Analysis of chord-length distributions

Transcription

1 Acta Crystallographica Section A Foundations of Crystallography ISSN Analysis of chord-length distributions C. Burger a *² and W. Ruland b Receied 4 December Accepted March a Max-Planck-Institut fuèr Kolloid- und GrenzflaÈchenforschung, D-444 Potsdam, Germany, and b Philipps-UniersitaÈt Marburg, Fachbereich Chemie, D-353 Marburg, Germany. Correspondence cburger@sunysb.edu # International Union of Crystallography Printed in Great Britain ± all rights resered A closed-form analytical solution for the inersion of the integral equation relating small-angle scattering intensity distributions of two-phase systems to chord-length distributions is presented. The result is generalized to arbitrary deriaties of higher order of the autocorrelation function and to arbitrary projections of the scattering intensity (including slit collimation). This inerse transformation offers an elegant way to inestigate the impact of certain features, e.g. singularities, in the chord-length distribution or its higher-order deriaties on the scattering cure, e.g. oscillatory components in the asymptotic behaior at a large scattering ector. Seeral examples are discussed.. Introduction Chord-length distributions (CLD) play an important role in the qualitatie and quantitatie ealuation of the small-angle scattering (SAS) of two-phase systems, in both the dilute and the dense case (Me ring & Tchoubar-Vallat, 965, 966; Porod, 967; MeÂring & Tchoubar, 968; Wu & Schmidt, 97). The relationship between the CLD g r and the autocorrelation function (CF) r is gien by g r ˆl p r > ; where l p is the aerage chord length (Porod length). The CLD contains the complete structural information obtainable from non-normalized SAS intensities of statistically isotropic twophase systems. The behaior of g r in the icinity of small r is de ned by the general structure of the interface (edges, ertices, curature; Ciccariello et al., 98; Ciccariello & Benedetti, 98; Ciccariello, 993; Sobry et al., 994; Ciccariello & Sobry, 995). Singularities in g and g at nite alues of r are related to particularities of the surface in the icinity of minimal or maximal diameter of monodisperse particles in dilute solution (Wu & Schmidt, 974) or of the structural unit in periodic dense two-phase systems, respectiely. In the case of lamellar two-phase systems, the appropriate one-dimensional equialent of the CLD to be used for the ealuation of structural parameters is the interface distribution function (Ruland, 977, 978). The CLD is determined by the integral transform (Me ring & Tchoubar-Vallat, 965, 966) g r ˆ 8 3 s 4 l p k sin z I s Š ds; z ² Present address: Department of Chemistry, State Uniersity of New York at Stony Brook, Stony Brook, NY 794, USA. where I is the SAS intensity (in unsmeared pin-hole collimation), s ˆ sin is the absolute alue of the scattering ector, the scattering angle, the waelength, k is a normalization factor (Porod inariant), z ˆ rs and the deriaties of sin z =z are with respect to z. The possibility of an inersion of (), i.e. the integral transform to be used to obtain I s from g r, has been discussed by Me ring & Tchoubar (968), but an explicit expression for this transform was not found by those authors. In this study, we present a closed-form solution to a generalization of this problem, namely the transformation of n r for arbitrary n (which can be specialized to n ˆ, but is also alid for n ˆ ) to an arbitrary projection of the corresponding scattering cure (so that slit-smeared scattering cures in the in nite-slit-length approximation are also included in the approach). Examples of its application are gien that demonstrate the usefulness of this transformation for the solution of arious problems. The paper closes with a short reiew of the requirements for an accurate determination of g r from experimental data.. Basic relationships For the purpose of the mathematical treatment deeloped in this paper, we consider the function r to be de ned for positie alues of r only, and continuously differentiable at r ˆ. It is shown in Appendix A that this de nition of r does not restrict the information contained in r if the appropriate transforms are used. We start with the relationship I s ˆk r sin z 4r dr; z where k ˆ V (Porod, 95, 95a,b) andv is the irradiated olume of the sample, j is the (constant) 3 48 Burger and Ruland Chord-length distributions Acta Cryst. (). A57, 48±49

2 Table The kernels ;n and 3;n for n up to 4. n ;n 3;n cos z z sin z sin z z z cos z sin z cos z z cos z z sin z 3 z sin z z z 3 sin z z cos z 4 z = cos z z 4 z 4 cos z z sin z Table The kernel ;n for n up to 4. n ;n J z J z = J z H z J z H z 3 zj z J z z= J z H z J z H z Š 4 z = J z z= J z =4 3 z J z H z J z H z Š electron density (small-angle X-ray scattering, SAXS) or scattering-length density (small-angle neutron scattering, SANS) within phase j and is the olume fraction of one phase. This de nition of k implies that the interfacial boundaries of the phases are in nitely sharp. If we designate a d-dimensional projection by fg d, we obtain and fig s ˆ R ˆ k R I s y = dy r r J z dr fig s ˆ R yi s y = dy ˆ k R r cos z dr; 5 where J is the Bessel function of the rst kind of zero order. fig is equialent to the SAS intensity measured with a slit system of `in nite' length. fig is related to the lattice size component of a Debye±Scherrer line pro le. If we simplify the notation by putting k ˆ, I 3 ˆ I, I ˆfIg and I ˆfIg, we nd the general expression I d s ˆR r K d z d d;r ; 6 where K d is the general transformation kernel K d z ˆ d= z= d= J d= z and the olume element in d-dimensional space is gien by d d;r ˆ d= d= rd dr: The inersion of (6) is r ˆR I d s K d z d d;s ; where d d;s is the reciprocal-space analog to d d;r. 3. Relationship between I d and the deriaties of c From (9), we obtain n r ˆR s n I d s K n d z d d;s; where the superscript n represents the nth deriatie with respect to r or z, respectiely. For the inersion of the integral transform (), we expect a relationship of the type s n I d s ˆR n r Kd;n z d d;r ; where Kd;n is the inerse kernel to K n d. It is shown in Appendix A that a general expression for the inerse kernel can be found: Kd;n z ˆ z n d = d F n d n =; d n = z 4 ; where d n ˆ d n = d is the Pochhammer symbol and F is a hypergeometric function. Tables and show explicit expressions for d ˆ,, 3, and alues of n up to 4. For odd alues of d, the inerse kernel is composed of trigonometric functions. Een alues of d result in combinations of Bessel functions J and Strue functions H, both of the rst kind. It is readily eri ed that the inerse kernels Kd;n follow a three-termed recurrence relation Kd;n z ˆ n zkd;n z d n 3 zkd;n 3 z Š; d;n z 3 which can be used for n > to extend these tables. It should also be noted that symbolic software packages such as Mathematica or Maple hae no problems nding explicit expressions for speci ed d and n. Howeer, especially for the case where n ˆ, the practical use of these explicit expressions is limited since the computational effort of calculating a single Strue H function, being itself a hypergeometric function of the F type, is comparable to calculating the complete K;n kernel using the generic expression (), and analytical properties such as asymptotic expansions are also better inestigated with the generic kernel. The equialent of the integral transform () has been discussed by MeÂring & Tchoubar (968) and the problem of its inersion has been mentioned, but a solution was not found. A similar problem has been considered by Ciccariello (995) for the case d ˆ 3 and odd n which can be reformulated in the form (). A solution has been gien, but it is complicated and lengthy. The general kernel of the inerse transformation (), especially presented in its concise form (), has been unknown until now. Acta Cryst. (). A57, 48±49 Burger and Ruland Chord-length distributions 483

3 4. Relationship between I d and the chord-length distribution The relationship between and the chord-length distribution g (Me ring & Tchoubar-Vallat, 965, 966, Me ring & Tchoubar, 968) is gien by (). The Porod length l p is the aerage chord length of the two-phase system de ned by l p ˆ R lg l dl ˆ hl i hl i ; 4 where hl j i is the number aerage of the chord length within phase j. The non-oscillatory component in Kd; is related to the Porod asymptote. We consider the relationship s I 3 s ˆ r K3; z d d;r ˆ r z d d;r r z cos z z sin z d d;r ˆ 4 s l p Since, by de nition, g r dr R g r dr ˆ ; 3 g r cos z z sin z dr5: 5 6 we obtain 3 s 4 I 3 s ˆlp G 3 s Š: 7 From (7) and (5), we nd Porod's law (Porod, 95a,b) for I 3 : lim s! 3 s 4 I 3 s ˆ=l p 8 and the relationship G 3 s ˆ 3 s 4 l p I 3 s ˆ R g r z = 3; z Š dr; which is the inersion of the relationship g r ˆ 8 R G 3 s K 3 z ds: 9 For arbitrary alues of d, the non-oscillatory component of Kd; z is d z d. This leads to G d s ˆ 3 d = s d d =Š l pi d s ˆ g r zd d K d; z dr; which is the inersion of the integral transform g r ˆ 4= d =Š d= G d s K d z ds: For arbitrary alues of d, Porod's law is gien by 3 d = s d lim s! d =Š I d s ˆ : 3 l p The equialent of the integral transform () for d ˆ ; ; 3 has already been gien by Me ring & Tchoubar (968), whereas the inerse transform () has been unknown until now. 5. Applications 5.. Spheres We consider the CF of a sphere of diameter D and its higher-order deriaties with respect to x ˆ r=d: s ˆ 3x x3 x 4 s ˆ 3x x 5 s 3 ˆ 3 x x Š 6 s 4 ˆ 3 x x Š: 7 is the Heaiside unit-step function [ x ˆ x ] and the Dirac function. Equations (4)±(7) proide an instructie example how to apply the inerse transformation if singularities are present which can be reduced to functions upon multiple differentiation. As our inerse transformation () holds for arbitrary differentiation order n, we are at liberty to differentiate our CF or CLD as often as we wish until the resulting n is in a form suitable to perform the integration. Since anrintegration oer a function triially reduces to the kernel, x t K t dt ˆ K x, functions are preferred targets for this multiple differentiation process. If we encounter a situation where n consists of a linear combination of terms where their functions appear at different n, like it is in (6) and (7), we may een split this linear combination into single terms and treat each term separately with a different n, this technique being justi ed by the fact that both the differentiation as well as our inerse integral transform are linear operations. While we beliee this intuitie argument is rigorous and, thus, suf cient, the same result can also be shown in a more formal way inoling integration by parts. Hence, applying () to the terms (` shells') in s 3 and s 4, we obtain I s;d ˆ 6d= D d d= z 3 Kd;3 z z 4 Kd;4 z Š; 8 where z ˆ Ds. Ford ˆ 3, this reduces to the well known Rayleigh expression for the scattering of a sphere. For d ˆ, we obtain an analytical expression for the scattering of a sphere using in nite slit collimation. Clearly, the sphere represents a special case, since its CLD can be completely reduced to functions so that the exact solution for the scattering cure can be expressed in terms of 484 Burger and Ruland Chord-length distributions Acta Cryst. (). A57, 48±49

4 inerse kernels. In the general case, there will be additional terms which cannot be reduced to functions. In this case, functions can only be produced at the singularites of the CLD and the application of the described technique to only these terms leads to the asymptotic behaior of the scattering cure. 5.. Ellipsoids of reolution The rst oscillatory components of the SAS of an ellipsoid of reolution hae been deried by Wu & Schmidt (973a). In this section, we show a way to determine all components of the asymptotic behaior of the SAS. The direction-dependent CF of an ellipsoid of reolution e x is obtained from the CF of a sphere by an af ne deformation e x ˆ s Tx 9 using the tensor k = T k = A: 3 =k Since x ˆ r=d and det T ˆ, the olume of the ellipsoid is equal to that of the corresponding sphere (V ˆ D 3 =6) independent of the deformation parameter. The semiaxes of the ellipsoid are a ˆ b ˆ D= = and c ˆ D=; < de nes a prolate, > an oblate ellipsoid, respectiely. Using polar coordinates (polar angle, ˆ cos ), we nd e x ˆ s xf Š; 3 where f ˆ Š = : 3 The third and fourth deriaties of e with respect to x are e 3 x; ˆ 3 f Š 3 f xf Š x xf Šg 33 e 4 x; ˆ 3 f Š 3 f xf Š f xf Šg: 34 The terms (` shells') in (33) and (34) are e 3 x; Š ˆ 3 f Š 3 xf Š 35 e 4 x; Š ˆ 3 f Š 4 xf Š: 36 The spherical aerage is obtained by (note that d ˆ sin d) h n e x; i! n x ˆ e R n e x; d: 37 For the spherical aerage of the shell in s 3, we nd the expressions 3 e 3 x Š ˆ x 4 3 x Š = x = x Š ˆ 3= expfi sgn Š =4g x 4 j 3 j = x = x = Š = x = x Š: The spherical aerage of the shell in 4 e is gien by 38 Applying () to the shells in e 3 the parameter y ˆ Ds, we obtain where I e;3 y ˆ 4 e x Š ˆ x 3 e x Š : 39 4 y 4 = x e 3 x Š and 4 e and introducing yk33 xy x K 34 xy dx ˆ y 4 = 3 e x Š F x; y dx; 4 F x; y ˆ4x yk33 xy x K 34 xy ˆ 4x 4 4x cos xy xy 4 cos xy xy 8 sin xy : 4 y From the rst non-oscillatory term of F x; y, we obtain the Porod term I e;3;p : I e;3;p y y 4 4 x e 3 x Š dxa = ˆ y 4 3 = arccos 3= : 4 Accordingly, the aerage chord length is l p ˆ 4D 3 3 = arccos 3= : 43 From the second non-oscillatory term of F x; y, we obtain the Kirste & Porod (96) term I e;3;kp : I e;3;kp y ˆ y 4 3 dx y e x Š A x = ˆ y arccos 3 3 3= : 44 = The oscillatory terms I e;3;osc are gien by the integral transform which results in I e;3;osc y ˆ y 4 = e 3 x Š 4x cos xy 4 cos xy xy 8 sin xy y dx; 45 Acta Cryst. (). A57, 48±49 Burger and Ruland Chord-length distributions 485

5 y 4 I e;3;osc y ' X a n 9 n =4 y nˆ = n j 3 j = cos = y n= sgn =4Š X b n sin y n= : 46 nˆ 3 yš n Explicit expressions for the coef cients a n and b n are gien in Appendix D Relationship between oscillations in I d and discontinuities in g and its deriaties Wu & Schmidt (974) hae reported that discontinuities in g and g lead to oscillatory components I osc of the type I osc s /s 4 X k j k sin D k s k D k s k ; 47 r ˆ D k is considered to be a position where g r or g r has a discontinuity. k and j k are parameters that are determined by the principal curatures and other properties of the interface at the point of contact of the chord of length D k. An essential result of their paper is the statement that the exponents k are expected to show alues in the range k. The results of x3 offer the possibility of examining this statement Integer exponents. Finite discontinuities in g r and its deriaties are characterized by the appearance of Dirac functions in the next-higher deriatie. If, for example, g n r shows a step of height g n k at r ˆ D k, then g n r contains a function of weight g n k at r ˆ D k. Considering the intensity component I k related to ˆ l p n k,we nd g n k s 4 I k s /s n r kg n k K d;n 3 z k ; 48 where z k ˆ D k s. Taking into account that the oscillatory components of K3;n are proportional to z cos z and z sin z for odd alues of n, and to z sin z and z cos z for een alues of n, we nd that nite discontinuities in the nth deriatie of g r result in exponents k with the alues n and n, and that the corresponding phases k as de ned in (47) can only take the alues or =. Generally, nite discontinuities are located at D k alues which correspond to minimal or maximal dimensions of particles (Wu & Schmidt, 973a). If we consider the method used to derie 3 in x5., nite discontinuities appear when the shell of the aeraging sphere touches the shell of the function 3 x Š at a distance r ˆ D k at which the principal radii of curature (i.e. the reciprocal absolute alues of the Figure Prolate ellipsoid of reolution (dark) and spherical aeraging shell (light) touching it at the pole. The principal radii of curature of the prolate ellipsoid are both smaller than the radius of the aeraging shell. Figure Oblate ellipsoid of reolution (light) and spherical aeraging shell (dark) touching it at the pole. The principal radii of curature of the oblate ellipsoid are both larger than the radius of the aeraging shell. Figure 3 Oblate ellipsoid of reolution (dark) and spherical aeraging shell (light) touching it at the equator. One principal radius of curature of the oblate ellipsoid is equal to and the other is smaller than the radius of the aeraging shell. 486 Burger and Ruland Chord-length distributions Acta Cryst. (). A57, 48±49

6 principal curatures themseles) of the latter are either both smaller (Fig. ) or both larger (Fig. ) than D k Non-integer exponents. Non-integer exponents can be related to in nite discontinuities of the type x k or x k, where k < andxˆr=d k. As we hae seen in x5., in nite discontinuities with k ˆ= appear when the shell of the aeraging sphere touches the shell of the function 3 x Š at a distance r ˆ D k at which one principal radius of curature of the latter is equal to D k and the other is either smaller (Fig. 3) or larger (Fig. 4) than D k. Wu & Schmidt (974) hae shown that this type of discontinuity appears in all conex particles of reolution. If we use the method gien in x5. for general ellipsoids, we obtain in 3, apart from two nite discontinuities at the smallest and the largest diameter, a logarithmic singularity of the type ln j xj. This type of singularity has already been discussed by Wu & Schmidt (973b). It should be mentioned that under certain conditions both a nite step and a logarithmic singularity can occur (Ciccariello, 99a,b). In 4, the logarithmic singularity produces a discontinuity of the type j xj sgn x. The corresponding D k is de ned by the intermediate diameter of the ellipsoid. Thus, in nite discontinuities with integer alues of k appear in n with n > 3 when the shell of the aeraging sphere touches the shell of the function 3 x Š at a distance r ˆ D k at which one principal radius of curature of the latter is larger and the other is smaller than D k (see Fig. 5). These obserations suggest that the only non-integer exponents to be expected in the SAS of monodisperse conex particles are of the type k ˆ n = Generation of general non-integer exponents. We consider a hypothetical CLD of the type g x /x x ; 49 where x ˆ r=d max. g x is supposed to be non-zero only in the interal x. We consider alues larger than zero. For ˆ, we obtain the CLD of a sphere of diameter D max. The scattering intensity I (ˆ I 3 ) corresponding to (49) is gien by I s ˆF 5=; 4 z 4 ; 5 where z is D max s. For large alues of s, we obtain the asymptote proportional to z a 4 z a cos z = a 3 sin z = 6 z 4 z 5 cos z = O ; 5 z 6 where the coef cients a k are functions of. If<<, the rst deriatie g x / x x x 5 shows an in nite discontinuity at x ˆ with an arbitrary noninteger exponent. For larger alues of, in nite discontinuities appear in the corresponding deriaties of higher order. It is of interest to note that (49) is equialent to the CLD of a polydisperse dilute system of spheres with a number distribution of diameters h D of the type h D x /x x ; 53 where x ˆ D=D max. There is, apparently, no monodisperse system of particles corresponding to a chord-length distribution as gien by (49). Figure 4 Prolate ellipsoid of reolution (light) and spherical aeraging shell (dark) touching it at the equator. One principal radius of curature of the prolate ellipsoid is equal to and the other is larger than the radius of the aeraging shell. Figure 5 General ellipsoid of reolution (dark) with three different semi-axes a 6ˆ b 6ˆ c and spherical aeraging shell (light) touching it. One principal radius of curature of the ellipsoid is smaller and the other is larger than the radius of the aeraging shell. Acta Cryst. (). A57, 48±49 Burger and Ruland Chord-length distributions 487

7 6. Experimental requirements for an accurate determination of CLDs The determination of g r from experimental data requires not only a ery high accuracy of the SAS intensity, especially at large s alues (Porod asymptote), but also that one is fully aware of all the necessary corrections to be taken into account. These are collimation effects, the subtraction of the scattering due to density uctuations within the phases, and the elimination of the effect of the nite width of the phase boundaries (Ruland, 97). Furthermore, the statistical structure of the phase boundaries (Ruland, 987; Semeno, 994) can produce supplementary effects. In general, a comprehensie approach taking into account all these effects in a single model which works on the original data and has proper error propagation is preferable to a stepwise application of subsequent corrections where fundamental problems such as ill-posedness and data cut-offs can quickly render the result of one step of the chain meaningless, and usually error propagation is also lost. It is for example highly questionable if any meaningful Porod analysis is still possible after a separate collimation desmearing step. Without a proper treatment of these corrections, the determination of Porod's asymptote is rather arbitrary. In this context, it should be noted that log±log plots of uncorrected intensities ersus s produce, in many cases, fractional exponents which can be misinterpreted as the scattering from fractal structures (Ruland, ). APPENDIX A Deriation of the inerse kernel We assume r to be de ned for positie alues of r only, and continuously differentiable at r ˆ. We consider the integral transform n r ˆR where the kernel is gien by with z ˆ rs and s n I d s K n d z d d;s; 54 K d z ˆ d= z= d= J d= z ; d d;s ˆ d= d= sd ds is the olume element in reciprocal space. We want to determine the inerse kernel Kd;n with the property s n I d s ˆR n r Kd;n z d d;r ; 57 where d d;r ˆ d= d= rd dr: The relationship between the kernels can be de ned by 58 K n d rs d= d= s d d;n rs d= d= rd dr ˆ s s : 59 A closed-form solution for arbitrary n can be obtained by representing K n d z as a Meijer's G function d z K d z ˆF 4 ˆ d G z 4 ; d= 6 K n d z ˆ n d= G z n = 3 4 ; 6 ; =; d n = where the deriatie order n now is an arbitrary parameter. The integral of the product of two G functions results in another G function. Dirac's function can be represented by G. The inerse kernel is assumed to be representable by a G function, the coef cients of which hae to be determined such that all parameters of G anish in the result of (59). This is the case for Kd;n z ˆ n d= G 3 z d n = 4 n=; d =; d= z d n =; d n = 4 ˆ z n d n F d = : 6 APPENDIX B Alternatie deriation of the inerse kernel For d ˆ, an equialent expression can be obtained by considering the complete function c de ned by c r ˆ jrj : Using Fourier transform theory, we nd 63 c n r ˆ R s n I s cos n z ds 64 s n I s ˆ R c n r cos n z dr: 65 For m, the deriaties of c r are gien by m c m c r ˆ m jrj r ˆ m jrj This results in Pm kˆ m k k r sgn r Pm m k k r : kˆ Burger and Ruland Chord-length distributions Acta Cryst. (). A57, 48±49

8 " R s m I s ˆ m m r cos z dr Pm kˆ k s k m k " R s m I s ˆ m m r sin z dr Pm kˆ # 68 # k s k m k : 69 It can be shown that the aboe expressions are equialent to s n I s ˆR n r K;n z d ;r ; 7 proided that i.e. R m r dr ˆ m ; m ˆ : 7 7 The latter condition has to be considered as a criterion of conergence for r. Comparable results can be found for other alues of d using relationships of the type I 3 s I s 73 I s ˆ R I 3 s y = dy: 74 APPENDIX C Asymptotic expansion of the inerse kernel The asymptotic expansion of the inerse kernel Kd;n is split into a non-oscillatory and an oscillatory component: d;n ˆ d;n;nos d;n;osc: For the non-oscillatory part, we nd Kd;n;nos z ˆ n d n z d n d 3 F ; n ; 3 n 4 z 75 : 76 For n ˆ orn ˆ, this expression anishes identically due to the dierging function in the denominator of the prefactor. Thus, there are no non-oscillatory parts for n <. Since n is an integer, one of the parameters of the 3 F function always becomes a negatie integer. Thus, the hypergeometric function reduces to a polynomial in z of the order n = for een n and n 3 = for odd n, respectiely. Explicitly, we nd d;n;nos z ˆ n d z d n n n ( n! d n 3 n z d 3 d n 3 n 4 n 5 n z 4! ) O z 6 : 77 The building law of this expression and its termination are obious. For the oscillatory component of the asymptotic expansion of the inerse kernel Kd;n, we nd d;n;osc z ˆ n = d= =z d = cos z d n =4Š d d 4n 3 cos z d 3 n =4Š 8z O d 3...z cos z d 5 n =4Š : 78 This series terminates for d ˆ after the rst and for d ˆ 3 after the second term, respectiely. It can be shown that, when both the non-oscillatory and the oscillatory series of the asymptotic expansion terminate, the asymptotic expansion is equal to the exact solution. Thus, for odd alues of d, the inerse kernel can be expressed in simple trigonometric form, see Table. The same behaior is found e.g. in Bessel functions J d=, which hae a simple trigonometric representation for odd integer d. For an een d including d ˆ, the asymptotic expansion (78) has an in nite number of terms. This case is more complicated and the exact solution cannot be expressed in trigonometric form. Howeer, it can be represented in terms of Bessel and Strue functions of the rst kind of integer orders. Explicit expressions are listed in Table. APPENDIX D Asymptotic behaior of the ellipsoid of reolution The function I e;osc of the ellipsoid of reolution is composed of the real and the imaginary parts, respectiely, of the components J k y 4 I e;osc y ˆRe J J 3 Š Im J Š; 79 where the functions J k y are the Fourier transforms of f k x, and R J k ˆ f k x exp ixy dx 8 = Acta Cryst. (). A57, 48±49 Burger and Ruland Chord-length distributions 489

9 Table 3 The coef cients a n and b n de ned in (88) of the asymptotic expansion of the ellipsoid of reolution. n a n b n 6 6= 57= 3 3= 3 43= = 3 899= = = = f x ˆ 4x e 3 x Š 8 f x ˆ8 y 3 e x Š 8 f 3 x ˆ 4 xy 3 e x Š : 83 The Fourier transforms can be obtained as an asymptotic expansion according to Erde lyi (956) of the type J k ˆ B k;n A k;n O y N ; 84 where A k;n ˆ XN n = n! y nˆ n = j 3 j = expfi = y n= sgn k x 85 n xˆ = k x ˆ x = = j 3 j = expfi sgn Š =4gf k x 86 B k ; N ˆ XN y n expfi y n Š= f k x : 87 n xˆ The result is y 4 I e;osc y ' X a n 9 n =4 y nˆ n = j 3 j = cos = y n= sgn Š=4 X b n sin y n= : nˆ 3 yš 88 n The rst six coef cients of expansion (88) are shown in Table 3. APPENDIX E Application of the inerse kernel to one-dimensional and two-dimensional two-phase systems In our treatment so far, we considered a three-dimensional isotropic two-phase system, and the spatial dimension d described the projection the corresponding intensity distribution was subject to. Let us limit the following considerations to the unprojected (unsmeared) case, but consider two-phase systems of lower dimension, namely systems of in nitely high cylinders or prisms (d ˆ ) and systems of in nitely extended lamellae (d ˆ ). In these lower-dimensional two-phase systems, we are interested in the CFs d, the CLDs g d / d and the intensity distributions I d of the two-dimensional cross section perpendicular to the cylinders or the one-dimensional section in the direction of the lamella normals, respectiely I d s ˆR d r K d z d d;r ; 89 which is different from the oerall three-dimensional CLD of the spherically aeraged system. A spherical aerage of I d leads to 8 9 >< h s s I s 3 i! >= I s ˆ hi s s 3 i! >: >; I 3 s 8 9 >< s I s >= ˆ s I s >: >; ˆ d= s d 3 d= I d s : 9 I 3 s The inerse kernel deried in this paper allows us to establish a relationship between the (in pinhole collimation) experimentally measured spherically aeraged intensity I and the subdimensional CLDs d and their higher deriaties n d. d r ˆ I d s K d z d d;s 9 n d r ˆ s n I d s K n d z d d;s 9 ˆ s n d= d= I s K n sd 3 d z d d;s 93 and the inersion is n d= n d= s 3 d n I s ˆR n d r Kd;n z d d;r : 94 It is een possible to combine a subdimensional two-phase system with a projection of the intensity distribution, but in this case the general inerse kernel can only be gien in terms of a Meijer's G function that cannot be further simpli ed. References Ciccariello, S. (99a). Phys. Re. A, 44, 975±983. Ciccariello, S. (99b). J. Appl. Cryst. 4, 59±55. Ciccariello, S. (993). Acta Cryst. A49, 398±45. Ciccariello, S. (995). J. Math. Phys. 36, 9±46. Ciccariello, S. & Benedetti, A. (98). Phys. Re. B, 6, 6384±6389. Ciccariello, S., Cocco, G., Benedetti, A. & Enzo, S. (98). Phys. Re. B, 3, 6474±6485. Ciccariello, S. & Sobry, R. (995). Acta Cryst. A5, 6±69. ErdeÂlyi, A. (956). Asymptotic Expansions, pp. 49±5. New York: Doer. Kirste, R. G. & Porod, G. (96). Kolloid. 84, ±7. MeÂring, J. & Tchoubar, D. (968). J. Appl. Cryst., 53±65. MeÂring, J. & Tchoubar-Vallat, D. (965). C. R. Acad. Sci. 6, 396± Burger and Ruland Chord-length distributions Acta Cryst. (). A57, 48±49

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