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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

(a) Show that the ground-state energy of the hydrogen atom can be written \(E_{1}=-k e^{2} /\left(2 a_{0}\right),\) where \(a_{0}\) is the Bohr radius. (b) Explain why, according to classical physics, an electron with energy \(E_{1}\) could never be found at a distance greater than \(2 a_{0}\) from the nucleus.
A hydrogen atom has a radius of about \(0.05 \mathrm{nm}\) (a) Estimate the uncertainty in any component of the momentum of an electron confined to a region of this size. (b) From your answer to (a), estimate the electron's kinetic energy. (c) Does the estimate have the correct order of magnitude? (The ground-state kinetic energy predicted by the Bohr model is \(13.6 \mathrm{eV} .\) )
A proton and a deuteron (which has the same charge as the proton but 2.0 times the mass) are incident on a barrier of thickness \(10.0 \mathrm{fm}\) and "height" \(10.0 \mathrm{MeV} .\) Each particle has a kinetic energy of $3.0 \mathrm{MeV} .$ (a) Which particle has the higher probability of tunneling through the barrier? (b) Find the ratio of the tunneling probabilities.
What are the de Broglie wavelengths of electrons with the following values of kinetic energy? (a) \(1.0 \mathrm{eV}\) (b) $1.0 \mathrm{keV} .
An electron moving in the positive \(x\) -direction passes through a slit of width \(\Delta y=85 \mathrm{nm} .\) What is the minimum uncertainty in the electron's velocity in the \(y\) -direction?
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