Pcell _hexagonal dataset: |
These data are taken from Gatta et al. (2011). This is an example of an isothermal compression study of a mineral showing a iso-symmetric first order phase transition a little above 3.6GPa.
Because this is a first-order transition with a step in the volume at the transition, the appropriate way to analyse the data is to split the data into two groups and fit the data for the two phases separately. If you have a continuous transition, then use the tools for modeling the transition as part of the EoS. See the example on continuous transitions, and the description of the methods used in EoSFit to describe those types of transition.
See Pcell_hexagonal for further details about this dataset.
A macro file (Pcell_hexagonal.mcr) is available in the software distribution package together with the dataset file. If you run the macro, it will do all of the commands and calculations for you!! Note: to ensure that the macro runs correctly, close EosFit7c and start it again immediately before running the macro.
Dataset: |
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Read the dataset |
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It is always strongly recommended to open a log file where all of your calculations and results will be written and stored. More information. Type read Pcell_hexagonal to read the dataset. The file name for the dataset file can be provided with or without extension. If no extension is provided the program will try to load the file with the default extension for dataset files (*.dat see file types ) . You could also just type read and then the program will ask you for the filename. The input data are then listed. |
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Estimate the parameters using ffcon |
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You need reasonable estimates of the EoS parameters to start least-squares. First, you need to decide which EoS you will use....we will use a Birch-Murnaghan EoS.
Pay attention on the sudden increase of the F values from about 59 to about 45 between row 11 and 12. This is a clear indication of a "jump" in the Ffplot maybe due to a phase transition. The best way of handling a phase transition is to split the dataset in two or more parts. This is done with the use/omit command |
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Select the data to use |
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Type Omit to discard data that you don't want to use and keep only the data for what we have identified as the low-P polymorph that is clearly represented by all the data lines between row 1 and 11. Therefore you want to omit all the data lines from 12 to 19. Type 12,-19 to omit all the data included in the interval between data line 19 and 54. Check that all the data from data lines 19 to 54 are flagged with 0 at the beginning of the line. For further information on omit command use please refer to the Fitting EoS. |
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Input the thermoelastic coefficients |
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Type input then type reasonable parameters for a reasonable EoS for the low-P polymorph. Type pr to chose the appropriate EoS Type 2 to select the Birch-Murnaghan (refer to Equations of State) Type 3 to select a 3nd-order EoS Input 201.0 as the approximate V0,298 Input 60 as the best estimate of K0,298 Check the summary of the active EoS and make sure that you have typed in the correct parameters. If the parameters are correct, type exit to quit the INPUT EOS> subroutine |
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Check the dataset |
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Type list to visualize the dataset It is always a good idea to check that the dataset and all the input parameters have been read correctly. Since the EoS parameters have been input, the program will output the Vcal and deltaV based on the current EoS parameters compared to the values contained in the dataset. You can see by looking at deltaP and deltaV that the fit for the low pressure polymorph is not too bad, but the one for the high pressure one is completely wrong. This is a further obvious confirmation of what already seen with the Ffcon output. But this is not a problem because you will only fit the low-P polymorph for now. For further information on list command use please refer to Fitting EoS. |
Fitting EoS |
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Set up the refinement |
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Type fit to start performing the fitting of the dataset. Because you already typed in the EoS parameters now you can just change them by typing P and inputing the new parameter or you can just use the loaded values, as we do here.... Make sure that you have typed in the correct parameters and then type N if you do not want to change the active EoS parameters For all EoS the scaling parameters are V0,298 and K0,298 and for the Birch-Murnaghan 3rd-order EoS the value of K'0,298 is also refineable: The program now asks whether each parameter should be refined: Type Y to refine V0,298 Type Y to refine K0,298 Type Y to refine K'0,298 Then you set the weighting scheme. We recommend to use all available experimental uncertainties in weighting your EoS fits. Type Y to use esd(P) in weights Type Y to use esd(V) in weights |
Fitting EoS results |
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After selection of the weighting scheme, the program performs least-squares refinement of the parameter values. It lists the results from every least-squares cycle. If you have problems with the refinement, examine this output to see which variable has the largest shifts, or is oscillating in value. That usually indicates the problem. In the example, the refinement converged rapidly (shift/error=0) and stopped after 3 cycles. W-CHI^2 is the weighted chi-squared statistic. It is a measure of the fit of the EoS to your data. The value of 0.72 indicates a good fit to the data. For further information on fitting results use please refer to the Fitting EoS. |
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Hit Enter: you can see the good fit in the listing of the data points. delP is very close in absolute value to that of the estimated uncertainty in the measured pressure values for the lowP polymorph. For the high-P polymorph you can see the high values of delP confirming once again that the phase transition has been correctly bracketed in the pressure interval between 3.5 and 3.9GPa. Type Y to update the EoS parameters Type N to not perform further calculations At this point it is a good idea to save your results: Type save and enter a filename ('Pcell_hexagonal_lowP') in the file browser. All of the refined parameters and their variance-covariance matrix will be saved to the file Pcell_hexagonal_lowP.eos, so that you can read them in later to perform more calculations or output the data for plotting (N.B. the .eos file can also be read in Eosfit7 GUI). |
Further calculation: changing the pressure range |
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Change the pressure range |
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Now we need to change the pressure range to be able to work on the high pressure polymorph. If you remember from before, the high-P polymorph data range goes from data line 12 to 19 (if you don't remember just type list to see the entire dataset and look at the use flags where 0 means not in use and 1 means in use, see Fitting EoS). Type use and then enter 12,-19 to include data from line 12 to line 19 (high-P polymorph). Type omit and then enter 1,-11 to omit data from line 1 to 11 (low-P polymorph) Now you might want to re-check what reliable estimate of the EoS parameters for the high-P polymorph would be therefore Type ffcon and enter 2 for a Birch-Murnaghan EoS
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Input the thermoelastic coefficients |
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You now can refine only the selected part of the dataset (only data for the high-P polymorph) changing the EoS parameters to those just obtained from the estimate using the ffcon command. Type input to input the new EoS parameters and formalism Type pr to chose the appropriate EoS Type 2 to select the Birch-Murnaghan (refer to Equations of State) Type 3 to select a 3rd-order EoS Hit return (<CR>) to accept the V0,298 value from the previous refinement Type 45 as the new K0,298 estimate Type 4 as the new K'0,298 estimate Note that we can start with the value of 4 even if the actual value will be greater at the end of the refinement because the program will refine the K'0,298 starting from 4. Quit the INPUT EOS> subroutine by typing exit Type list to check the delP and delV values for the high pressure polymorph. The results are certainly not as good as for the low P polymorph but at least we can be sure that we are not using completely wrong starting estimated parameters.Therefore, Type fit and Type N to start refinement of the EoS parameters without any change. Type Y to refine each of the parameters and Type Y to assign weights from both V and P |
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Refinement of the high pressure polymorph data |
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After refinement you can see that:
All of these results indicate that this is a not unreasonable fit to the data for the high-P polymorph. Type Y to update the EoS parameters. The least-squares results will be stored for future calculations. Type N to 'further calculations?' You are now back at the command prompt and you can use your refined parameters to do other calculations. For further information on fitting results use please refer to Fitting EoS. At this point it is a good idea to save your results: Type save and hit return <CR> to save the EoS parameters into an *.eos file. In the browser window chose a directory and enter filename ('Pcell_hexagonal_highP'). All of the refined parameters and their variance-covariance matrix will be saved to the file Pcell_hexagonal_highP.eos, so that you can read them in later to perform more calculation or output the data for plotting (N.B. the .eos file can also be read in Eosfit7 GUI). |
Output for plotting: |
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In this section we will generate a file with the calculated 3rd-order Birch-Murnaghan EoS that we determined by fitting P-V data for both the low-P and the high-P polymorphs. This file can be used for plotting p-V and other curves in external programs. First recover the 3rd-order parameters for the low-P polymorph by typing load Pcell_hexagonal_lowP. If you do not remember whether the eos parameter values are correct, type list. This will give you a listing of Pobs, Pcalc, and delP. Are all delP values small? If 'yes' proceed. If 'no' find the right eos file! Type pvcal The program will now prompt for the EoS parameters. Hit return (<CR>) to keep the same obtained from the fitting (type Y to chose a different *.eos file or change the EoS parameters) Type the filename for the output file Pcell_hexagonal_lowP. The extension *.cal will be used as default unless differently specified (see file types) Input the pressure limits 0,7,0.1 (pressure interval 0 to 7 GPa, with step of 0.1 GPa) Repeat the procedure for the high-pressure phase: Load Pcell_hexagonal_highP pvcal ....send the output to Pcell_hexagonal_highP and use the same pressure limits 0,7,0.1 The files Pcell_hexagonal_lowP.cal and Pcell_hexagonal_highP.cal will now contain all of the properties of the EoS plus their errors over these pressure ranges. They are a text files, so you can look at them with Notepad, and easily import them into spread sheets and graphics programs.For further information on Pvcal and Tvcal commands and outputs please refer to Output for plotting. |
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