Problem 71 - College Physics

Question

Ion microscopes. Just as electron microscopes make use of the wave properties of electrons, ion microscopes use the wave properties of atomic ions, such as helium ions \(\left(\mathrm{He}^{+}\right),\) to image materials. A helium ion has a mass 7300 times that of an electron. In a typical helium ion microscope, helium ions are accelerated by a high voltage of \(10-50 \mathrm{kV}\) and focused on the material to be imaged. At these energies, the ions don't travel very far into the material being imaged, so this type of microscope is used primarily for surface imaging of biological structures. A different method of imaging has been proposed that is sensitive to the entire thickness of the material. This method uses helium ions with much greater energies (in the MeV range), which can pass all the way through biological samples such as cells. In this second type of ion microscope, the energy lost as the ion beam passes through different parts of a cell can be measured and related to the distribution of material in the cell, with thicker parts of the cell causing greater energy loss. How does the wavelength of a helium ion compare to that of an electron accelerated through the same potential difference? A. The helium ion has a longer wavelength because of its greater mass. B. The helium ion has a shorter wavelength because of its greater mass. C. The wavelengths are the same because the kinetic energy is the same. D. The wavelengths are the same because the electric charge is the same.

Step-by-Step Solution

Answer

B. The helium ion has a shorter wavelength because of its greater mass.

Step 1: Understanding the Question

We need to compare the wavelengths of a helium ion and an electron both accelerated through the same potential difference. This requires the understanding of how mass affects the wavelength of a particle.

Step 2: Using de Broglie Wavelength Formula

The de Broglie wavelength for any particle is calculated using \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

Step 3: Expressing Momentum in Terms of Kinetic Energy

The momentum \( p \) can be expressed in terms of the kinetic energy (KE) as \( p = \sqrt{2m \cdot KE} \). When particles are accelerated through a potential difference \( V \), their kinetic energy is \( KE = qV \). For the helium ion and electron, \( q \) is their charge.

Step 4: Calculating Wavelength Ratio

The de Broglie wavelength for a particle can be written as \( \lambda = \frac{h}{\sqrt{2m \cdot qV}} \). Given that both particles are accelerated through the same voltage, the major differing factor would be their mass \( m \). With \( m_{He^+} = 7300 \times m_e \), the helium ion's wavelength is \( \lambda = \frac{h}{\sqrt{2m_{He^+} \cdot eV}} \), and for an electron, \( \lambda = \frac{h}{\sqrt{2m_e \cdot eV}} \). The ratio is dominated by the mass term.

Step 5: Comparing Wavelengths

Since the mass \( m_{He^+} \) is much larger than \( m_e \), the denominator in the helium ion wavelength formula becomes larger, making \( \lambda_{He^+} \) smaller than \( \lambda_{e} \). Thus, the helium ion has a shorter wavelength due to its larger mass.

Key Concepts

de Broglie wavelength

The de Broglie wavelength is a concept that highlights the wave-like nature of particles. In essence, any moving particle can be associated with a wave, and this wavelength is known as the de Broglie wavelength. Its formula is given by \[ \lambda = \frac{h}{p} \]where \( \lambda \) is the wavelength, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s})\), and \( p \) is the momentum of the particle.

• Momentum \( p \) is calculated using \( p = \sqrt{2m \cdot KE} \), where \( m \) is mass and \( KE \) is kinetic energy.
• For charged particles accelerated by a potential \( V \), \( KE = qV \) where \( q \) is the charge.

The de Broglie wavelength shows us that even matter has wave-like behaviors, which is crucial in fields like quantum mechanics and microscopy techniques.

helium ions

Helium ions, specifically \( \text{He}^+ \), are helium atoms that have lost an electron, resulting in a positive charge. This charge allows them to be manipulated similarly to electrons when exposed to electric fields, a principle used in ion microscopes.

• Helium ions have a much greater mass, about 7300 times that of an electron. This significant mass difference leads to different properties when accelerated.
• The high mass of helium ions results in shorter de Broglie wavelengths compared to electrons, making them ideal for specific imaging techniques.

Ion microscopes that utilize helium ions can produce high-resolution images, especially useful for surface imaging in materials science and biology.

electron microscopes

Electron microscopes use the principles of wave-particle duality to magnify objects much smaller than what traditional light microscopes can handle. In these devices, electrons, rather than photons, are used to form images.

• Due to their smaller de Broglie wavelengths, electrons can achieve higher resolution imaging than visible light.
• They operate by generating an electron beam and focusing it onto a sample, magnifying structures to the nanometer scale.

Electron microscopes are foundational in fields such as materials science, nanotechnology, and biology, enabling scientists to see the intricate details of cell structures and materials with exceptional clarity.

wave properties of particles

The wave properties of particles, as proposed by Louis de Broglie, suggest that particles like electrons and ions exhibit both particle and wave characteristics. This concept is an integral part of quantum mechanics.

• Particles have an associated wavelength depending on their momentum, which affects how they interact with matter.
• This dual nature is particularly important in microscopic imaging, enabling the use of technologies such as electron and ion microscopes.

Understanding the wave properties of particles helps explain phenomena such as diffraction and interference at the atomic and subatomic levels, illustrating the complex nature of matter in the quantum realm.