# Time Evolution of the Density Matrix under RF Pulses

In pulsed NMR experiments the time evolution of the nuclear spins system is conditioned by the spin interactions and the application of appropriate RF pulses, these pulses are responsible for inducing changes in the populations of the energy levels of the spin system and the creation of coherences between the states. The general equation of motion for the density matrix is the Liouville/von Neumann Eq. 4.43 which has the solution 4.44 when the Hamiltonian is time independent. As seen when studying the motion of noninteracting spins 4.3, it is convenient to analyze the evolution of the spin system in the rotating frame introduced in 4.3.1. In this frame {x', *y', Z*} with its *Z* axis coinciding with the *z* axis of the laboratory fixed frame but rotating around it with a uniform angular velocity ^, which at resonance takes the value ^ = — y B* _{0} e_{z},* the Hamiltonian becomes simplified and when relaxation processes can be disregarded in short time lapses the Hamiltonian may become time independent in those lapses allowing 4.44 as a valid solution. The transformation to the rotating frame of the relevant operators is carried out through the unitary operator

with *ш* the rotation frequency and *I _{z}* the

*z*component of the total spin operator. The density matrix in the rotating frame is

and the Liuoville/von Neumann equation becomes where

To evaluate *H _{e}* we note that the system Hamiltonian

*H*is composed of three contributions which are, respectively, the Zeeman term H

_{0}, the RF term H

_{RF}and the spin interactions term

*Hi*, leading to

The terms in *RH _{RF} R*

^{—1}oscillating with frequency 2ш are not effective in inducing magnetization changes and can be neglected. At resonance, where

*ш = у B*the term arising from

_{0},*RH*R

_{0}^{-1}, is canceled by the term

*шНI*in Eq. 4.54, giving for the effective Hamiltonian the form

_{z}with ш_{1} = *у* B_{1}. *H _{e}* given by Eq. 4.56 is composed of two terms, one originated by the RF pulses —

*ш*and the other arising from the spin interactions

_{1}НI_{x}*H'*. The time dependence of

_{i}*H*comes from the term

_{e}*H'*and arises due to the modulation of spin interactions created by the molecular motions and reorientations. When the RF pulses are hard pulses, the case discussed here, and while an RF pulse is being applied the term —

_{i}*ш*dominates

_{1}НI_{x}*H*and the spin interactions term can be neglected. During this period a solution of the type Eq. 4.44 for Eq. 4.53 applies. In absence of RF pulses

_{e}*H'*which is time dependent in general can for certain time lapses be replaced by an averaged Hamiltonian enabling also a solution of the type Eq. 4.44 to be applicable (Kimmich, 1997). Under these circumstances where

_{i}*He*takes different time-independent forms during different time intervals r

_{1}, r

_{2}... t, the solution of Eq. 4.53 can be written as

An example of this method appears in the next chapter. When * H_{e }*is time dependent and cannot be substituted in successive time intervals by time-independent approximations, specific solution methods for Eq. 4.53 with a time-dependent

*must be used, which fall outside the scope of this introduction, and the reader is referred to more specialized literature (Ernst et al., 1992; Kimmich, 1997; Slichter, 1992).*

**H**_{e}