rutile dataset: |
Data are taken from Zaffiro et al. (2019). This is an example of high-temperature thermal expansion data at room pressure.
See in the example datasets section rutile.dat for further details about this dataset.
A macro file (rutile.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: rutile |
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Set the log file |
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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. Type log to open the log and chose the appropriate directory using the file browser, then Type the log file name rutile and the program will create the log file "rutile.log" The file name for the log file can be provided with or without extension. If no extension is provided the program will choose the default extension for log files (*.log, see file types). Instead of using the file browser is possible to directly type log followed by the filename. In this case all the file will be placed in the working directory lastly used. Whenever you want you can change the log file. For further information on log files handling please refer to EoS Calculations. |
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Read the dataset |
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Type read to read the dataset and type the dataset name rutile 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 ) . For further information on log files handling please refer to Fitting EoS. The program will output the title of the dataset together with all the available input data types, in this case T,P,a,c, and V. Because the crystal system = tetragonal in the data file, the program can calculate any cell parameter, d-spacing or lattice spacing from the input data, which you can then fit. Type V to select volume The data are listed, along with information about which esd's have been read from the file. All the calculations will be performed in terms of volume. You can always select other inputs. To select other input use the command select and type the new input (see Fitting EoS) |
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Input the thermoelastic coefficients |
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Type input then type reasonable parameters for a reasonable EoS parameters. Type th to chose the appropriate T-EoS parameters Enter 1 to select the Berman EoS (refer to Thermal EoS) Enter a blank for the reference temperature (Tref) Input 1.0 for Vo, because you have volumes scaled to the V=1.0 at Tref Input 1 as alpha0 (a guess) Input 1 as alpha1 (also a guess) Input a blank for gamma0 and q; they cannot be used with only TV data and their values do not affect the results. Check the summary of the active EoS and make sure that you have typed in the correct parameters. If all is ok, type x or exit to exit from the input utility |
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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. Moreover in this case since the T-EoS parameters have already been input, the program will output the Vcal and deltaV based on the current T-EoS parameters as well together with the values contained in the dataset. Despite the input parameters being a guess, they are give close enough Vcal values that refinement should proceed without problems. |
Fitting EoS |
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Setup refinement |
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Type fit to start performing the fitting of the dataset. Remember that since you already typed in the EoS parameters now you can just change them by typing Y and inputing the new parameter or you can just Hit return (<CR>) and carry on with the fitting after you have checked the summary of the active EoS and make sure that you have typed in the correct parameters For further information on fit command use please refer to the Fitting EoS. In order to select the parameters that you want to refine Type N to not refine V0 because it is fixed by the scaling of the data Type Y to refine alpha0 Type Y to refine alpha1 Type N to not use the weights on P because they are not relevant Y twice to use the weighting scheme based on the volume e.s.d.'s and the T esd's. The program will now start the refinement cycles until convergence. |
Fitting EoS results |
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When convergence is reached the program will output the results together with the relevant statistical parameters from the fit (see Fitting EoS). In this case the refinement is converged (shift/error of the parameters is zero) so the fit to the data cannot be improved. However, the fit is poor, as you can see with the weighted-chi-squared value (W-CHI^2) = 16.8, much greater than 1.0 Hit the enter (when it says 'Input <CR> to Continue' to see a listing of the measured volumes compared to those calculated with the refined EoS parameters for the Berman model. |
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Look at the next-to-last column of this table. It lists the difference of Vmeasured from Vcalculated, divided by the esd(Vmeasured). You see lots of values, much larger than 1. For a good fit, we expect 2/3 of these values to be less than 1, and 95% of them to be less than 2. This is Gaussian statistics. In particular, look at the very last value, for the datum collected at 15K (-258 C). It is not fitted by 23 esd's!! This indicates that the saturation in thermal expansion is not fit by the Berman model. If you try to fit the Fei 1995 model or the Modified HP 1998 models, you will get similar results. Further, the best-fit parameters for these models actually describe increasing thermal expansion coefficients with decreasing T below about 50 -100K. This is unphysical behaviour! It happens because these models do not represent the principles of thermodynamics, that the thermal expansion coefficient is zero at 0 K. Therefore we must use a model that obeys thermodynamics! Hit 'enter' twice to exit from the fitting commands |
Further calculation: changing the EoS parameters |
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Change the EoS formalism and parameters to refine |
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We will now refine the Kroll model for thermal expansion, which has the correct thermodynamic behaviour. Type input and then th to change the EoS parameters and the EoS formalism for temperature data Enter 4 to select the the Kroll model, then you only need to change the alpha1 value to 0, therefore Hit return (<CR>) to keep the reference T at 298K or 24.9 C Hit return (<CR>) to keep the V0 as 1.00 Enter 6.9 for the value of Kp (taken from the literature) Enter 1.0 set the alpha0 (a guess again) Enter 330 for the Einstein temperature (value from Holland and Powell, 2011) Leave the values of Gamma0 and q as they are. They cannot affect the results of this refinement. The program will output the all the parameters set for the data fit. Check the summary of the active EoS and make sure that you have typed in the correct parameters. If all is ok, type x or exit |
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Fit the parameters to the dataset |
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Type Fit to start the refinement process Type N to refine V0 because it is fixed by the scaling of the data Type Y to refine alpha0 so this is the only refineable parameter in this model. Type N to not use the weights on P because they are not relevant Y twice to use the weighting scheme based on the volume e.s.d.'s and the T esd's. The program will now start the refinement cycles. It converges immediately because there is only one parameter being refined. The W-CHI^2 is now only 4.5 instead of 16.8. Also, look at the Vobs and Vcalc and the differences scaled by the esd(V). the differences are all a lot smaller than with the Berman model, and the lowest-T datum (the last one) is also fit well. This is a much better model than the Berman model. For further information on fitting results use please refer to Fitting EoS. At the end of the listing, type Y to update the parameters and then N to 'Further Calculations?' |
Checking the properties |
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First save the refined EoS: Type save, then navigate to the folder where you want to save the eos file (the browser opens in the current folder where your log file is located), and save the file. Now we want to check on the behaviour of these parameters and the Kroll model. Now it is easier to work in K, so type tscale and enter K to have the input and output temperatures in Kelvin. type props to start the utility to calculate the properties of the EoS. Leave the EoS parameters unchanged. Enter some temperatures: 298 to check that the room T properties are ok. Note that the program lists values for a PV EoS like K, Kp etc. This is because we use Kp in the thermal expansion equations, but the values output are not used and are meaningless. 100: Note that the alpha (volume thermal expansion) is much less than at 298K 50: and alpha is smaller still. 30: and the alpha is now almost zero, as required by thermodynamics. You can also fit the Salje model to this data, to obtain very similar results. However, for the Salje equation you need to refine the V0 parameter, as the equation uses Tref = 0K |
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Output for plotting: |
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You can use the tvcal command if you need to plot the data as function of T. It will create a .cal file with the EoS parameters and list of volumes and other properties as a function of Temperature. You can then import this text file into any plotting program Type Tvcal The program will now prompt for the EoS parameters. Hit return (<CR>) to keep the same obtained from the fitting Enter the filename in the browser. Type the temperature limits 0,1200,20 (temperature interval 0 to 1200 K, with step of 20K) You can open this file with any text editor, and check that the thermal expansion drops to zero as T goes to 0K For further information on Pvcal and Tvcal commands and outputs please refer to Output for plotting. |
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