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# Class Note for GEOS 596A at UA

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Date Created: 02/06/15

142 symmetrical intensity distribution 4 equations 7 and 8 is verified as follows as 2 IVlts2y2gtqltydy Equation 7 postulates 4 Seems 8001182 y2 qy dy cos 2m39s ds 0 o 2n SoosfsIsJo2msds 11 Proceeding as in the case of equation 9 one finds for the lefthand side 2 cos 27m 4 So 212 dz Sogs qQz2 s W 18 The integral in s yields 712 S gz cos 99qz sin 90 cos 2am cos pdltp 0 If as postulated in 4 9z COS ltPQZ Sin ltP fz this integral yields Acta Cryst 1964 17 142 FOURIER TRANSFORM METHODS FOR THE SLITHEIGHT CORRECTION vz f zJ 0 2372 which confirms equation 7 Equation 8 is easily verified by combining the above procedure with that adopted for equation 10 I am indebted to Dr H Tompa for stimulating discussions during the course of this work References DU MOND I W M 1947 Phys Rev 72 83 GUINIER A amp FOURNET G 1947 J Phys Radium 8 345 GUINIER A amp FOU39RNET G 1955 SmallAngle Scat tering of X rays New York Wiley London Chapman amp Hall HOSEMANN R 1951 Ergebn exakt Naturw 24 142 KRANJC K 1954 Acta Cryst 7 709 KRATKY 0 POROD G amp KAHOV39EC L 1951 Z Elektrochem 55 53 POROD G 1951 Kolloidzschr 124 83 POROD G 1952 Kolloidzschr 125 51 109 SCHWARTZ L 1950 Th orie des Distributions Actualit s Scientifiques Paris Hermann Sm Ex V 1960 Acta Cryst 13 378 SYNE Ex V 1962 Acta Cryst 15 642 WATSON G N 1952 A Treatise on the Theory of Bessel Functions Cambridge University Press The Effect of Thermal Motion on the39Estimation of Bond Lengths from Diffraction Measurements BY WILLIAM R BUSING AND HENRI A LEVY Chemistry Division Oak Ridge National Laboratory Oak Ridge Tennessee U SA Received 25 March 1963 Diffraction studies of crystals locate the centroids or maxima of the distributions of atoms under going thermal motion and separations computed from these positions cannot in general be inter preted directly as interatomic distances Methods are presented for calculating the mean separation of two atoms given the isotropic or anisotropic temperature factor coefficients In order to apply these methods it is necessary that the joint distribution which describes the motion of the atoms in question be known or assumed The atomic coordinates resulting from a crystal struc ture analysis represent the maximum or the centroid of a distribution of scattering density arising from the combined effects of atomic structure and thermal displacement It has been common practice to compute an interatomic distance as the distance between a pair of these atomic positions With improvement in the accuracy of experimental techniques it has be Operated for the US Atomic Energy Commission by Union Carbide Corporation come clear that this estimate is valid only in the limit of negligibly small thermal displacements For exam ple a discrepancy between spectroscopic and diffrac tion estimates of the C C distance in benzene has been shown by Cox Cruickshank amp Smith 1955 1958 to arise from the large rotatory oscillation of this molecule about its hexad axis Cruickshank 1956a 1961 has discussed in detail the effect of the oscillations of a rigid molecule on the positions of maxima in a density distribution and consequently on the estimation of bond lengths The present authors Busing amp Levy WILLIAM R BUSING AND HENRI A LEVY 1957 have discussed the 0 H distance in CaOH2 using two methods to account for the effect of libra tion of the OH ion While it is clear that errors introduced by neglect of thermal effects will frequently be appreciable it is seldom possible on the basis of available information to make the rigorously appropriate corrections Spe cifically a knowledge of the correlation in thermal displacements of the two atoms that is their joint distribution is needed and this in general would require a detailed analysis of the dynamics of the atomic system However it frequently happens that useful estimates can be made from simplified models of the vibrating system and these estimates may serve as acceptable approximations to the actual sys tem it is also possible to place rigorous upper and lower bounds upon the corrections It is proposed that a suitable measure of an inter atomic distance is the mean separation the average being taken over the joint distribution of the two atoms This approach to the problem is different in principle from that taken by Cruickshank 1956a 1961 Cruickshank seeks to establish the equilibrium position of the point of maximum scattering density of an individual atom the correction being applied to the apparent position of maximum density It is believed that the present treatment in dealing with the mean separation of a pair of atoms in contrast with the equilibrium position of an individual often has the advantage of being closer to the physical problem of interest Further the use of the centroid rather than the density maximum is more nearly in accord with the atomic coordinates determined ex perimentally at least if they are obtained by least squares refinement A consequence of the use of the density maximum in Cruickshank s treatment is the need for peakshape parameters of the atom at rest no such parameters enter into the present treatment A general expression will be presented for the mean interatomic separation in terms of the parameters describing the distribution of instantaneous spearation Equations for upper and lower bounds to this mean will be derived The result will then be specialized for two simple joint distributions expressing the desired quantities in terms of the parameters of the individual atomic distributions The application of the correction to the case of rigidbody rotary oscillation is next considered Finally the evaluation of the pertinent quantities in terms of temperature factor coefficients and in terms of principalaxis displacements is given The mean separation of two atoms Let the distribution of the instantaneous separation S of two atoms be described by the function 9S So E 98 in which So represents the separation of the mean positions of the two atoms For convenience and without loss of generality consider a cylindrical coordinate system with origin at 80 and cylinder 143 axis z in the direction of So Then since SSos and SSoz2w2 3 where w is the radial compo nent of s in this system a Taylor Maclaurin expan sion of 8 yields SSozw22So The remaining linear and quadratic terms vanish identically The mean interatomic separation obtained by averaging 8 over the distribution 98 is then SSoE 2So since 20 if So is chosen as described above The error in terminating the series approximately g3 g2 z2S2 will be satisfactorily small in cases of interest If 98 is symmetric with respect to inversion through the origin this remainder vanishes and the next term is of the order of E283 It is clear that W is the relative vector displacement of two atoms projected on the plane normal to the line of mean positions The value of 202 will now be expressed in terms of quantities which are experimen tally accessible namely the mean components of dis placements of the atoms individually While this can be done explicitly only by making a specific choice of the joint distribution function of the displacements of the individual atoms the upper and lower bounds to 3 may be derived without such a specification Let WB and WA be the projected instantaneous displace ment of the two atoms so that WWB WA and w2w 3 2WBWAwi A wellknown theorem of statistics places bounds upon the second term of the righthand member namely 4273397121 s wBwA s iv 3272 It follows that wia cit W12 52 s macaw a we and Solwi2So S S S Sow2U2So Since the quantities involved are all observable it is possible to place upper and lower bounds upon the mean separation of two atoms without any assump tions as to their correlated motion The lower bound corresponds to highly correlated parallel displacements of the two atoms and the upper bound to highly cor related antiparallel displacements Neither of these extremes is to be expected to describe the actual displacements of atoms in crystals We next present expressions for 339 in two simple special cases which are of course intermediate be tween the foregoing extremes Riding motion Let the vector separation be independent of the posi tion of one of the atoms A and let this position be 144 expressed in terms of r A the displacement of A from its mean position Then the second atom B at a posi tion similarly defined by r B will be distributed accord ing to the convolution of the distribution function QArA of the first atom with that of the separation vector 9sgrB rA for independence of rA and r B r A implies that the joint distribution of A and B is 9JI A 1 8 QArAQrB I A and hence 9131 3 E SdI AQJI A 1 13 S QArAQrB rAdrA the convolution of 9A with g It is shown in the Appen dix that the meansquare components are additive on convolution so that tan 172 and So w wi2So The riding case may be expected to provide a useful model of atomic motion in a number of real situations For example if atom B is much lighter than atom A and is strongly linked only to A the lighter atom may be thought to ride on the heavier in the manner described This situation occurs in hydrogenous crystals frequently Strictly one must exclude the possibility of rigidbody rotations of A and B about a center removed from A and this will be discussed below A related application of this case is the following let A represent not an individual atom but the center of mass of a reasonably rigid isolated molecule and let B represent an atom of the molecule Atom B may then be described as riding on the motion of the center of mass If the motion of the center of mass or of atom A should be very small a possible condition at low temperature then the motion of B would be described by this case with Ei tO N orbcorrelated motion Let the positions of atoms A and B be distributed independently then their joint distribution is QJU A 1 8 QAI AQB1 B Let their relative displacement be srB rA The distribution of s is given by 98 S QJI A SrAdrA S QArAQBSrAdrA which is the convolution of 9A with the inversion of 93 through its origin Again invoking the general properties of convoluted distributions derived in the Appendix we have EFFECT OF THERMAL MOTION ON THE ESTIMATION OF BOND LENGTHS and SSo wiw 2So This case is a description of the motion of non interacting atoms and may be a reasonable approxima tion for nonbonded atoms in a molecular crystal Molecular libration The case of molecular libration may correspond to largeamplitude correlated motion and therefore may often require appreciable corrections to interatomic distances The mean separation may be computed in this situation provided the principal axes and ampli tudes of rotation are known Unfortunately it is not possible in general to derive this information di rectly from observed quantities A procedure usually applicable to rigidbody molecular motion has been described by Cruickshank 1956b Assume that such a procedure has been used to determine an axis defined by the unit vector 21 about which a group of atoms involving the interatomic vector S oscillates with mean square amplitude Then So the observed value of this interatomic dis tance will be foreshortened by an amount which depends on wz where w is the component perpendicular to S of the change in S produced by a small rotation 99 Since dSdgvagtltS is itself perpendicular to S it follows that wZS2 sin2 zprp2 where 1 is the angle subtended by a and S Then SSoltp280 sin21p2 where 80 has been substituted for the approximately equal quantity S For independent oscillations about more than one principal axis the corrections are additive Note on the correction of angles Thermal displacements may be expected to produce distortions in apparent angles as well as distances A sensible measure of an angle is its mean value over the joint distribution describing correlated motion of the three atoms involved but the evaluation of this mean angle is a more complex problem than that of computing mean distances It is to be emphasized that angles computed by triangulation from mean separations are not in general proper measures of mean angles An extreme example is the case of a linear molecule undergoing pure bending motion tri angulation from mean separations will yield a non linear configuration while averaging over the proper joint distribution must of course give a 180 angle Unless the atomic motion is correctly analyzed in detail it appears preferable to compute angles from the uncorrected distances Evaluation in terms of temperature factor coefficients Since the characteristic parameters of atomic distribu tions are usually available in the form of temperature WILLIAM R BUSING AND HENRI A LEVY factor coefficients the expression for w2 will be evalu ated in these terms For this purpose it is noted that w293 22 where 7 is the instantaneous displacement of the atom from its centroid and z is the component of displacement parallel to the interatomic vector Let the anisotropic temperature factor be repre sented by the matrix 3 in which the elements 16 are coefficients in the quadratic form 3 M Z ijkikj h k 7 7 1 In matrix notation Mi1 h in which the column matrix h represents the three reciprocalaxis coordinates k k l Let g be the metric matrix with components gig aha the scalar products of the unit cell vectors The quantity h Bh 2 nzhg 1h is the mean square component of displacement of the atom in the direction corre sponding to h We desire this quantity for the direc tion of the interatomic vector separation So which will ordinarily be described in terms of the direct lattice components 3 So ZSOiat 731 The reciprocal axis components are given by the ma trix product 980 where So represents the column matrix with components Sm Hence h N 22 9303930V 2yz209150 or in algebraic notation 3 3 22 Z ijUt Uj27I2Z SeaUr 2397391 i1 where dw U1 31310307 k 1 II The value of 72 is proportional to the sum of the eigenvalues and hence to the trace of the matrix g Busing amp Levy 1958 or ii Trpg2yz2 and I172 may be calculated In algebraic notation 7392 iajai27z2 17391 If the thermal motion of the atoms is known only in terms of isotropic temperature factors M Bsin 0102 the above expressions reduce to W 2B87z2 145 Evaluation in terms of principal axis displacements It may happen that a description of the atomic dis tributions in terms of principal axis displacements is available Let 3 represent the mean square displace ment parallel to principal axis 73 of the distribution and let 91 be the direction cosine of So with respect to this axis Then 3 2E1 7 M5 Examples and discussion It is again emphasized that proper application of the correction discussed in this paper except for evaluat ing the bounds calls for physical insight into the dynamics of the crystal and this is not always pos sible to obtain The considerations involved are illus trated in the following examples Calcium hydroxide This crystal Busing amp Levy 1957 contains discrete 0 H groups presumably ions separated from each other and from Ca by distances great enough to preclude strong bonding The hydrogen atom shows pronounced anisotropic displacements which are considerably greater than those of oxygen In this situation and in view of the 16fold greater mass of oxygen it is reasonable to suppose that three independent modes of motion ob tain 1 translational oscillations of the ion as a whole 2 rotary oscillations about a pair of axes substantially passing through 0 and normal to 0 H and 3 stretching of 0 H All of these modes are encompassed by postulating that the displacements of H with respect to O are uncorrelated with those of O in other words H rides upon 0 Making use of the reported roomtemperature parameters the fol lowing values for the 0 H distance are obtained Uncorrected 0936 A Lower bound 0956 Riding motion 0983 Upper bound 1051 For the reasons given above the value 0983 A is judged the best however it is of interest that the mean separation cannot be shorter than 0956 A whatever the nature of the atomic motion The closest H H distance in CaOH2 also provides an instructive example The structure suggests that these atoms are in van der Waals contact If it is supposed that the contact is a soft one that is that the motion of one atom does not appreciably affect the motion of its neighbor then the assumption of noncorrelated displacements is appropriate If on the other hand the contact is assumed to be hard so that the atoms strongly repel each other and undergo correlated parallel motion then the mean distance 146 approaches the lower bound which in this case is the same as the uncorrected separation The values ob tained follow Uncorrected distance or lower bound 2201 A Uncorrelated motion 2235 Upper bound 2279 It is of interest that even the upper bound is less than a nominal van der Waals contact of 234 A Benzene The structure of benzene Cox Cruickshank amp Smith 1958 illustrates the treatment of rigidbody oscillation and rotation The thermal displacements are strongly anisotropic with principal components normal to the plane of the ring radial to the ring and tangential to the ring The large difference in magni tude between the tangential components and the others together with the lack of restraining contacts in the structure suggests a rotary oscillation of the molecule in its own plane Most of the remaining motion may be assigned to translational oscillations of the molecule which are isotropic in the molecular plane although this is no doubt something of an over simplification On this basis the mean separation of two adjacent carbon atoms is computed as follows We have 80 1378 and for libration about the hexad axis sin 1 1 and p274 4o58n2137803903671378 Then w20390367 and the correction is 0013 A A similar treatment of libration about axes in the plane of the molecule yields an additional correction of 0001 A and a mean distance of 31392 A Because hexagonal molecular symmetry is assumed the lower bound is equal to the uncorrected value 1378 A and the upper bound 156 A is clearly too large to correspond to a true separation The value 1892 is in agreement with Cruickshank s treatment which makes parallel assumptions as to the nature and amplitude of the molecular rotation EFFECT OF THERMAL MOTION ON THE ESTIMATION OF BOND LENGTHS APPENDIX Properties of convoluted distributions Given two normalized distributions 91r and 921 each with centroid at r0 their convolution is pr S 91sgzrsds The negative sign defines the con volution proper and is pertinent to the riding case of the text the positive sign defines the convolution of 91 with the inversion of 92 and is pertinent to the case of uncorrelated motion Let 2 be any Cartesian component zrk where k is a unit vector Then averaging over the convolution 2 2 ll Srk2grdr rk2S 91s92r 1 sdsdr u k2 i 2uk s k s k291sdsdu 92mg uk i sk291sds du 92uS ll me3 z izi since 212z0 Because 2 is any Cartesian component the same consideration applies to x and 3 hence r2x2y2z2r ir and References BUSING W R amp LEVY H A 1957 J Chem Phys 26 563 BUSING W R amp LEVY H A 1958 Acta Cryst 11 450 Cox E G CRUICKSHANK D W J amp SMITH J A S 1955 Nature Land 175 766 Cox E G CRUICKSHANK D W J amp SMITH J A S 1958 Proc Roy Soc A 247 1 CRUICKSHANK D W J 1956a Acta Cryst 9 757 CRUICKSHANK D W J 1956b Acta Cryst 9 754 CRUICKSHANK D W J 1961 Acta Cryst 14 896

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