Precise Measurements of the Masses of Cs, Rb and Na – A New Route to the Fine Structure Constant

Hyperfine Interactions 132: 177–187, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

177

Precise Measurements of the Masses of Cs, Rb andNa – A New Route to the Fine Structure Constant

SIMON RAINVILLE�, MICHAEL P. BRADLEY, JAMES V. PORTO��,JAMES K. THOMPSON and DAVID E. PRITCHARDResearch Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA; e-mail: [email protected]

Abstract. We report new values for the atomic masses of the alkali 133Cs, 87Rb, 85Rb, and 23Nawith uncertainties � 0.2 ppb. These results, obtained using Penning trap single ion mass spec-trometry, are typically two orders of magnitude more accurate than previously measured values.Combined with values of h/matom from atom interferometry measurements and accurate wavelengthmeasurements for different atoms, these values will lead to new ppb-level determinations of the molarPlanck constant NAh and the fine structure constant α. This route to α is based on simple physics.It can potentially achieve the several ppb level of accuracy needed to test the QED determinationof α extracted from measurements of the electron g factor. We also demonstrate an electronic cool-ing technique that cools our detector and ion below the 4 K ambient temperature. This techniqueimproves by about a factor of three our ability to measure the ion’s axial motion.

Key words: single ion mass spectrometry, ppb, cesium, rubidium, sodium, electronic cooling, feed-back.

1. The fine structure constant

If QED correctly predicts the g-2 anomaly of the electron, the fine structure con-stant, α, is known to 3.8 ppb [1, 2].† In order to test QED at this accuracy, andto remove theoretical uncertainties highlighted by the recent readjustment of thetheoretical value of g-2 [3], other measurements of α at this level of accuracy areneeded. Unfortunately, the next most precise measurements are far from this accu-racy: α has been measured to 56 ppb using the AC Josephson effect [4, 5], to 37 ppbusing a beam of neutrons [6], and to 24 ppb from the Quantum Hall effect [7] (seeFigure 1). There is obviously a great need for a QED-independent determinationof α at the ppb level: not only could it probe QED an order of magnitude moreaccurately, but it could possibly suggest the source of the discrepancies in theseother measurements.

� Corresponding author.�� Currently at NIST, Gaithersburg, MD 20899-8424, USA.† Groups at the Universities of Washington and Harvard plan to reduce this error to about 1 ppb.

178 S. RAINVILLE ET AL.

Figure 1. Current measurements of the fine structure constant α with uncertainty below 100ppb are plotted with respect to the 1998 CODATA recommended value [12]. The measurementof the g-2 factor of the electron combined with QED calculations yields a value of α with anuncertainty of (3.8 ppb) [1, 2]. The relative uncertainties of the other measurements are: 24 ppbfrom the Quantum Hall effect [7], 37 ppb from neutron beam interferometry [6], and 56 ppbfrom the AC Josephson effect and the magnetic moment of the proton [4, 5]. The methoddiscussed here (this work) currently yields a preliminary value at 29 ppb.

A route to α that appears likely to yield a value at the ppb level is opened by therelationship (in SI units)

α2 = 2R∞c

103

Mp

mp

me(NAh), (1)

where mx and Mx represent the mass of particle x in SI and atomic units respec-tively, and NAh is known as the molar Planck constant. The Rydberg constant, R∞has been measured to an accuracy of about than 0.008 ppb [8], the ratio of the massof the proton to the electron mp/me is known to 2 ppb [9], and we have determinedthe mass of the proton Mp (in atomic units) to 0.5 ppb [10] (Van Dyck et al. haverecently reported a value of Mp accurate to 0.14 ppb which agrees with ours [11]).The speed of light c is a defined constant. Thus a measurement of NAh, the “molarPlanck constant”, at the ppb level can determine α to about 1 ppb.

The molar Planck constant can be precisely determined from the quantum rela-tionship between de Broglie wavelength and momentum:

λ = h

mv�⇒ λv = h

m= 103NAh

M. (2)

Note that to determine NAh we need to measure the particle’s atomic mass M aswell as the wavelength of the light λ and the particle’s velocity v. This simple

PRECISE MEASUREMENTS OF CS, RB AND NA 179

physics underlies the neutron-based measurement of α [13], where our value ofthe neutron’s atomic mass [10] leads to the value for α mentioned in Figure 1.Chu’s group at Stanford University has measured the recoil velocity v for Cs atomsabsorbing photons of laser light at the D1 line to 10−6 using an atom interferom-eter [14]. They now project an accuracy of 10 ppb using a vertically configuredapparatus like the one used to reach ppb measurements of the local gravitationalacceleration constant g. Hänsch’s group at the Max-Planck-Institut in Garching hasmeasured the wavelength of the cesium D1 line (λ) with precision of 0.12 × 10−9

[15]. Combining these results with our sub-ppb measurement of the atomic massof Cs should give NAh to about 10 ppb. From Equation (1), this will lead to a newdetermination of α to about 5 ppb.

The simplicity of the physics involved in this route to α recommends it as thepreferred check of QED and the other α measurements. The most complicatedphysics is in the Rydberg constant, where the theory is more than adequate at theppb level. By comparison, the quantized Hall effect may involve unknown solidstate or sample geometry corrections, and ongoing programs to determine α fromthe fine structure of an atom involve new atomic energy level calculations withrequired ppt certainty, a much more uncertain proposition.

The possibility of redundancy in the experimental determination of NAh wouldgreatly enhance the confidence in determinations of α from Equation (1). Themass ratio mp/me would be the only quantity without more than a single directmeasurement at the ppb level (a recent value of me/m12C extracted from theoryand boundstate electron g factor measurements in hydrogenic 12C has confirmedthe value to about 2 ppb [16]). This is not a trivial point since it would take aconsiderable weight of evidence to believe that disagreement between the QEDand NAh determinations of α signifies some error in QED.

2. Experimental setup

Our experimental apparatus and procedure for measuring ion mass ratios have beendescribed earlier in the literature [17]. We will briefly outline here the generalfeatures of this experimental setup.

We trap a single ion in an orthogonally compensated Penning trap (of charac-teristic size d = 0.549 cm) [18, 19] placed in a highly uniform magnetic fieldof 8.5 T. The harmonic axial oscillation of the ion in the trap is detected with asuperconducting resonant circuit (niobium coil) having a stable Q of about 50000.The heart of our recently improved detector is a DC SQUID which has a technicalnoise floor 10 times smaller than the RF SQUID it replaced. The trapping voltage isadjusted to match the ion’s axial frequency ωz, with the resonance frequency of thedetection circuit (∼160 kHz). Typical frequencies for the other two normal modesof motion of an Ar+ ion in our trap are 3 MHz for the trapped cyclotron frequencyω′

c, and 4 kHz for the magnetron motion ωm. The cyclotron and magnetron modesare observed and cooled indirectly by coupling them to the axial mode with a tilted


quadrupole RF field applied on the split guard rings of our trap. With the properamplitude–time product, such a coupling pulse (“π -pulse”) can phase-coherentlytransfer all the energy of the cyclotron motion into the axial mode, in a mannerqualitatively and formally equivalent to the Rabi two-state problem [20].

The absence of direct damping of the radial modes avoids the possibility offrequency shifts due to such coupling and means that the cyclotron resonance haspractically zero linewidth. In addition, this procedure allows us to work with nearlyideal fields (pure quadrupole electric field and uniform magnetic field) which helpsto keep the calculated systematic errors below a few parts in 1011.

The trapped cyclotron frequency ω′c is obtained by directly exciting the ion’s

cyclotron mode, allowing it to evolve “in the dark” for a delay time T , and applyinga “π -pulse”. The axial ring down is then observed to extract the accumulated phase.Since phases are only defined between 0 and 2π , a series of measurements is madewith different delay times T to determine the proper phase unwrapping. With aphase error of about ±12 degrees, a measurement of one minute gives a precisionof ∼ 1–2 × 10−10. From this, we can extract the free space cyclotron frequencyωc = qB/m (where q is the ion charge, B is the magnetic field strength, and m

is the mass) using the invariance relationship (valid even if the trap is tilted withrespect to the B field or not perfectly cylindrical):

ωc = qB/mc =√(ω′

c)2 + (ωz)2 + (ωm)2. (3)

We don’t directly measure the magnetron frequency ωm when taking data, but weuse the relation

ωm ≈ ω2z

2ωc

(1 + 9

4sin(θm)

), (4)

where θm (typically, < 0.003 rad) is the measured angle between the magneticfield and the trap axis obtained by actually measuring the magnetron frequencyonce. The precision of a single ion cyclotron frequency measurement is limited bythe ∼ 2.5 × 10−10 short term fluctuations of the magnetic field.

Finally, a mass ratio is determined by alternately measuring the cyclotron fre-quencies of two ions during one night (when the magnetic field fluctuations arereduced). A typical set of data is shown in Figure 2. The final precision on the massratio (typically, ∼ 1 × 10−10) is currently limited mainly by the drift (few ppb perhour) of our magnetic field during the time it takes to make and isolate a new singleion. In this setup, the ions are produced in the trap by ionizing neutral gas with anelectron beam. Obviously, this method is not selective and unwanted ions are alsoproduced and trapped in the process. A crucial point for us is to be able to isolatethe ion of interest as quickly as possible. The case of Cs3+ was particularly difficultsince the making of a single Cs3+ would produce about 100 and 10 of the singlyand doubly charged states respectively. Furthermore, the hydrocarbons C5H+

6 andC3H+

x used as comparison species break apart into many unwanted fragments underthe electron-beam bombardment. The overall effect of these factors was to increase


Figure 2. Typical night of data. The solid line is a second order polynomial fit to the field drift.The 360◦ bar shows the magnitude in Hz of a 360◦ error in phase unwrapping. Also shown isa bar representing the magnitude of a ppb change in frequency.

the time to make and isolate a single ion to 20–30 minutes from the 5–10 minutesrequired for previous measurements.

3. Results of the alkali measurements

To determine the masses of the alkali atoms 133Cs, 87,85Rb, and 23Na, we measuredthe free-space cyclotron frequency ratios r ≡ ωc2/ωc1 listed in Table I. The ref-erence ions were selected because of the similar mass to charge ratios (aiding inthe reduction of systematic errors) and because we have previously measured theatomic masses of each of the consituent atoms.

A cyclotron frequency ratio r of two different ions was determined by a runmeasuring a cluster of ωc values for an ion of type A, then for type B, etc. Ina typical 4-hour run period (1:30–5:30 am when the nearby electrically-poweredsubway was not running), we recorded about 5 alternations of ion type (Figure 2).The measured free-space cyclotron frequencies exhibited a common slow drift. Wefit a common polynomial �(t) plus a frequency difference to the data. From thiswe obtained the frequency ratio rn and the uncertainty σn for a single night. Theaverage order of �(t) was 3 and was chosen using the F-test criterion [21] as aguide.

The distribution of residuals from the polynomial fits had a Gaussian centerwith a standard deviation σresid = 0.28 ppb and a background (≈ 2% of the points)of non-Gaussian outliers, as in our earlier measurements [10]. As before we choseto handle the non-Gaussian outliers using a robust statistical method to smoothlydeweight them [23].


Table I. Measured ion cyclotron frequency ratios, corrected for systematics

A/B ωc/(2π) (MHz) Nights ωc[A]/ωc[B]

133Cs+++/CO+2 2.968 5 0.992 957 580 983 (135)

133Cs++/C5H+6 1.977 4 0.993 893 716 487 (427)

87Rb++/C3H+8 2.994 2 1.013 992 022 591 (266)

87Rb++/C3H+7 3.028 3 0.990 799 127 824 (174)

85Rb++/C3H+7 3.064 2 1.014 106 122 230 (164)

85Rb++/C3H+6 3.100 2 0.990 367 650 976 (285)

23Na+/C+2 5.578 2 1.043 943 669 690 (076)

23Na++/C+ 11.155 2 1.043 944 716 614 (098)

Figure 3. Example of Variation of Mass Ratio from Night to Night. A measurement of theneutral mass of 133Cs is extracted from each night’s run of cyclotron frequency ratio measure-ments and plotted in ppb relative to our final published value of the neutral mass of 133Cs.The open and closed circles are from frequency ratio measurements of Cs+++/CO+

2 andCs++/C5H+

6 , respectively. The error bars on each night’s measurement are extracted from thelow order polynomial fit to both ion’s cyclotron frequencies and reflects the distribution of thecyclotron frequency measurements during that night. The shaded region represents the onesigma confidence interval arrived at in the final analysis.

As shown in Table I and Figure 3, we measured each frequency ratio on morethan a single night. For ratios involving Cs and Rb the measured ion mass ra-tios were distributed from night to night with a scatter larger than the uncertaintypredicted from the statistical scatter within a single night (χ2

ν ≈ 5). By contrastχ2ν ≈ 0.8 for ratios involving Na. None of the earlier data taken using this apparatus

[10] exhibited these excess night-to-night variations. A search for the source ofthese fluctations is discussed elsewhere [24] and was unsuccesful. To account for


Table II. Measured neutral alkali masses

Species MIT mass (u) ppb 1995 mass (u) [22] ppb deltaσ1995

133Cs 132.905 451 931 (27) 0.20 132.905 446 800 (3200) 24.0 1.6

87Rb 86.909 180 520 (15) 0.17 86.909 183 5 00 (2700) 31.0 −1.1

85Rb 84.911 789 732 (14) 0.16 84.911 789 300 (2500) 29.0 0.2

23Na 22.989 769 280 7 (28) 0.12 22.989 769 670 0 (2300) 9.8 −1.7

this excess scatter, the uncertainties in the weighted average of the ion mass ratiosinvolving Cs++/C5H+

6 and Cs+++/CO+2 were increased by factors of 2.6 and 2.2,

respectively, so that χ2ν = 1. Since the Rb measurements all had similar m/q, we

assumed that the night-to-night fluctuations involving the Rb ratios were drawnfrom a common statistical distribution. Therefore, we increased the uncertaintiesfor the Rb ion ratios by a factor of 2.2 so that the overall Rb χ2

ν was reduced to 1.For Na, χ2

ν ≈ 0.8 so the uncertainties were not adjusted.After correcting for molecular binding and electron ionization energies [25], we

obtained a set of neutral mass difference equations. We added to this the set of massdifference equations used to determine the atomic masses in [10]. A global leastsquare fit to this overdetermined set of linear equations gave the neutral massesof the alkali metals (see Table II) with uncertainties σod as well as the previouslypublished neutral masses with χ2

ν = 0.83. The previously published masses wereessentially unchanged [10]. Uncertainties in M[16O] and M[H] (the only atomsother than 12C in the ratios of Table I) contributed less than 0.1 ppb uncertainty tothe alkali masses.

The use of two distinct reference ions gave a check on systematics by pro-viding two independent values for each neutral mass. For Rb and Cs χ2

ν is lessthan 1. However, because of the larger uncertainty on M[Cs] from Cs++/C5H+

6we quote a final uncertainty of 0.20 ppb (cf. σod(Cs) = 0.16 ppb). For 87,85Rb wequote σod(87,85Rb) as the final uncertainties. For the neutral masses from Na++/C+and Na+/C+

2 , the statistical uncertainties are 0.09 and 0.07 ppb, respectively. The0.2 ppb disagreement of the two values may be evidence for a systematic at the0.1 ppb level. To reflect this we assigned M[23Na] a 0.12 ppb uncertainty (cf.σod(Na) = 0.06 ppb) which spans both independent measurements.

Table II quotes the final values for M[133Cs], M[87Rb], M[85Rb] and M[23Na]obtained from the global least square fit to the overdetermined set of mass dif-ference equations with uncertainties from the above discussion. Also included inTable II are the alkali masses from the 1995 mass evaluation [22]. Our valuesdiffer from the 1995 values by typically 1.5σ1995, which suggests that the uncer-tainties on the masses from the 1995 evaluation were slightly underestimated.Our value for M[133Cs] lies within the uncertainty of the recent measurement


of M[133Cs] reported by the SMILETRAP collaboration [26] to 0.6 ppb. UsingR∞ [8], mp/me [9], the preliminary value of the photon recoil shift [27], fD1

for the photon recoil transition [28], and our values for M[133Cs] and Mp weobtain α−1 = 137.035 992 2 (40). This preliminary value is shown in Figure 1with an uncertainty of 29 ppb. The method discussed here holds promise for manyindependent values of α at a precision of few ppb.

4. Electronic cooling

We have recently developed techniques to cool the detector and ion below the 4 Kambient temperature of the coupling coil and trap environment. This is done withelectronic cooling [29]. Not only does this technique greatly improve our signal-to-noise, but it also reduces the amplitude of the thermal motion of the ion, animportant source of error in our measurements. These significant improvementswill be crucial in our efforts to improve our accuracy beyond 10−10. In particular,we plan to make simultaneous measurements of the cyclotron frequencies of twodifferent ions to alleviate the scatter due to magnetic field fluctuations between themeasurements which are now the limit on our precision. Our first approach to dothis will be to put two ions in the same trap at the same time.

The essence of electronic cooling is to measure the thermal noise, phase shift thesignal and then apply it to the coupling coil in such a way that it cancels the thermalexcitation in the coil. The key is that our DC SQUID has technical noise muchlower than 4 K and can measure the current in the coil well in a time shorter thanthe thermalization time of the coil (Q0/ω ∼ 30 ms). This feedback also decreasesthe apparent quality factor Q of the coil. Figure 4 shows the thermal noise of thecoil at different gain settings. Analyzing these data, we find that the thermal energyin the coil, corresponding to the area under the peak, is reduced below 4 K bythe factor Q/Q0, as expected from the detailed solution of the circuit (assuming aparallel LRC coupling coil where the resistor R = Q0ω0L has the usual Johnsonnoise current).

Another effect of this feedback is to effectively reduce the impedance presentedto the ion by the detector, thereby increasing the damping time of the ion. Thisreduces the bandwidth of our signal, increasing our signal-to-noise ratio (the John-son noise is a constant current/

√Hz). This translates directly into a better ability

to estimate the parameters of the axial oscillation of the ion. With this technique,we can now measure the phase of the cyclotron motion with an uncertainty aslow as 5 degrees, which is more than a factor of 2 improvement. Our ability todetermine the amplitude of the ion signal has also improved, again by more thana factor of 2, and we can measure the frequency of the axial motion with 4 timesbetter precision. The better phase noise allows us to obtain the same precision on acyclotron measurement in a shorter time. This will be very important in the futuresince we would have to acquire data for 10 minutes to reach a precision of 10−11

with the previous phase noise (∼ 12 degrees), or 100 minutes for 10−12 ! We can


Figure 4. Thermal profile of the detector coil as a function of the quality factor Q adjustedwith the gain of the feedback. The thermal energy in the coil (area under the peak) is propor-tional to Q/Q0, where Q0 is the Q of the detector coil without feedback. This shows that thefeedback does indeed reduce the thermal fluctuations in the coil.

also use the improved signal-to-noise to reduce the cyclotron amplitude we use,which in turn reduces the frequency shifts due to relativity and field imperfections.Finally, this technique gives us the ability to arbitrarily select the damping time ofthe ion by changing the gain of the feedback.

The improved detector opens some exciting possibilities for our experiment.For example, we can now probe and characterize the electrostatic anharmonicitiesof our trapping potential much more effectively. It also opens the door for us tovery high precision at small mass-to-charge ratio, (e.g., 6,7Li, 3He, 3H) where weused to suffer from excessively short ion damping times.

Before every measurement, we cool the ion’s motion by coupling it to our de-tection circuit until it comes into equilibrium with the detector. (Only the axialmotion is coupled to the detector, but we cool the two radial modes using themode coupling field mentionned in Section 2.) This remaining “4K” motion ofthe ion adds vectorially to the displacement from our cyclotron drive pulses andhence prevents us from establishing an exactly reproducible amplitude and phase ofmotion with each excitation pulse. This effectively adds random noise to the phasewe measure. Since it is the same Johnson noise that drives the ion’s thermal motionand is added to the ion image current to form our detected signal, these two sourcesof noise both contribute to our measurement error (phase noise). Moreover, thethermal cyclotron amplitude fluctuations cause relativistic mass variations and alsocombine with field imperfections to introduce random fluctuations of the cyclotronfrequency which can be of the order of few parts in 10−11. After magnetic fieldfluctuations, this is the dominant source of noise in our measurements, and willbe the main obstacle to higher precision in our simultaneous measurements of two


different ions. With the electronic cooling technique described above, the ion’smotion should now come into equilibrium with the colder detector thereby greatlyreducing these problems, and allowing a precision better that 10−11 in a reasonableamount of time.

The first practical result of this improved detector was a new calibration of ourcyclotron amplitude. Knowing the absolute ion cyclotron radius is important to beable to calculate and correct (if necessary) the cyclotron frequency we measure forfrequency shifts due to relativity and trap imperfections. Also knowing the ion-ion separation will be crucial when we make measurements on two ions in thesame trap. Before this work, there was a factor of two uncertainty in our amplitudecalibration, despite our having worked hard on three different methods to determineit. During the month of July 2000, we accurately measured the relativistic cyclotronfrequency shift (a natural law) versus cyclotron radius for Ne++ and Ne+++ ionsusing the feedback technique described above to maintain adequate S/N over abroad range of cyclotron radii. The measurements on each ion separately led totwo independent amplitude calibrations with an uncertainty of 1.6% and 1.8%,respectively. The agreement between the two calibrations was about 3%. Thesetwo measurements were done at very different cyclotron frequencies so that anysystematic error would affect them differently. Thus we feel confident that we nowknow the absolute amplitude of motion of our ions to about 3%, a tremendousimprovement.

Acknowledgements

This work is supported by the National Science Foundation and a NIST PrecisionMeasurements Grant.

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Precise Measurements of the Masses of Cs, Rb and Na – A New Route to the Fine Structure Constant