Static Longitudinal Field Relaxation

Relaxonomy --> here

---

When the muon spin polarization <math>\vec{P}</math> is initially in the same direction as the applied magnetic field <math>\vec{B}</math>, we call that the <math>z</math> direction. This is called the longitudinal field (LF) geometry. The relaxation of <math>\vec{P}</math> is then usually described by

<math> g_{zz}(t) \; \equiv \; \langle P_z(0) \, P_z(t) \rangle </math>

where the lower case <math>g</math> is used (instead of the more general <math>G</math>) to designate a static relaxation function, just like in ZF. (Dynamic cases will be treated later.)

In the limit where <math>B \gg </math> any random local magnetic fields (RLMF), this formulation is valid. (At last, a "low-bogosity" case!) However, in modest applied fields (B <math> \sim </math> RLMF) it is subject to the same caveats as the ZF case, which see.

The "decoupling" effect of LF was observed in the same experiment where ZF "Kubo-Toyabe relaxation" was first observed:

Error creating thumbnail: File missing

<math>g^{\rm GKT}_{zz}(t)</math> in MnSi at 285 K for LF = 0, 10 and 30 Oe.

It is slightly ironic that the ZF/LF relaxation by nuclear dipole moments predicted by Kubo and Toyabe was first observed in MnSi at 285 K, since at lower temperatures MnSi exhibits a rich magnetic behavior due to the strong paramagnetic moments of the Mn ions. At room temperature these huge moments fluctuate fast enough to decouple from both the muon and the Mn nuclei, leaving their spins to influence each other directly.