# Solid Echo Pulse Sequence

The solid echo pulse sequence is composed of two n/2 RF pulses, the first one along the *X'* axis of the rotating frame and the second one along the *Y'* axis. This pulse sequence generates an echo on the detected signal that appears after the second pulse and separated from it of a time interval equal to the pulse separation. The signal starting from the echo top onward reproduces the FID that is generated by a single n/2 pulse. The use of this pulse sequence in the determination of the absorption spectra of the spin Hamiltonian presents advantages over the single n/2 pulse sequence specially for wide-frequency-range spectra associated to fast FIDs. After a n/2 RF pulse the signal detecting electronics stays nonoperational for some microseconds and if the FID is fast this loss of signal is not acceptable. With the solid echo pulse sequence the signal to be Fourier transformed starts at the echo top that may be placed sufficiently apart from the RF pulses allowing the recording of the FID without any data loss. To carry out all the calculations the same spin system considered for the one pulse sequence will be used. The effect of the first n/2 RF pulse along the *X'* axis on the density matrix was determined before giving rise to density matrix reported in Eq. 5.46 that at *t _{2} = т* where

*т*is the pulse separation takes the value

At t_{2} = *т* a n/2 RF pulse along *Y'* is applied to the spin system, this is achieved with a radio-frequency (RF) term in the Hamiltonian of the same formas used for the *X'* pulse but dephased by n/2 as follows

and the contribution to effective Hamiltonian in the rotating frame at resonance becomes

Neglecting the oscillating terms that are not effective in changing the magnetization and also the contribution from the interaction terms *К** -* as done for the

*X*

*pulse, the effective Hamiltonian while the*

**'***Y'*pulse is ON becomes

*H*^ — y k

_{e}*B*

_{1}I_{y}. While the

*Y*

*pulse is ON the density matrix evolves according to*

**'**

where *ю _{1} = *y

*B*and t

_{i}_{3}=

*t — t*t

_{i}—_{2}. Using the following relation obeyed by the spin 1 operators (Kimmich, 1997)

and evaluating *p' (t _{3} =* which corresponds to a n/2

*Y'*pulse leads to the result

At t_{3} = the RF pulse is switched off and the density matrix enters

a new evolution period with the effective Hamiltonian given by Eq. 5.42. The solution of the von Neumann equation in the rotating frame in this new period yields the result

where t_{4} = *t — t _{1} — t_{2} —* t

_{3}. Using Eq. 5.45 and the following relation obeyed by the spin 1 operators (Kimmich, 1997)

the density matrix given by Eq. 5.54 takes the form

The transverse magnetization in the time interval after the second pulse becomes

When *t*_{4}* = т* in Eq. 5.57 the transverse magnetization reaches a maximum value independent of the strength of the quadrupolar interaction given by showing that an echo forms at this time. The time dependence of the complex magnetization starting from the echo maximum onward is identical to the one obtained after just one п/2 *X'* pulse justifying its use in obtaining the Hamiltonian absorption spectra through Fourier transform of the FID. Once more the expected decay of the transverse magnetization is not observed in Eq. 5.57 as the simplified Hamiltonian considered for the spin system excludes relaxation.

**5.6 Experimental Details**

NMR equipment operates in the RF range that includes the bandwidths of radio stations operating in most countries.^{3}

All NMR spectrometers include a magnetic field source (e.g., permanent magnet, conventional or superconductor electromagnet) and an emitter/receiver RF system, usually referred to as the NMR console. Modern NMR setups include computer systems for setup control and data acquisition. One important element of the emitter/receiver unit is the RF antenna that allows for the sample's irradiation and signal reception. In Fig. 5.1 is presented a cartoon that illustrates some main basic features of the coupling between the external magnetic field, the nuclear spin magnetization and the RF in resonance with the Larmor frequency.

The solenoid coil is the emitter/receiver RF antenna inside which the sample is placed.^{b} The shape and number of coil windings that define the coil's self-inductance and the high-power capacitors are used to form a RLC resonance electric circuit tuned at the resonance frequency of the emitter/receiver frequency and also the nuclear spins' Larmor frequency. The coil axis is set perpendicular to the main, static Zeeman magnetic field in order for the RF field components (e.g., B* _{L})* to lie in the plane perpendicular to the Zeeman field. In fact, the RF field is linearly polarized along the

^{a}For this reason RF shielding is of crucial importance in all NMR equipment. ^{b} Depending of the probe head design the RF coils can be solenoidal or saddle coils.

**Figure 5.1 **Illustrative cartoon of the emitter/receiver coil, the magnetic field, and the coupling between the static magnetic field, the RF field, and the nuclear spin magnetization. The path of the magnetization vector tip illustrates the effect of the applied RF field on the nuclear magnetization for a nucleus with a positive gyromagnetic ratio (y *>* 0). Cartoon generated with FEATPOST software (Goncalves, 2004).

coil axis. It can be considered as the combination of the rotating fields in opposite directions. The only component in resonance with the Larmor frequency is that rotating in the same direction as the Larmor precession, as described in Chapter 4. According to Bloch's equations (Eq. 4.19) the magnetization precesses around the vertical axis and always perpendicular to the RF field and the angle between the nuclear spin magnetization and the external magnetic field increases with time. If the RF is a pulse with length *t**_{p} *= x/(

*2*yB

*at the end of the pulse the magnetization will be in the plane perpendicular to B*

_{l})_{0}. Such pulse is referred to as a

*п/2-pulse.*An RF pulse with length 2

*t*

*will leave the nuclear spin magnetization aligned antiparallel to B*

**p**_{0}and is referred to as a

*n-pulse*(Farrar and Becker, 1971).

The same coil is used to detect the sample's response signal as the result of the nuclear spin relaxation processes as the result of the evolution of the magnetization back to its equilibrium state (e.g., aligned parallel to B_{0}), in the absence of any applied RF.

The precession of * M* components perpendicular to B

_{0}will induce a electromotive force in the coil and a time-decaying AC current will be generated in the resonance circuit with the Larmor frequency After signal amplification and rectification the low-frequency components can be analyzed and viewed on an oscilloscope or equivalent. The detected signal is the FID. The time evolution of magnetization components aligned with Bо will not be detected in the coil.

The amplitude of the signal detected in the coil depends on several factors (Abragam, 1961). Important factors are the quality factor of the resonance electrical circuit, the value of B_{0}, and the filling factor (e.g., the * volume of the sample/volume of the coil* ratio). For a good experiment it is important to adjust the system to obtain the largest signal amplitude possible. The

*ratio can be a limiting condition in any NMR experiment. In particular, this ratio depends on B^*

**signal/noise**^{2}and for magnetic fields below 0.2 T is it hard to obtain good NMR FIDs (Abragam, 1961; Noack, 1986).

Due to its gyromagnetic ratio hydrogen proton spin has the largest Larmor frequency for a given magnetic field in comparison with all other nuclear spins. For a 7.05 T magnetic field the ^{1}H Larmor frequency corresponds to approximately 300 MHz.^{a}

'The Larmor frequency for 1 T is 42.577 MHz.