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This is tutorial for K-12 students. This involves the analysis of a relevant research problem in nuclear materials - the segregation of chromium (Cr) and helium (He) to grain boundaries in iron (Fe). This project will introduce a student to using spreadsheet tools to extract information about this important reaction in nuclear energy applications. This is part of a program at Pacific Northwest National Laboratory designed to get high school students involved with STEM-related projects.

Author(s): Mark A. Tschopp, Fei Gao (PNNL), Joanna Sun (student, PNNL)


Here are some questions that may be needed to understand what we are doing in this example.

  • How does radiation damage materials? Here is a link to Radiation Material Science that gives a relatively good basic explanation of how radiation affects materials.
  • What is a point defect? Here is a link to find out about point defects.
  • What is a vacancy? It is a type of point defect. Here is a link to find out about a vacancy defect.
  • What is an interstitial atom? This is another type of point defect in a material. Here is a link to find out about interstitial defects. This is not used in the present example.
  • How are atoms arranged in a metal? Here is a link to find out about crystal structures. In this example, iron (Fe) is a body-centered cubic structure.
  • What is a formation energy? Here is a link to find out about grain boundaries. Grain boundaries join two perfect lattices of different orientations.
  • What is segregation? Here is a link to find out about segregation to grain boundaries.
  • Why Chromium in Iron? Here is a link to find out about Chromium steels, otherwise known as stainless steels. These are commonly used in the nuclear industry for their corrosion resistance.

Cohesive energy

What is the cohesive energy?

Let's start with the cohesive energy of a particular type of element: iron. For instance, iron is often in a body-centered cubic structure with 2 atoms/unit cell. Why 2? Think of a cube with atoms at the corners (8 of them) and an atom in the middle. Each of the eight atoms at the corners is shared with 7 other neighboring cubes, so each of the corner atoms counts as 1/8th of an atom. Hence, 8 * 1/8 + 1 = 2/unit cell. The length of the sides of the cube is called the lattice spacing.

Q: So if we had 10 lattice spacings in each direction, how many atoms would we have? A: 10 x 10 x 10 = 1000 cubes (or unit cells). Since there are 2 atoms per unit cell, this would result in 2000 total atoms.

Let's assume these are iron atoms. Now if we relaxed these atoms in our atomistic codes, we would get a total energy of -8026.08262 eV. The cohesive energy is merely the energy of each atom.

Q: What is the cohesive energy of iron? A: -8026.08262 eV / 2000 atoms = -4.012986 eV/atom

Chromium is also a body-centered cubic structure. For 10 x 10 x 10 unit cells of Cr (2000 atoms), the total energy is -7672.56624 eV and the cohesive energy is -3.83628312 eV/atom.

Formation energy

What is a formation energy?

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