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µSR in MnSi:
A Universal Prototype?
Canadian Inst. for Advanced Research
Dept. of Physics & Astronomy,
Univ. of British Columbia, Vancouver, BC, Canada V6T 1Z1
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Manganese silicide (MnSi) is such an interesting material from the
point of view of magnetism theory that it seems surprising that
it should have played such a central role in so many important
discoveries of new µSR effects and techniques;
but many of our most heavily-utilized methods were first
demonstrated on MnSi. It therefore joins copper, quartz,
silicon and nitrogen as one of the main ``prototype samples''
- Local Field in an Ordered Helimagnet
Below Tc = 29.5K, MnSi orders magnetically
into a rather exotic structure: the magnitude of the local magnetization
is constant but its direction is a function of position:
there is a constant axial component and a transverse component
that spirals about that axis with a long (compared to the lattice spacing)
wavelength. Fortunately, ÁSR can measure the magnitude
of the local field regardless of its direction.
Although one expects a crystallographically unique site for the
µ+ in MnSi, there are evidently two
magnetically distinct versions. The earlier experiment revealed
only the lower-field site's Bhf(T).
- Knight Shift in a Paramagnet
The muon Knight shift is proportional to the local spin density
and therefore varies linearly with the susceptibility in a
``Jaccarino plot'' just as does the 55Mn Knight shift,
for the same reasons. By comparing the slopes of these plots
one can determine the hyperfine couplings of the muon and
the Mn nucleus with the polarized conduction electrons.
Temperature is an implicit parameter in the plots.
- Kubo-Toyabe ``Relaxation'' in Zero Field
Since MnSi is so emphatically magnetic, it might seem the least
likely material in which to observe static nuclear dipolar
relaxation; nevertheless, it was the first substance in which
the now-familiar ``static gaussian Kubo-Toyabe relaxation function''
(predicted by Kubo & Toyabe many years earlier)
was ever observed. This is because the paramagnetic spin density
fluctuates so fast at room temperature that it is effectively
decoupled from the muon and the 55Mn nuclei,
leaving the latter to generate random static dipolar fields
that produce a gaussian distribution at the muon
in zero external field (ZF).
At 61.2K one is able to detect a very slow fluctuation of the
55Mn nuclear spins via its effect on the muon's
ZF Kubo-Toyabe relaxation function Gzz(t):
the ``1/3 tail'' (due to the component of the muon polarization
that is parallel to the local field in a completely random disribution)
slowly relaxes as the local field fluctuates so that it is no longer
parallel to that polarization component.
This gets more interesting later. . . .
- Critical Divergence of T1-1
At lower T the muon is strongly relaxed by the
large fluctuating fields of the paramagnetic conduction electron
Where 55Mn NMR had failed to get within 100K of the
critical temperature, µSR was able to follow the
critical slowing down of spin density fluctuations down to
within half a degree of Tc and confirm
Moriya's self-consistent renormalization (SCR) theory.
This picture shows the same results along with a little more detail
about the SCR predictions.
- Muon-Nuclear Double Relaxation
Recall this picture from above. Now consider what happens
when the paramagnetic spin density fluctuates more slowly
and causes the 55Mn nuclear moments to fluctuate
(relax) faster, while at the same time beginning to have a
direct effect on the muon spins. . . .
The unmistakeable characteristic signature of ``double relaxation''
of this type is when the muon polarization function falls below
the (high temperature) static gaussian ZF Gzz(t)
at early times but rises above same at late times.
This clearly cannot be due to a simple multiplication of
Gzz(t) by an additional exponential
relaxation function representing the direct effect of the
Excellent fits are possible over the entire temperature range
in ZF, using a constant value for the static dipolar width
(due to Mn nuclear moments) and allowing the 55Mn
and µ+ relaxation rates to vary
- Critical Divergence Revisited
However, there is one further effect to account for.
In order to obtain more reliable measurements of
at the higher temperatures, a longitudinal magnetic field was applied
to effectively quench the effect of the 55Mn nuclear moments.
Unfortunately, at certain fields this introduces yet another
relaxation mechanism. . . .
Knowing that the muon relaxation at high T is dominated by
nearly-static 55Mn nuclear dipoles whereas that at low T
is dominated by fluctuating paramagnetic moments, and in between both
must be taken into account, a much better determination of the
critical divergence of T1-1(µ)
was possible. At the same time, by equating the fluctuation rate of the
nuclear dipolar fields in GKT(t) with the
relaxation rate of the 55Mn nuclei, it was possible to
obtain the critical divergence of T1-1(Mn)
more than an order of magnitude closer to Tc
than was possible using NMR.
- Quadrupolar Muon-Nuclear ALCR
Armed with this information, the authors were finally able to extract
reliable values for both T1-1(µ)
and T1-1(Mn) over the entire range of
temperatures and fields studied. See
R. Kadono et al., Phys. Rev. B 48, 16803 (1993).
Avoided Level Crossings in MnSi at several temperatures.
Temperature dependence of the resonant fields.