Bloch Lorentzian lineshape

The Lorentzian lineshape is obtained for liquid samples under ideal high-resolution NMR conditions10 and is readily derived from the Bloch equations.31 The normalized Lorentzian lineshape function in the frequency domain is given by... 

An ESR line is not infinitely sharp it has a shape and width due to spin relaxation. The equations of motion for Mx, My, and M in the presence of an applied field Ho and including the spin relaxation processes discussed above are called the Bloch equations. The solution to these equations predicts a Lorentzian line with a halfwidth at halfheight of Lorentzian lineshapes are indeed often found for free radicals in liquids. In this case T2 can be determined from the linewidth. The Bloch equations also predict how the ESR signal intensity will vary with increasing microwave power. The ESR signal increases, reaches a maximum, and then decreases with increasing microwave power this behavior is called power saturation. From an analysis of the power saturation curve of ESR intensity versus microwave power, it is possible to determine Ti. 

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

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