3.091 Introduction to Solid State Chemistry. Lecture Notes No. 4 THE NATURE OF CRYSTALLINE SOLIDS
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1 3.091 Introduction to Solid Stte Chemistry Lecture Notes No. 4 THE NATURE OF CRYSTALLINE SOLIDS In n ssembly of toms or molecules solid phse is formed whenever the intertomic (intermoleculr) ttrctive forces significntly exceed the disruptive therml forces nd thus restrict the mobility of toms, forcing them into more-or-less fixed positions. From energy considertions it is evident (s discussed in LN-2) tht in such solids the toms or molecules will lwys ttempt to ssume highly ordered structures which re chrcterized by symmetry. Depending on the nture of the cting intertomic forces, ll solids my be subdivided into: () Ionic solids (NCl) (b) Covlent solids (Dimond) (c) Metllic solids (Fe, Ni, etc.) (d) Vn der Wls solids (ice, solid He) Solids s we encounter them in nture my or my not reflect the internlly ordered rrngement in their ppernce. We find, for exmple, well-formed qurtz crystls, grnets, dimonds nd snowflkes which re ll chrcterized by flt bounding plnes which intersect t chrcteristic ngles. On the other hnd, we lso observe rounded stones nd mn-mde cst solid objects with no externl evidence of internl order (fig. 1). snow flke wter ice single crystl q (het) melt crystls q (het) c c 2 1 c 4 c 3 c5 poly crystlline solid consisting of intergrown rndomly oriented single crystls Figure 1 Crystls, internl order nd externl ppernce 1
2 To understnd the externl ppernce of the solid stte it is necessry to consider the formtion of solids from different phses. Solids, for exmple, re formed upon cooling of liquids (melts) - by freezing or solidifiction; this solidifiction process normlly proceeds in totl confinement nd the resulting cst structure will hve n externl ppernce which reflects in detil the confining geometry (nd not the internl order). Moreover, depending on solidifiction conditions, the solid body my be either single crystl or polycrystlline. Polycrystlline solids (in excess of 95% of the solid stte encountered) my be thought of s n ssembly of microscopic single crystls with rndom orienttion held together like mze structure by the interwoven irregulr shpes of the individul crystls. A typicl exmple of n unconfined phse trnsformtion is the formtion of snowflkes where the externl boundries of the solid hve ssumed crystlline ppernce, reflecting in detil the internlly ordered moleculr (H 2 O) rrngement. Another unconfined formtion of solid is precipittion from solution (sugr crystls, CuSO 4 nd the like). Similrly, the formtion of crystls from the vpor phse leds to bodies which externlly reflect elements of internl order. 1. CRYSTAL STRUCTURE From the erlier discussion it should be pprent tht, when strong intertomic forces exist, toms tend to pck closely together - the closeness of pcking being prticulrly pronounced in the solid stte. In this cse toms cn be regrded s hrd spheres nd the problem of close pcking cn be treted s one in which the whole ssembly hs tendency towrd efficient pcking. A little thought or few simple experiments with ping-pong blls quickly convince us tht regulr rrngements of the spheres generlly led to more compct ssemblies thn irregulr rrngements (fig. 2). The sme principle pplies to rrngement of toms in the solid stte. Where strong ttrctive forces re exerted we find tht the toms or molecules concerned rrnge themselves 2
3 Figure 2 The pcking density of ordered systems. in regulr three-dimensionl pttern. It is this regulrity which is the bsis of crystllinity in mterils: i.e., crystl structure is nothing more thn n orderly rry of toms or molecules. This definition of crystl is distinct from the populr concept bsed on observtion of externl symmetry of crystls, often seen during the study of elementry chemistry, in which some crystls pper cubic, others needle-shped nd so on. The regulr externl shpe is obtined only when the conditions of crystlliztion re fvorble to development of flt, geometric fces. In most instnces, prticulrly with metls, these conditions re bsent, nd the crystls hve irregulr surfces even though the internl rrngement is perfectly geometric. Atomic rrys in crystls re conveniently described with respect to three-dimensionl net of stright lines. Consider lttice of lines, s in fig. 3, dividing spce into equl-sized prisms which stnd side-by-side with ll fces in contct, thereby filling ll spce nd leving no voids. The intersections of these lines re points of spce lttice, i.e., geometricl bstrction which is useful s reference in describing nd correlting symmetry of ctul crystls. These lttice points re of fundmentl importnce in describing crystls for they my be the positions occupied by individul toms in crystls or they my be points bout which severl toms re clustered. Since prisms of mny different shpes cn be drwn through the points of spce lttice to prtition it into cells, the mnner in which the network of reference lines is drwn is rbitrry. It is not necessry tht the lines be drwn so tht toms lie only t corners of unit prisms. In fct, it is more convenient to describe some crystls with respect to 3
4 Figure 3 The spce lttice prisms in which toms lie t prism centers or t the centers of prism fces s well s t prism corners. An importnt chrcteristic of spce lttice is tht every point hs identicl surroundings: the grouping of lttice points bout ny given point is identicl to the grouping bout ny other point in the lttice. In other words, if we could move bout in the lttice, we would not be ble to distinguish one point from nother becuse rows nd plnes ner ech point would be identicl. If we were to wnder mong the toms of solid metl or chemicl compound, we would find the view from ny lttice point exctly the sme s tht from ny other. There re fourteen spce lttices (fig. 4). Tht is, no more thn fourteen wys cn be found in which points cn be rrnged in spce so tht ech point hs identicl surroundings. Of course, there re mny more thn fourteen wys in which toms cn be rrnged in ctul crystls; thus there re gret number of crystl structures. Too often the term lttice is loosely used s synonym for structure, n incorrect prctice which is frequently confusing. The distinction cn be clerly seen if we 4
5 7 Crystl systems 14 Brvis Lttices cubic =b=c α=β=γ=90 o tetrgonl =b c α=β=γ=90o c orthorhombic b c α=β=γ=90 o c b rhombohedrl =b=c α=β=γ 90 ο α hexgonl =b c α=β=90 o γ=120o c monoclinic b c α=γ=90o β b c β triclinic b c α β γ 90o b c Figure 4 The 7 crystl systems nd the 14 Brvis lttices 5
6 remember tht spce lttice is n rry of points in spce. It is geometricl bstrction which is useful only s reference in describing nd correlting symmetry of ctul crystls. A crystl structure, however, is the rrngement of toms or molecules which ctully exists in crystl. It is dynmic, rther thn sttic, rrngement nd is subject to mny imperfections. Although ny crystl structure hs n inherent symmetry which corresponds to one of the fourteen spce lttices, one, two or severl toms or molecules in the crystl structure my be ssocited with ech point of the spce lttice. This symmetry cn be mintined with n infinite number of different ctul rrngements of toms - mking possible n endless number of crystl structures. To specify given rrngement of points in spce lttice, it is customry to identify unit cell with set of coordinte xes, chosen to hve n origin t one of the lttice points (fig. 5). In cubic lttice, for exmple, we choose three xes of equl length tht z c β γ α b y x Figure 5 Specifiction of Unit Cell prmeters re mutully perpendiculr nd form three edges of cube. Ech spce lttice hs some convenient set of xes, but they re not necessrily equl in length or orthogonl. Seven different systems of xes re used in crystllogrphy, ech possessing certin 6
7 chrcteristics s to equlity of ngles nd equlity of lengths. These seven crystl systems re tbulted in Tble I (to be considered in conjunction with fig. 4). Tble I. The Seven Crystl Systems System Prmeters Interxil Angles Triclinic b c α β γ Monoclinic b c α = γ = 90 β Orthorhombic b c α = β = γ Tetrgonl = b c α = β = γ Cubic = b = c α = β = γ = 90 Hexgonl = b c α = β = 90, γ = 120 Rhombohedrl = b = c α = β = γ 90 The network of lines through the points of spce lttice (fig. 3) divides it into unit cells (see lso fig. 4). Ech unit cell in spce lttice is identicl in size, shpe nd orienttion to every other unit cell. It is the building block from which the crystl is constructed by repetition in three dimensions. The unit cells of the fourteen spce lttices re shown in fig. 4. All crystl structures re bsed on these fourteen rrngements. The body-centered cubic, fce-centered cubic nd hexgonl lttices re common nd of prime importnce in metls. Some of the metls ssocited with nucler pplictions, such s urnium nd plutonium, hve crystl structures which re more complicted 7
8 thn these three reltively simple types. In generl, crystlline cermics lso re more complex. 2. UNIT CELLS VS PRIMITIVE CELLS In the literture we often find reference to unit cells nd to primitive cells. The primitive cell my be defined s geometricl shpe which, when repeted indefinitely in three dimensions, will fill ll spce nd is the equivlent of one tom. The unit cell differs from the primitive cell in tht it is not restricted to being the equivlent of one tom. In some cses the two coincide. For instnce, in fig. 4 ll fourteen spce lttices re shown by their unit cells. Of these fourteen, only seven (which re those?) re lso primitive cells. Primitive cells re drwn with lttice points t ll corners, nd ech primitive cell contins the equivlent of one tom. For instnce, simple cubic unit cell hs n tom t ech corner. However, t ny of these given corners, this tom must be shred with seven other identicl cubes which fill the volume surrounding this point. Thus there is effectively only 1/8 of the tom which cn be ssigned to tht prticulr unit cell. Since there re eight corners in cube, there is the equivlent of one tom, nd thus the primitive cell nd unit cell coincide. Continuing, consider the body-centered cubic (BCC) lttice. In this cse there is one tom t the center of the cube nd one tom contributed by the eight corners. This cell, then, hs two toms nd, to void confusion, should be termed unit cell. In the fce-centered cubic lttice there re six fce toms, but ech fce tom is shred by two cells. Consequently, ech fce contributes 1/2 n tom. The fces thus contribute three toms nd the corners one, for totl of four toms in the unit cell. The fce-centered cubic structure (FCC) cn lso be considered s four interpenetrting simple cubic cells. 8
9 In the study of crystls the primitive cell hs limited use becuse the unit cell more clerly demonstrtes the symmetricl fetures of lttice. In other words, the unit cell cn usully be visulized redily wheres the primitive cell cnnot. For exmple, the cubic nture of the fce-centered cubic lttice is immeditely pprent in the unit cell, but it is not nerly so obvious in the rhombohedrl primitive cell. 3. PACKING OF ATOMS A crystl structure is regulr rry of toms rrnged on one of the fourteen spce lttices. The lest complicted crystl structures re those hving single tom t ech lttice point. Polonium hs the simplest structure, being simple cubic. In norml metls, the toms (or positive ions) re held together by cloud of free electrons so tht ech tom tends to be ttrcted eqully nd indiscrimintely to ll its geometriclly nerest neighbors by the free electrons pssing between them. This condition fosters the formtion of closely pcked structures of the types which cn be demonstrted by efficiently pcking uniformly-sized spheres into given volume. You cn gin better grsp of pcking by conducting n experiment s follows. Assume we hve quntity of smll spheres which we re required to efficiently pck into box ( two-dimensionl pproch cn be mde using hndful of pennies). After some shuffling, it is obvious tht the closest possible pcking is obtined when the spheres re in contct nd their centers occupy positions which correspond to the pices of equilterl tringles (fig. 6). It is lso evident tht there re two sets of Figure 6 Close pcked tomic rrngements 9
10 tringles one set with vertices pointing wy from the observer (points up) nd the other set with vertices pointing towrd the observer (points down). When second lyer is dded, there is closest-pcking if the spheres in this new lyer rest in the hollows formed by the spheres of the first lyer. The centers of the spheres in the second lyer will lie bove the centers of the points-up tringles or bove the centers of the points-down tringles, but not both simultneously. Which set is used is immteril. For our discussion, however, ssume the second lyer is centered on the points-up tringles. When we strt dding third lyer, the spheres will gin rest in the hollows formed by the spheres in the second lyer. And gin we hve the option of plcing the third lyer on the points-up or on the points-down tringles. If we center the third lyer on the points-down tringles, we find the third lyer is directly bove the first lyer. If dditionl lyers re dded using n lternte stcking sequence (i.e., lterntely centering the lyers on the points-up nd the points-down tringles), the sequence cn be written s ABABABAB... This rrngement of spheres, trnslted to n rrngement of toms, is the hexgonl close-pcked (HCP) structure - very importnt, but not discussed in detil here. Mny elements hving covlent bonding form rrngements in which the coordintion number is (8 N), where N is the number of vlence electrons. Wht, then, is the coordintion number of the HCP structure just demonstrted? The geometry of the structure shows tht ny one tom hs twelve equidistnt neighbors. It is pprent tht if, in ny lyer, given sphere is plced in the djoining lyer, it fits in the hollow formed by three spheres nd consequently is tngent to three spheres in the djoining lyer. Thus ny given tom in the HCP structure is tngent to twelve other toms six in its own lyer nd three ech in two djoining lyers. 10
11 When the third lyer of spheres ws dded in the bove discussion, we ssumed this lyer ws centered on the points-down tringles of the second lyer. Wht hppens if the centers of the points-up tringles of the second lyer re used insted? The distribution of spheres in the third lyer is the sme s in the first two lyers, but does not lie directly bove either of these two lyers. If fourth lyer is dded, centered on the points-up tringles of the third lyer, we find the fourth lyer is directly bove the first lyer nd duplictes it completely. The stcking sequence for this structure cn be written s ABCABCABC... This rrngement hs the sme density of pcking nd the sme coordintion number s the HCP structure. However, it is the fce-centered cubic (FCC) structure. The HCP nd FCC crystl structures hve the sme density of pcking nd the sme coordintion number. Therefore we might expect the behvior of the two HCP nd FCC structures to be very much like with regrd to physicl nd mechnicl properties. This, however, in most instnces is not the cse. The mximum density of pcking is found only in the HCP nd FCC crystl structures. Why the metllic bond does not lwys produce one or the other of these two densest rrngements of toms is s yet subject of intensive studies. The BCC unit cell contins two toms, nd the coordintion number is eight. There is prtil compenstion for this in the fct tht there re six next-nerest neighbors t distnces only slightly greter thn tht of the eight nerest neighbors. Some chrcteristics of cubic structures re given in Tble II. An interesting exmple of the type of crystl structure obtined in covlently bonded elements which obey the (8 N) coordintion number rule (where N is the number of vlence electrons) is the dimond structure. This structure is found in crbon, germnium, silicon, nd tin t low tempertures nd in certin compounds. Ech tom 11
12 hs four nerest neighbors configurtion tht is vriously clled dimond cubic, body-centered tetrhedrl or tetrhedrl cubic. TABLE II. Chrcteristics of Cubic Lttices Simple Body-Centered Fce-Centered Unit Cell Volume Lttice Points Per Cell Nerest Neighbor Distnce Number of Nerest Neighbors Second Nerest Neighbor Distnce 2 Number of Second Neighbors Mny compounds crystllize in vritions of cubic forms. Rock slt (NCl), for exmple, is typicl of mny oxides, fluorides, chlorides, hydrides nd crbides. It is sometimes considered s simple cubic with lternte N nd Cl toms on the cube corners. Actully the structure is two interpenetrting FCC lttices one of N nd the other of Cl nd the corner of one is locted t point 1/2, 0, 0 of the other. Mny other oxides, fluorides nd some intermetllic compounds hve the fluorite (CF 2 ) structure. This is FCC with C t the cube corners nd fce centers nd F t ll qurter points long the cube digonls. 12
13 4. LATTICE PLANES AND DIRECTIONS It is desirble to hve system of nottion for plnes within crystl or spce lttice such tht the system specifies orienttion without giving position in spce. Miller indices re used for this purpose. These indices re bsed on the intercepts of plne with the three crystl xes i.e., the three edges of the unit cell. The intercepts re mesured in terms of the edge lengths or dimensions of the unit cell which re the unit distnces from the origin long the three xes. For instnce, the plne tht cuts the x-xis t distnce from the origin equl to one-hlf the x-dimension of the cell is sid to hve n x-intercept equl to 1/2, nd if it cuts the y-xis t 1/2 the y-dimension of the cell, the y-intercept is 1/2, regrdless of the reltive mgnitudes of the x- nd y-dimensions. If plne is prllel to n xis, it intercepts the xis t infinity. To determine Miller indices (hkl) of plne, we tke the following steps: 1. Find the intercepts on the three xes in multiples or frctions of the edge lengths long ech xis. 2. Determine the reciprocls of these numbers. 3. Reduce the reciprocls to the three smllest integers hving the sme rtio s the reciprocls. 4. Enclose these three integrl numbers in prentheses, e.g., (hkl). A cube hs six equivlent fces. If we hve definite orienttion nd wish to discuss one specific plne of these six, it is possible to specify this plne by using the proper Miller indices. Prentheses re used round the Miller indices to signify specific plne. On the other hnd, it is often dvntgeous to tlk bout plnes of form i.e., fmily of equivlent plnes such s the six fces of cube. To do this it is customry to use the Miller indices, but to enclose them in curly brckets (brces). Thus the set of cube fces cn be represented s {100} in which {100} = (100) + (010) + (001) + (100) + (010) + (001) 13
14 This nottion thus provides shorthnd scheme to void writing the indices for ll six cube fces. The utility of the scheme is even more evident in the cse of the (110) plnes i.e., the dodechedrl plnes (in cubic system), where {110} = (110) + (101) + (011) + (110) + (101) + (011) + (110) + (101) + (011) + (110) + (101) + (011) The equivlent form for the orthorhombic system is {110} = (110) + (110) + (110) + (110) {101} = (101) + (101) + (101) + (101) {011} = (011) + (011) + (011) + (011) The octhedrl plnes for the cube re {111} = (111) + (111) + (111) + (111) + (111) +(111) + (111) + (111) Direction indices re defined in different mnner. A line is constructed through the origin of the crystl xis in the direction under considertion nd the coordintes of point on the line re determined in multiples of lttice prmeters of the unit cell. The indices of the direction re tken s the smllest integers proportionl to these coordintes nd re closed in squre brckets. For exmple, suppose the coordintes re x = 3, y = b nd z = c/2, then the smllest integers proportionl to these three numbers re 6, 2 nd 1 nd the line hs [621] direction. As further exmples, the x-xis hs direction indices [100], the y-xis [010] nd the z-xis [001]. A fce digonl of the xy fce of the unit cell hs direction indices [110], nd body digonl of the cell hs direction indices [111]. Negtive indices occur if ny of the coordintes re negtive. For exmple, the -y xis hs indices [010]. A full set of equivlent directions, i.e., directions of form, re indicted by crets: <uvw>. 14
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