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

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

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Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

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Method

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Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

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Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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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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