deletions | additions       diff --git a/methods.tex b/methods.tex  index e69de29..868bc2f 100644  --- a/methods.tex  +++ b/methods.tex 
...
\section{Methods}  \subsection{Sample preparation and mass spectrometry}  Two sections of the vein were cut parallel to the main growth axis and, using a Dremel drill with a dental bit, samples for isotopic analysis drilled at 5mm intervals in the direction of growth, Figure 2. Care was taken to avoid using undue pressure so as to minimize any frictional heating of the sample. Analyte CO_{2} is produced by reacting 6-8mg of sample powder with 102\% ortho-phosphoric acid in vacuo at 25$^{\circ}$C and for a period of 12 hours. The evolved CO_{2} is then dried, collected by cryo-distillation into a calibrated volume manometer to check reaction yields and then stripped of potential hydrocarbon contaminants before collection in a Louwers-Hapert valved glass gas tube. The drying stage involves freezing the CO_{2} into a glass spiral trap at liquid nitrogen temperatures before sublimation at -120$^{\circ}$C, passing the gas through a second trap at -120$^{\circ}$C whilst freezing with liquid nitrogen into the manometer. We strip any potential hydrocarbon contaminants from the CO_{2} by cryo-distillation into the valved gas tube via a 20cm x 4mm i.d. glass tube packed with porapak Q ion exchange resin and held at a temperature of -20$^{\circ}$C.  The sample gases were analysed for their clumped isotope values, $\delta$^{45} - $\delta$^{49} on the UEA MIRA dual-inlet isotope ratio mass spectrometer (Dennis, 2015). All analyses are made at a major beam (m/z=44) intensity of 7.5×10^{-8}A with simultaneous data acquisition for each cardinal mass of the CO_{2} molecule (m/z = 44 - 49). Each measurement consists of 4 acquisitions, each of 20 reference-sample gas pairs. Before analysis and between each acquisition the sample and reference gas volumes and signal strengths are balanced to within 1\%. A sample or reference cycle consists of a 10s ‘dead time’ after switching of the changeover valve followed by a 20s integration period. The total measurement time, including sample and reference gas balancing is approximately 90 minutes. The integration time is 1600s each for the sample and reference gas. Internal precisions ($\pm$1$\sigma$σ) for $\delta$^{45} and $\delta$^{46} are better than 0.001‰, for $\delta$^{47} better than 0.008‰, for $\delta$^{48} better than 0.03‰ and for $\delta$^{49} better than 10‰. The reference gas used in MIRA is CO_{2} produced by reaction of BDH marble chips with 85\% ortho-phosphoric acid and subsequently equilibrated with water at 20$^{\circ}$C for a period of 1 month. This is to ensure that the $\Delta$_{47} value of the reference gas is in equilibrium at the laboratory temperature. The nominal composition of the reference gas is: $\delta$^{13}C = 2.007‰_{VPDB}; $\delta$^{18}O = 34.899‰_{VSMOW}, and $\Delta$_{47} = 0.94‰_{URF}. To ensure a robust calibration of scale compression and transfer function between the local reference frame for $\Delta$_{47} and the universal reference frame (URF) both 1000$^{\circ}$C heated and 20$^{\circ}$C water equilibrated reference gas samples are measured on a daily basis (Dennis et al., 2011). Data quality and long term stability of measured values is monitored by daily measurement of two laboratory standards that bracket the range of $\Delta$_{47} values for samples in this study: UEACMST ($\Delta$_{47} = 0.384±0.013‰, n= ) and UEAHTC ($\Delta$_{47} = 0.562±0.014‰, n=). Based on the analyses of standards our best estimate of the external precision for sample analysis is $\pm$0.014‰.  The MIRA response is flat with respect to the calculated $\Delta$_{47} and $\Delta$_{48} values of samples as a function of their bulk isotopic composition as represented by their $\delta$^{47} and $\delta$^{48} values \citep{Huntington:2009to}. Not-with-standing this we regularly check for linearity by measurement of 1000$^{\circ}$C heated cylinder CO_{2} (BOC) that is depleted in $\delta$^{47} with respect to the reference gas by approximately 65‰.  \subsection{Data handling and calculation of $\Delta$ values}  The clumped isotope $\Delta_{i}$ value of a sample is defined as:  \[{\Delta _i} = \left( {\frac{{{R_i}}}{{R_i^*}} - 1} \right) \times 1000\]     where $R_{i}$ is the measured ratio of isotopologue i to the non-isotopically substituted isotopologue and $R_{i}^{*}$ is the expected ratio assuming a stochastic distribution of all isotopes over all possible sites in the lattice (reference). For CO_{2} we are largely concerned with the isotopologue ^{13}C^{18}O^{16}O (i = 47) but also determine $\Delta$ values for ^{13}C^{18}O^{17}O (i = 48) and ^{13}C^{18}O^{18}O (i = 49).  The ratios $R_{i}$ and $R_{i}^{*}$ are determined from the measured $\delta^{i}$, $\delta^{13}C$ and $\delta^{18}O$ values of the sample CO_{2}. For $R^{47}$:  \[{R_{47}} = \left( {\frac{{\delta _{sam - wrg}^{47}}}{{1000}} - 1} \right) \times R_{47}^{wrg}\]     where $R_{47}^{wrg}$ is the 47/44 ratio of the working reference gas (wrg) and is determined as:  \[R_{47}^{wrg} = 2 \cdot R_{13}^{wrg} \cdot R_{18}^{wrg} + 2 \cdot R_{18}^{wrg} \cdot R_{17}^{wrg} + R_{13}^{wrg} \cdot {(R_{17}^{wrg})^2}\]     Note that implicit in this treatment is an assumption that the mass spectrometer working reference gas has a stochastic distribution of isotopes. It is self evident that this is incorrect since the working reference gas has been equilibrated with water at the laboratory temperature. However, since the $\Delta$ values are <1‰ we can make this assumption and carry out a later linear transformation of the data to take account of the actual reference gas $R_{47}$ value without introducing any significant errors.  The ratios $R_{13}$, $R_{17}$ and $R_{18}$ are determined from the $\delta^{13}C_{VPDB}$, $\delta^{17}O_{VSMOW}$ and $\delta^{18}O_{VSMOW}$ values of the working reference gas:  \[{R_{13}} = \left( {\frac{{\delta _{wrg - VPDB}^{13}}}{{1000}} + 1} \right) \times R_{13}^{VPDB}\]     and similar equations for $R_{17}$ and $R_{18}$. The reference gas ratios are $R_{13}^{VPDB}$ = 0.0112372 (reference), $R_{17}^{VSMOW}$ = 0.0004023261 (reference) and $R_{18}^{VSMOW}$ = 0.0020052 (reference).  Similarly the $R_{47}^{*}$ ratio for a sample corresponding to a stochastic distribution of the isotopes is given by:  \[R_{47}^* = 2 \cdot R_{13}^{sam} \cdot R_{18}^{sam} + 2 \cdot R_{18}^{sam} \cdot R_{17}^{sam} + R_{13}^{sam} \cdot {\left( {R_{17}^{sam}} \right)^2}\]     with $R_{13}$, $R_{17}$ and $R_{18}$ determined using equation 4 and the measured $\delta^{13}C$, $\delta^{17}O$ and $\delta^{18}O$ values for the sample.  Substitution of $R_{47}$ (equation 2) and $R_{47}^{*}$ (equation 5) into equation 1 allows determination of $\Delta_{47}$. Evaluation of $\Delta_{48}$ and $\Delta_{49}$ follows the same steps as above. The complete data reduction algorithm and it’s implementation in a Mathematica program for data obtained on the MIRA mass spectrometer is included in the supplementary information.  Finally, using the heated and water equilibrated gas standards we determine a transfer function between the local scale, that is for measurements made with respect to the mass spectrometer working reference gas, and the absolute reference frame (ARF) \citep{Dennis:2011jd}. All the results are reported with respect to the ARF. The full data set including results reported on both the local and ARF scales are included with the supplementary information.  \subsection{Temperature estimation using $\Delta_{47}$}  Using the clumped isotope composition of carbonate minerals as a geothermometer is a young and developing technique. Critical to it's successful application is a robust calibration between $\Delta_{47}$ and temperature. At present there exist several different calibrations (references). These are illustrated in Figure 4. There is a range in both the temperature sensitivity (gradient) and offset of the different calibrations. The differences between calibrations are laboratory dependent and measurements of samples when converted to temperature only make geological sense when the local $\Delta$_{47} - T relationship is used. For this study we have used the temperature calibration determined at UEA using biogenic carbonates (bivalves and foraminifera) and travertine samples collected from sites with well characterized temperatures:  \[{\Delta _{47}} = \frac{{0.0389 \times {{10}^6}}}{{{T^2}}} + 0.2139\]  where T is in Kelvins. As with most previous studies the calibration has been made over a restricted temperature range (0-56$^{\circ}$C). For the most part this range is below the temperature of most of the samples reported in this study. We have confidence, however, that the calibration can be extrapolated to higher temperatures. In Figure 4 we have plotted data obtained in our laboratory for the $\Delta_{47}$ value of a Carrara marble sample that had been crushed and experimentally re-crytallised at 600$^{\circ}$C and 1000MPa in a solid media apparatus before rapidly quenching (Bernasconi, 2014 pers. comm.) Our extrapolated calibration passes very close to the mean of these high temperature data. Also, our calibration lies very close to the theoretical calculations of the temperature dependence of heavy isotope clumping in calcite \citep{Guo:2009fy}. These observations suggest that the UEA calibration is robust over the range of temperatures expected in the upper crust.   \section{Results}  The results of isotopic and clumped isotope analysis of all samples and standards are reported in Table 1 and plotted in Figure 5 and 6.  The spatial distribution of isotope values and temperature for DLR7 is plotted in Figure 5(a) and for DLR7i in Figure 5(b). Because the two sections were cut at different locations across the vein we don't expect a direct correlation between them. There are, however, similarities. DLR7 has a distinctive, repeating asymetric saw-tooth pattern along the growth direction that is most marked for $\delta$^{18}O and T($\Delta$_{47}). Four regions of section DLR7, separated by the gray bands in Figure 5(a), are identified. The bands mark step changes in T($\Delta$_{47}) between adjacent samples. Each section is characterized by a rising temperature from a minimum of 40$^{\circ}$ to a maximum of 90$^{\circ}$C. This pattern is mirrored by antithetic changes in $\delta$^{18}O, falling from a maximum value of -8‰ to a minimum of -10‰_{VPDB}. The pattern of change in $\delta$^{13}C is not so marked or consistent but is still apparent. For example between 10 and 25mm there is a positive covariation of $\delta$^{18}O and $\delta$^{13}C with an inverse correlation between 30 and 50mm.   In contrast, the asymmetric, sawtooth pattern observed in section DLR7 is not readily apparent in section DLR7i. None-the-less it is still possible to identify regions where there is a marked pattern of variation for both T($\Delta$_{47}) and $\delta$^{18}O, Figure 5(b). Between 10 and 20, 55 and 60, and 65 and 75mm T($\Delta$_{47}) rises whilst there is an inverse fall in the $\delta$^{18}O value. Between 25 and 50mm in section DLR7i there is an apparent breakdown in the inverse relationship between T($\Delta$_{47}) and $\delta$^{18}O. In this region temperatures for the most part are greater than 70$^{\circ}$C whilst $\delta$^{18}O remains constant and close to -9.8‰_{VPDB}. As with section DLR7 the carbon isotope data does not show a marked covariation with either temperature or oxygen isotope composition. The exception is in the section between 10 and 20mm where there is a marked inverse relationship between $\delta$^{13}C and $\delta$^{18}O with a 2‰ positive trend in carbon isotope values matched by a 2‰ negative trend in the oxygen isotope values. This trend, however, is heavily levered by a single point with $\delta$^{13}C = 1.5‰_{VPDB} and $\delta$^{18}O = -7.5‰_{VPDB}.  Plotting the bulk isotope compositions composition of the two sections shows the vein to occupy a restricted range of $\delta$^{13}C and $\delta$^{18}O values between +1.5 to +3.7‰_{VPDB} and between -7 to -10‰_{VPDB} respectively, Figure 6(a). The plot has a degree of structure, but given the limited data set it is hard to determine the significance of this. For the most part $\delta$^{18}O and $\delta$^{13}C are decoupled with a horizontal band of data covering a range of $\delta$^{18}O between -8 and -10‰_{VPDB} and little or no variation in $\delta$^{13}C ($\approx$+3.5‰_{VPDB}). There is a subset of the data that shows a positive 1:1 correlation between $\delta$^{13}C and $\delta$^{18}O and an outlier of three points with relatively high $\delta$^{18}O values (-7 to -8‰_{VPDB}) and low $\delta$^{13}C values (1.5 to 2‰_{VPDB}).