Crystalline diffraction problem

[furtivefelon]

There is one problem in the problem set i don't know how to approach it..
Problem:
While an artificial diffraction grating is used to study a spectrum of a light beam, electromagnetic waves with wavelengths much less than the wavelength of the visible light are used to study the natural diffraction
grid, namely a crystalline structure. Electromagnetic waves in this range are called the x-rays. The condition for constructive interference is given by Bragg’s law:
2dsin- = m+, m = 1, 2, 3, …

If the wavelength and diffraction angle are measured, the Bragg equation can be used to calculate the spacing between atomic planes.
In our problem, the crystalline of NaCl with density D = 2.16 g/cm3 is exposed to x-rays with the glancing angle - = 60.0o. The reflected rays form the second order maximum.
(a) Find the wavelength of x-rays used in the experiment.
(b) Explain why the visible light cannot be used to obtain the diffraction pattern on crystals.

i don't understand what equation relates the interplanar spacing and density together, or am i on the wrong track?

Thanks alot!

 

[Nate810]

The equation relating the interplanar spacing and density together is Bragg's law. It states that the wavelength of the x-rays used in the experiment can be calculated by 2dsin- = m+, where m is the order of the diffraction pattern, d is the interplanar spacing, and is the glancing angle. In order to answer (a), you will need to calculate the wavelength of x-rays using the equation above. In order to answer (b), you can explain that visible light has a much longer wavelength than x-rays, making it less effective at diffracting off of crystalline structures and thus not suitable for obtaining a diffraction pattern.

 

[quicknote]

I understand your confusion and will be happy to provide some guidance on this problem. First, let's review the basics of diffraction and Bragg's law.

Diffraction is the phenomenon where a wave, such as light or electromagnetic waves, bends and spreads out when passing through an opening or around an obstacle. This can create an interference pattern, where the waves either reinforce or cancel each other out.

Bragg's law, named after physicist William Henry Bragg, describes the conditions for constructive interference in diffraction. It states that when a wave is incident on a crystal at a certain angle, the diffracted waves will constructively interfere if the path difference between the waves is an integer multiple of the wavelength. This can be expressed as 2dsinθ = mλ, where d is the spacing between atomic planes, θ is the angle of incidence, m is the order of the diffraction peak, and λ is the wavelength.

Now, to answer your question about the relationship between interplanar spacing and density, it is important to understand that the spacing between atomic planes in a crystal is determined by its lattice structure, not its density. The density of a crystal is simply a measure of how closely packed the atoms are within the crystal lattice. Therefore, there is no direct equation that relates interplanar spacing and density.

In this problem, you are given the density of NaCl and the glancing angle of the x-rays. From this, you can use Bragg's law to calculate the wavelength of the x-rays. Simply rearrange the equation to solve for λ and plug in the given values. This will give you the answer to part (a) of the problem.

As for part (b), visible light cannot be used to obtain diffraction patterns on crystals because its wavelength is much larger than the interplanar spacing in most crystals. This means that the path difference between the diffracted waves would not be an integer multiple of the wavelength, resulting in a very weak or no interference pattern. X-rays, on the other hand, have much smaller wavelengths and can interact with the crystal lattice, producing a clear and observable diffraction pattern.

I hope this explanation helps you approach the problem with more clarity. Remember to always use the appropriate equations and understand the concepts behind them to solve scientific problems effectively. Keep up the good work!