# Are Bloch equations empirical?

Are Bloch equations (which describe the evolution of macroscopic magnetization) empirical? If so, under what assumptions do they hold?

$$\frac {d M_x(t)} {d t} = \gamma (\mathbf{M} (t) \times \mathbf {B} (t) ) _x - \frac {M_x(t)} {T_2}$$ $$\frac {d M_y(t)} {d t} = \gamma (\mathbf{M} (t) \times \mathbf {B} (t) ) _y - \frac {M_y(t)} {T_2}$$ $$\frac {d M_z(t)} {d t} = \gamma (\mathbf{M} (t) \times \mathbf {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1}$$

I wouldn't call them empirical but rather phenomenological. These equations relate our model of how spins behave in magnetic fields to observables such as macroscopic magnetization. One of the things required then is the ability to define a sensible M. What you usually do is to say $$M = \frac{1}{V} \Sigma \mu_i ,$$ where $M$ is the magnetization made up of the sum of magnetic moment of all spins i in the volume V. What you want is a V large enough to contain a few spins but small enough for the external fields over it to be constant. (Amongst other things external $B_0$ is defining the magnitude of $M_0$). The set of spins in the volume is called a spin isochromat.
You can derive transverse and longitudinal relaxation from a QM basis. (Slichter: 'Principles or MR' or worse and less digestible: Bloembergen Purcell Pound: PhysRev 73 p670 (1948)) but beyond the validity of QM theory and the ones above no further assumptions have to be satisfied. [I mean, yes, if you vary temperature, $M_0$ will change but let's not be silly]