Heisenberg’s uncertainty principle in the PTOLEMY project: a theory update

A. Apponi,1, 2 M. G. Betti,3, 4 M. Borghesi,5, 6 A. Boyarsky,7 N. Canci,8 G. Cavoto,3, 4 C. Chang,9, 10 V. Cheianov,7

Y. Cheipesh,7 W. Chung,11 A. G. Cocco,12 A. P. Colijn,13, 14 N. D’Ambrosio,8 N. de Groot,15 A. Esposito,16

M. Faverzani,5, 6 A. Ferella,8, 17 E. Ferri,5 L. Ficcadenti,3, 4 T. Frederico,18 S. Gariazzo,19 F. Gatti,20 C. Gentile,21

A. Giachero,5, 6 Y. Hochberg,22 Y. Kahn,10, 23 M. Lisanti,11 G. Mangano,12, 24 L. E. Marcucci,25, 26 C. Mariani,3, 4

M. Marques,18 G. Menichetti,26, 27 M. Messina,8 O. Mikulenko,7 E. Monticone,19, 28 A. Nucciotti,5, 6 D. Orlandi,8

F. Pandolfi,3 S. Parlati,8 C. Pepe,19, 28, 29 C. Perez de los Heros,30 O. Pisanti,12, 24 M. Polini,26, 31, 32 A. D. Polosa,3, 4

A. Puiu,8, 33 I. Rago,3, 4 Y. Raitses,21 M. Rajteri,19, 28 N. Rossi,8 K. Rozwadowska,8, 33 I. Rucandio,34 A. Ruocco,1, 2

C. F. Strid,35 A. Tan,11 L. K. Teles,18 V. Tozzini,36 C. G. Tully,11 M. Viviani,25 U. Zeitler,15 and F. Zhao11

(PTOLEMY Collaboration)1INFN Sezione di Roma 3, Roma, Italy2Universita di Roma Tre, Roma, Italy

3INFN Sezione di Roma 1, Roma, Italy4Sapienza Universita di Roma, Roma, Italy

5INFN Sezione di Milano-Bicocca, Milan, Italy6Universita di Milano-Bicocca, Milan, Italy

7Instituut-Lorentz, Universiteit Leiden, Leiden, The Netherlands8INFN Laboratori Nazionali del Gran Sasso, L’Aquila, Italy

9Argonne National Laboratory, Chicago, IL, USA10Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA

11Princeton University, Princeton, NJ, USA12INFN Sezione di Napoli, Napoli, Italy

13Nationaal instituut voor subatomaire fysica (NIKHEF), Amsterdam, The Netherlands14University of Amsterdam, Amsterdam, The Netherlands

15Radboud University, Nijmegen, The Netherlands16School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA

17Universita di L’Aquila, L’Aquila, Italy18Instituto Tecnologico de Aeronautica, Sao Jose dos Campos, Brazil

19INFN Sezione di Torino, Torino, Italy20Universita di Genova e INFN Sezione di Genova, Genova, Italy

21Princeton Plasma Physics Laboratory, Princeton, NJ, USA22Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem, Israel

23University of Illinois Urbana-Champaign, Urbana, IL, USA24Universita degli Studi di Napoli Federico II, Napoli, Italy

25INFN Sezione di Pisa, Pisa, Italy26Universita degli Studi di Pisa, Pisa, Italy

27Center for Nanotechnology Innovation @NEST, Istituto Italiano di Technologia, Pisa, Italy28Istituto Nazionale di Ricerca Metrologica (INRiM), Torino, Italy

29Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Torino, Italy30Uppsala University, Uppsala, Sweden

31Istituto Italiano di Tecnologia, Graphene Labs, Genova, Italy32University of Manchester, Manchester, United Kingdom

33Gran Sasso Science Institute (GSSI), L’Aquila, Italy34Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas (CIEMAT), Madrid, Spain

35Johannes Gutenberg-Universitat Mainz, Germany36Istituto Nanoscienze–CNR, NEST–Scuola Normale Superiore, Pisa, Italy

We discuss the consequences of the quantum uncertainty on the spectrum of the electron emittedby the β-processes of a tritium atom bound to a graphene sheet. We analyze quantitatively theissue recently raised in [1], and discuss the relevant time scales and the degrees of freedom that cancontribute to the intrinsic spread in the electron energy. We propose an experimental signature totest this theoretical understanding. We also perform careful calculations of the potential betweentritium and graphene with different coverages and geometries. With this at hand, we propose possibleavenues to mitigate the effect of the quantum uncertainty.

I. INTRODUCTION

Neutrinos are one of the most elusive particles knownto us, and many questions regarding their nature remain

unanswered. On the one hand, it is now well assessed thatat least two of the three standard neutrinos are massive [2–7], and the values of their squared mass differences areknown with good precision (see, e.g., [8–10]). On the

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other hand, we still do not know their absolute mass scale(i.e. the mass of the lightest neutrino) and mass ordering(i.e. whether the lightest neutrino is mostly within thefirst or third leptonic family). Moreover, the existence ofa cosmic neutrino background is a robust predictions ofthe current cosmological paradigm, and it is expected tocarry a plethora of information about the early stages ofthe Universe [11]. Despite many indirect evidences, it hashowever yet to be observed directly.

One of the best ways to determine the absolute neutrinomass scale is by studying the spectrum of the electronemitted by β-decay close to its maximum kinetic energy,the so-called endpoint. The best up to date bound onthe effective neutrino mass, m2

ν =∑i |Uei|

2m2i [12], is

mν < 0.8 eV at 90% C.L., as obtained by the KATRINexperiment using gaseous molecular tritium [13, 14]. Asfar as the cosmic neutrino background is concerned, in-stead, one could in principle detect it via the processof neutrino capture [15–23]. In this case, if the processhappens in vacuum, the energy of the emitted electronis expected to be larger than the endpoint by twice mν .However, a finite experimental resolution broadens theobserved spectrum, turning the β-decay contribution intothe main source of background, which could hide theabsorption peak.

The proposed PTOLEMY experiment [24] is expectedto address both points. In particular, the goal is to studythe spectrum of electrons produced by the decay andabsorption processes of atomic tritium:1

3H→ 3He+ + e− + νe , (1a)

νe + 3H→ 3He+ + e− . (1b)

The proposed target substrate is graphene [24], which canefficiently store atomic tritium by locally binding it tocarbon atoms, hence allowing a large target mass in asmall scale experiment, and providing a good voltage refer-ence for an electromagnetic spectrometer. Thanks to thecorresponding large event rate, this should substantiallyimprove on the existing bounds for the absolute massscale, by accurately measuring the normalization of thespectrum, which is sensitive to the lightest neutrino mass.Indeed, it is expected for PTOLEMY to have the sensitiv-ity to measure an effective mass as small as mν = 50 meValready at the early stages of the experiment, with atarget mass of 10 mg [24]. This is almost completelyindependent on the experimental energy resolution.

To detect cosmic neutrinos from the emitted electronspectrum, instead, one needs to resolve the peaks comingfrom the capture process (1b) from the contribution com-ing from the standard β-decay (1a). To do that, both alarge event rate and a precise determination of the elec-tron energy are required. For the former, a much largertarget mass is required and PTOLEMY would ideally

1 For the possibility of using a heavier emitter with an endpointsimilar to tritium see [25].

have 100 g of tritium, while for the latter, it is expectedto achieve an energy resolution as small as 100 meV [24].

Nonetheless, as first pointed in [1] and recently rean-alyzed in [26], since a condensed matter substrate is im-plicitly present in the processes of Eqs. (1), Heisenberg’sprinciple implies an intrinsic uncertainty in the electron en-ergy. Indeed, since the initial tritium is spatially confined,it has an associated spread in its momentum, which inturns propagates to a spread in the energy of the emittedelectron. Using the graphene–tritium binding potentialsavailable in literature [27, 28], this spread is expected tobe about an order of magnitude larger than the desiredexperimental resolution. Consequently, to still be able todetermine whether or not the absorption process has beendetected, one would need sufficiently accurate theoreticalpredictions for the electron spectrum, along the lines ofwhat was done for the KATRIN experiment [29–33]. This,in turns, requires a detailed knowledge of the initial andfinal state of the reaction. Due to the intricacies of thecondensed matter state, this is a task substantially harderthan originally expected.

In this work we provide a theoretical update on theissue reported above. This is done by both spelling out theproblem in a firm quantitative way, as well as by proposingpossible avenues to mitigate it. First, we quantitativelyreview the issue, with emphasis on the role played bydifferent final states of the reaction. We also propose anexperimental test to verify the understanding behind thepredicted spread in the electron energy. We then discussthe relevant degrees of freedom that could influence thespectrum of the emitted electron, both before and after thereaction. Possible avenues to circumvent the problem arealso discussed. In particular, a solution to this issue shouldlikely come from a judicious tuning of the initial state forthe reactions in Eq. (1), rather that from the inclusion offurther degrees of freedom in the final state. We presentpossible ways to engineer the initial tritium wave functionusing geometries alternative to flat graphene.

Conventions: Throughout this manuscript we work innatural units, ~ = c = 1.

II. QUANTUM SPREAD IN A NUTSHELL

Here we briefly review the key points of the argumentformulated in [1]. We also compute the expected electronrate, under certain simplifying assumptions. For the sakeof the current argument, it suffices to consider the caseof a single neutrino.

Let us work at finite volume, V , and consider an initial3H atom with wave function ψi(xT), and a final 3He+ withwave function ψf (xHe). Both the electron and neutrino

wave functions are plane waves, ψβ(xβ) = eikβ ·xβ/√V

and ψν(xν) = eikν ·xν/√V . On large enough distances,

the weak interaction Hamiltonian is roughly constant, andthe location of the final decay products is the same as the

3

initial tritium:2

Hw = g δ (xT − xHe) δ (xT − xβ) δ (xT − xν) . (2)

Correspondingly, the transition matrix element from theinitial state i to a specific final state f is

Mfi =g

V

∫d3xψi(x)ψ∗f (x) e−i(kβ+kν)·x . (3)

Using Fermi’s golden rule, the corresponding transitionprobability rate is

dΓfi = (2π)∣∣Mfi

∣∣2δ(Ei − Ef − Eβ − Eν)dρf , (4)

where Ei is the energy of the initial bound tritium (restmass + negative binding energy), Ef the energy of thefinal helium (depending on the particular final state underconsideration), and Eβ and Eν the relativistic electronand neutrino energies. Moreover, dρf is the phase spaceof the final decay products, which also depends on theparticular final state.

For the typical graphene–tritium potential, the ini-tial wave function is roughly gaussian, localized in spacewithin a distance λ, which is of order of a few fractions ofan A, but whose precise value depends on the details ofthe substrate.3 Since the initial state is not a momentumeigenstate, exact momentum conservation is spoiled: itwill only be satisfied up to a spread of order ∼ 1/λ.

Two qualitatively different scenarios can now arise.4

On the one hand, the 3He+ might remain bound to thegraphene, ending up in some discrete level of its potential.In particular, on the short time scales over which the β-processes happen, it will be subject to the same bindingpotential as the initial tritium, as explained in Section III.Among all these final states, when the 3He+ remains inthe ground state is when the outgoing electron can havekinetic energy as high as possible. For events where alsono lattice vibrational modes are excited, this correspondsto Kmax

β = Q −mν , where Q ≡ m3H −m3He+ −me '18.6 keV [36] is the Q-value in vacuum.5 However, inthis scenario the final helium wave function, ψf , is itself

2 The coupling g can be related to the microscopic theory of weakinteractions by [34]

|g|2 = G2F

∣∣Vud∣∣2 (f2V ∣∣MF

∣∣2 + f2A∣∣MGT

∣∣2) ,with GF the Fermi constant, Vud the entry of the CKM matrix,fV and fA the vector and axial couplings of the nucleon, andMF and MGT the so-called Fermi and Gamow–Teller matrixelements. We are neglecting the relativistic Coulomb form factorfor simplicity, since it is approximately one. Here we use fV = 1,fA = 1.25, |MF | = 1 [34] and |MGT | = 1.65 [35].

3 For the sake of the current argument, we consider an isotropiccase, where all direction are equivalent, as discussed in [1]. In amore realistic scenario λ would be a matrix—see also Section IV.

4 We focus here on the two extreme scenarios. An infinite number ofpossible final states will interpolate between these two instances.

5 Here m3H and m3He+ are the atomic masses, see also [24].

localized in space. The corresponding matrix element atthe endpoint will then be exponentially small, Mfi ∼e−(λkβ)2 1, due to the large separation of scales betweenthe electron momentum and the typical atomic size. Thismakes the events close to the endpoint (correspondingto electrons with highest possible momentum) extremelyunlikely. This instance is completely analogous to whathappens in the Mossbauer effect (see, e.g., [37]).

In the second scenario the 3He+ is freed (or almostfreed) from the graphene sheet, i.e. it is excited close toor above the absolute zero of the potential. In this case,the maximum electron kinetic energy is Q−ε0−mν , withε0 the ground state binding energy of the initial tritium.The matrix element near the endpoint is given byMfi ∼e−λ

2|kHe+kβ |2 . Close to the maximum energy allowedby the process, this is still exponentially suppressed but,moving to smaller energies, the momentum of the outgoinghelium can compensate that of the electron, kHe ' −kβ ,making the probability for this final state sizeable. Verysimilar arguments hold for the rate of absorption of acosmic neutrino.

The result is an overall distortion of the electron spec-trum, which not only changes its emitted energy, but italso makes the absorption peaks either disappear underthe β-decay part of the spectrum or extremely rare. InFigure 1 we quantify the above observations. The detailsfor the calculation are reported in Appendix A.

Note that, when no vibrational mode is excited, therecoil energy of graphene is completely negligible due toits large mass. The electrons emitted in this instance (redlines in Figure 1) are therefore more energetic than theendpoint in vacuum. The maximum allowed energy islarger precisely by an amount equal to the recoil energythat the 3He+ would have in vacuum, Krec ' 3.44 eV.6 Inparticular, this feature could be used to test the physicalpicture presented here, by looking for events where theelectron has energy significantly larger than the endpointin vacuum.

III. TIME SCALES, LENGTH SCALES ANDRELEVANT DEGREES OF FREEDOM

Let us now discuss the interplay between the differentscales present in the problem at hand. This discussionwill partially overlap with what presented in [26].

Two distinct physical effects need to be taken intoaccount when considering the presence of the multipledegrees of freedom in a substrate accommodating theradioactive isotope:

6 A similar effect should also be present in the KATRIN setup [33].In that case, the best one can do is to transmit the momen-tum to the molecule as a whole. With respect to vacuum,this should increase the maximum electron energy by roughly

m3Hm3H

+m3He+Erec ' 1.72 eV.

4

FIG. 1. Event rates for different configurations as a functionof the outgoing electron kinetic energy, measured with respectto the endpoint in vacuum at zero neutrino mass, K0

β ≡m3H −m3He+ −me −Krec, with Krec ' 3.44 eV. We assumea target mass of 100 g, corresponding to NT ' 2 × 1025

atoms of tritium. The presence of the graphene in the initialstate shifts the event rate to different energies and makes theabsorption peaks (CNB) much rarer, and hidden under theβ-decay part of the spectrum. For illustrative purposes wehave set mν = 0.2 eV, taken the initial wave function to be theground state of the full coverage graphane potential presentedlater in Section IV, and convoluted with an experimentalresolution described by a gaussian with full width at halfmaximum ∆ = 0.05 eV. Moreover, we only considered the twoextreme scenarios: the free helium (solid and dashed blue lines)and the helium bound to the ground state with no emission ofvibrational modes (solid and dashed red lines). Intermediateinstances will populate the regions between the blue and redlines, resulting in a smooth total rate. For comparison, wereport the rate expected if the process were to happen invacuum, i.e. with an initial free atomic 3H (gray lines).

1. Initial state effects. Quasi-particle degrees of free-dom, present before the decay, will affect the ex-perimental resolution through their thermal andzero-point motion, which contribute to the broad-ening of the initial tritium momentum.

2. Final state effects. The process of radioactive decaymay create one or several quasi-particle excitationssuch as a phonons, electron-hole pairs or plasmons.Each of them subtracts a fraction of energy fromthe electron and, being indistinguishable instances,causes further broadening.

The first effect can be illustrated by considering thesituation of a tritium atom chemically attached to a car-bon atom in a free-standing graphene, as also discussedin [26]. Since the tritium is four times lighter than thecarbon, we can neglect its effect on the vibrational spec-trum of the lattice. The velocity of the tritium atom canbe written as vT = vC + vTC, where vC is the velocity ofthe carbon, and vTC that of the tritium relative to thecarbon to which it is attached. The uncertainty on the

tritium velocity then reads

(∆vT)2

= (∆vC)2

+ (∆vTC)2. (5)

Here ∆vTC is the uncertainty in the tritium velocity dueto the localization of its wave function near a carbon atomwith which it forms a chemical bond (see the previoussection). We have neglected correlations between thezero-point vibrations of the tritium–carbon bond andzero-point motion of the carbon to which the tritium isattached (such an approximation can be justified along thelines of the Born–Oppenheimer approximation [38]). Theeffect of the initial state phonons in graphene is encodedin the thermal energy contribution (see, e.g., [39]), givenby

(∆vC)2

=∑Q

∑s

ωQ,s

2mCNcoth

(ωQ,s

2T

), (6)

with mC the mass of a carbon atom, N the number ofcarbon atoms in a sample, Q the Bloch momentum of thegraphene phonon. Moreover, s enumerates the phononpolarization branch, and ωQ,s its dispersion relation. Fortemperatures larger than the Debye temperature (thebandwidth of the phonons), the phonon contribution re-duces to the Dulong–Petit law, (∆vC)2 = T/mC. Thecontribution decreases steadily with decreasing temper-ature reaching, in the zero temperature limit, its intrin-sic quantum value, (∆vC)2 = ω/2mC, where ω is theBrillouin-zone average phonon frequency, which is on thesame order as the phonon bandwidth. The initial statevibrational modes will then contribute to the broadeningof the tritium velocity, ∆vT. This phenomenon is, how-ever, suppressed with respect to ∆vTC by factor m3H/mC,causing an order ∼ 10 % effect—see also [26].

The second class of effects, the final state ones, leadsto the broadening of the spectrum due to the followingreason: each quasi-particle will generally carry away someamount of energy bounded by its bandwidth. Even focus-ing on the softest quasi-particle, the flexural phonons [40],the branch for this channel has a span of ∼ 0.1 eV. Thisensures that the creation of one such quasi-particle leadsto the loss of required energy resolution.

What degrees of freedom can contribute to final statebroadening? The key quantity to consider is the timeover which the β-decay “happens”, since this is when theemitted electron decouples from the rest of the system.This time scale can be determined from the formation timeof the process, i.e. from the time it takes to separate theelectron wave function from the helium one. Close to theendpoint the electron carries a momentum kβ ' 139 keV.

Its de Broglie wavelength is then roughly 0.01 A, makingit point-like compared to the 3He+ atom, whose radius isrHe ∼ 1 A. The formation time is then simply given by

tβ ∼merHe

kβ∼ 10−18 s . (7)

Final state degrees of freedom which are excited overtime scales significantly larger than this are essentially

5

irrelevant for what concerns the spectrum of the observedelectron, in agreement with the so-called sudden approxi-mation (see also [26]). An example of degrees of freedomwhose excitation happens over times scales shorter thantβ , and that could hence influence the final spectrum, areelectrons located at small distances from the decay point,or the vibrational modes of the nearby carbon atom. In-deed, their excitation happens via the propagation of theCoulomb potential of the 3He+ atom, which travels atthe speed of light.

Along similar lines, since the β electron velocity is muchlarger than both the Fermi velocity and then 3He+ veloc-ity, vβ vF vHe, the helium atom and the electrons ofthe graphene can be considered as frozen on the positionsthey occupy when the reaction happens. In particular,this means that the 3He+ atom will experience the samepotential as the initial 3H. The change to the new poten-tial, in fact, follows the rearrangement of the grapheneelectrons induced by the positive charge, which happensover times significantly larger then tβ (see, again, [26]).7

Let us conclude this section with a comment. Onemight wonder whether those degrees of freedom that canindeed be excited in the final state of the reaction (atomicelectrons, lattice vibrations, etc.) can help alleviate theproblem of the exponential suppression of the matrix ele-ment discussed in the previous section, by compensatingfor the electron momentum in other ways. This is not thecase, as we now argue. Consider a set of possible degreesof freedom, whose mass we represent with mi. The totalmass of the system after the reaction (excluding the βelectron) will be M = m3He+ +

∑imi. If we denote their

positions as xi, the center of mass of the system and thedistances from the helium atom are

R =m3He+xHe +

∑imixi

M, ri = xHe − xi . (8)

After the decay, the plane wave of the β electron, whichis located at the same position as the helium atom, canhence be rewritten as

eikβ ·xβ = eikβ ·R∏i

eimiM kβ ·ri , since xβ = xHe . (9)

Assuming that the typical separation between the differentcomponents is of atomic size, a ∼ 1 A, this tells us thatlight degrees of freedom for which (mi/M)kβa . 1 can beexcited to discrete levels with sizeable probability, sincethey will not suffer from the exponential suppression de-scribed in the previous section. Heavy degrees of freedom,instead, for which (mi/M)kβa 1, must be liberated, or

7 Since Coulomb interactions are long range, the rearranginggraphene electrons and the 3He+ ion, despite being slow, couldactually play a role, by contributing to the corrections to the sud-den approximation. Indeed, such a subleading effect has alreadyproved to be of some relevance in [29, 30]. The inclusion of thiseffect is beyond the scope of the present work.

else the matrix element is strongly suppressed. Wheneverthis happens, a mechanism like the one explained in theprevious section will cause an intrinsic spread in the elec-tron energy. In Appendix B, we show this in the simplesetting of atomic tritium.

IV. THE SUBSTRATE–TRITIUMINTERACTION POTENTIAL

As already explained, the most substantial contributionto the intrinsic uncertainty on the electron energy comesfrom the localization of the initial tritium wave function.To tackle the problem at hand, it is then crucial to havedetailed knowledge of the initial state of the reactions.We therefore start by studying the interaction betweenhydrogen and graphene, given that the former has thesame chemical properties of tritium. We then also considerdifferent possible graphene derived materials. These couldbe alternatives to manipulate the tritium potential—i.e.the initial state of our reactions—and optimize it in orderto mitigate the intrinsic quantum effects.

A. Tritium on extended graphene

Let us start by first assessing the form of the potentialbetween hydrogen and extended graphene, similar to whatconsidered in [1]. The binding of the hydrogen happensto a carbon site with a C3 symmetry. The potential hastwo contributions, one perpendicular and one parallel tothe sheet. The former is simply the binding potential,while the latter is the hopping potential, which controlsthe mobility of the atom along the surface of the graphenesheet. These potentials depend sensibly on the pristinestatus of graphene (i.e. its local structure, doping andhydrogen coverage), and several ab initio studies areavailable in the literature.

Consider first the case of a single isolated atom bindingto the flat graphene. The binding energy with respectto the atomic state has been evaluated to be around0.7− 0.8 eV, arising from a potential minimum locatedaround 1.1 A. There is then a barrier of 0.2− 0.3 eV, anda van der Waals well, with minimum at about 2.5− 3 A,and a very shallow depth estimated to be 5− 7 meV (seeFigure 2, upper panel, thick black line) [27, 28, 41–48].It was also shown that the binding energy is stronglydependent on the local curvature of the sheet [47, 49–51].In particular, it increases up to an additional 1.5 eV onvery convex surfaces, as the exterior of small fullerenes [52]or on “spikes” of crumpled carbon sheets forming ongiven substrates [53]. Conversely, it decreases withinconcavities, as in the interior of carbon nanotubes. (SeeFigure 2, upper panel, colored lines [49].)

Due to a cooperative effect, the hydrogen atoms havea tendency to dimerize (see Figure 2, lower panel, greenline) [53] on the graphene surface. The formation of largeclusters of atoms bound on the same side, however, is

6

limited by the consequent creation of curvature on thesheet, which destabilizes the structure hence decreasingthe binding energy per atom (Figure 2, lower panel, greenshade [54]). In this respect, a more favorable high coveragesetup is that of dimers separated by vacant sites (red line),bearing weak global curvature, or even graphane [55](blue lines), where the sites of the triangular sublatticeare occupied on different sides of the sheet. Even in thiscondition there is a dependence on the coverage of oneside with respect to the other (blue shades), and thebinding energies are in the range 4− 6 eV [56].

The PTOLEMY proposal would like to achieve a highcoverage of tritium, to maximize the event rate. In light ofthe considerations above, fully (half) occupied graphaneare favorable conformations: these give a stoichiometry C:3H = 1:1 (2 :1), meaning a 20 (11)% of gravimetric loadingof tritium—200 (110) g of tritium per kg of material.

The hopping potential of hydrogen on graphene (i.e.its ability to move along the surface) is even less wellcharacterized in the literature, and also likely to be de-pendent on the curvature and other local features of thesheet. As prototypical examples, we considered two caseswith different coverage and local conformation (gener-ated as described in Figure 3). These represent bothpartially saturated graphane and the crumpled surface ofsupported graphene with partial covalent bonding to itssubstrate (see also Figure 2). We evaluated the hoppingprofiles within the framework of the Density FunctionalTheory (DFT, details reported in Appendix D) findinga hopping barrier around 2 eV. Its average value turnsout lower than the average desorption energy (comparewith Fig. 2), and the profiles can be asymmetric, due tothe irregular and disordered conformation of the hoppingsites. Additionally, the minima between barriers appearshallower than in the desorption profile and modulateddepending on the specific path. All this shows a depen-dence of the hopping barrier on the local geometry of thesheet, indicating the possible existence of specific pathsalong which the hopping, and hence the mobility of thetritium, is particularly favored. We now discuss one suchpossibility.

B. Tritium in nanotubes

The ideal setup to try and detect the cosmic neutrinobackground is one where the initial atomic tritium wavefunction is an eigenstate of momentum, as it would hap-pen in vacuum. In this case no intrinsic quantum effectscontribute to the uncertainty on the electron energy, whichis then dominated by the experimental resolution. All theconsiderations reported in the previous sections indicatethat flat graphene is not the optimal substrate to hostthe atomic tritium while still hoping to detect the neu-trino background. However, the modulation of bindingpotentials and barriers operated by the local puckeringand curvature of the sheet specifically suggests that con-cave sites might be favorable conformations to realize

FIG. 2. Graphene–hydrogen binding potential as a functionof the distance from the binding site. Upper panel: Bind-ing potential for a single hydrogen atom for different localcurvatures (puckering) of the binding site—see Figure 5 for adefinition. Flat graphene corresponds to d = 0 (thick blackline), while d > 0 corresponds to convex sites (outward puck-ering, as in the spikes of crumpled supported sheets shown inthe lower inset) and d < 0 to concave ones (inward puckering,as within the nanotube shown in the upper inset). The po-tentials are parametrized as an interpolation between a vander Waals potential at large distances and a binding potentialat short distances. The depth of the latter depends on thecurvature—see Appendix C for details—which has been variedbetween d = −0.35 A and d = +0.35 A. Tritium is expectedto have the same chemical properties as hydrogen. Lowerpanel: Effect of the hydrogen coverage. Same side bindinghas a positive cooperative effect for dimers (red line) or clusterup to a small value of the coverage (green line). As the clusterssize increases (i.e. when more atoms are added on the sameside) the binding destabilizes due to mechanical distortion(green shade). Conversely, two sides coverage is more stable(blue lines) although still dependent on coverage (blue shade).Representative structures are reported. Energy profiles are ob-tained with a standard Density Functional Theory calculation,as reported in Appendix D.

the desired almost-free tritium state. In this section wepropose one possible configuration: the interior of carbonnanotubes.

To check the feasibility of this proposal, we used DFTto evaluate the potential felt by a hydrogen atom inside ananotube of diameter between 4 A and 5 A, the smallest

7

FIG. 3. Hopping energy profiles on graphane at differentcoverage as a function of the reaction path (in arbitrary units),i.e. the path connecting the initial and final positions ofthe hydrogen atom. The hopping paths are indicated in thestructures reported as insets, in corresponding colors. Bothstructures are hydrogenated on the lower side with alternatingsites, for a coverage of 50%. On the upper side, instead,the alternating sites are only partially hydrogenated, for acoverage of 62% (blue) and 75% (red). The occupied sites arerepresented by the blue dots in the insets.

synthesized so far [57], as reported in Figure 4. For thebare case, we studied a nanotube with a diameter of 4.8 Aand we find that the atom is almost completely free tomove along the axis of the tube, with a potential thatis essentially flat with weak modulations coming fromroughly 70 meV barriers. However, the inspection of theenergy profile orthogonal to the axis reveals that, besidethe central minimum where the hydrogen is almost freeto slide in the center of the nanotube, there is a secondminimum corresponding to a configuration where theatom is bound to the internal surface of the tube.

The possibility for the atom to bind to the tube is clearlynot ideal, as it would prevent its free motion along the axis.This can be prevented by passivating the carbon nanotube,for example, with hydrogen bound to the external surface.In this case, due to the increase in the number of atoms,we study a nanotube with a diameter of 3.7 A. Indeed, wefind that, for a tube passivated in this way, the potentialalong the direction perpendicular to the tube does notfeature the minima near the walls anymore—see, again,Figure 4. We also observe an increase in the periodicmodulation of the potential along the tube axies, simplydue to the smaller size of the tube. Larger tubes, as thoserealized in lab, should feature an essentially flat potentialat large separations. Therefore, for single tritium atoms ina nanotube, this would realize the almost ideal situationof a free motion, at least in one dimension.

Nonetheless, if more than one tritium atom is presentin each nanotube, the recombination is not completelyprevented: the energy profiles clearly show a markedpotential well for the formation of the molecule as the twoatoms get close. However, the spin resolved calculations

show that the electronic spin ground state of the moleculeis the singlet, with null magnetization, while the electronicspin ground state of the separated atoms is the triplet.Indeed, when the initial spin configuration of the pair isS = 1, there is a small barrier that must be overcomein order to form a molecule. In particular, if the pair isforced to be in a triplet configuration all the way, ourcalculation shows the emergence of a ∼ 1 eV barrierpreventing recombination. In the next section we discusshow this mechanism could be implemented to preventdimerization of the 3H.

Before that, let us give a rough estimate of the amountof tritium that could be stored in carbon nanotubes.8

Assuming that a recombination barrier has been achieved,we see from Figure 4 that the minimum distance betweentritium atoms is ∼ 3 A, leading to a stoichiometric ratioof C : 3H of at least 10 : 1 or 20 : 1 (depending on thetube). This implies a gravimetric loading that is an orderof magnitude smaller than for graphene, i.e. between 10and 20 g of tritium per kg of material.

C. Magnetic fields to prevent dimerization

The peculiar spin ground state of the molecular hydro-gen (and tritium) offers a way to possibly prevent therecombination of two atoms. As mentioned, the molec-ular hydrogen ground state is an electronic spin singlet(S = 0), while the preferred spin state for atoms far apartis the triplet (S = 1). It follows that a sufficiently highexternal magnetic field could force the pairs to be in spintriplet, hence preventing the molecular binding. Indeed,it has already been shown that in vacuum an externalfield of 4−5 T was capable of stabilizing atomic hydrogenat the temperature a few Kelvin [59, 60]. Moreover, thelow dimensionality of our systems might help stabilizingthe atomic tritium against dimerization. In fact, the re-combination happening when the barrier is overcome islikely to be a nonadiabatic process happening through aspin state transition and the restrain to move in a singledirection might reduce the accessible pathways to thisprocess, consequently reducing its occurrence probability.As a consequence, this should increase the life time of theatomic tritium. Recall that, indeed, when the hydrogenenergy profile along the nanotube is evaluated forcing aspin triplet configuration, hence somewhat emulating theeffect of a magnetic field, a barrier disfavoring dimeriza-tion appears (see again previous Section and Figure 4).

We further note that, by conservation of angular mo-mentum, the introduction of a magnetic field parallel tothe axis of the nanotube would induce a net polarizationof the triutium sample, thus favoring the emission of elec-trons along the axis. This in turn would increase thefavorable event rate.

8 Assuming a diameter of 10 nm, one can store around 1012 nan-otubes per cm2 [58].

8

FIG. 4. Potential for a hydrogen atom inside a carbon nanotube. Left panels: Potential along the tube axis as a function ofthe relative distance between two atoms (see insets) and for different spin configurations. When the atoms are well separatedthe potential is almost flat, with small barrier depending on the spin configuration. When they get close to each other theytend to bind into a molecule, provided that their spin is (or can flip to) a singlet configuration (red and blue lines). If a tripletconfiguration is forced on the pair, a potential barrier prevents dimerization (black line). In the lower panel, the hydrogenatednanotube potential shows a pronounced modulation because the radius of the hydrogenated nanotube is smaller than the bareone. In this case the hydrogen/tritium is affected by the atomic structure of the tube. This effect is less noticeable the greaterthe radius of the nanotube, as in the top panel. Right panels: Potential perpendicular to the tube axis. For a naked tubetwo minima are present, one at the center of the tube and one close to its walls. The latter corresponds to the binding of theatoms to one of the carbon sites, which would prevent its motion along the axis. If the tube is hydrogenated, only the centralminimum is left. Illustrative structures are reported in the insets. All energy profiles are measured with respect to gaseousatomic hydrogen.

We remark that the recombination probability alsodepends on the concentration of hydrogen (tritium) in thesystem (surface or tube) and its mobility, both of whichshould be kept low to increase the half life of the atomicstate. Clearly, these needs conflict with other needs inthe PTOLEMY experiment: high tritium concentrationto improve the event rate, and high tritium mobility tomake it as close as possible to a momentum eigenstate.Therefore, a delicate balance between these parameters isneeded, that could be achieved by properly choosing thematerial and the environmental conditions.

V. CONCLUSION

The localization of the initial tritium atom on thegraphene sheet induces an intrinsic quantum spread in

the energy spectrum of the electron emitted followingeither β-decay or neutrino capture. For the simplest caseof a single tritium atom bound to a flat graphene sheet,this uncertainty is predicted to be at least an order ofmagnitude larger than the energy resolution expected inthe PTOLEMY experiment [1, 26].

In this work we determine quantitatively the expectedrate as a function of the electron kinetic energy, undera set of simplifying assumptions. Building on what washighlighted in [1], we explain the important role played bythe fate of the final 3He+ atom, in particular whether itends up in a continuous or discrete state of the potential.In the first case, the neutrino capture peak is hidden underthe β-decay part of the spectrum. In the second case,instead, and when no additional degrees of freedom areexcited, the neutrino capture peak remains well separatedfrom the β-decay continuum, but its rate is exponentially

9

suppressed, making this event highly unlikely.Nevertheless, the possibility of final states where the

3He+ atom remains bound to the graphene allows forelectron energies up to a few electronvolts higher than invacuum. Interestingly, this could be used as an experi-mental signature to test our understanding: in a setuplike the one proposed by PTOLEMY one should observeelectrons considerably more energetic than in vacuum.

We also perform a careful study of the tritium–graphenepotential, and especially of its dependence on coverage aswell as local geometry of the sheet. We propose carbonnanotubes as a possible solution to reduce the intrinsicquantum uncertainty. Inside a carbon nanotube passi-vated with hydrogen a tritium atom would be free to slidealong the axis, hence realizing in one dimension a situa-tion analogue to what happens in vacuum. Moreover, theintroduction of an external magnetic field, which forces apair of triutium atoms in a triplet configuration, mightbe able to prevent the formation of molecules.

ACKNOWLEDGMENTS

We are grateful to N. Arkani-Hamed, J. Maldacena,A. Pilloni and R. Rattazzi for important discussions.A.E. is a Roger Dashen Member at the Institute forAdvanced Study, whose work is also supported by theU.S. Department of Energy, Office of Science, Office ofHigh Energy Physics under Award No. DE-SC0009988.Further support and computational resources are pro-vided by EU under FETPROACT LESGO (AgreementNo. 952068) and by the Italian University and Re-search Ministry, MIUR under MONSTRE-2D PRIN2017KFMJ8E. G.M. acknowledges the IT center of the Uni-versity of Pisa, the HPC center (Franklin) of the IIT ofGenova, and the allocation of computer resources from

CINECA, through the ISCRA C projects HP10C9JF51,HP10CI1LTC, HP10CY46PW, HP10CMQ8ZK. We re-ceived support from the Amaldi Research Center fundedby the MIUR program “Dipartimento di Eccellenza”(CUP:B81I18001170001). TF acknowledges Fundacaode Amparo a Pesquisa do Estado de Sao Paulo (FAPESP)grants 2017/05660-0 and 2019/07767-1, and ConselhoNacional de Desenvolvimento Cientıfico e Tecnologico(CNPq) grants 308486/2015-3 and 464898/2014-5 (INCT-FNA).

Appendix A: Event rates

Let us show concretely how the arguments of Section IIcome about. For the sake of the present work, it is enoughto consider the case of a single neutrino. Let us start bycomparing the rate obtained when the 3He+ is freed fromthe graphene to that obtained when it remains in theground state of the binding potential. For simplicity weassume an isotropic wave function for the ground state,

ψ0(x) =1

π3/4λ3/2e−

x2

2λ2 . (A1)

The width is taken from the harmonic approximation ofthe graphane potential at maximum coverage (Figure 2)around its minimum, λ = (mκ)−1/4, with m the atomic3H or 3He+ mass, and κ ' 29 eV/A

2is the fitted spring

constant. When the helium remains in the ground state,the matrix element is

Mβ,gs =g

Ve−λ

2|kβ+kν |2/4 , (A2)

and the corresponding event rate—i.e. the transitionprobability rate times the number of tritium atoms—is

dRβ,gs = NT|g|2

V 2e−λ

2|kβ+kν |2/2(2π)δ(m3H −m3He+ − Eβ − Eν)V d3kβ(2π)3

V d3kν(2π)3

= NT|g|2

(2π)5e−λ

2|kβ+kν |2/2kνEνd3kβdΩν ,

(A3)

with NT the number of tritium atoms, and Ων the solidangle of the outgoing neutrino. In the second equalitywe have integrated over the neutrino energy using theδ-function. Focusing on the near-endpoint region, we cannow neglect the neutrino momentum in the exponentialand perform the final integrals, obtaining

dRβ,gs

dEβ= NT

|g|2

2π3e−λ

2k2β/2kνEνkβEβ . (A4)

The maximum electron energy for this final state isEmaxβ = Q+me−mν . Note, importantly, that here we are

considering the graphene sheet as infinitely massive, andtherefore neglecting its recoil energy. This corresponds tothe event where no vibrational modes are excited.

When the helium is, instead, freed from the graphene,its wave function is a plane wave. The matrix element is

Mβ,f =g

V 3/223/2π3/4λ3/2e−λ

2|kHe+kβ+kν |2/2 , (A5)

and the event rate is given by

10

dRβ,f = NT|g|2

V 323λ3π3/2e−λ

2|kHe+kβ+kν |2(2π)δ

(m3H − ε0 −m3He+ −

k2He

2m3He+− Eβ − Eν

)V d3kHe

(2π)3

V d3kβ(2π)3

V d3kν(2π)3

= NT|g|2

32π13/2λ3e−λ

2|kHe+kβ+kν |2d3kHekνEνd3kβdΩν , (A6)

where again we used conservation of energy to fix theenergy of the neutrino. Here ε0 = U0− 3

2

√κ/m ' 5.76 eV

is the ground state binding energy. Now we again neglectthe neutrino momentum in the exponent and, integratingover the electron and helium solid angles, we obtain

dRβ,fdEβ

= NT|g|2λ2π7/2

Eβ

×∫ kmax

He

0

dkHe e−λ2(kHe−kβ)2kHekνEν ,

(A7)

where we have neglected exponentially small terms inthe integral. The maximum helium momentum is ob-tained requiring that Eν ≥ mν , which returns kmax

He =√2m3He+(Q+me −mν − ε0 − Eβ). Correspondingly,

the maximum energy that the electron can have in thisconfiguration is Q+me −mν − ε0.

The rates for neutrino absorption are given by

dRCNB

dEβ=dσCNB

dEβNTnνvνfc , (A8)

where nν ' 56 cm−3 is the neutrino density, while fc isthe so-called clustering factor. The latter satisfies fc ≥ 1,and for sufficiently large masses follows the empirical law,fc ' 76.5 (mν/eV)2.21 [24]. The absorption cross section,σCNB, can be found in similar ways as shown above, andwe will not report the details here.

The reason why the absorption peak for a free he-lium ends up below the β-decay spectrum is that, whilethe maximum energy allowed for the absorption is stilllarger than the maximum energy for the β-decay by 2mν ,the most likely one happens when the kHe = kβ , whichgives an energy that, compared to the maximum energy

for the β-decay in this final state, is smaller by roughlymeQ/m3He+ ' 3.4 eV. We also notice that, the only in-stance where the absorption peak is well separated fromthe rest of the spectrum is the case when the electronhas the maximum allowed energy, i.e. when the heliumremains bound in its ground state. This event, however,happens with very small probability.

Appendix B: Recoil of heavy and light degrees offreedom

Here we show how other degrees of freedom can recoilafter the emission of the electron. We do that with asimple toy example: a free tritium atom. We denote asR the center of mass coordinate and as r the relativedistance of the atomic electron from the decaying/decayednucleus. The initial wave function will be given by

ψi(R, r) ∝ φ(Z=1)0 (r) , (B1)

where φ(Z=1)0 is the ground state hydrogenic wave function.

We also took the (free) center of mass to be at rest. Afterthe β-decay, the final wave function will be

ψf (R, r,xβ) ∝ eiP ·Reikβ ·xβφ(Z=2)n (r) , (B2)

where we neglect the neutrino momentum. Here φ(Z=2)n is

the wave function for some excited state of the hydrogenicatom with two positive charges. The weak matrix elementforces the β electron to be produced in the same locationas the initial tritium and final helium. The matrix elementthen reads

Mfi ∝∫d3Rd3r e−i(P+kβ)·R φ

(Z=1)0 (r)φ(Z=2)

n (r) e−i me

me+m3He+kβ ·r

∝ δ(P + kβ)

∫d3r φ

(Z=1)0 (r)φ(Z=2)

n (r) e−i me

me+m3He+kβ ·r

.

(B3)

where we used the fact that, since the β electron is onthe same position as the helium nucleus, we can writexβ = R + me

me+m3He+r.

The equation above is telling us that, on the one hand,

momentum is conserved by the recoil of the system asa whole. On the other hand, since the hydrogenic wavefunctions are localized over a distance of the order ofthe Bohr radius, a, while me

me+m3He+kβa 1, the atomic

11

electron can be excited to some higher discrete level whilestill be bound—the matrix element does not suffer fromthe exponential suppression discussed in Section II.

Appendix C: Analytic form of the binding profiles

The analytical H–C binding profile of Figure 2(a) isobtained combining two Morse potentials, one for thechemical binding and one for the van der Waals interac-tion, in a way such that the former dominates at smalldistances and the latter at large distances:

V (r) = uch(r)f(r) + uvdW

(1− f(r)

), (C1)

with

uch(r) =(εb + εoff

) [(e− r−rchαch − 1

)2

− 1

]+ εoff ,

uvdW(r) = εvdW

[(e− r−rvdW

αvdW − 1

)2

− 1

], (C2)

f(r) =1

er−Rσ + 1

.

Here rch ' 1.1 A and rvdW ' 2.7 A are respectively thechemical and van der Waals C–H distances, while thechemical and van der Waals widths are αch ' 0.2 A andαvdW ' 1.5 A. Moreover, while the depth of the van derWaals potential is fixed to be εvdW ' 6 meV, the one of thechemical potential is a function of the puckering distances,d (see Figure 5), specifically εb '

(0.8 + 4.45(d/A)

)eV

and εoff '(0.5 − 0.2(d/A) − 2.5(d/A)2

)eV. These ex-

pressions are obtained by fitting the dependence of thebarrier height and of the hydrogen binding energy on thecurvature from [49]. The parameter of the function f(r)are tuned to reproduce the correct barrier height for flatgraphene, and are given by σ ' 0.15 A and R ' 2.0 A.

Note that from this one can compute the value of thespring constant corresponding to the minimum of thepotential, obtaining

κ ' 2(εb + εoff)/α2ch . (C3)

This can be strongly reduced for large values of negatived, i.e. within concavities.

FIG. 5. Definition of the puckering distance d, i.e. the out-of-plane displacement of a carbon atom with respect to planedefined by its three neighbors, here used as a measure of thelocal curvature of the sheet. We indicate with d > 0 convexitiesand with d < 0 concavities.

Appendix D: Density Functional Theory calculation

We carried out the DFT calculations with QuantumEspresso [61–63], which uses a plane wave basis set. Thepseudopotentials were taken from the standard solid-statepseudopotential efficiency library [64–68] with cutoffs of60 Ry and 480 Ry for the wave functions and the density.The exchange-correlation potential was treated in theGeneralized Gradient Approximation, as parametrizedby the Perdew–Burke–Ernzerhof formula [69], with thevan der Waals (vdW)-D2 correction as proposed byGrimme [70]. For the Brillouin Zone integrations, we em-ployed a Marzari–Vanderbilt smearing [71] of 4×10−3 Rywith a Monkhorst–Pack [72] k-point grid with 12× 12× 1points for self-consistent calculations of the charge densityand for geometry optimization of the graphene sheets. Forthe carbon nanotubes ((0,6) naked and (0,4) H-passivated)we used supercells with the exact periodicity of 5 repeatedunits along the z-axis and Γ point self-consistent calcula-tions. We optimized the geometrical structures relaxingthe atomic positions until the components of all the forceson the ions are less than 10−3 Ry/Bohr (the supercells ofthe nanotubes were also relaxed along the direction of theaxis of the tube). For the 2D graphene-like geometrieswe employed 4× 4 and 5× 5 supercells, starting with aunit cell with a lattice constant a0 = 2.46 A [73, 74]. Weconsidered a supercell with about 18 A of vacuum alongthe orthogonal direction to avoid interaction between theperiodic images. Analogously, we left approximately 12 Aof lateral free space between nanotubes. Besides spin-unrestrained calculations, in order to mimic the effect ofan external magnetic field, we evaluated the energy profileof the hydrogen atom along the carbon nanotube addinga penalty function to the total energy to restrain the spinconfigurations to the initial ones (singlet or triplet).

The minimum energy paths of Figure 3 have beenobtained using the climbing image version [74, 75] ofthe Nudget Elastic Band (NEB) method as implementedin the QE code. This method is a modification of theregular NEB method [76, 77] to converge rigorously tothe highest saddle point (transition state) on the energysurface containing the initial and the final chosen states.

We use the VESTA [78], Xcrysden [79] and VMD[80]graphics software tools to visualize the geometrical struc-ture, to produce the plots and to generate the startingstructures of the nanotubes.

12

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