Measurement of the Mean Free Path of a Neutral Atom.

Max Born. Eine direkte Messung der freien Weglänge neutraler Atome.
Phys. ZS., 21, p. 518, 1920.

The idea of the method used by the author is extremely simple. There is a Knudsen-type monochromatic beam of silver; probes are placed in it at specified distances, reaching to the center of the beam; the relative thicknesses of the silver deposited on the probes are determined with the aid of a photometer, and from this the mean free path is then easily calculated.

A quartz tube about 3 cm in diameter is connected by means of a ground joint to the glass parts of the apparatus; at the bottom of this tube a piece of silver is placed, or else a brass cylinder is inserted into the tube along its axis, having at the bottom along its axis a small tube 3 mm in diameter; this cylinder carries within it four partitions, spaced 1 cm apart and having in their centers openings 5 mm in diameter. The probes were glass plates in the form of squares. They were arranged in the following way: the first square glass plate was placed on the first partition so that its center coincided with the axis of the beam; the second plate, placed on the second partition, was turned by \(90^\circ\) relative to the first, etc., so that each square glass plate intercepted only the corresponding part of the silver beam.

On the outside, against the walls of the quartz tube, there adjoins a brass box with solid carbon dioxide; directly at the bottom of the tube there is an electric furnace, by means of which the piece of silver can be melted and formed into a sphere.

The pressure was measured with MacLeod and Knudsen manometers.

At pressure \(p = 0\) the decrease of blackening on four successive squares was negligible, but already a pressure \(p = 5.8 \cdot 10^{-3}\) mm Hg produces a strong decrease of blackening with distance (collisions of silver atoms with air molecules). These collisions can also be judged from the fact that the boundary of the deposit at this pressure is blurred, whereas in the first case (\(p = 0\)) it is extremely sharp.

As I have already mentioned above, comparison of the relative thicknesses of the deposits was carried out with a photometer.

From the preliminary results the author calculates the mean free path. If \(D_{10}\) is the thickness of the silver layer on the first quadrant at the highest vacuum, then the layer thickness at mean free path \(\lambda\) will be

\[ D_1 = D_{10} e^{-\frac{z_1}{\lambda}}, \]

where \(z_1\) is the distance of the first quadrant from the source of the beam; likewise for the second quadrant (probe plate) we have:

\[ D_2 = D_{20} e^{-\frac{z_2}{\lambda}}, \]

From these two formulas:

\[ \lambda = \frac{Z_2 - Z_1} {\lg\left(\frac{D_1}{D_2}\cdot\frac{D_{20}}{D_{10}}\right)} . \]

For the author \(Z_2 - Z_1 = 1\) cm; the expression \(\dfrac{D_1}{D_2}\cdot\dfrac{D_{20}}{D_{10}}\) is equal to 1.8 or 1.5, and for \(p = 5.8 \cdot 10^{-3}\) mm he obtains \(\lambda = 1.7\) cm, or \(p\lambda = 9.9 \cdot 10^{-3}\)

“ \(p = 4.? \cdot 10^{-3}\) ” ” ” \(\lambda = 2.4\) ” ” \(p\lambda = 10.8 \cdot 10^{-3}\)

Bearing in mind that preliminary results are given here, \(p\lambda\) must be considered sufficiently constant.

Further, the author, taking into account the result found for \(p\lambda\) and using Maxwell’s expression for the mean free path \(\lambda\), calculates the distance \(\delta\) between the centers of an air molecule and a silver atom at the moment of impact and finds \(\delta = 2.6 \cdot 10^{-8}\) cm, which agrees quite well with the order of magnitude of the diameter of an atom.

T. Molodoi.