The Effect of RF Pulses

Likewise if we calibrate the pulse width to give the maximum peak height in the spectrum (90° pulse), we can calculate the pulse amplitude in hertz  

All pulse rotations are counterclockwise when viewed with the B vector pointing toward you. 

At the end of the 90° pulse with B on the x axis, the net magnetization is on the —/ axis, and we have no z component. We will refer to this spin state as —Iy. Because the z component of net magnetization results from the population difference between the a and states, we can say that there is no population difference at the end of a 90° pulse (Fig. 6.7). With the 90° pulse, we have effectively converted the population difference into coherence. If we record the FID right after this pulse, we would get a normal spectrum with a positive absorptive peak. 

At the end of a 360° pulse, the net magnetization vector has made one complete rotation around the B vector and lands on the +z axis, exactly where it started (Fig. 6.7, right). The spin state is identical to the equilibrium state, I,. We have the equilibrium (Boltzmann) population distribution, represented with one open circle in the upper state and one filled circle in the lower state. If we collect an FID right after the 360° pulse, we will see no spectrum. 

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This section summarizes primarily the classical description of NMR based on the vector model of the Bloch equations. Important concepts like the rotating frame, the effect of rf pulses, and the free precession of transverse magnetization are introduced. More detailed accounts, still on an elementary level, are provided in textbooks [Deri, Farl, Fukl]. 

The Bloch equations describe the motion of the transverse magnetization in the static magnetic field in terms of a precession around the axis of the field. Similarly (2.2.65) describes a rotation of the density matrix around the z-axis by an angle (Wo(t to)- The effects of rf pulses are consequently described by rotations of the density matrix around axes in the transverse plane. For instance, a rotation around the y-axis by an angle a is expressed by... 

In accordance with the works we assume that for analysing the effect of RF pulses on the quadrupolar spin system, it is sufficient to consider only three t)q)es of interactions quadrupolar interaction, homonuclear dipole-dipole interaction and the nuclear-spin interaction with the magnetic component of the RF field. Before the initiation of the multi-pulse sequence the quadrupolar system is described by two Hamiltonians quadrupolar Hq and the part of the homonuclear dipole-dipole Hamiltonian Hj secular in relation to the quadrupole Hamiltonian, the sum of which can be regarded as the effective Hamiltonian of the spin system independent of the time factor. 

We can now use these explicit forms to calculate the effect of RF pulses on the deviation matrix densities for spin 1 /2 nuclei. In the case of a r/2 pulse with the magnetic field aligned with the x-direction of the rotating frame, we have ... 

We can summarize the results of this section saying that the effects of RF pulses can be described in the rotating frame in terms of the rotation operators (usually around the X, y, —x,and —y axes) applied to the deviation density matrix starting from thermal equilibrium. The evolution of the system after or between the RF pulses is described as a free precession aroimd the z-axis, with a frequency that depends on the frequency offset and is therefore different for nuclei experiencing distinct local fields (due to chemical shifts, for example). 

For an ensemble of nuclei with spin / > 1/2 experiencing no quadrupolar interaction, such as in an isotropic liquid or a crystal with cubic symmetry, the equations describing the time evolution of the density matrix under action of static and RF magnetic fields are a natural extension of the / = 1/2 case. The Hamiltonian contains only the Zeeman and RF terms the effects of RF pulses are described by rotations of the spin operators around the transverse axes in the rotating frame, whereas free evolution corresponds to rotations around the z-axis. 

The effects of RF pulses are analysed in the rotating frame, precessing around the z-axis with frequency The effective Hamiltonian for a RF pulse applied along the x-direction in this case is given simply by ... 

The Cartesian product operators are the most common operator basis used to understand pulse sequences reduced to one or two phase combinations. This operator formalism is the preferred scheme to describe the effects of hard pulses, the evolution of chemical shift and scalar coupling as well as signal enhancement by polarization transfer. The basic operations can be derived from the expressions in Table 2.4. The evolution due to a rf pulse, chemical shift or scalar coupling can be expressed by equation [2-8]. 

More recently, pulsed ENDOR methods have been introduced. There are two commonly used ENDOR pulse sequences, both of which are based on the impact of RF pulses on the intensity of a spin echo that is formed from a series of three pulses at the microwave frequency. These techniques are sometimes called ESE-ENDOR. In Mims ENDOR the pulse at the RF frequency is applied between the second and third microwave pulses whereas in Davies ENDOR the RF pulse is applied between the first and second microwave pulses. The Mims ENDOR experiment is particularly effective for weakly coupled nuclei, but has some blind spots (frequencies that cannot be observed). It is often advantageous to combine data from both ENDOR methods. 

These experiments yield T2 which, in the case of fast exchange, gives the ratio (Aoi) /k. However, since the experiments themselves have an implicit timescale, absolute rates can be obtained in favourable circumstances. For the CPMG experiment, the timescale is the repetition time of the refocusing pulse for the Tjp experiment, it is the rate of precession around the effective RF field. If this timescale is fast witli respect to the exchange rate, then the experiment effectively measures T2 in the absence of exchange. If the timescale is slow, the apparent T2 contains the effects of exchange. Therefore, the apparent T2 shows a dispersion as the... 

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