In an experiment a beam of silver atoms emerges from an...

In an experiment a beam of silver atoms emerges from an oven, which contains silver vapor at $T=1200 \mathrm{~K}$. The beam is collimated by being passed through a small circular aperture. (a) Give an argument to show that it is not possible, by narrowing the aperture $a$, to decrease indefinitely the diameter of the spot, $D$, on the screen. (b) If the screen is at $L=1$ meter from the aperture, estimate numerically the smallest $D$ that can be obtained by varying $a$. (You may assume for simplicity that all atoms have the same momentum along the direction of the beam and have a mass of $M_{\mathrm{Ag}}=1.8 \times 10^{-22} \mathrm{~g}$ ). FIGURE CAN'T COPY.

In an experiment a beam of silver atoms emerges from an oven, which contains silver vapor at $T=1200 \mathrm{~K}$. The beam is collimated by being passed through a small circular aperture.
(a) Give an argument to show that it is not possible, by narrowing the aperture $a$, to decrease indefinitely the diameter of the spot, $D$, on the screen.
(b) If the screen is at $L=1$ meter from the aperture, estimate numerically the smallest $D$ that can be obtained by varying $a$. (You may assume for simplicity that all atoms have the same momentum along the direction of the beam and have a mass of $M_{\mathrm{Ag}}=1.8 \times 10^{-22} \mathrm{~g}$ ).
FIGURE CAN'T COPY.

Key Concepts

Diffraction

When a beam of matter waves, such as silver atoms with an associated de Broglie wavelength, passes through a finite aperture, it experiences diffraction. This phenomenon causes the beam to spread out after passing through the aperture, meaning that reducing the aperture size increases the diffraction angle. Consequently, even if one tries to limit the beam’s width by narrowing the aperture, diffraction sets a lower bound on the size of the spot that can form on a screen placed at a distance.

Heisenberg Uncertainty Principle

The uncertainty principle states that the uncertainties in position and momentum are inversely related. By confining the atoms to a very narrow aperture, the position is precisely known, which in turn introduces a large uncertainty in their transverse momentum. This increased momentum uncertainty results in a larger spread in the angles at which the atoms emerge, preventing indefinite reduction of the beam spot size on the screen.

de Broglie Wavelength

Every moving particle exhibits wave-like properties described by its de Broglie wavelength, which is given by the ratio of Planck’s constant to its momentum. This wavelength is essential in understanding the diffraction effects observed when the beam passes through the aperture and thus determines the scale of the interference and the minimum achievable spot size.

Geometrical and Diffraction Limit Scaling

The spot size on a screen due to a diffracted beam is governed by an interplay between the aperture size, the de Broglie wavelength, and the distance from the aperture. The relationship, often approximated as the spot diameter being proportional to the product of the distance from the aperture and the ratio of the wavelength to the aperture size, shows that as the aperture is made smaller to reduce the initial beam size, the increased diffraction counteracts this reduction, setting a lower limit for the spot size.

Transcript

Please give Ace some feedback