Corrections to Liquid Drop Model / Semi Empirical BE Formula

Introduction and Liquid Drop Model of the Nucleus

In this section, the speaker introduces the concept of recurring peaks for small mass numbers in the binding energy curve, which correspond to alpha particles. They also mention the liquid drop model of the nucleus and its similarities with a drop of liquid.

Recurring Peaks and Liquid Drop Model

• The recurring peaks in the binding energy curve correspond to alpha particles (carbon-12).

• The nucleus can be compared to a drop of liquid due to similar properties.

• The liquid drop model helps explain the binding energy of a nucleus.

• The binding energy per nucleon graph obtained from this model matches experimental data.

Modifications to Improve Accuracy

This section discusses how modifications or additions can be made to the binding energy expression to improve its accuracy. These modifications will lead to a more accurate curve in the binding energy per nucleon graph.

Improving Accuracy with Modifications

• The previous expressions for binding energy can be further modified for greater accuracy.

• Two additional energy terms will be introduced into the binding energy expression.

• Including these terms will result in a more accurate curve on the binding energy per nucleon graph.

• The process will involve deriving a semi empirical binding energy formula for a given nucleus.

Stability and Asymmetry Energy

This section explains how stability is affected by the total number of neutrons and protons in a nucleus. It introduces asymmetry energy as an important factor influencing stability.

Neutrons, Protons, and Stability

• Nuclei tend towards configurations where neutrons and protons are approximately equal in number.

• Nuclei with significantly higher numbers of neutrons or protons are usually unstable.

• Unstable nuclei undergo radioactive decay to achieve a more balanced neutron-proton ratio.

• Asymmetry energy refers to the energy difference between configurations with different neutron-proton ratios.

Symmetry Energy and its Application

This section introduces symmetry energy and explains how it can be applied to the binding energy expression. The concept of energy levels in the nucleus is also discussed.

Energy Levels and Symmetry Energy

• Neutrons and protons arrange themselves in energy levels within the nucleus, similar to electrons in an atom.

• The energy levels for neutrons and protons are approximately similar.

• The difference in energy between two configurations with the same mass number but different neutron-proton ratios can be calculated using symmetry energy.

• Symmetry energy contributes to the overall binding energy of a nucleus.

Difference in Energies for Different Configurations

This section explores how the energies of different configurations with the same mass number but varying neutron-proton ratios differ. It demonstrates this difference using specific examples.

Comparing Energies of Different Configurations

• Two configurations with the same mass number (12) are compared.

• In one configuration, neutrons and protons are equal (6 each), while in the other, there are 10 neutrons and 2 protons.

• The difference in neutron-proton count is 8, which requires converting 4 protons into neutrons.

• Converting protons into neutrons increases their energies due to higher-energy levels occupied by neutrons.

Calculation of Energy Difference

This section focuses on calculating the change in energy when converting protons into neutrons. It provides an expression for this difference based on half of the neutron-proton difference and the energy level difference.

Calculating Energy Difference

• To create an excess of 8 neutrons, 4 protons need to be converted into neutrons.

• The change in energy for each converted proton is equal to 2e (energy level difference).

• The total energy difference can be expressed as half of the neutron-proton difference multiplied by 2e.

Conclusion

This section concludes the discussion on energy differences between configurations with varying neutron-proton ratios. It summarizes the derived expression for energy difference.

Summary

• The change in energy when converting protons into neutrons is determined by the neutron-proton difference and the energy level difference.

• The expression for this change in energy is half of the neutron-proton difference multiplied by 2e.

Timestamps are approximate and may vary slightly depending on the video version.

Total Energy Difference between Configurations

In this section, the total energy difference between two configurations is discussed, specifically focusing on the energy per nucleon and the number of neutrons.

Calculation of Total Energy Difference

• The total energy difference between two configurations is determined by multiplying the energy per nucleon by the new number of neutrons.

• The energy difference per nucleon occurs when a proton gets converted to a new neutron. It can be calculated using the expression: (1/8) * (n - Z)^2 * e, where n represents the number of neutrons and Z represents atomic number.

Deriving an Expression for Energy Difference

• The expression for energy difference can be simplified as follows: 1 - 8a - Z - Z^2 = (1/8) * (a - 2Z)^2 * e, where a represents mass number.

• This energy difference destabilizes the nucleus and decreases its binding energy.

Contributions to Binding Energy

This section discusses different contributions to the binding energy of a nucleus, including asymmetry energy and pairing energy.

Asymmetry Energy

• Asymmetry energy contributes towards decreasing the binding energy of a nucleus. It is represented by E and has a negative symbol due to its destabilizing effect on stability.

• The expression for asymmetry energy is inversely proportional to the mass number (A). It can be represented as E = ae^(−3/4), where 'a' is a constant of proportionality.

Pairing Energy

• Pairing energy arises from observations that nuclei with even numbers of protons and neutrons are more stable compared to those with odd numbers.

• The variation in energy or stability between different versions of nucleons is directly proportional to A^(−3/4).

• Pairing energy (P) can be positive, negative, or zero depending on the evenness or oddness of the nucleus. It contributes towards the binding energy.

Final Expression for Binding Energy

This section presents the final expression for the binding energy of a nucleus, incorporating contributions from various terms.

Final Expression

• The binding energy of a nucleus can be calculated using the semi-empirical binding energy formula.

• The formula includes terms derived from the liquid drop model (volume energy, surface energy, and coulombic energy), as well as additional terms for asymmetry energy and pairing energy.

• The complete expression for binding energy is given by: B = av - asA + acZ^2/A^(1/3) - aeA^(−3/4) ± ap(N-Z)^2/A + δ(N,Z), where av, as, ac, ae, and ap are constants representing different energies.

Conclusion

The transcript provides an overview of calculating total energy difference between configurations and discusses contributions to the binding energy of a nucleus through asymmetry and pairing energies. The final expression for binding energy incorporates these factors along with other terms derived from the liquid drop model.

New Section

This section discusses the expression for binding energy per nucleon and compares it with experimental data.

Plotting the Binding Energy Curve

• The expression for binding energy per nucleon is obtained by dividing the entire expression by capital A.

• A program in Scilab is used to replicate the semi-empirical expression and plot the binding energy curve.

• The constants a1, a2, a3, a4, and a5 are used in the program to calculate the binding energy.

• The program also includes expressions for asymmetry energy and pairing energy.

New Section

This section compares the obtained binding energy curve with the experimental data.

Comparison of Binding Energy Curves

• The obtained binding energy curve consists of volume energy, Coulomb back energy, surface energy, pairing energy, and asymmetry energy.

• The new binding energy curve shows better accuracy compared to the older one obtained from the liquid drop model.

• Peaks in the binding energy curve correspond to alpha particles and other particles for small mass numbers.

• The peak of the binding energy curve is around mass number 56, after which it starts decreasing.

• The comparison with experimental data shows that these features are replicated accurately.

New Section

This section provides access to code and resources related to replicating the binding energy curves.

Accessing Code and Resources

• The code used to obtain and plot both the original binding energy curve from liquid drop model and the corrected one is available on 8physics.com.

• Images of the obtained curves are also provided on this website.

• Users can access and use this code to replicate similar curves.

New Section

This section concludes by stating that both the semi-empirical expression and the experimental data show a good resemblance in the binding energy curve.

Validity of the Liquid Drop Model

• The obtained binding energy curve from the semi-empirical expression and the experimental data have a very accurate representation.

• This suggests that the liquid drop model, along with its additional corrections, has validity in explaining the behavior of nuclei.

The transcript is already in English.