Sub-GeV Dark Matter Detection with Electron Recoils in Carbon Nanotubes
SSub-GeV Dark Matter Detection with Electron Recoils in Carbon Nanotubes
G. Cavoto, ∗ F. Luchetta, and A.D. Polosa
Sapienza University of Rome, Piazzale Aldo Moro 2, I-00185 Rome, Italy
Directional detection of Dark Matter particles (DM) in the MeV mass range could be accomplishedby studying electron recoils in large arrays of parallel carbon nanotubes. In a scattering processwith a lattice electron, a DM particle might transfer sufficient energy to eject it from the nanotubesurface. An external electric field is added to drive the electron from the open ends of the arrayto the detection region. The anisotropic response of this detection scheme, as a function of theorientation of the target with respect to the DM wind, is calculated, and it is concluded that nodirect measurement of the electron ejection angle is needed to explore significant regions of the lightDM exclusion plot. A compact sensor, in which the cathode element is substituted with a densearray of parallel carbon nanotubes, could serve as the basic detection unit.
Introduction.
Two-dimensional targets for direc-tional dark matter searches have been recently stud-ied in [1] and [2]. Array of carbon nanotubes consid-ered in [1] could work as highly transmitting channelsfor carbon ions recoiled by DM particles with masses M χ > ◦ . Interstices among carbon nanotubes(CNT) are also found to cooperate to enlarge the effec-tive channeling angle of the array: the maximal recoilangle a carbon ion can have to be channeled by the arrayand eventually be detected is computed in [3].If instead electron recoils are considered, graphenesheets are potentially very good directional targets forDM in the MeV mass range [2]. Most of the ideas onsub-GeV DM and on the possibilities for exploring andrevealing it are summarized in the report [4]; more specif-ically see Refs. [5]–[11].The electrons or ions recoiling against the hitting DMparticles on two-dimensional layers are emitted by thematerial, with reduced internal rescatterings, differentlyfrom crystalline or gaseous targets.The energy price to pay to extract valence band, π -orbital electrons from graphene is in the order of feweVs (the work function being φ wf ≈ . ). Dueto the nature of the scattering with electrons in thegraphene structure, electron recoils tend to follow thesame direction of the incident DM . Graphene layers, ori-ented perpendicularly to the DM wind, tend to emit elec-trons in the same direction, which should be immediatelycollected/detected. A measurement of the recoil anglewould provide clear directional information enhancing In addition to this, more energy is needed to eject an ionizedcarbon nucleus, necessary condition to be channeled in the nan-otube array. the capabilities of background rejection.In this note we follow the suggestion by Hochberg et al. [2] of using electron recoils from both π and sp − orbitals in graphene, but again we resort to thewrapped configuration provided by carbon nanotubes( single-wall carbon nanotubes are essentially graphenesheets wrapped on a cylindrical surface). This allows toreach a higher density of target material, i.e. smaller de-tectors, which in turn could be more easily handled andoriented in the DM wind direction. With carbon nan-otubes we find the same directional behavior of electronrecoils it is found in [2] for graphene layers. The target scheme.
When electron recoils are con-sidered, the nanotube walls cannot work as reflectingsurfaces, as they were considered in [1, 3], capable ofchanneling ions having transverse energies lower thanthe reflecting potential barriers at the boundaries (in theorder of few hundreds of eVs). On the contrary, elec-trons injected in the body of nanotubes (or in the in-terstices) by DM-electron scatterings, might go throughthe walls, with definite transmission coefficients, and un-dergo multi-scattering events, crossing several nanotubes,before exiting from the array. As a benchmark for ouranalysis, we have used the results of some experimentalstudies on the determination of transmission coefficientsof electrons through graphene planes [12–16]. More infor-mation from experiments of this kind would be extremelyuseful for determining directly also reflection and absorp-tion coefficients.The addition of an electric field E , coaxial with nan-otube parallel axes works to drive the ejected electronsto the detection region in the direction opposite to thesubstrate, where the nanotubes have been deposited on— see Fig. 1. Following [1, 3], we consider to align thenanotube axes in the DM wind direction in order to getmost of the recoils in that direction. The alignment canbe kept fixed by a continuous mechanical tracking sys-tem.We assume that the carbon nanotube array is engi-neered as a forest of metallic nanotubes, on a conductingplate. An opposite electrode makes an electric field E directed to the former with field lines concentrated as on a r X i v : . [ h e p - ph ] N ov sharp edges, at the nanotube ends. If R is the averagedistance between the axes of two nanotubes in a squarearray and r < R is the nanotube radius, the electric fieldintensity will increase at the extremity as in E (cid:48) ≈ R r E (1)With R (cid:39)
50 nm and r (cid:39) E (cid:48) ≈
500 kV/cm at the ends ofthe nanotubes with E ≈
10 kV/cm – the typical elec-tric fields used to collect ejected electrons (see discussionbelow). E (cid:48) must not be large enough to produce fieldemission electrons, a potentially important background.As from the Fowler-Nordheim theory of field emissionfrom metallic carbon nanotubes (see for example [17]),the characteristic field emission currents are j ( µA ) (cid:39) . E (cid:48) φ / exp (cid:18) − . φ / E (cid:48) (cid:19) coth (cid:18) . φ / E (cid:48) r (cid:19) (2)where E (cid:48) is expressed in V/nm, the work function is φ (cid:39) r is in nm. Theexpected current per nanotube at 500 kV/cm is negligibe, j ≈ exp( − ) µA , even when multiplied by the wholenumber of nanotubes in the array. We observe here thata further reduction of the electric field at the nanotubeends can be reached by decreasing the average relativedistance R among the seeds on which the nanotubes aregrown.As reminded above, electron recoils are mainly for-ward, keeping track of the DM direction. We call N + = N ( θ w ) the number of electrons reaching the detection re-gion as a function of the angle θ w between the averagedirection of the DM wind and the carbon nanotube (par-allel) axes, oriented along the direction from the closedbottom to the open ends. We also define N − = N (180 ◦ )and we will seek for θ w angles giving the largest asym-metry A ( θ w ) = N + − N − N + + N − (3)On the basis of the results obtained in [2], we expect A to be maximal in correspondence of θ w (cid:39) ◦ where theemission cross section is higher. However, the electronswhich will most likely reach the driving electric field arethose recoiled along the axes of nanotubes — see thediscussion below on the low energy transmission throughcarbon nanotubes.According to our simulations, an asymmetry as largeas A ∼ . A could bemeasured with 5 σ experimental significance by countinga total number of about 60 events, in absence of back-ground.To observe the anisotropy A , there is no need to mea-sure the ejection angle of recoiling electrons. Only an FIG. 1: Scheme of the basic unit of the carbon nanotube ar-ray. The grey sectors represent the sections of two close-bynanotubes. The detection apparatus is located on the side ofthe open ends of the array (on the right of the scheme). Elec-trons can also be transmitted through the nanotubes walls orreflected/absorbed. efficient electron counting and mechanical tracking sys-tem is needed. This might allow to consider a detectionapparatus in which, for instance, the carbon nanotubesarray target is replacing cathode of a compact device.We will illustrate in what follows how we reach theseconclusions.
Trajectories and the absorption coefficient.
The basic unit of a carbon nanotube array is sketchedin Fig. 1. A collision with a DM particle might generatea ‘top’ electron, which aims in the direction towards theopen ends of the nanotubes, where the detection appa-ratus is located, or a ‘bottom’ electron, which is insteaddirected towards the substrate, where it is always ab-sorbed. Top electrons might be reflected, transmitted orjust stopped/absorbed by the nanotube walls.In the conditions described, electrons are in the the 1-10 eV energy range, i.e. , they have negligible resolutionpower for atoms and nuclei. These long wavelength elec-trons (between 4 and 12 ˚A) interact with portions of thegraphene (or nanotube surfaces) producing a diffractionpattern in transmission, as discussed in [18] and [19]. Thelargest intensity is expected in the forward direction, ascan be seen from Fig. 9 in [19] and Fig. 4 in [18]. Sec-ondary maxima are found at angles θ between the inci-dent electron wave-vector k and the final one k (cid:48) givenby sin( θ/
2) = (cid:0) λ/ l (cid:1) (cid:112) m + m + m m in the elasticapproximation | k | ∼ | k (cid:48) | — here l is the bound length l (cid:39) .
14 nm and m , are integers.Almost everywhere in the range 1 < E <
10 eV, theprevious equation has only the solution m , m = 0 and θ = 0, corresponding to forward transmission. In the up-per part of the energy range E (cid:38) z component since k (cid:48) − k = q with q inthe graphene plane. To the best of our knowledge, thereare no experiments giving precise information about therelative weight of transmissions versus reflections fromgraphene monolayers. Specular reflections of low energyelectrons are expected to occur at low energies.Significant changes in the size of momentum, | k (cid:48) | (cid:28) | k | , correspond to ‘absorptions’. The condition | ( k − q ) / k | (cid:28) q is the momentum exchanged with thelattice) suggests that electrons grazing the surface of nan-otubues can be more easily absorbed.Indeed in the experiments reported in [12–16] it isfound that the largest part of the electron beam imping-ing orthogonally to the graphene monolayer (depositedon a surface with holes) is transmitted or reflected.The probability of transmission T , reflection R andabsorption C are introduced ( T + R + C = 1). In thefollowing, as suggested in [12–16] we will keep T + R to be the dominant fraction thus varying only the small C = 1 − ( T + R ) value. The chosen values of C are sug-gested in the quoted references; in particular in [16], C ismeasured to be C ∼ − for electrons with E (cid:46) P on it are chosen. An electron is ejected from P with some random direction. Electrons facing thesubstrate/open-ends are labelled as the ‘bottom’/‘top’ones. Positions and velocities can be computed at ev-ery step of the simulation (using an Euler algorithm).This allows to reconstruct the whole trajectory. At ev-ery intersection of the trajectory with any nanotube inthe array, a transmission/reflection/absorption is decidedwith probabilistic weights T, R, C . Some few trajectoriesterminate neither on the side where the open ends ofthe nanotube array are, nor on the substrate: there is alimited number of ‘side’ electrons, similarly to what wasfound for ‘side’-ions in the simulations discussed in [3].The calculation are repeated for an arbitrary number ofinitial electrons.The average distance spanned by recoiled electrons inquasi-parallel directions to nanotube axes (within 0 ◦ ÷ ◦ ), is several hundreds µ m, in the range of absorptioncoefficients we are considering — the length of alignedcarbon nanotubes being ≈ µ m. Results.
Following [2], we consider now the colli-sion of DM particles with graphene electrons in both π and sp − orbitals, with cross section given by Eq. (10),in the Appendix. This depends on the | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | factor, which measures the probability for the recoiledelectron to have 3-momentum k (cid:48) when the value of theexchanged 3-momentum with the graphene lattice is q and (cid:96) is the lattice momentum in the Brillouin zones.In our description, single-wall carbon nanotubes corre-spond to wrapped graphene planes. Thus, first of all,we have reproduced, with perfect agreement, the resultsillustrated in [2] and then considered periodic boundaryconditions on graphene planes to get the appropriate (cid:101) ψ functions.The diameter of a nanotube is of the order of 2 R =100 ˚A and we consider electrons having recoil kinetic en- ergies in the range ∼ −
10 eV, corresponding to deBroglie’s wavelengths 4 < λ <
12 ˚A. At λ , the nanotubecurved surface is almost identical to the tangent plane,being R (cid:29) λ . It is therefore reasonable to assume thatelectrons do not resolve the curvature of the nanotubes,as if they were locally interacting with graphene planes.The differential rates obtained with Eq. (23) are dis-played in Fig. 2 for an indicative value of the DM massof M χ = 5 MeV. To compute the rates we have as-sumed a Maxwell-Boltzmann velocity distribution f ( v )for the DM particle velocities in the laboratory frame(see Eq. (25)).The three curves reported in Fig. 2, are relative tothree different orientations θ w of the DM wind averagedirection ¯ v with respect to the carbon nanotube parallelaxes (both sp and π –electrons are considered).Consider a graphene plane oriented orthogonally to theDM wind. Consequent electron recoils will be orientedin the same direction. In the case of carbon nanotubes,given the curvature of the surface the DM will collideon, the effect is slightly modified with respect to whatfound on graphene sheets. Despite the fact that largestelectron recoil rates are for θ w (cid:39) ◦ (and electron re-coils are in the forward region) the largest fraction ofelectrons collected in the detection region correspondsto those ejected with a small angle with respect to CNTaxes. In this sense, the detector is ‘ directional ’: the largernumber of countings is expected when the CNT axes arein the direction of Cygnus.The asymmetry defined in Eq. (3), in absence of back-ground, is A (0) ≈ .
4. Changing C by a an orderof magnitude, A changes by ≈ π − orbitals and from sp − hybridized orbitals.To obtain a 5 σ evidence of a non-zero asymmetry, wecompute the exposure, in units of the target mass timesthe data acquisition time for a fixed value of the absorp-tion coefficient and M χ = 5 MeV. This quantity expressesthe amount of target mass needed or the number of yearsof exposure to appreciate a statistically significant asym-metry. We find that M · t (kg · day) (cid:39)
16 (4)Varying C by an order of magnitude, M · t varies by a ap-proximately a factor of 2. Calculations are done includingboth electrons from π − orbitals and from sp − hybridizedorbitals and with a σ eχ ∼ − cm . There are not many papers on the experimental determinationof transmission coefficients of graphene at these energies. Has-sink [12] measure a transmittance of 0 . − . −
30 eV. There is no direct experimental information on ab-sorption coefficient and in our simulations we adopt several plau-sible values.
FIG. 2: Differential rates of ejected electrons per year per kg, distributed in the recoil energy E R for M χ = 5 MeV. We includeboth sp and π –orbital electrons. The three curves reported are relative to three different orientations of the DM wind maindirection with respect to the carbon nanotube parallel axes. The plot reported here is found with Eq. (23) and the addition ofthe absorption probability at every hit with the CNT. E R corresponds to the kinetic energy of the electrons emitted from thesurface of the CNTs, having overcome the work function φ wf . We have to underscore here that in order to measurea certain degree of asymmetry A , we do not need to pre-cisely measure the electron recoil direction. We only needto count the electrons reaching the detection region. A compact apparatus.
We consider an array ofsingle-wall metallic carbon nanotubes positioned in vac-uum and in a uniform electric field directed parallel toCNT axes. CNTs are held at a fixed negative poten-tial. Field lines will concentrate on the open ends of thisCNT cathode, like on sharp edges, as described in Fig. 1and commented in the Introduction. Electrons ejectedby collisions with DM particles will travel in vacuum re-gions among (or within) CNTs and will eventually reachthe region where the electric field is intense. Once there,electrons will be further accelerated in an electric field ofseveral kV/cm towards the anode where a silicon diodeis located, as in a hybrid light sensor (HPD or HAPD).The signal produced by a collision with a single DMparticle is expected to be represented by single electroncount . Therefore, the detector has to be devised to dis-criminate between single and multi-electron signals. Thismight be obtained with HPD-type sensors, having an in-trinsically low gain fluctuation, when coupled to a verylow electronic noise amplification stage. Notice that inthis configuration, given the very low rate of interaction,neither fast nor highly segmented sensors are required.On the other hand, we expect photons from radioac-tivity to convert into the CNT target array. This would generally produce electrons with keV or higher ener-gies. These events are expected to extract several elec-trons from the CNT cathode. Therefore the signal-to-background discrimination, at this level, is that betweensingle-electron and multi-electron counts.The detection element can be replicated to reach therequired target mass. Eventually, two arrays of elementscan be installed on a system that is tracking the Cygnusapparent position. Two CNT arrays can be installed ina back to back configuration: in one the open ends arein the direction of the Cygnus (where the DM wind isexpected to come from). A different counting rate isthen expected on the two arrays, maximally exploitingthe anysotropy of the detection apparatus. More sophis-ticated schemes might require the use of magnetic andelectric fields, such as the one sketched in [2].We conclude that the anisotropic response studied inthis note allows to use existing technology with the sub-stitution of the photocathode element only, and makingthem blind to light. This makes our proposal easy totest experimentally and scalable to a large target mass.For the sake of illustration, assume a 1 × sub-strate coupled to a single photo-diode channels. Onthis substrate a number of 10 , 10 nm diameter CNTscan be grown. Since the surface density of a graphenesheet is 1/1315 gr/m , a single-wall CNT weights about50 × − grams. This is equivalent to ∼
10 mg on asingle substrate. In the case of HPD, O(10 ) units per100 g CNT are needed. In principle, the system is scal-able at will, since the target mass does not need to beconcentrated in a small region.Single electrons counts can be triggered by environ-ment neutrons as well. This is a well known sourceof background afflicting all direct DM search experi-ments and the screening techniques are the standardones. Thermal neutrons have scattering lengths of fewfermis with electrons in graphene, but they have notenough energy to extract them efficiently from the ma-terial. A neutron moderation screen, as those currentlyused in these kind of experiments, has to be includedwhen devising the apparatus. We assume that workingwith compact units as HPDs, this kind of screening mightbe achieved more easily than with other configurations.Another source of single electron counts, which belongsto similar configurations too, is the electron thermo-emission. This can strongly be attenuated by coolingthe device down to cryogenic temperatures. However, asnoted in [21], the thermionic electron current from an ef-fective surface of 1 m of graphene should definitely benegligible at room temperatures being proportional to j ≈ T exp ( − φ wf /kT ) (5)This is essentially due to the fact that the work-function φ wf in graphene is almost three times as large than thetypical work-function of photocathodes.As for the field emission, this has also been studied in[18] where it is found that its starts being significant forelectric fields above 1V/nm, way larger than the ones weconsider, see (2). Conclusions.
We have shown that single wall car-bon nanotube arrays might serve as directional detectorsalso for sub-GeV DM particles, if an appropriate externalelectric field is applied and electron recoils are studied.An appreciable anisotropic response, as large as A ∼ . et al. [2] on DM scatter-ing on graphene planes and adapted to the wrapped con-figuration of single wall carbon nanotubes. The fact that with a coefficient β = 115 . A/m K − . carbon nanotubes, and interstices among them in the ar-ray, almost behave as empty channels is still an essentialfeature to obtain the results of the calculations describedhere. The mean free paths attainable in these configura-tions are definitely higher if compared to dense targetsas graphite or any crystal. We also observe that, in thedetection scheme proposed, differently from [1], small ir-regularities in the geometry of nanotubes are inessential.For comparison with previous work, we present theexclusion plot, see Fig. 3, which can be obtained withthe detection configuration here proposed. We performa full calculation including π and sp − electrons. The FIG. 3: We compare our results with those obtained byHochberg et al. [2]. Calculations are done including bothelectrons from π − orbitals and from sp − hybridized orbitals.The exposure of 1 kg × year is used. latter figure summarizes the potentialities of the schemeproposed. They result to be very much comparable towhat found in [2], although with rather different appara-tus and practical realization. To conclude, we notice thatthe device here described might be used alternatively asa detector of heavier DM particles. Just by changing thedirection of the electric field, one could count positivecarbon ions recoiled out of and channeled by the carbonnanotubes (or within the interstices among them), as inthe original proposal [1] [3]. Acknowledgements.
We are very grateful to YonitHochberg for several comments and suggestions on themanuscript and to Chris Tully for informative discus-sions. We also thank Maria Grazia Betti, Carlo Mari-ani and Francesco Mauri for several useful hints on thephysics of CNTs. We thank an anonymous referee forextremely useful comments and suggestions. G.C. ac-knowledges partial support from ERC Ideas ConsolidatorGrant CRYSBEAM G.A. n.615089.
Appendix: DM-electron scattering.
In this Ap-pendix we report the essential formulae we have used toobtain the results in the text. We have adapted the ex-pressions in [2] to the configuration with CNTs.The M χ DM mass needed to eject electrons fromgraphene is about 3 MeV at the galactic escape velocity.In the χe − scattering process, part of the momentum isexchanged with the crystal lattice — this is especiallytrue for exchanged momenta smaller than the inverseof the average spacing between atoms a = 0 .
14 nm =0 .
721 keV − .The energy released by the DM particle in the scatter-ing process is∆ E = E i ( (cid:96) ) + φ wf + k (cid:48) / m e (6)where − E i ( (cid:96) ) is the energy of the electron in the valenceband (depending on the lattice momentum (cid:96) ), − φ wf isthe work function of graphene and k (cid:48) is the momentumof the ejected electron. Let q be the DM four-momentumdifference before and after the scattering. One finds that∆ E = v · q − q M χ (7)Therefore the cross section of the χe − scattering is dσ (cid:96) = 1 F | M eχ ( q ) | d p (cid:48) (2 π ) ε (cid:48) d k (cid:48) (2 π ) E (cid:48) | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | × (2 π ) δ (cid:18) E i ( (cid:96) ) + φ wf + k (cid:48) m e − v · q + q M χ (cid:19) (8)where F is the flux F = 4 εE | v | = 4 M χ m e | v | and M eχ ( q )is the amplitude of the transition process. Electrons in π − orbitals have rather soft kinetic energies which allow E (cid:39) m e . Here p (cid:48) and ε (cid:48) are related to the DM afterthe collision with the electron. The factor | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | measures the probability for the recoiled electron to have3-momentum k (cid:48) for q , (cid:96) fixed [2] (and has dimensions ofeV − ). We also consider electrons from sp − hybridizedorbitals. The explicit forms of E i ( (cid:96) ) and (cid:101) ψ ( q − k (cid:48) , (cid:96) )are different for the two cases [2]; see also [20]. Thecalculations are done separately for the four orbitals(three in the sp − hybridized configuration). Electrons in π − orbitals are more weakly bound and more sensible tolight DM particles. At higher recoil energies σ -electronsdominate.The definition is used | M eχ ( α m e ) | πm e M χ ≡ σ eχ µ eχ (9)where σ eχ is the cross section of the non-relativistic χe − elastic scattering and µ eχ is the reduced mass ofthe electron-DM system. In most calculations we use σ eχ (cid:39) − cm as a benchmark value for the cross sec-tion. It is found σ (cid:96) = σ eχ µ eχ π ) | v | (cid:90) d p (cid:48) d q | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | × δ (cid:18) E i ( (cid:96) ) + φ wf + k (cid:48) m e − v · q + q M χ (cid:19) (10)The Dirac delta function defines a minimum speed for χ to eject the electron v min = ∆ E | q | − | q | M χ (11) If the minimum speed were higher than the Milky Wayescape velocity ( v min > v esc + v = 550 + 220 Km / sec)the process would simply be forbidden. If we assume that | q | (cid:28) M χ , then ∆ E | q | (cid:39) v min < v esc + v (12)which in turn means | q | (cid:38) . v esc + v (cid:39) . | q | < a − = 8 . . (cid:46) | q | (cid:46) . R = N C ρ χ M χ A cu (cid:90) (cid:96) ∈ B d (cid:96) (2 π ) d v f ( v ) v σ (cid:96) (15)with A cu = √ a unit cell area of the graphene and B the first Brillouin zone of the reciprocal lattice . N C is the number of carbon atoms per kg, N C =5 × kg − . ρ χ /M χ is the DM number density, with ρ χ (cid:39) . being the local density. Finally f ( v )is the velocity distribution to be defined below.In the specific case of single wall carbon nanotubes,which is of interest in this paper, periodic boundary con-ditions are imposed on the argument (cid:96) of | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | .If x is the coordinate on the boundary of the nanotube,and r is its radius, at fixed altitude z , the conditionexp( i ( x + 2 πr ) (cid:96) x ) = exp( ix(cid:96) x ) (16)leaves (cid:96) y continuous in the integral (15) whereas a dis-crete sum on (cid:96) x has to be taken (cid:90) d (cid:96) → (cid:88) n (cid:90) d(cid:96) y (17)where (cid:96) x = n/r , in the first Brillouin zone. In the follow-ing this substitution is understood.Replacing the cross section formula for σ (cid:96) in (15) it isfound R = N (cid:90) d (cid:96) (cid:90) d k (cid:48) d q | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | × (cid:90) v max v min ( (cid:96) ,k (cid:48) ,q ) d v f ( v ) δ ( v min | q | − v · q ) (18)where N = 12(2 π ) N C ρ χ A cu M χ σ eχ µ eχ (19) Observe that (cid:82) (cid:96) ∈ B d l (2 π ) = A cu . and v min ( (cid:96) , k (cid:48) , q ) = E i ( (cid:96) ) + φ wf + k (cid:48) m e | q | − | q | M χ (20)In Eq. (18) v max is computed solving the inequality v ( v − v cos θ ) ≤ ( v − v ) (21)and d v = v dv d cos θ dφ (22)We can turn to the differential rate in the recoil energy.Differentiating the recoil energy of the electron E r = k (cid:48) M χ one has dRd ln E r = E r dRdE r = k (cid:48) dRdk (cid:48) which allows to writethe differential cross section distribution dRd ln E r = N (cid:90) k (cid:48) d Ω k (cid:48) (cid:90) d (cid:96) (cid:90) q max q min dφ q dq | q | | (cid:101) ψ ( q − k (cid:48) , (cid:96) ) | × (cid:90) v max v min ( (cid:96) ,k (cid:48) ,q ) d Ω v dv v f ( v ) (cid:90) d cos θ q δ (cid:16) cos θ q − v min v (cid:17) (23)where θ q is the angle between q and v (and φ q the az-imuthal angle around v ) δ ( v min | q | − v · q ) = 1 | q | v δ (cid:16) cos θ q − v min v (cid:17) (24)The Maxwell-Boltzmann distribution of velocities f ( v )in the Galaxy is f ( v ) = α e − ( v − v v θ ( v esc − | v − v | ) (25)where v (cid:39)