Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries

Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries

Y. V. Stadnik y.stadnik@unsw.edu.au    V. V. Flambaum School of Physics, University of New South Wales, Sydney 2052, Australia
August 2, 2019
Abstract

We calculate the proton and neutron spin contributions for nuclei using semi-empirical methods, as well as a novel hybrid ab initio/semi-empirical method, for interpretation of experimental data. We demonstrate that core-polarisation corrections to ab initio nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. We derive constraints on the spin-dependent P,T-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We derive a limit on the CPT and Lorentz-invariance-violating parameter GeV, which improves on the most stringent existing limit by a factor of 8, and demonstrate sensitivities to the parameters and at the level GeV, which is a one order of magnitude improvement compared to the corresponding existing sensitivities. We extend previous analysis of nuclear anapole moment data for Cs to obtain new limits on several other CPT and Lorentz-invariance-violating parameters: GeV, , GeV and .

I Introduction

The violation of the fundamental symmetries of nature is an active area of research. Atomic and molecular experiments, which probe -odd and ,-odd interactions, provide very sensitive tests of the Standard Model (SM) and physics beyond the SM Khriplovich1991_PNC-Book (); Ginges_PhysRep2004 (); Roberts2015-Review(PNC) (). Measurements and calculations of the Cs - parity nonconserving (PNC) amplitude stand as the most precise atomic test of the SM electroweak theory to date, see e.g. Bouchiat1982Cs (); Wood1997_Cs-PNC (); Dzuba1989Cs (); Blundell92Cs (); Kozlov01Cs (); Safronova02Cs (); Ginges02Cs (); Roberts2012_Cs-pnc (). Experimental searches for nuclear anapole moments are ongoing in Fr Aubin2013_Fr-AM-project (), Yb Tsigutkin2009_Yb-PNC (); Tsigutkin2010_Yb-pnc () and BaF Demille2008 (); DeMille2014-BaF(Zeeman) (). At present, Hg provides the most precise limits on the electric dipole moment (EDM) of the proton, quark chromo-EDM and ,-odd nuclear forces, as well as the most precise limits on the neutron EDM and quantum chromodynamics (QCD) term from atomic or molecular experiments Griffith2009improved (); Swallows2013 (), while ThO provides the most precise limit on the electron EDM Baron2014 (). Most recently, it was suggested that EDM measurements in molecules with ,-odd nuclear magnetic quadrupole moments may lead to improved limits on the strength of ,-odd nuclear forces, proton, neutron and quark EDMs, quark chromo-EDM and the QCD term Flambaum2014-MQM ().

Field theories, which are constructed from the principles of locality, spin-statistics and Lorentz invariance, conserve the combined symmetry. The violation of one or more of these three principles, presumably from some form of ultra-short distance scale physics, opens the door for the possibility of -odd physics. Some of the most stringent limits on -odd and Lorentz-invariance-violating physics come from searches for the coupling between a background cosmic field, , and the spin of an electron, proton, neutron or muon, Berglund1995 (); Bluhm2000a (); Hou2003 (); Cane2004 (); Heckel2006 (); Heckel2008 (); Bennett2008CPT (); Altarev2009 (); Gemmel2010 (); Brown2010 (); Peck2012 (); Allmendinger2014 (). For further details on the broad range of experiments performed and a brief history of the improvements in these limits, we refer the reader to the reviews of Kostelecky1999 (); Kostelecky2011data () and the references therein.

Other very important unanswered questions in fundamental physics are the strong problem, namely the puzzling observation that QCD does not appear to violate the combined charge-parity () symmetry, see e.g. Weinberg1976 (); Weinberg1978 (); Peccei1977a (); Peccei1977b (); Wilczek1978 (); Moody1984 (), and dark matter and dark energy, see e.g. Bertone2005 (); Spergel2007 (); Agnese2013 (); Riess1998 (); Perlmutter1999 (). A particularly elegant solution to the strong problem invokes the introduction of a pseudoscalar particle known as the axion Peccei1977a (); Peccei1977b () (see also Kim1979 (); Shifman1980a (); Zhitnitsky1980 (); Dine1981a ()). It has been noted that the axion may also be a promising cold dark matter candidate. Thus axions, if detected, could resolve both the dark matter and strong problems Kim2010 (); Kawasaki2013 (); Brambilla2014 (); Baer2014 (); Stadnik2014_Axions-review (). The decay of supersymmetric axions to produce axions has also been suggested as a possible explanation for dark radiation Jeong2012 (); Graf2013a (); Graf2013b (); Queiroz2014 ().

Many tests of the fundamental symmetries of nature and searches for axion, weakly-interacting massive particle (WIMP) and topological defect dark matter involve couplings of the form between a field or operator and the spin angular momentum of a proton () or neutron (), or depend explicitly on the spin angular momenta of the nucleons involved. We point out that in experiments, which measure nuclear spin-dependent (NSD) properties, the contribution of non-valence nucleon spins cannot be neglected, due to polarisation of these nucleons by the valence nucleon(s). Nuclear many-body effects have previously been considered in association with the interpretation of atomic clock experiments Flambaum2003Nuc (); Flambaum2006a (); Berengut2011K (), nuclear-sourced EDMs and NSD-PNC interactions mediated via -boson exchange between electrons and the nucleus (see e.g. Berengut2011K ()), static spin-gravity couplings Flambaum2009a (); Kimball2014_Nuc () and long-range dipole-dipole couplings Kimball2014_Nuc ().

In the present work, we calculate the proton and neutron spin contributions for a wide range of nuclei, which are of experimental interest in tests of the fundamental symmetries of nature and searches for dark matter, including axions, WIMPs and topological defects, using semi-empirical methods, as well as a novel hybrid ab initio/semi-empirical method. We then demonstrate that core-polarisation corrections to ab initio nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. As an illustration of the importance of many-body effects in such studies, we revisit the experiments of Refs. Gemmel2010 (); Allmendinger2014 (), in which a He/Xe comagnetometer was used to place constraints on the and Lorentz-invariance-violating parameter , which quantifies the interaction strength of a background field with the spin of a neutron. We show that, due to nuclear many-body effects, the He/Xe system is in fact also quite sensitive to proton interaction parameters. By reanalysing the results of Ref. Allmendinger2014 (), we derive a limit on the parameter that is the world’s most stringent by a factor of 8. Likewise, by reanalysing the results of Ref. Cane2004 (), in which a He/Xe comagnetometer was also used, we demonstrate improved sensitivities to the parameters and by one order of magnitude. From existing data in Ref. Tullney2013 (), in which experiments were performed with a He/Xe comagnetometer, we derive constraints on the spin-dependent ,-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We also extend our previous analysis of nuclear anapole moment data for Cs Roberts2014 () to obtain new limits on several other and Lorentz-invariance-violating parameters.

Ii Nuclear Theory

The nuclear magnetic dipole moment can be expressed (in the units of the nuclear magneton ):

(1)

where and are the expectation values of the total proton and neutron spin angular momenta, respectively, while is the expectation value of the total proton orbital angular momentum. In the present work, we consider nuclei with either one valence proton or one valence neutron (even-even nuclei are spinless due to the nuclear pairing interaction).

We start by considering the contribution of the valence nucleon alone. Assuming all other nucleons in the nucleus are paired (and ignoring polarisation of the nuclear core for now), the spin and nuclear magnetic dipole moment are due entirely to the total angular momentum of the external nucleon: . In this case, the nuclear magnetic dipole moment is given by the Schmidt (single-particle approximation) formula

(2)

with

(3)
(4)

The gyromagnetic factors are: for a valence proton and for a valence neutron. We present the values for from Eq. (3) (“Schmidt model”) in Tables 1 and 2.

Schmidt model Minimal model Preferred model
Nucleus
H 0.500 0.500 0.000 0.500 0.000
Li 0.500 0.443 0.057 0.436 0.064
F 0.500 0.483 0.017 0.480 0.020
Na -0.300 -0.078 -0.222 -0.051 -0.249
Al 0.500 0.378 0.122 0.363 0.137
Cl -0.300 -0.226 -0.074 -0.217 -0.083
K -0.300 -0.272 -0.028 -0.268 -0.032
K -0.300 -0.290 -0.010 -0.289 -0.011
Ga 0.500 0.311 0.189 0.289 0.211
Br 0.500 0.338 0.162 0.319 0.181
Rb -0.357 -0.305 -0.052 -0.299 -0.058
Rb 0.500 0.389 0.111 0.376 0.124
Nb 0.500 0.434 0.066 0.426 0.074
I 0.500 0.290 0.210 0.265 0.235
Cs -0.389 -0.297 -0.092 -0.286 -0.103
La -0.389 -0.276 -0.113 -0.262 -0.127
Pr 0.500 0.445 0.055 0.438 0.062
Tb -0.300 -0.099 -0.201 -0.075 -0.225
Ho 0.500 0.324 0.176 0.303 0.197
Tm 0.500 0.179 0.321 0.140 0.360
Tl 0.500 0.376 0.124 0.361 0.139
Tl 0.500 0.377 0.123 0.363 0.137
Bi -0.409 -0.251 -0.158 -0.232 -0.177
Fr -0.409 -0.268 -0.141 -0.251 -0.158
Fr -0.409 -0.263 -0.146 -0.246 -0.164
Table 1: Expectation values and for selected odd-proton nuclei. Nuclear spin and parity assignments, and experimental values of were taken from Ref. lbnl_nuclear_isotopes ().
Schmidt model Minimal model Preferred model
Nucleus
He 0.500 0.500 0.000 0.500 0.000
Be 0.500 0.422 0.078 0.413 0.087
C -0.167 -0.174 0.007 -0.174 0.008
Ne -0.300 -0.108 -0.192 -0.085 -0.215
Si 0.500 0.356 0.144 0.339 0.161
Ar 0.500 0.43 0.07 0.43 0.07
Ge 0.500 0.390 0.110 0.377 0.123
Sr 0.500 0.413 0.087 0.403 0.097
Zr 0.500 0.435 0.065 0.428 0.072
Te 0.500 0.391 0.109 0.378 0.122
Xe 0.500 0.379 0.121 0.365 0.135
Xe -0.300 -0.252 -0.048 -0.246 -0.054
Ba -0.300 -0.267 -0.033 -0.263 -0.037
Ba -0.300 -0.278 -0.022 -0.275 -0.025
Yb -0.167 -0.151 -0.015 -0.150 -0.017
Yb -0.357 -0.140 -0.217 -0.114 -0.243
Hg -0.167 -0.153 -0.014 -0.151 -0.016
Hg 0.500 0.356 0.144 0.339 0.161
Pb -0.167 -0.162 -0.005 -0.161 -0.005
Table 2: Expectation values and for selected odd-neutron nuclei. Nuclear spin and parity assignments, and experimental values of were taken from Ref. lbnl_nuclear_isotopes ().

Experimentally, the Schmidt model is known to overestimate the magnetic dipole moment in most nuclei. The simplest explanation for this is that the valence nucleon polarises the core nucleons, reducing the magnetic dipole moment of the nucleus. The degree of core polarisation can be estimated using experimental values of the magnetic dipole moment, and improved estimates for and can hence be obtained.

The reduction in nuclear magnetic dipole moment from the Schmidt value to the experimental value can proceed by a number of mechanisms. The simplest and most efficient way is to assume that the internucleon spin-spin interaction transfers spin from the valence proton (neutron) to core neutrons (protons):

(5)

where and are the Schmidt model values (one of which is necessarily zero). In general, there is also polarisation of the proton (neutron) core by the valence proton (neutron), but transfer of valence proton (neutron) spin to core proton (neutron) spin does not change the result. Note that the denominator in (5) is a large number, so the required change in and to obtain the experimental value is minimal. We present the values for and from Eq. (5) (“minimal model”) in Tables 1 and 2.

It is also possible for a reduction in nuclear magnetic dipole moment to occur by different mechanisms, for instance, by transfer of the spin angular momentum of a valence proton (neutron) to core proton (neutron) orbital angular momenta, or in a more unlikely manner by transfer of valence proton (neutron) spin angular momentum to core neutron (proton) orbital angular momenta.

The preferred model of Refs. Flambaum2006a (); Berengut2011K () is intermediate to the two previously mentioned “extreme models”. In this model, it is assumed that the total projections of proton and neutron angular momenta, and , are separately conserved, and that the projections of total spin and orbital angular momenta, and , are also separately conserved (which corresponds to the neglect of the spin-orbit interaction). In this case

(6)
(7)

where for a valence proton and for a valence neutron, with the Schmidt model value for the spin of the valence nucleon, given by (3). From Eqs. (1), (6) and (7), we find

(8)
(9)

We present the values for and from Eqs. (8) and (9) (“preferred model”) in Tables 1 and 2.

In the present work, we develop a new and alternate hybrid method, in which semi-empirical core-polarisation corrections are applied to ab initio nuclear shell model calculations from Refs. Engel1991 (); Ressell1997 (); Toivanen2009 (). We use the results of the many-body calculations for , and from Refs. Engel1991 (); Ressell1997 (); Toivanen2009 () as the input values (instead of the Schmidt model values) and improve them using the known experimental values of . Minimal model corrections [from Eq. (5)] to the proton and neutron spin angular momentum expectation values of the available nuclei are seen to generally reduce discrepancies in proton and neutron spin expectation values from different calculations, as shown in Table 3.

Nucleus Ref. ab initio model
Te Ressell1997 () Bonn A 0.287 0.001 0.274 0.014
Te Ressell1997 () Nijmegen II 0.323 -0.0003 0.297 0.026
I Ressell1997 () Bonn A 0.075 0.309 0.071 0.313
I Ressell1997 () Nijmegen II 0.064 0.354 0.100 0.318
I Toivanen2009 () Bonn-CD 0.030 0.418 0.108 0.340
Xe Ressell1997 () Bonn A 0.359 0.028 0.337 0.050
Xe Ressell1997 () Nijmegen II 0.300 0.0128 0.308 0.005
Xe Toivanen2009 () Bonn-CD 0.273 -0.0019 0.256 0.015
Xe Ressell1997 () Bonn A -0.227 -0.009 -0.196 -0.040
Xe Ressell1997 () Nijmegen II -0.217 -0.012 -0.187 -0.042
Xe Engel1991 () QTDA -0.236 -0.041 -0.235 -0.042
Xe Toivanen2009 () Bonn-CD -0.125 -0.00069 -0.122 -0.004
Cs Toivanen2009 () Bonn-CD 0.021 -0.318 -0.076 -0.221
Table 3: Expectation values and for selected nuclei after correcting ab initio nuclear shell model spin expectation values via the minimal model correction scheme. For nuclei, where more than one calculation has been performed, we take the average of the final values of and for computing limits in the present work.

Iii Application I: Dark Matter Searches

Proton and neutron spin contents are important for interpretations of experimental data from various dark matter detection schemes, which are based on effects involving couplings to nuclear spins. WIMP dark matter can undergo elastic, spin-dependent scattering off nuclei, see e.g. Wilczek1985 (); DarkSide2012 (); Picasso2012 (); Zeplin2012 (); Fornasa2012 (); ArDM2013 (); EdelweissII2013 (); LUX2013 (); Cresst2014 (); DEAP-3600 (); Simple2014 (); SuperCDMS2014 (); Green2014 (). Axions can induce oscillating nuclear Schiff moments via hadronic mechanisms Graham2011 (); Graham2013 (); Stadnik2014 (), which can be sought for either directly through nuclear magnetic resonance-type experiments (CASPEr) Budker2013C () or oscillating atomic EDMs Stadnik2014 (). Axions can interact directly with nuclear spins via the time-dependent spin-axion momentum coupling , where is the axion mass Graham2013 (); Stadnik2014 (); Sikivie2014Atoms (), induce the time-dependent nuclear spin-gravity coupling and oscillating nuclear anapole moments Stadnik2014 (); RobertsCosmicPNC-long2014 (). Magnetometry techniques can also be used to search for monopole-dipole and dipole-dipole axion exchange couplings Mainz2013exp (); ArvanitakiGeraci2014 (). Topological defect dark matter, which consists of axion-like pseudoscalar fields, can interact with nuclear spins via the time-dependent coupling , where is the pseudoscalar field comprising the topological defect Pospelov2013 (), and can give rise to transient nuclear-sourced EDMs Stadnik2014defects (). Both of these effects can be sought for using GNOME Pustelny2013GNOME (). One may use Tables 1, 2 and 3 for the interpretation of dark matter searches based on all of the mentioned schemes, as well as for tests of the fundamental symmetries of nature.

Iv Application II: Comagnetometer Experiments

We first revisit the experiments of Refs. Gemmel2010 (); Allmendinger2014 (), in which a He/Xe comagnetometer was used to place constraints on the Standard Model Extension (SME) - and Lorentz-invariance-violating parameter Colladay1997-SME (); Colladay1998-SME(LV) (), which quantifies the interaction strength of a background field with the spin of a neutron. The observed quantities are the amplitudes of sidereal frequency shifts, and , which in the case of the He/Xe system are related to the SME parameters via Kostelecky1999 ():

(10)

where , and are the gyromagnetic ratios of He and Xe, respectively, with , and is the angle between Earth’s rotation axis and the quantisation axis of the spins. Within the Schmidt model, in which only valence neutrons participate in the spin-dependent coupling , it was determined that Allmendinger2014 ():

(11)
(12)

However, in a non-single-particle model, proton spins also contribute. From our spin content values for Xe in Table 3 and the values for the well-studied case of He from Ref. Chupp1990 (), we find, using Eq. (10), instead of Eqs. (11) and (12):

(13)
(14)

which gives the following limits () on , where , within the preferred model:

(15)
(16)

Note that (16) improves on the world’s best proton-coupling limit of Brown2010 () by a factor of 8 (Table 4). Thus in this case, the He/Xe system is sensitive not only to neutron SME parameters, but also has reasonable sensitivity to analogous proton parameters.

Parameter Ref. Brown2010 () Ref. Allmendinger2014 () This work
/ GeV
/ GeV
Table 4: Comparison of limits () on the SME parameters and .

Similarly, we reanalyse the results of Ref. Cane2004 (), in which a He/Xe comagnetometer was also used to place constraints on the SME parameters , and Colladay1997-SME (); Colladay1998-SME(LV) (), among others. The observed quantities are again the amplitudes of sidereal frequency shifts:

(17)

where

(18)
(19)

Noting that the dominant contributions are from nucleons near the Fermi surface ( MeV from the surface), taking the nucleon depth well to be MeV for both protons and neutrons, and using our spin content values for Xe in Table 3 and the values for He from Ref. Chupp1990 (), along with the experimental data in Ref. Cane2004 (), we find the following results (all of which are consistent with zero):

(20)
(21)

where the uncertainties in the coefficients of and are a factor of several, while the uncertainties in the coefficients of and are an order of magnitude. We note that the corresponding sensitivities to the parameters and are at the level GeV, which is a one order of magnitude improvement on the best corresponding proton-coupling sensitivities derived in Peck2012 ().

Likewise, we revisit the experiment of Ref. Tullney2013 (), in which a He/Xe comagnetometer was used to place constraints on the spin-dependent ,-violating interaction of a bound neutron with nucleons. The spin-dependent monopole-dipole coupling potential between two nucleons is given by Moody1984 ():

(22)

where is the dimensionless scalar coupling constant of the nucleon inside the spin-unpolarised sample, is the dimensionless pseudoscalar coupling constant of the spin-polarised bound nucleon , is the unit vector from the bound nucleon to the unpolarised nucleon, is the spin of the polarised bound nucleon and is the one-boson-exchange range. The resulting shift in the weighted frequency difference is given by (using results of derivations from Refs. Tullney2013 (); Zimmer2010 ()):

(23)

with:

(24)

where is the number density of nucleons in the unpolarised sample, and are the thicknesses of the cylindrical spin-polarised and unpolarised samples, respectively, is the finite gap between the two samples and is a correction function accounting for the finite sizes of the two samples Tullney2013 ().

Combining the experimental data of Tullney2013 () with our spin content values for Xe in Table 3 and the values for He from Ref. Chupp1990 (), we obtain the 95 confidence level upper limits on the parameters and shown in Figures 1 and 2, respectively. For some of the other limits on these parameters, we refer the reader to Refs. Kimball2014_Nuc (); Youdin1996 (); Baesler2007 (); Glenday2008 (); Serebrov2010 (); Petukhov2010 (); Hoedel2011 (); Raffelt2012 (); Jenke2012 (); Bulatowics2013 (); Chu2013 ().

Figure 1: confidence level upper limit on as a function of the one-boson-exchange range . Solid black line corresponds to limits derived in our present work. Shaded orange region indicates ‘classical’ region of axion masses.
Figure 2: confidence level upper limit on as a function of the one-boson-exchange range . Solid black line corresponds to limits derived in our present work. Dashed purple line corresponds to limits obtained from Schmidt model in Ref. Tullney2013 (). Shaded orange region indicates ‘classical’ region of axion masses.

V Application III: Tests of Fundamental Symmetries

Consider the following Lorentz-invariance-violating terms in the SME Lagrangian (in the natural units ) Kostelecky1999 ():

(25)

where and are background fields, is the fermion wavefunction with , , and are Dirac matrices, and the two-sided derivative operator is defined by: . The first term in (25) is -odd, while the second term is -even. In the non-relativistic limit, the Lagrangian (25) gives rise to the following interaction Hamiltonian

(26)

where is the fermion mass, is the fermion spin and is the fermion momentum operator. In our previous work Roberts2014 (), we showed in the single-particle approximation that the first term in (26) gives rise to nuclear anapole moments associated with valence nucleons Flambaum1984 () (see also Stadnik2014 ()). Experimentally, the nuclear anapole moment manifests itself as a NSD contribution to a PNC amplitude. Hence from the measured and calculated (within the SM) values of the anapole moments of Cs and Tl, we were able to extract direct limits on the parameter .

In the single-particle approximation, the nuclear anapole moment contribution from interaction (26) is

(27)

where is the Fermi constant of the weak interaction, , and the dimensionless constants and are given by

(28)
(29)

where is the fine-structure constant, and are the mass and magnetic dipole moment of the unpaired nucleon ( and ), respectively, and we take the mean-square radius , with fm, and the atomic mass number. Combining the measured values for the nuclear anapole moment of  Wood1997_Cs-PNC (); Flambaum1997-AM-Murray () and  Vetter1995_Tl-PNC (); Khriplovich1995 (), with the values and from nuclear theory Dmitriev1997 (); Dmitriev2000 (); Haxton2001c (); Haxton2002 () (see also Ginges_PhysRep2004 ()), and with Eq. (29), we extract limits on the parameter in the single-particle approximation (Table 5).

Ref. Roberts2014 () This work
Parameter Model Cs Tl Cs Tl
/ GeV s.p.
s.p.
/ GeV m.b.
m.b.
/ GeV m.b.
m.b.
Table 5: New limits (, in laboratory frame) on the SME parameters , , and . s.p. denotes single-particle (Schmidt model) limit and m.b. denotes many-body (hybrid model) limit.

We now leave the single-particle approximation and consider nuclear many-body effects. For a single-particle state, the angular momenta factors in (27) can be rewritten as

(30)

Hence, unlike NSD-PNC effects arising from -boson exchange between electrons and the nucleus Ginges_PhysRep2004 (), we cannot simply average over the spins of the single-particle proton and neutron states without explicitly considering the angular momenta of each individual nucleon. To circumvent this difficulty, we make use of the following approximation. Note that for single-particle states with , the prefactors before in Eq. (30) are . For non-light nuclei, most nucleons have . Also, the deviations of the prefactors in (30) from are of opposite sign for . Thus for nuclei with valence nucleon(s), which have , we can approximately sum over the proton and neutron spin angular momenta that appear in (30) to give the many-body generalisation of formula (27):

(31)

From Eq. (31), we extract limits on the parameters , , and for Cs, for which , using the calculated spin content values in Table 3. The limits are presented in Table 5. For Tl, where , Eq. (31) is not a good approximation and so we do not present many-body model limits in this case. We note that the limits in Table 5 are weaker than those that would be obtained indirectly from the most stringent limits on and , if one assumes a static background cosmic field. These corresponding upper limits are roughly as follows: GeV, , GeV and , assuming that the typical speed of Earth relative to the static background cosmic field is .

Acknowledgements.
We would like to thank Dmitry Budker and Lutz Trahms for useful discussions and for motivating this work. We are particularly grateful to Alan Kostelecký for pointing out that the methods in our previous work Roberts2014 () could be extended to extract limits on the SME parameter and that the methods of our present work could be extended to extract sensitivities on the SME parameters and . Y. V. S. would like to thank Pierre Sikivie for useful discussions and for suggesting several further nuclei of interest in axion dark matter searches. The authors would also like to thank Ben Roberts for useful discussions. This work was supported in part by the Australian Research Council and by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. V. V. F. would also like to acknowledge the Humboldt foundation for support through the Humboldt Research Award and the MBN Research Center for hospitality.

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