3. Other types of close-packed structures:ABAC......ABABCBCAC,etc(Eachlayerbelongs to ahexagonal2Dlattice!(bcporbce)A24.Body-centred cubicpackingEach unit cell has twospheres(atoms)(0,0,0),(0.5,0.5,0.5)
3. Other types of close-packed structures: ABAC., ABABCBCAC., etc. (Each layer belongs to a hexagonal 2D lattice!) Each unit cell has two spheres(atoms). 4. Body-centred cubic packing (bcp or bcc) A2 (0,0,0), (0.5,0.5,0.5)
8.2.2 Packing density1)ccp---fcc4R= √2aaRa=4R/V2cell =α3 =(4R/ /2)3 =16/2RThe volume of the unit cellThe total volume of the four spheres in the unit cell= 4 ×(4元R /3) = 16元R3 /3spheresPacking coefficient:/ Vcell = 元 /(3/2) = 74.05%spheresNote: The hexagonal close packing (hcp) of identical spheresgives the same packing density. (74.05%)
Note: The hexagonal close packing (hcp) of identical spheres gives the same packing density. (74.05%) a 4R/ 2 R a 3 3 3 Vcell a (4R/ 2) 16 2R 4 (4 /3) 16 /3 3 3 Vspheres R R Vspheres /Vcell /(3 2) 74.05% 8.2.2 Packing density 1) ccp -fcc The volume of the unit cell : The total volume of the four spheres in the unit cell: Packing coefficient: 4R 2a
hcp structureα=b=2R, =232R2RC= 4V2R/ /3 =cacoSy=(4R//2)3=8/2Rcell2Rsphere = 2(4元R3 /3)= 8元R /3/Vcel = 元 /(3/2) = 74.05%spher
hcp structure 2R 2R c b 2R a a b 2R， 2/3 2 3 3 Vcell ca cos ( 4R / 2 ) 8 2R 2 4 3 8 3 3 3 V R / R / sphere （ ） Vsphere/Vcell /( 3 2 ) 74.05% c 4 2R / 3
2) Body-centred cubic packing (bcp or bcc)Two spheres in a unit cell14R=/3a=a=R64R0元RCelspheres9R64pheresa/3元 =68.02%8 Thus bcp has a lower density than ccpbcpisnot a close-packed structure!
2) Body-centred cubic packing (bcp or bcc) • Thus bcp has a lower density than ccp. • bcp is not a close-packed structure! a R 3 3 3 9 3 64 3 4 Vspheres 2( )R ; Vcell a R 68.02% 8 3 ) 9 3 64 ) /( 3 4 / 2 ( 3 3 Vspheres Vcell R R R a a R 3 4 4 3 Two spheres in a unit cell
8.2.3Intersticesa) octahedral holes in ccp:a=4R//2Holeradius:1a/2 -R = 0.414 RFor a close-packed structure formed from identical spheresof radius R, the octahedral hole size is 0.414R
8.2.3 Interstices For a close-packed structure formed from identical spheres of radius R, the octahedral hole size is 0.414R. a) octahedral holes in ccp a/2 R = 0.414 R Hole radius: a 4R / 2 a