arXiv:2010.03588v1 [hep-ph] 7 Oct 2020 - Stanford University

Searching for New Interactions at Sub-micron Scale Using the Mossbauer Effect

Giorgio Gratta,1 David E. Kaplan,2 and Surjeet Rajendran2

1Department of Physics, Stanford University, Stanford, California 94305, USA2Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA

(Dated: October 9, 2020)

A new technique to search for new scalar and tensor interactions at the sub-micrometer scaleis presented. The technique relies on small shifts of nuclear γ lines produced by the coupling be-tween matter and the nuclei in the source or absorber of a Mossbauer spectrometer. Remarkably,such energy shifts are rather insensitive to electromagnetic interactions that represent the largestbackground in searches for new forces using atomic matter. This is because nuclei are intrinsi-cally shielded by the electron clouds. Additionally, electromagnetic interactions cause energy shiftsby coupling to nuclear moments that are suppressed by the size of the nuclei, while new scalarinteractions can directly affect these shifts. Finally, averaging over unpolarized nuclei, further re-duces electromagnetic interactions. We discuss several possible configurations, using the traditionalMossbauer effect as well as nuclear resonant absorption driven by synchrotron radiation. For thispurpose, we examine the viability of well known Mossbauer nuclides along with more exotic onesthat result in substantially narrower resonances. We find that the technique introduced here couldsubstantially improve the sensitivity to a variety of new interactions and could also be used, inconjunction with mechanical force measurements, to corroborate a discovery or explore the newphysics that may be behind a discovery.

I. INTRODUCTION

Light, weakly coupled particles emerge in many the-ories of physics beyond the standard model. Examplesof such particles include scalars such as moduli [1] andrelaxions [2–4] that are tied to solutions of the hierar-chy problem as well as mediators between the standardmodel and the dark sector. These particles can be ex-perimentally probed by searching for the new forces theymediate between standard model particles. Current lim-its [1, 5] on such forces are a strong function of the range(i.e. mass) of the new particle. Roughly, for distancesgreater than tens of microns, the strength of the newforce is constrained to be weaker than gravity. But, atdistances below the micron scale, forces that are manyorders of magnitude larger than gravity are still allowed.

There are two fundamental reasons for this sharp de-pendence of the sensitivity on the range. First, as therange decreases, only a progressively thinner sliver of ma-terial is at the correct distance to probe the interaction,so that the force becomes weaker. Second, and more im-portantly, in the case of experiments based upon the in-teraction between atomic matter [6], electromagnetic ef-fects such as Casimir forces [7] and gradient interactions,such as produced by patch potentials, increase rapidly atshort distances, resulting in overwhelming backgrounds.While conductive shields can mitigate both effects, prac-tical considerations related to the small distances in-volved and the finite conductivity of materials limit theireffectiveness in the very short range regime. Indeed,while to-date no experiment using interactions betweenatoms has reported the discovery of a new force at thesub-mm scale, it is reasonable to ask whether these tech-niques would offer sufficient redundancy and cross checksto support a positive claim of extraordinary importance.The use of neutrons, with a charge radius that is much

smaller than that of atomic matter, has also been pur-sued in scattering experiments [8]. Although in this casethe systematics are very different, the sensitivity is sub-stantially lower due to the difficulty in obtaining suitableneutron sources.

It is thus interesting to investigate sensing platformsthat might naturally suppress the large electromagneticbackgrounds while retaining sensitivity. In this paper, wepoint out that there is an enticing possibility to probe thedirect coupling of nuclei to scalar and tensor interactions.This can be achieved by studying, by means of Mossbauerspectroscopy [9], the very small energy shifts expectedwhen nuclei are exposed to new scalar and tensor interac-tions. The magnitude of such a shift only depends uponthe strength of the interaction and is unsuppressed bynuclear moments. At the same time, electromagnetic ef-fects on nuclear energy levels are significantly suppressed,being shielded by the electron clouds. Moreover, elec-tromagnetic effects can shift nuclear energy levels onlythrough multipole effects, and these are suppressed bythe size of the nucleus. Further, these multipole mo-ments are set by the spin of the nucleus. In a samplewhere the spins are not aligned, these effects will averagedown, unlike the effects of the signal from a new scalarinteraction. We note that this method will not be usefulin searching for new vector forces, since those are anal-ogous to electromagnetism, and their effect on nuclearenergy levels will be suppressed accordingly. One maytake the point of view that such a limitation can actuallybe exploited to measure the spin of the new mediator bycomparing a search with this technique with one usingatomic matter.

In the following, we investigate different Mossbauer se-tups and sources to search for such scalar and tensorforces. We begin in section II with a conceptual overviewof the setup, estimating the likely systematics. We then

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discuss three possible experimental realizations in Sec-tion III. In section IV we estimate the effects of the newinteractions on nuclear energy levels and compute thepotential reach of the experimental approach. Finally, insection V, we conclude.

II. CONCEPT

In the scheme proposed here, generically illustrated inFigure 1, photons are emitted by a Mossbauer source.The resonant re-absorption of these photons by an ab-sorber is tested when the source or the absorber are per-turbed by an “attractor” that generates the new inter-action. A change in the re-absorption cross-section as afunction of the distance d between the “attractor” layerand the source or the absorber would reveal the existenceof the new interaction.

In the rest of the paper, we will use the terms “source”and “absorber”, even if, in some cases, the two maybeidentical. Further, the device generating the new inter-action will be called “attractor” irrespective of the signof the interaction. While, at least in principle, the at-tractor can be setup to perturb either the source or theabsorber, it will arbitrarily assumed that the absorber isperturbed. The choice between the two possibilities willdepend on the technical details of a design. Typically thenew interaction is applied across a planar gap, althoughother geometries are possible. Likewise, in Figure 1, theeffect is notionally illustrated by measuring the absorp-tion of photons with a detector that is co-planar withthe absorber, but other schemes whereby the incoherentre-emission from the absorber is used are also possible.The attractor can be though of as a self-supporting slab,as illustrated in Figure 1, positioned at variable distancesfrom the absorber by means of piezoelectric actuators. Inthis fashion one can plausibly adjust the distance downto a fraction of a micron, for properly planarized surfacesin vacuum. However, in a different scheme, a solid layercan be condensed on the surface of the absorber, with thethickness of the layer setting the distance scale probed.Xenon may be an ideal choice for this application, owingto the high freezing temperature and large atomic mass.The use of separated isotopes would also allow to test theinfluence on the measurement of the number of neutronsand the nuclear spin.

The relationship between the new force and the energyshift, ∆E, is discussed in Section IV. The resonant ab-sorption cross-section of a photon of energy Eγ is givenby

σ0(Eγ , E) =2π

E2

1

1 + β

2I ′ + 1

2I + 1

(Γ/2)2

(E − Eγ)2

+ (Γ/2)2 (1)

where Γ and E are, respectively, the natural line widthand energy of the resonance, β the internal conversioncoefficient and I (I ′) the spin of the ground (excited)state. The observed width may be larger than the natu-ral one, because of non-homogeneous condensed matter

FIG. 1. Conceptual sketch of the type of experiment pro-posed. The distance d between attractor and absorber canbe adjusted to probe the effect of a new interaction. Theattractor may not even be a self-supporting foil but, rather,a layer of a solid grown onto the absorber. The distance tbetween source and absorber and the exact geometry of thetwo Mossbauer foils and of the photon detector are inessentialand can be optimized differently for different experimental re-alizations.

conditions around the nuclei of interest. Such a spreadis often modeled as a Gaussian function, to be convolvedwith the expression (1). It is clear that, for the presentpurpose, the system should approximate the natural linewidth as closely as possible (and the nuclides with thenarrowest Γ provide the best probes). Apart from linebroadening, in some cases condensed matter effects mayresult in line shifts that may be different in sources andabsorbers. These play no role in the measurements pro-posed here, since they can be calibrated-out with theattractor in a retracted position.

Sensitivities to ∆E in the range 10−15 eV to 10−17eVare found in Section IV to be achievable from the linewidths and counting statistics in systems using knowntechnology. Here we examine the limits imposed on thesemeasurements by the backgrounds produced by electro-magnetic coupling to the attractor. These would ariseas shifts of the transition energies, δEbkgd, that cannotbe easily distinguishable from the shift, ∆E, due to thesignal. δEbkgd for a single nucleus, can be estimated forCasimir forces, patch potentials and magnetic impuri-ties. The electric field due to the Casimir effect be-tween two plates that are separated by a distance d is

δECasimirbkgd ≈

√4π240

πd2 , so that the resulting energy shift

to a nucleus through the electric quadrupole moment is

≈√

4π240

παEMr2N

d3 where rN is the size of the nucleus and

αEM is the fine-structure constant. Taking, conserva-tively, rN to be the radius of the tantalum nucleus andd ≈ 10−8 m, an energy shift δECasimir

bkgd ≈ 10−13 eV is ob-tained. Electric fields from electrostatic patch potentials

3

roughly scale as V0/d where V0 ≈ 100 mV for distances

d ≈ 10 nm [10], resulting in a shift δEpatchbkgd ≈ 10−15 eV.

Finally, ferromagnetic domains in the attractor shouldcontain fewer than ≈ 100 polarized spins in the 10 nmscale when the attractor is at d ≈ 10−8 m, inducing

line shifts δEMagbkgd < 10−10 eV. Since, at first order,

these backgrounds couple to magnetic dipole and electricquadrupole moments of nuclei, for unpolarized samplesthe overall shift will be averaged down as δEbkgd/

√Nγ ,

where Nγ is the number of nuclei participating in the res-onant absorption experiment (i.e. the number of eventsin the experiment). Typical experiments discussed in thefollowing use Nγ ' 1013, so that, even the largest back-

ground shift discussed here, δEMagbkgd, is expected to aver-

age down to a sufficient level not to limit the sensitivity.The nucleus-to-nucleus difference in δEbkgd will result ina small line broadening, however such broadenings shouldbe measurable independently from the energy shifts.

Second order electromagnetic shifts can arise throughthe chemical/isomeric shift in the nucleus. Unlike thefirst order effects discussed above, these do not averagedown with the number of events and arise due to theoverlap between the electron clouds and the finite size ofthe nucleus. The intrinsic value of the resulting shift isdominated by the inner electrons that are closer to thenucleus. This shift becomes a background to the mea-surement only if its magnitude changes with the distancebetween the attractor and the absorber, as it can be thecase when the electric field from the attractor polarizesthe electron clouds. It can be verified that this effect isdominated by the outermost electrons. The contributionof the outermost electron clouds to chemical/isomericshift is ≈ 5 × 10−9 eV for iron-like elements [11, 12].An electric field E0 will change the overlap of the out-ermost s-electron by ≈ αEM (E0aB/ωe)

2where aB is the

Bohr-radius and ωe the binding energy of the electron.Taking typical values aB ≈ 0.05 nm and ωe ≈ 10 eV, it isestimated that this energy shift is dominated by Casimir

forces and is ≈ 3 × 10−15(

10 nmd

)4eV. It is likely that

smaller shifts than this can be obtained with a judiciouschoice of the chemistry of the absorber. For example,if the absorber is made of an ionic compound of ironwhere iron is doubly oxidised, its outermost s-electronswould have a reduced overlap with the nucleus. Sincethese electrons are the ones that are most easily polar-ized, the shift is likely to decrease. Given the uncer-tainties, we will conservatively take this line shift to be

10−14(

10 nmd

)4eV. While this number was calculated for

iron, due to the dependence of the effect on chemicalproperties of the absorber, we will use it as a limit to thesensitivity of other nuclei as well. At 10 nm, this shiftis larger than the statistical sensitivity that could poten-tially be reached by the experiment, but since it dropsrapidly with distance, it quickly becomes sub-dominant.Note that lattice imperfections can give rise to first ordereffects wherein defects cause an intrinsic polarization ofthe electron cloud which can then couple to these electric

fields at first order. However, these do average down andare sub-dominant to the effects discussed above.

Two types of temperature effects need to be consid-ered in analysing systematics for the proposed measure-ments [13]: the second order Doppler shift and the Lamb-Mossbauer effect. The second order Doppler shift can beunderstood as the dilation of proper time for nuclei un-dergoing thermal vibrations in the lattice, resulting ina shift, δEtemp, dependent on the temperature. Unlikethe linear counterpart, this effect does not average outin the lattice. The temperature dependence [14, 15] is

≈ 10−11 (T/300 K)3

eV/K, resulting in shifts that canbe caused by temperature drifts. There are two possiblestrategies to mitigate these effects. The first is to runthe experiment at low temperatures (e.g.≈ 4 K), wherethe value of the shift is smaller than the line shifts ofinterest. The second consists in monitoring the temper-atures of the source and the absorber and correct forthe effect. A combination of these approaches could alsobe pursued. For example, if the system is cooled to≈ 30 K, the average temperature of the system needsto be known to within 100 mK over the course of themeasurement to keep systematic shifts below 10−15 eV.An additional consideration that arises from the secondorder Doppler shift is the possibility that the Debye tem-perature of the lattice may be changed by electromag-netic backgrounds when the attractor is brought nearthe absorber. For the electric field to change the Debyetemperature, it has to cause relative motion between theatoms in the lattice, analogous to a tidal effect. Further,since the atoms are neutral, the leading order force onthem is through a dipole moment. This effect shouldalso be suppressed by the mass of the nucleus and theintrinsic stiffness i.e. the Debye frequency of the sys-tem. Thus, the fractional change to the line width is at

most δE/E ≈(E′(dad

)2 1MθD

)2

where E′ is the electric

field over a distance d, da the electric dipole moment ofthe atom, M the mass of the nucleus and θD the Debyetemperature of the lattice. In deriving this estimate, wehave made use of the fact that terms that are linear inthe intrinsic “velocity” of the nucleus in the lattice donot cause line shifts. It can be verified that this effect issignificantly smaller than the second order chemical shiftdiscussed above.

The second effect of the temperature is through theLamb-Mossbauer factor which changes the fraction of re-coilless emission from the source. While this is not a lineshift, it will change the number of resonant events, simu-lating a shift of the emission or absorption lines. At tem-

peratures T � θD, this fraction is e−3 E

θD

(1+ 2π2T2

3θ2D

)[16].

For Mossbauer emissions E ≈ 10 keV, θD ≈ 400 K andT ≈ 30 K, the temperature coefficient of the resonantrate is ≈ 10−4 K−1. For this effect not to exceed thestatistical fluctuations over the 1013 events/month con-sidered in Sec. IV, is sufficient to limit or measure tem-perature fluctuations at the 3 mK level. This systematic

4

Nuclide E (eV) T1/2 Γ (eV) Γ/E

5726Fe 14,413 98.3 ns 4.7 × 10−9 6.4 × 10−13

7332Ge 13,328 2.92 µs 1.6 × 10−10 1.2 × 10−14

18173 Ta 6,237 6.05 µs 7.5 × 10−11 1.2 × 10−14

6730Zn 93,300 9.07 µs 5.0 × 10−11 5.4 × 10−16

4521Sc 12,400 318 ms 1.4 × 10−15 1.13 × 10−19

10747 Ag 93,125 44.3 s 1.03 × 10−17 1.1 × 10−22

10345 Rh 39,753 56.1 min 1.36 × 10−19 3.4 × 10−24

18976 Os 30,814 5.8 hr 2.2 × 10−20 7.0 × 10−25

TABLE I. Properties of some nuclides of interest [17] orderedby the half life of the Mossbauer transition T1/2. E and Γ arethe energy and the natural line width of such transition, thelatter calculated from the half-life. The four nuclides abovethe line have relatively short half lives and are mostly dis-cussed in the context of traditional Mossbauer setups, whilenuclides below the line are though of as more aggressive op-tions, requiring substantial R&D and the use of excitation bysynchrotron radiation.

can also be mitigated by continuously normalizing theresonant rate, e.g. using two absorbers, one of which isnot perturbed by the attractor.

III. EXPERIMENTAL REALIZATION

We investigate the potential of traditional sources withmodest lifetimes and natural line widths 10−11 . Γ .10−9 eV. Those are listed above the line in Table I. Forthese nuclides, the traditional Mossbauer effect has beenexperimentally demonstrated with line widths compara-ble to the natural ones. As discussed in Section IV,we find that traditional Mossbauer experiments basedon these nuclides can be used as competitive probes fornew forces with range as short as 100 nm and, possibly,shorter.

Nuclides in the second part of Table I could plausiblyfurther improve the sensitivity of the technique, owingto the exceedingly narrow natural line widths. For thisreason, these nuclides may be also competitive with forcemeasurements for distances larger than ' 100 nm. How-ever, substantial work is required to take advantage ofthese very sharp resonances, possibly in conjunction withthe use of synchrotron radiation for their excitation.

A. Traditional Mossbauer Technique

In the traditional Mossbauer technique, the appropri-ate nuclide NM is produced directly in the isomeric stateN∗M from a different progenitor nuclide NP, using a betaor EC decay with a convenient half-life. Hence, the sourceis intrinsically non-homogeneous. The absorber can beentirely made of the nuclide NM in its ground state so

that, typically, there is an energy shift between emissionand absorption lines. More parameters for the four nu-clides at the top of Table I are shown in Table II, wherethe natural line widths are compared with experimentallyobtained values (in the 57Fe case by commercial sources).In our application, the attractor has to be brought inclose proximity to the source or the absorber. Sincesource preparation is complex, here we concentrate onthe absorber that needs to be thick enough to provide fullresonant absorption, to achieve the best constrast, while,at the same time, no thicker than the range at which thenew interaction is tested. The resonant mean-free path,`, can be derived from Eq. 1. An efficient absorber for57Fe can be made by coating some inert substrate witha layer sufficiently thin to reach attractor distances be-low 100 nm. From the values of ` in Table II, it appearsthat similar properties can be achieved with 181Ta usinga few-layer array of absorbers and attractors. 73Ge and67Zn are not suitable for the study of short distance in-teractions, unless an arrangement can be found to applythe attractor to the source.

In order to maximize the statistical power of the mea-surement, it is also important to use a system in whichthe product ( 1

1+β )η is maximized. Here η is the fraction

of decays of the progenitor feeding the Mossbauer state.Since low energy transitions have larger internal conver-sion coefficients, the desire for a large 1

1+β is at odds

with the other requirement that the Mossbauer transitionshould have low energy, so that a substantial fraction ofdecays is recoilless at room temperature. The recoillessfraction is not shown in Table II because it depends onthe matrix of the source and the absorber, but for 57Fe,73Ge and 181Ta in metallic matrices can be assumed to bearound 70% at 300 K. The Mossbauer transition in 67Znhas a much larger energy and hence a substantial recoil-less fraction can only be achieved at low temperature, aswas done in [18].

In an ideal experiment, similar statistics would be ac-quired without attractor, with source and absorber inperfect resonance, and with the attractor potentiallyshifting the line. Because of the shift between source andabsorber due to the different matrices, some mechanismto scan the lines is required. This is usually achievedusing Doppler drive systems where constant velocity isoften achievable for the majority of the stroke. An alter-native technique may consist in producing a shift withan external magnetic field, which can be properly tunedand operated statically during data taking. The shiftproduced in this way is ≈ 10−7 eV/T. However, whilethis solution may be ideal when the excitation is pro-vided by synchrotron radiation and only sub-line widthshifts are required, when different matrices in the sourceand absorber require larger shifts, the efficiency may bereduced because magnetic splitting will produce severallines, only one of which can be used in the measurement.

From the discussion above, it appears that the mostcommon source, 57Fe, is also best suited for the techniquedescribed here, using a traditional Mossbauer setup. In

5

Mossbauer Decay Parent Properties

NM E Γ ΓEXP ΓEXP/Γ Ref. β ` η NP Decay Half

(eV) (eV) (eV) (nm) mode life (d)5726Fe 14,413 4.7 × 10−9 ≈ 5 × 10−9 ≈ 1 [19] 8.56 48 0.89 57

27Co EC 2727332Ge 13,328 1.6 × 10−10 1.6 × 10−10 ≈ 1 [20] 1.12 × 103 3.2 × 104 1 73

33As EC 80.318173 Ta 6,237 7.5 × 10−11 5.5 × 10−10 7.5 [21] 70.5 180 1 181

74 W EC 121.26730Zn 93,300 5.0 × 10−11 7.5 × 10−11 1.5 [18] 0.87 3.3 × 103 1 67

31Ga EC 3.2

TABLE II. For the first four nuclides NM in Table I, left columns: decay energy (E), natural line width Γ and line width ΓEXP

derived from data reported in the references listed. In the case of 57Fe, ΓEXP is that of a commercial supplier, indicated in thereference. β is the internal conversion coefficient, as already mentioned, ` is the resonant mean free path, in a sample of pureNM, calculated from Eq. 1, and η is the fraction of decays of the parent nuclide NP landing in the Mossbauer state. Rightcolumns: properties of the commonly used parent nuclides.

section IV the sensitivity is computed for the cases of 57Feusing an activity of 100 mCi, and a ±10◦ collimation,resulting in a 0.03 sr solid angle and a 6% line broadening,as would be the case for the usual Doppler tuning.

B. Coherent Synchrotron Light Excitation

Since the advent of synchrotron light sources, thepossibility has arisen of directly exciting the isomericstate from the ground state by the absorption of a pho-ton [22, 23]. For the purpose of interest here, this has theadvantage that source and target can be made out of thesame material and resonant absorption is achieved auto-matically, with a minimum amount of line shift requiredto scan the resonance and probe the signal from a newinteraction. Hence, the use of a magnetic field to shift theline, as mentioned above, is expected to be the techniqueof choice. More importantly, the use of excitation by syn-chrotron radiation opens the possibility of using nuclidesfor which there is no obvious production scheme with thetraditional method. Because of the homogeneous natureof the source, it is also possible that ultra-sharp lines,such as those resulting from the nuclides in the secondpart of Table I, can be eventually approached.

Other peculiar properties of synchrotron light exci-tation may not be relevant here. While in all casesMossbauer spectroscopy relies on the coherent recoil ofan entire crystal, the radiation emitted after synchrotronlight excitation is also coherent in the sense of superra-diance [24], whereby the emission can be considered asa wave common to the entire crystal. In this picture,the exponential decay of the isomeric state should bethought of as the decrease in the amplitude of the waveand sufficiently large energy shifts between source andabsorber manifest themselves in “quantum beats” mod-ulating the exponential decay. In practice, energy shiftsat the threshold of detection, generally contemplated inthe measurements discussed here, do not produce appre-ciable modulation and their detection has to be based onthe more mundane overall rate decrease. Furthermore,the drastic background reduction afforded by the fastsynchrotron pulses, allowing for the observation of the

resonance as nuclei de-excite after the end of the pulse,is only applicable to cases where the lifetimes are shorterthan the repetition rate of the synchrotron light.

The main limitation of the the excitation by syn-chrotron radiation derives from the achievable statistics.This is because of the substantial mismatch between thespectral density of the synchrotron radiation and the ex-ceedingly sharp absorption line of the isomeric transi-tions. Dedicated monochromators exist at least at theAPS (Argonne National Lab, USA) [25] and SPring-8(Japan) [26] with ≈ 1 meV linewidth and peak inten-sities of 5 × 1013 γ/s at 14.4 keV (BL09XU beamline ofSPring-8). With this state of the art equipment, 5×10−6

(1.4 × 10−12) of the available photons are useful to ex-cite the 57Fe (45Sc) transition, resulting in an integratedrate of 1014 month−1 (6 × 105 month−1), after account-ing for the gamma conversion factor β. The first fig-ure already exceeds the rate of 1013 month−1 expectedfor the 100 mCi traditional source collimated to ±10◦

discussed in the previous section. While an experimentat a synchrotron radiation facility is more involved thana traditional Mossbauer one, shorter run times, consis-tent with typical beam line usage appear realistic andthe possibility of magnetic tuning may represent an ap-pealing option. While an alternative approach to furtherincreasing the resonant event rate is offered in the follow-ing section, it is also possible that the advent of properX-ray lasers [27] will result in sources with narrower bandand higher spectral density.

C. Three-State Synchroton Light Excitation

The poor coupling between the energy spectrum of thesynchrotron and the isomeric transition may be mitigatedusing a three-level system, as illustrated in Figure 2. Thehigher energy state N∗∗ is excited by synchrotron radia-tion with good efficiency, owing to its substantial width.N∗∗ then spontaneously decays into the Mossbauer stateof interest, N∗M . While this process does not involve thecoherence mentioned above and, hence, it is not expectedto produce quantum beats, as discussed this is not im-portant for the current application.

6

FIG. 2. Three level scheme discussed in the text. The groundstate NM is excited to N∗∗

M by a (relatively) broad-band syn-chrotron light photon, γS . N∗∗

M then de-excites to the verynarrow Mossbauer state N∗

M , which decays back to the groundstate with the emission of the photon γM.

The mechanism requires a state N∗∗ with sizeable am-plitudes connecting it to both N∗M and the ground stateN. This appears to be the case at least for 189Os, with aN∗∗ state of JP = 7/2−, energy 216.67 keV and half life0.4 ns. The excitation energy is challenging but not out ofreach for synchrotron radiation sources. Importantly, thestrength of the transition N∗∗ → N∗M, k∗ = 100, is compa-rable to that of the transition N∗∗ → N, k0 = 34.3, mean-ing that the state N∗∗ is accessible with similar proba-bility from the ground and the Mossbauer states. Thegain in rate with respect to direct synchrotron radiationexcitation would be given by

A ≈ Γ∗∗

Γ

I∗∗

I∗Mk∗k0

(k∗ + k0)2(2)

where Γ∗∗ is the line width of the N∗∗ state and I∗M(I∗∗) the spectral density of synchrotron radiation sourcefor the transition energy from the ground state to N∗M(N∗∗). Using parameters for the BL09XU (BL8W) beam-line at SPring-8, I∗M ≈ 5 × 1013 s−1meV−1 (I∗∗ ≈1.7× 104 s−1meV−1) and A ≈ 3.4× 104, indicating thatsome advantage may exist.

States N∗∗ with lower energies exist in 5726Fe (136 keV),

7332Ge (67 keV) and 181

73 Ta (136 keV), although more workis needed to understand the strength of the transitionsto N and N∗M.

IV. REACH

In this section, the reach of the method discussed inprobing a variety of scalar and tensor forces is evaluated.The following interactions are considered:

L ⊃ yqφqq+φ

fγF 2µν +

φ

fgG2µν +

hµνfT

FµσFνσ + gφh2 (3)

where φ and hµν are new scalar and tensor interactions,respectively, that couple to quarks (q), electromagnetism(Fµν), gluons (Gµν) and the Higgs (h). In the setupshown in Figure 1 where the attractor and the absorberare parallel thin plates separated by a distance d, the at-tractor produces a potential whose value at the absorberis:

φ, hµν ≈gNnd

2

2e(4)

where gN is the coupling per-nucleon and n the numberdensity of nucleons in the material. This expression as-sumes that the absorber is placed at a distance d = λwhere λ is the range of the new force, causing the poten-tial to drop by e−d/λ = e. The scalar φ shifts nuclearenergy levels in the absorber, changing the frequencyof the Mossbauer line. The tensor hµν does not signifi-cantly change nuclear energy levels. Instead, it changesthe frequency of the emitted γ as it propagates out ofthe source towards the absorber. In the following, weestimate these effects (sub-sections IV A and IV B) andcompute the reach (sub-section IV C) of the experiment.

A. Scalar

For each of the scalar couplings in Eq (3), the effec-tive coupling gN between a nucleon and φ as well as theenergy shift ∆E induced in the nucleus due to φ needto be estimated. Because of the non-perturbative na-ture of nuclear physics, these effects cannot be calculatedfrom first principles, and, instead, are derived from phe-nomenological models. These should be regarded as pro-viding the rough order of magnitude of the effect. Gen-erally, the estimates of gN are more reliable than thoseof ∆E. Since this is an experiment where a new field isproduced and detect, the dependence of the fundamentallagrangian parameters (yd, g, fγ , fg, fT ) on gN and ∆E is

yd, g, f−1γ , f−1

g , f−1T ∝

√gN∆E.

1. Yukawa Modulus

As an example of the Yukawa couplings in Eq. (3), wefocus on φ that couples to the down quark. Given theterm ydφdd, the effective nucleon coupling gNφNN canbe estimated from lattice methods, yielding gN ≈ 9.5yd[28].

To estimate the energy shift in the nuclear levels dueto φ, we calculate the change in the 1-pion exchange po-tential and equate it to the energy shift ∆E. Using theparametrization in [29],

7

∆E = ∆Vπ ≈15

4

m2π

m2N

1

r≈ 2.5ydφ, (5)

where the relationship between the pion mass and is as-sumed the down quark mass is used, and is assumed thatnuclear level transitions change the relative distance be-tween nucleons (of mass mN ) by r ≈ 1 fm.

2. Alpha Modulus

The coupling φfγF 2µν shifts the fine structure constant

α → α − α2φfγ

. To derive an effective nucleon coupling

gN , we compute the contribution of the electromagneticfield to the mass of the nucleus. In a nucleus with Zprotons and A nucleons, φ shifts the self energy of the

electromagnetic field by − Z2α2φA1/3fγrp

, where rp ≈ 1.2 fm is

the proton radius. From this, the coupling per-nucleon

gN = Z2α2

A4/3fγrpis extracted.

Next, the energy shift ∆E of the nuclear level causedby φ is estimated. Mossbauer nuclei such as 57Fe and181Ta have nearly degenerate low lying energy levels im-plying that the nuclei are highly deformed. Further,their intrinsic quadrupole moments are comparable to(A1/3rp

)2. The Mossbauer transitions are between states

that have different spins. Thus, their quadrupole mo-ments are O (1) different and it is reasonable to assumethat the effective location of the transitioning nucleonchanges by the size of the nucleus itself. ∆E is taken tobe equal to the change in the Coulomb energy, yielding

∆E = − Zα2

A1/3fγrpφ.

3. Gluon Modulus

The coupling φfgG2µν shifts the QCD structure constant

αs → αg − α2sφfg

. We use the rough argument that the

mass mN of a nucleon should be mN ≈ αsrp

. This yields

αs ≈ 5 and the effective nucleon coupling gN ≈ α2s

fgrp.

To estimate the energy shift in the nuclear levels, the

1-pion exchange potential Vπ = g2

4m2π

m2N

1rp

is used. The

change in αs shifts g2 by4πα2

sφfg

yielding an energy shift

∆E ≈ 0.08 GeVfg

φ.

4. Higgs Portal

The Higgs portal coupling gφh2 emerges naturally intheories such as the relaxion that can solve the hierarchyproblem [2, 30, 31]. This coupling shifts nucleon energylevels since it effectively acts as a down quark Yukawamodulus, leading to an energy shift ∆E ≈ 2.5gmd

m2hφ. Its

10-8 10-7 10-6

10-15

10-14

10-13

10-12

10-11

10-10

1010

1012

1014

1016

1018

λ (m)

y d α

FIG. 3. Sensitivity to the down quark Yukawa modulus yd,as a function of the range, λ. The green and the black linesare the projected sensitivities for 57Fe (1013 total decays) and181Ta (3 × 1014 total decays) respectively, assuming naturalline width in both cases. For comparison, the correspondingvalue of α (strength of force relative to gravity) is shown onthe right. The envelope of current limits is indicated by theblue region and the dashed red line, representing the limitobtained in [5] where a large background from Casimir inter-actions was subtracted.

10-8 10-7 10-6 10-5

10-11

10-9

10-7

10-5

104

106

108

1010

1012

1014

1016

1018

λ (m)

f γ-1(GeV

-1)

α

FIG. 4. Sensitivity to the alpha modulus fγ , as a function ofthe range, λ. The remainder of the description is the same asin Figure 3.

effective coupling to nucleons, gN , can be estimated vialattice methods and is gN ≈ 10−5 g

GeV [32].

B. Tensor

The tensor hµν in (3) couples to the electromagneticcontribution to the mass of the nucleus. From thismass Z2α

A1/3rp, we calculate the coupling per-nucleon gN =

Z2αA4/3fT rp

. When a photon of energy ω traverses a po-

tential difference ∆h, its energy changes by ∆E = ω∆hfT

,

where, for the present purpose ∆h = h.

8

10-8 10-7 10-6

10-14

10-13

10-12

10-11

10-10

1010

1012

1014

1016

1018

λ (m)

f g-1(GeV

-1)

α

FIG. 5. Sensitivity to the gluon modulus fg, as a function ofthe range, λ. The remainder of the description is the same asin Figure 3.

10-8 10-7 10-6

1

10

100

1000

104

1010

1012

1014

1016

1018

λ (m)

g(eV)

α

FIG. 6. Sensitivity to the Higgs portal g, as a function of therange, λ. The remainder of the description is the same as inFigure 3.

C. Sensitivity

Using Eq. (4) for the scalar and tensor fields pro-duced, the energy shifts for the various terms in Eq. (3)are computed using the approximations discussed above.The reach of the experiment for each coupling is cal-

culated as ∆E = Γ√Nγ

+ 10−14(

10 nmd

)4eV. The sec-

ond term in this expression arises from the systematiclimit which is dominated by the chemical shift causedby Casimir forces, as discussed in Sec. II. The sensitivityof the technique is estimated disregarding the statisti-cal fluctuations due to possible non-resonant absorption,as the resonant photon statistics can be obtained by de-tecting the re-emission of the Mossbauer photons by theabsorber or through the use of pulsed synchrotron exci-tation schemes, at least for the short lifetime cases.

57Fe and 181Ta are used as examples, assuming natu-ral line-width are achieved. For 57Fe, that is taken as theconservative case, we assume a total of 1013 decays whichcan be obtained with a commercial 100 mCi source oper-ating for a month, with ±10◦ angular collimation. 181Tais treated as a more aggressive case, assuming a 1 Ci

source operating for a month with O (1) angular cover-age, such as may be realistic using magnetic tuning. Thisyields a total of 3 × 1014 decays. In both cases the pa-rameters β and η from Table II are used; in addition, itis assumed that the total number of decays is evenly splitbetween times when the attractor, made out of gold, isnear and far from the absorber. The corresponding small-est measurable energy shifts are ∆E ≈ 10−15 eV for 57Feand ∆E ≈ 10−17 eV for 181Ta. The sensitivity is limitedby the chemical shift due to Casimir interactions at dis-tances around 10 nm and becomes statistics limited atlarger distances.

These results are plotted in Figures 3, 4, 5, 6 and 7 forthe down quark Yukawa, alpha and gluon moduli, Higgsportal and the tensor coupling, respectively. In each fig-ure, the solid green and black lines represents the reachfor 57Fe and 181Ta, respectively. The solid blue and reddashed lines represent the current experimental limits onthe relevant couplings. The red dashed line is the limitfrom obtained in [5] where a large electromagnetic back-ground from Casimir interactions was subtracted using acommon Casimir shield. The solid blue line is the limitwithout this background subtraction. For each plot, thestrength relative to gravity, α is also shown to the right.The conversion to α of the parameter gN used in the com-putations of yd (Figure 3), fg (Figure 5) and g (Figure 6)is straight forward, since those depend solely on QCD.The analogous conversion for fγ (Figure 4) and fT (Fig-ure 7) also depends on the atomic number of the nucleus.For a given nucleus this is a well defined relationship andthus our projected sensitivities can be accurately calcu-lated. But, to convert the limits in [5] in terms of ourparameters, detailed knowledge of the nuclei used in thatsetup is necessary. We took the parameters of silicon, asa representative example.

As is evident from these figures, the Mossbauer ap-proach can substantially improve the sensitivity to shortdistance forces in the range 10−8m - 10−7m, beyond cur-rent laboratory detection schemes. It can also signifi-cantly extend constraints on forces in the range 10−7m -10−6m with a robust natural background suppression in-stead of relying on the subtraction of large backgrounds.

With the continuous evolution of synchrotron radia-tion sources and the possibility of implementing novelexcitation schemes (see section III C), it may be possibleto access a newer class of Mossbauer nuclei (see TableI). It is interesting to speculate on the possible reachof such systems, for example considering the possibilitythat the 12 keV level of 45Sc is directly excited by syn-chrotron radiation. Using the estimates in Sec. III B,≈ 105 Mossbauer photons are expected, yielding an en-ergy sensitivity ≈ 10−17 eV, comparable to that of theprojected sensitivity of 181Ta in the above figures. Thethree level synchrotron excitation scheme could poten-tially be used to access the 30 keV level of 189Os, result-ing in as many as ≈ 102 Mossbauer photons, correspond-ing to sensitivity to energy shifts as small as 10−21 eV. Ifsuccessfully developed, this excitation scheme could place

9

10-8 10-7 10-6

10-11

10-10

10-9

10-8

10-7

108

1010

1012

1014

1016

1018

λ (m)

f T-1(GeV

-1)

α

FIG. 7. Sensitivity to a new tensor force between matter andlight, fT , as a function of the range, λ. The remainder of thedescription is the same as in Figure 3.

competitive limits in the micron range, where the abilityof Casimir and patch potentials to shift the nuclear linesare sufficiently suppressed.

These results also show that there are interesting dif-ferences in the potency of this Mossbauer approach inrelation to the direct measurement of forces betweenmatter as a way to search for new interactions. TheMossbauer technique is sensitive both to how the newinteraction couples to a nucleon (and thus searched forin measurement of forces) and to the way the new interac-tion changes nuclear levels. We have already commentedon how the latter fact prevents this approach from beingsensitive to new vector interactions. This fact is also ofimportance to scalar interactions since different scalarschange the nuclear level splittings differently. For exam-ple, the Yukawa and alpha moduli are more efficient inchanging the nuclear energy differences as opposed to thegluon modulus whose effects are suppressed by ratios ofthe pion and nucleon masses. Thus, in the event thata new interaction is discovered, this method provides aunique way to probe the microphysics of the new inter-action.

In addition to these laboratory probes, there are alsoastrophysical limits on the light particles considered, aris-ing from the possibility that such particles may be emit-ted by astrophysical systems causing them to cool morerapidly than observed [33]. Roughly, these limits are atyd ≈ 10−13, g ≈ 10 eV, fg ≈ 109 GeV and fγ , fT ≈1010 GeV. However, these limits are not robust to pertur-bations of the underlying model - in particular, if thereare additional highly suppressed but long-ranged inter-actions between the standard model and these particles,the astrophysical limits can completely disappear [34].Similarly, density dependent effects can also eliminatethese limits [35]. Given the very different nature of stel-lar environments and a terrestrial experiment, there areno model independent astrophysical limits in this part ofparameter space.

V. CONCLUSIONS

While, over the years, the Mossbauer effect has hada tremendous impact in the fields of material scienceand chemistry, except for the notable case of the clas-sic experiment verifying the frequency shift of photonsin the Earth’s gravitational field [18, 36], it has nothad significant applications to the physics of fundamen-tal interactions. The comparison with sensing platformsbased on optical and atom interferometers may be of in-terest. While the intrinsic energy resolution of a singleMossbauer transition is substantially higher than that ofatoms, the much smaller cross sections of nuclear transi-tion results in modest rates. In addition, resonant tech-niques are substantially more complex and less exploredin the case of nuclear physics, because of the relative lackof suitable instrumentation. Thus, the Mossbauer effectcan be competitive only in sensing applications whereatom-based sensors encounter difficulties. This is the casefor material science and chemistry where the Mossbauereffect is used to study nano-scale properties of lattices.The search for short distance forces falls into the samecategory - substantial electromagnetic effects at shortdistances inhibits the use of atom-based sensors. Thesecluded nature of the nucleus inside an electron cloud,along with the suppressed interactions between nuclearmoments and electromagnetism, makes the Mossbauereffect ideal for probing short distance forces.

The requirements for this application is different fromthose needed for material science and chemistry. In thelatter cases, substantial line shifts are produced and theuse of different matrices for source and absorber is re-quired. For applications to fundamental interactions theinterest is in obtaining the narrowest possible lines andthe smallest possible intrinsic shift between source andabsorber.

Excitingly, we have shown that a first round of com-petitive measurements is possible using the traditionalMossbauer technique as well as nuclear resonance withsynchrotron radiation excitation. Further technical im-provements in both areas may extend the sensitivity wellbeyond the current state of the art. At the shortest dis-tances, the Mossbauer approach presented here appearsto be limited by chemical shifts due to Casimir interac-tions. However, since this approach is tailored to thediscovery of scalar interactions by measuring a line shiftrather than the displacement of an object, it might bepossible to mitigate this background. For example, onemay consider placing the absorber between two attrac-tors, measuring the distance between the absorber andthe attractors and feeding back to null out the Casimirforce on the absorber. Unlike for the case of conventionalforce sensing experiments, a new scalar interaction fromthe attractors will simply add, further shifting the line.Since the background is a second order effect, even a par-tial cancellation results in substantial improvements.

It may also be possible to mitigate the backgroundby adopting a differential measurement with two differ-

10

ent nuclides, e.g. 57Fe and 181Ta. The absorber couldbe made of an alloy/compound containing both, so thatboth are subject to the same local molecular structureaffected by the electric coupling to the attractor. Theline shift produced by the new force is generally expectedto be different from that originating from the change inmolecular fields for the two nuclides, so that, effectively,one nuclide can be used as a “co-electrometer” to mea-sure the effect of electric fields, separately from the newinteraction. The ability to potentially implement sucha differential measurement strategy is a key differencebetween this approach and conventional force measure-ments. In the latter case, once electromagnetic back-grounds dominate, the ability to look for a new interac-tion is effectively blocked. With the Mossbauer setup,it is possible to get additional information, potentially

providing a path towards better sensitivity.

VI. ACKNOWLEDGEMENTS

One of us (GG) is grateful to E.E. Alp (ANL) foran introduction to synchrotron radiation excitation ofMossbauer states and to U. Bergmann, A. Halavanau,C. Pellegrini (SLAC) for discussions on X-ray lasers. Wealso gratefully acknowledge some preliminary discussionswith J. Schiffer (ANL) and the help of P. Vogel (Caltech)in understanding several nuclear physics phenomena. Fi-nally, we thank E.E. Alp and P. Vogel for their carefulreading of the manuscript.

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