From e47df094215986d9a29c6caf90435a823bac496b Mon Sep 17 00:00:00 2001 From: Sam W Date: Wed, 7 Nov 2018 11:21:39 +0000 Subject: [PATCH] Added year 2 experiment 0 laboratory report. Added addition book to common additional textbooks bibliography. --- bibliographies/additionaltextbooks.bib | 15 + year2/0/0.bcf | 2009 +++++++++++++++++ year2/0/0.pdf | Bin 0 -> 910413 bytes year2/0/0.run.xml | 92 + year2/0/0.tex | 237 ++ .../images/chlorophyll-a-decomposition-2.png | Bin 0 -> 230403 bytes .../images/chlorophyll-a-decomposition-3.png | Bin 0 -> 232633 bytes year2/0/images/chlorophyll-a-graph-2.png | Bin 0 -> 208333 bytes .../images/chlorophyll-a-residuals-plot-2.png | Bin 0 -> 190573 bytes .../images/chlorophyll-a-residuals-plot-3.png | Bin 0 -> 192195 bytes .../images/chlorophyll-a-residuals-plot-4.png | Bin 0 -> 194542 bytes year2/0/images/decomposition-2.png | Bin 0 -> 305410 bytes year2/0/images/decomposition-3.png | Bin 0 -> 306462 bytes year2/0/images/graph-2.png | Bin 0 -> 258221 bytes year2/0/images/residuals-plot-2.png | Bin 0 -> 188432 bytes year2/0/images/residuals-plot-3.png | Bin 0 -> 190659 bytes year2/0/images/residuals-plot-4.png | Bin 0 -> 192804 bytes year2/0/specific.bib | 17 + 18 files changed, 2370 insertions(+) create mode 100644 year2/0/0.bcf create mode 100644 year2/0/0.pdf create mode 100644 year2/0/0.run.xml create mode 100644 year2/0/0.tex create mode 100644 year2/0/images/chlorophyll-a-decomposition-2.png create mode 100644 year2/0/images/chlorophyll-a-decomposition-3.png create mode 100644 year2/0/images/chlorophyll-a-graph-2.png create mode 100644 year2/0/images/chlorophyll-a-residuals-plot-2.png create mode 100644 year2/0/images/chlorophyll-a-residuals-plot-3.png create mode 100644 year2/0/images/chlorophyll-a-residuals-plot-4.png create mode 100644 year2/0/images/decomposition-2.png create mode 100644 year2/0/images/decomposition-3.png create mode 100644 year2/0/images/graph-2.png create mode 100644 year2/0/images/residuals-plot-2.png create mode 100644 year2/0/images/residuals-plot-3.png create mode 100644 year2/0/images/residuals-plot-4.png create mode 100644 year2/0/specific.bib diff --git a/bibliographies/additionaltextbooks.bib b/bibliographies/additionaltextbooks.bib index 82b6875..16997bd 100644 --- a/bibliographies/additionaltextbooks.bib +++ b/bibliographies/additionaltextbooks.bib @@ -6,3 +6,18 @@ edition = {2} } +@book{hughes-hase-uncertainties, + title = {Measurements and their Uncertainties}, + author = {Ifan G. Hughes and Thomas P. A. Hase}, + year = {2010}, + publisher = {Oxford University Press}, + address = {Oxford} +} + +@book{bransden-qm, + title = {Quantum Mechanics}, + author = {B. H. Bransden and C. J. Joachain}, + year = {2000}, + publisher = {Pearson Education Limited}, + edition = {2} +} diff --git a/year2/0/0.bcf b/year2/0/0.bcf new file mode 100644 index 0000000..d76eab4 --- /dev/null +++ b/year2/0/0.bcf @@ -0,0 +1,2009 @@ + + + + + + output_encoding + utf8 + + + input_encoding + utf8 + + + debug + 0 + + + mincrossrefs + 2 + + + minxrefs + 2 + + + sortcase + 1 + + + sortupper + 1 + + + + + + + alphaothers + + + + + labelalpha + 0 + + + labelnamespec + shortauthor + author + shorteditor + editor + translator + + + labeltitle + 0 + + + labeltitlespec + shorttitle + title + maintitle + + + labeltitleyear + 0 + + + labeldateparts + 0 + + + labeldatespec + date + year + eventdate + origdate + urldate + nodate + + + julian + 0 + + + gregorianstart + 1582-10-15 + + + maxalphanames + 3 + + + maxbibnames + 999 + + + maxcitenames + 2 + + + maxitems + 3 + + + minalphanames + 1 + + + minbibnames + 1 + + + mincitenames + 1 + + + minitems + 1 + + + singletitle + 0 + + + sortalphaothers + + + 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page size to A4. +\usepackage{graphicx} %Allow import of images. +\usepackage{floatrow} %Positioning of table & figure captions. +\usepackage{siunitx} %SI Units formatting. +\usepackage[version=4]{mhchem} %Chemical Equations. +\usepackage{amsmath} %Mathematical alignments. +\usepackage{amssymb} %Mathematical symbols e.g. therefore. +\usepackage[parfill]{parskip} %Blank lines between paragraphs. +\usepackage{array} %Table centring. +\usepackage[autostyle]{csquotes} %Recommended for babel. +\usepackage[british]{babel} %British localisation. +\usepackage[backend=biber,style=chem-rsc]{biblatex} %Bibliography. +\usepackage[extreme,mathspacing=normal,leading=normal]{savetrees} %Reduce whitespace. + +%Configure settings. +\numberwithin{equation}{subsection} %Number equations based on their section +\bibliography{../../bibliographies/reference,../../bibliographies/coretextbooks,../../bibliographies/additionaltextbooks,specific} %Use Bibliography of reference books. +\bibliography{specific} +\floatsetup[table]{style = plaintop} %Put captions on top of tables. +\floatsetup[figure]{style = plain} %Ensure figure captions are below figures. +\sisetup{range-phrase = --,range-units = single} %Use - for ranges and only have a single unit shown after. +\sisetup{separate-uncertainty = true,multi-part-units=single} %Use \pm for uncertainties. +\setlength{\parindent}{0cm} % Don't indent paragraphs. + +%Document Headings. +\begin{document} +\title{Experiment 0: Analysis of the absorption spectrum of plant pigments} +\author{Sam White (Author) and Emma Winful (Laboratory Partner)} +\date{02/11/2018} +\maketitle + +\begin{abstract} + The UV-Visible spectrum of the pigments contained within a plant leaf + obtained and then the sum of various Gaussian curves was used to generate a + fit line for the spectrum. It was found that modelling the absorptions at + different wavelengths with Gaussian curves yielded poor results and it is + likely that the use of Voigt curves will give better agreement with the + experimental data due to the inclusion of lifetime broadening effects in + addition to Doppler broadening effects on the spectra linewidths. +\end{abstract} + +\section{Results and Analysis} +A number of Gaussian curves were added together to generate a fit line for the +UV-Visible spectroscopic data collected. Gaussian curves were used since it was +assumed that the major contributor to the spectral linewidth would be Doppler +broadening due to the Doppler effect, thus resulting in a Gaussian form for the +absorption peaks due to the molecular speeds being distributed according to the +Maxwell-Boltzmann distribution.\autocite{atkins-physical} + +%Gaussian curves were used since the central limit +%theorem states that the distribution of the sum of a sequence of independent and +%identically distributed random variables will converge to the normal +%distribution in the limit as the number of variables tend to infinity. This +%hence means that we can assume that any sufficiently large sample will have +%sample statistics which are normally distributed and hence will be in the form +%of a Gaussian curve. + +%Within UV-Visible spectroscopy the exact absorption frequency is affected by a +%number of factors such as due to the Doppler effect and lifetime +%broadening\autocite{atkins-physical}, hence leading to a distribution of +%absorption frequencies. This absorption frequency distribution can be taken to +%be normal by the central limit theorem since the samples of materials used +%within UV-Visible spectroscopy contain a very large number of individual +%molecules (and hence chromophores) thus satisfying the large sample requirement +%and hence why Gaussian curves were used to generate fit lines for the obtained +%data. +\subsection{Chlorophyll \textit{a} Sample Data} +Within the data for Chlorophyll \textit{a} three clear absorption peaks can be +identified. Due to this three separate Gaussian functions were added together +to produce an overall curve to fit to the data as shown in figure +\ref{fig:chl-a-decomposition}. + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{images/chlorophyll-a-decomposition-3.png} + \caption{Graph showing the generated fit line for the chlorophyll \textit{a} UV-Visible + absorption data and the component Gaussian curves used to generate the fit.}\label{fig:chl-a-decomposition} +\end{figure} + +The parameters for each Gaussian curve shown in the fit statistics box on the +figure correspond to those in equation \ref{eq:gaussian} where $x$ is used to +differentiate between the different curves. + +\begin{equation}\label{eq:gaussian} + f_{ax}(\lambda) = A_{ax} e^{-\frac{1}{2} \left(\frac{\lambda - \lambda_{ax}}{w_{ax}}\right)^2} +\end{equation} + +The residuals for this fitting ($\chi_{1}$) were then plotted as a histogram and +their residuals ($\chi_{2}$) relative to another fitted Gaussian curve (used as +the residuals should occur independently and randomly, thus by the central limit +theorem they should be normally distributed) were also plotted (as shown in +figure \ref{fig:chl-a-residuals}). These residuals ($\chi_{2}$) were plotted as +a ratio of the uncertainty of the absorption residual value for which they +corresponded (i.e. as a ratio of the uncertainty in $\chi_{1}$) with this +uncertainty estimated to be $\sqrt{N}$ where $N$ is the number of occurrences of +the residual value. Furthermore the yellow area on the plot represents a +distance within one standard deviation to the fitted Gaussian and the green area +represents a distance between one and two standard deviations. + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{images/chlorophyll-a-residuals-plot-4.png} + \caption{Histogram showing the frequency of each residual value for the fit + in figure \ref{fig:chl-a-decomposition} with a plot of the residuals for + this with respect to a fitted Gaussian curve.}\label{fig:chl-a-residuals} +\end{figure} + +\subsection{Collected Experimental Data} +In the raw data collected the absorption decayed to between \num{0.056} and +\num{0.051} for all wavelengths greater than \SI{700}{\nano\meter}. This was +identified as a systematic error and was corrected by subtracting an absorbance +of $0.051$ from all of the collected data. + +The absorption peaks for the UV-Visible spectrum obtained are less well defined +than those for the pure chlorophyll \textit{a} spectrum and when a fit line was +constructed using three Gaussian curves the corresponding fit to the data was +quite poor with the maxima of the Gaussian curves used for the fit clearly not +matching the maxima in the experimental data. + +It was found that plant pigments naturally contain another form of chlorophyll +called chlorophyll \textit{b} in addition to chlorophyll +\textit{a}.\autocite{berkeley-pigments} Chlorophyll \textit{b} absorbs UV +radiation at a slightly lower frequency than chlorophyll \textit{a} (principle +peak absorption at \SI{642}{\nano\meter})\autocite{chlorophyll-uv-vis} and has +two absorption peaks in total within the \SIrange{550}{750}{\nano\meter} region. +To account for the presence of chlorophyll \textit{b} in the sample two +additional Gaussian curves were fitted to the data (represented by violet +dash-dotted lines in figure \ref{fig:exp-decomposition}) to create the fit line +seen. In figure \ref{fig:exp-decomposition} the three Gaussian curves +representing the contribution to the absorption caused by the chlorophyll +\textit{a} are shown as as blue dashed lines. + +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{images/decomposition-3.png} + \caption{Graph showing the generated fit line for the experimental UV-Visible absorption data and +the five component Gaussian curves used to generate the fit.}\label{fig:exp-decomposition} +\end{figure} +\begin{figure}[H] + \centering + \includegraphics[width=0.9\textwidth]{images/residuals-plot-4.png} + \caption{Histogram showing the frequency of each residual value for the fit + in figure \ref{fig:exp-decomposition} with a plot of the residuals for + this with respect to a fitted Gaussian curve.}\label{fig:exp-residuals} +\end{figure} + +\section{Discussion} +The $\chi^2$ value for the experimental plot in figure +\ref{fig:exp-decomposition} is much lower than that for the chlorophyll +\textit{a} plot shown in figure \ref{fig:chl-a-decomposition} ($\num{9.909e-5}$ +compared to $\num{4.171e-3}$) which suggests a better fit. In addition to this +is can clearly be seen that the fit line on the experimental plot follows the +data more closely than that for the chlorophyll \textit{a} plot where the peaks +on the fit line do not completely coincide with the peaks in the data points +which, while resulting in a lower $\chi^2$ value (in comparison to setting the +$\lambda_{ax}$ value exactly to the peak wavelength), does not accurately follow +the assumed absorption model (i.e. that the total absorption results from the +linear combination of various Gaussian curves generated due to chromophores +absorbing radiation at a small range around a specific wavelength). + +Despite this it can be seen from the residual plots (figures +\ref{fig:chl-a-residuals} and \ref{fig:exp-residuals}) that the residuals for +the chlorophyll \textit{a} fit line appear to be better described by a Gaussian +curve (used to fit the residuals due to application of the central limit +theorem) than the residuals for the experimental plot fit line. This can be seen +from the lower $\chi^2$ value, and also from how $\chi_{2}/\sqrt{N}$ has a +smaller magnitude maximum and minimum in addition to how it is more concentrated +within two standard deviations (the criteria for "reasonable +agreement")\autocite{hughes-hase-uncertainties} of zero. This is unexpected +since from the visual appearance of the fit lines the residuals for the +experimental plot were expected to exhibit random behaviour better modelled by +the normal distribution than those for the chlorophyll \textit{a} plot since the +fit line for the chlorophyll \textit{a} plot is clearly inadequate in places. +This suggests that that Gaussian curves used to generate the fit for the +experimental data may not accurately describe the underlying physical system. + +%The fit residuals for the data were modelled by a Gaussian curve since (in the +%absence of any systematic errors) they should result from random errors, which +%are (by definition) independent and random, hence their distribution must +%converge to the normal distribution. However for both data sets the mean for +%residuals is close to +%zero, as expected since there should be an equal probability for random +%fluctuations above the fit line as for below the fit line. + +In figure \ref{fig:chl-a-decomposition} it can be seen that the fit curve decays +faster than data (most apparent by inspection of the bottom of the major +absorption peak). In the fitting used for the experimental data this is +accounted for by the wider Gaussian curve used to represent the principle +chlorophyll \textit{a} absorption (i.e. by the relatively large value of +$w_{b1}$), however it is possibly unrealistic to expect that the width of the +principle absorption peak for chlorophyll \textit{b} will be almost double the +width of the curve for chlorophyll \textit{a} (seen from how $w_{a1} = +\num{7.36(4)}$ while $w_{b1} = \num{15.06(4)}$). Furthermore the third Gaussian +curve used to describe the chlorophyll \textit{a} absorption in the experimental +data is very wide ($w_{a3} = \num{49.50(3)}$) compared to the rest of the curves +which while allowing the generation of a better fit curve for the data is also +unlikely to represent the underlying absorption distribution at that wavelength. +This (in conjunction with the irregularities in the fitting of the residuals to +a normal distribution) may suggest that the model of using Gaussian curves for +the absorption is not fully correct and a different waveform (which doesn't +decay as fast as a Gaussian curve) might better represent the absorption of the +physical system. + +This is likely to be the cause of the slight discrepancies between the peak +wavelengths of the Gaussian curves used for fitting the chlorophyll \textit{a} +data and the corresponding curves used to fit the chlorophyll \textit{a} +absorptions in the experimental data, although the fact that these discrepancies +are quite small (the largest is \SI{10.3}{\nano\meter}) suggests that Gaussian +curves are not totally inadequate for modelling UV-Visible absorptions. + +Spectral linewidths are affected by lifetime broadening effects in addition to +Doppler broadening effects which were +considered.\autocite{atkins-physical,bransden-qm} Lifetime broadening effects +contribute an Lorentzian curve component for the spectral +lines,\autocite{bransden-qm} thus explaining why Gaussian curves appear to be +unable to fully describe the absorption peaks. Voigt curves are a composite of +Gaussian and Lorentzian curves, thus the use of linear combinations of Voigt +curves to fit the absorption data collected is likely to result in an improved +fit over that which was obtained with Gaussian curves. + +%The chlorophyll \textit{a} fitted curve can be seen to not fit the sample data +%very well with the forms of the absorption peaks not being particularly good +%matches with the Gaussian curves. In addition it can be seen that the fit curve +%decays quicker to the right hand-side than the experimental data. These factors +%may suggest that some chlorophyll \textit{b} impurity was present in the sample +%since they are accounted for the in fitting of the experimental data with five +%Gaussian curves which clearly had a much higher amount of chlorophyll \textit{b} +%present. + +\printbibliography + +\end{document} +%Note: gq means : reformat the text included in the motion +% vim:tw=80 diff --git a/year2/0/images/chlorophyll-a-decomposition-2.png b/year2/0/images/chlorophyll-a-decomposition-2.png new file mode 100644 index 0000000000000000000000000000000000000000..4e264e922c0179e25799ba4df7a54642e1bb3e6b GIT binary patch literal 230403 zcmeFZhd-A6|30p%RJ1e*6%D(x6)JnnPDUj~X7-5Om6Sqcl%4EN_AW_L;j*_#TxMj8 zzQ@_P@6Y%9c>I2U!RLBE?(RG9%XOXS>-8MRaXgRL`9MWUdJ7dD6$J&w7FihyRSF8q z1PY3E$2P3TPgYzvmEvFPFUw0yQ2ZqR7gZAf1V7nmC3D_}g5t0|@z1Kw?Fz^7LrPm& zMM=uOO=~F*QXTc)dyax)KZUHs8Fk0@(JlvD+RC3lCtJ2`JTToI>h3P{WS1zL#-XT$ z{o!@YY?V>A88?rvCdC}x9=Ray643;6po`QWYg_rLM)jMb~I|Myq-LmO1q{P$Om 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