Useful model-free approximations in SANS

We here present a number of useful approximations and limiting cases for small and large values of the scattering vector, \(q\). These approximations are important for the analysis and understanding of SANS data.

The Debye formula

For a randomly oriented particle of general shape, Debye was able to show a highly useful expression for the average value of the complex phase

\begin{equation} \label{eq:debye} \left\langle \exp(i {\mathbf q}\cdot{\mathbf r}) \right\rangle = \dfrac{\sin(qr)}{qr}. \end{equation}

This formula is often used for calculating the form factor of particles in solution; the proof is sketched below.

Consider a \({\bf q}\), fixed in the laboratory system, and let us define the \(z\) axis along \({\bf q}\). We consider the particles fixed in space and randomly oriented. Hence, the position, \(\bf r\), of one particular nucleus can be found anywhere on a sphere of radius \(r\), described by the spherical coordinates, \((\theta, \phi)\). We can now write \({\bf q}\cdot{\bf r} = q r \cos(\theta)\). The average value of the complex exponential function is given by the spherical integral

\begin{equation}\label{dummy25288786} \left\langle \exp(i {\mathbf q}\cdot{\mathbf r}) \right\rangle = \dfrac{1}{4\pi} \displaystyle\int d\theta d\phi \sin(\theta) \exp(i {\mathbf q}\cdot{\mathbf r}) = \dfrac{1}{2} \displaystyle\int_0^\pi d(\cos\theta) \exp(i qr \cos\theta) = \dfrac{\sin(qr)}{qr} . \end{equation}

The Guinier approximation

We will now present an important approximation for particle structure factors in small-angle scattering. Here, we show the Guinier approximation for small values of \(q\).

Let us first study the hard-sphere form factor equation \eqref{eq:sans_spheres} for small values of \(q\), e.g. \(q R \ll 1\). We expand the trigonometric functions to 5th(!) order in \(q R\), reaching

\begin{equation}\label{eq:sphereguinier} P_{\rm sphere}(q) \approx 1-\dfrac{1}{5}(q R)^2 .\, \end{equation}

This function has the same series expansion (to second order) as the expression \(\exp(-(q R)^2/5)\), which is what we define as the Guinier approximation for the sphere form factor.

To generalize from this example, we can cast the equation in terms of the radius of gyration. The radius of gyration of the sphere is easily calculated by converting \eqref{eq_rg} into an integral, reaching

\begin{equation} \label{eq:Rg_sphere} R_{\rm g}^2 = 3 R^2 / 5 . \end{equation}

This leads to

\begin{equation}\label{eq:guinier} P(q) \approx \exp\left(-\dfrac{1}{3}(q R_{\rm g})^2\right) . \end{equation}

In fact, this is the general Guinier approximation valid for any particle shape. We will, however, not show it in the general case here.

The Porod law

We here present the important Porod law, which is valid for large \(q\) values.

We again consider the sphere form factor equation \eqref{eq:sans_spheres}, this time for large values of \(q\), e.g. \(q R \gg 2 \pi\). To leading value of \(q R\), this becomes

\begin{equation}\label{dummy994030239} P_{\rm spheres}(q) \approx \dfrac{9 \cos^2(q R)}{(q R)^{4}} . \end{equation}

Now, for large \(q\), even very small variations in the particle size, \(R\), will lead to a variation of the cosine argument \((q R)\) by an amount comparable to, or larger than, \(2\pi\). For such a sample, we will observe approximately the average value of the cosine, i.e., \(\left\langle \cos^2(q R) \right\rangle = 1/2\), leading to \(P(q) \approx 9/(2q^4R^4)\) or

\begin{equation}\label{eq:porod} P(q) \propto q^{-4} .\, \end{equation}

This is, in fact, the generally valid Porod law for small-angle scattering. We will not show the Porod law directly here. Intuitively, however, it is reasonable to generalize from the example of the spheres. Here, the real-space structure (the particle radius) probed at a scattering vector of \(q\) is of the order \(\pi/q\), as judged from the position of the first "dip" in the expression for the particle form factor. Hence, at much larger values of \(q\), the measurement is sensitive to smaller real-space structures. However, the only small real-space structures visible are sharp surfaces: The boundary between two different scattering length densities. And the reflectivity from a surface is in fact proportional to \(q^{-4}\), as will be shown in the page on Neutron reflectivity.