For crystal diffraction experiments (discussed in Section $39.1$), wavelengths on the order of $0.20$ nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is a photon.

Verified step by step guidance

1. Step 1: Recall the relationship between the energy of a photon and its wavelength, which is given by the equation: , where $h$ is Planck's constant ($6.626\; imes\; 10^\{-34\}\; \backslash (\backslash text\{\; J·s\}\backslash )$), $c$ is the speed of light ($3.00\; imes\; 10^8\; \backslash (\backslash text\{\; m/s\}\backslash )$), and $\backslash (\backslash text\{\lambda \}\backslash )$ is the wavelength in meters.
2. Step 2: Convert the given wavelength from nanometers to meters. Since $1\; \backslash (\backslash text\{\; nm\}\backslash )\; =\; 10^\{-9\}\; \backslash (\backslash text\{\; m\}\backslash )$, the wavelength $\backslash (\backslash text\{\lambda \}\backslash )\; =\; 0.20\; \backslash (\backslash text\{\; nm\}\backslash )$ becomes $\backslash (\backslash text\{\lambda \}\backslash )\; =\; 0.20\; imes\; 10^\{-9\}\; \backslash (\backslash text\{\; m\}\backslash )$.
3. Step 3: Substitute the values of $h$, $c$, and $\backslash (\backslash text\{\lambda \}\backslash )$ into the energy equation: . Perform the calculation to find the energy in joules.
4. Step 4: Convert the energy from joules to electron volts (eV). Use the conversion factor $1\; \backslash (\backslash text\{\; eV\}\backslash )\; =\; 1.602\; imes\; 10^\{-19\}\; \backslash (\backslash text\{\; J\}\backslash )$. Divide the energy in joules by this factor to obtain the energy in electron volts.
5. Step 5: The final result will give the energy of the photon in electron volts. Ensure the units are consistent throughout the calculation and verify the result for accuracy.

Key Concepts

De Broglie Wavelength

The De Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept is crucial for understanding the wave-particle duality of matter and is particularly relevant in diffraction experiments.

Energy of a Photon

The energy of a photon can be calculated using the equation E = hc/λ, where E is the energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship shows that the energy of a photon is inversely proportional to its wavelength, meaning shorter wavelengths correspond to higher energy photons. This concept is essential for determining the energy associated with the given wavelength of 0.20 nm.

Electron Volts (eV)

An electron volt (eV) is a unit of energy commonly used in the field of physics, particularly in atomic and particle physics. It is defined as the amount of kinetic energy gained by a single electron when it is accelerated through an electric potential difference of one volt. Converting energy from joules to electron volts is often necessary when dealing with subatomic particles, making it a key concept in the context of photon energy calculations.