<< Chapter < Page Chapter >> Page >
Diffraction pattern obtained for electrons diffracted by crystalline silicon is shown. The diffraction pattern has a bright spot at the center of a circle with brighter and darker regions occurring in a symmetric manner.
This diffraction pattern was obtained for electrons diffracted by crystalline silicon. Bright regions are those of constructive interference, while dark regions are those of destructive interference. (credit: Ndthe, Wikimedia Commons)

Electron wavelength versus velocity and energy

For an electron having a de Broglie wavelength of 0.167 nm (appropriate for interacting with crystal lattice structures that are about this size): (a) Calculate the electron’s velocity, assuming it is nonrelativistic. (b) Calculate the electron’s kinetic energy in eV.

Strategy

For part (a), since the de Broglie wavelength is given, the electron’s velocity can be obtained from λ = h / p size 12{λ = h/p} {} by using the nonrelativistic formula for momentum, p = mv. size 12{p= ital "mv"} {} For part (b), once v size 12{v} {} is obtained (and it has been verified that v size 12{v} {} is nonrelativistic), the classical kinetic energy is simply ( 1 / 2 ) mv 2 . size 12{ \( 1/2 \) ital "mv" rSup { size 8{2} } } {}

Solution for (a)

Substituting the nonrelativistic formula for momentum ( p = mv size 12{p= ital "mv"} {} ) into the de Broglie wavelength gives

λ = h p = h mv . size 12{λ = { {h} over {p} } = { {h} over { ital "mv"} } } {}

Solving for v size 12{v} {} gives

v = h . size 12{v = { {h} over {mλ} } } {}

Substituting known values yields

v = 6 . 63 × 10 –34 J s ( 9.11 × 10 –31 kg ) ( 0 . 167 × 10 –9 m ) = 4 . 36 × 10 6 m/s . size 12{v = { {6 "." "63 " times " 10" rSup { size 8{"–34"} } " J " cdot " s"} over { \( 9 "." "11 " times " 10" rSup { size 8{"–31"} } " kg" \) \( 0 "." "167 " times " 10" rSup { size 8{"–9"} } " m" \) } } =" 4" "." "36 " times " 10" rSup { size 8{6} } " m/s"} {}

Solution for (b)

While fast compared with a car, this electron’s speed is not highly relativistic, and so we can comfortably use the classical formula to find the electron’s kinetic energy and convert it to eV as requested.

KE = 1 2 mv 2 = 1 2 ( 9.11 × 10 –31 kg ) ( 4.36 × 10 6 m/s ) 2 = (86.4 × 10 –18 J) ( 1 eV 1.602 × 10 –19 J ) = 54.0 eV alignl { stack { size 12{"KE "= { {1} over {2} } ital "mv" rSup { size 8{2} } } {} #=" 0" "." 5 \( 9 "." "11 " times " 10" rSup { size 8{"–31"} } " kg" \) \( 4 "." "36 " times " 10" rSup { size 8{6} } " m/s" \) rSup { size 8{2} } {} # =" 8" "." "64 " times " 10" rSup { size 8{"–18"} } " J " cdot { {"1eV"} over {1 "." "60 " times " 10" rSup { size 8{"–19"} } " J"} } {} #=" 54" "." "0 eV" "." {} } } {}

Discussion

This low energy means that these 0.167-nm electrons could be obtained by accelerating them through a 54.0-V electrostatic potential, an easy task. The results also confirm the assumption that the electrons are nonrelativistic, since their velocity is just over 1% of the speed of light and the kinetic energy is about 0.01% of the rest energy of an electron (0.511 MeV). If the electrons had turned out to be relativistic, we would have had to use more involved calculations employing relativistic formulas.

Electron microscopes

One consequence or use of the wave nature of matter is found in the electron microscope. As we have discussed, there is a limit to the detail observed with any probe having a wavelength. Resolution, or observable detail, is limited to about one wavelength. Since a potential of only 54 V can produce electrons with sub-nanometer wavelengths, it is easy to get electrons with much smaller wavelengths than those of visible light (hundreds of nanometers). Electron microscopes can, thus, be constructed to detect much smaller details than optical microscopes. (See [link] .)

There are basically two types of electron microscopes. The transmission electron microscope (TEM) accelerates electrons that are emitted from a hot filament (the cathode). The beam is broadened and then passes through the sample. A magnetic lens focuses the beam image onto a fluorescent screen, a photographic plate, or (most probably) a CCD (light sensitive camera), from which it is transferred to a computer. The TEM is similar to the optical microscope, but it requires a thin sample examined in a vacuum. However it can resolve details as small as 0.1 nm ( 10 10 m size 12{"10" rSup { size 8{ - "10"} } `m} {} ), providing magnifications of 100 million times the size of the original object. The TEM has allowed us to see individual atoms and structure of cell nuclei.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Berger describes sociologists as concerned with
Mueller Reply
what is hormones?
Wellington
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Basic physics for medical imaging. OpenStax CNX. Feb 17, 2014 Download for free at http://legacy.cnx.org/content/col11630/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Basic physics for medical imaging' conversation and receive update notifications?

Ask