Symposium on Teaching Crystallography
Two Crystallographic Laboratory and Computational Exercises for Undergraduates Leslie Lessinger Barnard College, New York, NY 10027 In our chemistry curriculum, we rely heavily on laboratory exercises and the readings associated with each experiment to supplement the coverage of X-ray diffraction and crystal chemistry given in lectures. The following two exercises, offered in the advanced laboratorv course taken bv iuniors and seniors, are designed as an i&oduction to thk Iundamental ideas and methods of crvstalloaa~hvand to convev some important features of inorganic and organic crystal structures. In the two afternoons allotted for laboratory work and the two weeks given for supplementary reading, answering assigned questions, calculations, and writing UD results, we try to maximize both the direct, hands-on experimental and computational experience of the student and the knowledee she obtains bv readine additional material closely connGted with the laboratory work she has done. The chief merit of the exercises described below lies not so much in any novelty of the ideas covered as in the careful choice of the two crystals, study of which we believe can provide a complete and coherent introduction to a large number of basic concepts in crystallography.
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The Crystal Structure of NIO
Experimental A common undergraduate physical chemistry experiment comprises recording and interpreting X-ray powder diffraction patterns, usually those of cubic crystals, such as the
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simple NaC1, KF, or b - ~ nor~ the , somewhat more complex Cu20 or CoFe20a (a spinel) (1-5). In our variant of this standard exercise, designed with the objectives described above, we use NiO, an extremely interesting substance, to which an entire chapter is devoted in the excellent book on solids by Moore ( 6 ) . In the laboratory, our students use samples of NiO powder contained in 0.2-mm-internal-diameter capillaries, record the CuKo X-ray diffraction pattern on 35-mm X-ray film that they have cut, punched, and loaded into a 57.3-mm-diameter powder camera, and process the film in an ordinary 35-mm processing tank. (Caution: X-rays are extremely dangerous. Although the students turn the X-ray generator on and off and attach and remove the powder camera themselves, this is done only under the direct supervision of the faculty instructor, who takes care to point out the potential hazards in using the apparatus and to explain the many safety interlocks, lights, and shutter mechanisms of the generator, the use of the Azfiroff safety beam tunnel, and the lead glass beam stop of the powder camera.) The positions of the 10 pairs of lines on the photograph (shown on page 68 of Aziroff and Donohue (4b), who use NiO simply to illustrate indexing the powder pattern of a cubic crystal) are measured accurately on a traveling microscope film reader with a 0.01-mm vernier scale. Finally, the students make at least two measurements of the density of NiO powder, using a 10-mL or 25-mL pycnom-
eter and CCld as thecalibrating andmeasuringliquid. This is a difficult measurement. T o ensure accuracy, sufficient NiO (at least 2 g) must be used; CC14wets NiO much better than does Hz0 and greatly reduces errors caused by air kibbles. T o emphasize that density is a temperature-dependent property, students are instructed to find the density of CCla by applying the formula p(CCI,) = 1.63255 - 1.9110 X 10-~(t' C )
- 0.690 X 10-'(t
DC)2g/mL
We draw attention to the experimental desien of the Straumanis arrangement, in which the film is punched so as to beself-calibratine, thedistance hrtween thecenters of the holes c ~ r r e s ~ o n d i n i20 t o= 180° exactly. The experimental resolution is alu,nss sufficient to show dear nlnl splitting of ~ line 10, and someiimes a partially resolved doublet on line 9, and the student must use the a propriate wavelength X = 1.5405A for C u K q , X = 1.5443 for CuKaz, or the intensity-weighted average X = 1.5418 A, to calculate d for each line. As always, we stress the estimation of experimental uncertainties and their propagation through the calculation of the derived auantities of interest. In this experiment. the distance measirements can be very accurate; giving good values for the cubic cell edge a and volume V. By far the largest uncertainty is in the density. Incidentally, values for the densitv of NiO cited in the literature are almost all erroneous.blder editionsof the CRCHandbook (7) gavep = 7.45 gIcm3, a very poor value; newer editions give p = 6.67 g/ cm3, which is still not very good. We take as reference values thosegiven by Swanson and Tatge (a), p = 6.806 g/cm3, cubic cell edge length a = 4.177 A a t 26 "C.
d:
Calculations and Analysis Following the procedures illustrated clearly by AzHroff (4aL . .,measured distances are converted to aneles 8 usina the experimentally determined distance:angle raiio, ~ r a g g ' law i A = 2dm sin @M(which the students derive) is applied to c a l c u l a t ~ s p a c ihhaL, ~ s and a cubic index chart used to find an approximate value of the single parameter a, which will give integral Miller indices h, k, 1 for each spacing according to the relation
Once the lines are indexed, averaging the accurate values of the cubic cell edge length a calculated for each line gives results usually in the range a = 4.177 f 0.005 A. Solution of the crystal structure then follows closely the logical sequence of analysis set forth hy Buerger (9).Calculation of the unit cell volume V, combined with the measured density p and the formula weight F W for NiO, gives the number of formula units of NiO per unit cell, Z = ~ V N A I F W . Assuming Z integral gives an unequivocal value Z = 4. The indexing procedure shows not only that NiO diffraction can be indexed on a cuhic cell, but also, from the systematic absence of reflections with h, k, 1not all even or all odd, that the cell is face-centered. (The students are asked to explain
Figure 1. (Left)The sphalerite structure (F43m). (Right) The halite structure (F&m).
the general phenomenon of systematically absent reflections.) A general introduction is given to the information on space group symmetry contained in the International Tables (10). and the students are then expected to find those face-centered cubic space groups that contain at least two sets of fourfold equipoints upon which four Ni2+ and four 0 2 - could he positioned (F23, Fm3, F432, F43m, Fm3m). Comparison of all possible positionings of four Ni2+and four 0 2 - on two sets of equipoints shows that these give rise to only two distinct crystal structures, the sphalerite or zinc hlende (6'-ZnS) structure (or its inverse) and the halite or rock salt ( ~ a ~structure,'hoth i ) shown id Figure 1. These two possibilities can be distinguished only by consideration of the intensities of diffraction. (A table of intensities is given in Swanson and Tatge (81.) This can he done with a complete calculation, involving scattering factors dependent on sin #/A, multiplicities, and Lorentz and polarization corrections, or simply by considering the pattern of relative intensities expected for proiection reflections hkO of the two different strictures. ~ u e r g k r(9) gives complete details for both methods, either of which serves to establish that NiO has the halite (NaCl) crystal structure. Crystal Chemlstry The examination of and comparison between the halite structure. in which each atom or ion has sixfold octahedral coordination, and the sphalerite structure, in which each atom or ion has fourfold tetrahedral coordination, leads naturally to a little further discussion of coordination geometries and some simple crystal chemistry (11). Ni(II), in its compounds and complexes with lipand;, most often shows sixfold (ctnhedral coordination, but both fourfold tetrahedral coordination and fourfold square planar coordination are also known. Furthermore, the inadequacy of the radius ratio concept as a general explanatory principle in crystal chemistry has been pointed out by Wells (12). The correct crystal structure of NiO must therefore be determined experimentally, as outlined above. Some simple AB crystal structure tvnes that it mieht be instructive to consider here ,. " are the CsCl structure, a primitive cuhic arrangement, in which each atom or ion has eightfold cubiccoordination; the hexagonal NiAs structure, in which each Ni is coordinated octahedrally hy six As, while each As is surrounded by six Ni at the corners of a trigonal prism; and the wurtritestructure, a hexagonal pnlymorph of ZnS, in which each atom or ion is fourfold tetrahedrally coordinated, just as in the sphalerite (zinc blende) structure. Nonstoichiometry First amone the s ~ e c i acharacteristics l of NiO that make it particularly suitable as a subject of investigation in this type of experiment is the fact that it can be either stoichiometric (a daltonide) or noustoichiometric (a berthollide), depending on the method of preparation (6). Green NiO, formula Nil ooOl m,iseasily soluble in dilute acid and isa good insulator. Black NiO, average composition Niog80100, is insoluble even in concentrated acid and is a p-type semiconductor. The nonstoichiometrv is due not to an excess of interstitial oxygen hut rather to "acancies at sites normally occupied by Ni2+. Overall electroneutrality is achieved by having two Ni3+ ions in the crystal for every vacant Ni2+site, as shown in Figure 2. The interesting electrical properties to which this nonstoichiometry gives rise are more fully discussed by Moore ( 6 ) . Since the comnosition of the comoound can make such a significant difference in its properties, we ask the students to assess if thev can determine whether their sample of NiO is stoichiometiic or nonstoichiometric. ~ s s u m i =n 4~exact~ lv. ". thev. calculate the formula weight - of the NiO in their sample, and estimate its uncertainty, using their experimental data. Error analvsis thus becomes something more than a routine exercise, b"t rather is intimately hound up with the
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48 1
possibility of answering a significant scientific question, whether the law of definite proportions holds or is sometimes violated (which is possible only in solids). Reasonable estimates of the detectable limit of nonstoichiometry in NiO, which is governed by the relatively large uncertainty in the density measurement, show that it would have to be far in excess of that actually found in blackNiO to be observable using our experimental methods and apparatus. We expect the students to reach this conclusion in no uncertain terms. Polymorphism
Second, NiO a t room temperature is actually rhombohedral, not cubic. Rooksby (13,using the very high resolution obtainable with a 19-cm-diameter powder camera, discovered that many of the lines were doubled or in some instances tripled, in addition to the ala2 splitting. He indexed all the lines on a rhombohedral unit cell with edge length up. = 2.9520 A and angle a = 60°4.2' a t 20 OC. The relation between the two cells can be understood by considering Figure 3, which shows the same set of lattice points described by a face-centered cubic cell of edge length ac and by a primitive rhombohedral unit cell of edge length aR = ac/2112 and rhombohedral angle a = 60' exactly. (Students are asked to prove these geometrical relationships.) Extension
or compression along one of the four threefold axes of a cube makes that axis unique and reduces the symmetry from cuhic to rhombohedral. NiO is only very slightly distorted from cubic, so the usual apparatus has insufficient resolution to observe the phenomenon. If the crystal is cooled, the distortion increases. If the crystal is heated, the distortion decreases smoothly, and above 250 O C the angle a = 60' and NiO has the strictly cubic halite crystal structure. The need to include such solid-solid transformations on the phase diagrams of substances should be pointed out, and other well-known examples of substances exhibiting such transformations, such as sulfur, carbon, and water, noted. The relatively common occurrence of polymorphic arrangements of atoms in the solid state can be pointed out, and conveniently illustrated further by showing the relationships and difference between overall atomic dispositions in the two homoeotectic ZnS polymorphs, sphalerite (cubic) and wurtzite (hexagonal), in both of which each atom is fourfold tetrahedrally coordinated, but which differ in layer stacking. (Note that either the Zn or the S atoms in these structures, considered separately, have in sphalerite the face-centered cubic (cubic close-packed) arrangement and in wurtzite the hexagonal close-packed arrangement, another pair of homoeotectic structures, each with 12-coordinate close packing, but again differing in layer stacking.)
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Maanetic Prooerties
The polymorphism of NiO is intimately connected with its maanetic ~ r o ~ e r t i eThe s . [Ar13d8electronic configuration of ~ i 2 ; means each ion has two-unpaired spins. At room temperature the spins in NiO, coupling hy superexchange through the p electrons of the 0 2 - ions, form an ordered antiferromagnetic array, with layers of Ni2+ ions having net spin magnetic moments all in the same direction alternating with layers of Ni2+ ions having net spin magnetic moments all in the opposite direction, as shown in Figure 4. Thus there is one unique threefold axis, and the structure is rhombohedral. The quasi-cubic magnetic unit cell has an edge length just twice that of the chemical unit cell. The solid state rhombohedral-cubic transition temperature of 250 O C is correspondingly the Nee1 temperature-, above which antiferromaanetic ordering is lost and Xi0 displays ordinary paramagnetism. ~ x ~ e r i m e n tmethods al for measuring magnetic
A Figure 2. Schematicdepictionof balanced vacanciesand Ni3+ ions in nonstoichiornetric NiO.
MAGNETIC UNlT C E L L
Figure 3. Facecemered cubic unit cell and prlrnitlve rhornbohedral unit cell.
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Journal of Chemical Education
UNlT CELL
Figure 4. Antiferromagnetic supersmcture of NiO. Only Ni2+ ions are shown. Arrows indicate net spin orientation.
susceptibility x and the characteristically different temperature dependences of x for antiferromagnetic and paramagnetic substances can be described here. This is also a useful point at which to introduce a general discussion of diffraction by neutrons, which are scattered by nuclei and also interact with unpaired electron spins, by contrast with X-rays, which are sensitive only to the electron density distribution. Uses of NiO Finally, NiO is an industrially significant material, used, for example, in the ceramics industry for making frits and glazes and in the manufacture of ferrites for electronics applications. I t is also a convenient starting reagent for producing the wide variety of nickel compounds used in the plating, chemical, and catalyst industries, since NiO dissolves readily in mineral acids, without the evolution of hydrogen, which would occur if metallic Ni were used. The Crystal Structure of &Furnark Acid,
HOOCCH=CHCOOH This second exercise is concerned with illustrating the application of crystallography to organic molecular structures, and its primary emphasis is on the variety of mathe-. matical relations and calculations that are applied to obtain chemical information from diffraction data. Rather than have students work on complex structures using ready-tohand, sophisticated crystallographic program packages as "black boxes", we prefer to choose a very small structure, which displaysmost of the features we wish to illustrate, and to have the necessary calculations performed by programs the students write themselves, offering what we believe to be excellent and interesting practical exercises in computer programming. Crystallization Methods and Polymorphism Slow evaporation of aqueous alcoholic solutions of fnmaric acid yields crystals of the a polymorph (often twinned), which is monoclinic space group P21/c, a = 7.619 A, b = 15.014 A, c = 6.686 6 = 112.0°, Z = 6. This structure was solved by Brown (14). Sublimation of fumaric acid, on the other hand, results in the polymorph which is triclinic, space group Pi, a = 5.264 b = 7.618 c = 4.487 A, a = 106.85", 3! = 86.33O, y = 134.94', Z = 1. This structure was solved by Bednowitz and Post (15). The identity (within experimental uncertainty) of the monoclinic a-axis length and the triclinic b-axis length results from a corresponding structural feature in the two polymorphs, the linking togetber of molecules of fumaric acid to form chains by a double hydrogen bonding system between adjacent carboxylic acid groups, which is discussed further below.
A,
$
A,
CrystallographicSymmetfy and Molecular Symmetry Consideration of these two polymorphs gives the students further practice in use of the International Tables (10) and familiarity with two of the lower symmetry space groups commonly encountered among molecular crystal structures. Students are asked to demonstrate that in the triclinic polymorph the molecule must lie on a crystallographic center of symmetry (thus requiring that the locations of only two C, two 0, and two H atoms be determined), while in the monoclinic polymorph the six molecules per unit cell compose a set of four lying on the fourfold general position plus a set of two lying on centers of symmetry. The distinction between molecular symmetry (a center of inversion, planarity, and so on), which may be exhibited in a crystal but which has to be checked to see whether it holds t o within experimental uncertainty, and exact, required crystallographic symmetry, usually lower than the maximum possible molecular symmetry to allow for efficient packing, is noted. An additional exercise is to explain why in crystals of malenic acid, HOOCCHzCOOH, which have the space group PI (16),the
molecules cannot lie on centers of inversion, and therefore the unit cell must contain two molecules. Solving the Structure Althoueh. because of time limitations. we discuss onlv briefly t h l drocedures for solving crystal structures, actuall; solvine the structure of one or both nolvmorohs of fumaric acid w&d be an excellent exercise ;or advanced undereraduate research course or for eraduate trainine in crvstalyographic methods. Methods forspace group deGrminkion, includine information from svstematicallv absent reflections and the use of intensity statistics, couid be illustrated. The fumaric acid structures lend themselves well to the application of a variety of solution techniques: (1) packing considerations, including hydrogen bonding; (2) deconvolution of the Patterson function; (3) the use of rotation and translation functions with a rigid search model; (4) direct phase determination, employing either symbolic addition or multisolution tangent formula methods. Fourier Synthesis Focussing their attention now only on the triclinic polymorph, using the structure factor amplitudes (Fhkd observed by and the phases (signs) calculated by Bednowitz and Post (15), students are asked to write a program to calculate Fourier summations for the electron densitv. for which the completely general expression is
X
exp (;ahbl)exp [-2ri(hx
+ k y + lr)]
Assuming Friedel's law, they can show that this reduces to
Furthermore, for a centrosymmetric space group such as Pi, in which p(x, y, z) = p(-x, -y, -21, they can show that the imaginary component Bhk, of every structure factor F h k l = Ahkr iBhu is exactly zero, which is equivalent to having the phase ahat restricted to two possible values, 0 or a,so that F h k i becomes a signed real number, and the electron density equation simplifies further to
+
The advantage of a triclinic crystal in the context of this introductory exercise is that students need not be concerned with generating phases for reflections related by symmetry to a unique reflection set, beyond the application of Friedel's law, which is completely general t o the extent that anomalous scatterine can be neelected. Thev can concentrate on " the core of the exercise, computing and interpreting Fourier smtheses of electron densitv. - . which are ex~licatedverv nicely by Waser (17). Among the practical details the students must deal with in their programs are the input, storage, and retrieval of the indices, magnitudes, and signs of the structure factors; formation of the appropriate loops to calculate the Fourier summations; choice of an appropriate or convenient portion of the unit cell over which to perform the calculations; choice of a suitable grid spacing on which t o represent the continuous electron density function; scorage~ofthe results until they are output; and choice of a scale factor which will give a convenient and useful ranee of numbers when the resuks are printed out. Althoueh a full three-dimensional calculation would be feasible, we prefer t o have the students calculate three proVolume 65 Number 6 June 1988
483
Figure 5. Projectlonsof the eiectrA density of pfumaric acid, along a (top), along b (middle), and along c (bottom). F(000) has b w n omitted. Contours are shown drawn on ail three projections, and interpreted for the two resolved projections.
jections only, using hkO, hO1, and Okl data, with observed magnitudes and calculated phases. In addition to teaching what projections are and why only selected subsets of data are needed for the calculations, the computations are drasticallv shortened and simolified. and the results. shown in ~ig;re 5, can be displayed on only three sheets'of printed outnut. acid was carefullv chosen from amone the . B-fumaric . small set of crystals that have two fully resolved projections, so that complete three-dimensional information on the mo484
Journal of Chemical Education
lecular geometry and intermolecular interactions is available from projection data alone. The third projection in 8fumaric acid is not resolved, but shows the orientation of the molecular plane, almost exactly edge on. Note that the use of fractional coordinates referred to the triclinic axis system of the crystal means that displaying the projections in arectangular format with arhitrary length scales along the two axes does not matter, since it leaves the peak positions unchanged.
Once satisfactory maps of the three projections have been orinled and contours drawn, students must interpret them in terms of atomic positions, bringing to bear their knowledge of the connectivity in the molecule of fumaric acid, the trans geometry about the central C=C double bond, the fact that the molecular center coincides with a crystallographic center of inversion, and the expected difference in peak electron density among 0, C, and H. Two of the three projections exhibit complete resolution of C and 0 atoms, as shown in Figure 5, and this is sufficient to give three-dimensional coordinates x ., v. z for those atoms. Once approximate values of atomic coordinates are found from the positions of the maxima on the grid points of the initial maps, these can be refined in several ways, for examvle, by calculatine maps with higher - -arid resolution in the hciniiyof the peais, or; moreelegantly, by usinga technique such as the parabolic interpolation method explained by Stout and Jensen (IS). Differences will remain between the coordinates obtained by students and those determined by Bednowitz and Post (15), since the latter were found not by Fourier refinement of projection data but by least-squares refinement of a three-dimensional model against the complete data set. Hvdroeen atoms do not show uv in these Fourier svntheses, Hometimes to the dismay of the students. We ask t&emto discuss why this might be so, considering factors such as the resolution of this data compared to the length of C-H and O-H bonds, the electron densitvat the nucleus of a bonded hydrogen atom compared to that a t the nuclei of carbon and oxygen, and the magnitudes of the errors in this data, a sense of which is given by noting that the final crystallographic residual R = 0.062 (15). I t is stressed that hydrogen atoms can be located experimentally from X-ray diffraction data using difference Fourier syntheses after the remainder of the structure has been determined and refined. The imvortant application of neutron diffraction for the more acc&ate location of hydrogen nuclear positions in crystals is also pointed out.
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Geometric Calculations Once the fractional coordinates x. v. z of the atoms have been determined, all geometrical quantities of organic chemical interest, such as bond lengths, bond angles, torsion angles, least-square planes, and so on, can be &lculated from them. Since 0-fumaric acid is triclinic, the students must learn to perform calculations in a general nonorthogonal coordinate system, defined by the unit cell edges a, b, and c, which give the length scales in the three axis directions along unit vectors r , j, and 6, respectively, the angulafrelations amone which are defined bv the scalar vroducts 1.1 = cos . . j.6 = COSm, 6.; = cos 8. ~ g e s calcula~ons e are most easily done using vectors. If the?vector from atom A to atom B is given by rAB = (xg - x ~ ) a l +( . y ~- Y A ) ~ J(ZB- zA)c6, then B A to B is simply the length of this the distance ~ A from vector, which can be calculated using scalar products from ) ' ~ angle . GBAC defined by the equation ~ A =B ( ~ A B ' ~ A BThe the atoms B-A-C is given by the equation GBAc= COS-I ( ~ A B T A C I ~ AUncertainties B ~ A C ) . in both fractional coordinates and unit cell parameters can be propagated through these calculations to give uncertainties in the derived bond leneths and bond aneles .. (IS). . . Students can now determine, from their calculated values and associated uncertainties. first. whether or not the C-C single bond length is distingkshable from the C=C double bond leneth: second. whether or not the two carbon-oxveen bonds are of different lengths, that is, whether or not igere are distinct carbonyl and hydroxyl groups comprising the carboxyl group; third, all further distances and angles rele-
vant to a complete description of the hydrogen bonding in this structure, which is discussed more extensively below. (A comnarison of the values found for corresvondini eeometrical quantities in molecules of the same s\bstancg fumaric acid. in three different crvstalloaa~hic - . environments in the two polymorphs, is instructive.) Wdrogen Bonding All aspeds of hydrogen bonding in the solid state are clearly discussed by Hamilton and Ibers (19).The hydrogen bonding found in 8-fumaric acid conforms to a pattern common with carboxylic acids, dimerization around a center of inversion. Identification of an O--H.. .O hydrogen ~. bond is clearly made here hy comparison of the observed 0.. .0 distance, 2.673 A, to the sum of the following three values (assumine a linear O-H.. .O eeometrv): 11) the O-H bond length, 1.100 A; (2) the van d e r k a a l s k d i u s of H, 1.20 A; (3) the van der Waals radius of 0 , 1.40 A, giving a total 0.. .O distance of 3.60 A expected if there were no hydrogen bondine interaction between the oxveen atoms. (An excellent di&ssion of the concept of van-ier Waals radius, together with an extensive table of values, is aiven bv Bondi W).) One can also ask whether the c ~ n t ~ ~ s ~ m ~ e t rrelated ically uair of hvdrozen bonds found in this structure are themselves sy&eiric, in the sense that the hydrogen atoms are equidistant from two oxygen atoms and both c a r b o n a x v bond lengths are identical because of complete velectron delocalization over the OCO group, or asymmetric, with hvdroeen atoms closer to one oxveen-than the other and a localizedC=O double b o n d d i ~ t i & ~shorter than the C-0 single bond. This question can be addressed by ascertaining whether both the two carbon--oxygen bond lengths and the two CCO angles, respectively, are essentially the same or measurably different. Of course, direct location of the acidic hvdroaen . - atom, either from a difference Fourier svnthesis or, more accur&ely, using neutron diffraction, would be powerful additional evidence.
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Acknowledgment I would like t~ thank my students a t Barnard College for their patience, their criticisms, and their enthusiasm as these experiments were developed and Miriam Rossi of Vassar College for giving me theopportunity to present this work. 1. DaniekF.;Williams.J. W.;Bender,P.;Albrty,R.A.;Cornwell. C. O.;Hmiman,J. E. In Elpoimenrat Physied Chemistry, 7th ed.: McCraw-Hill: New York. 1910:
-,.
+
FL, annul. 8. Swanaon, H. E.;Tabe,E.. Eds. In Standard X-ray Diffraction Povdor Pnflerw; NationalBureaualStandadsCireulsr 539: Washinaton. DC, 1953;Vol.1.m 4749. 9. Bucrmr, M.J. In Crystal Structure Analysis; Wilcy: NewYark,1960:Chspier 12. 10. Hshn, T., Ed. Infermofionol Tabla for Crystallography:hidel: Booton. 1983;Val. A, space-GroupSvmmetry. 11. Kreba, H. In Fundomontola oflnorgonic Crystal Chemistry; Mffirsw-Hill:London. 1968:Chapten 12-16. 12. Wells, A.F. J. Chem. Edue. I9SZ,59,630-633. 13. Rwksby. H. P. Nofure 1943,152,304, 14. Brown. C. J. Acto Cwt. 1966,21,15. t 586571. 15. Bednowits A. L.;Pmt. B. Aeta C ~ y s 19E6.2I. 16. Goedkoop. J.A.:MseDihvry,C. H.Ada Crysf. 1951.10.125-127. 17. Waser, J. J. Chsm. Edue. 1968,45,44M51. IS. Stout, G. H.: Jenaen. L. H. X-ray SLwcture Doterminotion. A Placlicd Guide: Maemillan: New York, 1968. IS. Hamilton, W.C.: I b r a , J. A.HydrogenBondinginSolids;Benjamin:New York. 1968. 20. Band!. A . J. Phw. Chom. 1964,68.441451.
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