What Are the Miller Indices for the Bragg Reflections?

[cepheid]

Homework Statement

An alkali halide is studied with the Debye-Scherrer technique and Cu $K_{\alpha}$ radiation. The Bragg angles for the first five lines (in degrees) are 10.83, 15.39, 18.99, 22.07, and 24.84. Calculate

(a) The lattice parameter
(b) The Miller indices for the planes producing the mentioned diffraction beams
(c) The Miller index for the line(s) producing the largest allowable Bragg angle.

Added in by the prof as an afterthought:

$$\textrm{Take} \ \ \lambda[\textrm{Cu} \ K_{\alpha}] = 1.542 \ \ \textrm{\AA}$$

Homework Equations

As far as I can tell they are:

The Bragg condition for diffraction:

$$2d \sin \theta = n \lambda$$

A previously derived result that for a cubic lattice with lattice constant a, the spacing between successive (hkl) places is given by:

$$d = a[h^2 + k^2 + l^2]^{-\frac{1}{2}}$$

The Attempt at a Solution

I'm pretty lost here. Is the lattice parameter (cubic unit cell dimension) 'a' supposed to be arrived at somehow independently of the diffaction data?

The Debye-Scherrer technique was never discussed in class. I looked it up and discovered that rather than varying the angle of incidence of a single beam on the crystal face, the beam is shone at a sample of many ground up crystals of random orientations in order to cover all incidence angles simultaneously (or something like that!) This leads me to wonder how you can tell which line is due to which angle, but I'm not going to look into it further. I think the whole idea is that the specific experimental technique used to produce the data is not relevant to the problem, which is why the prof didn't have a problem introducing terminology he hadn't defined (although he doesn't have a problem doing that anyway! :grumpy: )

As for part b, the logical thing to do would seem to be to try and use the data to calculate n. But how is that done without knowing what d is? Which requires knowing a...

I don't even understand how x-ray diffraction works. Is each line associated with a different angle (one of the five given)? If so, does each line correspond to reflection off the *same* atomic plane, but just a different incidence angles? Or is each line produced by reflections from different planes with different Miller indices!

 

[cepheid]

Hey everyone,

Don't worry about it. It's too late. The prof has handed out solutions though. I can post them here if anyone else is curious.

Thanks.

 

[uskalu]

could you please post the solution!
Thanks

 

[cepheid]

No, I can't, because this thread is three years old!

 

[paranormal]

?As a scientist, it is important to have a solid understanding of the concepts and techniques involved in a problem before attempting to solve it. In this case, it seems like you may need to review the principles of x-ray diffraction and the Debye-Scherrer technique before tackling this problem.

To answer your questions, the lattice parameter (a) can be calculated using the Bragg condition for diffraction, as you have correctly identified. However, in order to do this, you will need to know the value of n for each of the five Bragg reflections. This can be determined by considering the value of the Miller indices for each reflection.

The Miller indices for the planes producing the diffraction beams can be calculated using the formula provided in the problem. Each line corresponds to a different angle, and each angle corresponds to a different set of Miller indices. Therefore, each line is produced by reflections from different planes with different Miller indices.

In order to determine the Miller index for the line producing the largest allowable Bragg angle, you will need to consider the value of n for each of the five reflections and determine which one corresponds to the largest angle.

Overall, it is important to have a clear understanding of the concepts and techniques involved in a problem before attempting to solve it. It may also be helpful to discuss the problem with your classmates or professor to gain additional insights and clarify any confusion.