# 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*/*T*1)

where *S* is signal intensity (normalised to M_{0}), *TR* is the repetition time and *T*1 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 M_{z}) after a 90° pulse, the magnetisation in this direction is described by this equation:

M_{z}(t) = M_{0}(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 M_{0}, or 63.2% (i.e. when t = *τ*). On the following graph, the recovery is normalised to M_{0}.

For transverse decay (M_{xy}) after a 90° pulse, the equation is

M_{xy}(t) = M_{0}(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 M_{0}, 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.