Theoretical and computational study of the energy dependence of the muon transfer rate from hydrogen to higher-Z gases
Dimitar Bakalov, Andrzej Adamczak, Mihail Stoilov, Andrea Vacchi
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Theoretical and computational study of the energydependence of the muon transfer rate from hydrogen tohigher-Z gases
Dimitar Bakalov a, ∗ , Andrzej Adamczak b , Mihail Stoilov a , Andrea Vacchi c a Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of sciences,Tsarigradsko chauss´ee 72, Sofia 1784, Bulgaria b Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342Krakow, Poland c Istituto Nazionale di Fisica Nucleare, Sezzione di Trieste, Via A. Valerio 2, 34127 Trieste,Italy
Abstract
The recent PSI Lamb shift experiment and the controversy about proton sizerevived the interest in measuring the hyperfine splitting in muonic hydrogenas an alternative possibility for comparing ordinary and muonic hydrogen spec-troscopy data on proton electromagnetic structure. This measurement criticallydepends on the energy dependence of the muon transfer rate to heavier gasesin the epithermal range. The available data provide only qualitative informa-tion, and the theoretical predictions have not been verified. We propose a newmethod by measurements of the transfer rate in thermalized target at differenttemperatures, estimate its accuracy and investigate the optimal experimentalconditions.
Keywords: muonic hydrogen, muon transfer, proton radius, hyperfinestructure, laser spectroscopy
1. Introduction
The laser spectroscopy measurement of the hyperfine structure of the groundstate of the muonic hydrogen atom was first proposed more than two decadesago [1], motivated by the understanding that in some sense it would be com-plementary to the top accuracy measurements of the hyperfine splitting (HFS)in ordinary hydrogen [2]. Subsequent studies revealed, however, some severedifficulties to this goal: the absence of a tunable near-infrared (NIR) laser withsufficient power that necessitates the use of a multi-pass optical cavity to en-hance the stimulation of transitions between the hyperfine levels, which in turnmakes inapplicable the initially proposed experimental method. In the following ∗ Corresponding author
Preprint submitted to Elsevier December 3, 2017 ears an alternative experimental method was proposed that exploits specificfeatures of the muon transfer reaction at epithermal collision energies [3] and iscompatible with the use of a multi-pass cavity. Also, the recent progress in thedevelopment of pulsed NIR sources of monochromatic radiation and IR optics[4] has brought the produced laser power close to the needed magnitudes. Allthis made the experimental project look feasible, though still remaining a signif-icant challenge. What supplied the missing motivation to start the work on theproject were the results of the recent muonic hydrogen Lamb shift experimentat PSI [5] and the discovered incompatibility of the values of the charge radiusof the proton extracted from muonic hydrogen spectroscopy on the one hand,and ordinary hydrogen spectroscopy and electron-proton scattering data, on theother. The few years of intense search for an explanation of this discrepancyhave not lead to any conclusion, and the most direct way to throw light onthe subject — next to an independent new measurement of the proton chargeradius — appears to be the measurement of the hyperfine splitting in muonichydrogen and extracting from it the proton Zemach radius: confirming the valueobtained from hydrogen spectroscopy would probably give more weight to theexplanations supposing methodology uncertainties in the hydrogen-based pro-ton charge radius, while in the opposite case the search for new physics willhave good reasons to be intensified.With this motivation in mind, as a first stage of the preparation for themeasurement of the HFS in muonic hydrogen, we started the thorough studyof the efficiency and accuracy of the method proposed for this experiment inRef. [3]. Its physical background is simple: the muonic hydrogen atoms thatabsorb a photon of the resonance frequency, undergo a hyperfine para-to-orthotransition. When de-excited back to the para spin state in a collision witha H molecule, these atoms are accelerated by ∼ . µ − p atoms.In Ref. [3] it was proposed to observe the transfer of the muon from muonichydrogen to the nucleus of a heavier gas added to the hydrogen target forwhich the transfer rate is energy-dependent in the epithermal range. Thereare experimental evidences that for some gases (e.g., oxygen, argon, etc.) thisis indeed the case [6, 7, 8]. The goal of our investigations is to obtain reliablequantitative data on the energy dependence of the rate of muon transfer to thesegases — and possibly other as well — in order to select the optimal chemicalcomposition and physical parameters (pressure, temperature) of the hydrogengas target that will provide the highest accuracy in the future measurementsof the hyperfine splitting of ( µ − p ) s and the Zemach radius of the proton. Inthe present paper we focus at the main physical processes with µ − p atoms in amixture of hydrogen and a higher-Z gas, and deduce the optimal experimentalconditions for the measurement of the energy dependence of the muon transferrate to the admixture gas. The results have been obtained with a Monte Carlosimulation code that uses the cross sections and rates for the scattering µ − p +H ,which were calculated in Refs. [9, 10]. The code has been verified on the many2receding occasions (see, e.g., [8, 11, 12]).
2. Preceding results and novelty of the proposed measurements
The muon transfer in collisions of muonic hydrogen atoms with atoms ofhigher-Z gas admixtures has been actively investigated for many years becauseof the important impact of hydrogen impurities on the observable rates of spe-cific reactions such as nuclear muon capture [13], muonic molecule formation[14], etc. In the limit of low collision energies, theory predicts in the lowest or-der approximation a flat energy dependence of the muon transfer rate [15, 16],while more refined calculations show that for some gases its value may vary byup to an order of magnitude for collision energies below 1 eV [17]. Some exper-imental data, e.g., on transfer to sulfur [6], consist of a single data point, theaverage rate of muon transfer at certain pressure and temperature, obtained asthe disappearance rate of muonic hydrogen atoms and determined from the timedistribution of the events of de-excitation of the muonic atoms of the heaviergas. In the case of oxygen [6], argon [8] and neon [7], however, the time distribu-tions of these events clearly display two different time ranges with substantiallydifferent disappearance exponents for the hot and the thermalized atoms. Thiswas interpreted as an evidence for a non-flat energy dependence of the muontransfer rate [11], and a step-function was introduced to describe qualitativelythe energy dependence of the transfer rate to oxygen, argon and neon. The ear-lier estimates of the efficiency of the experimental method in [1, 3] were obtainedusing this step-function approximation. Its accuracy, however, is far from whatis needed for the planning and optimization of the muonic hydrogen hyperfineexperiment because the contribution from the atoms with energies in the wholeepithermal range is averaged over their energy distribution and is accounted forwith a single parameter — the disappearance slope for the “unexpected delayed”events [6] — which incorporates the uncertainties due to the strongly model-dependent energy distribution of the epithermal atoms [18, 19, 20]. To solve theproblem, we propose to perform instead a series of measurements of the muontransfer rate from thermalized muonic hydrogen atoms at different temperaturesin an as broad as possible temperature range. The numerical technique of inves-tigating the energy dependence of the muon transfer rate, which is presented inSect. 4, relies substantially on the assumption that the muonic hydrogen atomsare thermalized and their energy distribution — the Maxwell-Boltzmann distri-bution — contains no uncertainties. Our study in Sect. 3 is also focused on theprocesses in a mixture of hydrogen and higher-Z gases at thermal equilibrium.
3. Processes involving the muonic hydrogen atoms in a mixture ofhydrogen and higher-Z gases
When slowed down and stopped in a mixture of hydrogen and higher-Zgases, a negative muon is captured by the Coulomb field of the nuclei and formsexotic muonic atoms in which one (or more) of the electrons are ejected and3eplaced by the muon. We do not consider here the processes of muon capturein an excited state of the exotic atom and the subsequent de-excitation bycompeting Auger and radiative transitions and in non-elastic collisions becausefor the hydrogen target densities of interest (pressures above 10 atm at roomtemperature) the whole cascade down to the ground 1 s state takes no morethan about 1 ns. This choice of the target density is determined by the specificfeatures of the available gas containers: simulations (to be reported elsewhere)with the FLUKA code [21] have shown that too few muons are stopped in thetarget at lower densities. So, to a good approximation, the description of thehistory of a muonic atom can start from the ground 1 s state.In the general case (when the target gas is a natural mixture of hydrogen anddeuterium), the following processes take place: elastic scattering of the muonicatoms, spin-flip, muon exchange between the hydrogen isotopes, and rotational-vibrational transitions in target molecules; formation of the ppµ , pdµ , and ddµ molecules; nuclear fusion in pdµ and ddµ ; nuclear muon capture; muon decay,and muon transfer from muonic hydrogen atoms to atomic states of higher-Z el-ements. The part of muons that are transferred to muonic deuterium at naturaldeuterium concentration is small and has little impact on the process of muontransfer to higher-Z nuclei [22]; nuclear fusion therefore can also be neglected.The muons that are captured in muonic molecules drop out of the sequence ofprocesses leading to transfer to higher-Z gases; since muonic molecule formation[14] is much slower that muon transfer even at low admixture gas concentration(see Sect. 3.3) it will not be taken into account. As for the muons directlycaptured in muonic atoms of the higher-Z admixture gas, their fast cascade de-excitation produces prompt X-rays that can be clearly distinguished from thedelayed X-rays that follow the transfer of the muon from µ − p . With all this inmind, we restrict our consideration to the following subset of processes:1. thermalization of the muonic hydrogen atoms via elastic scattering of µ − p by H molecules;2. depolarization of the muonic hydrogen atoms in spin-flip scattering by H molecules;3. muon transfer from the hydrogen to higher-Z muonic atom;4. disappearance of the muon by either a muon decay or nuclear muon cap-ture. Part of the energy released in the cascade de-excitation of µ − p atoms is trans-formed into kinetic energy of the latter, and the initial energy distribution in theground 1 s state is spread over a broad interval up to the keV range [18, 19, 20],with the para ( F = 0) and ortho ( F = 1) spin states populated statistically. Aphenomenological distribution of the initial kinetic energy that has given goodresults in simulations of preceding experiments is the sum of two Maxwelliandistribution density, one corresponding to the target gas temperature, and an-other one with mean energy of 20 eV; it has been used in the present work too.Thermalization and depolarization occur in elastic and spin-flip scattering with4he surrounding hydrogen and higher-Z atoms and molecules. The energy lossin collisions with the light H is the main mechanism of thermalization, so thatthe rate of thermalization is little sensitive to the admixture concentration anddepends only on the hydrogen density φ (or pressure P ) and — through themolecular cross sections — on the temperature T . The same holds for the rateof depolarization.The process of thermalization is best illustrated with the time evolutionof the average kinetic energy of ( µ − p ) s , E ( t ; P, T ). Figure 1 presents theMonte Carlo simulations for E ( t ; P, T ) in pure hydrogen for a set of relevantpressures P , at T = 300 K. The plot clearly shows that the time needed ( m - p) thermalization in H at T=300 Kand different gas pressures [ atm ] t [ ns ] a v e r age ( m- p ) s ene r g y [ e V ]
40 20 10 5
Figure 1: The average ( µ − p ) s energy E ( t ; P, T ) versus time t , for pressures P =5, 10, 20,and 40 atm (0.51, 1.01, 2.03, and 4.05 MPa). for the average energy E ( t ; P, T ) to approach the equilibrium energy is ap-proximately inverse proportional to the pressure (i.e., density): E ( t ; kP, T ) ∼ E ( t/k ; P, T ) , k >
0. In particular, at room temperature and 20 atm, the ( µ − p ) s atoms are “practically” thermalized already at t = 300 ns. This conclusion isconfirmed by the plots in Fig. 2 of E ( t ; P, T ) at fixed pressure, for a set of val-ues of the temperature and the corresponding hydrogen density φ , in the unitsof liquid hydrogen density (LHD). When the density is fixed, the thermaliza-tion time is practically the same for all discussed temperatures, which is shownin Fig. 3. At φ = 0 . µ − p ) s atoms are thermalized after about 150 ns,for all the temperatures. For times 5 ns . t .
150 ns, the time evolution ofenergy differs due to the different thermal energies of H molecules. Thus, thegas density is the parameter that determines the thermalization time at small5 -2 -1 ( m - p) thermalization in H at P=35 atmand different gas temperatures t [ ns ] a v e r age ( m- p ) s ene r g y [ e V ] T=300 K, f =0.04T=150 K, f =0.08T=70 K, f =0.17 Figure 2: Time evolution of the average ( µ − p ) s energy E ( t ; P, T ) at a fixed pressure P =35 atm (3.55 MPa), for a set of temperatures T . φ is the H gas density in units of LHD. admixtures of higher- Z gases. The same conclusion can be drawn for the timeof quenching the F = 1 state, since the collisions of ( µ − p ) s atoms with H molecules establish an effective mechanism of the downwards spin flip.The ( µ − p ) s atom depolarization (i.e., the de-population of the ortho F = 1spin state) proves to be much faster than the thermalization process: the plotin Fig. 4 of the time evolution of Q F =1 ( t ; P, T ), the part of all ( µ − p ) s atomsthat occupy the ortho F = 1 hyperfine level of the ground state (normalizedby Q F =1 (0; P, T ) = 3 / t nanoseconds, where the rough estimate of t reads: t [ns] ∼ × T [K] / P [atm] . (1) The proposed experimental method consists in measurements of the rateof muon transfer from thermalized ( µ − p ) s atoms at time t > t ; the muonsthat have decayed or been captured by this time are lost. The part of “lostmuons” is (1 − exp( − λ ∗ t )) /λ ∗ , λ ∗ = λ dec + λ cap where λ dec = 0 . µ s − and λ cap = 0 . − are the free muon decay rate and the nuclear muon capture rate6 -2 -1 ( m - p) thermalization in H at f =0.045and different gas temperatures t [ ns ] a v e r age ( m- p ) s ene r g y [ e V ] T=300 KT=150 KT=70 K
Figure 3: Time evolution of the mean ( µ − p ) s energy E ( t ; φ, T ) at a fixed density φ = 0 . T . in ( µ − p ) s , both of them — independent of hydrogen gas density or temperature.For t = 500 ns, the losses are ∼
20% and increase with t . In the experiment for the measurement of the energy dependence of therate of muon transfer to a higher-Z admixture nucleus we have the followingparameters at our disposal to control the processes: the pressure and tempera-ture of the target, the start-of-counting time t and the concentration c of theadmixture, defined as c = n Z / ( n H + n Z ), where n H and n Z are the numberconcentrations, i.e. the number of species (atoms or molecules) of hydrogen andhigher-Z element per cubic centimeter. The choice of optimal values of theseparameters should guarantee that the maximal number of muon transfer eventstake place from thermalized ( µ − p ) s atoms.The Monte Carlo simulations may have, in principle, only restricted validitybecause for most of the possible admixture gases only the average muon transferrate is known. Oxygen is one of the very few elements for which there existexperimental and theoretical data about the energy dependence of the transferrate. We chose to present the results of the simulations for a mixture of hydrogenand oxygen for this reason, and also because it will be the primary target of theoncoming experimental investigations.In the experiment, it is crucial to have a large number ( µ − p ) s atoms at timesmuch larger than the time width of the prompt peak. In order to study this7 population of ( m - p) (F=1) in H at T=300 Kand different gas pressures [ atm ] t [ ns ] Q F = ( t; P , T )
40 20 10 5
Figure 4: Time evolution of the population Q F =1 ( t ; P, T ) of the F = 1 state in hydrogen atroom temperature T = 300 K, for a set of pressures P =5, 10, 20, and 40 atm (0.51, 1.01,2.03, and 4.05 MPa). problem, the Monte Carlo simulations have been performed for a fixed tem-perature and pressure of H +O mixture and various oxygen concentrations.Figure 5 shows the time evolution of the relative population of ( µ − p ) s atomsat T = 300 K and P = 35 atm. The curve for c = 0%, i.e. for pure H , deter-mines the absolute upper limit on the number of surviving ( µ − p ) s determinedby the decay and capture rate λ ∗ . Any admixture of oxygen decreases this num-ber. At c & P = 35 atm and T = 300 K have shown that the fraction of muon stops intarget practically does not depend on the concentration if c . thermalized ( µ − p ) s out of a fixed number of muon stops in H .The results of the Monte Carlo simulations for various oxygen concentrations atfixed temperature and pressure, shown in Fig. 6, clearly demonstrate that thestatistical uncertainty of the experimental data on the muon transfer rate —which is inverse proportional to the squared root of the transfer events — maybe drastically reduced by careful planning of the experiment. Our Monte Carloprediction of the sharp dependence of the number of muon transfer events onthe concentration of oxygen, which should be recalculated for any other targettemperature and gas admixture, is a central result of the present paper.8 population of p m in H +O at T=300 K, P=35 atmand different atomic concentrations of oxygent [ ns ] Q ( t; P , T )
1% 0.1% 0.01% 0%
Figure 5: Number Q ( t ; P, T ) of the ( µ − p ) s that have survived in H +O mixture at T =300 K, P = 35 atm (3.55 MPa), and oxygen concentrations c = 0%, 0.01%, 0.1%, and 1% upto time t . Q ( t ; P, T ) is normalized to the number of thermalized depolarized ( µ − p ) s atomsat time t . (Here for simplicity we have set t = 0).
4. Determining the energy dependence of the muon transfer rate fromexperimental data
In this last section we present the mathematical model of the proposed ex-periment and derive the equations that allow for determining the energy depen-dence of the muon transfer rate from the experimental data on the temperaturedependence of the muon transfer rate in thermalized mixture of hydrogen anda higher-Z gas. We consider the general case when the rate λ of muon transferin collisions of the muonic hydrogen atom with an atom of the heavier admix-ture does depend on the center-of-mass (CM) collision energy ε of the species: λ = λ ( ε ). The observable rate of muon transfer Λ o is the average of λ over thethermal distribution of the collision energy ρ ( ε ):Λ o = Z λ ( ε ) ρ ( ε ) dε . In thermal equilibrium the distribution ρ ( ε ) is nothing but the Maxwell-Boltzmanndistribution: ρ ( ε ) = ρ MB ( ε ; T ) = 1 ε T ρ ( ε/ε T ) ,ρ ( x ) = 2 √ π √ x exp( − x ) , ε T = k B T , -4 -3 -2 -1 atomic concentration of oxygen [ % ] m uon t r an s f e r e v en t s / m uon s t op Figure 6: The number of muon transfer events from thermalized ( µ − p ) s in H at T = 300 Kand P = 35 atm, normalized to the number of thermalized and depolarized muonic hydrogenatoms at t = t = 0 versus the number concentration c of oxygen. where T is the temperature and k B is Boltzmann’s constant.Of primary importance is the behavior of λ ( ε ) in the thermal and nearepithermal energy range since the efficiency of the adopted experimental methodfor the measurement of the µ − p hyperfine splitting depends on the variationsof λ ( ε ) in this range [3]. We use a polynomial approximation for λ ( ε ): λ ( ε ) = N a X i =1 λ i N a Y j =1 ,j = i ε − ε j ε i − ε j , λ i = λ ( ε i ) , (2)where the values ε i , i = 1 , . . . , N a will be referred to as “reference energies”. Aslong as the transfer rate λ ( ε ) is not expected to have any anomalous behavioroutside the above range (e.g., abrupt decrease or growth at higher energies) weevaluate the integral for the observable muon transfer rate at temperature T with a Gauss-type quadrature of appropriate rank N G :Λ( T ) = Z λ ( ε ) ρ ( ε ; T ) dε = Z ρ ( x ) λ ( ε T x ) dx = N G X n =1 w n λ ( ε T x n ) , (3)where w n and x n are the weights and nodes of the Gauss quadrature formulawith weight function ρ ( x ). Eqs. (2,3) establish a linear relation between Λ( T )10nd the values λ i :Λ( T ) = N G X n =1 w n N a X i =1 N a Y j =1 ,j = i k B T x n − ε j ε i − ε j λ i . (4)This way, by measuring the values of the observable muon transfer rate Λ k =Λ( T k ) in thermalized target gas at temperatures T k , k = 1 , . . . , N a , one will geta system of N a linear equationsΛ k = N a X i =1 M ki λ i , M ki = N G X n =1 w n Y j = i k B T k x n − ε j ε i − ε j . (5)These equations can be resolved for the values λ i of the transfer rate at thereference energies: λ i = N a P k =1 M − ik Λ k , and thus the parameters in the polynomialapproximation for λ ( ε ) of Eq. (2) are expressed in terms of the experimentaldata { Λ k } : λ ( ε ) = N a X i =1 N a Y j =1 ,j = i ε − ε j ε i − ε j . N a X k =1 M − ik Λ k (6)While these transformations are quite straightforward, it is worth briefly dis-cussing the uncertainty of the parameters λ i = λ ( ε i ), induced by the experimen-tal uncertainty of Λ k = Λ( T k ). If assuming that Λ k are normally distributedaround ¯Λ k with standard deviation ∆ k , the parameters λ i will be normallydistributed with standard deviation δ i , where δ i = N a X k =1 (cid:0) M − ik (cid:1) ∆ k . (7)The specific values of δ i depend on the choice of target temperatures ( T k , k =1 , . . . , N a ) (that may be predetermined by the available cooling devices) and ofthe reference energies ε i (whose optimal choice will be done after simulationsof the muonic hydrogen HFS experiment). Consider as an example a seriesof three measurements at 70 K (cooling with liquid nitrogen), 195 K (liquidcarbon dioxide) and 300 K (room temperature), and the set of reference energies ε = 0 .
006 eV, ε = 0 .
05 eV, and ε = 0 .
12 eV. For simplicity, we assumethat all experimental uncertainties are equal: ∆ k = ∆, k = 1 , . . . ,
3. Theaccuracy of the proposed method is assessed with the ratios δ i / ∆ that show howmuch the uncertainties of the calculated values λ i exceed the uncertainty of theexperimental data Λ k . On Fig. 7 we plot (with thick solid curves) the responseof δ i / ∆ to the variations of each of the three reference energies ε j , j =1,2,3, inthe neighborhood of the values listed above (denoted with dashed vertical lines)while keeping the other two fixed. We see that:11 the uncertainty of λ ( ε ) increases very fast with the reference energy ε . Thisis not surprising: the contribution to the observable transfer rate Λ( T )from the “code” of the Maxwell-Boltzman distribution ρ MB ( ε ; T ) (plottedwith dotted lines for the selected temperatures T k ) decreases exponentiallywith energy; • the uncertainty δ i is little sensitive to variations of the reference energies ε j , j = i .Eq. 7 allows for estimating the accuracy goals ∆ k in the measurements ofΛ k that will lead to uncertainties δ i of the calculated muon transfer rates λ i within the limits, required for planning and optimizing the muonic hydrogenHFS experiment [3]. The example discussed above shows that, with a relativelymodest statistics of ∼ muon transfer events, the transfer rate λ ( ε ) can bedetermined in the whole range of epithermal energies of interest with uncertaintyof the order of 1% – far below the uncertainty of the currently available dataabout the energy dependence of the muon transfer rates.The numerical results are independent of the rank N G of the Gauss quadra-ture formulae provided that N a ≤ N g . The nodes and weights of two lowerrank quadratures are given in Table 1. Table 1: Examples of quadrature formula for averaging over the Maxwell-Boltzmann energydistribution N g Nodes x n Weights w n
5. Conclusions
Using a realistic model of the processes with muonic hydrogen atoms in theground state that occur in mixture of hydrogen and a higher-Z gas, we havefound by means of Monte Carlo simulations the experimental conditions (ad-mixture concentration and time gate) for preselected pressure and temperaturewhich optimize the efficiency of the measurement of the temperature depen-dence of the average rate of muon transfer from muonic hydrogen to the higher-Z admixture. We have also developed a computational technique allowing todetermine from these data the energy dependence of the muon transfer rate andgive an estimate of the experimental uncertainty of the latter. These results willbe implemented in the methodology of the oncoming experiment FAMU [23],to be performed in 2015 at the RIKEN-RAL muon facility.12 e e e T=70 KT=195 K T=300 K e i [ eV ] un c e r t a i n t y r a t i o d i / D d / D d / Dd / D Figure 7: Dependence of the uncertainty ratios δ i / ∆ on the reference energies ε j , j = 1 , , δ i / ∆ to variations of ε j in the neigh-borhood of the value denoted by a dashed vertical line while keeping fixed the other tworeference energies ε k , k = j . The dotted curves are the Maxwell-Boltzman energy distributiondensities for temperatures T = 70 K, 195 K, and 300 K. Acknowledgments
This work has been partly supported under the bilateral agreement of theBulgarian Academy of sciences and the Polish Academy of Sciences.
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