Symposium on Teaching Crystallography
Crystallographic Determination of Molecular Parameters A Physical Chemistry Laboratory Experiment James H. Loehlin and Alexandra P. Norlon' Wellesley College, Wellesley, MA 02181
Crystallographic methods provide the details of molecular structure. Many chemistry texts show ball-and-stick pictures. electron densitv maps. and stereo diagrams based on crystsl structure det&minations. Most chemistry texts have sections devoted to crystals and the determination of their structures by diffraction methods. Many of these eo well beyond introducing the 14 Bravais lattices and the ~ r law.aBy the ~ time ~ theycomplete their undergraduate training, most chemists have heard the statement: "This configuration has been proved by X-ray crystallographic analysis." Yet probably few have an idea of how that crystal structure determination is done. Though laboratory experience is often the hest way to learn. few crvstalloera~hicexperiments are simple enoueh for most undkgrad;aies to perform. ~ x p e r i m e n t designed s for the ohvsical cbemistw laboratorv are usuallv restricted to dem&trating the diffraction phenomenon for a cubic crystal using the powder method (1-4). Measurement of the diffraction angles, 28, and use of the Bragg law leads to the identification of the Bravais lattice and the determination of the single lattice constant needed to define the cubic unit cell. This may allow the determination of interatomic distances, since the atomic positions in most very simple crystals, such as NaC1, many pure metals, and even the somewhat more complex diamond, are fixed a t crystallographic special positions in the unit cell by symmetry constraints. Thus. unit cell dimensions alone mav be used to calculate interatomic distances. For example, "the 1.5445-A carboncarbon bond length in diamond is just (,/3)/4 times the 3.5669-A lattice constant. Most atom positions, however, are not fixed by the crystal's symmetry. The information needed to determine the atom positions and displacements caused by thermal motion comes from the relative intensities of the diffracted reflections. Several other experiments require specialized equipment, considerable crystallographic expertise on the partof the instructor, and frequently several weeks of class or lahoratorv time. A sinele crvstal camera can he utilized to determine the lattice coktants of an orthorhombic crystal and determine restrictions on nossible soace erouvs . from the observed systematic extinctions (51, b A this requires both personnel and equipment available only in crystallographic laboratories. Several possibilities exist for advanced undergraduate students where a large block of time and the necessary specialiled equipment and/or computer programs are available (fi-8). They do closel\~simulate real rrystal structure determinations; hut most-chemistry students will probably not take a course where these are used. This experiment attempts to overcome the shortcoming of the simpler experiments by using both diffraction-angle and diffraction-intensitv information to determine the lattice constant and a lattice independent molecular parameter, while still emolovine diffraction . " -standard X-rav. powder . techniques. The experiment requires only a little more stu486
Journal of Chemical Education
dent time than the typical lattice constant determination hut illustrates the way diffracted intensities are used to obtain atom positions in the unit cell. Crystal Structure and the X-Ray Method Only a very cursory treatment of crystallographic theory will be -given as backeround for this experiment. More thorough treatments are given in other articles and texts on the Relations siml~lifiedfor the cubic case are suhiect (9-121~. given here. Crystals are characterized by a regularly repeating structure in three dimensions. The pattern of the repeat, one point in space representing each of the repeat units, is referred to as the lattice. The lattice noints are related bv translational symmetry-translation from one lattice point to anv other leaves it indistineuishable from the orieinal. In the cubic crystal system there are three possible iattices: primitive. P. bodv-centered. I. and face-centered. F. which have one,twb, and four lattice points per cubic unit cell. The structure that undergoes the translational repeat is referred to as the motif. The motif must contain an integral number of empirical formula units (often one in simple crvstals. as here).-Since the motif may itself have symmetr;, only a fraction of it, the asymmetric unit may actually need to he determined. Excellent introductions to periodic symmetry concepts are available (10,13,14). The analysis of the crystal structure by X-ray diffraction is done in two parts. First, the diffraction geometry provides the information needed to determine the lattice eeometrv. Bragg's law is used to relate the observed diffractFon angles 2%to the d spacing of the planes giving rise to the diffraction, and these are then used to determine the lattice constant (or constants in the case of noncubic crvstals). Usine h. k. and 1 for the generalized Miller indices,-the Bragg Gw' may be formulated as follows: A = 2dhk,sin Bhkl
(1)
This differs from the usual textbook eauation in that all diffraction is assumed to be first order, thus eliminating n from the equation. Second-order diffraction is interpreted as first-order diffraction from planes whose indices are twice those of the actual crystal planes, etc.; for example, the 840
Presentations: 185th Natl. Mtg.; Am. Chem. Soc., Div. Chem. Educ.#69, Seattle. WA. 1983; also Loehlin. J. H. "An Experimental Introduction to Crystallography for Chemists," Am. Cryst. Assn. 1985 Mtg., Stanford Univ.. CA #PD 18 & "Teaching Crystallographic Concepts in the Undergraduate Laboratory," Am. Cryst. Assn. 1987 Mtg., Univ. of Texas. Austin, TX, #F 7. Present address: 62-203 Lawrence Berkeley Laboratory, Berkelev. ~ ,CA . 94720. -~ Most Dooks of crystallographic Interest are avai ab e from Polycrystal Book Service. P. 0.Box 3439, Dayton, OH 45401
'
~
~
reflection is really fourth-order diffraction from the 210 planes. For cubic crystals, the spacing of individual lattice planes is related by geometry to the single lattice constant, ao, by d,,, = o
d \ i m
(2)
Second, the intensity of each X-ray line is related to the positions of atoms in the motif via the structure factor, F. The structure factors squared are proportional to the diffraction intensities, ~
where k includes some corrections discussed below. The form of the structure factor equation is
where f., is the atomic scatterinefactor and x,., ..,-v,.z, .... ...are the coordin&s of the mth atom in-the cubic unit cell. To keeo the experiment within manageable bounds, a cubic crystal was ciosen to retain the simplest possible metric problem, with a single lattice constant to be determined. Substances forming crystals with- the potassium chloroplatinate structure, space group Fm3m ( I 5 Val. I, p 338), were selected since they have only a single latticeindependent structural parameter to be found-the metalhalogen bond length in the anion. (The inclusion of a scale factor and whether isotropic or anisotropic thermal vihration parameters are varied will add one to five additional parameters).
.-"~
F l-o m K,SIF.shucture. Tothe ien is one unit cell of the lanlce with the - 1.The ~ lance polnn c rcled n the center s one motrf To the r ght is a unflce I of !he compound snowing only those atoms bong *holly or partially oms de lne cell. Tne swium atoms are rhadea and the Sf moms are so a blac*. ~
~
C o v e r Glass
. b
gee-
Steps in X-Ray Structure Determination The steps in the determination of the structure of acrvstal by X-ray biffraction methods are given in outline form below. Though written for small molecule structures, most of the steps apply to crystals of macromolecules as well. The list is included to provide a comparison with the various steps of this experiment as given in the following section. A. Obtain compound and grow crystals. B. Prepare and mount crystal sample. C. Find and measure positions of diffracted beams of X-rays. D. From diffraction angles determine lattice constants. E. Use systematic extinctions to give space group information. F. Measure intensities of available reflections. G . Reduce intensity data to structure factor magnitudes. 1. Average equivalent reflections. 2. Correct for diffraction phenomena (Lp and multiplicity). 3. Take square root to give lFi. H. Begin phase determination. 1. Patterson method. 2. Direct methods. I. Recognizelidentifypart of structure. J. Complete determinationfrom partial structure. 1. Phase data set on basis of known fragment. 2. Generate Fourier electron density map. 3. Find rest of structure. K. Refine variable parameters to get "best fit" to data.
In this experiment all steps hut A and H are included. Default values mav he used in several steps to expedite the analysis. Experimental Details A standard Philips powder diffractometer Type 150-100 with Ni filtered Cu K , radiation was used. Other powder diffractometers or powder cameras with a way to measure intensities should be suitable also. Where departmental diffraction facilities are lacking, the experiment could be done using data or a diffractogram obtained from another institution where such facilities exist. (Geology departments and industrial laboratories involved in materials analysis and testing often use powder diffractometers similar to ours.) The steps below follow the outline in the previous section.
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M i c r o s c o p e Shde
surf ace
F l g ~ r e2. An ' e x p msa' v ew 01 tne arrangement for preparation of a samp e with a tendled surface. hole that me sample is prepared u p s m a w n ; m e cover g1855. mpeo to the sample holaer, s.ppons the oonom of tne samp e in its experimental position
Some of the alternative possibilities tried are indicated in Results and Conclusions. Potassium hexafluorosilicate and potassium hexachloroplatinate both are satisfactory compounds. The KnSiFs structure is shown in Figure 1.I t is the better material since, in addition to being less expensive, the lower atomic number elements in it lead to less X-ray absorption and extinction. KzPtCls provides a nice alternative. Other commercially available isostructural compounds we tried were not satisfactory. Commercial KzSiF,$ was ground with a mortar and pestle and passed through a 150-mesh sieve. Powder samples of K2SiF6tend to orient preferentially against a flat surface leading to distorted intensity measurements. Mineralogists use several techniques to circumvent this difficulty. We have ohtained satisfactory results using the method of Bloss et al. (If?), which clumps the crystal powder with a polymer s p a \ , I~rforeplacing i r inro a srandard A1 snm[,le holdcr. This, howe\.er, decreases the diffraction intensitv and also increases background scattering. For this experiient we have found that using a sheet of paper, fastened to the microscope slide against which the sample is formed, provides a textured surface on a scale comparable to the crystal size (see Fig. 2). The 100 faces of the crystals orient against this rough surface in a much more random manner andlead to peak intensities similar to those calculated from the known structure, without including any foreign material in the X-ray beam. A single diffraction scan was made for both diffraction angle and intensity measurement at a scan speed of l.OO/min from about 16" to 91" 2.9. The recorder output includes a marker a t every half degree allowing a precision of about Waltz & Bauer, Inc., 375 Fairfield Ave., Stamford,CT 06902. Volume 65
Number 6 June 1988
487
0.05O. Compressed diffractograms using Cu K, radiation witha Ni foil filter for K8iFn and for KgPtClc are shown in F ~ibur , largest peaksarel 11,220,222, Figure 3. F O ~ K - S ~the and 400. At angles - larger - than 50' noticeable 01-02 splittine.is observed. The diffraction lines are indexed by calculating sin2 Oil sin2O1 for each and then multiplying by the smallest number that will convert the set of ratios to integers. These integers, Ni, are used with a table of the possible integers for the three cubic lattices to classify the crystal as face-centered rather than primitive or body-centered. They also provide the Miller indices, since Ni = hi2 hi2 li2.The Bragg equation is then used to calculate the lattice constant from each line. Where the Kel and K,p lines are resolved (above -50°), the appropriate wavelengths are used rather than the average value below that.
+ +
XKi XK., XK,,
A = 1.5405 A = 1.5443 A = 1.5418
An averaged a 0 value is obtained from the independently calculatedvalues of an;. Usina-. peak . ~ositions>45' aives best results. Peak heights have been used as measures of intensity and give satisfactory results if a correction is included to compensate for the decrease in peak height once the K,1 peaks begin to be resolved from the K,2 ones. Multiplication of the CUK,I peak height by 1.50 will give an intensity comparable to that where the twoK- oeaks are unresolved (15. Vol. 111. D 71). The observed strucGre factors, Fo(hkl),must be determined for each uniaue lane (hkl) from the observed diffraction intensities. F& powde; data from cubic crystals the followine. corrections must be a . o-~ l i e d Certain . noneauivalent planes in a cubic powder give lines that are superbosed on one another. One choice is to eliminate these from the data set since a priori it is unclear how much each set of planes contributes to the observed peak. For KzSiFs, we have chosen instead to calculate the individual intensities by assigning to each the appropriate fraction of the total intensity based on the observed single-crystal data. The resulting fractions are shown in Table 1. The second step in the data reduction is to account for the number of planes of each type in a crystal. Each intensity is
Table 1.
KzPtCls
Miller Indices 511, 442, 551. 553. 820. 860. 555. 911. 755, 860.
333 800 711 731 844 822 751 753 933, 1000
771
Table 2.
B Figure 3. Experimemal powder diffractograms for (A) K2SIFsand (El KPtCls. m e chart speed was slowed down from normal experimental conditions. One cycle ol the event marker on the len bordar equals one degree 28.
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Journal of Chemical Education
Fractlonlng of Superposed Lines K2SiF8Fractions 0.918. 0.963. 0.968. 0.992, 0.895, 0.720, 0.544, 0.878, 0.805, 0.927,
0.082 0.037 0.032 0.008 0.105 0.280 0.456 0.122 0.187, 0.073
K2PtCieFractions 0.857, 0.806, 0.648, 0.487. 0.508. 0.320, 0.361. 0.474,
0.008
Multiplicity of Cubic Powder Lines h0 0 hhO
6 12
hhh hkO hhk hkl
8 24 24 48
0.143 0.194 0.352 0.513 0.492 0.680 0.639 0.526
divided hv the multinlicitv that is shown in Table 2. For example the 111refiecti4 belongs to the h h h class, and, since there are eightplanes of this type that are equivalent in a cubic crystal, we divide its intensity by 8. The 6 4 2 peak on the other hand belongs to the set h k 1with 48 planes, so its intensity is divided by 48. The result is a set of intensities that are com~arableto those for diffraction from individual planes in a single crystal. This set is then corrected for Lorentz and polarization (Lp) effects (15, Vol. 11, p 266) and the square root taken to give a set of observed structure factor magnitudes, IFoI. For our diffractometer the Lp correction that multiplies the intensity is
+
Lp = (1 cos22O)lsin 28
(5)
where 28 is the observed diffraction angle. For Dehye-Scherrer geometry the term in the denominator should be (sin2 8 cos 8). The structures used here have a center of symmetry that simplifies the phase determination. We have only to decide whether the positive or negative square root is correct in obtaining Fofrom the intensity I. In other words, a or must somehow be assigned to each IF01 to give Fo. This is the famous "Ohase ~rohlem"in crvstalloara~hv. .. . . We have bvpassed the general methods of phase determination by uiing the facr that the high svmmetrvof the FCC unit cell reauires that the K and Si a&& lie a t fixed positions in the cell:~his is based on the required agreement of the experimental crystal density with the lattice constant and formula weight. Students determine the density by suspension of recrystallized salt in CC14-CHBr3 (15, Vol. 111, p 18). Though there are actually two ways to assign the origin to a center of svmmetrv. we have laced the Si atom a t the oriein rather than 2 t f i center of'the unit cell. This results in &ere being an S i F P ion a t each lattice ~ o i n and t shows the face-centered arrangement more cleariy. The K atoms must lie at the points V4, l/p, Vp and -%, -I/& -l/4. With a partial solution, the structure determination is completed asit would he for any other crystal structure. This partial structure is used to assign probable phases to the structure factors in calculating an electron-density map for the crystal. A structure factor is calculated for each reflection based on the partial model and its phase, + o r -,is then assigned to the Fo. The electron density in the z = 0 plane is calculated by summing over all of the observed reflections (including index permutations, khl, lhk, klh, lkh, hlk) using the equation (15, Vol. I, p 512),
+
The Fourier map generated from these phased Fos clearly shows the nositions of not onlv the Si and K atoms but also the F atoms located octahedrally around the Si (see Fig. 4). Tbedistance of the fluorinesfrom the silicon can he estimated from the map. Symmetry constraints force the F atoms to lie on the cell axes with only a single undetermined parameter, the Si-F distance (actually the fractional cell coordinate, constant, is used). The F atom posix = bond lenathilattice tions around the origin Si are x, 0,O; 0, x, O; 0,O, x; -x, 0,O; 0, -x, 0 and 0.0, -x. The finalstep in thestructuredetermination is tovary the adiustahle varameters to obtain the best least-squares fit of the Fo with'the calculated structure factor, F,, based on the model (eq 4 or 7). One additional factor which must be included at this point is the thermal motion of the atoms in the structure. These serve to "smear out" the electron distribution and the effect is usually corrected for by including temperature factor terms in the equation giving the calculated nt,ructure factors. . . ~ ~ - ~ The ~ simnlest thermal vibrations are isotropic, with one adjustable parameter. Though these ma?. be used for all atomsin thisstructure,n better model has two ~
~
anisotropic thermal parameters for the halide atoms, thus better accounting for the differences in motion along and perpendicular to the bonds. The form of this equation for K2SiF6is F, = 41fs,e-2.'lFsiih2+k'+l2,Ia;
+ 2fkcos (rhl2) cos (rk12) X cos (r1/2)e-2'2m'k2tk2t1'1ii02 + 2fF(cos2rhr + cos 2rkx + cos 2alx)e- ~ r z ~ ~ h z + k ~ + ! ' l l ~ ~
+
+ +
+
where the UF(h2 kZ 12)is replaced hy UFlh2 UFz(k2 12)in the anisotropic case. The mean square amplitude of vibration, (u2) = U. The dimensions of u are those of X and ao, usually A. For values of atomic scattering factors see ref 15. Vol. 111.DO 202ff). The least:s&ares'minimization attempts to get a best fit between the observed and calculated structure factors bv minimizing the sum of squares of the differences between the observed and calculated structure factors, 2(IFol - IF,Il Sf12where SF is a scale factor to he determined. Since the form of the function for F, is not linear in the variables, several cycles of minimization are usually necessary. The quality . . of the fit between the data and the model is given bv a discrepancy index, R, which is summed over all the data..
R = X tl(SR.IF& - lF,ll)lX (SF.IFd)
(8)
Most colleges and universities with an active crystallography lab prot,ahly have computer program packages designed to do crystallographir computations of various types, includine the data reduction and least-souares refinement done h&e. We have used several generat& of XTAL crystallographic programs (17) for this experiment. Since many colleges do not have resident crystallographers, however, and
Figure 4. A Fowler map based on FO WIG? phases from F, fw Ihe partial s l r u ~ t ~ rlncludina e onlv the Si and K atoms. me ma0 am . reoresents . . i o n of the dnlt cell face with a contour drawn in around the Si and F atom P€6iIiOnJ. For clarow, only parltive e ectron density values are shown. Tne map shows the F atom at x = 0.21, y = 0. and omsr relalea to a by symmetry.
-
~~
~
Volume 65 Number 6 June 1988
489
Table 3. Resuns for Various SubstancesTrled Exprimenta Values Compound
a0
x
R
U(
Thwml Parameten' S i W,
*
(f
'Fw K2RCI., vslues am U
