6.5 De broglie’s matter waves (Page 5/10)

University physics volume 3 Page 5 / 10

Neutron scattering

Suppose that a neutron beam is used in a diffraction experiment on a typical crystalline solid. Estimate the kinetic energy of a neutron (in eV) in the neutron beam and compare it with kinetic energy of an ideal gas in equilibrium at room temperature.

Strategy

We assume that a typical crystal spacing a is of the order of 1.0 Å. To observe a diffraction pattern on such a lattice, the neutron wavelength

λ

must be on the same order of magnitude as the lattice spacing. We use [link] to find the momentum p and kinetic energy K . To compare this energy with the energy

E_{T}

of ideal gas in equilibrium at room temperature

T = 300 K,

we use the relation

K = \frac{3}{2} k_{B} T,

where

k_{B} = 8.62 \times 10^{−5} eV / K

is the Boltzmann constant.

Solution

We evaluate pc to compare it with the neutron’s rest mass energy

E_{0} = 940 MeV :

p = \frac{h}{λ} \Rightarrow p c = \frac{h c}{λ} = \frac{1.241 \times 10^{−6} eV \cdot m}{10^{−10} m} = 12.41 keV .

We see that $p^{2} c^{2} ≪ E_{0}^{2}$ so $K ≪ E_{0}$ and we can use the nonrelativistic kinetic energy:

K = \frac{p^{2}}{2 m_{n}} = \frac{h^{2}}{2 λ^{2} m_{n}} = \frac{{(6.63 \times 10^{−34} J \cdot s)}^{2}}{(2 \times 10^{−20} m^{2}) (1.66 \times 10^{−27} kg)} = 1.32 \times 10^{−20} J = 82.7 meV .

Kinetic energy of ideal gas in equilibrium at 300 K is:

K_{T} = \frac{3}{2} k_{B} T = \frac{3}{2} (8.62 \times 10^{−5} eV / K) (300 K) = 38.8 MeV .

We see that these energies are of the same order of magnitude.

Significance

Neutrons with energies in this range, which is typical for an ideal gas at room temperature, are called “thermal neutrons.”

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Wavelength of a relativistic proton

In a supercollider at CERN, protons can be accelerated to velocities of 0.75 c . What are their de Broglie wavelengths at this speed? What are their kinetic energies?

Strategy

The rest mass energy of a proton is

E_{0} = m_{0} c^{2} = (1.672 \times 10^{−27} kg) {(2.998 \times 10^{8} m/s)}^{2} = 938 MeV .

When the proton’s velocity is known, we have

β = 0.75

and

β γ = 0.75 / \sqrt{1 - {0.75}^{2}} = 1.714 .

We obtain the wavelength

λ

and kinetic energy K from relativistic relations.

Solution

λ = \frac{h}{p} = \frac{h c}{p c} = \frac{h c}{β γ E_{0}} = \frac{1.241 eV \cdot μm}{1.714 (938 MeV)} = 0.77 fm

K = E_{0} (γ - 1) = 938 MeV (1 / \sqrt{1 - {0.75}^{2}} - 1) = 480.1 MeV

Significance

Notice that because a proton is 1835 times more massive than an electron, if this experiment were performed with electrons, a simple rescaling of these results would give us the electron’s wavelength of

(1835) 0.77 fm = 1.4 pm

and its kinetic energy of

480.1 MeV / 1835 = 261.6 keV .

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Check Your Understanding Find the de Broglie wavelength and kinetic energy of a free electron that travels at a speed of 0.75 c .

$λ = 1.417 pm;$ $K = 261.56 keV$

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Summary

De Broglie’s hypothesis of matter waves postulates that any particle of matter that has linear momentum is also a wave. The wavelength of a matter wave associated with a particle is inversely proportional to the magnitude of the particle’s linear momentum. The speed of the matter wave is the speed of the particle.
De Broglie’s concept of the electron matter wave provides a rationale for the quantization of the electron’s angular momentum in Bohr’s model of the hydrogen atom.
In the Davisson–Germer experiment, electrons are scattered off a crystalline nickel surface. Diffraction patterns of electron matter waves are observed. They are the evidence for the existence of matter waves. Matter waves are observed in diffraction experiments with various particles.

Conceptual questions

Which type of radiation is most suitable for the observation of diffraction patterns on crystalline solids; radio waves, visible light, or X-rays? Explain.

X-rays, best resolving power

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Speculate as to how the diffraction patterns of a typical crystal would be affected if $γ -rays$ were used instead of X-rays.

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If an electron and a proton are traveling at the same speed, which one has the shorter de Broglie wavelength?

proton

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If a particle is accelerating, how does this affect its de Broglie wavelength?

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Why is the wave-like nature of matter not observed every day for macroscopic objects?

negligibly small de Broglie’s wavelengths

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What is the wavelength of a neutron at rest? Explain.

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Why does the setup of Davisson–Germer experiment need to be enclosed in a vacuum chamber? Discuss what result you expect when the chamber is not evacuated.

to avoid collisions with air molecules

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Problems

At what velocity will an electron have a wavelength of 1.00 m?

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What is the de Broglie wavelength of an electron travelling at a speed of $5.0 \times 10^{6} m/s$ ?

145.5 pm

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What is the de Broglie wavelength of an electron that is accelerated from rest through a potential difference of 20 keV?

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What is the de Broglie wavelength of a proton whose kinetic energy is 2.0 MeV? 10.0 MeV?

20 fm; 9 fm

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What is the de Broglie wavelength of a 10-kg football player running at a speed of 8.0 m/s?

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(a) What is the energy of an electron whose de Broglie wavelength is that of a photon of yellow light with wavelength 590 nm? (b) What is the de Broglie wavelength of an electron whose energy is that of the photon of yellow light?

a. 2.103 eV; b. 0.846 nm

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The de Broglie wavelength of a neutron is 0.01 nm. What is the speed and energy of this neutron?

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What is the wavelength of an electron that is moving at a 3% of the speed of light?

80.9 pm

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At what velocity does a proton have a 6.0-fm wavelength (about the size of a nucleus)? Give your answer in units of c .

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What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 fm?

$2.21 \times 10^{− 20} m/s$

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Find the wavelength of a proton that is moving at 1.00% of the speed of light (when $β = 0.01) .$

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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progressive wave

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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

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Source: OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4

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