Quantum atomic sensors have shown great promise for vacuum metrology. Specifically, the density of gas particles in vacuum can be determined by measuring the collision rate between the particles and an ensemble of sensor atoms. This requires preparing the sensor atoms in a particular quantum state, observing the rate of changes of that state, and using the total collision rate coefficient for statechanging collisions to convert the rate into a corresponding density. The total collision rate coefficient can be known by various methods, including quantum scattering calculations using a computed interaction potential for the collision pair, measurements of the postcollision sensoratom momentum recoil distribution, or empirical measurements of the collision rate at a known density. Observed discrepancies between the results of these methods call into question their accuracy. To investigate this, we study the ratio of collision rate measurements of colocated sensor atoms, ^{87}Rb and ^{6}Li, exposed to natural abundance versions of H_{2}, He, N_{2}, Ne, Ar, Kr, and Xe gases. This method does not require knowledge of the test gas density and is, therefore, free of the systematic errors inherent in efforts to introduce the test gas at a known density. Our results are systematically different at the level of 3% to 4% from recent theoretical and experiment measurements. This work demonstrates a modelfree method for transferring the primacy of one atomic standard to another sensor atom and highlights the utility of sensoratom crosscalibration experiments to check the validity of direct measurements and theoretical predictions.
I. INTRODUCTION
The total collision rate coefficients can be obtained from quantum scattering calculations (QSC).^{18–22} The quantum scattering computations used to determine the collision cross sections rely on electronic potential energy surfaces (PESs), which must be determined by ab initio quantum chemistry computations. However, sufficiently accurate PESs may be limited to molecular species with a small number of active degrees of freedom. Alternatively, empirical estimates of the total collision rate coefficients, $ \u27e8 \sigma tot \u2009 v \u27e9 ( T )$, can be obtained from measurements with a gas of known density.^{23,24} This approach is limited to gas species compatible with the operation of existing orifice flow pressure standards.
A third method involves using the sensor atom collision recoil energy (or momentum) distribution to determine the total collision rate coefficient using the quantum diffractive collision universality law.^{11–13,25} The validity of this latter approach was previously shown for heavy collision partners, including Rb + N_{2} and Rb + Rb collisions,^{26} but deviations from the universality law were found for lowmass test species.^{27} The results of these three methods have been found to agree at the level of a few percent, with the exception of a few special cases.^{23,24} Understanding the reasons for the observed disagreements is essential.
The main goal of this work is to report the results of a fourth method for validating prior theoretical and experimental work. We perform direct experimental comparisons of the total collision rate coefficients of colocated cold ensembles of ^{87}Rb and ^{6}Li atoms exposed to the test gas species, H_{2}, He, Ne, N_{2}, Ar, Kr, and Xe. We also present a comparison of the total collision rate coefficients when the sensor atoms are exposed to a sample of parahydrogen. This same method was used to compare $ \u27e8 \sigma tot \u2009 v \u27e9 Rb + H 2$ and $ \u27e8 \sigma tot \u2009 v \u27e9 Li + H 2$ in a previous study, where the partial pressure of H_{2} was varied by heating a nonevaporable getter (NEG).^{27} This allowed us to determine the ratio of the total collision rate coefficients: $ \u27e8 \sigma tot \u2009 v \u27e9 Li + H 2 / \u27e8 \sigma tot \u2009 v \u27e9 Rb + H 2 = 0.83 ( 5 )$ for an ambient temperature of 300(2) K. The uncertainty in the ratio is the statistical uncertainty (type A) derived from fitting the loss rate data.
This method of comparing the total collision rate coefficients is modelindependent and is free of some of the systematic errors inherent in other methods. In particular, the fact that the two sensor atom traps are colocated ensures that they are both exposed to the same baseline environment and the same test gas density. These conditions cannot be guaranteed in measurements comparing a cold atom vacuum standard to a calibrated ion gauge^{12,13} or to a flowmeter and dynamic expansion system where colocation is not possible.^{23,24} In addition, this colocation measurement provides a systematicerrorfree method for transferring the primacy of one atomic standard to another sensor atom.
II. EXPERIMENTAL PROCEDURE
In this work, the ratio $ R X ( T )$ is measured directly by confining Rb and Li atoms to the same spatial region. Prior to introducing any test gas, X, to the vacuum system, the sensor atom ensembles exhibit baseline trap loss rates. These baseline loss rates are due to collisions with common residual background gases, such as H_{2}, and intrinsic trap loss mechanisms.
The advantage of this method of determining R_{X} is that an exact knowledge of the test gas density or residual background composition and the background density is not required. Instead, it is sufficient that these densities remain constant for each measurement pair. This technique contrasts with prior experimental work that relied on another pressure standard to set a known test gas density.^{23,24}
The apparatus used to perform these loss rate measurements of both Rb and Li atoms is very similar to one described in previous work, although the current iteration is a complete rebuild.^{27,28} In the rebuilt version, an additional vacuum chamber with a variable leak valve, ion gauge, and residual gas analyzer (RGA) was connected near the oven section of our apparatus. This allows us to control the partial pressure $ P X = n X k B T$ of the test gas X introduced through the leak valve, where $ k B$ is Boltzmann's constant.
The temperature of the gases inside the vacuum was inferred from a measurement of the room temperature by a single digital thermometer above the apparatus. We assume that the test gas is at thermal equilibrium with the room. As discussed in the Appendixes, the common temperature variation of the total collision rate coefficients renders their ratio insensitive to temperature variations. Specifically, even an uncertainty of ±10 K on an ambient temperature of 300 K results in an uncertainty of R_{X} of less than 0.06%. Thus, there is no requirement to measure the temperature with high accuracy or precision.
A schematic of the apparatus is shown in Fig. 1. When introducing a new gas, we start by evacuating a small reservoir connected to the variable leak valve and the scroll pump. Subsequently, we purge the reservoir by filling the reservoir with the test gas at a pressure of approximately 10^{5} Pa and evacuating it again for 2–3 purge cycles. The reservoir is then filled to a pressure $ P r \u223c 10 5$ Pa and sealed. We then vary the partial pressure of the test gas in the magnetic trap chamber, in the range from $ 10 \u2212 8$ to $ 10 \u2212 7$ Pa, by changing the small leak rate using the variable leak valve. The RGA is used to verify the leaked gas is free of contaminants.
The oven section and magnetic trap section, where the trap loss measurements are performed, are connected by a 12 mm long, 6 mm diameter differential pumping tube with an estimated hydrogen conductance of 0.85 l/s.^{29} We turn off the ion pump in the oven section (where the pressure is P_{1} as shown in Fig. 1) to prevent sputtering and pressure fluctuations due to Argon instability when pumping heavy noble gases.^{30} The much lower pressure in the magnetic trap section eliminates the need to do the same for the ion pump located there. Instead, we leave this second ion pump on, as it lowers the baseline pressure achieved when not flowing the test gas, thus improving the lower bound on the measured loss rates and enhancing the experimental precision achieved in R_{X}. Because the statistical estimate for the linear slope, R_{X}, is optimized by making measurements that are spaced far apart in the independent variable (here the Rb loss rate), we lower the baseline pressure to expand the span of the loss rate measurements, which are bounded from above at a pressure where the loss is so high that the signaltonoise becomes severely compromised. In addition, we verified that turning off this ion pump does not alter the measured value of R_{X}.
For both species, colocated magnetooptic traps (MOTs) capture atoms from thermal, parallel atomic beams that are decelerated and cooled by a Zeeman slower.^{28} To ensure a similar number of atoms are loaded for each experimental shot, the Zeeman slowing light is extinguished when the fluorescence from the MOT reaches a set value $ V MOT$. The atoms are then optically pumped to the desired hyperfine state, and all light is extinguished, at which point they are confined by the quadrupole magnetic field. We then ramp the magnetic field gradient in 20 ms to the hold gradient used for the MT phase. After a variable hold time in the MT, the atoms are recaptured in the MOT, and their fluorescence $ V MT ( t )$ is recorded using a photodiode. To improve the signal to noise, we determine the recaptured fraction $ f recap ( t ) = V MT ( t ) / V MOT$ as described previously.^{27} This quantity is insensitive to the residual variations in the number of atoms loaded into the MOT.
Glancing collisions, which do not induce trap loss, can lead to heating of the ensemble and, consequently, influence the measured loss rate in a nontrivial way (see Sec. 1 of Appendix A). To minimize this effect, the Li and Rb atoms are confined in a very shallow magnetic trap for these measurements. The Rb atoms are confined with an axial magnetic field gradient of $ b \u2032 = 35$ G/cm, and the trap depth is limited to $ U Rb / k B = 125 \u2009 \mu K$ by applying radio frequency (RF) radiation repeatedly sweeping between 20 and 40 MHz for 2 s at the end of the variable hold time. We verified that sweeping the RF from 0 to 40 MHz completely empties the trap. The RF induces spatially localized transitions between the trapped, weakfield seeking $  F = 1 , m F = \u2212 1 \u27e9$ and the untrapped $  F = 1 , m F = 0 \u27e9$ and antitrapped, strongfield seeking $  F = 1 , m F = 1 \u27e9$ magnetic Zeeman states. More details are provided in Sec. 1 of Appendix A. For ^{6}Li, the atoms are confined in an axial gradient $ b \u2032 = 100$ G/cm. We do not apply any RF for ^{6}Li, since the field dependence of the magnetic moment of the $  F = 1 / 2 , m F = \u2212 1 / 2 \u27e9$ state limits the trap depth to $ U max / k B = 314 \u2009 \mu K$.^{27}
Because the optimal Zeeman + MOT loading and MT trapping parameters are so different for the two sensor atoms, we alternate between loss rate measurements instead of making a simultaneous measurements of a MT loaded with both sensor atoms. Since we require that $ n X$ be the same for each sensor atom loss rate measurement, we perform measurements at a series of different but constant leak rates. To mitigate the effect of any slow drift of the background gas density, we minimize the time between sensor atom measurements by performing twopoint measurements, alternating between trapped species. We measure the recapture fraction at t = 0 and t = t_{1}, and we compute the loss rate as $ \Gamma = ln \u2009 [ f recap ( t = t 1 ) / f recap ( t = 0 ) ] / t 1$. We interlace the measurements for the two species such that the twopoint lifetime measurement for the two species is overlapped in time. Thus, a single $ ( \Gamma Rb , \Gamma Li )$ point is collected as follows:

Measure $ f recap ( t = 0 )$ for Li;

Measure $ f recap ( t = 0 )$ for Rb;

Measure $ f recap ( t = t 1 )$ for Li;

Measure $ f recap ( t = t 1 )$ for Rb.
To optimize the sampling, we choose a hold time for each species such that $ t 1 \u2261 1.5 / \Gamma \xaf$, where $ \Gamma \xaf$ is an estimate of the Rb or Li loss rate that is updated based on the average of the previous five measurements. In addition, the order in which the t = 0 and t = t_{1} measurements are taken is randomized to mitigate the effect of systematic drifts of the MOT to MT transfer efficiency. We also observed that the decay of the recapture fraction, even at the very lowest background pressures where nonlinearities in the decay would be most apparent, is purely exponential (see Sec. 3 of Appendix A). The consequence is that a twopoint measurement is sufficient to determine the total loss rate at any test gas pressure.
Finally, we note that for this study, we use ^{6}Li instead of ^{7}Li sensor atoms. Because the ^{6}Li atom is a composite Fermion, when we prepare and magnetically trap a spinpolarized sample in the $  F = 1 / 2 , m F = \u2212 1 / 2 \u27e9$ state at a temperature below 300 μK, intratrap collisions do not occur. This is because swave collisions are forbidden due to the Pauli principle, and pwave collisions are frozen out at this temperature. Because collisions between sensor atoms can lead to evaporation losses that confound the determination of the losses due to collisions with background particles, such collisions are problematic for precision vacuum metrology. Efforts to minimize sensor atom collisions have been described previously.^{13,17,26,27} Here, these efforts are unnecessary because the ^{6}Li ensemble is a true experimental realization of the idealized model of a noninteracting trapped sensor atom ensemble.
III. DISCUSSION OF RESULTS
In Fig. 2, we plot the ordered pairs $ ( \Gamma Rb , \Gamma Li )$ for different values of the H_{2} and He densities in a 2D scatterplot where the abscissa is the Rb loss rate and the ordinate is the Li loss rate. The measurements for H_{2} and He were chosen as examples because they represent the highest and lowest values of R_{X}, respectively. We fit these points to a linear model using orthogonal distance regression (ODR) and extract the slope R. The individual points are weighted by the statistical uncertainty, which we estimate by characterizing the variation in $ f recap$ at constant pressure and using uncertainty propagation to find the uncertainty in Γ. In Fig. 2, these uncertainties are suppressed for visual clarity, except for a few selected points.
Figures 2(b) and 2(c) show the same data as a time series, together with the partial pressures measured by the RGA. For each gas under test, the partial pressure was increased and decreased several times in random order during each experimental run to minimize the effects of any systematic, slow variation in the background pressure on the measurement. We also repeated each measurement on three different days and observed the same values for R_{X} to within 1%.
The results of our measurements are summarized in Table I and Fig. 3, where they are compared to recent theoretical and experimental results.^{22–24,27} We note that our measurements utilize ^{6}Li and ^{87}Rb atoms, and thus we report $ R 6 , 87 \u2261 ( \u27e8 \sigma tot \u2009 v \u27e9 6 Li + X / \u27e8 \sigma tot \u2009 v \u27e9 87 Rb + X )$, whereas prior theoretical and experimental results are for ^{7}Li and ^{87}Rb atoms, providing $ R 7 , 87 \u2261 ( \u27e8 \sigma tot \u2009 v \u27e9 7 Li + X / \u27e8 \sigma tot \u2009 v \u27e9 87 Rb + X )$.
Species .  (This work) .  $ R 7 , 87 KT$ (Ref. 22) .  $ R 7 , 87 BExp$ (Refs. 23 and 24) .  $ R 6 , 87 SExp$ (Ref. 27) .  $ R 6 , 87 ( th . )$ (Ref. 27) .  $ R 6 , 87 C 6$ .  

$ R 6 , 87 meas$ .  $ R 6 , 87$ .  
pH_{2}  0.828(6)  0.827(6)  0.82(3)  0.869  0.801(3)  
H_{2}  0.825(5)  0.824(5)  0.82(3)  0.83(5)  0.869  0.801(3)  
He  0.729(6)  0.729(6)  0.69(2)  0.73(2)  0.775(1)  
Ne  0.798(5)  0.797(5)  0.78(10)  0.74(2)  0.778(2)  
N_{2}  0.798(5)  0.796(4)  0.764(14)  0.75(3)  0.789(1)  
Ar  0.798(5)  0.796(5)  0.768(4)  0.721(13)  0.788(1)  
Kr  0.798(5)  0.795(5)  0.769(4)  0.78(2)  0.797(2)  
Xe  0.803(4)  0.799(4)  0.779(8)  0.77(2)  0.803(2) 
Species .  (This work) .  $ R 7 , 87 KT$ (Ref. 22) .  $ R 7 , 87 BExp$ (Refs. 23 and 24) .  $ R 6 , 87 SExp$ (Ref. 27) .  $ R 6 , 87 ( th . )$ (Ref. 27) .  $ R 6 , 87 C 6$ .  

$ R 6 , 87 meas$ .  $ R 6 , 87$ .  
pH_{2}  0.828(6)  0.827(6)  0.82(3)  0.869  0.801(3)  
H_{2}  0.825(5)  0.824(5)  0.82(3)  0.83(5)  0.869  0.801(3)  
He  0.729(6)  0.729(6)  0.69(2)  0.73(2)  0.775(1)  
Ne  0.798(5)  0.797(5)  0.78(10)  0.74(2)  0.778(2)  
N_{2}  0.798(5)  0.796(4)  0.764(14)  0.75(3)  0.789(1)  
Ar  0.798(5)  0.796(5)  0.768(4)  0.721(13)  0.788(1)  
Kr  0.798(5)  0.795(5)  0.769(4)  0.78(2)  0.797(2)  
Xe  0.803(4)  0.799(4)  0.779(8)  0.77(2)  0.803(2) 
The dataset is observed to separate into two groups: the higher mass collision partners for which the ratio is almost constant [with an average value $ R 6 , 87 = 0.797 ( 2 )$ for Ne, N_{2}, Ar, Kr, and Xe], and the lower mass collision partners, H_{2} and He, for which the $ R 6 , 87$ values are distinct from this constant value.
The $ R 6 , 87$ values reported here for H_{2} and Ne are consistent with the quantum scattering calculations (QSC) of Kłos and Tiesinga^{22} (KT). Specifically, the measured value for H_{2}, $ R 6 , 87 = 0.824 ( 5 )$, agrees with the KT QSC value, $ R 7 , 87 KT = 0.82 ( 3 )$. It is also in agreement with our previous measurement, $ R 6 , 87 SExp = 0.83 ( 5 )$. Our earlier experimental measurement also employed colocated sensor ensembles but relied on a different source for the H_{2} gas (the H_{2} outgassing induced by heating a nonevaporable getter pump). The measured value for He, $ R 6 , 87 = 0.729 ( 6 )$, agrees with the updated BExp value,^{23,24} $ R 7 , 87 BExp = 0.73 ( 2 )$, and is within 2 standard deviations of the KT QSC value, $ R 7 , 87 KT = 0.69 ( 2 )$. Finally, the value for Ne, $ R 6 , 87 = 0.797 ( 5 )$, is in agreement with the KT QSC value, $ R 7 , 87 KT = 0.78 ( 11 )$, but is outside 2 standard deviations of the BExp measurement of $ R 7 , 87 BExp = 0.74 ( 2 )$. In contrast, the $ R 6 , 87$ values for the heavier species (N_{2}, Ar, Kr, and Xe) are systematically larger than either the KT QSC or the BExp experimental values (see Table I). We discuss these discrepancies and potential causes here and in the Appendix.
A unique advantage of our experimental technique and the data we have obtained is that the Li and Rb sensor atoms are exposed to exactly the same test gas pressure, since the traps are colocated. In the experimental work of Barker et al,^{23,24} $ \u27e8 \sigma tot \u2009 v \u27e9 ( T )$ was deduced from measurements of the loss rate at test gas densities, n_{X}, created by an orifice flow standard. We hypothesize that pressure gradients may have existed between the orifice flow standard output and the locations of the two separate CAVS sensor assemblies connected to the standard. Indeed, the ambient temperatures reported for these two sensor ensembles were different, indicating variations in the local test environments. Such gradients could result in a systematic error in the inferred pressure and, thus, the $ \u27e8 \sigma tot \u2009 v \u27e9 ( T )$ values reported. As the lightest noble gas, He has the highest conductance through the vacuum system and does not exhibit significant outgassing or surface adsorption, so pressure gradients are expected to be smallest for this species. This may account for the close agreement between the experimental values. In contrast, we observe larger discrepancies between the $ R 7 , 87 BExp$ and the $ R 6 , 87$ values reported here for the heavier species (see Table I), consistent with this hypothesis.
We also had the opportunity to measure $ R 6 , 87$ for >99.9% pure parahydrogen (pH_{2}) created using a closed cycle He fridge and a (FeOH)O magnetic converter at 14 K.^{32} This gas was produced and provided to us by the group of Professor Takamasa Momose of the University of British Columbia. We found no statistical difference in the value of R for parahydrogen compared to normal hydrogen, as shown in Table I. Normal hydrogen is a mixture of ortho and parahydrogen with nuclear spins, I = 1 and 0, respectively. In the $ X \u2009 1 \Sigma g +$ ground molecular state, the wavefunction symmetry demands that I + J is even, where J is the rotational quantum number of the rovibronic state. Thus, orthohydrogen populates only the odd J rotational levels, and parahydrogen populates only the even rotational levels. For glancing collisions, there is insufficient anisotropy in the three body PESs to result in a noticeable change in rotational state required for an paraH_{2} to orthoH_{2} conversion (or viceversa). Thus, we expected the $ R 6 , 87$ (and $ \u27e8 \sigma tot \u2009 v \u27e9 Li + X , \u27e8 \sigma tot \u2009 v \u27e9 Rb + X$) to be the same for normal H_{2} and paraH_{2}, as our measurements confirm.
In the Appendixes, we discuss the systematic errors (type B errors) associated with the $ R 6 , 87$ values reported here. These are summarized in Table II. The largest systematic error is associated with the corrections due to the finite trap depth confining the sensor atoms deviating the measured value of $ \Gamma loss = n \u27e8 \sigma loss \u2009 v \u27e9$ from the target $ \Gamma = n \u27e8 \sigma tot \u2009 v \u27e9$. This systematic error has been removed from our quoted values of $ R 6 , 87$ in Table I, and it is well below the 1% level for all collision partners for the trap depths employed in this study. Thus, we are confident that the discrepancies we observe indicate the need for the refinement of both theoretical and experimental determinations of $ \u27e8 \sigma tot \u2009 v \u27e9$.
Error source .  Estimate, $ \delta R R$ . 

$ \Gamma Loss meas$ deviation from $ \Gamma tot$  <0.5% 
Ensemble heating induced error  <0.004% 
2body intratrap loss uncertainty  <0.05% 
Ambient temperature uncertainty  <0.012% 
Baseline gas composition variation  <0.14% 
Error source .  Estimate, $ \delta R R$ . 

$ \Gamma Loss meas$ deviation from $ \Gamma tot$  <0.5% 
Ensemble heating induced error  <0.004% 
2body intratrap loss uncertainty  <0.05% 
Ambient temperature uncertainty  <0.012% 
Baseline gas composition variation  <0.14% 
In Appendix B, we hypothesize that the systematic 3.4(7)% difference between the measured $ R 6 , 87$ and computed $ R 7 , 87 KT$ values for X = N_{2}, Ar, Kr, and Xe is due to the incomplete suppression of the effects of glory oscillations via the Maxwell–Boltzmann (MB) averaging of $ \u27e8 \sigma tot \u2009 v \u27e9$ for Li–X collisions. By contrast, the glory scattering is almost completely erased by the MB averaging for Rb–X collisions. We support this by illustrating the very different glory oscillation behavior between Li–Xe and Rb–Xe, comparing the computed $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ values from the potentials reported by KT to a semiclassical (SC) interaction potential consisting solely of longrange van der Waals terms, $ \u27e8 \sigma tot \u2009 v \u27e9 SC$. The ratio of these total collision loss rate coefficients, $ \u27e8 \sigma tot \u2009 v \u27e9 SC : \u27e8 \sigma tot \u2009 v \u27e9 KT$, is 1.0 for Rb–X (X = N_{2}, Ar, Kr, and Xe) collision partners, indicating a near complete erasure of glory oscillation effects. The corresponding ratio is $ \u2243 1.02$ for Li–X collisions, indicating some information about the glory scattering persists in these $ \u27e8 \sigma tot \u2009 v \u27e9$. Thus, the Li–X $ \u27e8 \sigma tot \u2009 v \u27e9$ values are conveying some information about the details of the core portion of the interaction potentials. The 3.4(7)% discrepancy could be a manifestation of some small inaccuracy of the potential used by KT to compute the Li–X $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ values.
The hypothesis that the potentials used to compute the Li $ + X$ cross sections may need refinement is supported both by our experimental results and by the experimental values of $ \u27e8 \sigma tot \u2009 v \u27e9$ for ^{7}Li collisions with the heavier collision partners reported by Barker et al.^{23} The majority of the experimental values reported by Barker et al. are systematically 2%–4% higher than the theoretical predictions. The results presented here, with their attendant high precision, offer an exciting opportunity to examine the shape of the core repulsion for Li collisions experimentally.
IV. CONCLUSIONS
The collisioninduced loss rate of cold, trapped atoms from a shallow trap due to a test gas, X, can be written as: $ \Gamma ( T ) = n X \u27e8 \sigma tot \u2009 v \u27e9 ( T )$, where n_{X} is the density of the test gas, and $ \u27e8 \sigma tot \u2009 v \u27e9 ( T )$ is the total collision rate coefficient characterizing the collision interaction. Knowledge of the total collision rate coefficient allows the measured loss rate to be used to measure the gas density directly. Here, we have measured the loss rates of ^{87}Rb atoms and of ^{6}Li atoms from shallow magnetic traps when exposed to natural abundance versions of H_{2}, N_{2}, He, Ne, Ar, Kr, and Xe gases at T = 298(2) K. The loss rates were recorded in pairs, $ ( \Gamma Rb + X , \Gamma Li + X )$, for a series of gas densities, $ n X$, for each gas species, X. These loss rates, measured at trap depths $ U Li < 0.314$ mK and $ U Rb < 0.125$ mK, were used to compute the collision rate coefficient ratio, $ R X ( T ) = \u27e8 \sigma tot v \u27e9 Li + X / \u27e8 \sigma tot v \u27e9 Rb + X$. The innovation and advantages of this method are (i) the cold trapped ensembles are colocated in the vacuum system, ensuring that the two trapped ensembles experience the same static background gas environment and the same test gas densities free from any systematic errors that may be encountered for ensembles probing the vacuum at different locations, and (ii) the ratiometric method means that the density of the test gas need not be measured or known to obtain $ R X ( T )$. The only requirement is that the test gas density remains constant over the measurement duration (tens of seconds) for each density setting. This method provides an experimental technique for transferring the known or calibrated loss rate coefficient for one species pair (A–X) to a second species pair (B–X), where X is the test gas, and A and B are two different trapped sensor atom ensembles. This experimentally determined loss rate ratio, $ R X ( T )$, also provides systematicerror free measurements that can be compared directly to ab initio computations of the loss rate coefficients $ \u27e8 \sigma tot v \u27e9 ( T )$. We believe these measurements can be used to guide refinements of the theoretical potential energy surfaces describing the collision interactions.
The measurements indicate that the loss rate ratios are constant for the heaviest species tested (X = N_{2}, Ar, Kr, and Xe) with an average of $ \u27e8 R 6 , 87 \u27e9 = 0.7965 ( 9 )$. This is remarkably close to the purely C_{6} prediction $ \u27e8 R 6 , 87 C 6 \u27e9 = 0.795 ( 5 )$ for ^{6}Li + X:^{87}Rb $ + X$. This finding appears to be consistent with the universality hypothesis that asserts that the total collision rate coefficient for heavy collision partners is principally determined by the longrange van der Waals interactions.^{11} By contrast, the Kłos and Tiesinga (KT) quantum scattering computation ratio predicts $ \u27e8 R 7 , 87 KT \u27e9 = 0.770 ( 3 )$, and the BExp measurements yield $ \u27e8 R 7 , 87 BExp \u27e9 = 0.756 ( 21 )$ for these same test species. The possible experimental sources of systematic error in our measurements have been examined in this paper, and we do not believe that they are large enough to explain these discrepancies. In particular, we find that the deviation of the measured $ \Gamma loss$ from the true $ \Gamma tot$ due to the finite shallow trap depth and cold atom ensemble energy distributions is <0.5%, ensemble collisioninduced heating uncertainties are less than 0.004%, unaccounted 2 and 3body intratrap collisions <0.05%, uncertainty associated with variation of the ambient temperature of the test environment <0.012%, and drift of the background gas density <0.14%.
Because our systematic errors are so small and because the majority of the experimental values of $ \u27e8 \sigma tot \u2009 v \u27e9$ for ^{7}Li collisions with test gases reported by Barker et al.^{23} are systematically 2%–4% higher than the theoretical predictions, we hypothesize that the observed systematic differences in our $ R 6 , 87$ values compared to the $ R 7 , 87 KT$ may arise from errors in the potential energy surfaces used in the KT quantum scattering computations for Li sensor atoms. Indeed, a 3.4% discrepancy is not large for such complex, multielectron systems and underscores the difficulty of quantifying the uncertainties in ab initio computations. The agreement observed between the observed R_{X} and the ab initio KT predictions^{22} for H_{2} may indicate that the potential energy surface computed for this much simpler system is more reliable. We believe this work provides an exciting opportunity to help refine the theoretical models and associated experimental measurements for atombased vacuum metrology.
ACKNOWLEDGMENTS
We acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC grants RTI201600120, RGPIN201904200, RGPAS201900055) and the Canadian Foundation for Innovation (CFI project 35724). This work was done at the Center for Research on UltraCold Systems (CRUCS) and was supported, in part, through computational resources and services provided by Advanced Research Computing at the University of British Columbia.
AUTHOR DECLARATIONS
Conflict of Interest
K.W.M and J.L.B. have US patents 8,803,072 and 11,221,268 issued.
Author Contributions
Erik Frieling: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Riley A. Stewart: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). James L. Booth: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Kirk W. Madison: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: ERROR ESTIMATES
1. Effect of the sensor ensemble energy distribution
Here, we discuss the systematic deviation of the R values arising from the finite trap depth and the sensor ensemble energy distribution.
For the purpose of determining the ratio of total rate coefficients R, we measure the sensor atom loss rates at exceedingly shallow trap depths, where the loss rate $ \Gamma loss ( U ) \xaf$ approaches the total collision rate $ \Gamma tot$. To characterize the deviation of the loss rate from the total collision rate due to the nonzero trap depth U and finiteenergy distribution $ \rho ( E )$ of the trapped ensemble, we use the trapdepth dependence model based upon ab initio calculations performed by Kłos and Tiesinga (KT).^{22} We note that a trapdepth dependence model based on quantum diffractive universality provides similar estimates for the expected deviation in the Rb loss rate.^{11,13}
We calculate the deviation of the measured loss rate from the total collision rate by integrating $ \u27e8 \sigma loss ( U ) v \u27e9$ over the experimentally measured energy distributions, as in Eq. (A2), for both trapped species. From this, we compute the deviation in R and present the results in Table IV. Across all gas species investigated here, the expected deviation δR relative to the value of R at zero trap depth, due the effects of a finite trap depth U and energy distribution $ \rho ( E )$, is less than 0.51%. The largest deviations are associated with gas species where the loss rate decreases fastest with increasing trap depth. Generally, these correspond to species with a large polarizability and/or small collisional energies at room temperature.^{27}
2. Collisioninduced heating
Collision events with the background thermal gas, which include the test gas species X, result in momentum exchange between the collision partners, thereby redistributing the energies of the trapped particles. This leads to heating of the sensor atom ensemble, modifying the energy distribution $ \rho ( E )$.^{11,17,26,27,35} As the average loss rate over the entire ensemble depends on this distribution [Eq. (A2)], heating modifies the observed loss rate.^{17,26,27,35} We note that, given the experimentally realized trap depth and energy distributions listed in Table III, the fraction of collisions that do not result in loss is less than 0.6% across all species, significantly suppressing heating effects. Previous studies accounted for heating by experimentally measuring the time evolution of the trapped distribution.^{17,26,27,35}
Species .  $ T / k B$ ( $ \mu K$) .  $ E min / k B$ ( $ \mu K$) . 

Rb  54.0(1)  46.1(1) 
Li  $ 1500 ( 500 )$  116.7(8) 
Species .  $ T / k B$ ( $ \mu K$) .  $ E min / k B$ ( $ \mu K$) . 

Rb  54.0(1)  46.1(1) 
Li  $ 1500 ( 500 )$  116.7(8) 
Bk. species .  $ \delta \u27e8 \sigma loss v \u27e9 / \u27e8 \sigma tot \u2009 v \u27e9$ (Rb/Li) (%) .  $ \delta R / R$ (%) . 

H_{2}  −0.151/−0.017  0.129 
He  −0.058/−0.005  0.051 
Ne  −0.216/−0.023  0.185 
N_{2}  −0.316/−0.033  0.270 
Ar  −0.323/−0.035  0.280 
Kr  −0.449/−0.049  0.383 
Xe  −0.588/−0.064  0.500 
Bk. species .  $ \delta \u27e8 \sigma loss v \u27e9 / \u27e8 \sigma tot \u2009 v \u27e9$ (Rb/Li) (%) .  $ \delta R / R$ (%) . 

H_{2}  −0.151/−0.017  0.129 
He  −0.058/−0.005  0.051 
Ne  −0.216/−0.023  0.185 
N_{2}  −0.316/−0.033  0.270 
Ar  −0.323/−0.035  0.280 
Kr  −0.449/−0.049  0.383 
Xe  −0.588/−0.064  0.500 
For this work, we estimate the residual effects of heating by utilizing the trapdepth dependence of the loss rate coefficient $ \u27e8 \sigma loss v \u27e9 ( U )$ to determine the postcollisional energy probability density function P_{t}. Our formulation to account for heating follows a rigorous treatment discussed previously^{17,35} and recently applied to correct the measurements of Barker et al.^{24} In general, the postcollisional energy of a trapped sensor particle depends on the relative velocity and scattering angle of the collision event. However, in the limit where the distribution of relative velocities is dominated by the contribution from the collisional partner in the background thermal gas, P_{t} is approximately independent of the sensor particle energy. For a background thermal gas X composed of particles of mass $ m X$ at temperature $ T X$, corrections to this approximation are on the order of $ m bg T S / m S T bg$, which, across all trapped and test species partners investigated in this work, contribute corrections less than 1%.^{17} These are neglected in the following analysis. We also neglect possible intratrap thermalization collisions between trapped sensor atoms in the Rb ensemble owing to low intratrap densities.^{36} Such intratrap thermalization collisions are forbidden for the cold ^{6}Li ensemble.
In accordance with our exceedingly low trap depths (and hence small number of heating collisions), we find that the loss rates $ \Gamma loss \xaf$ derived from a twopoint measurement scheme, including ensemble heating, are modified by less than 0.004% across all test species for the Rb ensemble and less than 0.0004% for the Li ensemble given our experimental conditions. Table V presents the combined relative deviation $ \delta R / R$ due to ensemble heating. Across all gas species, the expected deviations due to ensemble heating are less than 0.004%.
Bk. species .  $ \delta \Gamma loss \xaf / \Gamma loss \xaf$ (Rb/Li) ( $ 10 \u2212 4$%) .  $ \delta R / R$ ( $ 10 \u2212 4$%) . 

H_{2}  −2.43/−0.04  −2.39 
He  −0.357/−0.004  −0.353 
Ne  −5.01/−0.06  −4.95 
N_{2}  −10.43/−0.14  −10.29 
Ar  −11.28/−0.16  −11.12 
Kr  −21.12/−0.29  −20.83 
Xe  −36.17/−0.50  −35.67 
Bk. species .  $ \delta \Gamma loss \xaf / \Gamma loss \xaf$ (Rb/Li) ( $ 10 \u2212 4$%) .  $ \delta R / R$ ( $ 10 \u2212 4$%) . 

H_{2}  −2.43/−0.04  −2.39 
He  −0.357/−0.004  −0.353 
Ne  −5.01/−0.06  −4.95 
N_{2}  −10.43/−0.14  −10.29 
Ar  −11.28/−0.16  −11.12 
Kr  −21.12/−0.29  −20.83 
Xe  −36.17/−0.50  −35.67 
3. Densitydependent losses
In order to calculate an upper bound on the resulting shift in R, we perform a MonteCarlo simulation of the sequence of measurements used to construct R. First, we draw ten samples from a uniform distribution of background pressures corresponding to onebody loss rates between 0.07 and 0.4 s^{−1}, corresponding to a experimentally relevant range of onebody loss rates (see Fig. 2). At each pressure, the loss rate for the Rb and Li ensembles is calculated from the atom number at two trapped times t = 0 and $ 1.5 / \Gamma $, as determined by Eq. (A17), with the $ \beta \u0303$ values fixed for each species to the values specified above. We then determine an inferred value of R, given the simulated loss rate measurements at the ten randomly distributed test gas pressures, using ODR and a linear model for the relationship between $ \Gamma Li$ and $ \Gamma Rb$, as performed for the experimental measurements. This process is then repeated until we converge to an approximately constant distribution of R values, corresponding to the possible R values one would obtain for a given experimental measurement and the twobody loss rate coefficients $ \beta \u0303$ for the two species. From this distribution, we observe a relative deviation in the average value of R of less than −0.05% due to twobody loss.
4. Test and background gas temperature
Bk. species .  C_{6} ( $ E h a 0 6$) (Rb/Li) .  $ \delta R / R 0$ (%) . 

H_{2}  148.8/82.8  0.004 
He  44.7/22.5  0.006 
Ne  88.2/43.8  0.008 
N_{2}  349/184  0.008 
Ar  337/174  0.008 
Kr  499/260  0.01 
Xe  782/411  0.012 
Bk. species .  C_{6} ( $ E h a 0 6$) (Rb/Li) .  $ \delta R / R 0$ (%) . 

H_{2}  148.8/82.8  0.004 
He  44.7/22.5  0.006 
Ne  88.2/43.8  0.008 
N_{2}  349/184  0.008 
Ar  337/174  0.008 
Kr  499/260  0.01 
Xe  782/411  0.012 
5. Contributions of other background gases
Throughout this work, we have assumed that the composition and density of the residual background gases other than the test gas remains static throughout the measurement. In this section, we relax this assumption and estimate the contribution to the uncertainty in R_{X} due to changes in the background gas density. These changes can either be random shottoshot fluctuations, which contribute only to the statistical uncertainty, or changes where the density of the background gas is correlated to the test gas density, which contribute to the systematic uncertainty in R.
In this section, we estimate this systematic contribution. Correlated changes in the density of residual background gases and the test gas could occur due to contamination of the test gas before it is leaked in or due to the changing gas load affecting the behavior of the ion pumps, NEGs or the turbo pump. We begin by deriving an expression for the systematic error in R in terms of the ratio of the R values for the test gas to the residual background gases, and the estimated correlated change in density at the location of the atoms. Then, we explain how we use the RGA readings to estimate the density of the background and test gases at the location of the MT/MOT.
This analysis is limited to gases for which we know R_{i}. Fortunately, the only mass numbers (apart from those corresponding to the test gas X) that show up in significant quantities on the RGA are 2 (H_{2}), 40 (Ar), and 28 (N_{2}), along with those corresponding to the test gas species we introduced.
Test gas .  $ \Delta P X$ (Torr) .  $ \delta R ( P ) / R$ (%) .  $ \delta R ( P ) / R$ (w/o He) (%) . 

H_{2}  $ 3.4 \xd7 10 \u2212 9$  −0.067  $ 8.8 \xd7 10 \u2212 4$ 
He  $ 7.2 \xd7 10 \u2212 9$  0.002  
Ne  $ 1.0 \xd7 10 \u2212 8$  −0.023  0.335 
N_{2}  $ 4.0 \xd7 10 \u2212 9$  −0.018  0.021 
Ar  $ 8.1 \xd7 10 \u2212 9$  −0.009  0.120 
Kr  $ 9.4 \xd7 10 \u2212 9$  −0.012  0.062 
Xe  $ 9.5 \xd7 10 \u2212 9$  −0.008  0.135 
Test gas .  $ \Delta P X$ (Torr) .  $ \delta R ( P ) / R$ (%) .  $ \delta R ( P ) / R$ (w/o He) (%) . 

H_{2}  $ 3.4 \xd7 10 \u2212 9$  −0.067  $ 8.8 \xd7 10 \u2212 4$ 
He  $ 7.2 \xd7 10 \u2212 9$  0.002  
Ne  $ 1.0 \xd7 10 \u2212 8$  −0.023  0.335 
N_{2}  $ 4.0 \xd7 10 \u2212 9$  −0.018  0.021 
Ar  $ 8.1 \xd7 10 \u2212 9$  −0.009  0.120 
Kr  $ 9.4 \xd7 10 \u2212 9$  −0.012  0.062 
Xe  $ 9.5 \xd7 10 \u2212 9$  −0.008  0.135 
Since Helium has by far the lowest R value, it provides a relatively large contribution to $ \delta R P$. We include a column showing the values of $ \delta R P$ with the contribution of Helium removed. From Table VII, it is clear that the contribution of these pressure fluctuations is well below the statistical error, $ \u2264 0.8 %$.
APPENDIX B: A DISCUSSION OF THE ORIGINS OF THE OBSERVED 3.4% SYSTEMATIC DISCREPANCIES BETWEEN $ R 6 , 87$ AND $ R 7 , 87 KT$
In this work, we have observed that the $ R 6 , 87 = \u27e8 \sigma tot \u2009 v \u27e9 6 Li + X / \u27e8 \sigma tot \u2009 v \u27e9 87 Rb + X$ value is remarkably constant for heavier collision partners, X = N_{2}, Ar, Kr, and Xe. Indeed, the corresponding $ R 7 , 87 KT = \u27e8 \sigma tot \u2009 v \u27e9 7 Li + X / \u27e8 \sigma tot \u2009 v \u27e9 87 Rb + X$ values based on the total collision rate coefficients computed by Kłos and Tiesinga (KT) also demonstrate this same behavior. However, the average of the $ R 6 , 87$ values for these four heavy collision partners, $ \u27e8 R 6 , 87 \u27e9 = 0.7965 ( 9 )$, is systematically 3.4(7)% larger that the corresponding average value, $ \u27e8 R 7 , 87 KT \u27e9 = 0.770 ( 5 )$. The $ R 6 , 87$ reported here have been corrected for finite trap depth effects (an effect that is less than 0.5% in all cases), and the $ R 7 , 87 KT$ have been matched to our experimental ambient temperature following the KT prescription.^{22} The remaining systematic measurement errors (type B), as shown in Appendix A, have been estimated to be less than 0.15% in this work and the estimated statistical errors on our individual $ R 6 , 87$ are less than 0.8%. Thus, we face the conclusion that the systematic 3.4% discrepancy is significant and must be due to some other, unaccounted for source.
This systematic discrepancy could be the result of systematic errors in the C_{6} coefficients reported in the literature. For example, each of the Li + X (X = N_{2}, Ar, Kr, and Xe) C_{6} coefficients could be systematically underestimated by approximately $ 8 % \u2013 10$% or the Rb–X C_{6} coefficients could be systematically overestimated by $ 8 % \u2013 10$%. This explanation seems unlikely. An intriguing possibility is that the discrepancies arise from systematic errors in the potential energy surfaces (PES) used in the quantum scattering computations (QSC) which are the basis of the $ R 7 , 87 KT$ estimates. Accurate determination of the interaction potential (especially at short range) is difficult, especially when the number of electrons in the collision complex is large, as it involves solving a computationally complex manybody quantum problem. In many cases, exact diagonalization is not possible and approximations are employed which may not accurately reflect the true potential.
To illustrate this point, it is helpful to examine the case of H_{2} collisions with the sensor atoms. Recent work investigated the variation of the ab initio Li + H_{2} potentials that resulted from various levels of approximation, allowing the authors to converge on a PES that had a small underlying uncertainty.^{18} For this case, the corresponding quantum scattering computation value, $ R 7 , 87 KT = 0.82 ( 3 )$, agrees with our value here $ R 6 , 87 = 0.824 ( 5 )$ and with our previous measurement, $ R 6 , 87 SExp = 0.83 ( 5 )$.
However, the next simplest collision partner to model, He, yielded $ R 6 , 87 = 0.729 ( 6 )$ compared to $ R 7 , 87 KT = 0.69 ( 2 )$, a $ 2 \sigma $ discrepancy. The $ R 6 , 87$ experimental result does, however, agree with the value deduced from the measurements by Barker et al., $ R 7 , 87 BExp = 0.73 ( 2 )$.
In their simplest form the potential energy functions, describing the collision dynamics consist of a short range core region valid for interspecies separations, $ r < r c$, smoothly connected to a longrange van der Waals attraction, $ \u2212 C 6 / r 6 \u2212 C 8 / r 8 \u2212 C 10 / r 10$ for $ r > r c$. Ab initio calculations of the van der Waals coefficients, C_{n}, have been performed for a wide range of collision partners.^{22,31,37,38} Of these sources, only Derevianko^{38} and KT^{22} reported an explicit uncertainty estimates for the C_{6} dispersion coefficients. The Derevianko C_{6} coefficient uncertainties are less than 0.6% for Li + (Ar, Kr, Xe) and for Rb + (Ar, Kr, Xe) and KT reported uncertainties in C_{6} for Li + N_{2} and Rb + N_{2} of less than 0.2%. Dervianko et al. do not report values for the C_{8} and C_{10} dispersion coefficients and KT take these from Jiang.^{31} The core portion of the potential is a challenge to model and, like the dispersion coefficients, estimating its accuracy is difficult. Indeed, in the absence of direct experimental data to compare against, the accuracy of model core potentials is very difficult to establish. Instead of the potential accuracy, the variations between the results of different computational models are usually reported. Thus, it is possible that the observed systematic difference between the computed $ R 7 , 87 KT$ and the measured $ R 6 , 87$ values is due to a systematic error in the core portion of the PES used in the theoretical predictions, where the effects of this error are different for the Rb + X collisions and the Li + X collisions. We hypothesize that these differences may arise from the effects of glory scattering on the $ \u27e8 \sigma tot \u2009 v \u27e9$

The number of oscillations per unit velocity decreases with a decreasing reduced mass of collision partners.

The amplitude of the oscillations increases with a decreasing reduced mass of collision partners.
These features are illustrated in Fig. 6, which shows scaled plots of quantum scattering computations (QSC) of $ \sigma ( v )$ for Li + Xe and Rb + Xe collisions, based on the interaction potentials published by Kłos et al.^{22} We observe the larger oscillation amplitude for the Li + Xe cross sections compared to those for Rb + Xe, and the lower oscillation rate with speed for Li + Xe.
To underscore the difference in the behavior between Li + Xe and Rb + Xe collisions, the products, $ ( \rho X ( v , T X , m X ) \u2009 \xb7 \u2009 \sigma ( v ) \u2009 \xb7 \u2009 v )$, as a function of v for these species are shown in Fig. 7. One observes that the Rb + Xe plot approximates the ideal case with many small amplitude oscillation over the velocity range selected by the MB function. The Li + Xe plot, however, shows only a few largeamplitude oscillations over the same velocity range. Thus, one expects that the details of the core portion of the potential will not contribute significantly to $ \u27e8 \sigma tot \u2009 v \u27e9 Rb + Xe$. By contrast, the details of the core will likely persist for the Li + Xe collisions. Thus, one might expect that changing the details of the core used in the QSC used to compute $ \sigma ( v )$ will lead to changes in the value of $ \u27e8 \sigma tot \u2009 v \u27e9$.
Thus, it is plausible that a large portion of the discrepancies between the measured values of $ R 6 , 87$ presented here and the values derived from the theoretical QSC $ \u27e8 \sigma tot \u2009 v \u27e9$ of Kłos and Tiesinga^{22} used in $ R 7 , 87 KT$ arise from details of the core of the interaction potentials between Li and its collision partners used in the theoretical estimates. While it is beyond the scope of this paper to investigate the detailed properties of the potentials, two tests were carried out.
Species .  $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ .  $ \u27e8 \sigma tot \u2009 v \u27e9 SC$ .  $ \u27e8 \sigma tot \u2009 v \u27e9 SC / \u27e8 \sigma tot \u2009 v \u27e9 KT$ . 

Li + N_{2}  2.65  2.70  1.019 
Li + Ar  2.34  2.37  1.015 
Li + Kr  2.15  2.20  1.025 
Li + Xe  2.25  2.30  1.025 
Rb + N_{2}  3.46  3.49  1.007 
Rb + Ar  3.045  3.035  0.997 
Rb + Kr  2.80  2.79  0.999 
Rb + Xe  2.88  2.89  1.003 
Species .  $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ .  $ \u27e8 \sigma tot \u2009 v \u27e9 SC$ .  $ \u27e8 \sigma tot \u2009 v \u27e9 SC / \u27e8 \sigma tot \u2009 v \u27e9 KT$ . 

Li + N_{2}  2.65  2.70  1.019 
Li + Ar  2.34  2.37  1.015 
Li + Kr  2.15  2.20  1.025 
Li + Xe  2.25  2.30  1.025 
Rb + N_{2}  3.46  3.49  1.007 
Rb + Ar  3.045  3.035  0.997 
Rb + Kr  2.80  2.79  0.999 
Rb + Xe  2.88  2.89  1.003 
The $ \u27e8 \sigma tot \u2009 v \u27e9$ values have statistical uncertainties (type A) associated with them: for the Li + X $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ values these are less than 1% for the collision partners listed in Table VIII. For Rb + X (X = Ar, Kr, or Xe) the statistical uncertainties on $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ are less than 0.4% and 1.7% for Rb + N_{2}. We approximate the corresponding uncertainty in $ \u27e8 \sigma tot \u2009 v \u27e9 SC$ to be less than 0.25% based on the statistical uncertainty equal to 0.6% on the C_{6} coefficients reported by Derevianko^{38} for Li + X and Rb + X. This leads to the estimate that the statistical uncertainties in the Li + X $ \u27e8 \sigma tot \u2009 v \u27e9 SC : \u27e8 \sigma tot \u2009 v \u27e9 KT$ ratio are $ \u2264 1$%. Thus, if the effects of glory scattering we removed by MB averaging for the ratio values should be distributed around 1.0, as seen for the Rb + X results. By contrast, the Li + X ratios $ \u27e8 \sigma tot \u2009 v \u27e9 SC$: $ \u27e8 \sigma tot \u2009 v \u27e9 KT$ show a systematic 2% offset, supporting the hypothesis that some information about the glory scattering is contained in the Li + X $ \u27e8 \sigma tot \u2009 v \u27e9$ values.
Next, the KT description of the interaction potential for Li + Xe^{22} stitches together the shortrange or core portion of the potential with the longrange van der Waals potential at an interspecies location labeled r = r_{c}. In KT's work,^{22}^{,} $ r c = 18.5 \u2009 a 0$ for Li + Xe. Three different r_{c} values were selected, and the QSC for $ \u27e8 \sigma tot \u2009 v \u27e9$ were carried out to observe the shift in its value. The results are summarized in Table IX. We observe variations of less than of 1% even for very large changes in this parameter, indicating that shifting this parameter is not effective at changing the glory contributions to $ \u27e8 \sigma tot \u2009 v \u27e9$. We believe these tests rule out the longrange portion of the potentials as being responsible for the discrepancies observed between our reported $ R 6 , 87$ and $ R 7 , 87 KT$. We hypothesize that the core potentials described by Kłos et al.^{22} may need to be modified by adjusting the width and location of the potential minimum, and/or by adjusting the shape of the very short range repulsion curve to increase the computed $ \u27e8 \sigma tot \u2009 v \u27e9$ values. If this hypothesis is correct, then this provides a rare and exciting opportunity to investigate the shape of the shortrange portion of the interaction potential experimentally.