3.1 $\mu$SR vs. NMR

In a transverse field $\mu$SR experiment, one measures the internal magnetic field distribution of a superconductor in the vortex state. Prior to the development of this technique, the internal field distribution could be studied with NMR, which in principle provides the same information as $\mu$SR.

The basic principles of the NMR technique are as follows: The interaction of the magnetic moment of a nucleus ${\bf \mu}_N \! = \! -\gamma_N \hbar {\bf I}$ (where ${\bf I}$ is a nonzero nuclear spin and $\gamma_N$ is the gyromagnetic ratio characteristic of the nucleus) with the local magnetic field ${\bf B}$ at its site is described by the Hamiltonian

$$H = - \gamma_N \hbar {\bf I} \cdot {\bf B} \, ,$$ (1)

which splits the nuclear energy levels into 2I+1 lines with energies

$$E_m = - \gamma_N \hbar B m \, ,$$ (2)

where m is an integer or a half-integer in the range $-I \! < \! m \! < \! I$, depending on whether I is an integer or a half-integer. The energy separation between adjacent nuclear energy levels is then

$$\Delta E_{m, m+1} = E_m - E_{m+1} = \gamma_N \hbar B \equiv \Delta E \, .$$ (3)

Nuclear magnetic dipole transitions may be excited among these levels by applying a RF field of an appropriate frequency. In particular, when the frequency $\omega$ of the RF field is such that the energy $\hbar \omega$ is equal to the energy separation $\Delta E$ between the quantum states of the nuclear spin, there is an absorption of energy. The resulting resonance can be detected and the local field identified as $B \! = \! \omega/ \gamma_N$.Since the distances between similar nuclei in a superconductor are small relative to the separation of the vortices, the magnetic field distribution associated with the vortex lattice is sampled by measuring the fields at the sites of the nuclei. As long as the nuclei are uniformly distributed, the sampling is volume-weighted.

An NMR technique which is a close parallel to $\mu$SR is ``pulsed NMR'', in which one observes time-dependent transverse nuclear polarization or so-called ``free induction decay'' of the nuclear polarization. In this form of NMR, an RF pulse is applied to rotate the nuclear spins $90^{\circ}$ from the direction of the local magnetic field ${\bf B}$. When the RF field is switched off, the nuclear spins perform a free precession around the local field ${\bf B}$ and relax back to their initial direction along ${\bf B}$. The precession is detected by a pickup coil. The frequency of the nuclear spin precession is a measure of the local field (i.e. $B \! = \! \omega/ \gamma_N$). In this pulsed NMR technique, the different precession frequencies are observed simultaneously without variation of the RF or DC magnetic fields [44]. However, there are several limitations and added difficulties associated with the NMR technique which are overcome in a $\mu$SR experiment.

The first problem is that because the skin depth of the RF field probe is small, NMR only probes the surface of the sample. For most high-T_c samples, the surface has many imperfections, which results in strong vortex-line pinning and a highly disordered vortex lattice--a topic which will be discussed in the next section. To measure the field distribution in the bulk, magnetically aligned powders are often used (see for example Ref. [51]). However, the vortex-lattice structure in the bulk of a small crystallite is also likely to be affected by the strong pinning of the vortices at the crystallite surface. Furthermore, it is difficult to align all of the crystallites with the applied field. Due to mass anisotropy, the field distribution will not be the same in crystallites with different orientations. To account for the misaligned fraction, one must measure the NMR spectrum before and after alignment, and then subtract out the random powder contribution by using a predetermined percentage of alignment. The penetration depth of the RF field also limits the range over which the vortex lattice can be sampled.

On the other hand, $\mu$SR is a bulk probe that is easily performed on single crystals, so that one can measure the ``true'' magnetic field distribution in the bulk. The importance of having this capability will become clear when the structure of the vortex-lattice is discussed in some detail later in this thesis.

Other problems with NMR are the additional sources of line broadening in the measured internal field distribution (i.e. line broadening sources which are negligible in $\mu$SR). For instance, in the high-T_c materials the linewidths originating from the copper and oxygen nuclei are very broad due to quadrupole interactions and chemical shifts (or metallic Knight shifts). A field distribution which corresponds more closely to that of the vortex lattice is obtained with the $\mu$SR technique. Since the muon is a spin 1/2 particle, it has no quadrupole interaction. Also, $\mu$SR is sensitive over relaxation times as small as 10 ns compared to $10~\mu$s for NMR. The signal relaxation in $\mu$SR due to the inhomogeneous field distribution of the vortex lattice typically occurs over the first few $\mu$s.