Structure Factor
What is Structure Factor?
Structure factor [1]is a term that is often used in condensed matter physics. And it is also called static structure factor. It is a mathematical expression of how a material scatters. It has two mathematical expressions in use. The first one is $S(\bold{q})$, which is more generally valid, and the other is usually written as $F$ or $F_{hkl}$, and is only valid for crystal systems.
$F_{hkl}$ is not a special case of $S(\bold{q})$;
$S(\bold{q})$ gives the scattering intensity, and is most useful for disordered systems. While $F_{hkl}$ gives the amplitude. It is the squared modulus $|F_{hkl}|^2$ that gives the scattering intensity. $F_{hkl}$ is defined for a perfect crystal. For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed.
In this post, I’m going to focus on the first expression , $S(\bold{q})$.
Scattering
What is scattering?
Atoms or molecules which are exposed to light absorb light energy and re-emit light in different directions with different intensity. This phenomenon is an example of Scattering[2].
Scattering may also refer to particle-particle collisions between molecules, atom, electrons, photons and other particles.
When radiation is only scattered by one localized scattering center, this is called single scattering. It is very common that scattering centers are grouped together; in such cases, radiation may scatter many times, in what is known as multiple scattering.
Because the location of a single scattering center is not usually well known relative to the path of the radiation, the outcome, which tends to depend strongly on the exact incoming trajectory, appears random to an observer. This type of scattering would be exemplified by an electron being fired at an atomic nucleus. In this case, the atom’s exact position relative to the path of the electron is unknown and would be unmeasurable, so the exact trajectory of the electron after the collision cannot be predicted. Single scattering is therefore often described by probability distributions.
With multiple scattering, the randomness of the interaction tends to be averaged out by a large number of scattering events, so that the final path of the radiation appears to be a deterministic distribution of intensity.
Models of light scattering can be divided into three domains based on a dimensionless size parameter, $\alpha$ which is defined as:
$$ \alpha = \pi D_p/\lambda $$
where $\pi D_p$ is the circumference of a particle and $\lambda$ is the wavelength of incident radiation. Based on the value of $\alpha$, these domains are:
- $\alpha \ll 1$: Rayleigh scattering (small particle compared to wavelength of light)
- $\alpha \approx 1$: Mie scattering (particle about the same size as wavelength of light, valid only for spheres)
- $\alpha \gg 1$: geometric scattering (particle much larger than wavelength of light)
::: warning Question What type of scattering of my study belongs to? :::
::: warning Question What is wavelength of neutron scattering? :::
Neutron scattering
Neutron scattering[3] can be referred to the man-made experimental techniques that use the natural process for investigating materials.
“Fast neutrons” have a kinetic energy above 1 Mev[4]
Unlike an x-ray photon with a similar wavelength, which interacts with the electron cloud surrounding the nucleus, neutrons interact primarily with the nucleus itself. Important elements like carbon and oxygen are quite visible in neutron scattering. Thus neutrons can be used to analyze materials with low atomic numbers, including proteins and surfactants.
Scattering almost always presents both elastic and inelastic components. The fraction of elastic scattering is determined by the Debye-Waller factor. Depending on the research question, most measurements concentrate on either elastic or inelastic scattering.
SANS
Through the use of cold (i.e. long wavelength) neutrons and tight beam collimation, the SANS instruments are able to probe structure on a length scale ($d$) ranging from 1 nm to nearly 10,000 nm. The neutron wavelength($\lambda$) and scattering angle ($\theta$) determine the length scale probed through the relationship.
$$ d \approx \frac{\lambda}{\theta} \frac{(\text {wavelength})}{(\text {scatteringangle})} $$
::: tip
Wave basics[wiki]
$\lambda$, the wavelength can be measured between any two points with the same phase, such as between crests(on top), or troughs (on bottom), or corresponding zero crossings as shown.
The wavelength $\lambda$ of a sinusoidal waveform traveling at a constant speed $v$ is given by
$$ \lambda=\frac{v}{f} $$
where $v$ is called the phase speed(magnitude of the phase velocity) of the wave and $f$ is the wave’s frequency.
Mathematical representation
Travelig sinusoidal waves are often represented mathematically in terms of their velocity $v$ (in the $x$ diection), frequency $f$ and wavelength $\lambda$ as:
$$ y(x, t)=A \cos \left(2 \pi\left(\frac{x}{\lambda}-f t\right)\right)=A \cos \left(\frac{2 \pi}{\lambda}(x-v t)\right) $$
where $y$ is the value of the wave at any position $x$ and time $t$, and $A$ is the amplitude of the wave. They are also commonly expressed in terms of wavenumber $k$ ($2\pi$ times the reciprocal of the wavelength) and angular frequency $\omega$ ($2\pi$ times the frequency) as:
$$ y(x, t)=A \cos (k x-\omega t)=A \cos (k(x-v t)) $$
in which wavelength and wavenumber are related to velocity and frequency as :
$$ k=\frac{2 \pi}{\lambda}=\frac{2 \pi f}{v}=\frac{\omega}{v} $$
or
$$ \lambda=\frac{2 \pi}{k}=\frac{2 \pi v}{\omega}=\frac{v}{f} $$
In the second form given above, the phase $(kx-\omega t)$ is often generalized to $(\mathbf{k} \cdot \mathbf{r}-\omega t)$, by replacing wavenumber $k$ with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector $\bold{r}$. In that case , the wavenumber $k$, the magnitude of $\bold{k}$, is still in the same relationship with wavelength as shown above, with $v$ being interpreted as scaler speed in the direction of the wave vector. :::
Fundamentals
$$ \bold{q}=\bold{k}{\mathrm{f}}-\bold{k}{\mathrm{i}} $$
where $\bold{k}_f$ and $\bold{k}_i$ denote the wave vectors of the incident and scattered plane waves, respectively, the result of a scattering experiment is usually expressed by giving the intensity distribution in q-space, $I(q)$. In the majority of scattering experiments on polymers the radiation frequency remains practically unchanged. Then we have
$$ \left|\bold{k}{\mathrm{f}}\right| \approx\left|\bold{k}{\mathrm{i}}\right|=\frac{2 \pi}{\lambda} $$
and $|q|$ is related to the Bragg scattering angle $\vartheta_{\mathrm{B}}$ by
$$ |\bold{q}|=\frac{4 \pi}{\lambda} \sin \vartheta_{\mathrm{B}} $$
$\vartheta_{\mathrm{B}}$ is identical to half of the angle enclosed by $\bold{k}_i$ and $\bold{k}_f$
Basic Equations
Two different functions can be used for representing scattering data in reduced form.
The first one, denoted $\sum(\bold{q})$, di the differential scattering cross-section per unit volume
$$ \Sigma(\bold{q})=\frac{1}{\mathcal{V}} \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}=\frac{1}{\mathcal{V}} \frac{I(\bold{q}) A^{2}}{I_{0}} $$
In light scattering experiments this function is called the ** Rayleigh ratio**. While the effect of the volume is removed, $\sum(\bold{q})$ remains dependent on the scattering power of the particles in the sample, which varies with the applied radiation. For X-rays, the scattering power is related to the electron densities, for light scattering to the associated refractive indices and for neutron scattering to the scattering length densities.
This dependence on the applied radiation is eliminated in the second function which, however, can only be employed if the scattering can be treated as being due to just one class of particles. In polymer systems these can be identified with the monomeric unit. For equal particles the scattering properties can be described by the interference function $S(q)$, also called scattering function or scattering law, which is defined as
$$ S(\bold{q})=\frac{I(\bold{q})}{I_{\mathrm{m}} \mathcal{N}_{\mathrm{m}}} $$
Here $\mathcal{N}_{\mathrm{m}}$ represents the total number of particles/monomers in the sample and $I_m$ is the scattering intensity produced by one particle, if placed in the same incident beam. The interference function expressed the ratio between the actual intensity and the intensity that would be measured, if all particles in the sample were to scatter incoherently.
What is correlation ?
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables.
What the use of correlation function?
Correlation is a measure of how similar signals are.[Ref] It’s a automatic way to calculate how similar signals are, rather than compare with your observation. Find some pattern, can be used in radar system.
How to calculate correlation function?
$$ Corr_{x,y} = \sum_{n= 0}^{N-1}x[n]y[n] $$
$$ Corr_norm_{x,y} = \frac{\sum_{n= 0}^{N-1}x[n]y[n]}{\sqrt{\sum_{n=0}^{N-1}x^2[n] \sum_{n=0}^{N-1}y^2[n]}} $$
The possible normalised correlation value is between -1 and 1.
Cross Correlation[ref]
How to apply correlation function to structure factor?
What is pair correlation function?
It is also called radial distribution function.[ref] This quantity can directly be compared with the structure factor which is obtained from experiments by means of neutron or X-ray scattering.
$\theta$ space and $Q$ space.
The pair correlation function quantifies how the particle of interest is surrounded by other particles.
For example, the pair correlation function or radial distribution function has a constant value for all radial distances for a rarefied gas in which there is no internal structure. In contrast, for a solid system in which molecules are almost regularly located, the pair correlation function has sharply peaked values at the positions of particles, and is nearly zero at the positions where particles are seldom located.
The radial distribution function, $g(r)$ in a system of particles, describes how density varies as a function of distance from reference particle.
If a given particle is taken to be at the origin $O$, if $\rho = N/V$ is the average number density of particles, then the local time-averaged density at a distance $r$ from $O$ is $\rho g(r)$.
::: tip 球の表面積の公式 半径$r$の球の表面積は $S = 4 \pi r^2$. [ref] :::
::: tip 径向分布函数
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The derivation of structure factor
Many textbooks discuss the various structure factors. The Fourier transform of $S(Q,E)$ is in the time domain[7].
$$ \mathrm{S}(\mathrm{Q}, \mathrm{t})=\int_{-\infty}^{+\infty} \mathrm{d} \mathrm{E} \exp \left(\frac{\mathrm{i} \mathrm{Et}}{\hbar}\right) \mathrm{S}(\mathrm{Q}, \mathrm{E}) $$
$S(Q,t)$ is the time-dependent density-density correlation function also called time-dependent structure factor.
$S(Q,E)$ is measured by most quasielastic/inelastic neutron scattering spectrometers such as the triple axis, the backscattering and the time-of-flight instrument measure. Note that $S(Q)$ is also expressed as:
$$ \mathrm{S}(\mathrm{Q})=\mathrm{S}(\mathrm{Q}, \mathrm{t}=0)=\int \mathrm{S}(\mathrm{Q}, \mathrm{E}) \mathrm{d} \mathrm{E} $$
Elastic scattering does not really mean with energy transfer $E=0$ (zero peak and zero width); it rather means integrated over all energy transfers(summing up over all energy modes).
$S(Q)$ is the density-density correlation function.
$$ \mathrm{S}(\mathrm{Q})=<\mathrm{n}(-\mathrm{Q}) \mathrm{n}(\mathrm{Q})> $$
Another form of the density-density correlation function $S(Q)$ is related to the pair correlation function $g(\vec{r})$ through the space Fourier transform:
$$ \mathrm{S}(\mathrm{Q}) = 1 + \overline{\mathrm{N}} \int \mathrm{d}\vec{\mathrm{r}} \exp(-\mathrm{i} \vec{\mathrm{Q}} \cdot \vec{\mathrm{r}})[\mathrm{g}(\vec{\mathrm{r}})-1] $$
Here $\overline{\mathrm{N}}=\mathrm{N} / \mathrm{V}$ is the particle number density. Note also that the scattering lengths have still not been separated out. These will be averaged of each scattering unit to form the contrast factor which will be multiplying $S(Q)$.
[1] https://en.wikipedia.org/wiki/Structure_factor ↩︎ [2] https://en.wikipedia.org/wiki/Scattering ↩︎ [3] https://en.wikipedia.org/wiki/Neutron_scattering ↩︎ [4] An electronvolt (symbol eV, also written electron-volt and electron volt) is the amount of kinetic energy gained (or lost) by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. ↩︎ [5] https://www.nist.gov/ncnr/neutron-instruments/small-angle-neutron-scattering-sans↩︎ [6] The angstrom or ångström is metric unit of length equal to $10^{-10}$m. [7] SANS BookProbing Nanoscale Structures - The SANS Toolbox ↩︎