De Broglie Wavelength Of Electrons At Different Energies
de Broglie wavelength of electrons at different energies and comparison to light
Identify the core assignment task: The problem asks to calculate the de Broglie wavelength in Angstroms for electrons at two specified energies, 100 eV and 15 keV, and then compare these wavelengths with visible light, commenting on the advantage of electron microscopes based on this comparison. The problem also provides useful formulas, constants, and hints regarding the calculation of wavelength from electron energy, including the relationships involving Planck's constant, electron mass, and the kinetic energy of electrons.
Calculate the de Broglie wavelength of an electron at 100 eV and 15 keV, converting the energies to joules, calculating the electron velocities, and then computing the wavelengths in Angstroms. Follow with a comparative analysis explaining how the small wavelengths of high-energy electrons enable high-resolution imaging in electron microscopes, vastly surpassing the resolution limits set by visible light wavelengths.
Paper For Above instruction
The wave-particle duality of electrons, a fundamental principle of quantum mechanics, underpins the functionality of electron microscopes. The de Broglie wavelength of an electron is inversely proportional to its momentum, and thus its kinetic energy, which enables extremely high resolutions in electron microscopy. This paper compares the de Broglie wavelengths of electrons at specific energies—100 eV and 15 keV—and contextualizes their significance relative to visible light, emphasizing the technological advantage conferred by their shorter wavelengths.
Introduction
The wave-particle duality concept introduced by Louis de Broglie in 1924 posits that particles such as electrons exhibit both particle-like and wave-like properties. The wavelength associated with a particle, known as the de Broglie wavelength, is given by the relation:
λ = h / p
where h is Planck's constant and p is the momentum of the particle. For electrons accelerated to specific energies, this wavelength can be calculated and compared to visible light to understand the potential resolution in microscopy applications.
Calculations
Given the formulas and constants, the calculations follow a systematic approach. First, convert the electron energies into joules. 1 eV equals 1.602 × 10-19 joules. For the 100 eV case, the energy in joules is:
100 eV: E = 100 × 1.602 × 10-19 = 1.602 × 10-17 J
Similarly, for 15 keV:
15 keV: E = 15,000 × 1.602 × 10-19 = 2.403 × 10-15 J
Calculating electron velocities
Using the kinetic energy relation:
E = (1/2) m v²
where m is the electron mass, 9.109 × 10-31 kg, and solving for v:
v = √(2E / m)
For 100 eV:
v = √(2 × 1.602 × 10-17 / 9.109 × 10-31) = √(3.524 × 10-17 / 9.109 × 10-31) = √(3.866 × 1013) ≈ 6.22 × 106 m/s
For 15 keV:
v = √(2 × 2.403 × 10-15 / 9.109 × 10-31) = √(4.206 × 10-15 / 9.109 × 10-31) = √(4.618 × 1015) ≈ 6.80 × 107 m/s
Calculating de Broglie wavelengths
Using λ = h / p = h / (m v), where h = 6.626 × 10-34 Js:
For 100 eV:
λ = 6.626 × 10-34 / (9.109 × 10-31 × 6.22 × 106) ≈ 6.626 × 10-34 / 5.666 × 10-24 ≈ 1.17 × 10-10 meters
For 15 keV:
λ = 6.626 × 10-34 / (9.109 × 10-31 × 6.80 × 107) ≈ 6.626 × 10-34 / 6.194 × 10-23 ≈ 1.07 × 10-11 meters
Converting to Angstroms (1 Å = 10-10 m):
- At 100 eV: λ ≈ 1.17 Å
- At 15 keV: λ ≈ 0.107 Å
Discussion and Comparison to Visible Light
Visible light wavelengths typically range from approximately 400 to 700 nanometers (nm), or 4000 to 7000 Å. The calculated electron wavelengths at 100 eV and 15 keV are around 1.17 Å and 0.107 Å, respectively, which are orders of magnitude shorter than visible light.
This significant reduction in wavelength allows electron microscopes to achieve atomic-scale resolution, surpassing the resolving power of optical microscopes limited by the diffraction of visible light. The short de Broglie wavelengths of high-energy electrons result in minimal diffraction and enable detailed imaging of structures at the atomic level, which is impossible with conventional light microscopes.
Therefore, electron microscopes exploit the wave properties of electrons at high energies to achieve resolutions that significantly exceed the limitations of visible light microscopy, revolutionizing materials science, biology, and nanotechnology.
Conclusion
Calculating the de Broglie wavelength of electrons at different energies demonstrates how high-energy electrons possess extremely short wavelengths, facilitating high-resolution imaging. The stark contrast in wavelength scales between electrons and visible light underscores the advantage of electron microscopes. This capability enables scientists to observe and analyze materials at atomic resolutions, which is critical for advancements across numerous scientific disciplines.
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