Question

The neutrons in a parallel beam, each having kinetic energy \(\frac{1}{40}\) eV (which approximately corresponds to room temperature), are directed through two slits \(0.50 \mathrm{~mm}\) apart. How far apart will the peaks of the interference pattern be on a screen \(1.5 \mathrm{~m}\) away?

Short answer

Question: Neutrons with a kinetic energy of 1/40 eV pass through two narrow slits separated by 0.5 mm. At a distance of 1.5 m from the slits, the neutrons generate an interference pattern on a screen. Calculate the distance between adjacent peaks in the interference pattern. Answer: To find the distance between adjacent peaks in the interference pattern, follow these steps: 1. Calculate the wavelength of neutrons using their kinetic energy and the de Broglie wavelength equation. 2. Determine the angular separation between peaks using the Young's double-slit experiment formula. 3. Compute the distance between peaks on the screen using the angular separation and the distance from the screen. By applying these steps, we can calculate the distance between adjacent peaks in the interference pattern.

Step-by-step solution

Find the wavelength of neutrons

To find the wavelength of neutrons, we can use the de Broglie wavelength equation: \(\lambda=\frac{h}{p}\), where \(h\) is the Planck constant (\(6.626 \times 10^{-34} \mathrm{J\cdot s}\)) and \(p\) is the momentum of the neutron. First, we need to find the momentum of the neutron using its kinetic energy. The kinetic energy of the neutron is given as: \(KE=\frac{1}{40}\, \text{eV} \approx \frac{1}{40} \times 1.602\times 10^{-19} \mathrm{J}\). The relation between kinetic energy, momentum, and mass of a particle can be written as: \(KE=\frac{p^2}{2m}\), where \(m\) is the mass of the particle. The mass of a neutron is \(m = 1.675\times 10^{-27} \mathrm{kg}\). We can solve the equation for \(p\): \(p = \sqrt{2m \times KE}\), and then compute the wavelength.

Calculate the angular separation between peaks

After finding the wavelength, we can use the formula for the angular separation between peaks in the Young's double-slit experiment: \(\sin{\theta} = \frac{m\lambda}{d}\), where \(m\) is an integer representing the interference order (we will use the first order, \(m=1\) for the distance between adjacent peaks), and \(d\) is the distance between slits (given as \(0.5 \mathrm{mm} = 0.0005 \mathrm{m}\)). Then, we can find the angle \(\theta\).

Calculate the distance between peaks on the screen

Finally, we will find the distance between the peaks on the screen using the angular separation and the distance from the screen, which can be found using the formula: \(y = L\tan{\theta}\), where \(y\) is the distance between peaks, and \(L\) is the distance between the slits and screen, which is given as 1.5 m. Compute the distance \(y\) to get the final result.