Determination of the London penetration depth with the tunnel diode oscillator technique

Using a distribution of the Meissner currents over the surface of an infinitely long superconducting slab with a rectangular cross section, the magnetic moment of the slab is calculated, taking into account corrections associated with a small but finite value of the London penetration depth $\lambda$. Since these corrections determine the shift of the resonant frequency in the tunnel diode oscillator technique, formulas for determination of $\lambda$ within this technique are derived for the slab. These formulas are valid for any aspect ratio of its cross section, and they differ from those that are often used in analyzing experimental data. Namely, it is shown that sharp edges of the slab can cause the large frequency shift proportional to the change in the value of $\lambda^{2/3}$. Although this result complicates the extraction of a temperature dependence of $\lambda$ from the frequency shift, it also opens up new possibilities in determining the London penetration depth. In particular, under certain conditions, it is possible not only to measure the changes in $\lambda$ with the temperature, but also to estimate its absolute value.