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The impedance dependence of a tuning-fork-shaped quartz oscillator on gas pressure is theoretically analyzed on the basis of a string-of-beads model of the quartz oscillator. The theoretical results are numerically compared with experimental data. The analysis has been already carried out in molecular flow and viscous flow region. In this paper, the impedance behavior in the intermediate flow region is formulated by making use of the theory of slip effect. Furthermore, by the formal extension of this formula to the molecular flow region, a single formula describing the impedance dependence on gas pressure over the whole pressure region is obtained. This formula is found to be in good agreement with experimental data. However, there is still a slight discrepancy (<20%) between the two in the intermediate flow region. This shows the limit of the slip theory. For this reason, the Millikan’s empirical formula is adopted in the intermediate flow region. The unified formula obtained thusly is in numerical agreement with experimental data for various kinds of gases.

The theory of a friction vacuum gauge using a tuning-fork-shaped quartz oscillator is presented. The dependences of electric impedance and resonance frequency of the quartz oscillator on gas pressure are theoretically analyzed on the basis of "a string-of-beads" model for the quartz oscillator. It is shown that the impedance change results from the velocity term of the drag acting on an oscillator, while the resonance frequency shift results from the acceleration term of the drag. The drag is calculated by means of the kinetic theory of gases in lower pressure ( molecular flow) region, and by means of fluid mechanics in higher pressure ( viscous flow) region. These theoretical results are found to be in quantitative agreement with experimental results. In particular, it is shown that the increase of impedance in the neighborhood of atmospheric pressure results from the small vibration of fluid ( surrounding gas) rather than turbulent flow or the emission of sound. Also, theoretical predictions about the dependences of impedance and resonance frequency on gas species are experimentally verified. Furthermore, the pressure dependence of the impedance in intermediate pressure region is formulated by taking into account a "slip" effect of gas molecules at the surface of an object. By the formal extension of this formula to molecular flow region, a single formula describing the pressure dependence of the impedance over the whole pressure region is obtained. This formula is found to be in numerical agreement with experimental data.

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