## Mastering Physics Solutions: Using X-ray Diffraction

Using X-ray Diffraction

Part A = 0.2130nm
Part B = 69.9°
Part C = No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.

Solution Below:

When an x-ray beam is scattered off the planes of a crystal, the scattered beam creates an interference pattern. This phenomenon is called Bragg scattering. For an observer to measure an interference maximum, two conditions have to be satisfied:

The angle of incidence has to be equal to the angle of reflection.

The difference in the beam’s path from a source to an observer for neighboring planes has to be equal to an integer multiple of the wavelength; that is,

2dsin(θ) = mλ (for m = 1, 2, …)

The path difference 2dsin(θ) can be determined from the diagram. The second condition is known as the Bragg condition.

Part A

An x-ray beam with wavelength 0.200nm is directed at a crystal. As the angle of incidence increases, you observe the first strong interference maximum at an angle 28.0°. What is the spacing d between the planes of the crystal?

This is very straightforward. Just use the formula given in the intro:

2dsin(θ) = mλ

2d * sin(28) = 1 * 0.200
d = 0.2130 nm

0.2130 nm

Part B

Find the angle at which you will find a second maximum.

Just use the same formula, only change m to “2″ and use the value of d that you found in Part A:

2d * sin(28) = 1 * 0.200
2 * 0.2130 * sin(θ) = 2 * 0.200
sin(θ) = 0.938967
θ = 69.9°

69.9°

Part C

Will you observe a third maximum?

• Yes, because all crystals have at least three planes.
• Yes, because the diffraction pattern has an infinite number of maxima.
• No, because the angle of a third maximum is greater than 180°.
• No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.
• The answer is “No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.”

Note that this depends on the inputs for your particular version of the problem. You can use the calculator in Part B to solve for the 3rd maxima, just make “m” equal to 3

No, because the existence of such a maximum produces an unphysical result such as the sine of an angle being greater than one.

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