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The BCS theory for supercondictivity says that the effect of variation of lattice ion mass (M) and its effect on transition temperature is given as $T_{c} \space\alpha\space M^{-\beta}$ . The experimental agreement with the theoretical value of $\beta=0.5$ is very good for Zinc, Lead and Mercury who have 5,3 and 7 stable isotopes each. On the other hand Molybdenum which has 33 isotopes, has a value $\beta=0.33$ and other elements like Os, Ru, Zr too do not agree with the value of $\beta$ from theory, so much so that some of them have a $\beta$ value of zero, i.e. the transition temperature $T_{c}$ doesn't depend on the mass of the ions on the lattice. Regarding this, I have two questions.

1) Looking at the small number of isotopes of zinc , lead and mercury, can we actually say that we have verified the isotope mass effect? I ask this since we have less number of data points in these three cases, than lets say, Molybdenum.

2) Why don't we see the isotope effect i.e. the dependence of transition temperature on the mass of ions at lattice sites in many elements (and compounds)?

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  • $\begingroup$ 1) Check literature: dx.doi.org/10.1103/PhysRev.78.487 and dx.doi.org/10.1103/PhysRev.78.477 (found from the Wikipedia page on superconductivity / ctrl+F: isotope effect) to see how the isotope effect is verified for more than three points as you claim. 2) The isotope effect is not supposed to be valid for alloys, which unfortunately represent the large majority of superconductors. There might well be generalisation of the isotope effect for non-pure materials, but I'm not aware of such a generalisation. $\endgroup$ – FraSchelle Apr 16 '15 at 6:44
  • $\begingroup$ @FraSchelle I have stated an example of Molybdenum which is not an alloy. Rather I am not making any statement for alloys. Literature says that the isotope effect is not valid for many 'elements' including Molybdenum. $\endgroup$ – Abhijit Apr 16 '15 at 6:49
  • $\begingroup$ Indeed, my comment is in fact wrong in several points : 1) the two above references just correspond to the isotope effect for mercury, and you already mentioned this in your answer. 2) it might well exist pure materials not following the BCS isotope effect, since there are a lot of materials not following the BCS mechanism for superconductivity. It would be interesting to note how far from the BCS prediction is Molybdenum. For instance, I wonder if the BCS prediction is valid in some range of isotope mass. $\endgroup$ – FraSchelle Apr 16 '15 at 6:56
  • $\begingroup$ As for your point 2, I'm pretty sure the transition temperature varies with the mass of the ions constituting the lattice, but it might well be the case that it does not follow the BCS prediction. Would be interesting to see discussion about that in literature. Would you please give us the reference of your statement Literature says that the isotope effect is not valid for many 'elements' including Molybdenum ? Thanks in advance. $\endgroup$ – FraSchelle Apr 16 '15 at 7:00
  • $\begingroup$ @FraSchelle The $\beta$ value for Mo is 0.33. 1) My 1st question was not 'how' you calculate the beta factor. Its rather trivial to do straight line fit once we have been given data. My question is that, say we have two set of readings to verify a certain phenomenon (let us assume in this case zinc vs Mo) , then we usually take the set of reading with more number of data points and assume it to be true. In this case we are doing exactly the opposite. 2) I don't think the range for isotope mass that you propose is correct, because the standard deviation in case of say (again) Mo is very low. $\endgroup$ – Abhijit Apr 16 '15 at 7:06

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