Gas equilibration gas modulation refractometry (GEq-GAMOR) for assessment of pressure with sub-ppm precision

A novel realization of Gas Modulation Refractometry (GAMOR) that outperforms the original realization [Single Cavity Modulated GAMOR (SCM-GAMOR)], is presented. The reference measurements are carried out by equalizing the pressures in the two cavities. By this, the time it takes to reach adequate conditions for the reference measurements has been reduced. This implies that a larger fraction of the measurement cycle can be devoted to data acquisition, which reduces white noise and improves on short-term characteristics. The presented realization also encompasses a new cavity design with improved temperature stabilization and assessment. This has contributed to an improved long-term characteristics. The system was characterized with respect to a dead weight piston gauge. For short integration times (up to 10 min) it can provide a response that exceeds that of the original SCM-GAMOR system by a factor of two. For longer integration times, and up to 18 hours, the system shows, at 4303 Pa, an integration time independent Allan deviation of 1 mPa (corresponding to a precision defined as twice the Allan deviation, of 0.5 ppm). This implies that the novel system shows a significant improvement with respect to the original realization for all integration times (by a factor of 8 for an integration time of 18 hours). When used for low pressures, it can provide a precision in the sub-mPa region; for the case with an evacuated measurement cavity, the system provided, up to ca. 40 measurement cycles (ca. 1.5 hours), a white-noise limited noise of 0.7 mPa sqrt(cycle), and minimum Allan deviation of 0.15 mPa. Furthermore, over the pressure range investigated, i.e. in the 2.8 - 10.1 kPa range, it shows, with respect to a dead weight piston gauge, a purely linear response. This implies that the system can be used for transfer of calibration over large pressure ranges with exceptional low uncertainty.


I. INTRODUCTION
Fabry-Perot cavity (FPC) based refractometry is a sensitive technique for assessment of gas refractivity, density, pressure, and gas flows. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] The technique is built on the principle that the frequency of a cavity mode in an FPC is shifted when gas with a given refractivity is let into the cavity. 1-3, 15 Recent works have indicated that the technique has the potential to replace current pressure standards, in particular in the 1 to 100 kPa range. [18][19][20][21][22][23][24] With the revision of the SI-system in May 2019, in which the Boltzmann constant will be defined as a fixed value without any uncertainty, the performance would, in principle, be limited only by the accuracies of quantum calculations of gas parameters and the determination of the gas temperature.
Ordinary FPC refractometry is in general limited by the stability of the length of the cavity; 17, 25-27 a length change of 1 pm of a 30 cm long cavity during a measurement corresponds, for N2, to an uncertainty in the assessment of pressure of 1 mPa. To achieve this degree of precision with FPC-based refractometry an exceptional mechanical stability is required.
A means to alleviate this is to utilize a dual FP cavity (DFPC) in which two cavities are bored in the same spacer block; in this case one cavity serves as the measurement cavity (in which gas is let in and pumped out) while the other is the reference cavity. 3, 13-18, 26, 27 For each cavity, a laser is locked to one of the cavity modes and the beat frequency of the two lasers is measured. Any change in length of the cavity spacer that is common to the two cavities will then cancel in the beat frequency and not affect the assessment.
Despite this, since the two cavities in a DFPC set-up can drift dissimilarly, the highest performance still requires extraordinary stable conditions. As a means to remedy this, Gas Modulation Refractometry (GAMOR) has recently been developed. 28,29 By performing repeated reference assessments with the measurement cavity being evacuated (while the reference cavity is held at a constant pressure), the methodology can significantly reduce the influence of the long-term drifts (i.e. eliminating its linear parts) that are mainly caused by length changes of individual cavities. This allows it to benefit from the high precision FPC-based refractometry has at short time scales also at long time scale. It was recently shown that a GAMOR instrumentation with a nontemperature-stabilized cavity spacer, could, when referenced to a dead weight piston gauge, reduce the influence of drifts of the cavities three orders of magnitude with respect to a DFPC refractometer with conventional (static) detection. This system could demonstrate a (1σ ) sub-ppm precision for pressure assessment in the 5 kPa range. 28 On short time scales, the assessment of refractivity from a given density of gas by the GAMOR instrumentation realized was limited by white noise. Although, in general, such noise can be averaged down, averaging can only be performed when the gas pressures have become fully equilibrated. Since it takes considerable time for the measurement cavity to reach equilibrium during the emptying stage, the performance of the original GAMOR realization on short time scales was limited by the gas evacuation process.
On long time scales, it was restricted by temperature gradients in the cavity spacer. The differences in temperature between the temperature probes and the walls of the cavities resulted in uncertainties in the assessments of the gas temperature to which the pressure is proportional.
This indicates that there are possibilities to improve on the precision of the GAMOR methodology. Regarding its short-term characteristics, it would be beneficial if the time it takes for the system to reach stable conditions for the reference assessments could be shortened, so that the reference signal could be averaged over a longer time. For the long-term performance, the precision could be improved if the signal could be averaged over longer time before being influenced by any drift.
We present in this work a novel realization of GAMOR, in part based on its original realization, 28 with some parts upgraded, but also encompassing a new gas modulation methodology, that together address these concepts. To improve on the shortterm characteristics, we have developed and realized a new modulation methodology termed Gas Equilibration GAMOR (GEq-GAMOR) that shortens the time it takes to reach equilibrium conditions for the reference assessments. In this methodology, instead of evacuating the measurement cavity for the reference measurement, the gas in the measurement cavity is let into the reference cavity, to equalize the pressure in the two cavities. This implies that in GEq-GAMOR the amount of gas in both cavities are modulated within each measurement cycle. Since gas equilibration at finite pressures (above the molecular regime) is a faster process than evacuation of a cavity filled with gas down to vacuum, this allows for assessment (averaging) of the beat frequency during the reference assessment over a longer time. This reduces the noise picked up in an individual measurement cycle, which thereby contributes to an improved white noise characteristics of the system. To improve on the long-term stability, we have implemented a temperature stabilized cavity spacer with an improved temperature measurement capability.
The performance of the novel system is compared to that of the first realization of GAMOR, 28 henceforth referred to as Single Cavity Modulated GAMOR (SCM-GAMOR). It is shown that GEq-GAMOR can provide a two times better precision under short-term conditions and that the drifts have been reduced significantly (more than 8 times for integration times for 500 measurement cycles, or 18 hours). When used for low pressures, it is demonstrated that it can provide a ( 2σ ) precision in the sub-mPa region; for an evacuated measurement cavity, and for up to ca. 40 measurement cycles (ca. 1.5 hours), the system exhibits a white-noise of 0.7 mPa (cycle) 1/2 , and a minimum Allan deviation of 0.15 mPa. Finally, the system is characterized with respect to its linearity with respect to a dead weight piston gauge over the 2.8 -10.1 kPa range.

II. THEORY
Since GEq-GAMOR has much in common with SCM-GAMOR, the theoretical description of the former can most conveniently be based on that of the latter, which, in turn, can be assessed from expressions for ordinary (i.e. unmodulated) DFPC-based refractometry. To provide a comprehensible description of GEq-GAMOR, the basis for SCM-GAMOR is therefore reviewed in the Supplementary material [URL] ("Derivation of expressions that relate the change in beat frequency between the two laser fields to the refractivity of the gas in the measurement cavity in SCM-GAMOR and GEq-GAMOR").
Based on this, the defining expressions for GEq-GAMOR are derived and presented. In addition, the last part of the Supplementary material [URL] contains a list of the entities used together with their designation ("Nomenclature"). For both methodologies, it is assumed that, for each measurement cycle, the measurement cavity is filled with gas from an external source to a pressure of Ext P , whose refractivity, 1 Ext n − , is to be assessed by refractometry.

A. SCM-GAMOR
An expression that relates the change in beat frequency between the two laser fields to the change in refractivity of the gas in the measurement cavity in SCM-GAMOR Preferably, the two beat frequencies, (
There are a few possible remedies to this. The one presented here is to first As is shown in the Supplementary material [URL], the latter entity is given by and where we have expressed the relative deformations of the two cavities due to the gas under gas equilibrium conditions, where, in turn, r ε is the refractivity-normalized deformation coefficient of the reference cavity, and where we have used ε ∆ as a short hand notation for m r ε ε − .
This implies that, for the case with GEq-GAMOR, the refractivity of the gas that have been let into the measurement cavity can be assessed from Eq. (2) in which Eq

C. Assessment of gas density and pressure
The density of the gas that have been let into the measurement cavity, Ext ρ , can then be assessed from the refractivity by where R A and B ρ  are the molecular polarizability and the first density virial coefficient, respectively. The latter is given by 2 ( where T is the temperature of the gas, A N is the Avogadro's number, B k is the Boltzmann constant, and ( ) p B T is the first pressure virial coefficient. 26

III. Experimental setup
The setup for GEq-GAMOR, which is illustrated in Fig. 1, is based to a large degree on that for SCM-GAMOR. 28 28 The left out parts for arm 2 are identical to those of arm 1.
For improved locking bandwidth, the light from each laser is sent through a fiber- The outputs from the photo detectors are connected to a commercial digital servo module based on a field programmable gate array (FPGA, Toptica, Digilock 110). In this, to produce a PDH error signal for the locking, the reflected signal is demodulated at 12.5 MHz before it passes through one of two proportional-integral-derivative (PID) servos.
One provides a slow feedback (with a bandwidth up to 100 Hz), which controls the piezoelectric transducer of the fiber laser, while the other gives a fast feedback (with a bandwidth up to around 100 kHz), which is connected to a voltage controlled oscillator (VCO) that produces an RF frequency of around 110 MHz for the frequency tuning of the first order output of the AOM. The transmission signal is used to activate feedback to the laser to enable automatic re-locking during controlled mode jumps.
The output of the FPGA that is routed to the laser is limited to a voltage corresponding to a change of the laser frequency of two FSR. When the feedback voltage reaches this limit, the automatic re-locking routine of the module relocks the laser to an adjacent mode with a frequency closer to the center of its working range. The relocking process is fast, it takes typically a tenth of a second, and it does not influence the refractivity assessment (since no mode jumps take place during data acquisition). Hence, it allows for a dynamic range that is not limited by the tunability of the lasers.
The output signal of the beat frequency detecting photo detector is sent to a frequency counter (Freq. Counter, Aim-TTi instruments, A TF960). The beat frequency is acquired with a rate of 5 Hz. As the frequency counter is limited to 6 GHz, temperature tuning is used to initially (i.e. before the measurement series) set the frequencies of the two lasers so that the beat frequency is in the center of the range of the frequency counter.
After this manual (coarse) setting, the automatic re-locking routine keeps the beat frequency between 2 and 6 GHz under all measurement conditions.
As an upgrade compared to the previous GAMOR system, 28 the temperature of the cavity spacer is monitored by six Pt-100 sensors (RS Pro PT100 Sensor, 457-3710) that are placed in holes drilled into the cavity spacer for monitoring of the temperature and its distribution in the spacer. The holes were bored so that the distance between the sensors and the measurement cavity wall is 5 mm. The sensors are connected to two DAQ (data acquisition) modules (National instruments universal analog input module, NI-9219). The probes and the DAQ modules were calibrated at RISE to within a combined uncertainty of 10 mK. During operation, both the (temporal) stability and the (spatial) gradients in temperature of the cavity were assessed to be below 5 mK. To monitor drifts in the DAQ modules a 100 Ω standard reference resistor is simultaneously monitored.
The DFPC is connected to a gas and vacuum system whose main parts are displayed in Fig. 2. The system comprises three parts: a gas supply unit (the leftmost part of the system depicted in Fig. 2), a pressure stabilizing unit, providing a stable pressure (the center part of the system), and the refractometer (the rightmost part of the system).

IV. Methodology
Before any measurement series was initiated, the wavelengths of the lasers

A. The valve switching and gas modulation procedure
As for SCM-GAMOR, to achieve gas modulation in the system, GEq-GAMOR is realized by periodically changing the states of the various valves in the system. Figure. 3 shows a schematic illustration of the valve states for the three different states that GEq-GAMOR comprises.   In State II, as is shown in Fig. 3(b), the valves 1 and 3 are closed whereafter valve 2 is opened. As is illustrated in section II in Fig. 4 This also indicates that there is no need to continuously or actively monitor the number of mode jumps during any gas filling or evacuation process; the status of the system provides, at any time, enough information to deduce any mode jump in any cavity.
In State III, to ensure gas purity (i.e. to reduce the influence from outgassing and gas leaks), as is shown in Fig. 3(c), the state begins by opening valve 3 whereby, as is shown in section III in Fig. 4(a), both cavities are evacuated.
After this, state I of the next cycle starts.
For this particular case, when the pressure to be assessed is provided by a dead weight piston gauge (as is shown by Fig. 2), the filling of the piston gauge (as was

B. The GAMOR feature
The beat signal between the two laser fields is in practice measured during the entire measurement cycle. It can therefore be seen as a continuous function of time, i.e. as ( ) f t . A schematic representation of a possible ( ) f t cycle is given by the black curve in Fig. 4(c). As has been alluded to above, because of drifts of the cavity spacer, this signal will not be a complete replica of the gas pressure [schematically displayed by the red curve in Fig. 4(a)]. To alleviate this, the estimated empty measurement cavity beat  where the molecular polarizability additionally has been recalculated for the actual wavelength used. The two refractivity-normalized deformation coefficients, m ε and r ε , have, in this work, been set to zero since the DFPC has not yet been fully characterized.
The average of the values of the cavity-drift-corrected shift in beat frequency, , during the part of the cycle when the cavity pressure, m P , has been equalized with respect to the external pressure, Ext P , which are marked by the large colored box at the end of section I of Fig. 4(d), represents the (0,0 ) Ext f → ∆ entity. This is then used to assess the refractivity according to the Eqs. (1), (4), and (5), as well as the density and pressure, Ext ρ and Ext P , by additional use of the Eqs. (6) and (7).
Hence, by this procedure, the effect of the linear drift of the cavities is efficiently eliminated from the assessment of gas refractivity, density, and pressure.

A. Stability and precision
To evaluate the stability and precision of the refractometer it was compared to a dead weight piston gauge (RUSKA 2465A) set to a pressure of 4303 Pa ( Ext P ). Figure 5 shows the pressure difference P ∆ between the pressure assessed by the refractometer, shows, by the blue markers, the case for the SCM-GAMOR system taken over 20 h (data adapted from Silander et al. 28 ) while panel (b) displays, by the red markers, that for the GEq-GAMOR system measured over 100 h. As a help to guide the eye, the black solid curves represent the moving mean over 10 samples. The dashed horizontal lines represent two standard deviations of the pressure differences, i.e. ±2σ.
The data taken by the GEq-GAMOR system [ Fig. 5(b)] show that the combined 2σ stability of the system (representing the fluctuations between the deadweight piston gauge and the refractometer) over 4 days was within ±5 mPa (or ±1 ppm). This is a significant improvement from the original realization of GAMOR [the SCM-GAMOR system, Fig. 5(a)], which showed ±16 mPa (or ±4 ppm) over 20 h (data adapted from Silander et al. 28 ). This indicates that the GEq-GAMOR system has a standard deviation measured over 100 h that is more than a factor of three smaller than that of the SCM-GAMOR system assessed over 20 h. This improvement is attributed to the upgraded temperature control and assessment. It can be seen by the leftmost data points, which represent the short-term response of the system, that the novel GEq-GAMOR system, provides, for averaging times below 10 measurement cycles, in comparison with the original SCM-GAMOR system, a reduction in noise by a factor of 2, from 3 to 1.5 mPa (cycle) 1/2 . This is attributed to the longer integration time in the reference pressure assessment, which reduces white-noisetype fluctuations.
To verify this assumption, the data for the GEq-GAMOR system (the red markers), which were integrated for 40 s, were re-evaluated with a reduced integration time (10 s), corresponding to the smaller box in Fig. 4(d). The resulting set of data is presented by the green markers in Fig. 6 (the second set of data counted from above). A comparison between these two GEq-GAMOR data sets (green and red) indisputably shows that the integration time affects the short-term response of the system. Assuming that both curves are affected by the same amount of flicker noise for the shortest time scales as for the longer, i.e. 1 mPa (see below), the residual white-noise contribution to the short-term response in the two data sets can be estimated to be 2 and 1 mPa (cycle) 1/2 , respectively. This is in agreement with the expected improvement of a factor of 2 originating from the square root of the decreased integration time.
Regarding the long-term response, the data show that the GEq-GAMOR system does not exhibit the same amount of drifts for longer averaging times (above 20 cycles) as the SCM-GAMOR system does. For integration times in the 300 -500 cycle (11 -18 hours) interval, the Allan deviation of the GEq-GAMOR system is 6 -8 times lower than for the SCM-GAMOR system. This reduction in drift is attributed to the improved temperature measurement and control in the GEq-GAMOR setup.
The data show though that, for a pressure of 4303 Pa, the GEq-GAMOR system is limited by flicker noise for averaging times above 10 cycles. The flicker noise, expressed in terms of an Allan deviation, is 1 mPa (which corresponds to 0.25 ppm) for averaging times up to 500 cycles (or 18 h). This corresponds to a precision, defined as twice the Allan deviation, of 0.5 ppm. Although the origin of this flicker noise has not been irrefutably identified, one possible reason is that it originates from the non-linear parts of the drifts that the GAMOR methodology does not eliminate. 33 The purple markers, which represent a measurement series with an evacuated However, when assessments are performed with finite (non-zero) pressures, the white noise is higher than when the measurement cavity is empty. For example, for a pressure of 4303 Pa, the white noise was found to be around 6 mPa Hz -1/2 . The additional noise is believed to originate from a combination of fluctuations in pressure from the dead weight piston gauge and temperature (including its assessment).

B. Linearity
To assess the linearity of the system a series of measurements were performed for a set of piston gauge weights. Figure 7(a) presents, by the individual markers, the external pressure assessed by use of the refractometer, denoted R P , as a function of the estimated pressure of the piston gauge, dw P , calculated by use of Eq. (4) in Silander et al. 28 , for seven different weights. The data series were taken over a period of 8 days in a non-consecutive order (2841, 4303, 7225, 10148, 3426, 5764, and 8687 Pa). Fig. 7(b) displays the averages of the same set of data in relative units. Regarding the data in Fig. 7(a), the black curve shows the best second order fit to the data of the type 2 dw dw p a bP cP = + + .
The fitted function agrees very well with the data. It addition, it provides a value of the c coefficient of 3 × 10 -10 Pa -1 . This shows that the GEq-GAMOR system provides a response that has a very small (virtually insignificant) non-linear component; the value of the non-linear term is, for each pressure, smaller than the estimated noise level. It can therefore be concluded that the GEq-GAMOR system, evaluated by the theory and the evaluation procedures described above, does not exhibit any noticeable systematic nonlinear dependence over the pressure range addressed. Hence, it suffices to evaluate its performance with respect to its linear response.
A corresponding linear fit to the data in Fig. 7(a) The fit also reveals a non-zero value of the constant term. The cause for this is presently unknown, but possible causes can be an incorrect piston weight or the assessments of the hood and residual pressure.

VI. Summary, Conclusions, and Outlook
A novel technique for refractometry, based on Gas Modulation Refractometry (GAMOR), that outperforms the original realization of GAMOR, here referred to as Single Cavity Modulated GAMOR (SCM-GAMOR), has been developed, realized, and scrutinized. It is based upon the fact that the reference measurements, which in SCM-GAMOR are performed by evacuating the measurement cavity, instead are carried out by equalizing the pressures in the two cavities. This new methodology is therefore referred to as Gas Equilibration GAMOR (GEq-GAMOR).
By this, the time it takes to reach adequate conditions for the reference measurements has been reduced. This implies that a larger fraction of the measurement cycle time can be devoted to acquisition of data, in particular during the reference assessments. In addition, the residual gas pressure assessment, which for GEq-GAMOR is performed during state I, can be averaged significantly longer than what is the case for SCM-GAMOR (in which the same entity had to be measured under non-equilibrium conditions in state II while the system is being pumped down).
Both these features reduce the white noise and improve on the short-term characteristics.
Yet another advantage of GEq-GAMOR is that the pressure during the reference measurement (i.e. the equilibration pressure) does not need to be assessed with the same accuracy as the residual pressure in the measurement cavity in SCM-GAMOR.
The system presented also incorporates a new cavity design with improved temperature stabilization and assessment. This has contributed to an improved long-term response of the GAMOR methodology.
A characterization of the novel GEq-GAMOR system, by use of a dead weight piston gauge, shows that it can provide a short-term response that exceeds that of the original SCM-GAMOR system by a factor of two. For longer integration times (above 10 cycles, ca. 20 min), and for a pressure of 4303 Pa, the system is limited by flicker noise, which, when expressed in terms of an Allan deviation, is 1 mPa, which, in turn, for this pressure, corresponds to 0.25 ppm. This thus corresponds to a precision (defined as twice the Allan deviation) of 0.5 ppm. For the case with an evacuated measurement cavity, it was found that the system is white-noise limited up to around 40 cycles (ca. 1.5 hours), with a white-noise of 0.7 mPa (cycle) 1/2 , and that it exhibits a minimum Allan deviation (for averaging times in the 40 -80 cycle range, corresponding to 1.5 -3 hours) of 0.15 mPa. For the longest integration times considered, and again for a pressure of 4303 Pa, it was found that the GEq-GAMOR system could reduce long-term drifts with respect to that of the original SCM-GAMOR system significantly; for 18 hours, by a factor of 8.
The linearity of the system was assessed by a comparison with a dead weight piston gauge. It was found that the GEq-GAMOR system provides a linear response (with respect to the piston gauge) over the entire pressure range investigated (2.8 -10.1 kPa), with no evidence of any systematic non-linearity. This range can be extended downwards into the Pa region with an expected precision in the sub-mPa region. This implies that the system can be used for transfer of calibration for pressures outside the range in which it was characterized.
Outgassing and leaks will contaminate the gas over time and affect its refractivity. 14 This will degrade the accuracy of density and pressure measurements. As the GAMOR principle involves periodical evacuation of the cavities, as is discussed in some detail elsewhere, the effect of gas contamination in GAMOR methodologies is assumed to be small. 33 If though still non-negligible, it can be quantified by altering the modulation period and interpolating the modulation period to zero. This provides a system that in practice is not affected by leakages or outgassing. 33 This work has mainly been focused on improving the precision and long-term stability of GAMOR. The accuracy of the instrumentation has not yet been addressed. In this case, it was found that the system has a finite (systematic) discrepancy with respect to the dead weight piston gauge, with a linear response that shows a deviation of 0.228 % from the ideal response (unity). This discrepancy can be attributed to a number of causes of which one is the un-characterized DFPC (the deformation of the cavity due to the pressure of the gas, which has not yet been assessed).
In the future, the accuracy of the system can be improved by a characterization with respect to both the physical deformation of the cavity, i.e. the value of m ε and ε ∆ , and the gas constants. This can, for example, be done by the use of two gases assessed under the same conditions, of which one is He, for which the molar refractivity (together with the relevant density and pressure virial coefficients) is known. Alternatively, the system can be calibrated in terms of the combination of molar refractivity and physical elongation of the cavity by the use of a pressure standard.
Since it has been shown that it is possible to construct a GAMOR setup from offthe-shelf components with a relatively simple cavity design, this opens up for a new class of pressure standards, with no moving actuators, that are traceable to the SI system through frequency and temperature. The potential benefits of these standards, in comparison to current mercury manometer and dead weight piston gauges, are substantial, both in reduced maintenances and operating cost.

SCM-GAMOR
In SCM-GAMOR (Single Cavity Modulated GAMOR), the measurement cavity is alternatingly evacuated and filled with gas whose refractivity (density or pressure) is to be assessed while the reference cavity is held at a constant pressure. This implies that the frequency of the laser that addresses the measurement cavity (henceforth referred to as the measurement laser) alternates between two frequencies, one when the measurement cavity is being evacuated (referred to as the reference assessment) and another when it contains gas (referred to as the filled measurement cavity assessment).
To relate such a change in frequency to the corresponding change in refractivity, let us first, to set the nomenclature, assume that the frequency of the laser that addresses the measurement cavity when it is fully evacuated (to a pure vacuum), (0) m ν , can be written as Since, for practical reasons, the measurement cavity is not evacuated to a pure vacuum during the reference assessment, we will assumed that, under this assessment, it contains a residual amount of gas that has a refractivity of 1 respectively. Here, This implies that when gas is evacuated from (or filled into) the measurement cavity the measured beat frequency alternates between, for the reference assessment, , which, in this case, is solely given by the shift in frequency of the measurement laser, can be expressed as where we have introduced This shows that a change in the refractivity in the measurement cavity, from ( ) where (0,0) f  , referred to as the interpolated empty measurement cavity beat frequency, is given by Eq. (SM-11) evaluated at the same time instance as (0, ) g f is measured.

GEq-GAMOR
For the case when GEq-GAMOR (Gas Equilibration GAMOR) is used, the procedure for the filled measurement cavity assessment is identical to that in SMC-GAMOR, i.e. with the measurement cavity being filled with the gas whose refractivity is to be assessed while the reference cavity is being evacuated. This implies that the frequency of the This expression shows that the beat frequency under gas equilibrium conditions is dissimilar to the evacuated measurement cavity beat frequency measured in SCM-GAMOR when both cavities are evacuated, given by Eq. (SM-5). This implies that also the shift in beat frequency between the filled measurement cavity assessment and the reference assessment is dissimilar to that measured in SCM-GAMOR. Hence, Eq.
(SM-10) is not valid for GEq-GAMOR in its present form.
There are a few means to accommodate this. The one used here, which is simple and convenient, is to recalculate which fully evacuated measurement cavity beat frequency (henceforth referred to as the estimated empty measurement cavity beat frequency and denoted ( ) , which in practice is never measured, by using the fact that Eq f should be exposed to an interpolation according to Eq. (SM-11).

Useful expressions for GAMOR when the refractometer is connected to an external pressure source
For the case when the pressure in the measurement cavity during the filled measurement cavity assessment is equilibrated with that of an external pressure source (or pressure standard) with a pressure of Ext P , with a corresponding refractivity of 1 Ext n − , Eq. (SM-9) , when used for SCM-GAMOR, can be written as