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\begin_body

\begin_layout Chapter
Quasi--two-dimensional molecular magnets [Cu(HF
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2
\end_layout

\end_inset

)(pyz)
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\begin_layout Plain Layout
2
\end_layout

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]
\emph on
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LatexCommand label
name "chap:cuhf2pyz2x"

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status open

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\align center
\begin_inset Graphics
	filename figures/cuhf2pyz2x/structure/CuHF2pyz2PF6-structure.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
The structure of [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]PF
\begin_inset script subscript

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6
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, as an example of the [
\emph on
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\emph default
(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
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 series.
 Copper ions are joined in a 2D square lattice by pyrazine ligands to form
 Cu(pyz)
\begin_inset Formula $_{\text{2}}^{\text{2+}}$
\end_inset

 sheets; the 2D layers are joined in the third dimension by HF
\begin_inset Formula $_{\text{2}}^{-}$
\end_inset

 groups, making a pseudocubic 3D structure; and this structure is stabilised
 by a PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 anion at the centre of each cubic pore.
 For clarity, hydrogen atoms attached to pyrazine rings have been omitted,
 and only one PF
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\end_inset

 anion is shown.
\begin_inset CommandInset label
LatexCommand label
name "fig:M(HF2)(pyz)2X-structure"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The synthesis of the [
\emph on
M
\emph default
(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
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\emph default
 system
\begin_inset CommandInset citation
LatexCommand citep
key "Manson2006-CuHF2pyz2BF4-chemcomm,goddard2008-2DHMexchange,Manson2009-Cu-SbF6"

\end_inset

 represented the first example of the use of a bifluoride building block
 to make a three-dimensional coordination polymer.
 This class of materials possesses a highly stable structure due to the
 exceptional strength of the bifluoride hydrogen bonds.
 The structure of the [
\emph on
M
\emph default
(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
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\end_inset

]
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 system
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LatexCommand citep
key "Manson2006-CuHF2pyz2BF4-chemcomm,Manson2009-Cu-SbF6"

\end_inset

 comprises infinite 2D [
\emph on
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\emph default
(pyz)
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\end_inset

]
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\end_inset

 sheets which lie in the 
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\end_inset

-plane, with bifluoride ions (HF
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\end_inset

)
\begin_inset Formula $^{-}$
\end_inset

 above and below the metal ions acting as bridges between the planes, to
 form a pseudocubic network.
 The 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset

 anions occupy the body-centre positions within each cubic pore.
 An example structure, for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
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\end_inset

]PF
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6
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\end_inset

, is shown in 
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reference "fig:M(HF2)(pyz)2X-structure"

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.
 Samples are produced in polycrystalline form via aqueous chemical reactions
 between 
\emph on
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\end_inset

 salts and stoichiometric amounts of ligands.
 Preparation details for the compounds are reported in Refs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Manson2006-CuHF2pyz2BF4-chemcomm,Manson2009-Cu-SbF6,Manson2011-NiHF2pyz2X"

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.
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status open

\begin_layout Plain Layout
Samples studied in this chapter were synthesised by J.
 L.
 Manson and J.
 A.
 Schleuter.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In this section we consider those materials where the 
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 cations are Cu
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 3d
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\begin_inset Formula $S=\frac{1}{2}$
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 centres.
 It is thought that the magnetic behaviour of these materials results from
 the 
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\end_inset

 orbital of the Cu at the centre of each CuN
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\end_inset

F
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 octahedron lying in the CuN
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 plane, so that the spin exchange interactions between neighbouring Cu
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 ions occur through the 
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-bonded pyz ligands
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LatexCommand citep
key "Manson2006-CuHF2pyz2BF4-chemcomm"

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.
 The interplane exchange through the HF
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 bridges connecting two Cu
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 ions should be very weak as these bridges lie on the 4-fold rotational
 axis of the Cu 
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\end_inset

 magnetic orbital, resulting in limited overlap with the fluorine 
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 orbitals.
 Therefore to a first approximation, the magnetic properties of [
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(HF
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)(pyz)
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]
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 can be described in terms of a 2D square lattice.
\end_layout

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Measurements for 
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= BF
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, ClO
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 and SbF
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 were made using the MuSR spectrometer at ISIS, whilst PF
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, AsF
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\end_inset

, NbF
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 and TaF
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\end_inset

 were measured using GPS at PSI.
\end_layout

\begin_layout Section
Long-range magnetic order
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\begin_inset Formula $-14(2)$
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\begin_inset Formula $4.32(3)$
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\begin_layout Plain Layout
\begin_inset Formula $0.23(3)$
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\begin_inset Formula $12.8$
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\begin_layout Plain Layout
\begin_inset Formula $1\times10^{-2}$
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
SbF
\begin_inset script subscript

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6
\end_layout

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\end_layout

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\begin_layout Plain Layout
\begin_inset Formula $3.51(2)$
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\begin_inset Formula $2.14(2)$
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.8(3)$
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $13.3$
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\begin_inset script subscript

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6
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\end_layout

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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.33(7)$
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\begin_layout Plain Layout
\begin_inset Formula $2.36(5)$
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</cell>
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\begin_layout Plain Layout
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\begin_layout Plain Layout
\begin_inset Formula $4.28(1)$
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\begin_layout Plain Layout
\begin_inset Formula $0.33(3)$
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\begin_layout Plain Layout
\begin_inset Formula $2.0(4)$
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<cell alignment="center" valignment="top" usebox="none">
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- 
\end_layout

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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-
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\end_inset
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<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
TaF
\begin_inset script subscript

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6
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<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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- 
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\begin_inset Formula $4.22(1)$
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\begin_layout Plain Layout
\begin_inset Formula $0.25(1)$
\end_inset

 
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\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.5(3)$
\end_inset

 
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<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
- 
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-
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Caption

\begin_layout Plain Layout
Fitted parameters for molecular magnets in the [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
 family.
 The first parameters shown relate to fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

, which allow us to derive frequencies at 
\begin_inset Formula $T=0$
\end_inset

, 
\begin_inset Formula $\nu_{i}$
\end_inset

; probabilities of stopping in the various classes of stopping site, 
\begin_inset Formula $p_{i}$
\end_inset

, in percent; and phases associated with fitting the oscillating components
 
\begin_inset Formula $\phi_{i}$
\end_inset

.
 Then, the temperature dependence of 
\begin_inset Formula $\nu_{i}$
\end_inset

 is fitted with 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta-2"

\end_inset

, extracting values for the Néel temperature, 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, critical exponent 
\begin_inset Formula $\beta$
\end_inset

 and parameter 
\begin_inset Formula $\alpha$
\end_inset

.
 Finally, the quoted 
\begin_inset Formula $J/k_{\mathrm{B}}$
\end_inset

 is obtained from pulsed-field experiments
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

, and the ratio of inter- to in-plane coupling, 
\begin_inset Formula $J_{\perp}/J$
\end_inset

, is obtained by combining 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 and 
\begin_inset Formula $J$
\end_inset

 with formulae extracted from quantum Monte Carlo simulations (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:exchange-anisotropy"

\end_inset

, and Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "goddard2008-2DHMexchange"

\end_inset

).
 Dashes in the 
\begin_inset Formula $\lambda_{3}$
\end_inset

 column for the SbF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 and TaF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 compounds indicate that no fast-relaxing component was used to fit those
 data.
 Dashes in the 
\begin_inset Formula $J/k_{\mathrm{B}}$
\end_inset

 and 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 columns for NbF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 and TaF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 indicate a lack of pulsed-field data for these materials.
 
\begin_inset CommandInset label
LatexCommand label
name "tab:Cu(HF2)(pyz)2X-magnetism-parameters"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/cuhf2pyz2x/summarygraphs/summary-tetra.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Data and the results of fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

 for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
 magnets with tetrahedral anions 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset

.
 From left to right: (a) and (d) show sample asymmetry spectra 
\begin_inset Formula $A(t)$
\end_inset

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

 along with a fit to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

; (b) and (e) show frequencies as a function of temperature [no data points
 are shown for the second line because this frequency 
\begin_inset Formula $\nu_{2}$
\end_inset

 was held in fixed proportion to the first, 
\begin_inset Formula $\nu_{1}$
\end_inset

 (see text)]; and (c) and (f) show relaxation rates 
\begin_inset Formula $\lambda_{i}$
\end_inset

 as a function of temperature.
 In the 
\begin_inset Formula $\nu(T)$
\end_inset

 plot, error bars are included on the points but in most cases they are
 smaller than the marker being used.
 The blue solid line representing 
\begin_inset Formula $\nu_{1}$
\end_inset

 in (b) and (e) corresponds to the blue filled circles in the third column
 of graphs [(c) and (f)] for that component's relaxation, 
\begin_inset Formula $\lambda_{1}$
\end_inset

, whilst the red dashed line and red unfilled circles correspond to 
\begin_inset Formula $\nu_{2}$
\end_inset

 and 
\begin_inset Formula $\lambda_{2}$
\end_inset

, respectively.
 The green filled triangles correspond to the fast relaxation 
\begin_inset Formula $\lambda_{3}$
\end_inset

.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Cu(HF2)(pyz)2X-tetrahedral-magnetism-graphs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/cuhf2pyz2x/summarygraphs/summary-octa.eps
	scale 97

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Example data and fits for [Cu(HF
\begin_inset Formula $\text{_{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
 magnets with octahedral anions 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset

.
 From left to right: (a), (d), (g), (j) and (m) show sample asymmetry spectra
 
\begin_inset Formula $A(t)$
\end_inset

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

 along with a fit to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

; (b), (e), (h), (k) and (n) show frequencies as a function of temperature
 [no data points are shown for the second line because this frequency 
\begin_inset Formula $\nu_{2}$
\end_inset

 was held in fixed proportion to the first, 
\begin_inset Formula $\nu_{1}$
\end_inset

 (see text)]; and (c), (f), (i), (l) and (o) show relaxation rates 
\begin_inset Formula $\lambda_{i}$
\end_inset

 as a function of temperature.
 In the 
\begin_inset Formula $\nu(T)$
\end_inset

 plot, error bars are included on the points but in most cases they are
 smaller than the marker being used.
 The blue solid line representing 
\begin_inset Formula $\nu_{1}$
\end_inset

 in (b), (e), (h), (k) and (n) corresponds to the blue filled circles in
 the third column of graphs [(c), (f), (i), (l) and (o)] for that component's
 relaxation, 
\begin_inset Formula $\lambda_{1}$
\end_inset

, whilst the red dashed line and red unfilled circles correspond to 
\begin_inset Formula $\nu_{2}$
\end_inset

 and 
\begin_inset Formula $\lambda_{2}$
\end_inset

, respectively.
\begin_inset CommandInset label
LatexCommand label
name "fig:Cu(HF2)(pyz)2X-octahedral-magnetism-graphs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The main result of our measurements on these systems is that, below a critical
 temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, oscillations in the asymmetry spectra 
\begin_inset Formula $A(t)$
\end_inset

 are observed at two distinct frequencies, for all materials in the series.
 This shows unambiguously that each of these materials undergoes a transition
 to a state of LRO.
 Example asymmetry spectra are shown in the left-hand column of Figs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-tetrahedral-magnetism-graphs"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-octahedral-magnetism-graphs"

\end_inset

.
 They were found to be best fitted with a relaxation function 
\begin_inset Formula 
\begin{eqnarray}
A(t)= & A_{0} & \left[p_{1}\mathrm{e}^{-\lambda_{1}t}\cos(2\pi\nu_{1}t+\phi_{1})\right.\nonumber \\
 &  & \left.+p_{2}\mathrm{e}^{-\lambda_{2}t}\cos(2\pi P_{2}\nu_{1}t+\phi_{2})+p_{3}\mathrm{e}^{-\lambda_{3}t}\right]\nonumber \\
 &  & +A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t},\label{eq:M(HF2)(pyz)2X-A(t)-fit}
\end{eqnarray}

\end_inset

 where 
\begin_inset Formula $A_{0}$
\end_inset

 represents the contribution from those muons which stop inside the sample
 and 
\begin_inset Formula $A_{\mathrm{bg}}$
\end_inset

 accounts for a relaxing background signal due to those muons that stop
 in the silver sample holder or cryostat tails, or with their spin parallel
 to the local field.
 Of those muons which stop in the sample, 
\begin_inset Formula $p_{1}$
\end_inset

 indicates the weighting of the component in an oscillating state with frequency
 
\begin_inset Formula $\nu_{1}$
\end_inset

; 
\begin_inset Formula $p_{2}$
\end_inset

 is the weighting of a lower-frequency oscillating state with frequency
 
\begin_inset Formula $\nu_{2}$
\end_inset

; and 
\begin_inset Formula $p_{3}$
\end_inset

 represents the weighting of a component with a large relaxation rate 
\begin_inset Formula $\lambda_{3}$
\end_inset

.
 All parameters were initially left free to vary.
 The second frequency was found to vary with temperature in fixed proportion
 to 
\begin_inset Formula $\nu_{1}$
\end_inset

 via 
\begin_inset Formula $\nu_{2}=P_{2}\nu_{1}$
\end_inset

 for each material.
 The parameter 
\begin_inset Formula $P_{2}$
\end_inset

 was identified by fitting the lowest-temperature 
\begin_inset Formula $A(t)$
\end_inset

 spectra where 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

 would be expected to most accurately describe the data, and subsequently
 held fixed during the fitting procedure.
 Phase factors 
\begin_inset Formula $\phi_{i}$
\end_inset

 were also found to be necessary in some cases to obtain a reliable fit.
 The parameters resulting from these fits are listed in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Cu(HF2)(pyz)2X-magnetism-parameters"

\end_inset

, and data with fits are shown in Figs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-tetrahedral-magnetism-graphs"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-octahedral-magnetism-graphs"

\end_inset

.
 We also note here that the discontinuous nature of the change in all fitted
 parameters and the form of the spectra at 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 strongly suggest that these materials are magnetically ordered throughout
 their bulk.
\end_layout

\begin_layout Standard
The frequencies and relaxation rates as a function of temperature extracted
 from these fits are shown in the central column of Figs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-tetrahedral-magnetism-graphs"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-octahedral-magnetism-graphs"

\end_inset

.
 The muon precession frequency, which is proportional to the internal field
 in the material, can be considered an effective order parameter for the
 system.
 Consequently, fitting extracted frequencies as a function of temperature
 to the phenomenological 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-2}
\end{equation}

\end_inset

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

 to be extracted.
 Our results fit well with a previous observation
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

 that the compounds divide naturally into two classes: those with tetrahedral
 anions 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

=
\emph on
 
\emph default
BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, ClO
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 and those with octahedral anions 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= 
\emph on
A
\emph default
F
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

.
 The tetrahedral compounds have lower transition temperatures 
\begin_inset Formula $T_{\mathrm{N}}\lesssim2\mathrm{\; K}$
\end_inset

, as compared to the octahedral compounds' 
\begin_inset Formula $T_{\mathrm{N}}\gtrsim4\;\mathrm{K}$
\end_inset

; and the tetrahedral compounds also display slightly lower oscillation
 frequencies than their octahedral counterparts
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

.
\end_layout

\begin_layout Standard
This difference has been explained
\begin_inset space ~
\end_inset

 in terms of differences in the crystal structure between the two sets of
 compounds.
 Firstly, the octahedral anions are larger than their tetrahedral counterparts.
 Secondly, the pyrazine rings are tilted by differing amounts with respect
 to the normal to the 2D layers: those in the octahedral compounds are significa
ntly more upright.
 Since the Cu 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 orbitals point along the pyrazine directions, these tilting angles might
 be expected, to first order, to make little difference to nearest-neighbour
 exchange because such rotation is about a symmetry axis as viewed from
 the copper site.
 However, it may be that the different direction of the delocalised orbitals
 above and below the rings through which exchange probably occurs, possibly
 in conjunction with hybridisation with the anion orbitals, results in an
 altered next-nearest neighbour or higher-order interactions, changing the
 transition temperature.
\end_layout

\begin_layout Standard
Within the tetrahedral compounds, the difference in the weighting of the
 oscillatory component (
\begin_inset Formula $p_{1}+p_{2}$
\end_inset

) in 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 and ClO
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 probably results from the difficulty in fitting the fast-relaxing component.
 Even with little change in the size of the oscillations, any error in assigning
 the magnitude of this component will affect the proportion of the 
\begin_inset Formula $A(t)$
\end_inset

 signal attributed to them.
 This difficulty is partly due to the resolution-limited nature of ISIS
 arising from the pulsed beam structure.
 In the octahedral compounds, 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= SbF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 and TaF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 do not appear to have a resolvable fast-relaxing component, and consequently
 
\begin_inset Formula $p_{3}$
\end_inset

 was set to zero during the fitting procedure.
 This is reflected by dashes in the 
\begin_inset Formula $p_{3}$
\end_inset

 and 
\begin_inset Formula $\lambda_{3}$
\end_inset

 columns in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Cu(HF2)(pyz)2X-magnetism-parameters"

\end_inset

.
\end_layout

\begin_layout Standard
The fact that two oscillatory frequencies are observed points to the existence
 of at least two magnetically distinct classes of muon site.
 In general it is found that 
\begin_inset Formula $p_{1}\approx p_{2}$
\end_inset

 for these materials, making the probability of occupying the sites giving
 rise to magnetic precession approximately equal.
 The weightings 
\begin_inset Formula $p_{1,2}$
\end_inset

 were found to be significantly less than the weighting 
\begin_inset Formula $p_{3}$
\end_inset

 relating to the fast-relaxing site.
 This, in combination with the magnitude of the fast relaxation 
\begin_inset Formula $\lambda_{3}(T=0)\gtrsim10\mathrm{\; MHz}$
\end_inset

, suggests that this term should not be identified with the 
\begin_inset Formula $\frac{1}{3}$
\end_inset

-tail which results from muons with spins parallel to their local field.
 (If that were the case then we would expect 
\begin_inset Formula $(p_{1}+p_{2})/p_{3}=2$
\end_inset

, which is not observed.) It is likely that each of the components, 
\begin_inset Formula $p_{1}$
\end_inset

, 
\begin_inset Formula $p_{2}$
\end_inset

 and 
\begin_inset Formula $p_{3}$
\end_inset

, therefore reflect the occurrence of a separate class of muon site in this
 system.
 The possible positions of these three classes of site are investigated
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:[M(HF2)(pyz)2]X-muon-sites"

\end_inset

.
\end_layout

\begin_layout Standard
The temperature evolution of the relaxation rates 
\begin_inset Formula $\lambda_{i}$
\end_inset

 is shown in the right-hand columns of Figs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-tetrahedral-magnetism-graphs"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(HF2)(pyz)2X-octahedral-magnetism-graphs"

\end_inset

.
 In the fast-fluctuation limit, the relaxation rates are expected
\begin_inset CommandInset citation
LatexCommand cite
key "hayano1979-muonrelaxation"

\end_inset

 to vary as 
\begin_inset Formula $\lambda\propto\Delta^{2}\tau$
\end_inset

, where 
\begin_inset Formula $\Delta=\sqrt{\gamma_{\murm}^{2}\langle(B-\left\langle B\right\rangle )^{2}\rangle}$
\end_inset

 is the second moment of the local magnetic field distribution (whose mean
 is 
\begin_inset Formula $\left\langle B\right\rangle $
\end_inset

) in frequency units, and 
\begin_inset Formula $\tau$
\end_inset

 is the correlation time.
 In all measured materials, the relaxation rate 
\begin_inset Formula $\lambda_{1}$
\end_inset

, corresponding to the higher oscillation frequency, starts at a small value
 at low temperature and increases as 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 is approached from below.
 This is the expected temperature-dependent behaviour and most likely reflects
 a contribution from critical slowing down of fluctuations near 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 (described e.g.
\begin_inset space \space{}
\end_inset

in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Pratt2007-critical-behaviour-CoGly"

\end_inset

).
 In contrast, the relaxation rate 
\begin_inset Formula $\lambda_{2}$
\end_inset

 (associated with the lower frequency) starts with a higher magnitude at
 low temperature and decreases smoothly as the temperature is increased.
 This is also the case for the relaxation rate 
\begin_inset Formula $\lambda_{3}$
\end_inset

 of the fast-relaxing component.
 This smooth decrease of these relaxation rates with temperature has been
 observed previously in magnetic materials
\begin_inset CommandInset citation
LatexCommand citep
key "Lancaster2003-PrO2,Lancaster2007-YMnO3pressure"

\end_inset

 and seems to roughly track the magnitude of the local field.
 It is possible that muon sites responsible for 
\begin_inset Formula $\lambda_{1}$
\end_inset

 and 
\begin_inset Formula $\lambda_{3}$
\end_inset

 lie further from the 2D planes than those sites giving rise to 
\begin_inset Formula $\lambda_{2}$
\end_inset

, and are thus less sensitive to 2D fluctuations, reducing the influence
 of any variation in 
\begin_inset Formula $\tau$
\end_inset

.
 The temperature evolution of 
\begin_inset Formula $\lambda_{1}$
\end_inset

 and 
\begin_inset Formula $\lambda_{3}$
\end_inset

 might then be expected to be dominated by the magnitude of 
\begin_inset Formula $\Delta$
\end_inset

, which scales with the size of the local field and would therefore decrease
 as the magnetic transition is approached from below.
 
\end_layout

\begin_layout Standard
The need for nonzero phases 
\begin_inset Formula $\phi_{i}$
\end_inset

 has been identified in previous studies of molecular magnets
\begin_inset CommandInset citation
LatexCommand citep
key "Lancaster2006-CuPzN,Lancaster2004-CuX2(pyz)-and-0DMMs,Lancaster2006-Cu(HCO2)2(pyz),Lancaster2007-Cupyz2ClO42"

\end_inset

, but never satisfactorily explained.
 One possible explanation for these might be that the muon experiences delayed
 state formation.
 However, the simplest model of this can be ruled out as the phases appear
 not to correlate with 
\begin_inset Formula $\nu_{i}$
\end_inset

.
 Such a correlation would be expected since a delay of 
\begin_inset Formula $t_{0}$
\end_inset

 before entering the precessing state would give rise to a component of
 the relaxation function 
\begin_inset Formula $a_{i}(t)=\cos\left[2\pi\nu_{i}\left(t+t_{0}\right)\right]=\cos\left(2\pi\nu_{i}t+\phi_{i}\right)$
\end_inset

, with 
\begin_inset Formula $\phi_{i}\propto\nu_{i}$
\end_inset

, which is not observed.
 This does not completely rule out delayed state formation, as 
\begin_inset Formula $t_{0}$
\end_inset

 could be a function of temperature (although this seems unlikely at these
 temperatures).
 Nonzero phases are also sometimes observed when attempting to fit data
 with cosinusoidal relaxation functions from systems having incommensurate
 magnetic structures.
 The phase then emerges as an artefact of fitting, as a cosine with a 
\begin_inset Formula $\frac{\pi}{4}$
\end_inset

 phase shift approximates the zeroth-order Bessel function of the first
 kind, 
\begin_inset Formula $J_{0}(\omega t)$
\end_inset

, which is obtained from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

of an incommensurately-ordered system
\begin_inset CommandInset citation
LatexCommand cite
key "Major1986-ZF-muSR-Cr,Amato1997-muSR-heavy-fermions"

\end_inset

.
 The Bessel function arises because the distribution of fields seen by muons
 at sites is asymmetric.
 However, attempts to fit the data with a pair of damped Bessel functions
 produced consistently worse fits than fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

, suggesting that a simple incommensurate structure is not a satisfactory
 explanation.
 It is also possible that several further magnetically-inequivalent muon
 sites exist, resulting in multiple, closely-spaced frequencies which give
 the spectra a more complex character which is not reflected in the fitting
 function.
 The simpler relaxation function would then obtain a better fit if the phase
 were allowed to vary.
 This has been observed, for example, in LiCrO
\begin_inset Formula $_{\text{2}}$
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Sugiyama2009-LiCrO2"

\end_inset

.
 A final possibility is that the distribution of fields at muon sites is
 asymmetric for another reason, perhaps arising from a complex magnetic
 structure.
 This may give rise to a Fourier transform which is only able to be fitted
 with phase-shifted cosines.
 However, the mechanism by which this would occur is unclear.
\end_layout

\begin_layout Section
Nonmonotonic field dependence of 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:nonmonotonic-CuHF2pyz2BF4"

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/cuhf2pyz2x/CuHF2pyz2BF4-high-field/nonmonotonic.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Sample TF 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

data measured for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 in an applied field of 2
\begin_inset space ~
\end_inset

T are shown in (a) and (b).
 Data are shown in the `rotating reference frame', rotating at 
\begin_inset Formula $\gamma_{\murm}\times1.9\mathrm{\; T}=\mathrm{257\; MHz}$
\end_inset

, nearly cancelling out spin precession induced by the 
\begin_inset Formula $2\mathrm{\; T}$
\end_inset

 applied transverse field.
 (c) The evolution of the magnetic broadening 
\begin_inset Formula $\sigma$
\end_inset

 with 
\begin_inset Formula $T$
\end_inset

, showing a magnetic transition at 1.98
\begin_inset space ~
\end_inset

K in 2 T.
 (d) The 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram from Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

 showing the nonmonotonic behaviour at low applied magnetic field.
 In the key, HC is heat capacity, theory represents the results of computational
 modelling, and 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 shows our results from TF measurements (see main text).
 
\begin_inset CommandInset label
LatexCommand label
name "fig:nonmonotonic-TN(B)-BF4"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Although the interplane exchange coupling 
\begin_inset Formula $J_{\perp}$
\end_inset

 is expected to have a large amount of control of the thermodynamic properties
 of these materials, it may be the case that single-ion anisotropies are
 also responsible for deviations in the behaviour of our materials from
 the predictions of the 2DSLQHAF model.
 In particular, these anisotropies have been demonstrated to show a crossover
 to magnetic behaviour consistent with the 2D 
\begin_inset Formula $XY$
\end_inset

 model
\begin_inset CommandInset citation
LatexCommand cite
key "Xiao2009-XY-q2DHA"

\end_inset

.
 It was recently reported
\begin_inset CommandInset citation
LatexCommand cite
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

 that [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 exhibits an unusual nonmonotonic dependence of 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 as a function of applied magnetic field 
\begin_inset Formula $B$
\end_inset

 [see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:nonmonotonic-TN(B)-BF4"

\end_inset


\begin_inset space ~
\end_inset

(d)].
 This behaviour was explained as resulting from the small 
\begin_inset Formula $XY$
\end_inset

-like anisotropy of the spin system in these systems.
 The physics of the unusual field-dependence then arises due to the dual
 effect of 
\begin_inset Formula $B$
\end_inset

 on the spins, both suppressing the amplitude of the order parameter by
 polarising the spins along a given direction, and also reducing the phase
 fluctuations by changing the order parameter phase space from a sphere
 to a circle.
 A more detailed explanation for the behaviour
\begin_inset CommandInset citation
LatexCommand cite
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

 reveals that the energy scales of the physics are controlled by a Kosterlitz--T
houless-like mechanism, along with the interlayer exchange interaction 
\begin_inset Formula $J_{\perp}$
\end_inset

.
\end_layout

\begin_layout Standard
The measurement of the 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 reported in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

 was made by observing a small anomaly in specific heat.
 In order to test whether the phase boundary could be determined using muons,
 transverse-field (TF) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

measurements were carried out using the LTF instrument at S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm 
\end_layout

\end_inset

S.
 In these measurements, the field is applied perpendicular to the initial
 muon spin direction, causing a precession of the muon-spins in the sum
 of the applied and internal field directed perpendicular to the muon-spin
 orientation (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:transverse-field"

\end_inset

).
 Example TF spectra measured in a field of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{2}{
\backslash
tesla}
\end_layout

\end_inset

 are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:nonmonotonic-TN(B)-BF4"

\end_inset


\begin_inset space ~
\end_inset

(a) and (b).
 The spectra are well described by a function 
\begin_inset Formula 
\begin{equation}
A(t)=A(0)\mathrm{e}^{-\sigma^{2}t^{2}/2}\cos(2\pi\nu t+\phi),
\end{equation}

\end_inset

where the phase factor depends on the details of the detector geometry,
 and 
\begin_inset Formula $\sigma$
\end_inset

 is proportional to the second moment of the internal field distribution
 via 
\begin_inset Formula $\sigma^{2}=\gamma_{\murm}^{2}\langle B^{2}\rangle$
\end_inset

.
 Upon cooling through 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 a large increase in 
\begin_inset Formula $\sigma$
\end_inset

 is observed, as shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:nonmonotonic-TN(B)-BF4"

\end_inset


\begin_inset space ~
\end_inset

(c).
 This approximately resembles an order parameter, and the discontinuity
 at the onset of the increase is identified with 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 by fitting 
\begin_inset Formula $\sigma$
\end_inset

 with the above-
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 relaxation adding in quadrature to the additional relaxation present below
 the transition.
 The resulting point at 
\begin_inset Formula $T_{\mathrm{N}}(B=2\mathrm{\; T})=1.98(4)\mathrm{\; K}$
\end_inset

 is shown to be consistent with the predicted low-field phase boundary in
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:nonmonotonic-TN(B)-BF4"

\end_inset


\begin_inset space ~
\end_inset

(d).
 A further point, identifiable by its vertical rather than horizontal error
 bar, was found by performing a field scan at a fixed temperature of 
\begin_inset Formula $T=1.8\mathrm{\; K}$
\end_inset

.
 The field-dependence of the relaxation rate shows a sharp increase at the
 transition, at 
\begin_inset Formula $B=1.5\pm0.3\mathrm{\; T}$
\end_inset

.
\end_layout

\begin_layout Standard
Points derived from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 measurements possibly lie slightly lower in 
\begin_inset Formula $T$
\end_inset

 than both that predicted by theory, and the line predicted on the basis
 of the specific heat measurements.
 The theoretical calculations use 
\begin_inset Formula $J/k_{\mathrm{B}}=5.9\mathrm{\; K}$
\end_inset

 and 
\begin_inset Formula $J_{\perp}/J=2.5\times10^{-3}$
\end_inset

, whilst our estimates suggest 
\begin_inset Formula $J/k_{\mathrm{B}}=6.3\mathrm{\; K}$
\end_inset

 and 
\begin_inset Formula $J_{\perp}/J=0.9\times10^{-3}$
\end_inset

.
 Performing these calculations for a purely 2D system results in the entire
 curve shifting to the left
\begin_inset CommandInset citation
LatexCommand cite
key "Sengupta2009-nonmonotonic-TN-B-CuHF2pyz2BF4"

\end_inset

, and consequently the leftward shift of our data points is consistent with
 our finding of increased exchange anisotropy.
 It is clear that the TF 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

technique may be used in future to measure the 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram and enjoys some of the same advantages it has in zero field
 over specific heat and magnetic susceptibility in anisotropic systems.
\end_layout

\begin_layout Section
Muon response in the paramagnetic phase
\begin_inset CommandInset label
LatexCommand label
name "sec:[Cu(HF2)(pyz)2]X-paramagnetic"

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="13" columns="6">
<features booktabs="true" tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0pt">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\emph on
M
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\emph on
X
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $r_{\text{\murm--F}}$
\end_inset

 (nm) 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $p_{1}$
\end_inset

 (%) 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

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

 (MHz) 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T$
\end_inset

 (K)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.1038(1)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $77(1)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.29(1)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $5.1$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.1081(2)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $70(1)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.37(1)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $5.1$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
PF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.1044(2)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $74(2)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.29(2)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $5.2$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
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</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Muon--fluorine dipole--dipole interaction fitted parameters in the family
 [
\emph on
M
\emph default
(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
, extracted from fitting data to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu"

\end_inset

.
 In addition to separation by metal ion, Cu compounds are grouped by the
 temperature at which the measurement was made: those compounds measured
 over a range of temperatures appear in both sections of the table.
\begin_inset CommandInset label
LatexCommand label
name "tab:Cu(HF2)(pyz)2X-muF-parameters"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, the character of the measured spectra changes considerably and lower-frequency
 oscillations characteristic of the dipole--dipole interaction between muons
 and fluorine nuclei
\begin_inset CommandInset citation
LatexCommand cite
key "lancaster2007-muF"

\end_inset

 are observed.
 The Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 electronic moments, which dominate the spectra for 
\begin_inset Formula $T<T_{\mathrm{N}}$
\end_inset

, are disordered in the paramagnetic regime and fluctuate very rapidly on
 the muon time scale.
 They are therefore motionally narrowed from the spectra, leaving the muon
 sensitive to the quasi-static nuclear magnetic moments.
\end_layout

\begin_layout Standard
A muon and nucleus interact via the two-spin Hamiltonian 
\begin_inset Formula 
\begin{equation}
\hat{H}=\sum_{i>j}\frac{\mu_{0}\gamma_{i}\gamma_{j}\hbar}{4\uppi r^{3}}\left[\bm{S}_{i}\cdot\bm{S}_{j}-3\left(\bm{S}_{i}\cdot\hat{\bm{r}}\right)\left(\bm{S}_{j}\cdot\hat{\bm{r}}\right)\right],\label{eq:dipole-coupling-Hamiltonian}
\end{equation}

\end_inset

 where the spins 
\begin_inset Formula $\bm{S}_{i,j}$
\end_inset

 with gyromagnetic ratios 
\begin_inset Formula $\gamma_{i,j}$
\end_inset

 are separated by the vector 
\begin_inset Formula $\bm{r}$
\end_inset

.
 This gives rise to a precession of the muon spin, and the muon-spin polarisatio
n along a quantisation axis 
\begin_inset Formula $z$
\end_inset

 varies with time as 
\begin_inset Formula 
\begin{equation}
D_{z}(t)=\frac{1}{N}\left\langle \sum_{m,n}\left|\left\langle m\left|\sigma_{q}\right|n\right\rangle \right|^{2}\mathrm{e}^{\mathrm{i}\omega_{m,n}t}\right\rangle _{q},
\end{equation}

\end_inset

where 
\begin_inset Formula $N$
\end_inset

 is the number of spin states, 
\begin_inset Formula $\left|m\right\rangle $
\end_inset

 and 
\begin_inset Formula $\left|n\right\rangle $
\end_inset

 are eigenstates of the total Hamiltonian 
\begin_inset Formula $\hat{H}$
\end_inset

, 
\begin_inset Formula $\sigma_{q}$
\end_inset

 is the Pauli spin matrix corresponding to the direction 
\begin_inset Formula $q$
\end_inset

, and 
\begin_inset Formula $\left\langle \right\rangle _{q}$
\end_inset

 represents an appropriately-weighted powder average.
 The vibrational frequency of the muon--fluorine bond exceeds by orders
 of magnitude both the frequencies observable in a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

experiment, and the frequency appropriate to the dipolar coupling in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:dipole-coupling-Hamiltonian"

\end_inset

; the bond length probed via these entangled states is thus time-averaged
 over thermal fluctuations.
 Fluorine is an especially strong candidate for this type of interaction
 firstly because it is highly electronegative causing the positive muon
 to stop close to fluorine ions, and secondly because its nuclei are 100%
 
\begin_inset Formula $^{\text{19}}$
\end_inset

F, which has 
\begin_inset Formula $I=\frac{1}{2}$
\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/cuhf2pyz2x/Fmu/BF4-Fmu.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Data taken at 
\begin_inset Formula $T=15\mathrm{\; K}\gg T_{\mathrm{N}}=1.4\mathrm{\; K}$
\end_inset

 for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

, showing 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations along with a fit to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-A(t)"

\end_inset

.
 The inset shows the energy levels present in a simple system of two 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 spins, along with the allowed transitions.
\begin_inset CommandInset label
LatexCommand label
name "fig:F-mu-BF4"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Data were fitted to a relaxation function 
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}(p_{1}\mathrm{e}^{-\lambda_{\text{F--\murm}}t}D_{z}(t)+p_{2}\mathrm{e}^{-\sigma^{2}t^{2}})+A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t},\label{eq:F-mu-A(t)}
\end{equation}

\end_inset

 where the amplitude fraction 
\begin_inset Formula $p_{1}\approx70\%$
\end_inset

 reflects the muons stopping in a site or set of sites near to a fluorine
 nucleus, which result in the observed oscillations 
\begin_inset Formula $D_{z}(t)$
\end_inset

; the weak relaxation of the muon spins is crudely modelled by a decaying
 exponential.
 The fraction 
\begin_inset Formula $p_{2}\approx30\%$
\end_inset

 describes those muons stopping in a class of sites primarily influenced
 by the randomly-orientated fields from other nuclear moments, giving rise
 to a Gaussian relaxation with 
\begin_inset Formula $\sigma\approx0.3\mathrm{\; MHz}$
\end_inset

.
 Example data and a fit are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:F-mu-BF4"

\end_inset

, whilst parameters extracted by fitting this function to data from each
 compound are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Cu(HF2)(pyz)2X-muF-parameters"

\end_inset

.
\end_layout

\begin_layout Standard
Fits to a variety of different 
\begin_inset Formula $D_{z}(t)$
\end_inset

 functions were attempted, including that resulting from a simple 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

bond (previously observed in some polymers
\begin_inset CommandInset citation
LatexCommand cite
key "lancaster2009-FmuF-PTFE"

\end_inset

) and the better-known 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
FmuF
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

complex comprising a muon and two fluorine nuclei in linear symmetric configurat
ion, which is seen in many alkali fluorides
\begin_inset CommandInset citation
LatexCommand cite
key "Brewer1986-FmuF"

\end_inset

.
 This latter model was also modified to include the possibilities of asymmetric
 and nonlinear bonds.
 Previous measurements
\begin_inset CommandInset citation
LatexCommand cite
key "lancaster2007-muF"

\end_inset

 made in the paramagnetic regime of [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

 suggested that the muon stopped close to a single fluorine in the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 group and also interacted with the more distant proton.
 This interaction is dominated by the F--
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm
\backslash
 
\end_layout

\end_inset

 coupling and, for our fitting, the observed muon--fluorine dipole--dipole
 oscillations were found to be well described by a single 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

interaction damped by a phenomenological relaxation factor.
 For such 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

entanglement, the time evolution of the polarisation is described by 
\begin_inset Formula 
\begin{equation}
D_{z}(t)=\frac{1}{6}\left[1+\sum_{j=1}^{3}u_{j}\cos\left(\omega_{j}t\right)\right],\label{eq:F-mu}
\end{equation}

\end_inset

where 
\begin_inset Formula $u_{1}=2$
\end_inset

, 
\begin_inset Formula $u_{2}=1$
\end_inset

 and 
\begin_inset Formula $u_{3}=2$
\end_inset

.
 The frequencies 
\begin_inset Formula $\omega_{j}=j\omega_{\mathrm{d}}/2$
\end_inset

, where 
\begin_inset Formula $\omega_{\mathrm{d}}=\mu_{0}\gamma_{\murm}\gamma_{\mathrm{F}}\hbar/4\uppi r^{3}$
\end_inset

, in which 
\begin_inset Formula $\gamma_{\mathrm{F}}=2\pi\times2.518\times10^{8}\mathrm{\; MHz\, T^{-1}}$
\end_inset

 is the gyromagnetic ratio of a 
\begin_inset Formula $^{\text{19}}$
\end_inset

F nucleus
\begin_inset CommandInset citation
LatexCommand cite
key "NMR-Encyclopedia2002"

\end_inset

, and 
\begin_inset Formula $r$
\end_inset

 is the muon--fluorine separation.
 These three frequencies arise from the three transitions between the three
 energy levels present in a system of two entangled 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 particles (see inset to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:F-mu-BF4"

\end_inset

).
 The fact that the relaxation function is similar in all materials in the
 series, including [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

 which is the only compound studied without fluorine in its anion, (the
 only difference being a slight lengthening of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm
\end_layout

\end_inset

 --F bond, and with no significant change in oscillating fraction) suggests
 that the muon site giving rise to the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations in all systems is near the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 bridging ligand.
\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
material 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $r_{0}\mathrm{\;(nm)}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $a\mathrm{\;(10^{-7}\, nm\, K^{-2}})$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
[Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.10376(3)$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.96(6)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
[Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.10842$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.7212$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
PVDF
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "lancaster2009-FmuF-PTFE"

\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.10914$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.9488$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Fitted values obtained by fitting muon--fluorine bond lengths with a 
\begin_inset Formula $T^{2}$
\end_inset

 scaling law, as 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Fmu-r-scaling"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "tab:Fmu-r-scaling"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The temperature evolution of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

signal was studied for 
\begin_inset Formula $T\leq300\mathrm{\; K}$
\end_inset

 in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 and [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

.
 In both cases, the dipole--dipole oscillations disappear gradually in a
 temperature range 
\begin_inset Formula $150\lesssim T\lesssim250\mathrm{\; K}$
\end_inset

, with oscillations totally absent in the centre of this range, followed
 by reappearing as temperature is increased further.
 Plots of 
\begin_inset Formula $A(t)$
\end_inset

 spectra at a variety of temperatures are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\begin_inset space ~
\end_inset

(a) and (d).
 The data were initially fitted to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-A(t)"

\end_inset

, with all parameters left free to vary.
 The temperature-evolution of the muon--fluorine bond length, 
\begin_inset Formula $r_{\text{F--\murm}}$
\end_inset

, can be seen in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\begin_inset space ~
\end_inset

(b) and (e).
 The spectra were also fitted with
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}\left(p_{1}\mathrm{e}^{-\lambda_{0}t}+p_{2}\mathrm{e}^{-\sigma^{2}t^{2}}\right)+A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t},\label{eq:F-mu-vanished-A(t)}
\end{equation}

\end_inset

a sum of an exponential and a Gaussian relaxation, which might be expected
 to describe the data in the region where the oscillations vanish.
 Both this relaxation and that extracted from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-A(t)"

\end_inset

 are plotted in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\begin_inset space ~
\end_inset

(c) and (f), labelled 
\begin_inset Formula $\lambda_{0}$
\end_inset

 and 
\begin_inset Formula $\lambda_{\text{F--\murm}}$
\end_inset

 respectively.
\end_layout

\begin_layout Standard
This bond length appears to grow and then shrink by nearly 20% over the
 
\begin_inset Formula $100\mathrm{\; K}$
\end_inset

 range where the oscillations fade from the spectra and reappear.
 This variation is significantly larger than any variation in crystal lattice
 parameters which would be expected.
 Since the oscillations visibly disappear from the measured spectra, results
 from fitting with an oscillatory relaxation function are artefacts of the
 fitting procedure: since the frequencies scale with 
\begin_inset Formula $1/r^{3}$
\end_inset

, increasing bond length together with the associated relaxation rate fits
 the data with a suppressed oscillatory signal.
 This can be approximately quantified by examining the ratio 
\begin_inset Formula $Q=2\uppi\lambda_{\text{F--\murm}}/\omega_{\mathrm{d}}$
\end_inset

, where a large value indicates that the function relaxes significantly
 before a single 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillation is completed.
 The shaded regions in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset

 show where 
\begin_inset Formula $Q>2$
\end_inset

, which acts as an approximate bound on where the parametrisation in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-A(t)"

\end_inset

 would be expected to fail.
 In the low-
\begin_inset Formula $T$
\end_inset

 region where 
\begin_inset Formula $Q<2$
\end_inset

, the bond lengths appear to scale roughly as 
\begin_inset Formula $T^{2}$
\end_inset

, which has previously been observed in fluoropolymers
\begin_inset CommandInset citation
LatexCommand cite
key "lancaster2009-FmuF-PTFE"

\end_inset

.
 Parameters extracted from fitting to
\begin_inset Formula 
\begin{equation}
r_{\text{F--\murm}}=aT^{2}+r_{0}\label{eq:Fmu-r-scaling}
\end{equation}

\end_inset

 are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Fmu-r-scaling"

\end_inset

.
\end_layout

\begin_layout Standard
The observation in these two samples of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations which disappear and then reappear is puzzling.
 While a definitive mechanism has not been identified, it is probably possible
 to rule out an electronically mediated effect since, for 
\begin_inset Formula $T\gg T_{\mathrm{N}}$
\end_inset

, the Cu moment fluctuations will be outside the muon time-window.
 An explanation could involve nearby nuclear moments, possibly influenced
 by a thermally-driven structural distortion or instability.
\end_layout

\begin_layout Standard
A similar study of [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]SbF
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\begin_inset space ~
\end_inset

(g), (h) and (i).
 In this material, the oscillations appear not to vanish over the temperature
 range studied, though a brief disappearance at 
\begin_inset Formula $T\approx\SI{200}{\kelvin}$
\end_inset

 cannot be ruled out.
 Instead, the oscillations show an apparently monotonic increase in damping
 with temperature, and the fitted bond length does not follow 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Fmu-r-scaling"

\end_inset

.
 The shaded region in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\begin_inset space ~
\end_inset

(h) and (i) has no upper bound, though we cannot rule out a constraint at
 
\begin_inset Formula $T>\SI{250}{\kelvin}$
\end_inset

.
 The pure relaxation 
\begin_inset Formula $\lambda_{0}$
\end_inset

 is omitted because there is no region where the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations are sufficiently damped for 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-vanished-A(t)"

\end_inset

 to be a good parametrisation.
\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/cuhf2pyz2x/Fmu/waterfalls.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
(caption on next page)
\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 ERT
status open

\begin_layout Plain Layout


\backslash
ContinuedFloat
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
(figure on previous page) (a) Muon--fluorine dipole--dipole oscillations
 in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 over a temperature range 
\begin_inset Formula $2\leq T\leq300\mathrm{\; K}$
\end_inset

.
 Asymmetry spectra are displaced vertically so as to approximately align
 with the temperature scale on plots (b) and (c).
 Ticks on the 
\begin_inset Formula $y$
\end_inset

-axis of (a) denote 1% asymmetry.
 Plot (b) shows the fitted value for 
\begin_inset Formula $r_{\text{F--\murm}}$
\end_inset

 as a function of temperature.
 The line shown is a fit to the low-
\begin_inset Formula $T$
\end_inset

 points with a 
\begin_inset Formula $T^{2}$
\end_inset

 scaling law, 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Fmu-r-scaling"

\end_inset

.
 The upper 
\begin_inset Formula $x$
\end_inset

-axis shows values of the dipole frequency, 
\begin_inset Formula $\nu_{\mathrm{d}}$
\end_inset

, which correspond to the lower 
\begin_inset Formula $x$
\end_inset

-axis values of 
\begin_inset Formula $r_{\text{F--\murm}}$
\end_inset

.
 Plot (c) shows fitted relaxation rates 
\begin_inset Formula $\lambda_{\text{F--\murm}}$
\end_inset

 and 
\begin_inset Formula $\lambda_{0}$
\end_inset

, referring to the relaxation of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

function 
\begin_inset Formula $D_{z}(t)$
\end_inset

 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-A(t)"

\end_inset

, and the pure relaxation in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:F-mu-vanished-A(t)"

\end_inset

.
 Shaded regions indicate temperatures where 
\begin_inset Formula $Q=2\uppi\lambda_{\text{F--\murm}}/\omega_{\mathrm{d}}>2$
\end_inset

, roughly parametrising the disappearance of the oscillations.
 (d), (e) and (f) follow (a), (b) and (c), but show data for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

.
 (g), (h) and (i) similarly, but for [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]SbF
\begin_inset Formula $_{\text{6}}$
\end_inset

 over the range 
\begin_inset Formula $5\leq T\leq250\mathrm{\; K}$
\end_inset

.
 The 
\begin_inset Formula $5\mathrm{\; K}$
\end_inset

 
\begin_inset Formula $A(t)$
\end_inset

 plot is omitted because the background is raised substantially by approach
 to the transition to LRO.
\begin_inset CommandInset label
LatexCommand label
name "fig:CuHF2pyz2X-Fmu-At-waterfall"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Muon site determination
\begin_inset CommandInset label
LatexCommand label
name "sec:[M(HF2)(pyz)2]X-muon-sites"

\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/cuhf2pyz2x/dipolepdfs/CuHF2pyz2X-nu-pdfs.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions of muon precession frequencies at positions
 close to likely muon stopping sites in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default
 with Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 moments 
\begin_inset Formula $\mu=\mu_{\mathrm{B}}$
\end_inset

.
 The graphs show dipole fields near the fluorine or oxygen atoms in the
 negative anions; near the fluorine atoms in the bifluoride ligands; and
 near the carbon and nitrogen atoms, as a proxy for proximity to the pyrazine
 ring.
 The type of line indicates the compound for which the calculation was performed.
 The muon site is constrained to be close to particular atoms, indicated
 by line color.
 The shaded areas indicate ranges of fitted frequencies as 
\begin_inset Formula $T\rightarrow0$
\end_inset

; two frequencies 
\begin_inset Formula $\nu_{1,2}^{\mathrm{\mathrm{BF_{4}}}}$
\end_inset

 represent those observed where 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, and 
\begin_inset Formula $\nu_{1,2}^{\mathrm{PF_{6}}}$
\end_inset

 those observed in the 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

 analogue.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:2DMM-pdfs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Combining the data measured above and below the transition in these materials
 allows us to attempt to construct a self-consistent picture of possible
 muon sites.
 The observed dipole--dipole observations above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 suggest that at least one muon stopping site is near a fluorine ion.
 We consider three classes of probable muon site: Class
\begin_inset space ~
\end_inset

I sites near the fluorine ions in the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 groups, Class
\begin_inset space ~
\end_inset

II sites near the pyrazine rings, and Class
\begin_inset space ~
\end_inset

III sites near the anions at the centre of the pseudocubic pores.
 Comparison of Tables
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Cu(HF2)(pyz)2X-magnetism-parameters"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Cu(HF2)(pyz)2X-muF-parameters"

\end_inset

 show that the dominant amplitude component for 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

 arises from dipole--dipole oscillatory component 
\begin_inset Formula $p_{1}\mathrm{e}^{-\lambda t}D_{z}(t)$
\end_inset

 and from the fast-relaxing component 
\begin_inset Formula $p_{3}\mathrm{e}^{-\lambda_{3}t}$
\end_inset

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

, and that these are comparable in amplitude.
 It is therefore plausible to suggest that these two signals correspond
 to contributions from the same Class
\begin_inset space ~
\end_inset

I muon sites near the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 groups.
 Moreover, the analysis of the 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

 spectra in the previous section implies that this site lies 
\begin_inset Formula $r_{\text{\murm--F}}\approx\SI{0.11}{\nm}$
\end_inset

 from an F in the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 groups.
 The remainder of the signal (the oscillating fraction below 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 and the Gaussian relaxation above) can also be identified, suggesting that
 the sites uncoupled from fluorine nuclei (Classes
\begin_inset space ~
\end_inset

II and/or III) result in the magnetic oscillations observed for 
\begin_inset Formula $T<T_{\mathrm{N}}$
\end_inset

.
\end_layout

\begin_layout Standard
Further, the evidence from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations makes the occurence of Class
\begin_inset space ~
\end_inset

III muon sites unlikely.
 The fact that spectra observed for 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

 in the 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= ClO
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 material are nearly identical to those in all other compounds, in which
 
\emph on
X
\emph default
 contains fluorine, suggest that the muons do not stop near the anions.
 Moreover, as discussed in Secs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:[Cu(pyz)2(pyo)2]X2"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:[Cu(pyo)6]X2"

\end_inset

 
\begin_inset Note Note
status open

\begin_layout Plain Layout
999 kludged multiref 
\end_layout

\end_inset

below, no 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations are observed in [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

(pyo)
\begin_inset Formula $_{\text{2}}$
\end_inset

]
\emph on
X
\emph default

\begin_inset Formula $_{\text{2}}$
\end_inset

 (
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

= BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

, PF
\begin_inset Formula $_{\text{6}}^{-}$
\end_inset

) or [Cu(pyo)
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

](BF
\begin_inset Formula $_{\text{4}}$
\end_inset

)
\begin_inset Formula $_{\text{2}}$
\end_inset

, suggesting that muons do not stop preferentially near these fluorine-rich
 anions either.
 We can therefore rule out the existence of Class
\begin_inset space ~
\end_inset

III muon sites and propose that the magnetic oscillations measured for 
\begin_inset Formula $T<T_{\mathrm{N}}$
\end_inset

 most probably arise due to Class
\begin_inset space ~
\end_inset

II sites found near the pyrazine ligands.
\end_layout

\begin_layout Standard
Below 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, the measured muon precession frequencies allow us to determine the magnetic
 field at these Class
\begin_inset space ~
\end_inset

II muon sites via 
\begin_inset Formula $\nu=\gamma_{\murm}B/2\pi$
\end_inset

.
 Simulating the magnetic field inside the crystal therefore allows us to
 compare these 
\begin_inset Formula $B$
\end_inset

-fields with those predicted for likely magnetic structures and may permit
 us to constrain the ordered moment.
 For the case of our ZF measurements in the antiferromagnetic state, the
 local magnetic field at the muon site 
\begin_inset Formula $\bm{B}_{\mathrm{local}}$
\end_inset

 is given by 
\begin_inset Formula 
\begin{equation}
\bm{B}_{\mathrm{local}}=\bm{B}_{\mathrm{dipole}}+\bm{B}_{\mathrm{hyperfine}}.
\end{equation}

\end_inset

The spin density giving rise to the contact hyperfine field 
\begin_inset Formula $\bm{B}_{\mathrm{hyperfine}}$
\end_inset

 is particularly difficult to estimate accurately in complex molecular systems,
 but it is probable for insulating materials such as these that the spin
 density on the copper ion is well localised and so the hyperfine contribution
 is ignored in this analysis.
 The dipole field 
\begin_inset Formula $\bm{B}_{\mathrm{dipole}}$
\end_inset

 can then be considered the sole contribution to local magnetic field, and
 evaluated as described in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:dipole"

\end_inset

.
\end_layout

\begin_layout Standard
Although these materials are known to be antiferromagnetic from their negative
 Curie--Weiss temperatures and zero spontaneous magnetisation at low temperature
s
\begin_inset CommandInset citation
LatexCommand citep
key "Manson2009-Cu-SbF6,Manson2006-CuHF2pyz2BF4-chemcomm"

\end_inset

, their magnetic structures are unknown.
 Dipole-field simulations were therefore performed for a variety of trial
 magnetic structures with 
\begin_inset Formula $\mu=\mu_{\mathrm{B}}$
\end_inset

.
 Results are analysed using the Bayesian scheme outlined in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:dipole"

\end_inset

.
 We begin by allowing the possibility that the magnetic precession signal
 could arise from any of the possible classes of muon site identified above.
 Random positions in the unit cell were generated and dipole fields calculated
 at these.
 To prevent candidate sites lying too close to atoms we constrain all sites
 such that 
\begin_inset Formula $r_{\murm\text{--\emph{A}}}>\SI{0.1}{\nm}$
\end_inset

 where 
\begin_inset Formula $A$
\end_inset

 is any atom.
 Possible Class
\begin_inset space ~
\end_inset

I muon sites were identified with 
\begin_inset Formula $r_{\text{\murm--F}}=r_{0}\pm\SI{0.01}{\nm}$
\end_inset

 (where 
\begin_inset Formula $r_{0}$
\end_inset

 is the muon--fluorine distance established from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations) and possible Class
\begin_inset space ~
\end_inset

II sites were selected with the constraint that 
\begin_inset Formula $\num{0.10}\leq r_{\text{\murm--C,N}}\leq\SI{0.12}{\nm}$
\end_inset

.
 The predicted probability density function (pdf) of muon precession frequencies
 (resulting from the magnitudes of the calculated fields) are plotted in
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:2DMM-pdfs"

\end_inset

, with the observed frequencies superimposed.
 Results are shown for a trial magnetic structure comprising copper spins
 lying in the plane of the pyrazine layers and at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{45}{
\backslash
degree}
\end_layout

\end_inset

 to the directions of the pyrazine chains, and with spins arranged antiferromagn
etically both along those chains and along the HF
\begin_inset Formula $_{\text{2}}$
\end_inset

 groups.
 This candidate structure is motivated by analogy with [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

](ClO
\begin_inset Formula $_{\text{4}}$
\end_inset

)
\begin_inset Formula $_{\text{2}}$
\end_inset

, which also comprises Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 ions in layers of 2D pyrazine lattices
\begin_inset CommandInset citation
LatexCommand cite
key "Tsyrulin2010-Cu(pz)2(ClO4)2"

\end_inset

, and with the parent phases of the cuprate superconductors
\begin_inset CommandInset citation
LatexCommand cite
key "Kastner1998-cuprates-review"

\end_inset

, which are also two-dimensional Heisenberg systems of 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 ions.
 Other magnetic structures investigated give qualitatively similar results.
 From 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:2DMM-pdfs"

\end_inset

 it is clear that the only sites with significant probability density near
 to the observed frequencies are those lying near the anions (i.e.
\begin_inset space \space{}
\end_inset

Class
\begin_inset space ~
\end_inset

III sites) which do not appear to be compatible with our data.
 The more plausible muon sites correspond to higher frequencies than those
 observed.
 Our conclusion is that it is likely that the Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 moments are rather smaller than the 
\begin_inset Formula $\mu_{\mathrm{B}}$
\end_inset

 assumed in this initial calculation.
\end_layout

\begin_layout Standard
If we accept that the muon sites giving rise to magnetic precession are
 near the pyrazine groups then we may use this calculation to constrain
 the size of the copper moment.
 Since 
\begin_inset Formula $\nu$
\end_inset

 is obtained from experiment, what we would like to know is 
\begin_inset Formula $g(\mu\vert\nu)$
\end_inset

, the pdf of copper moment 
\begin_inset Formula $\mu$
\end_inset

 given the observed 
\begin_inset Formula $\nu$
\end_inset

.
 This can be obtained from our calculated 
\begin_inset Formula $f(\nu/\mu)$
\end_inset

 using Bayes' theorem, which yields 
\begin_inset Formula 
\begin{equation}
g(\mu\vert\nu)=\frac{\frac{1}{\mu}f(\nu/\mu)}{\int_{0}^{\mu_{\mathrm{max}}}\frac{1}{\mu'}f(\nu/\mu')\,\mathrm{d}\mu'}\,,
\end{equation}

\end_inset

where a prior probability is assumed for the copper moment that is uniform
 between zero and 
\begin_inset Formula $\mu_{\mathrm{max}}$
\end_inset

.
 We take 
\begin_inset Formula $\mu_{\mathrm{max}}=2\mu_{\mathrm{B}}$
\end_inset

, although the results are insensitive to the precise value of 
\begin_inset Formula $\mu_{\mathrm{max}}$
\end_inset

 as long as it is reasonably large.
 When multiple frequencies 
\begin_inset Formula $\{\nu_{i}\}$
\end_inset

 are present in the spectra, it is necessary to multiply their probabilities
 of observation in order to obtain the chance of their simultaneous observation,
 so we evaluate 
\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:g(mu)}
\end{equation}

\end_inset

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

 is the error on the fitted frequency.
 Results are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:mu_Cu-pdf"

\end_inset

, along with the dipole-field pdfs which gave rise to them.
 By inspection of the pdfs, the copper moment is likely to be 
\begin_inset Formula $\mu\lesssim0.5\mu_{\mathrm{B}}$
\end_inset

.
 The dipole-field simulations results also lend weight to our contention
 that the oscillatory signal cannot arise from the sites that also lead
 to the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

component above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

.
 If this were the case then the most likely moment on the copper would be
 
\begin_inset Formula $\mu_{\mathrm{Cu}}\lesssim0.2\mu_{\mathrm{B}}$
\end_inset

, which seems unreasonably small.
 Moment sizes of 
\begin_inset Formula $\mu\lesssim0.5\mu_{\mathrm{B}}$
\end_inset

 were also observed for the 2DSLQHA system La
\begin_inset Formula $_{\text{2}}$
\end_inset

CuO
\begin_inset Formula $_{\text{4}}$
\end_inset

 (a recent estimate
\begin_inset CommandInset citation
LatexCommand cite
key "Reehuis2006-neutron-high-B-La2CuO4"

\end_inset

 from neutron diffraction gave 
\begin_inset Formula $[0.42\pm0.01]\mu_{\mathrm{B}}$
\end_inset

), despite the predictions of 
\begin_inset Formula $0.6\mu_{\mathrm{B}}$
\end_inset

 from spin wave theory and Quantum Monte Carlo
\begin_inset CommandInset citation
LatexCommand cite
key "Chakravarty1989-2DHA-moment-size"

\end_inset

.
 It was suggested in that case
\begin_inset CommandInset citation
LatexCommand cite
key "Chakravarty1989-2DHA-moment-size"

\end_inset

 that disorder might play a role in reducing the moment sizes; an additional
 possible mechanism for this suppression is ring exchange
\begin_inset CommandInset citation
LatexCommand cite
key "Motrunich2005-ring-exchange-triangular-lattice,Katanin2002-ring-exchange-La2CuO4"

\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/cuhf2pyz2x/dipolepdfs/CuHF2pyz2BF4-muCu-pdfs.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Probability density functions for muons in the putative oscillating sites
 near the pyrazine rings in [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

)(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

]BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 (a) for muon precession frequency 
\begin_inset Formula $\nu$
\end_inset

 assuming that the moment on the copper site 
\begin_inset Formula $\mu=\mu_{\mathrm{B}}$
\end_inset

, created fom a histogram of dipole fields evaluated at points satisfying
 the constraints detailed in the text; and (b) for the moment on the copper
 sites given the frequencies actually observed, evaluated using the pdfs
 in (a) and 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:g(mu)"

\end_inset

.
 Lines represent trial magnetic structures.
 All exhibit antiferromagnetic coupling through both the bifluoride and
 pyrazine exchange paths, whilst (i) has copper moments pointing at 
\begin_inset Formula $45^{\circ}$
\end_inset

 to the pyrazine grid, (ii) has moments along one of the pyrazine grid direction
s (
\begin_inset Formula $\bm{a}$
\end_inset

 or 
\begin_inset Formula $\bm{b}$
\end_inset

), and (iii) has copper moments pointing along the bifluoride axis (
\begin_inset Formula $\bm{c}$
\end_inset

).
\begin_inset CommandInset label
LatexCommand label
name "fig:mu_Cu-pdf"

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
One limitation of this analysis is that the mechanism for magnetic coupling
 of copper ions through the pyrazine rings is postulated to be via spin
 exchange, in which small magnetic polarisations are induced on intervening
 atoms
\begin_inset CommandInset citation
LatexCommand cite
key "Lloret1998-spin-pol-pyz-pyd"

\end_inset

.
 Density functional theory calculations estimate that these are small, with
 the nitrogen and carbon moments estimated at 
\begin_inset Formula $\mu_{\mathrm{C}}\approx0.01\mu_{\mathrm{B}}$
\end_inset

 and 
\begin_inset Formula $\mu_{\mathrm{N}}\approx0.07\mu_{\mathrm{B}}$
\end_inset

, respectively
\begin_inset CommandInset citation
LatexCommand cite
key "Middlemiss2008-CuHF2pyz2BF4-DFT"

\end_inset

.
 However, their effect may be non-negligible: they may be significantly
 closer to the muon site than a copper moment, and dipole fields fall off
 rapidly, as 
\begin_inset Formula $1/r^{3}$
\end_inset

.
 Further, since much of the electron density in a pyrazine ring is delocalised
 in 
\begin_inset Formula $\uppi$
\end_inset

-orbitals, the moments may not be point-like, as assumed in our dipole-field
 calculations.
 This may also lead to overlap of spin density at the muon site and result
 in a nonzero hyperfine field.
\end_layout

\end_body
\end_document
