# Calculating Structure Factor for Sodium Chloride Crystal Structure

• captainjack2000
In summary, sodium chloride has a face-centred cubic structure with sodium atoms at (0,0,0), (0.5,0.5,0), (0,0.5,0.5), and (0.5,0,0.5) and chlorine atoms at (0.5,0,0), (0,0.5,0), (0,0,0.5), and (0.5,0.5,0.5). The structure factor for this structure is given by S=Σbjexp(iK.r), and for this specific case, it can be calculated by adding the number of electrons in sodium (10) and chlorine (18) for each
captainjack2000

## Homework Statement

Sodium chloride has a face-centred cubic structure which can be regarded as a primitive cubic structure with a basis of sodium atoms at fractional coordinatese (0,0,0) (0.5,0.5,0) (0,0.5,0.5) and (0.5,0,0.5) and chlorine atoms at (0.5,0,0) (0,0.5,0) (0,0,0.5) and (0.5,0.5,0.5). The sodium ion has 10 electrons and the chlorine ion 18. Calculate the three different values of this structure factor. Show that the non zero values satify the relationship h+k=2n and k+l=2n where n is an integer.

## Homework Equations

I know that the structure factor is given by S=Sigma[bjexp(iK.r)] and that Shkl=Sbasis * Sffc

## The Attempt at a Solution

Sbasis = N(sodium) +N(chloride)exp(-i*PI*h)
=10+18 = 28 if h, k l are all even
=10-18 = -8 if h,k, l are all odd

I know this is wrong because the answers should be 112, -36 and 0
But my notes also say that If h,k,l are all odd the Shkl=4N and if they are all even Shkl=4N and if one or more is odd and the others are even Shkl=0. I can see that you get the required numbers by using these three cases with the numbers calculated above but I don't know how it all fits together and you can justify multiplying them by four.

Also, I have no Idea how to show the relationships h+k=2n and k+l=2n

Because you simply have list of atoms I don't think you need to break it down into $$S = S_{basis} \times S_{fcc}$$ (although I suppose it is possible to do it that way?).

Using the sum and substituting in the list of vectors you should get:

$$\sum_j = n_j exp(iK.r_j) = n_{Na}( 1 + e^{i\pi(h + k)} + ... ) + n_{Cl} ( e^{i \pi h} + e^{i \pi k} + ...)$$

where $$n_{Na}, n_{Cl}$$ are number of electrons in Sodium and Chlorine respectively. There should be 8 parts to the sum, and using $$e^{i \pi 2n} = 1$$ and $$e^{i \pi (2n+1)} = -1$$ where n is an integer, you should get the desired result.

To get you started, i think for h, k, l all even you should get $$S_{hkl} = 4(n_{Na} + n_{Cl})$$. There should be different three values S can take.

## 1. What is the structure factor?

The structure factor is a measure of the arrangement of atoms or molecules within a crystal lattice. It describes the scattering of X-rays or neutrons by the crystal lattice, providing information about the positions and types of atoms present.

## 2. How is the structure factor calculated?

The structure factor is calculated by taking the Fourier transform of the crystal lattice, which converts the information about the positions and types of atoms into a mathematical representation. This calculation involves complex mathematical equations and requires specialized software or programs.

## 3. What does the structure factor tell us about a crystal?

The structure factor provides information about the unit cell size, symmetry, and atomic arrangement within a crystal. It can also reveal any defects or impurities present in the crystal lattice.

## 4. How is the structure factor used in materials science?

The structure factor is a crucial tool in materials science as it allows scientists to determine the atomic or molecular structure of a material. This information is important for understanding the properties and behavior of a material, such as its strength, conductivity, or optical properties.

## 5. Can the structure factor be measured experimentally?

Yes, the structure factor can be measured experimentally through techniques such as X-ray or neutron diffraction. These methods involve scattering a beam of X-rays or neutrons off a crystal and measuring the intensity and direction of the scattered radiation, which can then be used to calculate the structure factor.

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