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The FµF Spin System
Canadian Inst. for Advanced Research
Dept. of Physics & Astronomy,
Univ. of British Columbia, Vancouver, BC, Canada V6T 1Z1
This page is under construction at
The moral of this story is: muons are never alone.
Their spins affect neighbour spins exactly as much as vice versa,
and no spin that causes the muon spin to move can ever remain static.
Whenever there is a local interaction between a small number of
spins, they evolve together. This seems obvious in retrospect,
but it took us a while to notice. . . .
- Classical Dipolar ``Relaxation'' by RLMF:
In transverse field (TF), as long as the
random local magnetic fields (RLMF)
due to nuclear dipole moments
are small compared to the applied field Ho,
their only effect will be to add or subtract
their components parallel to Ho
so that different muons precess at different frequencies
and get out of phase with each other after a while.
Such ``dephasing'' is technically not true
relaxation, since the muon polarization can be
fully recovered by a suitable 180o pulse
of RF power; hence the terminology ``T2''
as opposed to ``T1'' and all that.
If the RLMF change with time, whether due to nuclear
spin fluctuations unrelated to the muon or to muon diffusion
between sites with different static RLMF, then true relaxation
can take place. Abragam provided a general relaxation function
Gxx(t) from which the muon hop rate
can be extracted given a knowledge of the static width of the
RLMF distribution (assuming the latter has a gaussian shape).
In weak longitudinal field (LF) and (by extension)
zero field (ZF) the situation is more complicated:
all three components of the RLMF contribute to muon
``relaxation'' and the distinction between ``T1''
and ``T2'' becomes blurred. The static dipolar
ZF relaxation function Gzz(t) worked out
many years ago by Kubo and Toyabe can be ``dynamicized''
by several methods including that of Kehr, to produce
dynamic functions that can be fitted to extract muon hop rates
Analogous functions can be derived for lorentzian
distributions of RLMF, which are more appropriate for
dilute systems of strong moments.
However, the lorentzian distribution is fundamentally
non-physical, having an infinite second moment;
in reality there is always a ``cutoff'' (maximum possible)
local field so that fast enough hopping will cause the
relaxation rate to decrease eventually. In a truly
Lorentzian field distribution there is no such
- Quantum Mechanics Comes to
In the early 1980's Peter F. Meier and Moreno Celio showed that
the ``static 1/3 tail'' of Gzz(t)
for completely static muons in copper was in fact not
static but exhibited small oscillations due to the fact that
the muon forms a nearly closed spin system with its 6 nearest neighbour
Cu nuclei and that system evolves quantum mechanically together.
This effect was important because the resultant ``droop'' in the
was being interpreted as evidence for slow muon hopping,
the temperature dependence of which was crucial to the theory of
quantum diffusion to which that work was contributing.
Shortly after that the error of ignoring quantum mechanics was
dramatically illustrated by the first observation of
ZF oscillations in LiF, a manifestation of the
system of 3 spin-1/2 particles formed by the
F- µ+ F-
or just FµF ion
as the muon becomes strongly ``hydrogen bonded''
to two neighbouring fluorine ions.
The time evolution of this spin system under symmetric
dipole-dipole interactions is easily derived and matches
the observed behaviour exactly (after an overall additional
relaxation is imposed due to other nearby spins and/or
other mechanisms. We would never be able to think about
Gzz(t) the same way again!
[Although some still try.]
- FµF - an Ubiquitous Ion:
It was quickly shown that the FµF ion
forms not only in the ``rock salt'' alkali fluorides
but in the ``fluorites'' (alkaline earth fluorides),
in ZnF2, YF3 and in fact
in all the insulating metal fluoride compounds.
Moreover, the bonds between the muon and the fluorines
are always almost exactly the same length, indicating that
the formation of this ion takes energetic precedence
over any considerations of lattice symmetry or structure.
If you are doing an experiment on such a material,
you may rest assured that the muon is in an FµF ion,
whether the magnetic properties of the system allow you to
observe it easily or not.