Workbook, 1st Edition This workbook is designed to instruct those neutron users new to the Quasi-Elastic Neutron Scattering technique about basic QENS data reduction. The learning approach adopted is guided analysis of real data to illustrate some of the challenges posed by actual problems and datasets. The worked examples chosen also illustrate the kind of information one can extract from a QENS experiment once data reduction is complete and analysis begins. The reduction and analysis process is specifically underpinned by the Mantid package. Mark Telling NBIAW, November 2014
Introduction This workbook is designed to instruct those neutron users new to the Quasi-Elastic Neutron Scattering technique about basic QENS data reduction. The learning approach adopted is guided analysis of real data to illustrate the challenges posed by actual problems and datasets. The worked examples chosen (translational diffusion of water molecules and side group rotation in peptides) also illustrate the kind of information one can extract from a QENS experiment once data reduction is complete and analysis begins. The reduction and analysis process is specifically underpinned by the Mantid package. Comments and suggestions about this workbook are most welcome and should be sent to: mark.telling@stfc.ac.uk All the data files referred to in this workbook can be downloaded as a.zip file from: www.isis.stfc.ac.uk/groups/muons/downloads/downloads4612.html under Related Links The Mantid project provides a framework that supports high-performance computing and visualization of scientific data. Mantid has been created to manipulate and analyse neutron and muon data, but could be applied to many other techniques. The framework is open source and supported on multiple target platforms (Windows, Linux, Mac). Tailor made data reduction and analysis procedures can be formulated using Python scripting. However, this tutorial focuses on the reduction and analysis of QENS data collected from any neutron instrument via a Graphical User Interface (GUI). This interface has been specifically designed to guide the user through the data reduction and analysis procedure. NB: All procedures have been tested using the Windows 7 installation and Mantid 3.2 Mark Telling NBIAW, 10-14 th November 2014 1
Mantid can be downloaded from, download.mantidproject.org For reliability choose the Latest Stable Build. However, as the interface is developing rapidly, the Nightly Development Build may contain additional features and bug fixes that are useful for data analysis. The page, www.mantidproject.org/main_page contains a wealth of information and supplementary information regarding the Mantid project and its functionality. Mark Telling NBIAW, 10-14 th November 2014 2
Getting Started To launch the main Mantid interface type: MantidPlot (Unix) or click on the Windows desktop icon The worked examples presented in this workbook will use two dedicated GUI s. These are launched by following: Interfaces > Indirect > Convert To Energy or Interfaces > Indirect > Indirect Data Analysis from the tool bar at the top of the main MantidPlot window. Before starting you should ensure that Mantid is configured for your needs. In particular you should add paths to directories which contain the worked example data. To do this, click on File > Manage User Directories. The window below will appear. Data can be added as a path by browsing or typing the directory name in the field below Data Search Directories and then clicking Add Directory. Once a path has been added, data stored in that location can be accessed simply by entering the run number. The Manage User Directories window Mark Telling NBIAW, 10-14 th November 2014 3
Basic Mantid Data Manipulation This section explains: the concept of workspaces how to export data how to use QtiPlot fitting how to overlay data I. Workspaces Mantid utilises workspaces to hold data. These come in several forms, but the most common is the Matrix Workspace which represents XYE data for one, or several, spectra. The data contained in a workspace can viewed as a matrix or as a table, and graphed in many ways including line graphs, contour plots and 3D graphs. Interaction with workspaces is typically through a Graphical User interface. Matrix Workspaces are typically created by executing one of Mantid's 'Load' algorithms or are the output of algorithms which take a Matrix Workspace as input. In addition to data, workspaces hold a workspace history which lists the algorithms that have been applied to the data. Two types of workspace are listed in the Workspace List pane in the figure below NB: In the following discussion workspace IRIS00057252 has been created by loading raw data file via the LOAD button at the top of the Workspaces window and IRS57252_graphite002_red has been created using the Interfaces > Indirect > Convert To Energy GUI. These two methods of loading data will be discussed later Mark Telling NBIAW, 10-14 th November 2014 4
The workspace list window a) Workspace: IRIS00057252 Workspace, IRIS00057252, holds within it raw neutron counts / s for each individual IRIS detector. To examine the data collected in a single detector, right click on workspace IRIS00057252 and select Plot Spectrum. The following input fields appear. Raw data plotting options Mark Telling NBIAW, 10-14 th November 2014 5
On IRIS, as way of example, there are 114 detectors hence ID numbers: 1-114 (1,2 = neutron monitors, 3-53 = graphite analyser detectors, 54-107 = mica analyser detectors, 107-114 = diffraction detectors). Enter a detector (ID) number and click OK to plot the associated raw data. The raw counts / s for detector 5 are shown below. Alternatively, entering '5,10,20' plots spectra 5,10 and 20 on one graph Raw data (counts / s) collected in IRIS detector no. 5 (left) and in detectors 5,10,20 (right) NB: Click on the small arrows beside the workspace name to expand the entry and see specific workspace information. b) Workspace: IRS57252_graphite002_red Contains corrected or _REDuced data converted from counts/s to counts/energy transfer. Workspace IRS57252_graphite002_red contains only 10 spectra/histograms since the raw data in workspace IRIS00057252 has been reduced and corrected such that a) only detectors viewing the graphite analyser are included (i.e. 3 to 53) in the analysis and b) the raw counts in every 5 detectors have been added together to give 10 output spectra. Mark Telling NBIAW, 10-14 th November 2014 6
II. Exporting Data To export the data contained within any listed workspace click on the Algorithms tab at the bottom of the workspace list pane. The options shown below should appear. Select SaveAscii from the dropdown menu (beside Execute) and click Execute. The SaveAscii Input Dialog box should appear. Select a directory (for the written data file) and specify a file name. The workspace to be exported can be selected from the dropdown list next to InputWorkspace. The user can also, for example, define which workspace spectra i.e. indices (Min and Max) to export, choose the type of data separator used in the file (CSV is usually a reliable option), add comments and decide whether to WriteXError. The Algorithms options Mark Telling NBIAW, 10-14 th November 2014 7
The SaveAscii dialog box III. QtiPlot Enabling QtiPlot allows data sets and workspaces to be analysed using a suite of inbuilt Origin fitting algorithms. To enable QtiPlot follow View > Preferences > Fitting. The following window should appear. Select Enable QtiPlot Fitting The Choose Default Settings window Mark Telling NBIAW, 10-14 th November 2014 8
When QtiPlot fitting has been enabled, and a data file has been plotted, Analysis appears in the list of options above the main MantidPlot window toolbar. Tool bar options at the top of MantidPlot. Analysis appears when QtiPlot is enabled The fitting options found under Analysis are shown below The QtiPlot analysis options Mark Telling NBIAW, 10-14 th November 2014 9
IV Overlaying Data To overlay data, select the plot window in which you wish to compare data. Right click on the plot window and select Add/Remove Curve. A list of available data sets will appear. As an example, in the figure below we have the option of adding the data contained in the workspaces IRIS_57252_H20_290K_002-sp-10 and IRIS_57252_H20_290K_002-sp-20 to the plot window of IRIS_57252_H20_290K_002-5. To do so, first select the two spectra in the left hand pane and then click the right pointing arrow. Reversing this process removes data from a plot window. Add / Remove Curves options window The plot style (line, scatter, line + scatter etc) of the added data set can be selected using the New Curves Style drop down menu at the top of the Add/Remove Curve window. Mark Telling NBIAW, 10-14 th November 2014 10
Example One: translational diffusion Aims To reduce and correct (using Mantid s in-built analytical and numerical integration algorithms) QENS data collected from water (H 2 O @ 290 K) To perform line width analysis and determine the diffusion coefficient of H 2 0 at 290K To compare the result with literature To consider the limitations imposed by the experimental data on the analysis Data files File Name Sample Proton Current IRIS_57242_H20_280K_002 Water 280 K 300.0 uamps IRIS_57246_H20_290K_002 Water 290 K 300.0 uamps IRIS_57249_H20_300K_002 Water 300 K 300.0 uamps IRIS_57252_H20_310K_002 Water 310 K 300.0 uamps IRIS_57081_Van_RT_002 Van Cylinder 300.0 uamps IRIS_56068_Empty_Annular_RT_002 Empty Annular Can 300.0 uamps Mark Telling NBIAW, 10-14 th November 2014 11
Task One: optimal sample thickness To minimise multiple scattering effects (and hence complex correction procedures which will not be discussed here) a sample s thickness should be limited such that it scatters only 10 15 % of the incident beam. For more info about multiple scattering see: Bée M., Quasi- Elastic Neutron Scattering, 1988, Adam Hilger, Chapter 4, p 107) Using the Beer-Lambert relation below, calculate the optimal thickness of a water sample I trans = I incident exp(-nt) n = number density = total scattering cross section / formula unit t = sample thickness 1. Calculate the thickness of the water sample Task Two: Detector efficiency correction file Different detectors will have slightly different efficiencies. The result is an observable fluctuation in measured neutron intensity across a detector bank. Ideally, a wholly elastic scattering sample should have the same measured intensity in all detectors. To correct for such discrepancy, data is collected from a sample that only scatters neutrons elastically; usually a thin walled vanadium cylinder ( ~ 10% scatterer). This measurement should be performed before the actual experiment begins and then used to create a detector calibration file that is applied to all subsequent measurements during analysis. Mark Telling NBIAW, 10-14 th November 2014 12
To create a detector calibration file using Mantid: 1. Launch the GUI: Interfaces > Indirect > Convert To Energy 2. Select the correct spectrometer 3. Click on the tab Calibration 4. Browse for the Vanadium Cylinder data file and Open 5. Select Create RES File (default limits are loaded but ensure Start/End < 0.5) 6. The calibration algorithm works by subtracting a background level and integrating over the elastic peak for each of detector. The resulting integrals, or areas, are then compared and a correction factor generated. 7. To see the resulting detector calibration and RES files instrument Run_No _AnalyserReflection_calib instrument Run_No _AnalyserReflection_res check Plot result and Save Result and then Run Calibration The IRIS57081_graphite002_calib file Mark Telling NBIAW, 10-14 th November 2014 13
Q. How might the calibration plot differ if you created a detector calibration file using data collected from a 2mm thick flat vanadium plate oriented at 90 o to the incident beam? A note on Create Res File : Choosing this option creates a workspace labelled Instrument Run_No _AnalyserReflection_res which will be used later for QENS line width analysis. The algorithm takes the experimental vanadium calibration data in each detector, efficiency corrects the raw counts and sums all the spectra. The result / output is a single high statistic instrument resolution file. Q. What are the possible assumptions / limitations imposed by creating a RES file in this manner? Task Three: Applying the detector efficiency correction file to the data 1. Select tab Energy Transfer 2. Load data file IRIS_57246_H20_290K_002 in Run Files input field 3. Load the detector calibration file and check Use Calib File 4. In Mapping select individual the mapping option allows you to define how you group the detectors. The option individual treats all 51 IRIS spectra individually (surprise surprise!) 5. Select contour from the Plot Output options 6. Click Run Energy Transfer and the contour plot below should appear (a new workspace with detector efficiency corrected H20 290K data in it should appear with the name ****_red ) 7. Reduced the empty sample can data (56068) in the same way Mark Telling NBIAW, 10-14 th November 2014 14
Efficiency corrected data : individual spectra : IRIS57246_graphite002_red Tip: to visualise a workspace in 3D double click on the workspace name and the option 3D plot will appear in the main taskbar Q. How could you test that the detector efficiency corrections were being applied correctly? Try it! Task Four: Absorption corrections and empty can subtraction The main correction to be applied to neutron scattering data is that for neutron absorption both in the sample and its container, when present. For flat plate geometry, theses corrections can be analytical and have been discussed, for example, by Carlile [C J Carlile, Rutherford Laboratory report, RL-74-103 (1974)]. The situation for cylindrical geometry is more complex and requires numerical integration. These techniques are well known, used in liquid and amorphous diffraction and are described in the ATLAS manual [A K Soper, W S Howells & A C Hannon, RAL Report RAL-89-046 (1989) H H Paalman & C J Pings, J Appl Phys Mark Telling NBIAW, 10-14 th November 2014 15
33 2635 (1962)]. The routines used here have been developed from the corrections programs in the ATLAS suite and take into account the wavelength variation of both the absorption and the scattering cross-sections for the inelastic flight paths.the application of theoretical absorption corrections to experimental data is a 2-stage process. Stage 1: Generate correction files based on user input sample and sample-can geometry and composition information: 1. Launch the Indirect Data Analysis GUI 2. Click on Calculate Corrections 3. Input type File and select detector efficiency corrected H 2 0 file 4. Select Use Can this option will incorporate the effect of scattering from the aluminium sample can. Deselect if you just want to treat the sample 5. The water data was collected using an aluminium annular can. Enter the 3 radii and step size (see For Reference below). NB: The calculation cuts the cylinder into several annuli where step-size is the radial increment. There are a minimum number of annuli, n, required for reliable results (20) where n is determined from n = (r2-r1)/step. If n < 20 then an error will occur. The smaller the step the better the result but computing time increases. 6. Calculate number density and absorption / scattering cross sections for sample and sample can. Enter calculated parameters. Number density in atoms/å 3. NB In is possible to generate cross section information by entering the samples chemical formula. For example, H2O is entered: H2-O but check the values generated in the Result Log 7. Select Plot Output wavelength, tick Save Result and click Run 8. Workspace ****_cyl_abs should be created. This work space contains neutron scattering and absorption data for both sample and sample-can as a function of wavelength. Mark Telling NBIAW, 10-14 th November 2014 16
Stage 2: Apply correction file to experimental data and subtract scattering intensity from empty sample-cell 1. Select tab Apply Corrections 2. Tick all relevant check boxes: a. Tick Use Corrections to apply the ****_cyl_abs correction file to the data b. Tick Use Can to subtract empty sample-can scattering intensity c. Tick Scale Can By if exact number densities / neutron scattering properties are unknown. The Scale value will adjust the intensity of the empty cell data to avoid over correction (which may result in an inverted peak at E = 0) 3. Enter empty sample-can file or select workspace 4. Select Plot Contributions to see relative intensities and Plot Output > contour 5. Select Save Result and click Run 6. Two new workspaces are created. Depending on the level of correction performed the workspaces will have different labels. For fully corrected data (i.e absorption and empty cell subtraction) the extensions will be: ****_correct_runno_rqw, ****_correct_runno_red ****_correct_runno_result NB: For just empty can subtracted data correct will be replaced by subtract For Reference: Aluminium has a Face Centre Cubic crystal structure with a lattice constant of 4.0495 Å and a density of 2.7 gcm -3 Mark Telling NBIAW, 10-14 th November 2014 17
Total scattering and absorption cross sections can be found at: http://www.ncnr.nist.gov/resources/n-lengths/ For annular geometry enter values of r1, r2 and r3 where: r1 = inner sample radius r2 = outer sample radius r3 = outer sample radius + (2 x can wall thickness) For the sample can use: Inner sample radius = 1cm Sample thickness = use calculated value from Task One Sample can wall thickness = 0.06cm Beam width = 2 cm Task Five: Line-width analysis in energy There are several approaches to QENS line width analysis; the method chosen is really a matter of personal preference and can include basic least squares fitting or Bayesian probability (D.Sivia, Physica B, 202, 332, 1994). Here we will use the former, and the ConvFit option, to convolve N lorentzian functions with the instrument resolution file (and a background if necessary) to get the best fit to the experimental data. To perform line width analysis: Mark Telling NBIAW, 10-14 th November 2014 18
1. In the Indirect Data Analysis GUI click on ConvFit 2. Select your corrected water data file 3. Select the _res file create in Task Two 4. Select Fit Type One Lorentzian Q. Do you need a background term? Include one for the purpose of this demonstration 5. Adjust your starting parameters - fitting range (blue lines), background level etc. If you want to Fix a parameter for all fits, right click on the Property name and select Fix 6. Manually adjusting the red lines allows you to guess a starting line width and amplitude for your fit. To activate them, and visualise the effect of broadening/narrowing the lorentzian, click Plot Guess you can also adjust the parameters of the lorentzian by using the Lorentzian 1 or 2 Property box 7. Select Plot Output (Single Run) and click Run this will plot the fit for just one spectrum in a new plot window with the residual; the spectrum shown being determined by the value in the first Spectra Range input field. Suggestion: choose spectrum 10 8. If the resulting fit is poor adjust your fit parameters accordingly 9. When satisfied with the fit of one spectrum select the complete spectral range for fitting, Plot Output : FWHM, check Save Result and click Run Sequential Fit 10. A plot of line width vs. Q will appear. The plot data (data, fits, result, residual) for a sequential 1L fit will be written to workspace: *****_graphite002_conv_1lfitl_ start _to_ end _Workspace 11. All fit parameters will be written to table *****_graphite002_conv_1lfitl_0_to_50_parameters Mark Telling NBIAW, 10-14 th November 2014 19
12. Graphs of line width and amplitude will be written to *****_graphite002_conv_1lfitl_0_to_50_results Q. From a data analysis point of view, what are the limitations of the data that has been collected? Q. From an experimental point of view, what could you do to improve the results of your analysis? Task Six: Fitting By modelling the Q dependence of the QENS line width, diffusion coefficients, residence times and jump lengths associated with diffusive motions of molecules can be extracted. Several models are proposed in the literature and each considers deviations from the continuous diffusion model, or Fick s Law, for specific environments when the mechanism of the diffusion needs to be considered at smaller and smaller length scales i.e. larger and larger Q. For a comprehensive overview see Bée M., Quasi-Elastic Neutron Scattering, 1988, Adam Hilger, Chapter 5, p 148. As way of examples, Model Function Fick s law ( hwhm)( Q) DQ 2 2 2 l Q 6 The Chudley-Elliot Jump diffusion model (1961) 1 sin( Ql) ( hwhm)( Q) 1 Ql Mark Telling NBIAW, 10-14 th November 2014 20
Diffusive modes in water: i) Singwi and Sjölander (1960) 2 2 1 exp( Q D o R l ( hwhm)( Q) 1 2 o 1 Q D o 2 1 ) ii) Teixeira (1985) 2 2 DQ l ( hwhm)( Q) ; D 2 1 DQ 6 o o av Use the Teixeira model, fit your line width data and extract root mean jump length, L = <l 2 > 0.5, the residence time, o, and ultimately the translational diffusion constant, D. To code and fit a function in Mantid: 1. Click on the plot of line width vs. Q 2. Click on Analysis in the main Mantid window tool bar 3. Select Fit Wizard 4. Select Category > User defined 5. Enter a name for your new function 6. Enter the equation into the blank input field. For example, y=mx+c might be coded: Name: Linear Fit Parameters: m, c Entry on equation field: (m*x)+c 7. Save the file and click on Fit to launch the Fit Wizard dialogue box 8. Click Fit Mark Telling NBIAW, 10-14 th November 2014 21
Q. How do your values compare with values reported in the literature? Q. A single lorentzian (and background) fit to S(Q,) is a greatly oversimplified description of the water diffusion problem. Why? What does such a simple model neglect? Suggestions: Further Analysis Process the 280, 300 and 310K data sets and extract mean squared jump lengths and residence times. Do the results follow the trends presented in the Teixeira paper? Create a plot to compare of line width vs. Q vs. temperature and associated fits. Mark Telling NBIAW, 10-14 th November 2014 22
Example Two: localised diffusion While QENS line width analysis can tell us a lot about the motion, or diffusive properties, of molecules the Q dependence of the intensity of only those neutrons scattered elastically (i.e. the intensity at E = 0) also contains a wealth of information; from the geometry of a localised motion to transition temperatures. For more detailed information about the science driving the following worked analysis please see: pubs.rsc.org/en/content/articlelanding/2011/sm/c1sm05603d Aims To reduce and correct elastic incoherent scattering data collected as a function of temperature from the lyophilised protein, apoferritin. To identify transition temperatures via analysis of the elastic scattering intensity To investigate the effect of hydration upon the protein via analysis of the mean squared displacement To investigate the Q-dependence of the line width associated with a localised mode To determine the elastic incoherent structure factor (EISF) associated with the dynamic process Mark Telling NBIAW, 10-14 th November 2014 23
Data files File Name Temp Sample Start Temp / Increment Proton Current IRS38113 10 K Dry Apo 300.1 IRS38114 to IRS38156 Dry Apo Start Temp 15K T=5K 30.1 IRS38157 300 K Dry Apo 1687.9 IRS56016 290K Empty Flat Plate Cell 526.1 OSI65560 5K Hydrated Apo h=0.14 65 OSI65562 to OSI65608 Hydrated Apo h=0.14 Start Temp 10 K T=10K 65 OSI65610 to OSI65636 Hydrated Apo h=0.14 Start Temp 245 K T=5K 65 OSI73075 4K Empty Flat Plate Cell 65 OSI65437 10K Hydrated Apo h=0.31 80 OSI65439 to OSI65495 Hydrated Apo h=0.31 Start Temp 20 K T=10K 65.1 OSI73075 4K Empty Flat Plate Cell 160.1 Task One: Detector efficiency correction file 1. Taking the DRY (aka LYOPHILISED) PROTEIN data only! Follow the procedure set out in Example 1: Task 2. However, this time create a detector efficiency file from the sample itself measured at low temperature. NB: This approach should ONLY be used if the temperature of the sample is low enough that you are sure that ALL scattering is elastic! The advantage of this approach, however, is that Mark Telling NBIAW, 10-14 th November 2014 24
you have a direct measure of absorption effects from both sample and sample can as well as correction for the orientation of the sample can should a flat plate cell be used. Q. What might be the disadvantages of this method when creating a RES file from experimental flat sample-can data? Task Two: Applying the detector efficiency correction file to the data 1. Create a suitable detector mapping file (****.map) to process individual detectors BUT such that the dead detectors seen in the efficiency plot are removed from all subsequent analysis. A mapping file should have the following structure: 3 ; number of spectra to be created 1 ; spectrum 1 13 ; number of detectors to be added 3 4 5 6 7 8 9 10 11 12 13 14 15 ; detector nos 2 ; spectrum 2 5 16 17 18 19 20 3 5 21 22 23 24 25 The mapping file above will create 3 spectra. These spectra will be the sum of counts in detectors 3-15, 16-20 and 21-25 Mark Telling NBIAW, 10-14 th November 2014 25
3. Check that the mapping file is applied correctly 4. Apply the detector efficiency correction and mapping file to all the lyophilised data in the Data Files list above so you have a _red file for 38113 > 38157 and also for the empty cell Task Three: Empty sample can subtraction 1. Follow the procedure set out in Stage 2, Example 1: Task 4 to subtract the empty cell from all detector efficiency corrected data files Task Four: Analysis of the elastic scattering intensity Since energy must be conserved during the scattering process, any onset of QENS broadening must result in an associated decrease in elastic scattering intensity. Monitoring the elastic scattering intensity, I(Q,T, 0), as a function of temperature and momentum transfer therefore allows transition temperatures to be identified. To analyse the intensity of the elastic peak from our lyophilised protein: 1. Select the Indirect Data Analysis GUI 2. Click on the Elwin tab 3. Select the _subtract_ data files which correspond to temperatures 15K to 290K 4. The blue lines show the range over which integration of the elastic line will be performed Q. Does this range seem suitable? Remember ONLY neutrons scattered elastically should be included in the analysis Mark Telling NBIAW, 10-14 th November 2014 26
7. Check Save Result and click Run 8. Three workspaces are written: Inst First Run No _AnalyserReflection_to_ Last Run No _eq1 = intensity vs. Q vs. temp Inst First Run No _ AnalyserReflection _to_ Last Run No _eq2 = ln(intensity) vs. Q 2 vs. temp (for mean squared displacement analysis) Inst First Run No _ AnalyserReflection _to_ Last Run No _elf = intensity vs. Q vs. temp NB: if only one data file is processed using ELWIN the output workspaces will be named inst Run No _ AnalyserReflection _eq'n' and inst Run No _ AnalyserReflection _elf NB: depending on when a data file was collected, the x-axis in the.elf workspace will be either the last 3 digits of the run number (i.e. 38113 > 113) OR the temperature associated with the measurement. Sample temps will be used if the raw data file has a temperature log associated with it. For example, when loading 38113, Elwin looks for SE Log Name: sample. However, this log name does not exist in the raw data log information (right click on the irs38113_graphite002_red and open Sample Logs) In contrast, irs27252_graphite002_red does contain a temperature log Name called sample 9. Select the _elf file > right click > Plot Spectrum With Errors > enter 3 spectrum values of your choice (suggestion: 10,30,40) and click OK. Alternatively, plot the _elf file in 3D Q. What do you notice? Is the elastic intensity constant as a function of run number? Q. Is the data too noisy? If so, how could you reduce the noise? Mark Telling NBIAW, 10-14 th November 2014 27
Task Five: Mean square displacement, <r(t) 2 >, analysis The mean square displacement, <r(t) 2 >, of atoms in a material can be extracted from elastic neutron scattering intensities, I(Q,T, 0), by fitting, S inc 1 2 2 ( Q, T, 0) I( Q, T, 0) exp( Q r( T) ) (1) 3 Q Fit exp 2 r 3 2 Strictly speaking, this form is valid for harmonic oscillations or equivalent atoms. Any deviation from harmonic behaviour is noted as the activation of an-harmonic behaviour. Experimentally, harmonic to an-harmonic behaviour is usually identified by an inflexion in the temperature dependence of <r(t) 2 >. Equation (1) is only valid in the low-q regime. However, <r(t) 2 > still proves itself to be a useful tool for relative changes in <r(t) 2 > as a function of, for example, hydration level, temperature and / or sample. Mark Telling NBIAW, 10-14 th November 2014 28
To extract <r(t) 2 > from the elastic scattering created in Task Four: 1. Select tab MSD Fit 2. Browse for and load the _eq2 file 3. Select one of the higher temperature spectra to display using Plot Spectrum (suggestion : 30) 4. Eq (1) above will be fitted to data points delineated by the blue lines. Q. Does the fit range in Q look suitable? If not, adjust the range by modifying StartX and EndX in the MSD tab Property field or drag the blue lines on the plot 5. Check Plot Result and Save Result 6. Click on Run to fit the data on screen OR Run Sequential to fit all temperatures 7. The fit of Eqn (1) to the data on screen is shown 8. Three workspaces are created: Inst First Run No _graphite002_to_ Last Run No _msd_workspaces contains individual workspaces with raw data, associated fits and residuals Inst First Run No _graphite002_to_ Last Run No _msd contains the initial amplitude (A0) and gradient (A1) vs run no/temp Inst First Run No _graphite002_to_ Last Run No _msd_parameters table of A0 and A1 vs run number / temperature 16. To see <r 2 > vs. temperature - right click on Inst First Run No _graphite002_to_ Last Run No _msd and select Plot Spectrum With Errors. Right click on the plot window that appear, select Add/Remove Curve and remove workspace _a0 Mark Telling NBIAW, 10-14 th November 2014 29
Q. Is there a change in the slope of <r(t) 2 >? If so, does it broadly correspond with the temperature at which the elastic intensity starts to decrease? NB: if no temperature information is included in the output files then the temps will have to be entered manually by viewing the table in workspace Inst First Run No _graphite002_to_ Last Run No _msd_parametersand changing the x-axis values Task Six: <r(t) 2 > and the effect of hydration 1. Now repeat the data reduction, ELWIN and MSD Fit analysis detailed above using the hydrated protein data sets. Plot the resulting <r(t) 2 > from the hydrated materials on top of the lyophilised protein response. Q. What do you notice? Q. Should you actually compare the three experimental data sets? Q. If so, what is this result telling you about the protein? Task Seven: Line width analysis in time Experimentally, the measured scattering function, S meas inc(q,), is a convolution of S inc (Q,) and the resolution function of the neutron instrument, R(Q,). For spectrometers operating in Q- space, S meas inc(q,w) = S inc (Q,) x R(Q,). In its simplest form the instrument resolution approximates to a Gaussian or Lorentzian function of finite width, res (usually quoted as full width at half maximum). As we saw in Example 1, using either a measured or theoretical R(Q,), least squares fitting or Bayesian analysis routines can be used to isolate the intensities and widths of the spectral contributions to S inc (Q,). Here, however, we will Mark Telling NBIAW, 10-14 th November 2014 30
adopt an analysis method which relies on the Fourier transform of the measured scattering function. The merits of fitting in the time regime are discussed in V. Arrighi, J. S. Higgins, A. N. Burgess and W. S. Howells, Macromolecules, 1995, 28, 4622 4630. Using Fast Fourier Transform (FFT) methods, the measured QENS and resolution spectra are converted to the time domain. Deconvolution of R(Q,) and S meas inc(q,), is achieved by simply dividing the Fourier response of the sample by that of the resolution. The result is the time-dependent intermediate scattering function, I(Q,t). In the simplest case, a single relaxation process will manifest itself in the time domain as a simple exponential, I(Q,t) = A o (Q) + [1 - A o (Q)]exp((t/). Here, is the relaxation time and A o (Q) is the Elastic Incoherent Structure Factor (EISF). A system that exhibits a distribution of relaxation rates, however, may be better described using the Kohlrausch Williams Watt (KWW), or stretched exponential, form i.e. I(Q,t) = A o (Q) + [1 - A o (Q)]exp(-(t/ KWW ) ) It should be noted that here KWW is an effective relaxation time which is dependent upon both T and, or more correctly the temperature dependence of the spectral shape of the distribution. Non-exponential behaviour is immediately apparent should the stretching parameter fall below unity. Another advantage of converting S(Q,) to I(Q,t) is for direct comparison with data collected using NSE. Such comparison can allow relaxation phenomena to be followed out to much longer time scales and thus modelled with much greater certainty; at least at certain momentum transfer vectors. In this task we will look to analyse data taken at 300K in the time domain. Analysis is a 2 stage process i.e. first we FFT the spectra, changing energy to time, and then we fit. Mark Telling NBIAW, 10-14 th November 2014 31
Stage 1: 1. Using the Convert To Energy GUI, reduce data file 38157 (lyophilised protein, 300K) such that a) run 38113 (lyophilised protein, 10K) is used to create the detector efficiency file and b) a suitable mapping file is produced that collates every 5 detectors. 2. Reduce the empty sample-can data and the base temp sample file as above 3. Using the Indirect Data Analysis GUI, subtract the empty sample-cell data from both reduced sample files 4. Select Indirect Data Analysis > Fury 5. Browsed for the 300K '_subtract_' data file and Plot Input 7. For Resolution, browse for the 10K '_subtract_' data file 8. Adjust E low, E high and E width to be -0.5, 0.01 and 0.5 respectively 9. Check Save Result and click Run 10. The resulting workspace from Fury analysis is named 'inst'run No'_AnalyserReflection_iqt' I(Q,t) data for spectra 1,4,7, 10 from IRS38157_graphite002_iqt Mark Telling NBIAW, 10-14 th November 2014 32
Stage 2: There are several options to fit I(Q,t) curves a) the FuryFit GUI, b) creating a user defined fit function in Fit Wizard or c) using an external program such as DAVE FuryFit: The following details how to perform I(Q,t) analysis using the FuryFit GUI a GUI very similar in operation to the ConvFit analysis tool described earlier. 1. Select File or workspace IRS38157_graphite002_iqt. A plot of the data in spectrum N will appear (default for N is spectrum 0). Select spectrum 5 2. Select Fit Type: 1 Stretched Exponential 3. Click Constrain Intensities to insure that the sum of the initial amplitudes of all individual fit components is 1 4. Define the range of the fit by either entering values in the StartX / End X fields or dragging the blue lines on the plot 5. Define the background level by either entering values in the LinearBackground field or by dragging the green base line up to the required level 6. Selecting Plot Guess automatically adds a fit to the data based only the initial values associated with each fit function Property. 7. Once initial values are set, click Run to fit the spectrum displayed in the GUI only NB Selecting Plot Output launches a new window (when Run is pressed) showing the single fit, data and residuals in greater detail and only over the x and y fit range entered 8. Function parameters can be fixed by right clicking on the parameter name and selecting Fix. Fixed parameters are un-fixed by following the same procedure Mark Telling NBIAW, 10-14 th November 2014 33
9. Once happy with the fit of a single spectrum, select which fit parameter to plot (e.g. tau, intensity or beta), select Save Result and click Run Sequential Fit to analyse all the spectra 10. Three workspaces are created: Inst Run No _graphite002_fury_1s_s0_to_9_result contains fit parameter values for all spectra in WORKSPACE form which can be plotted individually Inst Run No _graphite002_fury_1s_s0_to_9_parameters contains fit parameter values for all spectra in TABLE form Inst Run No _graphite002_fury_1s_s0_to_9_workspace contains data (ID 0), fits (ID 1) and residuals (ID 2) for each spectrum. Expand the workspace (click on the side arrow) to see individual spectrum workspaces. To create the multi-plot below, select four workspaces > right click > Plot Spectrum > enter, 0-1 (for data and fit only) I(Q,t) data and sequential fits for spectra 1,4,7, 10 from IRS38157_graphite002_iqt Mark Telling NBIAW, 10-14 th November 2014 34
11. Examine the fits and fit parameters. Are they reasonable? Are any fit parametrs constant (within error) as a function of Q? Could any be fixed at a single value to reduce uncertainty in the other parameters? NB If beta is constant, within error, then it can be constrained at a mean value by either a) FIXING the parameter OR b) clicking Constrain Beta Over All Q. The latter creates the output files detailed above BUT with the additional description _1Smult 12. Analyse the I(Q,t) data using FuryFit and ensure you can broadly replicate the plots in the reference SoftMatter paper. NB: for consistency with the data in the SoftMatter paper fix beta at 0.64 Fit Wizard: Analysis can also be performed using the Fit Wizard GUI described earlier. As such only additional information required to analyse the data will be given below. 1. Using irs38157_graphite002_iqt, create a plot showing all I(Q,t) curves > right click, Plot Spectrum, enter spectrum ID numbers : 1-10. Adjust axis values to: x = 0 to 0.1. y = 0.82 to 1.02 2. Click on the graph that appears and select Main Menu > Analysis > Fit Wizard 3. Create and save a new User function with the form : (1-A)*exp(-(x/t)^B))+A 4. Select Fit With Selected User Function and click Fit >> 5. In Custom Output >> click One Table For All Fits, click Parameter Table and then << Fit. A new table named Parameters should appear 6. Select Curve ***-sp-1, change fit range to x = 0 to 0.1 and click Fit. The fit parameters should be appended to the Parameter table 7. Repeat for Curves ****-sp-2 to ****-sp-10 8. Manipulate the fit parameters and plot the Parameter table values to replicate the plots in the reference SoftMatter paper Mark Telling NBIAW, 10-14 th November 2014 35
DAVE: For completeness, the following details how to perform I(Q,t) analysis using the very versatile IDL based package, DAVE, which can be downloaded here:www.ncnr.nist.gov/dave/ Taking a monkey-see-monkey-do approach to speed up analysis 1 In Mantid, click on the Algorithms tab 2 In the Execute input field, type SaveDaveGrp and click Execute 3 A new dialogue box will appear which will allow us to convert the Mantid 'inst'run No'_AnalyserReflection_iqt' file into a format DAVE can read. Select the relevant '_iqt' file, check ToMicroEv and enter a filename for the written dataset 4 Click Run 5 Launch DAVE by typing Dave 6 In the DAVE Gui > File > Preferences : Set Data And Working Directory 7 Load your I(Q,t) file by following: Data Input/Output > Read Dataset From > ASCII 8 Click on OK when the Specify Axes Labels For Data window appears 9 Data Input/Output > Write Dataset As > DAVE - a file with the extension.dave is created in your working directory 10 Analysis > 1D Peak Fitting (PAN). The PAN GUI will launch and the data in the.dave file should be visible in the plot window 11 In the PAN GUI, define the fit range by following File > Preferences and check / enter the following: Mark Telling NBIAW, 10-14 th November 2014 36
9. Click Accept and the Dismiss 10. To write a function: Select Function > Userfunction. Create a simple exponential decay function. Save your expression Q. Do you need to add a background to your equation? What does any background (or asymptotic I(Q,t) level) intensity tell you? 11. Click Accept > enter suitable start values > click Apply And Dismiss 12. Click Fit Current Group Q. Does a simple exponential provide a suitable description of the data? Note: linear axes can be deceptive! Examine the data with logx and logy axes by following Plot_Options > XLog and YLog. Focus the plot range by following File > Preferences. Suggestion: x = 0.01 to 0.1 and y = 0.87 to 1.0 13. Click Fit All Groups and examine the goodness of fit at different Q values. 14. Click Clear All Curves and create a new stretched exponential function. Refit the data Mark Telling NBIAW, 10-14 th November 2014 37
Q. Is the stretched exponential description more appropriate? What does the stretched exponential form tell you about the internal mobility of the protein? 15. To examine the resulting fit parameters click > Plot Fit Parameter and select the parameter to view. For example, for the simple exponential model Parameter #0 will probably be the background level and Parameter #1 will probably be the decay time, Q. Compared to the water example would you say that the relaxation rate,, is Q dependent? Q. What does a Q independent relaxation rate suggest about the motion being probed? Q. To improve the fit / reduce errors could you fix one of the parameters in your model? Refit the data with or fixed. To perform a new fit, first click on Clear All Curves and then follow the procedure above Task Eight: Fitting the Elastic Incoherent Structure Factor (EISF) The EISF, A o (Q), provides information about the geometry of a particular localised motion and can be extracted from I(Q,t) if the decay curve reaches a plateau at 'long' decay times. From the I(Q,t) analysis above, the Q dependence of the background level gives reasonable access to the EISF; or, in the case of a partially mobile sample as is the case here, an effective structure factor. The theoretical Q dependence of elastic scattering intensity has been computed for many different types of localised motions. For a comprehensive list see Bée M., Quasi-Elastic Neutron Scattering, 1988, Adam Hilger, Chapter 6, p 200). As examples: Mark Telling NBIAW, 10-14 th November 2014 38
A o,ch3 (Q) = 1/3[1 + 2j o (Qr 3)] the theoretical EISF expected for a proton undergoing a localised 3-fold jump rotation. r = 1.032 Angstrom and j o is the zero-order Bessel function. A o,jump (Q) = 1/2[1+j o (Qd)] the theoretical EISF response predicted for protons undergoing 2-site jumps of distance, d A o,diff (Q) = (3j 1 (Qr))/Qr 2 the theoretical EISF response predicted for protons diffusing on a sphere of radius, r, Use the equations above to model the EISF response generated from your I(Q,t) analysis. NB: when fitting the model EISF functions remember that your experimental EISF is in fact an effective EISF since only a certain percentage of the sample is actually mobile the theoretical EISF functions assume the entire sample is mobile! The theoretical expressions therefore have to be modified to reflect this : 1. From your Mantid I(Q,t) FuryFit analysis, plot the Q-dependence of your background parameter 2. Fit the data using FitWIzard and the three EISF expressions above; suitably modified to describe an effective EISF Q. Which of the three models provide the best description of the experimental data? Q. From your fits, what percentage of the protons in the material is mobile? Q. What would you need to do to truly ascertain the model which best describes the experimental data? Mark Telling NBIAW, 10-14 th November 2014 39
Suggestions: Further Analysis By modelling the temperature dependence of for the lyophilised material, determine a mean activation energy, E a, ave, for the protein dynamics observed. How does this compare to E a, ave determined from other dry proteins? Mark Telling NBIAW, 10-14 th November 2014 40
Example Three: Dental Cements Right...now we have all the tools to explore QENS data we will focus on data collected from dental cements; data which is so far is unanalysed and the 'result' unknown. The rational behind this work is given in the proposal (raw data directory) submitted to gain access to beam time on the IRIS instrument. Use the various data analysis methods explored in Examples One and Two to gain insight into the role of water in the solidification of dental cements. Don t forget to make use of advice from all the experimental team! Data Files File Name Temp Description 54638-40 2 K 28 day old HEMA cement 54635-36 310 K 54641-54698 15 300 K Elastic Window Scan Data 54435, 54498, 54499 310 K Liquid POLY 54436, 54497 300 K 54437, 54438 10 K 54439-54496 15 300 K Elastic Window Scan Data 53593 310 K Liquid HEMA 53594 300 K 53595 + 53596 10 K 53583 290 K Vanadium Mark Telling NBIAW, 10-14 th November 2014 41