Lamb Formula
Definition and meaning of Lamb Formula in chemistry.
The Lamb formula calculates the leading-order diamagnetic contribution to nuclear magnetic shielding in closed-shell atoms and molecules. It is expressed as σ = (α²/3) Σ ⟨1/r⟩, where α is the fine-structure constant, the sum runs over all electrons in the system, and r is each electron's distance from the nucleus (in atomic units).
In more detail
The Lamb formula provides the nonrelativistic, nonrecoil contribution to the induced magnetic field surrounding a nucleus, arising from the orbital circulation of electrons in response to an applied magnetic field. Derived by physicist Willis Lamb in 1941 and later generalized by Norman Ramsey for molecules, this formula is fundamental to calculating NMR chemical shifts. The shielding constant σ depends on the electron density near the nucleus, specifically the expectation value of the inverse distance between electrons and the nucleus. It represents the core contribution to total nuclear shielding, which also includes paramagnetic and spin-orbit contributions in more complete treatments.
Key facts
| Formula | σ = (α²/3) Σ ⟨1/r⟩ (α = fine-structure constant) |
|---|---|
| Applies to | Closed-shell atomic systems |
| Contribution type | Diamagnetic shielding |
| Field | Physical Chemistry |
For helium (a closed-shell atom with two electrons), the Lamb formula calculates the diamagnetic shielding of the nucleus by summing the inverse expectation distances of both 1s electrons, giving σ approximately 59.97 ppm. Because helium is a spherically symmetric S-state atom with no orbital angular momentum, it has no significant paramagnetic contribution, so this diamagnetic term accounts for essentially the entire observed nuclear shielding of helium-3, a value used as a reference standard in precision magnetometry.
Frequently asked questions
Why is the Lamb formula limited to closed-shell systems?
The formula assumes spherically symmetric electron distributions responding diamagnetically to the external magnetic field. In open-shell systems with unpaired electrons, paramagnetism dominates and requires separate treatment.
How does the Lamb formula relate to NMR chemical shifts?
The Lamb formula calculates the core diamagnetic shielding, which is combined with other contributions to determine the total chemical shift observed in NMR spectra. Higher electron density near the nucleus increases shielding, shifting resonance frequencies upfield.