Binding energy of nucleus and nucleons

Binding energy of nucleus and nucleons

Binding Energy

The nucleons are held together within a nucleus by strong attractive forces among the nucleons. One has to apply some energy in order to break the nucleus into its constituents. This energy required to decompose the nucleus into its constituents is known as binding energy of nucleus.

Experimentally, it is found that the mass of any permanently stable nucleus is less than the sum of masses of the constituent particles. The decrease in mass is known as mass defect. It is denoted by Δm.

The mass Δm disappears as an equivalent energy given by Einstein’s mass-energy relation :E=mc2 is liberated. This energy is called the binding energy of nucleus and is responsible for holding the nucleus together in the nucleus.

If M is the experimentally determined mass of a nucleus having z-protons and each of mass mp and N neutrons each of mass mn. Then mass defect is given by

Δm = (Zmp + Nmn) – M

So, B.E. = [(Zmp + Nmn) – M ]c2
The Binding energy is a measure of nuclear stability. Greater the binding energy, greater will be the stability of nucleus. A nucleus having the least possible energy equal to binding energy is said to be in the ground state. If the nucleus has energy greater than Emin, it is said to be in the excited state. If E = 0, the nucleus dissociates into its constituent particles.

Illustration of binding energy of nucleus :

Let us take an example of the deuteron to calculate binding energy. The nucleus of deuterium is called deuteron  and is made up of a proton and a neutron. If M is the mass of deuteron nucleus and mp and mn are the masses of proton and neutrons respectively, then mass defect

Δm = [(Zmp + Nmn) – M]

= (1.0086654 + 1.0072764) – 2.0135534

= 0.0023884 a.m.u.

So, B.E. = 0.0023884 × 931

= 2.23 MeV

Thus, deuteron conposes of  neutron and photon which are held together with energy equal to 2.23 MeV. In fact, when a γ – ray photon with energy 2.23 MeV or more collides with deuteron, the latter breaks down into proton and neutron. This process is known as photo disintegration.

Binding Energy per Nucleon Table

Mass Defect

Total Binding Energy

Average Binding Energy

O-16 0.1371127.627.976

Stability of nucleus and binding energy:

Binding energy per nucleon is the average energy which is we must supply to take out a nucleon from the nucleus.

B.E. per nucleon = Total binding energy of a nucleus/The number of nucleons it contains

The stability of a nucleus depends upon binding energy per nucleon rather than the total binding energy. Hence, knowledge of binding per nucleon is more important than the total binding energy of nucleus. If we plot a graph between binding energy per nucleon and the mass number for various nuclei, we obtain the graph as follows:

Binding energy curve
Variation of binding energy per nucleon with mass number

A few peaks are seen at low values of mass number A are for lighter nuclei He, C, O, which are comparatively stable nuclei in their neighborhoods.

Results from B.E. vs. A graph :

  1. Binding energy per nucleon for light nuclei such as 2He2 is very small. Then it increases rapidly with mass number up to A = 20 and the curve possesses peaks corresponding to nuclei 2He4, 6C12 and 8O16. Te peaks indicate that these nuclei are more stable than those in their neighborhood.
  2. After A = 20, binding energy per nucleon increases gradually and for mass numbers between 40 and 120, it becomes more or less flat. For mass numbers between 40 and 120, it becomes more or less flat. For A = 56 (26O56), binding energy per nucleon is maximum and is equal to 8.8 MeV.
  3. Then after binding energy per nucleon falls slowly with A, dropping to 7.6 MeV at highest mass number 240. Evidently, nuclei of intermediate mass number (40 – 120) are the most stable. This low value of binding energy per nucleon in case of heavy nuclei is unable to control over the coulomb’s repulsion between protons. This causes fission of heavy nuclei and they disintegrate emitting α particles have extra stability. some other particles like β and γ are also emitted. The process of disintegration of heavy nuclei is radioactivity.
  4. Thus, binding energy per nucleon has low value for both light and very heavy nuclei. In order to obtain higher values of binding energy per nucleon, the higher nuclei may unite together to form a heavier nucleus (fusion) or heavier nucleus may split into lighter nuclei (fission). In both processes, greater the value of binding energy per nucleon results in the liberation of energy.

Significance of binding energy curve :

  1. The binding energy curve rises slowly as A increases has a peak value at the middle at A = 56 (26O56) and then falls slowly. The fact that binding energy exists at all means that the nuclei more complex than single proton of hydrogen can be stable. Such stability in turn, accounts for the existence of various elements and hence explains the reasons for the existence of different forms of matter.
  2. The cause of release of energy in the fusion of light nuclei into heavier ones is explained by the the increase of binding energy per nucleon with mass number. Such a release of energy explains how sun stars get their energy.
  3. On the other hand, breaking of heavier nuclei into lighter ones (fission) also releases energy. We can use it for production of electric energy in nuclear reactors.

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