LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

Transcription

1 LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges Wh re Crstl Plnes & Directions Importnt? Mterils structures nd properties re relted to them! Crstls deform on specific plnes nd directions the don t just brek rndoml 2 VIDEO 1

2 Wh re Crstl Plnes & Directions Importnt? Mterils structures nd properties re relted to them! direction stress plne After Crstls deform on specific plnes nd directions The don t just brek rndoml 3 How do we epress Plnes nd Directions in Crstls? We use Miller indices: hkl or uvw re the generic letters we use. hkl nd uvw re clled indices. The will be numbers tht re relted to coordinte sstems. No comms between the numbers. h represents the plne perpendiculr to the -is; k represents the plne perpendiculr to the -is; l represents the plne perpendiculr to the -is. u represents the vector prllel to the -is; v represents the vector prllel to the -is; w represents the vector prllel to the -is. Negtive vlues re epressed with br over the number E.: -3 is epressed s 3 ( br 3 ) Crstllogrphic directions: [1 11] ; one step in + dir.; one step in dir.; one step in + dir. 4 [010] ; ero step in dir.; one step in + dir.; ero step in dir. 2

3 Point Coordintes c 000 1,1,1 b Epress s frctions of unit vectors. Point coordintes for unit cell corner re 1,1,1 (or, b, c) Wht bout this position? 1, 1, 0 (or, b, 0) 2 2 For the bove 1,1,1 coordinte wht is its direction? 5 Let s Relte Miller Indices to Vectors M crimes hve both directions nd mgnitude. Vector from Despicble Me Representtion of lttice nd unit cell Where is the origin position in crstl? Wht do ou look for? Vector points in specific direction (hence ou need n origin) Vector hs unit of length or mgnitude 6 We will use vectors to define directions nd lengths in crstl sstems 3

4 Directions in Crstls Directions nd their multiples re identicl [010] [020] [030] [110] [220] [330] E.: [220] 2 [110] ( Trnsltionl Smmetr ) Trnsltion: integer multiple of lttice constnts identicl position in nother unit cell: (111), (222), (333), etc. 7 How to ppl Miller Indices for Directions Drw vector nd define the til s the origin. Dt Determine the length of the vector projection in unit cell dimensions, b, nd c. [111] [ 2 0 1] P Emple in clss [021] Remove frctions b multipling b the smllest possible fctor. 8 Enclose in squre brckets In cubic crstls, directions nd their negtives re equivlent but NOT the sme. 2 [110] b c [0 2 1] Point P (hed) Origin (til) 4

5 In clss emple #1: Wht re the indices of the line/vector connecting points O nd P? Wht re the indices of the line connecting points Q nd R? Q P O R 9 In clss emple #1: SOLUTION Wht re the indices of the line/vector connecting points O nd P? Wht re the indices of the line connecting points Q nd R? Q P 2 b c Point P Origin O [1 2 1] O R 4 b c Point R Origin Q [2 4 3] 10 5

6 In clss emple #2: In cubic unit cell, drw correctl vector with indices [146]. O 11 In clss emple #2: SOLUTION In cubic unit cell, drw correctl vector with indices [146]. This step is the opposite of clering frctions! Select our origin. Put it wherever ou wnt to. indices [1 4 6] Div. b O These frctions denote how fr to step in the,, or directions (w from the origin). 12 NOTE: It would be wise to select the origin so tht ou cn complete the desired steps within the cell tht ou re using! 6

7 Fmilies of Directions In cubic sstems, directions tht hve the sme indices re equivlent regrdless of their order or sign. [010] [001] [100] [010] We enclose indices in crts rther thn brckets to indicte fmil of directions 13 [100] [001] The fmil of < 100 > [100], [100] [010], [010] [001], [00 1] directions is: Fmilies of Directions In non cubic sstems, directions tht hve the sme indices re not necessril equivlent. [010] CUBIC = b = c [010] [001] b ORTHORHOMBIC c b c [010] [001] [010] c TETRAGONAL b c [010] [001] [010] 14 7

8 Crstllogrphic Plnes A specific direction is norml (90 o ) to its specific, equivlent plne. For emple [100] is norml to (100) but [100] is not norml to (010) 15 Adpted from Fig. 3.9, Cllister 7e. Miller Indices for Plnes Specific crstllogrphic plne: (hkl) Fmil of crstllogrphic plnes: {hkl} (hkl), (lkh), (hlk) etc. In cubic sstems, plnes hving the sme indices re equivlent regrdless of order or sign. AND directions re norml to the plnes 16 8

9 PROCEDURES FOR INDICES OF PLANES (Miller indices) 1. Identif the coordinte intercepts of the plne (i.e., the coordintes t which the plne intersects the,, nd es). If plne is prllel to n is (DOES NOT INTERSECT IT), the intercept is tken s infinit (). If the plne psses through the origin, consider n equivlent plne in n djcent unit cell or select different origin for the sme plne. 2. Tke reciprocls of the intercepts. 3. Cler frctions to the lowest integers. 4. Cite specific plnes in prentheses, (h k l), plcing brs over negtive indices. 17 MILLER INDICES FOR A SINGLE PLANE Intercept 1 Reciprocl 1/ 1/1 1/ Cler INDICES (010) 18 The cube fces re from the {100} fmil of plnes (100), (010), (001), (100), (010), (001), 9

10 MILLER INDICES FOR A SINGLE PLANE cont d Intercept 1 1 Reciprocl 1/1 1/1 1/ Cler INDICES (110) 19 The {110} fmil of plnes (110), (011), (101), (110), (011), (101) (110), (110), (101), (101), (011), (011) MILLER INDICES FOR A SINGLE PLANE cont d Intercept Reciprocl Cler INDICES 1/2 1/2 2/1 2/1 1/ (220) 20 10

11 Crstllogrphic Plnes emple b c 1. Intercepts 1/2 1 3/4 2. Reciprocls 1/(½) 1/1 1/(¾) 2 1 4/3 3. Reduction Miller Indices (634) c b 21 Generl Rules for Crstl Directions, Plnes, nd Miller Indices,, nd re the es (on n rbitrr origin). In some crstl sstems the es re not mutull perpendiculr. Unit cell, b, c nd α, β, γ re lttice prmeters. length of unit cell long side of unit cell. c h, k, l re the Miller indices for plnes nd directions. E., (hkl) nd [hkl] b Geometr of generl unit cell 22 11

12 Comment: HCP Crstllogrphic Directions In generl we cn define Miller indices just like we do for the other crstls. However, sometimes in engineering prctice, 4-indice sstem is used. It is clled Miller-Brvis indices. There re equtions to convert. ( 1 ) c 120 = b c α = β = 90 ; γ = 120 ( 2 ) 23 Comment: HCP Crstllogrphic Directions Miller Brvis indices (i.e., uvtw) re relted to the direction indices (i.e., UVW) s follows. [ UVW ] [ uvtw] Fig. 3.8(), Cllister 7e. 1 u 2 U V 3 1 v 2 V U 3 t U V w W 24 I WILL NOT TEST YOU ON THIS!!! I onl show it becuse some of ou will end up working with hegonl metls like Ti or Mg fter ou grdute. 12

13 Comment: HCP Crstllogrphic Directions DIRECTIONS (UVW) (uvtw) 3 3 [120] [110] [0110] [1100] 2 2 [110] [100] [1120] [2110] 1 [210] 1 [1010] I WILL NOT TEST YOU ON THIS!!! I onl show it becuse some of ou will end up working with hegonl metls once ou grdute. 25 Summr Miller Directions Indices for direction re enclosed in squre brckets. Negtive vlues re epressed with br over the number. An emple of Miller Direction: [1 11] one step in + dir.; one step in dir.; one step in + dir. Miller Plnes Intercepts for specific crstllogrphic plne re enclosed in prenthesis. When identifing the coordinte intercepts of the plne (i.e., the coordintes t which the plne intersects the,, nd es): If plne is prllel to n is (DOES NOT INTERSECT IT), the intercept is tken s infinit (). If the plne psses through the origin, consider n equivlent plne in n djcent unit cell or select different origin for the sme plne