## Friday, March 19, 2021

### Interpretation of isotope effects can be subtle

Isotopic substitution has provided significant insights into molecular and solid-state physics. This involves the substitution of particular atoms in a compound by the same chemical element with a different nuclear mass (i.e. a nuclear isotope). An example is hydrogen/deuterium substitution which has shown the significant role that quantum nuclear motion can play in hydrogen bonding, particularly in strong hydrogen bonds. Of particular relevance to the discussion below is that isotopic substitution does not only change vibrational frequencies but can also change bond lengths.

A key piece of evidence on the road to the BCS theory of superconductivity in 1957 was the observation of an isotope effect. In 1950 a shift in the transition temperature of mercury was observed, suggesting that superconductivity resulted from electron-phonon interactions, as argued by Frohlich that same year. In particular, the magnitude of the shift was consistent with theoretical work by Herbert Frohlich. (Whether he predicted or postdicted the observed effect is a matter of debate, as discussed by Jorge Hirsch.) BCS theory gives that $\Delta T_c/T_c = - {1/over 2} \Delta M /M$, which arises from the fact that phonon frequencies scale with $1/\sqrt{M}$, consistent with the mercury experiments.

However, in the 1960s there were many observations of “anomalous” isotope effects, particularly in transition metals, that were inconsistent with the prefactor in this equation. These anomalies were resolved by going beyond the BCS theory and allowing for strong-coupling effects. Following the discovery of cuprate superconductors in 1986, isotope effects were observed in some cuprates. However, the consensus now is that these observations do not support an electron-phonon mechanism for superconductivity but rather are due to structural changes due to the isotope substitution. For example, isotopic substitution changes the zero-point energy, and that can alter the unit cell volume and the hopping parameter t in a Hubbard model.

This illustrates that there are subtleties in interpreting isotope experiments. This is because there are both structural and dynamical isotope effects. Changes in isotope can lead to changes in structure, such as bond lengths or lattice constants, and even in changes in crystal symmetry. These structural changes arise because the equilibrium structure of the system is that which minimises the total energy of the system. The contribution to this energy from the zero-point energy of the atomic vibrations changes with isotope substitution and with bond lengths. Dynamical effects are those that involve exchange of phonons such as in superconductivity.

I am not sure how to sharpen this structural/dynamical or static/dynamical distinction. Or is it secondary and primary effects?

In the next post, I will discuss observations of isotope effects in spin-crossover materials and how that relates to recenttheoretical work with my collaborators.