Continuous transition dataset

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Akermanite dataset:


These data are taken from McConnell et al. (2000). This is an example of an isothermal compression study of a mineral showing a continuous phase transition a little above 1.7 GPa 


Because this is a continuous transition with no step in the volume at the transition, the appropriate way to analyse the data is to use a Landau model and fit the data for the two phases together together. If your data show a step in volume at the transition, then the transition is first order: See the example on first-order transitions, and the description of the methods for more details.


See Akermanite dataset for further details about this dataset.


A macro file (akermanite.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 Akermanite 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 dataset includes cell parameters and volumes. We will work with the volumes, so answer V when asked which data to select.


The input data are then listed.





Estimate the high-symmetry EoS parameters 


You always need reasonable estimates of the EoS parameters to start least-squares, and this is especially true when you model a phase transition, because of strong correlations between parameters.


So, first we will estimate and fit the cell parameters of the high-symmetry phase. If you look at, or plot, the cell parameters and volume it is obvious that the transition is at about 1.7 GPa.


Therefore, type omit 1 -11 so that we only use high-symmetry phase data.



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


Now, you need to decide which EoS you will use....we will use a Birch-Murnaghan EoS.


  • V0,298 can be guessed 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 90 GPa. This is our estimate for  K0,298.
  • K'0,298    is very close to 4, indicated by small increase values of F.



Fit the high-symmetry EoS parameters


Type Input to input the estimated values of the EoS parameters of the high-pressure phase as a normal EoS for pressure. Use a Birch-Murnagahn 2nd-order EoS and the parameters you just estimated above.


Then type fit and fit the high-symmetry phase data. 


You should get  K0,298 = 94.1(6) GPa and V0,298 = 306.78(7).


Look at the listing of P and V: You will see that the high-pressure data are fit well, but the low-pressure volumes are larger than the volumes calculated from the EoS. 


This difference gets bigger at lower pressures. This is the volume spontaneous strain due to the phase transition.It is 0.46/306.8 or 0.15% at room pressure.



Input the transition coefficients


Type input then type tran to start the input of transition coefficients. You will see that the 'normal' eos coefficients for the high-pressure phase are saved.


Input 1 to select the Landau P-only model


For the parameter input:


Input 1 to make the high-pressure phase the high-symmetry phase.


Input 1.718 as the transition pressure.


Input 1.5 as scaling parameter for the spontaneous strain (estimated above from the room-pressure V)


Input nothing for the first power-law term, as we can only guess its value and 1/2 is a reasonable guess.


Input nothing for the next two questions as there is no evidence of softening or spontaneous strain in the high-symmetry phase. The value of aH = 0.0 will mean no softening will be calculated for the high-symmetry phase. See the theory section for details.


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



Check the dataset


First we need to activate the low-pressure dataset. 


Type use 1 -11 or use all


In the listed dataset, you should see that Vcal is now nearer to Volume for the low-pressure data, indicating that our estimates of the transition parameters are ok.


.


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



Fitting the EoS with transition coefficients





Set up the refinement 

Type fit to start performing the fitting of the dataset.


Because you already typed in the EoS parameters 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


Because there are a lot of parameters to refine, the least-squares may become unstable. Therefore it is safer to proceed in several steps. We already have a good fit for the high-pressure phase, so we will keep V0,298 and K0,298 fixed to start. 



The program now asks whether each parameter should be refined:

Input N to fix V0,298

Input N to fix  K0,298



For the transition parameters in the low-symmetry phase:

Input N to fix Ptr (because refinement is not stable)

Input Y to refine aL

Input Y to refine βL



Then you set the weighting scheme. We recommend to use all available experimental uncertainties in weighting your EoS fits. 

Input Y to use esd(P) in weights

Input Y to use esd(V) in weights



After selection of the weighting scheme, the program performs least-squares refinement of the parameter values.





Fitting results


The results of refining the transition parameters are displayed in the same way as for conventional EoS. For further information on fitting results use please refer to the Fitting EoS.


You can do more cycles of fitting and refine the high-symmetry phase EoS parameters at the same time as the transition parameters, but you will find that the fit to the data does not improve very much in this case.


You can now do all of the calculations possible for a 'normal EoS', including save to save your refined parameters to an eos file, and pvcal to output calculated P,V and other properties to a cal file.