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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 = 3 2 k B T , where k B = 8.62 × 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 = h λ p c = h c λ = 1.241 × 10 −6 eV · 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 = p 2 2 m n = h 2 2 λ 2 m n = ( 6.63 × 10 −34 J · s ) 2 ( 2 × 10 −20 m 2 ) ( 1.66 × 10 −27 kg) = 1.32 × 10 −20 J = 82.7 meV .

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

K T = 3 2 k B T = 3 2 ( 8.62 × 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 × 10 −27 kg ) ( 2.998 × 10 8 m/s ) 2 = 938 MeV . When the proton’s velocity is known, we have β = 0.75 and β γ = 0.75 / 1 0.75 2 = 1.714 . We obtain the wavelength λ and kinetic energy K from relativistic relations.

Solution

λ = h p = h c p c = h c β γ E 0 = 1.241 eV · μm 1.714 ( 938 MeV ) = 0.77 fm
K = E 0 ( γ 1 ) = 938 MeV ( 1 / 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 × 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 × 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

as a free falling object increases speed what is happening to the acceleration
Success Reply
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
ganesh
you are correct
Priyojit
law of multiple
Wahid
if we heated the ice then the refractive index be change from natural water
Nepal
can someone explain normalization condition
Priyojit Reply
please tell
Swati
yes
Chemist
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Gitanjali
The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
Practice Key Terms 5

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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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