Question

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of \(9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Short answer

The characteristic wavelength of the electron is \(7.68 \times 10^{-11} \mathrm{m}\), which is within the range of the size of an average atom (generally 0.1 nm to 0.5 nm, or \(1 \times 10^{-10}\ \mathrm{m}\) to \(5 \times 10^{-10}\ \mathrm{m}\)). Therefore, the wavelength is comparable to the size of atoms.

Step-by-step solution

Recall the de Broglie wavelength formula

The de Broglie wavelength formula allows us to relate the wavelength of an electron to its momentum. The formula is given as: \[\lambda = \frac{h}{p}\] where \(\lambda\) is the wavelength of the electron, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), and \(p\) is the momentum of the electron.

Calculate the momentum of the electron

We can calculate the momentum of an electron using its mass and velocity, using the formula: \[p = m_{e}v\] where \(m_e\) is the mass of the electron (\(9.11 \times 10^{-31} \mathrm{kg}\)) and \(v\) is the given velocity (\(9.47\times10^{6}\ \mathrm{m/s}\)). The momentum of the electron can then be calculated as: \[p = (9.11 \times 10^{-31} \mathrm{kg}) (9.47 \times 10^{6} \mathrm{m/s}) = 8.62357 \times 10^{-24} \mathrm{kg\:m/s}\]

Calculate the wavelength of the electron

Now that we have the momentum of the electron, we can plug it into the de Broglie wavelength formula to find the wavelength: \[\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} \mathrm{Js}}{8.62357 \times 10^{-24}\ \mathrm{kg\:m/s}} = 7.68 \times 10^{-11} \mathrm{m}\]

Compare the wavelength with the size of atoms

The size of an average atom is generally around 0.1 nm to 0.5 nm which is equal to \(1 \times 10^{-10}\ \mathrm{m}\) to \(5\times 10^{-10}\ \mathrm{m}\). Our calculated electron wavelength is \(7.68 \times 10^{-11} \mathrm{m}\), which is within the range of the size of an average atom. So, the characteristic wavelength of this electron is comparable to the size of atoms.

Key concepts

Electron Microscope

The electron microscope is a powerful tool that uses a beam of electrons to create an image of a specimen, achieving much higher magnification and resolution than a traditional light microscope. Unlike light, which has a longer wavelength, electrons can be accelerated to such high speeds that their de Broglie wavelength becomes much shorter. This allows electron microscopes to resolve structures as small as individual atoms, making them essential in fields like materials science, biology, and nanotechnology.

When an electron beam is directed at a sample, the electrons interact with the atoms within that sample, and these interactions produce various signals that can be detected and translated into an image. There are different types of electron microscopes, including transmission electron microscopes (TEM) and scanning electron microscopes (SEM), each offering unique capabilities for visualizing the microscopic world.

Momentum of an Electron

The momentum of an electron is a fundamental concept in physics that describes the quantity of motion of the electron. It is calculated as the product of the mass of the electron and its velocity. For non-relativistic speeds, the formula is simple:
\[p = m_e \times v\]
where \(p\) is momentum, \(m_e\) is the mass of an electron, and \(v\) is its velocity. The momentum is directly proportional to the velocity; thus, as an electron accelerates, its momentum increases. This is crucial to understand when discussing de Broglie wavelengths, as the momentum influences the wavelength associated with a moving particle like an electron.

Planck's Constant

Planck's constant, symbolized by \(h\), is a vital physical constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately \(6.626 \times 10^{-34} \text{Js}\). Planck's constant is the proportionality factor in the formula \(E = h \times f\), where \(E\) is energy and \(f\) is frequency.

This constant also plays a pivotal role in determining the de Broglie wavelength of particles, encapsulating the idea that on a quantum level, particles like electrons exhibit wave-like properties. The smaller the constant, the less pronounced the wave-like behavior becomes, highlighting how on the scale of the very small, waves and particles are not distinct categories, but rather, intertwined aspects of quantum objects.

Size of Atoms

Atoms are the basic building blocks of matter, and their sizes are typically on the order of picometers (\(10^{-12} \text{m}\)). To give a sense of scale, a typical atom's diameter might range from 0.1 to 0.5 nanometers (nm) or \(10^{-10} \text{to} 5 \times 10^{-10} \text{m}\).

The actual size of an atom can vary depending on the element and its isotopes; however, the electron's de Broglie wavelength calculated in textbook problems, which may lie within these ranges, suggests that the electron microscope can be used to observe entities as small as atoms. Understanding the size of atoms is fundamental to analyzing the behavior and interactions of matter at the microscopic level, especially when comparing it to the wavelengths of particles used to probe them, like electrons in electron microscopy.