Time Constant Tool: y = 1-exp(-x/400)
An exponential decay may be turned into an exponential recovery by subtracting the decay equation from 1.
The construction of this recovery curve is simply 1 (representing full recovery,) minus exponential decay. Hey presto: exponential recovery! This represents recovery after a 90° pulse. The 90° pulse saturates the spin system (sets all longitudinal magnetisation to zero). The equation of this curve is, therefore, the equation of saturation recovery:
S = 1 - exp(-TR/T1)
where S is signal intensity (normalised to M0), TR is the repetition time and T1 is the longitudinal relaxation time.
Or: what value does τ take? (See the introduction. In the example above, τ = T1 = 400.) This depends on the sample / voxel under investigation. For longitudinal recovery (of Mz) after a 90° pulse, the magnetisation in this direction is described by this equation:
Mz(t) = M0(1-e-t/τ).
We define τ (which in this case is T1) to be equal to the time for the magnetisation to recover to (1-e-1) of M0, or 63.2% (i.e. when t = τ). On the following graph, the recovery is normalised to M0.
For transverse decay (Mxy) after a 90° pulse, the equation is
Mxy(t) = M0(e-t/τ)
and so τ (which in this case is T2) is equal to the time for the magnetisation to decay to e-1 of M0, or 36.8% (i.e. when t = τ).
Navigate below to the next equation in the list: 1 - 2·exp (-x/400).
Remember, T1, T2 and T2* are time constants.