Crystal Structure. Dragica Vasileska and Gerhard Klimeck

Transcription

1 Crystl Structure Drgic Vsilesk nd Gerhrd Klimeck

2 Crystl Structure Issues tht re ddressed in this lecture include:. Periodic rry of toms. Fundmentl types of lttices 3. Index system for crystl plnes 4. Simple crystl structures 5. Imging of tomic structure 6. Non-idel structures

3 . Periodic Arry of Atoms Crystls re composed of periodic rry of toms: The structure of ll crystls cn be described in terms of lttice, with group of toms ttched to ech lttice point clled bsis: bsis lttice crystl structure

4 The lttice nd the lttice trnsltion vectors,, nd 3 re primitive if ny two points stisfy: r' r u u u33 where u, u nd u 3 re integers. The primitive lttice trnsltion vectors specify unit cell of smllest volume. A lttice trnsltion opertor is defined s displcement of crystl with crystl trnsltion opertor. T u u u33 To describe crystl, it is necessry to specify three things:. Wht is the lttice. Wht re the lttice trnsltion vectors 3. Wht is the bsis

5 A bsis of toms is ttched to lttice point nd ech tom in the bsis is specified by: r x y z where 0 x j, y j, z j. j j j3 The bsis consists of one or severl toms. The primitive cell is prllelepiped specified by the primitive trnsltion vectors. It is minimum volume cell nd there is one lttice point per primitive cell. The volume of the primitive cell is: j ( ) V c 3 Bsis ssocited with primitive cell is clled primitive bsis nd contins the lest # of toms.

6 . Fundmentl types of lttices To understnd the vrious types of lttices, one hs to lern elements of group theory: Point group consists of symmetry opertions in which t lest one point remins fixed nd unchnged in spce. There re two types of symmetry opertions: proper nd improper. Spce group consists of both trnsltionl nd rottionl symmetry opertions of crystl. In here: T group of ll trnsltionl symmetry opertions R group of ll symmetry opertions tht involve rottions.

7 The most common symmetry opertions re listed below: C two-fold rottion or rottion by 80 C 3 three-fold rottion or rottion by 0 C 4 four-fold rottion or rottion by 90 C 6 six-fold rottion or rottion by 80 σ reflection bout plne through lttice point i inversion, I.e. rottion by 80 tht is followed by reflection in plne norml to rottion xis. Two-dimensionl lttices, invrint under C 3, C 4 or C 6 rottions re clssified into five ctegories: Oblique lttice Specil lttice types (squre, hexgonl, rectngulr nd centered rectngulr)

8 Three-dimensionl lttices the point symmetry opertions in 3D led to 4 distinct types of lttices: The generl lttice type is triclinic. The rest of the lttices re grouped bsed on the type of cells they form. One of the lttices is cubic lttice, which is further seprted into: - simple cubic (SC) - Fce-centered cubic (FCC) - Body-centered cubic (BCC) Note tht primitive cells by definition contin one lttice point, but the primitive cells of FCC lttice contins 4 toms nd the primitive cell of BCC lttice contins toms.

9 The primitive trnsltion vectors for cubic lttices re: simple cubic fce-centered cubic body-centered cubic z y x 3 ( ) ( ) ( ) x z z y y x 3 ( ) ( ) ( ) z y x z y x z y x 3

10 3. Index System for Crystl Plnes Miler Indices The orienttion of crystl plne is determined by three points in the plne tht re not colliner to ech other. It is more useful to specify the orienttion of plne by the following rules: Find the intercepts of the xes in terms of lttice constnts, nd 3. Tke reciprocl of these numbers nd then reduce to three integers hving the sme rtio. The result (hkl) is clled the index of plne. Plnes equivlent by summetry re denoted in curly brckets round the indices {hkl}.

11 Miller indices for direction re specified in the following mnner: Set up vector of rbitrry length in the direction of interest. Decompose the vector into its components long the principl xes. Using n pproprite multiplier, convert the component vlues into the smllest possible whole number set. [hkl] squre brckets re used to designte specific direction within the crystl. <hkl> - tringulr brckets designte n equivlent set of directions.

12 The clcultion of the miller indices using vectors proceeds in the following mnner: We re given three points in plne for which we wnt to clculte the Miller indices: P (0), P (0) nd P 3 (0) We now define the following vectors: r 0ijk, r i0jk, r 3 ij0k nd clculte the following differences: r - r xi (-y)j (-z)k r - r i - j 0k r 3 -r i j - k We then use the fct tht: (r-r ). [(r -r ) (r 3 -r )] A. (B C) 0

13 We now use the following mtrix representtion, tht gives The end result of this mnipultion is n eqution of the form: 4x4yz The intercepts re locted t: x3, y3, z6 The Miller indices of this plne re then: () 0 0 ) ( z y x C C C B B B A A A C B A

14 The seprtion between djcent plnes in cubic crystl is given by: The ngle between plnes is given by: l k h d ( )( ) cos l k h l k h l l k k h h θ

15 4. Simple Crystl Structures There re severl types of simple crystl structures: Sodium Chloride (NCl) -> FCC lttice, one N nd one Cl tom seprted by one hlf the body digonl of unit cube. Cesium Chloride -> BCC lttice with one tom of opposite type t the body center Hexgonl Closed pcked structure (hcp) Dimond structure -> Fcc lttice with primitive bsis tht hs two identicl toms ZnS -> FCC in which the two toms in the bsis re different.

16 5. Imging of Atomic Structure The direct imging of lttices is ccomplished with TEM. One cn see, for exmple, the density of toms long different crystlogrphic directions.

17 6. Non-idel Crystl Structures There re two different types of non-idelities in the crystlline structure: Rndom stcking The structure differs in stcking sequence of the plnes. For exmple FCC hs the sequence ABCABC, nd the HCP structure hs the sequence ABABAB. Polytypism The stcking sequence hs long repet unit long the stcking xis. Exmples include ZnS nd SiC with more thn 45 stcking sequences. The mechnisms tht induce such long rnge order re ssocited with the presence of spirl steps due to disloctions in the growth nucleus.