**Introduction**

The phenomological equations proposed by Felix Bloch in 1946 have had a profound effect on the development of magnetic resonance, on the ways in which the experiments are described and on the analysis of line widths and saturation behavior. Here we will describe the phenomological model, derive the Bloch equations and solve them for steady-state conditions.

**Bloch Equations**

When a magnetic field is applied to a spin, the spin quantization axis is defined by the field direction. Spin magnetic moments aligned with the field are only slightly lower in energy than those aligned opposed to the field. If we consider ensemble of spins, the vector sum of all the spin magnetic moment or macroscopic magnetization:

At equilibrium is in the direction of the field . If somehow is tilted away from there will be a torque that causes to precess about with the equation of motion

where . In addition to the precessional motion , there are two relaxation effects.

If is the equilibrium magnetization along and is the *z*-component magnetization under non-equilibrium conditions, then we assume approach with first-order kinetics:

where is the chararcteristic time to reach the equilibrium. Since this process involve transfer of energy from the spin system to the surrounding lattices, is called the *spin-relaxation time*.

There is a second kind of relaxation process besides process. Suppose that is somehow tilted down from the z-axis toward the xy-axis and the precessional motion is started. Each individual magnetic moment undergoes this precessional motion, but the individual spins may precess at slightly different rates that can be caused by either local shielding that makes small variations in or the effective g-factor may vary slightly through the sample. Thus an ensemble of spins that all start out in phase will gradually lose phase coherence. The characteristic time for this process is called *transverse relaxation time* , and that the transverse magnetization components decay to the equilibrium value of zero accordingly:

We have to notice that the dephasing of the transverse magnetization does not affect ; a process involves no energy transfer but, being a spontaneous process, does involve an increase in the entropy of the spin system.

The approach to equilirium by a process, in which approaches , also causes and to approach zero. Thus, the of eq. (4) must include both of the effects of spin-lattice relaxation as well as the dephasing of the transverse magnetization. Transfer relaxation is often much faster than spin-lattice relaxation and is then determined mostly by spin dephasing. However, in general we should write:

where is the spin dephasing relaxation time, and is the observed transverse relaxation time.

**Derivation of Bloch Equation**

Combining eqs. (2)-(4), we get:

In a magnetic resonance experiment, we apply not only a static field in the -direction but also an oscillating radiation field in in the -plane, so that the total field is:

Note there are other possible ways to impose a time-dependent . The one described in eq. (7) corresponds to a circularly polarized field initially aligned along the -axis and rotating about the -axis in a counterclockwise direction.

Inserting the into eq. (7) and separating the results into their components, then:

It is convenient to write and as:

or

This is equivalent to transformation into a coordinate system that rotates with the oscillating field; is that part of which is in phase with and is the part which is 90 out of phase. Differentiating eq. (13) and substituting eqs. (8) and (9), we get:

Similarly, we obtain:

Equations (15-17) are the *Bloch equations* in the rotating coordinate frame.

**Steady-state Solution**

In a continuous wave (CW) magnetic resonance experiment, the radiation field is continuous and is changed only slowly compared with the relaxation rates (*slow passage* conditions). Thus a steady-state solution to eqs (15-17) is appropriate. Setting the derivatives to zero and solving the three simultaneous equations, we get:

where is Larmor frequency and corresponds to the frequency of the energy level transition.

With and , we get the two components of ac susceptibilities:

is the ac susceptibility component in-phase with the driving field . In general a response that is exactly in phase with a driving signal does not absorb power from the signal source and in spectroscopy corresponds to *dispersion*. In contrast, an out of phase corresponds to *absorption*. In magnetic resonance, it is usually the absorption, or , that is detected.

When the microwave or radiofrequency power, proportional to , is small so that , Eqs. (21) and (22) becomes

Equation (24) of corresponds to the classical Lorentzian line shape function. The absorption curve will be a Lorentzian line with the half-width of the half height: