Zeitschrift for Physik A

Z. Physik A 287, 1-6 (1978)

9 by Springer-Verlag 1978

A Direct Determination of the Proton-Electron Mass Ratio G. G~irtner* and E. Klempt Institut ftir Physik der Johannes Gutenberg-Universit~it Mainz, West Germany Received March 18, 1978 The cyclotron resonance of protons and electrons in a magnetic field of 5.7 Tesla produced by a superconducting solenoid has been measured. Protons and electrons were alternately confined in an electrostatic quadrupole trap. The quotient of the cyclotron frequencies provides a first direct determination of the proton-electron mass ratio. The result of MJMe= 1836.1502 (53) agrees with other more precise but indirect determinations of this quantity.

I. Introduction

The electron-proton mass ratio represents one of the possible input data to a least squares adjustment of the fundamental constants [-1]. Therefore an experimental determination of this ratio is not only required for a meaningful interpretation of atomic spectra, but a precise knowledge of this ratio also provides a link between quite different fields of physics which are represented by their own fundamental constants like the fine structure constant, the value of 2e/h from Josephson effect, the proton gyromagnetic ratio, or the Faraday constant. Experimentally, the electron proton mass ratio has been determined from 2 independent types of experiments. In one group of experiments the ratio of the gfactor of the hydrogen atom gs(H) and the g-factor of protons in a spherical sample of pure water gp(H20 ) is determined [2]. Taking into account the calculable corrections for bound state effects and the known gfactor of free electrons [33, a value for the magnetic moment of the (shielded) proton in units of the Bohr magneton of/~p//~B=0.001520993136(21 ) 0.014 ppm is derived. This result of Phillips, Cooke and Kleppner is supported by an independent experiment by Lainbe [-4]. The second type of experiments yields a determination of the magnetic moment of the (shielded) proton in units of the nuclear magneton ffp/#N. This number can be obtained by a measurement of

the cyclotron frequency of free protons in a magnetic field which is mapped by detection of the spin resonance of protons in a spherical sample of water. The most recent results [5-9] have experimental errors varying from 0.43 ppm to 7.2 ppm. A value of #pitON =2.7927740(11) 0.38ppm is recommended in the least squares adjustement of the fundamental constants of 1973 [1]. Combining these 2 results an electron proton mass ratio of Mp/Me= 1836.1515(7) 0.38 ppm is derived. The cyclotron motion of the protons in those experiments which aim at a determination of #'p//~N is, however, easily influenced by stray electric fields which are difficult to control. Therefore former determinations yielded results which where incompatible with each other by several standard deviations 1-10]. Hence a direct determination of the electron-proton mass ratio in one experiment allows a valuable cross check of these quite different experiments and seems to be justified even if it is of lower precision. t

t

* Part of the dissertation E70

II. The Method

The experimental approach chosen here consists in the measurement of the cyclotron frequencies of electrons and protons in the same magnetic field; the ratio of these 2 frequencies yields, of course, directly the proton-electron mass ratio. For this purpose elec-

0340-2193/78/0287/0001/$01.20

2

G. Ggrtner and E, Klempt: A Direct Determination of the Proton-Electron Mass Ratio

ughin g pump

I/lJJIJ~heating coil

cryostat

quadrupole trap

~C

~

47cm

Varian-UHV-Volve IZ

50 ~

=,:s

I Iongetterpump

Fig. 1. The experimental set up

trons and protons are alternately stored in the magnetic field of a superconducting solenoid by use of an electrostatic quadrupole trap (see Fig. 1), a technique which has been described in detail by Dehmelt [11]. The trap consists of 2 opposite end caps of hyperbolic shape with distance 2R and a toroidal electrode. A voltage Vo across caps and torus leads to the quadrupole potential U--V o (r 2-2zz)/2R z where the z-axis is given by the magnetic field ~ = ( 0 , 0, Bz). In this field configuration charged particles of mass M exhibit 3 characteristic frequencies: the electrostatic quadrupole field provides the focussing forces for a harmonic oscillation parallel to the symmetry axis with a frequency 27r v~ = (2e Vo/MR2)1/2.

(1)

The magnetic field leads to the well known cyclotron frequency vc = e 9B / M c . But the electric field gradient of the trapping potential causes a shift of this cyclotron frequency and the cyclotron motion is now characterized by the frequency v B = Vc/2 + (v2/4 + v2/2) 1/2

(2)

which is nearly proportional to the trapping potential and coincides with the unperturbed cyclotron frequency for Vo--+0. This unperturbed cyclotron frequency may therefore be found by an extrapolation of the shifted cyclotron frequency to zero trapping potential. The ratio of the unperturbed cyclotron frequencies of protons and electrons then yields the proton-electron mass ratio. Finally the g x ~ drift leads to the magnetron oscillation with frequency v~B = Vc/2 - (v~/4 + v2/2) 1/2.

(3)

The sum of the frequencies vB and VEB amounts to the unperturbed cyclotron frequency as well. The electrons to be trapped are created by ionization of the rest gas by a pulse of primary electrons from a heated tungsten wire traversing the penning trap. To create protons hydrogen leaked into the apparatus through a heated palladium leak. By ionization of the rest gas inside the quadrupole trap protons and H~ ions were stored. The ratio of trapped protons and H~- ions was typically 3:1. At a rest gas pressure of 10 -9 Tort storage times of 1 rain for protons and of 1 h for electrons were achieved. The number of trapped particles was estimate to be a few thousands. Electrons or protons are detected through their harmonic oscillation parallel to the magnetic field. The capacity of the 2 end caps formed together with an external inductivity a resonant circuit which was externally excited. A linear change of the quadrupole voltage swept the harmonic oscillation of the trapped particles through this resonance and led to a detectable decrease of the amplitude of the resonant circuit. The filtered and amplified signal was averaged over several detection periods to improve the signal to noise ratio. A complete cycle was initiated by a pulse of electrons which ionized the rest gas and created the electrons or protons to be trapped, a storage time for cooling, an interaction time where the protons or electrons were exposed to rf fields in order to excite the cyclotron resonance, a linear sweep of the quadrupole voltage which allowed a detection of the trapped particles, and finally a short electric pulse which cleared the quadrupole trap. Excitation of the cyclotron frequency led to an increase of the temperature of the proton or electron cloud and thus to a detectable loss of trapped particles.

G. G~irtner and E. Klempt: A Direct Determination of the Proton-Electron Mass Ratio

III. The Cyclotron Resonance of Protons

80

M o s t data were taken at a magnetic field strength of 5.70 Tesla because at high magnetic fields the influence of the quadrupole trapping voltage and of space charge effects is less pronounced. In one set of data 29 p r o t o n cyclotron resonance curves at electrostatic trapping voltages from 20-45 V, at different RF field strengths and different space charge densities were measured. The analysis of these data will be discussed in some detail. Figure 2 shows a typical cyclotron resonance at a trapping voltage of 20 V. The solid line represents a Gaussian line fit with a line width of 180 H z at a frequency of 87,026,432 + 2 0 Hz. Nearly identical results were obtained by fitting a Lorentzian line shape. The result of the fit corresponds to a fractional half width at half m a x i m u m of 1 p p m and a fractional statistical error of 0.24 ppm. Yet several systematic effects have to be taken into a c c o u n t which will increase the final error by a factor of 10.

70

3

60 ppm

50 40 30 20 10 0

frequency 87d25.400

'

87626.200

87627.000 kHz'

Fig. 2. Proton signal amplitude as function of radio frequency

the electric quadrupole potential. This nonlinearity correction is estimated to 1 H z at a trapping voltage of 20 V and to 7 H z at 40 V.

a) Decrease in Time of Magnetic Field Strength The magnetic field strength of the superconducting solenoid decreased linearly in time, as the junction of the superconducting short plug has a small but finite resistance. This decrease of field strength was monitored over a time periode of 600 h - w h i c h included the 2 0 h during which these data were t a k e n - a n d was found to be 0.1165(5) ppm/h. All data were corrected for this decrease in field strength. After refilling the cryostat with liquid N 2 or He, changes of the magnetic field of the order of 0 . 4 p p m were observed. Therefore an overall error of 0 . 4 p p m is assigned to the stability of the magnetic field.

c) Line Shifts Due to the RF Field Strength With increasing RF field strength a power broadening of the cyclotron resonance but no systematic line shifts were observed. Therefore cyclotron resonances of different line widths were taken into a c c o u n t according to their statistical weight. These values are presented in Table 1.

b) Nonlinearity of the Shifted Cyclotron Frequency

d) Space Charge Effects

The square r o o t term of the shifted cyclotron frequency (1) leads to a nonlinear dependence of vB on

The mutual interaction of the trapped p r o t o n s m a y not only result in a line broading but also in a shift of

Table 1 Trapping

Uncorrected cyclotron frequency"

Nonlinearity of the shifted cyclotron frequency

Space charge

Inhomogeniety of the magnetic field

Corrected cyclotron frequency

20.0 V 25.0 V 30.0 V 35.0 V 40.0 V 45.0 V

87,026,426 (20) Hz 87,023,736 (50) Hz 87,021,016 (49) Hz 87,018,084 (133) Hz 87,015,023 (211) Hz 87,012,060 (159) Hz

1 (1) Hz 2 (2) Hz 3 (3) Hz 4 (4) Hz 6 (6) Hz 7 (7) Hz

85 111 137 164 192 220

- 13 - 37 - 57 - 74 - 88 -- 101

87,026,499 (88) Hz 87,023,812 (27) Hz 87,021,099 (156) Hz 87,018,178 (224) Hz 87,015,133 (228) Hz 87,012,189 (289) Hz

(85) Hz (111) Hz (137) Hz (164) Hz (192) Hz (220) Hz

(13) Hz (37) Hz (57) Hz (74) Hz (88) Hz (101) Hz

a The decrease of the magnetic field in time has been taken into account. Fit: Cyclotron frequency: 87,037,865(230) Hz.- Slope: -565 (8.2) Hz/Volt.-X2= 3.5 (4 degrees of freedom)

4

G. Ggrtner and E. Klempt: A Direct Determination of the Proton-Electron Mass Ratio

e) Inhomogeneity of the Magnetic Field 87040.

IM*p

A correction for the inhomogeneity of the magnetic field is necessary as electrons and protons may experience different magnetic field strengths, if the diffusion time from trap center towards the torus electrode is different. This correction was estimated by solving a diffusion equation for trapped protons and electrons, where the diffusion constant was fixed by the observed trapping time. Taking into account a magnetic field inhomogeneity of 5 ppm over the trap volume this procedure yielded inhomogeneity corrections for protons of -(0.46+_0.46)ppm at 20V trapping voltage up to -(1.55 + 1.55)ppm at 45 V in relation to the effective field seen by the electrons.

87030.. \.

\.

87020.

\

"\

87010. f ) Uncertainty in the Electric Field Strength Vo

0.0 10.0 2~0.0 310.0 iO.OVolt Fig. 3. Shifted proton cyclotron resonance as function of trapping potential

the line center. This is not in conflict with the conjecture of Dehmelt [7], that because of the symmetry of the center of mass motion with respect to particle exchange no space charge shifts of the cyclotron frequency should be observed, as this argument holds only for idealized circumstances. As has been shown by Liebes and Franken [12], the electric field of a charged cloud leads to a frequency shift of the cyclotron resonance which is linear in the space charge density, But as the space charge density is not homogeneous, the maximum possible line shift will be always smaller than the minimum line width which may be obtained at a particular space charge. Hence a correction of half of the minimum line width at a trapping voltage of 20 V is applied to all data at this trapping voltage and an error of the same amount is assigned to the correction. A model of the space charge effect based on a diffusion equation of trapped protons gave similar frequency shifts as estimated above. This model was used to calculate the space charge correction at higher trapping potentials where an increase of the proton density was observed. A direct determination of the frequency shift as function of the space charge density supported the estimate given above but suffered from the difficulty of a sufficient precise determination of the p and H~densities.

The trapping voltage was monitored by a digital voltmeter with a precision of + 30mV. Contact potentials should be smaller than 10 mV. Surface charges were excluded by a repeated heating of the apparatus to 400~ Inhomogeneity corrections of the quadrupole trap should contribute less than the equivalent of 20mV. Nevertheless we assign an error of +_100mV corresponding to 0.64ppm to our knowledge of the externally supplied DC electric field strength. Table 1 presents data and corrections which have been applied and the result of a linear least squares fit to the corrected data (see also Fig. 3). Of course, the final results depend on the corrections which were adopted. Therefore different corrections were applied to the raw data. But within reasonable limits of the corrections the analysis yielded always results well within the quoted error. The various errors and their contributions to the final error are summarized in Table 2.

Table 2. Errors in the determination of the cyclotron frequencies of electrons and protons

Statistical error Systematic error due to Space charge effects Magnetic field inhomogeneity Magnetic field instability Electrostatic trapping potential Relativistic mass correction Combined error for M / M e

Electrons and

Protons

0.3 p p m

0.7 p p m

0.6 p p m 0.4 p p m 1.0 p p m 1,3 p p m

2.2 p p m 0.9 p p m 0.4 p p m 0.7 p p m 2.6 p p m 2.9 p p m

G. G/irtner and E. Klempt: A Direct Determination of the Proton-Electron Mass Ratio

IV. The Electron Cyclotron Resonance At a magnetic field strength of 5.7 Tesla the cyclotron frequency of electrons amounts to 160 GHz. The detection of the cyclotron resonance therefore required a phase locked operation of an oscillator at this frequency. This was achieved in a 3 stage set up. The stability of the frequency was based on a frequency synthesizer (Rohde & Schwarz, XUC, stability better than 10-8), which gave an output frequency of 900 MHz. The 15 th harmonic of this frequency was phase locked to a carcinotron (Rohde u. Schwarz, SMC) operating at 13 G H z which was then used to phase lock a backwards wave oscillator (Siemens, R W O 80) operating at 80 GHz. Finally this frequency was doubled by use of a crystal diode. An upper limit of 10 -8 for the - 1 0 db bandwidth of the 160 G H z output frequency was established by careful checks of the different intermediate and output frequencies by means of a spectrum analyzer. On irradiation of the cyclotron resonance of the trapped electrons the increase of their mean energy was observed by the accompanied loss of electrons in the Boltzmann tail. The full width at half m a x i m u m of the electron cyclotron resonance varied typically from less than 1 p p m to 20 ppm, depending on the R F power seen by the electrons. With increasing power, trapped electrons may acquire an energy of a few eV, which then leads to a relativistic mass shift of the cyclotron resonance. The energy absorption of electrons initially at rest has been calculated by H a k kenberg and Weenink [13]. Using a modified ansatz allowing for an energy distribution of trapped electrons, the line shape of the cyclotron resonance has been calculated, details of the calculations may be found in [143. The fit of an electron cyclotron resonance using the calculated line shape gave statistical errors of typically less than 0.3 ppm. The presence of the quadrupole trapping voltage of 10V shifts the cyclotron resonance only by 0.03 ppm. Space charge effects and corrections due to the anharmonicity of the potential are therefore negligible. But an error of i p p m (corresponding to an uncertainty of 0.5 eV in the energy of the trapped electrons) has been assigned to the relativistic mass correction. Corrected for the decrease in time of the magnetic field the final answer for the electron cyclotron resonance frequency reads ve = 159,814.59 (20) M H z with a fractional error of 1.3 p p m (see Table 2). As has been shown by Dehmelt etal. [15], cyclotron resonances of very high precision may be obtained by use of cooled electrons and a bolometric detection scheme which avoids relativistic corrections. But it

should be pointed out that the final error of this experiment is dominated by the error in the determination of the proton cyclotron resonance. Therefore a refined set-up for detection of the cyclotron resonance of electrons did not seem to be necessary at the present stage of the experiment.

V. Results and Summary Electrons and protons were stored alternately in a magnetic field of 5.7 Tesla by use of an electrostatic quadrupole trap and their cyclotron frequencies have been determined. The quotient of proton cyclotron frequency vp = 87,037.87 (23) K H z and of electron cyclotron frequency ve

=

159,814.59 (20) M H z

provides the first direct determination of the proton-electron mass ratio: M J M e = 1836.1502 (53).

This number has to be compared to the currently accepted value [1] of M p / M e = 1836.1515 (7).

The value obtained here confirms the results of the 2 more precise but indirect determinations of this quantity, even if it is of lower precision. But there is evidence that the intrinsic accuracy of this method has not yet been fully exploited and that a considerable improvement may be obtained in future. We appreciate advice and help of Dr. H. Kilp in setting up the microwave arrangement. We wish to thank Professor Dr. G. Gr~iff for encouragement and continuous support.

References 1. Cohen, E.R., Taylor, B.N.: J. Phys. Chem. Ref. Data 2, 663 (1973) 2. Phillips, W.D., Cooke, W.E., Kleppner, D.: In: Atomic Masses and Fundamental Constantes 5. Paris 1975 3. Van Dyck, R.S., Schwinberg, P.B., Dehmelt, H.: Phys. Rev. Lett. 38, 310 (1977) 4. Lambe, E.B.D.: Ph.D. Thesis, Princeton (1959) and Polarisation, Mati~re et Rayonnement, Soc. Fr. de Phys., 441 (1969) 5. Fystrom, D.O.: Phys. Rev. Lett. 25, 1469 (1970)

6

G. Ggrtner and E. Klempt: A Direct Determination of the Proton-Electron Mass Ratio

6. Mamyrin, B.A., Aruyev, N.N., Alekseenko, S.A.: In: Atomic Masses and Fundamental Constants 4. New York 1972 7. Luxon, J.L., Rich, A.: Phys. Rev. Lett. 29, 665 (1972) 8. Gubler, H., Miinch, S., Staub, H.H.: Helv. Phys. Acta 46, 772 (1973) 9. Petley, B.W., Morris, K.: J-Phys. AT, 167 (1974) 10. Taylor, B.N., Parker, W.H., Langenberg, D.N.: Rev. Mod. Phys. 41, 375 (1969) 11. Dehmelt, H.: Adv. At. Mol. Phys. 3, 53 (1967) 12. Liebes, S., Franken, P.: Phys. Rev. 116, 633 (1959) 13. Hakkenberg, A., Weenink, M.: Physica 30, 2147 (1964)

14. Giirtner, G.: Ph.D. Thesis, Mainz 1977 15. Dehmelt, A.G., Walls, F.L.: Phys. Rev. Lett. 21, 127 (1968)

G. G~irtner E. Klempt Institut ftir Physik der Johannes Gutenberg-Universit~it Mainz Saarstrage 21 D-6500 Mainz Federal Republic of Germany