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Superconductivity and Superfluidity
Unfortunately, no direct experiments of the effect of isotopic substitution on the
electronic properties (such as the electronic specific heat and the Fermi energy), exist for
metals substantial for our consideration.
Let us consider what should be expected in such measurements. A convenient choice
for the superconductor is mercury, as it has many isotopes and their isotope effect has
been carefully measured back in the 50s of the last century as aforementioned.
The linear dependence of the critical temperature of a superconductor on its Fermi
energy (Eq.(6.20)) and also the existence of the isotope effect suggests the dependence of
the ion density in the crystal lattice from the mass of the isotope. Let us consider what
should be expected in such measurements.
Even then, it was found that the isotope effect is described by Eq.(10.4) in only a few
superconductors. In others, it displays different values, and therefore in a general case it
can be described by introducing of the parameter a:
Mi a T c = Const. (10.5)
At taking into account Eq.(6.20), we can write
T c ∼ E F ∼ n 2/3
e (10.6)
The parameter l which characterizes the ion lattice obtains an increment ∆l with an
isotope substitution:
where M i and ∆M i are the mass of isotope and its increment.
∆l
l
= − a 2 · ∆M i
M i
, (10.7)
It is generally accepted that in an accordance with the terms of the phonon mechanism,
the parameter a ≈ 1 2
for mercury. However, the analysis of experimental data [35]-[36]
(see Figure 4.3) shows that this parameter is actually closer to 1/3. Accordingly, one can
expect that the ratio of the mercury parameters is close to:
)
(
( ∆l
l
∆M i
M i
) ≈ − 1 6 . (10.8)
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