NORDITA-2018-117LCTP-18-25

Paleo-detectors: Searching for Dark Matter with Ancient Minerals

Andrzej K. Drukier,1, ∗ Sebastian Baum,1, 2, † Katherine Freese,1, 2, 3, ‡ Maciej Gorski,4, § and Patrick Stengel1, ¶

1The Oskar Klein Centre for Cosmoparticle Physics, Department ofPhysics, Stockholm University, Alba Nova, 10691 Stockholm, Sweden

2Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden3Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA

4National Centre for Nuclear Research, 05-400 Otwock, Swierk, Poland

Recently, we proposed paleo-detectors as a method for the direct detection of Weakly InteractingMassive Particle (WIMP) dark matter. In paleo-detectors, one would search for the persistenttraces left by dark matter–nucleon interactions in ancient minerals. Thanks to the large integrationtime of paleo-detectors, relatively small target masses suffice to obtain exposures, i.e. the productof integration time and target mass, much larger than what is feasible in the conventional directdetection approach. Here, we discuss the paleo-detector proposal in detail, in particular, a rangeof background sources. For the classes of target materials considered here, we identify (broadlyspeaking) two different background regimes. For low-mass WIMPs with masses mχ . 10 GeV,the largest contribution to the background budget comes from nuclear recoils induced by coherentscattering of solar neutrinos. For heavier WIMPs, the largest background source is nuclear recoilsinduced by fast neutrons created by heavy radioactive contaminants, particularly 238U; neutronscan arise in spontaneous fission or from α-particles created in 238U decays. We also discuss thechallenges of mineral optimization, specifically the determination of readily available minerals fromrocks in deep boreholes which are able to record persistent damage from nuclear recoils. In orderto suppress backgrounds induced by radioactive contaminants, we propose to use minerals foundin marine evaporites or in ultra-basic rocks. We estimate the sensitivity of paleo-detectors to spin-independent and spin-dependent WIMP-nucleus interactions. In all interaction cases consideredhere, the sensitivity to low-mass WIMPs with masses mχ . 10 GeV extends to WIMP-nucleon crosssections many orders of magnitude smaller than current upper limits. For heavier WIMPs withmasses mχ & 30 GeV cross sections a factor of a few to ∼ 100 smaller than current upper limits canbe probed by paleo-detectors.

I. INTRODUCTION

Recently, we proposed paleo-detectors [1] as a new ap-proach to the direct detection of Dark Matter (DM). Cur-rent strategies for the direct detection of Weakly Inter-acting Massive Particle (WIMP) DM employ large targetmasses, which are instrumented to observe nuclear recoilsin real time [2–4]. Such experiments have set impressiveupper limits on the strength of interactions of WIMPswith atomic nuclei [5–12], but so far failed to report con-clusive evidence for DM. The long-standing exception isthe DAMA/LIBRA experiment [13–15], reporting evi-dence for an annual modulation signal [4, 16, 17] com-patible with DM for more than a decade [18–26]. How-ever, the claimed signal is in tension with null-resultsfrom other direct detection experiments [22, 27–30].

Currently, there are two major trends in the evolu-tion of direct detection: experiments primarily aimed atWIMPs with masses mχ & 15 GeV plan to utilize largertarget masses, approaching the 100 t scale [31–35]. Suchexperiments use liquid noble gas targets and are sensi-tive to nuclear recoils with energies larger than O(1) keV.

∗ adrukier@gmail.com† sbaum@fysik.su.se‡ ktfreese@umich.edu§ maciej.gorski@ncbj.gov.pl¶ patrick.stengel@fysik.su.se

For lighter WIMPs with masses mχ . 15 GeV, the mainchallenge is to observe nuclear recoils with small ener-gies. Experiments using new approaches have been pro-posed, including cryogenic bolometric detectors that aimfor target masses of a few kg and recoil energy thresholdsof O(100) eV [36, 37].

Further, there is a major effort [38–46] to develop di-rectional detectors [16], where the ability to determinethe direction of the nuclear recoil would allow for pow-erful background rejection. Other recent proposals fordirect detection include nm-scale detectors [47, 48] andconcepts using molecular biology [49, 50].

Paleo-detectors are radically different from conven-tional direct detection experiments: Instead of instru-menting a (large) target mass in a laboratory in order toobserve WIMP-induced nuclear recoils in real time, wepropose to examine ancient minerals for traces of WIMP-nucleus interactions recorded over timescales as large as1 Gyr [1]. Recoiling nuclei leave damage tracks in cer-tain classes of minerals, so-called solid state track detec-tors (SSTDs). Once created, these tracks are preservedover time-scales much larger than a billion years. WIMP-induced recoils would leave damage tracks with lengthsup to O(500) nm in typical natural targets. Hence, inpaleo-detectors the challenge is to reconstruct such nano-scale features in natural minerals instead of anO(1) num-ber of phonons, electrons, or photons as typically donein conventional direct detection experiments.

In this paper, we give a detailed description of the

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paleo-detector proposal. In Sec. II, we describe the calcu-lation of the WIMP-induced spectrum of damage tracks.In particular, we show a semi-analytical calculation ofthe track length as a function of recoiling nucleus, targetmaterial, and recoil energy in Sec. II B and compare itwith the usually employed numerical calculation.

In Sec. III we discuss some generalities of solid statetrack detectors and a number of possible read-out meth-ods. We identify two realistic, though ambitious, read-out scenarios. The first scenario is particularly gearedtowards searching for low-mass WIMPs with massesmχ . 10 GeV. Using helium-ion beam microscopy, weestimate that target masses of O(10) mg can be read outwith ∼ 1 nm spatial resolution. Reconstructing tracksas short as O(1) nm, corresponding to O(100) eV nuclearrecoil energy thresholds, allows for thresholds compara-ble to conventional direct detection experiments employ-ing cryogenic bolometric detectors. However, assuminga 1 Gyr integration time, the exposure in paleo-detectorsfor O(10) mg of target material is ε = 0.01 kg Myr. Inorder to achieve the same exposure with a conventionaldirect detection experiment, one would need to observea target mass of 103 kg for 10 yr.

The second read-out scenario is more suitable for heav-ier WIMPs, mχ & 10 GeV, sacrificing some spatial reso-lution for larger exposure. Using Small-Angle X-ray scat-tering (SAXs), spatial resolutions of 15 nm are achiev-able. We estimate a feasible target mass of O(100) g forpaleo-detectors. The corresponding exposure for an in-tegration time of 1 Gyr is ε = 100 kg Myr. The nuclearrecoil energy threshold corresponding to O(10) nm longtracks is of the order of keV, similar to what is achievablein conventional direct detection experiments using liquidnoble gas targets. However, the required target mass toachieve ε = 100 kg Myr with an integration time of 10 yrwould be 107 kg in conventional direct detection experi-ments. Note that other read-out methods exist beyondwhat is discussed here.

In Sec. IV we discuss the most relevant backgroundsources for DM searches with paleo-detectors. For theclasses of target materials considered here, we identify(broadly speaking) two different background regimes:For low-mass WIMPs with masses mχ . 10 GeV, thelargest contribution to the background budget comesfrom nuclear recoils induced by coherent scattering ofsolar neutrinos. For heavier WIMPs, the largest back-ground source is nuclear recoils induced by fast neutrons.Depending on the target’s chemical composition, thedominant source of neutrons is either spontaneous fissionof heavy radioactive contaminants, or (α, n)-reactions ofthe α-particles from the decays of the heavy contami-nants with the material’s nuclei. Note that cosmogenicbackgrounds can be avoided for paleo-detectors by usingminerals obtained from sufficient depth. Conventionaldirect detection experiments must be operated in acces-sible underground laboratories. The deepest current lab-oratory, CJPL located in China, has an overburden of∼ 2.4 km rock. For paleo-detectors we envisage using

comparatively small target volumes . (10 cm)3, whichcan be obtained from much larger depths. For exam-ple, target materials may be obtained from the bore-cores of ultra-deep boreholes used for oil exploration andgeological R&D. Existing boreholes have depths up to12 km [51]. Cosmogenic backgrounds are exponentiallysuppressed with depth and will play virtually no rolefor target materials obtained from depths larger than∼ 5 km.

In Sec. V we discuss which minerals are best suited astargets for paleo-detectors. In order to suppress back-grounds induced by radioactive contaminants, we pro-pose to use minerals found in marine evaporites or inultra-basic rocks. Such minerals have significantly lowerconcentrations of radioactive contaminants, e.g. ura-nium, than typical minerals found in the Earth’s crust.For searches for WIMPs with masses mχ & 10 GeV wherethe background budget is dominated by neutrons fromradioactive processes, target minerals containing hydro-gen, e.g. hydrites, are particularly useful. This is be-cause hydrogen is an effective moderator of fast neutrons,reducing the number of neutron-induced nuclear recoilevents.

In Sec. VI, we present sensitivity projections forpaleo-detectors in the benchmark read-out scenarios de-scribed above. Sensitivity projections for canonical Spin-Independent (SI) WIMP-nucleus interactions have al-ready been presented in Ref. [1]. Here, we present resultsfor a larger selection of target materials. In addition, weshow the prospects of paleo-detectors for exploring Spin-Dependent (SD) WIMP-nucleus interactions. For easeof comparison with existing limits and future projectionsof conventional direct detection experiments, we presentsensitivity projections in the proton-only and neutron-only coupling scenarios usually employed in the directdetection literature. Both in the SI and the two SDscenarios considered here, paleo-detectors can probe low-mass WIMPs with masses mχ . 10 GeV even for WIMP-nucleon cross sections many orders of magnitude belowcurrent upper bounds. For heavier WIMPs mχ & 30 GeVthe projected sensitivity is better than current limits bya factor of a few to ∼ 100, depending on the type of inter-action and target material. Note that since the dominantbackground source for such WIMP masses is radioactiv-ity, the sensitivity may be improved substantially withrespect to what is presented here if target materials withlower concentrations of uranium than assumed in thiswork can be obtained.

We reserve Sec. VII for a summarizing discussion.

Before beginning the discussion of paleo-detectors, wewould like to acknowledge the earlier work of [52–57].These authors examined samples of ancient muscovitemica, initially for signatures from magnetic monopolesand then for WIMP-induced nuclear recoil tracks, usingatomic force microscopy to investigate the target sam-ple after cleaving and chemical etching. The progress inread-out technology which allows for the reconstructionof nano-scale features in target materials in the years

3

since their experiment allows the projected sensitivity ofour proposal to extend far beyond the upper limits ob-tained in Ref. [54]. Further improvements in our proposalwith respect to the earlier work include using a variety oftarget materials beyond muscovite mica, and proposingto use materials obtained from great depth, thus shieldedfrom cosmic rays.

II. DM SIGNAL

The differential nuclear recoil rate per unit target massfor elastic scattering of WIMPs with mass mχ off nucleiT with mass mT is(

dR

dER

)T

=2ρχmχ

∫d3v vf(v, t)

dσTdq2

(q2, v) . (1)

Here, ER is the recoil energy of T , ρχ the local WIMPdensity, f(v, t) the WIMP velocity distribution anddσT /dq

2 the differential WIMP-nucleus scattering crosssection with the (squared) momentum transfer q2 =2mTER. Taking into account canonical spin-independent(SI) and spin-dependent (SD) interactions only, the dif-ferential scattering cross section can be written as [58–62]

dσTdq2

(q2, v) =1

v2

[σSIT (0)F 2

SI(q2)

4µ2T

+SSDT (q2)

2J + 1

]Θ(qmax−q),

(2)where µT is the reduced mass of the WIMP-nucleus sys-tem, σSI

T (0) the spin-independent WIMP-nucleus scatter-ing cross section at zero momentum transfer, FSI the nu-clear form factor for SI scattering, SSD

T parameterizes theSD WIMP-nucleus scattering cross section, and J is thenuclear spin. The Heaviside Θ-function accounts for themaximal momentum transfer qmax = 2µT v and can betraded for a lower cutoff vmin =

√mTEr/2µ2

T in the ve-locity integral. Note that more general WIMP-nucleusscattering operators can lead to velocity dependence dif-ferent from dσT /dq

2 ∝ v−2, see e.g. the non-relativisticeffective field theory approach to direct detection [63, 64].

With Eq. (2) the differential recoil rate in Eq. (1) canbe written as(

dR

dER

)T

=2ρχmχ

[σSIT (0)F 2

SI(q2)

4µ2T

+SSDT (q2)

2J + 1

]×∫vmin

d3vf(v, t)

v.

(3)

For this work, we adopt a Maxwell-Boltzmann distribu-tion truncated at the galactic escape velocity vesc andboosted to the Earth’s rest frame for the velocity distri-bution as in the Standard Halo Model [17]. The remain-ing velocity integral in Eq. 3 can be calculated analyti-

cally, cf. Refs. [65, 66],∫vmin

d3vf(v, t)

v

=1

Nesc

1

2vE

[erf

(v+√2σv

)− erf

(v−√2σv

)]−(v+ − v−√2πvEσv

)e− v

2esc

2σ2v

,

(4)

with the normalization factor accounting for the trunca-tion of the Maxwell-Boltzmann distribution

Nesc = erf

(vesc√2σv

)−√

2

π

vescσv

e− v

2esc

2σ2v , (5)

and

v± ≡ min (vmin ± vE , vesc) . (6)

We adopt fiducial values of σv = 166 km/s for the ve-locity dispersion and vesc = 550 km/s for the escape ve-locity. Since the integration time of paleo-detectors ismuch larger than the orbital period of the Earth aroundthe Sun, we can neglect the relative motion of the Earthwith respect to the Sun. Then, the speed of the detectorwith respect to the galactic rest frame is given by

vE = v , (7)

and we adopt v =√

2σv + 13 km/s = 248 km/s.Note that for integration times O(1) Gyr the relative

speed of the Sun with respect to the galactic rest framemay become time dependent. In addition, including themotion of the Earth may have (minor) impact on thesensitivity: the Earth’s orbital motion would lead to abroadening of the WIMP induced recoil (and in turntrack length) spectra. We leave the investigation of sucheffects for future work.

A. WIMP-Nucleus Interaction Cross Section

In Eq. (2) we parameterized the differential WIMP-nucleus cross section in terms of the SI and SD crosssections. The zero-momentum WIMP-nucleus SI crosssection can be further parameterized as

σSIT (0) =

4

πµ2T [ZT fp + (AT − ZT ) fn]

2, (8)

where AT (ZT ) is the number of nucleons (protons) in thetarget nucleus and fp (fn) is the effective WIMP-proton(WIMP-neutron) coupling. Many WIMP models ex-hibit nearly isospin-conserving WIMP-nucleon couplingsfp ≈ fn, leading to the typical σSI

T (0) ∝ A2T enhance-

ment in the WIMP-nucleus scattering cross section. Thecouplings fp (fn) are related to the WIMP-nucleon crosssections σSI

p,n (the quantity in which experimental results

4

and projections are often quoted) by σSIp,n = 4µ2

p,nf2p,n/π

with µp (µn) the WIMP-proton (WIMP-neutron) re-duced mass.

We use the Helm nuclear form factor [67–69] for SIscattering

FSI(q2) = 3

sin(qrn)− qrn cos(qrn)

(qrn)3e−(qs)

2/2 , (9)

with the effective nuclear radius r2n ≈ c2 + 73π

2a2 − 5s2

where a ≈ 0.52 fm, c ≈ (1.23A1/3T − 0.6) fm, and s ≈

0.9 fm.For SD interactions it is useful to decompose the in-

teractions into an isoscalar (ST00), an isovector (ST11), andan interference term (ST01) [61, 62]

SSDT (q2) = a20S

T00(q2) + a21S

T11(q2) + a0a1S

T01(q2) , (10)

with the effective isoscalar and isovector couplings

a0 ≡ ap + an , a1 = ap − an . (11)

The effective SD WIMP-nucleon couplings are related tothe WIMP-nucleon scattering cross section as σSD

p,n(0) =

12µ2p,na

2p,n/π.

Where available1 we use the structure functions STij astabulated in [62]. For target nuclei where such structurefunctions do not exist we use the simplified parameteri-zation of the SD WIMP-nucleon cross section

SSDT (q2) =

(2J + 1)

π

(J + 1)

J2F 2SD(q2)

[ap〈STp 〉+ an〈STn 〉

]2,

(12)where 〈STp 〉 and 〈STn 〉 are the proton and neutron spin,respectively, averaged over the nucleus. We use values forthe 〈STi 〉 as tabulated in Ref. [61]. Note that there areconsiderable differences for these values when calculatedin different nuclear models; we follow the recommenda-tions given in Ref. [61] for which set of values to use foreach isotope.

For the nuclear structure function, we use the formgiven in Ref. [68]

F 2SD(qrn) =

0.047 , for 2.55 ≤ qrn ≤ 4.5 ,

sin2(qrn)/(qrn)2 , else ,

(13)

with the effective nuclear radius rn = 1.0A1/3T fm.

B. Track-Length Estimate

From the differential recoil rate for given target nu-clei we compute the associated spectrum of damage track

1 For the isotopes 19F, 23Na, 27Al, 29Si, 39K, 73Ge, 93Ni, 123Te,125Te, 127I, 129Xe, 131Xe, 207Pb

lengths. In obtaining our results, track lengths are ob-tained using numerical calculations described below. Toillustrate the relevant characteristics of the track lengthcalculation, we begin by presenting a semi-analytic ap-proximation here. The stopping power for a recoiling nu-cleus T incident on an amorphous target V with atomicnumber density nV can be estimated by [70](

dE

dx

)TV

= nVπa2TV γTVCTV

S(εTV ) , (14)

with the reduced energy

εTV =µ(TV )

mT

aTV E

αZTZV. (15)

ZT/V denotes the number of protons in T/V , CTV =εTV /E, γTV = 4µTV where µ(TV ) is the reduced massof the T–V system, and α is the fine-structure constant.The reduced stopping power S can be derived from ascreened interatomic Coulomb potential with screeninglength

aTV = 0.8853a0/(√

ZT +√ZV

)2/3, (16)

where a0 is the Bohr radius. For an average of variousinteratomic potentials, the reduced stopping power canbe parameterized as [70]

S(ε) ≈ 1

2

ln(1 + ε)

ε+AεB+ k√ε , (17)

where the parameters A = 0.14120, B = 0.42059 andk = 0.15 yield a reasonable fit to data for a wide varietyof T/V combinations. Note that the first term arisesfrom nuclear stopping and dominates when ε 1, whilethe second term arises from stopping due to the electronsassociated with the target nuclei.

For a composite material, the stopping power is ob-tained by summing over the contributions from differentconstituents V ,

dE

dxT=∑V

(dE

dx

)TV

, (18)

and the track length for a recoiling nucleus with energyER is

xT (ER) =

∫ ER

0

dE

(dE

dxT(E)

)−1. (19)

The track length spectrum for WIMP-induced nuclearrecoils within a target mineral is then given by a sumover constituent nuclei

dR

dx=∑T

ξTdERdxT

(dR

dER

)T

, (20)

where ξT is the mass fraction of the nuclei.

5

10−1 100 101 102

ER [keV]

100

101

102

103xT

[nm

]Nchwaningite

HOSiMn

10−1 100 101 102

ER [keV]

−0.2

0.0

0.2

(xS

RIM

T−x

ana

T)/x

SR

IMT

Nchwaningite

H/10

OSiMn

FIG. 1. Top: Track length xT as a function of nuclear re-coil energy ER for the different target nuclei in Nchwaningite[Mn2+

2 SiO3(OH)2·(H2O)] calculated with SRIM. Bottom: Rela-tive difference of xanaT (ER) from the semi-analytic calculationand xSRIM

T calculated with SRIM. Note that for hydrogen therelative difference is rescaled by a factor 1/10.

While illustrative of the electronic and nuclear stop-ping of recoiling nuclei in amorphous targets, theparametrization of the stopping power used in the semi-analytic approximation outlined above, in particular thefunctional form of the reduced stopping power Eq. (17)and the values of the numerical coefficients appearingtherein, are only matched to previous experimental datafor particular nuclei and energy ranges. In obtaining ourresults we instead use the stopping power obtained fromthe SRIM code [71, 72] and calculate the associated tracklengths using Eq. (19). Although the SRIM code improvesthe semi-analytic treatment by taking into account datafrom a more complete collection of nuclei at much widerranges of energy, these results strictly still hold for amor-phous targets only, i.e. neglect effects from the crystallinenature of our targets such as channeling [73, 74].

We show the track length as a function of recoilenergy for the different target nuclei in Nchwaningite[Mn2+

2 SiO3(OH)2·(H2O)] in the top panel of Fig. 1. Notethat lighter nuclei give rise to significantly longer tracksthan heavier nuclei for the same recoil energy because thestopping power increases with the charge of the nucleus,cf. Eqs. (14)-(17). Such figures look similar for othertarget nuclei, with quantitative differences mainly arising

from varying molecular number densities between differ-ent target minerals. In general, lower molecular numberdensities yield longer tracks for the same recoiling nucleusand recoil energy, cf. Eq. (14).

The top panel of Fig. 1 also indicates the small-est nuclear recoils which could be detected with paleo-detectors: If tracks as short as O(1) nm can be recon-structed, paleo-detectors are sensitive to nuclear recoilsas small as O(100) eV. Read-out methods with somewhatworse spatial resolution, say O(10) nm, would correspondto a recoil energy threshold of order keV.

In the bottom panel of Fig. 1 we show the relativedifference of the track length spectra calculated with thesemi-analytic approximation outlined above and with theSRIM code. In the most relevant recoil energy range0.1 keV . ER . 100 keV both results differ by less than10 % for nuclei other than hydrogen. We find larger dif-ferences outside of this energy range as well as for low-Znuclei such as hydrogen. Note that the stopping powerin the semi-analytic calculation is not fit to data outsideof this energy range or for low-Z recoils, while the resultsof the SRIM code still describe experimental data well inthis region.

In Fig. 2 we show the track length spectra induced byWIMPs in different target materials together with back-ground spectra discussed in the following section. Weshow spectra induced in two marine evaporites (MEs),Halite (NaCl) and Epsomite [Mg(SO4)·7(H2O)] and twoultra-basic rocks (UBR), Olivine [Mg1.6Fe2+0.4(SiO4)] andNickelbischofite [NiCl2 · 6(H2O)].2 For both MEs andUBRs this selection contains one mineral with and onewithout hydrogen. Note that we do not include tracksfrom hydrogen in the track length spectra shown in Fig. 2,cf. the discussion in Sec. III. Also, heavier WIMPs giverise to harder track length spectra than lighter WIMPsbecause heavier WIMPs induce more energetic nuclearrecoils.

Note that the track length discussed above is strictlythe range of nuclei in the material. In the following, wewill assume that this range coincides with the measuredtrack length modulo errors induced by the finite spatialresolution of the particular read out method. In princi-ple, this assumption can be violated either if permanentdamage only arises in part of a nucleus’ range, or if thenucleus loses energy via hard scatterings such that theshape of the track deviates significantly from a straightline. Both cases could lead to reconstructed tracks ap-pearing much shorter than the range calculated above.Measured track lengths shorter than the calculated rangecould lead to diminished sensitivity of paleo-detectors, al-though such effects can be partly mitigated by improvedspatial resolution or larger exposures.

2 Here and in the remainder of this paper, for brevity, we referto minerals in marine evaporite deposits as “marine evaporites”(MEs) and to minerals in ultra-basic rocks as “ultra-basic rocks”(UBRs).

6

100 101 102 103

x [nm]

10−4

10−1

102

105

108dR/dx

[nm−

1kg−

1M

yr−

1]

Halite; σSIp = 10−45 cm2, C238 = 0.01 ppb

5 GeV50 GeV500 GeV

νn234Th

100 101 102 103

x [nm]

10−4

10−1

102

105

108

dR/dx

[nm−

1kg−

1M

yr−

1]

Epsomite; σSIp = 10−45 cm2, C238 = 0.01 ppb

5 GeV50 GeV500 GeV

νn234Th

100 101 102 103

x [nm]

10−4

10−1

102

105

108

dR/dx

[nm−

1kg−

1M

yr−

1]

Olivine; σSIp = 10−45 cm2, C238 = 0.1 ppb

5 GeV50 GeV500 GeV

νn234Th

100 101 102 103

x [nm]

10−4

10−1

102

105

108

dR/dx

[nm−

1kg−

1M

yr−

1]

Nickelbischofite; σSIp = 10−45 cm2, C238 = 0.1 ppb

5 GeV50 GeV500 GeV

νn234Th

FIG. 2. Track length spectra from recoiling nuclei in Halite (NaCl; top left), Epsomite [Mg(SO4)·7(H2O); top right], Olivine[Mg1.6Fe2+0.4(SiO4); bottom left], and Nickelbischofite [NiCl2 ·6(H2O); bottom right] induced by the neutrino (ν) and neutron(n) background and WIMPs with mχ = 5, 50, 500GeV, assuming σSI

p = 10−45 cm2. The vertical gray line indicates thetrack length of 72 keV 234Th nuclei from (238U → 234Th + α) decays. Halite and Epsomite are MEs for which we assume a238U concentration of C238 = 0.01 ppb in weight while for the UBRs Olivine and Nickelbischofite we assume C238 = 0.1 ppb.See Sec. IV for a discussion of the background sources.

In any case, our studies using SRIM indicate that cor-rections to the tracks lengths due to inconsistent appear-ance of persistent damage or to significant scattering ofnuclei are small. In particular, they play a role only atthe end of the track when the nucleus has already lostmost of its energy and traversed most of its range. Theeffect of such corrections is a question which currentlycannot be answered quantitatively; detailed experimen-tal studies are required for each target material, whichwe leave for future work. To the best of our knowledge,reliable estimates exist only for the particular case of re-constructing tracks in muscovite mica after cleaving andchemical etching [57].

III. READ-OUT METHODS

In order to be suitable for paleo-detectors, target min-erals must satisfy a few basic criteria. Foremost, nucleiwith energies of order ER = 0.1 − 100 keV should give

rise to damage tracks in the material, and such tracksmust persist over sufficiently long time scales. Materialswith such properties are commonly referred to as SolidState Track Detectors (SSTDs) [75–78].

The exact mechanism for track formation by ionsin solids is not fully understood; popular models in-clude the thermal spike model [79] and the ion explosionmodel [80]. Which ions leave permanent tracks in whichmaterials remains a question that can only be answeredsemi-empirically. In general, materials must be insula-tors (or poor semiconductors) with electrical resistivitylarger than ∼ 2000 Ω cm to record tracks, see Ref. [78]for a more detailed discussion of track formation crite-ria. Much of research using SSTDs has employed opti-cal microscopy to read out tracks after chemical etching.Tracks from ions with charge Z & 10 and energies largerthan a few keV are easily etchable in virtually all insulat-ing materials studied. For tracks from lighter nuclei, inparticular α-particles (4He2+ ions), the situation is lessclear. Proton (1H+) tracks have been demonstrated to

7

be etchable only in plastics [78]. However, tracks fromα-particles with ∼ 5 MeV energies have been measured inAl2O3:C,Mg crystals without etching using confocal laserscanning and structured illumination microscopy [81, 82].In this work we remain agnostic about the question oflow-Z particles leaving tracks in the targets, but will in-stead consider two scenarios: with low-Z tracks, wherewe assume that all ions leave tracks, and without low-Ztracks where we assume that only ions with Z ≥ 3 leaveobservable tracks.3 Which scenario applies will have tobe determined experimentally for each combination oftarget material and read-out method.

Besides recording tracks in the first place, materialsalso have to preserve tracks over geological timescalesof 0.1 − 1 Gyr in order to be suitable for use as paleo-detectors. The fading, or annealing, of tracks is thoughtto be due to the diffusion of atoms from the vicinity ofthe damaged region into the tracks. The annealing timescale is exponentially suppressed with the temperature Tof the material [77, 78]

tann ∝ eEann/kT , (21)

where Eann is the so-called activation energy for anneal-ing and k the Boltzmann constant. Typically, Eann ∼kTm, where Tm is the melting temperature. For mostminerals the annealing time at room temperature is muchlarger than 1 Gyr. For typical low melting-point crystals,e.g. Calcite, annealing times are t & 109 yr at room tem-perature. Because of their high melting temperatures,refractory materials are of particular interest becausethey typically display large Eann and in turn annealingtimes, e.g. for Diopside (CaMgSi2O6) the annealing timehas been estimated to be t ∼ 1059 yr at room tempera-ture [77]. Note that reported temperatures at the bottomof ultra-deep boreholes vary from ∼ 100 C to ∼ 300 C,depending on the depth and local geology. Thus, depend-ing on the temperatures at the sites from where targetmaterials for paleo-detectors will be obtained, track an-nealing may play a role for low melting-point materials.In refractory materials, annealing times are larger than1 Gyr even at temperatures of a few 100 C.

Damage tracks which persist over geological time scaleshave been studied extensively for fission track dating,typically making use of tracks caused by the Z ∼ 50fragments from spontaneous fission of 238U. A commonlyused mineral for fission track dating is muscovite mica,which can be cleaved into thin slices. Fission tracks arethen read out with e.g. Electron Microscopy (EM) or af-ter chemical etching with optical microscopes. Note thatthe ∼nm resolution of EM allows the imaging of fissiontracks without prior enlargement by chemical etching.

3 For paleo-detectors the most relevant distinction is if hydrogennuclei leave tracks or not. Recovering tracks from α-particlesor not has almost no effect on the projected sensitivity of thematerials considered here.

However, chemical etching is often used for EM read-out in order to stabilize tracks against thermal annealingcaused by the incident electron beam. [77]

Previous searches for DM-induced nuclear recoil tracksused techniques similar to those of fission track dating,employing an Atomic Force Microscope (AFM) to scanthe surfaces of cleaved and etched samples of muscovitemica [54]. These past experiments were limited by boththe small throughput allowed by the extensive time re-quired for sample preparation and the manifestly two-dimensional nature of the ion track reconstruction.

We propose to read out samples with recently devel-oped Helium Ion beam Microscopy (HIM) [83], whichhas spatial resolution similar to EM. However, it causesless sample damage than EM [84] and is capable ofsub-surface imaging to depths of O(100) nm. Furtherprogress with respect to the two-dimensional readoutwith EM or AFM can be made by using a Focused IonBeam (FIB) of either neon or gallium ions for samplepreparation (for example, see [85]). The FIB can bothbe used for initial sample preparation and for realizingthree dimensional reconstruction of samples by sequen-tially imaging a surface layer with HIM and then sput-tering away the read-out layer using the FIB [86]. Muchfaster ablation of the read-out layer is achievable by us-ing pulsed lasers in addition to the FIB [87–89]. Weestimate that combining HIM with pulsed laser ablation,target volumes of a few mm3, corresponding to massesof O(10) mg, should be possible. Although the targetmasses which can be read out with HIM are particularlysmall, we see from Fig. 2 that, for mχ . 10 GeV, WIMP-induced recoils yield . 10 nm long nuclear recoil tracksat higher rates when compared to heavier WIMPs. Thus,the ∼nm spatial resolution possible with HIM read-outis ultimately more relevant for sensitivity to low massWIMPs since, as with conventional DD experiments,paleo-detectors become limited by the threshold of thedetector rather than its exposure.

Alternatively, for higher mass WIMPs with signifi-cantly longer induced nuclear recoil tracks, Small AngleX-ray scattering (SAXs) at synchrotron facilities shouldbe capable of fully three dimensional imaging of bulksamples with minimal sample preparation [90]. Ion trackshave been revealed without etching in crystalline materi-als using SAXs, although only after imaging the samplealong the direction of the recorded track [91]. Also, SAXstomography has achieved ∼ 15 nm three-dimensionalspatial resolution [92] but not for resolving damage fromion tracks, which cause only small variations in electrondensity of the target material. Thus, while we are propos-ing a particularly challenging application of SAXs andthe available beam time can be limited, we estimate atotal target volume of a few tens of cm3, correspond-ing to O(100) g, can be imaged at synchrotron facilities.The ∼ 104 increase in exposure relative to HIM allowsfor SAXs to probe much lower WIMP-nucleon scatteringcross sections for WIMPs with mχ & 10 GeV despite theloss of spatial resolution.

8

Even larger target volumes can be handled by Con-focal Laser Scanning Microscopy (CLSM) or StructuredIllumination Microscopy (SIM), although at the cost ofworse spatial resolution of the order of 100 nm. Reliableread out of α-particle tracks with sufficient spatial res-olution to allow for energy spectroscopy of the incidentα-flux has been demonstrated with CLSM and SIM with-out prior chemical etching in particular target materials(Al2O3:C,Mg) [81, 82]. Somewhat better spatial reso-lution may be achievable with Ultra-Violet Microscopy(UVM).

Read-out methods which can process comparativelylarge target volumes such as CLSM, SIM, or UVMare particularly useful to pre-screen target materials forpaleo-detectors and identify sub-volumes which are lowin radioactively induced backgrounds. Such sub-volumescould then be interrogated for traces of WIMP interac-tions with higher resolution read-out methods such asSAXs or HIM.

IV. BACKGROUND REJECTION

In order to be sensitive to DM signals, a number ofbackground sources must be mitigated or controlled toa sufficient level in paleo-detectors. Most of the back-ground sources are the same as in the conventional directdetection approach. However, the relative importance ofthe respective background sources is different in paleo-detectors and conventional direct detection experimentsdue to three reasons:

1) Paleo-detectors use long exposure times[. O(109) yr] and small target masses [. O(1) kg]while conventional direct detection experimentsuse short exposure times [. O(10) yr] and largetarget masses [. O(104) kg].

2) The experimental observable in paleo-detectors is1−500 nm long damage tracks from recoiling nuclei,while conventional direct detection experiments ob-serve the ionization charge, the scintillation light,or the heat (phonons) produced by the recoilingnucleus.

3) Paleo-detectors measure events integrated over theexposure time while conventional experiments haveprecise timing information.

In the remainder of this section we discuss the dominantsources of backgrounds and how to mitigate or controlthem. Note that natural crystal imperfections in ourtarget materials are either single-site or span across theentire (mono-)crystalline volume and are thus easily re-jected. All relevant backgrounds are damage tracks fromcharged particles being stopped in the material.

A. Cosmogenic

Cosmic rays can scatter off target nuclei, leading torecoils similar to WIMP-induced nuclear recoils. Sim-ilar to conventional direct detection experiments, suchbackground can be mitigated by using target samplesobtained from far below the Earth’s surface. The domi-nant cosmogenic background will then be due to muonsinteracting with nuclei in the vicinity of the target vol-ume, giving rise to fast neutrons, which in turn scatteroff target nuclei, inducing nuclear recoils.

In conventional direct detection experiments, suchneutron-induced nuclear recoils can be mitigated fur-ther by rejecting coincident events: fast neutrons havemean free paths of O(1) cm in typical detector mate-rials and usually scatter multiple times in the detectorvolume. Note that because the neutron’s mass is muchsmaller than the mass of typical target nuclei, neutronslose only a small fraction of their energy in a single scat-tering event. Due to the lack of timing information,such suppression strategies cannot be employed in paleo-detectors.

However, conventional detectors must be operated in(large) underground laboratories. The deepest such lab-oratories currently available (Snolab, Canada and CJPL,China) have an overburden of . 2 km of rock corre-sponding to . 7 km.w.e. (km water equivalent). Atsuch depths the neutron flux is O(10−2) m−2 yr−1 =O(103) cm−2 Gyr−1 [93]. While a neutron flux of thismagnitude is acceptable for a conventional direct detec-tion experiment it would pose a severe problem for paleo-detectors due to the large exposure and the lack of tim-ing information. However, the neutron flux is exponen-tially suppressed with the overburden. Target materialsfor paleo-detectors can be obtained from depths muchlarger than 2 km, e.g. from ultra-deep boreholes as usedfor geological R&D and oil exploration. The neutronflux at depths of 5 , 7.5 , 10 km rock overburden is oforder 1 , 10−4 , 10−8 cm−2 Gyr−1, making cosmogenicbackground negligible for materials obtained from depthslarger than & 5 km. Note that cores from existing bore-holes containing either MEs (for example, see [94]) orUBRs (for example, see [95]) with ages & 500 Myr havebeen analyzed from depths as large as 12 km [51].

Due to the comparatively small size of the target sam-ple, near-surface storage of the minerals between whenthey are obtained from deep in the Earth and readout does not lead to problematic levels of irradiationwith cosmic rays. For example, the induced neutronflux in a ∼ 50 m deep storage facility is smaller than∼ 0.2 cm−2 yr−1.

B. Radioactive Decays

Any target sample used for paleo-detectors will be con-taminated by traces of radioactive materials. The miti-gation of the corresponding background is crucial for the

9

Nucleus Decay mode T1/2

238Uα 4.468× 109 yr

SF 8.2× 1015 yr234Th β− 24.10 d

234mPaβ− (99.84 %)

1.159 minIT (0.16 %)

234Pa β− 6.70 d234U α 2.455× 105 yr230Th α 7.54× 104 yr226Ra α 1600 yr222Rn α 3.8325 d

TABLE I. Half-lives T1/2 of selected nuclei in the 238U de-

cay chain [96–100]. For 238U we denote both the half-livescorresponding to α-decays and spontaneous fission (SF). For234mPa we denote in addition the branching ratio for β−-decays and isomeric transitions (IT).

success of the proposed search. Similar to conventionaldirect detection experiments, it is crucial to select materi-als with the lowest possible concentrations of radioactivematerials. We discuss typical contaminations of targetmaterials further in Sec. V. The most relevant contami-nant is 238U. As a benchmark value, we will assume 238Uconcentrations of C238 = 0.01 ppb (part per billion) inweight in the following discussion.

The 238U decay chain is

238Uα−→ 234Th

β−−→ 234mPaβ−−→ 234U

α−→ 230Thα−→ 226Ra

α−→ 222Rnα−→ . . . −→ 206Pb .

(22)

We quote the half-lives for the most relevant decays inTab. I.

In paleo-detectors, there are qualitative differences be-tween α-decays and β/γ-decays: α-decays are (N →N ′ + α) 2-body decays, giving rise to a mono-energeticnuclear recoil and an α-particle. For the α-decays ofheavy nuclei in the uranium and thorium decay chains,the energies of the α-particles are a few MeV, and the in-duced nuclear recoils have energies of 10−100 keV. Boththe recoiling nucleus and the α-particle lose their energymostly via ionization and elastic scattering off other nu-clei, giving rise to a damage track in the target material.β (γ) decays are 3-body (2-body) decays where the nu-cleus emits an electron and a neutrino (a photon). Sincethe mass of the electrons/neutrinos/photons is negligiblewith respect to the mass of the nucleus, the light decayproducts carry most of the excess energy of the decay.The recoil energy of the daughter nucleus from the decayis EN

′

R . O(10) eV. The emitted electron is relativisticand does not deposit enough energy in a material to cre-ate a persistent damage track. Similarly, γ-rays lose theirenergy by scattering off electrons or by electron pair cre-ation, again giving rise to relativistic electrons which donot induce tracks.

In summary, α-decays give rise to a (heavy) recoilingnucleus with energies 10−100 keV and an α-particle witha few MeV of energy, both of which give rise to potentially

observable tracks. β and γ-decays on the other handgive rise to relativistic electrons which leave no persistentdamage in materials and recoils of the (heavy) nuclei withenergies of order 10 eV. Such low-energy recoils give riseto unobservably short damage tracks. Hence, only α-decays give rise to potential background events.

Recalling the 238U decay chain, cf. Eq. (22) and consid-ering the half-lives of the involved nuclei listed in Tab. Iwe find that the half-life of the initial 238U decay is com-parable or somewhat larger than the integration time ofpaleo-detectors. The subsequent decays are much faster;the longest half-life of the nuclei in the decay chain is for234U. Note that the accumulated half-life of the decaysafter 222Rn not listed in Tab. I is < 23 yr and the decaychain contains 4 α-decays between 222Rn and the stable206Pb.

Thus, almost all 238U nuclei which underwent the ini-tial (238U → 234Th + α) decay will have decayed fur-ther along the decay chain to stable 206Pb. Such eventswill manifest in the mineral as a sequence of 8 spatiallyconnected ER = O(100) keV recoils of the heavy daugh-ter nuclei in the decay chain accompanied by 8 α-tracks.Note that the typical range of an α-particle with ener-gies of order MeV is larger than a few µm in the targetminerals of interest.

The characteristic pattern of nuclear recoil tracks canbe used to efficiently mitigate 238U decay events evenunder the pessimistic assumption that the damage trackfrom the α-particles does not create sufficient damage inthe target material to be resolved when reading out thematerial.

However, due to the relatively long half-life of 234U,the second α-decay in the 238U decay chain, there willbe a population of events in the target sample whichhas undergone only a single α-decay. This backgroundposed a significant problem [101, 102] in the original at-tempt to search for DM induced damage tracks in ancientMica [54–57]. In [54] the authors used atomic force mi-croscopy (AFM) to search for damage tracks after cleav-ing and chemically etching a Mica sample. In Mica, α-tracks are not etchable, hence, only the recoil track of the72 keV 230Th nucleus from the (238U → 234Th + α) de-cay would be detectable. Due to etching the sample andusing AFM as read-out which can only scan the surfaceof the material, the track length resolution in Ref. [54]was relatively poor. Thus, the characteristic length ofthe 234Th induced track could not be used for rejectionof that background. However, the advances in read-outtechnology in the last decades now allow to efficientlymitigate the single-α background.

The number of single-α decays in a target sample isgiven by

N2381α (t) = N0

238

λ238λ234 − λ238

(e−λ238t − e−λ234t

), (23)

where λ238 (λ234) is the decay constant of 238U (234U)related to the half-life as λ = ln(2)/T1/2 and t is the age

of the mineral. The initial number of 238U atoms in the

10

sample, N0238 is given by

N0238 = MTC

238NA/mmol238 , (24)

where MT is the mass of the target mineral, C238 thefraction of 238U in the target material in weight, NA theAvogadro constant, and mmol

238 the molar mass of 238U.For T 234

1/2 . t . T 2381/2 , the age of most minerals of

interest for paleo-detectors, the number of single-alphaevents per unit target mass is well approximated by

n2381α ≡N238

1α

MT' N0

238

MT

λ238λ234

= 109 kg−1(

C238

0.01 ppb

).

(25)

Such large numbers of background events require effec-tive background suppression in order to retain sensitivityto hypothetical DM signals. In principle, there are twooptions to mitigate the single-α background dependingon the read-out scenario discussed in Sec. III: 1) in thewith low-Z tracks scenario, the coincident detection of anα-induced track and the damage track from the recoil-ing 234Th nucleus make background suppression trivial.The α-particle has a characteristic energy of 4.2 MeV,giving rise to tracks longer than 10µm in all target ma-terial considered. Such tracks are 1− 2 orders of magni-tude longer than WIMP-induced tracks. 2) in the withoutlow-Z tracks scenario, only the track of the 234Th nucleusis visible. Then, one has to rely on the monochromatic72 keV recoil energy of the 234Th nuclei, cf. Fig. 2 for thecorresponding track lengths. Note that the finite tracklength resolution will turn mono-chromatic recoils intoa Gaussian spectrum (after reconstruction) with widthgiven by the track length resolution. As we will see inSec. VI, the impact on the projected sensitivity of paleo-detectors is negligible since the corresponding narrow fea-ture in track length is easily rejected for high-resolutionread-outs such as HIM. For read-out methods with worseresolution such as SAXs, the single-α background depre-ciates the sensitivity to low mass WIMPs, while the sen-sitivity to heavier WIMPs is retained.

C. Neutrons

A small fraction of the 238U atoms in the target sam-ple will undergo spontaneous fission (SF) instead of α-decays. The fission event itself is an easily rejected back-ground: Spontaneous fission gives rise to two (heavy)daughter nuclei which recoil against each other and sub-sequently decay to stable nuclei. Such events will give riseto signatures even more spectacular than the 8 α-decaysin the usual 238U decay chain discussed above.

However, SF of 238U nuclei also gives rise to ∼ 2 fastneutrons with typical energies of ∼ 1 MeV. Fast neutronslose their energy predominantly via elastic scattering offtarget nuclei. The mean free path for fast neutrons in

typical target minerals is of the order of a few cm, makingit difficult to connect the induced recoil of a target nu-cleus with the SF event. Furthermore neutrons lose onlya small fraction of their energy when elastically scatter-ing off the typically much heavier target nuclei due to thescattering kinematics. Typically, neutrons will undergo100−1000 elastic scatterings before losing enough energyto not be able to give rise to nuclear recoils with energiessimilar to WIMP induced recoils. In conventional directdetection experiments, such multiple scatterings are usedto reject neutron induced recoil events. However, sincepaleo-detectors have no timing information, such an ap-proach cannot be used for background rejection.

The number of spontaneous fission events per unit tar-get mass in a target mineral of age t from 238U contami-nation is given by

n238SF =N0

238

MT

(1− e−λ238t

) T 2381/2

T 238; SF1/2

, (26)

with the initial number of 238U atoms in the sample, N0238

given by Eq. (24), MT the mass of the target mineral,λ238 = ln(2)/T 238

1/2 the decay constant of 238U, and T 2381/2

(T 238; SF1/2 ) the (SF) half-life of 238U reported in Tab. I.

For mineral ages short compared to the 238U half-life,the number of SF events is well approximated by

n238SF ' 2× 106 kg−1(

C238

0.01 ppb

)(t

1 Gyr

). (27)

Fast neutrons are also produced in so-called (α, n)-reactions, emission of fast neutrons from interactions ofα-particles with heavier nuclei. Although only a smallfraction of α-particles will lead to neutron emission,(α, n)-reactions yield a sizeable contribution to the neu-tron flux in the target since the number of α-particlesis ∼ 107 times larger than the number of neutrons fromSF events. Depending on the composition of the targetmaterial, the neutron spectrum can either be dominatedby neutrons from SF events or by neutrons from (α, n)interactions. Generally, the lighter the target nuclei are,the more relevant the (α, n) contribution becomes. How-ever, the strong dependence of the (α, n) cross section onthe nuclear structure of the target isotopes makes generalstatements difficult.

We use the SOURCES-4A code [103] to calculate the neu-tron spectra from SF of all nuclei in the 238U decay chainand (α, n) reactions induced by α-particles from the 238Udecay chain. From these spectra, we calculate the in-duced nuclear recoil spectra using the neutron-nucleoncross sections tabulated in the JANIS4.0 database [104].4

We take only elastic neutron-nucleon scattering into ac-count. Thus, the obtained background is a conservative

4 We use values from TENDL-2017 [105–108] for neutron-nucleoncross sections.

11

estimate: including additional processes such as inelasticscattering, neutron absorption, or (n, α) processes lowerthe background, since neutrons lose a larger fraction oftheir energies in such interactions than in elastic scatter-ings, or are absorbed.

Due to the scattering kinematics, neutrons lose onlya small fraction of their energy when elastically scatter-ing off heavy nuclei. However, when scattering off lighttargets, the energy transfer is much more efficient: Onaverage, a fast neutron from SF or (α, n)-reactions willgive rise to ∼ 4 (∼ 7) nuclear recoils with ER & 10 keV(ER & 1 keV) when scattering off nuclei with mT ∼1 GeV. When scattering off nuclei withmT ∼ 10 GeV, thesame neutrons give rise to ∼ 20 (∼ 50) recoils with ener-gies ER & 10 keV (ER & 1 keV). For even heavier nuclei,with mT ∼ 100 GeV, fast neutrons induce ∼ 30 (∼ 200)recoils with energies ER & 10 keV (ER & 1 keV). Becauseof the kinematic match of the neutron and proton (i.e. Hnuclei) masses and because the neutron-hydrogen elasticscattering cross section is large compared to those of mostheavier nuclei, fast neutrons will scatter efficiently off hy-drogen in a target and lose a large fraction of their energyin each interaction. This leads to a large reduction of thenumber of energetic neutron induced nuclear recoils, inparticular recoils of nuclei heavier than hydrogen, evenif hydrogen comprises only a relatively small fraction ofthe target molecules, as can be seen by comparing theleft and right panels of Fig. 2. In addition, depending onthe target material and read-out method, the hydrogenrecoils themselves may not give rise to observable tracks,cf. the discussion in Sec. III.

On the other hand, (α, n) cross sections are typicallylarge for the lightest target nuclei with Z ≥ 3. Thus,target minerals containing lithium or beryllium are notwell suited for paleo-detectors.

We show the track length spectra induced by neutronsin Fig. 2 together with the WIMP-induced spectra. Triv-ially, the neutron induced background is lower in materi-als with lower concentration of 238U. However, the differ-ence between target minerals with and without hydrogenis much larger, suppressing the neutron induced back-ground by more than two orders of magnitudes betweenotherwise similar target minerals.

D. Neutrinos

Neutrinos emitted in the Sun, supernovae explosions,and cosmic ray interactions in the atmosphere can coher-ently scatter off the target nuclei giving rise to nuclearrecoils; the same process which gives rise to the neutrinofloor for DD experiments [109]. The differential recoilspectrum per unit target mass induced by neutrinos isgiven by [109, 110](

dR

dER

)T

=1

mT

∫Eminν

dEνdσ

dER

dΦνdEν

, (28)

where Eminν =

√mTER/2 is the minimum neutrino

energy required to induce a nuclear recoil with en-ergy ER, similar to the maximum momentum transferin Eq. (2). The differential cross section for coherentneutrino-nucleus scattering is

dσ

dER(ER, Eν) =

G2F

4πQ2WmT

(1− mTER

2E2ν

)F 2(ER) ,

(29)where GF is the Fermi coupling constant, [QW ≡(AT − ZT ) −

(1− 4 sin2 θW

)ZT ] with the weak mixing

angle θW , and F (ER) is the nuclear form factor. Wetake the neutrino flux dΦν/dEν from Ref. [110]; forEν . 20 MeV the neutrino flux is dominated by solarneutrinos, for 20 MeV . Eν . 30 MeV by the diffusesupernova neutrino background (DSNB), and for largerenergies Eν & 30 MeV by atmospheric neutrinos. Thenuclear recoil spectrum due to the neutrino backgroundis converted to an ionization track length spectrum anal-ogously to the WIMP induced recoil spectra; we showthe resulting spectrum together with those induced byWIMPs and neutrons in Fig. 2 for a selection of targetmaterials.

E. Background Discussion

In Fig. 2 we compare the track length spectra inducedby the respective backgrounds with those induced byWIMPs. A more detailed comparison including the un-certainty of the background and the effects of finite tracklength resolution on the spectra is shown for Epsomitein Fig. 3 for low-mass WIMPs and in Fig. 4 for heavierWIMPs. In the upper panels of these figures, we indi-cate the neutrino, neutron, and single α-decay inducedbackgrounds with the solid, dashed, and dashed-dottedblack lines, respectively. The shaded bands around thelines show the systematic and statistical errors, added inquadrature, of the respective backgrounds. Colored linesshow the WIMP-induced spectra for a variety of WIMPmasses and benchmark WIMP-nucleon scattering crosssections, cf. the captions of Figs. 3 and 4. In addition, thebottom panels show the ratio of the number of signal tobackground events for each WIMP benchmark togetherwith the relative uncertainty of the total background.

The relevant background quantity is not the totalnumber of events, but the uncertainty of the numberof background events in the signal region. Here, weassume that backgrounds induced by neutrinos sufferfrom much larger uncertainties than radioactivity in-duced backgrounds. The main uncertainty on the neu-trino induced backgrounds stems from the normaliza-tion of the neutrino fluxes. Today’s fluxes are knownto an accuracy of a few percent for solar neutrinos,while the DSNB and atmospheric neutrino fluxes are onlyknown within some tens of percent [110]. However, paleo-detectors would measure the time-integrated neutrino in-duced background over a period of as large as a billion

12

10−1

100

101

102

103

104

105

Eve

nts

/1

nm

σx = 1 nm

Epsomite; ε = 0.01 kg Myr; C238 = 0.01 ppb

ν; Σν = 100 %n; Σn = 1 %

1 GeV; 10−42 cm2

5 GeV; 10−42 cm2

10 GeV; 10−42 cm2

15 GeV; 10−42 cm2

100 101 102 103

x [nm]

10−2

100

102

S/B

with low-Z tracks

10−1

100

101

102

103

104

105

Eve

nts

/1

nm

σx = 1 nm

Epsomite; ε = 0.01 kg Myr; C238 = 0.01 ppb

ν; Σν = 100 %n; Σn = 1 %1α; Σ1α = 1 %

1 GeV; 10−42 cm2

5 GeV; 10−42 cm2

10 GeV; 10−42 cm2

15 GeV; 10−42 cm2

100 101 102 103

x [nm]

10−2

100

102

S/B

without low-Z tracks

FIG. 3. Binned recoil spectra (upper panels) and ratio of signal (S) to background (B) events per bin (lower panels) inEpsomite [Mg(SO4) ·7(H2O)], assuming an exposure of 0.01 kg Myr and a 238U concentration of C238 = 0.01 ppb in weight.The colored lines are for WIMPs with (5, 10, 15, 50) GeV mass assuming a WIMP-nucleon cross section of σSI

n = 10−42 cm2

as indicated in the legend. Upper panels: The solid, dashed, and dash-dotted black lines are for the neutrino (ν), neutron(n), and single-α (1α) induced background spectra, respectively. The gray shaded bands around the respective lines indicatethe statistical and systematic errors, added in quadrature, of the respective backgrounds. Note that the uncertainty bands ofmultiple backgrounds overlap where the shaded area appears darker. Lower panels: Ratio of signal (S) to total background(B) events per bin. The gray shaded area indicates the relative uncertainty of the total background per bin, i.e. the ratio ofthe (quadratic sum of the statistical and systematic errors of all components)/(sum of all background events) per bin. Thesignal-to-noise ratio, as defined in Eq. (32), per bin is obtained by dividing S/B for the respective signal with the relativeuncertainty of the background. The left panels show spectra calculated with low-Z tracks, i.e. under the assumption thattracks from low-Z nuclei, including hydrogen, are visible. Thus, the single-α induced background can be rejected based on theappearance of the α-tracks. The spectra in the right panels are calculated without low-Z tracks. Thus, we assume that low-Znuclei such as hydrogen and helium (α-particles) do not leave visible tracks. Hence, the monochromatic 234Th recoils from thesingle-α background appear as broadened spectrum due to the finite track length resolution.

years, entailing further uncertainties from extrapolatingthe current neutrino fluxes over such time scales. As inRef. [1] we account for such uncertainties by assuming arelative systematic error of Σν = 100 % for the number ofneutrino induced background events. Note that a spec-tral analysis would allow one to significantly reduce thiserror since such an analysis would use a control regiondominated by neutrino induced events to measure thenormalization of the neutrino induced background [111].It should be kept in mind however that the normalizationof the neutrino induced backgrounds from the differentcomponents of the solar neutrino fluxes, DSNB, and at-mospheric neutrinos may fluctuate independently suchthat each component must be measured individually.

The normalization of the radioactive backgrounds onthe other hand is determined solely by the initial con-centration of radioactive materials in the target min-eral and the exposure time. There is no external sourcewhich may fluctuate with time as for the neutrino in-duced backgrounds. Hence, the normalization can be

predicted much better, e.g. from measuring the numberof full 238U decay chains in the target mineral. We as-sume a systematic uncertainty of 1 % on the single-α andneutron induced backgrounds.

For the assumed benchmark 238U concentration ofC238 = 0.1 ppb (C238 = 0.01 ppb) for UBR (ME) miner-als, we can identify two (broadly speaking) backgroundregimes. For tracks shorter than some tens of nm, thebackground uncertainty is dominated by recoils inducedby solar neutrinos. For longer tracks, the neutrino in-duced background is dominated by DSNB and atmo-spheric neutrinos, both of which have much smaller fluxesthan solar neutrinos. Thus, for tracks longer than sometens of nm, neutron induced events are the dominantcontribution to the background uncertainty. Note thatthe value of the track length below (above) which solarneutrinos (neutrons) dominate the background budget istarget dependent.

The single-α background plays a role only if α-trackscannot (reliably) be reconstructed by the chosen read-

13

101

102

103

104

105

106

107

108

109E

vent

s/

10n

mσx = 15 nm

Epsomite; ε = 100 kg Myr; C238 = 0.01 ppb

ν; Σν = 100 %n; Σn = 1 %

5 GeV; 10−42 cm2

50 GeV; 10−47 cm2

500 GeV; 10−46 cm2

5 TeV; 10−45 cm2

101 102 103

x [nm]

10−4

10−2

100

S/B

with low-Z tracks

101

102

103

104

105

106

107

108

109

Eve

nts

/10

nm

σx = 15 nm

Epsomite; ε = 100 kg Myr; C238 = 0.01 ppb

ν; Σν = 100 %n; Σn = 1 %1α; Σ1α = 1 %

5 GeV; 10−42 cm2

50 GeV; 10−47 cm2

500 GeV; 10−46 cm2

5 TeV; 10−45 cm2

101 102 103

x [nm]

10−4

10−2

100

S/B

without low-Z tracks

FIG. 4. Same as Fig. 3 but for an exposure of 100 kg Myr and assuming a track length resolution of σx = 15 nm. WIMP signalare shown for masses of (5, 50, 500, 5000) GeV and assuming WIMP-nucleon cross sections of (10−42, 10−47, 10−46, 10−47) cm2,respectively. Note that the bin-width is 10 nm instead of 1 nm in Fig. 3 and that the scales of the axes are different.

out method. However, the impact on the signal-to-noiseratio remains small as can be seen by comparing the leftand right bottom panels of Figs. 3 and 4, respectively.

V. MINERAL OPTIMIZATION

As discussed in Sec. IV, one of the major sourcesof backgrounds in paleo-detectors is decays of radioac-tive contaminants, in particular 238U. In order to sup-press this background, materials with as low 238U con-centration as possible must be chosen as target materi-als. The typical uranium concentration in the Earth’scrust is of the order of parts per million (ppm) inweight, which would lead to unacceptably large back-grounds. Much lower concentrations of uranium arefound in the Earth’s mantle and in seawater. Thus,promising target materials for paleo-detectors are ultra-basic rocks (UBRs), formed in the Earth’s mantle, andmarine evaporites (MEs), formed at the bottom of evap-orating oceans.5 The most common example of an UBRis Olivine [Mg1.6Fe2+0.4(SiO4)], and the most common MEsare Halite (NaCl) and Gypsum [Ca(SO4)·2(H2O)]. Notethat further purification of the materials may arise fromchemical expulsion of contaminants during the growth

5 Recall that for brevity, we refer to minerals in marine evaporitedeposits as “marine evaporites” and to minerals in ultra-basicrocks as “ultra-basic rocks”.

of crystals. However, the effect of such purification cannot be quantified in general, see e.g. [112].6 Here, we usebenchmark values for the 238U concentrations of 0.01 ppb(parts per billion) in weight for MEs and 0.1 ppb forUBRs.

As discussed in Sec. IV C, neutron induced back-grounds are further suppressed by the presence of hydro-gen in the material since hydrogen is an effective modera-tor of fast neutrons. Suppression of the neutron inducedbackground is particularly relevant to maintain sensitiv-ity to WIMP masses mχ & 10 GeV. Examples of UBRs

containing hydrogen are Nchwaningite [Mn2+2 SiO3(OH)2·

(H2O)] and Nickelbischofite [NiCl2 ·6(H2O)], while MEscontaining hydrogen are e.g. Gypsum [Ca(SO4)·2(H2O)]or Epsomite [Mg(SO4)·7(H2O)].

Further gains in sensitivity can be made by choos-ing materials which are optimal for particular ranges ofWIMP masses or WIMP-nucleus interaction types. Forexample, for low-mass WIMPs one would preferentiallyuse minerals with low mass density and low-mass targetnuclei. In such targets, tracks are relatively long, suchthat recoils induced by low-mass WIMPs are more easilyread out. On the other hand, light nuclei have com-paratively large (α, n) cross sections, rendering neutron-induced backgrounds in such targets challenging.

6 Once obtained, concentrations of radioactive trace elements intarget samples of interest for paleo-detectors can be measuredreliably to levels as low as ∼ 10−15 in weight using e.g. induc-tively coupled plasma mass spectroscopy.

14

For probing SI interactions of WIMPs with massesmχ & 10 GeV, it is advantageous to have as many heavytarget nuclei in the target material as possible since theWIMP-nucleus cross section is coherently enhanced withthe number of nucleons, σSI

T ∝ A2T . However, MEs and

UBRs containing both hydrogen and nuclei heavier thanNi are quite rare.

For probing SD interactions, the target material mustcontain nuclei with large nuclear spin. Interesting tar-get elements for WIMP-proton SD interactions are forexample H, B, F, Na, Al, K, or Mn. Note that outof the first two, hydrogen is only useful as a target inthe with low-Z tracks scenario. Boron is a good targetfor WIMP-proton SD scattering, however, it also leadsto relatively large backgrounds from (α, n)-interactions.Examples of UBRs containing such nuclei are Nch-waningite [Mn2+

2 SiO3(OH)2 · (H2O)] and Phlogopite[KMg3AlSi3O10F(OH)], while Borax [Na2(B4O5)(OH)4 ·8(H2O)] and Mirabilite [Na2(SO4)·10(H2O)] are examplesof MEs.

The situation is worse for WIMP-neutron SD scatter-ing, as natural targets do not usually contain sizeablefractions of target isotopes with unpaired neutrons. Themost interesting target elements are Mg, Si and Zr. Notethat O offers some sensitivity to WIMP-neutron scat-tering as well, although the abundance of the relevantisotope 17O is small. Examples of UBR targets are Bad-deleyite [ZrO2] and Phlogopite [KMg3AlSi3O10F(OH)],Example of MEs are Cattiite [Mg2.92Fe0.01(PO4)2.01 ·22.05(H2O)] and Epsomite [Mg(SO4)·7(H2O)].

Besides the materials mentioned above and on whichwe will focus in this work, other classes of materials maybe promising for particular applications. For example,natural plastics such as Evenkite (C24H50) are excellentSSTDs. Since they are comprised of relatively light tar-get nuclei, natural plastics would be interesting targetsfor low-mass WIMPs mχ . 10 GeV in particular. How-ever, little is known about typical uranium concentra-tions in natural plastics such that we do not considerthem as targets in this work. Another interesting mate-rial is diamond, samples of which can be extremely ra-diopure [113]. However, the diamond sample would haveto be much larger than the few cm mean free path of fastneutrons in order for the sample to not be contaminatedby nuclear recoils induced by neutrons originating fromthe surrounding material. In reverse, the background ofneutron induced recoils may be suppressed in general ifa target mineral of linear dimensions smaller than a fewcm is embedded in a uranium-poor environment, e.g. iceor pure salt deposits. In such a situation, the neutronsoriginating within the sample would mostly scatter out-side of the target volume, while the average neutron fluxthrough the sample would be reduced due to the lowerradioactivity in the surrounding material.

VI. SENSITIVITY ESTIMATION

From the track length spectra for the signal and thedifferent background components discussed in Secs. IIand IV, respectively, we estimate the sensitivity of paleo-detectors using a simple cut-and-count analysis.

For each WIMP mass hypothesis, we account for thefinite track length resolution by sampling the spectra andsmearing the track length of each event as

xsmear = xtrue + ∆x(σx) , (30)

where xsmear (xtrue) is the smeared (true) track lengthand ∆x a random number with dimensions of lengthdrawn from a normal distribution with standard devi-ation σx given by the track length resolution of the re-spective read-out method.

We then assume a signal region

xmin < xsmear < xmax , (31)

and count the number of signal (background) events S(B) in that signal window. The signal-to-noise ratio iscalculated as

SNR =S√

Bν + Σ2νB

2ν +BSF + Σ2

SFB2SF + . . .

, (32)

where Σi is the relative systematic error of the back-ground i and the “. . .” indicate additional backgroundcomponents, e.g. the single-α events.

The number of signal events is proportional to theWIMP-nucleon interaction cross section. For each sig-nal region satisfying

σx/2 ≤ xmin ≤ xmax − 2σx , xmax ≤ 103 nm , (33)

we find the smallest interaction cross section for which

SNR > 3 and S ≥ 5 , (34)

thus finding the signal region for each mass hypothesisyielding the best projected sensitivity in terms of thesmallest WIMP-nucleon cross sections which could beprobed.

In the remainder of this section, we present sensi-tivity projections for different benchmark assumptions:We use a high resolution (σx = 1 nm), low through-put (ε = 0.01 kg Myr) scenario, and a low resolution(σx = 15 nm), large throughput (ε = 100 kg Myr) sce-nario. The first scenario may for example be realizedby reading out 10 mg of a 1 Gyr old sample with HeliumIon beam Microscopy (HIM). The latter scenario couldbe realized by reading out 100 g of a 1 Gyr old samplewith Small Angle X-ray scattering (SAXs); see Sec. IIIfor a discussion of read-out methods. For both scenarioswe discuss sensitivity projections both under the assump-tion that all nuclei give rise to reconstructible tracks (withlow-Z tracks) and under the assumption that only nucleiwith Z ≥ 3 yield reconstructible tracks (without low-Z

15

100 101 102 103 104

WIMP mass mχ [GeV]

10−48

10−46

10−44

10−42

10−40S

IW

IMP

-nu

cleo

ncr

oss

sect

ionσ

SIn

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 1 nm; ε = 0.01 kg Myr

HaliteGypsumEpsomiteOlivine

NchwaningiteNickelbischofiteXe ν-floorcurrent limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−48

10−46

10−44

10−42

10−40

SI

WIM

P-n

ucl

eon

cros

sse

ctio

nσ

SIn

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 15 nm; ε = 100 kg Myr

HaliteGypsumEpsomiteOlivine

NchwaningiteNickelbischofiteXe ν-floorcurrent limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−48

10−46

10−44

10−42

10−40

SI

WIM

P-n

ucl

eon

cros

sse

ctio

nσ

SIn

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 1 nm; ε = 0.01 kg Myr

HaliteGypsumEpsomiteOlivine

NchwaningiteNickelbischofiteXe ν-floorcurrent limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−48

10−46

10−44

10−42

10−40

SI

WIM

P-n

ucl

eon

cros

sse

ctio

nσ

SIn

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 15 nm; ε = 100 kg Myr

HaliteGypsumEpsomiteOlivine

NchwaningiteNickelbischofiteXe ν-floorcurrent limits

FIG. 5. Sensitivity projection for spin-independent (SI) WIMP-nucleon scattering cross sections. The different colors are fordifferent target materials: Three examples of MEs, Halite (NaCl), Gypsum [Ca(SO4)·2(H2O)], and Epsomite [Mg(SO4)·7(H2O)]and three examples of UBRs, Olivine [Mg1.6Fe2+0.4(SiO4)], Nchwaningite [Mn2+

2 SiO3(OH)2 ·(H2O)], and Nickelbischofite [NiCl2 ·6(H2O)]. For reference, the light gray line indicates the conventional neutrino floor for Xe direct detection experiments [114].The shaded area shows current direct detection limits [5, 9, 11, 12]. For the left (right) panels we assume a track lengthresolution of σx = 1 nm (σx = 15 nm) and an exposure of ε = 0.01 kg Myr (ε = 100 kg Myr). The top panels are for the low-Ztracks scenario where we include tracks from all nuclei. The bottom panels are for the without low-Z tracks scenario, for whichwe assume only tracks with Z ≥ 3 to give rise to reconstructible tracks. We use 238U concentrations of C238 = 0.01 ppb inweight for MEs and C238 = 0.1 ppb for UBRs.

tracks). In particular, in the latter case we assume thathydrogen nuclei and α-particles (He nuclei) do not giverise to reconstructible tracks. We note that our analysisonly extends down to mχ ' 0.5 GeV since WIMPs withmass any lower do not give rise to a significant number ofnuclear recoil tracks & 1 nm in any of our target minerals.

As discussed in Sec. IV E, we assume a relative sys-tematic error of Σν = 100 % for the neutrino inducedbackground. For the backgrounds induced by radioactiv-ity, in particular the neutron and single-α decay inducedbackgrounds, we assume a relative systematic error ofΣn = Σ1α = 1 %.

A. Spin-Independent Interactions

In Fig. 5 we show sensitivity projections for Spin-Independent (SI) WIMP-nucleus interactions in 6 differ-ent target minerals. We pick three examples of MEs:The two very common evaporites Halite (NaCl) and Gyp-sum [Ca(SO4) · 2(H2O)] and the less common evapor-ite Epsomite [Mg(SO4) ·7(H2O)]. Halite is an examplefor an evaporite not containing hydrogen while Gypsumand Epsomite do. Likewise, we choose three examplesof UBRs: The very common Olivine [Mg1.6Fe2+0.4(SiO4)],and two minerals containing hydrogen, Nchwaningite[Mn2+

2 SiO3(OH)2 ·(H2O)] and Nickelbischofite.

Comparing the different panels of Fig. 5, we can first

16

note, that in general, for low-mass WIMPs with massesmχ . 10 GeV good spatial resolution is crucial, while forheavier WIMPs large exposure is more relevant. Thisis because lighter WIMPs give rise to less energetic re-coils and hence shorter tracks than heavier WIMPs. Asdiscussed in Sec. IV E, for the assumed 238U concen-trations of C238 = 0.01 ppb (C238 = 0.1 ppb) for MEs(UBRs), the dominant background source for WIMPswith mχ . 10 GeV are solar neutrinos. Hence, forthis mass regime, the sensitivity of MEs and UBRs aswell as targets containing or not containing hydrogenis comparable. We find that the typical sensitivity ex-tends to SI WIMP-nucleon cross sections as small asσSIn ∼ 10−43 cm2 for WIMP masses mχ ∼ 1 GeV and

to cross sections approximately one order of magnitudesmaller for WIMPs with masses mχ ∼ 10 GeV.

For heavier WIMPs with masses mχ & 10 GeV we findappreciable differences between targets containing or notcontaining hydrogen as well as between MEs and UBRs.This is because for such WIMP masses, the backgroundis dominated by neutron induced recoils. Since the neu-tron flux is proportional to the 238U concentration in thetarget material, MEs are more promising targets assum-ing that uranium concentrations in MEs are one order ofmagnitude lower than in UBRs. Further, the sensitivityis significantly increased in minerals containing hydrogen.If tracks from low-Z nuclei, in particular hydrogen, arereconstructed during read-out and cannot be differenti-ated from tracks from heavier nuclei, only MEs contain-ing hydrogen have projected sensitivities beyond currentlimits for mχ & 10 GeV, cf. the top right panel of Fig. 5.If hydrogen nuclei leave no reconstructible tracks or if hy-drogen tracks can reliably be differentiated from tracksfrom heavier nuclei, both UBRs and MEs have projectedsensitivities better than current limits, cf. the bottomright panel of Fig. 5. The sensitivity of MEs containinghydrogen such as Gypsum or Epsomite are approximatelya factor 10− 100 better than current experimental limitsfor WIMP masses mχ & 100 GeV.

Significant improvements with respect to these pro-jected sensitivities could be made by better control ofthe systematic error of the neutron-induced background,or by using materials with lower concentration of radioac-tive materials such as 238U. For uranium concentrationsC238 . 1 ppt (parts per trillion) in weight, paleo detec-tors could probe WIMP-nucleon cross sections within afactor 10 of the Xe neutrino floor indicated in Fig. 5.

B. Spin-Dependent Interactions

We present sensitivity projections to Spin-Dependent(SD) WIMP-nucleon interactions for two different cases:neutron-only and proton-only couplings, correspondingto ap = 0 and an = 0, respectively, in Eqs. (10), (12).In principle, the sensitivity of paleo-detectors to SD in-teractions is quite dependent on the ratio of the effec-tive WIMP-proton and WIMP-neutron couplings ap/an,

since typical target minerals will contain both elementssensitive to WIMP-proton and to WIMP-neutron SD in-teractions. We choose to present sensitivities for theneutron-only and proton-only cases here in order to allowfor easy comparison to limits and projections from directdetection experiments.

1. Neutron-Only

In Fig. 6 we show the sensitivity to SD neutron-onlycouplings for the MEs Cattiite [Mg2.92Fe0.01(PO4)2.01 ·22.05(H2O)], Epsomite [Mg(SO4)·7(H2O)], and Mirabilite[Na2(SO4) · 10(H2O)] and the UBRs Baddeleyite(ZrO2), Nickelbischofite [NiCl2 · 6(H2O)], and Phlogo-pite [KMg3AlSi3O10F(OH)]. Note that as before we as-sume 238U concentrations of C238 = 0.01 ppb in weightfor the MEs and C238 = 0.1 ppb for the UBRs Badde-leyite and Nickelbischofite. For Phlogopite, we assumeC238 = 0.01 ppb, since large samples of Phlogopite havebeen found with such 238U concentrations [115], althoughtypical concentrations of uranium in Phlogopite are muchlarger.

Comparing the different panels, we find similar behav-ior as for SI interactions, cf. the discussion in Sec. VI A.Comparing the different targets minerals we find strongerdependence on the particular chemical composition. Thisis because the WIMP-nucleus cross section for any partic-ular target nucleus is no longer controlled by the numberof protons and neutrons, but by the nuclear spin and howmuch of the spin is carried by the neutrons. Generally, itis difficult to find good target materials with much of thenuclear spin carried by the neutrons as this would requirean unpaired neutron. Such isotopes are quite rare in na-ture, for example, 17O only makes up 0.04 % of naturaloxygen. The most promising isotopes which are rela-tively common in nature appear to be 25Mg, comprising10 % of natural magnesium, and 29Si, comprising 5 % ofnatural silicon. Another isotope with large nuclear spinis 91Zr, comprising 11% of natural zirconium, however,zirconium is relatively rare in nature.

The best sensitivity out of the minerals shown in Fig. 6is found in the MEs Cattiite and Epsomite, which bothcontain Mg. For lighter WIMPs, the sensitivity of Phl-ogopite, which contains magnesium and silicon, is com-parable. For heavier WIMPs, Phlogopite is not com-petitive since it contains only a small fraction of hy-drogen, yielding too little suppression of the neutron-induced background. For comparison, we also show theME Mirabilite and the UBRs Baddeleyite and Nickel-bischofite. Mirabilite and Nickelbischofite do not displaycompetitive sensitivity to neutron-only interactions sincethey contain too few target elements with nuclear spincarried by neutrons. Baddeleyite does contain zirconium,however, it does not contain hydrogen, rendering neutroninduced backgrounds too large.

Despite the difficulties in finding good target miner-als for neutron-only interactions, we can note that the

17

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35S

DW

IMP

-neu

tron

cros

sse

ctio

nσ

SDn

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 1 nm; ε = 0.01 kg Myr

CattiiteEpsomiteMirabiliteBaddeleyite

NickelbischofitePhlogopiteXe ν-floor σSD

n

σSDn current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-n

eutr

oncr

oss

sect

ionσ

SDn

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 15 nm; ε = 100 kg Myr

CattiiteEpsomiteMirabiliteBaddeleyite

NickelbischofitePhlogopiteXe ν-floor σSD

n

σSDn current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-n

eutr

oncr

oss

sect

ionσ

SDn

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 1 nm; ε = 0.01 kg Myr

CattiiteEpsomiteMirabiliteBaddeleyite

NickelbischofitePhlogopiteXe ν-floor σSD

n

σSDn current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-n

eutr

oncr

oss

sect

ionσ

SDn

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 15 nm; ε = 100 kg Myr

CattiiteEpsomiteMirabiliteBaddeleyite

NickelbischofitePhlogopiteXe ν-floor σSD

n

σSDn current limits

FIG. 6. Same as Fig. 5 but for SD scattering assuming neutron-only couplings. We show three example of MEs: Cattiite[Mg2.92Fe0.01(PO4)2.01 ·22.05(H2O)], Epsomite [Mg(SO4)·7(H2O)], and Mirabilite [Na2(SO4)·10(H2O)] and three examples ofUBRs: Baddeleyite (ZrO2), Nickelbischofite [NiCl2·6(H2O)], and Phlogopite [KMg3AlSi3O10F(OH)]. As before, we assume 238Uconcentrations of C238 = 0.01 ppb in weight for MEs and C238 = 0.1 ppb for the UBRs Baddeleyite and Nickelbischofite. ForPhlogopite we assume C238 = 0.01 ppb, since large chunks of Phlogopite with linear dimension of order 1 m have been foundin natural deposits [115]. The shaded area shows current upper limits from direct detection experiments [7, 116], and the lightgray line indicates the neutrino-floor for SD neutron-only interactions in Xe [117].

sensitivity of paleo-detectors to low-mass WIMPs mχ .10 GeV extends to σSD

n ∼ 10−38 cm2 for mχ ∼ 10 GeVand cross sections approximately one order of magni-tude larger for mχ ∼ 1 GeV. For heavier WIMPs, paleo-detectors still promise sensitivity better than current up-per limits from direct detection, but only in ME targetminerals containing hydrogen and elements with sizeablefraction of the spin carried by neutrons, such as magne-sium.

2. Proton-Only

Compared to the neutron-only case, it is much easierto find suitable target materials for SD WIMP-protoninteractions. Elements with unpaired protons and size-

able nuclear spin are common in nature, e.g. H, B, F,Na, Al, K, or Mn. In Fig. 7, we show the sensitivityfor three examples of MEs, Borax [Na2(B4O5)(OH)4 ·8(H2O)], Epsomite [Mg(SO4) · 7(H2O)], and Mirabilite[Na2(SO4) · 10(H2O)], and three UBRs, Nchwaningite[Mn2+

2 SiO3(OH)2·(H2O)], Nickelbischofite [NiCl2·6(H2O)],and Phlogopite [KMg3AlSi3O10F(OH)]. As before, we as-sume 238U concentrations of C238 = 0.01 ppb in weightfor the MEs Borax, Epsomite, and Mirabilite as well asfor Phlogopite, and C238 = 0.1 ppb for Nchwaningite andNickelbischofite.

When comparing the different panels in Fig. 7, thesame arguments as for SI interactions in Sec. VI A apply.However, note that for proton-only SD interactions, thedifference between the with low-Z tracks and the withoutlow-Z tracks scenarios is particularly large. This is be-

18

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35S

DW

IMP

-pro

ton

cros

sse

ctio

nσ

SDp

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 1 nm; ε = 0.01 kg Myr

BoraxEpsomiteMirabiliteNchwaningite

NickelbischofitePhlogopite

C3F8 ν-floor σSDp

σSDp current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-p

roto

ncr

oss

sect

ionσ

SDp

[cm

2]

with low-Z tracksΣν = 100 %; Σn = 1 %

σx = 15 nm; ε = 100 kg Myr

BoraxEpsomiteMirabiliteNchwaningite

NickelbischofitePhlogopite

C3F8 ν-floor σSDp

σSDp current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-p

roto

ncr

oss

sect

ionσ

SDp

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 1 nm; ε = 0.01 kg Myr

BoraxMirabiliteNchwaningite

Phlogopite

C3F8 ν-floor σSDp

σSDp current limits

100 101 102 103 104

WIMP mass mχ [GeV]

10−43

10−41

10−39

10−37

10−35

SD

WIM

P-p

roto

ncr

oss

sect

ionσ

SDp

[cm

2]

without low-Z tracksΣν = 100 %; Σn = 1 %; 1ασx = 15 nm; ε = 100 kg Myr

BoraxMirabiliteNchwaningite

Phlogopite

C3F8 ν-floor σSDp

σSDp current limits

FIG. 7. Same as Fig. 5 but for SD scattering assuming proton-only couplings. We show three example of MEs: Borax[Na2(B4O5)(OH)4 ·8(H2O)], Epsomite [Mg(SO4) ·7(H2O)], and Mirabilite [Na2(SO4) ·10(H2O)] and three examples of UBRs:Nchwaningite (Mn2+

2 SiO3(OH)2 ·(H2O)), Nickelbischofite [NiCl2 ·6(H2O)], and Phlogopite [KMg3AlSi3O10F(OH)]. As before,we assume 238U concentrations of C238 = 0.01 ppb in weight for MEs and C238 = 0.1 ppb for the UBRs Nchwaningite andNickelbischofite. For Phlogopite we assume C238 = 0.01 ppb, since large samples of Phlogopite with such concentrations havebeen found in natural deposits [115]. The shaded area shows current upper limits from direct detection experiments [6, 118],and the light gray line indicates the neutrino-floor for SD neutron-only interactions in C3F8 [117].

cause hydrogen nuclei, i.e. protons, are excellent targetsfor SD WIMP-proton scattering of low-mass WIMPs andin addition, recoiling hydrogen nuclei have large rangesin typical target materials, cf. Fig. 1. The resulting sen-sitivity in the with low-Z tracks scenario extends to crosssections as small as σSD

p ∼ 10−42 cm2 for WIMPs withmasses of a few GeV. Due to the long range of hydrogen,the sensitivity to low-mass WIMPs is retained when us-ing read-out methods with worse spatial resolution, e.g.SAXs; cf. the top right panel of Fig. 7. The importanceof the hydrogen induced tracks also explains why all tar-get materials shown in Fig. 7 have similar sensitivity.

In the without low-Z tracks scenario, the behavioris more similar to the SI and SD neutron-only cases.The best targets, Cattiite, Epsomite and Phlogopite, arethose containing a sizeable fraction of heavier target el-

ements with large nuclear spins and unpaired protons inaddition hydrogen, e.g. B, F, or Na, and having low con-centrations of 238U. The sensitivity to low-mass WIMPsextends to σSD

p ∼ 10−40 cm2 for mχ ∼ 10 GeV and ap-proximately one order of magnitude smaller cross sec-tions for mχ ∼ 1 GeV WIMPs. For heavier WIMPs withmasses mχ & 30 GeV, the sensitivity is a factor 10− 100better than current limits.

Note that as discussed for SI interactions at the end ofSec. VI A, the sensitivity of paleo-detectors to SD WIMP-nucleus interactions may be improved significantly if sys-tematic uncertainties of the backgrounds are smaller thanwe have assumed here, or if target materials with lowerconcentrations of radioactive contaminants are available.

19

VII. DISCUSSION

In this article, we have given a detailed discussion ofour paleo-detector proposal. Paleo-detectors constitute aradically different approach to direct detection than con-ventional experiments: instead of instrumenting a (large)target mass in the laboratory and searching for WIMP-induced nuclear recoils in real time, we propose to ex-amine ancient minerals for the traces of WIMP-nucleoninteractions. This proposal rests upon the principle ofsolid state track detectors: in certain minerals, a recoil-ing nucleus will induce a damage track which, once cre-ated, persists over geological time-scales. Since paleo-detectors record WIMP-nucleon interactions over timescales as long as ∼ 1 Gyr, reconstructing tracks in rela-tively small target masses suffices to achieve exposures(the product of target mass and integration time) ordersof magnitude larger than what seems feasible in conven-tional direct detection experiments.

Here, we have identified two realistic, though ambitiousread-out scenarios: Reconstructing tracks with helium-ion beam microscopy combined with ablation of read-outlayers with pulsed lasers, target masses of O(10) mg maybe investigated with spatial resolutions of ∼ 1 nm. Trans-lated to conventional direct detection quantities, suchresolution would allow for nuclear recoil energy thresh-olds O(100) eV, comparable to what is achieved in cryo-genic bolometric detectors. Sacrificing some spatial res-olution for larger target masses, we propose to read outO(100) g of material with small-angle X-ray scattering,allowing for spatial resolutions of 15 nm. Such spatialresolution would allow for nuclear recoil energy thresh-olds of O(1) keV, similar to conventional direct detectionexperiments using liquid noble gas targets. However, thecorresponding exposures are larger by a few orders ofmagnitude than what conventional direct detection tech-niques envisage in the next decades.

We have described the most relevant backgroundsources in detail in Sec. IV. One major advantage ofpaleo-detectors compared to conventional direct detec-tion experiments is that the comparatively small tar-get masses can be obtained from depths much greaterthan those of the underground laboratories in which con-ventional detectors must be operated. This makes cos-mogenic backgrounds all but negligible. The dominantbackground sources in paleo-detectors will be nuclear re-coils induced by neutrons from radioactive processes andby neutrinos. Broadly speaking, we identify two back-ground regimes: For low-mass WIMPs mχ . 10 GeV,the background budget is dominated by solar neutrinos.For heavier WIMPs, the dominant source of backgroundsare neutron induced nuclear recoils. In our backgroundcalculation we include both the neutrons induced by thespontaneous fission of heavy radioactive contaminants,in particular 238U, and the neutrons induced by (α, n)-reactions of α-particles from decays of heavy radioactivecontaminants with the nuclei comprising the target ma-terial.

The selection of target minerals for paleo-detectors isheavily informed by our background study. Suitable min-erals must record tracks, preserve the tracks over geologi-cal time-scales, and be as pure from radioactive contami-nants as possible in order to reduce backgrounds inducedby radioactivity to an acceptable level. We identify twoclasses of material as suitable for paleo-detectors: Min-erals found in marine evaporite deposits and mineralsfound in ultra-basic rocks. Both type of minerals aremuch purer than typical materials found in the Earth’scrust. When searching for heavier WIMPs where thebackground is dominated by neutron-induced recoils, itis advantageous to use minerals which in addition con-tain hydrogen. This is because hydrogen is an effectivemoderator of fast neutrons, thus, the presence of hydro-gen significantly lowers the background induced by theneutrons from radioactivity.

In Sec. VI we present the projected sensitivity of paleo-detectors for a range of target materials and for the twodifferent read-out scenarios discussed above. While wehave already presented prospects for probing canonicalspin-independent WIMP-nucleus interactions in Ref. [1],we present results here for a larger selection of miner-als. In addition, we present projected sensitivities forspin-dependent WIMP-nucleus interactions in the usualproton-only and neutron-only interaction benchmark sce-narios used by conventional direct detection experiments.In all cases, WIMP-nucleon cross section many ordersof magnitude smaller than current experimental upperbounds can be probed for light WIMPs with massesmχ . 10 GeV. For heavier WIMPs where the dominantbackground source is neutron induced recoils, the pro-jected sensitivity strongly depends on the presence of hy-drogen. For the uranium concentrations assumed in thiswork, WIMP-nucleon cross sections a factor of a few to∼ 100 smaller than current experimental upper boundscan be probed with paleo-detectors using target mineralswhich contain hydrogen. Note that significant improve-ments with respect to these projections are possible; forexample, if target materials with uranium concentrationsof less than approximately one part per trillion in weightare available, the background for heavier WIMPs wouldbe dominated by nuclear recoils induced by atmosphericand supernova neutrinos. In such a situation, the sensi-tivity of paleo-detectors would extend to cross sectionsmore than one order of magnitude smaller than what isshown here. The sensitivity may also be improved by us-ing more sophisticated analysis techniques than the sim-ple cut-and-count approach employed here, for examplea spectral analysis [111].

We would like to comment on some interesting possi-bilities arising from the paleo-detectors approach, whichwe leave for future work. For example, using a series oftarget materials of different ages it would be possible toobtain information of the time-variability of nuclear recoilevents over scales as long as 1 Gyr. In the case of WIMPDM, such an approach would allow for studies of the sub-structure of the DM halo: The age of the oldest available

20

minerals is larger than the period of rotation of the Sunaround the galactic center. While sub-structure such asultra-compact mini-halos [119, 120] or tidal streams [121–123], typically renders conventional direct detection ex-periments less sensitive due to a decrease in the localDM density, in paleo-detectors, the signal may be en-hanced from the Earth passing through overdense DMregions. Using target materials which recorded WIMPsignals over different times, it may be possible to obtaininformation about such sub-structure of the DM halo.

The sensitivity and exposure time also makes paleo-detector interesting for a host of applications beyondWIMP DM searches. Examples include studying thetime-variability of the fluxes of cosmic rays, or of neu-trinos from the Sun or supernovae. Another examplewould be the study of proton decay facilitated by thelarge exposure.

In order to pave the way towards paleo-detectors, weplan to carry out a number of feasibility studies in thenear future. We intend to use natural minerals obtainedfrom close to the surface to demonstrate the reconstruc-

tion of fossil tracks with the read-out methods describedin this work. In order to demonstrate the feasibility ofpaleo-detectors for DM searches, we will create signalssimilar to those which WIMP DM may induce by irradi-ating target samples with neutrons.

ACKNOWLEDGMENTS

The authors would like to thank F. Avignone, J. Blum,J. Collar, J. Conrad, D. Eichler, R. Ewing, T. Ed-wards, A. Ferella, A. Goobar, B. Kavanagh, C. Kelso,D. Snowden-Ifft, L. Stodolsky, H. Sun, K. Sun, andC. Weniger for useful discussions. SB, AKD, KF, and PSacknowledge support by the Vetenskapsradet (SwedishResearch Council) through contract No. 638-2013-8993and the Oskar Klein Centre for Cosmoparticle Physics.SB, KF, and PS acknowledge support from DoE grantDE-SC007859 and the LCTP at the University of Michi-gan.

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