fix(final check): This fixes references and consistency

Fixes #214
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00_Introduction/introduction.md

@@ -64,15 +64,16 @@ Substances with Arrhenius-like temperature dependence,
 shown by the blue line from the bottom left to top right of the figure,
 are considered strong liquids.
 The visibly convex curves are describes as fragile liquids. \
-*Figure from @Lubchenko2007 used with permission © Annual Reviews*](../00_Introduction/figures/angell.png){#fig:angell}
+*Figure from @Lubchenko2007 used with permission © Annual Reviews*
+](../00_Introduction/figures/angell.png){#fig:angell width=80%}
 
 The crystal growth rates of molecular crystals
 are two orders of magnitude slower than those of alloys
 and six orders of magnitude slower than pure metals (@fig:growth_rates),
 yet there is no understanding of why this is the case.
 Understanding the crystal growth of these materials
 is inherently tied to understanding the lack of crystal growth,
-also known as glass formation. [@cite]
+also known as glass formation. [@Berthier2011;@Cavagna2009]
 Many materials including
 organic molecules,[@Alba-Simionesco1999] metals, [@Wang2004] and phase-change materials [@Wuttig2007]
 have applications in both the crystal and amorphous glassy state.
@@ -84,12 +85,12 @@ the crystal growth rate is maximum,
 while the downward arrows indicate the glass transition temperature.
 The data for silver is from molecular dynamics calculations
 while all other results are from physical measurements. \
-*Figure from @Orava2014 used with permission AIP Publishing*
-](../00_Introduction/figures/molecular_growth_rates.png){#fig:growth_rates}
+*Figure from @Orava2014 used with permission © AIP Publishing*
+](../00_Introduction/figures/molecular_growth_rates.png){#fig:growth_rates width=80%}
 
 The most notable and well studied molecular liquid is ortho-terphenyl,
-chosen for having both an incredibly slow growth rate [@cite]
-and being a highly fragile liquid. [@cite]
+chosen for having both an incredibly slow growth rate [@Orava2014]
+and being a highly fragile liquid. [@Angell2000;@Chong2004]
 There are many experiments on ortho-terphenyl,
 [@Cicerone1996;@Andreozzi1997;@Chang1994;@Fujara1992;@Mapes2006]
 using a range of techniques to better understand
@@ -240,7 +241,7 @@ like fewer neighbour interactions to consider.
 Additionally fewer particles are needed to remove finite size effects
 since particles only extend in two dimensions.
 Finally, the analysis of data in 2D is much easier to visualise.
-One of the reasons @fig:dynamic_heterogenaeties is used so widely
+One of the reasons @fig:dynamic_heterogeneities is used so widely
 to describe dynamic heterogeneities
 is that it so succinctly captures the idea,
 something made possible by the 2D nature of the simulation.
@@ -265,7 +266,9 @@ while the positions are defined by
 the distance $d$ from the center of the central particle,
 and the angle $\theta$ between the centers of the two radial particles.
 
-![This shows the construction of the trimer molecule.](../01_Methods/figures/trimer.pdf)
+![This shows the construction of the trimer molecule.
+The variant used within this thesis has $d=1, r=0.637556,$ and $\theta=180^\circ$
+](../01_Methods/figures/trimer.pdf)
 
 # Project Outline
 

01_Methods/Computational_Methods.md

@@ -69,7 +69,7 @@ should be treated with caution. [@Press1988]
 
 ![When RANDU generates 'pseudo-random' the coordinates of points in 3D, all the points
 lie on one of 15 planes shown.
-](../Projects/random_number_generators/presentation/figures/randu.png){#fig:rand width=80%}
+](../Projects/random_number_generators/presentation/figures/randu.png){#fig:randu width=80%}
 
 There are also classes of bugs which are more subtle,
 results which are different yet neither is definitively right.
@@ -244,7 +244,7 @@ Importantly when using version control
 is to keep the size of each change fairly small
 as it makes this retrospective analysis much easier.
 Smaller changes are also a lot easier
-for someone else to review. [@sec:peer-review]
+for someone else to review.
 
 The minimum documentation for a project should be a README file,
 a first port of call for anyone coming across the project.
@@ -376,7 +376,7 @@ All the figures within the thesis
 are drawn from the projects from which they were created,
 which contain the code to recreate them.
 These projects are [Dynamics](https://github.com/malramsay64/Dynamics),
-used for the results in @sec:Dynamics and @sec:Dynamics_Analysis,
+used for the results in @sec:Dynamics and @sec:Glassy_Dynamics,
 [Machine_Learning](https://github.com/malramsay64/Machine_Learning) used to generate the results in @sec:Machine_Learning,
 and [Crystal_Melting](https://github.com/malramsay64/Crystal_Melting)
 used for the results in @sec:Crystal_Melting and @sec:Melting_Behaviour.

01_Methods/Dynamics.md

@@ -60,7 +60,7 @@ I used the function `freud.density.RDF` from
 the freud python package [@Harper2016]
 for the analysis of the radial distribution function.
 The radial distribution is averaged over 1000 frames
-at the melting point shown in @fig:radial_distribution_function.
+at the melting point shown in @fig:radial_distribution.
 
 ![The radial distribution function of the Trimer liquid at a pressure $P=13.50$ and a
 temperature $T=1.50$. The radial distribution is taken as an average over 100 configurations of

01_Methods/Machine_Learning.md

@@ -122,7 +122,7 @@ $$ BACC = \frac{1}{2} \left[ \frac{TP}{P} + \frac{TN}{N} \right] $$
 and is a common metric when dealing with small imbalances
 in the size of the datasets. [@Brodersen2010;@Kelleher2015]
 
-**               | Positive | Negative |
+| Positive | Negative |
 ------           |--        |--        |
 Predict Positive | TP       | FP       |
 Predict Negative | FN       | TN       |

01_Methods/Molecular_Dynamics.md

@@ -13,7 +13,7 @@ used within this thesis.
 The investigation of liquids resistant to crystallisation
 necessitates simulations over long time scales
 to fully characterise this behaviour.
-Additionally, as described in @sec:computational_tractibility
+Additionally, as described in @sec:computational_tractability
 the size of the simulation needs to be large enough
 to avoid the impact of finite size effects,
 with 500 particles recommended as the lower limit. [@Kikugawa2015;@Moultos2016;@Maginn2018]
@@ -145,7 +145,7 @@ HOOMD-blue [@Anderson2008;@Nguyen2011] is a more recent software package
 designed for GPU computation.
 It makes full use of the GPU by avoiding communication overheads
 and using data structures and algorithms designed for this hardware,
-an example being the SARU pseudo-random number generator. [@sec:random-numbers-in-hoomd]
+an example being the SARU pseudo-random number generator (@sec:random_hoomd).
 While HOOMD-blue is on the cutting edge of technology,
 there is not the same diversity of literature to draw upon that LAMMPS has.
 This also means that there are not the library of extensions,
@@ -311,7 +311,7 @@ The periodic boundary conditions of molecular dynamics simulations
 make the calculation of distances somewhat more challenging.
 When given an initial position $\vect{r_0}$ and a final position $\vect{r_t}$
 with the periodic boundary conditions there are two possible directions
-the particle could have moved in each direction. [@fig:periodic_distance]
+the particle could have moved in each direction.
 The same is also true for rotational motion.
 The standard method of handling this is the minimum image convention,
 where the distance is calculated for the shortest of the two paths.
@@ -443,7 +443,7 @@ is an algorithm for generating statistically random values
 that is, their distribution matches that of true randomness,
 while being possible to exactly replicate a sequence.
 
-### Random Numbers in HOOMD-blue
+### Random Numbers in HOOMD-blue {#sec:random_hoomd}
 
 Random numbers are used in Hoomd
 for the initialisation of translational and rotational velocities.

01_Methods/Simulation_Conditions.md

@@ -10,9 +10,9 @@ The simulations use the parameters $\tau = 1.0$ and $\tau_P = 1.0$
 for the MTK thermostat and barostat respectively
 which correspond to the rate the temperature and pressure
 are restored to their desired values.
-A step size of \si{0.005} is used for all simulations.
+A step size of \num{0.005} is used for all simulations.
 In @sec:methods_dynamics we describe the simulations
-used for the analysis of Dynamics in @sec:Dynamics and @sec:Glassy_dynamics,
+used for the analysis of Dynamics in @sec:Dynamics and @sec:Melting_Behaviour,
 and in @sec:methods_melting we describe
 the simulations for
 machine learning in @sec:Machine_Learning

02_Dynamics/conclusion.md

@@ -1,4 +1,4 @@
-# Conclusion
+# Conclusion {#sec:dynamics_conclusion}
 
 In this chapter we have established that the Trimer model
 is an appropriate model for the dynamics of ortho-terphenyl.
@@ -11,7 +11,7 @@ by the presence of dynamic heterogeneities
 which are spatially correlated,
 finding regions of fast and slow relaxations.
 These are both behaviours observed in ortho-terphenyl.
-Finally in @sec:sed_breakdown we do not observe
+Finally in @sec:trans_rot_decoupling we do not observe
 the same decoupling of translational and rotational motion
 observed by @Chang1994.
 However, there are simulation studies where the calculation

02_Dynamics/introduction.md

@@ -95,12 +95,9 @@ The expected shape of structural relaxation is an exponential decay,
 meaning the observed non-exponential shape needs to be explained.
 The non-exponential shape is the average over a range of particles
 which gives rise to two possible explanations for non-exponentially; [@Richert1994]
-
-1. each particle is undergoing the same non-exponential relaxation process, or
-2. the relaxation of each particle remains exponential,
-however non-exponential is a distribution of
-   relaxation timescales.
-
+each particle is undergoing the same non-exponential relaxation process, or
+the relaxation of each particle remains exponential,
+however non-exponential is a distribution of relaxation timescales.
 This was initially investigated by simulation studies,
 which found the presence of spatially heterogeneous dynamics within the liquid. [@Hurley1995]
 This idea is encapsulated in @fig:dynamic_heterogeneities
@@ -169,7 +166,7 @@ Within experiments, the diffusion constant is often measured using
 \ce{1H}-NMR [@Chang1994;@Chang1994a;@Fujara1992;@Mapes2006;@Andreozzi1997]
 which allows for the comparison with simulations.
 
-### Rotational Diffusion Constant
+### Rotational Diffusion Constant {#sec:rotational_diffusion}
 
 While the values calculated for the translational diffusion constant
 in simulations and experiments are relatively comparable,
@@ -209,7 +206,7 @@ the 2nd degree Legendre polynomial
 chosen for matching the spectroscopic rotation [@Brodka1992]
 with the form;
 
-$$ R_2(t) = \frac{1}{2} \langle 3(\hat{\vect{n}}(t) \cdot \hat{\vect{n}}(0))^2 -1 \rangle. $$ {#eg:rot_relax}
+$$ R_2(t) = \frac{1}{2} \langle 3(\hat{\vect{n}}(t) \cdot \hat{\vect{n}}(0))^2 -1 \rangle. $$ {#eq:rot_relax}
 
 ### Rotational Motion through Jump Dynamics {#sec:intro_jump_dynamics}
 
@@ -286,8 +283,8 @@ This decoupling has been observed in further studies
 of supercooled liquids
 and remains an unexplained phenomenon. [@Debenedetti2001;@Fujara1992;@Cicerone1996;@Ediger2000]
 
-![Translational (□, ●, △)
-and Rotational (◇,◆) diffusion coefficients
+![Translational ($\square$, $\CIRCLE$, $\vartriangle$)
+and Rotational ($\lozenge$,$\blacklozenge$) diffusion coefficients
 of ortho-terphenyl.
 The coupling of these quantities breaks down below 290K,
 where translational diffusion is faster relative to the structural relaxation. \
@@ -304,12 +301,12 @@ we expect the Trimer model to display.
 These are;
 
 1. the non-Arrhenius temperature dependence characteristic of a fragile liquid
-2. the two-step structural relaxation described in @sec:structural_realaxation
+2. the two-step structural relaxation described in @sec:structural_relaxation
 3. The presence of dynamic heterogeneities
 4. Jump dynamics in the rotational relaxation
 5. Decoupling of translational and rotational motion
 
 The rest of this chapter is about characterising these dynamic quantities
 for the Trimer and comparing them to the expected results for ortho-terphenyl.
-@Sec:structural_realaxation describes 1, and 2, @sec:dynamic_heterogeneities describes
-3, while 4 and 5 are described in @sec:stokes_einstein_debye.
+@Sec:structural_relaxation describes 1, and 2, @sec:dynamic_heterogeneities describes
+3, while 4 and 5 are described in @sec:sed.

02_Dynamics/standard_dynamics.md

@@ -31,10 +31,10 @@ inline with expectations.
 ![The intermediate scattering function of the trimer molecule
 over a range of temperatures at a pressure $P=13.50$.
 Note the logarithmic scale on the time axis.
-](../Projects/Dynamics/figures/scattering_function.svg){width=81% #fig:intermediate_scattering_function}
+](../Projects/Dynamics/figures/scattering_function.svg){width=81% #fig:isf}
 
 The structural relaxation of the Trimer molecule
-is shown in @fig:intermediate_scattering_function.
+is shown in @fig:isf.
 At low temperatures ($T=1.25$ to $T=1.4$)
 the relaxation of the intermediate scattering function
 takes place through a two-step process,
@@ -55,12 +55,12 @@ where the fragility $m$ has been found to be 220.
 ](../Projects/Dynamics/figures/scattering_function_summary.svg){width=80% #fig:isf_relaxation}
 
 The timescales of the structural relaxation $\tau_S$
-are shown in @fig:isf_relaxtion,
+are shown in @fig:isf_relaxation,
 which include the relaxation from both
 high ($P=13.5$) and low ($P=1.0$) pressure simulations.
 These structural relaxation times
 are excellent examples of the non-Arrhenius
-temperature dependence of the dynamics in ortho-terphenyl (@sec:vtf).
+temperature dependence of the dynamics in ortho-terphenyl (@sec:intro_vtf).
 The fragility $m=220$ of the Trimer liquid
 is much higher than that of ortho-terphenyl,
 which makes it an even more interesting candidate.
@@ -113,7 +113,7 @@ Vogel--Tammann--Fulcher relation. {#tbl:glass_transition_temp}
 
 ## Dynamic heterogeneities {#sec:dynamic_heterogeneities}
 
-The non-exponential structural relaxation in @fig:intermediate_scattering_function
+The non-exponential structural relaxation in @fig:isf
 also hints at the presence of dynamic heterogeneities.
 As the temperature decreases from 2.5 to 1.30,
 the maximum value of non-Gaussian parameter increases,
@@ -191,7 +191,7 @@ while coloured areas have lots of motion.
 
 The translational diffusion is characterised by the
 Mean Squared Displacement (MSD) of particles shown in @fig:msd.
-The two-step relaxation process seen in the structural relaxation (@fig:structural_relaxation)
+The two-step relaxation process seen in the structural relaxation (@fig:isf_relaxation)
 is also seen in the MSD.
 At lower temperatures,
 the transition from the initial ballistic region
@@ -241,7 +241,8 @@ and the diffusion constant
 are a component of confirming that
 the observed dynamical behaviour is as expected.
 The figures showing the rotational relaxation times
-and the diffusion constant are shown in @sec:extra_dynamics.
+and the diffusion constant are shown in @fig:rotational_relaxation and
+@fig:diffusion_constant.
 While the ratios between the quantities
 are displayed in @fig:trans_rot_trimer.
 This is interesting

03_Glassy_Dynamics/conclusion.md

@@ -1,4 +1,4 @@
-# Conclusion
+# Conclusion {#sec:conclusion_dynamics_understanding}
 
 The introduction of new molecular relaxations in @sec:molecular_relaxation
 provides a method of describing the relaxation
@@ -21,7 +21,7 @@ many of them need to be observed.
 The distribution of these jumps is much wider than for Brownian dynamics,
 with particle motion dominated by
 relaxations which have a long timescale but low probability.
-When averaging over many relaxation times as in @fig:jump_heterogeneties,
+When averaging over many relaxation times as in @fig:jump_heterogeneities,
 the distribution returns to a Gaussian.
 The increasing length scale of the jumps
 is likely related to the increasing length scale

03_Glassy_Dynamics/introduction.md

@@ -3,7 +3,7 @@
 ## Breakdown in Stokes--Einstein--Debye
 
 The Stokes--Einstein--Debye (SED) theory of dynamics
-describes a series of relations (@eq:sed) between dynamics quantities.
+describes a series of relations (@eq:sed_proportionality) between dynamics quantities.
 These relations describe the dynamics of a wide range of liquids,
 however, they start to break down in fragile liquids
 close to the glass transition temperature.
@@ -55,7 +55,7 @@ and the average diffusion constant
 
 $$ \frac{D_\text{fast} + D_\text{slow}}{2} \approx \frac{D_\text{fast}}{2} $$
 
-we can how there are different relationships. [@cite]
+we can see how these relaxations behave differently.
 Here the presence of dynamic heterogeneities results in
 a decoupling of translational and rotational motion
 by the choice of parameters.

03_Glassy_Dynamics/stokes_einstein_debye.md

@@ -62,7 +62,7 @@ The progression of time is described by the colour of the point.
 The particle spends long periods of time in one configuration
 before quickly moving to the next,
 which typically requires both a translational and rotational motion.
-](../Projects/Dynamics/figures/molecule_trajectory_fast.svg){width=48% #fig:molecule_trajectory_fast}
+](../Projects/Dynamics/figures/molecule_trajectory_fast.svg){width=80% #fig:molecule_trajectory_fast}
 
 With the motion of particles so heavily influenced by Jump dynamics
 we can investigate the role they play in dynamic heterogeneities.
@@ -137,7 +137,8 @@ play an important role in the breakdown of the SED dynamics.
 The presence of a length scale dependence
 of the rotational and translational motion
 would clear up some of the confusion in the literature
-of the decoupling of rotations and translations. [@cite]
+of the decoupling of rotations and translations.
+[@Chong2009;@Tarjus1995;@Lombardo2006;@Sengupta2013;@Kawasaki2019;@Jose2006]
 
 When calculating the translational diffusion constant
 we describe the long timescale behaviour of the mean-squared-displacement
@@ -184,7 +185,7 @@ there are two main assumptions made about the dynamics;
 1. the motion is comprised of many small independent steps, and
 2. translational and rotational motions are independent.
 
-@Sec:jump-dynamics investigates the first of these assumptions,
+@Sec:jump_dynamics investigates the first of these assumptions,
 finding it not to hold for fragile liquids,
 so here we look towards the second.
 The observations of decoupling of translational and rotational motion
@@ -235,7 +236,7 @@ Large values of $\gamma$ required large values for both
 the translational component $\Delta r$
 and the rotational component $\Delta \theta$.
 The coupling parameter of the Trimer molecule is shown in @fig:gamma,
-having a very similar shape to that of the non-Gaussian parameter (@fig:non_gaussian).
+having a very similar shape to that of the non-Gaussian parameter (@fig:non-gaussian).
 As the temperature drops,
 the coupling between the rotational and translational motions grow
 and the timescale of coupling increases.

04_Machine_Learning/conclusion.md

@@ -1,4 +1,4 @@
-# Conclusion
+# Conclusion {#sec:conclusion_ml}
 
 Machine learning is a valuable tool that can be used
 for the understanding of crystal structures in molecular dynamics simulations.

05_Crystal_Melting/conclusion.md

@@ -1,4 +1,4 @@
-# Conclusion
+# Conclusion {#sec:conclusion_melting}
 
 In this chapter we have studied
 the crystal melting behaviour of the Trimer molecule.

05_Crystal_Melting/introduction.md

@@ -103,7 +103,7 @@ their liquid position to their crystal position
 requires interacting with other particles in the liquid,
 which gives the diffusional timescale to motion.
 With the diffusion constant of the trimer molecule
-begin described by the Vogel--Tammann--Fulcher relation, (@eq:vtf)
+begin described by the Vogel--Tammann--Fulcher relation, (@eq:VTF)
 the exponential dependence on temperature
 drastically reduces the possible rate of crystal growth as the temperature drops.
 The Wilson--Frenkel theory describes growth