Course description

Nuclei are made up of protons and neutrons. Let Zmp and Nmn be the individual masses of Z protons and N neutrons and which are combined to form a nucleus. Accurate measurement of nuclear masses by mass spectrometers (Fig. 10 a) show that the actual mass of a nucleus, containing Z protons and N neutrons, is less than (Zmp + Nmn), i.e., the mass of a nucleus is always less than the sum of the masses of the constituent protons and neutrons in the free state (Figs 10 b, c). The difference in masses

Zmp + Nmn – actual nuclear mass = ∆m

is called the mass defect?

According to Einstein, the mass defect (i.e., decrease in mass) is due to the release of energy (converted into energy) when the particles (protons and neutrons) combine to form the nucleus. The energy released can be calculated from the Einstein relationship, ∆E = ∆m.c2, where ∆m is the mass defect and c is the velocity of light. This energy is called the Binding Energy (B.E) of the nucleus and, which is a result of forces that hold the nucleus together (Fig. 10 d).

Therefore, mass defect is known, nuclear binding energy can be calculated by converting that mass to energy by using E = mc2. If a nucleus is to be broken into its constituent particles, the minimum energy required is the binding energy. Moreover, the magnitude of the binding energy of a nucleus determines its stability against disintegration. If the B.E. is large, the nucleus is more stable. The binding energy of nuclei is usually a positive number, since most nuclei require net energy to separate them into individual protons and neutrons.

 Let us calculate the mass defect and hence the binding energy of alpha particle which consists of two protons and two neutrons (Fig. 10 e). The sum of the masses of the two protons and two neutrons is 4.03188 amu and the measured mass of alpha particle is 4.00153 amu. Therefore, the mass defect, Δm = 4.03188 - 4.00153 = 0.0304 amu. Since, 1 amu = 1.66054 × 10-27 kg then converting it to energy we will get 

1.66054 × 10-27 kg × (3 × 108 m/s)2 = 1.49448 × 10-10 J 

Therefore, the energy equivalent of 1 amu (u) of mass is 933 MeV.

Since, E = mc2 then,

mass (m) = E/c2 = 933 MeV/c2 = 1 u = 1.66054 × 10-27 kg.

Hence the energy corresponds to the missing alpha particle mass (i.e., mass defect) is (0.0304 × 933) MeV = 28.36 MeV and this is the binding energy of alpha particle. 

Nuclear binding energy can also apply to situations when the nucleus splits into fragments composed of more than one nucleon; in these cases, the binding energies for the fragments, as compared to the whole, may be either positive or negative, depending on where the parent nucleus and the daughter fragments fall on the nuclear binding energy curve. If new binding energy is available when light nuclei fuse, or when heavy nuclei split, either of these processes result in the release of the binding energy. This energy, available as nuclear energy, can be used to produce nuclear power or build nuclear weapons. When a large nucleus splits into pieces, excess energy is emitted as photons, or gamma rays, and as kinetic energy, as a number of different particles are ejected.

The nuclear binding energies and forces are on the order of a million times greater than the electron binding energies of light atoms like hydrogen. The binding energy of nucleons in the nucleus of an atom amounts for most nuclei (i.e. Z > 5) to around 8 MeV per nucleon. Most naturally occurring nuclides have even number of protons and even number of neutrons. Nuclides with even number of protons and neutrons are generally more stable than those with odd numbers of these subatomic particles. Nuclides which have magic numbers (2, 8, 20, 28, 50, 82 and 126) of protons or neutrons are especially stable.

Stellar Energy Production: Both fission and fusion reactions have the potential to convert a small amount of mass into a large amount of energy and could conceivably account for the energy sources of stars. However, stars are made from light elements (mostly hydrogen and helium). Thus, fission cannot be initiated in stars as a source of energy, but fusion is quite possible if the right conditions prevail. As we shall see, these conditions can be found in the cores of stars, and thermonuclear fusion is the primary source of stellar energy (Figs 11 a, b).

Fig. 11 (c), indicates how stable atomic nuclei are; the higher the curve the more stable the nucleus. Notice the characteristic shape, with a peak near A=60. These nuclei (which are near iron in the periodic table and are called the iron peak nuclei) are the most stable in the Universe. The shape of this curve suggests two possibilities for converting significant amounts of mass into energy.

Fission Reactions: From the curve of binding energy, the heaviest nuclei are less stable than the nuclei near A = 60. This suggests that energy can be released if heavy nuclei split apart into smaller nuclei having masses nearer A=60. This process is called fission. It is the process that powers atomic bombs and nuclear power reactors.

Fusion Reactions: The curve of binding energy suggests a second way in which energy could be released in nuclear reactions. The lightest elements (like hydrogen and helium) have nuclei that are less stable than heavier elements up to A~60. Thus, sticking two light nuclei together to form a heavier nucleus can release energy. This process is called fusion, and is the process that powers hydrogen (thermonuclear) bombs and (perhaps eventually) fusion energy reactors. In both fission and fusion reactions the total masses after the reaction are less than those before. The "missing mass" appears as energy, by the famous Einstein equation, E = mc2. 

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