Section 1: Crystal Structure


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1 Phsics 927 Section 1: Crstl Structure A solid is sid to be crstl if toms re rrnged in such w tht their positions re ectl periodic. This concept is illustrted in Fig.1 using twodimensionl (2D) structure. T C Fig.1 A B A perfect crstl mintins this periodicit in both the nd directions from  to +. As follows from this periodicit, the toms A, B, C, etc. re equivlent. In other words, for n observer locted t n of these tomic sites, the crstl ppers ectl the sme. The sme ide cn be epressed b sing tht crstl possesses trnsltionl smmetr. The trnsltionl smmetr mens tht if the crstl is trnslted b n vector joining two toms, s T in Fig.1, the crstl ppers ectl the sme s it did before the trnsltion. In other words the crstl remins invrint under n such trnsltion. The structure of ll crstls cn be described in terms of lttice, with group of toms ttched to ever lttice point. For emple, in the cse of structure shown in Fig.1, if we replce ech tom b geometricl point locted t the equilibrium position of tht tom, we obtin crstl lttice. The crstl lttice hs the sme geometricl properties s the crstl, but it is devoid of n phsicl contents. There re two clsses of lttices: the Brvis nd the nonbrvis. In Brvis lttice ll lttice points re equivlent nd hence b necessit ll toms in the crstl re of the sme kind. On the other hnd, in nonbrvis lttice, some of the lttice points re nonequivlent. C C 1 A A 1 B 1 B Fig.2 In Fig.2 the lttice sites A, B, C re equivlent to ech other. Also the sites A 1, B 1, C 1, re equivlent mong themselves. However, sites A nd A 1 re not equivlent: the lttice is not invrint under trnsltion AA 1. 1
2 Phsics 927 NonBrvis lttices re often referred to s lttice with bsis. The bsis is set of toms which is locted ner ech site of Brvis lttice. Thus, in Fig.2 the bsis is represented b the two toms A nd A 1. In generl cse crstl structure cn be considered s crstl structure = lttice + bsis. The lttice is defined b fundmentl trnsltion vectors. For emple, the position vector of n lttice site of the two dimensionl lttice in Fig.3 cn be written s T=n 1 +n 2, (1.1) where nd re the two vectors shown in Fig.3, nd n 1,n 2 is pir of integers whose vlues depend on the lttice site. T Fig.3 So, the two noncolliner vectors nd cn be used to obtin the positions of ll lttice points which re epressed b Eq.(1). The set of ll vectors T epressed b this eqution is clled the lttice vectors. Therefore, the lttice hs trnsltionl smmetr under displcements specified b the lttice vectors T. In this sense the vectors nd cn be clled the primitive trnsltion vectors. The choice of the primitive trnsltions vectors is not unique. One could equll well tke the vectors nd = + s primitive trnsltion vectors (see Fig.3). This choice is usull dictted b convenience. Unit cell. In the cse of rectngulr two dimensionl lttice the unit cell is the rectngle, whose sides re the vectors nd. If the unit cell is trnslted b ll the lttice vectors epressed b Eq.(1), the re of the whole lttice is covered once nd onl once. A primitive unit cell is the unit cell with the smllest re which produces this coverge. In the two dimensionl cse the re of the unit cell is given b S=. The choice of the unit cell is not unique. For emple, the prllelogrm formed b the vectors nd in Fig.3 is lso n cceptble unit cell. The choice is gin dictted b convenience. The re of the unit cell bsed on vectors nd is the sme s tht bsed on vectors nd. WignerSeits unit cell. The primitive cell m be chosen s shown in Fig.4. (i) Drw lines to connect given lttice point to ll nerb lttice points. (ii) At the midpoint nd norml to these lines, drw new lines (plnes in 3D). The smllest volume enclosed is the WignerSeit primitive cell. All the spce of the crstl m be filled b these primitive cells, b trnslting the unit cell b the lttice vectors. Fig.4 2
3 Phsics 927 The unit cell cn be primitive nd nonprimitive (or conventionl). The unit cell discussed bove is primitive. However, in some cses it is more convenient to del with unit cell which is lrger, however, it ehibits the smmetr of the lttice more clerl. c 2 c 1 Fig.5 Vectors nd cn be chosen s primitive trnsltion vectors for the lttice shown in Fig.5. In this cse the unit sell is prllelogrm. However, the lttice cn lso be regrded s djcent rectngles, where the vectors c 1 nd c 2 cn be considered s primitive trnsltion vectors. The unit cell in this cse is lrger, however it ehibits the rectngulr smmetr more clerl. In the first cse we hve just one tom in unit cell, wheres in the second cse we hve lttice with bsis. The bsis consists of the two toms: one tom is locted in the corner of the unit cell nd nother tom in the center of the unit cell. The re of the conventionl unit cell is lrger b fctor of two thn the re of the primitive unit cell. Crstl lttices re clssified ccording to their smmetr properties, such s inversion, reflection nd rottion. Inversion center. A cell hs n inversion center if there is point t which the cell remins invrint under trnsformtion r > r. All the Brvis lttices re inversion smmetric. NonBrvis lttices m or m not hve n inversion center depending on the smmetr of the bsis. Reflection plne. A cell hs reflection plne if it remins invrint when mirror reflection in this plne is performed. Rottion is. This is n is such tht, if the cell rotted round the is trough some ngle, the cell remins invrint. The is is clled nfold if the ngle of rottion is 2 /n. Onl 2, 3, 4, nd 6fold es re possible. There re five Brvis lttice tpes in two dimensions shown in Fig.6. For ech of them the rottion es nd/or mirror plnes occur t the lttice points. However, there re other loctions in the unit cell with comprble or lower degrees of smmetr with respect to rottion nd reflection. For nonbrvis lttices we hve to tke into ccount the smmetr of the bsis which is referred s pointgroup smmetr. The point group smmetr includes ll possible rottions, reflections nd inversion, which leve the bsis invrint. Point groups re denoted b numericl nd m. The numericl indictes how mn positions within the bsis re equivlent b rottion smmetr. A single m shows tht the bsis hs mirror plne smmetr. (In two dimensions it is mirror is). E.g., 3m mens tht there re 3 equivlent sites within the unit cell nd there is one mirror plne. In two dimensions there re 10 point groups. When we combine the rottion smmetr of the point group with the trnsntionl smmetries, we obtin spcegroup smmetr. 3
4 Phsics 927 Fig.6 4
5 Phsics 927 All the lttice properties we discussed for two dimensions cn be etended to three dimensions. The lttice vectors re in this cse T=n 1 +n 2 +n 3 (1.2) where, nd re the primitive trnsltion vectors, nd (n 1,n 2,n 3 ) re triplet of integers whose vlues depend on prticulr lttice site. The unit cell in three dimensions is prllelepiped, whose sides re the primitive trnsltion vectors (see Fig.7). Here gin the choice of the unit cell is not unique, lthough ll primitive unit cells hve equl volumes. The unit cell fills ll spce b the repetition of crstl trnsltion opertions. The volume of the unit cell represented b prllelepiped with sides, nd is given b V=. (1.3) Also, it is sometimes more convenient to del with nonprimitive or conventionl cells, which hve dditionl lttice sites either inside the cell or on its surfce. β α γ Fig.7 In three dimensions there re 14 different Brvis crstl lttices which belong to 7 crstl sstems. These sstems re triclinic, monoclinic, orthorhombic, tetrgonl, cubic, hegonl nd trigonl. The crstl lttices re shown in Fig.8. In ll the cses the unit cell represents prllelepiped whose sides re, nd. The opposite ngles re clled α, β nd γ. The reltionship between the sides nd the ngles determines the crstl sstem. A simple lttice hs sites onl t the corners, bodcentered lttice hs one dditionl point t the center of the cell, nd fcecentered lttice hs si dditionl points, one on ech side. Note tht in ll the nonsimple lttices the unit cells re nonprimitive. The volume of the primitive unit cell is equl to the volume of the conventionl unit cell divided b the number of sites. Ech of the 14 lttices hs one or more tpes of smmetr properties with respect to reflection nd rottion. Reflection: The triclinic structure hs no reflection plne, the monoclinic hs one plne midw between nd prllel to the bsis plne, nd so forth. The cubic cell hs nine reflection plnes: three prllel to the fces, nd si other, ech of which psses through two opposite edges. Rottion: The triclinic structure hs no is of rottion (do not tke into ccount 1fold is), the monoclinic hs fold is norml to the bse. The cubic cell hs three 4fold is norml to the fces nd four 3fold is, ech pssing through two opposite corners. 5
6 Phsics 927 Triclinic ( α β γ) Monoclinic ( α=β=90 o γ) c α β b γ simple bsecentered Orthorombic ( α=β=γ=90 o ) simple bsecentered bodcentered fcecentered Tetrgonl ( = α=β=γ=90 o ) simple bodcentered Cubic ( = = α=β=γ=90 o ) simple bodcentered fcecentered Trigonl ( = = α=β=γ 90 o ) Hegonl ( = α=β=90 o γ=120 o ) Fig.8 6
7 Phsics 927 Most common crstl structures Bodcentered cubic (bcc) lttice: r 2 Primitive trnsltion vectors of the bcc lttice (in units of lttice prmeter ) re = ½½½; =  ½½½; = ½½½. The primitive cell is the rhombohedron. The pcking rtio is 0.68, defined s the mimum volume which cn be filled b touching hrd spheres in tomic positions. Ech tom hs 8 nerest neighbors. The conventionl unit cell is cube bsed on vectors = 001; = 010; = 001. It is twice big compred to the primitive unit cell nd hs two toms in it with coordintes r 1 = 000 nd r 2 = ½½½. The bcc lttice hve lkli metls such s N, Li, K, Rb, Cs, mgnetic metls such s Cr nd Fe, nd nd refrctor metls such s Nb, W, Mo,T. Fcecentered cubic (fcc) lttice: r 3 r 4 1 r 2 Primitive trnsltion vectors of the bcc lttice (in units of lttice prmeter ) re = ½½0; = 0½½; = ½ 0½. The primitive cell is the rhombohedron. The pcking rtio is Ech tom hs 12 nerest neighbors. The conventionl unit cell is cube bsed on vectors = 001; = 010; = 001. It is 4 times bigger thn the primitive unit cell nd hs 4 toms in it with coordintes r 1 = 000; r 2 = ½½0; r 3 = 0½½; r 4 = ½0½. The fcc lttice hve noble metls such s Cu, Ag, Au, common metls such s Al, Pb, Ni nd inert gs solids such s Ne, Ar, Kr, Xe. 7
8 A B C Phsics 927 Hegonl closedpcked (hcp) lttice: The hcp structure hs =, α=β=90 o nd γ=120 o with bsis of two toms, one t 000 nd the other t Along with the fcc structure, the hcp structure mimies the pcking rtio, mking it closepcked structure (fcc) hcc structure fcc structure A closedpcked structure is creted b plcing ler of spheres B on top of identicl closepcked ler of spheres A. There re two choices for third ler. It cn go in over A or over C. If it goes in over A the sequence is ABABAB... nd the structure is hcp. If the third ler goes in over C the sequence is ABCABCABC... nd the structure is fcc. In perfect hcp structure the rtio of the height of the cell to the nerest neighbor spcing is (8/3) ½. In prctice the ( / ) rtio is lrger thn for most hegonl crstls. Emples of nominll hcp crstls include the elements from Column II of the Periodic Tble: Be, Mg, Zn, nd Cd. Hcp is lso the stble structure for severl trnsition elements, such s Ti nd Co. Dimond structure is dopted b solids with four smmetricll plced covlent bonds. This is the sitution in silicon, germnium, nd gre tin, s well s in dimond. Dimond hs the trnsltionl smmetr of fcc lttice with bsis of two toms, one t 000 nd the other t ¼¼¼. Dimond structure represents two interpenetrting fcc sublttices displced from ech other b one qurter of the cube digonl distnce. 8
9 Phsics 927 Inde sstem for crstl directions nd plnes Crstl directions. An lttice vector cn be written s tht given b Eq.(1.2). The direction is then specified b the three integers [n 1 n 2 n 3 ]. If the numbers n 1 n 2 n 3 hve common fctor, this fctor is removed. For emple, [111] is used rther thn [222], or [100], rther thn [400]. When we spek bout directions, we men hole set of prllel lines, which re equivlent due to trnsntionl smmetr. Opposite orienttion is denoted b the negtive sign over number. For emple: [111] [011] [011] Crstl plnes. The orienttion of plne in lttice is specified b Miller indices. The re defined s follows. We find intercept of the plne with the es long the primitive trnsltion vectors, nd. Let s these intercepts be,, nd, so tht is frctionl multiple of, is frctionl multiple of nd is frctionl multiple of. Therefore we cn mesure,, nd in units, nd respectivel. We hve then triplet of integers ( ). Then we invert it (1/ 1/ 1/) nd reduce this set to similr one hving the smllest integers b multipling b common fctor. This set is clled Miller indices of the plne (hkl). For emple, if the plne intercepts,, nd in points 1, 3, nd 1, the inde of this plne will be (313). (313) 1 The Miller indices specif not just one plne but n infinite set of equivlent plnes. Note tht for cubic crstls the direction [hkl] is perpendiculr to plne (hkl) hving the sme indices, but this is not generll true for other crstl sstems. Emples of the plnes in cubic sstem: (100) (110) (111) 9
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