Multiple Aging Mechanism Pitfall.
The temperature dependence of aging rate depends on the processes used during a resonator's fabrication. Since the aging rates due to most known aging mechanisms have strong (e.g., exponential) dependencies on temperature, care must be taken in the use and interpretation of accelerated aging tests. Especially when two or more aging mechanisms are present, simple accelerated aging tests can lead to misleading or meaningless results [57,135].
For example, adsorption and desorption of contamination is believed to be a significant aging mechanism in many resonators. The halflife, t_{1/2}, of adsorbed molecules can be expressed as
(2) 
In Eq. 2, E_{d} is sometimes found to increase with decreasing coverage, and after a monolayer is formed, it is generally also different for a second adsorbed layer. Although Eq. 2 gives the halflife of the desorption process, the details of the time dependence of desorption are not specifically included. This detail is supplied by an analysis such as the ones by Landsberg [7] and Glang, et al.[9].
If, for instance, a resonator's aging is determined by the desorption of two species of contaminants, one with E_{d} = 30 kcal/mol and the other with E_{d} = 40 kcal/mol, then accelerated aging at, for example, 150° C will provide meaningless results. At 150° C, the molecules with E_{d} = 30 kcal/mol, which are the molecules that determine the aging at room temperature, are desorbed within a fraction of a second. The subsequent aging is then determined by the E_{d} = 40 kcal/mol molecules, which at room temperature do not contribute measurably to the aging. The 150° C aging results, therefore, will not shed any light on the aging that can be expected at room temperature. Data at several temperatures is necessary to reveal the presence of multiple aging mechanisms and their effect on the aging at the normal operating temperatures.
Brief Review of the Theory of Rate Processes.
The following tutorial discussion of rate processes [136] is included in this review paper because understanding rate processes can be important in our attempts to understand aging and accelerated aging results.
The various aging mechanisms are specific cases of rate processes. Many studies of chemical rate processes have been reported and are summarized in textbooks on physical chemistry and chemical rate processes. Early studies of chemical reaction rates were on gas and solution systems.
For gas reactions, the reaction rate is often controlled by collisions between reacting gas atoms and molecules. Arrhenius proposed that the reaction rate could be expressed as:
Reaction Rate = K e^{Ea/RT}  (3) 
Boltzmann later developed a statistical thermodynamic theory of gas reactions that showed that E_{a} in the Arrhenius proposal ought to depend on temperature because it had a part associated with the distribution of system energy (entropy) as well as the part associated with changes in the average system energy (thermal energy). According to this theory E_{a} should really be F_{a}, the change in the free energy of the reaction process (rather than the change in the enthalpy). This more rigorous theory helped to make sense of a larger set of experimental results, including those of a few reactions which were actually slower at higher temperatures. These particular reactions had rates controlled by the distribution of thermal energy rather than by the average change in the thermal energy. Further studies of gas phase reaction rates have included quantum effects for simple systems.
Experimental studies have shown that reactions in solutions are more complicated than gas phase reactions. Experimental studies have also shown that reactions involving the interaction of a gas or dissolved reactant with a surface are generally more complicated than either gas or solution phase reactions. In spite of the additional complexity, the gas phase ideas of reaction rates controlled by an Arrhenius or Boltzmann rate law and by collisions between reacting atoms, molecules, free radicals, and surfaces, have been very useful in developing an understanding of general reaction rates.
This brief history of reaction rate science suggests that oversimplification can easily lead to an inadequate understanding of the processes being considered. For example, it was found experimentally that many gas and solution phase reactions have Arrhenius activation energies that produce a doubling of the rate for every ten degrees Celsius increase in reaction temperature. If no other evidence is available, this activation energy can be used to design experiments to efficiently estimate the actual activation energy for the system of interest. Credible long term estimates of crystal resonator and crystal oscillator aging can only be determined from these experimentally determined activation energies and not by the general rule of thumb.
For many gas and solution systems, the dependence of the concentration of a reactant or resultant species on time is exponential, giving a reaction rate that can approach zero, as a reacting species is completely used up in the reaction. In systems for which one of the reacting species has a very large excess concentration, so that the concentration does not change very much during the entire reaction process, changing the concentration of that species may not measurably change the reaction rate.
A Logtime Law of Chemisorption, Oxidation, and Stress Relief.
A review of chemisorption on metal surfaces and oxidation of metal surfaces [7], two processes which could easily affect the dependence of the frequency of a crystal resonator on time, showed how most of the available experimental data were consistent with a logarithmic rate law. The source of such a law was also proposed. In spite of some uncertainties, most of the available metal chemisorption and oxidation experimental data could be described by:

(4) 
At least that part of the aging of crystal devices associated with chemisorption of contaminant gases on the crystal surfaces, or on the electrodes, or with the oxidation of the electrode material, or the quartz itself, might have a logarithmic rate law. Some quartz resonator designers have recognized that many resonators age a fixed amount per decade in time, which is a logarithmic rate dependence [4,6,68,71,89,95,103,137141].
The atomistic/molecular model of the chemisorption process with a logarithmic rate law was expressed in terms of the specialized language of adsorption chemistry, but can be stated in much simpler and more general terms. The simpler and more general statements are that: 1) the adsorption does not depend on any concentrations (zero order), and 2) that the Arrhenius activation energy is a linear function of the amount of adsorption. Both of these statements can be expressed as:

(5) 
Before examining several implications of Eq. 5, we should note that thermal stress relaxation in metals such as zinc [142] has also been reported to follow a logarithmic rate law [7], as does the oxidation of many metals [8]. Consequently, even if the logarithmic rate law were established as valid for crystal aging, other evidence would be needed to separate the effects of the chemisorption, oxidation, and thermal stress relaxation parts of the aging in the resonator [72]. The authors do not know about any other proposed mechanism for the aging of crystal resonators for which a basis as good as that of the logarithmic time mechanism has been developed.
Implication of the Logtime Aging Law.
An important question is: under what conditions can Eq. 5 be used to understand and control the aging of resonators? A careful examination of the consequences of Eq. 5 is one place to start developing an answer to this question.
Integrating Eq. 5 from t_{0} to t and q(t_{0}) to q(t) gives:

(6) 
The t_{0} defined here is not the same quantity as the t_{0} defined in Eq. 4; t_{0} in Eq. 4 is equal to (1/B) in Eq. 6. In Eq. 4 the starting t and starting q were both 0. In Eq. 6, q(t_{0}) may also depend on temperature. This reversible temperature dependence is not time dependent and is the temperature coefficient of q at t=t_{0}.1
Eq. 6 can be expressed in the following simpler form by defining combinations of variables as:

(7) 
Eq. 6 and Fig. 5 show that at times short compared to 1/B, q(t)q(t_{0}) is approximately zero. At times long compared to 1/B, q(t)q(t_{0}) is approximately D ln[B(t t_{0})] or D ln(B) + D ln(tt_{0}). The two parameters of interest in this aging model are D, the per decade change in q for long times, and (1/B), the characteristic time at which the logtime behavior begins to appear. For the simple model described in Eq. 6,

(8) 
Therefore, BD is activated in an Arrhenius sense, if C, E_{a0}, and E_{a1} are independent of soak temperature.
It can be inferred from experimental results on high quality 8MHz fundamental mode quartz resonators that B is a strongly decreasing function of temperature and that D is a strongly increasing function of temperature [137]. The authors are not aware of any reports on the dependence of BD on temperature.
Aging with Two Simultaneous Logtime Mechanisms.
There are no factors in Eq. 6 that could reverse the direction
of the resonator aging. Since several authors have reported measurements
that show a change in the direction of aging at some particular time [6,57,87,93,138,143],
either a more complicated aging model must be found or there must be more
than one simple logtime aging mechanism active in such resonators. For
example, Fig. 6 shows schematically how two simple logarithmic aging mechanisms
acting at the same time in a particular resonator can produce aging that
changes direction at a particular time.
Fig 6.
For this case the early logarithmic slope and time of the maximum positive change are mostly determined by the mechanism with the larger B. The time to zero aging has a component from both mechanisms. The long term logarithmic time slope is the sum of the logtime slopes of the two mechanisms. If the two long term slopes could be made equal (compensation), the aging would rise to a fixed value and then stop.
For some cases it may turn out that there is only a limited amount of possible aging in the particular device for one of the mechanisms. This case is not consistent with Eq. 4, and has not been reported in analyses of crystal aging.
In addition to some partly exponential time dependencies in diffusion, exponential time dependencies can arise from first order rate processes associated with adsorptiondesorption [9,68,72]. These kinds of processes can have an exponential time dependence if the activation energy does not depend on the amount of adsorption/desorption. A time dependence more complicated than exponential or logarithmic can arise if the activation energy also depends on the amount of adsorption/desorption.
In one report [68], an exponential aging model and simultaneous power law (diffusion) model fitted some isothermal data better than two different simultaneous logtime models. The significance of this result for other crystal fabrication technologies or for the majority of the filters made with the same technology is not known.
In another report [72], aging data were fitted better by logtime or by square root time (diffusion) models than by exponential time models, depending on the particular resonator being studied. In this work the logtime model was the best overall model.
Two experimental accelerated aging techniques, isothermal aging and thermal step stress aging, have been used to study the aging of crystal resonators (and the aging of other devices as well). These two techniques are discussed in the next two sections.
In this experimental technique each device (resonator, oscillator, etc.) is stored at only one of a set of temperatures and the frequency is measured at several times of particular interest [137,144]. Nearly all reported accelerated aging experiments have been of this type. The measurement times are selected to optimize the cost and resulting use of the data. Measurement time choices depend on many factors, such as the need for fixture calibration, the number of samples, and the type of device. For determining the parameters of a logtime model for a filter application, with large numbers of samples and the need for frequent fixture calibrations, a logarithmic spacing of measurement times is particularly efficient. For an oscillator aging study, closer spaced measurements may often be desirable, e.g., to characterize frequency jumps.
Fig. 7 shows a schematic plot of some typical isothermal aging data for a negative aging device at temperatures of T_{1} and T_{2} (where T_{2} > T_{1}). The important information here is the general shape of the curves, the times at which the rapid changes begin, and the slopes of the rapidly changing parts.
A measurement temperature, which may be different from the soak temperature, for practical reasons, is selected to make the frequency measurement as accurate as possible. This measurement temperature is often a turnover temperature for high stability resonators, or can be room
temperature for other types of resonators. The resonator
parameters may be measured directly or the resonator frequency may be measured
in a test oscillator. Both kinds of measurements may be made in a temperaturecontrolled
chamber.
Fig7
For the most accurate measurements, the frequency is corrected to a fixed temperature to reduce the effects of small unavoidable temperature variations. It is important to show that the aging caused by the thermal shocks connected with the change from the soak temperature to the measurement temperature can be neglected or removed from the data.
The time dependent data is fitted (by least squares, for example) to a particular aging law to derive statistical estimates and associated confidence limits for the values of the aging law parameters or of the aging at a particular time of interest. For the logtime law the parameters are B and D; particular times of interest could be 1 year, 5 years, 20 years, etc.
Each member of a group of resonators is assigned to a different soak temperature, preferably by statistical experiment design rules. For each resonator soak temperature experiment the data are fitted to the selected aging law as above. The aging law parameters or frequency shifts calculated at a particular time of interest and their confidence limits can then be fitted to a selected law of the temperature dependence. An Arrhenius temperature dependence law often turns out to be a useful choice.
The temperature dependence fit gives estimates and confidence limits of device aging or aging law parameters at any temperature of interest. For this temperature dependence fit, there are three sources of variability: 1) the choice of the wrong aging model, 2) the variability of the parameter estimates from each device, and 3) the error of each of the estimates.
The results of the two fittings, i.e., over time and over temperature, can only have meaning within the applicability of the assumed aging and temperature dependence laws. Since the results are highly statistical in nature, this technique can only characterize a particular fabrication technology, with the statistical meaning of the final predictions dependent on the validity of the choice of aging model, the number of samples chosen, and the number and ranges of times and temperatures selected for the design of the experiment. Within this context, statistical aging predictions can be made for any particular member of the characterized device group.
Thermal Step Stress (Differential Thermal Analysis).
In this experimental technique each device (resonator,
oscillator, etc.) is stored for a fixed time interval at a series of increasing
temperatures; the frequency is measured after each temperature soak [137,144].
Fig. 8 shows a schematic plot of typical thermal step stress data for a
negatively aging device. The important information here is the general
shape of the curve, the temperature at which rapid change begins, and the
slope of the rapidly changing part. A measurement temperature is selected
to make the frequency measurement as accurate as possible. This measurement
temperature is often a turnover temperature or room temperature. For the
most accurate measurements, the frequency is corrected to a fixed temperature
to reduce the effect of small unavoidable temperature variations on the
results. It is important to show that the aging caused by the thermal shocks
connected with the changes from the soak temperatures to the measurement
temperature can be neglected or removed from the data. This data can be
plotted versus temperature to provide a curve, the shape of which is a
signature of the distribution of aging mechanisms for that particular device.
Fig8
For an assumed aging model, this data can also be used to determine estimates and confidence limits for the aging law parameters. For the logtime law, Eq. 5 gives an expression for the slope of frequency versus time as a function of temperature. After each temperature step, the change in frequency during that step is the slope for that temperature, as

(9) 

After the parameters of the aging law have been determined, Eq. 9 can be used to calculate the dependence of frequency on time at any temperature of interest. These calculations can be compared with isothermal results on similar devices.
The advantage of the thermal step stress technique over the isothermal technique is that the aging parameters for a particular device can be derived in a short time, e.g., two weeks for a 16 hour time step. When the thermal step stress results compare well with the isothermal results, the thermal step stress technique can quickly provide useful estimates of aging parameters.
If the high temperature parts of the thermal step stress results show that extraneous high temperature aging mechanisms are present, it may be necessary to use a longer time step at lower temperatures to obtain the desired data. These extraneous high temperature mechanisms may include melting processes or phase changes for some of the materials in the device being tested. The thermal step stress results provide a useful way to quickly reveal the presence of these mechanisms so that their contribution to the device long term aging can be considered and resolved.
Other Aging Characterization Techniques.
There may also be apparent resonator aging mechanisms, i.e., drift, which exhibit linear time dependence. For example, resonator frequency changes due to continuous low level radiation from space or background sources might be linear.
Crystal resonators may have many kinds of aging mechanisms. Since the various aging mechanisms can cause both positive and negative aging, compensation among the various mechanisms is possible. This compensation may occur during the early history of the resonator and then become ineffective later; or, occasionally, it may occur years after the aging starts. For this reason great care must be taken to characterize the aging mechanisms associated with a particular resonator fabrication technology over a sufficiently wide range of temperatures and time before aging predictions can be validated or qualified. After the fabrication technology is qualified to produce lowaging resonators for the required times and temperatures of the application, subsequent production must be monitored to assure that the manufacturing process remains in qualification.