excitation pulse needs to be followed by a gOy refocusing pulse. The magnetisation is refocused at the time 2T after the first r.f. pulse.
90x 90y , • V FID , • ... • • • V
Quadrupole Echo time
Figure 2.8: The quadrupole echo pulse sequence consists of two 90° d. pulses, which are out of phase by 90° . The echo signal appears at a time T after the second 90° r.f. pulse.
The quadrupole echo pulse sequence will be used in chapter 9 to measure alignment of molecules in shear flow.
2.3.4 Two-Dimensional Spectroscopy
If two or more spins are coupled, the properties of such a system cannot be fully characterised by a conventional one-dimensional ( ID) spectrum . Two d imensional (2D) spectroscopy is a general concept which makes it possible to acquire more information about the system under investigation. It was fi rst pro posed by Jeener in 1973 [22] and the experimental realisation followed in 1 974 by Ernst et al. [23] .
j{ (P )
j{(e )
j{ (m )
j{ (d )
Preparation Evolution Detection
time
'p t I 'm t2
Figure 2.9: Basic scheme for a two-dimensional experiment.
In Ernst"s terminology [12] . a 2D spectrum represents a signal function S(Wt , W2) of two independent frequency variables. In the classic 2D time-domain experi ment, the signal s(t 1 , t}.) is measu red as a function of two independent time vari ables and is converted into a 2D frequenc�' spectru m by a 2D FT. The pulse train consists of the sequence '·preparation-evolution-mixing-detection" , as shown in figure 2.9. In the preparation period Tp. t he spin system is prepared in a coherent
2 . .3. SPIN MANIPULA.TIONS 23 non-equilibrium state, which will evolve in the subsequent periods. In the sim plest case, the preparation period is just one d. pulse. During the evolution period t l , the spin system evolves freely under the infl uence of a specific Hamiltonian . The evolution during t l determines the frequencies in the wl-domain. The t l -domai n i s sam pled by carrying out a series o f experiments with systematic incrementa tion of t 1 . The mixing period Tm may consist of one or more pulses separated
by intervals. During the mixing period , the spins may change their magnetisa tion , orientation or other parameters which are characteristic of the system u nder investigation . The evolution of the spins during the detection period t2 fi nally determines the frequencies in the w2-domain.
Ernst distinguishes between three types of 2D NMR spectroscopy: separation of interactions, correlation methods and exchange. If the Hamiltonian is com posed of terms of different physical origin, it is often possible to decompose the complex ID spectr u m by the choice of two suitable effective Hamiltonians 1{(e) and 1{(d) for the evolution and detection period. I n the 2D spectrum, the spectrum due to 1{(e) is then along the Wl axis, while the spectrum due to 1{(d) is along the
W2 axis. With correlation methods, one can measure the interaction between spins. The simplest experiment is based on the sequence 90x-tl-(,8)-t2 . This experiment is termed correlation spectroscopy
(
COSY)
. The transfer of coherence is induced by the pulse of flip angle {J. From the appearance of cross-peaks in the 2D correlation spectra the coupling parameters can be identified . With exchange spectroscopy, dynamic processes, such as chemical exchange or spin diffusion, can be measured . The fundamental idea is the labelling of spins before exchange takes place, such that after the mixing time, the magnetisation can be traced back. One particular 2D exchange experiment, due to Spiess et al. [24, 25, 26] ' provides a relevant exam ple. Here the Wl and W2 dimensions are dominated by dipolar or quadru polar interactions in which the frequency offset relates directly to local bond angle. The mixing period consists of the storage of Zeeman order along the magnetic field direction so that the recall of magnetisation at a later time reveals angular correlations arising from the angular reorientation of polymer segments which occurred over the mixing time Tm .I n chapter 4 this idea o f exchange will be extended to a different frequency space. The equivalent Wt and W2 dimensions correspond to the movements Zt and Z2 over two well defi ned time intervals, which are separated by the mixing time.
2 . 3 . 5 Signal Averaging and Phase Cycling
In many cases the signal acquired in one NMR experiment is not strong enough to extract meaningful data. One repeats therefore one experiment N times, adding
24 CHAPTER 2. INTROD UCTION TO NMR AND nvfAGING
up the signal from the individual scans. This increases the signal amplitude by a factor of N , while the noise level increases by a factor of N l /2. Thus, the signal to-noise ratio increases by a factor of N l/2• Between two scans, one has to wait for the magnetisation to come to thermal equilibri u m . This is usually the case after times of 2-3 Tl . The Tl value for protons in most liquids used in this thesis is less than 1 s. Therefore, repetition times of 1-1.5 s are considered long enough for the experiments in this thesis.
The NMR signal is usually overlapped by background interferences. These interferences can have various sources, such as d .c. offset in the amplifier of the receiver. By changing the phases of the d. pulses between different scans and adding or subtracting the signal in the correct way, one can su ppress these in terferences. Hoult and Richards [27] describe a four-step phase cycling which nullifies artifacts like d.c. offset in the receiver stage and imperfect phase settings of the rJ. pulses. For most experiments in this thesis, it was sufficient to employ a two-step phase cycle, which only nullifies the d .c. offset in the receiver. T his two-step phase cycling is shown in figure 2 . 1 0 .
• . ..
(b)
90_x :. • • •(c)
. .. .. . . .... . . . ... ... . ... . . ' .... .... . .. •• .. . . . .. . . . . ... .... ". .' ... . ... . . . time time timeFigure 2 . 10: (a) The F I D after a 90x r .f. pulse is acquired with a d.c. offset on the receiver.
(b) The F I D after a 90_x r.f. pulse. (c) Su btraction of the signals acquired in (a) a nd (b) cancels out the d.c. offset, but adds the F I Ds.
2.4. INTROD UCTION TO ;V/VIR /J'vIA.GING 25
2.3 . 6 The E ffect of Magnetic Field Gradients
In NMR spectroscopy, one is usually concerned to get the magnetic field i nside the sample as homogeneous as possible. Several layers of shim coils and specially designed pulse sequences ensu re that the broadening due to field inhomogeneities is smaller that the natu ral linewid th. However, if one wants to extract the spatial dependence of molecules in the sam ple, one deliberately has to vary the magnetic field across the sam pie. These magnetic field variations are created by specially designed gradient coils through which large, switch able currents can be passed .
Because the field variations due to the switch able gradients are much smaller than the static field Ba, only the component of the gradient parallel to Ba needs to be considered . The components of the field gradient can then be written as
Gx oBz (2.60) ox Gy oBz (2.61) ay Gz = oBz (2.62) oz
Because the magnetic field varies across the sample, the Larmor frequency will depend on the spatial coordinate r. The local Larmor frequency w(r) is given by
w (r) = , Bo + ,G · r . (2.63)
Equation 2.63 is the basic equation for NMR imaging. In the next section we will use this equation to calculate the NMR signal of a sample in a linearly varying magnetic field .