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MMaatteerriiaallss OOff SSccii.. && 
TTeecchhnnoollooggyy 
 CCrryyssaattaallllooggrraapphhiicc PPllaanneess && 
DDiirreeccttiioonnss 
 By – Boricha Paras H (130280105004)
3.9 Crystallographic Directions iinn CCuubbiicc 
SSyysstteemm 
► DDeetteerrmmiinnaattiioonn ooff tthhee ddiirreeccttiioonnaall iinnddiicceess iinn ccuubbiicc 
ssyysstteemm:: 
FFoouurr SStteepp PPrroocceedduurree ((TTeexxtt BBooookk MMeetthhoodd)) 
11.. DDrraaww aa vveeccttoorr rreepprreesseennttiinngg tthhee ddiirreeccttiioonn wwiitthhiinn tthhee uunniitt 
cceellll ssuucchh tthhaatt iitt ppaasssseess tthhrroouugghh tthhee oorriiggiinn ooff tthhee xxyyzz 
ccoooorrddiinnaattee aaxxeess.. 
22.. DDeetteerrmmiinnee tthhee pprroojjeeccttiioonnss ooff tthhee vveeccttoorr oonn xxyyzz aaxxeess.. 
33.. MMuullttiippllyy oorr ddiivviiddee bbyy ccoommmmoonn ffaaccttoorr ttoo oobbttaaiinn tthhee tthhrreeee 
ssmmaalllleesstt iinntteeggeerr vvaalluueess.. 
44.. EEnncclloossee tthhee tthhrreeee iinntteeggeerrss iinn ssqquuaarree bbrraacckkeettss [[ ]].. 
ee..gg.. [[uuvvww]] 
uu,, vv,, aanndd ww aarree tthhee iinntteeggeerrss
CCrryyssttaallllooggrraapphhiicc DDiirreeccttiioonnss 
1. Vector repositioned (if necessary) to pass 
through origin. 
2. Read off projections in terms of 
unit cell dimensions a, b, and c 
3. Adjust to smallest integer values 
4. Enclose in square brackets, no commas 
[uvw] 
z 
y 
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] 
-1, 1, 1 
families of directions <uvw> 
x 
Algorithm 
where overbar represents a 
negative index 
=> [ 111 ]
Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm 
[120] 
[111] 
[110]
Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm
a Head anndd TTaaiill PPrroocceedduurree ffoorr ddeetteerrmmiinniinngg 
MMiilllleerr IInnddiicceess ffoorr CCrryyssttaallllooggrraapphhiicc DDiirreeccttiioonnss 
11.. FFiinndd tthhee ccoooorrddiinnaattee ppooiinnttss ooff hheeaadd aanndd ttaaiill 
ppooiinnttss.. 
22.. SSuubbttrraacctt tthhee ccoooorrddiinnaattee ppooiinnttss ooff tthhee ttaaiill 
ffrroomm tthhee ccoooorrddiinnaattee ppooiinnttss ooff tthhee hheeaadd.. 
33.. RReemmoovvee ffrraaccttiioonnss.. 
44.. EEnncclloossee iinn [[ ]]
Indecies of Crystallographic Directions iinn CCuubbiicc SSyysstteemm 
Direction A 
Head point – tail point 
(1, 1, 1/3) – (0,0,2/3) 
1, 1, -1/3 
Multiply by 3 to get smallest 
integers 
3, 3, -1 
A = [33Ī] 
_ _ 
B = 
[403] 
C = [???] D = [???] 
Direction B 
Head point – tail point 
(0, 1, 1/2) – (2/3,1,1) 
-2/3, 0, -1/2 
Multiply by 6 to get smallest 
integers
Indices of Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc 
SSyysstteemm 
A = [???] 
Direction C 
Head Point – Tail Point 
(1, 0, 0) – (1, ½, 1) 
0, -1/2, -1 
Multiply by 2 to get the smallest integers 
_ _ 
C = 
[0I2] 
Direction D 
Head Point – Tail Point 
(1, 0, 1/2) – (1/2, 1, 0) 
1/2, -1, 1/2 
Multiply by 2 to get the smallest 
integers 
_ B= [???] 
D = [I2I]
Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm 
[210]
Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm
THANKS…

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crystallographic planes and directions

  • 1. MMaatteerriiaallss OOff SSccii.. && TTeecchhnnoollooggyy  CCrryyssaattaallllooggrraapphhiicc PPllaanneess && DDiirreeccttiioonnss  By – Boricha Paras H (130280105004)
  • 2. 3.9 Crystallographic Directions iinn CCuubbiicc SSyysstteemm ► DDeetteerrmmiinnaattiioonn ooff tthhee ddiirreeccttiioonnaall iinnddiicceess iinn ccuubbiicc ssyysstteemm:: FFoouurr SStteepp PPrroocceedduurree ((TTeexxtt BBooookk MMeetthhoodd)) 11.. DDrraaww aa vveeccttoorr rreepprreesseennttiinngg tthhee ddiirreeccttiioonn wwiitthhiinn tthhee uunniitt cceellll ssuucchh tthhaatt iitt ppaasssseess tthhrroouugghh tthhee oorriiggiinn ooff tthhee xxyyzz ccoooorrddiinnaattee aaxxeess.. 22.. DDeetteerrmmiinnee tthhee pprroojjeeccttiioonnss ooff tthhee vveeccttoorr oonn xxyyzz aaxxeess.. 33.. MMuullttiippllyy oorr ddiivviiddee bbyy ccoommmmoonn ffaaccttoorr ttoo oobbttaaiinn tthhee tthhrreeee ssmmaalllleesstt iinntteeggeerr vvaalluueess.. 44.. EEnncclloossee tthhee tthhrreeee iinntteeggeerrss iinn ssqquuaarree bbrraacckkeettss [[ ]].. ee..gg.. [[uuvvww]] uu,, vv,, aanndd ww aarree tthhee iinntteeggeerrss
  • 3. CCrryyssttaallllooggrraapphhiicc DDiirreeccttiioonnss 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvw] z y ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 families of directions <uvw> x Algorithm where overbar represents a negative index => [ 111 ]
  • 4. Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm [120] [111] [110]
  • 6. a Head anndd TTaaiill PPrroocceedduurree ffoorr ddeetteerrmmiinniinngg MMiilllleerr IInnddiicceess ffoorr CCrryyssttaallllooggrraapphhiicc DDiirreeccttiioonnss 11.. FFiinndd tthhee ccoooorrddiinnaattee ppooiinnttss ooff hheeaadd aanndd ttaaiill ppooiinnttss.. 22.. SSuubbttrraacctt tthhee ccoooorrddiinnaattee ppooiinnttss ooff tthhee ttaaiill ffrroomm tthhee ccoooorrddiinnaattee ppooiinnttss ooff tthhee hheeaadd.. 33.. RReemmoovvee ffrraaccttiioonnss.. 44.. EEnncclloossee iinn [[ ]]
  • 7. Indecies of Crystallographic Directions iinn CCuubbiicc SSyysstteemm Direction A Head point – tail point (1, 1, 1/3) – (0,0,2/3) 1, 1, -1/3 Multiply by 3 to get smallest integers 3, 3, -1 A = [33Ī] _ _ B = [403] C = [???] D = [???] Direction B Head point – tail point (0, 1, 1/2) – (2/3,1,1) -2/3, 0, -1/2 Multiply by 6 to get smallest integers
  • 8. Indices of Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm A = [???] Direction C Head Point – Tail Point (1, 0, 0) – (1, ½, 1) 0, -1/2, -1 Multiply by 2 to get the smallest integers _ _ C = [0I2] Direction D Head Point – Tail Point (1, 0, 1/2) – (1/2, 1, 0) 1/2, -1, 1/2 Multiply by 2 to get the smallest integers _ B= [???] D = [I2I]
  • 9. Crystallographic DDiirreeccttiioonnss iinn CCuubbiicc SSyysstteemm [210]

Editor's Notes

  1. Lecture 2 ended here