Small-angle scattering from multiphase fractals
aJoint Institute for Nuclear Research, Dubna 141980, Russian Federation, bHoria Hulubei National Institute of Physics and Nuclear Engineering, RO-077125 Bucharest-Magurele, Romania, and cLaboratory for Advanced Studies of Membrane Proteins, Moscow Institute of Physics and Technology, 141700 Dolgoprudniy, Russian Federation
*Correspondence e-mail: firstname.lastname@example.org
Small-angle scattering (SAS) intensities observed experimentally are often characterized by the presence of successive power-law regimes with various scattering exponents whose values vary from −4 to −1. This usually indicates multiple fractal structures of the sample characterized by different size scales. The existing models explaining the crossover positions (that is, the points where the power-law scattering exponent changes) involve only one contrast parameter, which depends solely on the ratio of the fractal sizes. Here, a model that describes SAS from a multiphase system with a few contrast parameters is described, and it is shown that the crossover position depends on the scattering length density of each phase. The Stuhrmann contrast variation method is generalized and applied to experimental curves in the vicinity of the crossover point beyond the Guinier region. The contrast variation is applied not to the intensity itself but to the model parameters, which can be found by fitting the experimental data with the suggested interpolation formula. The model supplements the existing two-phase models and gives the simple condition of their inapplicability: if the crossover point depends on the contrast then a two-phase model is not relevant. The developed analysis allows one to answer the qualitative question of whether one fractal `absorbs' another one or they are both immersed in a surrounding homogeneous medium like a solvent or solid matrix. The models can be applied to experimental SAS data where the absolute value of the scattering exponent of the first power-law regime is higher than that of the subsequent second power-law regime, that is, the scattering curve is `convex' near the crossover point. As is shown, the crossover position can be very sensitive to contrast variation, which influences significantly the length of the fractal range.
One of the most effective and reliable experimental methods for investigating fractal structures (Mandelbrot, 1982; Gouyet, 1996) of materials at nanoscales is small-angle scattering (SAS) (Guinier & Fournet, 1955; Glatter & Kratky, 1982; Feigin & Svergun, 1987), which measures the differential cross section of an irradiated sample versus the scattering vector magnitude q [q = (4π/λ)sinθ, where θ is half the scattering angle and λ is the wavelength of the incident radiation]. SAS allows one to obtain the edges of the fractal region in reciprocal space and the fractal dimension (Bale & Schmidt, 1984; Schmidt & Dacai, 1986; Martin & Hurd, 1987; Teixeira, 1988; Schmidt, 1991). The borders of linear dependence of the intensity in a double logarithmic scale show the edges of the fractal region, and its slope τ, being the power-law exponent of SAS intensity, yields the fractal dimension: , where
Here the notations D and Ds are adopted for the fractal dimension of mass and surface fractals, respectively. The value of the fractal dimension can lie within the ranges and for mass and surface fractals, respectively. These restrictions lead to a simple physical interpretation of experimental data: if then the measured sample is a mass fractal; if then it is a surface fractal. Note that this rule of thumb should be used with caution, because the appearance of the power-law behaviour can arise from factors other than the fractal structure of the irradiated sample. For instance, the scattering intensity from spatially and rotationally uncorrelated identical discs yields an exponent equal to −2. Another example is an `occasional' power-law behaviour in a quite narrow scattering vector range caused by polydispersity. This is a general and well known problem of the small-angle scattering method, which cannot give us a well defined and unambiguous structure of samples in real space. In order to resolve this general ambiguity and reveal the sample structure in real space, one needs to involve additional information from other experimental techniques or to know the conditions under which the system has been created.
SAS data (X-ray or neutron) often show a succession of power-law regimes whose scattering exponents take arbitrary values from −4 to −1 (Ehrburger-Dolle et al., 2001; Schaefer & Justice, 2007; Kravchuk et al., 2009; Schneider & Goritz, 2010; Singh et al., 2011; Schneider et al., 2012; Golosova et al., 2012). On a double logarithmic scale, the scattering intensity looks like connected straight-line segments. This type of behaviour is usually associated with two-phase multilevel structures, and the structural characteristics are revealed most often by using the Beaucage (1995) model. However, experimental SAS articles have reported the dependence of the scattering intensities on the scattering length densities of surrounding solvent or solid matrix in which the studied structures are immersed. In particular, the crossover positions qc (that is, the points where the power-law scattering exponent changes) can depend on the contrast value (Lebedev et al., 2005). This indicates a multiphase structure of the sample, and thus a method capable of extracting the structural properties of each phase is needed.
Let us explain the problem in more detail. The multilevel structures suggested by Beaucage (1995) assume that particles composing a multilevel system are complex structures themselves within their size scale. An example is a mass fractal with fractal dimension D1 composed of structural units each of which is another mass fractal with the dimension D2. Then the scattering intensity on a double logarithmic scale shows the `knee' behaviour, where one straight line with the slope -D1 passes into another straight line with the slope -D2 at the crossover point qc. This model is two phase, that is, it assumes that the primary units of the above construction are homogeneous with a constant scattering length density and all the construction is embedded in a solid matrix with another constant scattering length density. Hence, there is only one contrast parameter (that is, the difference between the scattering length densities), and the scattering intensity is proportional to the squared value of the contrast, but its form remains the same with the contrast variation. In particular, the value of the crossover position is independent of the contrast value and is determined only by the ratio of the fractal sizes.
On the other hand, multiphase materials are ubiquitous in nature, and one can expect that the presence of more than two phases could be even more relevant. In a recent article (Cherny et al., 2012), the crossover between Porod and mass fractal regimes has been explained for multiphase systems. Here, we extend these results and develop a model that describes SAS from a multiphase system with a few contrast parameters. As is known, the method of contrast variation (Stuhrmann, 1970) is successfully used in X-ray and neutron scattering for investigating multiphase systems. This powerful method studies the intensity at zero scattering vector and radii of gyration, and thus it is applicable only in the Guinier region. However, the Guinier region often lies beyond the resolution of the experimental device. We suggest an extension of this method and show how it can be applied to experimental curves in the vicinity of the crossover point beyond the Guinier region. The contrast variation is applied to the model parameters, which can be found by fitting the experimental data with a simple interpolation formula.
There are essential differences between the model suggested here and the Beaucage model. First, the structure of the material in real space is assumed to be different. In contrast to the real-space structure of the Beaucage model described above, we consider two possible different structures (see Fig. 3 below):
Type I. A fractal with scattering length density and dimension D2 is embedded in a larger fractal with density and dimension D1, and they are immersed in a homogeneous medium (solvent or solid matrix) with density .
Type II. Two non-overlapping fractals with dimensions D1 and D2 and scattering length densities and are immersed in a homogeneous medium with density .
Second, the suggested model is applicable if the scattering intensities are `convex' in the vicinity of the crossover point on a double logarithmic scale, that is, a steeper line at small values of scattering vector passes into a flatter line at the crossover point qc. This means that the fractal with higher power-law exponent dominates at small values of q, while its contribution to the scattering intensity is suppressed at large values of q. Here we show how in practice to distinguish between structures of type I and II and determine the densities and by using the contrast variation method.
We confirm our conclusions by performing numerical calculations with randomly oriented and uniformly distributed three-phase systems containing balls and mass fractals, embedded in a solvent or solid matrix. The basic model for the numerical calculation of mass fractals is the generalized Cantor fractals with controllable fractal dimension (Cherny et al., 2010b, 2011).
The article is organized as follows. In §2, we introduce some notations and emphasize some important issues about SAS from two- and three-phase systems. In §3, we derive a simple fitting formula describing SAS from multiphase systems and show how the contrast variation method can be used in the vicinity of the crossover point. The crossover position is estimated through the model parameters, and the obtained results are discussed. In §4, we show numerically that the developed model describes well the transition from Porod to mass fractal regimes.
Let us consider a sample consisting of microscopic scatterers with a scattering length bj. Then the differential cross section of elastic scattering is given by (Feigin & Svergun, 1987) , where
is the total scattering amplitude and is the total volume irradiated by the incident beam. Here, the scattering length density is defined with the help of Dirac's δ function: , where are the spatial positions of the scatterers. Note that a constant shift of the scattering length density in the overall sample is important only for very small values of q, unattainable with the SAS technique. This property is convenient to exclude the `background' density of the sample.
We consider first a two-phase sample composed of rigid homogeneous objects with scattering length density ρ immersed in a solid matrix of density . Therefore, by subtracting the `background' density , we can consider the system as if the objects were `frozen' in a vacuum and had the density . This is called the scattering contrast. In addition, the objects are supposed to be of the same volume and shape, and their spatial positions and orientations are uncorrelated. Then the scattering intensity (that is, the cross section per unit volume of the sample) is given by
where n is the object concentration in the sample, V is the volume of each object and is the normalized scattering amplitude of the object,
obeying the condition F(0) = 1. It is also called the normalized form factor. The brackets stand for the mean value given by the ensemble averaging over all orientations of the objects. If the probability of each orientation is the same, then this mean value can be calculated by averaging over all directions of the scattering vector , that is, by integrating over the solid angle in the spherical coordinates , and :
If the object is a mass fractal of the total length L, composed of p small structural units of the size l, then its normalized form factor can be estimated qualitatively by the formula
A surface fractal can be constructed, say, from balls with different radii r, whose centres are spatially uncorrelated inside the fractal. If the radii are taken at random with the distribution proportional to 1/rDs+1 at small r and the parameter Ds obeys the condition then one can show that Ds is indeed the fractal dimension of the total ball surfaces. Certainly, in practice, the restriction should be imposed, and the fractal properties appear only within this range. Now the qualitative estimation formula takes the form
(see Fig. 1b). We emphasize that here L is of the order of the largest radius of the balls but not the total fractal length, and l is the smallest radius. Equations (6) and (7) explicitly show that the scattering intensity in q space is characterized by three main ranges: Guinier at , fractal at and Porod at . Note that a simple three-dimensional set like a ball does not exhibit fractal behaviour at all; here the Porod range follows the Guinier range. Indeed, the fractal region `collapses' if we put formally l = L in either of the relations (6) or (7).
We make a few remarks about the scattering intensities [equations (6) and (7)]. First, the intensity in the Guinier range is actually parabolic: , where Rg is the radius of gyration. This parabolic behaviour of the intensity is ignored in the above estimations for the sake of simplicity. Second, the relation (6) is applicable at if the minimal distance between the structural units composing the mass fractal is of the order of their size. Otherwise, if the size is much smaller than the minimal distance then there exists a plateau (`shelf') between the end of the fractal range and the beginning of the Porod range (Cherny et al., 2010b). Third, the fractal region of the mass fractal (Fig. 1a) is determined by the maximal and minimal distances between the centres of the structural units, while the fractal region of the surface fractal (Fig. 1b) is determined by the largest and smallest radii present in the fractal. For a mass fractal in the fractal region, the interparticle spatial correlations between the structural units play an important role, and the fractal behaviour can be understood by analogy with optics (Cherny et al., 2011), while for a surface fractal, the spatial correlations between the structural units are not important for its fractal properties.
The three-phase systems consist generically of homogeneous structures with scattering length densities and , immersed in a homogeneous medium with density . By subtracting the `background' density , we arrive at the same structural units with the contrasts and , respectively, in a vacuum (see Fig. 2). Effectively, this procedure allows us to reduce the three-phase system to the two-phase system.
Let us calculate the scattering intensity from a macroscopic number of the objects shown in Fig. 2, whose spatial positions and orientations are supposed to be uncorrelated. In accordance with the general relation (2), the intensity is given by , where and n are the scattering amplitude of the objects and their concentration in the sample, respectively. The amplitude can be written down with the help of the normalized form factors and and volumes V1 and V2 of the space regions 1 and 2, respectively, shown on the left in Fig. 2. Indeed, it is given by the sum + , where is the scattering amplitude of region 2, and - is the amplitude of the region that remains after excluding region 2 from 1. Finally, we derive
where we denote and introduce by definition the `contrast parameter'
This dimensionless parameter determines the relative contributions of the corresponding form factors, and thus it controls completely the form of the resulting scattering intensity, while the average scattering length density of the object determines the intensity at zero q.
Let us analyse and further simplify the general relation (8) [or (9)] for the scattering intensity from the three-phase system. Suppose that the systems 1 and 2 shown in Fig. 2 are mass or surface fractals, obeying the power-law behaviour in the corresponding fractal ranges in accordance with equations (6) or (7). We assume that the following conditions are satisfied:
(a) Their fractal ranges in q space overlap.
Then the scattering intensity exhibits the `knee' convex structure on a double logarithmic scale. Indeed, at low q the first term on the right hand side of equation (9) dominates because of condition (c). On the other hand, the first term is proportional to F1, which decreases faster than F2 according to condition (b). At the crossover point qc, the two terms become approximately equal, and at high values of q the second term dominates.
The physical picture can be simplified further if we notice that the interference terms, proportional to + , can be neglected, provided one of the amplitudes dominates. This obvious statement can easily be proved rigorously by means of the Cauchy–Schwarz inequality , valid for arbitrary . Indeed, if for example, the second amplitude still dominates, , then . Neglecting the interference terms in equation (8) yields
Although this approximation is valid beyond the crossover position, the `crossover range' where the amplitudes are of the same order of magnitude always looks rather small on a double logarithmic scale. Therefore, we can use equation (11) even quite close to the crossover point qc. We show numerically in the next section that omitting the interference term is indeed a very good approximation.
where the exponents and are given by equation (1). This is the interpolation formula for the scattering intensity from a macroscopic number of objects of type `fractal inside fractal', type I described in the Introduction (see Fig. 3a).
One can consider a macroscopic number of non-overlapping fractals with concentrations n1 and n2, which are all spatially uncorrelated (system of type II shown in Fig. 3b). The scattering intensity for this type of system can be obtained in the same manner as equation (12):
Finally, we have in the particular case when both regions are in the scattering fractal regime
The first and second terms in the expression for scattering intensity [equation (11) or (13)] are of the same order of magnitude in the vicinity of the crossover point, and, therefore, equating these terms yields the crossover position. Note that the form factors (6) and (7) contain all the qualitative regions in the intensity curve – the Guinier, fractal and Porod regions – and thus this curve together with equation (11) or (13) allows us to obtain the crossover positions for all types of transition: Guinier–fractal, Guinier–Porod, fractal–fractal and fractal–Porod. For the fractal–fractal transition one can use directly equations (12) and (14) and obtain
where δ is the dimensionless `contrast parameter'
The parameter δ plays an important role, because it determines the relative `weights' of the squared normalized form factors in equations (11) and (13) [and correspondingly equations (12) and (14) in the fractal ranges]. As a consequence, it governs the form of the scattering intensity. Note that the suggested model works only if and simultaneously or inversely and . Otherwise only one form factor dominates for all values of q and there is no transition on the scattering curve at all. Within the model, one can obtain only the `convex' behaviour near the transition point on a double logarithmic scale.
where we put by definition
One can prepare a number of samples, which differ only in the density of solvent or solid matrix, and measure their SAS intensities; this is the method of contrast variation, initially suggested by Stuhrmann (1970). Fitting formula (17) in conjunction with equations (18) and (19) suggests the extension of this method when it is applied not for the scattering intensity itself but for the coefficients a1 and a2. Indeed, fitting each intensity curve with equation (17) yields the values of dimensions D1 and D2 (through the exponents and ) and the values of these coefficients at given values of . Since , one can find the value of as the contrast match point in the plot of a11/2 versus and determine the value of as well. A structure of type II exhibits a similar dependence of the coefficient a2, and the values of density and can be found in the same way. If a2 is independent of , we deal with a structure of type I and cannot extract more information than the contrast value itself unless other methods are used.
Very often, a priori information about the fractal system under investigation is known, mainly from other experimental techniques or by knowing the conditions under which the system has been created. One can then use the found coefficients like to obtain the unknown features. For example, if the concentration n1 is known from the sample preparation, the size L1 of region 1 can be estimated from the border of the Guinier range (see Fig. 1). Then the fractal volume V1 can be calculated as well.
Note that the coefficients a1 and a2 are given by the asymptotes of the functions and , respectively, beyond the crossover regime. Therefore, their values can be found from a Kratky-like plot on a double logarithmic scale (see Fig. 6 below).
In order to confirm the above results, we consider an `exactly solvable' three-phase system of type I (see Fig. 2): a mass fractal of the density (region 2) embedded in a ball of the density (region 1). A ball of radius R1 can be considered as the limiting case of surface fractal with surface dimension Ds1 = 2. This specific structure of type I is shown in Fig. 4. In the range , the scattering intensity of a ball is characterized by the Porod exponent, equal to 4. Therefore, the scattering intensity of the system should exhibit the transition from the Porod range to that of mass fractal. The ball should be big enough to absorb the fractal completely. We choose R1 = L231/2/2, where L2 is the fractal size, and the centres of the ball and fractal coincide. The volume of region 1 is obvious, , and its form factor is given by F1(q) = F0(q R1), where
The mass fractal is the generalized Cantor fractal with controllable fractal dimension. Its construction and properties are described in detail by Cherny et al. (2010b) (see also Cherny et al., 2010a, 2011). The initial ball of radius L2/2 is replaced by eight balls of radius , where is a scaling factor whose value lies between 0 and 1/2. Then this `creating and scaling' operation is applied to each of the resulted balls again and again (see Fig. 4). The number m of operations is called the fractal iteration. Then the total fractal volume is given by . The fractal dimension of the volume, in the limit of infinite fractal iterations, is . The dimension takes a value between 0 and 3 when the scaling factor varies from 0 to 1/2. This fractal belongs to the class of deterministic fractals exhibiting exact self-similarity. The form factor of the generalized Cantor fractal of the mth iteration can be calculated analytically (Cherny et al., 2010b):
Substituting the above form factors F1 and F2 into equation (9) yields the scattering intensity of the three-phase system. If the average is taken over the solid angles in accordance with equation (5), we obtain not a smooth (like shown in Fig. 1) but an oscillating curve where the oscillations are superimposed on the power-law regimes. A scattering curve like this is typical for deterministic fractals (Cherny et al., 2011). The smoothing effect, observed experimentally, arises as a consequence of randomness involved in the fractals. Indeed, in a physical system scatterers almost always have different sizes. Therefore, a more realistic description should involve size polydispersity. Here we consider an ensemble of three-phase fractals with different sizes taken at random. For the model, we choose the size distribution to be lognormal. It is characterized by the average size value and the size relative dispersion (for details, see Cherny et al., 2010b, 2011). Thus, the average in equation (8) is taken over both angles and sizes. Polydispersity obviously smears the intensity curves, and the oscillations become smoother (Cherny et al., 2011).
The results are shown in Fig. 5. For numerical simulations, we take the following values of the model parameters: the fractal iteration number m = 5 and the scaling parameter (which leads to the fractal exponent , coinciding with the fractal dimension). It is convenient to measure the densities and in units of and the scattering intensity in units of . We put for the relative dispersion of the lognormal distribution and choose for all the plots. Then the value of α depends on the ratio . It can be seen clearly from Fig. 5(a) that the interference terms in equation (9) play a minor role even in the Guinier range, where . This is a result of the condition .
Because of instrumental limitations, experimental SAS data may contain only a part of the whole scattering curve. The most common case is a curve showing only the transition between two power-law regimes, while the Guinier region and the Porod region at high q are usually missing. Nevertheless, one can exploit the interpolation formula (17) even in this case.
We remark on some features of the model scattering intensities. First, the crossover position (15) can be very sensitive to contrast variation, which influences significantly the length of the fractal range (see Fig. 5c). While for the length of the fractal range is large enough, for it becomes so small that, once the intensity has been obtained experimentally, this range can hardly be interpreted as a `true' fractal behaviour of the system. This suggests the necessity of contrast variation experiments to reveal the structure of a multiphase system in real space. Second, at the end of the fractal range, one can see a `shelf' (Cherny et al., 2010b), which is nothing else but the Guinier range of the smallest structural units composing the fractal (the balls in our model). Such behaviour is explained fairly well within the Beaucage model.
The transition of the scattering intensity from one regime to another can be more clearly observed when we represent the contrast variation data in a Kratky-like plot. At low and high q values in Figs. 6(b) and 6(c), respectively, one can clearly see plateaus corresponding to the coefficients a1 and a2 in equation (17), to which the Stuhrmann method of contrast variation can be applied. The further we are, in the plateau region, from the crossover point, the better the approximation (see the previous section). The oscillations observed beyond the plateau regions damp with increasing polydispersity.
We have explained the origin and positioning of the crossover between successive power-law regimes in SAS experimental data for multiphase fractal systems of type I and II (see Fig. 3). The crossover positions and the contributions of the different structural levels to the total scattering intensity are controlled by the dimensionless `contrast parameter' δ given by equation (16), which depends on the relative values of the scattering length density of each phase and their volumes and concentrations.
A simple estimation of the crossover position (15) is obtained. As is shown numerically in §4, the crossover position can be very sensitive to contrast variation, which influences significantly the length of the fractal range. The developed analysis is applicable for the scattering intensities that are `convex' in the vicinity of the crossover point on a double logarithmic scale and exhibits variation of the crossover point with contrast variation.
From a practical point of view, the main result of this paper is equations (17)–(19). They can be used to fit experimental SAS data and distinguish between the type I system (one fractal absorbs the other fractal) and the type II system (non-overlapping fractals) as described in §3. Once experimental intensity curves for a number of values of are available, one can fit the data with equation (17) and obtain the exponents and and the coefficients a1 and a2 as a function of the surrounding density . Then plotting a11/2 versus yields the scattering density of the first fractal as the contrast match point, and the quantity . Plotting a21/2 versus allows us to distinguish between types I and II: if a2 is independent of , we deal with a structure of type I; otherwise we deal with a structure of type II. In the latter case, the parameters and can be found as well. Thus, the Stuhrmann (1970) variation method is applied not to the scattering intensity at zero angle but to the coefficients a1 and a2.
We emphasize that the suggested multiphase model does not contradict the results of two-phase models such as Beaucage's model (Beaucage, 1995), but our model completes and supplements them. Indeed, the contrast variation method suggests a clear and transparent criterion for understanding whether a two-phase model is relevant to a specific power-law crossover, observed experimentally, or not. If the crossover point qc depends on the surrounding scattering length density then it cannot be explained in the framework of a two-phase model. Note that the crossover position can be very sensitive to the contrast variation (see the discussion at the end of §4).
As is discussed in the Introduction, the presence of the power-law behaviour in the scattering intensity is a sufficient but not necessary condition for the existence of fractals in the sample. The power-law behaviour can be `occasional' and appears as a result of polydispersity in a rather narrow range in momentum space. We emphasize that even in this case the above analysis allows us to distinguish between structures of types I and II.
In the previous articles (Cherny et al., 2010a,b), simple polydisperse models of mass fractals were considered, for which the scattering intensities are similar to that shown in Fig. 1(a). Here we develop a model that is quite close to real physical structures with rather complex distributions of scattering length density and whose behaviour is essentially different from the `standard' one shown in Fig. 1(a). Contrast variation experiments are therefore needed to reveal the real space structure of the investigated systems.
The authors are grateful to José Teixeira for valuable discussions. The authors acknowledge support from JINR-IFIN-HH projects and from JINR grant No. 13-302-02.
Bale, D. H. & Schmidt, P. W. (1984). Phys. Rev. Lett. 53, 596–599. CrossRef CAS Web of Science
Beaucage, G. (1995). J. Appl. Cryst. 28, 717–728. CrossRef CAS Web of Science IUCr Journals
Cherny, A. Yu., Anitas, E. M., Kuklin, A. I., Balasoiu, M. & Osipov, V. A. (2010a). J. Surf. Invest. 4, 903–907. Web of Science CrossRef
Cherny, A. Y., Anitas, E. M., Kuklin, A. I., Balasoiu, M. & Osipov, V. A. (2010b). J. Appl. Cryst. 43, 790–797. Web of Science CrossRef CAS IUCr Journals
Cherny, A. Yu., Anitas, E. M., Osipov, V. A. & Kuklin A. I. (2011). Phys. Rev. E, 84, 036203. Web of Science CrossRef
Cherny, A. Yu., Anitas, E. M., Osipov, V. A. & Kuklin, A. I. (2012). J. Phys. Conf. Ser. 393, 012031. CrossRef
Ehrburger-Dolle, F., Hindermann-Bischoff, M., Livet, F., Bley, F., Rochas, C. & Geissler, E. (2001). Langmuir, 17, 329–334. Web of Science CrossRef CAS
Feigin, L. A. & Svergun, D. I. (1987). Structure Analysis by Small-Angle X-ray and Neutron Scattering. New York, London: Plenum Press.
Glatter, O. & Kratky, O. (1982). Small-Angle X-ray Scattering. London: Academic Press.
Golosova, A. A., Adelsberger, J., Sepe, A., Niedermeier, M. A., Lindner, P., Funari, S. S., Jordan, R. & Papadakis, C. M. (2012). J. Phys. Chem. 116, 15765–15774. CAS
Gouyet, J. F. (1996). Physics and Fractal Structures. Berlin: Springer.
Guinier, A. & Fournet, G. (1955). Small-Angle Scattering of X-rays. New York: Wiley.
Kravchuk, K. V., Gomza, Yu. P., Pashkova, O. V., V'yunov, O. I., Nesin, S. D. & Belous, A. G. (2009). J. Non-Cryst. Solids, 355, 2557–2561.
Lebedev, D. V., Filatov, M. V., Kuklin, A. I., Islamov, A. Kh., Kentzinger, E., Pantina, R., Toperverg, B. P. & Isaev-Ivanov, V. V. (2005). FEBS Lett. 579, 1465–1468. Web of Science CrossRef PubMed CAS
Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. San Francisco: Freeman.
Martin, J. E. & Hurd, A. J. (1987). J. Appl. Cryst. 20, 61–78. CrossRef CAS Web of Science IUCr Journals
Schaefer, D. W. & Justice, R. S. (2007). Macromolecules, 40, 8501–8517. Web of Science CrossRef CAS
Schmidt, P. W. (1991). J. Appl. Cryst. 24, 414–435. CrossRef CAS Web of Science IUCr Journals
Schmidt, P. W. & Dacai, X. (1986). Phys. Rev. A, 33, 560–566. CrossRef PubMed Web of Science
Schneider, G. J. & Goritz, D. (2010). J. Chem. Phys. 132, 154903. Web of Science CrossRef PubMed
Schneider, G. J., Hengl, W., Brandt, K., Roth, S. V., Schuster, R. & Göritz, D. (2012). J. Appl. Cryst. 45, 430–438. Web of Science CrossRef CAS IUCr Journals
Singh, M., Sinha, I., Singh, A. K. & Mandal, R. K. (2011). J. Nanopart. Res. 13, 69–76. Web of Science CrossRef CAS
Stuhrmann, H. B. (1970). Acta Cryst. A26, 297–306. CrossRef IUCr Journals Web of Science
Teixeira, J. (1988). J. Appl. Cryst. 21, 781–785. CrossRef Web of Science IUCr Journals
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.