How Does X-Ray Diffraction Determine Atomic Spacing in Crystals?

[CollectiveRocker]

Here's an interesting problem: Find the atomic spacing in a crystal of rock salt (NaCl). The density of rock salt is 2.16 * 10^3 kg/m^3 and the average masses of the Na and Cl atoms are respectively 3.82 * 10^-26 kg and 5.89 * 10^-26 kg. Now I know that density (Rho) = mass/volume and 2d sin x = nλ, where n = 1,2,3, ... d is the distance between atomic molecules, λ is the wavelength, x = angle of incidence. How do I use this info to find out what d is? Is there another formula I'm missing?

 

[Gonzolo]

Find out the number of atoms in a cubic meter from the density equation and the NaCl structure (keep the Na mass to Cl mass ratio). You don't need Bragg's law for this. You can use Braggs's law to verify if you have access to NaCl x-ray spectra, it's probably from lambda = 1.54 Ang, so plug this in, with n = 1 and x from the spectra and the d should be the same.

Now you owe me a cookie.

 

[Gokul43201]

You don't need bragg's law for this problem...it doesn't help in any way. You're not given any data about -X-ray diffraction spectra, are you ?

What you do need to know is the structure of the NaCl crystal : it's a cubic lattice with Cl atoms at the corners and face centers and the Na atoms at edge centers and body center (this is the same as 2 interpenetrating FCCs).

Find the mass of the unit cell, by calculating the number of units of NaCl per unit cell. From the density, you can calculate the volume of the unit cell, and hence the unit cell edge. Now use the structure to determine the inter-atomic spacing.

 

[vlamir]

CollectiveRocker,
More than two years I participate in Physics Forums, but never saw similar thread.
Really, it is very interesting and perspective theme.
By the way to tell, I trust calculations on density more, than to calculations on diffractional parameters. For example, at calculation of crystal structure of boron, I have found out, that calculation on diffractional parameters gives the error up to 25 %.
See
http://vlamir.nsk.ru/B_&_N_&_O_e.pdf
However, I use own hypothesis about a structure of atoms.
For me, it will be interesting for finding out, what ideas concerning the geometrical form of atoms you have.

 

[Gokul43201]

You get large errors with XRD, only if your sample is strained or you don't have a large enough amount of the sample to get accurate results.

If you do careful parameter fitting with good data, you can easily get within 5% of the right numbers.

 

[vlamir]

It may be assumed, experimental data on XRD from directories of 1995-2000 editions are recommended, as the most exact. The point is how we understand process of interaction X-rays with atoms of crystal.
In one of my calculations, I have tried to construct transitions from cubic lattices into hexagonal and vice versa at the constant sizes and the form of atoms. These transitions, it agrees with XRD, occur at change of temperature of crystal. At that, it has turned out, that the density of substance changes very sharply. In experiment it is not observed.
Hence, at X-Ray Diffraction on cubic and hexagonal phases of the same crystal, there is a radiation of quantums from different sites on surface of atoms (or, if you want, reflection X-rays from different sites of surface of atoms).
In modern XRD – theory this feature is not taken into account.