Event rates for the scattering of weakly interacting massive particles from 23 Na and 40 Ar
aa r X i v : . [ nu c l - t h ] S e p Event rates for the scattering of weakly interacting massive particlesfrom Na and Ar R. Sahu ∗ , V.K.B. Kota † National Institute of Science and Technology, Palur Hills, Berhampur-761008, Odisha, India and Physical Research Laboratory, Ahmedabad 380 009, India (Dated: September 23, 2020)Detection rates for the elastic and inelastic scattering of weakly interacting massive particles(WIMP) off Na are calculated within the framework of Deformed Shell Model (DSM) based onHartree-Fock states. First the spectroscopic properties like energy spectra and magnetic momentsare calculated and compared with experiment. Following the good agreement for these, DSM wavefunctions are used for obtaining elastic and inelastic spin structure functions, nuclear structurecoefficients etc. for the WIMP- Na scattering. Then, the event rates are also calculated with agiven set of supersymmetric parameters. In the same manner, using DSM wavefunctions, nuclearstructure coefficients and event rates for elastic scattering of WIMP from Ar are also obtained.These results for event rates and also for annual modulation will be useful for the upcoming andfuture WIMP detection experiments involving detectors with Na and Ar.
I. INTRODUCTION
There is now universal agreement among the cosmolo-gists, astronomers and physicists that most of the massof the universe is dark [1–3]. There are overwhelmingevidences to believe that the dark matter is mostly non-baryonic. Also, data from the Cosmic Background Ex-plorer (COBE) [4] and Supernova Cosmology project [5]suggest that most of the dark matter is cold. The non-baryonic cold dark matter is not yet observed in earth-bound experiments and hence its nature is still a mys-tery. Axions are one of the candidates for dark matterbut they are not yet observed [1, 6]. However, the mostpromising nonbaryonic cold dark matter candidates arethe Weakly Interacting Massive Particles (WIMP) whicharise in super symmetric theories of physics beyond thestandard model. The most appealing WIMP candidateis the Lightest Supersymmetric Particle (LSP) (lightestneutralino) which is expected to be stable and interactsweakly with matter [1, 7].There are many experimental efforts [8–11] to detectWIMP via their scattering from the nuclei of the detectorproviding finger-prints regarding their existence. Some ofthese are Super CDMS SNOLAB project, XENON1T,PICO-60, EDELWEISS and so on; see for example[11–14]. Nuclei Na, Ar, Ga, Ge, As, I, Cs and
Xe are among the popular detector nu-clei; see [10, 11, 15] and references there in. Our fo-cus in this paper is on Na and Ar. The SodiumIodide (NaI) Advanced Detector (NAID) array experi-ment is a direct search experiment for WIMP operated byUK Dark Matter Collaboration in North Yorkshire [16];the NaI contains Na. Similarly, the DAMA/NaI andDAMA/LiBRA [17] experiments investigated the pres-ence of dark matter particles in the galactic halo using ∗ [email protected] † [email protected] the NaI(Tl) detector. In these experiments, the predictedannual modulation was not yet confirmed [11]. Other re-lated experiments with NaI detectors are ANAIS [18] andDM-Ice [19]. Also, there are the important DARKSIDE-50 [20] and DEAP-3600 [21] experiments using lquid Ar-gon (with Ar) as detector.Let us add that direct detection experiments are ex-posed to various neutrino emissions. The interactionof these neutrinos especially the astrophysical neutrinoswith the material of the dark matter detectors known asthe neutrino floor is a serious background source. Re-cently the coherent elastic scattering of neutrinos off nu-clei (CE ν NS) has been observed at the Spallation Neu-tron Source at the Oak Ridge National Laboratory [22]employing the technology used in the direct detection ofdark matter searches. The impacts of the neutrino flooron the relevant experiments looking for cold dark matterwas investigated for example in [23].There are many theoretical calculations which de-scribe different aspects of direct detection of dark mat-ter through the recoil of the nucleus in WIMP-nucleusscattering. For elastic scattering, we need to considerspin-spin interaction coming from the axial current andalso the more dominant scalar interaction. For inelasticpart, scalar interaction practically does not contribute.The scalar interaction can arise from squark exchange,Higgs exchange, the interaction of WIMPs with gluonsetc. Suhonen and his collaborators have performed aseries of truncated shell model calculations for this pur-pose [24–28]. In these studies, for example they havecalculated the event rates for WIMP-nucleus elastic andinelastic scattering for Kr and
Te [28] and also I, , Xe and
Cs [26]. In addition, recently Vergadoset al [29] examined the possibility of detecting electronsin the searches for light WIMP with a mass in the MeVregion and found that the events of 0.5-2.5 per kg-y wouldbe possible. Few years back full large-scale shell-modelcalculations are carried out in [15, 30] for WIMP scat-tering off , Xe, I, Ge, F, Na, Al and Sinuclei. Finally, using large scale shell model [31] andcoupled cluster theory [32] WIMP-nucleus and neutrino-nucleus scattering respectively, with Ar, are studied.In recent years, the deformed shell model (DSM),based on Hartree-Fock (HF) deformed intrinsic stateswith angular momentum projection and band mixing,has been established to be a good model to describethe properties of nuclei in the mass range A=60-90 [33].Among many applications, DSM is found to be quite suc-cessful in describing spectroscopic properties of mediumheavy N=Z odd-odd nuclei with isospin projection [34],double beta decay half-lives [35, 36] and µ − e conversionin the field of the nucleus [37]. Going beyond these ap-plications, recently we have studied the event rates forWIMP with Ge as the detector [38]. In addition to theenergy spectra and magnetic moments, the model is usedto calculate the spin structure functions, nuclear struc-ture factors for the elastic and inelastic scattering. Fol-lowing this successful study, we have recently used DSMfor calculating the neutrino-floor due to coherent elasticneutrino-nucleus scattering (CE ν NS) [23] for the candi-date nuclei Ge, Ga, As and
I. We found thatthe neutrino-floor contributions may lead to a distortionof the expected recoil spectrum limiting the sensitivityof the direct dark matter search experiments. In [10],DSM results for WIMP scattering from I, Cs and
Xe are described in detail. To complete these stud-ies that use DSM for the nuclear structure part, in thepresent paper we will present results for WIMP- Naelastic and inelastic scattering and WIMP- Ar elasticscattering. Now we will give a preview.Section II gives, for completeness and easy reading ofthe paper, a brief discussion of the formulation of WIMP-nucleus elastic and inelastic scattering and event rates. InSection III the DSM formulation is described with exam-ples drawn from As spectroscopic results. In SectionIV, spectroscopic results and also the results for elas-tic and inelastic scattering of WIMP from Na are pre-sented. Similarly, WIMP- Ar elastic scattering resultsare presented in Section V. The results in Sections IV andV are the main results of this paper. Finally, concludingremarks are drawn in Sect. VI.
II. EVENT RATES FOR WIMP-NUCLEUSSCATTERING
WIMP flux on earth coming from the galactic halois expected to be quite large, of the order 10 per cm per second. Even though the interaction of WIMP withmatter is weak, this flux is sufficiently large for the galac-tic WIMPs to deposit a measurable amount of energy inan appropriately sensitive detector apparatus when theyscatter off nuclei. Most of the experimental searches ofWIMP is based on the direct detection through their in-teraction with nuclei in the detector. The relevant theoryof WIMP-nucleus scattering is well known as available inthe papers by Suhonen and his group and also in ourearlier papers mentioned above [24–26, 28, 38]. For com- −10−8−6−4−20246 − − − o xx − + + + − − − − − + + − + + + o oo o x xx xx xx xx x E n e r g y ( M e V ) As E=−30.23 (MeV)Q=28.02K=3/2 − xx − ,3/2 + + ,3/2 − FIG. 1: HF single-particle spectra for As corresponding tolowest prolate configuration. In the figure, circles representprotons and crosses represent neutrons. The HF energy E inMeV, mass quadrupole moment Q in units of the square of theoscillator length parameter and the total azimuthal quantumnumber K are given in the figure. pleteness we give here a few important steps. In the caseof spin-spin interaction, the WIMP couples to the spinof the nucleus and in the case of scalar interaction, theWIMP couples to the mass of the nucleus. In the ex-pressions for the event rates, the super-symmetric part isseparated from the nuclear part so that the role playedby the nuclear part becomes apparent.
A. Elastic scattering
The differential event rate per unit detector mass fora WIMP with mass m χ can be written as [1], dR = N t φ f dσd | q | d v d | q | (1)Here, φ which is equal to ρ v/m χ is the dark matter fluxwith ρ being the local WIMP density. Similarly, N t stands for the number of target nuclei per unit mass and f is the WIMP velocity distribution which is assumed tobe Maxwell-Boltzmann type. It takes into account thedistribution of the WIMP velocity relative to the detector(or earth) and also the motion of the sun and earth. If weneglect the rotation of the earth about its own axis, then v = | v | is the relative velocity of WIMP with respect tothe detector. Also, q represents the momentum transferto the nuclear target which is related to the dimensionless Elastic scattering − − − − (9/2 − ) (13/2 − ) (17/2 − ) − − − − − − − E n e r g y ( M e V ) EXPT. DSM As FIG. 2: Comparison of DSM results with experimental datafor As for collective bands with negative parity. The exper-imental values are taken from [40] variable u = q b / b being the oscillator lengthparameter. The WIMP-nucleus differential cross sectionin the laboratory frame is given by [24–26, 28, 38] dσ ( u, v ) du = 12 σ (cid:18) m p b (cid:19) c v dσ A ( u ) du ; (2)with dσ A ( u ) du =( f A ) F ( u ) + 2 f A f A F ( u ) + ( f A ) F ( u )+ (cid:2) Z (cid:0) f S + f S (cid:1)(cid:3) | F Z ( u ) | + (cid:2) ( A − Z ) (cid:0) f S − f S (cid:1)(cid:3) | F N ( u ) | + 2 Z ( A − Z ) (cid:2) ( f S ) − ( f S ) (cid:3) | F Z ( u ) || F N ( u ) | . (3)where F Z ( u ) and F N ( u ) denote the nuclear form factorsfor protons and neutrons respectively. In Eq. (3), thefirst three terms correspond to spin contribution com-ing mainly from the axial current and the other threeterms stand for the coherent part coming mainly from thescalar interaction. Here, f A and f A represent isoscalarand isovector parts of the axial vector current and sim-ilarly f S and f S represent isoscalar and isovector parts of the scalar current. The nucleonic current parameters f A and f A depend on the specific SUSY model employed.However, f S and f S depend, beyond SUSY model, on thehadron model used to embed quarks and gluons into nu-cleons. The normalized spin structure functions F ρρ ′ ( u )with ρ , ρ ′ = 0,1 are defined as F ρρ ′ ( u ) = X λ,κ Ω ( λ,κ ) ρ ( u )Ω ( λ,κ ) ρ ′ ( u )Ω ρ Ω ρ ′ ;Ω ( λ,κ ) ρ ( u ) = q π J i +1 ×h J f k A X j =1 [ Y λ (Ω j ) ⊗ σ ( j )] κ j λ ( √ u r j ) ω ρ ( j ) k J i i (4)In the above equation ω ( j ) = 1 and ω ( j ) = τ ( j ); notethat τ = +1 for protons and − j λ is the spherical Bessel function. The static spin matrixelements are defined as Ω ρ (0) = Ω (0 , ρ (0). Now, the eventrate can be written as h R i = Z − dξ Z ψ max ψ min dψ Z u max u min G ( ψ, ξ ) dσ A ( u ) du du (5)In the above, G ( ψ, ξ ) is given by G ( ψ, ξ ) = ρ m χ σ Am p (cid:18) m p b (cid:19) c √ πv ψe − λ e − ψ e − λψξ (6)Here, ψ = v/v , λ = v E /v , ξ = cos ( θ ). Parametersused in the calculation are the following: the WIMP den-sity ρ = 0 . Gev/cm , σ = 0 . × − cm , massof proton m p = 1 . × − kg. The velocity of thesun with respect to the galactic centre is taken to be v = 220 Km/s and the velocity of the earth relative tothe sun is taken as v = 30 Km/s. The velocity of theearth with respect to the galactic centre v E is given by v E = p v + v + 2 v v sin( γ ) cos ( α ) where α is the mod-ulation angle which stands for the phase of the earth onits orbit around the sun and γ is the angle between thenormal to the elliptic and the galactic equator which istaken to be ≃ . ◦ . Using the notations, X (1) = F ( u ), X (2) = F ( u ), X (3) = F ( u ), X (4) = | F Z ( u ) | , X (5) = | F N ( u ) | , X (6) = | F Z ( u ) || F N ( u ) | the event rateper unit mass of the detector is given by h R i el =( f ) D + 2 f A f A D + ( f A ) D + (cid:2) Z (cid:0) f S + f S (cid:1)(cid:3) D + (cid:2) ( A − Z ) (cid:0) f S − f S (cid:1)(cid:3) D + 2 Z ( A − Z ) (cid:2) ( f S ) − ( f S ) (cid:3) D , (7)where D i being the three dimensional integrations ofEq.(5), defined as D i = Z − dξ Z ψ max ψ min dψ Z u max u min G ( ψ, ξ ) X ( i ) du (8)The lower and upper limits of integrations given in Eq.(5)and (8) have been worked out by Pirinen et al [28] andthey are ψ min = cv (cid:18) Am p Q thr µ r (cid:19) / (9) ψ max = − λξ + s λ ξ + v esc v − − v v − v v sin ( γ ) cos ( α )(10)With the escape velocity v esc from our galaxy to be 625km/s, the value of v esc /v − − v /v appearing in Eq.(10) is 7 . v /v ) sin ( γ )is 0 . u min and u max are Am p Q thr b and 2( ψµ r bv /c ) , respectively. Here, Q thr is the detec-tor threshold energy and µ r is the reduced mass of theWIMP-nucleus system. B. Inelastic scattering
In the inelastic scattering the entrance channel andexit channel are different. The inelastic scattering crosssection due to scalar current is considerably smaller thanthe elastic case and hence it is neglected. Hence, wefocus on spin dependent scattering. The inelastic eventrate per unit mass of the detector can be written as h R i in = ( f ) E + 2 f A f A E + ( f A ) E (11)where E , E and E are the three dimensional integra-tions E i = Z − dξ Z ψ max ψ min dψ Z u max u min G ( ψ, ξ ) X ( i ) du . (12)The limits of integration for E , E and E are [26, 28] u min ( max ) = 12 b µ r v c ψ " ∓ s − Γ ψ (13)where Γ = 2 E ∗ µ r c c v (14)with E ∗ being the energy of the excited state. ψ max issame as in the elastic case and the lower limit ψ min = √ Γ. The parameters like ρ , σ etc. have the same valuesas in the elastic case. III. DEFORMED SHELL MODEL
The nucleonic current part has been separated fromnuclear part in the expression for the event rates forelastic and inelastic scattering given by Eqs. (7) and (11) respectively with X ( i ) giving the nuclear structurepart. However, the D i ’s and E i ’s depend not only onthe nuclear structure part but also on the kinematicsand assumptions on the WIMP velocity. The evalua-tion of X ( i ) depends on spin structure functions and theform factors. We have used DSM for the evaluation ofthese quantities. Here, for a given nucleus, starting witha model space consisting of a given set of single particle(sp) orbitals and effective two-body Hamiltonian (TBME+ spe), the lowest energy intrinsic states are obtained bysolving the Hartree-Fock (HF) single particle equationself-consistently. We assume axial symmetry. For exam-ple, Fig. 1 shows the HF single particle spectrum for Ascorresponding to the lowest prolate intrinsic state. Usedhere are the spherical sp orbits 1 p / , 0 f / , 1 p / , and0 g / with energies 0.0, 0.78, 1.08, and 3.20 MeV, respec-tively, while the assumed effective interaction is the modi-fied Kuo interaction [39]. Excited intrinsic configurationsare obtained by making particle-hole excitations over thelowest intrinsic state. These intrinsic states χ K ( η ) do nothave definite angular momenta. Hence, states of goodangular momentum are projected from an intrinsic state χ K ( η ) and they can be written as, ψ JMK ( η ) = 2 J + 18 π √ N JK Z d Ω D J ∗ MK (Ω) R (Ω) | χ K ( η ) i (15)where N JK is the normalization constant. In Eq. (15), Ωrepresents the Euler angles ( α , β , γ ) and R (Ω) which isequal to exp( − iαJ z )exp( − iβJ y )exp( − iγJ z ) representsthe general rotation operator. The good angular mo-mentum states projected from different intrinsic statesare not in general orthogonal to each other. Hence theyare orthonormalized and then band mixing calculationsare performed. This gives the energy spectrum and theeigenfunctions. Fig. 2 shows the calculated energy spec-trum for As as an example. In the DSM band mixingcalculations used are six intrinsic states [23]. Let us addthat the eigenfunctions are of the form | Φ JM ( η ) i = X K,α S JKη ( α ) | ψ JMK ( α ) i . (16)The nuclear matrix elements occurring in the calculationof magnetic moments, elastic and inelastic spin struc-ture functions etc. are evaluated using the wave functionΦ JM ( η ). For example the calculated magnetic momentsfor the 3 / , 3 / and 5 / states are (in nm units)1.422, 1.613 and 0.312 compared to experimental val-ues [40] 1.439, 0.98 and 0.98 respectively. The calculatedvalues are obtained using bare gyromagnetic ratios andthe results will be better for the excited states if we take g pℓ = 0 . g nℓ = 0 . g ps = 4 and g ns = −
3. The neutronspin part is small and hence donot appreciably contributeto the magnetic moments of the above three states. Useof effective g -factors are advocated in [41]. Spectroscopic results −18−14−10−6−2 + + + o + + + + + + + + + o o x xx x E n e r g y [ M e V ] Na E=−66.19 (MeV)Q=17.32K=3/2 + FIG. 3: HF single-particle spectra for Na corresponding tolowest configuration. In the figure, circles represent protonsand crosses represent neutrons. The HF energy E in MeV,mass quadrupole moment Q in units of the square of the os-cillator length parameter and the total azimuthal quantumnumber K are given in the figure.
IV. RESULTS FOR WIMP- NA SCATTERING
The nuclear structure plays an important role in study-ing the event rates in WIMP-nucleus scattering. Hence,we first calculate the energy spectra and magnetic mo-ments within our DSM model for Na. Agreement withexperimental data will provide information regarding thegoodness of the wave functions used. This in turn willgive us confidence regarding the reliability of our pre-dictions on event rates. These spectroscopic results arepresented in Section IV-A. Let us add that in SectionsIV-B and C the value of the oscillator length parameter b is needed and it is taken to be 1.573 fm for Na. Inour earlier work in the calculation of transition matrixelements for µ − e conversion in Ge [37], we had takenthe value of this length parameter as 1.90 fm. Assum-ing A / dependence, the above values of the oscillatorparameter is chosen for Na.
A. Spectroscopic results
In the Na calculations, O is taken as the inert corewith the spherical single particle orbitals 0 d / , 1 s / and0 d / generating the basis space. ”USD” interaction ofWildenthal with sp energies − . − . . + + + + (11/2 + ) + + + + + E n e r g y [ M e V ] EXPT. DSM Na FIG. 4: Comparison of deformed shell model results with ex-perimental data for Na for yrast band which is of positiveparity. The experimental values are taken from [40] interaction is known to be quite successful in describingmost of the important spectroscopic features of nuclei inthe 1 s d -shell region [42]. For this nucleus, the calcu-lated lowest HF single particle spectrum of prolate shapeis shown in Fig. 3. The odd proton is in the k = 3 / + de-formed single particle orbit. The excited configurationsare obtained by particle-hole excitations over this lowestconfiguration. We have considered a total of five intrinsicconfigurations. As described above, angular momentumstates are projected from each of these intrinsic configu-rations and then a band mixing calculation is performed.The band mixed wave functions S JKη defined in Eq. (16)are used to calculate the energy levels, magnetic momentsand other properties of this nucleus.The calculated levels are classified into collective bandson the basis of the E2 transition probabilities betweenthem. The results for lowest positive parity band for Naare shown in Fig. 4. The experimental data are from Ref.[40]. For this nucleus, the ground state is 3 / + which isreproduced in our calculation. A positive parity bandbuilt on 3 / + has been identified for this nucleus. Thisband is quite well reproduced by the DSM calculation.An analysis of the wave functions shows that this bandmainly originates from the lowest HF intrinsic configu-ration shown in Fig. 3. However, there are admixturesfrom the good angular momentum states coming fromother intrinsic configurations. The wavefunction com-ing from the lowest HF intrinsic configuration slightly in-creases in value with increased angular momentum. Thisshows that the collectivity of this band does not changeappreciably at higher angular momentum. Since we areconsidering WIMP-nucleus scattering from ground stateand low lying positive parity states, the negative paritybands are not important for the present purpose. Spectroscopic results −4 −3 −2 −1 F F ProtonNeutron u | F ( u ) | F ρρ ’ FIG. 5: Spin structure functions and squared proton and neu-tron form factors for Na for the ground state.
In the calculation of the event rates, spin plays an im-portant role. Hence, magnetic moment of various low-lying levels in Na are calculated. The result for theground state of the lowest K = 3 / + band is 2.393 nmand the corresponding available experimental data valueis 2.218 nm. The contribution of protons and neutronsto the orbital parts are 0.957 and 0.262 and to the spinparts are 0.267 and 0.014, respectively. This decomposi-tion gives better physical insight. The calculated valueof magnetic moment for the ground state agrees quitewell with experimental data [40]. Let us add that thereare no experimental data for the magnetic moments ofthe excited states. An approach with state-dependentgyromagnetic moments, as advocated for example in [41]reproduces better the experimental magnetic moments.The DSM spectroscopic results are also in good agree-ment with the full shell model calculations reported in[15, 30]. B. Results for elastic scattering
The DSM wave functions given by Eq. (16) are usedto calculate the normalized spin structure functions givenin Eq. (4) and also the squared nuclear form factors forthese nuclei. Their values are plotted in Figs. 5 as a func-tion of u . The static spin matrix elements Ω and Ω forthe ground state of Na have values 0.727 and 0.652respectively. They compare well with other theoreticalcalculations for Na given in [15, 30]. An analysis of thenormalized spin structure functions for Na in Fig. 5shows that the values of F , F and F differ betweenu=0.4-3. Out side this region they are almost degener-ate. The form factors for proton and neutron in Naare almost identical up to u = 2. Afterwards they dif-fer and beyond u=2.6 the neutron form factor becomeslarger than proton form factor.The nuclear structure dependent coefficients given inEq. (8) are plotted in Fig. 6 for Na, as a function D D D D D
00 0 05 55 55 10 1010 1010 m χ D FIG. 6: Nuclear structure coefficients plotted as a functionof the WIMP mass in GeV for Na. The graphs are plottedfor three values of the detector threshold Q thr namely Q thr =0 , ,
10 keV. The thickness of the graphs for each value of Q thr represents the annual modulation. m χ E ve n t r a t e [ y − k g − ] FIG. 7: The event rates in units of yr − kg − as a functionof dark matter mass in GeV for Na at detector threshold Q th = 0 , keV The thickness of the curves represent theannual modulation.
Results for elastic scattering F ρρ ’ i n u F F F FIG. 8: Spin structure function in the inelastic channel5 / + → / + for Na. of the WIMP mass for different values of the detectorthreshold. Since Ω and Ω are of same sign, D i s are allpositive. The peaks of the nuclear structure coefficientsoccur at around m χ ∼
30 GeV at zero threshold energy.The peaks shift towards higher values of m χ as we goto larger threshold energy. The thickness of the graphsrepresents annual modulation. Annual modulation haslargest value near the peaks of the graphs. Annual mod-ulation provides strong evince regarding the observationof dark matter since the back ground does not exhibitsuch modulation; see [11] for a recent review on annualmodulation measurements. As seen from Figs. 6 and 7, Na shows larger modulation compared to heavier nucleilike I, Cs and
Xe [10].The event detection rates for these nuclei have beencalculated at a particular WIMP mass by reading outthe corresponding values of D i s from the Fig. 6 andthen evaluating Eq. (7) for a given set of supersymmetricparameters. The event detection rates for different valuesof m χ have been calculated using the nucleonic currentparameters f A = 3 . e − f A = 5 . e − f S = 8 . e − f S = − . × f S . These results are shown in Fig.(7) for detector threshold energy Q th = 0, 10 keV for Na. For Na, the peak occurs at m χ ≃
30 GeV. Theevent rate decreases at higher detector threshold energybut the peak shifts to the higher values of m χ occurringat ∼
50 GeV.
C. Results for inelastic scattering Na has 5 / + excited state at 440 KeV above theground state 3 / + . Therefore, we consider inelastic scat-tering from the ground state for this nucleus to the5 / + state. The static spin matrix elements for the in-elastic scattering to the J = 5 / + are Ω = − . = − . and Ω are of same sign. The inelastic spin structure E E E m χ (GeV) E i [ y − k g − ] FIG. 9: Nuclear structure coefficients E n in the inelastic chan-nel 5 / + → / + for Na. The thickness of the graphs rep-resents annual modulation. functions are given in Fig. (8). In the figures, F , F and F are shown. The spin structure functions almostvanish above u=4. With the value of u lying between 1to 4, the spin structure functions differ from each other.The nuclear structure coefficients E n are shown in Fig.9 for this nucleus. The inelastic nuclear structure coef-ficients do not depend on the detector threshold energy.Hence the event rate can be calculated by reading the val-ues of E i from the graph and using the nucleonic currentparameters. The modulation for the inelastic scatteringcase is much smaller than the elastic case. V. RESULTS FOR WIMP- AR ELASTICSCATTERING
The event rates for WIMP- Ar elastic scattering arecalculated using the nuclear wave functions generatedthrough our DSM calculation. In our calculation, the ac-tive spherical single particles orbitals are taken as 0 d / ,0 d / , 1 s / , 0 f / , 0 f / , 1 p / and 1 p / with O as theinert core. An effective interaction named sdpf − u anddeveloped by Nowacki and Poves [43] with single parti-cle energies − . . − . . . . .
479 MeV, respectively for the above seven orbitalshas been used. As discussed earlier, we first generate thelowest HF intrinsic state by solving the axially symmet-ric HF equation self-consistently. Then, we generate theexcited configurations by particle-hole excitations. Wehave considered a total of 9 intrinsic states. Good an-gular momentum states are projected from each of theseintrinsic states and then a band mixing calculation isperformed. The band mixed wave functions defined in D D D m χ m χ FIG. 10: Nuclear structure coefficients plotted as a functionof the WIMP mass m χ in GeV for Ar. The graphs areplotted for three values of the detector threshold Q thr namely Q thr = 0 , ,
10 keV. The thickness of the graphs for each valueof Q thr represents the annual modulation. m χ E ve n t r a t e [ y − k g − ] FIG. 11: The event rates in units of yr − kg − as a function ofdark matter mass m χ in GeV for Ar at detector threshold Q th = 0 , keV The thickness of the curves represent theannual modulation.
Eq. (16) are used in calculating the elastic event ratesand nuclear structure coefficients for the ground state ofthis nucleus. Note that the ground state is a 0 + state as Ar is a even-even nucleus and inelastic scattering fromground needs excited 1 + state. However, the 1 + stateslie very high in energy and hence only elastic scatteringof WIMP from Ar is important. The oscillator lengthparameter b for this nucleus is taken to be 1.725 f m .We have presented the nuclear structure dependent coef- ficients defined in Eq. (8) in Fig. 10 for this nucleus as afunction of the WIMP mass for different values of the de-tector threshold. Since Ar is a even-even nucleus, thereis no spin contribution from the ground state. Hence, wehave only D , D and D corresponding to the proton,neutron and proton-neutron form factors as defined inEq. (7). Their values are slightly larger compared to thecorresponding quantities in Na but the modulation issmaller. The peaks occur at around m χ = 35 GeV. How-ever for larger values of Q thr , the peaks shift towards thelarger m χ . The event rates for WIMP- Ar scatteringis plotted as a function of the dark matter mass in Fig.(11) for Q thr = 0 and 10 keV. The event rates are cal-culated using the same supersymmetric parameters as in Na. The values are smaller than in Na. This is be-cause Ar is a even-even nucleus and hence there is nospin contribution to the event rates in the ground state.At Q thr = 0, the peak occurs at 35 GeV. For Q thr = 10keV, the peak shifts to 45 GeV. VI. CONCLUSION
Deformed shell model is used to calculate first the eventrates for the elastic and inelastic scattering of WIMPfrom Na. Spectroscopic properties of this nucleus arecalculated within DSM to check the suitability of themodel. We have also calculated magnetic moments forthe lowest level in this nucleus since spin plays an im-portant role in the calculation of detection rates. Be-fore Na analysis, we have compared the DSM resultsalso for As for further confirmation of the goodness ofDSM for spectroscopy. After ensuring the good agree-ment with experiment, we calculated the spin structurefunctions, form factors, nuclear structure coefficients andthe event rates for WIMP- Na elastic and inelastic scat-tering. In addition, event rates for elastic scattering ofWIMP from Ar are also presented. Results in Figs. 7and 11 for event rates and in Figs. 6,7 and 9-11 for theannual modulation should be useful for the upcoming andfuture experiments detecting WIMP involving detectorswith Na and Ar. Let us add that the present studyusing DSM for the nuclear structure part is in additionto the results presented for WIMP scattering from Gein [38] and from I, Cs and
Xe in [10]. Finally,we hope that these and those obtained using other theo-retical models for nuclear structure may guide the exper-imentalists to unravel the fundamental mysteries of darkmatter particles.
Acknowledgments
Thanks are due to Prof. T.S. Kosmas for his interestin this work. R. Sahu is thankful to SERB of Depart-ment of Science and Technology (Government of India)for financial support. [1] G. Jungman, M. Kamionkowski and K. Griest,
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