Matter waves: Electrons emerge from an electron gun with a speed of 2.0 × 10⁶ m/s and then pass through a pair of thin parallel slits. Interference fringes with a spacing of 2.7 mm are detected on a screen far from the double slit and fairly close to the center of the pattern. What would the fringe spacing be if the electrons were replaced by neutrons with the same speed? (m_el = 9.11 × 10^-31 kg, m_neutron = 1.67 × 10^-27 kg) A) 1.5 µm B) 4.9 µm C) 0.93 nm D) 1.1 µm E) 1.5 nm

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Quizplus · Accepted Answer

The fringe spacing is given by:$$d = \frac{\lambda L}{a}$$where:d is the fringe spacingλ is the wavelength of the matter waveL is the distance between the slits and the screena is the distance between the slitsThe wavelength of the matter wave is given by:$$\lambda = \frac{h}{p}$$where:h is Planck's constantp is the momentum of the particleThe momentum of the particle is given by:$$p = mv$$where:m is the mass of the particlev is the speed of the particleSubstituting these equations into the equation for the fringe spacing, we get:$$d = \frac{hL}{mv}$$Since the speed and distance between the slits and the screen are the same for both electrons and neutrons, the fringe spacing will be inversely proportional to the mass of the particle. Therefore, the fringe spacing for neutrons will be smaller than the fringe spacing for electrons.The mass of a neutron is about 1836 times greater than the mass of an electron. Therefore, the fringe spacing for neutrons will be about 1836 times smaller than the fringe spacing for electrons.The fringe spacing for electrons is 2.7 mm. Therefore, the fringe spacing for neutrons will be about 2.7 mm / 1836 = 1.5 µm.

Matter Waves: Electrons Emerge from an Electron Gun with a Speed