ( total number of nucleons ) and atomic number Z ( number of protons ) is called a nuclide and is denoted by ZXA, where X denotes the chemical symbol of the element. If two nuclides have the same atomic number Z, but different atomic mass number, A they are called isotopes of an element. An element may have several isotopes. The unstable isotopes are called radioactive. Nuclides with same atomic mass number, A but different atomic number, Zs are called isobars.
In light nuclei the number of neutrons is approximately equal to the number of protons (N» Z). As A increases the number of neutrons becomes larger than the number of protons for stable nuclei. The neutron excess reduces the electrostatic repulsion energy between the protons making the nucleus stable.
The nucleus is bound by ‘binding energy’ *, defined as the energy required to break the nucleus into constituent particles.
A nucleus can be assumed to be sphere. Since the volume of the nucleus is proportional to A, we have the following relation for the radius of the nucleus,
R = r0 A1/3 (n6)
where the value of the constant r0 = 1.4 fm.
Liquid drop model of the nucleus: It was proposed by Weiszacker. In this model it is assumed that the nucleus behaves like a liquid drop. This model could successfully explain the dependence of the binding energy of the nucleus on mass number, A and nuclear fission. The liquid drop model leads to the following expression for the mass of the nucleus which is called the semi empirical mass formula.
M = Zmp + (A-Z) mn - b1A + b2 A2/3 + b3 Z2 A-1/3 + b4 (A- 2Z)2 A-1 + b5 A-3/4 (n7)
where the constants are obtained by fitting experimental data. b1 =14.0MeV, b2 = 13.0MeV, b3 = 0.58MeV, b4 = 19.3MeV. The constant b5 is given by the scheme,
|
N |
Z |
b5 |
|
even |
even |
-33.5MeV |
|
odd |
odd |
+33.5MeV |
|
odd |
even |
0 |
|
even |
odd |
0 |
The first two terms in eq. (n7) is the total mass of the nucleons. The third and successive terms are corrections to the first two terms.
The term -b1A is known as volume energy term. The force that holds the nucleons together ( short range together with saturation property ) is analogous to molecular forces holding a liquid drop. Therefore, this energy is proportional to the number of nucleons.
The molecules on the surface of a liquid drop are less tightly bound compared to those which are deep inside. Therefore the volume energy has been overestimated; the correction to it is proportional to the surface area 4 R2 = +b2A2/3 ( surface energy ).
The term b3Z2A-1/3 is due to the Coulomb repulsion energy ( Ec ) between the protons. Since
Ec = constt. Z (Z -1)/R
= b3 Z2 A-1/3 for large Z
The last two terms in the eq. (n7) are due to quantum mechanical effect. It is observed that as ˝ N -Z˝ increases, nuclei become increasingly unstable. The term b4 ( A - 2Z )2 A-1 = b4 ( N - Z )2 A-1 has been included to take into account this factor.
The last term can be understood by considering the stability of the nucleus in terms of various combinations of Z and N. It is observed that even-even nuclei are most stable whereas odd-odd nuclei are least stable. The stability of even-odd or odd-even nuclei lies in between these extremities.
Magic numbers and shell model: Very strong experimental evidence shows that when a nucleus contains 2, 8, 20, 50, 82, or 126 protons or neutrons the so called magic numbers a sharp departure in gross properties of the nuclei occurs. At these numbers nuclei are found to be particularly stable and numerous. The nucleons that complete the magic numbers are also found to have higher binding energies. A similarity with extranuclear atomic structure suggests the existence of shells inside the nucleus. A quantum mechanical model developed is capable of explaining not only magic numbers but many other nuclear properties such as spin, magnetic moment and energy levels.