6805: 1094 Activities 3

Write your name and answers on this sheet and hand it in when requested.

Calculating a form factor: Fourier transform

Goal: To derive some Fourier transform results and verify that a nuclear shape yields a diffraction pattern.

1. Slide 4 of the slides on Nuclear Sizes (05-Sep "before") says that the cross section for elastic electron scattering is proportional to the square of the Fourier transform (FT) of the charge distribution. Working collectively, reduce the three-dimensional FT integral to a single integral over r:
   
   
   
   
   
   
   
   
   
   
2. Open the Mathematica notebook form_factor_fourier_transform.nb from the "in class" link. The notebook contains a model function for the nuclear density (which we'll use instead of the charge density) and an example of an FT and its inverse. Change the density plot to calcium-40. What is it about the shape that looks like a liquid drop?
   
   
   
3. The next section of the notebook gives an example of the FT of a radial function. Ask questions about any part you don't understand. Replace the test function everywhere with the density from the first section (renaming testFT appropriately). What does the cross section plot look like? (Note: this is plotted against q rather than θ, but they are related: q² = k²(1-cos θ).)
   
   
   
   
4. In the last section, replace the test function again and verify that the inverse FT works. What is the integral you did? Did you recover the original function (according to the table)?
   
   
   
   
   
   

Discussion questions on selected Decadal summary topics

Goal: Check your understanding of some details in the Decadal Survey slides. Discuss with your neighbor(s) and ask questions!

1. The r-process of nucleosynthesis is a rapid neutron capture process. Why does it have to be rapid? In the figure on slide 3, the r-process trajectory goes straight up at certain neutron numbers (actually it is a zig-zag). Speculate on why this happens. (Hint: see r_process_path.png.)
   
   
   
   
   
   
2. Slide 4 shows large "shell effects" in the tin region. Where do the shell effects occur in the tin region? What is the shell effect (e.g., are these nuclei more or less bound than would be expected from the liquid drop formula)?
   
   
   
   
   
   
3. In the superheavy region figure, have these nuclei been found experimentally?
   
   
   
   

Nuclear saturation density

Goal: Use the empirical liquid drop radius formula to estimate the density inside of a nucleus.

1. To get a rough estimate, assume the nucleus is spherical with a sharp surface and constant density, with the radius given by R = (1.2 fm) A^1/3. What is the density in nucleons/fm³? [Hint: the density is A/V, where V is the volume.]
   
   
   
   
   
   
2. How does the value compare to what we assumed in the FT notebook?
   
   
   
3. What is the dependence of the density on A? How is this like a liquid drop?
   
   
   

SEMF activities: Symmetric nuclear matter (SNM), pure neutron matter (PNM), and neutron stars

Goal: Explore some implications of the liquid drop SEMF as given on slide 2 of the Nuclear Masses 1 slides ("in class" for 05-Sep).

1. Use the formula for the binding energy B, to take the SNM limit (N=Z, both becoming infinite, Coulomb turned off) and the PNM limit (Z=0, N becoming infinite) of B/A (the binding energy per nucleon). Are these well-defined limits? What do you get for each?
   
   
   
   
2. Open the Mathematica notebook semi_empirical_mass_formula.nb from the "in class" link. Compare the definitions to the formulas on the slides. Ask about Mathematica features you don't understand. Does the formula for B agree with the slides?
   
   
   
   
3. Evaluate the definition and then look at the "Limiting values" section. Which limit corresponds to SNM and which to PNM? What goes wrong with the third limit? Explain what it means that the SNM result is positive while the PNM result is negative.
   
   
   
   
4. We'll come back to this, but for now just run the Beta decay parabola section and compare to the plot on slide 4. Identify all the stable nuclei here.
   
   
   
   
5. The "Test of pairing: even-odd staggering" is an attempt to reproduce the graph on slide 5. There are similarities, but slide 5 has a jump that is not reproduced. Speculate on what is missing in our SEMF calculation.
   
   
   
   
6. Fill in the details for the estimate of the minimum size of a neutron star on slide 10. Why do we neglect Coulomb, surface, and pairing energies? Why do we solve for B=0? Carry out the calculation using the constants in the Mathematica notebook and G from the web. How do your answers for A, R, and M compare to those on the slide?
   
   
   
   
   
   
   
   

Exploring the Table of Nuclides

Goal: derive some physics from the interactive table.

1. Go to the Interactive Table of Nuclides (under "Online references"). There are two rows of options for the color code at the top, with the default being "half-life". If you click on a square (you may need to zoom), you will get a value for the half-life, or else "STABLE". What is the highest A isotope that is stable? [Hint: It is not lead-208, but it is close!]
   
   
   
   
2. Switch from "half-life" to "Decay mode". Characterize the pattern: for what type of isotopes is the dominant decay mode beta decay (and of which kind of beta decay), alpha decay, spontaneous fission, proton emission, neutron emission?
   
   
   
   
   
   
   
   
3. Switch to "S_n", which is the one-neutron separation energy (called S_1n in the slides). Check S_1n for ¹¹Li. Why is the value consistent with this being a halo nucleus? What does it mean that the values of neighboring nuclei are negative? What is the definition of the neutron dripline in terms of S_1n?