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\begin_body

\begin_layout Chapter
Low-moment magnetism in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\emph on
M
\emph default
OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 (
\emph on
M
\emph default

\begin_inset space ~
\end_inset

= Li, Na)
\begin_inset CommandInset label
LatexCommand label
name "chap:Ba2MOsO6"

\end_inset


\end_layout

\begin_layout Standard
Metal oxides containing 5d transition metal ions provide a wealth of novel
 magnetic behaviour in which orbital, charge and spin degrees of freedom
 play a rôle.
 The physics of the 5d transition metals differs from their more familiar
 3d counterparts in a number of ways.
 Firstly, the 5d orbitals are significantly larger in spatial extent, allowing
 them to behave more like free electrons in many systems and reducing the
 effect of electron correlations.
 Moreover, spin--orbit coupling in 5d ions is larger because of their higher
 atomic number (with atomic number 
\begin_inset Formula $Z$
\end_inset

, spin--orbit interaction scales as 
\begin_inset Formula $Z^{4}$
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "blundell-magnetism-in-condensed-matter-book"

\end_inset

).
 In 3d transition metal compounds, the spin--orbit coupling is typically
 rather smaller than the crystal-field splitting, but the larger spin--orbit
 interaction in 5d oxides contributes to a substantially different balance
 between energy scales.
\end_layout

\begin_layout Standard
In this context, osmium compounds provide a number of interesting examples.
 Osmium has oxidation states ranging from Os
\begin_inset script superscript

\begin_layout Plain Layout
4+
\end_layout

\end_inset

 (5d
\begin_inset script superscript

\begin_layout Plain Layout
4
\end_layout

\end_inset

) to Os
\begin_inset script superscript

\begin_layout Plain Layout
7+
\end_layout

\end_inset

 (5d
\begin_inset script superscript

\begin_layout Plain Layout
1
\end_layout

\end_inset

).
 The behaviour of compounds such as OsO
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

, SrOsO
\begin_inset script subscript

\begin_layout Plain Layout
3
\end_layout

\end_inset

 and BaOsO
\begin_inset script subscript

\begin_layout Plain Layout
3
\end_layout

\end_inset

 (all containing Os
\begin_inset script superscript

\begin_layout Plain Layout
4+
\end_layout

\end_inset

) is dominated by the large spatial extent of the orbitals; the electrons
 are delocalised, giving rise to Pauli paramagnetism
\begin_inset CommandInset citation
LatexCommand citep
key "Greedan1968-OsO2"

\end_inset

.
 On the other hand, the much smaller Os
\begin_inset script superscript

\begin_layout Plain Layout
7+
\end_layout

\end_inset

 ion gives rise to rather different behaviour: double perovskites of the
 form Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(Na,
\begin_inset space ~
\end_inset

Li)OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 [containing Os
\begin_inset script superscript

\begin_layout Plain Layout
7+
\end_layout

\end_inset

 ions arranged on a face-centred cubic (fcc) lattice] exhibit Mott-insulating,
 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 local-moment behaviour
\begin_inset CommandInset citation
LatexCommand citep
key "Stitzer2002-Ba2NaOsO6-growth,Stitzer2003-Os-triple-perovskites"

\end_inset

.
 The insulator barium sodium osmate (Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

) has drawn particular attention
\begin_inset CommandInset citation
LatexCommand citep
key "Erickson2007-Ba2NaOsO6-PRL"

\end_inset

 due to its seemingly contradictory combination of negative Weiss temperature
 (
\begin_inset Formula $\approx\SI{-10}{\kelvin}$
\end_inset

) and yet weak ferromagnetic (FM) moment (
\begin_inset Formula $\approx0.2\mu_{\mathrm{B}}$
\end_inset

/formula
\begin_inset space ~
\end_inset

unit) below 
\begin_inset Formula $T_{\mathrm{c}}\approx\SI{7}{\kelvin}$
\end_inset

.
 This contrasts with the isostructural Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, which has both a negative Weiss temperature and no ferromagnetic moment
\begin_inset CommandInset citation
LatexCommand citep
key "Stitzer2002-Ba2NaOsO6-growth,Sleight1962-Os-Rh-perovskites"

\end_inset

.
 The apparent contradiction in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 raises the possibility that it is a canted antiferromagnet, although first-prin
ciples density functional theory electronic structure calculations indicate
 that a ferromagnetic state has lower energy
\begin_inset CommandInset citation
LatexCommand citep
key "Xiang2007-Ba2NaOsO6-PRB"

\end_inset

.
 The large hybridisation between the osmium 5d and oxygen 2p orbitals leads
 to a very large crystal-field splitting (several eV) between the 
\begin_inset Formula $e_{g}$
\end_inset

 and 
\begin_inset Formula $t_{2g}$
\end_inset

 bands.
 It has been suggested that the low moment arises from a partial cancellation
 of orbital and spin angular momenta in the occupied spin-up 
\begin_inset Formula $t_{2g}$
\end_inset

 band
\begin_inset CommandInset citation
LatexCommand citep
key "Xiang2007-Ba2NaOsO6-PRB"

\end_inset

.
 An alternative mechanism
\begin_inset CommandInset citation
LatexCommand citep
key "Lee2007-Ba2NaOsO6-EPL"

\end_inset

 involves the frustration of both the antiferromagnetic (AFM) interactions
 and the orbital ordering by the fcc lattice, resulting in a delicate balance
 of interactions which favours ferromagnetism for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, but antiferromagnetism for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
 The high-temperature paramagnetic moment 
\begin_inset Formula $\mu_{\mathrm{eff}}\approx0.6\mu_{\mathrm{B}}$
\end_inset

 in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is significantly lower than the 
\begin_inset Formula $1\mu_{\mathrm{B}}$
\end_inset

 which would be expected for a free electron, and is thus indicative of
 substantial spin--orbit coupling
\begin_inset CommandInset citation
LatexCommand citep
key "Erickson2007-Ba2NaOsO6-PRL"

\end_inset

.
 The magnetic ground state of Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\emph on
M
\emph default
OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is thus a matter of significant speculation.
\end_layout

\begin_layout Standard
In view of these competing explanations, further experimental data which
 can distinguish between different magnetic configurations are desirable.
 An obvious solution to this would be to solve the magnetic structures by
 neutron diffraction, but these materials are not amenable to this for several
 reasons: firstly, osmium is a relatively strong absorber of neutrons; secondly,
 there is a significant probability that the 
\emph on
M
\emph default

\begin_inset space ~
\end_inset

=
\begin_inset space ~
\end_inset

Na compound is a ferromagnet, meaning that the magnetic peaks and structural
 Bragg peaks would coincide; and thirdly, the ordered moment is likely to
 be low, as we shall see, which would make detecting magnetic order difficult,
 especially if it is indeed a ferromagnet.
 Muon-spin relaxation provides a local probe of magnetism in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\emph on
M
\emph default
OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, and the novel Bayesian technique explained in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:dipole-bayes"

\end_inset

 allows bounds to be extracted on moment size and configuration.
 The muon data, in combination with measured magnetisation, are consistent
 with the development of long-range FM order with a reduced moment in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, and a reduced moment is also likely for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
The work comprising this chapter has been submitted for publication, and
 is available as a preprint as indicated in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Steele2011-Ba2MOsO6"

\end_inset

.
\end_layout

\begin_layout Section
Structure and magnetisation
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/structure/Ba2NaOsO6-structure.eps
	width 8cm

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Double-perovskite crystal structure of Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, comprising alternating OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 octahedra (top left and bottom right) and NaO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 octahedra, with barium ions occupying the remaining space.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-structure"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The double-perovskite structure of Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-structure"

\end_inset

; note that sodium and osmium ions inhabit alternate oxygen octahedra.
 The structure is described by the space group 
\begin_inset Formula $\text{Fm\bar{3}m}$
\end_inset

, with cubic lattice constant 
\begin_inset Formula $a=\SI{0.82870(3)}{\nm}$
\end_inset

.
 Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is isostructural, slight differences arising from the fact that Li
\begin_inset script superscript

\begin_layout Plain Layout
+
\end_layout

\end_inset

 is smaller than Na
\begin_inset script superscript

\begin_layout Plain Layout
+
\end_layout

\end_inset

, and thus the size difference between the Li and Os octahedra is somewhat
 less pronounced.
 The lattice constant is reduced to 
\begin_inset Formula $a=\SI{0.81046(2)}{\nm}$
\end_inset

.
 Full details of both structures can be found in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Stitzer2002-Ba2NaOsO6-growth"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/magnetisation/Ba2MOsO6-SQUID.eps

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Magnetisation measured at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{2}{
\backslash
kelvin}
\end_layout

\end_inset

 in a SQUID magnetometer for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\emph on
M
\emph default
OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
 Error bars are shown, but in most cases are smaller than the marker size.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2MOsO6-SQUID"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Magnetisation measurements using a SQUID magnetometer are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2MOsO6-SQUID"

\end_inset

, and are consistent with earlier work
\begin_inset CommandInset citation
LatexCommand citep
key "Stitzer2002-Ba2NaOsO6-growth,Erickson2007-Ba2NaOsO6-PRL"

\end_inset

.
 Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 exhibits a magnetisation curve typical of a ferromagnet with a small amount
 of hysteresis, approaching a saturation value of 
\begin_inset Formula $\approx0.2\mu_{\mathrm{B}}$
\end_inset

/formula
\begin_inset space ~
\end_inset

unit, but still displaying a slight upward trend in magnetisation even at
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{7}{
\backslash
tesla}
\end_layout

\end_inset

.
 This does not rule out canted antiferromagnetism, as the initial part of
 the curve could represent the alignment of net FM domains whilst the slow,
 near-linear increase in magnetisation represents a slow canting of spins
 in those domains as field is increased.
\end_layout

\begin_layout Standard
Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 initially displays the linear magnetisation typical of a slowly-canting
 antiferromagnet, but a spin-flop transition at around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{5.5}{
\backslash
tesla}
\end_layout

\end_inset

 with obvious hysteresis is also evident.
 Pulsed field magnetometry on Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 suggests that this compound does not reach a saturated magnetisation value
 even at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{45}{
\backslash
tesla}
\end_layout

\end_inset

, and also reveals no additional transitions up to this high field (see
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2LiOsO6-pulsed-B"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/magnetisation/Ba2LiOsO6-pulsed-B.eps
	lyxscale 200

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Magnetisation measured at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{4}{
\backslash
kelvin}
\end_layout

\end_inset

 in pulsed field for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, calibrated against the SQUID magnetisation data shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2MOsO6-SQUID"

\end_inset

.
 Data from a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{45}{
\backslash
tesla}
\end_layout

\end_inset

 shot (dashed line) are noisier than those from a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{25}{
\backslash
tesla}
\end_layout

\end_inset

 shot (solid line) but show no evidence for additional magnetic transitions.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2LiOsO6-pulsed-B"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Experimental methods
\end_layout

\begin_layout Standard
Crystallites of Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 and Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 were grown using a flux method
\begin_inset CommandInset citation
LatexCommand citep
key "Stitzer2002-Ba2NaOsO6-growth"

\end_inset

.
 Powders of Os (99.8%), Ba(OH)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\begin_inset Formula $\cdot$
\end_inset

8H
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O (98%) and high-purity NaOH
\begin_inset Formula $\cdot$
\end_inset

H
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O (99.996%) or LiOH (99.995%) and KOH (99.99%) were mixed in the ratio 1:2.1:300
 or 1:2.1:140:75 respectively.
 These mixtures were each placed in an alumina crucible inside a thick quartz
 tube which was inserted into a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{600}{
\backslash
celsius}
\end_layout

\end_inset

 pre-heated tube furnace where it was held for 3 days.
 The furnace was then rapidly cooled to room temperature and small single
 crystals were harvested from each crucible.
\begin_inset Foot
status open

\begin_layout Plain Layout
Samples studied in this chapter were synthesised by D.
 Prabhakaran.
\begin_inset CommandInset label
LatexCommand label
name "fn:footnote-prabhak"

\end_inset


\end_layout

\end_inset

 The muon experiments used a very large number of these crystals without
 alignment.
 The measurements were performed using the GPS instrument at PSI.
 Samples were wrapped in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{25}{
\backslash
um}
\end_layout

\end_inset

 Ag foil and mounted on a flypast holder comprising two silver prongs.
\end_layout

\begin_layout Section
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 results
\end_layout

\begin_layout Subsection
Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sub:Ba2NaOsO6-muon"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/musr/Ba2NaOsO6-At.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Muon data above (
\begin_inset Formula $T=\SI{8}{\kelvin}$
\end_inset

) and below (
\begin_inset Formula $T=\SI{3}{\kelvin}$
\end_inset

) the magnetic transition at 
\begin_inset Formula $T_{\mathrm{c}}=\SI{6.8}{\kelvin}$
\end_inset

 for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, along with fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Ba2NaOsO6-At"

\end_inset

 for the data below the transition, and a Gaussian relaxation for data above.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-At"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Typical zero-field 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

spectra measured above and below the magnetic transition for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-At"

\end_inset

.
 Two damped oscillations are visible at low temperature, demonstrating the
 existence of a transition to a state of long-range magnetic order.
 Data below the transition were fitted to 
\begin_inset Formula 
\begin{equation}
A(t)=\left[\sum_{i=1}^{2}A_{i}e^{-\lambda_{i}t}\cos\left(2\pi\nu_{i}t\right)+A_{3}e^{-\lambda_{3}t}\right]+A_{\mathrm{bg}}e^{-\lambda_{\mathrm{bg}}t},\label{eq:Ba2NaOsO6-At}
\end{equation}

\end_inset

representing two oscillatory components, labelled 1 and 2; a fast-relaxing
 term, labelled 3, to account for the rapid initial depolarisation; and
 a slow-relaxing term, labelled bg, which accounts for the background arising
 from muons which stop in the silver sample holder or cryostat tails and
 the 
\begin_inset Formula $\frac{1}{3}$
\end_inset

 of muons which stop inside the sample with spins parallel to the local
 magnetic field.
 The frequencies of the two oscillatory components were initially left free
 but, after observing that they varied in proportion with one-another, they
 were held in fixed proportion 
\begin_inset Formula $P_{2}=\nu_{2}/\nu_{1}$
\end_inset

 during fitting.
 The existence of two oscillatory components most likely indicates two crystallo
graphically-similar muon sites, perhaps one nearer to the magnetic Os ion
 and the other nearer to the non-magnetic Na ion.
 The ratio of the probabilities of stopping in the two oscillating states
 
\begin_inset Formula $A_{1}/A_{2}=0.19\pm0.02$
\end_inset

, and 
\begin_inset Formula $A_{3}/A_{2}=0.92\pm0.01$
\end_inset

.
 Further, 
\begin_inset Formula $\nu_{1}(T\to0)=\SI{3.9(1)}{\MHz}$
\end_inset

 and 
\begin_inset Formula $P_{2}=0.40\pm0.05$
\end_inset

.
 The relaxation rates 
\begin_inset Formula $\lambda_{1}\approx\SI{4}{\MHz}$
\end_inset

 and 
\begin_inset Formula $\lambda_{2}\approx\SI{2}{\MHz}$
\end_inset

 do not vary significantly as a function of temperature, whilst 
\begin_inset Formula $\lambda_{3}$
\end_inset

 decreases from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.6(1)}{
\backslash
MHz}
\end_layout

\end_inset

 at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.5}{
\backslash
kelvin}
\end_layout

\end_inset

 to zero at the transition temperature 
\begin_inset Formula $T_{\mathrm{c}}$
\end_inset

.
 The background term varies somewhat between experimental runs, but its
 amplitude and relaxation rate remain approximately constant, with 
\begin_inset Formula $A_{\mathrm{bg}}/A_{2}\approx0.75$
\end_inset

 and 
\begin_inset Formula $\lambda_{\mathrm{bg}}\approx\SI{0.03}{\MHz}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/nuT/Ba2NaOsO6-nuT.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
The frequency 
\begin_inset Formula $\nu(T)$
\end_inset

 extracted from fits to 
\begin_inset Formula $A(t)$
\end_inset

 spectra as a function of sample temperature 
\begin_inset Formula $T$
\end_inset

, along with fits to various models.
 Data are only shown for the larger frequency 
\begin_inset Formula $\nu_{1}$
\end_inset

 because 
\begin_inset Formula $\nu_{2}$
\end_inset

 was held in fixed proportion during fitting.
 The mean-field fit follows Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Whitaker1989-mean-field-magnetisation"

\end_inset

 and the Landau-type parameterisation of the FM equation of state is due
 to Kuz'min 
\begin_inset CommandInset citation
LatexCommand citep
key "Kuzmin2008-ferromagnet"

\end_inset

.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-nuT"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Since the muon-precession frequency is proportional to the local magnetic
 field, which is itself proportional to sample magnetisation, the precession
 frequency 
\begin_inset Formula $\nu(T)$
\end_inset

 acts as an effective order parameter for the system.
 The temperature evolution of this order parameter thus allows some insight
 into the nature of the magnetically-ordered state.
 This evolution is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-nuT"

\end_inset

, along with fits to various different parametrisations.
 The temperature dependence of 
\begin_inset Formula $\nu(T)$
\end_inset

 fitted rather poorly to an analytic expression for a mean-field 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 magnet
\begin_inset CommandInset citation
LatexCommand citep
key "Whitaker1989-mean-field-magnetisation"

\end_inset

,
\begin_inset Formula 
\begin{equation}
\frac{M}{M_{\mathrm{S}}}=\frac{\mathrm{e}^{g_{J}\mu_{\mathrm{B}}J\lambda M/k_{\mathrm{B}}T}-\frac{1}{2}}{\mathrm{e}^{g_{J}\mu_{\mathrm{B}}J\lambda M/k_{\mathrm{B}}T}+\frac{1}{2}}
\end{equation}

\end_inset

where 
\begin_inset Formula $\frac{M}{M_{\mathrm{S}}}$
\end_inset

 is the magnetisation as a fraction of the saturation magnetisation; 
\begin_inset Formula $J$
\end_inset

 and 
\begin_inset Formula $g_{J}$
\end_inset

 are the angular momentum and its associated 
\begin_inset Formula $g$
\end_inset

-factor, respectively; and 
\begin_inset Formula $\lambda M$
\end_inset

 is the mean field, proportional to the magnetisation.
 This might be expected since assuming that all moments are bathed in a
 uniform mean field is usually inappropriate in the case of magnetism driven
 by short-range exchange interactions.
 In particular, the estimated transition temperature 
\begin_inset Formula $T_{\mathrm{c}}=\SI{7.8(2)}{\kelvin}$
\end_inset

 is significantly higher than both the other fitting functions used (as
 is clear from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-nuT"

\end_inset

) and than the previous estimate of the transition temperature, 
\begin_inset Formula $T_{\mathrm{c}}=\SI{6.8(3)}{\kelvin}$
\end_inset

, derived from heat capacity.
 Fitting extracted frequencies as a function of temperature to the phenomenologi
cal function 
\begin_inset Formula 
\begin{equation}
\nu(T)=\nu(0)\left[1-\left(\frac{T}{T_{\mathrm{N}}}\right)^{\alpha}\right]^{\beta}\label{eq:nuT1-T-over-TN-alpha-beta}
\end{equation}

\end_inset

allows an estimate of the critical temperature and the exponent 
\begin_inset Formula $\beta$
\end_inset

 to be extracted.
 Leaving all parameters unconstrained does not result in a stable fit.
 However, fixing 
\begin_inset Formula $\beta=0.367$
\end_inset

, corresponding to the 3D Heisenberg model, yields 
\begin_inset Formula $T_{\mathrm{c}}=\SI{7.2(2)}{\kelvin}$
\end_inset

.
 A similarly good fit can be obtained using a Landau-type parametrisation
 of the equation of state of a ferromagnet (following Eq.
\begin_inset space ~
\end_inset

(18) of Kuz'min
\begin_inset CommandInset citation
LatexCommand citep
key "Kuzmin2008-ferromagnet"

\end_inset

):
\begin_inset Formula 
\begin{equation}
\frac{M^{2}}{M_{\mathrm{S}}^{2}}=\frac{\left[\kappa^{2}+4\left(1-\kappa\right)\frac{1-\left(T/T_{\mathrm{C}}\right)^{3}}{1+p\left(T/T_{\mathrm{C}}\right)^{3/2}}\right]^{\frac{1}{2}}-\kappa}{2\left(1-\kappa\right)}^{\frac{1}{2}},
\end{equation}

\end_inset

where 
\begin_inset Formula $T$
\end_inset

 is temperature and 
\begin_inset Formula $T_{\mathrm{C}}$
\end_inset

 is Curie temperature; and adjustable parameters 
\begin_inset Formula $p=0.25$
\end_inset

 and 
\begin_inset Formula $\kappa=0.18$
\end_inset

 were used, appropriate for Fe (a metallic ferromagnet).
 This yields the same estimate of the transition temperature as the phenomenolog
ical 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

.
 The agreement with these latter two models shows that the behaviour is
 consistent with three-dimensional FM order.
\end_layout

\begin_layout Subsection
Extracting magnetic moment with Bayesian analysis
\begin_inset CommandInset label
LatexCommand label
name "sub:Ba2NaOsO6-Bayes"

\end_inset


\end_layout

\begin_layout Standard
Dipole-field simulations were performed for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 using a variety of magnetic models, both FM and AFM (ignoring any canting)
 with moments orientated along 
\begin_inset Formula $[0\,0\,1]$
\end_inset

, 
\begin_inset Formula $[0\,1\,1]$
\end_inset

 and 
\begin_inset Formula $[1\,1\,1]$
\end_inset

.
 In the AFM cases, no propagation vector is used but instead different Os
 moments within the unit cell are orientated in opposite directions: those
 with a fractional coordinate 
\begin_inset Formula $x=0$
\end_inset

 are antiparallel to those with 
\begin_inset Formula $x=0.5$
\end_inset

, giving rise to 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 FM sheets.
 The exact magnetic structure is unknown, but varying the nature of the
 AFM order does not make a large difference to results, and hence an exhaustive
 search appears not to be necessary.
 Since the muon site is unknown, we adopt a probabalistic approach, and
 attempt to identify likely constraints on muon position.
 The metal ions in this system are all positively charged, meaning that
 the muon is unlikely to stop near them.
 As discussed in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:dipole"

\end_inset

, muons have been shown to stop around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{0.1}{
\backslash
nm}
\end_layout

\end_inset

 from an O
\begin_inset script superscript

\begin_layout Plain Layout
2
\begin_inset Formula $-$
\end_inset


\end_layout

\end_inset

 ion in other oxides
\begin_inset CommandInset citation
LatexCommand citep
key "Brewer1991-muon-1A-from-O,Li1991-muon-in-YBCO"

\end_inset

.
 Thus, in these calculations, positions in the unit cell were generated
 at random and dipole fields calculated at sites which were both approximately
 this distance from an oxygen ion (
\begin_inset Formula $\num{0.09}\leq r_{\mu\text{--}\mathrm{O}}\leq\SI{0.11}{\nm}$
\end_inset

) and not too close to a positive ion (
\begin_inset Formula $r_{\mu\text{--}+}\geq\SI{0.1}{\nm}$
\end_inset

).
 The magnitudes of the resulting fields were then converted into muon precession
 frequencies, and the resulting histogram yields the probability density
 function (pdf) 
\begin_inset Formula $f(\nu/\mu)$
\end_inset

, evaluated as a function of precession frequency 
\begin_inset Formula $\nu$
\end_inset

 divided by osmium moment 
\begin_inset Formula $\mu$
\end_inset

 (since the precession frequency scales with the osmium moment).
 A subset of the results is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes"

\end_inset

: the FM and AFM 
\begin_inset Formula $[1\,1\,1]$
\end_inset

 configurations were identified as most likely using DFT calculations; whilst
 the AFM 
\begin_inset Formula $[0\,0\,1]$
\end_inset

 shows a case with a pdf which differs markedly from those, and shows the
 extent of variation between these different spin configurations---and that
 this variation is quite small.
 The high-field tails, representing the muon sites approaching 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{0.1}{
\backslash
nm}
\end_layout

\end_inset

 from an osmium ion, extend to around 
\begin_inset Formula $\SI{200}{\MHz}/\mu_{\mathrm{B}}$
\end_inset

.
\end_layout

\begin_layout Standard
This analysis ignores contributions from the Lorentz field and demagnetising
 field, though these are likely to cancel each-other to some extent.
 Further, we can estimate an approximate upper bound on their magnitude
 by evaluating 
\begin_inset Formula $\mu_{0}M$
\end_inset

, giving 
\begin_inset Formula $\frac{\gamma_{\murm}}{2\uppi}\left|\bm{B}_{\mathrm{L}}+\bm{B}_{\mathrm{demag}}\right|\lesssim\SI{2}{\MHz}$
\end_inset

.
 The importance of the contact hyperfine field, 
\begin_inset Formula $\bm{B}_{\mathrm{hyperfine}}$
\end_inset

, is extremely difficult to evaluate (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:hyperfine-fields"

\end_inset

) and it is neglected.
 However, the Os
\begin_inset script superscript

\begin_layout Plain Layout
7+
\end_layout

\end_inset

 ion is both small and positively charged, meaning that not only is the
 spatial extent of its spin-polarised electron cloud small, but the muon
 is likely to stop far from it.
 Thus, this contribution should not be significant, and would be even further
 reduced if its magnetic moment is low.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/dipole/Ba2MOsO6-nu-distributions.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions of muon precession frequency 
\begin_inset Formula $\nu$
\end_inset

 per unit magnetic moment 
\begin_inset Formula $\mu$
\end_inset

 under various assumed magnetic structures.
 Results are shown for three different magnetic structures: the direction
 associated with each line indicates the moment direction; AFM configurations
 possess FM planes along the 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 direction, as described in the text.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-Bayes"

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/bayes-fields/Ba2NaOsO6-bayes-pdfs.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Schematic representation of the convolution-like procedure of 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:bayes-multiple-frequencies"

\end_inset

.
 The column of graphs on the left shows the pdf of muon precession frequencies
 in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 given FM
\begin_inset space ~
\end_inset


\begin_inset Formula $[1\,1\,1]$
\end_inset

 structure assuming various different values of the osmium moment 
\begin_inset Formula $\mu$
\end_inset

.
 Light grey shaded regions correspond to the observed frequencies together
 with their uncertainties, and the green shading then shows the integrated
 probability of this frequency being observed.
 The graph on the right, rotated by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{90}{
\backslash
degree}
\end_layout

\end_inset

, then shows the pdf of moment 
\begin_inset Formula $\mu$
\end_inset

 along the (vertical) 
\begin_inset Formula $x$
\end_inset

-axis.
 The green impulses then represent the two green areas on the corresponding
 left graph multiplied together.
 If the osmium has no moment, then muon precession would not be observed.
 The broad maximum at 
\begin_inset Formula $\mu\approx0.2\mu_{\mathrm{B}}$
\end_inset

 corresponds to the slow increase in probability of the lower frequency
 in tandem with the decrease in probability of the other.
 By 
\begin_inset Formula $\mu=0.4\mu_{\mathrm{B}}$
\end_inset

, it is no longer possible to observe the lower frequency and thus the probabili
ty of this moment is zero.
 At 
\begin_inset Formula $\mu=\mu_{\mathrm{B}}$
\end_inset

, neither frequency is present at the muon sites examined.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-Bayes-howto"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
We then follow the Bayesian scheme outlined in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:dipole-bayes"

\end_inset

, modified slightly due to the presence of multiple frequencies in the spectra.
 When multiple frequencies 
\begin_inset Formula $\nu_{i}$
\end_inset

 are observed, it is necessary to multiply their probabilities of observation
 in order to obtain the chance of their simultaneous observation, and so
 we evaluate the pdf of the magnetic moment 
\begin_inset Formula $\mu$
\end_inset

 given the set of observed frequencies 
\begin_inset Formula $\{\nu_{i}\}$
\end_inset

,
\begin_inset Formula 
\begin{equation}
g(\mu\vert\{\nu_{i}\})\propto\prod_{i}\int_{\nu_{i}-\Delta\nu_{i}}^{\nu_{i}+\Delta\nu_{i}}f(\nu_{i}/\mu)\,\mathrm{d}\nu_{i}\,,\label{eq:bayes-multiple-frequencies}
\end{equation}

\end_inset

 where 
\begin_inset Formula $\Delta\nu_{i}$
\end_inset

 is the error on the fitted frequency, and 
\begin_inset Formula $f(\nu/\mu)$
\end_inset

 is the pdf of frequency per unit moment as described previously.
 The effect of this operation is somewhat like a convolution, and is shown
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-howto"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/dipole/Ba2NaOsO6-bayes-pdfs.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions of osmium moment 
\begin_inset Formula $\mu$
\end_inset

 in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 given the set of frequencies observed in muon experiments, 
\begin_inset Formula $\{\nu_{i}\}$
\end_inset

.
 Results are shown for three different magnetic structures: the direction
 associated with each line indicates the moment direction; AFM configurations
 possess FM planes along the 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 direction, as described in the text.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-mu-Bayes"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The results of this are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-mu-Bayes"

\end_inset

, and show that for all collinear magnetic structures considered a low moment
 is overwhelmingly likely.
 Furthermore, for the FM case the most probable 
\begin_inset Formula $\mu\approx0.2\mu_{\mathrm{B}}$
\end_inset

, consistent with the magnetisation measurements.
 The lowest possible moment is set by the largest value of 
\begin_inset Formula $\nu/\mu$
\end_inset

 in the high-field tail, and is very close to zero.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/dipole-canting/Ba2NaOsO6-cant-process.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
(a) Schematic representation of two initially-AFM osmium moments being counter-r
otated by an angle 
\begin_inset Formula $\alpha$
\end_inset

 from an initial direction 
\begin_inset Formula $\hat{\bm{m}}$
\end_inset

 about a unit normal 
\begin_inset Formula $\hat{\bm{n}}$
\end_inset

, maintaining a projection 
\begin_inset Formula $0.2\mu_{\mathrm{B}}$
\end_inset

 along the direction of magnetisation within the sample.
 (b) Probability density function of osmium moment 
\begin_inset Formula $\mu$
\end_inset

 as a function of canting angle 
\begin_inset Formula $\alpha$
\end_inset

 in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 as moments are canted from FM order along the 
\begin_inset Formula $[0\,0\,1]$
\end_inset

 direction to AFM order along 
\begin_inset Formula $[0\,1\,0]$
\end_inset

, rotating about 
\begin_inset Formula $[1\,0\,0]$
\end_inset

.
 Density of black denotes the pdf of osmium moment size: vertical cross-sections
 are therefore pdfs displayed like those in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes"

\end_inset

.
 A vertical cross-section at 
\begin_inset Formula $\alpha=\SI{90}{\degree}$
\end_inset

 corresponds to the solid red line in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes"

\end_inset

.
 The similar vertical section at 
\begin_inset Formula $\alpha=\SI{0}{\degree}$
\end_inset

 does not correspond to the dashed, light blue line in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes"

\end_inset

 because the AFM moments are ordered in 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 FM sheets, which breaks the symmetry between 
\begin_inset Formula $\bm{b}$
\end_inset

- and 
\begin_inset Formula $\bm{c}$
\end_inset

-orientated moments.
 The solid and long-dashed red lines indicate the median value of moment
 for a given angle, and the interquartile range, respectively.
 The short-dashed blue lines show the canted moment 
\begin_inset Formula $\mu=(0.2\pm0.02)\mu_{\mathrm{B}}/\sin\alpha$
\end_inset

 required to obtain the desired magnetisation of 
\begin_inset Formula $\approx0.2\mu_{\mathrm{B}}$
\end_inset

, and thus the probability density between them gives rise to a pdf of canting
 angle as shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-cant2"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-Bayes-cant"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/dipole-canting/Ba2NaOsO6-cant.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions of osmium moment 
\begin_inset Formula $\mu$
\end_inset

 in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 given the set of frequencies observed in muon experiments, 
\begin_inset Formula $\{\nu_{i}\}$
\end_inset

.
 Results are shown for three different sets of magnetic structures: AFM
 moments initially pointing antiparallel along the direction 
\begin_inset Formula $\hat{\bm{m}}$
\end_inset

 are counter-rotated as shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-cant"

\end_inset


\begin_inset space ~
\end_inset

(a) around a unit vector 
\begin_inset Formula $\hat{\bm{n}}$
\end_inset

.
 AFM configurations possess FM planes along the 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 direction, as described in the text.
 The moment size is scaled with their rotation to ensure that the net moment
 is 
\begin_inset Formula $0.2\mu_{\mathrm{B}}$
\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2NaOsO6-Bayes-cant2"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Though this lends plausibility to the suggested low moment on the osmium
 site, it does not explicitly test for the alternative, i.e.
 a canted antiferromagnetic structure with a net moment.
 Thus, an alternative method of testing the hypothesis was attempted, considerin
g possible canted structures which would give rise to the 
\begin_inset Formula $\mu=0.2\mu_{\mathrm{B}}$
\end_inset

 seen in bulk measurements.
 In these simulations, moments are counter-rotated by an angle 
\begin_inset Formula $\alpha$
\end_inset

, subject to the constraint that 
\begin_inset Formula $\mu\sin\alpha=0.2\mu_{\mathrm{B}}$
\end_inset

, illustrated in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-cant"

\end_inset


\begin_inset space ~
\end_inset

(a).
 The intermediate evaluation of pdfs of moment size for various different
 angles 
\begin_inset Formula $\alpha$
\end_inset

 is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-cant"

\end_inset


\begin_inset space ~
\end_inset

(b), and the results are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes-cant2"

\end_inset

.
 They are again consistent with a reduced moment 
\begin_inset Formula $\mu$
\end_inset

, certainly lower than the high-temperature paramagnetic moment of 
\begin_inset Formula $0.6\mu_{\mathrm{B}}$
\end_inset

.
 
\end_layout

\begin_layout Standard
Thus, this probabilistic methodology suggests that the small magnetic moment
 of ordered Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is likely to be true weak-moment ferromagnetism, rather than an artefact
 of a canted magnetic structure with larger moments.
 This is not an exhaustive study of possible magnetic configurations but,
 given the results found in the variety of trial structures examined, it
 is unlikely that a combination of a pathological magnetic structure and
 muon site could give rise to the observed precession frequencies and yet
 disguise a significantly larger magnetic moment on the osmium site.
\end_layout

\begin_layout Subsection
Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/musr/Ba2LiOsO6-At.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Example 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 spectra above (
\begin_inset Formula $T=\SI{8.4}{\kelvin}$
\end_inset

) and below (
\begin_inset Formula $T=\SI{1.5}{\kelvin}$
\end_inset

) the transition temperature 
\begin_inset Formula $T_{\mathrm{N}}=\SI{8}{\kelvin}$
\end_inset

 for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
 For the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.5}{
\backslash
kelvin}
\end_layout

\end_inset

 data, the dashed line is a Kubo--Toyabe fit, the solid line is a damped
 oscillation plus a fast-relaxing component.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2LiOsO6-At"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
For Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

, a single, heavily-damped oscillation is present at low temperature.
 Example data above and below the transition are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2LiOsO6-At"

\end_inset

.
 The data fit to an exponentially-damped Kubo--Toyabe function,
\begin_inset Formula 
\begin{equation}
A(t)=A_{1}p_{\mathrm{KT}}\mathrm{e}^{-\lambda_{1}t}+A_{2}\mathrm{e}^{-\lambda_{2}t}+A_{\mathrm{bg}},
\end{equation}

\end_inset

where the first component contains 
\begin_inset Formula $p_{\mathrm{KT}}$
\end_inset

, the Kubo--Toyabe relaxation defined in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Kubo-Toyabe"

\end_inset

; the second component represents a relaxation; and the third component
 is a non-relaxing background.
 The Kubo--Toyabe function has a fitted field width 
\begin_inset Formula $\Delta/2\uppi=\SI{0.8(1)}{\MHz}$
\end_inset

 at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.5}{
\backslash
kelvin}
\end_layout

\end_inset

.
 As discussed in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:zero-field"

\end_inset

, this indicates a static but spatially-disordered spontaneous field.
 Similarly, it is possible to fit the data with a damped oscillation plus
 a fast-relaxing Gaussian component,
\begin_inset Formula 
\begin{equation}
A(t)=A_{1}\cos\left(2\uppi\nu t\right)\mathrm{e}^{-\lambda_{1}t}+A_{2}\mathrm{e}^{-\sigma_{2}^{2}t^{2}}+A_{\mathrm{bg}},
\end{equation}

\end_inset

where the first component represents a damped oscillation, the second a
 pure relaxation and the third component a non-relaxing background.
 The large relaxation on the oscillating component, 
\begin_inset Formula $\lambda_{1}/2\uppi=\SI{0.84(5)}{\MHz}\approx\nu=\SI{1.2(1)}{\MHz}$
\end_inset

, indicates a wide field distribution, and thus also points to the presence
 of significant disorder.
 Following either argument, it seems that the long-range magnetic ordered
 state present in the sample is static, but significantly disordered.
 At higher temperature the relaxation function can be modelled with a Gaussian,
 signifying the paramagnetic state.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/musr/Ba2LiOsO6-sigmaT.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
The width 
\begin_inset Formula $\sigma$
\end_inset

 of an initial Gaussian relaxation fitted over the first 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1}{
\backslash
us}
\end_layout

\end_inset

 of data as a function of temperature 
\begin_inset Formula $T$
\end_inset

.
 Since this would be expected to scale with the internal field of the magnet,
 this can be used to approximately identify the transition to long-range
 order, which occurs at around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{8}{
\backslash
kelvin}
\end_layout

\end_inset

.
 The line is a guide to the eye.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2LiOsO6-sigmaT"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The fast initial relaxation is associated with the onset of long-range order,
 but how exactly is unclear.
 One possibility is that muons stop at a site with a very small average
 field, but a wide distribution of fields, and thus this relaxation would
 be expected to scale with the order parameter.
 Focussing only on the early-time relaxation and plotting the relaxation
 rate allows a crude parametrisation of the development of static fields
 as a function of temperature.
 This is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2LiOsO6-sigmaT"

\end_inset

, and displays the magnetic transition at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{8}{
\backslash
kelvin}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
The moments in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 therefore appear to be static but show a larger degree of disorder than
 those of the Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 for 
\begin_inset Formula $T<T_{\mathrm{N}}$
\end_inset

.
 This makes applying the Bayesian technique more difficult, firstly because
 identifying a single internal field is problematic, and secondly because
 simple magnetic structures do not give rise to such distributions of fields,
 and thus can only be an approximation to the true structure.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/ba2moso6/dipole/Ba2LiOsO6-bayes-pdfs.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions of osmium moment 
\begin_inset Formula $\mu$
\end_inset

 given the approximate local field extracted from experimental data in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
 Results are shown for three different magnetic structures: the direction
 associated with each line indicates the moment direction; AFM configurations
 possess FM planes along the 
\begin_inset Formula $[1\,0\,0]$
\end_inset

 direction, as described in the text.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ba2LiOsO6-mu-Bayes"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The local field at the muon site can be estimated from either the extracted
 Kubo--Toyabe field distribution width or the damped oscillation frequency
 and both are indeed similar in value.
 Performing a Bayesian analysis assuming that the fitted oscillation frequency
 at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.5}{
\backslash
kelvin}
\end_layout

\end_inset

 gives the field at the muon site allows the osmium moment size in this
 lithium analogue to be estimated.
 The distributions of muon precession frequency at muon sites near oxygen
 and far from positive ions are similar to those for Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2NaOsO6-Bayes"

\end_inset

, whilst inputting the inferred frequency gives rise to pdfs of osmium moment
 shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ba2LiOsO6-mu-Bayes"

\end_inset

.
 This analysis therefore yields a likely low moment 
\begin_inset Formula $\lesssim0.2\mu_{\mathrm{B}}$
\end_inset

, consistent with that measured in the spin-flop state seen in Figs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Ba2MOsO6-SQUID"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Ba2LiOsO6-pulsed-B"

\end_inset

.
\end_layout

\begin_layout Section
Discussion
\end_layout

\begin_layout Standard
Thus, both bulk probes in large applied field and a Bayesian analysis of
 zero-field 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 suggest that a low moment is common to both Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\emph on
M
\emph default
OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 compounds studied here.
 This agreement between techniques acts as both corroboration of the weak
 ferromagnetism proposed in the sodium osmate, and suggests that the lithium
 compound is not simply an unexciting AFM structural analogue since it too
 possesses a reduced moment.
 It is also a preliminary vindication of the Bayesian dipole-field technique,
 given the agreement between the muon-estimated moments and those inferred
 from SQUID and pulsed-field magnetisation measurements.
\end_layout

\begin_layout Standard
In contrast to BaIrO
\begin_inset script subscript

\begin_layout Plain Layout
3
\end_layout

\end_inset

, which achieves a low ferromagnetic moment due to charge-density wave formation
\begin_inset CommandInset citation
LatexCommand citep
key "Brooks2005-BaIrO3"

\end_inset

, in these compounds the origin must be rather different.
 It has been postulated
\begin_inset CommandInset citation
LatexCommand citep
key "Xiang2007-Ba2NaOsO6-PRB,Lee2007-Ba2NaOsO6-EPL"

\end_inset

 that the d
\begin_inset script superscript

\begin_layout Plain Layout
1
\end_layout

\end_inset

 spin-moment of Os is compensated by the 
\begin_inset Formula $t_{2g}$
\end_inset

 
\begin_inset Formula $L=1$
\end_inset

 orbital moment induced by the very strong spin--orbit coupling.
 Partial orbital quenching by the environment is thought to destroy the
 perfect compensation of the moment, leading to a small remaining magnetic
 moment.
 However, an important complication
\begin_inset CommandInset citation
LatexCommand citep
key "Lee2007-Ba2NaOsO6-EPL"

\end_inset

 is that the antiferromagnetic coupling between OsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 clusters takes place on an fcc lattice.
 It is therefore strongly frustrated and hence may be responsible for the
 sensitivity of the ground state to minor chemical changes, explaining the
 difference between the magnetic properties of isostructural Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

LiOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 and Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

.
\end_layout

\end_body
\end_document
