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Exploring Sphere Arrangements in Cubic Unit Cells: Types and Coordination Numbers

This text discusses the arrangement of spheres in various cubic unit cells, specifically focusing on Primitive, Body-Centered, Face-Centered, and End-Centered configurations. It covers the packing fractions for each arrangement and details the volumes occupied by spheres in unit cells. Additionally, it examines coordination numbers associated with these configurations, highlighting the highest packing fraction achieved in face-centered cubic structures. Understanding these arrangements is essential for applications in materials science and crystallography.

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Exploring Sphere Arrangements in Cubic Unit Cells: Types and Coordination Numbers

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  1. There are not more than 4 ways of arranging spheres in any shape of unit cell These arePrimitive, Body Centered,Face Centered & End Centered

  2. Primitive Cubic Unit Cell shape view Simple Cubic 1 3 2 4 5 7 6 8 Unit Cell arrangement view a = 2r

  3. Layer arrangement view

  4. Primitive Cubic

  5. Volume occupied by a sphere in the unit cell Total volume occupied by all the spheres in the unit cell Primitive Cubic

  6. Packing Fraction Fraction of the Unit cell’s volume occupied by the spheres Primitive Cubic

  7. Coordination number 6 Primitive Cubic

  8. Body Centered Cubic Unit Cell shape view Unit Cell arrangement view

  9. Layer arrangement view

  10. a > 2r Body Centered Cubic

  11. Body Centered Cubic

  12. Packing Fraction Volume occupied by a corner sphere in the unit cell Volume occupied by the central sphere in the unit cell Total Volume occupied by the spheres in the unit cell Packing Fraction Body Centered Cubic

  13. Coordination number 8 Body Centered Cubic

  14. Face Centered Cubic Unit Cell shape view Unit Cell arrangement view

  15. a Face Centered Cubic

  16. Face Centered Cubic

  17. Packing Fraction Volume occupied by a corner sphere in the unit cell Volume occupied by a face centered sphere in the unit cell Total Volume occupied by the spheres in the unit cell Packing Fraction Highest Packing Fraction of all shapes and of all arrangements Face Centered Cubic

  18. Coordination number y-z plane x-y plane x-z plane z x y Face Centered Cubic

  19. Coordination number a/2 a/2 Face Centered Cubic

  20. End Centered

  21. Radius Ratios Minimum rc/ra for various coordination numbers Coordination number - 3 2ra O B 30o A

  22. Coordination number - 4 (0,0,0) A B (a/4, a/4,a/4) z-axis Radius Ratios

  23. Coordination number - 4 (square planar) or 6 (octahedron) A B Radius Ratios

  24. Coordination number - 8 (cube) Radius Ratios

  25. Final Radius Ratios Co-ordination No. Radius Ratio, rc/ra 2 <0.155 [0.155, 0.225) 2 or 3 [0.225, 0.414) 2 or 3 or 4 Td [0.414, 0.732) 2 or 3 or 4 Td, 4 sq. pl or 6 Oh [0.732, 0.99) 2 or 3 or 4 Td, 4 sq. pl or 6 Oh or 8 Radius Ratios

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