NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

Nuclear magnetic resonance (NMR) is a very subtle technique which uses the spin of the nuclei as a probe for their environment. Nuclear magnetic resonance therefore can be used for measuring many microscopic properties of solids.

While nuclear spin resonance takes advantage of the lifted degeneracy of a nuclear spin in an external magnetic field, in nuclear quadrupolar resonance this degeneracy is lifted by the interaction of the quadrupolar momentum of the core with its environment. But the discussion of the contributions to the hamiltonian of a spin in a solid can be found in another chapter of this work. Here we want to restrict our attention to the terms which are of a technical interest.

In nuclear spin resonance one applies a static field in the z-direction and a rf field in the x-y-plane. The terms have the form:

$\displaystyle \mathcal H _{Z}$	=	$\displaystyle - \gamma_\mathrm{N} \hbar \mathrm{\bf I} \mathrm{ \bf B },$	(2.1)
$\displaystyle \mathcal H _{RF}$	=	$\displaystyle - 2 \gamma_\mathrm{N} \hbar B _{1} I_x \mathrm{cos} \omega t,$	(2.2)

For the time being we would like to restrict our attention to the first of these two terms, which are -- apart from the $\mathcal H _{Q}$ -- orders of magnitude larger than the others. Our first goal is now to describe the dynamics of a non-interacting spin system affected only by (

). This has been done by many authors [#!abragam!#,#!slichter!#] and I will only give a rough outline.

As one knows from quantum mechanics the state $\left\vert \Psi \right>$ of a spin in an external magnetic field can be described by a projection operator on an eigenstate of the hamiltonian (

$\begin{displaymath}\mathrm{ \bf P } := \left\vert \Psi \right> \left< \Psi \right\vert. \end{displaymath}$

(2.3)

$\begin{displaymath}\mathrm{ \bf P }(t) = e^{-\frac{i}{\hbar}\int_{0}^{t}\mathcal... ...bf P }(t) e^{\frac{i}{\hbar}\int_{0}^{t}\mathcal H (t')dt'}. \end{displaymath}$

(2.4)

The problem of solving Schrödinger's equation is now reduced to the evaluation of the expression (

). This has been done for example in [#!kind!#]. In considering only the Zeemann interaction one obtains the expectation values

$\displaystyle \left< \mathrm{ \bf I}_x (t) \right>$	=	$\displaystyle \left< \mathrm{ \bf I}_x (0) \right> \mathrm{cos} \omega_0 t + \left< \mathrm{ \bf I}_y (0) \right> \mathrm{sin} \omega_0 t$	(2.5)
$\displaystyle \left< \mathrm{ \bf I}_y (t) \right>$	=	$\displaystyle - \left< \mathrm{ \bf I}_x (0) \right> \mathrm{sin} \omega_0 t + \left< \mathrm{ \bf I}_y (0) \right> \mathrm{cos} \omega_0 t.$	(2.6)

They are rotating around the z-axis with the larmor frequency $\omega_{0} = \gamma_\mathrm{N} \mathrm{\bf B}_{z}$ given by the external static field. Therefore one can take the Zeemann part of the hamiltonian into account by simply changing into a coordinate system which rotates with the same frequency around the z-axis.

It is now easy to generalize to a system of N identical, non-interacting spins. Its density operator takes the form

$\begin{displaymath}\mathrm{ \bf P } = \mathrm{ \bf P }_{1} \otimes \mathrm{ \bf P }_{2} \otimes \ldots \otimes \mathrm{ \bf P }_{N}. \end{displaymath}$

(2.7)

Since also in an ensemble the expectation values of an operator on a single spin depend only on the density operator of this spin

$\begin{displaymath}\left< \mathrm{ \bf A } \right> = \mathrm{ Tr } \left( \mathr... ... P}_{n} \right) = N \cdot \mathrm{Sp}(\mathrm{\bf A} \rho ), \end{displaymath}$

(2.8)

$\begin{displaymath}\rho _{jk} := \frac{1}{N}\sum_{n=1}^{N}P_{n_{jk}}. \end{displaymath}$

(2.9)

With $\rho_{jk}$ we can again solve our equation of motion $i\hbar \frac{d\rho}{dt} = \left[\mathcal H , \rho \right]$ . But we should recall the fact that there are many mechanisms such as local field fluctuations which may slowly destroy the phase coherence of the spin ensemble. The characteristic time scale in which this coherence decays is usually shorter than T₂, the characteristic time of the spin-spin-interactions. Therefore equation (

) only holds for a short time $t \ll T_2$ , when there is no significant loss of phase coherence, or for a long time when we have completely arbitrary phases (random phase approximation, RPA). With the first assumption one replaces $\omega _{0}$ of (

) and (

) by the ensemble average and obtains again the oscillating off-diagonal elements. In the RPA the off-diagonal elements vanish anyway.

We now want to apply the rf field to the spin system which means that we start to take (

) into account. But we don't want to go through all of the formalism, instead we'll just quote the results as they are obtained for example in [#!kind!#]:

In the case of resonant irradiation, $\omega = \omega_0$ , the expectation values of the spin operators behave in such a way that the magnetisation in the rotating coordinate system behaves like

M_z	=	$\displaystyle \left\vert{\bf M}\right\vert \mathrm{cos}\omega_1t$	(2.10)
M_x'	=	0	(2.11)
M_y'	=	$\displaystyle \left\vert{\bf M}\right\vert \mathrm{sin}\omega_1t,$	(2.12)

Contrarily to the slower process of the spin-spin-interactions the decay caused by different larmor frequencies is completely reversible. A very popular pulse sequence which uses this fact is the so called spin echo. It is a sequence of two pulses. It is best understood in a classical picture where we describe the spins as magnetic momenta. The first pulse ( $\frac{\pi}{2}$ -pulse) then brings the magnetization into the x-y-plane. The second pulse ( $\pi$ -pulse) -- which comes after a time called $\tau$ -- reflects all the momenta at the origin; an echo after another interval $\tau$ will be the answer of the spin system to this sequence.

**Figure:** (a) M₀ is in thermal equilibrium lying along the z-direction, (b) magnetization immediately after a $\frac{\pi}{2}$ -pulse, (c) an element of magnetization $\delta M$ has precessed an extra angle owing to the magnetic field inhomogeneity, (d) effect of a $\pi$ -pulse, (e) at time $2\tau$ all elements of the magnetization have refocused along the +y-direction. image source: [#!slichter!#], p. 41
$\includegraphics[width=10cm]{spinecho.eps}$

In a quantum mechanical picture of energy levels one is inducing transitions between different energy levels by irradiating the spin system with the frequency $\nu = \Delta E / h$ . Therefore the echo intensity is proportional to the number of transitions (spins) with this transition frequency. But in pulsed NMR, as we do it in our laboratory, it is, due to the uncertainty principle, no longer possible to irradiate with a well-defined frequency. The resolution is limited by $\Delta \nu \gtrsim 1 / \Delta t$ , where $\Delta t$ is the duration of the longer of the two pulses in an echo sequence for the longer pulse makes the preciser selection in frequency.

With this knowledge it is possible to measure spectra, which may be a very helpful tool to interpret contributions to the total hamiltonian. Unfortunately it can also be quite difficult to separate single effects which form a spectrum. We will come back to this later. At the moment we quickly pay some attention to two more dynamical and easily to measure properties of a spin in a solid.

We already introduced the spin-spin-ralaxation rate T₂. It is the characteristic time during which the magnetization in the x-y-plane gets irreproducibly lost. It's therefore the time constant with which the echo intensity decays as a function of $\tau$ (of $2\tau+$ length of the second pulse). This can very easily be measured.

There is another governing time constant in a spin system. It is called T₁, the spin-lattice-relaxation rate. It is the time during which the spin system relaxes to thermal equilibrium with the lattice after a perturbation. In order to measure T₁ one again takes advantage of the fact that the echo sequence only measures one single level. If this level is empty, there will be no echo. Figure

shows an outline of how a T₁-measurement has do be designed. But we will go into detail later.

**Figure:** typical pulse sequence for a T₁-measurement: first one destroys all the magnetization with the comb sequence; after the recovery time one measures with a normal echo sequence how much of the magnetization is already restored.
$\includegraphics[width=10cm]{comb_sequence.eps}$

In usual solids T₁ is much larger than T₂. That means that the hamiltonian describing the lattice almost commutes with the hamiltonian describing the spin system. It is then often convenient to describe the total system by more than one temperature. The irradiation of a comb sequence is then equivalent to only heating the spin system [#!kind!#].

NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

Basic Theory