We have seen that in the nutation measurement the effect of the static magnetic field inhomogeneity ÖB can be overcome by using a sufficiently strong rf field. However, for a sufficiently larger rf field the effect of the rf field inhom ogeneity, 5B rf, becomes significant. Solomon [8] first proposed a method called rotary echo method by which the effect of the rf field inhomogeneity can be eliminated. As is shown in Fig.4-3, when a resonant rf
field is applied on an ensemble of two-level systems, the M vectors will precess
in the y-z plane and, at time t = T, the angle of precession is 0 = x T- Due to the
inhomogeneity in the rf field, the M vectors will fan out in the y-z plane
resulting in a faster decay of nutation signal. If at t = x, we perform a 180°
phase shift on the rf field so that Brf is suddenly reversed. From then on, the M
vectors will precess at the same rate as before, but in the opposite direction. Thus those centres that experienced the largest Brf and hence have precessed the farthest before, now precess fastest in the opposite direction. Therefore, at t =
2t, the M vectors refocuses along the z axis and cause an echo. In this way, the
effect of 5Brf is minimized which allows the spin relaxation effects including spin-spin relaxation and spin-lattice relaxation to be measured. The rotary echo decays with the time constant given by Eq.(4.2). The reason that the rotary echo is sometimes called nutation echo is obvious from the above discussion.
B
180°
tim e ---►
Figure 4-3. The rotary echo or nutation echo, (a) As the F is applied along the x axis, the M vectors precess in the y-z plane, some moving faster than others because of inhomogeneity in RF field, (b) Phase reversal of the F vector causes the M vectors to precess in the direction opposite that in (a), (c) The faster moving M vectors catch up to the slower moving M vectors to form an echo along the z axis.
Fig.4-4 shows a typical rotary echo observed in the 5.4 MHz transition using the Raman heterodyne technique. The nutation has an oscillation period of -2 9 fisec corresponding to a Rabi frequency o f %/27t ~ 34.5 kHz. The magnitude of this oscillation would continue to decrease, but a
n
phase shift occurring at -6 0 0 |nsec results in a rotary echo at -1200 jisec. It is worth noting that the observation of rotary echo indicates the presence of an inhomogeneity in the rf field. However, the relative small magnitude of the rotary echo implies that the rf field inhomogeneity is small in our experiments.C/3
'S
3 X)1— cd cd C bJQ • *-H on <D c -oo
Ui <D <DX
§i
QC 1 |180° phase shift0
600
1200
10-is)
Figure 4-4. Rotaty echo of the 5.4 MHz NMR transition measured using the Raman heterodyne technique.
4.4 Free-Induction-Decay (FID): Measurement of T
2*
FID [9,10] occurs after an ensemble of two-level systems at equilibrium are subjected to a pulsed excitation for a period of T so that %T = 7t/2. Fig.4-5 shows a typical experimental FID trace using a ~5.9 jisec 71/2 pulse. The FID signal decays exponentially with a time constant of ~ 88 jisec corresponding to a linewidth of ~ 3.6 kHz.
f|90° pulse I__________I__________I__________I________
0
100
200
300
400 t (ns)
Figure 4-5. Free-Induction-Decay of the 5.4 MHz NMR transition measured using the Raman heterodyne technique.
In the vector model (see, Fig.4-6) the Bloch vector M corresponding to a two-level system at equilibrium lies along the z axis. If a strong resonant pulse
(X »r inh)
is applied, the field vectorF
lies nearly along the x axis for all centreswithin the inhomogeneous distribution. Following the
n/2
pulse, M lies alongthe y axis. As transverse relaxation occurs, the coherence decays. If a homogeneous broadened transition is studied then the field vector
F
= 0 after the 7C/2 pulse and the coherence decays freely in a time of T2 as predicted by the Bloch equation, Eq.(1.22). However, for an inhom ogeneous broadened transition after then/2
pulse the field vectorF
* 0 due to the presence ofinhomogeneity. The field vector
F
lies along the z axis and has a differentmagnitude for the different centres within the inhomogeneous distribution. In this case the Bloch vectors of different centres process in the x-y plane at different frequencies and as a consequence interfere destructively with each other. This results in a more rapidly decaying signal than in the homogeneous
case. Therefore, in general, the FID signal decays in a time T ?* which includes contributions from both the pure coherence decay term T-> and the inhom ogeneity term due to both static m agnetic field and crystal inhomogeneities.
time --->
(a) t = T
(b) t >T
Figure 4-6. The free-induction-decay. (a) At the end of a k / 2 pulse
along the x axis the M vectors lie along the y axis, (b ) Dephasing
occurs as the M vectors precess at differing rates in the x-y plane.