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Chapter 10: Problem 52

X rays of wavelength 2.63 A were used to analyze a crystal.The angle of first- order diffraction \((n=1 \text { in the Bragg }\) equation) was 15.55 degrees. What is the spacing between crystal planes, and what would be the angle for second-order diffraction \((n=2) ?\)

Short Answer

Expert verified
The spacing between the crystal planes is \(4.98Å\), and the angle for the second-order diffraction is \(31.62°\).

Step by step solution

01

Write down the given information

Given information: - Wavelength of X-rays: \(\lambda = 2.63 Å\) - Angle of first-order diffraction: \(\theta_1 = 15.55°\) - First-order diffraction: \(n_1 = 1\)
02

Apply Bragg's Law to find spacing between crystal planes

Bragg's Law states that \(n \lambda = 2d\sin{\theta}\), where \(n\) is the order of diffraction, \(\lambda\) is the wavelength of X-rays, \(d\) is the spacing between crystal planes, and \(\theta\) is the angle of diffraction. From the given information, we have: \(n_1 = 1\), \(\lambda = 2.63 Å\), and \(\theta_1 = 15.55°\) Substitute the given values in Bragg's Law: \(1 \cdot 2.63 = 2d \cdot \sin{15.55°}\) Now, we will solve for \(d\), the spacing between crystal planes.
03

Calculate the spacing between crystal planes (d)

To find the spacing between crystal planes, \(d\), rearrange the equation and solve for \(d\): \(d = \frac{1 \cdot 2.63}{2 \cdot \sin{15.55°}}\) After calculating, we get the spacing between the crystal planes: \(d = 4.98Å\)
04

Determine the angle of second-order diffraction

Now, we need to find the angle for the second-order diffraction (\(\theta_2\)), with \(n_2 = 2\). We will use Bragg's Law for this purpose: \(2 \lambda = 2d\sin{\theta_2}\) To find the angle \(\theta_2\), we will rearrange the equation and substitute the known values.
05

Calculate the angle for second-order diffraction (\(\theta_2\))

Rearrange the equation to solve for \(\theta_2\): \(\theta_2 = \sin^{-1}{\frac{2 \lambda}{2d}}\) Now, substitute the given values of \(\lambda = 2.63Å\) and \(d = 4.98Å\): \(\theta_2 = \sin^{-1}{\frac{2 \cdot 2.63}{2 \cdot 4.98}}\) After calculating, we get the angle for the second-order diffraction: \(\theta_2 = 31.62°\)
06

Final result

The spacing between the crystal planes is \(4.98Å\), and the angle for the second-order diffraction is \(31.62°\).

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Most popular questions from this chapter

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