The difference between microscopic and macroscopic cross sections.

[CaptainMayhem]

Hello there folks. So I'm doing some homework, and one of the questions asks for a comparison of the microscopic absorption cross sections of B, Cd, Gd, and Hf with the fission cross section of U-235. I've found the fission cross section in my trusty 17th edition of Nuclides and Isotopes, but the next part of the question confused me. It then asks for the macroscopic cross sections assuming density is derived from the periodic table.

I'm not sure how to figure out the micro/macroscopic cross sections. There is only one given value per nuclide. Help?!

 

[Astronuc]

Microscopic cross-sections (σ) are given for each nuclide usually in units of barns, where 1 b = 1 x 10^-24 cm². It is essentially the effective area for the particular reaction.

The macroscopic cross-section (Σ) has units of cm^-1 is derived from the microscopic section according to

Σ = N σ, where N is the atomic density of a particular nuclide.

The reaction rate for a give interaction = Σ Φ, where Φ is the neutron flux.

These forumla work well for average or thermal neutron energies. If a wide enough energy spectrum is considered then one has to use integrals over the appropriate energy range.

 

[raining Pete]

Hi there! I can try to help you out with this question. The microscopic absorption cross section refers to the probability of an individual atom of a specific element (in this case B, Cd, Gd, and Hf) absorbing a neutron. This value is usually given in units of barns (1 barn = 10^-24 cm^2). The fission cross section of U-235 is also given in barns.

To compare these values, you can simply divide the microscopic absorption cross section of each element by the fission cross section of U-235. This will give you a ratio that shows how likely each element is to absorb a neutron compared to U-235.

As for the macroscopic cross sections, these refer to the total probability of a bulk material (in this case, the elements B, Cd, Gd, and Hf) absorbing a neutron. This value is dependent on the density of the material. To calculate the macroscopic cross section, you can use the following formula:

Macroscopic cross section = (microscopic cross section * number density of atoms)/density of material

The number density of atoms can be calculated using the Avogadro's number and the atomic weight of the element. The density of the material can be found on the periodic table.

I hope this helps! Let me know if you have any other questions. Good luck with your homework!