International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 2.6, pp. 89-104

## Section 2.6.1. X-ray techniques

O. Glattera

### 2.6.1. X-ray techniques

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#### 2.6.1.1. Introduction

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The purpose of this section is to introduce small-angle scattering as a method for investigation of nonperiodic systems. It should create an understanding of the crucial points of this method, especially by showing the differences from wide-angle diffraction. The most important concepts will be explained. This article also contains a collection of the most important equations and methods for standard applications. For details and special applications, one must refer to the original literature or to textbooks; the reference list is extensive but, of course, not complete.

The physical principles of scattering are the same for wide-angle diffraction and small-angle X-ray scattering. The electric field of the incoming wave induces dipole oscillations in the atoms. The energy of X-rays is so high that all electrons are excited. The accelerated charges generate secondary waves that add at large distances (far-field approach) to give the overall scattering amplitude. All secondary waves have the same frequency but may have different phases caused by the different path lengths. Owing to the high frequency, it is only possible to detect the scattering intensity – the square of the scattering amplitude – and its dependence on the scattering angle.

The angle-dependent scattering amplitude is related to the electron-density distribution of the scatterer by a Fourier transformation. All this holds for both wide-angle diffraction and small-angle X-ray scattering. The main difference is that in the former we have a periodic arrangement of identical scattering centres (particles), i.e. the scattering medium is periodic in all three dimensions with a large number of repetitions, whereas in small-angle scattering these particles, for example proteins, are not ordered periodically. They are embedded with arbitrary orientation and with irregular distances in a matrix, such as water. The scattering centres are limited in size, non-oriented, and nonperiodic, but the number of particles is high and they can be assumed to be identical, as in crystallography.

The Fourier transform of a periodic structure in crystallography (crystal diffraction) corresponds to a Fourier series, i.e. a periodic structure is expanded in a periodic function system.

The Fourier transform of a non-periodic limited structure (small-angle scattering) corresponds to a Fourier integral. In mathematical terms, it is the expansion of a nonperiodic function by a periodic function system.

So the differences between crystallography and small-angle scattering are equivalent to the differences between a Fourier series and a Fourier integral. It may seem foolish to expand a non-periodic function with a periodic function system, but this is how scattering works and we do not have any other powerful physical process to study these structures.

The essential effect of these differences is that in small-angle scattering we measure a continuous angle-dependent scattering intensity at discrete points instead of sharp, point-like spots as in crystallography.

Another important point is that in small-angle scattering we have a linear increase of the signal (scattered intensity) with the number of particles in the measuring volume since intensities are adding. Amplitudes are adding in crystallography, so we have a quadratic relation between the signal and the number of particles.

In addition, there is a loss of information in small-angle scattering experiments caused by the averaging over all orientations in space. The three-dimensional structure is represented by a one-dimensional function – the dependence of the scattered intensity on the scattering angle. This is also true for powder diffraction. To recover the structure uniquely is therefore impossible.

The computation of the scattering function for a known structure is called the solution of the scattering problem. This problem can be solved exactly for many different structures. The inversion, i.e. the estimation of the structure of the scatterer from its scattering functions, is called the inverse scattering problem. This problem cannot be solved uniquely.

The description and solution of the scattering problem gives information to the experimenter concerning the scattering functions to be expected in a special situation. In addition, this knowledge is the starting point for the evaluation and interpretation of experimental data (solution of the inverse problem).

There are methods that give a rough first-order approximation to the solution of the inverse scattering problem using only a minimum amount of a priori information about the system to obtain an initial model. In order to improve this model, one has to solve the scattering problem. The resulting theoretical model functions are compared with the experimental data. If necessary, model modifications are deduced from the deviations. After some iterations, one obtains the final model.

It should be noticed that it is possible to find different models that fit the data within their statistical accuracy. In order to reduce this ambiguity, it is necessary to have additional independent information from other experiments. Incorrect models, however, can be rejected when their scattering functions differ significantly from the experimental data.

What type of investigations can be performed with small-angle scattering? It is possible to study monodisperse and polydisperse systems. In the case of monodisperse systems, it is possible to determine size, shape, and, under certain conditions, the internal structure. Monodispersity cannot be deduced from small-angle scattering data and must therefore be assumed or checked by independent methods.

For polydisperse systems, a size distribution can be evaluated under the assumption of a certain shape for the particles (particle sizing).

All these statements are strictly true for highly diluted systems where the interparticle distances are much larger than the particle dimensions. In the case of semi-dilute systems, the result of a small-angle scattering experiment is influenced by the structure of the particles and by their spatial arrangement. Then the scattering curve is the product of the particle scattering function and of the interparticle interference function. If the scattering function of one particle is known, it is possible to evaluate information about the radial distribution of these particles relative to each other. If the system is dense, i.e. if the volume fraction of the particles (scattering centres) is of the same order of magnitude as the volume fraction of the matrix, it is possible to determine these volume fractions and a characteristic length of the phases. The most important practical applications, however, pertain to dilute systems.

How are small-angle scattering experiments related to other scattering experiments? Small-angle scattering uses radiation with a wavelength in the range 10−1 to 100 nm, depending on the problem and on the source used. This range is similar for X-rays and neutrons, but neutrons interact with the nuclei of the atoms whereas X-rays interact with the electrons. The scattering efficiency increases linearly with the atomic number for X-rays. The dependence is much more complicated for neutrons and does not show a systematic trend. The essential fact in neutron scattering is the pronounced difference in the scattering power between hydrogen and deuterium, which is important for varying the contrast between the particles and the matrix.

The wavelength and the scattering efficiency limit the range of small-angle scattering experiments to systems in the size range of a few nanometres up to about 100 nm. Special instruments permit the study of larger particles (Bonse & Hart, 1967; Koch, 1988). These instruments need a high intensity of the primary beam (synchrotron radiation) and are not very common.

Particles in the size range from 100 nm up to some micrometres can be investigated by static light-scattering techniques (Glatter, Hofer, Jorde & Eigner, 1985; Glatter & Hofer, 1988a,b; Hofer, Schurz & Glatter, 1989). Particles exceeding this limit can be seen in an optical microscope or can be studied with Fraunhofer diffraction (Bayvel & Jones, 1981).

Electron microscopy is in competition with all these scattering methods. It has the marked advantage of giving real pictures with rather high resolution but it has the inherent disadvantage that the preparation may introduce artefacts. Small-angle scattering, on the other hand, is a method to study macromolecules in solution, which is a very important advantage for biological samples and for polymers.

Crystallography gives more information about the particle (atomic structure) and can be applied to relatively large systems. It is possible to study particles as large as proteins and viruses if good crystals of these substances are available. The experiment needs synchrotron radiation for large molecules like proteins or viruses, i.e. access to a large research facility is necessary. Small-angle X-ray scattering with conventional generators is a typical next-door technique with the advantage of ready availability.

Small-angle scattering has developed into a standard measuring method during recent decades, being most powerful for the investigation of submicrometre particles.

#### 2.6.1.2. General principles

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In this subsection, we are concerned with X-rays only, but all equations may also be applied with slight modifications to neutron or electron diffraction. When a wave of X-rays strikes an object, every electron becomes the source of a scattered wave. All these waves have the same intensity given by the Thomson formula where is the primary intensity and a the distance from the object to the detector. The factor is the square of the classical electron radius (e2/mc2 = 7.90 × 10−26[cm2]). The scattering angle 2 is the angle between the primary beam and the scattered beam. The last term in (2.6.1.1) is the polarization factor and is practically equal to 1 for all problems dealt with in this subsection. should appear in all following equations but will be omitted, i.e. the amplitude of the wave scattered by an electron will be taken to be of magnitude 1. is only needed in cases where the absolute intensity is of interest.

The amplitudes differ only by their phases ϕ, which depend on the positions of the electrons in space. Incoherent (Compton) scattering can be neglected for small-angle X-ray scattering. The phase ϕ is 2π/λ times the difference between the optical path length of the wave and an arbitrary reference wave (with λ being the wavelength). The direction of the incident beam is defined by the unit vector s0 and of the scattered beam by s. The angle between these two unit vectors (scattering angle) is 2. The path difference between the rays through a point P and an arbitrary origin O is −r(ss0). The phase is if we define the scattering vector h as This vector bisects the angle between the scattered beam and the incident beam and has length . We keep in mind that may be replaced by in small-angle scattering. We now introduce the electron density ρ(r). This is the number of electrons per unit volume at the position r. A volume element dV at r contains ρ(r) dV electrons. The scattering amplitude of the whole irradiated volume V is given by We see that the amplitude A is the Fourier transform of the electron-density distribution ρ. The intensity I(h) of the complex amplitude A(h) is the absolute square given by the product of the amplitude and its complex conjugate A*, where is the convolution square (Bracewell, 1986): The intensity distribution in h or reciprocal space is uniquely determined by the structure in real space.

Until now, we have discussed the scattering process of a particle in fixed orientation in vacuum. In most cases of small-angle scattering, the following situation is present:

–The scatterers (particles or inhomogeneities) are statistically isotropic and no long-range order exists, i.e. there is no correlation between points at great spatial distance.

–The scatterers are embedded in a matrix. The matrix is considered to be a homogeneous medium with the electron density ρ0. This situation holds for particles in solution or for inhomogeneities in a solid. The electron density in equations (2.6.1.3)–(2.6.1.5) should be replaced by the difference in electron density Δρ = ρ − ρ0, which can take positive and negative values.

The average over all orientations leads to (Debye, 1915) and (2.6.1.4) reduces to the form or, with to γ is the so-called correlation function (Debye & Bueche, 1949), or characteristic function (Porod, 1951). The function p(r) is the so-called pair-distance distribution function PDDF (Guinier & Fournet, 1955; Glatter, 1979). The inverse transform to (2.6.1.9) is given by or by The function p(r) is directly connected with the measurable scattering intensity and is very important for the solution of the inverse scattering problem. Before working out details, we should first discuss equations (2.6.1.9) and (2.6.1.10).

The PDDF can be defined as follows: the function p(r) gives the number of difference electron pairs with a mutual distance between r and r + dr within the particle. For homogeneous particles (constant electron density), this function has a simple and clear geometrical definition.

Let us subdivide the particle into a very large number of identical small volume elements. The function p(r) is proportional to the number of lines with a length between r and r + dr which are found in the combination of any volume element i with any other volume element k of the particle (see Fig. 2.6.1.1 ). For r = 0, there is no other volume element, so p(r) must be zero, increasing with as the number of possible neighbouring volume elements is proportional to the surface of a sphere with radius r. Starting from an arbitrary point in the particle, there is a certain probability that the surface will be reached within the distance r. This will cause the p(r) function to drop below the parabola and finally the PDDF will be zero for all , where D is the maximum dimension of the particle. So p(r) is a distance histogram of the particle. There is no information about the orientation of these lines in p(r), because of the spatial averaging.

 Figure 2.6.1.1 | top | pdf |The height of the p(r) function for a certain value of r is proportional to the number of lines with a length between r and r + dr within the particle.

In the case of inhomogeneous particles, we have to weight each line by the product of the difference in electron density Δρ, and the differential volume element, dV. This can lead to negative contributions to the PDDF.

We can see from equation (2.6.1.9) that every distance r gives a sin(hr)/(hr) contribution with the weight p(r) to the total scattering intensity. I(h) and p(r) contain the same information, but in most cases it is easier to analyse in terms of distances than in terms of sin(x)/x contributions. The PDDF could be computed exactly with equation (2.6.1.10) if I(h) were known for the whole reciprocal space.

For h = 0, we obtain from equation (2.6.1.9) i.e. the scattering intensity at h equal to zero is proportional to the area under the PDDF. From equation (2.6.1.11), we find (Porod, 1982), i.e. the integral of the intensity times is related to the mean-square fluctuation of the electron density irrespective of the structure. We may modify the shape of a particle, the scattering function I(h) might be altered considerably, but the integral (2.6.1.13) must remain invariant (Porod, 1951).

#### 2.6.1.3. Monodisperse systems

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In this subsection, we discuss scattering from monodisperse systems, i.e. all particles in the scattering volume have the same size, shape, and internal structure. These conditions are usually met by biological macromolecules in solution. Furthermore, we assume that these solutions are at infinite dilution, which is taken into account by measuring a series of scattering functions at different concentrations and by extrapolating these data to zero concentration. We continue with the notations defined in the previous subsection, which coincide to a large extent with the notations in the original papers and in the textbooks (Guinier & Fournet, 1955; Glatter & Kratky, 1982). There is a notation created by Luzzati (1960) that is quite different in many details. A comparison of the two notations is given in the Appendix of Pilz, Glatter & Kratky (1980).

The particles can be roughly described by some parameters that can be extracted from the scattering function. More information about the shape and structure of the particles can be found by detailed discussion of the scattering functions. At first, this discussion will be about homogeneous particles and will be followed by some aspects for inhomogeneous systems. Finally, we have to discuss the influence of finite concentrations on our results.

#### 2.6.1.3.1. Parameters of a particle

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Total scattering length. The scattering intensity at h = 0 must be equal to the square of the number of excess electrons, as follows from equations (2.6.1.7) and (2.6.1.12): This value is important for the determination of the molecular weight if we perform our experiments on an absolute scale (see below).

Radius of gyration. The electronic radius of gyration of the whole particle is defined in analogy to the radius of gyration in mechanics: It can be obtained from the PDDF by or from the innermost part of the scattering curve [Guinier approximation (Guinier, 1939)]: A plot of log[I(h)] vs h2 (Guinier plot) shows at its innermost part a linear descent with a slope tan α, where (see Table 2.6.1.1).

 Table 2.6.1.1| top | pdf | Formulae for the various parameters for h (left) and m (right) scales

The radius of gyration is related to the geometrical parameters of simple homogeneous triaxial bodies as follows (Mittelbach, 1964):

Radius of gyration of the cross section. In the special case of rod-like particles, the two-dimensional analogue of is called radius of gyration of the cross section . It can be obtained from where is the PDDF of the cross section or it can be calculated from the innermost part of the scattering intensity of the cross section : with (see Table 2.6.1.1).

Radius of gyration of the thickness. A similar definition exists for lamellar particles. The one-dimensional radius of gyration of the thickness can be calculated from or from the innermost part of the scattered intensity of thickness : with (see Table 2.6.1.1 and §2.6.1.3.2.1).

Volume. The volume of a homogeneous particle is given by This equation follows from equations (2.6.1.12)–(2.6.1.14). Such volume determinations are subject to errors as they rely on the validity of an extrapolation to zero angle [to obtain ] and to larger angles ( extrapolation for Q). Scattering functions cannot be measured from h equal to zero to h equal to infinity.

Surface. The surface S of one particle is correlated with the scattering intensity of this particle by Determination of the absolute intensity can be avoided if we calculate the specific surface (Mittelbach & Porod, 1965)

Cross section, thickness, and correlation length. By similar equations, we can find the area A of the cross section of a rod-like particle and the thickness T of lamellar particles by but the experimental accuracy of the limiting values and is usually not very high.

The correlation length is the mean width of the correlation function γ(r) (Porod, 1982) and is given by

The maximum dimension D of a particle would be another important particle parameter, but it cannot be calculated directly from the scattering function and will be discussed later.

Persistence length . An important model for polymers in solution is the so-called worm-like chain (Porod, 1949; Kratky & Porod, 1949). The degree of coiling can be characterized by the persistence length (Kratky, 1982b). Under the assumption that the persistence length is much larger than the cross section of the polymer, it is possible to find a transition point h+ in an I(h)h2 vs h plot where the function starts to be proportional to h. There is an approximation depending on the length of the chain (Heine, Kratky & Roppert, 1962). For further details, see Kratky (1982b).

Molecular weight. Particles of arbitrary shape. The particle is measured at high dilution in a homogeneous solution and has an isopotential specific volume and mol. electrons per gram, i.e. the molecule contains electrons if M is the molecular weight. The number of effective mol. electrons per gram is given by where is the mean electron density of the solvent. The molecular weight can be determined from the intensity at zero angle I(0):(Kratky, Porod & Kahovec, 1951), where P is the total intensity per unit time irradiating the sample, a [cm] is the distance between the sample and the plane of registration, d [cm] is the thickness of the sample, c [g cm−3] is the concentration, and NL is Loschmidt's (Avogadro's) number.

Rod-like particles. The mass per unit length Mc = M/L, i.e. the mass related to the cross section of a rod-like particle with length L, is given by a similar equation (Kratky & Porod, 1953):

Flat particles. A similar equation holds for the mass per unit area Mt = M/A:

Abscissa scaling. The various molecular parameters can be evaluated from scattered intensities with different abscissa scaling. The abscissa used in theoretical work is . The most important experimental scale is m [cm], the distance of the detector from the centre of the primary beam with the distance a [cm] between the sample and the detector plane. withThe angular scale 2θ with was used in the early years of small-angle X-ray scattering experiments. The formulae for the various parameters for m and the h scale can be found in Table 2.6.1.1, the formulae for the 2θ scale can be found in Glatter & Kratky (1982, p. 158).

#### 2.6.1.3.2. Shape and structure of particles

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In this subsection, we have to discuss how shape, size, and structure of the scattering particle are reflected in the scattering function I(h) and in the PDDF p(r). In general, it is easier to discuss features of the PDDF, but some characteristics like symmetry give more pronounced effects in reciprocal space.

#### 2.6.1.3.2.1. Homogeneous particles

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Globular particles. Only a few scattering problems can be solved analytically. The most trivial shape is a sphere. Here we have analytical expressions for the scattering intensity and for the PDDF (Porod, 1948)where R is the radius of the sphere. The graphical representation of scattering functions is usually made with a semi-log plot [logI(h) vs h] or with a log–log plot [logI(h) vs log h]; the PDDF is shown in a linear plot. In order to compare functions from particles of different shape, it is preferable to keep the scattering intensity at zero angle (area under PDDF) and the radius of gyration [slope of the main maximum of I(h) or the second moment of p(r)] constant.

The scattering function of a sphere with R = 65 is shown in Fig. 2.6.1.2 [dashed line, logI(0) normalized to 12]. We see distinct minima which are typical for particles of high symmetry. We can determine the size of the sphere directly from the position of the zeros and (Glatter, 1972). or from the position of the first side maximum (. The minima are considerably flattened in the case of cubes (full line in Fig. 2.6.1.2). The corresponding differences in real space are not so clear-cut (Fig. 2.6.1.3 ). The p(r) function of the sphere has a maximum near r = R = D/2 (x 0.525) and drops to zero like every PDDF at r = D, where D is the maximum dimension of the particle – here the diameter. The p(r) for the cube with the same is zero at r 175. The function is very flat in this region. This fact demonstrates the problems of accuracy in this determination of D when we take into account experimental errors. In any case, this accuracy will be different for different shapes.

 Figure 2.6.1.2 | top | pdf |Comparison of the scattering functions of a sphere (dashed line) and a cube (continuous line) with same radius of gyration.
 Figure 2.6.1.3 | top | pdf |Distance distribution function of a sphere (dashed line) and a cube (continuous line) with the same radius of gyration and the same scattering intensity at zero angle.

Any deviation from spherical symmetry will shift the maximum to smaller r values and the value for D will increase [ and constant!]. A comparison of PDDF's for a sphere, an oblate ellipsoid of revolution (axial ratio 1:1:0.2), and a prolate ellipsoid of revolution (1:1:3) is shown in Fig. 2.6.1.4 . The more we change from the compact, spherical structure to a two- and one-dimensionally elongated structure, the more the maximum shifts to smaller r values and at the same time we have an increase in D. We see that is a very informative function. The interpretation of scattering functions in reciprocal space is hampered by the highly abstract nature of this domain. We can see this problem in Fig. 2.6.1.5 , where the scattering functions of the sphere and the ellipsoids in Fig. 2.6.1.4 are plotted. A systematic discussion of the features of can be found elsewhere (Glatter, 1979, 1982b).

 Figure 2.6.1.4 | top | pdf |Comparison of the p(r) function of a sphere (continuous line), a prolate ellipsoid of revolution 1:1:3 (dash-dotted line), and an oblate ellipsoid of revolution 1:1:0.2 (dashed line) with the same radius of gyration.
 Figure 2.6.1.5 | top | pdf |Comparison of the I(h) functions of a sphere, a prolate, and an oblate ellipsoid (see legend to Fig. 2.6.1.4).

Rod-like particles. The first example of a particle elongated in one direction (prolate ellipsoid) was given in Figs. 2.6.1.4 and 2.6.1.5. An important class is particles elongated in one direction with a constant cross section of arbitrary shape (long cylinders, parallelepipeds, etc.) The cross section A (with maximum dimension d) should be small in comparison to the length of the whole particle L: The scattering curve of such a particle can be written as where the function is related only to the cross section and the factor 1/h is characteristic for rod-like particles (Kratky & Porod, 1948; Porod, 1982). The cross-section function is This function was used in the previous subsection for the determination of the cross-section parameters , A, and . In addition, we have where is the zero-order Bessel function and (Glatter, 1982a). The function is the PDDF of the cross section with The symbol * stands for the mathematical operation called convolution and the symbol means averaging over all directions in the plane of the cross section. Rod-like particles with a constant cross section show a linear descent of p(r) for if . The slope of this linear part is proportional to the square of the area of the cross section, The PDDF's of parallelepipeds with the same cross section but different length L are shown in Fig. 2.6.1.6 . The maximum corresponds to the cross section and the point of inflection gives a rough indication for the size of the cross section. This is shown more clearly in Fig. 2.6.1.7 , where three parallelepipeds with equal cross section area A but different cross-section dimensions are shown. If we find from the overall PDDF that the particle under investigation is a rod-like particle, we can use the PDDF of the cross section to obtain more information on the cross section (Glatter, 1980a).

 Figure 2.6.1.6 | top | pdf |Distance distributions from homogeneous parallelepipeds with edge lengths of: (a) 50 × 50 × 500 Å; (b) 50 × 50 × 250 Å; (c) 50 × 50 × 150 Å.
 Figure 2.6.1.7 | top | pdf |Three parallelepipeds with constant length L (400 Å) and a constant cross section but varying length of the edges: continuous line 40 × 40 Å; dash-dotted line 80 × 20 Å; dashed line 160 × 10 Å. (a) p(r) function. (b) f(r) = p(r)/r.

Flat particles. Flat particles, i.e. particles elongated in two dimensions (discs, flat parallelepipeds), with a constant thickness T much smaller than the overall dimensions D, can be treated in a similar way. The scattering function can be written as where is the so-called thickness factor (Kratky & Porod, 1948) or which can be used for the determination of , T, and . In addition, we have again: and PDDF's from flat particles do not show clear features and therefore it is better to study (Glatter, 1979). The corresponding functions for lamellar particles with the same basal plane but different thickness are shown in Fig. 2.6.1.7(b). The marked transition points in Fig. 2.6.1.7(b) can be used to determine the thickness. The PDDF of the thickness can give more information in such cases, especially for inhomogeneous particles (see below).

Composite structures – aggregates, subunits. The formation of dimers can be analysed qualitatively with the p(r) function (Glatter, 1979). For an approximate analysis, it is only necessary to know the PDDF of the monomer. Different types of aggregates will have distinct differences in their PDDF. Higher aggregates generally cannot be classified unambiguously. Additional information from other sources, such as the occurrence of symmetry, can simplify the problem.

Particles that consist of aggregates of a relatively large number of identical subunits show, at low resolution, the overall structure of the whole particle. At larger angles (higher resolution), the influence of the individual subunits can be seen. In the special case of globular subunits, it is possible to determine the size of the subunits from the position of the minima of the corresponding shape factors using equation (2.6.1.39) (Glatter, 1972; Pilz, Glatter, Kratky & Moring-Claesson, 1972).

#### 2.6.1.3.2.2. Hollow and inhomogeneous particles

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We have learned to classify homogeneous particles in the previous part of this section. It is possible to see from scattering data [I(h) or p(r)] whether a particle is globular or elongated, flat or rod-like, etc., but it is impossible to determine uniquely a complicated shape with many parameters. If we allow internal inhomogeneities, we make things more complicated and it is clear that it is impossible to obtain a unique reconstruction of an inhomogeneous three-dimensional structure from its scattering function without additional a priori information. We restrict our considerations to special cases that are important in practical applications and that allow at least a solution in terms of a first-order approximation. In addition, we have to remember that the p(r) function is weighted by the number of excess electrons that can be negative. Therefore, a minimum in the PDDF can be caused by a small number of distances, or by the addition of positive and negative contributions.

Spherically symmetric particles. In this case, it is possible to describe the particle by a one-dimensional radial excess density function Δρ(r). For convenience, we omit the Δ sign for excess in the following. As we do not have any angle-dependent terms, we have no loss of information from the averaging over angle. The scattering amplitude is simply the Fourier transform of the radial distribution: and(Glatter, 1977a). These equations would allow direct analysis if A(h) could be measured, but we can measure only I(h). ρ(r) can be calculated from I(h) using equation (2.6.1.10) remembering that this function is the convolution square of ρ(r) [equations (2.6.1.5) and (2.6.1.8)]. Using a convolution square-root technique, we can calculate ρ(r) from I(h) via the PDDF without having a phase problem' like that in crystallography; i.e. it is not necessary to calculate scattering amplitudes and phases (Glatter, 1981; Glatter & Hainisch, 1984; Glatter, 1988). This can be done because ρ(r) differs from zero only in the limited range (Hosemann & Bagchi, 1952, 1962). In mathematical terms, it is again the difference between a Fourier series and a Fourier integral.

Details of the technique cannot be discussed here, but it is a fact that we can calculate the radial distribution ρ(r) from the scattering data assuming that the spherical scatterer is only of finite size. The hollow sphere can be treated either as a homogeneous particle with a special shape or as an inhomogeneous particle with spherical symmetry with a step function as radial distribution. The scattering function and the PDDF of a hollow sphere can be calculated analytically. The p(r) of a hollow sphere has a triangular shape and the function f(r) = p(r)/r shows a horizontal plateau (Glatter, 1982b).

Rod-like particles. Radial inhomogeneity. If we assume radial inhomogeneity of a circular cylinder, i.e. ρ is a function of the radius r but not of the angle ϕ or of the value of z in cylindrical coordinates, we can determine some structural details. We define as the average excess electron density in the cross section. Then we obtain a PDDF with a linear part for r > d and we have to replace Δρ in equation (2.6.1.46) by with the maximum dimension of the cross section d. The p(r) function differs from that of a homogeneous cylinder with the same only in the range . A typical example is shown in Fig. 2.6.1.8 . The functions for a homogeneous, a hollow, and an inhomogeneous cylinder with varying density are shown.

 Figure 2.6.1.8 | top | pdf |Circular cylinder with a constant length of 480 Å and an outer diameter of 48 Å. (a) Homogeneous cylinder, (b) hollow cylinder, (c) inhomogeneous cylinder. The p(r) functions are shown on the left, the corresponding electron-density distributions (r) on the right.

Rod-like particles. Axial inhomogeneity. This is another special case for rod-like particles, i.e. the density is a function of the z coordinate. In Fig. 2.6.1.9 , we compare two cylinders with the same size and diameter. One is a homogeneous cylinder with density , diameter d = 48 and length L = 480, and the other is an inhomogeneous cylinder of the same size and mean density , but this cylinder is made from slices with a thickness of 20 and alternating densities of and , respectively. The PDDF of the inhomogeneous cylinder has ripples with the periodicity of 40 in the whole linear range. This periodicity leads to reflections in reciprocal space (first and third order in the h range of the figure).

 Figure 2.6.1.9 | top | pdf |Inhomogeneous circular cylinder with periodical changes of the electron density along the cylinder axis compared with a homogeneous cylinder with the same mean electron density. (a) p(r) function; (b) scattering intensity; continuous line inhomogeneous cylinder; dashed line homogeneous cylinder.

Flat particles. Cross-sectional inhomogeneity. Lamellar particles with varying electron density perpendicular to the basal plane, where ρ is a function of the distance x from the central plane, show differences from a homogeneous lamella of the same size in the PDDF in the range , where T is the thickness of the lamella. An example is given in Fig. 2.6.1.10 where we compare a homogeneous lamellar particle (with ) with an inhomogeneous one, being a three-step function alternating between the values +1, −1, +1.

 Figure 2.6.1.10 | top | pdf |p(r) function of a lamellar particle. The full line corresponds to an inhomogeneous particle, ρt(x) is a three-step function with the values +1, −1, +1. The broken line represents the homogeneous lamella with .

Flat particles. In-plane inhomogeneity. Lamellae with a homogeneous cross section but inhomogeneities along the basal plane have a PDDF that deviates from that of a homogeneous lamella in the whole range . These deviations are a measure of the in-plane inhomogeneities; a general evaluation method does not exist. Even more complicated is the situation that occurs in membranes: these have a pronounced cross-sectional structure with additional in-plane inhomogeneities caused by the membrane proteins (Laggner, 1982; Sadler & Worcester, 1982).

Contrast variation and labelling. An important method for studying inhomogeneous particles is the method of contrast variation (Stuhrmann, 1982). By changing the contrast of the solvent, we can obtain additional information about the inhomogeneities in the particles. This variation of the contrast is much easier for neutron scattering than for X-ray scattering because hydrogen and deuterium have significantly different scattering cross sections. This technique will therefore be discussed in the section on neutron small-angle scattering.

A method for distance determination with X-rays by heavy-atom labelling was developed by Kratky & Worthmann (1947). These ideas are now used for the determination of distances between deuterated subunits of complex macromolecular structures with neutron scattering.

High-resolution experiments. A special type of study is the comparison of the structures of the same molecule in the crystal and in solution. This is done to investigate the influence of the crystal field on the polymer structure (Krigbaum & Kügler, 1970; Damaschun, Damaschun, Müller, Ruckpaul & Zinke, 1974; Heidorn & Trewhella, 1988) or to investigate structural changes (Ruckpaul, Damaschun, Damaschun, Dimitrov, Jänig, Müller, Pürschel & Behlke, 1973; Hubbard, Hodgson & Doniach, 1988). Sometimes such investigations are used to verify biopolymer structures predicted by methods of theoretical physics (Müller, Damaschun, Damaschun, Misselwitz, Zirwer & Nothnagel, 1984). In all cases, it is necessary to measure the small-angle scattering curves up to relatively high scattering angles ( nm−1, and more). Techniques for such experiments have been developed during recent years (Damaschun, Gernat, Damaschun, Bychkova & Ptitsyn, 1986; Gernat, Damaschun, Kröber, Bychkova & Ptitsyn, 1986; I'anson, Bacon, Lambert, Miles, Morris, Wright & Nave, 1987) and need special evaluation methods (Müller, Damaschun & Schrauber, 1990).

#### 2.6.1.3.3. Interparticle interference, concentration effects

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So far, only the scattering of single particles has been treated, though, of course, a great number of these are always present. It has been assumed that the intensities simply add to give the total diffraction pattern. This is true for a very dilute solution, but with increasing concentration interference effects will contribute. Biological samples often require higher concentrations for a sufficient signal strength. We can treat this problem in two different ways:

–We accept the interference terms as additional information about our system under investigation, thus observing the spatial arrangement of the particles.

–We treat the interference effect as a perturbation of our single-particle concept and discuss how to remove it.

The first point of view is the more general, but there are many open questions left. For many practical applications, the second point of view is important.

The radial distribution function. In order to find a general description, we have to restrict ourselves to an isotropic assembly of monodisperse spheres. This makes it possible to describe the situation by introducing a radial interparticle distribution function P(r) (Zernicke & Prins, 1927; Debye & Menke, 1930). Each particle has the same surroundings. We consider one central particle and ask for the probability that another particle will be found in the volume element dV at a distance r apart. The mean value is (N/v) dV; any deviation from this may be accounted for by a factor P(r). In the range of impenetrability , we have P(r) = 0 and in the long range . So the corresponding equation takes the form The second term contains all interparticle interferences. Its predominant part is the hole' of radius D, where [P(r) − 1] = −1. This leads to a decrease of the scattering intensity mainly in the central part, which results in a liquid-type pattern (Fig. 2.6.1.11 ). This can be explained by the reduction of the contrast caused by the high number of surrounding particles. Even if a size distribution for the spheres is assumed, the effect remains essentially the same (Porod, 1952). Up to now, no exact analytical expressions for P(r) exist. The situation is even more complicated if one takes into account attractive or repulsive interactions or non-spherical particle shapes (orientation).

 Figure 2.6.1.11 | top | pdf |Characteristic types of scattering functions: (a) gas type; (b) particle scattering; (c) liquid type.

If we have a system of spheres with known size D, we can use equation (2.6.1.37) for in equation (2.6.1.53), divide by this function, and calculate P(r) from experimental data by Fourier inversion. The interference term can be used to study particle correlations of charged macromolecular solutions (Chen, Sheu, Kalus & Hoffmann, 1988).

If there are attractive forces, there will be a tendency for aggregation. This tendency may, for instance, be introduced by some steps in the procedure of preparation of biological samples. Such aggregation leads to an increase of the intensity in the central part (gas type). In this case, we will finally have a polydisperse system of monomers and oligomers. Again, there exist no methods to analyse such a system uniquely.

Elimination of concentration effects – liquid type. In most cases, the interference effect is a perturbation of our experiment where we are only interested in the particle scattering function. Any remaining concentration effect would lead to errors in the resulting parameters. As we have seen above, the effect is essential at low h values, thus influencing I(0), , and the PDDF at large r values.

The problem can be handled for the liquid-like type in the following way. We measure the scattering function I(h) at different concentrations (typically from a few mg ml−1 up to about 50 mg ml−1). The influence of the concentration can be seen in a common plot of these scattering curves, divided by their concentrations. For large h values, these curves are identical. In the low-h range, the curves must be extrapolated to zero concentration. It depends on the problem as to whether a linear fit is sufficient or whether a second-degree polynomial has to be used. The extrapolation can be performed in a standard I(h)/c versus h plot or in a Zimm plot versus h (Cleemann & Kratky, 1960; Kirste & Oberthür, 1982). The Zimm plot should be preferred when working with highly concentrated solutions (Pilz, 1982).

As mentioned above, the innermost part of the scattering function is lowered and the apparent radius of gyration decreases with increasing concentration. The length of the linear range of the Guinier plot can be extended by the interference effect for non-spherical particles. Thus, an elongated linear Guinier plot is no guarantee of the completeness of the elimination of the concentration effect. Remaining interparticle interferences cannot be recognized in reciprocal space.

The PDDF is affected considerably by interparticle interference (Glatter, 1979). It is lowered with increasing distance r, goes through a negative minimum in the region of the maximum dimension D of the particle, and the oscillations vanish at larger r values. This is shown for the hard-sphere model in Fig. 2.6.1.12 . The oscillations disappear when the concentration goes to zero.

 Figure 2.6.1.12 | top | pdf |Distance distribution – hard-sphere interference model. Theoretical p(r) functions: continuous line θ = 0; short-dashed line θ = 0.25; dash-dotted line θ = 0.5; long-dashed line θ = 1.0. Circles: results from indirect transformation: θ = 0.5, h1R = 2.0. 2% statistical noise. Dmax = 300 Å, ΔRg = 0.5%, ΔI0 = 1.2%.

The same behaviour can be found from experimental data even in the case of non-spherical data (Pilz, Goral, Hoylaerts, Witters & Lontie, 1980; Pilz, 1982).

In some cases, it may be impossible to carry out experiments with varying concentrations. This will be the case if the structure of the particles depends on concentration. Under certain circumstances, it is possible to find the particle parameters by neglecting the innermost part of the scattering function influenced by the concentration effect (Müller & Glatter, 1982).

Aggregates – gas type. When the particles show a tendency to aggregation with increasing concentration, we can follow the same procedures as discussed for the liquid type, i.e. perform a concentration series and extrapolate the I(h)/c curves to zero concentration.

However, in most cases, the tendency to aggregation exists at any concentration, i.e. even at very high dilution we have a certain number of oligomers coexisting with monomers. There is no unique way to find the real particle parameters in these cases. It is not sufficient just to neglect the innermost part of the scattering function because that leads to an increasing loss of essential information about the particle (monomer) itself.

#### 2.6.1.4. Polydisperse systems

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In this subsection, we give a short survey of the problem of polydispersity. It is most important that there is no way to decide from small-angle scattering data whether the sample is mono- or polydisperse. Every data set can be evaluated in terms of monodisperse or polydisperse structures. Independent a priori information is necessary to make this decision. It has been shown analytically that a certain size distribution of spheres gives the same scattering function as a monodisperse ellipsoid with axes a, b and c (Mittelbach & Porod, 1962).

The scattering function of a polydisperse system is determined by the shape of the particles and by the size distribution. As mentioned above, we can assume a certain size distribution and can determine the shape, or, more frequently, we assume the shape and determine the size distribution. In order to do this we have to assume that the scattered intensity results from an ensemble of particles of the same shape whose size distribution can be described by , where R is a size parameter and denotes the number of particles of size R. Let us further assume that there are no interparticle interferences or multiple scattering effects. Then the scattering function I(h) is given by where is a constant, the factor takes into account the fact that the particle volume is proportional to , and is the normalized form factor of a particle size R. In many cases, one is interested in the mass distribution [sometimes called volume distribution ]. In this case, we have

The solution of these integral equations, i.e. the computation of or from I(h), needs rather sophisticated numerical or analytical methods and will be discussed later.

The problems of interparticle interference and multiple scattering in the case of polydisperse systems cannot be described analytically and have not been investigated in detail up to now. In general, interference effects start to influence data from small-angle scattering experiments much earlier, i.e. at lower concentration, than multiple scattering. Multiple scattering becomes more important with increasing size and contrast and is therefore dominant in light-scattering experiments in higher concentrations.

A concentration series and extrapolation to zero concentration as in monodisperse systems should be performed to eliminate these effects.

#### 2.6.1.5. Instrumentation

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X-ray sources are the same for small-angle scattering as for crystallographic experiments. One can use conventional generators with sealed tubes or rotating anodes for higher power. For the vast majority of applications, an X-ray tube with copper anode is used; the wavelength of its characteristic radiation (Cu Kα line) is 0.154 nm. Different anode materials emit X-rays of different characteristic wavelengths.

X-rays from synchrotrons or storage rings have a continuous wavelength distribution and the actual wavelength for the experiment is selected by a monochromator. The intensity is much higher than for any type of conventional source but synchrotron radiation is available only at a few places in the world. Reviews on synchrotron radiation and its application have been published during recent years (Stuhrmann, 1978; Holmes, 1982; Koch, 1988). In these reviews, one can also find some remarks on the general principles of the systems including cameras and special detectors.

#### 2.6.1.5.1. Small-angle cameras

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General. In any small-angle scattering experiment, it is necessary to illuminate the sample with a well defined flux of X-rays. The ideal condition would be a parallel monochromatic beam of negligible dimension and very high intensity. These theoretical conditions can never be reached in practice (Pessen, Kumosinski & Timasheff, 1973). One of the main reasons is the fact that there are no lenses as in the visible range of electromagnetic radiation. The refractive index of all materials is equal to or very close to unity for X-rays. On the other hand, this fact has some important advantages. It is, for example, possible to use circular capillaries as sample holders without deflecting the beam. There are different ways of constructing a small-angle scattering system. Slit, pinhole, and block systems define a certain area where the X-rays can pass. Any slit or edge will give rise to secondary scattering (parasitic scattering). The special construction of the instrument has to provide at least a subspace in the detector plane (plane of registration) that is free from this parasitic scattering. The crucial point is of course to provide the conditions to measure at very small scattering angles.

The other possibility of building a small-angle scattering system is to use monochromator crystals and/or bent mirrors to select a narrow wavelength band from the radiation (important for synchrotron radiation) and to focus the X-ray beam to a narrow spot. These systems require slits in addition to eliminate stray radiation.

Block collimation – Kratky camera. The Kratky (1982a) collimation system consists of an entrance slit (edge) and two blocks – the U-shaped centre piece and a block called bridge. With this system, the problem of parasitic scattering can be largely removed for the upper half of the plane of registration and the smallest accessible scattering angle is defined by the size of the entrance slit (see Fig. 2.6.1.13 ). This system can be integrated in an evacuated housing (Kratky compact camera) and fixed on the top of the X-ray tube. It is widely used in many laboratories for different applications. In the Kratky system, the X-ray beam has a rectangular shape, the length being much larger than the width. Instrumental broadening can be corrected by special numerical routines. The advantage is a relatively high primary-beam intensity. The main disadvantage is that it cannot be used in special applications such as oriented systems where the two-dimensional scattering pattern has to be recorded. For such applications, any type of point collimation can be used.

 Figure 2.6.1.13 | top | pdf |Schematic drawing of the block collimation (Kratky camera): E edge; B1 centre piece; B2 bridge; P primary-beam profile; PS primary-beam stop; PR plane of registration.

Slit and pinhole cameras. The simplest way to build a camera is to use two pairs of slits or pinholes at a certain distance apart (Kratky, 1982a; Holmes, 1982). The narrower the slits and the larger the distance between them, the smaller is the smallest attainable scattering angle (sometimes called the resolution'). Parasitic scattering and difficult alignment are the main problems for all such systems (Guinier & Fournet, 1955). A slit camera that has been used very successfully is that of Beeman and co-workers (Ritland, Kaesberg & Beeman, 1950; Anderegg, Beeman, Shulman & Kaesberg, 1955). A rather unusual design is adopted in the slit camera of Stasiecki & Stuhrmann (1978), whose overall length is 50 m! A highly developed system is the ORNL 10 m camera at Oak Ridge (Hendricks, 1978).

Standard-size cameras for laboratory application are commercially available with different designs from various companies.

Bonse–Hart camera. The Bonse–Hart camera (Bonse & Hart, 1965, 1966, 1967) is based on multiple reflections of the primary beam from opposite sides of a groove in an ideal germanium crystal (collimator and monochromator). After penetrating the sample, the scattered beam runs through the groove of a second crystal (analyser). This selects the scattering angle. Rotation of the second crystal allows the measurement of the angle-dependent scattering function. The appealing feature of this design is that one can measure down to very small angles without a narrow entrance slit. The system is therefore favourable for the investigation of very large particles (D 350 nm). For smaller particles, one obtains better results with block collimation (Kratky & Leopold, 1970).

Camera systems for synchrotron radiation. Small-angle scattering facilities at synchrotrons are built by the local staff and details of the construction are not important for the user in most cases. Descriptions of the instruments are available from the local contacts. These small-angle scattering systems are usually built with crystal monochromators and focusing mirrors (point collimation). All elements have to be operated under remote control for safety reasons. A review of the different instruments was published recently by Koch (1988).

#### 2.6.1.5.2. Detectors

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In this field, we are facing the same situation as we met for X-ray sources. The detectors for small-angle scattering experiments are the same as or slightly modified from the detectors used in crystallography. Therefore, it is sufficient to give a short summary of the detectors in the following; further details are given in Chapter 7.1 . If we are not investigating the special cases of fully or partially oriented systems, we have to measure the dependence of the scattered intensity on the scattering angle, i.e. a one-dimensional function. This can be done with a standard gas-filled proportional counter that is operated in a sequential mode (Leopold, 1982), i.e. a positioning device moves the receiving slit and the detector to the desired angular position and the radiation detector senses the scattered intensity at that position. In order to obtain the whole scattering curve, a series of different angles must be positioned sequentially and the intensity readings at every position must be recorded. The system has a very high dynamic range, but – as the intensities at different angles are measured at different times – the stability of the primary beam is of great importance.

This drawback is eliminated in the parallel detection mode with the use of position-sensitive detectors. Such systems are in most cases proportional counters with sophisticated and expensive read-out electronics that can evaluate on-line the accurate position where the pulses have been created by the incoming radiation.

Two-dimensional position-sensitive detectors are necessary for oriented systems, but they also have advantages in the case of non-oriented samples when circular chambers are used or when integration techniques in square detectors lead to a higher signal at large scattering angles.

The simplest and cheapest two-dimensional detector is still film, but films are not used very frequently in small-angle scattering experiments because of limited linearity and dynamic range, and fog intensity.

Koch (1988) reviews the one- and two-dimensional detectors actually used in synchrotron small-angle scattering experiments. For a general review of detectors, see Hendrix (1985).

#### 2.6.1.6. Data evaluation and interpretation

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After having discussed the general principles and the basics of instrumentation in the previous subsections, we can now discuss how to handle measured data. This can only be a very short survey; a detailed description of data treatment and interpretation has been given previously (Glatter, 1982a,b).

Every physical investigation consists of three highly correlated parts: theory, experiment, and evaluation of data. The theory predicts a possible experiment, experimental data have to be collected in a way that the evaluation of the information wanted is possible, the experimental situation has to be described theoretically and has to be taken into account in the process of data evaluation etc. This correlation should be remembered at every stage of the investigation. Before we can start any discussion about interpretation, we have to describe the experimental situation carefully.

All the theoretical equations in the previous subsections correspond to ideal conditions as mentioned in the subsection on instrumentation. In real experiments, we do not measure with a point-like parallel and strictly monochromatic primary beam and our detector will have non-negligible dimensions. The finite size of the beam, its divergence, the size of the detector, and the wavelength distribution will lead to an instrumental broadening as in most physical investigations. The measured scattering curve is said to be smeared by these effects. So we find ourselves in the following situation.

The particle is represented by its PDDF p(r). This function is not measured directly. In the scattering process it is Fourier-transformed into a scattering function I(h) [equation (2.6.1.9)]. This function is smeared by the broadening effects and the final smeared scattering function is measured with a certain experimental error . In the case of polydisperse systems, the situation is very similar; we start from a size-distribution function D(R) and have a different transformation [equations (2.6.1.54), (2.6.1.55)], but the smearing problem is the same.

#### 2.6.1.6.1. Primary data handling

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In order to obtain reliable results, we have to perform a series of experiments. We have to repeat the experiment for every sample, to be able to estimate a mean value and a standard deviation at every scattering angle. This experimentally determined standard deviation is often much higher than the standard deviation simply estimated from counting statistics. A blank experiment (cuvette filled with solvent only) is necessary to be able to subtract background scattering coming from the instrument and from the solvent (or matrix in the case of solid samples). Finally, we have to perform a series of such experiments at different concentrations to extrapolate to zero concentration (elimination of interparticle interferences).

If the scattering efficiency of the sample is low (low contrast, small particles), it may be necessary to measure the outer part of the scattering function with a larger entrance slit and we will have to merge different parts of the scattering function. The intensity of the instrument (primary beam) should be checked before each measurement. This allows correction (normalization) for instabilities.

It is therefore necessary to have a so-called primary data-handling routine that performs all these preliminary steps like averaging, subtraction, normalization, overlapping, concentration extrapolation, and graphical representation on a graphics terminal or plotter. In addition, it is helpful to have the possibility of calculating the Guinier radius, Porod extrapolation [equations (2.6.1.24)], invariant, etc. from the raw data.

When all these preliminary steps have been performed, we have a smeared particle-scattering function with a certain statistical accuracy. From this data set, we want to compute I(h) and p(r) [or D(R)] and all our particle parameters. In order to do this, we have to smooth and desmear our function . The smoothing operation is an absolute necessity because the desmearing process is comparable to a differentiation that is impossible for noisy data. Finally, we have to perform a Fourier transform (or other similar transformation) to invert equations (2.6.1.9) or (2.6.1.54), (2.6.1.55). Before we can discuss the desmearing process (collimation error correction) we have to describe the smearing process.

#### 2.6.1.6.2. Instrumental broadening – smearing

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These effects can be separated into three components: the two-dimensional geometrical effects and the wavelength effect. The geometrical effects can be separated into a slit-length (or slit-height) effect and a slit-width effect. The slit length is perpendicular to the direction of increasing scattering angle; the corresponding weighting function is usually called P(t). The slit width is measured in the direction of increasing scattering angles and the weighting function is called Q(x). If there is a wavelength distribution, we call the weighting function where and is the reference wavelength used in equation (2.6.1.2). When a conventional X-ray source is used, it is sufficient in most cases to correct only for the Kβ contribution. Instead of the weighting function W(λ′) one only needs the ratio between Kβ and Kα radiation, which has to be determined experimentally (Zipper, 1969). One or more smearing effects may be negligible, depending on the experimental situation.

Each effect can be described separately by an integral equation (Glatter, 1982a). The combined formula reads This threefold integral equation cannot be solved analytically. Numerical methods must be used for its solution.

#### 2.6.1.6.3. Smoothing, desmearing, and Fourier transformation

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There are many methods published that offer a solution for this problem. Most are referenced and some are reviewed in the textbooks (Glatter, 1982a; Feigin & Svergun, 1987). The indirect transformation method in its original version (Glatter, 1977a,b, 1980a,b) or in modifications for special applications (Moore, 1980; Feigin & Svergun, 1987) is a well established method used in the majority of laboratories for different applications. This procedure solves the problems of smoothing, desmearing, and Fourier transformation [inversion of equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)] in one step. A short description of this technique is given in the following.

Indirect transformation methods. The indirect transformation method combines the following demands: single-step procedure, optimized general-function system, weighted least-squares approximation, minimization of termination effect, error propagation, and consideration of the physical smoothing condition given by the maximum intraparticle distance. This smoothing condition requires an estimate Dmax as an upper limit for the largest particle dimension: For the following, it is not necessary for to be a perfect estimate, but it must not be smaller than D.

As for , we can use a function system for the representation of p(r) that is defined only in the subspace . A linear combination is used as an approximation to the PDDF. Let N be the number of functions and be the unknowns. The functions are chosen as cubic B splines (Greville, 1969; Schelten & Hossfeld, 1971) as they represent smooth curves with a minimum second derivative.

Now we take advantage of two facts. The first is that we know precisely how to calculate a smeared scattering function from I(h) [equation (2.6.1.56)] and how p(r) or D(R) is transformed into I(h) [equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)], but we do not know the inverse transformations. The second fact is that all these transformations are linear, i.e. they can be applied to all terms in a sum like that in equation (2.6.1.58) separately. So it is easy to start with our approximation in real space [equation (2.6.1.58)] taking into account the a priori information Dmax. The approximation to the ideal (unsmeared) scattering function can be written as where the functions are calculated from by the transformations (2.6.1.9) or (2.6.1.54), (2.6.1.55), the coefficients remain unknown. The final fit in the smeared, experimental space is given by a similar series where the are functions calculated from by the transform (2.6.1.56). Equations (2.6.1.58), (2.6.1.59), and (2.6.1.60) are similar because of the linearity of the transforms. We see that the functions are calculated from in the same way as the data were produced by the experiment from p(r). Now we can minimize the expression where M is the number of experimental points. Such least-squares problems are in most cases ill conditioned, i.e. additional stabilization routines are necessary to find the best solution. This problem is far from being trivial, but it can be solved with standard routines (Glatter, 1977a,b; Tikhonov & Arsenin, 1977).

The whole process of data evaluation is shown in Fig. 2.6.1.14 . Similar routines cannot be used in crystallography (periodic structures) because there exists no estimate for Dmax [equation (2.6.1.57)].

 Figure 2.6.1.14 | top | pdf |Function systems ϕv(r); Ψv(h); and χv(h) used for the approximation of the scattering data in the indirect transformation method.

Maximum particle dimension. The sampling theorem of Fourier transformation (Shannon & Weaver, 1949; Bracewell, 1986) gives a clear answer to the question of how the size of the particle D is related to the smallest scattering angle h1. If the scattering curve is observed at increments starting from a scattering angle h1, the scattering data contain, at least theoretically, the full information for all particles with maximum dimension D The first application of this theorem to the problem of data evaluation was given by Damaschun & Pürschel (1971a,b). In practice, one should always try to stay below this limit, i.e. taking into account the loss of information due to counting statistics and smearing effects. An optimum value for Δh = π/(6D) is claimed by Walter, Kranold & Becherer (1974).

Information content. The number of independent parameters contained in a small-angle scattering curve is given by with and being the lower and upper limits of h. In practice, this limit certainly depends on the statistical accuracy of the data. It should be noted that the number of functions N in equations (2.6.1.58) to (2.6.1.60) may be larger than because they are not independent. They are correlated by the stabilization routine. An example of this problem can be found in Glatter (1980a).

Resolution. There is no clear answer to the question concerning the smallest structural details, i.e. details in the function that can be recognized from an experimental scattering function. The limiting factors are the maximum scattering angle , the statistical error , and the weighting functions , and (Glatter, 1982a). The resolution of standard experiments is not better than approximately 10% of the maximum dimension of the particle for a monodisperse system. In the case of polydisperse systems, resolution can be defined as the minimum relative peak distance that can be resolved in a bimodal distribution. We know from simulations that this value is of the order of 25%.

Special transforms. The PDDF or the size distribution function D(R) is related to I(h) by equations (2.6.1.9) or (2.6.1.54), (2.6.1.55). In the special case of particles elongated in one direction (like cylinders), we can combine equations (2.6.1.41) and (2.6.1.43) and obtain This Hankel transform can be used in the indirect transformation method for the calculation of in (2.6.1.59). Doing this, we immediately obtain the PDDF of the cross section from the smeared experimental data. It is not necessary to know the length L of the particle if the results are not needed on an absolute scale. For this application, we only need the information that the scatterers are elongated in one direction with a constant cross section. This information can be found from the overall PDDF of the particle or can be a priori information from other experiments, like electron microscopy. The estimate for the maximum dimension (2.6.1.57) is related to the cross section in this application, i.e. the maximum dimension of the cross section must not be larger than .

The situation is quite similar for flat particles. If we combine (2.6.1.47) and (2.6.1.49), we obtain being the distance distribution function of the thickness. We have to check that the particles are flat with a constant thickness with maximum thickness . A is the area of the particles and would be needed only for experiments on an absolute scale.

#### 2.6.1.6.4. Direct structure analysis

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It is impossible to determine the three-dimensional structure ρ(r) directly from the one-dimensional information I(h) or p(r). Any direct method needs additional a priori information – or assumptions – on the system under investigation. If this information tells us that the structure only depends on one variable, i.e. the structure is in a general sense one dimensional, we have a good chance of recovering the structure from our scattering data.

Examples for this case are particles with spherical symmetry, i.e. ρ depends only on the distance r from the centre, or particles with cylindrical or lamellar symmetry where ρ depends only on the distance from the cylinder axis or from the distance from the central plane in the lamella. We will restrict our discussion here to the spherical problem but we keep in mind that similar methods exist for the cylindrical and the lamellar case.

Spherical symmetry. This case is described by equations (2.6.1.51) and (2.6.1.52). As already mentioned in §2.6.1.3.2.2, we can solve the problem of the calculation of ρ(r) from I(h) in two different ways. We can calculate ρ(r) via the distance distribution function p(r) with a convolution square-root technique (Glatter, 1981; Glatter & Hainisch, 1984). The other way goes through the amplitude function A(h) and its Fourier transform. In this case, one has to find the right phases (signs) in the square-root operation . The box-function refinement method by Svergun, Feigin & Schedrin (1984) is an iterative technique for the solution of the phase problem using the a priori information that ρ(r) is equal to zero for (Dmax/2). The same restriction is used in the convolution square-root technique. Under ideal conditions (perfect spherical symmetry), both methods give good results. In the case of deviations from spherical symmetry, one obtains better results with the convolution square-root technique (Glatter, 1988). With this method, the results are less distorted by non-spherical contributions.

Multipole expansions. A wide class of homogeneous particles can be represented by a boundary function that can be expanded into a series of spherical harmonics. The coefficients are related to the coefficients of a power series of the scattering function I(h), which are connected with the moments of the PDDF (Stuhrmann, 1970b,c; Stuhrmann, Koch, Parfait, Haas, Ibel & Crichton, 1977). Of course, this expansion cannot be unique, i.e. for a certain scattering function I(h) one can find a large variety of possible expansion coefficients and shapes. In any case, additional a priori information is necessary to reduce this number, which in turn influences the convergence of the expansion. Only compact, globular structures can be approximated with a small number of coefficients.

This concept is not restricted to the determination of the shape of the particles. Even inhomogeneous particles can be described using all possible radial terms in a general expansion (Stuhrmann, 1970a). The information content can be increased by contrast variation (Stuhrmann, 1982), but in any event one is left with the problem of how to find additional a priori information in order to reduce the possible structures. Any type of symmetry will lead to a considerable improvement. The case of axial symmetry is a good example. Svergun, Feigin & Schedrin (1982) have shown that the quality of the results can be further improved when upper and lower limits for ρ(r) can be used. Such limits can come from a known chemical composition.

#### 2.6.1.6.5. Interpretation of results

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After having used all possible data-evaluation techniques, we end up with a desmeared scattering function I(h), the PDDF p(r) or the size-distribution function D(R), and some special functions discussed in the previous subsections. Together with the particle parameters, we have a data set that can give us at least a rough classification of the substance under investigation.

The interpretation can be performed in reciprocal space (scattering function) or in real space (PDDF etc.). Any symmetry can be detected more easily in reciprocal space, but all other structural information can be found more easily in real space (Glatter, 1979, 1982b).

When a certain structure is estimated from the data and from a priori information, one has to test the corresponding model. That means one has to find the PDDF and I(h) for the model and has to compare it with the experimental data. Every model that fits within the experimental errors can be true, all that do not fit have to be rejected. If the model does not fit, it has to be refined by trial and error. In most cases, this process is much easier in real space than in reciprocal space. Finally, we may end up with a set of possible structures that can be correct. Additional a priori information will be necessary to reduce this number.

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#### 2.6.1.7.1. Simulations

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Simulations can help to find the limits of the method and to estimate the systematic errors introduced by the data-evaluation procedure. Simulations are performed with exactly known model systems (test functions). These systems should be similar to the structures of interest. The model data are transformed according to the special experimental situation (collimation profiles and wavelength distribution) starting from the theoretical PDDF (or scattering function). Experimental data points' are generated by sampling in a limited h range and adding statistical noise from a random-number generator. If necessary, a certain amount of background scattering can also be added. This simulated data set is subjected to the data-evaluation procedure and the result is compared with the starting function. Such simulation can reveal the influence of each approximation applied in the various evaluation routines.

On the other hand, simulations can also be used for the optimization of the experimental design for a special application. The experiment situation is characterized by several contradictory effects: a large width for the functions P(t), Q(x), and W(λ′) leads to a high statistical accuracy but considerable smearing effects. The quality of the results of the desmearing procedure is increased by high statistical accuracy, but decreased by large smearing effects. Simulations can help to find the optimum for a special application.

#### 2.6.1.7.2. Model calculation

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In the section on data evaluation and interpretation, we have seen that we obtain a rough estimate for the structure of the particles under investigation directly from the experimental data. For further refinement, we have to compare our results with scattering functions or PDDF's from models.

#### 2.6.1.7.3. Calculation of scattering intensities

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The scattering curves can be calculated semi-analytically for simple triaxial bodies and for models composed of some of these bodies. The scattering amplitude for regular bodies like ellipsoids, parallelepipeds, and cylinders can be calculated analytically for any orientation. The spatial averaging has to be performed numerically. Such calculations have been performed for a large number of different models by Porod (1948), Mittelbach & Porod (1961a,b, 1962), and by Mittelbach (1964). More complicated structures can be described by models composed of several such triaxial bodies, but the computing time necessary for such calculations can be hours on a mainframe computer.

Models composed only of spherical subunits can be evaluated with the Debye formula (Debye, 1915): where the spatial average is carried out analytically. Another possibility would be to use spherical harmonics as discussed in the previous section but the problem is how to find the expansion coefficients for a certain given geometrical structure.

#### 2.6.1.7.4. Method of finite elements

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Models of arbitrary shape can be approximated by a large number of very small homogeneous elements of variable electron density. These elements have to be smaller than the smallest structural detail of interest.

Sphere method. In this method, the elements consist of spheres of equal size. The diameter of these spheres must be chosen independently of the distance between nearest neighbours, in such a way that the total volume of the model is represented correctly by the sum of all volume elements (which corresponds to a slight formal overlap between adjacent spheres). The scattering intensity is calculated using the Debye formula (2.6.1.67), with .

The computing time is mainly controlled by the number of mutual distances between the elements. The computing time can be lowered drastically by the use of approximate values in (2.6.1.67). Negligible errors in I(h) result if values are quantized to Dmax/10000 (Glatter, 1980c). For the practical application (input operation), it is important that a certain number of elements can be combined to form so-called sub-structures that can be used in different positions with arbitrary weights and orientations to build the model.

The sphere method can also be used for the computation of scattering curves for macromolecules from a known crystal structure. The weights of the atoms are given by the effective number of electrons where is the apparent volume of the atom given by Langridge, Marvin, Seeds, Wilson, Cooper, Wilkins & Hamilton (1960).

Cube method. This method has been developed independently by Fedorov, Ptitsyn & Voronin (1972, 1974a,b) and by Ninio & Luzzati (1972) mainly for the computation of scattered intensities for macromolecules in solution whose crystal structure is known. In the cube method, the macromolecule is mentally placed in a parallelepiped, which is subdivided into small cubes (with edge lengths of 0.5–1.5 Å. Each cube is examined in order to decide whether it belongs to the molecule or to the solvent. Adjacent cubes in the z direction are joined to form parallelepipeds. The total scattering amplitude is the sum over the amplitudes from the parallelepipeds with different positions and lengths. The mathematical background is described by Fedorov, Ptitsyn & Voronin (1974a,b). The modified cube method of Fedorov & Denesyuk (1978) takes into account the possible penetration of the molecule by water molecules.

#### 2.6.1.7.5. Calculation of distance-distribution functions

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The PDDF can be calculated analytically only for a few simple models (Porod, 1948; Goodisman, 1980); in all other cases, we have to use a finite element method with spheres. It is possible to define an analogous equation to the Debye formula (2.6.1.67) in real space (Glatter, 1980c). The PDDF can be expressed as ) is the PDDF of a sphere with radius and electron density equal to unity, is the cross-term distance distribution between the ith and kth spheres (radii and ) with a mutual distance .

Equation (2.6.1.69) [and (2.6.1.67)] can be used in two different ways for the calculation of model functions. Sometimes, it is possible to approximate a macromolecule as an aggregate of some spheres of well defined size representing different globular subunits (Pilz, Glatter, Kratky & Moring-Claesson, 1972). The form factors of the subunits are in such cases real parameters of the model. However, in most cases we have no such possibility and we have to use the method of finite elements, i.e. we fit our model with a large number of sufficiently small spheres of equal size, and, if necessary, different weight. The form factor of the small spheres is now not a real model parameter and introduces a limit of resolution.

Fourier transformation [equation (2.6.1.10)] can be used for the computation of the PDDF of any arbitrary model if the scattering function of the model is known over a sufficiently large range of h values.

#### 2.6.1.8. Suggestions for further reading

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Only a few textbooks exist in the field of small-angle scattering. The classic monograph Small-Angle Scattering of X-rays by Gunier & Fournet (1955) was followed by the proceedings of the conference at Syracuse University, 1965, edited by Brumberger (1967) and by Small-Angle X-ray Scattering edited by Glatter & Kratky (1982). The several sections of this book are written by different authors being experts in the field and representing the state of the art at the beginning of the 1980's. The monograph Structure Analysis by Small-Angle X-ray and Neutron Scattering by Feigin & Svergun (1987) combines X-ray and neutron techniques.

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