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Chapter 28: Problem 6

The distance between atoms in a crystal of \(\mathrm{NaCl}\) is $0.28 \mathrm{nm} .$ The crystal is being studied in a neutron diffraction experiment. At what speed must the neutrons be moving so that their de Broglie wavelength is \(0.28 \mathrm{nm} ?\)

Short Answer

Expert verified
Answer: The neutrons must be moving at a speed of approximately \(1.44\times10^{6}\,\mathrm{m/s}\) for their de Broglie wavelength to be equal to the distance between atoms in the NaCl crystal.

Step by step solution

01

Recall the de Broglie wavelength formula

The de Broglie wavelength of particles, such as the neutrons in this experiment, is given by the formula: $$ \lambda = \frac{h}{mv} $$ where \(\lambda\) is the de Broglie wavelength, \(h\) is Planck's constant, \(m\) is the mass of the neutron, and \(v\) is the speed of the neutron.
02

Set the de Broglie wavelength equal to the distance between atoms

The distance between atoms in the \(\mathrm{NaCl}\) crystal is given as \(0.28\,\mathrm{nm}\). So, we want the neutron's de Broglie wavelength to be equal to this distance: $$ \lambda = 0.28\,\mathrm{nm} $$
03

Solve for the speed of the neutron

To find the speed of the neutron, we can use the de Broglie wavelength formula in the following way: $$ v = \frac{h}{m\lambda} $$ We know that \(h = 6.63\times10^{-34}\,\mathrm{Js}\) and the mass of a neutron \(m = 1.67\times10^{-27}\,\mathrm{kg}\). We are given that \(\lambda = 0.28\,\mathrm{nm} = 0.28\times10^{-9}\,\mathrm{m}\). Plug in the values into the equation: $$ v = \frac{6.63\times10^{-34}\,\mathrm{Js}}{(1.67\times10^{-27}\,\mathrm{kg})(0.28\times10^{-9}\,\mathrm{m})} $$
04

Calculate the speed of the neutron

Calculate the speed of the neutron by solving the equation: $$ v = \frac{6.63\times10^{-34}\,\mathrm{Js}}{(1.67\times10^{-27}\,\mathrm{kg})(0.28\times10^{-9}\,\mathrm{m})} \approx 1.44\times10^{6}\,\mathrm{m/s} $$ Hence, the neutrons must be moving at a speed of approximately \(1.44\times10^{6}\,\mathrm{m/s}\) for their de Broglie wavelength to be equal to the distance between atoms in the \(\mathrm{NaCl}\) crystal.

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Most popular questions from this chapter

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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