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NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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ISBN: 3486235966 ISBN: 3486235966 ISBN: 3486235966 ISBN: 3486235966

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Next: The Flow Cryostate Up: NMR Technique Previous: Basic Theory

Signal Detection

In the last section we have seen that it is very interesting to measure the magnetization in the x-y-plane. To do so one mounts the sample in a coil in which the rotating magnetic moment induces an alternant current. But as it is easily shown with a rough estimation the signal one has to expect is too small. For this reason one usually connects the coil to a tank circuit which is tuned to $\omega _{0}$ in order to multiply the voltage by the quality factor Q of the tank circuit. In our measurements we were always using tank circuits as shown in figure with either tuneable capacitors or constant ones.

**Figure:** Tank circuit as we used it in our measurements. The capacitors have to be as close as possible to the sample coil, which may in practice cause some problems.
$\includegraphics[width=8cm]{tank_circuit.eps}$

To avoid reflections of the signal not only the resonance frequency of the tank circuit should be tuned to $\omega _{0}$ but also the impedance should be 50 $\Omega$ because all the electronic devices are built to handle with 50 $\Omega$ . Proper tuning is possible as long as $\omega L / Q < 50\Omega < \omega L Q$ and the properties of the circuit are then described by:

C₁ + C₂	=	$\displaystyle \frac{1}{L \omega^2_0}$	(2.13)
C₁²	=	$\displaystyle \frac{1}{50 Q L \omega^3_0}$	(2.14)
R	=	$\displaystyle \frac{\omega L}{Q}.$	(2.15)

The nuclear signal coming from the tank circuit has to be preamplified before it is fed to the NMR spectrometer. In order to protect the preamplifier from the rf power of the excitation pulses (which are orders of magnitude higher in voltage than the nuclear signal), one usually uses a directional coupler, which separates the incoming high voltage signal from the weak answer of the spin system. But there is still a certain cross talking which has to be taken care of in choosing a preamplifier which has a fast recovery from saturation. Its amplification has to be sufficient to amplify the noise (Nyquist) of the probe head above the noise level of the mixer at the spectrometer input.

After passing the preamplifier the signal is fed into the double conversion spectrometer. It is the heart of an NMR spectrometer. It prepares the rf-pulses to be fed to the power amplifier and it receives and mixes the signal from the preamplifier. But it would go far beyond the frame of this work to go into details here. Only the basic concepts shall be discussed.

Let us assume the case of a single sharp resonance at $\omega _{0}$ . It is one of our goals to determine $\omega _{0}$ . The larmor frequencies are in general too fast do be analyzed directly. Therefore one often uses the technique of mixing the signal with a reference frequency $\omega$ . With

$\begin{displaymath}\mathrm{sin}(\omega_0 t)\mathrm{cos}(\omega t + \phi) = \frac... ...}{2}\mathrm{sin}\left( ( \omega_0 + \omega ) t + \phi)\right), \end{displaymath}$

(2.16)

one can easily see, that we will now have a signal at the sum and at the difference of the frequencies. The part high in frequency can easily be filtered out and only the part low in frequency remains. In doing so one effectively goes to a coordinate system which is rotating with $\omega$ . If we are in resonance it is the same frame as mentioned in the last section. This technique is called lock-in detection. Resulting shapes of free induction decays and echoes are given in figure

. The real part denotes the case $\phi = 90^\circ$ and the imaginary part the case of $\phi = 0^\circ$ . In the rotating frame $\phi$ can be interpreted as the angle under which one is looking at the spin system. The frequency of the oscillations in (e) and (f) is due to the lock-in detection given by the difference $\Delta \omega = \left\vert \omega - \omega_0 \right\vert$ . (Due to the finite pulse length also transitions with $\Delta \omega \neq 0$ may be induced.)

**Figure:** (a) the calculated real part shape of an echo if measuring in resonance; (b) the calculated imaginary part very close to resonance (it would completely disappear if $\omega-\omega_0=0$ ); (c) the real part of a free induction decay (fid) in resonance; (d) the imaginary part of an fid in resonance; (e) the calculated echo in case of broad off-resonant irradiation; (f) a fid under the same conditions.
$\includegraphics[width=12cm]{signal_shape.eps}$

With this knowledge we have a way of finding a resonance frequency. From a given signal we know $\Delta \omega$ by doing a real fourier transform, but we do not know the sign of the deviation. In order to do better one could measure in two orthogonal directions and make a complex fast fourier transform (fft) of the signal. This procedure is known as quadrature detection and is usually implemented by mixing with respect to the phase of the applied pulse. Our double conversion spectrometers offer two pulse input channels x' and y', two output channels x and y and a regulator of the angle $\phi$ between the x-y- and the x'-y'-system. For further details about the fourier transform please consult any text book about NMR. Here we only want to recall the principle that one has to choose the relative phase $\phi$ such that the absorbtion line and the dispersion line are separated as in the first two pictures of figure .

In NMR the signal-to-noise ratio is crucial. It is possible to improve this ratio -- at least as far as the so called white noise is considered -- by averaging many measurements. The gain in the ratio is equal to the square root of the number of averages. This enables us to even measure signals which would otherwise disappear in the noise. But it also brings the disadvantage of having a finite resolution and a finite length in time. The finite length limits the frequency resolution of the fft $\Delta\nu = 1 / \Delta t$ ; the finite resolution limits the range in frequency, which is less of a problem because it is limited by the width of the tank circuit anyway. In our case we used Le Croy transient recorders with a maximum of ten thousand channels.

But the application of quadrature detection enables us to even more sophisticated measurement techniques such as phase cycling. In many cases one has to cope with coherent noise such as the ringing of the resonant system or the free induction decays after the second pulse of an echo sequence. Fortunately this kind of noise is phase-correlated to the irradiation. In changing the phase of the pulses from +x' to -x' also the response changes sign from +x to -x. In other words: If we change the phase of the second pulse in an echo sequence every time we are sending it and average over an even number of sequences the noise completely disappears.

Next: The Flow Cryostate Up: NMR Technique Previous: Basic Theory



Einführung in die Festkörperphysik (Gebundene Ausgabe) von Charles Kittel

Siehe auch:

Teilchen und Kerne. Eine Einführung in die physikalischen Konzepte: Eine Einfuhrung in Die Physikalischen Konzepte (Springer Lehrbuch) von Bogdan Povh

Festkörperphysik. Einführung in die Grundlagen (Springer Lehrbuch) von Harald Ibach

Festkörperphysik von Neil W. Ashcroft

Atom- und Quantenphysik. Einführung in die experimentellen und theoretischen Grundlagen (Springer Lehrbuch) von Hermann Haken

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