1 / 17

Objectives

Objectives. By the end of this section you should: know how atom positions are denoted by fractional coordinates be able to calculate bond lengths for octahedral and tetrahedral sites in a cube be able to calculate the size of interstitial sites in a cube

hollie
Download Presentation

Objectives

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Objectives By the end of this section you should: • know how atom positions are denoted by fractional coordinates • be able to calculate bond lengths for octahedral and tetrahedral sites in a cube • be able to calculate the size of interstitial sites in a cube • know what the packing fraction represents • be able to define and derive packing fractions for 2 different packing regimes

  2. 1. 2. 3. 4. 0, 0, 0 ½, ½, 0 ½, 0, ½ 0, ½, ½ Fractional coordinates Used to locate atoms within unit cell Note 1: atoms are in contact along face diagonals (close packed) Note 2: all other positions described by positions above (next unit cell along)

  3. Octahedral Sites Coordinate ½, ½, ½ Distance = a/2 Coordinate 0, ½, 0 [=1, ½, 0] Distance = a/2 In a face centred cubic anion array, cation octahedral sites at: ½ ½ ½, ½ 0 0, 0 ½ 0, 0 0 ½

  4. Tetrahedral sites Relation of a tetrahedron to a cube: i.e. a cube with alternate corners missing and the tetrahedral site at the body centre

  5. Can divide the f.c.c. unit cell into 8 ‘minicubes’ by bisecting each edge; in the centre of each minicube is a tetrahedral site

  6. So 8 tetrahedral sites in a fcc

  7. Bond lengths important dimensions in a cube Face diagonal, fd (fd) = (a2 + a2) = a 2 Body diagonal, bd (bd) = (2a2 + a2) = a 3

  8. Bond lengths: Octahedral: half cell edge, a/2 Tetrahedral: quarter of body diagonal, 1/4 of a3 Anion-anion: half face diagonal, 1/2 of a2

  9. Sizes of interstitials fcc / ccp Spheres are in contact along face diagonals octahedral site, bond distance = a/2 radius of octahedral site = (a/2) - r tetrahedral site, bond distance = a3/4 radius of tetrahedral site = (a3/4) - r

  10. Summaryf.c.c./c.c.p anions 4 anions per unit cell at: 000 ½½0 0½½ ½0½ 4 octahedral sites at: ½½½ 00½ ½00 0½0 4 tetrahedral T+ sites at: ¼¼¼ ¾¾¼ ¾¼¾ ¼¾¾ 4 tetrahedral T- sites at: ¾¼¼ ¼¼¾ ¼¾¼ ¾¾¾ A variety of different structures form by occupying T+ T- and O sites to differing amounts: they can be empty, part full or full. We will look at some of these later. Can also vary the anion stacking sequence - ccp or hcp

  11. Packing Fraction • We (briefly) mentioned energy considerations in relation to close packing (low energy configuration) • Rough estimate - C, N, O occupy 20Å3 • Can use this value to estimate unit cell contents • Useful to examine the efficiency of packing - take c.c.p. (f.c.c.) as example

  12. So the face of the unit cell looks like: Calculate unit cell side in terms of r: 2a2 = (4r)2 a = 2r 2 Volume = (162) r3 Face centred cubic - so number of atoms per unit cell =corners + face centres = (8  1/8) + (6  1/2) = 4

  13. Packing fraction The fraction of space which is occupied by atoms is called the “packing fraction”, , for the structure For cubic close packing: The spheres have been packed together as closely as possible, resulting in a packing fraction of 0.74

  14. Group exercise: Calculate the packing fraction for a primitive unit cell

  15. Primitive

  16. Close packing • Cubic close packing = f.c.c. has =0.74 • Calculation (not done here) shows h.c.p. also has =0.74 - equally efficient close packing • Primitive is much lower: Lots of space left over! • A calculation (try for next time) shows that body centred cubic is in between the two values. • THINK ABOUT THIS! Look at the pictures - the above values should make some physical sense!

  17. Summary • By understanding the basic geometry of a cube and use of Pythagoras’ theorem, we can calculate the bond lengths in a fcc structure • As a consequence, we can calculate the radius of the interstitial sites • we can calculate the packing efficiency for different packed structures • h.c.p and c.c.p are equally efficient packing schemes

More Related