# Quantum Mechanical Analysis of Selected NMR Pulse Sequences

NMR pulse sequences play a central role in the modern NMR technique; they allow the manipulation of the nuclear spin Hamiltonian with great versatility, setting up the conditions necessary for the desired NMR observable to be detected. The pulse sequences to be discussed next include the one pulse sequence, which, as discussed in Section 4.6 for a n/2 pulse, gives rise to a *free induction decay *(FID) whose Fourier transform coincides with the spin Hamiltonian absorption spectrum, and the solid echo pulse sequence which is quite frequently used in 1D high-field NMR spectroscopy of anisotropic fluids.

## Spin System Subjected to a Single n/2 RF Pulse in Resonance

In high-field pulsed NMR spectroscopy the quantity experimentally recorded is the complex transverse magnetization *m(t) = M _{x}*(t) +

*iM*(t) (4.50), whose components in the lab frame [x, y, z] are related to those in the rotating frame [x',

_{y}*y', z*(4.3.3) by

^{/}]

where *p'(t)* is the density matrix in the rotating frame, *ш = *y B_{0}, and *N* the number density of target spins. Knowing *p'(t)* enables the determination of m(t) through Eq. 5.36. As discussed previously in 4.6.2 it is more convenient to analyze the evolution of the spin system in the rotating frame where the time evolution of the density matrix is simplified. In order to carry out explicitly all the calculations, an ensemble of spin 1 particles subjected to an axially symmetric quadrupolar interaction is considered in the analysis. As the spins are not interacting with each other, it is sufficient to consider just one spin. The time evolution of the density matrix starts at *t =* 0^{—}with the spin system in thermal equilibrium with the lattice, leading to the following density matrix in the rotating frame

At time *t =* 0 the RF *B _{1}* field is switched on for a short period of time sufficient to tilt the magnetization by 90

^{o}. In the rotating frame the time evolution of the density matrix is governed by Eq. 4.53 and while the B

_{1}field is on the effective Hamiltonian is

*H*^ —

_{e}*у hB*and the density matrix follows a solution of the type given in 4.44 leading to

_{1}I_{x }

where *ш _{1} = у* B

_{1}. Using the following relation obeyed by the spin operators (Kimmich, 1997)

and considering in Eq. 5.37 that the quadrupolar term in the Hamiltonian is much smaller than the Zeeman term, Eq. 5.38 gives the result

^{For t} = ^{t}1 = 22L_ ^{we} g^{et}

At *t = t _{1} =* П/

^{2}the RF pulse is switched off and the density matrix enters a new evolution time period with the effective Hamiltonian taking the form

Again due to the simplifying assumptions considered the effective Hamiltonian is time independent and a solution of the type 4.44 is verified leading to

where t_{2} *= t — t.* Using the results of Eq. 5.41 and Eq. 5.42, *p' *becomes

and using the relation obeyed by the spin 1 operators (Kimmich, 1997)

one obtains

Inserting Eq. 5.46 in Eq. 5.36 one obtains for the complex transverse magnetization

The complex magnetization given by Eq. 5.47 has a Fourier transform composed of two delta functions at the frequencies *Y B _{0} + ^eQVzz* and y B

*—*

_{0}*which are precisely the absorption frequencies of the spin Hamiltonian considered (see Eq. 5.33), in agreement with the discussion of the method in 4.6. The fact that the result Eq. 5.47 gives a transverse magnetization that oscillates but does not decay is a consequence of the simplified Hamiltonian considered in Eq. 5.37 that excludes relaxation.*

^{3}4eQV'_{zz}