Face-centered cubic crystal -p227, 7.23Special case: NaClUnit cell contains 4 lattice points, or 4NaClEach lattice point (LP) corresponds to a NaClPut Cl at (0,0,0), then a neighboring Na at (1/2,0,0)Onlywhenh.k.l are all evenorall odd candiffractions be observed!2i(hx,+hy,+t,) = 4[ fe, + fnehi ]Fhkl = 4j-lNowsumupoverallatomswithinaLP!Case 1: if h =2n (note we also have l-2n and k-2n)Strong diffraction!Fhk1 =4[ fc, + fNa ]Case2:ifh=2n+1(notewealsohave[=2n+1,k=2n+1)Weakdiffraction!Fhki =4[ fc, - fna]
Face-centered cubic crystal – Special case: NaCl p227, 7.23 • Unit cell contains 4 lattice points, or 4NaCl • Each lattice point (LP) corresponds to a NaCl. • Put Cl at (0,0,0), then a neighboring Na at (1/2,0,0). • Only when h,k,l are all even or all odd can diffractions be observed! Case 1: if h = 2n (note we also have l=2n and k=2n) Strong diffraction! Case 2: if h = 2n+1 (note we also have l=2n+1, k=2n+1) Weak diffraction! f e [ f f e ] h i C l N a i( h x ky lz ) j j π j j j π F 4 4 2 2 1 hkl [ f f ] 4 Cl Na F hkl [ f f ] 4 Cl Na F hkl Now sum up over all atoms within a LP!
Face-centered cubic crystal -Special case: ZnS (sphalerite)Unit cell contains 4 lattice points, or 4ZnSEach lattice point corresponds to a ZnSS(0,0,0), Zn(1/4,1/4,1/4)(different elements!)When h,k.l are all even or all odd, diffractions observable2e2mi(h+)+,)=4[s+fe(h++)/2 ]Fhkl=4.ej-1> (111),(200),(220),(311),(222),(400),(331),(420),(422),Case 1: if h+k+l = 4n, e.g., (220),(400),(440).Strongestdiffraction!Fhk1 = 4[ fs + fzmn ]Case 2: if h+k+l = 4n+2, e.g., (200),(222),(420),(442)FWeakestdiffraction!=4[fs- fzn]hkl
Face-centered cubic crystal – Special case: ZnS (sphalerite) • Unit cell contains 4 lattice points, or 4ZnS • Each lattice point corresponds to a ZnS. S(0,0,0), Zn(1/4,1/4,1/4) (different elements!) • When h,k,l are all even or all odd, diffractions observable, (111),(200),(220),(311),(222),(400),(331),(420),(422),. Case 1: if h+k+l = 4n, e.g., (220),(400),(440). Strongest diffraction! Case 2: if h+k+l = 4n+2, e.g., (200),(222),(420),(442). Weakest diffraction! f e [ f f e ] i( h k l )/ S Zn i( h x ky lz ) j j 2 j j j 2 2 1 Fhkl 4 4 π π [ f f ] 4 S Zn F hkl [ f f ] 4 S Zn F hkl
Face-centred cubiccrystal--Derivethesystemabsenceofdiamond!Specialcase:DiamondO,7-Fd3mLattice points: (0,0,0)+, (1/2,1/2,0)+, (0,1/2,1/2)+, (1/2,0,1/2)+Each LP contains two C atoms (i.e., structure motif=2C)C1-- (0,0,0), C2-- (1/4,1/4,1/4) (the same element)The other six C atoms within a unit cell can be derived as(1/2,1/2,0),(3/4,3/4,1/4) ; (0,1/2,1/2),(1/4,3/4,3/4);(1/2,0,1/2),(3/4,1/4,3/4)Such an arrangement of C atoms produces new translationsymmetry elements, i.e., screw axes and d glide planes, which inturn introduce special system absence of diffractions (in additionto the system absence from normal FCC lattice!!!!
• Lattice points: (0,0,0)+, (1/2,1/2,0)+, (0,1/2,1/2)+, (1/2,0,1/2)+ • Each LP contains two C atoms (i.e., structure motif =2C) C1- (0,0,0), C2- (1/4,1/4,1/4) (the same element) • The other six C atoms within a unit cell can be derived as (1/2,1/2,0), (3/4,3/4,1/4) ; (0,1/2,1/2), (1/4,3/4,3/4) ; (1/2,0,1/2), (3/4,1/4,3/4) • Such an arrangement of C atoms produces new translation symmetry elements, i.e., screw axes and d glide planes, which in turn introduce special system absence of diffractions (in addition to the system absence from normal FCC lattice !!!!!!! Special case: Diamond Oh 7 - Fd3m Face-centred cubic crystal - Derive the system absence of diamond!
Face-centred cubiccrystal--Derivethesystemabsence ofdiamondSpecialcase:DiamondO,7-Fd3mSuch an arrangement of C atoms produces new translationsymmetry elements, i.e., screw axes and d glide planes, which inturn introduce special system absence of diffractions (in additiontothe system absencefrom normalFCC lattice!!!d glideplane1/8,3/85/8,7/8Let's derivethe structural factor of diamond tounravelitssystemabsence
• Such an arrangement of C atoms produces new translation symmetry elements, i.e., screw axes and d glide planes, which in turn introduce special system absence of diffractions (in addition to the system absence from normal FCC lattice !!!!!!! Special case: Diamond Oh 7 - Fd3m Sideview topview 41 d glide plane 1/8,3/8, 5/8,7/8 Face-centred cubic crystal - Derive the system absence of diamond! • Let’s derive the structural factor of diamond to unravel its system absence
Face-centered cubic crystal -Special case: DiamondNow sum up over all atoms within a unit cell!82mi(hx+ky,+lz)Fnk!fceNow sum up over all atoms within a LP!i=l1125I七2元2元i2元i)2mi（hx,+hy,+z）1222"x2= fc[l+e+e+eej-1（h+)K-212元i02元i02元i022222J(1 + e(h+k+1)/2)=fc[l+e+e+e(Note: two carbon atoms within a lattice point: (0,0,0), (1/4,1/4,1/4)>Systemabsencehkh1k12元i（2元i2元222222a)[l+e=0or+e+eb) (1+emi(h+k+1)/2=0i.e., a) h,k,l are neither all even nor all odd! & b) h+k+l= 4n+2> Observable diffractions: (111), (220),(311),(400),(331),(422) &If h+k+l=4n, Fhki = 8fc, (220),(400)... strongest diffraction!
Face-centered cubic crystal – Special case: Diamond (Note: two carbon atoms within a lattice point: (0,0,0), (1/4,1/4,1/4)) System absence: i.e., a) h,k,l are neither all even nor all odd! & b) h+k+l = 4n+2 Observable diffractions: (111), (220),(311),(400),(331),(422) & If h+k+l=4n, Fhkl = 8fC , (220),(400). strongest diffraction! [1 ](1 ) [1 ] F ( )/ 2 ) 2 2 ) 2 ( 2 2 ) 2 ( 2 2 2 ( 2 ( ) 2 1 ) 2 2 ) 2 ( 2 2 ) 2 ( 2 2 2 ( 2 ( ) 8 1 hkl i h k l k l i h l i h k i C i h x ky lz j k l i h l i h k i C i h x ky lz i C f e e e e f e e e e f e j j j i i i Now sum up over all atoms within a unit cell! Now sum up over all atoms within a LP! b) (1 ) 0 ) [1 ] 0 or ( )/ 2 ) 2 2 ) 2 ( 2 2 ) 2 ( 2 2 2 ( i h k l k l i h l i h k i e a e e e