The Brillouin Zone-An Interface between Spectroscopy and Crystallography Sldney F. A. Kettle University of East Anglia, Norwich NR4 7TJ, U.K. Lars J. Norrby Royal Military College of Canada. Kingston. ON, Canada K7K 5L0 Most university students of chemistry will encounter the concept of the Brillouin zone, commonly in the context of a discussion of the electronic properties of metals. However, i t cannot be said that this concept is well understood by chemists. I t is a "difficult" one, perhaps closer to solid state ohvsics than t o chemistrv and possibly "obscured" by the ia&uaReof solid state phisirs. ~ h e r e f d r eit is not surprising that discussions of and references to Rrillouin zone theory in this Journal have been so rare over the years. The most recent one is that by Gerstein (I)in 1973. Yet it is important; it and the counterpart concept of the reciprocal lattice lie at the basis of all spectroscopic and structural measurements on crystalline solids. The purpose of the present article is to tread a readable path through many intermingled concepts: Wigner-Seitz unit cells, reciprocal space, k space, the Ewald construction and diffraction. In doing so we recognize that we shall only be able to explore these concepts deep enough to provide a firm conceptual basis. The reader may well approach this article from one of several different backgrounds. In particular we distinguish between the chemist who is a spectroscopist and the one who is more of a crystallographer. It is a teaching problem that the approach, definitions, and symbols used hy these two groups of chemists differ so much. As our title indicates, we seek to bring the two together. Toward this end we divide the middle of the article into two parts printed in parallel. One is written to be read first by the reader with a chemical spectroscopy background and the other should be read first by those more familiar with chemical crystallography. The two parts have been written so as to dovetail in either order. The final section brings the two approaches together and makes a unity of the subject matter. Before the division of the paths, however, we give a common discussion of the concept of lattices. Lattices A common confusion among students (and textbook writers!) is that between a crystal structure and its associated lattice. Every crystal structure has a lattice that is a representation of the translational symmetry of the structure--the translational subgroup of the space group. Note "a lattice", not "two interpenetrating lattices", which is an incorrect statement often encountered for NaCl and other simple ionic structures. (This and related problems have been treated in an article in this Journal by Brock and Lingafelter (Z).)The lattice divides the crystal into congruent regions called unit cells. A crystal structure is said to have a basis, a word that tends to mean different things for spectroscopists and crystallographers. T o the former the basis is the "chemical content" of a primitive unit cell. All spectrosopic analyses are based on such unit cells. Forexample, to spectroscopists KN03, crystallizing in space group Pnma with Z = 4, has a basis of four formula units. Application of all the pure translation operations of the space group 1022
Journal of Chemical Education
generates the entire crystal from such a basis. For a crystallographer the "basis" is just one formula unit of KN03, the chemical contents of the asymmetric unit, from which the crystal is generated by the entire set of symmetry operations-not just the pure translations-of the space group. Although unit cells will be frequently referred to, we shall not be much concerned with bases, the chemical content of unit cells, because we are principally concerned with the consequences of translational symmetry. The lattice of a crystal structure is specified as the real lattice, and distances within i t have dimensions of length in SI units of meter. However, we will use the Angstrom, which is a more convenient and natural unit in the world of molecules and crystals; 1A = 10-10 m. With every real lattice we will also have to associate a reciprocal lattice. Displacement within such a lattice have dimension of length-' in units of A-1; hence the use of the word "reciprocal". We will have much to say about the concept of the reciprocal lattice. I t is important to distinguish between a unit cell and a set of primitive translation vectors. I t is common to choose the edges of the unit cell so that they correspond to such a set of primitive translation vectors. The unit cell is then a simple parallelepiped. However, for any structure there is an infinite number of possihle choices of unit cell, primitive as well as nonprimitive. The latter may be body-centered, face-
Figure 1. The Wigner-Seitr unlt cell of a body-centered cubic lattice and its relation to the usual unit cell.
centered, or end-centered and are often encountered. However, the choice of a nonprimitive unit cell for a particular crvstal structure is one of convenience and not one of necessity. Furthermore, a unit cell does not have to be bounded t ~ y three oaira of oarallel faces. M'e now desrrihe one oarticularly convenienichoice of unit cell, the wigner-sei& unit cell, which often has more than six faces. A Wigner-Seitz (henceforth called W-S) unit cell is a orimitive unit cell showing the full rotational svmmetrv of the lattice, which is not the-case for all primitive "nit cells. I t is therefore also called the symmetrical unit cell. An important feature of lattices is that they are all centrosymme&c, although a given crystal structure may well not be. The definition of a W-S unit cell is that it comprises all that space around a given lattice point that is closer to that articular lattice ~ o i nthan t to anv other lattice m i n t . The W-sunit cell for the body-centered cubic lattice is shown in Fieure 1toeether with the usual nonmimitive choice of unit cell. Note tLat this W-S cell, a truncated octahedron, has six sauare and eight hexaeonal faces, i.e., seven pairs of oarallel faces. The us;al body-;entered unit cell coniains two lattice points while the W-S unit cell, being a ~rimitiveunit cell, eontains only one. Therefore its volume is-exactly one half of
that of the nonprimitive body-centered unit cell. W-S cells are most important, because, as we shall see, the W-S cell of the reciprocal lattice defines the first Brillouin zone of a given crystal structure. According to classical crystallography any lattice must be one of the 14 Bravais lattice types. Illustrations of W-S unit cells of all these may be found in a terse hut essential book by Koster (3). We now turn to the conEept of the reciprocal lattice, which students often find remote and unwelcome-but which is central to our discussion. Although it is possible to find a common origin, it is introduced differently in spectroscopy and crystallography. This is why we divide the next part of this article into two parallel columns giving the reader two choices. One is the reciprocal lattice as associated with the characters of the irreducible representations of space erouos. . Such irr.re~s.are of "ereat imoortance for the selection rules associated with all spectroscopic measurements on crystalline solids. The other is the reciporcal lattice as generated in a diffraction pattern of an X-ray, electron, or neutron diffraction experiment and used by crystallographers. We shall finally see that the reciprocal lattice provides a meeting ground for spectroscopy and crystallography.
The Chemlcal Spectroscopl8t's Column or Spectroscopy and the Reclprocal Lattlce Spectroscopists have a constant concern with selection rules. These selection rules dictate what will he spectroscopically observable and usually arise as symmetryconstrai& on integrals representing transition probabilities. It is then natural that spectroscopists, when studying the solid state, should be interested in the character tables of space groups just as they are interested in the character tables of point groups. In practice the crystals studied by spectroscopists alwavs have a basis. but we are not concerned with such base; in this artirle. This means that weahall not look at the distribution of atoms or molecules within a unit cell: therefore, in particular, we will not discuss factor groupn. Rather we will be concerned wirh the group theory of translational symmetry. Every spacegroup has an a subgroup thegroup of all the pure translations of the lattice. It is the general properties of this invariant aubyroup that are our immediate concern. In crystallography one uses radiation with wavelengths smaller than a typical lattice translation vector. However, in spectroscopy o&is almost always concerned with radiation of wavelengths that are very much greater than a typical lattice translation vector. Indeed, since one is commonly employing wavelengths of thousands of Angstroms, the wavelength can be taken as infinite to a first, quite good approximation. That is, we assume that the effect of the radiation is constant throuehout a crvstal irresoective of the size or shape of the crystar The effe"cts of theiadiation are translationally invariant and the only excitations that can be produced in the crystal are also translationally invariant. Translational invariance reauires that the character associated with any trans~ationalh~eration be unity, that corresoondine irr.re~.that - to the well known totally. symmetric . occurs in every point group chararter [able. This means that the character itself trivially ia l-it is the enumeration and labeling of these characters that creates a problem. For a real crystal contains an exceedingly large number of translational operations, typically of the order How is one to keep track of such a phenomenally large number of characters? I t is for this purpose that we have to take a detour into the realm of the reciprocal lattice.
The Chemlcal Crystallographer's Column or Crystallography and the Reclprocal Lattlce '1'0 the rrysrrtllographer a rrystal is thr endless repetition in three dimensions of a huildine brick that contains a symmetrical arrangement of atoms, ions, or molecules. "Endless" because crystals of the tiny size used in X-ray diffraction experiments typically contain about 1018 building bricks. These "building bricks" are properly called unit cells, but one must beware of misusing this name. For any crystal structure there is no unique unit cell; there is a manifold of choices. I t is a matter of convenience, and the choice is normally made between two or three different unit cells in anv eiven case. A maior ooint of difference between smctr&opists and crystailo&aphers is that the former invariably choose to work with a orimitive unit cell. The latter. however, may well choose inonprimitive cell that is cen: tered in some way and has a volume and content that is an integer multiple of that of a primitive cell. The lattice displays the three-dimensional periodicity of the crystalline material and is independent of the detailed atomic content of the unit cell. When studying the lattice on its own, it is convenient to replace each unit cell by a point that could be placed anywhere in the cell, normally at its center. Around 1850 the French crystallographer Auguste Bravais showed that the general symmetry of a three-dimensional arraneement of such points can be one of onlv 14. s Illustrations of these, or more Hence the l i ~ r a v a i lattices: often the conventional unit cells associated with them. are rommonly found in textt)ooks on general or inorganic chemistry or crystallography. We refer to a well-known crystallographic text by Buerger (5). Althoueh there are verv manv different crvstal structurea. the fact that their lattire~arelimitedto 14 isagreat simplifi: cation in the context of this article. In X-rav tor electnm or neutron) crystallography one is interested in both the relative positions of the diffraction peaks (or spots on a photographic film) and the relative intensities of these peaks or spots. The pattern of the peaks is determined by the lattice. while the intensities are largely determined by the contents of the unit cell. The lattice describes the set of translations that, when they operate on a unit cell, generate the crystal structure. These translations are the focus of our interest in
(Continued on page 1024, coi. I)
(Continued on page 1024, col. 2) Volume 67
Number 12 December 1990
1023
The Spectroscopld's Column
The Clydallographer's Column
Consider first an infinite one-dimensional lattice consisting of an infinite set of multiples of a primitive translation operation t; thus t , 2t, 3t, . . ., m t , where t is a vector. (Crystallographers prefer to use the notation a , and that is what we use in the Chemical Crystallographer's Column.) The lattice is perhaps more simply visualized as a row of regularly spaced dots each separated from the next by t. What happens a t the end of the row? This problem is neatly sidestepped by an assumption that the lattice forms an enormous circle (a "cyclic boundary condition") so that there is no end. One replaces the infinity of translational operations by a very large finite number, N. When we turn to a real crystal, where N is indeed extremely large, this kind of assumption will not affect the results just as the real occurrence of boundary surfaces of the crystal do not alter the hulk physics of it. A translation of Nt then takes us back to where we started, completes the circle, and is thus equivalent to the identity operation. An important property of these N different translational operations is that they all commute; the translational group is Abelian. That is, an operation a t followed by bt leads to the same result as taking the two operations in the opposite order or, equally well, by b)t. This means that performing only one operation (a each operation is in a class of its own (in the group theoretical meaning of "class") and so each irreducible representation of the translational erour, is sinelv deeenerate and onedimensional. (A note of'caut.ion: o &d b a r e integers here, while the crvstallonraphers' established use of these letters are as real lattice v%xs a and b.) In real cr.wtals we must consider translations in three directions (not necessarily perpendicular). However, such translations are mutually independent, and they all commute. The final result is indeuendent of the order in which the translation operations a;e applied. The translational suherouu of a real crvstal is therefore the direct oroduct of ~" three one-dimensional translational groups (and is Abelian). With the h c b o f a someclassical trieonometrv we will now he able to obtain a useful set of characters fuklling all the requirements we need to put on them.
this article. We will therefore not be concerned with intensities, which otherwise is a subject dear to the crystallogranher. ~I t is usual to define the lattice bvreference to three linearly independent primitive translation vectorsa,,a?,and al,so rhat a general lattice translation, T, may be written as
+
~
.
and cos 8
+ i sin 0 = eiB
Now, if we let different translations be represented hy different 0 angles and choose the symbol X, for the character representing an operation r and define
.
~
T = mal + map+ naaz where n ~nr, , and nl are inteaera. (Beware of oossihle confusion here: f6llowing convention these integers are denoted al, an, and aa in the Chemical Spectroscopist's Column.) The origin of the lattice is arbitrary hut is conveniently defined by n~ = nz = n3 = 0.The primitive vectorsa~,a%,andas relate a given point to neighboring equivalent points. These vectors may be mutually perpendicular as in the primitive cubic and the orthorhombic cases. Eouallv. - -, he " , someor all mnv -inclined to each other as in monoclinic and triclinic lattices. I t is a desire for mutallv ~ e r ~ e n d i c u aaxes l r that leads the crystallographer to work k i t h nonprimitive unit cells, as in the body-centered and face-centered cubic cases. Such nonprimitive unit cells have the advantage of displaying the full symmetry of the lattice in a much simpler way than do the corresponding primitive unit cells. The endpoints of the set of vectors, T, define an infinite set of points that forms the lattice. I t is common practice to choose the primitive unit cell as that subtended by thevectors al, an,and as. By simple vector algebra the volume of this or any alternative primitive unit cell is given by
.
i.e., a scalar quantity with dimensions of length3 in units of m3 or more conveniently in A3. Dlffractlon Several models dealing with the phenomenon of diffraction exist describine the interaction between the incident -~~~~~~~~~ radiation and the at& in the crystdine materialin slightly different wavs. Braee's model. which nrohahlv is the best known, pictGres difgction as &ecula;reflect;on. I t is as if the atoms create planes of parallel mirrors in the crystal, wbich reflect an incident X-ray beam if a certain condition between the angle of incidence, 0, wavelength, A, and distance, d, between these "mirrors" is met (Bragg's law) sin 0 = nAl(2d)
(The dot sign means that the operation at follows upon operation bt, and i t does not mean the scalar product of two vectors.) For the corresponding characters, el8" and ei8*,
The experimentalist observes the angle 0, which is related to the inverse of the distance between the "mirrors", l l d . I t is this inverse that leads the crystallographer into the realm of the reciprocal lattice, wbich becomes part of what is physically observed. Every lattice has a corresponding reciprocal lattice, which may be constructed in the following way. In the pattern of dots illustratine a lattice. is i t easv to discern manv different sets of equally"separated paralli planes through the dots. Choose one lattice point as the oriein. Erect from that ooint a perpendicular tobne of the s e t s 2 planes. The numger of lanes that intersect this perpendicular per unit length, per 1s . the significant quantity. Represent the perpendicular as a vector with a length proportional to this number, and mark the end of i t with a dot. Repeat this time and time again for each possible set of planes. You then arrive a t anew set of dots. No two dots will coincide, because each set of planes of the original lattice was chosen to be different from d l the other sets. This newset of dots forms another latticethe reciprocal lattice. Distances in this are proportional to
(Continued on page 1025, col. 1)
(Continued on page 1025, col. 2)
x, = cos 0 + i sin 0 and
x:
= cos 8 - i
sin 0
where x: represents the complex conjugate, we find that
In this way our characters X, correctly satisfy a requirement of all Abelian groups. We must also demonstrate that these characters combine ("multiply") isomorphically to the elements of the translational group. If we let two translations operate on a point x , we may write
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Journal of Chemical Educatiol~
x,
The Spectroscopist's Column
The Crystallographer's Column
where 8, and 06 represent the operations a t and bt, we have eiRo, ei8h
= eilRn+bh)
so they do indeed multiply correctly. Since 0 can assume an infinite number of different values, we are ahle to work with infinite or very large finite Ahelian groups like the translational group we are interested in. We may conclude that the characters of the irr.reps. of the translational group are of the form eiU.But such a character must depend not only on the translation in the real lattice, at, hut also on the particular representation under consideration. Clearly, this double dependence must appear in some explicit way. Furthermore, the exponent must be a dimensionless pure number. But a translation such as at is a vector and not a scalar and has dimensions of length. However, ifwe form the scalar product of at with another vecrur nf dimensions length-', i.e., a vector in reciprocal space, we obtain a dimensionless scalar. If we denote this reci~rucallattice vector k (which is the usual choice in solid state physics) we obtain 0 as a scalar
the numher of parallel planes per A of the original lattice. If the whole procedure is repeated starting from the reciprocal lattice. the orieinal lattice will he reconstructed. The relation bdtween tge real and the reciprocal lattice is truly reciprocal-and both lattices are "real". The reciprocal lattice has many properties paralleling those of the real lattice. Thus, i t may he generated from three primitive vectors b ~b2, , and bs. A general translation, G, within the reciprocal lattice will be of the form G = hb,
+ kb2 + lb3
where h, k, and 1 are integers. In the construction described above reciprocal lattice vectors were constructed perpendicular to the corresponding planes of the real lattice. Therefore the primitive reciprocal lattice vector bl is perpendicular to the plane defined by the two real lattice vectors a 2and as. T o obtain a relation, correct in magnitude and dimension, between the realand the reciprocal lattice, bl is defined as bl = (as X adlV
and we may write
where V, as above, is the volume of the real unit cell, so that x&t)
= eik-"'
-
which takes the value of 1when k - a t = 0 (or modulo 2a). I t is convenient to measure # in radians and to incorporate the consequent 2a factor in a formal definition of k 2rIt. Recalling that 0 varies with both translation and irr.rep, we see that the translation is contained in the factor a t and that the reciprocal space vector k labels the irr.rep. For the group of any finite number of translations there will he the same finite number of irr.reps. (character tables of finite groups are always square). Rather than using k itself we can label individual irr.reps. by an index, m, which runs O,1, . . .,N 1 ( N i s equivalent to 0, the identity operation). We may now write k,
and Equivalent definitions hold for bp and bs. In concise form these relations become
,
a.. . , b = 6.,,,
i~
= 1,2,3
where 6ijis the "Kronecker delta", which takes thevalue 1if i = j and 0 if i # j. This means that the dot (scalar) product of any two translations in the real and its reciprocal lattice simply becomes
= 2r(m/N)/t
where it proves convenient to include the N in the central factor so that (mlW is always less than unity. For the character x d a t ) we then have
Because the real crystal is three-dimensional, the translations in real and reciprocal space must he written as
i.e., an integer 0.1, 2. . .. Let us bring all this together and see how i t relates to a diffraction pattern. In a crystal each infinite set of lattice planes, specified by nl, nz, and n3 (with the same meaning as above), is associated with a single reciprocal lattice vector, specified by h, k, and 1 (the so-called Miller indices). The magnitude of a general reciprocal lattice vector, Ghhi, is inversely proportional to the separation between adjacent planes in the real lattice,
and so that
(Note: all crossterms of the type kl .apt*are zero.) We have now accomplished the goal and found "well-behaved" characters for the irr.reps. of the translational group. We have seen that the vector k may he regarded as either the label of an irr.rep. or with equal validity as a vector in reciprocal space. Let us explore the connection between the two in more detail. Just as for any crystal there is aunique lattice in real space corresponding to the t's, so in reciprocal space there is a unique lattice corresponding to the k's. Obviously the k's span a three-dimensional space. We can define a unit cell in (Continued on page 1026, col. I)
(Note that we here employ, as is common practice, the labels hhl instead of their real lattice counterparts for l/f,,l,,,2,,,3.) According to Bragg's law we need information on fl and X to determine lld. This can conveniently be done through vectors in reciprocal space. The magnitude of the wave vector of the incoming radiation is ikl = 11X. k is associated with the angle, 8, of the incident radiation. In spectroscopy this wave vector is generally multiplied by 2n, so that lkl = 2rIX. Reciprocal space is called "h space" or "phase space" by spectroscopists. The common denominator, literally, among these variantsof reciprocalspace is that they express quantities per unit of length. Let us return the Bragg's law, taking n = 1for simplicity sin B = Al(2d) = (1/2d)l(llh)
(Continuedon page 1026, col. 2) Volume 67
Number 12
December 1990
1025
The Cryslallographer's Column
The Spectroscopist's Column
reciprocal space that is akin to a unit cell in real space. However, a large period in real space will correspond to a small period in reciprocal space. There are some further interesting interchanges. If the unit cell of the real lattice is body-centered cubic, then that of the reciprocal lattice is face-centered cubic andviceversa. Just as in real space there is a manifold of possible choices of unit cell, so too in reciprocal space. However, in reciprocal space one always chooses a primitive unit cell explicitly showing the full rotational symmetry of the lattice. In real space this is the W-S unit cell. The W-S unit cell of a reci~rocallattice is called the Brillouin zone-strictly speaking the first Brillouin zone, hut the second and other Brillouin zones only become of relevance when one consider a basis (4). T o obtain a W-S unit cell in real space, we draw lines from the lattice point chosen as the origin t o equivalent points in space and then erect pemendicular bisecting planes. In reciprocal space the repiat disulnce is measured in uniu of 2rl h. The corresponding construction to ohtaio a W-S cell in reciprocal space then involves perpendicular bisecting planes at -=It and rlt. Repeated in three dimensions this construction gives rise to the (first) Brillouin zone, which may have more than three pairs of parallel boundary faces. In our earlier discussion k,... ran from 0 ti ouch -~~ - 2 r l ( N - I)/ N]t.Toaccommodateourchange toa W-S unit cell, wemust allow k, tosoan theeauivalent set of \,slues - r l ( N - 1)lNIl t through r [ i ~ 1 ) l ~ l tI .t is usual to think of k, running from -*It to rlt with only one of the two limits included in the Brillouin zone (because they are 2r/t apart), or -r/t 5 k, < rlt. Because N typically is of the order of 10'jin each dimension (yielding a total of 10'8 translations), these k, then also define another lattice in k space of the same symmetry as the reciprocal lattice bat with a much, much finer "mesh", a milion times smaller than the one defined by the k vectors. In three dimensions each vector will need three indices to be completely specified (corresponding to kl, kz, and k d . Each k, vector effectivelv eives the coordinates of a point within t h e ~ r i l l o u i nzone, each point corresponding to a unique irr.rep. We may picture the Brillouin zone as being packed with fine grains of sand. Each grain represents a unique k, and so a unique irr.rep. of the translational suhgroup of the space group. The Brillouin zone associated in this way with the translational group is akin to the character table of a point group in that it contains all the irr.reps. In order to find all irr.reps. for the whole space group, i t suffices to find these irr.repsof the translational subgroup, because, in large measure, the only additional data needed are the irr.rens. of the 32 ooint erouos associated with the 230 space groups. However, such a derivation, which is fairly straightforward for the symmorphic space groups hut gets quite cumbersome for the nonsymmorphic ones, would be beyond the scope of the present article. Because of the "infinite" wavelengths used in practical spectroscopic work, most first-order spectra relate to a wave vector k = 0, i.e., a t the center of the Brillouin zone. In order t o explore anything other than this center we need much shortbr wavelengths. If we were able to study a spectroscopic observable as we continually varied the wavelength of the incident radiation from "infinitv" ( k~= 0) throueh to one. ~ ~ ~ ~ ~ corresponding to an edge of the ~ r i l l o u i nzone, wewould be followine the dis~ersionof that s ~ e c t r o s c o ~variahle. ic In thG sectio; we have introduced the >eciprocal lattice and the Brillouin zone. We have seen how, spectroscopically, we may move from the center to the edge of this zone. T o go outside the first Brillouin zone we turn to the crystallographer. Those who havenot yet read "The Chemical Crystalloera~her'sColumn" printed in parallel with this should now 20 so before proceeding to the final section, ~
~
-
&
1026
Journal of Chemical Education
- .
.
CRYSTAL
X-RAY INCIDENT BEAM
\ ORIGINOF I
RECIPROCAL L&TICE
Fiaure 2. The Ewald construction in recl~rocalsmce of the diffractioncond" lions shown in two dimensions. (For lhe sake of clarity two reciprocal lattice , triangle have been omitted.) poinn lying within the kk,G Using the expression given above for l l d and 1/X, Bragg's law in reciprocal space hecomes sin 8 = (IG1/2)/k We will now use Fieure 2 to demonstrate that this form of " Rragg's law is equivalent to the following geometrical construction accordine to the German crvstalloera~herPaul P. Ewald. (1) Draw a three-dimensional reciprocal lattice dot pattern to represent the crystal. (2) Draw a vector corresponding to the incident wave vector k; = llh, so that it terminates on any reciprocal lattice point, which is taken to be the origin. (3) Draw a sphere of radius lkil going through this origin. This sphere is called the reflection or Ewald sphere. Because diffraction is an elastic event leaving the X-ray (or other) wavelength unchanged, the magnitude of the wave vector of the diffracted beam, 1141is equal to lkil. (4) It follows that k,may be represented by a vector originatingat the same place as ki (that actually represents the center of the ervstal) and terminatine- on the surface if the Ewald sohere. ( 5 ) Suppoae that k, happens to terminare at a rrciproral larticr point: hy cunstructiun k. ducu. Then k. - k. = G,a rrciprocnl lartice vector. In such s rase we ree from Figure 2 that
and the Bragg rundition for diffraction is fulfilledrxactly. Now, in a real d,fframun experiment the crystal is normally rotnted while r h ~ incident beam has a btationary dirretion. This is equivnlent r u the reciprocal lattice dot pattern rotating nround i t s norram. ( 6 , During such a n~tarnmthe Euald sphere will rweepa spherical volume of radius 2/h of the recipmcal lattice. This larger rphere is called the limiting sphere; one cannot without changing to a shorter A "see" anything of the reciprocal lattice beyond this limit. (7) . . Whenever the surfaceof the Ewald sohere durine the rotation coincides with a reciprocal lnrtice point, the eonditrons I'm diffraction ere fulfilledand an obscwation i~ posaihle. The dot pattern obtained by the endpoints of the reciprocal lattice translation vectors G is isomorphic with the dots ~hwicallvobserved in a diffraction experiment, be it of k~eitronsiX-rays or neutrons. Figure 3 shows a two-dimensional projection of such a dot pattern recorded in an electron diffraction experiment by Lundberg and Sundberg (6). We here see a genuine physical picture of the reciprocal lattice. Reciorocal soace is not some fantasv construction or "unreal" th;ng! Summarizing, i t is obvious that the reciprocal lattice is
The Crystallographer's Column
hand, is the exploration of just these other lattice points. The reciprocal lattice is a common meeting ground for spectroscopy and crystallography. But i t is not difficult to see why there isso little congruency between the usualpresentation of reciprocal space in these two areas of chemistry. Let us take a final detailed look a t this one more.
Edge of the , ~ ~ n l l o u mne i n
k=O /
'.
inelastic Neutron
Scattering
i
Figure. 4. A twc-dlrnensional reciprocal lattice showing the Brillouin zone as an IntBrtace between spectrosmpy and crystallography. Figure 3. Electron diffraction panern of monoclinic KNbtsOsrprojected along the crystallographic b axis.
central for the understanding of diffraction by crystals and the corresponding theory. There are many excellent accounts of the use of the reciprocal lattice in diffraction theory. We particularly recommend books by Buerger (7) and Vainstain (8). Those who have not yet read "The Chemical Spectroscooint'sColumnn printed in parallel with thisshould now doao before proceeding to the final section.
Finale In the Chemical Spectroscopist's section we met reciprocal space as a vehicle for the enumeration and labelinp. of the irreducible representations of the translational subg;oup of the space group. The Brillouin zone "contains" these irr. reps., and that is why we drew a parallel with a character table. Once a complete listing of the irr.reps. has been ohtained, there is no need to go farther out in reciprocal space, t o go outside the first Brillouin zone. The number of k, vectors within the Brillouin zone is identical to the number of unit cells in the real crystal; avery large number, typically 1018 (we ignore edge effects). In addition to its relevance to spectroscopy, the Brillouin zone is of ereat im~ortaucein manv other a s ~ e c t sof the theories of iolids su& as lattice dynamics, electrbnic properties, and Fermi surface, to name a few. The symmetry of anv delocalized wave function in a crystalline material is dicta;ed by the symmetry of the Brillouin zone. In the Chemical Crystallographer's section we met reciprocal space as, effectively, an observable detailing the outcome of the diffraction conditions. We were only concerned with integral reciprocal lattice vectors; there was nothing of importance between the lattice points. We see a connection between the spectroscopist and the crystallographer in the definition of the Wigner-Seitz unit cell of the reciprocal lattice--the Brillouin zone. As we have seen, i t comprises all points in reciprocal space that are closer t o the reciprocal lattice point at the center of the zone than to any other reciprocal lattice point. Spectroscopy is thus confined to a volume of reciprocal space that involves no other reciprocal lattice point than the origin. Crystallography, on the other
Figure 4 shows a two-dimensional reciprocal lattice of the t m e we met in the Chemical Cwstalloera~her'sColumn and s i b pictured in Figure 3. ~ r o u n dthe o;igin, which corresponds to the k = 0 condition of spectroscopy, we have drawn a Brillouin zone, constructed as in Figure 1.A typical X-ray wavelength is 1.5 A, and, as we have seen, the diffraction conditions are satisfied a t points i n k space outside the first Brillouin zone. Now, the wavelengths t ically used in Raman and infrared spectroscopy, -5,000 r a n d 50,000 A, respectively, are effectively infinite as far as primitive translation vectors are concerned (typically 3-10 A). The corresponding k values are of the order of to A-l, or essentiallv k = 0. The same areument holds for other forms of spectroscopy; wavelengths appropriate to EI'R are of the order of centimeters and to NMR of meters. S ~ e c t r o s c o ~ i s t s do study the center of the Brillouin zone! vibrational spect r o s c o ~ in r the form of inelastic neutron scatterine is oarticularlyi&resting, for the effective wavelength of the neutrons can be gradually shortened from "infinity" (k = 0) a t the center of the Brillouin zone until the edge of the zone is reached, where ikl = d a . In this way dispersion curves are obtained. We see how spectroscopists and crystallographers share a common interest ink soace but are usine different Darts of it. The spectroscopist works from k = 0 p; to the eige of the Brillouin zone. The crystallographer utilizes reciprocal lattice points outside the first Brillouin zone (and the more points the better). TheBrillouin zone is indeed a n interface between spectroscopy and crystallography! Acknowledgment We are indebted to Monica Lundberg and Margareta Sundberg, Arrhenius Laboratory, Inorganic Chemistry, University of Stockholm, for allowing us to reproduce the electron diffraction pattern of KNh130~3shown in Figure 3. LJN wishes t o acknowledge the receipt of a travel grant from the British Council and the University of Stockholm for a study leave. Furlher Reading Manv of the i m ~ o r t a naoints t made in the Chemical S ~ e c troscopks' column have been thoroughly developed by bther authors and we refer to authoritative treatments by Koster (9), Streitwolf (lo), and Franzen (11). ~pplicatibnsof reciprocal space and Brillouin zone theory to solid state Volume 67 Number 12 December 1990
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have been made in many excellent and books, and we refer LO those hy Kittel (12L Ziman (13L and Blakemore (1.1). Further reading in crystallographyis con.. . . .. tained in refs 5 and 8. 1. Gerstein, B. S. J . Chrm Educ. 1913,50,316322. 2. Broek, C. P.; Lingsfelfer, E. C. J. Cham Edue. 198O.57.552554. 3. K O ~ ~ ~F.Sport ~ , G . ~ ~ ~ ~ ~ ~ ~ ~ d T h ~ i r R eNSW p ~r0 ~~ k ~, 1 ~9 6~7 :t~ 22. 4. Durmsn, R.: Jeyssaonya. U. A,; Kettle, S. F. A,; Mahasuvershai, ti^^^. R; Norrby, L.J . J . Chem P h y r 1984.81,5247-5251.
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Literature Cited
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,' H,~wn.c.hl J C m r s m b . r d r > C r \ s i o l l n p r o p k ~ . M . . C r s x Hall I r a Y.>rk IE , 1 1 1 2 B Lu?dlvre. \I ..Sundherg. \I .I S w o Q o e r L'hrm 1986.d~.Ilci-.ll M .I . ; I m w r B e t l h 1381, Vo. I P 2,5 .r....K...o a r r
