Jvrg Waser
Gates and Crellin Laboratories of Chemistry' California institute of Technology Pasadena, 91 109
Pictorial Representation of the Fourier Method of X-Ray Crystallography
T h e impact of X-ray crvstallography on chemistry in the last few decades has been great,, and some iusight into the nature of this method is therefore of value to teachers and students alike. Indeed, in many texts of elementary chemistry (1) the Bragg equat,ion of X-ray diffraction (see Appendix) is developed, but litt,le or no indication is given of thc far more central position of the Fourier method in current cryst,allographic research. The reason is undoubtedly thc relative complexit,y of the mathematics involved. It is, however, possible to gain an understanding of the Fourier method mit,h the aid of diagrams such as can he generated by present-day computers. This coritention is developed in the main body of the present paper. A summary of the pertinent mathematics is relegated to an appendix, because the insight into the Fourier met,hod developed earlier by graphical methods does not require an understanding of the mat,hematical details. As a hist,orical sidelight, the man who first recognized (in 191,5) the connection between Fourier theory and X-ray diffraction and appreciated its importance was Sir William Henry Bragg, the father of the Sir William 1,awrence Bragg who derived (in 1912) as a graduate student the equat,ion bearing his name. It took until 1929 until experimental procedures and theoretical understanding had been perfect,ed sufficiently to apply present-day Fourier techniques, as they were initiated in that year by W. L. Bragg. Periodic Functions and the Fourier Theorem
Loosely stated, periodic functions are functions that repeat themselves as shown, for example, in Figure 1.
Figure 2. A rinuraidal wove. The wave is choracterired by the period Iolm called the wovelength), the amplitude, and the lateral displacement known or the phase a n d defined here or the distance between ordinate axis and the first peak of the wove.
waves must repent an integral number of t,imes, such as once, twice, threc times, et,c., including zero times. This last component in t,hc superposition is a wave in an extended sense only; it is a const,ant t,hat represents the average of the funct,ion. The way in which each wave cont,ributes t o the total function is determined by two quantities, onc of which is the amplitude, which describes the magnitude of the wave, and the other the lateral displacement known as the phase (Fig. 2). That is, if each wave contributes with the proper amplitude and phase, the sum total is t,he original function. This is what the Fourier theorem says. An example is shown in Figure 3. The formalism by which appropriat,c phases and amplitudes of the different waves may be determined from a knowledge of the original function is well known, and is described in many standard texts of calculus (2). But this is of no concern here, because t,he sit,uation is the other way around. The function we are interested in is unlinowri a t the beginning, but it may be constructed by adding together its component waves. Here is where X-ray diffraction enters, because it provides information about these component waves. T o see how this comes about, we begin by considering an essential property of crystals. The Periodic Nature of Crystals
Figure I . A periodic function. of t h e function.
Note the r e p e a t dirtonce Ithe period)
Fourier's theorem states that any (reasonable) periodic function can be reproduced by superimposing (or adding to one another) a suficient number of sinusoidal waves (Fig. 2), of a repent pattern that is identicnl with that of the funct,ion they are t,o represent,. That is, in the interval in which t,he original function repeats, also Imomn as the period of t,he function, the sinusoidal Contribution No. 3000.
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An important aspcct of crystals is that they are built on a pattern of atoms, molecules, or ions that is repeated periodically in three dimensions. The pattern is contained in a parallelepiped called the unit cell (5). For cubic crystals (example, NaC1) the unit cell is a cube (for which all edges are of equal length and enclose angles of 90°),but for crystals of lower symmetry than cubic the edge lengths may he unequal and the angles enclosed by the edges may be oblique. The crystal may be conceived of as a three-dimensional stack of unit cells, packed so that faces of adjoining cells match. The unit cell is representative of the structure of the crystal, and its periodic, three-dimensional repetition may he thought of as generating the crystal.
,
,
,
,.
period
n =O
0=3 i
A w
w
A y
A V
A V
A V
A w w
Figure 4. Model d o one-dimenrionol crystal. The crystal con.i& of the periodic repetition of o diatomic molecule (such os O?), 0%shown on the upper right. On the left ore the waver, the superposition of which approximates the electron density in the "crystal." The curves an the right show, in wccerrion, the wperporition of the top two, three, four, and Rve waver on the left. Considering the curves on the right for their information content, the top two curves are seen to indicate, in an over-all fashion, the presence of the molecules, the second curve showing the molecular outliner quite distinctly. In the bottom two curves the earlier single maximum i s resolved into two peak. that represent the two otomr. iUruolly no conrtont term ir added in representotions of this kind, because the important features of the R n d curves ore the existence of well resolved peokr above o general background of wiggler. These features would not b e changed b y adding a constant, which would only roise the entire r ~ w e . 1 Figure 3. Superpo3ition of waves t o reprerent a periodic function. Addition of the wove$ yields the function of Figure 1. The respective phores ore olro shown.
The atoms, molecules, or ions in a crystal contain electrons, and the spatial distribution of these electrons is described by what is called the electron density in the crystal. The electron density follows the repeat pattern just described, and has peak values a t places a t which atoms are located. It is periodic in three dimensions, and the unit-cell edges are the repeats, the periods, in the three different directions. The Fourier theorem applies to the electron density, being a periodic function, and the way this works shall first he described by considering onc-dimensional models.
ures 7-10. Again, the rcsolution is seen to improve as more waves of increasing frequency arc added. A point to remember is that the component waves of the electron density are all standing still and do not show any other motion. That is, they do not behave like waves on a piano st,ring, for which there is constant oscillation, or like waves in water, which travel. Real crystals are three-dimensional, and to represent their elect,ron densities requires waves that extend in many directions through the crystal and are not limited to a plane as in the two-dimensional example just discussed. This extension to three dimensions makes pictorial portrayal more difficult. One way to handle
Two Models of One-Dimensional 'Trystals"
The representation of the electron density in two one-dimensional "crystals" by the superposition of waves is shown in Figures 4 and 5. Portions of the "crystals" consisting of a periodic sequence of linear diatomic or triatomic molecules are shown in the upper right of the figures. The individual waves ( d h proper amplitudes and phases) are shown on the left, and the results of various superpositions on the right, as described in the captions. Note t,hat the representation is improved as waves of higher and higher frequency (shorter and shorter wavelength) make their contribution-there is higher ~esolutioninto t,he individual atomic electron densities. Two-Dimensional and Three-Dimensional Crystals
Consider next the model of a two-dimensional "crystal" containing in each unit cell one linear AB2 molecule (Fig. 6). The reprcscntation of its electron density by the superposition of waves is shovn in Fig-
Figure 5. A second one-dimenrionol crystal. The " c r y d d ' now consists of the periodic repetition of o lineor triotomic molecule, ar shown on the upper right. The curves ore ondogour to those in Figure 4. Note that it i s the same wave. that ore wperimpared in the two Rgurer. Only the amplitudes and the phases are different.
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Adding the two waver rhown on Figure 8. The two waves next in ,ire. the left improves the definition of the molecular outlines, but no strustural detoilr of the molecvler ore or yet visible. The reason is thot up to now all wore. of higher frequency thon the lowest hove been oriented more or less along the mountain ranges.
Figure 6. Model of the two-dimensional cryttol. Four unit cells of a "cryslolll ore rhown, each containing a linear, triotomic molecvle A h . The next four figure. rhow the representation of the electron denrity of this "crystar' b y the wperporition of wover.
Figure 7.
The four largest waver
The four wave.
with the largest
amplitudes required to represent the electron density of the "cryrtol'. of Figure 6 are shown on the left and their rum on the right. The resulting mountain ranges begin to rhow the outlines of the moleculer. The pairs (1.11 and on the far left characterize the of integers different waver. The first integer i s equal to the number of wove crests or troughs within the repeat distance along the x direction. The second integer does the some for the y direction.
ll,Ol, (0.11,
Figure 9. Three more wover. The three wove5 next in r i m are oriented tranrverrely to the mountain ranger, and their addition yields three distinct p e o k ~in each range, representing the atoms of the molecule. The bars on top d the ~econdintegers in the pairs of integers ot the for left represent minus signs and indicote thot o wove crest that interrectr the + x axis d m interrectr the - y axis. [It may b e noticed thot o wove crest that intenectr the - x oxis dro intersects the +y axis; for example, the wove 12.31 could also have been lobelled 0 %(2.31. Similarly, in Figures 7 ond 8 the orientation of the wover i s rvch that a given crest simultaneously ruts across either the +x ond the + y oier, or else the - x ond the - y ares; these waves could therefore olro have been labelled b y o pair of negative rother thon positive integers For example, 12.11 and (2.1) indicate the same wave.]
(2,2)
the situation is to pass many closely spaced, parallel planes through the crystal and to describe the electron density on each of these planes by its own set of twodimensional waves. An example will be given later (Fig. 13). The question may be asked, how it is possible to find the amplitudes and phases of the different waves that need to be added to show the electron density in any real crystal. This brings us to X-ray diffraction. X-Ray Diffraction by Crystals
A crystal irradiated by monochromatic X-rays (Xrays of a single wavelength) is capable of producing a great many diffracted beams, also called ~eJlect.ions, that are radiated from the crystal in different directions (Fig. 11). They may be recorded on photographic film, or by newer methods that use Geiger counters, proport,ional counters, or scintillation counters. 448
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Figure 10. Repre3entation by contour mapping. The mountain ranger and peaks of the diagram on the far right of Figure 9 areshown b y contour liner of conrtant altitude, altitude being synonymous with electron denrity here. This i s the common woy in which electron densities ore represented in X-ray dlffroction work, The increment in electron density from one contour line t o the next is cen.tant.
X-ray f
beom
Figure 11. Cryst01 diffraction of monoshromatic X-rays. A narrow beam of monoshromotis X-roys, delimited b y a set of pinholes, impinges on 0 crystal. Many diffracted beoms (or reflection4 may result, each of them depending on on oppropriote orientation of the cry~tal. To produce oll possible reflections the cryrtd must b e rotated .bout different axes. The reflections may b e recorded photographically or by other means. To prevent exposure to the direct X-ray beam o rmali circular portion of the fllm has been removed. The flgure is from C. W. Bunn'r book ( 3 ) (copyright b y Academic Press, New York, 19651.
These reflections and the waves discussed earlier are connected in an intimate way, but this relationship can only he stated here, because its derivation requires a considerable amount of physics. Each X-ray reflection is associated with one of the component waves of the electron density, and the intensity of the reflection is proportional to the square of the amplitude of the wave. X-ray diffraction offers, therefore, an experimental means t o find the amplitudes of the different waves. This relationship is again touched upon in the Appmdix. Two points still need to be discussed to complete the picture. The first is the question of how to associate a given X-ray reflection with the appropriate wave. This does not present a real problem, but an exposition of the details would require the complexities of lattice geometry. This is briefly discussed in the Appendix also. Fortunately, ingenious devices have become available in recent years which produce an arrangement of X-ray reflections on film that makes the assignment straightforward. An example, produced by the precession camera designed by M. J. Buerger (4), is shown in Figure 12. The second point is of major importance and is called the phase prohlem. The Phase Problem
This problem comes about because the intensity of an X-ray reflection is independent of the phase (or the lateral displacement) of the corresponding electron density wave and therefore contains no information about this phase. In the one- and two-dimensional examples given earlier the phase of each wave happens t o he either zero or one-half the wavelength, because of the centrosymmetric arrangement of the atoms. A change of phase of one-half a period has the effect of replacing the wave by its negative. That is, crests are
Figure 12. X-Ray reflections produced with the prcsenion camera. The auxiiiory lints and the integers indicate the w o v e with whish the reflections ore arrocioted. Strong rcfleclionr rvrh ar (0.2). (3.11. 13.i). and (3.31 indicate large amplitudes of the wove components of the electron density labelled b y the same pairs of integers, and rimilarly weak re. flections such 0 % (4.0) and (3,31 indicate small amplitudes. The array of spots is dong lines enclosing oblique angler, become the corresponding directions in the cryrtol are also not a t right angler to each other.- Reflection Ih,k) and have the same intensity and similarly for (h,kl and The picture was taken b y 1. V. A z r r r d (41, with a rryrtol of Foirk Reiditc, a mineral.
replaced by troughs, and troughs by crests. I n the centro~~mmetric case, it is therefore not known whether each wave, taken at zero phase, is t o be added or subtracted. With two possibilities for each wave there are 2 N possibilities for N waves, and this is an astronomical figure for even the relatively simple situation of one-hundred waves. For crystals that are not centrosymmetric there are no restrictions on the phases of the different waves, so that the number of possibilities is greatly increased, making the phase problem even more complicated. (The distinction between the centrosymmetric and non-centrosymmetric case is not obvious, but one aspect should be evident from the preceding discussion. This is that if there is a center of inversion, each component wave of the electron density must either have a crest or a trough a t the center; only in this way is each component itself symmetric relative t o the center, and this it must be, if the sum of the waves is to be centrosymmetric. This implies in turn that each component wave can only be added or subtracted. No such restrictions are present when there are no centers of symmetry.) Many methods of dealing with the phase problem of X-ray crystallography have been devised. None of them is surefire, but their application has been successful in a great many cases. The sophistication of approach that the phase problem requires is part of the attractiveness of research in this field. The summing of the different component waves of the electron density is no major problem, once the phases have been worked out, particularly not in the day of electronic computers. The centers of the atoms in the crystal show up as peaks in the resulting electron density function (Fig. 13). Volume 45, Number 7, July 1968
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insights. The fourth step is t o determine the locations of the atoms, by findiug the peaks in Fourier maps similar t o those of Figures 10 and 13. This provides only a first approximation t o the atomic coordinates, and the final step is a refinement procedure, by which those coordinate values are found which give the best possible fit t o all observed X-ray intensities. This then is the important averlue that has led, within the last few decades, t o a detailed understanding of the structure of many crystals and of many molecules, including that of some proteins, and that holds great promise for further great insights and triumphs in the future. Acknowledgment
The plottiug routine employed illthe preparation of Figures G , 7 , and S was adapted from the FORTRAN IV program "PLOT3D" written by D. L. Nelson, Departmeut of Physics, University of Maryland, College Park, Maryland 20742. I wish t o thank Mr. C. W. Wilson, Jr. for introducing me t o this program, Professor 1,. V. Azaroff for the X-ray film used in Figure 12, Professor Dornthy Hodgkin for the plate used in Figure 13, and Dr. R. E. Marsh for fruitful discussion and criticism. Appendix: Some of the Mathematical Details
Figure 13. The electron density in potosrivm benryl penicillin. The model of which a photograph is rhown in the upper portion of the flgure was obtained in the following manner. Electron density contours were calculated from X-ray data for o set of poroilel planer p o s e d through the cryctal. Contour mops of the results were drown on sheets of clear plortic and the sheets were stocked together, representing a threedimen.ionol electron denrity mop. The lower portion d the figure provides a key to the different otomr visible in this mop. The mop extend, only over a portion of the molecule, and port of o corboxyl group neor the top, o carbon and an oxygen atom near the bottom left, and part d a benzene ring near the bottom right are omitted. Note the partiwiorly large electron concentrotionr due to the potassium ion and the sulfur atom. Hydrogen otomt ore not visible in the present mop, but they con b e located quite well in contour mops bared on present-day high-prebv cision measurements. The flavre ir from reference 161 . . icowriaht . Princeton University Prer., 19491 and is of historical inter&, because the X-ray diffroction rewlts of which it i s the cviminotion were of maim importance in the elucidation of the structure of penicillin. The photograph wor token by Helen Murprott.
.. -
The charaelerisat,ion of X-ray reflections is considered first. The ihree-dimensiod lattice of a crystal may be decomposed iu many different ways into sets of parallel net planes. Each of these is designated by a triplet of integers (h', k', l'), c d e d Miller indices, in which the integem are inversely proportional to the intercepts of the net planes along the z, v, and a axes (Fig. 14). The integers h', k', and I' are always chosen to be as small as possible so that they have no common factor. For example, (4, 2, 2) would not be appropriate Miller indices, hut (2, 1, 1) would. A negative Miller index relates to an intercept on the negative side of an axis. Each set of planes is able to reflect. X-rays with the planes acting a8 if they were mirror planes, but the Bragg equation must be satisfied. This e q ~ a t i o nhas the form
2d sin 0 = nx
(1)
.
I n brief review, it is instructive t o see how the pieces of information discussed' earlier are put together in a typical crystal structure iuvestigation by X-ray diffraction. The Course of a Diffraction Study
Very bricfly, thc dimensious and the geometry of the uuit cell of the crystal under investigation can be dctermincd from mcasurcmcnts of the directions of thc diffracted beams. This is the first step in a typical investigation, and is based on the Bragg equation. All information pcrtaiuing t o thc location of the atoms in the unit ccll is contained iu the intensities of thc different rcflcctioris, as discussed. The second step is thcrcforc a carcfnl dctcrminatim of thesc intcnsitics. Thirdly, the appl.opriatc phascs arc worked out, by o r ~ cmcthod or mother. This is usually the most trouhlcs~~mc, hut als(~thc most interest,ing step, where tharo is corisidcri~hlc scopc for ingenious idcas and 450 / Journol of Chemicol Educofion
Figure 14. Miller indicer. The axis intercepts of the top plane rhown ore a, b, and c. In the present situation o is equal in length to the edge of the unit cube, b = -/2, and c = 013,sothat h8:k':l' = ~ i / a l : l i / b ~ : l l / c l= 1:2:3. The Miller indicer of the set of piones rhown ore therefore 11,2,31. The distance between neighboring planer i s called the spacing d. (The h. been chosen to orientation of the coordinate system i. unuwal and s make the intercepts fuiiy virible.)
of the X-rays; n is x positive integer. The reflection by the planes (h', k', 1') and charncterised by the int,eger n is called the reflection (h, k, I ) , where h = nh', k = nk', and 1 = 4 ' . For example, the reflection from the planes ( 1 , 2, 3 ) satisfying eqn. ( I ) with n = 2 is called the (2, 4, 6 ) reflection. Turning now to the Fourier representation of the electron density, the one-dimensional examples in Figures 4 nrld .i cat, be expressed by the Fourier sums pdx) = -1.16 cos 2 a z
+ 0.38 cos Z r ( 2 z ) + 0.30 cos Z r ( 3 x ) 0.56 ens Z r ( 4 x ) + 0.43 cos 2 r ( 5 z ) ( 2 )
for Fignre 4 and
+
+
pdx) = -1.45 cas 2 r z - 0.41 cos 2 4 2 2 ) 0.26 cos 2 s ( 3 a ) 0.30 cos Z r ( 4 z ) - 0.30 ros Z s ( 5 z ) (3)
for Figure 5. The two-dimensional example (Figs. 6-10) is represented by the two-dimensional series
+
pa(z,y) = [-0.145 cos2ax - 0.145 c o s 2 r y 0.249 cos2a X ( x y ) +0.144 cos 2 4 2 2y)l (-0.100 cos 2 4 s 11) 0.100 cos 2 r ( x Zy)] [-0.090 cos 2r(33- - 2 y ) 0.090 cos 2 r @ x - By) 0.077 cos 2 r ( 2 s - 2 y ) ] (4)
+
+
+
+
+
+
+
The terms have been grouped in brackets to correspond to the waves shown on the left of Figores 7-0. I n all these examples the coefficients have been scaled to produce graphs with reasonable ordinates, so that only their relative values are meaningfnl. The general, three-dimensional case will be farmnlated in a more compact form that uses summation symbols. The Fourier representation of the electron density p(z, I,, z ) in n crystal is
Each term in the triple sum represents a wave, of which F,., is the amplitude and a,,,is the phase. The indices r, s, mid 1 nln sepnrately from - m to m, at l e s t in theory, brlt in practice the coefficients become negligibly small for large values of T, s, or t, so that only a finite number of waves needs to be considered. The connection with thelsttice net-planes and X-ray reflections
discussed earlier is that the intensity of the ~.eflection(h, k, I ) is proportional to the square of thal coefficient in eqn. (3), the subscripts of which are r = h, s = k, and t = 1, or in other words to (Fskr)2. A closer look shows that t,he wave characterized by the triple ( h , k, I ) has crests and troughs that run parallel to the planes with the Miller indices (hi, k', 1') in such a way that there are n crests or troughs hetween neighboring planes, wheren is t,he integer that is eqnal l o t,he ratio hlh' = klk' = / / I , , the integer in the Rragg equation. Hence the wave (h, k, I ) is seen to be involved in an important, direct way in the X-ray reflection of the same index triple (h, k, I). I t is not altogether surprising that the amplitude of the wave and the intensity of the reflection are closely related. Literature Cited ( 1 ) For example, PAULING, L., "College Chemistry," (3rd ed.),
Freeman, San Francisco, 1964. T., A N D JOHN, F., "Introduction to ( 2 ) For esample, COURANT, CRICUIUS and Arialysis," V d . 1, Interscience (division of John Wiley & Sons, Inc.), New York, 1965. , Role in Nature ( 3 ) For example; BUNN,C. W . , " ' ~ r y s t ~ l sTheir and Science," Academic Press, New York, 1965, A N D HOLDEN,A,, A N D SINGER,P., "Crystal and Crystal Growth," Doubleday, New York, 1960. These two psperbacks are recommended for their readabilit,~,attractiveness, and wealth of interesting informstion. The first of them contains an extensive and clear exposition of X-ray diffraction. There is no discussion of X-rav diffraction in the nec-
other experiments. ( 4 ) BUERGER,M . J., "The Precession Method," John Wiley & Sons, Inc., New York, 1965. This is an excellent text. but on an advanced level. ( 5 ) AZAROFF,L. V., Acta Cryst., 10, 413 (1957).
+
1968 MCA Award Recipients The three recipients of the 1968 College Chemistly Teacher Award of the Rlanufacturing Chemists Association are: (left t u right) Dr. John C. Bailar, Jr., Universit,y of Illinois; Dr. Clark E. Bricker, Uuive~.sityof Kansm; and Dr. Leo Sehnbert., American University, Washington, 11. C. The a n m d awards were established bj- X C A in 10.V and have a dual purpose: to recognize and reward teachers in the field of chemistry who have been ontst,anding in their work with rtndergradi~atesand to pwvide public recognition of the importance of good teaching in the field of chemistry. THIS JOI.RX.\I.joins wilh their studenls, friends, and rolleag~esin co~lgratdsling Pmfessora Bsilsr, Brickel., m d Schubert. Volume 45,
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