For an electron to be maximally diffracted by a crystal, its wavelength must be equal to the spacing of the crystal’s planes. For a separation between planes of 0.250 nanometers, find the potential difference through which an electron must be accelerated from rest if it is to be maximally diffracted by these planes.
Let’s start by gathering some of the important information we’ve been given. We’re told that the distance that separates planes in the crystal structure is 0.250 nanometers, which we’ll call 𝑑.
We want to solve for a potential difference, which we’ll call capital 𝑉. It’s helpful to recall that potential difference is related to energy. To begin our solution, let’s recall a relationship between the energy of an object and its wavelength.
The energy of a particle is equal to the square of Planck’s constant of value of approximately 6.626 times 10 to the negative 34 meters squared kilograms per second divided by two times the mass of the object times its wavelength squared.
When we apply this relationship to our scenario, we can substitute in 𝑑 for 𝜆 because we’re told that for maximum diffraction to occur, the wavelength of the electrons must equal the separation distance which we’ve called 𝑑.
And, second, we substitute in the mass of the electron for 𝑚 mass, where that mass is equal to 9.1 times 10 to the negative 31st kilograms. We’re now ready to substitute in for these values and solve for 𝐸, which will give us a result in units of joules. With these numbers substituted in, when we calculate 𝐸, we find that is equal to 3.8597 times 10 to the negative 18th joules.
We’ve kept more significant figures here then is justified for a final answer, but we’ll keep them for this intermediate step in order to preserve accuracy. Now this result is in joules, but we want an answer in volts. We can use the fact that one electron volt is equal to 1.602 times 10 to the negative 19th joules.
So if we divide 𝐸 by that factor in joules, we’ll get a result in volts that an electron would experience. And when we do, we find that an electron would need to experience a potential difference of 21.4 volts in order to be maximally diffracted.