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Chapter 28: Problem 74

A beam of neutrons has the same de Broglie wavelength as a beam of photons. Is it possible that the energy of each photon is equal to the kinetic energy of each neutron? If so, at what de Broglie wavelength(s) does this occur? [Hint: For the neutron, use the relativistic energy-momentum relation \(\left.E^{2}=E_{0}^{2}+(p c)^{2} .\right]\)

Short Answer

Expert verified
Answer: The energy of each photon equals the kinetic energy of each neutron when the de Broglie wavelength is equal to the rest energy of the neutron (\(\lambda = E_0\)).

Step by step solution

01

Understand the de Broglie wavelength

The de Broglie wavelength is a characteristic of a particle that is related to its momentum, given by the formula: \(\lambda=\dfrac{h}{p}\), where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant, and \(p\) is the momentum of the particle.
02

Write the energy-momentum relation for photons

The energy-momentum relation for photons is given by \(E=pc\), where \(E\) is the energy, \(p\) is the momentum, and \(c\) is the speed of light.
03

Write the energy-momentum relation for neutrons

From the given hint, the energy-momentum relation for neutrons (relativistic) is \(E^{2}=E_{0}^{2}+(pc)^{2}\), where \(E\) is the total energy, \(E_0\) is the rest energy, and \(p\) is the momentum.
04

Relate the de Broglie wavelength to the momentum

Using the de Broglie wavelength formula \(\lambda=\dfrac{h}{p}\), we can express momentum \(p\) in terms of \(\lambda\). \(p=\dfrac{h}{\lambda}\)
05

Write the energy expressions for photons and neutrons in terms of de Broglie wavelength

Substitute the expression for momentum into the energy-momentum relations: For photons, \(E=\dfrac{hc}{\lambda}\). For neutrons, \(E^2=E_0^2 + \left(\dfrac{hc}{\lambda}\right)^2\)
06

Set the energy expressions equal and solve for the de Broglie wavelength

Equate the energies of the photons and neutrons: \(\dfrac{hc}{\lambda}=\sqrt{E_0^2 + \left(\dfrac{hc}{\lambda}\right)^2}\) Simplify and solve for \(\lambda\): \(\left(\dfrac{hc}{\lambda}\right)^2 = E_0^2 + \left(\dfrac{hc}{\lambda}\right)^2 - 2E_0\dfrac{hc}{\lambda}\) Divide by \(h^2c^2\), and rearrange to form a quadratic equation: \(\lambda^2 - 2E_0\lambda + E_0^2 = 0\)
07

Solve the quadratic equation for the wavelength(s)

Use the quadratic formula to solve for \(\lambda\): \(\lambda_\pm = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} = \dfrac{2E_0\pm\sqrt{(2E_0)^2 - 4E_0^2}}{2}\) Finally, we find the possible de Broglie wavelengths: \(\lambda_\pm = E_0\). Since \(E_0\) is a positive constant (the rest energy of the neutron), there is only one possible de Broglie wavelength, \(E_0\), where the energy of each photon is equal to the kinetic energy of each neutron.

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Most popular questions from this chapter

A marble of mass \(10 \mathrm{g}\) is confined to a box \(10 \mathrm{cm}\) long and moves at a speed of \(2 \mathrm{cm} / \mathrm{s} .\) (a) What is the marble's quantum number \(n ?\) (b) Why can we not observe the quantization of the marble's energy? [Hint: Calculate the energy difference between states \(n\) and \(n+1 .\) How much does the marble's speed change?]
A bullet leaves the barrel of a rifle with a speed of $300.0 \mathrm{m} / \mathrm{s} .\( The mass of the bullet is \)10.0 \mathrm{g} .$ (a) What is the de Broglie wavelength of the bullet? (b) Compare \(\lambda\) with the diameter of a proton (about \(1 \mathrm{fm}\) ). (c) Is it possible to observe wave properties of the bullet, such as diffraction? Explain.
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.]
An electron is confined to a one-dimensional box. When the electron makes a transition from its first excited state to the ground state, it emits a photon of energy 1.2 eV. (a) What is the ground-state energy (in electronvolts) of the electron? (b) List all energies (in electronvolts) of photons that could be emitted when the electron starts in its second excited state and makes transitions downward to the ground state either directly or through intervening states. Show all these transitions on an energy level diagram. (c) What is the length of the box (in nanometers)?
A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of \(16.4 \mathrm{m}\) downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected? (b) What is the de Broglie wavelength of the neutrons? (c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.
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