### X-ray Diffraction Exercise**Problem Statement:**An X-ray beam of wavelength \( 3.4 \times 10^{-10} \) meters makes an angle of \( 27^\circ \) with a set of planes in a crystal which results in first order constructive interference. Determine the plane spacing in nanometers. (Please include 2 decimal places).**Solution:**To solve this problem, you will need to use Bragg's Law which is represented by the equation:\[ n\lambda = 2d\sin\theta \]where:- \( n \) is the order of the interference (for first order, \( n = 1 \))- \( \lambda \) is the wavelength of the X-ray ( \( 3.4 \times 10^{-10} \) meters)- \( d \) is the distance between the crystal planes- \( \theta \) is the angle of incidence ( \( 27^\circ \) )Rearranging the formula to solve for \( d \):\[ d = \frac{n\lambda}{2\sin\theta} \]Step-by-step calculation:1. Substitute the known values into the equation:\[ d = \frac{(1)(3.4 \times 10^{-10} \, \text{m})}{2\sin(27^\circ)} \]2. Calculate \( \sin(27^\circ) \). Use a calculator to find:\[ \sin(27^\circ) \approx 0.454 \]3. Substitute this value back into the equation:\[ d = \frac{3.4 \times 10^{-10}}{2 \times 0.454} \]4. Perform the division:\[ d \approx \frac{3.4 \times 10^{-10}}{0.908} \approx 3.74 \times 10^{-10} \, \text{m} \]5. Convert the result from meters to nanometers. (1 \, \text{m} = 10^9 \, \text{nm}): \[ d \approx 3.74 \times 10^{-1} \, \text{nm} \]Thus, the plane spacing \( d \) is approximately \( 0.37 \, \text{nm} \).**Conclusion:**The plane spacing is \(

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An X-ray beam of wavelength 3.4 x 10-10 m makes an angle of 27° with a set of planes in a crystal which results in first order constructive interference. Determine the plane spacing in nanometers. (Please include 2 decimal places).

An X-ray beam of wavelength 3.4 x 10-10 m makes an angle of 27° with a set of planes in a crystal which results in first order constructive interference. Determine the plane spacing in nanometers. (Please include 2 decimal places).

Physics for Scientists and Engineers

10th Edition

ISBN:9781337553278

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter36: Wave Optics

Section: Chapter Questions

Problem 43AP

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