Question

A relation connecting the isotopic mass \(M\) of a superconductor with its critical temperature \(T_c\) is given by:

• A. \(M = kT_c\)
• B. \(M^{1/2}\,T_c = \text{a constant}\)
• C. \(M_c^{1/2} = \text{a constant}\)
• D. \(M_c^{2} = \text{a constant}\)

Hint:

Isotope effect: \(T_c \propto M^{-1/2}\), so \(M^{1/2}T_c\) stays constant across isotopes.

Answer 1

The Correct Option is B

Step 1: The isotope effect in superconductivity states that the critical temperature \(T_c\) depends on the isotopic mass \(M\) of the lattice ions through \[T_c \propto M^{-\alpha},\] with \(\alpha \approx \tfrac{1}{2}\) for many conventional (BCS) superconductors.

Step 2: Taking \(\alpha = \tfrac{1}{2}\) gives \[T_c \propto M^{-1/2} \quad\Longleftrightarrow\quad M^{1/2}\,T_c = \text{constant}.\]

Step 3: Physically, superconductivity is mediated by electron-phonon coupling. Lattice (phonon) frequencies scale as \(\omega \propto M^{-1/2}\), and since \(T_c\) tracks the characteristic phonon energy, \(T_c \propto M^{-1/2}\).

Step 4: Hence the correct relation is \(M^{1/2}T_c = \text{a constant}\).\[\boxed{M^{1/2}\,T_c = \text{constant}}\]