High pressure PV dataset

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PV_quartz dataset:


Data are taken from Angel et al. (1997). This is an example of an isothermal compression study.

See  PV_Quartz for further details about this dataset.


A macro file (quartz_fit.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: 




Read the dataset


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 PV_quartz 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 launch a file browser where you can type a filename and select a working directory.


The input data are then listed.


Estimate the parameters using ffcon


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.


  • V0,298 is simply taken as the room-pressure V
  • K0,298  is estimated from the f-F values. We generate them by using the ffcon command. We select the type of EoS (Birch-Murnaghan in this case), and the program generates a list of f and F values. Extrapolation to P=0, f=0 gives F about 37 GPa. This is our estimate for  K0,298.
  • K'0,298    is greater than 4, indicated by the increasing values of F.



Note: The program gives you the option to save the f-F values for a later use (i.e. for plotting)


Input the thermoelastic coefficients


Type input 


Type Pressure (or simply pr) to chose the appropriate EoS 


Type 2 to select the Birch-Murnaghan (refer to Equations of State


Type 2 to select a 2nd-order EoS


Input 112 as the approximate V0,298


Input 37 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 and then type exit to quit the INPUT EOS> subroutine




Check the dataset


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 already 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 that the fit at low pressure is not too bad, but it gets worse at higher pressures. This is because V0,298 is slightly wrong, and K'0,298 is really bigger than 4. But this is good enough to start fitting.


For further information on list command use please refer to Fitting EoS.



Fitting EoS





Set up the refinement 

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 Y and enter the new parameter (see input commands for further details) or you can just use the loaded values, as we do here.... 


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 all EoS the scaling parameters are V0,298 and  K0,298 . They should always be refined.  We have chosen the Birch-Murnaghan 2nd-order EoS. These are therefore the only EoS parameters that are refineable because  K'0,298 is fixed to 4.


The program now asks whether each parameter should be refined:

Type Y to refine V0,298

Type Y to refine K0,298



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



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 6 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 128 indicates a poor fit to the data. 



For further information on fitting results use please refer to the Fitting EoS.


Hit Enter:  you can see the poor fit in the listing of the data points. delP is much bigger than the estimated uncertainty in the measured pressure values. delP also shows a systematic pattern with pressure; negative at low pressures, and positive at high pressures. This indicates we have the wrong EoS model.


But we can use these results as a basis for further refinement.


Type Y to update the EoS parameters


Type Y to perform further calculations





Further calculation: changing the EoS parameters



Change to 3rd-order EoS




You now have the opportunity to change the EoS; we will increase the order of the Birch-Murnaghan EoS and then refine the value of K'0,298 as well.


At the prompt 'Do you want to change....' type Y to go in to the INPUT EOS> sub program.

 

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


Then hit return (<CR>) to accept all three parameter values. Note that you are now asked for a value of  K'0,298 . We will start with the value of 4, which was the default value for the previous refinement.



Type exit to quit the INPUT EOS> sub program and start refinement of the EoS parameters.


Type Y to refine each of the parameters and

Type Y to assign weights from both V and P


Refinement of 3rd order EoS


After refinement you can see that:

  • the weighted chi-squared statistic W-CHI^2 = 0.95, just less than 1, indicating a good fit to the data. 
  • the parameter esd's are much smaller than the previous fit
  • the value of K'0,298 deviates significantly from the implied value of 4 in the 2nd-order fit


And when you enter <CR> and get a listing of the data, you will see that the maximum delP value is -0.025, and most values of delP are of the same order as the uncertainties in P.


All of these results indicate that this is a reasonable fit to the data.


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 the program will prompt the stored parameters for you to review and will ask you if you are certain that you want to save them into the *.eos file. Type Y and the program will prompt you a file browser where you can type the filename ('quartz') and chose the directory where to store the *.eos file. 


All of the refined parameters and their variance-covariance matrix will be saved to the file quartz.eos, so that you can read them in at a later date (N.B. the .eos file can also be read in Eosfit7 GUI).


Refinement of 4th order EoS

Now we can explore whether the 4th-order EoS will be a better fit to the data.


Type fit  at the command prompt 


Type pr to change the EoS


Type 2 to select the Birch-Murnaghan (refer to Equations of State


Type 4 to select a 4th-order EoS


Then hit return (<CR>) to accept all three parameter values. Note that you are now asked for a value of  K''0,298 . We will start with the default value for the previous refinement.



Hit return (<CR>) to start refinement of the EoS parameters.


Type Y to refine each of the parameters and

Type Y to assign weights from both V and P



Look at the results and compare them to the 3rd-order fit. After refinement you can see that:

  • the weighted chi-squared statistic W-CHI^2 = 0.93, just less than 1, indicating a good fit to the data. But it has not improved significantly from 0.95
  • the parameter esd's are much bigger than the previous fit
  • the value of K''0,298 deviates by only 1 esd y from the implied value in the 3rd-order fit


Therefore we conclude that the 3rd-order fit was a better (statistically and realistically) fit than the 4th-order fit. The additional parameter in the 4th-order fit does not improve the fit, and makes the uncertainties in parameter values higher, because the value of the new parameter is not well-defined by the data.






Output for plotting:





In this section we will generate a file with the calculated 3rd-order Birch-Murnaghan EoS that we determined by fitting P-V data. this file can be used for plotting p-V and other curves in external programs.


First recover the 3rd-order parameters by typing load quartz


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 and will ask you if you want to change them.


Type N to keep the values obtained from the fitting (typer Y if you want to change them)


Type the filename for the output file quartz_bm3 in the browser window. The extension *.cal will be used as default unless differently specified (see file types)


Type the pressure limits 10.0,0.1 (pressure interval 0 to 10 GPa, with step of 0.1 GPa)


The file quartz_bm3.cal will now contain all of the properties of the EoS plus their errors over this pressure range. It is a text file, so you can look at it with Notepad, and easily import it into spread sheets and graphics programs.For further information on Pvcal and Tvcal commands and outputs please refer to Output for plotting.




EoS Calculations:



Calculate elastic parameters at any pressure


To calculate the properties of the refined EoS at one pressure, 


Type props


The program will now prompt for the EoS parameters and ask you if you want to change them. 

Type N to leave them unchanged


Type the pressure values at which the calculation will be performed 


After one calculation you can type other values  or


Hit return (<CR>) to exit the calculation



For further information on Props command please refer to EoS calculation.



Calculate pressure at any volume

To calculate the pressure at any volume from the refined EoS parameters,


Type pcal


The program will now prompt for the EoS parameters and ask you if you want to change them. 

Type N to leave them unchanged


You are asked if you want to input V, or V/V0. For this example, enter N to enter volumes.


Type the volume values at which the calculation will be performed 


After one calculation if done you can type other values  or


Hit return (<CR>) to exit the calculation



For further information on Pcal command please refer to EoS calculation