Muonic hydrogen cascade time and lifetime of the short-lived 2S

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Muonic hydrogen cascade time and lifetime of the short-lived 2S state

L. Ludhova,^1,2,*F. D. Amaro,³A. Antognini,⁴F. Biraben,⁵J. M. R. Cardoso,³C. A. N. Conde,³A. Dax,^2,6,†S. Dhawan,⁶ L. M. P. Fernandes,³T. W. Hänsch,⁴V. W. Hughes,^6,‡ P. Indelicato,⁵L. Julien,⁵P. E. Knowles,¹F. Kottmann,⁷ Y.-W. Liu,⁸J. A. M. Lopes,³C. M. B. Monteiro,³ F. Mulhauser,^1,§F. Nez,⁵R. Pohl,^2,4P. Rabinowitz,⁹ J. M. F. dos Santos,³

L. A. Schaller,¹C. Schwob,⁵D. Taqqu,² and J. F. C. A. Veloso^3,储

1Département de Physique, Université de Fribourg, 1700 Fribourg, Switzerland

2Paul Scherrer Institut, 5232 Villigen-PSI, Switzerland

3Departamento de Física, Universidade de Coimbra, 3000 Coimbra, Portugal

4Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany

5Laboratoire Kastler Brossel, ENS, UPMC and CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France

6Physics Department, Yale University, New Haven, Connecticut 06520-8121, USA

7Institut für Teilchenphysik, ETH Zürich, 8093 Zürich, Switzerland

8Physics Department, National Tsing Hua University, Hsinchu 300, Taiwan

9Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA

共

Received 21 November 2006; published 4 April 2007兲

Metastable 2Smuonic-hydrogen atoms undergo collisional 2Squenching, with rates which depend strongly on whether the␮pkinetic energy is above or below the 2S→2Penergy threshold. Above threshold, collisional 2S→2Pexcitation followed by fast radiative 2P→1Sdeexcitation is allowed. The corresponding short-lived

␮p

共

2S

兲

component was measured at 0.6 hPa H₂ room-temperature gas pressure, with lifetime ␶2S short

= 165₋₂₉⁺³⁸ns

共

i.e., ␭2Squench= 7.9_−1.6^+1.8⫻10¹²s⁻¹ at liquid-hydrogen density

兲

and population ␧2Sshort= 1.70_−0.56^+0.80%

共

per␮patom

兲

. In addition, a value of the␮pcascade time,T_cas^␮p=

共

37± 5

兲

ns, was found.

Muonic hydrogen共␮⁻p兲is a simple atomic system, sen- sitive to basic features of the electromagnetic and weak in- teractions. Of particular interest is its metastable 2S state, long sought after for measuring the␮p共2S兲-Lamb shiftL_␮. Vacuum polarization dominatesL_␮. It shifts the 2S level by 0.2 eV below the 2Plevel关1,2兴. A measurement ofL_␮is in progress at the Paul Scherrer Institute共PSI兲, Switzerland关3兴.

We report here the data analysis of a preliminary stage of this experiment made at low H₂gas pressurep_H

2= 0.6 hPa关4兴. Muons stopped in H₂ gas form highly excited␮^{p atoms} 关5兴. A cascade of both collisionally induced and radiative deexcitations leads to the 1Sground state or, with a probabil- ity␧2S 共few %兲, to the 2S state. The␮p共1S兲 kinetic energy distributionE_kin^1S has been measured atp_H

2= 0.06– 16 hPa关6兴, and cascade simulations show thatE_kin^1S andE_kin^2S do not differ significantly under our conditions关5兴.

The 2S state lifetime is, in the absence of collisions, es- sentially equal to the muon lifetime␶␮= 2.2␮^{s. In H}2 gas, collisional 2S quenching occurs, with different processes for kinetic energiesE_kin^2S above or below the 2S→2P transition

threshold which is共1 +m_␮p/m_H

2兲兩L_␮兩⬇0.3 eV in the labora- tory frame.

共i兲Most␮p共2S兲atoms are formed at energies above this threshold 关6兴 where a collisional 2S→2P Stark transition, followed by 2P→1S deexcitation with 1.9 keV K␣ ^{x ray} emission共the 2P lifetime is 8.5 ps兲,

␮^p共2S兲+ H₂→␮^p共2P兲+ H₂→␮^p共1S兲+ H₂+K␣^, 共1兲 leads to fast 2S depletion共collisional quenching兲. A lifetime

␶2Sshort⬃100 ns/p_H

2关hPa兴was predicted for thisshort-lived2S component 关7–9兴. This is too short to have been seen in previous searches forK␣x rays delayed with respect to the

␮^p共2S兲 formation time 关10–12兴. In this Rapid Communica- tion we report on the first measurement of␶_2S^shortand the cor- responding population␧_2S^short.

共ii兲Due to elastic collisions, a fraction of the␮^p共2S兲at- oms decelerates to energies below 0.3 eV where process共1兲 is energetically forbidden. This fraction is thelong-lived2S component. A recent experiment关13兴showed that its domi- nant quenching process is nonradiative deexcitation to the ground state, with lifetime ␶_2S^long⬇1.3␮s at 0.6 hPa and population␧_2S^long⬇1%.

The cascade timeT_cas^␮pis the mean delay between␮^p-atom formation and final deexcitation to the ground state关when a

␮p K-line x ray, other than from ␮p共2S兲decay, is emitted兴.

The T_cas^␮p value results from the average of the various cas- cade deexcitation processes and depends onp_H

2. It was cal- culated关5兴but never measured for␮^p.

In our experiment, muons stop in H₂ gas containing a small admixture of N₂共air兲, and we measure simultaneously the three time distributions

*Present address: Dipartimento di Fisica, Università degli Studi, via Celoria 16, 20133 Milano, Italy. Electronic address:

[email protected]

†Present address: Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.

‡Deceased.

§Present address: University of Illinois at Urbana-Champaign, Ur- bana, IL 61801, USA.

储Present address: Departamento de Física, Universidade de Aveiro, 3819-193 Aveiro, Portugal.

1 Published in "Physical Review A 75: 040501(R), 2007"

which should be cited to refer to this work.

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K␣^{, i.e.,}␮^p共n= 2→1兲x rays 共1.898 keV兲, K共⬎␤兲, i.e.,␮p共n⬎3→1兲x rays关2.45共2兲keV关14兴兴,

␮N, i.e.,␮N共n= 5→4兲x rays共3.08 keV关15兴兲. The␮^p共n= 3→1兲K␤^line共2.249 keV兲is not well separated fromK␣ andK共⬎␤兲. The 0.4共1兲% air admixture in the H2

was useful for calibrating x-ray energies and times. The␮N time distribution is similar to that of␮^p formation because the␮N cascade time is negligibly short共⬃10⁻¹⁰s兲 关16兴. The

␮^p cascade time will therefore show up as a time delay in theK␣ ^andK共⬎␤兲 distributions compared to ␮N. The sig- nature for 2Sradiative decay will be a tail inK␣not present in K共⬎␤兲. The muon transfer rates ␮p+ N→␮N +p are

⬃10³s⁻¹ for ␮p共1S兲 and ⬃10⁴s⁻¹ for ␮p共2S兲 关17兴, too small to affect our results.

The experiment was performed at the recently developed low-energy␮⁻ source attached to the␲E5 beamline at PSI 关18兴. It provides ⬃10³s⁻¹␮⁻ with energies of a few keV.

The muons were axially injected into a 1-m-long, 20-cm-bore solenoid operated at 5 T, containing the muon entrance detectors and the gas target共see关3兴兲. Two detectors, based on nanometer-thick carbon foils, signaled the arrival of slow muons关19兴. Muons were stopped in a 20-cm-long tar- get vessel filled with 0.6 hPa H₂ gas 共temperature 290 K兲, and ␮^p atoms were formed in a volume of 0.5⫻1.5

⫻20 cm³. Twenty large-area avalanche photodiodes 共LAAPD兲, each with sensitive area 13.5⫻13.5 mm², were used as x-ray detectors关20兴. Muon-decay electrons were de- tected by a set of plastic scintillators and also by the LAAPDs. More than 5⫻10⁵ events, each where an x ray was followed by an electron, were analyzed.

Calibration data were used for each LAAPD to deduce the energyE_x and timet_x 共relative to muon entrance兲of a mea- sured x ray. Typical resolutions共full width at half maximum兲 were⌬Ex/E_x⬇25% and⌬tx⬇35 ns for 2-keV x rays. Most

␮^p x rays were found in the time interval 300艋t_x艋600 ns, corresponding to the widely distributed muon slowing-down times.

The K␣^,K共⬎␤兲, and ␮N time distributions were deter- mined from a fit of theE_x spectra for different t_xintervals.

The usefult_x range 共0.2– 6.5␮s兲was divided into 28 inter- vals 共50, 100, and 500 ns wide兲. For each interval an E_x spectrum was produced. Three typical spectra are shown in Fig.1, one at times of ␮p formation and deexcitation 共top兲 and two at later times. The function fit to each spectrum is composed of several x-ray lines and a continuous back- ground. Each line is the sum of a Gaussian peak and a tail toward lower energies 共an LAAPD characteristic兲, with energy-dependent weights关4兴.

The lines fitted in Fig. 1 correspond to x rays from ␮^p 关K␣^{, K}␤^{, and} K共⬎␤兲兴, ␮^N 共main transitions at 1.67, 3.08, and 6.65 keV关15兴兲,␮^O共2.19, 4.02, and 8.69 keV兲, and␮^C 共4.89 keV兲. The intensities for theK␣^andK共⬎␤兲and 3.08-, 4.02-, 4.89-, and 6.65-keV lines were free parameters, whereas the relative intensities of the other lines as well as the positions and widths of all lines were fixed by requiring global consistency for all data 关4兴 and considering known

yields关21兴. The late-time 2-keV peak共see Fig.1, bottom兲is due to␮^p x rays from second-muonstops—i.e., muons en- tering the target at random times shortly after afirst muon which definedt_x= 0. The␮^C4→3共4.89 keV兲line at late times arises from␮^p atoms drifting to the polypropylene foils in front of the LAAPDs where muon transfer to C atoms oc- curred. The continuous background is due to x rays with energy ⬎10 keV, e.g., ␮C K- and L-line transitions, whose deposited charge was not fully amplified by the LAAPDs. It was well modeled by a sum of an exponential and a linear function, with two amplitudes as free parameters. The full details of the extensive background studies are found in关4兴. The fitted intensities共with statistical errors兲of the 28 en- ergy spectra are shown in Fig.2as a function of t_x, for the three lines ␮^N 共3.08 keV兲, ␮p K共⬎␤兲, and K␣. The late- time events are caused bysecond-muonstops. Thet_xspectra from different LAAPD positions show关4兴that for muons not stopped during their first pass through the gas target, a con- siderable fraction was reflected at the gold-plated back side of the vessel and stopped during the return pass. Conse- quently, the␮^{N spectrum}共showing the muon stop-time dis- tribution兲 was represented by a sum of two functions, each one the convolution of a Gaussian with an exponential共Fig.

2, top兲. The simultaneous fit of the three time spectra, using the common muon stop-time distribution, gives values for the intensitiesA_共␮N兲,A_共⬎␤兲, andA_␣.

0 2500 5000 7500 10000

0 50 100 150

0 5 10 15 20

1 2 3 4 5 6 7 8 9

t_x=[355-405]ns Kα

Kβ K(>β)

μN54 μN43

μO54 μO43

μC43

t_x=[855 - 955]ns

Kα K(>β)

μC43

Bgr

t_x=[2955 - 3455]ns Kα

K(>β)

μC43

Bgr

Energy[keV]

Events/50eVbins

FIG. 1.

共

Color online

兲

X-ray energy spectra

共

3 of 28

兲

for differ- entt_x intervals. The fit function is composed of the peaks for␮p K␣, K␤, K

共⬎␤兲

, ␮N, ␮O, and ␮C

共

4.89 keV

兲

, and a continuous background

共

Bgr

兲

.

2

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The K共⬎␤兲 and K␣ spectra are delayed and slightly broadened with respect to␮N due to the␮^pcascade time. It is not possible to extract the precise shape of the cascade time distribution from the data, but calculations关5兴indicate that the K共⬎␤兲 cascade has an approximately exponential time distribution, whereas the K␣ cascade has the same asymptotic form, but includes growth behavior at earlier times. The␮N fit function was therefore convoluted with an exponential 共parameter ␶cas兲 to obtain the K共⬎␤兲 function and further convoluted with a Gaussian共parameter␴cas␣ 兲for the K␣ function. In addition, free time offsets ⌬T_共⬎␤兲 and

⌬T_␣ 关for K共⬎␤兲 and K␣, with respect to ␮N兴 were intro- duced. The contribution of the short-lived␮p共2S兲 state was considered by adding to theK␣function a convolution of the

K共⬎␤兲distribution with an exponential共parameter^˜␶_2S^short兲and a relative population˜␧_2S^short共normalized for the fit toA_␣兲.

A simultaneous fit of the three time spectra was per- formed, resulting in␹²= 57.5 for 67 degrees of freedom. The fit functions and residuals are shown in Fig.2. The scatter of the residuals confirms that no relevant systematic deviations exist between the data and the model. The resulting cascade time slope is ␶cas=共26± 2兲ns. The time shift ⌬T_共⬎␤兲

=共0 ± 5兲ns is consistent with zero, as expected for the K共⬎␤兲 cascade, whereas ⌬T_␣=共13± 5兲ns and ␴cas␣

=共15± 2兲ns approximate a K␣ cascade time distribution which deviates from an exponential in the first⬃20 ns. The deduced mean cascade times T_cas^共⬎␤兲=␶cas+⌬T_共⬎␤兲

=共26± 5兲ns and T_cas^␣ =␶cas+⌬T_␣=共39± 5兲ns, weighted by the correspondingK-line yields, result in a mean␮^pcascade time T_cas^␮p=共37± 5兲ns. 关The relative K yields at p_H

2

= 0.6 hPa were deduced from 关14兴 as Y_␣= 0.821共12兲, Y_␤

= 0.061共9兲, and Y_共⬎␤兲= 0.118共11兲, and the K␤ cascade time was assumed to be equal toT_cas^共⬎␤兲.兴T_cas^␮pdepends only weakly on the fit function details because it essentially reproduces the center-of-gravity shifts of the K␣ ^and K共⬎␤兲 distribu- tions relative to␮^N.

The fit results for the 2S tail are ^˜␶_2S^{short= 133}₋₂₂^{+29ns and}

␧

˜_2S^short= 1.44_−0.46^+0.67%. As shown in Fig.3, the absence of the 2S tail, corresponding to˜␧_2S^short= 0, is excluded by 6␴. As a test, theK共⬎␤兲spectrum was also fit with a “2S” tail. Its ampli- tude ␧_共⬎␤兲 关normalized to A_共⬎␤兲兴 is compatible with zero 共within 0.7␴兲 for any^˜␶2Sshort, as expected. We conclude that our fit function correctly reproduces the muon stop, cascade, andsecond-muontime distributions. The difference between

␧

˜_2S^short and ␧_共⬎␤兲 is ˜␧_2S^short−␧_共⬎␤兲=共1.36± 0.28兲% at ^˜␶2Sshort

= 133 ns. A zero value of this difference is excluded by 艌4.4␴^{for any˜}␶2Sshort, confirming that the tail in theK␣^spec- trum can only come from 2Squenching.

The measured^˜␶_2S^shortand˜␧_2S^shortvalues have to be corrected for several effects. The necessary corrections were deduced from a Monte Carlo simulation of the experiment, based on the known distribution of E_kin^2S 关6兴 and on calculated cross sections 关9兴 for process 共1兲 and elastic collisions. It was

1 10 10² 10³ 10⁴

Events/50ns inc

ref

bg

μN

-2-1012

Residuals

1 10 10² 10³ 10⁴

Events/50ns

T^{> β}_cas= 0

μp K(> β)

-2-1012

Residuals

10 10² 10³ 10⁴ 10⁵

Events/50ns

2S

without 2S

μp Kα

-2-1012

0 500 1000

Residuals

2000 3000 4000 5000 6000

Time[ns]

FIG. 2.

共

Color online

兲

Fit functions

共

solid line

兲

and residuals

共

normalized by the errors

兲

for fits to the x-ray time spectra for the lines␮N

共

3.08 keV

兲 共

top

兲

,␮p−K

共⬎␤兲 共

middle

兲

, andK␣

共

bottom

兲

. Each point results from a fit of the x-ray energy spectrum for the correspondingt_x interval

共

50, 100, and 500 ns wide

兲

. The dotted lines in the␮N spectrum are the functions for incoming

共

inc

兲

, re- flected

共

ref

兲

, and second

共

bg

兲

muons. In theK

共⬎␤兲

spectrum the dotted line is the fit function without cascade time. In theK␣spec- trum the 2S tail—i.e., the contribution of the short-lived ␮p

共

2S

兲

state—is shown as the dotted line. The dashed line is the fit function minus the 2Stail.

0 1 2 3 4 5

0 50 100 150 200 250

τ^~short_2S [ns]

ε~short 2S[%]

1σ 2σ

6σ

6σ

FIG. 3.

共

Color online

兲

Relative population˜␧_2S^short

共

normalized to A_␣

兲

of the short-lived␮p

共

2S

兲

component versus its lifetime˜␶_2S^short

共

without systematic corrections

兲

. The dot represents the best fit.

3

http://doc.rero.ch

found that共i兲some␮^p共2S兲atoms reach the target walls be- fore being quenched; 共ii兲 the solid angle for K␣ detection varies with time due to the␮p共2S兲 motion;共iii兲 because of collisions, theE_kin^2S distribution depends on time and hence so do the mean cross sections and␮^p velocities. Those effects modify the 2S tail shape at the late times where the experi- ment is most sensitive. The resulting correction factors are 1.24± 0.07 for^˜␶_2S^short and 1.34± 0.09 for ␧˜_2S^short. In addition,

␧

˜_2S^short has to be multiplied by 共1 −␶_2S^short/␶_␮兲⁻¹ to account for muon decay and by Y_␣/共1 +␧_2S^long兲 to normalize to all ␮^p atoms. The final result for the radiative lifetime and popula- tion of the short-lived ␮p共2S兲 component at 0.6 hPa 共tem- perature 290 K兲is␶_2S^short= 165₋₂₉⁺³⁸ns and␧_2S^short= 1.70_−0.56^+0.80%.

The results of our experiment can be discussed as follows:

共i兲 The ␮^p cascade time and the 2S tail have been disen- tangled from the stop-time distribution for the first time. This has been made possible by the high statistics, the low gas pressure, and the good stop-time resolution. The measured

␮^pcascade timeT_cas^␮p=共37± 5兲ns is less than half the⬇90 ns value predicted for 0.6 hPa by cascade calculations关5兴. The measured value may be slightly affected by a possible differ- ence in the␮⁻atomic capture times of N₂and H₂predicted by some models关22兴: the muon energy from which capture occurs is expected to increase withZ 关23兴, so␮N atoms can form earlier than ␮^p atoms during the muon stopping, an effect which would result in an even lower measured T_cas^␮p value. A result which does not depend on such effects is the measured difference T_cas^␣ −T_cas^共⬎␤兲=共13± 4兲ns, also signifi- cantly smaller than the calculated value of⬇25 ns. The cal- culated cascade times may be too long since Coulomb deex- citations, which dominate the cascade at highnlevels even at low p_H

2 关5兴, may be accompanied by simultaneous Auger

transitions, an effect not yet considered.共ii兲 We have com- pared the measured radiative 2S lifetime with the results of calculations. Since the existing calculated cross sections ne- glect molecular effects, we allowed for the extreme cases where the H₂ molecule is taken as two separate H atoms or as a single atom. Analyzing the simulated data in the time region where the experiment is sensitive, we obtained a value of共178± 30兲ns which agrees well with the experimen- tal result ␶2Sshort= 165₋₂₉⁺³⁸ns. This confirms the validity of the cross sections calculated for␮p共2S兲+ H→␮p共2P兲+ H Stark transitions 关8,9兴. The quenching rate for the short-lived 2S component, when normalized to liquid-hydrogen atom den- sity 共LHD, 4.25⫻10²²atoms/ cm³兲, is ␭_2S^quench共LHD兲

= 7.9_−1.6^+1.8⫻10¹²s⁻¹,⬃20 times faster than for the long-lived one关13兴.共iii兲The sum of the measured relative populations

␧_2S^short= 1.70_−0.56^+0.80% and ␧_2S^long⬇1% 共extrapolated to 0.6 hPa, from 关13兴兲 is 共2.7± 0.8兲%, in agreement with the total 2S population ␧2S=共2.49± 0.17兲% at 0.6 hPa deduced directly from the measured␮p K-line yields 关14兴. We conclude that there is now good understanding of both the short- and long- lived␮p共2S兲dynamics.

We thank L.M. Simons and B. Leoni for setting up the cyclotron trap, O. Huot and Z. Hochman for technical sup- port, and T.S. Jensen for fruitful discussions. We also thank the PSI accelerator division, the Hallendienst, and the work- shops at PSI, MPQ, and Fribourg for their valuable help.

This work was supported by the Swiss National Science Foundation, the French-Swiss program “Germaine de Staël”

共PAI No. 07819NH兲, the BQR de l’UFR de physique fonda- mentale et appliquée de l’Université Paris 6, the Portuguese FCT and FEDER under Project No. POCTI/FNU/41720/

2001, and the U.S. Department of Energy.

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