In feldspars, mean tetrahedral T–O bond lengths (T

From this analysis, Al site occupancies, t, can be derived from
observed mean tetrahedral distances,

with the observed distance

with b

Finally, from an expression that converts the

The phase relations of alkali feldspars, (Na,K)[

In monalbite, similar to sanidine, there are two symmetrically
non-equivalent tetrahedral sites, T

Even though a multitude of albite crystal structure refinements have been
performed at ambient and elevated temperatures, few refinements are
available on samples that have been demonstrably equilibrated with respect
to their Al,Si distribution. Only Ribbe (1994) presented a diagram of Al
site occupancies vs. temperature based on four structure refinements: one
published and three unpublished. The crystals had been equilibrated in
reversed experiments as part of a high-pressure study by Goldsmith and
Jenkins (1985) aimed at clarifying the nature of the high- to low-albite
transition, which turned out to be smooth and continuous, being restricted
to the temperature range between ca. 590 and 720

For his presentation of the temperature variation in the Al,Si distribution
in Na-feldspar, Ribbe (1994) derived site occupancies t from mean tetrahedral
distances

When

It is our goal to utilise the equilibrated experiments of Goldsmith and
Jenkins (1985) and Waterwiese et al. (1995) to derive from reported
diffraction peak positions and lattice parameters the Al site occupancies of
Na-feldspar as a function of both temperature and pressure. From the
unit cell dimensions given by Waterwiese et al. (1995), values for t can be
easily obtained. Goldsmith and Jenkins (1985), however, provided only the
differences between two peak positions,

However, finding (t

Idealised projection of the feldspar framework along the

The length of an individual T–O bond in a feldspar structure primarily
reflects the Al content of the respective tetrahedral site. To some extent,
however, the bond length is modified by secondary factors related to the
atomic environment of the bond (Fig. 2). The following correlations have
been discussed in the literature:

Schematic drawing of the atomic environment of a T–O bond (heavy line).

Downs et al. (1996) performed procrystal electron density
calculations on low albite by placing spherically averaged wave functions at
the known atomic positions (independent atom model). The electron density obtained this way closely approximates the experimental electron density. From a
topological analysis, bonded pairs of atoms can be identified. According to
Bader (1990), a bond between two atoms is established if a saddle point
exists in the path between the pair of atoms where the gradient of the
electron density is zero and the curvature is positive along the path but
negative normal to it. The saddle point defines the bond critical point,

Gibbs et al. (2014) investigated experimental and calculated electron
density distributions in oxides, silicates and siloxane molecules and
established a power law between critical point density

The third factor that perturbs T–O bond lengths is the T–O–T angle (Fig. 2). Its influence has been the subject of some debate but has been shown to
be effective in studies by Gibbs et al. (1981), Wenk and Kroll (1984),
Geisinger et al. (1985) and Angel et al. (1990). The T–O–T angle influence
is linearised by considering the fractional

In a similar way, the fractional

The correlation observed by Liebau (1985) between T–O distances and

In our regression analysis, we did not consider the individual T–O distances
as the dependent variables but instead used the deviations between
individual and mean values, thereby separating intra- and inter-tetrahedral
variations:

Our analysis relies on a methodically homogeneous group of six X-ray structure refinements of low-albite crystals available from the literature (Table 1), deliberately omitting neutron structure refinements by Harlow and Brown (1980) and Smith et al. (1986).

Mean tetrahedral distances,

We use the notation

In the first regression run, we have chosen the first term on the right-hand
side of Eq. (6),

Distance differences

At this point it is appropriate to briefly discuss our choice of the Na
coordination by oxygen atoms on which the determination of

The regression results for the equation

Results of the regression analyses according to Eqs. (7) and (8). Errors given in parentheses refer to the last decimal places.

The second term on the right-hand side of Eq. (6),

The regression results given in Table 2 demonstrate small standard
deviations for the c

Figure 4 demonstrates the quality of the fit.

Equation (8) may be recast into the form of Eq. (1), with the observed bond
lengths,

A considerable number of structure refinements of Na-feldspars that were
heat-treated to produce variable states of disorder are available from the
literature. Meneghinello et al. (1999) performed refinements on albites
heated at

To characterise the Al,Si distribution in their
heated albites, Tribaudino et al. (2018) choose an order parameter that may be rewritten in terms of
Eq. (1):

When site occupancies of Na-rich feldspars are to be derived
from

The individual site occupancies listed in Table 3 are plotted versus the
difference

Al site occupancies t

Mean tetrahedral distances,

Continued.

Kroll and Ribbe (1983, 1987) related the length difference

Linear variation in

In order to transfer the values of

When confining pressure is applied to albite that is initially in a
disordered state (IA or HA), its Al,Si distribution will change towards
order. This was attributed by Orville (1967) and Goldsmith and Jenkins (1985) to the entropy and volume changes of LA to HA in terms of the
Clausius–Clapeyron equation

Calibration of

In order to estimate the pressure effect, we follow an approach
different from the Clausius–Clapeyron equation. It relies on the direct
comparison of equilibrated structural states attained at low pressures (LPs) and high
pressures (HPs). For this comparison, however, only LP samples equilibrated
at

Variation with temperature of

Waterwiese et al. (1995) have taken care of this problem by applying a
thermodynamic approach. Using Landau theory, Salje et al. (1985) have treated
the thermodynamic behaviour of Na-feldspar in terms of two coupled order
parameters,

We are now prepared to convert the

Variation in the thermodynamic order parameter

The HA regime extends over a temperature range that is more than twice as
large as the IA regime. Whereas in the IA region Al,Si order and disorder are the
dominant structural processes, displacive structural shearing dominates in the
HA region; that is, in the IA regime, the

From Fig. 9 we can judge the effect that increasing pressure has on the
state of order. The effect is relatively small for HA but is drastic in the
IA region. For example, intermediate albite in internal equilibrium at 700

The variation in the individual Al site occupancies with temperature is
displayed in Fig. 10. The phase transition HA

Variation in the individual Al site occupancies in Na-feldspar in
equilibrium with temperature at ambient pressure. Straight lines are drawn
to guide the eye. t

It is frequently stated in the literature and in textbooks as well that
monalbite (or analbite) is fully disordered. This does not apply,
however, as is seen in Fig. 10. The statement neglects the character of the
disordering process of albite. It is a convergent process in the sense that
the T

The transition from LA to HA at 1 bar occurs over practically the same
temperature range as it does at

There is only one feldspar other than low albite whose state of order we can
be confident of; that is low microcline (LM). Anorthite, which is also
usually taken to be fully ordered, is problematic, as shown by Angel et al. (1990), who found that the Val Pasmeda crystal had

Commercial software from PTC Inc. (2020) and OriginLab Corporation (2020) has been used for the calculations.

Data were taken from the literature as cited in Tables 1 and 3.

HK performed the calculations and prepared the manuscript with substantial contributions from the co-authors.

The authors declare that they have no conflict of interest.

Our thoughts are with our friend, the late Horst Pentinghaus, KIT Karlsruhe, who has contributed much to this work through enduring fruitful discussions and never-ending practical help. He passed away shortly before this work was finished.

The authors are grateful to the reviewers, Andrew G. Christy and an anonymous referee, for carefully reading and commenting the manuscript, which greatly helped to improve the original text.

This paper was edited by Andrew Christy and reviewed by Andrew Christy and one anonymous referee.