Question

What is the formla for the energy difference between the allowed nuclear spin states for `""^1 H` nuclei?

Answer 1

The formula is `ΔE = hν = γh(B_0-B_e)`.

Explanation:

In the presence of an applied external magnetic field, `""_1"H"` nuclei exist in two nuclear spin states of different energy.

A large magnet creates a homogeneous external magnetic field, `B_0`.

The energy difference `ΔE` between the spin states is proportional to the strength of `B_0`.

NMR spectroscopy records transitions between these spin states induced by a radio frequency electromagnetic field called the `B_1` field.

In practice, the electrons surrounding the nucleus produce a small magnetic field `B_e` that opposes the `B_0` field.

The field experienced by the proton is then `(B_0-B_e)`.

Finally, there is a proportionality constant `γ` (the gyromagnetic ratio) that is characteristic of a proton.

The final equation is

`ΔE = hν = γh(B_0-B_e)`, where

`ν` is the frequency of the `B_1` field.

The graph below shows how the energy difference between the two spin states varies with the applied field.