Hi there:

it is quite easy: equilibrium is reached, when the ratio of the number of nuclides is the same as the ratio of halflifes (or the inverse ratio of decay constants).

This basic fact can easily be derived from the differential equation of decay, and should be found in all introductory text. For the numbers: just look up the decay constants in a table, like at the kaery website or similar

I think our colleague asks why there is, for instance, rarely a secular equilibrium between Ra224 and Rn daughters (like in a typical sample (rock or soil) when measured in a lab. The reason why it is less than 1 (=there are fewer Rn progeny than Ra226) is that some Rn atoms escaped from the sample - because Rn is a gas, which diffuses through the matrix. It's called emanation rate, or coefficient, and varies widely depending on many parameters. The same applies to Ra224 from the Th232 series, even though the half-life of thoron is only 55 s (Rn has 3.8 days). In general, the mechanism of the transport of atoms and molecules through solid matrix is not understood. So the answer to your question - there is no simple relation to estimate whether it is above or below 1.

Using HPGe assembly, we can calculate specific activities of various radio isotopes in Bq/Kg and then see the ratios between any two isotopes to check the equilibrium condition. In case of uranium exploration, e.g., we can do with the ratio between the specific activities of 238U and 214Bi.

Thanks for all, i actually computed the ratio between 2 radioisotopes (parent/daughter) from U-238 series and Th-232 series and this ratio was 0.66 (U-238 series) and 1.1 (Th-232 )

what is the meaning of these results? or , what is the exemption value of this ratio below it there will not exist equilibrium?

Which 2 radioisotopes?

U-238 series: Pb-214 / Bi-214

Th-232 series: Ac-228 / Pb-212

Th in sample, for all practical purposes, is in secular equilibrium as fresh Th attains equilibrium within 50 years which is geologic instant, thus we get close to 1.0 ratio between Ac-228 and Pb-212. This is not so in case of U-238 but between Pb-214 and Bi-214 this ratio should have been 1.0 if daughters of radon attain equilibrium with radon.

It is very unlikely that Pb214-Bi214 (RaB and RaC), 28 m and 19 m half-lives, would not be in equilibrium.  If yes, then it is a major anomaly. What is more likely that you did not take into account the intensities (branching ratios) of individual gamma peaks. They are available on GOOGLE, 

Disequilibrium between Ac228 and Pb212 (ratio Pb/Ac less than one) could be due thoron escaping (emanating from your sample), How big is your sample? Is it crushed rock, or soil?

To measure gammas requires that you study the decay schemes carefully, as someone mentioned it to you above. You should complement your equipment with an alpha spectrometer. The decay schemes of alphas are much simpler.