hw04_solutions.pdf - Physics 5110 Spring 2020 Homework#4 Due in class Wednesday February 5th 1 Using the Bethe-Weizs\u00a8acker semi-empirical mass formula

# hw04_solutions.pdf - Physics 5110 Spring 2020 Homework#4...

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Physics 5110, Spring 2020 Homework #4 Due in class Wednesday February 5 th 1. Using the Bethe-Weizs¨ acker semi-empirical mass formula, plot the binding energy per nu- cleon of nuclei with A = 29 as a function of the nuclear charge Z . Let Z range from 8 to 21. What value of the nuclear charge corresponds to the most stable nucleus? Does this agree with observation? 2. Calculate the binding energy of the last neutron in 4 He and the last proton in 16 O. How do these compare with B/A for these nuclei? What does this tell you about the stability of 4 He relative to 3 He, and of 16 O relative to 15 N? Hint: the binding energy of the last neutron needed to form a nucleus ( A,Z ) is given by [ M ( A - 1 ,Z ) + m n - M ( A,Z )] c 2 . An analogous expression holds for the last proton. 3. Martin and Shaw , Problem 2.9. 4. Martin and Shaw , Problem 2.10. continued on the following page
5. Below is an atomic energy level diagram, for the simplified model in which electrons do not interact. The vertical axis is the energy, and the horizontal axis is the angular momentum quantum number l . The energy levels are labeled with the standard spectroscopic notation, wherein the number is the principal quantum number n and the letter represents the angular momentum quantum number as s = 0, p = 1, d = 2, f = 3, g = 4, h = 5 et cetera . The numbers in the column at the right represent the total number of electrons enclosed in the particular shell. Another basic result of quantum mechanics is the solution of the Schr¨ odinger equation for the 3D harmonic oscillator potential , V ( r ) = 1 2 2 r 2 The solution has energy levels given by 1 : E n,l = ¯ parenleftbigg 2 n + l - 1 2 parenrightbigg where n = 1 , 2 , 3 ... l = 0 , 1 , 2 ... for any n Create a diagram similar to the above for noninteracting fermions in the harmonic oscillator potential. Use ω = 1 . 5 × 10 20 sec - 1 to get nuclear-like energy levels, and include all states with energy less than or equal to 4 s . Include a column of multiplicities.

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• Fall '09
• Atom, Binding energy, Atomic orbital, Atomic nucleus

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