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\begin_body

\begin_layout Chapter
Other quasi--two-dimensional molecular magnets
\begin_inset CommandInset label
LatexCommand label
name "chap:other2dmm"

\end_inset


\end_layout

\begin_layout Standard
In addition to the [Cu(HF
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

)(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

]
\emph on
X
\emph default
 systems examined in the previous chapter, a wide range of related molecular
 materials can be fabricated which display quasi--two-dimensional magnetic
 interactions.
 This type of investigation of a variety of related materials allows a systemati
c evaluation not only of the molecular magnetism itself, but also the use
 of muon techniques to study it.
 In particular, the range of differing dimensionalities present in molecular
 systems and how this relates to the critical exponent 
\begin_inset Formula $\beta$
\end_inset

, often derived from 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 data, is examined.
\end_layout

\begin_layout Standard
This chapter reports the results of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 measurements on other quasi-2D systems.
 First, [Cu(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(pyo)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

]
\emph on
Y
\emph default

\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

, with 
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

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

, in which pyo ligands bridge Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

 planes.
 Then, the quasi-2D non-polymeric compounds [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

]
\emph on
Z
\emph default

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

, where 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset

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

, ClO
\begin_inset Formula $_{\text{3}}^{-}$
\end_inset

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

 are examined.
 I also investigate materials in which either Ni
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (
\begin_inset Formula $S=1$
\end_inset

) or Ag
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

) form the magnetic species in the quasi-2D planes rather than Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

.
\end_layout

\begin_layout Section
[Cu(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(pyo)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

]
\emph on
Y
\emph default

\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


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

\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/other2dmm/structure/Cupyz2pyo2BF42-structure.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Structure of [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

.
 Copper ions lie in 2D square layers, bound by pyrazine rings.
 Pyridine-
\emph on
N
\emph default
-oxide ligands protrude from the coppers in a direction approximately perpendicu
lar to these layers.
 Tetrafluoroboride ions fill the pores remaining in the structure.
 Ion sizes are schematic; copper ions are shown twice as large for emphasis,
 and hydrogens have been omitted for clarity.
\begin_inset CommandInset label
LatexCommand label
name "fig:Cu(pyz)2(pyo)2(BF4)2-structure"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In this section, I report the magnetic behaviour of another family of molecular
 systems which shows quasi-2D magnetism, but for which the interlayer groups
 are very different and arranged in a completely different structure, resulting
 in a 2D coordination polymer.
 This system is [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
Y
\emph default

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

, where 
\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

.
 As with the previous case, 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

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

 ions are bound in a 2D square lattice of [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

 sheets lying in the 
\begin_inset Formula $\bm{ab}$
\end_inset

-plane.
 Pyridine-
\emph on
N
\emph default
-oxide (pyo) ligands [shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:ligands"

\end_inset


\begin_inset space ~
\end_inset

(b)] protrude from the copper ions along the 
\begin_inset Formula $c$
\end_inset

-direction, perpendicular to the 
\begin_inset Formula $\bm{ab}$
\end_inset

-plane in the 
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

 material, but making an angle 
\begin_inset Formula $\beta-90\approx29^{\circ}$
\end_inset

 with the normal in 
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

.
 The anions then fill the pores remaining in the structure.
 The structure of [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

 is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cu(pyz)2(pyo)2(BF4)2-structure"

\end_inset

.
\end_layout

\begin_layout Standard
In a typical synthesis
\begin_inset Foot
status open

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

\end_inset

, an aqueous solution of Cu
\emph on
Y
\emph default

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

 hydrate (
\emph on
Y
\begin_inset Formula $^{-}$
\end_inset


\emph default

\begin_inset space ~
\end_inset

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

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

) was combined with an ethanol solution that contained a mixture of pyrazine
 and pyridine-
\emph on
N
\emph default
-oxide or 4-phenylpyridine-
\emph on
N
\emph default
-oxide.
 Deep blue-green solutions were obtained in each case, and when allowed
 to slowly evaporate at room temperature for a few weeks, dark green plates
 were recovered in high yield.
 Crystal quality could be improved by sequential dilution and collection
 of multiple batches of crystals from the original mother liquor.
 The relative amounts of pyz and pyo were optimised in order to prevent
 formation of compounds such as Cu
\emph on
Y
\emph default

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

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

 or [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

]
\emph on
Y
\emph default

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

.
\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Tabular
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<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 $\nu_{1}\mathrm{\,(MHz)}$
\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 $p_{1}$
\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 $p_{2}$
\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 $p_{3}$
\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 $T_{\mathrm{N}}\mathrm{\,(K)}$
\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 $\beta$
\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 $\alpha$
\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 $J/k_{\mathrm{B}}\mathrm{\:(K)}$
\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 $|J_{\perp}/J|$
\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(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

)
\begin_inset Formula $_{\text{2}}$
\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 $1.4(1)$
\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 $<10$
\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 $50$
\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 $50$
\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 $1.5(1)$
\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 $0.25(10)$
\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 $1.6(3)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
- 
\end_layout

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

\begin_layout Plain Layout
-
\end_layout

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

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

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

](PF
\begin_inset Formula $_{\text{6}}$
\end_inset

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


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.64(1)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $<10$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $50$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $50$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $1.91(1)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.22(2)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $1.1(3)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $4.5(5)$
\end_inset

 *
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.05(3)$
\end_inset


\end_layout

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

\begin_layout Plain Layout
[Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

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


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.599(5)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $15$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $30$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $55$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.649(5)$
\end_inset

 
\end_layout

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

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

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $1.7(1)$
\end_inset

 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $1.09(2)$
\end_inset

 
\begin_inset Formula $^{\dagger}$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $0.23(2)$
\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
Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

O
\begin_inset Formula $_{\text{8}}$
\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 $2.45(4)$
\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 $30$
\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 $70$
\end_inset


\end_layout

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

\begin_layout Plain Layout
-
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $7.8(3)$
\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 $0.19(2)$
\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 $3(2)$
\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 $52.7(3)$
\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 $\sim10^{-6}$
\end_inset


\end_layout

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

\end_inset


\begin_inset Caption

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

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

]
\emph on
Y
\emph default

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

, [Cu(pyo)
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

](BF
\begin_inset script subscript

\begin_layout Plain Layout
4
\end_layout

\end_inset

)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

 and Ag(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(S
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O
\begin_inset script subscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

).
 The first parameters shown relate to fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyz)2(pyo)2X2-A(t)"

\end_inset

 (for the first three rows) or 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Ag(pyz)2(S2O8)-A(t)"

\end_inset

 [for Ag(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(S
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O
\begin_inset script subscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

)].
 This allows us to derive frequencies at 
\begin_inset Formula $T=0$
\end_inset

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

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

, in percent.
 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"

\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

 (*
\begin_inset space ~
\end_inset

The value of 
\begin_inset Formula $g$
\end_inset

 in this compound varies strongly with angle, between 
\begin_inset Formula $g_{ab}=2.028$
\end_inset

 and 
\begin_inset Formula $g_{c}=2.255$
\end_inset

.
 The value of 
\begin_inset Formula $g$
\end_inset

 was thus approximated as 
\begin_inset Formula $(\frac{2}{3}g_{ab}+\frac{1}{3}g_{c})\pm(g_{c}-g_{ab})$
\end_inset

.
 
\begin_inset Formula $^{\dagger}$
\end_inset


\begin_inset space ~
\end_inset

The value of 
\begin_inset Formula $J$
\end_inset

 is extracted from heat capacity and susceptibility from Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Algra1978-Cupyo6X2"

\end_inset

.).
 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

).
 The dash in the 
\begin_inset Formula $p_{3}$
\end_inset

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

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

) reflects the fact that there is no third component in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Ag(pyz)2(S2O8)-A(t)"

\end_inset

.
 Dashes in the 
\begin_inset Formula $J/k_{\mathrm{B}}$
\end_inset

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

 columns for [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

 indicate a lack of pulsed-field data for this material.
\begin_inset CommandInset label
LatexCommand label
name "tab:other2dmm-params"

\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/other2dmm/summarygraphs-pyzpyo/summary-pyzpyo.eps

\end_inset

 
\begin_inset Caption

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

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

]
\emph on
Y
\emph default

\begin_inset Formula $_{\text{2}}$
\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:Cu(pyz)2(pyo)2X2-A(t)"

\end_inset

; (b) and (e) show the frequency 
\begin_inset Formula $\nu$
\end_inset

 as a function of temperature; and (c) and (f) show relaxation rates 
\begin_inset Formula $\lambda_{i}$
\end_inset

 as a function of temperature.
 The relaxation rates 
\begin_inset Formula $\lambda_{1}$
\end_inset

, associated with the oscillation, and 
\begin_inset Formula $\lambda_{3}$
\end_inset

, the fast-relaxing initial component, do not vary significantly with 
\begin_inset Formula $T$
\end_inset

 and are not shown.
 Only a slight trend in 
\begin_inset Formula $\lambda_{2}$
\end_inset

 in the 
\emph on
Y
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

=
\begin_inset space ~
\end_inset

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

 material [graph (f)] is observed.
\begin_inset CommandInset label
LatexCommand label
name "fig:Cu(pyz)2(pyo)2X2-magnetism-graphs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Samples were measured in the LTF apparatus at S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm 
\end_layout

\end_inset

S.
 Example data measured on [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
Y
\emph default

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

 are shown in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cu(pyz)2(pyo)2X2-magnetism-graphs"

\end_inset

, where oscillations are observed in 
\begin_inset Formula $A(t)$
\end_inset

 at a single frequency below 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

.
 Data were fitted to a relaxation function 
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}\left(p_{1}\cos(2\uppi\nu_{1}t)\mathrm{e}^{-\lambda_{1}t}+p_{2}\mathrm{e}^{-\lambda_{2}t}+p_{3}\mathrm{e}^{-\lambda_{3}t}\right)+A_{\mathrm{bg}}.\label{eq:Cu(pyz)2(pyo)2X2-A(t)}
\end{equation}

\end_inset

The small amplitude fraction 
\begin_inset Formula $p_{1}<10\%$
\end_inset

 for both samples refers to muons stopping in a site or set of sites with
 a narrow distribution of quasi-static local magnetic fields, giving rise
 to the oscillations; 
\begin_inset Formula $p_{2}\approx50\%$
\end_inset

 is the fraction of muons stopping in a class of sites giving rise to a
 large relaxation rate 
\begin_inset Formula $30\lesssim\lambda\lesssim60$
\end_inset


\begin_inset space ~
\end_inset

MHz and 
\begin_inset Formula $p_{3}\approx50\%$
\end_inset

 represents the fraction of muons stopping in sites with a small relaxation
 rate 
\begin_inset Formula $\lambda_{3}\approx1\;\mathrm{MHz}$
\end_inset

.
 The data from these compounds fit best with 
\begin_inset Formula $\phi=0$
\end_inset

, and it is thus omitted from this expression.
 Frequencies obtained from fitting the data to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyz)2(pyo)2X2-A(t)"

\end_inset

 were then modelled with 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

.
 The results of these fits are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cu(pyz)2(pyo)2X2-magnetism-graphs"

\end_inset

, and 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:other2dmm-params"

\end_inset

.
\end_layout

\begin_layout Standard
Our results show that [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

 has a transition temperature 
\begin_inset Formula $T_{\mathrm{N}}=1.5\pm0.1\mathrm{\; K}$
\end_inset

 and a quasi-static magnetic field at the muon site 
\begin_inset Formula $\nu_{1}(T=0)=1.4\pm0.1\mathrm{\; MHz}$
\end_inset

.
 No quantities other than 
\begin_inset Formula $\nu_{1}$
\end_inset

 show a significant trend in the temperature region 
\begin_inset Formula $0.1\leq T\leq1.6\mathrm{\: K}$
\end_inset

.
 Above the transition, purely relaxing spectra are observed, displaying
 no 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations.
 As suggested above, this makes the existence of muon sites near the anions
 unlikely.
 A critical exponent of 
\begin_inset Formula $\beta=0.25\pm0.10$
\end_inset

 is found, where the large uncertainty results in part from the difficulty
 in fitting the 
\begin_inset Formula $A(t)$
\end_inset

 data in the critical region.
\end_layout

\begin_layout Standard
Our results for [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

](PF
\begin_inset Formula $_{\text{6}}$
\end_inset

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

 show that the transition temperature is slightly higher at 
\begin_inset Formula $T_{\mathrm{N}}=1.72\pm0.02\mathrm{\; K}$
\end_inset

 and the oscillations occur at a lower frequency of 
\begin_inset Formula $\nu_{1}(T=0)=1.07\pm0.03\mathrm{\; MHz}$
\end_inset

.
 The relaxation rates 
\begin_inset Formula $\lambda_{2}$
\end_inset

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

 also decrease in magnitude as temperature is increased, settling on roughly
 constant values 
\begin_inset Formula $\lambda_{2}\approx0.6\mathrm{\; MHz}$
\end_inset

 and 
\begin_inset Formula $\lambda_{3}\approx15\mathrm{\; MHz}$
\end_inset

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

.
 No other quantities show a significant trend in the temperature region
 
\begin_inset Formula $0.2\leq T\leq1.7\mathrm{\: K}$
\end_inset

.
 Above the transition, relaxing spectra devoid of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations are again observed.
 The critical exponent 
\begin_inset Formula $\beta=0.22\pm0.02$
\end_inset

.
\end_layout

\begin_layout Standard
The small amplitude of the oscillations, common to both samples, might be
 explained in a number of ways.
 The materials may undergo long-range ordering but there may be an increased
 likelihood of stopping in sites where the magnetic field nearly precisely
 cancels.
 Alternatively, a range of similar muon sites may be present with a large
 distribution of frequencies, or alternatively the presence of dynamics,
 washing out any clear oscillations in large fractions of the spectra and
 instead resulting in a relaxation.
 Finally, it is not possible to exclude the possibility that only a small
 volume of the sample undergoes a magnetic transition; this may indicate
 the presence of a small impurity phase, possibly located at either grain
 boundaries or, given that this is a powder sample, near the crystallites'
 surfaces.
 The behaviour of fitted parameters in these materials is qualitatively
 similar to that reported in CuCl
\begin_inset Formula $_{\text{2}}$
\end_inset

(pyz), where there is also a relatively small precessing fraction of muons
 and little variation in relaxation rates as 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 is approached from below
\begin_inset CommandInset citation
LatexCommand cite
key "Lancaster2004-CuX2(pyz)-and-0DMMs"

\end_inset

.
\end_layout

\begin_layout Section
[Cu(pyo)
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

]
\emph on
Z
\emph default

\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:[Cu(pyo)6]X2"

\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/other2dmm/structure/Cupyo6BF4-structure.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Structure of [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

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

, viewed along the three-fold (
\begin_inset Formula $\bm{c}$
\end_inset

-) axis, after Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Reinen1979-Cupyo6X2"

\end_inset

.
 Copper ions are surrounded by octahedra of six oxygens, each part of a
 pyridine-
\emph on
N
\emph default
-oxide ligand; [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

 complexes space-pack with BF
\begin_inset Formula $_{\text{4}}^{-}$
\end_inset

 stabilising the structure.
 Ion sizes are schematic; copper ions are shown twice as large for emphasis,
 and hydrogens have been omitted for clarity.
 As indicated in the top left, the 
\begin_inset Formula $\bm{a}$
\end_inset

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

 directions lie in the plane of the paper, separated by 
\begin_inset Formula $\gamma=120^{\circ}$
\end_inset

, whilst the 
\begin_inset Formula $\bm{c}$
\end_inset

 direction is out of the page.
\begin_inset CommandInset label
LatexCommand label
name "fig:Cu(pyo)6(BF4)2-structure"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The next example is not a coordination polymer, but instead forms a three-dimens
ional structure of packed molecular groups.
 The molecular magnet [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

]
\emph on
Z
\emph default

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

, where 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

, ClO
\begin_inset Formula $_{\text{3}}^{-}$
\end_inset

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

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

 ions on a slightly distorted cubic lattice, located in [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

 complexes, and surrounded by octahedra of oxygen atoms
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2"

\end_inset

.
 The structure is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cu(pyo)6(BF4)2-structure"

\end_inset

.
 This approximately cubic structure, which arises from the molecules' packing,
 might suggest that a three-dimensional model of magnetism would be appropriate.
 In fact, although the observed bulk properties of [
\emph on
M
\emph default
(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

]
\emph on
X
\emph default

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

 where 
\emph on
M
\emph default

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

 
\begin_inset space ~
\end_inset

= Co
\begin_inset Formula $^{\text{2+}}$
\end_inset

, Ni
\begin_inset Formula $^{\text{2+}}$
\end_inset

 or Fe
\begin_inset Formula $^{\text{2+}}$
\end_inset

 are largely isotropic, the copper analogues display quasi--low-dimensional,
 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 Heisenberg antiferromagnetism
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2"

\end_inset

.
 Weakening of superexchange in certain directions, and thus the lowering
 of the systems' effective dimensionality, is attributed to lengthening
 of the superexchange pathways resulting from Jahn--Teller distortion of
 the Cu--O octahedra, which is observed in structural and EPR measurements
\begin_inset CommandInset citation
LatexCommand cite
key "Wood1980-Cupyo6X2,Reinen1979-Cupyo6X2"

\end_inset

.
 At high temperatures, (
\begin_inset Formula $T\gtrsim\mathrm{100\; K})$
\end_inset

, these distortions are expected to be dynamic but, as 
\begin_inset Formula $T$
\end_inset

 is reduced (to 
\begin_inset Formula $\approx50\mathrm{\; K}$
\end_inset

), they freeze out.
 The anion 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset

 determines the nature of the static Jahn--Teller elongation.
 The 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset


\emph on

\begin_inset space ~
\end_inset


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

 material displays ferrodistortive ordering which, in combination with the
 antiferromagnetic exchange, gives rise to 2D Heisenberg antiferromagnetic
 behaviour
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2,Burriel1990-Cupyo6BF4-susceptibility"

\end_inset

.
 By contrast, 
\emph on
Z
\emph default

\begin_inset Formula $^{-}$
\end_inset


\emph on

\begin_inset space ~
\end_inset


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

, NO
\begin_inset Formula $_{\text{3}}^{-}$
\end_inset

 (neither of which is investigated here) display antiferrodistortive ordering
\begin_inset CommandInset citation
LatexCommand cite
key "Wood1980-Cupyo6X2"

\end_inset

, which gives rise to quasi-1D Heisenberg antiferromagnetism
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2"

\end_inset

.
 All of the samples investigated were measured in the LTF spectrometer at
 S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm 
\end_layout

\end_inset

S.
\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/other2dmm/summarygraphs-pyo6/summary-Cupyo6BF42.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Example data and fits for [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

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

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

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

 and 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

, along with fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyz)2(pyo)2X2-A(t)"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyo)6(BF4)2-highT"

\end_inset

, respectively; (b) shows frequency as a function of temperature; and (c)
 shows relaxation rate 
\begin_inset Formula $\lambda_{2}$
\end_inset

 as a function of temperature; 
\begin_inset Formula $\lambda_{1}$
\end_inset

 was held fixed during the fitting procedure, and 
\begin_inset Formula $\lambda_{3}$
\end_inset

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

.
 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.
\begin_inset CommandInset label
LatexCommand label
name "fig:Cupyo6BF4-summary"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the 
\emph on
Z
\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset


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

 compound, below a temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, a single oscillating frequency is observed, indicating a transition to
 a state of long-range magnetic order.
 Example data above and below the transition, along with fits, are shown
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6BF4-summary"

\end_inset


\begin_inset space ~
\end_inset

(a).
 Data were fitted to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyz)2(pyo)2X2-A(t)"

\end_inset

, and the frequencies extracted from the procedure fitted as a function
 of temperature to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

, as shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6BF4-summary"

\end_inset


\begin_inset space ~
\end_inset

(b).
 This procedure identifies a transition temperature 
\begin_inset Formula $T_{\mathrm{N}}=0.649\pm0.005\mathrm{\; K}$
\end_inset

.
 The only other parameter found to vary significantly in the range 
\begin_inset Formula $\SI{20}{\milli\kelvin}\leq T\leq T_{\mathrm{N}}$
\end_inset

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

, shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6BF4-summary"

\end_inset


\begin_inset space ~
\end_inset

(c).
 The fitted parameters are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:other2dmm-params"

\end_inset

.
 We may compare this with the result of an earlier low-temperature specific
 heat study
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2"

\end_inset

 which found a very small 
\begin_inset Formula $\lambda$
\end_inset

-point anomaly at 
\begin_inset Formula $T_{\mathrm{N}}=\SI{0.62(1)}{\kelvin}$
\end_inset

, slightly lower than our result.
 Fitting the magnetic component of the heat capacity with the predictions
 from a two-dimensional Heisenberg antiferromagnet gives 
\begin_inset Formula $J/k_{\mathrm{B}}=\SI{-1.10(2)}{\kelvin}$
\end_inset

, and similar analysis of the magnetic susceptibility
\begin_inset CommandInset citation
LatexCommand cite
key "Algra1978-Cupyo6X2"

\end_inset

 yields 
\begin_inset Formula $J/k_{\mathrm{B}}=\SI{1.08(3)}{\kelvin}$
\end_inset

.
 Using these values, together with the muon estimate of 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Yasuda-J-Jprime"

\end_inset

, allows us to estimate the inter-plane coupling, 
\begin_inset Formula $J_{\perp}/J=\num{0.26(1)}$
\end_inset

.
 (Using the value of 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 from heat capacity results in an estimate 
\begin_inset Formula $J_{\perp}/J=\num{0.21(2)}$
\end_inset

.)
\end_layout

\begin_layout Standard
For temperatures 
\begin_inset Formula $T_{\mathrm{N}}<T\leq\SI{1}{\kelvin}$
\end_inset

, the spectra are well described by a relaxation function 
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}(p_{1}\mathrm{e}^{-\lambda t}+p_{2}\mathrm{e}^{-\sigma^{2}t^{2}})+A_{\mathrm{bg}},\label{eq:Cu(pyo)6(BF4)2-highT}
\end{equation}

\end_inset

 comprising an initial fast-relaxing component with 
\begin_inset Formula $\lambda\approx\SI{20}{\MHz}$
\end_inset

, and a Gaussian relaxation with 
\begin_inset Formula $\sigma\approx\SI{0.4}{\MHz}$
\end_inset

 corresponding to the slow depolarisation of muon spins due to randomly-orientat
ed nuclear moments.
\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/other2dmm/summarygraphs-pyo6/summary-Cupyo6ClO32.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Example data and fits for [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

](ClO
\begin_inset Formula $_{\text{3}}$
\end_inset

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

.
 Representative 
\begin_inset Formula $A(t)$
\end_inset

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

 and 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

, along with fits to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Cu(pyo)6(ClO3)2-A(t)"

\end_inset

, are shown in (a), whilst the value of the relaxation rate, 
\begin_inset Formula $\lambda$
\end_inset

, as a function of temperature is shown in (b).
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Cupyo6ClO32"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 
\emph on
Z
\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset


\emph default
= ClO
\begin_inset Formula $_{\text{3}}^{-}$
\end_inset

 material also shows evidence for a magnetic transition, although in this
 case oscillations in the muon asymmetry are not observed.
 Instead a discontinuous change in the relaxation which seems to point towards
 an ordering transition is seen.
 Example asymmetry spectra are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6ClO32"

\end_inset


\begin_inset space ~
\end_inset

(a) and data at all measured temperatures are well described with the relaxation
 function 
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}\mathrm{e}^{-\lambda t}+A_{\mathrm{bg}}\,.\label{eq:Cu(pyo)6(ClO3)2-A(t)}
\end{equation}

\end_inset

Evidence for a magnetic transition comes from the temperature evolution
 of 
\begin_inset Formula $\lambda$
\end_inset

 [
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6ClO32"

\end_inset

(b)], where the relaxation decreases with increasing temperature until it
 settles at 
\begin_inset Formula $T\approx\SI{0.3}{\kelvin}$
\end_inset

, on a value 
\begin_inset Formula $\lambda\approx\SI{0.5}{\MHz}$
\end_inset

.
 It is likely that this tracks the internal magnetic field inside the material,
 and is suggestive of 
\begin_inset Formula $T_{\mathrm{N}}=\SI{0.30(1)}{\kelvin}$
\end_inset

.
 
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/other2dmm/summarygraphs-pyo6/summary-Cupyo6PF62.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Sample asymmetry spectrum for [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

](PF
\begin_inset Formula $_{\text{6}}$
\end_inset

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

 measured at 
\begin_inset Formula $T=0.02\mathrm{\; K}$
\end_inset

.
 The spectra remain indistinguishable from this across the range of temperatures
 examined, 
\begin_inset Formula $0.02\mathrm{\; K}\leq T\leq1\mathrm{\; K}$
\end_inset

.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:Cupyo6PF62"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The final member of this family studied, 
\emph on
Z
\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset


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

, shows no evidence for a magnetic transition over the range of temperatures
 studied, 
\begin_inset Formula $\SI{0.02}{\kelvin}\leq T\leq\SI{1}{\kelvin}$
\end_inset

.
 An example spectrum is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Cupyo6PF62"

\end_inset

.
 The data resemble the above-transition data measured in the BF
\begin_inset Formula $_{\text{4}}$
\end_inset

 and ClO
\begin_inset Formula $_{\text{3}}$
\end_inset

 compounds and it therefore seems likely that the paramagnetic state persists
 to the lowest temperature measured.
\end_layout

\begin_layout Section
Ag(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

(S
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O
\begin_inset script subscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

)
\begin_inset CommandInset label
LatexCommand label
name "sec:Agpyz2S2O8"

\end_inset


\end_layout

\begin_layout Standard
The examples so far have used Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (3d
\begin_inset Formula $^{\text{9}}$
\end_inset

) as the magnetic species.
 An alternative strategy is to employ Ag
\begin_inset Formula $^{\text{2+}}$
\end_inset

 (4d
\begin_inset Formula $^{\text{9}}$
\end_inset

) which also carries an 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

 moment.
 This idea has led to the synthesis of Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

), which comprises square sheets of [Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

 units spaced with S
\begin_inset Formula $_{\text{2}}$
\end_inset

O
\begin_inset Formula $_{\text{8}}$
\end_inset


\begin_inset Formula $^{\text{2}-}$
\end_inset

 anions
\begin_inset CommandInset citation
LatexCommand cite
key "Manson2009-Agpyz2S2O8"

\end_inset

.
 Each silver ion lies at the centre of an elongated (AgN
\begin_inset Formula $_{\text{4}}$
\end_inset

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

) octahedron, where the Ag--N bonds are significantly shorter than the Ag--O.
 Preparation details can be found in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Matthews1971-Ag-Cu-Co-Ni-pyz-prep"

\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/other2dmm/summarygraphs-Agpyz2S2O8/summary-Agpyz2S2O8.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Example data and fits for Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

).
 From left to right: (a) shows 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:Ag(pyz)2(S2O8)-A(t)"

\end_inset

; (b) shows the frequency as a function of temperature; and (c) shows relaxation
 rates 
\begin_inset Formula $\lambda_{i}$
\end_inset

 as a function of temperature.
 Filled circles show the relaxation rate 
\begin_inset Formula $\lambda_{1}$
\end_inset

, which relaxes the oscillation.
 The filled triangles correspond to the relaxation 
\begin_inset Formula $\lambda_{2}$
\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:Ag2pyz2S2O8-magnetism-graphs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset

Measurements were made using the GPS instrument at S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm 
\end_layout

\end_inset

S.
 Example muon data, along with fits to various parameters, are shown in
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ag2pyz2S2O8-magnetism-graphs"

\end_inset

.
 Asymmetry oscillations are visible in spectra taken below a transition
 temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

.
 The data were fitted with a relaxation function
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}\left(p_{1}\cos(2\uppi\nu_{1}t+\phi_{1})\mathrm{e}^{-\lambda_{1}t}+p_{2}\mathrm{e}^{-\lambda_{2}t}\right)+A_{\mathrm{bg}},\label{eq:Ag(pyz)2(S2O8)-A(t)}
\end{equation}

\end_inset

comprising a single damped oscillatory component, a slow-relaxing component,
 and a static background signal.
 The onset of increased relaxation 
\begin_inset Formula $\lambda_{1}$
\end_inset

 as the transition is approached from below leads to large statistical errors
 on fitted values, as is evident in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ag2pyz2S2O8-magnetism-graphs"

\end_inset


\begin_inset space ~
\end_inset

(c).
 The relaxation 
\begin_inset Formula $\lambda_{2}$
\end_inset

 decreases with increasing temperature.
 Fitting to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

 allows the critical parameters 
\begin_inset Formula $\beta=0.19\pm0.02$
\end_inset

 and 
\begin_inset Formula $T_{\mathrm{N}}=\SI{7.8(3)}{\kelvin}$
\end_inset

 to be determined.
 Fitted values are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:other2dmm-params"

\end_inset

.
\end_layout

\begin_layout Standard
The in-plane exchange 
\begin_inset Formula $J$
\end_inset

 is too large to be determined with pulsed fields
\begin_inset CommandInset citation
LatexCommand cite
key "Manson2009-Agpyz2S2O8"

\end_inset

: 
\begin_inset Formula $M(B)$
\end_inset

 does not saturate in fields up to 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{64}{
\backslash
tesla}
\end_layout

\end_inset

.
 However, fitting 
\begin_inset Formula $\chi(T)$
\end_inset

 data allows an estimate of the exchange 
\begin_inset Formula $J/k_{\mathrm{B}}\approx\SI{53}{\kelvin}$
\end_inset

 (and thus, in conjunction with 
\begin_inset Formula $g$
\end_inset

 measured by EPR, the saturation field 
\begin_inset Formula $B_{\mathrm{c}}$
\end_inset

 is estimated to be 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{160}{
\backslash
tesla}
\end_layout

\end_inset

).
 Thus, 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=0.148\pm0.006$
\end_inset

.
 Estimation of the exhange anisotropy with 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Yasuda-J-Jprime"

\end_inset

 yields 
\begin_inset Formula $|J_{\perp}/J|\sim10^{-6}$
\end_inset

, but this very small ratio of ordering temperature to exchange strength
 is outside the range in which the equation is known to yield accurate results.
 The alternative method of parametrising the low dimensionality in terms
 of correlation length at the Néel temperature (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:exchange-anisotropy"

\end_inset

) yields 
\begin_inset Formula $\xi(T_{\mathrm{N}})/a=1000\pm300$
\end_inset

.
\end_layout

\begin_layout Section
[Ni(HF
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

)(pyz)
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

]
\emph on
X
\emph default

\begin_inset CommandInset label
LatexCommand label
name "sec:dog"

\end_inset


\end_layout

\begin_layout Standard
In order to investigate the influence of a different spin state on the magnetic
 cation in the [
\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
 architecture the [Ni(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
X
\emph default
 (
\emph on
X
\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset


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

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

) system has been synthesised
\begin_inset CommandInset citation
LatexCommand cite
key "Manson2011-NiHF2pyz2X"

\end_inset

.
 These materials are isostructural with the copper family discussed in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:cuhf2pyz2x"

\end_inset

, but contain 
\begin_inset Formula $S=1$
\end_inset

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

 cations.
\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/other2dmm/summarygraphs-NiHF2pyz2X/summary-Ni.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Example data and fits for 
\emph on
M
\begin_inset space ~
\end_inset


\emph default
= Ni magnets.
 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 rate 
\begin_inset Formula $\lambda_{3}$
\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 inset to (f) shows 
\begin_inset Formula $\nu_{2}\equiv\nu$
\end_inset

 plotted against 
\begin_inset Formula $\lambda_{3}\equiv\lambda$
\end_inset

 on a log--log scale.
 The black line shows 
\begin_inset Formula $\lambda=\nu^{2}$
\end_inset

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

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="10">
<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">
<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">
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\emph on
X
\emph default
 
\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset


\begin_inset space ~
\end_inset

(
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
kelvin}
\end_layout

\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 $\nu_{1}$
\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 $\nu_{2}$
\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 $\lambda_{3}$
\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 $p_{1}$
\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 $p_{2}$
\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 $p_{3}$
\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 $\beta$
\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 $\alpha$
\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
PF
\begin_inset Formula $_{\text{6}}$
\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 $6.0(4)$
\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 $12.0$
\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 $9.3$
\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 $90$
\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 $15$
\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 $10$
\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 $75$
\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 $0.25(10)$
\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 $2.8(2)$
\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
SbF
\begin_inset Formula $_{\text{6}}$
\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 $12.26(1)$
\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 $12.3(1)$
\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 $8.98(1)$
\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 $80$
\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 $25$
\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 $10$
\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 $65$
\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 $0.34(4)$
\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 $3.1(1)$
\end_inset

 
\end_layout

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

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Fitted parameters for 
\emph on
M
\begin_inset space ~
\end_inset


\emph default
= Ni magnets.
 Values of 
\begin_inset Formula $\nu_{1}$
\end_inset

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

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

 are given in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
MHz}
\end_layout

\end_inset

.
 Errors shown are statistical uncertainties on fitting and thus represent
 lower bounds.
 Errors on Ni...PF
\begin_inset Formula $_{\text{6}}$
\end_inset

 could not be estimated due to the fitting procedure (see text).
\begin_inset CommandInset label
LatexCommand label
name "tab:NiHF2pyz2X-params"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Data were taken using the GPS spectrometer at PSI.
 Example data are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ni(HF2)(pyz)2X-magnetism-graphs"

\end_inset

.
 Oscillations are observed at two frequencies below the materials' respective
 ordering temperatures.
 Data were 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)+p_{2}\mathrm{e}^{-\lambda_{2}t}\cos(2\pi P_{2}\nu_{1}t)\right.\nonumber \\
 &  & \left.+p_{3}\mathrm{e}^{-\lambda_{3}t}\right]+A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t}.\label{eq:Ni(HF2)(pyz)2X-A(t)-fit}
\end{eqnarray}

\end_inset

Of those muons which stop in the sample, 
\begin_inset Formula $p_{1}\approx25\%$
\end_inset

 indicates the fraction of the signal corresponding to the high-frequency
 oscillating state with 
\begin_inset Formula $\nu_{1}(T=0)\approx12.3\mathrm{\: MHz}$
\end_inset

; 
\begin_inset Formula $p_{2}\approx10\%$
\end_inset

 corresponds to muons stopping in the low-frequency oscillating state with
 
\begin_inset Formula $\nu_{2}(T=0)\approx9.0\mathrm{\; MHz}$
\end_inset

; and 
\begin_inset Formula $p_{3}\approx65\%$
\end_inset

 represents muons stopping in a site with a large relaxation rate 
\begin_inset Formula $\lambda_{3}(T=0)\approx70\mathrm{\; MHz}$
\end_inset

.
 The frequencies were observed to scale with one-another, and consequently
 the second frequency was held in fixed proportion 
\begin_inset Formula $\nu_{2}=P_{2}\nu_{1}$
\end_inset

 during the fitting procedure.
 The only other parameter which changes significantly in value below 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

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

, which decreases with a trend qualitatively similar to that of the frequencies.
 Fitting the extracted frequencies to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

 allows the transition temperature 
\begin_inset Formula $T_{\mathrm{N}}=12.25\pm0.03\mathrm{\: K}$
\end_inset

 and critical exponent 
\begin_inset Formula $\beta=0.34\pm0.04$
\end_inset

 to be extracted.
 In contrast to the copper family studied in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:cuhf2pyz2x"

\end_inset

, the relation 
\begin_inset Formula $\lambda\propto\nu^{2}$
\end_inset

 holds true, suggesting that a field distribution whose width diminishes
 with increasing temperature is responsible for the variation in 
\begin_inset Formula $\lambda$
\end_inset

, and that dynamics are relatively unimportant in determining the muon response.
 This is shown graphically in the inset to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Ni(HF2)(pyz)2X-magnetism-graphs"

\end_inset


\begin_inset space ~
\end_inset

(f), where a plot of frequency against relaxation rate lies on top of a
 line representing a 
\begin_inset Formula $\lambda=\nu^{2}$
\end_inset

 relationship.
 The phase 
\begin_inset Formula $\phi$
\end_inset

 required in previous fits (e.g.
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:M(HF2)(pyz)2X-A(t)-fit"

\end_inset

) is not necessesary in fitting these spectra, and is set to zero.
 Fitted parameters are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:NiHF2pyz2X-params"

\end_inset

.
\end_layout

\begin_layout Standard
Data for the 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

 compound was subject to similar analysis, fitting spectra below 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

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

\end_inset

, this time with 
\begin_inset Formula $p_{1}\approx15\%$
\end_inset

, 
\begin_inset Formula $\nu_{1}(T=0)\approx12.0\mathrm{\: MHz}$
\end_inset

; 
\begin_inset Formula $p_{2}\approx10\%$
\end_inset

, 
\begin_inset Formula $\nu_{2}(T=0)\approx9.3\mathrm{\: MHz}$
\end_inset

; and 
\begin_inset Formula $p_{3}\approx75\%$
\end_inset

, 
\begin_inset Formula $\lambda_{3}(T=0)\approx100\mathrm{\; MHz}$
\end_inset

.
 The phase 
\begin_inset Formula $\phi$
\end_inset

 again proved unnecessary.
 These spectra do not show as sharp a transition as the 
\emph on
X
\emph default

\begin_inset Formula $^{-}$
\end_inset


\begin_inset space ~
\end_inset

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

 compound, with the oscillating fraction of the signal decaying before the
 appearance of spectra whose different character indicates clearly that
 the sample is above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

.
 The available data do not allow reliable extraction of critical parameters,
 but it is possible to estimate 
\begin_inset Formula $\SI{5.5}{\kelvin}\leq T_{\mathrm{N}}\leq\SI{6.2}{\kelvin}$
\end_inset

 and 
\begin_inset Formula $0.15\leq\beta\leq0.4$
\end_inset

.
 Fitted values are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:NiHF2pyz2X-params"

\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/other2dmm/summarygraphs-NiHF2pyz2X/NiPF6-Alongt.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
The asymmetry at late times 
\begin_inset Formula $A(t>5\mathrm{\;\murm s)}$
\end_inset

 measured in Ni(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

PF
\begin_inset Formula $_{\text{6}}$
\end_inset

 along with a fit to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:NiPF6-Alatet-Fermi-step"

\end_inset

.
 Overlaid are regions corresponding to the range of values found by fitting
 the frequency of oscillations as a function of temperature with 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

 (
\begin_inset Formula $5.5\mathrm{\; K}\leq T_{\mathrm{N}}\leq6.2\mathrm{\; K}$
\end_inset

), and the value of the 
\begin_inset Formula $\lambda$
\end_inset

-like anomaly in specific heat capacity (SHC) measurements with a representative
 error of 
\begin_inset Formula $0.1\mathrm{\; K}$
\end_inset

 (
\begin_inset Formula $6.1\mathrm{\; K}\leq T_{\mathrm{N}}\leq6.3\mathrm{\; K}$
\end_inset

).
 The darker area represents the overlap between these regions.
\begin_inset CommandInset label
LatexCommand label
name "fig:NiPF6-Alatet"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Another method to locate the transition is to observe the amplitude of the
 muon spectra at late times as a function of temperature.
 The transition is then from zero in the unordered state to the `
\begin_inset Formula $\frac{1}{3}$
\end_inset

-tail' characteristic of LRO, described in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:zero-field"

\end_inset

.
 Spectra were fitted with the simple relaxation function 
\begin_inset Formula $A(t>5\mathrm{\;\murm s})=A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t}$
\end_inset

, and then the amplitudes obtained were fitted with a Fermi-like step function
 
\begin_inset Formula 
\begin{equation}
A(t>5\mathrm{\;\murm s},\; T)=A_{2}+\frac{A_{1}-A_{2}}{\mathrm{e}^{(T-T_{\mathrm{mid}})/w}+1}\,,\label{eq:NiPF6-Alatet-Fermi-step}
\end{equation}

\end_inset

which provides a method of modelling a smooth transition between 
\begin_inset Formula $A_{1}=A(T<T_{\mathrm{N}})$
\end_inset

 and 
\begin_inset Formula $A_{2}=A(T>T_{\mathrm{N}})$
\end_inset

.
 The fitted amplitudes and Fermi function are shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:NiPF6-Alatet"

\end_inset

.
 The fitted mid-point 
\begin_inset Formula $T_{\mathrm{mid}}=6.4\pm0.1\mathrm{\; K}$
\end_inset

, and width 
\begin_inset Formula $w=0.3\pm0.1\mathrm{\; K}$
\end_inset

.
 Spin relaxation peaks just above 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, and so one would expect that 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 lies at the lower end of this transition.
 Thus, the 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
muSR
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

analysis suggests 
\begin_inset Formula $T_{\mathrm{N}}=6.1\pm0.3\mathrm{\; K}$
\end_inset

 (i.e.
\begin_inset space ~
\end_inset


\begin_inset Formula $T_{\mathrm{mid}}-w\pm w$
\end_inset

).
 This is consistent with the estimate from 
\begin_inset Formula $\nu_{i}(T)$
\end_inset

 and the value 
\begin_inset Formula $T_{\mathrm{N}}=6.2\mathrm{\; K}$
\end_inset

 obtained from heat capacity
\begin_inset CommandInset citation
LatexCommand cite
key "Manson2011-NiHF2pyz2X"

\end_inset

.
\end_layout

\begin_layout Standard
Members of the 
\emph on
M
\emph default

\begin_inset space ~
\end_inset

= Ni family exhibit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset

oscillations rather like their copper counterparts, with a similar fraction
 of the muons in sites giving rise to dipole--dipole interactions.
 The results of these fits are shown along with those from Cu compounds
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Cu(HF2)(pyz)2X-muF-parameters"

\end_inset

.
 Because the nickel data were measured at temperatures different from those
 of the copper compounds and, as described in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:[Cu(HF2)(pyz)2]X-paramagnetic"

\end_inset

, the muon--fluorine bond length in the compound is sensitive to changes
 in temperature, care must be taken when comparing these values to those
 of the Cu family.
 Linearly interpolating the bond lengths 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

 at 
\begin_inset Formula $9\mathrm{\; K}$
\end_inset

 and 
\begin_inset Formula $29\mathrm{\; K}$
\end_inset

 to find an approximate value of the bond length at 
\begin_inset Formula $19\mathrm{\; K}$
\end_inset

 yields 
\begin_inset Formula $r_{\text{\murm--F}}(T=19\mathrm{\; K})=0.1062\pm0.0002\mathrm{\; nm}$
\end_inset

 (where the error represents a combination of the statistical errors on
 the fits and variation recorded in thermometry, and is thus a lower bound),
 very similar to that measured for [Ni(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

, 
\begin_inset Formula $r_{\text{\murm--F}}(T=19\mathrm{\; K})=0.1068\pm0.0004\mathrm{\; nm}$
\end_inset

.
\end_layout

\begin_layout Standard
In spite of being isostructural to the [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
X
\emph default
 systems, the dimensionality of these Ni variants is ambiguous.
 Susceptibility data fit acceptably to a number of models, and 
\emph on
ab initio
\emph default
 theoretical calculations are suggestive of one-dimensional behaviour, dominated
 by the exchange along the bifluoride bridges.
 This is discussed more fully in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Manson2011-NiHF2pyz2X"

\end_inset

.
\end_layout

\begin_layout Section
Discussion
\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/other2dmm/review/2dmagnets-rev.eps

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Quantification of the two-dimensionality of the materials in this thesis
 and comparison with other notable 2DSLQHA systems.
 Filled circles show materials investigated in this thesis; open circles
 show other molecular materials; and filled triangles show inorganic systems.
 The low dimensionality is parametrised firstly with the directly experimental
 ratio 
\begin_inset Formula $T_{\mathrm{N}}/J$
\end_inset

, and then with predictions derived from quantum Monte Carlo simulations
 for both the exchange anisotropy, 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 [
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Yasuda-J-Jprime"

\end_inset

], and the correlation length of an ideal 2D Heisenberg antiferromagnet
 with the measured 
\begin_inset Formula $J$
\end_inset

 at a temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 [
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Beard-correlation-length"

\end_inset

].
 The greying-out of the axis for 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 indicates where 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Yasuda-J-Jprime"

\end_inset

 is extrapolated beyond the range for which it was originally derived
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "yasuda2005-2DHA-JvsTN"

\end_inset

.
 Values of 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J$
\end_inset

 for the other materials were evaluated from Refs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Woodward2001-5BAP2CuBr4,Greven1995-Sr2CuO2Cl2,Aeppli1989-La2CuO4-Ba-doping"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:exchange-anisotropy-review"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:exchange-anisotropy-review"

\end_inset

 collects the results from this and the preceding chapter and shows how
 isolation between two-dimensional layers varies over a variety of systems;
 those presented in this thesis, molecular materials studied elsewhere,
 and inorganic materials.
 The primary axis is the experimental ratio 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J$
\end_inset

.
 Also shown are the the ratios of the in-plane and inter-plane exchange
 interactions, 
\begin_inset Formula $J_{\perp}/J$
\end_inset

, and the correlation length at the transition, 
\begin_inset Formula $\xi(T_{\mathrm{N}})/a$
\end_inset

, extracted from fits to quantum Monte Carlo simulations.
 Of the materials in this thesis, the least anisotropic is [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

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

, whose low transition temperature is caused by a small exchange constant
 rather than particularly high exchange anisotropy.
 Below this, [Cu(pyo)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

](PF
\begin_inset Formula $_{\text{6}}$
\end_inset

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

 exhibits a ratio 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J\approx0.4$
\end_inset

.
 The members of the [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
X
\emph default
 family with octahedral anions have 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J\approx0.33$
\end_inset

, rather more than the 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J\approx0.25$
\end_inset

 shown by their counterparts with tetrahedral anions (as has been noted
 previously
\begin_inset CommandInset citation
LatexCommand cite
key "goddard2008-2DHMexchange"

\end_inset

).
 This makes the latter comparable to highly 2D molecular systems [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

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

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

(pyz), and the cuprate parent compound La
\begin_inset Formula $_{\text{2}}$
\end_inset

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

.
 The prototypical inorganic 2D system Sr
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

 exhibits 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=0.177\pm0.009$
\end_inset

, and thus 
\begin_inset Formula $\xi(T_{\mathrm{N}})/a=280\pm90$
\end_inset

.
 The most 2D material investigated in these chapters, Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

), has 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J=0.148\pm0.006$
\end_inset

, implying 
\begin_inset Formula $\xi(T_{\mathrm{N}})/a=1000\pm300$
\end_inset

, with the added benefit that the magnetic field required to probe its interacti
ons is far closer to the range of fields achievable in the laboratory.
 By these measures, molecular magnets provide some excellent realisations
 of the 2DSLQHA, with Ag(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

) being the best realisation found to date.
\end_layout

\begin_layout Standard
Another method of examining the dimensionality of these systems is to consider
 their behaviour in the critical region.
 The critical exponent 
\begin_inset Formula $\beta$
\end_inset

 is a quantity frequently extracted in studies of magnetic materials, and
 it is often used to make inferences about the dimensionality of the system
 under study.
 In the critical region near a magnetic transition, an order parameter 
\begin_inset Formula $\Phi$
\end_inset

, identical to the (staggered) magnetisation, would be expected to vary
 as
\begin_inset Formula 
\begin{equation}
\Phi(T)=\Phi_{0}\left(1-\frac{T}{T_{\mathrm{N}}}\right)^{\beta}.\label{eq:critical-beta}
\end{equation}

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
differing from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

 in that the parameter 
\begin_inset Formula $\alpha$
\end_inset

, used to account phenomenologically for behaviour such as spin waves as
 
\begin_inset Formula $T\rightarrow0$
\end_inset

, is omitted.
 --keep for thesis
\end_layout

\end_inset

In simple, isotropic cases, the value of 
\begin_inset Formula $\beta$
\end_inset

 depends on the dimensionality of the system, 
\begin_inset Formula $d$
\end_inset

, and that of the order parameter, 
\begin_inset Formula $D$
\end_inset

.
 For example, in the 3D Heisenberg model (
\begin_inset Formula $d=3$
\end_inset

, 
\begin_inset Formula $D=3$
\end_inset

), 
\begin_inset Formula $\beta=0.367$
\end_inset

, whilst in the 2D Ising model (
\begin_inset Formula $d=2$
\end_inset

, 
\begin_inset Formula $D=1$
\end_inset

), 
\begin_inset Formula $\beta=\frac{1}{8}$
\end_inset

.
 Since the muon precession frequency is proportional to the local field,
 it is also proportional to the moment on the magnetic ions in a crystal,
 and can be used as an effective order parameter.
 However, 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:critical-beta"

\end_inset

 would only be expected to hold true in the critical region.
 The extent of the critical region (defined as that region where simple
 mean-field theory does not apply) can be parametrised by the Ginzburg temperatu
re 
\begin_inset Formula $T_{\mathrm{G}}$
\end_inset

, which is related to the transition temperature 
\begin_inset Formula $T_{\mathrm{c}}$
\end_inset

 by
\begin_inset CommandInset citation
LatexCommand cite
key "Chaikin-and-Lubensky"

\end_inset


\begin_inset Formula 
\begin{equation}
\frac{|T_{\mathrm{G}}-T_{\mathrm{c}}|}{T_{\mathrm{c}}}=\left[\left(\frac{\xi}{a}\right)^{d}\left(\frac{\Delta C}{k_{\mathrm{B}}}\right)\right]^{\frac{2}{d-4}},
\end{equation}

\end_inset

where 
\begin_inset Formula $d$
\end_inset

 is the dimensionality, 
\begin_inset Formula $\xi$
\end_inset

 is the correlation length and 
\begin_inset Formula $\Delta C$
\end_inset

 is the discontinuity in the heat capacity.
 Quantum Monte Carlo simulations suggest
\begin_inset CommandInset citation
LatexCommand cite
key "sengupta2003-bulkmeasuresbad"

\end_inset

 that 
\begin_inset Formula $\Delta C/k_{\mathrm{B}}\approx J_{\perp}/J$
\end_inset

.
 It follows that for 
\begin_inset Formula $d=3$
\end_inset

, where 
\begin_inset Formula $\Delta C/k_{\mathrm{B}}\approx1$
\end_inset

 we have 
\begin_inset Formula $\left|T_{\mathrm{G}}-T_{\mathrm{c}}\right|/T_{\mathrm{c}}\approx(\xi/a)^{-6}$
\end_inset

, giving rise to a narrow critical region.
 In two dimensions, we have 
\begin_inset Formula $\left|T_{\mathrm{G}}-T_{\mathrm{c}}\right|/T_{\mathrm{c}}\approx(\xi/a)^{-2}(\Delta C/k_{\mathrm{B}})^{-1}$
\end_inset

.
 Anisotropic materials with small 
\begin_inset Formula $J_{\perp}/J$
\end_inset

 only show a small heat capacity discontinuity, while 
\begin_inset Formula $\xi/a$
\end_inset

 grows according to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Beard-correlation-length"

\end_inset

.
 This leads to a 
\begin_inset Formula $\left|T_{\mathrm{G}}-T_{\mathrm{c}}\right|/T_{\mathrm{c}}$
\end_inset

 of order 1 for our materials, that is, a larger critical region for 2D
 (as compared to 3D) systems.
\end_layout

\begin_layout Standard
The large critical region in these materials allows meaningful critical
 parameters to be extracted from muon data.
 The simplest method of doing so is to fit the data to 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

, as has been performed throughout this thesis; alternatively, critical
 scaling plots can be used (e.g.
 Ref.
\begin_inset space ~
\end_inset


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

\end_inset

), which have been performed; the results of these were found to be unchanged
 within error.
 Since 
\begin_inset Formula $\beta$
\end_inset

 might be expected to give an indication of the dimensionality of the hydrodynam
ical fluctuations in these materials, a comparison between extracted 
\begin_inset Formula $\beta$
\end_inset

 and exchange anisotropy parametrised by 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J$
\end_inset

 is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:beta-vs-TNJ"

\end_inset

.
 Members of the [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
X
\emph default
 family show some correlation between the critical exponent and the effective
 dimensionality but overall, the relationship is weak.
 This is probably because 
\begin_inset Formula $\beta$
\end_inset

, which is not a Hamiltonian parameter, is not simply a function of the
 dimensionality of the interactions, but probes the nature of the critical
 dynamics (including propagating and diffusive modes) which could differ
 substantially between systems.
\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/other2dmm/review/beta-vs-TNoverJ.eps

\end_inset

 
\begin_inset Caption

\begin_layout Plain Layout
Critical exponent 
\begin_inset Formula $\beta$
\end_inset

, as commonly extracted from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:nuT1-T-over-TN-alpha-beta"

\end_inset

, plotted against the experimental ratio 
\begin_inset Formula $k_{\mathrm{B}}T_{\mathrm{N}}/J$
\end_inset

, indicative of exchange anisotropy.
 Bright filled circles indicate [Cu(HF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

]
\emph on
X
\emph default
 materials examined in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:cuhf2pyz2x"

\end_inset

, whilst darker circles indicate other materials studied in this work: [Cu(pyz)
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

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

\end_inset

); [Cu(pyo)
\begin_inset Formula $_{\text{6}}$
\end_inset

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

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

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

\end_inset

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

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

O
\begin_inset Formula $_{\text{8}}$
\end_inset

) (
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:Agpyz2S2O8"

\end_inset

).
 Open circles indicate other molecular materials: CuF
\begin_inset Formula $_{\text{2}}$
\end_inset

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

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

(pyz) (Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Manson2008-CuF2(H2O)2(pyz)"

\end_inset

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

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

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

 (Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Lancaster2007-Cupyz2ClO42"

\end_inset

).
\begin_inset CommandInset label
LatexCommand label
name "fig:beta-vs-TNJ"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Conclusions
\end_layout

\begin_layout Standard
A systematic study has been presented of muon-spin relaxation measurements
 on several families of quasi two-dimensional molecular antiferromagnet,
 comprising ligands of pyrazine, bifluoride and pyridine-
\emph on
N
\emph default
-oxide; and the magnetic metal cations Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

, Ag
\begin_inset Formula $^{\text{2+}}$
\end_inset

 and Ni
\begin_inset Formula $^{\text{2+}}$
\end_inset

.
 In each case 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 has been shown to be sensitive to the transition temperature 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, which is often difficult to unambiguously identify with specific heat
 and magnetic susceptibility measurements.
 These measurements have been combined with predictions of quantum Monte
 Carlo calculations to identify the extent to which each is a good realisation
 of the 2DSLQHA model.
 The critical parameters derived from following the temperature evolution
 of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 precession frequencies do not show a strong correlation with the degree
 of isolation of the 2D magnetic layers.
\end_layout

\begin_layout Standard
The analysis of magnetic ordering in zero applied field in terms of inter-layer
 coupling 
\begin_inset Formula $J_{\perp}$
\end_inset

 presented here does not take into account the effect of single-ion--type
 anisotropy on the magnetic order.
 This has been suggested to be important close to 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

 in several examples of 2D molecular magnet
\begin_inset CommandInset citation
LatexCommand cite
key "Xiao2009-XY-q2DHA"

\end_inset

 where it causes a crossover to 
\begin_inset Formula $XY$
\end_inset

-like behaviour.
 In fact, its influence is confirmed in the nonmonotonic 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram seen 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

.
 It is likely that this is one factor that determines the ordering temperature
 of a system, although, as shown in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Xiao2009-XY-q2DHA"

\end_inset

, it is a smaller effect than the interlayer coupling parametrised by 
\begin_inset Formula $J_{\perp}$
\end_inset

.
 The future synthesis of single crystal samples of these materials will
 allow the measurement of the single-ion anisotropies for the materials
 studied here.
\end_layout

\begin_layout Standard
The presence of muon--fluorine dipole--dipole oscillations allows the determinat
ion of some muon sites in these materials, although it appears from our
 results that these are not the sites that give rise  to magnetic oscillations.
 However, the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu
\backslash
 
\end_layout

\end_inset

 signal has been shown to be useful in identifying transitions at temperatures
 well above the magnetic ordering transition, which appear to have a structural
 origin.
 The fluorine oscillations hamper the study of dynamic fluctuations above
 
\begin_inset Formula $T_{\mathrm{N}}$
\end_inset

, which often appear as a residual relaxation on top of the dominant nuclear
 relaxation.
 It may be possible in future to use RF radiation to decouple the influence
 of the fluorine from the muon ensemble to allow muons to probe the dynamics.
\end_layout

\begin_layout Standard
The muon-spin precession signal, upon which much of the analysis presented
 here is based, is seen most strongly in the materials containing Cu
\begin_inset Formula $^{\text{2+}}$
\end_inset

 and is more heavily relaxed in the Ni
\begin_inset Formula $^{\text{2+}}$
\end_inset

 materials.
 This is likely due to the larger spin value in the Ni-containing materials.
 This is borne out by measurements on pyz-based materials containing Mn
 and Fe ions
\begin_inset CommandInset citation
LatexCommand cite
key "Lancaster2006-Mndca2pyz-and-FeNCS2pyz2"

\end_inset

, where no oscillations are observed, despite the presence of magnetic order
 shown unambiguously by other techniques.
 In the case of Mn-containing materials, magnetic order is found with 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 through a change in relative amplitudes of relaxing signals due to a differeren
ce in the nature of the relaxation on either side of the transition.
 It is likely, therefore, that muon studies of molecular magnetic materials
 containing ions with small spin quantum numbers will be most fruitful in
 the future.
\end_layout

\begin_layout Standard
Finally, the temperature dependence of the relaxation rates in these materials
 has been shown to be quite complex, reflecting the variety of muon sites
 in these systems.
 In favourable cases these data could be used to probe critical behaviour,
 such as critical slowing down, although the unambiguous identification
 of such behaviour may be problematic.
\end_layout

\begin_layout Standard
Despite these limitations on the use of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 in examining molecular magnetic systems of the type studied here, it is
 worth stressing that the technique still appears uniquely powerful in providing
 insights into the magnetic behaviour of these materials and will certainly
 be useful in the future as a wealth of new systems are synthesised and
 the goal of microscopically engineering such materials is approached.
\end_layout

\end_body
\end_document
