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Chapter 28: Problem 10

Neutron diffraction by a crystal can be used to make a velocity selector for neutrons. Suppose the spacing between the relevant planes in the crystal is \(d=0.20 \mathrm{nm}\) A beam of neutrons is incident at an angle \(\theta=10.0^{\circ}\) with respect to the planes. The incident neutrons have speeds ranging from 0 to \(2.0 \times 10^{4} \quad \mathrm{m} / \mathrm{s} .\) (a) What wavelength(s) are strongly reflected from these planes? [Hint: Bragg's law, Eq. \((25-15),\) applies to neutron diffraction as well as to x-ray diffraction.] (b) For each of the wavelength(s), at what angle with respect to the incident beam do those neutrons emerge from the crystal?

Short Answer

Expert verified
2. What are the given values for d and θ in this exercise? 3. Which steps should be followed to solve this exercise? 4. What is the angle of emergence for the calculated wavelength(s)?

Step by step solution

01

Calculate the Wavelength(s)

First, let's calculate the wavelength λ using Bragg's law. Since the integer n can take any value, we need to find the minimum value of n that will give a meaningful result. Bragg's law: nλ = 2d sinθ Rearrange to find λ: λ = (2d sinθ)/n
02

Calculate the Angles

Now, let's calculate the angle at which the neutrons emerge from the crystal for each of the wavelength(s) found in step 1. We know that the angle of emergence will be equal to the angle of incidence, θ. So, the angle of emergence with respect to the incident beam will be θ. In this case, θ = 10.0°.

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Most popular questions from this chapter

An electron is confined in a one-dimensional box of length \(L\). Another electron is confined in a box of length 2 \(L\). Both are in the ground state. What is the ratio of their energies $E_{2 l} / E_{L} ?$
What is the ground-state electron configuration of nickel (Ni, atomic number 28 )?
In the Davisson-Germer experiment (Section \(28.2),\) the electrons were accelerated through a \(54.0-\mathrm{V}\) potential difference before striking the target. (a) Find the de Broglie wavelength of the electrons. (b) Bragg plane spacings for nickel were known at the time; they had been determined through x-ray diffraction studies. The largest plane spacing (which gives the largest intensity diffraction maxima) in nickel is \(0.091 \mathrm{nm} .\) Using Bragg's law [Eq. ( \(25-15\) )], find the Bragg angle for the first-order maximum using the de Broglie wavelength of the electrons. (c) Does this agree with the observed maximum at a scattering angle of \(130^{\circ} ?\) [Hint: The scattering angle and the Bragg angle are not the same. Make a sketch to show the relationship between the two angles.]
(a) What are the electron configurations of the ground states of fluorine \((Z=9)\) and chlorine \((Z=17) ?\) (b) Why are these elements placed in the same column of the periodic table?
An electron is confined to a box of length \(1.0 \mathrm{nm} .\) What is the magnitude of its momentum in the \(n=4\) state?
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