Heisenberg's uncertainty as a limiting factor for neutrino mass detection in β-decay
aa r X i v : . [ h e p - ph ] J a n Heisenberg’s uncertainty as a limiting factor for neutrino mass detection in β -decay Yevheniia Cheipesh, ∗ Vadim Cheianov, and Alexey Boyarsky Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: January 28, 2021)In order to explore the potential of beta-decay experiments to detect relic neutrinos (C ν B), westudy the intrinsic irreducible uncertainty in energy of electrons resulting from the zero-point motionof the emitter. Motivated by the PTOLEMY proposal [1], we consider the case of Tritium adsorbedon graphene. We show that in this case the intrinsic uncertainty is larger than the target resolutionof the measurement. We discuss nevertheless that the ambitious goal of detecting C ν B could stillbe achieved with further refinement of the experimental design. One option is to substitute Tritiumwith a heavier emitter; another is to design an angle-resolved detection setup where the emitter hasa higher degree of mobility along the substrate.
Introduction. — The Cosmic Neutrino Background(C ν B) is a potential source of invaluable informationabout the early Universe. Due to their extremely weakinteraction with matter neutrinos were able to travelfreely in the pre-recombination age carrying the “pho-tographic” image of the universe at the neutrino decou-pling transition. The weakness of the neutrino-matterinteraction does, however, pose a challenge in readingthat image. Although indirect signatures of C ν B havebeen observed [2], direct detection of the relic neutrinoremains an outstanding problem. The concept of suchan experiment is well known [1, 3, 4]. The challengingpart is to reach an adequate number of the C ν B neutrinocapture events keeping sufficient energy resolution.The magnitude of the challenge is illustrated in FIG. 1,which shows the β -emission spectrum including a hypo-thetical relic neutrino contribution [5] of monoatomic Hin vacuum. One can see that the spectrum is dominatedby the spontaneous β -decay background, shown in red,while the signal due to the relic neutrino capture pro-cess comprises of a series of tiny peaks, shown in green.Not only are the predicted C ν B peaks quite miniscule,consisting of only a few events per year for 100 g of Tri-tium, but they also are positioned within a few tens ofmeV from the massive spontaneous decay background,which implies that the energy resolution of the experi-ment needs to be at the level of 10 −
20 meV . It is dueto the low event count of the sample combined with theinsufficient energy resolution caused by the excitation ofinternal motions of the Tritium molecule [6, 7], that thebest to date experiment, KATRIN [8], using Tritium ingaseous form falls short of achieving the required sensi-tivity.An experimental design aiming to simultaneously getrid of molecular vibrations, increase the amount of Tri-tium per unit of volume and the control over the emit-ted electrons as well as an outstanding energy resolutionof the apparatus has been proposed by the PTOLEMYcollaboration [1]. In this setup, atomic Tritium is de-posited on graphene sheets arranged into a parallel stack. ∗ [email protected] However, as we will show later, the proximity of Tritiumatoms to a solid state system also comes at a price of anuncertainty in the recoil energy due to various uncontrol-lable interactions. This leads to irreducible intrinsic limi-tations on the energy resolution that should be accountedfor along with the energy resolution of the apparatus.In general, the interaction of an adsorbed radioactiveatom with graphene is complicated and it gives rise toseveral effects each contributing to the broadening ofthe measured β -emission spectrum. In this paper, weonly focus on one which is arguably the simplest and thestrongest of all: the zero-point motion of an atom dueto the chemical bonding. We calculate the change in themeasurable spectrum induced by this effect and discussthe consequences for the possibility of C ν B detection.
Defining the problem — Following the proposals [1, 3,4] we consider the β -decay channels of atomic Tritium H → He + e + ¯ ν e ν e + H → He + e (1)The main goal of the C ν B detection experiments isto detect the electrons produced in the neutrino capturechannel (see FIG. 1) that depends on the mass of thelightest neutrino and the hierarchy [1, 9–11]. Since thecaptured relic neutrinos are soft, it has a shape of 2 nar-row peaks [12] separated from the end of the main part ofthe spectrum by double the mass of the lightest neutrino.The spectrum depicted on FIG. 1 is calculated for anisolated Tritium atom in the rest frame, where the recoilenergy is defined by the conservation laws. However, ifTritium is absorbed on a substrate, it can not be consid-ered at rest and the recoil energy of the nucleus acquiressome amount of uncertainty and so does the measuredspectrum of the emitted electron (see FIG. 3).Two complementary views on such an uncertainty arepossible, both leading to the same conclusion in thepresent context. In the “semiclassical” view the source ofthe uncertainty is the zero-point fluctuations of the ve-locity of the Tritium atom, which result in a fluctuatingdefinition of the centre of mass frame at the moment of β -decay. In the fully quantum view the uncertainty re-sults from quantum transitions of an atom into the highlyexcited vibrational states in the potential which confines m ν − . − .
05 0 0 .
05 0 . E el − ( Q − E rec ) [eV] d Γ d E e l [ y r − e V − ] Ideal detectorPTOLEMY
FIG. 1.
The spectrum of the β -decay in the centre of themass reference frame. The β -decay emission lines are coloredin red: dashed corresponds to a perfect detector and full to adetector that has a finite energy resolution b . The green linecorresponds to the C ν B emission line b . a for PTOLEMY experiment the energy resolution is taken to be10 meV b Assuming the normal hierarchy it to the graphene sheet. We shall begin our discussionwith the semiclassical picture.It follows from Heisenberg’s uncertainty principle thatan atom restricted to some finite region in space by thebonding potential cannot be exactly at rest. Even in thezero temperature limit it performs a zero-point motion sothat its velocity fluctuates randomly obeying some prob-ability distribution F ( u ) . For localized states, F ( u ) hasa vanishing mean and dispersion defined by the Heisen-berg uncertainty principle ∆ u ∼ ~ /m nucl λ nucl . Due tothese random fluctuations in the velocity of the nucleus,the observed velocity distribution of the emitted electronin the laboratory frame is given by the convolution˜ G ( v ) = Z du F ( u ) G ( v + u ) . (2)where G ( v ) is the velocity distribution of an electronemitted by a free Tritium atom at rest corresponding tothe energy distribution given by a Fermi Golden Rule (seeFIG. 1). The formal applicability condition of Eq. (2) isthat the energy level spacing for the out-states of the He + ion after the decay be much less than the typicalrecoil energy ∆ ε ≪ E rec . This condition is readily sat-isfied for the recoil energy in vacuum E rec = 3 .
38 eV . We shall revisit this argument when we turn to the fullyquantum picture.In the following analysis we will restrict ourselves tothe particular case of the Tritium atoms adsorbed onthe graphene following the PTOLEMY proposal. How-ever the obtained results are also valid for more generalbonding potentials (see the discussion at the end). In the zero temperature limit, the function F ( u ) ap-pearing in Eq. (2) is encoded in the wave function of thestationary state of a Tritium atom in the potential of theinteraction of the atom with graphene. Although such apotential has a rather complicated shape, as can be seenfrom multiple ab-initio studies [13–16], the large mass ofthe nucleus justifies the use of the harmonic approxima-tion near a local potential minimum U = 12 κ i,j r i r j + U where r i are the components of the atom’s displacementvector and κ is the Hessian tensor. Then, it follows that F ( u ) is a multivariate normal distribution F ( u ) = 1(2 π ) / √ det Σ exp − X i,j =1 u i Σ − i,j u j . (3)with zero mean and a covariance matrix Σ = ~ m / √ κ. To find the latter, we proceed to the analysis of the bond-ing potential near its minima.An adsorbed Tritium atom is predicted to occupy asymmetric position with respect to the graphene lattice,characterised by a C point symmetry group. For thisreason, the Hessian will generally have two distinct prin-cipal values, one corresponding to the axis orthogonal tographene and one to the motion in the graphene planeyielding two differnt potential profiles.According to the ab initio studies [13–16], the poten-tial that bonds the Tritium atom in the perpendiculardirection has two minima, a deep chemisorbtion mini-mum (in the range of 0 . − . . d (cid:2) (cid:6) A (cid:3) B i nd i n g e n e r g y λ osc FIG. 2.
Schematic profile of the potential that bonds theTritium atom in the direction perpendicular to the graphene. an atom is governed by the so-called migration poten-tial [18]. The stiffness of the migration potential in thecase of chemisorption smaller than the vertical stiffness,however is substantial, as can be seen from Table I. Thecase of a substrate producing a negligible migration po-tential will be discussed below.Introducing the normal displacement z of an atom rel-ative to the potential minimum, we can approximate the Potential Source κ, h eV / (cid:6) A i λ, (cid:2) (cid:6) A (cid:3) ∆ E, [eV][15] 2.15 0.16 0.60Chemisorption [13], GGA 4.62 0.13 0.73[13], vdW-DF 4.9 0.13 0.75Physisorption [16] 0.08 0.37 0.26[15] 0.09 0.34 0.28[13], GGA 0.18 0.29 0.33[13], vdW-DF 0.13 0.32 0.3[14], GGA 0.04 0.43 0.22[14], LDA 0.01 0.55 0.17Migration [18] 0.283 0.264 0.37TABLE I. Harmonic fit with the stiffness κ of the chemisop-tion, physisorption potentials and the migration potentialof the chemisorbed atom profiles near the minimum. λ = ~ / √ m nucl κ and ∆ E is the energy broadening of the emittedelectron estimated from Eq. (5). potential in the direction perpendicular to the grapheneas U ( z ) = κz / U . The uncertainty in the position ofthe nucleus is then characterised by the oscillator length λ = ~ / √ m nucl κ. The values of the constants κ and λ fordifferent potential minima obtained from the fitting ofthe theoretical bonding profiles [13–16] are given in Ta-ble I. The pronounced variability in the predicted valuesof the spriing constant κ is explained by the diversity ofapproximations used in different ab initio schemes. Note,however that the variability in the predicted values of theoscillator length is much less significant as λ ∼ κ − . Forthis reason one can crudely neglect the difference betweenthe strength of the lateral and normal confinement andconsider the function F ( u ) as approximately isotropic F ( u ) ≈ √ π ∆ u exp (cid:18) − u ∆ u (cid:19) . (4)We also note that, according to the Table I, the typicalpredicted oscillator length is about an order of magnitudeless than the typical length of the bond, which providesa posterior justification for the harmonic approximation. Estimate — We are now in a position to obtain anestimate for the uncertainty in the energy of an emittedelectron. By virtue of Heisenberg’s uncertainty principle,the variance of the velocity of the nucleus near a localpotential minimum is ∆ u ≈ ~ /m nucl λ. For an electronemitted at speed v el in the centre of mass frame the un-certainty of the energy measured in the laboratory frameis ∆ E ≈ m el v el ∆ u, which near the edge of the electronemission spectrum can be written as∆ E ≈ ~ cλ el γ (5)where λ ≡ ~ / √ m el κ and we have introduced the di-mensionless parameter γ = (cid:20) Q m el m c (cid:21) / (6) The values for ∆ E as predicted for Tritium adsorbedon graphene according to different ab initio calculationsare given in Table I. One can see that ∆ E is approx-imately 0 . ν B merged and completely over-lap with the continuum background and hence will notvisible in an experiment.To conclude our semiclassical analysis, we have foundthat adsorption of a β -emitter to graphene, or indeedany other surface, is bound to create an uncertainty inthe energy spectrum of an emitted electron. We have de-rived an estimate for such an uncertainty and, in case ofTritium adsorbed on graphene, found it to be prohibitivefor the observation of the relic neutrino signal. A fullyquantum derivation of the same result can be found inthe supplementary material. Discussion — Motivated by the outstanding technicalcharacteristics of the apparatus used to measure electronemission spectrum [1], we studied the intrinsic irreduciblelimitations on the energy resolution. One source of suchlimitation considered in this paper is zero-point motion ofthe emitter. Tritium absorbed on a substrate experiencesquantum zero motion, in accordance to Heisenberg un-certainty principle. As a result, an additional irreducibleuncertainty to the energy spectrum should be taken intoaccount. As a result, we observed, that the uncertainty inthe energy of the electrons emitted by Tritium absorbedon graphene is around 0 . − . ν B channel (see FIG. 3).An important conclusion of our work is that the de-pendence of the energy broadening on the stiffness ofthe bonding potential is ∆ E ∝ κ / . In particular, itmeans that a whole class of experimental designs thatuse bounded Tritium will have comparable limitation onthe energy resolution. The final electron spectrum inthe C ν B channel may be both continuous or discrete,depending on the bonding potential, but the overall en-velope will be Gaussian with the width ∆ E . This inagreement with the previous results for the molecularTritium [6] (the molecular potential is stronger than thebonding potential of graphene).This finding raises the question as to what possiblemodifications of the system could potentially improvethe visibility of C ν B. As was argued before, changing thestiffness of the bonding potential amounts to a very weakchange in ∆ E . An order of magnitude improvement in∆ E, which is needed for the PTOLEMY experiment re-quires a four orders of magnitude reduction in the valueof κ. Such a substantial deformation of the bonding po-tential may present a significant experimental challenge. − − . . . E el − ( Q − E rec ) [eV] d Γ d E e l [ y r − e V − ] β -spectrumC ν B − .
05 0 0 .
05 0 . . . . . . . . . . . . . . . . . . . . . . E el − ( Q − E rec ) [eV] γ × FIG. 3.
The estimate of the smearing of the electron emission spectrum due to the bonding of the Tritium atom tographene. Left panel: The electron emission spectrum for the physisorbed atomic Tritium ( λ osc = 0 . (cid:6) A). The red line depictsthe β -decay emission and the green line corresponds to the C ν B emission line. Right panel: The dependence of the emissionspectrum (blue lines correspond to the fixed event rate d Γ /dE el measured in yr − eV − ) both on the energy of electron andthe dimensionless parameter γ that characterizes the emitter, Eq. (6) (for the physisorbed Tritium γ ≈ × − ). One can seethat with increasing γ the two C ν B peaks merge with the continuum. C ν B peak is visibleC ν B peak is not visible < − o v e r l a p p i n g e v e n t s p e r y e a r > v e r l a p p i n g e v e n t s p e r y e a r
10 20 30 40 5000 . . . . m ν [meV] γ × FIG. 4.
Visibility of the C ν B peak depending on the massof the lightest neutrino and a dimensionless parameter γ thatcharacterizes the emitter, Eq. (6) (for the physisorbed Tritium γ ≈ × − ). The visibility is defined by the number of C ν Bevent that overlap with the continious spectrum.
A possible avenue is to search for substrates with sub-stantially weaker lateral potential. In the limiting caseof the perfect in-plane mobility, electrons emitted par-allel to the graphene will not have any additional un-certainty in the energy. Correspondingly, for the out-of-plane angles θ < θ max = arcsin (∆ E max / ∆ E ) theenergy uncertainty will be bounded by ∆ E max . Here∆ E denotes the energy uncertainty for the isotropic casewith finite mibility. Restricting the detection collectionto θ < θ max reduces the number of events by a factor η − ≈ πθ max / ◦ . As an example, for ∆ E max = 10 meVone obtains θ max ≈ ◦ , η ≈
10. This direction requires a full in-depth analysis which we leave for future studies.An alternative route to achieve a better performanceof the detector would be through the parameter γ. Thisparameter only depends on the internal properties of a β -emitter such as the mass of the nucleus and the energyreleased in the decay process. Therefore, to improve theenergy resolution of the experiment one needs to searchfor the β -emitter that minimizes γ while simultaneouslysatisfying other experimental constaints, e.g. sufficientlylong half-life time. The effect of the parameter γ on thevisibility of the C ν B peak is shown on the right-handpanel of FIG. 3 and FIG. 4. One can see that, e.g.,Tritium which has γ ≈ × , lies deep inside the regionwhere the obervation of the C ν B peak is impossible.The zero-point motion of a Tritium atom adsorbedon a substrate does not exhaust the list of mechanismsthat introduce uncertainty and errors into the beta-decayspectrum. Other mechanisms which we believe to be sig-nificant are the electrostatic interaction of the ionizedatom with the sheet, charge relaxation in graphene, X -ray edge singularity, etc. We leave these questions forfurther studies. Acknowledgements — We are grateful to Chris Tully,A.P. Colijn and the whole PTOLEMY collaborationfor fruitful discussions and feedback on the manuscriptthat allowed for its significant improvement. We alsothank Kyrylo Bondarenko and Anastasiia Sokolenko forthe useful discussion. YC is supported by the fund-ing from the Netherlands Organization for Scientific Re-search (NWO/OCW) and from the European ResearchCouncil (ERC) under the European Union’s Horizon2020 research and innovation programme. AB is sup-ported by the European Research Council (ERC) Ad-vanced Grant “NuBSM” (694896). VC is grateful tothe Dutch Research Council (NWO) for partial support, grant No 680-91-130. [1] E. Baracchini, M. Betti, M. Biasotti, A. Bosca, F. Calle,J. Carabe-Lopez, G. Cavoto, C. Chang, A. Cocco, A. Col-ijn, et al. , arXiv preprint arXiv:1808.01892 (2018).[2] B. Follin, L. Knox, M. Millea, and Z. Pan, Physicalreview letters , 091301 (2015).[3] Y. Li, Z.-z. Xing, and S. Luo, Physics Letters B ,261 (2010).[4] A. Faessler, R. Hodak, S. Kovalenko, and F. Simkovic,arXiv preprint arXiv:1304.5632 (2013).[5] J. Hamann, S. Hannestad, G. G. Raffelt, I. Tamborra,and Y. Y. Wong, Nuclear Physics B-Proceedings Supple-ments , 72 (2011).[6] L. Bodine, D. Parno, and R. Robertson, Physical ReviewC , 035505 (2015).[7] A. Faessler, R. Hod´ak, S. Kovalenko, and F. ˇSimkovic, in Quarks, Nuclei and Stars: Memorial Volume Dedicatedto Gerald E. Brown (World Scientific, 2017) pp. 81–91.[8] J. Wolf, K. Collaboration, et al. , Nuclear Instrumentsand Methods in Physics Research Section A: Acceler-ators, Spectrometers, Detectors and Associated Equip-ment , 442 (2010).[9] S. Mertens, T. Lasserre, S. Groh, G. Drexlin, F. Glueck,A. Huber, A. Poon, M. Steidl, N. Steinbrink, and C. Weinheimer, Journal of Cosmology and AstroparticlePhysics , 020 (2015).[10] S. S. Masood, S. Nasri, J. Schechter, M. A. T´ortola, J. W.Valle, and C. Weinheimer, Physical Review C , 045501(2007).[11] M. Betti et al. (PTOLEMY), JCAP , 047 (2019),arXiv:1902.05508 [astro-ph.CO].[12] Each of the peak corresponds to a separate flavour.[13] M. Moaied, J. Moreno, M. Caturla, F. Yndur´ain, andJ. Palacios, arXiv preprint arXiv:1405.3165 (2014).[14] D. Henwood and J. D. Carey, Physical Review B ,245413 (2007).[15] H. Gonz´alez-Herrero, E. Cort´es-del R´ıo, P. Mallet,J. Veuillen, J. Palacios, J. G´omez-Rodr´ıguez, I. Brihuega,and F. Yndur´ain, 2D Materials , 021004 (2019).[16] V. Ivanovskaya, A. Zobelli, D. Teillet-Billy, N. Rougeau,V. Sidis, and P. Briddon, The European Physical JournalB , 481 (2010).[17] We note, that we use the results of ab initio calculationsfor hydrogenated graphene. This is appropriate becauseHydrogen is chemically equivalent to Tritium.[18] D. Boukhvalov, Physical Chemistry Chemical Physics , 15367 (2010). Appendix A: Quantum derivation of the energy uncertainty
The aim of the fully quantum derivation is to underpin the semiclassical heuristic that was obtained in the maintext as well as demonstrating its limitations. We note that we will not keep track of the pre-factors ~ , c and willrestore them in the end. The rate of β -emission of an electron is given by the Fermi Golden Rule rule d Γ dE = X f π | h f | ˆ V | i i | δ ( E i − E f ) δ ( E − E f, el ) . (A1)Here the vector | i i represents the initial state of the system having the energy E i , the vector | f i , represents a finaleigenstate of the Hamiltonian having the energy E f = E f, el + E f, He where E f, el , is the kinetic energy of the outgoingelectron and E f, He , is the energy of the He + ion. The sum is performed over all such final states. The interactionpotential ˆ V is responsible for β -decay vertex and is for our purposes an ultralocal product of the creation andannihilation operators of the fields involved in the process.We make an assumption that the neutrino has zero kinetic energy. It is equivalent to restricting ourselves to regionnear the edge of the spectrum, which is exactly the region of interest to us. The energy conservation implies ~k m el + ~p m nucl = ˜ Q, (A2)where ~k , ~p - are two-dimensional final momenta of the electron and nucleus respectively. ˜ Q is the total energy of thenucleus before β -decay.The initial state of the system is a product of a plane wave state of an incoming relic neutrino, which it is safe todescribe as a plane wave with nearly zero momentum, and the lowest energy eigenstate of a Tritium atom in the localminimum of the bonding potential. As was discussed in the main text, such a state can be safely approximated as aground state of a harmonic oscillator with two distinct principal stiffness eigenvalues (see table I). The wave funcionof such a state has the form ψ i ( r ) ∝ exp − z λ ⊥ − ̺ λ k ! , (A3)where z stands for the orthogonal displacement and ̺ for the magnitude of the lateral displacement relative to thelocal potential minimum. Due to the in-plane symmetry of the graphene with respect to rotation, we can effectivelyrestrict ourselves to a two-dimensional space z, ̺ .The space of all possible final states | f i is quite large, and their wave functions may be quite complicated dueto the intricate interaction of the He + ion with the graphene sheet. However, as we shall see momentarily thedominant contribution to the sum in (A1) comes from the states which are amenable to the WKB approximation andare therefore analytically tractable. Introducing the notation ψ f ( r ) for the final state of the He + ion, we write thematrix element in (A1) as h f | ˆ V | i i ∼ Z d r ψ ∗ f ( r ) ψ i ( r ) e − i kr (A4)where k is the wave vector of the emitted electron at kinetic energy close to Q. Since the electron’s wave vector isquite large k ∼ ˚ A − the rapid oscillations suppress the integral in Eq. (A4) unless the state ψ f ( r ) also containsan oscillatory factor, which has a roughly opposite De Broglie wave vector near r = 0 , where the support of ψ i ( r )is concentrated. This implies that the kinetic energy of the ion needs to be on the order of 3eV, which exceeds thepredicted chemisorption binding energy [13–16] and is orders of magnitude greater than the vibrational quantum nearthe potential minimum ( ~ ω ∼ .
01 eV). Such highly excited states are generally characterised by a level spacing whichis much narrower than the vibrational quantum near the minimum. They are also well described by semiclassicalWKB wave functions, which on the scale of the oscillator length are indistinguishable from a plane wave.With these considerations in mind, the application of the Fermi Golden Rule to such states gives d Γ dE ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ dx Z ∞−∞ dy Z ∞−∞ dz exp − i ( k x + p x ) x − i ( k y + p y ) y − i ( k z + p z ) z − x λ k − y λ k − z λ ⊥ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A5)where we have extended the integration over z to −∞ . One can do it since the integrand is localized. k/p x,y,z , arerespectively the components of the electron and nucleus momenta that satisfy the energy conservation law | p | = r m nucl (cid:16) ˜ Q − E el (cid:17) | k | = p m el E el (A6)We re-scale coordinates ˜ r i = r i √ λ i and obtain d Γ dE ∝ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ d ˜ x Z ∞−∞ d ˜ y Z ∞−∞ d ˜ z exp (cid:16) − i √ λ k ( k x + p x ) − i √ λ k ( k y + p y ) − i √ λ ⊥ ( k ⊥ + p ⊥ )˜ z − ˜ x − ˜ y − ˜ z (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) , (A7)that can be brought to a Gauss integral d Γ dE ∝ e − λ ⊥ ( k ⊥ + p ⊥ ) − λ k ( k k + p k ) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ d ˜ x Z ∞−∞ d ˜ y Z ∞−∞ d ˜ z exp − (cid:18) ˜ x + iλ k ( k x + p x ) √ (cid:19) − (cid:18) ˜ y + iλ k ( k y + p y ) √ (cid:19) − (cid:18) ˜ z + iλ ⊥ ( k z + p z ) √ (cid:19) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A8)where k k /p k = k x /p x + k y /p y , p ⊥ /p ⊥ = k z /p z . Integrating Eq. A8 gives the Gaussian distribution d Γ dE ∝ e − λ ⊥ ( k ⊥ + p ⊥ ) − λ k ( k k + p k ) . (A9)
1. Variance of the energy shift
The distribution Eq. (A9) depends on the angles of the emitted nucleus and electron ϕ , . These angles are takenrelative to the axes perpendicular to the graphene substrate. d Γ dE ∝ e − λ ⊥ ( | k | cos ϕ + | p | cos ϕ ) − λ k ( | k | sin ϕ + | p | sin ϕ ) , (A10) . . . · − . . Q − E el [eV] ∝ d Γ d E e l [ y r − e V − ] ϕ nucl = π − ϕ el ϕ el =0.0 πϕ el =0.02 πϕ el =0.04 πϕ el =0.06 π . . . . . . Q − E el [eV] ∝ d Γ d E e l [ y r − e V − ] ϕ el = 0 ϕ nucl =1.0 πϕ nucl =0.98 πϕ nucl =0.96 πϕ nucl =0.94 πϕ nucl =0.92 π FIG. 5.
Distribution function (not normalized) of the energy of the electron near the edge of the spectrum. Electron andnucleus are emitted with the corresponding angles ϕ e/ nucl (relative to the axes perpendicular to the graphene substrate). Let us estimate the variance of this distribution for the normal emission of the electron d Γ dE ∝ e − λ ( k − p ) , (A11)where k = √ m el E el , p = r m nucl (cid:16) ˜ Q − E el (cid:17) .In order to obtain the variance, wee need to expand near the maximum of the distribution that corresponds to itsmean. If we write everything in terms of the deviation from the mean energy of the electron δE el = ˜ Q − E rec − E el k = r m el (cid:16) ˜ Q − E rec − δE el (cid:17) ≈ r m el (cid:16) ˜ Q − E rec (cid:17) (cid:18) − δE el
2( ˜ Q − E rec ) (cid:19) p = p m nucl ( E rec + δE el ) ≈ p m nucl E rec (cid:18) δE el E rec (cid:19) . (A12)Accounting to the fact that E rec ≈ m el m nucl ˜ Q , k ≈ q m el ˜ Q (cid:18) − δE el Q (cid:19) p ≈ q m el ˜ Q (cid:18) m nucl m el δE el Q (cid:19) . (A13)With this we obtain Gaussian distribution d Γ dE ∝ exp (cid:18) − λ m m el ˜ Q δE (cid:19) , with the variance with the restored units is σ = ~ λ q ˜ Qm el m nucl ..