Question

A nickel crystal is used as a diffraction grating for x-rays. Then the same crystal is used to diffract electrons. If the two diffraction patterns are identical, and the energy of each \(\mathrm{x}\) -ray photon is \(E=20.0 \mathrm{keV},\) what is the kinetic energy of each electron?

Short answer

The kinetic energy of each electron is approximately 38.0 eV.

Step-by-step solution

Understanding Bragg's Law

Both x-rays and electrons are diffracted according to Bragg's Law, which can be expressed as \( n\lambda = 2d\sin\theta \), where \( \lambda \) is the wavelength, \( d \) is the spacing between planes in the crystal lattice, \( \theta \) is the angle of diffraction, and \( n \) is the order of the diffraction. Given that the two diffraction patterns are identical, the effective wavelength for both electrons and x-rays must be the same.

Determine X-ray Wavelength

The wavelength of x-rays can be found using the energy-wavelength relation for photons: \( \lambda = \frac{hc}{E} \). Here, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J} \cdot \text{s}) \) and \( c \) is the speed of light \( (3.00 \times 10^{8} \, \text{m/s}) \). Given \( E = 20.0 \, \text{keV} = 20.0 \times 10^{3} \times 1.60 \times 10^{-19} \, \text{J} \), calculate \( \lambda \).

Use de Broglie Wavelength for Electrons

The de Broglie wavelength formula \( \lambda = \frac{h}{p} \) applies to electrons, where \( p \) is the momentum. Momentum in terms of kinetic energy \( K \) for electrons can be expressed as \( p = \sqrt{2mK} \), with \( m \) being the electron mass \( (9.11 \times 10^{-31} \, \text{kg}) \). Substitute \( \lambda = \frac{hc}{E} \) from Step 2 into the de Broglie wavelength equation to solve for \( K \).

Solve for Kinetic Energy of Electrons

Set \( \frac{h}{\sqrt{2mK}} = \frac{hc}{E} \). By squaring both sides, \( h^2 = 2mK \left(\frac{hc}{E}\right)^2 \). Simplifying gives \( K = \frac{h^2c^2}{2mE^2} \). Substitute known constants and calculate \( K \).

Key concepts

X-Ray Diffraction

X-ray diffraction is a powerful method used by scientists to study the arrangement of atoms within a crystal. When X-rays pass through a crystal, they can spread out and create patterns by bouncing off the crystal’s inner planes. These repeating patterns help reveal the crystal's structure. This is similar to how light waves can cause a rainbow when they pass through a prism.
Bragg's Law, which is written as \( n\lambda = 2d\sin\theta \), helps us determine the wavelengths involved in the diffraction process. Here, \( n \) represents the order of the diffraction, \( \lambda \) is the wavelength of the incident rays, \( d \) indicates the distance between crystal planes, and \( \theta \) is the angle at which the waves are diffracted. When the diffraction patterns of two different waves, such as X-rays and electrons, are identical, it shows that their effective wavelengths match.
By analyzing these patterns, scientists can gather critical insights into the molecular structures of various substances, making X-ray diffraction a valuable tool in chemistry and material science.

De Broglie Wavelength

The concept of the de Broglie wavelength extends wave-particle duality to electrons and other particles. Electrons don't just behave like small balls; they also have wave-like characteristics. The de Broglie wavelength \( \lambda = \frac{h}{p} \) reveals how these wave properties emerge. Here, \( h \) is Planck’s constant, and \( p \) is the momentum of the electron.
When electrons are diffracted by a crystal, they create patterns that match those from X-rays, given their wavelengths are identical. Since X-rays have known wavelengths due to their energy, we can apply this knowledge to the electrons using the formula for de Broglie wavelength. This allows us to determine the necessary conditions for wave interference or diffraction.
Understanding the de Broglie wavelength is essential for fields like quantum mechanics, as it emphasizes that every particle has both particle-like and wave-like properties, paving the way for innovations in technology and our understanding of the universe.

Kinetic Energy Calculation

Calculating the kinetic energy of electrons involves relating their motion to their energy. Given that the electrons' diffraction patterns match those of X-rays, they share the same wavelength. Using the de Broglie wavelength formula, we can find this connection through the relationship between momentum and kinetic energy.
The momentum \( p \) of an electron in motion is linked to kinetic energy by the formula \( p = \sqrt{2mK} \,\) where \( m \) represents the mass of the electron, and \( K \) is the kinetic energy. By substituting the de Broglie wavelength formula, electrons with the identical wavelength as X-rays \( \big( \lambda = \frac{hc}{E} \big) \) allows us to solve for the kinetic energy \( K\).With the expression \( K = \frac{h^2c^2}{2mE^2} \,\) scientists employ constants like Planck's constant \( h \,\) the speed of light \( c \,\) and the previously determined energy values to compute the kinetic energy of electrons. This calculation is vital in fields like electron microscopy and radiation physics, providing insights into the properties and interactions of small particles in the microscopic world.