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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 composes of  neutron and proton 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

Element
Mass Defect

(amu)
Total Binding Energy

(MeV)
Average Binding Energy

(MeV)
H-20.00242.231.12
He-40.030428.297.07
Li-70.043140.15
5.74
Be-90.062458.136.46
C-120.098992.157.679
O-16 0.1371127.627.976
Ca-400.3674342.04948.551
Fe-560.5286492.2668.79
Ag-1070.9825914.70758.549
Pb-2061.74121261.0577.869
U-2351.93541801.8577.667
U-2381.93411800.6477.566

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

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.

Frequently Asked Questions

How to calculate total binding energy of nucleus?

Binding energy is the result of the mass defect of proton and neutrons. So, it can be calculated by subtracting the experimental mass of nucleus from its expected mass.
Let us take helium (He-4) atom as an example. Helium atom has two protons and two neutrons. Thus its expected mass (in amu) can be calculated as:
2*(mass of proton) + 2*(mass of neutron)
= 2*1.0073 + 2*1.00867
= 4.03194 amu
But from experimental results, it was found that the mass of He-4 nucleus is 4.0016 amu.
So the mass defect or binding energy of He-4 nucleus is:
exected mass – experimental mass
= 4.03194 – 4.0016
= 0.03034 amu
This gives the total binding energy for helium nucleus. To convert amu into Mev, we can multiply it by 931 ( can be calculated by using formula E=mc2. Hence the binding energy of helium nucleus in terms of Mev is 28.3 Mev.
In this way we can calculate the binding energy of any atomic nucleus

What does binding energy measure?

Binding energy gives the amount of energy that holds the nucleons (protons + neutrons) together inside the nucleus. In other words, total binding energy gives the total energy required to break the nucleons apart.
Similarly, binding energy per nucleon gives the amount of energy hold by each nucleon. So, greater the binding per nucleon, more stable the atom will be.

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