Shell Model Description of Spin-Dependent Elastic and Inelastic WIMP Scattering off 119Sn and 121Sb

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Shell Model Description of Spin-Dependent Elastic and Inelastic WIMP Scattering off 119Sn and 121Sb © 2022 by the authors. Licensee MDPI, Basel, Switzerland. Published version Kasurinen, Joona; Suhonen, Jouni; Srivastava, Praveen C.; Pirinen, Pekka Kasurinen, J., Suhonen, J., Srivastava, P. C., & Pirinen, P. (2022). Shell Model Description of SpinDependent Elastic and Inelastic WIMP Scattering off 119Sn and 121Sb. Universe, 8(6), Article 309. https://doi.org/10.3390/universe8060309 2022 Citation: Kasurinen, J.; Suhonen, J.; Srivastava, P.C.; Pirinen, P. Shell Model Description of Spin-Dependent Elastic and Inelastic WIMP Scattering off 119Sn and 121Sb. Universe 2022,8, 309. https:// doi.org/10.3390/universe8060309 Academic Editor: Csaba Balazs Received: 19 April 2022 Accepted: 29 May 2022 Published: 31 May 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). universe Article Shell Model Description of Spin-Dependent Elastic and Inelastic WIMP Scattering off 119Sn and 121Sb Joona Kasurinen 1,*, Jouni Suhonen 1, Praveen C. Srivastava 2and Pekka Pirinen 1 1Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), 40014 Jyväskylä, Finland; [email protected] (J.S.); [email protected] (P.P.) 2Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India; [email protected].ac.in *Correspondence: [email protected] Abstract: In this work, we calculate the spin structure functions for spin-dependent elastic and inelastic WIMP scattering off 119 Sn and 121 Sb. Estimates for detection rates are also given. 119 Sn and 121 Sb are amenable to nuclear structure calculations using the nuclear shell model (NSM). With the possible exception of 201 Hg, they are the only such nuclei still unexplored theoretically for their potential of inelastic WIMP scattering to a very low excited state. The present calculations were conducted using a state-of-the-art WIMP–nucleus scattering formalism, and the available effective NSM two-body interactions describe the spectroscopic properties of these nuclei reasonably well. Structure functions were found to be high for both nuclei in the case of elastic scattering. Elastic scattering dominated at the zero momentum transfer limit. Detection rate calculations indicated that inelastic scattering was relevant for both nuclei, even surpassing elastic rates for some recoil energies. Keywords: dark matter; WIMP; direct detection; spin structure functions; nuclear structure 1. Introduction Galaxy rotation curves [ 1 – 4 ] and structure formation [ 5 , 6 ] indicate that either our present understanding of gravity is wrong or most of the matter in the universe is comprised of a dark component of unknown nature. Recent experiments [ 7 , 8 ] analyzing the cosmic microwave background (CMB) have given yet more credibility to standard cosmology in which there is around five times as much dark matter as there is regular matter. Competing theories do exist, such as modified Newtonian dynamics (MOND) [ 9 ] and Tensor-Vector-Scalar (TeVeS) gravity [ 10 ], f(R) gravity [ 11 ], as well as dark fluid [ 12 ] and negative mass [13], to name a few. Challenges to modified gravity theories are posed by weak lensing studies [14,15] and gravitational wave measurements [ 16 ]. Nevertheless, some of them are left standing, and the debate continues. Assuming dark matter is explained by some kind of undiscovered particles, the particles would likely be nonrelativistic and rather massive. This is the standard dark matter picture, where non-baryonic weakly interacting massive particles (WIMPs) are introduced. The theoretical motivation for WIMPs can be found in several frameworks ranging from Kaluza–Klein theories [ 17 , 18 ] to technicolor [ 19 , 20 ], little Higgs [ 21 , 22 ], and supersymmetry [ 23 ]. To detect these massive WIMPs, a direct way would be to observe their scattering from atomic nuclei. Hence, nuclei would be an excellent direct probe of the properties of dark matter [ 24 ]; therefore the search for good candidate nuclei is of paramount importance. The mechanism of the WIMP–nucleus interaction is unknown, so that ideal detectors would consist of nuclei that are sensitive to both coherent and incoherent interactions and also allow inelastic scattering of WIMPs. Unfortunately, such a combination of properties excludes almost all nuclei available for experiments. However, should one want to study Universe 2022,8, 309. https://doi.org/10.3390/universe8060309 https://www.mdpi.com/journal/universe Universe 2022,8, 309 2 of 10 both incoherent elastic and inelastic scattering off nuclei, the ideal target would be an odd-proton or odd-neutron nucleus, offering low-energy excited states and allowing spindependent scattering off nuclei. Such possible targets have been identified for some iodine, xenon, and cesium nuclei [ 25 – 28 ]. Beyond these, the nuclei 83 Kr and 125 Te, with a very low-energy first excited state, were studied in [ 29 – 31 ]. These studies have been conducted by exploiting the nuclear wave functions obtained by the use of the nuclear shell model (NSM). In [ 32 ], the microscopic interacting boson–fermion model (IBFM-2) was used for the first time to discuss spin-dependent WIMP–nucleus scattering off 125 Te, 129 Xe, and 131 Xe. The results were benchmarked against the earlier NSM results of [ 27 , 28 , 33 ]. In these studies, the one-body and leading long-range two-body WIMP–nucleus currents derived from the chiral effective field theory (c-EFT) [27] were employed. Very recently in [ 34 ], the IBFM-2 was applied to describe the elastic and inelastic WIMP scattering cross sections off 183 W. This was the first time that the scattering of WIMPs off a heavy deformed nucleus was described successfully. The used scattering formalism was updated by adopting the formalism of [ 35 ], where the contributions from all pion-exchange, pion-pole, and contact currents were taken into account. We use this same scattering formalism in the present work but compute the nuclear wave functions of the ground and first excited states of 119 Sn and 121 Sb by using the NSM. In these nuclei, the excitation energy of the first excited state (3 / 2 + state at 23.87 keV in 119 Sn and 7 / 2 + state at 37.13 keV in 121 Sb) is very low; thus, these nuclei are good candidates for direct detection of dark matter by inelastic scattering, where the background signal can be reduced by exploiting coincidence with the involved magnetic dipole (M1) transition to the ground state. For observation of inelastic WIMP–nucleus scattering, the target nuclei should have a low excited state below some 100 keV. These nuclei are heavy or very heavy and often deformed. To our knowledge, there are only three nuclei still unexplored for spin-dependent WIMP scattering to a low excited state that are at the same time suitable for NSM description owing to their (near) semi-magicity. Two of them are the presently discussed 119 Sn and 121 Sb. The third is 201 Hg, which is very heavy and hard to describe with adequate precision using presently available shell-model interactions. 2. Cross Section and Spin Structure There are many factors that influence the probability of WIMP–nucleus scattering. After stripping off extrinsic factors such as dark matter density and the size of the detector, we are left with the scattering cross section. Going deeper, we can take out more factors until what is left is just the nuclear physics. This is the idea of the spin structure functions. The cross section of spin-dependent WIMP–nucleus scattering can be derived from effective field theory (EFT). It is related to the structure functions by the following Equation [36]: dσ dq2=8G2 F (2J+1)v2SA(q), (1) where GF is the Fermi coupling constant, J is the ground state angular momentum, and v is the speed of the WIMP in the laboratory frame. SA is the axial structure function, which contains the nuclear physics. It is a function of momentum transfer q . Knowing the structure functions of a nucleus means knowing how the nuclear structure contributes to the scattering cross section. The structure functions were determined following the formalism of [ 27 , 35 ], in which two-body currents are included as effective one-body currents via a normal-ordering approximation. The values chosen for the low-energy couplings and nuclear density were c1=− 1.20 ( 17 )GeV−1 , c3=− 4.45 ( 86 )GeV−1 , c4= 2.69 ( 70 )GeV−1 , ˆ c6= 5.83, cD= − 8.0 . . . 2.0, and ρ= 0.11 ( 1 )fm−3 . See Table 1for a summary. The parameter choices were based on [ 34 , 35 , 37 ]. For ˆ c6 , having a range of values is not necessary, because the results are insensitive to it. As for the others, we calculated the smallest and largest theoretically possible values for the structure functions given these ranges of parameters. The calculation was performed for each of the four cases: elastic scattering off 119 Sn, inelastic scattering off 119 Sn, Universe 2022,8, 309 3 of 10 elastic scattering off 121 Sb, and inelastic scattering off 121 Sb. In addition, the same calculations were performed a second time using one-body currents only. Table 1. Parameters fed into the WIMP scattering program. Param. Values c1−1.37 . . . −1.03 GeV−1 c3−5.31 . . . −3.59 GeV−1 c41.99 . . . 3.39 GeV−1 ˆ c65.83 cD−8.0 . . . 2.0 ρ0.10 . . . 0.12 fm−3 3. Shell Model Calculations The nuclear structure is entered into the WIMP scattering program in the form of one-body transition densities (OBTD). These were obtained by a shell model calculation. The calculations were performed in the 50–82 model space with the realistic CD-Bonn interaction [38]. The CD-Bonn potential based on meson exchange is the charge-dependent one-bosonexchange nucleon–nucleon potential fitted for proton–proton data below 350 MeV. The reproduction of data is more accurate than with phase-shift analysis or other nucleon–nucleon potentials. The charge dependence of the present potential is based on the predictions by the Bonn full model for charge symmetry and charge-independence breaking. The nonlocality of the potential is represented in terms of the covariant Feynman amplitudes for one-boson exchange. The interaction was renormalized using the perturbative G-matrix approach [ 39 ]. The renormalization effectively takes into account the single-particle space outside the presently used valence space. The effective single-particle energies were taken as 0.0 (0 g7/2 ), 0.172 (1 d5/2 ), 2.55 (1 d3/2 ), 2.45 (2 s1/2 ), and 3.00 (0 h11/2 ) MeV, respectively. These effective energies can be viewed as phenomenological renormalization coming from our restricted valence space and three-body forces neglected in the bare shell-model Hamiltonian. Earlier shell model results with this interaction are reported in Refs. [ 40 , 41 ]. The shell model calculations were carried out using the codes NuShellX [42] and KShell [43]. The shell model calculation gave reasonable predictions for the lowest two states (see Figure 1). The most notable deviation from the experiment is that the first two states of 121 Sb were flipped. In addition, the energies were not replicated exactly. However, what is more relevant is that the magnetic dipole moments and electric quadrupole moments were correct, because this indicates that the wave functions contained the components that captured the nuclear spectroscopy relevant for WIMP–nucleus scattering. In Table 2, we have reported the leading configurations of the two lowest states of 119 Sn and 121 Sb. For 119 Sn, only the neutrons were active, and the wave functions of the lowest 1 / 2 + and 3 / 2 + states were quite fragmented with the leading components having the 1 d5/2 orbital filled and the 0 g7/2 orbital either filled or having two holes. The higher orbitals had varying degrees of occupation. In the case of the lowest 5 / 2 + and 7 / 2 + states in 121 Sb, the fragmentation was again strong with the neutron side possessing features similar to that of 119 Sn. On the proton side, the 5 / 2 + state was described as a proton in the 1d5/2 orbital, and the 7/2+state was characterized as a proton in the 0g7/2 orbital. Universe 2022,8, 309 4 of 10 Exp. NSM 0.0 0.2 0.4 0.6 0.8 1.0 1/2+ 3/2+ 7/2+ 3/2+ 5/2+ 1/2+ 3/2+ 7/2+ 5/2+ Energy (MeV) (a) Exp. NSM 0.0 0.2 0.4 0.6 0.8 1.0 5/2+ 7/2+ 3/2+ 1/2+ 9/2+ 7/2+ 9/2+ 7/2+ 5/2+ 3/2+ 1/2+ 9/2+ Energy (MeV) (b) Figure 1. Comparison of experimental and calculated states: ( a ) left panel is for 119 Sn, with experimental values from Ref. [ 44 ], and ( b ) right panel is for 121 Sb, with experimental values from Ref. [ 45 ]. Ground state and first excited state are flipped for 121Sb. Table 2. The configurations of the lowest-lying states of 119Sn and 121Sb. Nucleus JπConfiguration 119Sn 1/2+ν(g6 7/2d6 5/2d2 3/2s1 1/2h4 11/2)(17.5%) ν(g8 7/2d6 5/2d0 3/2s1 1/2h4 11/2)(14.4%) ν(g6 7/2d6 5/2d0 3/2s1 1/2h6 11/2)(10.9%) ν(g8 7/2d6 5/2d2 3/2s1 1/2h2 11/2)(5.5%) 3/2+ν(g8 7/2d6 5/2d1 3/2s0 1/2h4 11/2)(12.8%) ν(g6 7/2d6 5/2d1 3/2s2 1/2h4 11/2)(11.8%) ν(g6 7/2d6 5/2d1 3/2s0 1/2h6 11/2)(10.9%) ν(g6 7/2d6 5/2d3 3/2s0 1/2h4 11/2)(6.6%) ν(g8 7/2d6 5/2d1 3/2s2 1/2h2 11/2)(5.4%) 121Sb 5/2+π(d1 5/2)⊗ν(g6 7/2d6 5/2d2 3/2s0 1/2h6 11/2)(8.6%) π(d1 5/2)⊗ν(g6 7/2d6 5/2d2 3/2s2 1/2h4 11/2)(6.9%) π(d1 5/2)⊗ν(g8 7/2d6 5/2d2 3/2s0 1/2h4 11/2)(4.6%) 7/2+π(g1 7/2)⊗ν(g6 7/2d6 5/2d2 3/2s0 1/2h6 11/2)(9.6%) π(g1 7/2)⊗ν(g6 7/2d6 5/2d2 3/2s2 1/2h4 11/2)(4.7%) π(g1 7/2)⊗ν(g6 7/2d6 5/2d0 3/2s2 1/2h6 11/2)(4.4%) The shell model calculation produced reasonable nuclear moments. A summary of the calculated and experimental moments is presented in Table 3. For 119 Sn, the experimental values for the magnetic moments in the ground state and first excited state were − 1.0459 ( 5 )µN and + 0.633 ( 3 )µN , respectively, [ 44 ]. These were close to the calculated values − 1.213 µN and + 0.749 µN . The quadrupole moment of the first excited state was − 0.132 ( 1 )eb , which in our calculations was − 0.107 eb . Based on these comparisons, the shell model calculation produces wave functions that are reasonably realistic for the purposes of our WIMP calculation. For 121 Sb, the magnetic moments were + 3.3580 ( 16 )µN in the ground state and + 2.518 ( 7 )µN in the first excited state [ 45 ]. According to our shell model calculation, these values were + 3.681 µN and + 1.185 µN . The values are reasonable, though the second one was not as accurate. Quadrupole moments for the ground state and first excited state were experimentally − 0.543 ( 11 )eb and − 0.727 ( 16 )eb , and according to our shell model calculation, they were −0.439 eb and −0.559 eb, in reasonable agreement with the data. Universe 2022,8, 309 5 of 10 Table 3. Comparison between calculated and experimental nuclear moments in the lowest two states of 119Sn and 121Sb. Effective charges and g-factors were ep= 1.5e, en= 0.5e, ge f f l=gfree l, and gef f s=gfree s. State µ(µN)exp. µ(µN)NSM Q(eb)exp. Q(eb)NSM 119Sn 1/2+−1.0459(5) −1.213 - - 119Sn 3/2++0.633(3) +0.749 −0.132(1) −0.107 121Sb 5/2++3.3580(16) +3.681 −0.543(11) −0.439 121Sb 7/2++2.518(7) +1.185 −0.727(16) −0.559 It is also of interest to examine how well the computed wave functions described the electromagnetic transitions between the lowest few states. In Table 4, a comparison of the calculated and available experimental electric quadrupole (E2) and magnetic dipole (M1) transitions is shown. For 119 Sn the reduced E2 transition probabilities, B (E2), were in reasonable agreement with the data. Notably, the experimental reduced M1 transition probability, B (M1), was quite well reproduced by the calculations. This is relevant for the inelastic WIMP scattering to the 3 / 2 + state, since this scattering was mainly of the M1 type, mediated by the Pauli spin operator σ at the zero momentum exchange limit. The contrary was true for 121 Sb for which the computed B (M1) was roughly an order of magnitude too low; thus, the cross section of inelastic scattering to the 7 / 2 + state was likely underestimated. Table 4. Reduced transition probabilities and energies for electromagnetic transitions between the lowest few states in 119 Sn and 121 Sb. The effective charges and g-factors used were ep = 1.5e, en = 0.5e, gef f l=gfree l, and gef f s=gfree s. Nuc. Transition B(W.u.) exp. B(W.u.) NSM Eγ(keV) exp. Eγ(keV) NSM 119Sn E2: 3 2 + →1 2 +<0.70 0.40 23.870(8)123 E2: 5 2 + →1 2 +5(3)1.99 921.4(2)815 E2: 5 2 + →3 2 +17(6)2.75 897.5(2)693 M1: 3 2 + →1 2 +0.015 0.017 23.870(8)123 121Sb M1: 7 2 + →5 2 +0.01047(17)0.00072 37.1298(2)191 4. WIMP Scattering Results We express the structure functions in terms of recoil energy ER=q2/ 2 mA=u/b2mA , where u is a dimensionless variable, b is the harmonic oscillator length, q is the momentum transfer, and mA is the mass of the nucleus. The harmonic oscillator length b is a parameter describing the range of the harmonic oscillator wave functions that were used as the basis in the shell model calculations [46]. We break down SAby its isoscalar and isovector components as SA(u) = a2 0S00(u) + a0a1S01(u) + a2 1S11(u), (2) and then express the results as ”neutron-only” and ”proton-only” couplings Sn(u) = S00(u)−S01(u) + S11(u),Sp(u) = S00(u) + S01(u) + S11(u). (3) The functions Sρρ0(u) are defined in [ 31 ]. This division to “neutron-only” and “proton-only” structure functions is conventionally used, but is not entirely accurate when two-body currents are included (see [27] for discussion). The structure functions Sn and Sp for 119 Sn and 121 Sb are shown as functions of recoil energy ER in Figures 2and 3. The thickness of the Sn and Sp curves show the error resulting from the uncertainties in the EFT low-energy constants and nuclear density. Universe 2022,8, 309 6 of 10 0 200 400 600 800 10−6 10−5 10−4 10−3 10−2 10−1 119Sn elastic ER(keV) S Sn Sp Sn(1b) Sp(1b) (a) 0 200 400 600 800 10−6 10−5 10−4 10−3 10−2 10−1 119Sn inelastic ER(keV) S Sn Sp Sn(1b) Sp(1b) (b) Figure 2. Structure functions of nucleus 119 Sn: ( a ) Left panel shows neutron-only and proton-only structure functions in the case of elastic scattering (nucleus remains in the ground state 1 / 2 + ). ( b ) Right panel shows neutron-only and proton-only structure functions in the case of inelastic scattering (nucleus excites to the first excited state 3/2+). Sn,p(1b) are with one-body currents only. 0 200 400 600 800 10−6 10−5 10−4 10−3 10−2 10−1 121Sb elastic ER(keV) S Sn Sp Sn(1b) Sp(1b) (a) 0 200 400 600 800 10−6 10−5 10−4 10−3 10−2 10−1 121Sb inelastic ER(keV) S Sn Sp Sn(1b) Sp(1b) (b) Figure 3. Structure functions of nucleus 121 Sb: ( a ) Left panel shows neutron-only and proton-only structure functions in the case of elastic scattering (nucleus remains in the ground state 5 / 2 + ). ( b ) Right panel shows neutron-only and proton-only structure functions in the case of inelastic scattering (nucleus excites to the first excited state 7/2+). Sn,p(1b) are with one-body currents only. The elastic scattering structure functions were high for both nuclei: 119 Sn was similar to 125 Te and 129 Xe [ 27 , 32 , 47 ], while for 121 Sb, the functions were higher still. For the inelastic scattering from 119 Sn, there was a large dip between ER= 100 keV and ER= 700 keV . For 121 Sb, on the other hand, the inelastic scattering structure functions were on par with the elastic ones starting at around ER=50 keV. It could be safer to take the inelastic result for 121 Sb as a lower limit, since the calculated transition probability for the corresponding M1 transition was an order of magnitude smaller than the measured one. It can also be seen in the figures that for 121 Sb already the one-body contribution produced realistic structure functions, whereas for 119 Sn there were notable contributions by two-body currents to Sp in the elastic case for recoil energies below some 50 keV and in the inelastic case within several recoil energy intervals. The spin expectation values of the ground state are important in elastic scattering. They determine the structure functions in the zero momentum transfer limit by the following equation: SA(0) = (2J+1)(J+1) 4πJ   (a0+a1+δa1(0))hSpi+ (a0−a1−δa1(0))hSni   2. (4) The spin expectation values for 119 Sn were found to be hSpi= 0 and hSni= 0.31712. For 121Sb, they were hSpi=0.43887 and hSni=0.04041. The differential event rates can be estimated for a given WIMP mass. This was calculated following [ 28 , 34 ]. The results of the event rate calculations are shown in Figures 4–6 . It is assumed that coupling was only with the unpaired nucleon. This is justified by the Universe 2022,8, 309 7 of 10 structure functions of the unpaired nucleon being about an order of magnitude higher than those of the paired nucleon. 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 100 GeV 119Sn ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (a) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 300 GeV 119Sn ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (b) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 1 TeV 119Sn ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (c) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 10 TeV 119Sn ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (d) Figure 4. Differential event rates in 119 Sn with WIMP masses: ( a ) 100 GeV; ( b ) 300 GeV; ( c ) 1 TeV; and (d) 10 TeV. 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 100 GeV 121Sb ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (a) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 300 GeV 121Sb ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (b) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 1 TeV 121Sb ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (c) 0 200 400 600 800 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 mWIMP = 10 TeV 121Sb ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (d) Figure 5. Differential event rates in 121 Sb with WIMP masses: ( a ) 100 GeV; ( b ) 300 GeV; ( c ) 1 TeV; and (d) 10 TeV. Universe 2022,8, 309 8 of 10 0 20 40 60 80 100 120 140 160 180 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 mWIMP = 50 GeV 119Sn ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (a) 0 20 40 60 80 100 120 140 160 180 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 mWIMP = 50 GeV 121Sb ER(keV) dR/dER(events/kg/d/keV) elastic inelastic (b) Figure 6. Differential event rates for WIMPs of mass 50 GeV: (a) in 119Sn and (b) in 121Sb. For low mass WIMPs, event rates move quickly to zero for higher recoil energies. This is because WIMPs of a sufficiently low mass simply do not have enough momentum to make the much heavier nucleus recoil past a certain maximum recoil energy. The most dramatic change in the shape of the differential event rate graphs is seen when the WIMP mass passes the mass of the nucleus at around 120 GeV. For WIMP masses larger than this, the overall shape of the event rate curve stays the same, though the slope becomes steeper (when viewed through a linear scale). The event rate is close to inversely proportional to the WIMP mass in the low recoil energy region. This is most apparent when comparing the graphs for 1 TeV and 10 TeV WIMPs. In terms of the event rate, both nuclei performed similarly in the case of elastic scattering. 119 Sn had higher differential event rates for inelastic scattering at the zero momentum transfer limit, because the rates did not fall so steeply when approaching zero. However, for inelastic scattering there was a large dip between 100 keV and 700 keV. For 121 Sb, the inelastic differential event rates were low for small recoil energies, climbed fast to a peak near 40 keV, and then remained within an order of magnitude of their elastic counterparts. For WIMPs near the low end of the possible mass range, inelastic scattering was about an order of magnitude more important in 119 Sn than in 121 Sb. See Figure 6for the differential event rates of 50 GeV WIMPs. However, due to the previously mentioned underprediction of the M1 transition probability in 121 Sb, this conclusion may be incorrect. For masses nearing 10 GeV, elastic scattering was restricted to small recoil energies, and inelastic scattering becomes impossible. 5. Discussion Our calculations showed that the structure functions corresponding to the unpaired nucleon were consistently about an order of magnitude larger: neutron-only structure functions were more important for 119 Sn, and proton-only structure functions were more important for 121 Sb. This is expected, as similar results are seen in the literature for various other nuclei. When compared with other nuclei, the elastic scattering structure functions of 119 Sn were similar in shape and magnitude to those of 125 Te and 129 Xe [ 27 , 32 , 47 ]. The structure functions of 121 Sb were higher still, making 121 Sb a promising WIMP detector. In particular, the proton-only elastic scattering structure function of 121 Sb was about 0.4 near the zero momentum transfer limit. This is among the highest values seen in the literature. The most noticeable difference between the two nuclei is that there was a large dip in the inelastic structure functions of 119 Sn. After reaching its maximum at around 50 keV, both the proton-only and neutron-only structure functions plummeted several orders of magnitude. This dip was also seen in the corresponding event rates. 121 Sb had no such dip, and so for 121 Sb, the event rates for inelastic scattering were comparable to those of elastic scattering across a wide range of recoil energies. 119 Sn had better inelastic event rates below 100 keV, while the elastic event rates in this zone were similar. Therefore, 119 Sn had a better overall event rate than 121 Sb in the sub 100 keV region. For low mass WIMPs, this is the only region that matters, in which case