a-s:,'?tZ

-:.:

Figure 8 A primitive cell nay also be chosen following..his orocedwe: (1) draw lines to connrct a given latticipoirt to all nearbv Iattice points;.(2) at tlie midpointarJ ncmal to tbese lines, draw new iines or plalai Thesnallest volume encjosed ia tlis way is tle liigner-Seitzprnitire cell. -{i1 space mav be f;lled b1,these1els, jutas bv the cels of Fig. 7.

a Prirnitil'e cell is a minimum-vofu:E]e ceil.8 There is a densir)* of one latticepoint per primitive ceil.s There aie iattice points at the eighi corners of theparailelepiped, but each corner point is shared among the eight cells whichtouch tlere. The volume !'" of a primitive cerl defined by primiuie a_res a, b, c is

V"- la=b.cl , (3)

by elementary vector anall'sis. The basis associated r+j& a lattice point of aprimiiive cell may be calied a p::-a-:iu.,,e basis. No basa conr"ins feiver atomsttran a primiiive basis contains.

Alothe.*'ay of choosing a cel of equar 'oiume v" is shown in Fig. g.The cer formed in this rvay is isro*'n to phy-sicrsts as a \yigrrer-seitz primitir.ecell.

FL\DA}fEI{IiL TI?ES OF L{IrICES1O

Crystal lattices .can be carried into themserves not on-ly by the ratticetrarulations T of Eq. (2). but also b'r'ariou-s poir,t srn:rmetry op"r"uonr. etypical syrnmetry operation is tiat of rotation about an arcis which passesthrough a latLice poirt. Lattices can be for:nd such fhaf 6s6_, tr,,,o_, tluee_,fo'r-, and six-fold rotation axes are permissible, corresponding to rotations by2n, 2r/2, %r/3, 2r/4, and 2r/6 ra&ans and by integral *ulUpl", of theserotations. The rotation axes are cienoted by the slmbof, f, S, 3, +, and 6.

we cannot find a lattice that goes into itself under other rotations, sucha-sby 2tr/7 radians or 2n/s radiarrs. A single molecule can have any degree ofrotational rymmetry, but an infinite periodic lattice cannot. We can make acrystal from molecu-tres which individually have a five.fold rotatioo axis, butwe should not eryect the lattice to have a ffve-fold rotation axis. rn Fig. ga.

1v3-show what happens if we try to corrstruct a periodic lattice having €ve-

fold symmetry: the peaiagons do not fit togetJ-rer neatly. we see that *I ."o-€There ae muvrvays of crro;sngile primiti'e ues ad primitive cen foragivenrattice(Fig.7a).eThe nmtrer of atom in a pri:nitive ell is the nmber of atoms in the bisis.

-.-,. toT" origrnal i848 study by A- Bnvais is reprinted in his Erud6 crystalragraphigubs, ceutier-vill's' Pads, 1866; a cemm traslation appem io ostward's xtot"ixn io ooi6-,rii:xarcrqtn

90' (f897). For a firll dimsion of crystal .im*erry, se F. Seitz, Z. Krist, 88, 433 (I9S4):.g0, 21jg(1935); 9r' s6 (1935); 94, 100 (i936); and vol. t-of IntemtiorcI tobla for'r+ay *rrr;ig;;phv,5yry9 1*, Birmingham, 1952. A panicuialy redable discsion of sprc grorp-, i, gi'"eo UyF. C. Phillips, An inhodrction to agstallographrl, fVil"y, 1963, 3ril ed-

I Crystal Strcture t3

{

---^-\.t\H/: \v)\--/

Figure 9a A 6ve-fold axis of srmmen* camot eri5tin a lattice becaue it is not possible to fill aII space*-ith a connected arav of pmtagons. In Fig. 32 uegire u exmple of re.illa pentagonal packhg *iiicncies noi have t.he trulsledonai hvriace of a latdce.

Figrrre l0 The clvstal lattice is rotated bv aangle 9 about a lattice point. The vector a bcffiied into a, by the rctatioo. For special rzlueof 9 tle rotated lattie mbcids with the orfginallattice. For a squ*e lattice tlese special valusre 9 = |z ud multiplc ttrereof, so that tbepoint group of ttre sqwe lattie includes a fou-fold rotation. \!'herever tie rotated lattice coin_cides vith the original lattice, the vector a, _ avill be a lattice vecto!. Such a lattice vector sillnever be shorter than E becaue there is no latticevector shorter than t, except zero. Simila requirements determine the special values of 9 flr alpossibie lattict s.

I-igure gb Kepier's demonstrahon (Fidntron{€rrundi, 1619) that a seven-fold ads of slTrmei-,cff1ot eist in a lattice. (Cescttntelte lI'gie. \bt. 6ilecii. \lunich, 19.1O.)

a

: -.r .;1

i

il

- 1-€-i).:,,:::--ri'::t:< ?_!-

.':

aFigure ll Diagrams illustratiag foul-iT,

g:""pr: The dots denote equiva_Ient points. The point group I hro nosymmebl- elements, so here a point hcno other point equir.a.lent to iL For lz:ner:

N a minor plane: one dot on re_trtraoD across ttre minor plaae becomatt,e second doi. Wit-h a ts.o_fold aris 2,a rotation of r takes one dot into ,Jreotier. \Vith a two_fold axis ald ooeminor plaae ttrere is automa6call;. asecond

-minor plane nomal to tlel:: Tu

u'e have tle Point grouD znmqttt tou equjvaleDt poirts.

not cornbi'e five-foid point s)-mmetiv qith -.he required transrational s.!Tnmeh-)..The arg'"rment aqainst se'en-ford q'mnetn' is gi'en in Fig. gb. The lines of analgebraic argument are indjcated i. e,i". iif ^''

-{ lattice point group is defined aI the co'ectron of the s}mmetry opera_tiore rvhich. u.hen anplied abotd a lottice 1:oittf ,Iear.e the lattice inrariant.The possibte 'ot"uo*'n",=-il; ;';;'"i',rTr. ,r" can aiso ha'e mi*6y ys.flections m about a plane tluough"a r*tti"" foi.,t. r.rre inversion operation is

r#"# *: ::::"$"1 ;?T::: :" "':#:" in a prane n ormar,i u

" .o,u-

Locatiors of equir.alent O"J;;"'r;j* n ,o, a* poinr groups in Fig. il.The symmetrv oP"r"oo::-:l-the group ;;ry;" point into al the equivalentpositions. The poinrs themselr.es nr.ot not'U" tlJ"gfri ;;;;;:il ,r*mebT elements; thev may be ,d:r: trianglJs-or *i""rf* *[. ""'ffi"r'elements. The synmetr-y axes and pl"n", Jf a cube are shown in Fig. 12.Ta*Itimercia nal Lattice Tgp es

There is an unlimited number of possible rattices beca'se there is nofi.l*'Jff:;H *tle

lenetl's ", a "i,r'" r.ruce transration .,,"",o., o, oo

a.n d b.-A *.,".J - i"Li"tq :' iffil,Lll?;'l H ffi , '?;#X ;invariaat only under rotation of ri and Zo

^Uo,ri*y lattice poinlBut special lattices'of t]re obiique ti""; be invariant ,nder rotation of

""Y,,!;JJ! j;li'*!r,;X,-d",*io.."a""u*wemusti-poo.-JJ"u,,"

'nderone.,;;;;;#Jru[?,"'"T::,:;ffi ":*l#rnilffi :

at

IY

oa2mm

\@ffijffi

(e)

.')\. !i- - \, '.i/7' 'j.

\ :..-, r]-:,

':.1

rII r t5\'

--\ / 'J'i 5-- -.+" 1

(c) id)

Figue 1l (a) The t'hree plms of +Tmes)* Parallei to the faas of a cube (b) Tbe six diagonal

plme of s\EEetrY - " ""ot'tttt"l'il"t'-*"-; i"y'tJ*;i;1" cube' (d) Tbe fou ELad axes of a

.i*lt"i 'it" ,o ii"a *o or " "tut'

The ilvesion enter is not show

there are four distirct types of restriction' and each leads to what we may call

a special Iattice $'pe'^ t'vo dimensiors, the obliqueThus there are 6ve distinct lattice types m

lattice and the four qp."JJ*"".. Brrrais^ lattice is the common phrasell for

a distinct lattice $pe; *" "y that there n1s five Bravais lattices in two

dimersions'

rl\\e have not rucceeded in finding or corotmcting a defrnition w'hich stuts out "A Bavais

lattice is . . ."; the soues w";;ilfi';t ;tyl:t'"t *'* " et"'l* u*""'" The phrre "hm&mental

type of lattice" is more zugqestive'

I

rl--t ; -==-=J

'./ ,/.-.-.-.-----r .//

:r"j::_ ,::j: a:i_

. : ..1

o

aa..- . a a

(a) Square lattice

!ai=ibi; e=e9'(b) Hexagonal lattie

iaj = lbl; c= lme

a

(c) Rctangular latticelal* 1lh e: es"

(d) C€ntered reciangular tattice;ax* are sho'm ior both theprimitive ell and for therectangular unit c€ll, for\rhichiai *ib1;P=gO'

Figue 13

The point operation 4 requires a square lattice (Fig. 13a). The pointoperations 3 and 6 require a hexagonal lattice (Fig. 13b). This Lattice isinvariarrt nnder a rotation 2ri6 about an a:ris throug! a lattice point and rrormalto the plane.

There are important consequences if the mirror reflection m is preseat.We write the primitive translation vectors q b in terms of the unit vectorsi, i along tle Cartesian r, y axes:

t=azi*av9; b=b"x*b,9. {4)If the primitive vectors are mirrored in &e r axis, then a, b are tuansformedby the reflection operation into new vectors a', b' giveo by

a' : a"*. - ao9 ; b'=b.k-bui . (5)

If the lattiee is invaria-nt under the reflection, t}ten a', b' must be lattice vectors;that is, tley must be of the form n1a I nzb, rvhere n1 and n2 are integers.If we take

a=ai; b=b9; (6)then a' - a and b' : -b, so that the lattice is carried into itseX. The latticedefined by (6) is rectanguJar (Fig. 13c).

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. A second distinci possibility for a and b gives another tn;einvariant under reflection. Note tl-iat b, rr,ill be a lattice vector if

b':a--b;

Table I The ffve tp.o-dimensional latticc trpes

(The notaiion mrn mearls that two mirror lines are preseni)

This chorce gives a centered rectangular lattice (Fig. l3d).We have now exhausted the trvo-dimensional Bravajs lattices u.hich are

consistent *'ith the point-group operai;o;rs appried to lattice points. The fi,,.epossibilities in hvo dimensioru are summarized in Table t. rhe point s,-metr,-given is that of rhe bttice; an ach:a-l cn'stal siructur. *ay hrrre lor'e. srrrn-meky than its iattice. Thr's it is possible for a crvstal with a sq'are lattice tohave the operation 4 without having all of the operatioru 4mrr.

I Crystnl Stretwe

of lattice

(7)

then using (5) rve have

br'=a"-b"=br: bo' : ao - bu = -bu (8)

These equations have a soluhon lf. a, : A; b,: ]a,; thus a possible choice ofprimitive trarxlation vectors for a lattice *'ith mirror sl.rnmetrv is

a-ai; b=12ai4bui- (s)

LatticeConr enhoaal

cell

,{xa ofcoventional

celi

Point-group srmmetrv oflattice abouilatirce poilts

ObliqueSquareHexagooalPrirnitive lsctr n gulalCentered rectangular

ParaleioqramSquare

60' rhombusRectairgleRectarrgle

a-=b,9-gl'a-b, p:gO'a:b, 9: I20'a=b, 9= 90'a+b, 9:gO"

2

lrnm6mm2mm2m

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Table 2 The fourteen lattice types in three dimensions

Svstem

Nmberof

latticesin

svstemlattiesrmbols

Resbictiom ononventional ell

samdmgles

Triclinic a7b7 ca* B*t

\f snoclinig P,C albt'cd = y:90'+B

C)rthorhombic P,C, I,F a*b* ca=B=Y=90'

Teiragonal a =bt'ca-B-7=90"

Cubic PorscI or bccF or fcc

a=b=ca=B:y=90'

Trigonal a=b=ca- $- y<I2,0',;.90"

HeragonaJ a.: b; cc=B=9["T - 120"

Th r ee -D imznsional Inui c e Ty p es

In trvo dimensions the point gloups are associated rvith five di_fferent

lrpes of lattices. In three dimensions the point slrnmeu/4 groups requLe thefourteen different (one general arid thjrteen speciaj) lattice types show:a inFig. 14 and listed in Table 2. The general lattice t'?e is the h-iclinic lattice.

The fourteen lattice types are conveniently grouped into ,seven systems

sccolding to tle seven types of conventional unit cells: triclinic, monoclinic,orthorhombic, tetragonal, cubic, higonal, a"d hexagonal. The djvision intosystems is summarized conveniently in terms of tle special axial relatioos forthe conventional unit cells. The axes q b, c and angles a, F, y ue defined inFig. 15.

The urrit cells shorvn in Fig. 14 are the eonventional c.ells, and ttrey arenot a-hl,ays primitive cells. Sometimes a nonprimitive cell has a more obviousconnection with the point trynmety elements than has a primitive cell. lVenow discuss the various lattices by their classification in systerns.

1. In the triclinic system the single lattice $pe has a pfi161iye (P) unitcell, rvith tluee axes of unequal lengths and unequal angles.

Tetmlonal P

iat: _-

?:rl -:_-:1. i_:'_tsj,.'':7€if_;+: ::.-,

Figue 1{ The fourtea Bnrzis or space }afties- The alls shom se the conventional cells, whichue not always the prinitive cells.

Figure 15 Crystal ares a, b, c. The ugle a is indudedbetrveen b and c.

a

Monmlinic P Monrclinic C Triclin ic

Table 3 Charac{eristics of cubic lattices

(Tables of numbers of neighbors and distances in sc, bcc, fec, hcp, and diamond strucfuresare given on pp. 1037-1039 of J. Hirschfelder, C. F. Cutis, and R. B. Btrd" tr[oleculnrtheorg of gases and. Iiquids, \Vilev, 196{.)

Simple Bodv<atered Face<eqtered

\blume, conventiona-l eellI;ttice polnts per ceil\iolume, prirnitive c.ell

I;ttice poilts per r.nit volumeNumbei of neerest leighbo;s"Nearest-neighbor distance\umber of second neighborsSecond neigh'oor dis*ac-e

a3

2-!-3

2i6e6

Sr/2a1, _ 0.Wo

a

a3

4\-34/az

a/Zt'z - 0.7a7atla

a3

1

a3

l/62t)

a

u

" \sst neighbon m tle nsst la*tice poirts to dv givetr lattie point.

2. In the monoclirlic svstem there are tr.r'o lattiee qT)es, one uith a primi-tive unit ceii and the other *ith a nonprimitive conventional cell xfuch maybe base-centered (C) rvilh lattice points at tie centers of tf-ie rectaagular cellfacets in the c& plane.

3. In ttre orthorhombic svstem ti.ere are four lattice tvpes: one Lattice

has a primitive cell; one lattice is base-centeredl one a bodl'-centered(I : German Innen:entrierte); and one is face-center.d (F).

-1. In the tetragonal r-stem the simplest urit is a right square prism; thisis a primitive cell and is assoclated $-ith a tekagonal space Lattice. A secondtetragonal lattice tlpe is bod1.-centered.

5. In the cubic system there are tl-ree lattices: the simple cubic (sc) lattice,the body-centered cubic (bcc) lattice, and the face-centered cublc (fcc)lattice. The characteristics of the three cubic lattices are summarized ia Table 3.

A primitive cell of the body-centered cubic lattice is shown in Fig. 16;tlre primitive translation vectors are shown in Fig. 17. Trte primitive tmnsla-tion vectors of the face-centered cubic lattice are shown in Fig. 18. The primi-tive cells contain only one lattice poinl but ttre conventional cubic cellscontain two lattice points (bcc) or four lattice points (fcc).

6. In the trigonal system a rhombohedron is usually chosen as the primitivecell. The lattice is primitive.

7. In the hexagonal system the converitional cell chosen is a right prismbased on a rhombus rvith an angle of 60'. The lattice is prirnitive. The relaticnof the rhombic cell with a hexagonal prism is shown in Fig. 19.

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Figure 17 Primitive traslation vectors of the bodr <enteredcubic latdce: tiese r.ectors conoect the lattice point at theorigin to latice points at the body.centers. The primiti,,ecell is obtained on completing tbe rhombohe&on- h temof the cube eCge a the prinitil.e .saulation vector ue

^' = !G+ :. - i); s, =.-9,1-i + i + r)i, a,- ,c'=--(iij+ir.

Thae primitire t.o *"t.-*gto of 109"26, siti ech otler.

Figure 19 Relation of tie primitive ellir. the hexagonal system (heaw lines) toa prism of hexagonal symmetr,v. I{erea=b*c.

Figure t8 The rhombohedral primitive cell of tie face-centeredcubic crvstal. The primitive translation vectors a,, b,, c, conamtthe lattice point at tle origin with lattice points ,t th" i"". *ot"rr.As dra*a, the primitive vrctors ae:

a'=;(i+i); b'-_ it;+i), c =ft:+ll.The mgles betrveen tie ares ae 60".

3ta

Figue 90 This piaue intercepts ttre a, b, c u6 at 3a, 2b, 2c. Thereciprocals of tiese numbers re !, |, !. The smallat three iltegenharing tle eme Gtio ue 2, 3, 3, od tbu the \liller ildjces of theplme ue (233).

(200)

Figure 2l lr{iiler indices of some important planes into (100).

(100)

a cubic crystal The plane (200) is paajlel

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I CrytUI StrucAtre

POSITION AND ORIENTAIION OF PI-ANES IN CRYSTALS

The position and orientation of a crystal plane are d.etermined by anytlree points in the plane, provided the poitrs are not couinear. ff each of thepoints lies on a cn'stal axis, the plane may be qpecified by giving the positionsof the points along the axes in terms of tie lattice constants. I{ for example,dte atoms determining the plane have coordinates (4,0,0), (0, I,0), (0,0,2)relative to the a-xis vectors from some origrn, tle plane may t ,p""in"a Uythe three numbers 4, I,2.

But it is more usefi:l for structure analysis to specify the orientation of aplane by l{iller indices,l2 determined as in Fig. 20:

1. Find the intercepts on the ares a, b, c in terms of the lattice constants.The ares ma,v*.beprimitive or nonprimitive.

. 2. Ta&ezdre reciprocals of these numbers and tlien reduce to three integershaving the s,rne ralio. usually the smallest tlree integers. The result is en-closed in pareniheses: (hkl).

For the plane rvhose intercepts a:e 4, I,2 the reciprocals are ], 1, and {anc ihe rliller indices are (I42). If an intercept is at ininity, the correspond-ing index is zero. The \liller in&ces of some important planes in a cubiccrvstal are illustrated bv Fig. 2I.

The indices (hkl) ma,v denote a single plane or a set of parallel p1anes.r3If a plane cuts ari axis on the negative side of the origin, th. *.r-"rporrdirrg

T1:t u regati'e and is indicated bv placing a minus sign above th" rnde",t/rl1;. The cube faces o[ a cubic cn.sra] are (fOO), fOfOi, (001), (I00),(010)' a,'.d (001). Pianes equivaient br so:rnmetr.v may be denoted by currybracke's (braces) aro'nd irri]ler i'&ces; tle set of

"ou" faces is 1roo1. $e

ofien speaL simply of the 100 faces. If rve speal< of the (200) p1"o" *." *""r,a plane parallel to (r0o) but cutting the a a-xis at {a. The formauon of the (rl0),(111), a-rd (322) planes of an fcc cry-star skucture, starting &om (r00) planes

:

of atoms, is showr in Figs. 2la" b, c on p.24. iThe indices of a direction in a crystal are expressed as the set of the

smallest integers which have the ratio of the components of a vector in thedesired direction referred to the axes. The integers are written between squarebrackets [hkl] . In a cubic crystal t]re r a_ris is the t 100] direction; tn" _ y oi,is the [010] direction. often we spearr of the [hrcr] and equivarent air""Lo*,or simply of the [hkl] directions. In cubic crysials the direJtion [hkr] is arrvaysperpendicular to a plane (hkl) having tie sarne indices (problem :1, but thisis not generally true in other crystal svstems.

- 12

'{ t first sight the uefuLress of the lrriiler indices sems improbable, but cbapter 2 makes creutheir _cmrrnience

and elegance.

, . l3Other smbols are frequenLlr, used bv cn.stallographers; they may ue (, c, o in place of ouh, k, I.

Figue 2la A (110) plare of u fcc cn,sal

*."*.: a built up fron 1I00r ler.en.

'T ":d_S: "":onpmling pbotog:aphsue br.J. F. \icbolas, Atlzs of ^oi"i oycrystcJu-7rm. Gordou ud Biqcb, f S65i.

Figue 2lb .{ (l1l) plane of m fcc cn.stzlstructue, bxed on (100) laven.

Figure 2lc A (322) plane of m fcc wsalstructwe, bred on (100) layen. Th" to_entration of atoms tends to be lower inplanes of b.igh indices tban in planes of lowirdices.

I Cryst4l Stflcturc

Figrge 22 Tbe coordnates of the cenh-alpcilt of a ell ue | ] |, ir terns of thelengtl of the ues.

POSITION I\- THE CELL

The positions of a point in a cell is specified in terms of atc,mic clo.dinatesx, g, z ii7 *'hich each coordinate is a fraction of the axial l"ngtb, a. b, or c, inthe direction of the coor&nate, u.i.rl'r the origin taken at a corner of a cell. Thus(Fig. or 1 the coordinaies of tie centra-l point of a cell are i i *. The face_centered posiHons L'rclude l ] O; O i i; i 0 i. The coordinates of atoms ir fccand bcc lattjces are usuall-r giren ia terrns of the conventionai cubic cell.Tables of crvstal stuchrres usually spec4- the type and size of the cell, a;iri tienthe-v give tle values of the atomic coordinates r,r, !r,4 for eac! of the atomsin the cell.

25