Relaxation (NMR) - BPP Theory

BPP Theory

In 1948, Nicolaas Bloembergen, Edward Mills Purcell, and Robert Pound proposed the so-called Bloembergen-Purcell-Pound theory (BPP theory) to explain the relaxation constant of a pure substance in correspondence with its state, taking into account the effect of tumbling motion of molecules on the local magnetic field disturbance. The theory was in good agreement with experiments on pure substances, but not for complicated environments such as the human body.

This theory makes the assumption that the autocorrelation function of the microscopic fluctuations causing the relaxation is proportional to, where is called the correlation time. From this theory, one can get T₁、T₂ for magnetic dipolar relaxation:

,

where is the Larmor frequency in correspondence with the strength of the main magnetic field . is the correlation time of the molecular tumbling motion. is defined for spin-1/2 nuclei and is a constant with being the magnetic permeability of free space of the the reduced Planck constant, γ the gyromagnetic ratio of such species of nuclei, and r the distance between the two nuclei carrying magnetic dipole moment.

Taking for example the H₂O molecules in liquid phase without the contamination of oxygen-17, the value of K is 1.02×1010 s−2 and the correlation time is on the order of picoseconds = s, while hydrogen nuclei 1H (protons) at 1.5 teslas carry an Larmor frequency of approximately 64 MHz. We can then estimate using τ_c = 5×10−12 s:

(dimensionless)

= 3.92 s

= 3.92 s,

which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T₁ equals T₂.