## Abstract

We show that the rehydroxylation (RHX) method can be used to date archaeological pottery, and give the first RHX dates for three disparate items of excavated material. These are in agreement with independently assigned dates. We define precisely the mass components of the ceramic material before, during and after dehydroxylation. These include the masses of three types of water present in the sample: capillary water, weakly chemisorbed molecular water and chemically combined RHX water. We describe the main steps of the RHX dating process: sample preparation, drying, conditioning, reheating and measurement of RHX mass gain. We propose a statistical criterion for isolating the RHX component of the measured mass gain data after reheating and demonstrate how to calculate the RHX age. An effective lifetime temperature (ELT) is defined, and we show how this is related to the temperature history of a sample. The ELT is used to adjust the RHX rate constant obtained at the measurement temperature to the effective lifetime value used in the RHX age calculation. Our results suggest that RHX has the potential to be a reliable and technically straightforward method of dating archaeological pottery, thus filling a long-standing gap in dating methods.

## 1. Introduction

In an earlier paper (Wilson *et al.* 2009), we set out the principles of rehydroxylation (RHX) measurements on fired-clay bricks and tiles, and we noted that this method of dating should be applicable to archaeological pottery. Here we present the first results of applying RHX dating to three types of archaeological pottery, varying in age, type and depositional environment. We describe in detail the RHX dating methodology and issues surrounding sample equilibration, ceramic composition and type, sample lifetime temperature and data analysis. This work represents a significant advance in transferring RHX dating from brick and tile to archaeological pottery, and we show, for the first time, how RHX applies to pottery materials with varying firing history, composition and burial environment.

## 2. Rehydroxylation and the RHX kinetic model

We use the term RHX to describe the chemical recombination of fired-clay ceramics with environmental moisture. There is incontrovertible evidence that a slow progressive rehydration of this kind occurs generally in all clay ceramics fired up to temperatures of around 1150^{°}*C* (for a recent review, see Hamilton & Hall 2012). It appears that this is largely or partly a reversal of the dehydroxylation reaction which occurs in ceramic firing.

Wilson *et al.* (2003) showed evidence that moisture expansion in fired-clay bricks and the associated mass gain increases as the fourth root of the time since firing (the RHX power-law model). Further reports (Savage *et al.* 2008*a*,*b*; Wilson *et al.* 2009) have confirmed this power-law model, and in the case of moisture expansion over periods of more than 50 years (Hall *et al.* 2011). Thus, the RHX rate equation is given as
2.1where *y* is the fractional RHX mass gain at elapsed time *t* and *α* the RHX rate constant. For any material, *α* varies with temperature *T*. *α* also varies from one material to another, depending on the ceramic mineralogy, the maximum firing temperature and the duration of firing (Tosheva *et al.* 2010; Mesbah 2011).

The basis of RHX dating follows directly. A fired-clay ceramic sample gains mass progressively throughout its lifetime from the time of first firing. We assume that it does so according to the power law given in equation (2.1). If we can establish this lifetime fractional RHX mass gain *y*_{a} and can also determine the rate constant *α* for the sample, then the estimated RHX age *t*_{a} is simply given by the RHX age equation
2.2

We describe in the following sections how the quantities *y*_{a} and *α* are obtained, and discuss how the associated errors determine the uncertainty in the RHX age *t*_{a}.

## 3. Components of the sample mass

RHX dating is a gravimetric method in which the critical quantities are obtained by precision measurements of sample mass in well-defined operations. The total sample mass is made up of a number of components. For the purposes of RHX dating, we must distinguish carefully between various contributions to the total mass. A typical archaeological pottery sample has a complicated composition. Here, we denote the *as-received* mass by *m*_{r} and define this as the mass of the sample as it enters the RHX measurement procedure. At this point, the sample has been washed to remove all loose or friable material. We consider that the quantity *m*_{r} has five distinct components which we discuss in turn. These components are shown schematically in figure 1.

*Ceramic mass**m*_{cer}. This is the mass of the total inorganic mineral assemblage which remains intact after re-heating the sample to 500^{°}*C*until it reaches constant mass. We assume that this is essentially the mineral assemblage formed in the original firing process, but includes any long-term diagenesis of the mineral assemblage that is refractory. The mass gain/loss associated with diagenetic changes is small at most, and in any case has no impact on the RHX age estimate.*Non-refractory component mass**m*_{nrc}. We include here any substances*other than water*present in the as-received sample that contribute to the mass loss on re-heating to 500^{°}*C*. These may be organic materials such as food residues, microbiological contaminants, absorbed humic acids, etc. Minerals unstable below and at 500^{°}*C*contribute here also.*Type 0 (T0) water mass**m*_{w0}. We designate as T0 water all weakly held molecular water, which is removed by heating the sample to constant mass at 105^{°}*C*. This includes capillary water held in the open pores of the sample and may also include weakly bound adsorbed water. The distinction between these two types of water is not sharp, but the capillary water is held as a result of Laplace forces arising from the pore geometry and water surface tension; while the adsorbed water is held by physisorption on ceramic surfaces.*Type 1 (T1) water mass**m*_{w1}. We designate as T1 water all molecular water, which is not removed at 105^{°}*C*and which is removed during heating to 500^{°}*C**but excluding T2 water*. T1 water is likely to be chemisorbed molecular water, which is typically removed at 200–300^{°}*C*. We do not need to know the amounts of T0 and T1 water separately and denote the T0+T1 water together as T01. This is illustrated in figures 1 and 3.*Type 2 (T2) water mass**m*_{a}. This is the mass of water gained during RHX. It is assumed in RHX dating that this is the RHX mass acquired since firing during the entire lifetime of the sample (a period of time equal to the sample age*t*_{a}). RHX dating depends on measuring this quantity*m*_{a}accurately. It is assumed*operationally*that T2 water is removed by heating the sample to 500^{°}*C*until it reaches constant mass.

In the as-received sample, *m*_{r}=*m*_{cer}+*m*_{nrc}+*m*_{w0}+*m*_{w1}+*m*_{a}. The RHX measurement procedure then comprises several steps, also shown in figure 1.

The sample is dried at 105

^{°}*C*to constant weight, thus removing all T0 water. By definition T1 water and T2 water are not removed in this step. We assume that no non-refractory contaminant mass is lost either, although if it is, there is no impact on the RHX estimated date. At the end of this step, the sample mass*m*_{1}=*m*_{cer}+*m*_{nrc}+*m*_{w1}+*m*_{a}.We condition the sample in a controlled environment at the same relative humidity (RH) and temperature as used in step 4 in the microbalance procedure to determine the sample RHX rate constant

*α*. The temperature is approximately set to the effective lifetime temperature (ELT) of the sample. The mass increases at this stage because the sample takes up some T0 water and comes to equilibrium with the conditioning environment. The sample mass is now , where*m*^{′}_{w0}is the mass of T0 water regained by the end of this step.The sample is heated to 500

^{°}*C*. In this process T0, T1 and T2 water components are all lost, as are any non-refractory components. The mass at the end of this step is*m*_{3}=*m*_{cer}.The sample is transferred to the microbalance chamber at the same controlled humidity and temperature as used in step 2, and mass data

*m*(*t*) are recorded. The sample gains mass as it takes up T01 water to reach equilibrium with the microbalance environment and, at the same time, progressively gains mass as a result of the slow RHX process. After the uptake of T0 and T1 water is complete (the end of the so-called stage I at*t*=*t*_{1}), the continuing mass gain (owing to continuing uptake of T2 water) is the characteristic*t*^{1/4}power-law RHX process that provides us an estimate of the RHX rate constant*α*of the sample. At the end of the initial stage I transient when*t*=*t*_{1}, the sample mass is . Here,*m*_{w2}is the time-dependent mass gain arising from uptake of T2 water in the slow RHX process.

For *t*>*t*_{1} microbalance mass–time data are well represented by the RHX power-law model so that
3.1where the constant *m*_{4} is the intercept on the mass axis. Using segments of the mass–time data from the microbalance obtained in step 4, we test the gradient d*m*/d*t*^{1/4} until it becomes constant with time. This stabilized gradient d*m*/d*t*^{1/4}=*α*_{m}. For RHX dating, we need to know precisely the mass arising from the lifetime RHX of the sample *m*_{a}. The quantity *m*_{4} comprises the ceramic mass *m*_{cer}, together with the mass of T01 water that has been taken up during stage I. Thus, . It then follows that
3.2Thus, *m*_{a} can be calculated from experimental quantities *m*_{2} and *m*_{4} provided that (1) the mass *m*_{nrc} due to the non-refractory component is either known or assumed to be zero, and (2) the mass of T01 water which the sample contained at the end of step 2 (*m*^{′}_{w0}+*m*_{w1}) is the same as the mass of T01 water which the sample recovers in step 4 (). Given that (2) is true, we have *m*_{a}=(*m*_{2}−*m*_{nrc})−*m*_{4}, and if *m*_{nrc}=0, we have *m*_{a}=*m*_{2}−*m*_{4}.

Information about *m*_{nrc} for individual samples can be obtained through subsidiary experiments; and for many types of samples we consider that this component is absent, in which case we take the mass *m*_{nrc}=0. Alternatively, we can analyse the sample for non-refractory components, such as total organic carbon and carbonates, before and after reheating to 500^{°}*C*.

The assumption about T01 water is provisional but based on reasonable assumptions about the reversibility of adsorption and the stability of the microstructure during reheating at 500^{°}*C*. It should be noted that the quantities *m*_{r}, *m*_{1} and *m*_{3} are not used in the RHX dating calculation.

Finally, it is useful for comparison between different samples to normalize the quantity *m*_{a} by the reference mass *m*_{4} to obtain the quantities *y*_{a}=*m*_{a}/*m*_{4} and *α*=*α*_{m}/*m*_{4} which appear in equation (2.2).

## 4. Materials and experimental methods

In the previous section, we have set out the essential steps we take to obtain the gravimetric quantities needed to estimate an RHX age. Here, we provide practical details of the experimental procedures used in RHX dating measurements on a group of pottery samples and carried out using a recording microbalance of 5 g capacity. Mineralogy was quantified by X-ray diffraction (Bruker, D8) using semi-quantitative Rietveld refinement, and a combustion analyzer (Carlo Erba, NA2500) was used to determine organic carbon. Specific surface area was measured by BET nitrogen gas sorption (Micromeritics, Gemini 2360).

### (a) Pottery materials

The experimental data were obtained following a large number of exploratory experiments on a small group of well-sourced archaeological pottery samples. These are

— L1—An Anglo-Saxon fired-clay loomweight recovered at Bourne House Stables, Lambourn (West Berkshire, UK) in excavations in December 2007, and for which there is a radiocarbon date for associated bone material from the same archaeological context. Radiocarbon age (SUERC 33836): 1430±35 BP, calibrated date 560–660 AD (95.4% probability).

— S1—A samian-ware sherd from an excavation at Owslebury (Hampshire UK) in 1967 (Collis 1968), considered to have been made at La Graufesenque, France. The assigned date of 45–75 AD is based on typography and a maker’s stamp, Damonus (G Dannell, private communication).

— W1, W2, W3—Three Werra earthenware sherds from a wastepit excavated at Enkhuizen (The Netherlands) in 1979. The samples have inscribed dates of 1605 AD (Bruijn

*et al.*1992).

These samples are diverse in microstructure, mineralogy and age. Figure 2 shows the microstructure of the L and S materials. Werra composition is mainly quartz (around 65%) and feldspar (around 15%) with illite (less than 10%) and minor hematite. The Lambourn loomweight by comparison has less feldspar (around 5%) than Werra but contains more illite (approx. 45%) and no hematite. The samian ware is composed of quartz (around 20%), and Ca feldspar (around 70%), with minor hematite. We do not identify directly the amorphous clay remnants, the components in which the RHX reaction is likely to take place (Hamilton & Hall 2012).

The samian mineralogy suggests that it was well fired at a temperature of at least 1000^{°}C, and made using calcareous clay. The high firing temperature is clear also from the vitreous nature of the microstructure (figure 2). The presence of illite in the case of the loomweight indicates a firing temperature of no more than 900^{°}C, or 800^{°}C or less if the starting clay was calcareous (Cultrone *et al.* 2001). Thus, the loomweight material is most probably a low-fired ceramic, with a less vitreous microstructure. The imprint of a grass blade is visible in the SEM image, suggesting the possible presence of organic residue. Consistent with this, the organic carbon content was measured only for L1, which lost 0.67 wt.% carbon during a control test at 500^{°}C. The measurement uncertainty associated with the organic carbon analysis is ±0.02 wt.% and this produces an age uncertainty of ± 35 years. We believe the true uncertainty associated with this analysis is probably greater but remains unknown. A full assessment of both carbon and RHX measurement uncertainty (see §7) is beyond the scope of this paper, but is undoubtedly important in further development of the RHX method.

We tackle three specific perceived issues associated with applying the RHX method to excavated archaeological pottery. The Werra specimens come from a waterlogged site and we show that immersion in water during burial does not affect the RHX process. We also demonstrate that the RHX dating is applicable to both coarse, low firing-temperature ceramics (loomweight) and fine, vitrified ceramics with a slip layer (samian).

### (b) Sample preparation

First, an appropriately sized sample is cut to fit into the wire-loop sample holder in the microbalance chamber. We generally use samples in the range 0.1–2.5 g. Samples are wet-cut using a water-cooled machine saw (a tile saw, for example) to avoid generating heat which may cause some dehydroxylation. The sample is then thoroughly cleaned under running water to remove surface debris. Next, the wet sample is dried to constant mass at 105^{°}C in an air oven to remove T0 water (liquid capillary and adsorbed water). At the end of this step, the T0 water content is zero and the sample mass is *m*_{1}. The time required to achieve constant mass depends on sample size, on the quantity of water absorbed during handling and preparation, and on the capillary properties of the material. It is typically a few hours for a sample of 500 mg; and up to several days for a sample of 2–4 g.

### (c) Conditioning step

Once dry, the sample is conditioned at the same temperature and RH that are used in the later microbalance measurement. The temperature *T*, to which *α*(*T*) in equation (2.1) refers, is set to the estimated ELT, or to a value close to the ELT, say within ±2^{°}C. (The ELT, discussed in §6*a*, is typically about 8–11^{°}C for samples from north-west Europe). The RH is routinely set to 30 per cent. Conditioning may be carried out in the microbalance chamber. A typical set of mass gain data in figure 3*a* obtained for a sample of Werra ware (W2) shows that the time to reach equilibrium may take several days to achieve, although this depends on sample size and varies from one ceramic material to another. The final constant mass is *m*_{2}. We can achieve an equilibrium sample mass stable to better than 1 ppm over a period of 20 h.

### (d) Reheating or dehydroxylation step

The sample is heated to constant mass in a laboratory furnace at 500^{°}C, after which it is transferred directly to the microbalance chamber. To achieve constant mass, the sample may either be heated for a length of time known to be sufficient for materials of a particular type, or else it is tested repeatedly in the microbalance until a constant mass *m*_{3} is obtained. At the time of transfer, the sample mass *m*_{3}=*m*_{cer}, but this cannot be precisely measured as the sample is re-equilibrating with T01 water and also gaining mass through the RHX reaction while cooling during transfer. The quantity *m*_{3} is not required in the RHX dating calculation.

### (e) RHX step

The mass gain of the samples following reheating is measured using a recording microbalance (here, both a CI Electronics CiSorp Water Sorption Analyser and a Quantachrome Instruments Aquadyne DVS, each with a capacity of 5 g, weighing to 0.1 μ*g*) capable of maintaining tightly controlled environmental conditions of relative humidity (±0.1%) and temperature (±0.2^{°}C) in an enclosed weighing chamber.

In figure 3*b*, we show a complete microbalance dataset for the mass gain following dehydroxylation for the same Werra sample for which conditioning data are shown in figure 3*a*. The total mass *m* is plotted against time^{1/4}. Stages I and II are clearly seen and the linear fit of equation (2.1) through the stabilized stage II data together with its intercept on the *y*-axis are shown.

Stage II microbalance data for all of the experimental samples are shown in figure 4. The striking linearity of the stage II data, when the sample mass *m* is plotted versus *t*^{1/4}, is evident. The difference in slope *α*_{m} between samples is due to variation in sample mass and material properties. Since the RHX reaction is a chemically activated process, the stage II slope depends on the microbalance measurement temperature. For this reason, either the measurement temperature must be set to the calculated ELT of the sample, or else the measured slope must be adjusted to the ELT using the activation energy and the Arrhenius equation as described below.

## 5. Microbalance results

In table 1 we give for all the pottery samples the experimental values of components of mass, as defined in figure 1 and obtained from microbalance data, such as shown in figures 3*a*,*b* and 4. Column 2 gives the mass of each sample following conditioning, *m*_{2}, and column 3 the conditioned dehydroxylated mass *m*_{4}, the latter obtained as shown in figure 3*b* as the intercept at *t*=0. Column 4 gives the mass *m*_{a} of T2 water, that is, the mass of RHX water acquired during the sample lifetime. This is calculated as the difference *m*_{2}−*m*_{4}. Column 4 shows also the normalized value *y*_{a}=*m*_{a}/*m*_{4}. The values in column 4 show how small the amount of RHX water is, here around 1–3% of sample mass. This quantity must be measured accurately and must be separated from the T01 water and any contaminants.

Column 5 gives the RHX rate constant *α*_{m}, obtained as the stabilized stage II gradient of each dataset. We show also *α*=*α*_{m}/*m*_{4} as it is useful to normalize the RHX rate constant by sample mass for comparison between samples. Column 6 gives the relative standard error of the estimate of *α*. These determine the age error arising directly from the least-squares fit to the RHX stage II data.

In column 7, we give an estimate of the amount of T01 water *m*_{01} obtained as the difference *m*_{3}−*m*_{4}, and also the normalized fractional value *y*_{01}=*m*_{01}/*m*_{4}. The amount of T01 water (not required in the dating calculation) varies widely between the different materials. The high-fired samian ware with low specific surface area (table 2) and vitrified internal pore structure adsorbs little T01 water and the ratio *y*_{01} is 0.05. On the other hand, the Lambourn loomweight with high specific surface area has much more T01 water and *y*_{01} is 0.35. The Werra pottery, intermediate in firing temperature and in SSA, has *y*_{01} 0.2.

## 6. Temperature effects

As we have noted earlier, the RHX rate constant *α* varies markedly with measurement temperature *T*. The variation can be represented by the Arrhenius equation
6.1with an activation energy *E*_{a} which is a material property independent of temperature. Here, *α*_{0} is the RHX rate constant at some reference temperature *T*_{0},**R** is the gas constant=8.314 J (mol K)^{−1}, and *T* and *T*_{0} are absolute temperatures (K). We have also shown (Hall *et al.* in press) that the definition of the RHX activation energy given in Wilson *et al.* (2009) is consistent with the general format of solid state chemical reactions (for example, Vyazovkin & Wight 1997). It is not yet clear to what extent this activation energy varies from material to material and is probable that there is some dependence on mineralogy. We have determined the activation energies of the three pottery types used here by measuring the RHX rate constant, *α*, at at least three temperatures between 10 and 50^{°}C. The values are given in table 2 and lie in the range 43–83 kJ mol^{−1}.

### (a) Effective lifetime temperature

The temperature dependence of the RHX process means that in order to apply the RHX age equation we need to take the value of *α* appropriate to the environmental temperature which the sample has experienced over its lifetime. We call this *T*_{e}, so that we need *α*(*T*_{e}) to insert into equation (2.2) to obtain *t*_{a}. The temperature which the sample experiences over its lifetime is not constant and is subject to variations arising from factors such as changes of location; seasonal, annual and long-term climate variations; changes of burial depth with time, etc. But, however complicated the temperature history may be, there must exist a single ELT, *T*_{e}, for every sample. That there is such a temperature *T*_{e} follows from the consideration that for a sample of known age, *t*_{k}, and having a measured fractional lifetime mass gain *y*_{k}, there is a value of *α*=*α*_{k} which satisfies equation (2.2). Given information on *α*(*T*) for the sample, we can therefore find the value of *α*=*α*_{k} for which *T*=*T*_{e}. This is the operational definition of the ELT, *T*_{e}; the corresponding value of *α*, *α*_{e}, is the effective lifetime RHX rate constant for that sample.

It follows that in order to derive an RHX age for a sample whose age is not known, we must have a procedure to calculate or estimate the ELT, *T*_{e}, from what is known of the temperature history of the sample. We have described this procedure fully elsewhere (Hall *et al.* in press) and also in the electronic supplementary material, where we illustrate each stage in the ELT calculation. If the lifetime temperature, history of the sample is *T*_{h}(*t*) for equal intervals of time in 0≤*t*≤*t*_{a}, we calculate the quantity *α*(*t*_{h}) and then *α*_{e}=〈*α*(*t*_{h})^{4}〉^{1/4}. The quantity 〈*α*^{4}〉^{1/4} is the *fourth power mean* of *α*. This is computed by taking the fourth root of the mean of *α*^{4} over the entire temperature history. Once *α*_{e} is known, we obtain *T*_{e} directly from the Arrhenius equation. This expression for *α*_{e} is consistent with equation (2.2), which can be written as 4*y*^{3}*y*^{′}=*α*(*t*)^{4} where *y*^{′} denotes *dy*/d*t*. The right-hand side can be replaced by its mean value 〈*α*^{4}〉, so that the effective value *α*_{e}=〈*α*^{4}〉^{1/4}.

### (b) Lambourn loomweight

For this sample, we have an excellent excavation history with information on date and location, burial depth and soil type. The reconstruction of the temperature history is based on monthly surface air temperatures for the period 1914–2006 at the site estimated from the UK gridded records. These have been applied to the entire sample history with adjustment of the monthly mean by the reconstructed Northern Hemisphere temperature anomaly (Mann *et al.* 2008). We calculate the soil temperature at the burial depth assuming a linearly increasing burial depth with time. A use period of 20 years is included before burial, and a storage period of 4 years after excavation. From the complete temperature history, using the measured RHX activation energy 50.5 kJ mol^{−1}, we obtain an ELT of 9.1^{°}C (table 3).

### (c) Samian ware

The lifetime history of this sample is the most complicated. It was produced at La Graufesenque, France (G Dannell, private communication). The calculation of the ELT is based on the following: After production, the ware was transported by river through France to Britain over a period of 2 years. It was then in use for around 10 years (Peña 2007), before being discarded and eventually buried, say 20 years after the original firing. Increasing the use period from 20 to 50 years produces an ELT increase of only +0.004^{°}C and corresponds to a decrease in estimated RHX age of 0.04% or 0.8 year in 2000 years. Taking an extreme case and assuming the use-period temperature was 25^{°}C rather than 15^{°}C over a use period of 20 years produces an ELT increase of 0.015^{°}C, demonstrating what little effect use period has on ELT accuracy in older ceramics. Excavation history: Owslebury 1967, burial depth 0.6 m, light flinty soil above chalk bedrock.

### (d) Werra ware

The information on the provenance of these samples is comprehensive. The assigned date of firing is 1605 and they were discarded immediately as wasters. The 1979 excavation is well documented (Bruijn *et al.* 1992). The sherds were located in a clay pit at a depth of 1.8 m. At the time of the excavation, there was a high water table which caused daily flooding. From 1979 to 2011, the sherds were stored in a warehouse for which we have complete temperature records for the period 2008–2010. The only difficulty in constructing the ELT arises from the uncertain thermal effects of a glasshouse which stood over the excavation site for an unknown period of time. The seasonal mean surface air temperature for Enkhuizen for the period 1605–1979 is available from gridded reconstructions of northern European climate by Luterbacher *et al.* (2004). The ELT is calculated by taking the seasonal (quarterly) mean temperature reduced by the damping effects of ground burial for the period 1605–1979 and warehouse temperature for the period 1979–2011.

We consider that in these cases, the uncertainty in the estimates of ELT for all samples given in table 3 is ±0.2^{°}C (see the electronic supplementary material).

## 7. Age estimates and uncertainties for pottery samples

The estimated RHX ages of the pottery samples are given in table 3. Using the quantities *y*_{a} and *α* from table 1, we obtain the nominal age at 11^{°}C from the RHX age equation (equation (2.2)). This is then corrected using equation (A5) and the determined activation energy *E*_{a} to obtain the age at the estimated ELT.

Table 3 gives the calculated ELT values in column 2, and the values of *t*^{1/4}_{a}=(*y*_{a}/*α*) at the measurement temperature in column 3. Column 4 gives the nominal age of each sample as calculated from mass gain measurements made at 11.0^{°}C. The ages adjusted for the ELT are given in column 5 and these produce the RHX dates in column 6. For comparison, the assigned dates are given in column 7. Our RHX dates are in good agreement with the independently assigned dates. For the Werra sherds, we have exact assigned dates. For the loomweight, we have an associated age of the context from radiocarbon dating. For the samian-ware sample S1, we have dates of 45–75 AD based on a potter’s name.

There are two quantities which contribute to uncertainty in the RHX-estimated age *t*_{a}. These are *y*_{a} and *α*. Since *t*_{a} is proportional to (*y*_{a}/*α*)^{4}, errors in both *y*_{a} and *α* propagate strongly into *t*_{a}. When analysing a set of data, the gradient (*α*) and intercept (*m*_{4}) are calculated from data chunks of increasing length *n* (starting length depends on the number of data points in the set), from which the standard uncertainties (Kirkup & Frenkel 2006) of both *α* and *m*_{4} are calculated. Minimizing the norm of the residuals and the uncertainty (*u*) for *α* (generally much larger than that of *m*_{4}) is a good indicator of the goodness of fit to the data. Data chunk size is monotonically increased until it contains all of the stage II data. Uncertainty in the quantity *α* is calculated from the product of (*s*)/ and where , *s* is the L2 norm of the residuals and *x*_{i} are the measured values of the *n* data points. Taking the product of the uncertainty in alpha and the L2 norm of the residuals produces a quantity which is independent of chunk size, n. Plotting this value, which we call the error product, against produces a minimum value indicating optimum chunk size prior to the onset of data curvature. The gradient uncertainty, *u*, is presented in table 1 as a percentage of *α* and used to calculate the ± error bounds on the determined RHX age as shown in table 3. Plotting the residuals along with the data points is useful for detecting any curvature in the data during the analysis of each data chunk. An example using the data for W3 (Werra pottery) is shown in figure 5 and illustrates the usefulness of this procedure: viewing the data from the last point to the first, we plot the residuals of the optimum chunk length in the top right-hand graph and the corresponding fitted dataset is shown on the top left-hand graph. The demonstrated good fit indicates that only stage II data are being fitted and these are not contaminated by mass gain owing to incomplete uptake of T01 water. The bottom graphs show the next sequential set of data and we see the curvature characteristic of the transition of stage I to stage II, which is not sharp. The same procedure was applied to the data of figure 4 prior to obtaining the final dates for table 3. The estimated uncertainty of the ELT-corrected RHX age (table 3, column 5) and the corresponding RHX dates (column 6) is the combined uncertainty calculated for *α* and the ELT only. It is important to note that a true uncertainty, which is beyond the scope of this paper, is a combination of various effects and, in particular, measurement uncertainty which we have not fully evaluated.

## 8. Conclusions

The results presented here are a culmination of several pilot studies to transfer the method from ceramic brick to archaeological pottery. We have shown that the RHX method is capable of dating archaeological pottery but accuracy requires a full understanding of sample behaviour and history. Critical aspects of the experimental work include preparing and cleaning the sample, fully conditioning the sample at the temperature and RH used in determining the reaction rate constant, and completely dehydroxylating the sample to constant mass before measuring the RHX rate. Our analysis shows that for any sample, it is crucial to identify the data points which are due solely to RHX (T2) water, and to determine the ELT of the sample. The results in table 3 show that the ages determined by the RHX method are within 2–4% of the assigned ages (samian and Werra) and 2–12% for loomweight for which we have an assigned date only for bone in the same context rather than the ceramic itself. The close agreement between assigned and RHX ages gives confidence in the reliability of the method.

The results we present lead to several main conclusions.

— The RHX method is applicable to dating archaeological pottery.

— The RHX characteristics and behaviour of archaeological pottery of widely different types are shown to be consistent with the time

^{1/4}law described previously for brick and tile materials.— A gravimetric methodology allows the lifetime RHX water and the sample-specific RHX rate constant to be determined.

— A data analysis routine allows careful selection of data points in the stage II regime.

— Using the RHX power-law rate equation, the nominal sample age at the measurement temperature

*T*_{m}can be calculated from the gravimetric quantities*m*_{a}and*α*. Measurements made at three or more temperatures allow the calculation of an activation energy for the RHX process.— The ELT,

*T*_{e}, for the sample can be estimated from historical temperature records and a model of temperature variation with burial depth. The estimated sample age at*T*_{e}is then calculated using the measured RHX activation energy of the sample.

## Acknowledgements

We thank the Leverhulme Trust, UK EPSRC, UK NERC and the Universities of Manchester and Edinburgh for financial support; Diana King (Foundations Archaeology) for providing loomweight samples, Geoffrey Dannell (Study Group for Roman Pottery) for providing samples of samian ware and technical advice, and Martin Veen (Provinciaal Depot voor Archeologie, Provincie Noord-Holland) and Marc Harsveld for the Werra ware samples. We acknowledge use of historical temperature data from the UK Meteorological Office (UKCP09) and from the Royal Dutch Meteorological Institute KNMI. We thank Professor W. D. Hoff, Dr Cathy M. Batt and Dr Sarah-Jane Clelland for helpful comments.

## Appendix A. Sensitivity of the RHX age to the ELT

We relate the error in the calculated RHX age arising from an error in the estimated ELT . Here, *T*_{e} and *t*_{a} are the true ELT and the true age, and the primed quantities the estimated ELT and age. Then
(A1)
where *α*_{e} and *α*^{′}_{e} are the RHX rate constants at temperatures *T*_{e} and *T*^{′}_{e}. Since *t*_{a}=(*y*_{a}/*α*_{e})^{4}, the fractional age error
(A2)
But from equation (6.1), we have
(A3)
where *T*^{′}_{e} and *T*_{e} are absolute temperatures. Retaining only linear terms gives
(A4)
For typical temperatures 1/(**R***T*^{2}_{e})≈0.00149 mol kJ^{−1} K^{−1}. Therefore, we obtain the general formula for the fractional age error arising from an error in the estimated ELT
(A5)
or for the percentage age error 100*ϵ*/*t*_{a}≈−0.149*E*_{a}Δ*T*. The minus sign signifies that by underestimating the ELT (Δ*T*_{e} negative), we overestimate the age, so that *ϵ*/*t*_{a} is positive.

As a numerical illustration: for a ceramic sample with a measured RHX activation energy of 70 kJ mol^{−1}, an estimated ELT *T*^{′}_{e} of 12.0^{°}C but whose true ELT *T*_{e} is 12.5^{°}C (so that Δ*T*_{e}=−0.5^{°}C), we find the fractional age error from equation (A5) *ϵ*/*t*_{a}≈+0.052, and the percentage error +5.2 per cent. Thus, if the ELT is *underestimated* by 0.5^{°}C, the age of a sample with true age 1000 years is *overestimated* as 1052 years.

- Received February 17, 2012.
- Accepted June 8, 2012.

- This journal is © 2012 The Royal Society