An image of a biological sample is to have a resolution of 5 nm. (a) What is the kinetic energy

(a) First, we need to find the momentum of the electrons using the de Broglie wavelength formula

An image of a biological sample is to have a resolution of 5...

An image of a biological sample is to have a resolution of $5 \mathrm{~nm}$. (a) What is the kinetic energy of a beam of electrons with a de Broglie wavelength of $5.0 \mathrm{~nm}$ ? (b) Through what potential difference should the electrons be accelerated to have this wavelength? (c) Why not just use a light microscope with a wavelength of $5 \mathrm{~nm}$ to image the sample?

Key Concepts

Electron Acceleration and Potential Difference

When electrons are accelerated through a potential difference, they gain kinetic energy equal to the work done on them by the electric field, expressed as E = eV, where e is the electron charge and V is the potential difference. This concept is essential in determining the conditions required to generate electrons with a specific kinetic energy, and therefore a specific de Broglie wavelength, which is crucial in electron microscopy.

de Broglie Hypothesis

The de Broglie hypothesis is a fundamental concept in quantum mechanics that relates a particle's momentum to its wavelength through the equation ? = h/p, where ? is the wavelength, h is Planck’s constant, and p is the momentum of the particle. This principle underlies the wave-particle duality of matter, allowing particles such as electrons to exhibit wave-like behavior, which is critical in explaining phenomena like electron diffraction and interference.

Relationship Between Momentum and Kinetic Energy

In classical mechanics, the kinetic energy of a particle is connected to its momentum via the relation E = p²/(2m), where E is the kinetic energy, p is the momentum, and m is the mass of the particle. This relationship is used in deriving the kinetic energy from the particle’s wavelength, as given by the de Broglie relation, thereby bridging classical concepts with quantum mechanics.

Diffraction Limit and Resolution in Microscopy

The diffraction limit is a fundamental constraint on the resolution of imaging systems due to the wave nature of light, described by Abbe’s criterion, which states that the minimum resolvable detail is on the order of the wavelength of the illumination. In the context of imaging at very small scales, such as 5 nm resolution, traditional light microscopes face limitations because even light with a 5 nm wavelength would produce significant diffraction effects and practical challenges, making electron microscopes a more suitable choice due to their ability to achieve much shorter effective wavelengths through electron acceleration.

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