What Is the Packing Fraction in a Simple Cubic Crystal Structure?

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SUMMARY

The packing fraction in a simple cubic crystal structure is calculated to be π/6, approximately 0.5, indicating that half of the volume is occupied by atoms while the other half is empty space. This ratio is derived from the volume of spheres positioned at the corners of a cube, where each sphere has a radius corresponding to half the atom's diameter. Additionally, it is established that the largest impurity atom that can fit between the host atoms in this arrangement has a diameter of 0.4d, where d is the diameter of the host atoms.

PREREQUISITES
  • Understanding of geometric volume calculations, specifically for spheres and cubes.
  • Familiarity with the concept of packing fractions in crystal structures.
  • Knowledge of simple cubic arrangements in crystallography.
  • Basic algebra for manipulating equations related to volume and diameter.
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  • Study the derivation of packing fractions in different crystal structures, such as face-centered cubic (FCC) and body-centered cubic (BCC).
  • Learn about the implications of packing fractions on material density and properties.
  • Explore the concept of impurity atoms in crystal lattices and their effects on material characteristics.
  • Investigate the mathematical principles behind volume calculations for various geometric shapes.
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Students and professionals in materials science, crystallography, and chemistry who are interested in understanding the spatial arrangement of atoms in crystal structures and their implications on material properties.

Timiop2008
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Problem:
A)
In a simple cubic arrangement the atoms sit at the corners of imaginary cubes with their curved surfaces just touching. Show that the ratio of filled space to empty space between the atoms is pi/6 or about 0.5. This "packing fraction" is related to the density of the material. The volume of a sphere of radius r is 4/3 pi r³.

B)
If the atoms in the simple cubic arrangement in part A have diameter d, show that largest impurity atom that can just fit in between the host atoms has a diameter of 0.4d.

Please Help! I don't understand how to approach this Question!
 
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This is really just about calculating volumes. If there are 8 spheres at the corner of a cube what fraction of the volume of each sphere is in the cube. Compare that to the volume of the cube.
 

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