Isotopic shift
In NMR spectroscopy, isotopic effects on chemical shifts are typically small, far less than 1 ppm, the typical unit for measuring shifts.
The 1H NMR signals for 1H2 and 1H2H ("HD") are readily distinguished in terms of their chemical shifts.
The asymmetry of the signal for the "protio" impurity in CD2Cl2 arises from the differing chemical shifts of CDHCl2 and CH2Cl2.
Isotopic shifts are best known and most widely used in vibration spectroscopy, where the shifts are large, being proportional to the ratio of the square root of the isotopic masses.
Thus, the (totally symmetric) C−H and C−D vibrations for CH4 and CD4 occur at 2917 cm−1 and 2109 cm−1 respectively.
[1] This shift reflects the differing reduced mass for the affected bonds.
Isotope shifts in atomic spectra are minute differences between the electronic energy levels of isotopes of the same element.
They are the focus of a multitude of theoretical and experimental efforts due to their importance for atomic and nuclear physics.
From a nuclear physics perspective, isotope shifts combine different precise atomic physics probes for studying nuclear structure, and their main use is nuclear-model-independent determination of charge-radii differences.
[2] It is traditionally divided to a normal mass shift (NMS) resulting from the change in the reduced electronic mass, and a specific mass shift (SMS), which is present in multi-electron atoms and ions.
The NMS is a purely kinematical effect, studied theoretically by Hughes and Eckart.
[3] It can be formulated as follows: In a theoretical model of atom, which has a infinitely massive nucleus, the energy (in wavenumbers) of a transition can be calculated from Rydberg formula:
The above equations imply that such mass shift is greatest for hydrogen and deuterium, since their mass ratio is the largest,
The effect of the specific mass shift was first observed in the spectrum of neon isotopes by Nagaoka and Mishima.
[4] Consider the kinetic energy operator in Schrödinger equation of multi-electron atoms:
For a stationary atom, the conservation of momentum gives
, which gives the normal mass shift formulated above.
The second term in the kinetic term gives an additional isotope shift in spectral lines known as specific mass shift, giving
Using perturbation theory, the first-order energy shift can be calculated as
which requires the knowledge of accurate many-electron wave function.
term in the expression, the specific mass shift also decrease as
This difference induces a change in the electric charge distribution of the nucleus.
[5][6][7] Adopting a simplified picture, the change in an energy level resulting from the volume difference is proportional to the change in total electron probability density at the origin times the mean-square charge radius difference.
For a simple nuclear model of an atom, the nuclear charge is distributed uniformly in a sphere with radius
Similarly, calculating the electrostatic potential of an ideal charge density uniformly distributed in a sphere, the nuclear electrostatic potential is
Such a perturbation of the atomic system neglects all other potential effect, like relativistic corrections.
has radial and angular parts, but the perturbation has no angular dependence, so the spherical harmonic normalize integral over the unit sphere:
The explicit form for hydrogenic wave function,
Differentiation of the above equation gives the first order in
This equation confirms that the volume effect is more significant for hydrogenic atoms with larger Z, which explains why volume effects dominates the isotope shift of heavy elements.
