General principles

Glatter, O.

doi:10.1107/97809553602060000581

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

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 90-91

Section 2.6.1.2. General principles

O. Glatter^a

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 $[I_e(\theta)=I_pT_f{1\over a^2}{1+\cos ^22\theta \over 2},\eqno (2.6.1.1)]$ where [I_p] is the primary intensity and a the distance from the object to the detector. The factor [T_f] is the square of the classical electron radius (e²/mc² = 7.90 × 10⁻²⁶[cm²]). The scattering angle 2 $[\theta]$ 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. [I_e] 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. [I_e] 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 $[\theta]$ . The path difference between the rays through a point P and an arbitrary origin O is −r(s − s₀). The phase is $[\varphi = -{\bf h}\cdot {\bf r}]$ if we define the scattering vector h as $[{\bf h}=(2\pi/\lambda)({\bf s}-{\bf s}_0).\eqno (2.6.1.2)]$ This vector bisects the angle between the scattered beam and the incident beam and has length $[h=(4\pi/\lambda)\sin \theta]$ . We keep in mind that $[\sin\theta]$ may be replaced by $[\theta]$ 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 $[A({\bf h})=\textstyle\int\!\!\int\!\!\int\rho({\bf r})\exp (-i{\bf h \cdot r}){\,{\rm d}}V.\eqno (2.6.1.3)]$ 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*, $[I({\bf h})=A({\bf h})A({\bf h})^* =\textstyle\int\!\!\int\!\!\int\!\!\tilde{\rho}^{2}({\bf r})\exp (-i{\bf h\cdot r}){\,{\rm d}}V,\eqno (2.6.1.4)]$ where $[\tilde{\rho}^{2}({\bf r})]$ is the convolution square (Bracewell, 1986): $[\tilde{\rho}^{2}({\bf r})=\textstyle\int\!\!\int\!\!\int\!\!\rho({\bf r}_1)\rho ({\bf r}_1-{\bf r}){\,{\rm d}}V_1.\eqno (2.6.1.5)]$ 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 $[\langle\ \rangle]$ leads to $[\langle\exp (-i{\bf h\cdot r})\rangle={\sin hr\over hr}\eqno (2.6.1.6)]$ (Debye, 1915) and (2.6.1.4) reduces to the form $[I(h)=4\pi\int\limits^\infty _0 r^2\Delta\tilde{\rho}^2(r)\displaystyle{\sin hr\over hr}{\,{\rm d}}r\eqno (2.6.1.7)]$ or, with $[p(r)=r^2\Delta\tilde{\rho}^2(r)=r^2V{\gamma}(r),\eqno (2.6.1.8)]$ to $[I(h)=4\pi\int\limits^{\infty}_0 p(r) \displaystyle{\sin hr\over hr}{\,{\rm d}}r\semi\eqno (2.6.1.9)]$ 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 $[p(r)={1\over 2\pi^2}\int\limits^\infty _0 I(h)hr\sin (hr){\,{\rm d}}h\eqno (2.6.1.10)]$ or by $[V{\gamma}(r)={1\over 2\pi^2}\int\limits ^\infty _0 I(h)h^2 \displaystyle{\sin hr\over hr} {\,{\rm d}}h.\eqno (2.6.1.11)]$ 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 [r^2] 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 [r^2] parabola and finally the PDDF will be zero for all $[r \, \gt \, D]$ , 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) $[I(0)=I(\overline {\Delta\rho})^2V^2=4\pi\textstyle\int\limits^\infty _0p(r){\,{\rm d}}r,\eqno (2.6.1.12)]$ 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 $[V{\gamma}(0)={1\over 2 \pi^{2}}\int I(h)h^2{\,{\rm d}}h=V{\overline {\Delta\rho^2}}\eqno (2.6.1.13)]$ (Porod, 1982), i.e. the integral of the intensity times [h^2] 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). $[{\rm Invariant}\ Q=\textstyle\int\limits ^\infty _0 I(h)h^2 {\,{\rm d}}h.\eqno (2.6.1.14)]$

References

Bracewell, R. (1986). Fourier transform and its applications. New York: McGraw-Hill.Google Scholar

Debye, P. (1915). Zerstreuung von Röntgenstrahlen. Ann. Phys. (Leipzig), 46, 809–823.Google Scholar

Debye, P. & Bueche, A. M. (1949). Scattering by an inhomogeneous solid. J. Appl. Phys. 20, 518–525.Google Scholar

Glatter, O. (1979). The interpretation of real-space information from small-angle scattering experiments. J. Appl. Cryst. 12, 166–175.Google Scholar

Guinier, A. & Fournet, G. (1955). Small angle scattering of X-rays. New York: John Wiley.Google Scholar

Porod, G. (1951). Die Röntgenkleinwinkelstreuung von dichtgepackten kolloiden Systemen. I. Kolloid Z. 124, 83–114.Google Scholar

Porod, G. (1982). In Small-angle X-ray scattering, edited by O. Glatter & O. Kratky, Chap. 2. London: Academic Press. Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 90-91