Really, your lattices are all primitive, Mr. ~ravais!' Sidney F. A. Kettle School of Chemical Sciences, University of East Anglia, Norwich NR4 TTJ, U.K.
Lars J. ~ o r r b ~ ~ Department of Chemistry and Chemical Engineering, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5L0, Scientific and technological developments over the past few years have increased attention on the solid state. This is reflected in, for example, a recent issue of this Journal (October 1991)that contains no fewer than four articles on solid state chemistry. In a sense chemistry has been well prepared for this development, for fundamental aspects of ihesolid state have been part of undergraduate coukes for generations, namely the basic crystal structures of metals, Halts, and simple minerals. ~ o n c i ~surh t s as unit cells. lattices, crystal systems, and the close-packing of spheres are almost everywhere taught early in the chemistry program. However, over time there has been little change in the way these tooics have been oresented-tvoicallv. bv describine . the seven crystal syste6s and iilustrating'the 14 ~ r a v a ; lattices. with the relationshios between the axes a. b. c of be their u k t cells. ~ e ~ r e s e n t a t i vexamples e of this found in two well-regarded textbooks by Oxtoby and Nachtrieb (1)and by Alberty and Silbey (2).Given this solid tradition, one might assume that the basics of structural chemist6 are secure parts of the subject and that no review is needed. The purpose of this paper is to question such an assumption. w e d o so because the app;oach of solid-state physicists and solid-state spectroscopists is rather different to that traditionally followed by chemists and crystallographers. Our first concern is with the 14 Bravais lattices. ~
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Bravais Lattices Auguste Bravais (3, 4) published his classical work in 1850, some 60 years before X-ray diffraction was discovered. The work was based on his thorough knowledge of eeometrical crvstalloeraohv and eeneral mathematical eonsiderations ;bout t%e h1l"ing of space with parallelepipeds. His results became relevant to the later development of X-ray crystallography, a subject that in turn became highly relevant to chemistry So, Bravais and his lattices became part of every chemist's education. Bravais found that there are onlv five olane (two-dimensional) and 14 space (three-dimensiouaf) lattices. The 3D-lattices have since become known to chemists as the seven primitive and the seven centered Bravais lattices. As seen in Table 1, the latter are not evenly distributed amonmt the seven crystal ~ y s t e m s The . ~ unit cells traditionaliy associated with the Bravais lattices all have the full point = u p svmmetry (the holohedry) of the crystal system to h i i h ihey 'The title. which ~roDerlvshould have been "En effet.vos rgseaux sont tous -~~- o&nitifs. -, M. ~raiais!". -~ , was chosen as homaoe to the areat " Frencn pnysfclslana crysta ographer A~gLsteBrava~s,1811-1863,. 'Autnor to whom correspondence sno~ldoe aooressw 3Each cryaa system s characterize0 by a cena n point grow (its hoiohedry)of finite order, the operations ofwhich bring the lattice into self-coincidence;see Table 1. Every lattice is also brought into selfcoincidence by an infinite number oftranslations. ~ranslationalinvariance is perhaps the most important feature of crystalline materials and is, in a sense, the motivation for this paper. 4Thecrystal is assumed to be infinite,a valid assumption for most purposes and certainly if one is interested in the bulk physical prop erties ofthe crystal; but see footnote5. ~~
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Table 1. The Seven Crystal Systems, the 14 Bravais Lattices and the Seven Holohedries. System
Lattices
Cubic
Holohedry (Point Group)
Point Group Order
P, I, F
Oh
48
Hexagonal
P
D6h
24
Tetragonal
P, l
0 4 h
16
Trigonal'
P (= R)
D3d
12
Olthorhombic
P, I, C, F
02h
P. c
C2h
8 4
P
C! ~.
2-
Monoclinic Triclinic
P = primitive, I = body-centered, F = face-centered.C = end-centered
'The trigonalsystsm is alsocalled the rhombohedra1 because of the shape primitive unit cell;the symbol R may be encountered instead of P,
01 its
belong, although an exception is generally encountered for the hexagonal system. Given this background, it may come a s a surprise to learn that, for the solid state physicist, all 14 Bravais lattices are primitive and that primitive unit cells exist for each and every lattice. This is not to say that the chemist's view of the seven centered Bravais lattices is wrone. but perhaps that the future development of the subject would be served bv a chanw in oresentation. What. then. is the physicist's approach~howdoes it differ from the chemist's, and what are the conseauences of these differences? To explore this further we turn to the translations that generate the entire lattice from a unit cell. Translations An invariant subgroup of every crystallographic space group is the group of all primitive translations, a group here denoted T. These translations, when acting on the eutire crystal, serve to superimpose the crystal onitselfjlt is . this group of translations that is of p&icular concern to the solid-state physicist. From T the Brillouin zone is derived and from it, in turn, the energy levels associated with crystals (5).Ifthe energy levels are to be correct, the group of all translations must be chosen correctly. A difference between physicists and chemists is that physicists are interested in Brillouin zones and therefore in translations of unit cells. Chemists are interested in the contents of unit cells, in the atoms and the molecules (we simplify!). To explore this difference further we turn to the contents of a orimitive unit cell. There is. as Bravais recoenized. no sinile choice of the shape and symmetry of a primitive unit cell. However. the volume and the atomic contents of a primitive unit cell are constant for any given crystal swucture. The operations of the complete set of primitive translations, of T, on this unit cell Eontent the positions of all the atoms of the crystal. Indeed, the only condition a primitive unit cell has to fulfil is to generate the entire crystal when the operations of T are applied. ~
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tals are very large compared to typical translation vectors, all unit cells "behave the same" and the spectra can be interpreted in terms of the vibrations of a single, primitiue unit cell. But if a centered, n-times larger unit cell is chosen, the predicted number of vibrational modes will be ntimes too large and the spectral predictions correspondingly incorrect. Perhaps now t h a t chemists, too, are becoming increasingly interested in crystal energy levels, it is time to reconsider their traditional approach to the nonprimitive Bravais lattices. To do so, t h e meaning of "primitive translation" must be elaborated. Primitive Unit Cells A lattice is the periodic collection of points generated by all possible linear combinations of three primitive translation vectors PI, Pz, ~ 3 ,
[--
fJ/ -7@7-
a %
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The translation group, T, consists of all operations t, where nl, nz, ns are integers running through all positive and negative values5. The origin of the lattice and the identity operation are defined by: n,=nz=n3=O=to For each translation t there is also a translation -t. Therefore every lattice point is a center of
generators, vectors symmetric. symmetry; pl, p,, all The must lattices p3, primitive the be linearly are primitive centrolattice in-
dependent and a s short as possible. More specifically,they are al-p, 2 ways chosen such that the sum of ,,' , the squares of their lengths,p? + Pa D pzZ+ pa2, is a minimum. I n many cases pl, p,, and p3 are nonorthogonal. For every lattice there r I=,I are three such primitive generators that relate any given lattice point to t h r e e of i t s nearestFigure 1. The seven conventionally primitive unit cells showing three primitive translation vectors pi, p2, neighboring lattice points. For m which are equal to the crystallographer's usual lattice parameters a, b, c. Note that the "atoms'are the seven primitive Bravais latlattice points (see footmote 6). tices pl, p,, p3 correspond to the Legend: (A) Cubic, (6) Hexagonal, (C) Tetragonal, (D) Rhombohedra1 (Trigonal),(E) Orthorhombic, crystallographer,s usual unit (F) Monoclinic, and (G) Triclinic. edges a, b, c, see Figure 1. InFigure 2 we give diagrams of the primitive unit cells of the seven centered Bravais lattices However, when the set of all translations operates on a nonprimitive unit cell, the crystal is generated two or four where, again, pl, p,, p, define the cell edges. The seven times ouer for body-centered or face-centered cells, respeccentered Bravais lattices differ in that a, b, o do not define tively (and three times over for rhombohedra1 cells on hexthe directions of the primitive generators. However, pl, p,, p3 are easily expressed in terms of a, b, c , a s is shown in agonal axes). To avoid such a multiplication, one has to work with a subset of T: 112, 1/4, and 113, respectively. But in so doing, one obtains a n incomplete Brillouin zone and the density of states will be off be a factor of 2,4, or 3. This 'In reality, the total number of translation operations is finite and is perhaps most easily illustrated, in a n inverse sense, by identical to the number of primitive unit cells in the crystal. For crystals of the sizes used in solid state experimentaltechniques,this is a reference to vibrational spectroscopy. Because the wavevery large number, typically in the range 10'~-10~'. lengths used in studies of the vibrational spectra of crys+
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Figure 2. The seven conventionally centered unit cells showing a convenient choice of corresponding primitive unit cells. The three primitive translation vectors PI, p2. p3 are notequal to the crystallographer's usual lattice parameters a, b, c, but are chosen in accord with the definitions detailed in Table 2. Note that the "atoms' are lanice points (see footnote 6). Legend: (A) Body-centered cubic, (0) Face-centered cubic (C) Body-centered tetragonal, (D) Body-centered orihorhombic, (E) Face-centered onhorhombic, (F) End-centered orthorhombic, and (G)End-centered monoclinic. Table 2, w h i c h i s adapted f r o m B u r n s a n d Glazer (6). Apart from some flexibility t h a t m a y arise f r o m point group symmetry, t h e p r i m i t i v e translation vector set i s n o w defined for a l l space lattices. E v e r y p r i m i t i v e unit cell i s t h e n a parallelepiped a n d contains one lattice point. I t i s importa n t t o note t h a t w e are here dealing with empty lattices, t h a t is, lattices w i t h o u t a n y chemical content. Therefore t h e "atoms" in a l l o u r figures are lattice points a n d n o t r e a l In a l l body-centered a n d face-centered cases t h e p r i m i t i v e unit cells a r e rhombohedrally shaped a n d have lower symmetries t h a n t h e corresponding Bravais lattices. In o t h e r words, these unit cells do n o t show t h e full p o i n t
61t is a very common mistake among students-and textbook writers!-to fail to distinguish properly between lattice points and atoms. The unit cells associated with a lattice are always empty and have, in the oresent context of solid state science. one of seven svmmetries. ~ o i e~~, v e r filled . unit cells. as encountered in real crvstal'structures. ~~~,~~~~~~ ,~ have one of 230 symmetries. In most rextooow the Lnr cels of tne BBC and FCC Ian ces are exempl flea wlh the crysta strmdres of a-Fe and Cu, respectively. In both cases the basis, the atoms associated with each lattice point, is one single atom. This means that a model (of the kind normally used in teaching) of an empty unit cell and one containing atoms lookthe same. However, this only happens when the basis is a sinole atom. Proaressina to such a simole structdre as CsC ma6es tneo tference betweenis empty pr m i ve cdb c Jnlt cel ano the I I ea one very OOVOLS ~
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Table 2. Primitive Generators for the Seven Centered Bravais Lattices as Functions of the Lattice Parameters a, b, c of the Conventional Centered Unit Cells. The lengths d the unit cell edges are a =Jal,b = Ibl, c = Icl, and the lengths of the generators are pi= Ipil Capital letters in brackets indicate the corresponding diagram in Figure 2.
Body-centeredcubic (A) pl=(a+b-cY2 p,=(a+b+c)/2 p3=(a-b+cY2 a=b=c;pl=p2=p3. The primitive unit cell is rhombohedra1(a = 109.47')with Dad symmetry. Note that in Fig. 2A the 3-fold axis of the rhombohedral unit cell is along the body diagonal through the lattice point where p, ends.
Face-centeredcubic (6) pl = (a + bY2 p, = (b + dl2 p3 = (a + c)i2 a=b=c;pl=pz=p3 The primitive unit cell is rhombohedra1 (a = 60 ') with D3d symme*. Note that in Fig. 2B the 3-fold axis of the rhombo hedral unit cell is along the body diagonal through the origin.
Body-centeredtetragonal (C) pl=(a+b-c)/2 pz = ( a+ b + cY2 p3=(a-b+cY2 a=b+c;pl=pz=p3.~ The primitive unit cell is a distorted rhombohedron of C,symmetry.
Figure 3. The compression of an elongated rhombohedral unit cell (A) along ih threefold axis gives at one point in the defornlation the primitive unit cell of the FCC lattice (0). At this point the angle sub tended by adjacent "waist-band edges is 60'. As this angle increases, a trigonal lattice is again obtained until it reaches 90'(C), when a Drimitive cubic lattice is obtained. Yet more comDression oives a thoonal lattice aaain until the anale eauals 109.47..'the "tetrahedral &gk. At h I s po nt tne BCC atlice ' s generated(~,.The ~ l mate t compression through a lrlgonal attice gives, at an angle ot 120' the two-amensona hexagonal an ce (not snown,.
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Body-centeredorthorhombic (D) pl=(a+b-cY2 pa = (a + b + cX2 pB=(a-b+c)/2 a#b#e;pl=pz=p3. The primitive unit cell is a distorted rhombohedron of Cisymmetry.
Face-centeredorthorhombic (E) group symmetry of their lattices. In the orthorhombic endcentered case the primitive unit cell is ofD% symmetry, so the holohedry is retained, but the cell angles are no longer all 90'. Likewise, in the end-centered monoclinic case the primitive unit cell is of the holohedral symmetry ( C d , but the anele between D. and D . is not 90'. so the basic monoclinic &nditions a r l not ui(e1d. I n all seven cases the axes defmed bv the ~rimitivegenerators are inconvenient for. and so avoided Ly, the c ~ t a l l o g r a p h e r . The discussion above is echoed in the ambiguity of the primitive unit cell of the hexagonal crystal system. This unit cell is not hexagonal but of Du, symmetry. For that reason it is common to choose a three times larger unit cell (containing three lattice points instead of one) which does have a sixfold axis. Thus, it is triply-primitive and truly hexagonal; see Figure 1B.
Relations among Bravais Lattices The aooroach detailed above enables a rather different and, ho&ully, illuminating way of lookingat the relationshio among the Bravais lattices. This is nowhere better the trigonal (rhombohedral) primitive unit seen than cell shown i n Figure 3 and the sequence of different prim-
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p, = (a + b)/2 pz = (b+ dl2 p3 = (a + cU2 a+b+c;pl+p2#p3. The primitive unit cell is a distorted rhombohedron of Cisymmetry.
End-centeredorthorhornbic (F) pl = (a - bU2 p2=(a+b)/2 p3=0 a#b#c;p,=pz#p3. The primitive unit is cell is prismatic of symmetly DZh.The angle between pl and pz is neither 60' nor 90'.
End-centeredmonoclinic (G) p, = (a - bY2 h=(a+b)/2 Ps=c a f b f c ; fj#90';p1=p2#p3. The primitive unit is cell is prismatic of C2*symmetly. The angle between pl and p2 # 90'. The two-fold ads is along y.
itive unit cells generated a s it is compressed along the threefold axis. Note that in trigonal unit cells all edges are of equal length. In Figure 3A there is an angle of 45' between limbs of the zig-zag "waist band" around the middle of the cell containing six of its comers. As the cell is compressed along its &reefold axis a point will be reached where this angle becomes 60' (Fig. 3B). Each set of three adjacent waistband corners now define an equilateral triangle and thus a threefold axis. There are, then, three such threefold axes. Together with the threefold axis inherent in the original trigonal unit cell, there is a total of four threefold axes. We are in the cubic m s t a l svstem! Acomoarison between Flgures 2A and 3B shows that our rhombohedral cell is in fact the orimitive unit cell corresoondine toa FCC lattice. compre& the trigonal nnit cell hthe
