Fourier Analysis and Structure Determination Part Ill: X-ray Crystal Structure Analysis John P. Chesick Haverford College. Haverford, PA 19041 In Part I of this series1we have attempted to present some groundwork in the theory and application of Fourier transforms, and in Part 112we discussed pulse NMR with particular application to NMR imaging as a tool for structure determination. In this Part I11 we wish to continue the discussion to single crystal X-ray crystal structure analysis. X-ray diffraction as a tool for precise determination of molecular structure has been long and often discussed in a variety of sources. However, the historical development and usual presentations have not been in the framework of Fourier transform methods. We also find a common link between the NMR imaging and the traditional X-ray crystal structure analysis and feel that comparisons aid in the understanding of both techniques. The use of Fourier transforms in X-ray structure analysis has been most elegantly and recently described for the chemical audience by Dunitz3, and we attempt to follow his notation. Lipson and Cochran in a much older source' provide a useful discussion of X-ray diffraction developed from consideration of Fourier transforms, although their usage of optical analogues for solution of the crystallographic phase problem is now chiefly of historic interest.
made. The final unit cell axis selection is usuallv made to highlight any symmetry operations possible for thk unit cell. Another approach to thinking about the crystal was suagested in the &text of the disckion of ~ o u i i e transfo&s r in Part I.' Equation 1can be used to define a three-dimensional grid or lattice of points. If an impulse or delta function is associated with each lattice point, then the whole crystal may be seen as the three-dimensional convolution of this lattice with the unit cell. One brick convolved with a lattice function produces a three-dimensional brick pile of similarly aligned bricks. X-ray Scattering Vector and Phase
The simplest theory for the scattering of X-rays for such a crystal assumes that incident X-rays are scattered coherently without change in wavelength and with an amplitude proportional to the electron density at each point in the crystal. The electric vedors or amplitudes of the scattered X-rays are assumed to add linearly with due regard for phase shifts caused by different path lengths travelled. Figure 2 depicts the scattering of a plane X-ray wave by matter at points PI and Pg, with the reemitted or scattered
Crystal Lattice and Unit Cell A crystal is assumed to be an assembly of atoms in a unit cell, which is repeated by translation along three axes by integer multiples of the edge lengths of the unit cell to generate the crystal. The three axes need not be orthogonal. Thus three vectors a,, a2, as provide both the directions of the axes and the lengthsof theunit cell. Any point in the unit cell has a symmetry equivalent point in another unit cell related by translations along the cell axes. Such a translation to a symmetry equivalent point is given by the vector where n,,nzand na are integers. These translations by multiples of the unit cell edge vectors also perform the translation symmetry operations, which combine with the point group symmetry operations of the unit cell to generate the symmetry space group for the crystal. Fiaure 1 is a two-dimensional examnle. . . such as mizht be provided by a tile floor or wallpaper pattern. The vec& a1 and a ? in Figure 1 define a unit cell area. and the lines show the boundarjes of the repeated units. he lattice points may be found at the intersections of these lines. The rounded object repeated in Figure 1corresponds to a molecule or an aggregate of molecules, the "asymmetric unit" of the unit cell. There is one such asymmetric unit in the unit cell of Figure 1.The choice of the origin for the unit cell is arbitrary, and many choices for the unit cell axes might have been
Figure 1. Twc-dlmenalonalcrystal with one choice for a unit cell.
' Chaslck, J. P. J. Chem. Educ. 1989, 66,128. Chasick, J. P. J. Chem. Educ. 1989, 66, 283.
Dunih. J. D. X-ray Analysis and the Structure of Organic Mol.
cules; Cornell Unlversity: Imaca, NY, 1979; Chapter 1. Llpson. H.: Cochran. W. The Determination of Clystal Structures: Bell: London, 1968; Chapter 10.
Figure 2. lncidem and scanered wave amplhdes for selected source and observation directions. Volume 66
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wave having the same phase relative to the incident wave at both locations. P1 is at the coordinate origin, and the unit vector sodefines the direction for the incident X-ray beam. The unit vectors points in the direction of the X-ray detector. Figure 2 shows qualitatively how the differences in path lengths for both the incident and scattered X-ray beams depend on the location of the point P p relative to the origin PI.The scattered waves from the two points are shown with different amplitudes corresponding to different electron density values at the two points. The scattered wave amplitudes should then he added aleebraicallv to obtain the total amplitude seen hy the observeiin the diiection of s. Figure 2 thus contains the essentials of the ex~erimentshowing the codingof scatterer location (samplest;urture) in the relative phases and am~litudesadded to aive the resultant amplitude of the scaitered X-ray radiation. Figure 3 is a vector diagram, corresponding to Figure 2, to illustrate the calculation of the phase 6(R,r) of the wave from Pz relative to the wave from the origin point. We wish to compare the path length travelled and the phase for the wave produced at r relative to the path length travelled and the phase for a parallel wave front produced at the origin. From Figure 3 we have
-
.
= Zr(path difference)lX = 2m (s - so)/h
whereR I (s- so)/X;IRI = 2 sin OIL The scattered wave from a volume element at r has an amplitude proportional to the electron density d r ) , an angular frequency w which is the same as that of the X-ray source, and a phase 6 dependent on the location of the scattering center and on the geometry of the scattering experiment. All of this is wrappedup in eq 2 giving this scattered wave amplitude. The constant C contains structure-indenendent factors. The vector i is worthy of some further commentary. R and r are here defined in the same coordinate mace and therefore link crystal orientation and the directions of incident and scattered radiation. The direction of R, shown in Figure 3, is perpendicular to aplane which may heseen to be a "mirror" relatina the incident and the scattered X-ray beams. The angle [the Bragg angle, measures the deviation of both the incident and scattered beams from this mirror plane. ~~
We will consider further the internal factor in eo 3. This is ) is the structure-dependent part of tLe amplitude A ~ Rand defined as thestructure factor F(R). F(R) = J d r ) exp (iO(R,r))dr= J d r ) exp (i2rR .r)dr
(4)
Integration with d r as the increment implies a three-dimensional volume element, and at this point in our discussion the limits of the integral in eq 4 include the whole crystal sample. The location of the volume element of electron density a(r) is now defined in terms of the coordinate system of the unit cell by We have here the coordinate set ( x y , z ) in units of the unit cell edge lengths. Let us define another coordinate system that we will use for the vector R. This is done with the "reciprocal" axis basis vector set bl,bz,and bs. These vectors define axes that are perpendicular to the faces of the unit cell of the crystal. In vector notation these reciprocal coordinate system hasis vectors are defined through unit cell vector cross products and the unit cell volume V as
~~~
The Structure Factor
The linearity of addition of amplitudes of scattered waves from a h ) at different r values indicates that we should sum or integrate A(R,r) for values of r to include the whole volume of the crvstal samole. This would eive the total amplitude of the signal rececved by the detect; for a chosen R. An exercise in trigonometry will show that the addition of cos functions of different amplitudes and phases can be represented by the two component vector additions of complex numbers. The integration of eq 2 over the sample volume represents just such an addition of cos functions and hecorn& eq 3. A(R) = CJo(r) exp ( i b t + @(R,r)))dr The linear addition of waue forms representing X-rays scattered with amplitudes and phases that depend on the location of the scattering source guarantees results that will haue formal similarities to those we obtained in the discussion of NMR imaging2 414
scattering vectors: sand h a r e of unlt length.
Reciprocal Axes and Space
phase difference *(R,r)
~
Figwe 3. X-ray
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A consequence of this definition is that the a and b basis vectors are orthogonal as shown by the vector scalar product relations
For an orthorhomhic, tetragonal, or cubic crystal systems with unit cell axes at 90' angles to each other, the corresponding component vectors of the reciprocal and direct space systems are parallel with lbjl = l/1ajl. The pairs of basis vectors in reciprocal and direct space are actually the reciprocals of each other in magnitude for such right-angled crystal lattices. In this reciprocal space coordinate system, the vector R defining the scattering direction is The coordinate values (hl,hz,h3) are not necessarily integers at this stage of our exposition. The definitions of the space lattice vectors and the resulting orthogonality relation;, eq 7, between b and a give us in terms of the coordinates of R and r in the two related coordinate systems. Equation 9 really is the relation that provided the motivation for the definition of the reciprocal
lattice coordinatesystem in eq 6. We might have written e q 9 first and then used i t with the orthogonality conditions, eq 7, to get eq 6 defining the coordinate axis system in which the coordinates of R would be (hlrh2,h3). Substituting eq 9 into eq 4, we have F(R) = JJJs(r) exp (iZr(h,x + h g + h,z))dxdydz
(10)
Thus we see that the structure factor F(R), due to the addition of signals from all of the volume considered, is the inverse Fourier transform5 of the electron density function a ( r ) when the coordinate system for R is "properly defined. Equation 10 is identical in form to eq 12 of Part 112,which related the magnetic moment density MO(r) to the observed signal S(k)in the NMR imaging experiment. Both equations arose from the additions of wave forms of varying amplitudes and phases. In practice, however, there aretwo important differences which we meet in this comparison of NMR and X-ray structure determination. One of these is the set of constraints imposed by the lattice repeat struclureof a crystal. Structure Factors and Transforms
In the beeinnine we noted that a crvstal mav be considered to be therepeat of the unit cell in three dimensions.The electron densitv function for the wholr crvsral is the threedimensional convolution of the electron density function for theunit cell with the three-dimensional lattice function. The structure factor, the complex scattering amplitude F(R) resulting from the scattering of the incident X-ray beam by the electrons of the atoms in the crystal, was found t o be in eq 10 the inverse Fourier transform of the electron density function a(rl of the crvstal. The convolution theorem stated1 that &;Fourier tr&sform of a convolution is the product of the corresponding Fourier transforms of each of the functions convolved. Hence, we should find the inverse Fourier transformof thelattice. a n d m u l t i ~ l vit bv the inverse Fourier transform of the unit cell to f h d F(R) for the complete crystal. Lattice Transform
We can define the lattice as having a delta function scatterer a t each of the lattice points given by all choices for the integers nl, nz, and ns as the coordinate values (x,y,z) in eq 5. This defines the lattice electron density function as
reciprocal space. The scattering vector R must terminate a t onebf these-reciprocal lattice points for there to be non-zero amplitude for scattering. This reciprocal lattice also has the same - ~ - - - svmmetrv " as thecrvstal lattice in direct soace. The convolution theorem shows that the inverse ~ o u i i e transr form F(R) for a crvstal is the oroduct of the inverse Fourier t&fo& of the unit cell with'the inverse Fourier transform of the lattice, the reciprocal lattice. This product is zero except a t the reciprocal lattice points. Thus R cannot point t o anv arbitrarv direction for an observable scattering signal. Onlifor orientations of crystal and selections of scattering angle giving R a t the lattice points in reciprocal space can one see observahle intensity for the scattered X-rays. I t is the cleverness in construction and operation of the Weissenberg or precession X-ray diffraction cameras or of the fourcircle X-ray diffractometer that make3 possible the collection of a data set of measurements of the relative inlensiries of the scattering for these points in reciprocnl space. In other words. the X-rav diffraction exoeriment samoles the rnoynitude of the in;erse Fourier tiansform of the unit cefi a t reciorocal lattice ooints. lt'can also be shown most readily using Fourier transforms that a finite, rather than infinite, crystal lattice serves to broaden the &ular aperture for ohseived scattering, giving mots rather than points if photographic film is used as detector for the x-ray scattering ut' small crystal. Some common lartice imperfections have the same effect. It is demonstrable that the indices h,. .. h,.-.and h?. which were the coordinates in reciprocal space for non-zero values of the scatterine vector R. are the same as the Miller indices h,k,l used to zefine lattice planes in classical crystallography. Thus the F(R) that we obtained as the inverse Fourier transform of the electron density function for the crystal becomes ~
~A
~
Fh,, = JJJc(r) exp (i2dh.z + ky + 1z))dxdydz
where the integration is now over the volume of the unit cell, rather than over the volume of the whole crystal. Spherical Atoms
If chemical bonding effects are neglected, then s ( r ) is the linear superposition of the spherical electron densities of the individual atoms and the above integral is replaced by the sum of integrals for each of the atoms in the unit cell. This substitution and integration for each atom gives the computationally useful result F,,,, =
We also define the scattering vector R as before in terms of the reciprocal lattice axes so that for the lattice points eq 9 becomes (R .r)kt = hlnl
+ h,n, + h,n,
(11)
We still have not soecified anv condition for the coordinates of R in reciprocal space, (hl.h2,h:l),and R may point in any direction. The inverse Fourier transti~rmfor this lattice becomes a sum, and eq 10 gives the following result for this lattice transform
For summation limits from large negative to large positive values for the integers nl, nz, ns, eq 12 has the value 0 unless the argument (hlnl hzn2 h3n3) is an integer for all nl, n2, and n3 values. This requires that the coordinates hl, hz, and ha of R in the reciprocal coordinate system must all he integers.
+
+
Lattice Sampled Transforms
The Fourier transform of the lattice in direct (x,y,z) space turned out to he also a lattice in the previously defined
(13)
+
f, exp (i2r(hxj ky,
+ kj))
(14)
where (xj,yj,zj) are the coordinates of the atom j, the sum is over all atoms in the unit cell, and the atomic scattering fador f, is calculated as the inverse Fourier transform of the spherical atomic electron distrihution function as a function of the scattering angle 20. I t should he remembered that the mathematics of eq 14 was physically requiredby the addition of X-ray wave amplitudes produced by the scattering from each region of the sample. Thermal motion appears in a crystal as the vibration of atoms. This may be represented as the convolution of each
This identification of eq 10 as an hverse Fourier transform and designation of the integral ~ ( r=) JJJF(R). exp (-i2r(h,r
+ h o + h,z))dh,dh,fh,
as the Fourier transform corresponds to the sign conventions in the references on Fourier transforms, the NMR literature, and the previous discussions in Part I and Part II of this series. This is unfortunately the reverse of the sign convention employed in the X-ray crystallography literature, where eq 10 is described as the Fourier transform,and the above integral equation for a(,) is the inverse transform. The choice of labels is arbitrary: we chose to be consistent with our previous discussions. Volume 66
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atomic electron density function with a function representine the three-dimensional vibrational motion for the atom. ~ i c the r n in eq 14 should represent the inverse Fourier transform of this convolution. The convolution theorem tells us that this transform of a convolution may he replaced by the product of transforms. Each term in eq 14 then should acquire a new factor representing the inverse Fourier transform of the thermal motion function for the atom. Structure Determination
Equation 14 for Fhal shows that it should he simple to calculate the amplitude of scattering for each (h,h,l) from the o(r) of the crystal. However, it is crystal structure electron density function a ( r ) that is usually to be found, with lumps in u(r)defining the atom locations. The discrete Fourier transformation of Fhkl, eq 15, gives the desired electron density function
where Vis the volumeof the unit cell. Equation 15 isequivalent toea 13 in Part 112in the discussion of NMR imaaina. - We re& that Fhkl is a complex number, or a two-component vector, which can he also be displayed in terms of its magnitude and phase angle in the complex plane F,,
= IFhkd exp (iuhk,)
There are no phase-sensitive detectors for X-ray radiation as there are for the rf radiation ohsewed in the NMR experiment, and hoth of the components of the complex Fhkl cannot he measured. Only the intensity of the scattered X-ray beam can heohsewed. Thismeans that the X-ray diffraction experiment gives us a set of values for IFhkrI2 = F . F* for different combinations of (h,k,l),hut the relative phase information. the set of values for hkl. is exnerimentallv unrneasurabie. Equation 14 may be mkipulaied to give mathematicallveauivalent ea 16 which seoarates more clearlv the observable from the unohservahle.
-
&
The two component complex number Fhkl in eq 15 has been replaced by its observable amplitude and its unobservable relative phase angle ahkl in eq 16. The phase problem in Xray crystallography has been the history of methods used to ohtain u(r) when only amplitude data is available for Fhkl, missing the desired phase angles. This is nontrivial. A recent Nobel prize in chemistry was given toHauptmann and Karle for their work on this problem. Although the phase prohlem
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has provided sport and employment for crystallographers for decades and has been historicallv the chief roadhlock in the path of automatic crystal structure determination, we will take the topic no further. Conclusions
In summary, we may compare the two superficially dissimilar methods for structure determination which have been considered. In X-ray crystallography, nature determines that measurementsoftheamplitudeofscattered radiation are possible only a t discrete lattice points in a space reciprocal to that of the desired sample structure. In NMR imaging, evenly spaced points are selected in a reciprocal space for measurement of a radiation amplitude. This even spacing of data points permits the most rapid computations possible using fast Fourier transform algorithms; however, this measurement grid in reciprocal space is a t the control of the experimenter. This measurement grid also defines a lattice in the reciprocal space of the NMR imaging problem. The availability of phase-sensitive amplitude detection for the radio frequency signals measured in the NMR experiment means that there is no "phase prohlem" in the X-ray sense and that the calculation of the magnetic moment density function, the image, is in principle automaticand simply a matter of spending the computer time to perform the three-dimensional discrete Fourier transforms with a mass of data. The detection of X-ray signals is only possible using ohotoeraohic film or other radiation detectors sensitive to ihe en&& of the scattered X-ray beam, and all phase information is lost. Unfortunately, the phased information in the NMR experiment is subjectto phase distortions a t all stages of the operation. Gradient coil operation is usually less than ideal, and eddy cutrents may be induced by operation of the gradients which serve to provide phase twists in the magnetization for some sample voxels, for some values of the gradients and/or for some time domains. Phase information can be lost as nart of the sienal detection and amolification process. small signals varying in amplitude by factor of about 16,000 must be measured, requiring care in the measurement process. Phase distortion comprises the "phase prohlem" in the NMR imaging experiment. Data acquisition, management, processing, and display make significant demands on computer hardware and software. The NMR imaging application involves bringing together old concepts which are "deceptively ~ i m p l e "into ~ an integrated system.
a
WiIIcott, M. R., private communication.
