# Scattering amplitude of diffracted beam by a crystal

• Andreasdreas
In summary, the scattering amplitude F of a diffracted beam by a crystal is defined as the sum of the local electron concentration of the crystal in dV, multiplied by a plane wave. The integral is over all the volumeelements of the crystal, and the sum is over the set of all reciprocal lattice vectors. It can be shown that F is negligibly small when the wave vector of the diffraction beam, \delta \vec{k}, differs significantly from any reciprocal lattice vector, \vec{G}. This can be proven mathematically by considering the fact that the sum only goes over the reciprocal lattice vectors and not \delta \vec{k}.
Andreasdreas

## Homework Statement

The scattering amplidtude, F, of a, by a crystal, diffracted beam is defined to be:

(1) $F=\sum{_\vec{G}}\int n_{\vec{G}}e^{i(\vec{G}-\delta\vec{k})\cdot \vec{r}}\mathrm{d}V$

The integral is over all the volumeelements, dV, of the crystal.
$n_{\vec{G}}$ is the local electron concentration of the crystal in dV, $\vec{G}$ is a reciprocal lattice vector and the sum is over the set of all the reciprocal lattice vectors. $\vec{r}$ is the position vector of dV
further $-\delta\vec{k}=\vec{k}-\vec{k'}$ where $\vec{k}$ is the wave vector of the incomming beam and $\vec{k'}$ is the wave vector of the outgoing scattered beam.

The beam could be
fotons, electrons neutrons etc. It is descibed as a plane wave.

It is easy to see that $F=n_{\vec{G}}V$ when $\delta \vec{k}=\vec{G}$

But how can it be shown that F is negligibly small when $\delta \vec{k}$ differs sginificantly from any $\vec{G}$?

## Homework Equations

I thought some expression for the summation could be used.

Maybe

(2) $\sum{_m=0}^{M-1}x^m=\frac{1-x^{M}}{1-x}$

## The Attempt at a Solution

I can't find a away to make (1) fit into (2). And if i just look at (1) i really am lost.

## The Attempt at a Solution

I thought of using the fact that the sum only goes over all the reciprocal lattice vectors and not \delta \vec{k}. This means that for \delta \vec{k} to be significantly different from any \vec{G}, then F should be small for the term in the summation that involves \delta \vec{k}.But how can one prove this mathematically?

## What is the scattering amplitude of a diffracted beam by a crystal?

The scattering amplitude of a diffracted beam by a crystal is a measure of the intensity of the scattered radiation at a specific angle and wavelength. It is influenced by the atomic arrangement and properties of the crystal, as well as the properties of the incident radiation.

## How is the scattering amplitude calculated?

The scattering amplitude is calculated using the Bragg equation, which takes into account the wavelength of the incident radiation, the spacing between crystal planes, and the angle of incidence.

## What factors affect the scattering amplitude?

The scattering amplitude is affected by the atomic arrangement and properties of the crystal, the properties of the incident radiation (such as wavelength and polarization), and the angle of incidence of the radiation on the crystal.

## What is the significance of the scattering amplitude in crystallography?

The scattering amplitude is a crucial component in crystallography as it provides information about the atomic structure and properties of a crystal. By analyzing the intensity and pattern of diffracted beams, scientists can determine the arrangement of atoms within a crystal and use this information for various applications in materials science, chemistry, and biology.

## How does the scattering amplitude relate to the diffraction pattern of a crystal?

The scattering amplitude is directly related to the intensity and shape of the diffraction pattern produced by a crystal. The amplitude determines the intensity of the diffracted beams at different angles, which creates the characteristic pattern used to analyze the crystal's atomic structure.

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