The schematic drawing below (Fig. 2.42) shows the...

The schematic drawing below (Fig. 2.42) shows the experimental set up for the production of a well-collimated beam of sodium atoms for an atomic beam experiment. Sodium is present in the oven $S$, which is kept at the temperature $T=550 \mathrm{~K}$. At this temperature the vapor pressure of sodium is $p=6 \times 10^{-3}$ torr. The sodium atoms emerge through a slit in the wall of the oven. The hole is rectangular, with dimensions 10 mm $\times 0.1 \mathrm{~mm}$. The collimator $C$ has a hole of identical size and shape, and the sodium atoms which pass through $C$ thus constitute the atomic beam under consideration. The atomic mass of sodium is 23 . The distance $d$ in the figure is 10 cm . Fig. 2.42. (a) Compute the number $\phi$ of sodium atoms which pass through the slit in $C$ per second. (b) Derive an expression for the function $D(v)$ which describes the distribution of velocities of the particles in the beam in the sense that $D(v) d v$ is the probability that an atom passing through $C$ has a velocity in the range $(v, v+d v)$. (c) The region in which the beam propagates must, of course, be a reasonably good vacuum. Estimate (and give answer in torr) just how good the vacuum ought to be if the beam is to remain collimated for at least 1 meter. $(1$ torr $=1 \mathrm{mmHg})$. FIGURE CAN'T COPY.

The schematic drawing below (Fig. 2.42) shows the experimental set up for the production of a well-collimated beam of sodium atoms for an atomic beam experiment. Sodium is present in the oven $S$, which is kept at the temperature $T=550 \mathrm{~K}$. At this temperature the vapor pressure of sodium is $p=6 \times 10^{-3}$ torr. The sodium atoms emerge through a slit in the wall of the oven. The hole is rectangular, with dimensions 10 mm $\times 0.1 \mathrm{~mm}$. The collimator $C$ has a hole of identical size and shape, and the sodium atoms which pass through $C$ thus constitute the atomic beam under consideration. The atomic mass of sodium is 23 . The distance $d$ in the figure is 10 cm .
Fig. 2.42.
(a) Compute the number $\phi$ of sodium atoms which pass through the slit in $C$ per second.
(b) Derive an expression for the function $D(v)$ which describes the distribution of velocities of the particles in the beam in the sense that $D(v) d v$ is the probability that an atom passing through $C$ has a velocity in the range $(v, v+d v)$.
(c) The region in which the beam propagates must, of course, be a reasonably good vacuum. Estimate (and give answer in torr) just how
good the vacuum ought to be if the beam is to remain collimated for at least 1 meter. $(1$ torr $=1 \mathrm{mmHg})$.
FIGURE CAN'T COPY.

Key Concepts

Effusion of Gases

This concept involves the behavior of gas particles escaping through a small aperture from a container. In kinetic theory, when an aperture is small compared to the mean free path of the particles, the gas escapes in a process known as effusion. The rate at which particles exit is linked to the number density of the gas and the average speed of its particles. Calculating the flux through an aperture uses these principles, connecting molecular kinetics directly to experimental measurements in atomic beam setups.

Maxwell-Boltzmann Velocity Distribution

Central to understanding the speeds of particles in a thermalized gas, the Maxwell-Boltzmann distribution describes how the speeds of the particles are statistically spread out at a given temperature. The distribution function includes an exponential term that depends on the particle’s kinetic energy and temperature, and forms the basis for determining parameters like the most probable speed. This distribution is essential in deriving the velocity-dependent properties of atomic beams produced in such experiments.

Atomic Beam Collimation and Slit Geometry

Collimation refers to the alignment and restriction of particle trajectories to produce a beam with a narrow angular spread. In atomic beam experiments, carefully designed slits or collimators are used to filter atoms based on their direction. The geometry of the apertures, including their shape and size, plays a crucial role in determining the quality and intensity of the beam, ensuring that only atoms with trajectories close to the set beam direction contribute significantly.

Mean Free Path and Vacuum Requirements

The mean free path is the average distance a particle travels between collisions. In the context of atomic beams, maintaining a well-collimated beam over extended distances requires that the beam passes through a region with very low background pressure. This ensures that collisions with residual gas molecules are sufficiently rare to prevent scattering and beam degradation. Calculations involving the mean free path help set the necessary vacuum conditions for successful beam propagation.

Transcript

Please give Ace some feedback