NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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ISBN: 3110174855 ISBN: 3110174855 ISBN: 3110174855 ISBN: 3110174855

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Next: Results Up: Nuclear Relaxation Previous: Magnetization Recovery in a

Relaxation Mechanisms

Up until now we did not introduce any specific relaxation mechanisms. We only showed that the relaxation is induced by time-dependent fluctuations of the magnetic or the electric field at the place of the nucleus. In this section we are going to give two specific examples of relaxation mechanisms, namely the relaxation in a simple metal and the relaxation caused by paramagnetic impurities. They are both magnetic spin-lattice relaxation mechanisms. But first we have to transform equation () into a more applyable form.

The spin-lattice relaxation rate due to a fluctuating internal magnetic field is (see for example [#!kind!#,#!patrik!#,#!benno!#]):

$\begin{displaymath}\frac{1}{T_1} = \frac{\gamma^2_\mathrm{N}}{2} \int_{-\infty}^{\infty} dt \mathrm{cos} \omega_0 t \left<h_+(-t)h_-(0)\right>, \end{displaymath}$

(3.55)

with $\gamma_\mathrm{N}$ the nuclear gyromagnetic ratio, $h_\pm = h_x \pm i h_y$ and h_x,y the components of the field fluctuation. With the explicit form of the hyperfine interaction and with use of the fluctuation-dissipation theorem this equation can be further transformed.

In a simple metal we finally obtain the Korringa law (see [#!patrik!#]):

$\begin{displaymath}T_1TK^2 = \frac{\hbar}{4 \pi k_\mathrm{B}} \left( \frac{\gamma_e} {\gamma_\mathrm{N}} \right)^2. \end{displaymath}$

(3.56)

The left hand side of this equation is often called the Korringa constant.

In the case where there are magnetic moments in the sample the situation becomes much more complicated. A detailed theoretical analysis for a dilute magnetic alloy is given in [#!giovannini!#]. This model applies in the case where the coupling between the momenta of the impurities is small compared with the coupling to the external field. It has first been testet experimentally by McHenry et al. in La_1-xGd_xAl₂ (see [#!mchenry!#]). After the relaxation due to the paramagnetic impurities has been separated from the Korringa relaxation, they found very good coincidence of the experimental curves with the theoretical prediction.

There are four different mechanisms through which the spin of the impurities (S) can couple with the spin of the nucleus under investigation (I). There are the Benoit, de Gennes and Silhouette mechanism and the Giovannini and Heeger mechanism which both couple via the conduction electrons and rely on the RKKY model. In our case one would expect these mechanisms not be be dominant. On the other hand there is the direct dipolar coupling, wich is called either longitudinalor transversal. At high temperatures the longitudinalcoupling is expected to be dominant (see [#!abragam!#], chapter IX). We only want to quote the general result from [#!abragam!#] here:

$\begin{displaymath}\frac{1}{T_1}=\frac{9}{2} \left( \frac{\gamma_S\gamma_I\hbar\... ...ght)^2 \int_{-\infty}^{\infty}S_z(0) S_z(t)e^{-i\omega_It}dt, \end{displaymath}$

(3.57)

where all the symbols have their usual meaning. With I we denote the spin of the nucleus we are measuring at and with S the spin of the impurity. To evaluate this expression can be quite a messy task. The spins of the impurities obey a Boltzmann distribution. With this assumption one obtains after the evalution of the fourier transform in equation (

$\begin{displaymath}\frac{1}{T_1}=9 \left( \frac{\gamma_S\gamma_I\hbar S\cdot\mat... ... \tau_{S1}^2} \frac{\partial B_S(\alpha)} {\partial \alpha}, \end{displaymath}$

(3.58)

with

$\begin{displaymath}\alpha := \frac{g \mu_\mathrm{B}SH}{k_\mathrm{B} T}, \end{displaymath}$

(3.59)

$B_S(\alpha)$ the Brillouin function, and $\tau_{S1}$ the longitudinalrelaxation time of the impurity spin at the frequency $\omega_I$ . It is a measure for the field fluctuations seen in the I-nucleus. For completeness we quote also the result for transversedipolar coupling:

$\begin{displaymath}\frac{1}{T_1}=\frac{1}{2} \left( \frac{\gamma_S\gamma_I\hbar ... ...left( (1-2\mathrm{cos}^2\theta)+9\mathrm{sin}^4\theta \right). \end{displaymath}$

(3.60)

With $\tau_{S2}$ we denoted the longitudinalrelaxation time of the impurity spin at the frequency $\omega_I$ .

Next: Results Up: Nuclear Relaxation Previous: Magnetization Recovery in a



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