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\begin_body

\begin_layout Chapter
Muon-spin rotation and relaxation
\begin_inset CommandInset label
LatexCommand label
name "chap:musr"

\end_inset


\end_layout

\begin_layout Standard
In 
\emph on
muon-spin rotation and relaxation
\emph default
 (known together as 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR 
\end_layout

\end_inset

)
\begin_inset CommandInset citation
LatexCommand citep
key "Blundell1999-mureview"

\end_inset

 we take spin-polarised muons and fire them into a sample of interest, and
 the time-evolution of their spins is then monitored.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 is often compared to nuclear magnetic resonance (NMR) or electron spin
 resonance (ESR), with which it was named in analogy.
 However, a fortunate accident of nature, the parity-violating weak interaction,
 allows muons to be created in fully spin-polarised ensembles, and removes
 the requirement for a radio-frequency or microwave pulse to perturb the
 system to allow a measurement to be made.
 Thus, it is incorrect to refer to conventional 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 as a resonance experiment, although there does exist a technique known
 as 
\emph on
muon-spin resonance
\emph default
, another 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR 
\end_layout

\end_inset

, which does employ electromagnetic pulses to excite spins of muons, or
 those in the sample
\begin_inset CommandInset citation
LatexCommand citep
key "ISIS-RF-guide,Hitti1997-muSR-RF-Si,Cottrell1998-RF-pulsed-muSR"

\end_inset

.
\end_layout

\begin_layout Standard
In the following, I describe the principles of a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 experiment: the muon is introduced in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:the-muon"

\end_inset

; then, the experimental method is explored in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:the-musr-experiment"

\end_inset

, from production of the particles, through guiding to and implantation
 in the sample, to decay, and then detection of the decay products.
 The idealised experimental set-up is laid out in 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:experimental-set-up"

\end_inset

; this is followed by an overview of the kinds of relaxation functions used
 in this thesis, given in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:relaxation-functions"

\end_inset

; and finally, some practical considerations are given in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:experimental-considerations"

\end_inset

.
\end_layout

\begin_layout Standard
Full reviews of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 can be found in Refs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Blundell1999-mureview,Cox1987-muon-review,Roduner1997-muSR,Dalmas1997-musr,Schenck1985-muSR-book"

\end_inset

.
\end_layout

\begin_layout Section
The muon
\begin_inset CommandInset label
LatexCommand label
name "sec:the-muon"

\end_inset


\end_layout

\begin_layout Standard
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wide false
sideways false
status open

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property
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value
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mass, 
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SI{1.8835e-28}{
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SI{105.66}{
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\backslash
per c^{2}}
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spin, 
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mean lifetime, 
\begin_inset Formula $\tau$
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\begin_inset Caption

\begin_layout Plain Layout
The properties of the positive muon, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muon 
\end_layout

\end_inset

.
 In the table, 
\begin_inset Formula $c$
\end_inset

 is the speed of light, 
\begin_inset Formula $m_{\mathrm{e}}$
\end_inset

 is the electron mass, 
\begin_inset Formula $m_{\mathrm{p}}$
\end_inset

 is the proton mass, 
\begin_inset Formula $e$
\end_inset

 is the elementary charge, 
\begin_inset Formula $\mu_{\mathrm{B}}$
\end_inset

 is the Bohr magneton and 
\begin_inset Formula $\mu_{\mathrm{N}}$
\end_inset

 is the nuclear magneton.
\begin_inset CommandInset label
LatexCommand label
name "tab:muon-properties"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Muons are charged leptons---elementary particles belonging to the same family
 as electrons.
 The muon's properties are identical to those of the electron, except that
 it is around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
num{200}
\end_layout

\end_inset

 times heavier; like the electron, its antimatter and matter incarnations
 carry a charge 
\begin_inset Formula $\pm e$
\end_inset

, respectively, and both carry a spin 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

.
 It is unstable, with a mean lifetime of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{2.2}{
\backslash
us}
\end_layout

\end_inset

, making the muon the longest-lived unstable subatomic particle with the
 exception of the neutron; the muon lifetime exceeds that of pions or kaons,
 the next most stable subatomic particles, by one hundred times
\begin_inset CommandInset citation
LatexCommand cite
key "Nakamura2010-review-of-particle-physics"

\end_inset

.
 Its properties are summarised in 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:muon-properties"

\end_inset

.
 The only practical comparable probes for condensed matter physics are light
 atomic nuclei which undergo 
\begin_inset Formula $\upbeta$
\end_inset

-decay
\begin_inset CommandInset citation
LatexCommand cite
key "Kiefl2003-betaNMR"

\end_inset

, for example 
\begin_inset script superscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

Li.
 However, these ions can only be implanted near the surface and constitute
 a significantly larger defect in a crystal than a muon
\begin_inset CommandInset citation
LatexCommand cite
key "Kiefl2003-betaNMR"

\end_inset

.
 A review of the properties of common condensed matter probes is available
 in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:condensed-matter-probes"

\end_inset

.
\end_layout

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\begin_layout Plain Layout
\begin_inset Formula $>\SI{4.6e26}{y}$
\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 ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{2.197e-6}{
\backslash
second}
\end_layout

\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 ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{885}{
\backslash
second}
\end_layout

\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 ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{1.2}{
\backslash
second}
\end_layout

\end_inset


\end_layout

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

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
The properties of the positive muon, 
\begin_inset Formula $\muon$
\end_inset

, compared to other common probes in condensed matter physics, the neutron,
 n, electron, e
\begin_inset script superscript

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


\end_layout

\end_inset

, photon, 
\begin_inset Formula $\upgamma$
\end_inset

, and 
\begin_inset script superscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

Li
\begin_inset script superscript

\begin_layout Plain Layout
+
\end_layout

\end_inset

, an ion commonly used in 
\begin_inset Formula $\upbeta$
\end_inset

NMR
\begin_inset CommandInset citation
LatexCommand cite
key "Kiefl2003-betaNMR"

\end_inset

.
 Values are quoted in terms of the electron mass 
\begin_inset Formula $m_{\mathrm{e}}$
\end_inset

, the elementary charge 
\begin_inset Formula $e$
\end_inset

, and the Bohr magneton 
\begin_inset Formula $\mu_{\mathrm{B}}$
\end_inset

.
 Data are from Refs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Nakamura2010-review-of-particle-physics,Audi2003-nuclear-properties"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "tab:condensed-matter-probes"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Muons are produced naturally by interactions between high-energy particles
 with the atomic nuclei of gases in the upper atmosphere.
 In fact, there are around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
num{10000}
\end_layout

\end_inset

 muons passing through every square metre of the Earth's surface every minute
\begin_inset CommandInset citation
LatexCommand cite
key "Nakamura2010-review-of-particle-physics"

\end_inset

, and consequently several passing through you every second as you read
 this.
 These natural muons---dubbed 
\emph on
cosmic rays
\emph default
---were the first to be observed in 1936
\begin_inset CommandInset citation
LatexCommand citep
key "Anderson1936-muon-discovery"

\end_inset

.
 The fact that so many survive the journey from their production high in
 the atmosphere at around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{15}{
\backslash
kilo
\backslash
metre}
\end_layout

\end_inset

 altitude
\begin_inset CommandInset citation
LatexCommand cite
key "Nakamura2010-review-of-particle-physics"

\end_inset

 (roughly 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{50}{
\backslash
us}
\end_layout

\end_inset

 at the speed of light, and thus over 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
num{20}
\end_layout

\end_inset

 muon lifetimes), is a demonstration of relativistic time-dilation.
 This was confirmed in 1963, by identical muon-counting experiments atop
 Mount Washington and near sea level in Cambridge, Massachusetts, which
 found that the ratio of count rates was commensurate with the predictions
 of relativity
\begin_inset CommandInset citation
LatexCommand cite
key "Frisch1963-muon-time-dilation"

\end_inset

.
\end_layout

\begin_layout Section
The 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 experiment
\begin_inset CommandInset label
LatexCommand label
name "sec:the-musr-experiment"

\end_inset


\end_layout

\begin_layout Standard
The first 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 experiment
\begin_inset CommandInset citation
LatexCommand cite
key "Garwin1957-first-musr"

\end_inset

 was performed in 1957.
 Its purpose was not to examine a condensed matter system, but to test the
 Lee--Yang proposal that the weak interaction violated parity
\begin_inset CommandInset citation
LatexCommand cite
key "Lee1956-parity-violation-weak"

\end_inset

.
 This further corroborated the proposal, and demonstrated the principles
 that underpin all muon spectroscopy: without parity violation, we could
 neither produce our spin-polarised muons in the first place, nor glean
 anything useful about them from their decay.
\end_layout

\begin_layout Standard
Facilities producing muons for condensed matter exist at only a few locations
 around the world.
 The work contained in this thesis was performed primarily at ISIS, the
 synchrotron at the Rutherford Appleton Lab, Didcot, UK, and the Swiss Muon
 Source (S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
murm 
\end_layout

\end_inset

S) fed by the cyclotron at the Paul Scherrer Institut, Villigen, Switzerland.
\end_layout

\begin_layout Standard
The following section outlines the principles of a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 experiment, starting with the particle physics processes which lead to
 muon production, then describing successively the processes by which a
 muon is guided to the sample, embeds within it, interacts once embedded,
 and finally, the physics of muon decay which allows us to make measurements
 of its interactions.
\end_layout

\begin_layout Subsection
Muon production
\begin_inset CommandInset label
LatexCommand label
name "sub:muon-production"

\end_inset


\end_layout

\begin_layout Standard
The mass of the muon makes it impossible to produce one by nuclear fission,
 fusion or decay; the only way to produce muons is in a particle accelerator.
 A beam of high-energy protons is collided with a thin production target
 of a light element (taking ISIS as an example, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{800}{
\backslash
MeV}
\end_layout

\end_inset

 protons are collided with graphite).
 Pions are then produced by collision between the protons and nucleons in
 the target's atomic nuclei by processes such as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\mathrm{p+p} & \rightarrow & \mathrm{p+n+\uppi^{+}},\nonumber \\
\mathrm{p+n} & \rightarrow & \mathrm{p+p+\uppi^{-}},
\end{eqnarray}

\end_inset

where 
\begin_inset Formula $\mathrm{p}$
\end_inset

 denotes proton, 
\begin_inset Formula $\mathrm{n}$
\end_inset

 neutron and 
\begin_inset Formula $\mathrm{\uppi^{\pm}}$
\end_inset

 denote the positive and negative pion, respectively.
 This process stops a few percent of the protons, leaving the rest of the
 beam largely undeviated such that it can be used for other applications,
 often neutron production, downstream.
\end_layout

\begin_layout Standard
Charged pions comprise a down quark and an up antiquark, or vice-versa.
 They are the lightest charged hadrons, and consequently must decay via
 the weak interaction into leptons, which they do with a mean lifetime 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{26}{
\backslash
ns}
\end_layout

\end_inset

.
 Conservation of lepton number requires that an appropriate neutrino (
\begin_inset Formula $\upnu$
\end_inset

) be produced, giving rise to the reaction
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\mathrm{\uppi}^{+} & \rightarrow & l^{+}+\upnu_{l}\,,\label{eq:pi-decay-generic}
\end{eqnarray}

\end_inset

where 
\begin_inset Formula $l=\mathrm{\murm,\, e}$
\end_inset

; a muon or an electron.
 An accident of the weak interaction and conservation of angular momentum
 results in 
\begin_inset Formula $l=\murm$
\end_inset

 being preferred over 
\begin_inset Formula $l=\mathrm{e}$
\end_inset

 with a ratio of approximately 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
num{8000}
\end_layout

\end_inset

:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
num{1}
\end_layout

\end_inset

.
 A further upshot of conserving angular momentum is that the muons produced
 are fully spin-polarised.
\end_layout

\begin_layout Standard
The muons used for experiments in this thesis are positively charged antimuons.
 Negative muons 
\begin_inset Formula $\murm^{-}$
\end_inset

 usually fall into low atomic orbitals in extreme proximity to atomic nuclei.
 An order of magnitude estimate can be made by evaluating the Bohr radius
 with the muon mass substituted for the electron mass,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
a_{\murm}\approx\frac{4\uppi\epsilon_{0}\hbar^{2}}{m_{\mathrm{\murm}}e^{2}}=a_{0}\frac{m_{\mathrm{e}}}{m_{\murm}}=\SI{26}{\pico\metre}
\end{equation}

\end_inset

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

 is the permittivity of free space, 
\begin_inset Formula $\hbar$
\end_inset

 is the reduced Planck constant, 
\begin_inset Formula $m_{\mathrm{e}}$
\end_inset

 is the electron mass and 
\begin_inset Formula $e$
\end_inset

 is the fundamental electric charge.
 The muonic Bohr radius 
\begin_inset Formula $a_{\murm}$
\end_inset

 is then related to the electronic Bohr radius 
\begin_inset Formula $a_{0}$
\end_inset

 by the ratio of electron and muon masses.
 This results in a high probability density very near to the nucleus, and
 is a useful probe of nuclear charge distribution, or can be used to decrease
 the electric potential barrier which inhibits nuclear fusion
\begin_inset CommandInset citation
LatexCommand citep
key "Breunlich1989-mu-fusion"

\end_inset

.
 However, condensed matter physics is more concerned with electronic behaviour,
 and hence it is better to use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muon
\backslash
 
\end_layout

\end_inset

 which come to rest in areas of high electron density.
\end_layout

\begin_layout Standard
Most muons used in condensed matter are so-called 
\emph on
surface muons
\emph default
, produced from pions at rest in the surface of the production target.
 Conservation of energy and momentum allows the kinetic energy 
\begin_inset Formula $E_{\mathrm{k}}$
\end_inset

 of the muon to be simply evaluated:
\begin_inset Formula 
\begin{equation}
E_{\mathrm{k}}=\left(\frac{m_{\uppi}^{2}-m_{\murm}^{2}}{2m_{\uppi}}\right)c^{2}=\SI{4.119}{\MeV},
\end{equation}

\end_inset

where 
\begin_inset Formula $m_{\uppi,\murm}$
\end_inset

 are the pion and muon masses, respectively, and 
\begin_inset Formula $c$
\end_inset

 is the speed of light.
 The momentum of these surface muons is 
\begin_inset Formula $p_{\mathrm{surface}}=\SI{29.79}{\MeV}/c$
\end_inset

.
 Alternatively, production can proceed via the 
\emph on
decay channel
\emph default
, where pions which decay in flight are utilised to create muons with higher
 energy.
 This gives rise to a higher-energy beam which penetrates deeper into material
 before stopping, but with a broader energy and momentum distribution (typically
 
\begin_inset Formula $\num{70}<p_{\mathrm{decay}}<\SI{130}{\MeV}/c$
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "Roduner1997-muSR"

\end_inset

), and spin-polarisation 
\begin_inset Formula $<100\%$
\end_inset

.
\end_layout

\begin_layout Subsection
Muon beams
\end_layout

\begin_layout Standard
Muons are guided from the production target to the sample down an evacuated
 beampipe by a series of electromagnetic devices.
 As well as correctly guiding muons, it is important to discard as many
 other particles which may be produced at the target as possible; this reduces
 both spurious counts in experimental detectors and sample radiation damage.
\end_layout

\begin_layout Standard
Dipole steering magnets are used to alter the course of the muons.
 Because the 
\begin_inset Formula $g$
\end_inset

-factor of the muons is very nearly 2 (
\begin_inset Formula $g_{\upmu}=\num{2.002332}$
\end_inset

), magnetic fields rotate the muons' spin and trajectory at very nearly
 the same rate, ensuring that this manipulation does not affect the relative
 direction of their spin and motion.
 The radius of curvature 
\begin_inset Formula $r$
\end_inset

 in a magnetic field 
\begin_inset Formula $B$
\end_inset

 is a function of particle charge 
\begin_inset Formula $q$
\end_inset

 and momentum 
\begin_inset Formula $p$
\end_inset

,
\begin_inset Formula 
\begin{equation}
r=\frac{p}{qB}\,.
\end{equation}

\end_inset

This also allows particles with a specific 
\begin_inset Formula $p/q$
\end_inset

 ratio to be selected by placing a slit after the bending magnet, removing
 all uncharged particles such as neutrons and photons, and all negatively-charge
d particles such as electrons and negative muons, but positively-charged
 particles, such as positrons or positive pions, may pass if they have the
 correct momentum.
\end_layout

\begin_layout Standard
Particles can also be selected by velocity using a crossed-field separator,
 which employs perpendicular electric and magnetic fields.
 The magnetic field applies a force 
\begin_inset Formula $Bqv$
\end_inset

 perpendicular to both the particle velocity and the magnetic field, whilst
 the electric field of magnitude 
\begin_inset Formula $E$
\end_inset

 applies a force 
\begin_inset Formula $Eq$
\end_inset

 parallel to its field direction.
 Particles will continue undeflected if these forces are equal and, since
 only one depends on velocity, a single value of this is isolated.
 Applied to a beam in conjunction with momentum selection, positive particles
 other than muons are removed.
\end_layout

\begin_layout Standard
Quadrupole focusing magnets are used to focus a beam travelling with a given
 momentum in the 
\begin_inset Formula $z$
\end_inset

-direction along one direction in the 
\begin_inset Formula $xy$
\end_inset

-plane, whilst defocusing the beam in the remaining perpendicular direction.
 It is only possible to focus in one dimension in the 
\begin_inset Formula $xy$
\end_inset

-plane in a single quadrupole, but a series of units with alternating polarity
 can be used to give a net focussing effect.
 These can be used to collect muons near the production target, and to focus
 the beam onto the sample.
\end_layout

\begin_layout Standard
There are sometimes multiple instruments attached to one beamline, so electromag
netic kickers are used to direct muons to the appropriate apparatus.
 At ISIS, for example, the muons arrive in a pair of closely-spaced pulses
 (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:pulsed-muon-sources"

\end_inset

), and a large electric field can be used to deflect the first into one
 instrument before being rapidly removed for the arrival of the second pulse,
 allowing it to continue undeflected into a different experimental apparatus.
\end_layout

\begin_layout Standard
Electric fields do not couple to spin and consequently only change the muon
 momentum.
 This means that the muon direction of travel is rotated with respect to
 its spin in an electric field.
 This can be unintentional---a typical beamline with separators and kickers
 will rotate a muon-spin by a few degrees---or it can be used to rotate
 the spin with respect to the detector geometry before implantation, in
 a 
\emph on
spin rotator
\emph default
.
 This can be used to alter the experimental geometry, as examined in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:experimental-set-up"

\end_inset

.
\end_layout

\begin_layout Subsection
Muon stopping processes
\end_layout

\begin_layout Standard
The energy of a muon determines its likely penetration into matter.
 Extremely high-energy muons are very highly penetrating: neutrino detection
 experiments are often placed several miles underground in order to shield
 them from spurious events due to cosmic ray muons
\begin_inset CommandInset citation
LatexCommand citep
key "Fukuda1998-SuperK"

\end_inset

.
 Muons used in condensed matter research have a more moderate energy in
 order that they will stop in samples of typical size 
\begin_inset Formula $\lesssim\SI{1}{\cm\cubed}$
\end_inset

.
\end_layout

\begin_layout Standard
The 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{4}{
\backslash
MeV}
\end_layout

\end_inset

 surface muons have a typical stopping range in matter of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{110}{
\backslash
milli
\backslash
gram
\backslash
per
\backslash
centi
\backslash
metre
\backslash
squared}
\end_layout

\end_inset

, with a range width of approximately 20%
\begin_inset CommandInset citation
LatexCommand citep
key "Kilcoyne1999-inmuonscience"

\end_inset

.
 The processes which bring the muon to a halt occur in several distinct
 stages.
 First, over a period of approximately 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SIrange{0.1}{1}{
\backslash
ns}
\end_layout

\end_inset

, the muon is decelerated to an energy of a few 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
keV}
\end_layout

\end_inset

 by ionisation of atoms and scattering with electrons.
 Then, during a period lasting around a picosecond, the muon rapidly captures
 and loses successive electrons, forming fleeting muonium states (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:muon-interaction-with-sample"

\end_inset

), bringing its energy down to a few 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
eV}
\end_layout

\end_inset

.
 Finally, the muon (or muonium) thermalises, shedding almost all of its
 excess energy and coming to rest.
 Typical energies required to create a crystal vacancy are significantly
 larger than muon kinetic energy in the latter stages of its deceleration;
 the muon comes to rest ~
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
micro
\backslash
metre}
\end_layout

\end_inset

 from any sample damage it has caused, and these effects can be neglected.
 Since the stopping processes primarily involve electrostatic interactions,
 the muon implants with no significant reorientation of its spin.
 Futher information on these stopping processes can be found in Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Chappert1984-mu-pi-materials-research"

\end_inset

.
\end_layout

\begin_layout Standard
The `surface muon' nomenclature has the potential to be slightly confusing,
 because it refers to the position of the parent pions on the surface of
 the production target, rather than to the muon sites in the sample.

\emph on
 
\emph default
To enable studies near the surface of samples, or involving films or thin
 layers, 
\emph on
slow muons
\emph default
 are created
\begin_inset CommandInset citation
LatexCommand citep
key "Morenzoni2001-low-energy-muons"

\end_inset

.
 These are manufactured by taking conventional surface muons and reducing
 their momentum by first stopping them in a moderator, such as frozen argon,
 followed by subsequent re-acceleration with electric fields.
 Conversely, if deeper penetration into the sample is required, decay channel
 muons (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:muon-production"

\end_inset

) are used instead.
 This can be used, for example, to allow muons to traverse a reinforced
 pressure cell before coming to rest in the sample contained within it.
\end_layout

\begin_layout Subsection
Interaction with the sample
\begin_inset CommandInset label
LatexCommand label
name "sub:muon-interaction-with-sample"

\end_inset


\end_layout

\begin_layout Standard
The scenario in which 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 lends itself to simplest interpretation is the case of the muon interacting
 passively with its surroundings.
 In many magnetic and superconducting samples it is permissible to approximate
 the muon as an inert probe of local field.
 Muons possess a magnetic moment, and will therefore undergo Larmor precession
 in a magnetic field.
 Since the muon possesses 
\begin_inset Formula $S=\frac{1}{2}$
\end_inset

, it only experiences this simple dipolar coupling to its environment without
 the added complication of quadrupolar coupling, as seen in many NMR nuclei,
 or 
\begin_inset script superscript

\begin_layout Plain Layout
8
\end_layout

\end_inset

Li.
\end_layout

\begin_layout Standard
To take the simple example of an ordered ferromagnet with a single muon
 site, every muon in the ensemble experiences the same single value of magnetic
 field.
 This results in a coherent precession of the muon-spins, and is known as
 muon-spin rotation.
 In more complex materials, multiple magnetically-inequivalent sites may
 give rise to a number of precession frequencies, or a distribution.
 For example, in many materials randomly-orientated nuclear moments cause
 random magnetic fields at muon sites resulting in no coherent precession,
 but a slow depolarisation of muon spins.
 Further, fluctuations of the local field in the time domain can give rise
 to relaxation of the muon ensemble's polarisation caused by anything from
 critical fluctuations near a phase transition to passing `magnetic monopoles'
 in spin-ice
\begin_inset CommandInset citation
LatexCommand citep
key "Bramwell2009-monopoles"

\end_inset

.
\end_layout

\begin_layout Standard
It is also possible for the muon to interact quantum-mechanically with nearby
 spins.
 In an ordered magnet, the strength of the exchange interactions and size
 of electronic spins means that the muon does not perturb the spin system,
 and simply sees the field it generates.
 A two-way interaction requires a spin whose coupling to its environment
 is much weaker, the prototypical example being a fluorine nucleus.
 Through a quantum-entangled dipole--dipole interaction, the muon enters
 a bound 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
Fmu 
\end_layout

\end_inset


\begin_inset space ~
\end_inset

state (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:[Cu(HF2)(pyz)2]X-paramagnetic"

\end_inset

).
\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="4">
<features tabularvalignment="middle">
<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">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

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

\begin_layout Plain Layout
Mu
\end_layout

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

\begin_layout Plain Layout
H
\end_layout

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

\begin_layout Plain Layout
T
\end_layout

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

\begin_layout Plain Layout
mass (
\begin_inset Formula $m_{\mathrm{e}}$
\end_inset

)
\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{208}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{1837}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{5498}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
reduced mass (
\begin_inset Formula $m_{\mathrm{e}}$
\end_inset

)
\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.9952}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.9995}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.9998} 
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
Bohr radius (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
nm}
\end_layout

\end_inset

)
\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.05317}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.05295}
\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
num{0.05293}
\end_layout

\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
ionisation energy (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
eV}
\end_layout

\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 ERT
status open

\begin_layout Plain Layout


\backslash
num{13.540}
\end_layout

\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 ERT
status open

\begin_layout Plain Layout


\backslash
num{13.598}
\end_layout

\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 ERT
status open

\begin_layout Plain Layout


\backslash
num{13.603}
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Comparison of properties of muonium (Mu
\begin_inset space ~
\end_inset

= 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
muon 
\end_layout

\end_inset

e
\begin_inset script superscript

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


\end_layout

\end_inset

), protium (
\begin_inset script superscript

\begin_layout Plain Layout
1
\end_layout

\end_inset

H
\begin_inset space ~
\end_inset

= pe
\begin_inset script superscript

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


\end_layout

\end_inset

) and tritium (
\begin_inset script superscript

\begin_layout Plain Layout
3
\end_layout

\end_inset

H
\begin_inset space ~
\end_inset

= T
\begin_inset space ~
\end_inset

= pnne
\begin_inset script superscript

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


\end_layout

\end_inset

), illustrating that muonium can be viewed as a light isotope of hydrogen.
\begin_inset CommandInset label
LatexCommand label
name "tab:Mu-H-isotopes"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
As a positively-charged particle in matter, a further possibility is that
 the muon can capture an electron in a sample and form a hydrogen-like atom
 known as muonium (symbol Mu).
 Though the muon is a lepton, it can be considered a light proton in these
 circumstances.
 Since its mass is more than two orders of magnitude greater than an electron,
 but only nine times less than a proton, the atomic physics of Mu is startlingly
 similar to H.
 Some properties of muonium, hydrogen and tritium are compared in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "tab:Mu-H-isotopes"

\end_inset

.
 The effects of muonium formation are not examined in detail in this thesis,
 but an overview can be found in Refs.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Cox1987-muon-review,Cox2003-muonium-semiconductors"

\end_inset

.
\end_layout

\begin_layout Subsection
Muon decay
\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/musr/decay-positron-asymmetry/decay-positron-asymmetry.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
A polar plot showing the asymmetric emission directions of the positron
 in the aftermath of muon decay, averaged over all energies (see text).
 The thick yellow arrow represents the axis of the muon spin, which is also
 the most likely direction of positron emission.
\begin_inset CommandInset label
LatexCommand label
name "fig:decay-positron-asymmetry"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset

The muon decays via the three-body process
\begin_inset Formula 
\begin{equation}
\mathrm{\murm^{+}\rightarrow e^{+}+\upnu_{e}+\bar{\upnu}_{\murm},}
\end{equation}

\end_inset

creating a positron (antielectron), plus an electron-neutrino and a muon-antineu
trino to conserve lepton number.
 Like muon production, this too is a weak decay and consequently violates
 parity.
 This manifests in the positron being emitted preferentially in the direction
 of spin of the dying muon---and the positron is the only one of these three
 particles which we might hope to reliably detect.
 Unfortunately however, the extra degree of freedom introduced by the third
 decay product means that this process does not enjoy the simplicity of
 muon production.
 The emerging positron can vary in both energy and the angle at which is
 it emitted.
 The angle of emission is distributed about the muon spin direction in cardioid
 form,
\begin_inset Formula 
\begin{equation}
N(\theta)=N_{0}(1+a\cos\theta).\label{eq:cardioid}
\end{equation}

\end_inset

For the maximum possible positron energy of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{52.83}{
\backslash
MeV}
\end_layout

\end_inset

, 
\begin_inset Formula $a=1$
\end_inset

; integrating over all possible energies, 
\begin_inset Formula $a=\frac{1}{3}$
\end_inset

, leading to the less pronounced angular distribution shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:decay-positron-asymmetry"

\end_inset

.
 Nonetheless, this finite asymmetry still allows muon-spin precession to
 be followed arbitrarily precisely, given enough data.
\end_layout

\begin_layout Section
Experimental set-up
\begin_inset CommandInset label
LatexCommand label
name "sec:experimental-set-up"

\end_inset


\end_layout

\begin_layout Standard
In order to monitor the time-evolution of muon spin direction, scintillation
 detectors are set up around the sample to count the outgoing positrons
 as a function of angle and time.
 The simplest experimental configuration involves just two detectors: one
 forward (
\begin_inset Formula $\mathrm{F}$
\end_inset

) and one backward (
\begin_inset Formula $\mathrm{B}$
\end_inset

) of the initial muon-spin direction.
 The difference in count rate between these is then monitored as a function
 of time, and divided by the total count rate to normalise with respect
 to the ever-decaying population of muons remaining in the sample.
 The experimental quantity of interest is then the 
\emph on
positron asymmetry
\emph default
,
\begin_inset Formula 
\begin{equation}
A(t)=\frac{N_{\mathrm{F}}-\alpha N_{\mathrm{B}}}{N_{\mathrm{F}}+\alpha N_{\mathrm{B}}}\,,\label{eq:A(t)}
\end{equation}

\end_inset

where 
\begin_inset Formula $\alpha$
\end_inset

 is an empirical constant to take into account any slight inequivalencies
 between the forward and backward detectors.
 These can result from efficiency differences in the scintillators and electroni
cs, asymmetric sample positioning, difference in solid angle coverage of
 the detectors, and so on.
 
\begin_inset Formula $A(t)\propto p_{z}(t)$
\end_inset

; meaning that the measured quantity, 
\begin_inset Formula $A(t)$
\end_inset

 gives us direct access to the quantity of interest, 
\begin_inset Formula $p_{z}(t)$
\end_inset

, the component of the muon-spin polarisation along the 
\begin_inset Formula $z$
\end_inset

-axis of the apparatus.
 A schematic diagram of this experimental configuration is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:musr-experiment"

\end_inset

 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "fig:musr-experiment"

\end_inset

.
\end_layout

\begin_layout Standard
In an ideal apparatus, the initial asymmetry when all muon spins are polarised
 towards the 
\begin_inset Formula $\mathrm{F}$
\end_inset

 detector would be the energy-integrated value of 
\begin_inset Formula $a$
\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:cardioid"

\end_inset

, 
\begin_inset Formula $A(t=0)=a=\frac{1}{3}$
\end_inset

.
 However, various factors reduce this in reality, including the limited
 solid-angle coverage of the detectors; the exclusion of low-energy positrons
 which stop in the sample, cryostat or detector wrappings; and the effects
 of magnetic field on positron trajectories
\begin_inset CommandInset citation
LatexCommand citep
key "King2003-HiFi-plan,Lancaster2004-thesis"

\end_inset

.
 In fact, a typical initial asymmetry is around 
\begin_inset Formula $A(t=0)=25\%$
\end_inset

.
\end_layout

\begin_layout Standard
This configuration is known as the 
\emph on
longitudinal field
\emph default
 or LF configuration, where any magnetic field 
\begin_inset Formula $B_{\mathrm{long}}$
\end_inset

 is applied parallel to the initial muon spin direction.
 (Applying such a longitudinal field is not performed in the experiments
 featuring in this thesis.) The 
\emph on
zero-field
\emph default
 or ZF configuration is identical, but in the special case of no applied
 field.
 If muons were to stop in a sample with which they did not react at all,
 the `null' data measured in this geometry would be a flat line at 
\begin_inset Formula $A(t)=A_{0}$
\end_inset

.
 In addition to this geometry, there exists the 
\emph on
transverse field
\emph default
, or TF, configuration in which the field is applied perpendicular to the
 initial muon spin direction, leading to a precession of the spins around
 the applied field.
 This means that the `null' asymmetry function in an applied field 
\begin_inset Formula $B_{\mathrm{trans}}$
\end_inset

 would be a cosine, 
\begin_inset Formula $A(t)=A_{0}\cos\gamma_{\murm}B_{\mathrm{trans}}t$
\end_inset

.
 In this configuration, a larger number of detectors may be used to track
 the muon-spin direction as it rotates.
\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/musr/experiment/musr-ab.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Schematic diagram showing a simple 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 experiment where muons implant in a sample and experience a uniform 
\begin_inset Formula $\bm{B}$
\end_inset

-field directed perpendicular to the paper.
 (a) At 
\begin_inset Formula $t=0$
\end_inset

, one or more muons are injected into a sample from the right.
 Their spins, represented by the gold arrows, are directed antiparallel
 to their linear momentum.
 The muons implant at random positions throughout the sample.
 This image shows either a pulse of muons from a pulsed muon source, or
 the aggregate of many events at a continuous source.
 The sample is positioned between forward and backward detectors, F and
 B.
 (b) Moments later, positrons are being emitted along the direction of muon-spin
; these are represented by blue lines.
 The upper graph on the right shows the raw count rate.
 This is higher in the backward detector (blue line) than the forward one
 (red line), with the central green line showing the average count rate.
 The lower graph then shows the muon polarisation along the 
\begin_inset Formula $z$
\end_inset

-direction, 
\begin_inset Formula $p_{z}(t)$
\end_inset

, obtained from 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:A(t)"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:musr-experiment"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ContinuedFloat
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/musr/experiment/musr-cd.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
(continued) (c) A time nearly 
\begin_inset Formula $t=\uppi/\gamma_{\murm}B$
\end_inset

 later, the muon-spins have precessed through almost 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{180}{
\backslash
degree}
\end_layout

\end_inset

, giving rise to more counts in the forward positron detector than the backward
 one.
 The central average line has progressed, and it is now easily seen that
 it reflects the exponential decay of the muon population with 
\begin_inset Formula $\tau_{\murm}=\SI{2.2}{\us}$
\end_inset

.
 The asymmetry line, normalised for this decay, remains cosinusoidal.
 (d) At late times, only a few muons remain.
 The complete 
\begin_inset Formula $A(t)$
\end_inset

 trace to 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{4}{
\backslash
us}
\end_layout

\end_inset

 is shown.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Relaxation functions
\begin_inset CommandInset label
LatexCommand label
name "sec:relaxation-functions"

\end_inset


\end_layout

\begin_layout Standard
The asymmetry spectra 
\begin_inset Formula $A(t)$
\end_inset

 generated in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 are typically fitted in the time-domain.
 The fitting function used is known as a 
\emph on
relaxation function
\emph default
, and a diversity of forms reflects a diversity of underlying physics.
 In general, a muon responds to its local magnetic field 
\begin_inset Formula $\bm{B}$
\end_inset

 by precessing about the component of it transverse to the muon-spin direction
 with a Larmor frequency given by
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\omega=\gamma_{\murm}B
\end{equation}

\end_inset

where 
\begin_inset Formula $\gamma_{\murm}=2\uppi\times\SI{135.5}{\MHz\per\tesla}$
\end_inset

 is the muon gyromagnetic ratio and 
\begin_inset Formula $B=|\bm{B}|$
\end_inset

 is the magnitude of the magnetic field at the muon site.
 Semiclassically, we can consider this as the muon-spin precessing at the
 end of a cone of half-angle 
\begin_inset Formula $\theta$
\end_inset

, where 
\begin_inset Formula $\theta$
\end_inset

 is the angle between muon-spin direction and local field direction (see
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:spin-precession"

\end_inset

).
 The polarisation of the muon ensemble projected along the 
\begin_inset Formula $z$
\end_inset

-direction as a function of time will then be given by
\begin_inset Formula 
\begin{equation}
p_{z}(t)=\cos^{2}\theta+\sin^{2}\theta\cos\left(\gamma_{\murm}Bt\right).\label{eq:muSR-general-pzt}
\end{equation}

\end_inset

In this section, we will discuss first the case where the field bathing
 the muon originates entirely from the behaviour of the sample (
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:zero-field"

\end_inset

).
 Then, the effect of applied magnetic field will be considered.
 Applying field allows us to explore the phase diagram of the sample, but
 also has a direct effect on the muons and thus the asymmetry spectra obtained:
 the case where an external magnetic field is applied perpendicular to the
 initial muon-spin direction is briefly explored in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:transverse-field"

\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/musr/precession/precession.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
The semi-classical picture of a muon-spin, pointing along a direction 
\begin_inset Formula $\bm{s}_{\murm}$
\end_inset

, precessing around the local field 
\begin_inset Formula $\bm{B}$
\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:spin-precession"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
Fitting muon data in this thesis was performed using the Windows Muon Data
 Analysis (WiMDA) package
\begin_inset CommandInset citation
LatexCommand citep
key "pratt2000-wimda"

\end_inset

.
\end_layout

\begin_layout Subsection
Zero-field
\begin_inset CommandInset label
LatexCommand label
name "sub:zero-field"

\end_inset


\end_layout

\begin_layout Standard
In materials with spontaneous magnetic properties, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 can be used to probe samples without application of an external field.
 In an ordered magnet, 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:muSR-general-pzt"

\end_inset

 can give rise to spontaneous precession of the muon ensemble.
 Single-crystal experiments with muons are not considered in this thesis,
 and consequently we take a 
\emph on
powder average
\emph default
, imagining that an infinite number of randomly-aligned crystallites make
 up the sample under study.
 This is not a significant experimental hindrance because the precession
 frequency depends only on the magnitude of the local field, with only the
 amplitude of that precession depending on the direction.
 Taking a powder average over 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:muSR-general-pzt"

\end_inset

 yields
\begin_inset Formula 
\begin{equation}
p_{z}(t)=\frac{1}{3}+\frac{2}{3}\cos\left(\gamma_{\murm}Bt\right),
\end{equation}

\end_inset

due to the 
\begin_inset Formula $\frac{1}{3}$
\end_inset

 of crystallite orientations in which the local field will lie parallel
 to the initial muon-spin components.
 Thus, the onset of magnetic order can be seen by two effects: firstly,
 the presence of oscillations, and secondly the emergence of a 
\emph on

\begin_inset Formula $\mathit{\frac{\,1}{3\,}}$
\end_inset

-tail
\emph default
, a baseline at long times below which the spectra do not relax.
 An example of spectra above and below a magnetic transition in [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

]BF
\begin_inset script subscript

\begin_layout Plain Layout
4
\end_layout

\end_inset

, a material discussed in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "chap:cuhf2pyz2x"

\end_inset

, is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:eg-A(t)-CuBF4"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/cuhf2pyz2x/summarygraphs/At-BF4.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Example 
\begin_inset Formula $A(t)$
\end_inset

 spectra with fits above (
\begin_inset Formula $T=\SI{25}{\kelvin}$
\end_inset

) and below (
\begin_inset Formula $T=\SI{0.3}{\kelvin}$
\end_inset

) the magnetic transition temperature 
\begin_inset Formula $T_{\mathrm{N}}=\SI{1.4}{\kelvin}$
\end_inset

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

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

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

.
 The two spectra are approximately equal in initial asymmetry, but the presence
 of oscillations and `
\begin_inset Formula $\frac{1}{3}$
\end_inset

-tail' observed unambiguously identify the ordered phase.
 The slow oscillation observed for 
\begin_inset Formula $T>T_{\mathrm{N}}$
\end_inset

 is due to muon--fluorine dipole--dipole oscillations (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:[Cu(HF2)(pyz)2]X-paramagnetic"

\end_inset

).
 
\begin_inset CommandInset label
LatexCommand label
name "fig:eg-A(t)-CuBF4"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
For a single muon site which is magnetically equivalent throughout the crystal,
 this results simply in a cosinusoidal oscillation in asymmetry.
 In more complex static fields, the resulting asymmetry function is somewhat
 like a cosine Fourier transform of the local fields at the muon sites,
 but taking care to perform an appropriate angular average.
 A simple case which illustrates the importance of angular averaging in
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash
 
\end_layout

\end_inset

 spectra is that of a random local field at the muon site, normally-distributed
 about a mean of zero.
 This might occur, for example, in a spin-glass.
 The muon polarisation then follows the 
\emph on
Kubo--Toyabe function
\emph default
, 
\begin_inset Formula 
\begin{equation}
p_{\mathrm{KT}}(t)=\frac{1}{3}+\frac{2}{3}\left(1-\Delta^{2}t^{2}\right)\mathrm{e}^{-\Delta^{2}t^{2}/2},\label{eq:Kubo-Toyabe}
\end{equation}

\end_inset

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

 is the 
\emph on
field width
\emph default
 (the second moment of the field distribution, whose mean is 
\begin_inset Formula $\left\langle B\right\rangle $
\end_inset

, in frequency units), shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:Kubo-Toyabe"

\end_inset

.
 Note that this is not simply a Gaussian, as a naïve Fourier transform might
 lead us to imagine, though at early times it is well approximated by one.
 After an initial depolarisation, with the 
\begin_inset Formula $p_{z}(t)$
\end_inset

 reaching a minimum at 
\begin_inset Formula $\Delta t=\sqrt{3}$
\end_inset

, the 
\begin_inset Formula $\frac{2}{3}$
\end_inset

 of muon-spin components perpendicular to their local field dephase entirely,
 leaving the 
\begin_inset Formula $\frac{1}{3}$
\end_inset

-tail of parallel spin components.
\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/musr/kubo-toyabe/kubo-toyabe.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
The Kubo--Toyabe function in zero field [
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:Kubo-Toyabe"

\end_inset

].
\begin_inset CommandInset label
LatexCommand label
name "fig:Kubo-Toyabe"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Another prototypical relaxation function occurs in materials with an incommensur
ate magnetic structure, resulting in a field at the muon site which varies
 sinusoidally with the unit cell in which the muon stops.
 This distribution of local fields results in the muon relaxation following
 a Bessel function
\begin_inset CommandInset citation
LatexCommand citep
key "Major1986-ZF-muSR-Cr,Le1993-TMTSF2-X"

\end_inset

.
\end_layout

\begin_layout Standard
As well as static field distributions, the muon ensemble can be relaxed
 by dynamics present within the sample.
 This will usually give rise to an exponential relaxation of the form 
\begin_inset Formula $\mathrm{e}^{-\lambda t}$
\end_inset

.
 In the 
\emph on
fast-fluctuation limit
\emph default
, where 
\begin_inset Formula $1/\tau\gg\Delta$
\end_inset

 (i.e.
 the fluctuation rate, given as the reciprocal of the correlation time 
\begin_inset Formula $\tau$
\end_inset

, results in much faster depolarisation than the distribution of fields,
 
\begin_inset Formula $\Delta$
\end_inset

), then the relaxation rates are expected
\begin_inset CommandInset citation
LatexCommand cite
key "hayano1979-muonrelaxation"

\end_inset

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

.
 This is the case in most magnetic materials, where the effect of variation
 in field within a set of muon sites is small compared to the fluctuations
 in those fields caused by the dynamic behaviour of local moments.
\end_layout

\begin_layout Subsection
Transverse field
\begin_inset CommandInset label
LatexCommand label
name "sub:transverse-field"

\end_inset


\end_layout

\begin_layout Standard
As mentioned in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:experimental-set-up"

\end_inset

, the asymmetry function in an applied transverse field (TF), 
\begin_inset Formula $B_{\mathrm{trans}}$
\end_inset

, with muons stopped in an entirely inert sample would be a cosine.
 In a real sample, the cosine may have its frequency slightly shifted from
 that which would be expected in the applied field, and is modulated by
 a relaxing envelope dependent on the physics of the system being probed.
 This gives rise to the general TF relaxation function 
\begin_inset Formula 
\begin{equation}
A(t)=A_{0}\cos\left(\omega t\right)R(t),
\end{equation}

\end_inset

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

 is the initial asymmetry, 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $\omega\approx\gamma_{\murm}B_{\mathrm{trans}}$
\end_inset

 is the angular frequency of the oscillation,
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
 and 
\begin_inset Formula $R(t)$
\end_inset

 is some function describing the relaxation.
\end_layout

\begin_layout Standard
In a magnetic system, transverse fields small relative to the internal fields
 in the sample give rise to complex behaviour.
 Firstly, we must account for any change in internal field brought about
 by magnetisation in the sample but, further, these internal fields must
 be vector-summed with the field applied.
 In large applied field, the internal fields act as a small correction to
 the external field and thus, in a powder sample, we can consider the TF
 relaxation function as a cosine with 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $\omega=\gamma_{\murm}B_{\mathrm{trans}}$
\end_inset

 and a relaxation governed by the magnitude of the internal fields.
 This method is used to investigate the 
\begin_inset Formula $B$
\end_inset

--
\begin_inset Formula $T$
\end_inset

 phase diagram in a molecular system in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sec:nonmonotonic-CuHF2pyz2BF4"

\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/musr/flux-xy/fluxxy.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
(a) A schematic diagram of the magnetic field in the superconducting vortex
 lattice, and (b) the distribution of fields arising from this.
 Arrows link corresponding points on the two diagrams: the maximum and minimum
 fields at the vortex cores and furthest from them, respectively; and the
 modal field, at an intermediate position.
 Diagram follows Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Blundell1999-mureview"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:vortex-lattice"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/musr/superconductor/LiFe1-xNixAs-transition.eps

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 spectra for LiFe
\begin_inset script subscript

\begin_layout Plain Layout
0.99
\end_layout

\end_inset

Ni
\begin_inset script subscript

\begin_layout Plain Layout
0.01
\end_layout

\end_inset

As above and below the superconducting transition temperature 
\begin_inset Formula $T_{\mathrm{c}}=\SI{13.4}{\kelvin}$
\end_inset

 in a transverse field of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{10}{
\backslash
milli
\backslash
tesla}
\end_layout

\end_inset

.
 Relaxation of the spectrum taken at 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{5}{
\backslash
kelvin}
\end_layout

\end_inset

 results from the distribution of local magnetic fields arising from the
 superconducting vortex lattice.
 Data taken from Ref.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "Pitcher2010-LiFeAs-composition"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:LiFeAs-super-eg"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Another typical use of the TF geometry is to probe superconductivity.
 Type
\begin_inset space ~
\end_inset

II superconductors below their critical temperature 
\begin_inset Formula $T_{\mathrm{c}}$
\end_inset

 enter the mixed state when an external field 
\begin_inset Formula $B_{\mathrm{c1}}(T)<B<B_{\mathrm{c2}}(T)$
\end_inset

 is applied.
 This state is characterised by a vortex lattice of magnetic flux quanta,
 with a typical lattice spacing of order 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
si{
\backslash
nm}
\end_layout

\end_inset

.
 Since muons can stop at any set of crystallographically-equivalent sites,
 they decorate this lattice on a significantly smaller length scale, and
 can be considered to sample the entire vortex lattice.
 This results in the muons sampling a distribution of magnetic fields, highest
 at the centre of the vortices and lowest at positions furthest away from
 them.
 Thus, whilst 
\begin_inset Formula $R(t)\approx1$
\end_inset

 in a sample above the superconducting transition, the asymmetry function
 below the transition is given by the cosine Fourier transform of the new
 field distribution.
 This gives rise to a relaxing cosine which, in general, will have a frequency
 which differs slightly from the applied 
\begin_inset Formula $B_{\mathrm{trans}}$
\end_inset

 by the 
\emph on
diamagnetic shift
\emph default
.
 This is shown schematically in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:vortex-lattice"

\end_inset

, and with data taken by the author on Ni-doped LiFeAs in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:LiFeAs-super-eg"

\end_inset

.
\end_layout

\begin_layout Section
Experimental considerations
\begin_inset CommandInset label
LatexCommand label
name "sec:experimental-considerations"

\end_inset


\end_layout

\begin_layout Subsection
Continuous beam sources
\end_layout

\begin_layout Standard
Muon sources can be divided into two classes: continuous sources, where
 the muons arrive in a quasi-continuous stream; and pulsed sources (see
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:pulsed-muon-sources"

\end_inset

).
 In a continuous source, a single muon is in the sample at any given time.
 Muons arrive intermittently and implant in the sample.
 Veto detectors positioned behind the sample can be used to flag up muons
 which have flown past the sample and not implanted.
 The clock starts when a muon enters the detection apparatus, and stops
 when a decay positron is incident on one of the detectors.
 However, if a second muon arrives before a positron is detected, it is
 impossible to know which of the two subsequent decay positrons came from
 which muon, and both events must be discarded.
\end_layout

\begin_layout Standard
The requirement that only one muon be present at any given time restricts
 both the event rate and the low-frequency resolution of a continuous source.
 Since a muon may arrive and interrupt your experiment, increasing the rate
 of muon production eventually reduces the event rate of your experiment,
 because many events must be discarded.
 Similarly, increasing the time window over which each event is recorded
 in an attempt to increase the low-frequency resolution also increases the
 probability that a new muon will arrive and interrupt an event; thus, event
 rate and low-frequency resolution are in competition with one-another.
 A further consequence of a continuous source is high background; muons
 which neither embed in the sample nor set off the veto detector can result
 in spurious counts, especially at long times.
 In fact, this sets a practical limit of around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{10}{
\backslash
us}
\end_layout

\end_inset

 on experimental counting.
\end_layout

\begin_layout Subsection
Pulsed sources
\begin_inset CommandInset label
LatexCommand label
name "sub:pulsed-muon-sources"

\end_inset


\end_layout

\begin_layout Standard
A pulsed muon source delivers muons in large bunches rather than individually.
 All the muons implant in the sample approximately simultaneously, starting
 the experimental clock, and positron emission is timed with respect to
 the arrival of the pulse.
\end_layout

\begin_layout Standard
The chief disadvantage of a pulsed source is that a resolution limit is
 imposed on precession frequencies by the width of the pulse.
 The ambiguity in muon arrival time means that, roughly speaking, if the
 magnetic field being measured is such that the first muons to arrive have
 undergone one spin rotation by the time the last muons arrive, then those
 in between will find themselves at every intermediate stage in the precession,
 and any signal will be washed out.
 More precisely, a frequency resolution function can be extracted by taking
 the Fourier transform of the incoming pulse.
 In the simplest case of a single broad pulse, a monotonic reduction in
 signal amplitude occurs up to some cut-off frequency above which muon precessio
n cannot be reliably observed.
\end_layout

\begin_layout Standard
In spite of this restriction, it may still be possible to detect a transition
 to long-range magnetic order where frequencies are too high to resolve.
 Significant reductions in initial asymmetry below the transition are often
 observed because a reduced value of 
\begin_inset Formula $A(t)$
\end_inset

 corresponding approximately to the average value of the invisible oscillations
 is seen.
 Conversely, asymmetry at long times can increase due to the effect of the
 
\begin_inset Formula $\frac{1}{3}$
\end_inset

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

\end_inset

.
 It is also often possible to observe discontinuous variations in relaxation
 rates with temperature.
\end_layout

\begin_layout Standard
The limit on the rate of data acquisition at pulsed sources is imposed by
 the finite detector 
\emph on
dead time
\emph default
 (often around 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
SI{10}{
\backslash
ns}
\end_layout

\end_inset

).
 This is the time after detection of a positron during which further positrons
 will not be acknowledged.
 Consequently, rather than using just two detectors as in an idealised 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
muSR
\backslash

\end_layout

\end_inset

 experiment, large numbers of individual detectors are used.
 In LF configuration, these are grouped to act as two highly segmented detectors.
 For example, the MuSR spectrometer at ISIS has some 64 detectors split
 into two banks of 32, while the new CHRONUS instrument has 606 individual
 positron counters
\begin_inset CommandInset citation
LatexCommand citep
key "ISIS-muon-spectrometer-comparison-web"

\end_inset

!
\end_layout

\begin_layout Subsection
Sample mounting
\end_layout

\begin_layout Standard
Samples are mounted in such a way as to minimise the effect of mounting
 on the resulting asymmetry spectra.
 Large samples are usually mounted on a backing plate which will capture
 those few muons which do not implant in the sample.
 Small samples will often be mounted in a 
\emph on
flypast
\emph default
 configuration, where the sample is mounted on prongs with no backing; muons
 which do not implant in the sample thus fly past.
 At a continuous muon source, a veto counter detects this and discounts
 any positron detected as a result of this.
 At a pulsed source, a long flypast tube allows those muons which miss or
 pass through the sample to decay far from the positron counters.
 Small samples also sometimes require a quantity of 
\emph on
degrader
\emph default
 to be placed in front of them, to slow the muons to the cusp of stopping
 before they penetrate the sample, ensuring that they do not pass straight
 through.
\end_layout

\begin_layout Standard
When the sample is mounted on a backing plate, the materials most often
 used are silver (Ag) and haematite (Fe
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

O
\begin_inset script subscript

\begin_layout Plain Layout
3
\end_layout

\end_inset

).
 Silver has very little effect on muons: it is very weakly diamagnetic,
 does not superconduct under ordinary conditions, and possesses very small
 nuclear moments (
\begin_inset Formula $\sim\num{-0.1}\mu_{\mathrm{N}}$
\end_inset

).
 Thus, silver induces neither precession nor depolarisation of the muon
 spin ensemble.
 Haematite is sometimes used, for precisely the opposite reason: large,
 random internal magnetic fields all but instantly depolarise any muons
 which land in it.
\end_layout

\begin_layout Standard
Regardless of experimental geometry, samples are often wrapped in silver
 (Ag) foil packets.
 In ZF and LF experiments, samples are normally mounted on silver backing
 plates or prongs.
 In TF experiments, haematite is used preferentially to remove a background
 non-relaxing cosinusoidal signal which would reduce the sensitivity of
 the experiment to effects in the sample.
\end_layout

\begin_layout Subsection
Experimental determination of background terms
\end_layout

\begin_layout Standard
Many relaxation functions contain a 
\emph on
background term
\emph default
 of the form 
\begin_inset Formula $A_{\mathrm{bg}}$
\end_inset

 or 
\begin_inset Formula $A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t}$
\end_inset

 to account for a static or nearly-static offset common to all spectra.
 This often arises from muons stopping in the silver sample holder or cryostat
 tails because a sample either does not cover the full area of the muon
 beam, or is too thin to stop every incident muon.
 This underlying background must be identified by evaluating the constant
 
\begin_inset Formula $\alpha$
\end_inset

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

\end_inset

) to find the 
\emph on
full asymmetry
\emph default
, and then subtracting the 
\emph on
relaxing asymmetry
\emph default
 arising from the sample, leaving the background.
 It is possible to extract 
\begin_inset Formula $\alpha$
\end_inset

 by measuring a spectrum in transverse field mode when the sample is in
 a benign phase, i.e.
 paramagnetic, non-superconducting, 
\emph on
etc.
\emph default
, and then adjusting 
\begin_inset Formula $\alpha$
\end_inset

 until the spectrum is symmetric about the 
\begin_inset Formula $x$
\end_inset

-axis.
\end_layout

\begin_layout Standard
In addition to this, however, the background can also be affected by applied
 or stray magnetic fields (which alter the paths of both incident muons
 and decay positrons), detector efficiency, and so on.
 In practice, this means that backgrounds can vary slightly during the course
 of an experiment.
 However, it is often unnecessary to precisely constrain the background
 term where the quantity of interest is an oscillation frequency or a relatively
 fast relaxation.
 Consequently, unless a detailed analysis of slow relaxations or relative
 amplitudes is to be performed, it is often sufficient to use a phenomenological
 background term 
\begin_inset Formula $A_{\mathrm{bg}}\mathrm{e}^{-\lambda_{\mathrm{bg}}t}$
\end_inset

 which is allowed to vary slightly between experimental runs.
 Its magnitude is roughly constant, and any relaxation associated with it
 is typically found to be 
\begin_inset Formula $\sim\SI{0.01}{\MHz}$
\end_inset

; significant deviation from these conditions means it is probably incorrect
 to have identified this term as entirely background.
 If well-behaved, however, the significance of the background term is limited,
 and its behaviour is largely ignored in this thesis.
 For example, in Ba
\begin_inset script subscript

\begin_layout Plain Layout
2
\end_layout

\end_inset

NaOsO
\begin_inset script subscript

\begin_layout Plain Layout
6
\end_layout

\end_inset

 (see 
\begin_inset CommandInset ref
LatexCommand formatted
reference "sub:Ba2NaOsO6-muon"

\end_inset

), the background relaxation contributes around a third of the total signal
 in all runs, with a relaxation rate 
\begin_inset Formula $\lambda_{\mathrm{bg}}\approx\SI{0.03}{\MHz}$
\end_inset

, which is typical and thus disregarded.
\end_layout

\end_body
\end_document
