Using the HPx-eos

On this page we discuss key ideas that the user or would-be user should be familiar with before doing calculations with the HPx-eos. The discussion is general and does not relate to any particular calculation software. If you’re unfamiliar with the construction of the HPx-eos, read about it here or see our list of HPx-eos resources. Users of the THERMOCALC software should also read about using the HPx-eos in THERMOCALC.

Which phases are modelled by the HPx-eos?

Find out about the families of HPx-eos phases here.

Composition, mixing-on-sites and order-disorder

An HPx-eos describes a phase in terms of the ions appearing on mixing sites. For example, the end-members of the metabasite set HPx-eos for orthopyroxene are defined as follows (on a 6-oxygen basis):

end-memberM1 siteM2 sitetetrahedral site

Often, as for the orthopyroxene HPx-eos above, it is possible for two or more cations to mix on two or more sites. In this case, a full specification of the phase must include both its composition – the proportions of the different mixing elements involved, summing across all sites – and its state of order-disorder – the way that the elements are distributed over the mixing sites.

In the model orthopyroxene above, Mg and Fe appear on both M1 and M2, but the larger Fe(2+) ion has a tendency to partition onto the larger M2 site. The end-member “fm” represents full ordering of Fe onto M2 and Mg on M1. The fully disordered equivalent would have the same composition – a 1:1 mixture of enstatite + ferrosilite – but would have equal amounts of Mg and Fe on both M1 and M2. Note that this model also implicitly contains full disorder of Al and Si on the tetrahedral site. We could alternatively have chosen to model Al-Si ordering, by introducing two tetrahedral sites and a fully ordered end-member.

We caution that we don’t expect the compositions of phases, let alone the order-disorder, to be predicted very precisely by the HPx-eos. A paper discussing the uncertainties is in preparation.

Solvi and coexisting phases

In some phases, miscibility gaps in the form of solvi arise at lower temperatures. Such solvi making part of the phase’s compositional range inaccessible, and allow the possibility of two or more co-existing phases that have the same structure but different compositions – for example, Ca-rich clinopyroxene and pigeonite.

The HPx-eos incorporate solvi, as illustrated by phase C in Figure 1. Therefore, for phases that that co-exist across a solvus, one must use the same HPx-eos for both phases. A program such as Perple_X will automatically discover both of the stable compositions of the HPx-eos. In a less automated program, such as THERMOCALC, two phases must be named, both referring to the same HPx-eos, but with different compositions specified.

Relative stability

In most cases, the HPx-eos have been calibrated with the intention of accurately predicting the most stable phase assemblage under the specified conditions. However, this has not always been the case – sometimes we have been interested simply in modelling equilibria, without reference to whether or not they are the most stable equilibria possible.

In particular, the HPx-eos of Holland & Powell (2003) for Cbar1 and Ibar1 (Na-poor) plagioclase feldspar were only intended to model the Cbar1-Ibar1 transition, not to reproduce relative stabilities. Consequently the user must consider which to use where. The documentation for individual HPx-eos families draws attention to this sort of problem.

When the HPx-eos are calibrated, there is a risk of introducing unintended solvi and spurious stable compositions of phases, especially for complex and highly under-constrained phases such as melt. This might manifest as unexpected regions of two co-existing melts. We would appreciating hearing from you if you’ve experienced such problems.


The native calculation software for the HPx-eos is THERMOCALC. As of August 2019 we understand that the three major HPx-eos families are also available in Perple_X and Theriak/Domino, though we make no guarantees that these implementations are current or correct (we make benchmark calculations available for those wishing to test these or their own implementations).

In coming months we will make sure that the HPx-eos are correctly implemented in the ENKI modelling environment.

The various calculation programs have different strengths and primary purposes. Differences in the philosophies behind them can lead to aspects of HPx-eos construction seeming incomprehensible to those who use them in a program other than THERMOCALC.

For example, potential issues for users of Perple_X include:

  • Perple_X doesn’t make it easy to identify HPx-eos sets, so finding appropriate combinations of models may take some effort.
  • When we calibrate the HPx-eos and use them in THERMOCALC, we consider that the set of potential phases consist either of entire HPx-eos, representing solution phases, or of pure phases represented by individual end-members from the Holland & Powell dataset. But Perple_X treats all dataset end-members and all models for solution phases as potential members of the stable assemblage. This sometimes causes the HPx-eos to co-exist with their own end-members in Perple_X, if the end-member thermodynamic properties are modified within the HPx-eos (see notes on HPx-eos construction).

Please let us know if you’ve met any problems that you think we should mention here, for the benefit of other users.

Obtaining thermodynamic properties

The HPx-eos are essentially a calibration of Gph(P,T,x), the Gibbs energy of a phase as a function of pressure, temperature and composition. This function is of fundamental interest for predicting the most stable assemblage. The function Gas(P,T,X), the bulk Gibbs energy of the assemblage itself as a function of pressure, temperature and bulk composition, is the sum of the Gph(P,T,x) for the phases involved, weighted by the phase proportions.

Users are often interested not just in finding the most stable assemblage, but in its thermodynamic properties. These can be derived from the function Gas(P,T,X), and are printed by many calculation programs, including Perple_X and THERMOCALC (version 3.46 onwards). Users should be aware, however, that the derived properties are less well known than the Gas(P,T,X) functions themselves. This is because the nature of the calibration data makes it difficult to distinguish entropic (T-dependent) and volumetric (P-dependent) contributions to G.

Calibration and uncertainties

We aim to be transparent about how the HPx-eos are calibrated, and the uncertainties in calculations.

To find out more about the calibration, please see the paper in which the individual HPx-eos family of interest is published, as given on the page for that family. Over time we also mean to add more information to the Calibration and uncertainties page.

Uncertainty in HPx-eos is difficult to assess, owing to the complex structures of the data and calibration procedures involved. However, you should be aware that:

  • uncertainties may be quite different in different calculations.
  • in a calculated phase diagram representing a rock sample, uncertainty derived from the HPx-eos calibration is typically of a similar magnitude to the geological uncertainty derived from the assessment of equilibrium bulk composition.
  • the uncertainty in boundary positions in the above case may be of the order of ±100ºC or ±2 kbar.

This is an active field of investigation. We will provide updates on our Calibration and uncertainties page.

Still have questions?

See our list of HPx-eos resources, resource hub on phase equilibrium calculations and their uses, or pages relating to the THERMOCALC software. If you’re thinking of trying THERMOCALC, we’d recommend reading about using the HPx-eos in THERMOCALC.


Green et al (2016) J Metamorph Geol 34 845-892. Holland & Powell (1998) J Metamorph Geol 16 309-343. Holland & Powell (2011) J Metamorph Geol 29 333-383. Holland et al (2018) J Petrol 59 881-900. White et al (2007) J Metamorph Geol 25 511-527. White et al (2014) J Metamorph Geol 32 261-286.