Dependence Between the Dielectric Constant and the Minimum Ionizing Potential of a Gas

(K. T. Compton. Application of the electron theory of gaseous dielectrics to the calculation of minimum ionizing potentials. Physical Review VIII, p. 413, 1916).

The electron theory of dielectric polarization in its simplest form proceeds from the hypothesis of a quasi-elastic displacement of the electron in the dielectric molecule under the influence of the applied field1. On the other hand, the ionizing potential corresponds to the minimum work required for the complete extraction of an electron from the molecule. It is easy to see that peripheral electrons will be displaced especially strongly and can more easily than others be removed beyond the sphere of action of the atom. The relation between the dielectric constant and the ionizing potential of a given gas can be found quite simply for one-electron atoms, and also for those in which the orbit of the outer electron is considerably removed from the inner orbits.

From the equation of motion of an electron performing oscillations under the action of a quasi-elastic force, we find the frequency of oscillation of the electron:

\[ \nu_0=\frac{1}{2\pi\sqrt{mh}} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (1) \]

where \(m\) is the mass of the electron and \(h\) is a constant. On the other hand, the electron theory of dielectrics leads to the following relation, coinciding with the well-known Clausius–Massotti formula:

\[ \frac{\epsilon-1}{\epsilon+2}=\frac{4\pi hNe^2}{3} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (2) \]

where \(e\) is the charge of the electron, \(N\) the number of molecules in unit volume, and \(\epsilon\) the dielectric constant. For gases \(\epsilon\) is very close to unity; therefore one may put

\[ h=\frac{\epsilon-1}{4\pi Ne^2} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (3) \]

Substituting (3) into (1), we have:

\[ \nu_0=\frac{1}{\pi}\sqrt{\frac{\pi Ne^2}{m(\epsilon-1)}} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (4) \]

Basing himself on contemporary views of the nature of the photoelectric effect, Compton assumes that \(\nu_0\) is the minimum frequency capable of causing the ejection of an electron. According to Einstein’s hypothesis, the minimum energy

required for ionization of the atom is \(h\nu_0\), where \(h\) is the universal constant of radiation. Hence

\[ V_0 e = h\nu_0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) \]

\(V_0\) is the ionizing potential. Using (4), we have

\[ V_0 = \frac{300 h}{\pi e}\sqrt{\frac{\pi N e^2}{m(\varepsilon-1)}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (6) \]

or

\[ V_0 = \frac{0{,}194}{\sqrt{\varepsilon-1}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (7) \]

As indicated above, the theory has been developed only for the case of a one-electron atom of the hydrogen type. On the other hand, the experimental measurements of \(V_0\) and \(\varepsilon\) have undoubtedly been made with considerable error. All the more striking is the excellent agreement for almost all gases between \(V_0\), calculated by formula (7), and measured experimentally, as is evident from the table.

The extension of the theory to the case of many-electron atoms will apparently make it possible to arrive at a relation that is even more accurately justified by experiment.

S. Vavilov.

1. H. Lorentz, The Theory of Electrons, 1909. A refinement of the theory is Debye’s hypothesis on the existence in dielectrics of permanent dipoles (cf. Debye, Phys. Ztsch. 13, 97, 1912; Kroo ib. p. 246; Schrödinger, Wien. Ber. 21 II a 1937, 1912; Ratnowsky, Verh. d. deutsch. phys. Ges. p. 497, 1913). ↩