Voids in crystals

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Voids in crystals

  1. 1.  We have already seen that as spheres cannot fill entire space → the packingfraction (PF) < 1 (for all crystals) This implies there are voids between the atoms. Lower the PF, larger the volumeoccupied by voids. These voids have complicated shapes; but we are mostly interested in the largestsphere which can fit into these voids→ hence the plane faced polyhedron versionof the voids is only (typically) considered. The size and distribution of voids in materials play a role in determining aspectsof material behaviour → e.g. solubility of interstitials and their diffusivity The position of the voids of a particular type will be consistent with thesymmetry of the crystal In the close packed crystals (FCC, HCP) there are two types of voids →tetrahedral and octahedral voids (identical in both the structures as the voids areformed between two layers of atoms) The octahedral void has a coordination number 6 (should not be confused with 8 coordination!) In the ‘BCC crystal’ the voids do NOT have the shape of the regular tetrahedronor the regular octahedron (in fact the octahedral void is a ‘linear void’!!)Voids
  2. 2. SC The simple cubic crystal (monoatomic decoration of the simple cubic lattice) has large void in the centre of the unitcell with a coordination number of 8. The actual space of the void in very complicated (right hand figure below) and the polyhedron version of the void isthe cube (as cube is the coordination polyhedron around a atom sitting in the void)( 3 1) 0.732xrr= − =Actual shape of the void (space)!Polyhedral model (Cube)True Unit Cell of SC crystalVideo: void in SC crystalVideo: void in SC crystal
  3. 3. FCC Actual shape of the void is as shown below.Actual shape of voidPosition of some of the atoms w.r.t to the void
  4. 4. FCC The complicated void shown before is broken down in the polyhedral representation into two shapes the octahedronand the tetrahedron- which together fill space.Octahedra and tetrahedra in an unit cellQuarter of a octahedron belong to an unit cell4-Quarters forming a full octahedron4-tetrahedra in viewCentral octahedron in viewVideo of the constructionshown in this slideVideo of the constructionshown in this slide
  5. 5. VOIDSTetrahedral OctahedralFCC = CCPNote: Atoms are coloured differently but are the samecellntetrahedro VV241= celloctahedron VV61=¼ way along body diagonal{¼, ¼, ¼}, {¾, ¾, ¾}+ face centering translationsAt body centre{½, ½, ½}+ face centering translationsOVTVrvoid / ratom = 0.225rVoid / ratom = 0.414Video: voids CCPVideo: voids CCPVideo: atoms forming the voidsVideo: atoms forming the voids
  6. 6. FCC- OCTAHEDRAL{½, ½, ½} + {½, ½, 0} = {1, 1, ½} ≡ {0, 0, ½}Face centering translationNote: Atoms are coloured differently but are the sameEquivalent site for anoctahedral voidSite for octahedral voidOnce we know the position of a void then we can usethe symmetry operations of the crystal to locate theother voids. This includes lattice translations
  7. 7. FCC voids Position Voids / cell Voids / atomTetrahedral¼ way from each vertex of the cubealong body diagonal <111>→ ((¼, ¼, ¼))8 2Octahedral• Body centre: 1 → (½, ½, ½)• Edge centre: (12/4 = 3) → (½, 0,0)4 1
  8. 8. Size of the largest atom which can fit into the tetrahedral void of FCCCV = r + xRadius of thenew atomexre +=46225.0~1232 −=⇒=rxreSize of the largest atom which can fit into the Octahedral void of FCC2r + 2x = ara 42 =( ) 414.0~12 −=rxThus, the octahedral void is the biggerone and interstitial atoms (which areusually bigger than the voids) wouldprefer to sit here
  9. 9. VOIDSTETRAHEDRAL OCTAHEDRALHCP These voids are identical to the ones found in FCC (for ideal c/a ratio) When the c/a ratio is non-ideal then the octahedra and tetrahedra are distorted (non-regular)Important Note: often in these discussions an ideal c/a ratio will be assumed (without stating the same explicitly)Note: Atoms are coloured differently but are the sameCoordinates: (⅓ ⅔,¼), (⅓,⅔,¾)),,(),,,(),,0,0(),,0,0(: 8731328131328583sCoordinate
  10. 10. Octahedral voids occur in 1 orientation, tetrahedral voids occur in 2 orientationsThe other orientation of the tetrahedral voidNote: Atoms are coloured differently but are the same
  11. 11. Note: Atoms are coloured differently but are the sameFurther views
  12. 12. Note: Atoms are coloured differently but are the sameOctahedral voidsTetrahedral voidFurther views
  13. 13. HCP voids PositionVoids /cellVoids / atomTetrahedral(0,0,3/8), (0,0,5/8), (⅔, ⅓,1/8),(⅔,⅓,7/8)4 2Octahedral • (⅓ ⅔,¼), (⅓,⅔,¾) 2 1Voids/atom: FCC ≡ HCP→ as we can go from FCC to HCP (and vice-versa) by a twist of 60° around a central atom oftwo void layers (with axis ⊥ to figure) Central atomCheck belowAtoms in HCP crystal: (0,0,0), (⅔, ⅓,½)
  14. 14. AABFurther viewsVarious sections along the c-axisof the unit cellOctahedral voidTetrahedral void
  15. 15. Further views with some modelsVisualizing these voids can sometimes be difficult especially in the HCPcrystal. ‘How the tetrahedral and octahedral void fill space?’ is shown in theaccompanying videoVideo: Polyhedral voids filling spaceVideo: Polyhedral voids filling space
  16. 16.  There are NO voids in a ‘BCC crystal’ which have the shape of a regularpolyhedron (one of the 5 Platonic solids) The voids in BCC crystal are: distorted ‘octahedral’ and distorted tetrahedral However, the ‘distortions’ are ‘pretty regular’ as we shall see The distorted octahedral void is in a sense a ‘linear void’→ an sphere of correct size sitting in the void touches only two of the six atoms surrounding it Carbon prefers to sit in this smaller ‘octahedral void’ for reasons which we shallsee soonVoids in BCC crystal
  17. 17. VOIDSDistorted TETRAHEDRAL Distorted OCTAHEDRAL**BCCaa√3/2a a√3/2rvoid / ratom = 0.29 rVoid / ratom = 0.155Note: Atoms are coloured differently but are the same ** Actually an atom of correct size touches onlythe top and bottom atomsCoordinates of the void:{½, 0, ¼} (four on each face) Coordinates of the void:{½, ½, 0} (+ BCC translations: {0, 0, ½})Illustration on one face only
  18. 18. BCC voids PositionVoids /cellVoids/ atomDistortedTetrahedral• Four on each face: [(4/2) × 6 = 12] → (0, ½, ¼) 12 6DistortedOctahedral• Face centre: (6/2 = 3) → (½, ½, 0)• Edge centre: (12/4 = 3) → (½, 0, 0)6 3{0, 0, ½})Illustration on one face onlyOVTV
  19. 19. From the right angled triange OCM:41622aaOC +=54a r x= = +For a BCC structure: 3 4a r= (34ra = )xrr+=3445⇒ 29.0135=−=rxaa√3/2BCC: Distorted Tetrahedral Voidalculation of the size of the distorted tetrahedral void
  20. 20. 2axrOB =+=324rxr =+ raBCC 43: =1547.01332=−=rxDistorted Octahedral Voida√3/2aaaOB 5.02== aaOA 707.22==As the distance OA > OB the atom in thevoid touches only the atom at B (bodycentre).⇒ void is actually a ‘linear’ void*This implies:alculation of the size of the distorted octahedral void* Point regarding ‘Linear Void’ Because of this aspect the OV along the 3 axes can bedifferentiated into OVx, OVy & OVz Similarly the TV along x,y,z can be differentiated
  21. 21.  Fe carbon alloys are important materials and hence we consider them next The octahedral void in FCC is the larger one and less distortion occurs whencarbon sits there → this aspect contributes to a higher solubility of C in γ-Fe The distorted octahedral void in BCC is the smaller one → but (surprisingly)carbon sits there in preference to the distorted tetrahedral void (the bigger one) -(we shall see the reason shortly) Due to small size of the voids in BCC the distortion caused is more and thesolubility of C in α-Fe is small this is rather surprising at a first glance as BCC is the more open structure but we have already seen that the number of voids in BCC is more than that inFCC → i.e. BCC has more number of smaller voidsSee next slide for figures
  22. 22. A292.1=FeFCCrA534.0)( =octxFeFCCA77.0=CrCNVoid (Oct)FeFCCHA258.1=FeBCCr A364.0).( =tetdxFeBCCA195.0).( =octdxFeBCCFCCBCCFeBCCRelative sizes of voids w.r.t to atoms( . )0.155FeBCCFeBCCx d octr=( . )0.29FeBCCFeBCCx d tetr=Relative size of voids, interstitials and Fe atomNote the difference in size of the atomsSpend some time over this slideSize of the OVSize of Carbon atomSize of Fe atomCCP crystalSize of Fe atomBCC crystalSize of the OVSize of the TV0.71 ANr =0.46 AHr =Void (Tet)
  23. 23.  We had mentioned that the octahedral void in BCC is a linear one(interstitial atom actually touches only two out of the 6 atoms surrounding it) In the next slide we make a approximate calculation to see till what size will itcontinue to touch only two Fe atoms(these are ‘ideal’ simplified geometrical calculations and in reality other complications will have to be considered)
  24. 24. A258.1=FeBCCr22AaOA r x= + =2 63Arr x+ =raBCC 43: =2 61 0.63293Axr = − = ÷ ÷ Ignoring the atom sitting at B and assuming the interstitial atom touches the atom at A0.796AAOX x= =0.195ABOY x= = A364.0).( =tetdxFeBCC
  25. 25.  This implies for x/r ratios between 0.15 and 0.63 the interstitial atom has to pushonly two atoms (xcarbon/rFe)BCC ~ 0.6 This explains why Carbon preferentially sits in the apparently smaller octahedralvoid in BCC
  26. 26. rvoid / ratomSC BCC FCC DCOctahedral(CN = 6)0.155(distorted)0.414 -Tetrahedral(CN = 4)0.29(distorted)0.2251(½,½,½) & (¼, ¼, ¼)Cubic(CN = 8)0.732Summary of void sizes

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