# Difference between revisions of "Scattering"

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− | '''Scattering''' broadly refers to experimental techniques that use the interaction between radiation and matter to elucidate structure. In x-ray scattering, a collimated x-ray beam is directed at a sample of interest. The incident x-rays scatter off of all the atoms/particles in the sample. Because of the wavelike nature of x-rays (which are simply high-energy photons; i.e. electromagnetic rays), the scattered waves interfere with one another, leading to constructive interference at some angles, but destructive interference at other angles. The | + | [[Image:Standard-AgBH-gisaxs th000 spot3 60sec SAXS.png|thumb|200px|right|Example scattering pattern (for [[Material:Silver behenate|AgBH]]). The [[Scattering_features#Rings|rings]] of scattering at well-defined angles arise from scattering off of the lamellar order in the sample.]] |

+ | |||

+ | '''Scattering''' broadly refers to experimental techniques that use the interaction between radiation and matter to elucidate structure. In [[x-ray]] scattering, a collimated x-ray beam is directed at a sample of interest. The incident x-rays scatter off of all the atoms/particles in the sample. Because of the wavelike nature of x-rays (which are simply high-[[X-ray energy|energy]] photons; i.e. electromagnetic rays), the scattered waves interfere with one another, leading to constructive interference at some angles, but destructive interference at other angles. The end result is a [[Scattering features|pattern]] of scattered [[Scattering intensity|radiation]] (as a function of angle with respect to the direct beam) that encodes the microscopic, nanoscopic, and molecular-scale structure of the sample. | ||

+ | |||

+ | ==Geometry== | ||

+ | We define a vector in [[reciprocal-space]] as the difference between the incident and scattered x-ray beams. This new vector is the [[momentum transfer]], denoted by '''q''': | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | \mathbf{q} & = \mathbf{k}_o - \mathbf{k}_i \\ | ||

+ | & = k(\mathbf{s}_o - \mathbf{s}_i) \\ | ||

+ | & = \frac{2 \pi}{\lambda}(\mathbf{s}_o - \mathbf{s}_i) | ||

+ | \end{alignat} | ||

+ | </math> | ||

+ | |||

+ | The length of this vector is: | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | q = |\mathbf{q}| & = 2 k \sin { ( 2 \theta_s /2 ) } \\ | ||

+ | & = \frac{4 \pi}{\lambda} \sin{ \theta_s } | ||

+ | \end{alignat} | ||

+ | </math> | ||

+ | Where <math>\scriptstyle 2 \theta_s </math> is the scattering angle. | ||

+ | |||

+ | See also [[Geometry:TSAXS 3D]]. | ||

+ | |||

+ | ==Theory== | ||

+ | The mathematical form of scattering is closely related to the [[Fourier transform]]. The sample's [[realspace]] density distribution is Fourier transformed into an abstract 3D [[reciprocal-space]]; scattering probes this inverse space. The fundamental equation in scattering is: | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | I(\mathbf{q}) | ||

+ | & = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\ | ||

+ | \end{alignat} | ||

+ | </math> | ||

+ | Where the observed [[scattering intensity]] (''I'') in the 3D reciprocal-space ('''q''') is given by an ensemble average of the intensity for all (''N'') scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions (<math>\rho_n</math> denotes the scattering power) of the ''N'' entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density <math>\rho(\mathbf{r})</math>), we can write an integral over all of real-space: | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | I(\mathbf{q}) | ||

+ | & = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\ | ||

+ | \end{alignat} | ||

+ | </math> | ||

+ | The inner component can be thought of as the [[reciprocal-space]]: | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | |||

+ | F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\ | ||

+ | |||

+ | \end{alignat} | ||

+ | </math> | ||

+ | This is mathematically identical to the (three-dimensional) [[Fourier transform]]. | ||

+ | |||

+ | |||

+ | ==See Also== | ||

+ | * [[Bragg's law]] | ||

+ | * [[Diffraction]] | ||

+ | * [[Fourier transform]] | ||

+ | * Geometry: | ||

+ | ** [[Geometry:TSAXS 3D|TSAXS 3D]] | ||

+ | ** [[Geometry:WAXS 3D|WSAXS 3D]] |

## Latest revision as of 09:29, 23 January 2018

**Scattering** broadly refers to experimental techniques that use the interaction between radiation and matter to elucidate structure. In x-ray scattering, a collimated x-ray beam is directed at a sample of interest. The incident x-rays scatter off of all the atoms/particles in the sample. Because of the wavelike nature of x-rays (which are simply high-energy photons; i.e. electromagnetic rays), the scattered waves interfere with one another, leading to constructive interference at some angles, but destructive interference at other angles. The end result is a pattern of scattered radiation (as a function of angle with respect to the direct beam) that encodes the microscopic, nanoscopic, and molecular-scale structure of the sample.

## Geometry

We define a vector in reciprocal-space as the difference between the incident and scattered x-ray beams. This new vector is the momentum transfer, denoted by **q**:

The length of this vector is:

Where is the scattering angle.

See also Geometry:TSAXS 3D.

## Theory

The mathematical form of scattering is closely related to the Fourier transform. The sample's realspace density distribution is Fourier transformed into an abstract 3D reciprocal-space; scattering probes this inverse space. The fundamental equation in scattering is:

Where the observed scattering intensity (*I*) in the 3D reciprocal-space (**q**) is given by an ensemble average of the intensity for all (*N*) scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions ( denotes the scattering power) of the *N* entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density ), we can write an integral over all of real-space:

The inner component can be thought of as the reciprocal-space:

This is mathematically identical to the (three-dimensional) Fourier transform.

## See Also

- Bragg's law
- Diffraction
- Fourier transform
- Geometry: