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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

what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
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hii
Alvin
hi
Siddiquee
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
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vy beautiful equation
chandrasekhar
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when you consider systems consisting of fixed charges
Sherly
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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