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The scanning electron microscope (SEM) provides images by using secondary electrons produced by the primary beam interacting with the surface of the sample (see [link] ). The SEM also uses magnetic lenses to focus the beam onto the sample. However, it moves the beam around electrically to “scan” the sample in the x and y directions. A CCD detector is used to process the data for each electron position, producing images like the one at the beginning of this chapter. The SEM has the advantage of not requiring a thin sample and of providing a 3-D view. However, its resolution is about ten times less than a TEM.

Figure a shows a schematic of an electron microscope. Figure b shows an image of a shark tooth.
Schematic of a scanning electron microscope (SEM) (a) used to observe small details, such as those seen in this image of a tooth of a Himipristis , a type of shark (b). (credit: Dallas Krentzel, Flickr)

Electrons were the first particles with mass to be directly confirmed to have the wavelength proposed by de Broglie. Subsequently, protons, helium nuclei, neutrons, and many others have been observed to exhibit interference when they interact with objects having sizes similar to their de Broglie wavelength. The de Broglie wavelength for massless particles was well established in the 1920s for photons, and it has since been observed that all massless particles have a de Broglie wavelength λ = h / p . size 12{λ = h/p} {} The wave nature of all particles is a universal characteristic of nature. We shall see in following sections that implications of the de Broglie wavelength include the quantization of energy in atoms and molecules, and an alteration of our basic view of nature on the microscopic scale. The next section, for example, shows that there are limits to the precision with which we may make predictions, regardless of how hard we try. There are even limits to the precision with which we may measure an object’s location or energy.

Making connections: a submicroscopic diffraction grating

The wave nature of matter allows it to exhibit all the characteristics of other, more familiar, waves. Diffraction gratings, for example, produce diffraction patterns for light that depend on grating spacing and the wavelength of the light. This effect, as with most wave phenomena, is most pronounced when the wave interacts with objects having a size similar to its wavelength. For gratings, this is the spacing between multiple slits.) When electrons interact with a system having a spacing similar to the electron wavelength, they show the same types of interference patterns as light does for diffraction gratings, as shown at top left in [link] .

Atoms are spaced at regular intervals in a crystal as parallel planes, as shown in the bottom part of [link] . The spacings between these planes act like the openings in a diffraction grating. At certain incident angles, the paths of electrons scattering from successive planes differ by one wavelength and, thus, interfere constructively. At other angles, the path length differences are not an integral wavelength, and there is partial to total destructive interference. This type of scattering from a large crystal with well-defined lattice planes can produce dramatic interference patterns. It is called Bragg reflection , for the father-and-son team who first explored and analyzed it in some detail. The expanded view also shows the path-length differences and indicates how these depend on incident angle θ size 12{θ} {} in a manner similar to the diffraction patterns for x rays reflecting from a crystal.

An electron beam is striking at an angle theta on a crystal and the reflected rays are detected by a detector. A magnified view of the crystal is also shown with two rays of electrons striking the various layers of crystal at an angle theta and reflected back. A graph is shown of intensity variation versus theta. Intensity is along the y axis and theta is along the x axis.The shape of the curve is like a wave and each subsequent peak diminishes as we move out the x axis.
The diffraction pattern at top left is produced by scattering electrons from a crystal and is graphed as a function of incident angle relative to the regular array of atoms in a crystal, as shown at bottom. Electrons scattering from the second layer of atoms travel farther than those scattered from the top layer. If the path length difference (PLD) is an integral wavelength, there is constructive interference.

Let us take the spacing between parallel planes of atoms in the crystal to be d . As mentioned, if the path length difference (PLD) for the electrons is a whole number of wavelengths, there will be constructive interference—that is, PLD = ( n = 1, 2, 3, ) . Because AB = BC = d sin θ , we have constructive interference when = 2 d sin θ. This relationship is called the Bragg equation and applies not only to electrons but also to x rays.

The wavelength of matter is a submicroscopic characteristic that explains a macroscopic phenomenon such as Bragg reflection. Similarly, the wavelength of light is a submicroscopic characteristic that explains the macroscopic phenomenon of diffraction patterns.

Section summary

  • Particles of matter also have a wavelength, called the de Broglie wavelength, given by λ = h p size 12{λ = { {h} over {p} } } {} , where p size 12{p} {} is momentum.
  • Matter is found to have the same interference characteristics as any other wave.

Conceptual questions

How does the interference of water waves differ from the interference of electrons? How are they analogous?

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Describe one type of evidence for the wave nature of matter.

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Describe one type of evidence for the particle nature of EM radiation.

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

7.28 × 10 –4 m size 12{7 "." "28" times "10" rSup { size 8{–4} } " m"} {}

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What is the wavelength of an electron moving at 3.00% of the speed of light?

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At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = 10 15 m. size 12{"10" rSup { size 8{ - "15"} } " m"} {} )

6.62 × 10 7 m/s size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

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What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 cm (large enough for it to interfere with other billiard balls)?

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Find the wavelength of a proton moving at 1.00% of the speed of light.

1.32 × 10 –13 m size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

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Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b) What is its kinetic energy in eV?

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(a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic. (b) What is the neutron’s kinetic energy in MeV?

(a) 6.62 × 10 7 m/s size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

(b) 22.9 MeV

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What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?

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What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?

15.1 keV
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(a) Calculate the velocity of an electron that has a wavelength of 1 . 00 μm. size 12{1 "." "00 μm"} {} (b) Through what voltage must the electron be accelerated to have this velocity?

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The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

(a) 5.29 fm

(b) 4 . 70 × 10 12 J size 12{4 "." "70" times "10" rSup { size 8{ - "12"} } " J"} {}

(c) 29.4 MV

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The kinetic energy of an electron accelerated in an x-ray tube is 100 keV. Assuming it is nonrelativistic, what is its wavelength?

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Unreasonable Results

(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) 7.28 × 10 12 m/s size 12{7 "." "28" times "10" rSup { size 8{"12"} } " m/s"} {}

(b) This is thousands of times the speed of light (an impossibility).

(c) The assumption that the electron is non-relativistic is unreasonable at this wavelength.

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Questions & Answers

what is the difference between temperature and heat transfer?
Bonga Reply
temperature is the measurement of hotness or coldness of a body... heat transfer is the movement of heat from one body to another
U get it right
what is a simple barometer
Justina Reply
find the density of a fluid in which a hydrometer having a density of 0.750g/mL floats with 92.0% of its volume submerged.
Neshrin Reply
Uniform speed
(a)calculate the buoyant force on a 2.00-L Helium balloon.(b) given the mass of the rubber in the balloon is 1.50g. what is the vertical force on the balloon if it is let go? you can neglect the volume of the rubber.
Neshrin Reply
To Long
pleaseee. can you get the answer? I can wait till 12
a thick glass cup cracks when hot liquid is poured into it suddenly
Aiyelabegan Reply
because of the sudden contraction that takes place.
railway crack has gap between the end of each length because?
Aiyelabegan Reply
For expansion
Please i really find it dificult solving equations on physic, can anyone help me out?
Big Reply
what is the equation?
fersnels biprism spectrometer how to determined
Bala Reply
how to study the hall effect to calculate the hall effect coefficient of the given semiconductor have to calculate the carrier density by carrier mobility.
what is the difference between atomic physics and momentum
Nana Reply
find the dimensional equation of work,power,and moment of a force show work?
Emmanuel Reply
What's sup guys
cul and you all
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so what is going on here
hello peeps
Michelson Morley experiment
Riya Reply
how are you
am good
Calculate the final velocity attained, when a ball is given a velocity of 2.5m/s, acceleration of 0.67m/s² and reaches its point in 10s. Good luck!!!
Eklu Reply
vf=vi+at vf=2.5+ 0.67*10 vf= 2.5 + 6.7 vf = 9.2
s = vi t +1/2at sq s=58.5 s=v av X t vf= 9.2
how 2.68
v=u+at where v=final velocity u=initial velocity a=acceleration t=time
the answer is 9.2m/s
express your height in Cm
Emmanuel Reply
my project is Sol gel process how to prepare this process pls tell me
the dimension of work and energy is ML2T2 find the unit of work and energy hence drive for work?
Emmanuel Reply
Two bodies P and Quarter each of mass 1000g. Moved in the same direction with speed of 10m/s and 20m/s respectively. Calculate the impulse of P and Q obeying newton's 3rd law of motion
Shimolla Reply
the answer is 0.03n according to the 3rd law of motion if the are in same direction meaning they interact each other.
Practice Key Terms 1

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