discuss the different terms in the semi empirical mass formula.
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Volume termEdit
The term {\displaystyle a_{V}A} is known as the volume term. The volume of the nucleus is proportional to A, so this term is proportional to the volume, hence the name.
The basis for this term is the strong nuclear force. The strong force affects both protons and neutrons, and as expected, this term is independent of Z. Because the number of pairs that can be taken from A particles is {\displaystyle {\frac {A(A-1)}{2}}}, one might expect a term proportional to {\displaystyle A^{2}}. However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of pairs of particles that actually interact is roughly proportional to A, giving the volume term its form.
The coefficient {\displaystyle a_{V}} is smaller than the binding energy of the nucleons to their neighbours {\displaystyle E_{b}}, which is of order of 40 MeV. This is because the larger the number of nucleons in the nucleus, the larger their kinetic energy is, due to the Pauli exclusion principle. If one treats the nucleus as a Fermi ball of {\displaystyle A} nucleons, with equal numbers of protons and neutrons, then the total kinetic energy is {\displaystyle {3 \over 5}A\varepsilon _{F}}, with {\displaystyle \varepsilon _{F}} the Fermi energy which is estimated as 28 MeV. Thus the expected value of {\displaystyle a_{V}} in this model is {\displaystyle E_{b}-{3 \over 5}\varepsilon _{F}\sim 17\;\mathrm {MeV} }, not far from the measured value.
The term {\displaystyle a_{V}A} is known as the volume term. The volume of the nucleus is proportional to A, so this term is proportional to the volume, hence the name.
The basis for this term is the strong nuclear force. The strong force affects both protons and neutrons, and as expected, this term is independent of Z. Because the number of pairs that can be taken from A particles is {\displaystyle {\frac {A(A-1)}{2}}}, one might expect a term proportional to {\displaystyle A^{2}}. However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of pairs of particles that actually interact is roughly proportional to A, giving the volume term its form.
The coefficient {\displaystyle a_{V}} is smaller than the binding energy of the nucleons to their neighbours {\displaystyle E_{b}}, which is of order of 40 MeV. This is because the larger the number of nucleons in the nucleus, the larger their kinetic energy is, due to the Pauli exclusion principle. If one treats the nucleus as a Fermi ball of {\displaystyle A} nucleons, with equal numbers of protons and neutrons, then the total kinetic energy is {\displaystyle {3 \over 5}A\varepsilon _{F}}, with {\displaystyle \varepsilon _{F}} the Fermi energy which is estimated as 28 MeV. Thus the expected value of {\displaystyle a_{V}} in this model is {\displaystyle E_{b}-{3 \over 5}\varepsilon _{F}\sim 17\;\mathrm {MeV} }, not far from the measured value.
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A graphical representation of the semi- empirical binding energy formula . the binding energy per nucleon in Me V (highest numbers in yellow, in excess of 8.5 Me V per nucleon ) is plotted for various nuclides as a functions of Z, the atomic Numbers ( on the Y- axis ) vs. N, the neutron Numbers (on the X- axis ) .
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