ISSN 1063�7796, Physics of Particles and Nuclei, 2011, Vol. 42, No. 4, pp. 661–666. © Pleiades Publishing, Ltd., 2011.

661

11. CONVENTIONAL MÖSSBAUER SPECTROSCOPY

Consider a nucleus in an excited state with energyE and lifetime τ. If this nucleus is free (not bound in acrystal) the γ ray emitted upon deexcitation exhibitsthe energy E – ER where ER is the recoil energy of thenucleus after emitting the γ ray. In addition, because ofthe thermal motion of the free nucleus the energyspread of the emitted γ ray will usually be much largerthan the natural linewidth Γ = ћ/τ as determined bythe time�energy uncertainty principle. In the reverseprocess, to excite such a free nucleus to the state E aγ�ray energy of E + ER is necessary because the nucleusrecoils after absorption of the γ ray and, of course, theabsorption spectrum will also be Doppler broadenedby thermal motion. The recoil energy is given by ER =E2/(2Mc2) where M is the mass of the nucleus and c isthe speed of light in a vacuum. Doppler broadeningdue to thermal motion and the recoil energy ER aretypically in the meV range.

It was a great achievement reached by R.L. Möss�bauer [1] to discover that ER as well as Doppler broad�ening due to thermal motion can be avoided by imbed�ding the nuclei (atoms) in a crystalline lattice. Anexcited nucleus bound in a crystal can emit � with theprobability f, called the Lamb�Mössbauer factor � aγ ray with the full energy E and the natural linewidth(see Section 3.1).

To mention some examples of interesting Mössbauertransitions: 57Fe (E = 14.4 keV, Γ = 4.3 × 10–9 eV), 67Zn(E = 93.3 keV, Γ = 4.8 × 10–11 eV), 109Ag (E = 87.7 keV,Γ = 1.2 × 10–17 eV). The resonance cross section for aMössbauer γ ray which is emitted (with ER = 0) in a

1 The article is published in the original.

source and thereafter absorbed (again with ER = 0) bya target (absorber) is given by [1–3]

(1)

where λ is the wavelength of the γ photon2; s is a statis�

tical factor which is determined by nuclear spins, iso�topic abundance, etc. and can be considered to be ofthe order of unity. The experimental linewidth Γexp

takes line�broadening effects into account (see Sec�

tion 3.2). The maximal cross section = 2π(λ/2π)2

is only determined by λ and is typically in the rangebetween 10–17 and 10–19 cm2 and thus very much largerthan the cross section for weak interaction which is~10–44 cm2. The large cross section makes Mössbauerspectroscopy very interesting in many areas of physics.Mössbauer � if they could be produced, e.g., bybound�state β�decay (see Section 2) � would alsoexhibit these large resonance cross sections sinceMössbauer are characterized by low energies,where λ is much larger than the dimensions of anucleus. In this limit, the specific properties of theweak interaction come into play only via the naturallinewidth Γ, i.e. the lifetime of the resonant state.

2. USUAL β�DECAY AND BOUND�STATE β�DECAY

In the usual continuum�state β�decay (Cβ), a neu�tron in a nucleus transforms into a proton and theelectron (e–) and electron�antineutrino ( ) are emit�

2 Here we use the relation λ = h/p, where h is Planck’s constant,and p is the momentum.

σRγ 2π λ

2π�����⎝ ⎠

⎛ ⎞2

s2f2 ΓΓexp

�������,=

σRmax

νe

νe

νe

Mössbauer Antineutrinos: Recoilless Resonant Emission and Absorption of Electron Antineutrinos1

Walter PotzelPhysik�Department E15, Technische Universität München, D�85748 Garching, Germany

Abstract—Basic questions concerning phononless resonant capture of monoenergetic electron antineutrinos(Mössbauer antineutrinos) emitted in bound�state β�decay in the 3H–3He system are discussed. It is shownthat lattice expansion and contraction after the transformation of the nucleus will drastically reduce the prob�ability of phononless transitions and that various solid�state effects will cause large line broadening. As a pos�sible alternative, the rare�earth system 163Ho–163Dy is favoured. Mössbauer�antineutrino experiments couldbe used to gain new and deep insights into several basic problems in neutrino physics.

DOI: 10.1134/S1063779611040101

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PHYSICS OF PARTICLES AND NUCLEI Vol. 42 No. 4 2011

WALTER POTZEL

ted into continuum states. This is a three�body processwhere the e– and the exhibit broad energy spectra.

In the bound�state β�decay (Bβ), again a neutronin a nucleus transforms into a proton. The electron,however, is directly emitted into a bound�state atomicorbit [4]. This is a two�body process. Thus the emitted

has a fixed energy = Q + Bz – ER, determined

by the Q value, the binding energy Bz of the atomicorbit the electron is emitted into, and by the recoilenergy ER of the atom formed after the decay.

The reverse process is also possible: an electron�antineutrino and an electron in an atomic orbit areabsorbed by the nucleus and a proton is transformedinto a neutron. Also this is a two�body process. Therequired energy of the antineutrino is given by =

Q + Bz + where is the recoil energy of the atomafter the transformation of a proton into a neutron.

Due to the well�defined energies and Bβhas a resonant character [4, 5] which is partiallydestroyed by the recoil occurring after emission andabsorption of the antineutrino. The resonance crosssection (without Mössbauer effect) is given by [5] σ =

4.18 × 10–41 ρ( )/ft1/2 cm2, where g0 =

4π(ћ/mc)3|ψ|2 ≈ 4(Z/137)3 for low�Z, hydrogen�like

wavefunctions ψ; m is the electron mass, and ρ( ) is

the resonant spectral density, i.e., the number ofantineutrinos in an energy interval of 1 MeV around

For a super�allowed transition, ft1/2 ≈ 1000.

The 3H – 3He system has been considered as afavourable example [3, 6] for the observation of Möss�bauer The lifetime of 3H is τ = 17.81 y which corre�

sponds to a natural linewidth Γ = ћ/τ = 1.17 × 10–24 eV.The resonance energy is 18.59 keV; ft1/2 ≈ 1000. If gasesof 3H and 3He are used at room temperature the profilesof the emission and absorption probabilities are Dopplerbroadened and both emission as well as absorption of theelectron antineutrinos will occur with recoil. Thus theexpected resonance cross�section (without Mössbauereffect) is still tiny, σ ≈ 1 × 10–42 cm2. However, recoilfreeresonant antineutrino emission and absorption(Mössbauer effect) would increase the resonancecross�section by many orders of magnitude (see Sec�tion 1). To prevent the recoil and avoid Dopplerbroadening, 3H as well as 3He have been considered tobe imbedded in Nb metal lattices [6]. It has to be takeninto account that the ratio Bβ/Cβ = α = 6.9 × 10–3

with 80% and 20% of the Bβ events proceeding via theatomic ground state and excited atomic states, respec�tively [4]. In the case of the Mössbauer resonance onlythe fraction of transitions into (from) the ground�stateatomic orbit is relevant, i.e., α ≈0.005 in the source aswell as in the absorber. Following Eq. (1), the reso�

νe

νe Eνe

Eνe'

ER' , ER'

EνeEνe

' ,

g02 Eνe

res

Eνe

res

Eνe

res.

νe.

nance cross section for Mössbauer can be written as[1, 3]

(2)

i.e., for the 3H–3He system, ≈ 1.8 × 10–22 cm2, if

s2 ≈ 1, f 2 ≈ 1, and Γexp = Γ = 1.17 × 10–24 eV (no linebroadening).

In the following, the 3H–3He system in Nb metalwill be discussed as an example in more detail.

3. MÖSSBAUER BASIC QUESTIONS AND ANSWERS

3.1. Phononless Transitions

To make the Mössbauer effect possible lattice exci�tations have to be avoided. With two kinds of lat�tice�excitation processes have to be considered:

1. Momentum transfer due to emission/capture of a This process leads to the recoilfree fraction fr, well

known from conventional Mössbauer spectroscopy. Inthe Debye model and in the limit of very low temper�atures T, the recoilfree fraction fr is given by

(3)

where θ is the effective Debye temperature, kB is theBoltzmann constant, and E2/2Mc2 is the recoil energywhich would be transmitted to a free atom of mass M.

With θ ≈ 800 K for the 3H–3He system [7], ≈ 0.07for T → 0.

2. Lattice expansion and contraction. This processwhich is not present with conventional Mössbauerspectroscopy, occurs when the nuclear transformationtakes place during which the is emitted or absorbed:

The itself takes part in the nuclear processes trans�

forming one chemical element into a different one. 3Hand 3He are differently bound in the Nb lattice and usedifferent amounts of lattice space. Thus, the latticewill expand or contract when the weak processes occur.The lattice deformation energies for 3H and 3He in the

Nb lattice are = 0.099 eV and = 0.551 eV,respectively [8]. Assuming again, an effective Debyetemperature of θ ≈ 800 K one can estimate � in analogyto the situation with momentum transfer � the proba�bility fL that this lattice deformation will not cause lat�tice excitations:

(4)

νe

σR

νe 2π λ2π�����⎝ ⎠

⎛ ⎞2

s2α2f2 ΓΓexp

������� 1.822–×10 s2f2 Γ

Γexp

������� cm2≈=

σR

νe

νe:

νe

νe.

fr T 0→( ) E2

2Mc2���������� 2

2kBθ����������–

⎩ ⎭⎨ ⎬⎧ ⎫

,exp=

fr2

νe

νe

ELH EL

He

fLEL

He3

ELH

3

–kBθ

��������������������–⎩ ⎭⎨ ⎬⎧ ⎫

13–×10 .≈exp≈

PHYSICS OF PARTICLES AND NUCLEI Vol. 42 No. 4 2011

MÖSSBAUER ANTINEUTRINOS: RECOILLESS RESONANT EMISSION AND ABSORPTION 663

Thus, the total probability for phononless emissionand consecutive phononless capture of is

(5)

which is tiny and makes a real experiment with the3H–3He system extremely difficult.

3.2. Linewidth

With Mössbauer three types of line�broadeningeffects [9–11] are important.

1. Homogeneous broadening is caused by electro�magnetic relaxation, e.g., by spin�spin interactionsbetween the nuclear spins of 3H and 3He and with thespins of the Nb nuclei. Contrary to the claim in [12],such magnetic relaxations are stochastic processes.They lead to sudden, irregular transitions betweenhyperfine�split states originating, e.g., from magneticmoments (spins) of many neighbouring nuclei. Thusthe wave function of the emitted particle (photon, )is determined by random (in time) frequency changeswhich cause line broadening characterized by thetime�energy uncertainty relation [7]. As a conse�quence, the broadened line cannot be decomposedinto multiple sharp lines. For the system 3H–3He inNb metal, Γexp ≈ 4 × 1013, thus (Γ/Γexp)h ≈ 2.5 × 10–14

[7, 13, 14].2. Inhomogeneous broadening is caused by station�

ary effects in an imperfect lattice, e.g., by impuritiesand lattice defects, and, in particular, by the randomdistributions of 3H and 3He on the tetrahedral intersti�tial sites in Nb metal. Thus the periodicity of the latticeis destroyed and the binding energies EB of 3H and 3Hein the Nb lattice will vary. Typical values for EB are inthe eV range [8]. In conventional Mössbauer spectros�copy with photons, using the best single crystals, suchvariations in EB cause energy shifts of ~ 10–12 eV.Since, in the nuclear transformations, the energy isdirectly affected by EB, one has to expect that the vari�ations of the energy are much larger than 10–12 eV,

probably in the 10–6 eV regime. Thus, inhomogeneousline broadening is estimated to give (Γ/Γexp)i � 10–12

[7, 13], probably (Γ/Γexp)i ≈ 10–18 [14].3. Another contribution to line broadening in an

imperfect lattice is caused by relativistic effects. Anatom vibrating around its equilibrium position in a lat�tice exhibits a mean�square velocity ⟨v2⟩. According toSpecial Relativity Theory this results in a time�dilata�

tion, t' = t/ i.e., a reduction in fre�

quency (energy E) of the E' = E ≈E(1 – ⟨v2⟩/(2c2)). Thus, ΔE ≈ –⟨v2⟩E/(2c2). Beingproportional to ⟨v2⟩/c2, this reduction is called sec�ond�order Doppler shift (SOD). Due to the zero�

νe,

f2 fr2fL

2 78–×10 ,≈=

νe,

νe

νe

νe

1 v2⟨ ⟩– /c2

,

νe: 1 v2⟨ ⟩– /c2

point motion (zero�point energy) the contribution ispresent even in the low�temperature limit for T → 0.The essential point is, that a variation of the bindingenergies EB of 3H and 3He in an imperfect lattice willresult in a variation of the effective Debye tempera�tures and thus also in a variation of the zero�pointenergies. If the effective Debye temperature varies byonly 1 K, (ΔE/E) ≈ 2 × 10–14, corresponding to a line�shift of 3 × 1014 times the natural width Γ. Thus, thisbroadening effect is expected to give (Γ/Γexp)SOD ≈ 3 ×10–15 comparable to or even smaller than (Γ/Γexp)h [7].The broadening effect (Γ/Γexp)SOD is important but lessserious than (Γ/Γexp)i, the latter being mainly due tovariations of EB as discussed above.

Thus, there exist very serious difficulties to observeMössbauer with the system 3H–3He in Nb metal:The probability of phonon�less emission and detec�tion might be very low, f 2 ≈ 7 × 10–8; homogeneousand, in particular, inhomogeneous line broadeningmay cause (Γ/Γexp)i, to be as small as (Γ/Γexp)i ≈ 10–18.This is mainly due to the fact that inhomogeneities inan imperfect lattice directly influence the energy of the

[14]. In addition to the basic principles we have justdiscussed also enormous technological difficulties canbe expected [15] and possible alternatives to the 3H–3He system should be searched for [14].

4. ALTERNATIVE SYSTEM: 163Ho–163Dy

The system 163Ho–163Dy has been considered asthe next�best case [3, 14]. Compared to 3H–3He in Nbmetal, there exist, three major advantages:

(1) The Q value of 2.6 keV (i.e., the energy) isvery low, the mass of the rare earth nuclei is large. As aconsequence, according to Eq. (3), the recoilfree frac�tion is expected to be fr ≈ 1.

(2) The rare earth series is characterized by differ�ent numbers of electrons in the 4f shell, the outer 6s,6p shells remaining to a large extent unchanged.Therefore, the chemical behaviour due to the bondingelectrons of outer shells is altered only very slightly.Thus also the deformation energies of 163Ho and 163Dycan be expected to be similar. For this reason, the proba�bility f of phononless transitions could be larger than forthe 3H–3He system by ~7 orders of magnitude.

(3) Due to the similar chemical behaviour, rare�earth systems can also be expected to be less sensitiveto variations of the binding energies. In addition,because of the large mass, the relativistic effects men�tioned in Section 3.2 will be smaller.

The main disadvantage will be the large magneticmoments due to the 4f electrons of the rare�earthatoms. Electronic magnetic moments are typicallythree orders of magnitude larger than nuclearmoments. Thus, line broadening effects due to mag�

νe

νe

νe

664

PHYSICS OF PARTICLES AND NUCLEI Vol. 42 No. 4 2011

WALTER POTZEL

netic relaxation phenomena will be decisive for a suc�cessful observation of Mössbauer However, con�ventional Mössbauer spectroscopy gathered a lot ofinformation on the behaviour of rare�earth systems inthe past. In particular, it has been shown that an exper�imental linewidth of Γexp ≈ 5 × 10–8 eV for the 25.65 keVMössbauer resonance in 161Dy can be reached (seeSection 5). Thus it is possible to test and improvepromising source and target materials by investigationsusing rare�earth Mössbauer γ transitions and � in thisway � compensate the lack of technological knowledgeconcerning the fabrication of high�purity materials,preferably single crystals, in kg quantities [3]. Cer�tainly, further feasibility studies have to be performed.

5. INTERESTING EXPERIMENTS

If the emission and absorption of Mössbauer could be observed successfully, several basic questionsand interesting experiments could be addressed [7]:

(1) The states of the neutrinos νe, νμ, and ντ arecharacterized by a coherent superposition of masseigenstates and can change their identity when theypropagate in space and time, a process called flavouroscillation [16–21]. Mössbauer are characterizedby a very narrow energy distribution. The question hasbeen asked: Do Mössbauer oscillate? Consideringthe evolution of the neutrino state in time only, accord�ing to the Schrödinger equation neutrino oscillationsare characterized as a non�stationary phenomenon,and Mössbauer would not oscillate because of theirextremely narrow energy distribution: As stated by thetime�energy uncertainty relation Δt ΔE ≥ ћ, the timeinterval Δt required for a system (consisting, e.g., of asuperposition of mass eigenstates) to change signifi�cantly is given by Δt ≥ ћ/ΔE. If the uncertainty inenergy ΔE is small, as in the case of Mössbauer Δtwill be large. An evolution of the neutrino state inspace and time, however, would make oscillations pos�sible in both the non�stationary and also in the station�ary (Mössbauer ) case. In this way, Mössbauer experiments could lead to a better understanding ofthe true nature of neutrino oscillations [18–20]. Wenow discuss this aspect in greater detail.

Considering the evolution of the state in time

only, Mössbauer oscillations with an oscillation

length determined by will not be observed if therelative energy uncertainty fulfills the relation [20]

(6)

νe.

νe,

νe

νe

νe

νe,

νe νe

νe

νe

Δm232

ΔEE

�������14��

Δm232 c4

E2��������������,

where = – ≈ 2.4 × 10–3 eV2 is the “large”atmospheric mass�squared difference.

For the 3H–3He system (E = 18.6 keV) this wouldmean (ΔE/E)H–He � 1.8 × 10–12 which � according toSection 3.2 � most probably cannot be reached due toinhomogeneous line broadening. However, for the163Ho–163Dy system, Eq. (6) requires (ΔE/E)Ho–Dy �9.2 × 10–11 or ΔE � 2.4 × 10–7 eV.

For the 25.65 keV γ�transition in 161Dy (lifetime τ =29.1 × 10–9 s, i.e., natural width Γ = ћ/τ ≈ 2.26 × 10–8 eV)an experimental linewidth of Γexp ≈ 5 × 10–8 eV hasbeen achieved [22], which is ~ 5 times below the limitΔE � 2.4 × 10–7 eV. Thus, the 163Ho–163Dy system willprobably make it possible to perform Mössbauer experiments, and it looks encouraging that the ques�tion if Mössbauer oscillate can be answered exper�

imentally. For Γexp ≈ 5 × 10–8 eV, a significant changeof the state in time can occur only very slowly lead�

ing to a long oscillation path Lchange, since the isultrarelativistic:

(7)

Thus, for the 163Ho–163Dy system, Lchange ≈ 25 m.

For an evolution of the state in space and time,however, the oscillation length is given by

(8)

where L0 is given in meters, when the neutrino energyE and the mass�squared difference |Δm2| are in units ofMeV and eV2, respectively. With E = 2.6 keV for the163Ho–163Dy system, and ≈ 2.4 × 10–3 eV2, weobtain L0 ≈ 2.6 m, considerably shorter than Lchange.Thus in such a Mössbauer�neutrino experiment withΓexp ≈ 5 × 10–8 eV and if the evolution occurs in timeonly, instead of L0 the much longer Lchange would beobserved.

(2) If Mössbauer�neutrmo oscillations do occur,either due to the evolution of the neutrino state inspace and time or if Γexp is so large that Eq. (6) is nolonger fulfilled, ultra�short base lines (see Eq. (8))could be used to determine oscillation parameters. Forexample, for the determination of the still unknownmixing angle Θ13, a base line of only ~ 1 m (instead of~ 1500 m as required for reactor neutrinos [23]) wouldbe sufficient. In addition, very small uncertainties for

Θ13 and accurate measurements of = –

and = – could become possible [24].

(3) At present, only upper limits for the neutrinomasses are known. In addition, the mass spectrum

Δm232 m3

2 m22

νe

νe

νe

νe

Lchange�c ћΓexp

�������2π.

νe

L0 2.48 E

Δm2����������=

Δm232

Δm122 m2

2 m12

Δm232 m3

2 m22

PHYSICS OF PARTICLES AND NUCLEI Vol. 42 No. 4 2011

MÖSSBAUER ANTINEUTRINOS: RECOILLESS RESONANT EMISSION AND ABSORPTION 665

(hierarchy), i.e., the question if the small mass split�ting belongs to the two lightest or to the two heaviestneutrinos, could not yet be determined. Mössbauer

of the 163Ho–163Dy system could settle the questionconcerning the mass spectrum using a base line of~40 m. As shown in [25, 26], in such an experiment,the superposition of two oscillations with different fre�quencies can be observed: a low�frequency (so�called

solar neutrino) oscillation driven by = – ≈7.6 × 10–5 eV2 and a high�frequency (so�called atmo�

spheric neutrino) oscillation driven by = –

≈ 2.4 × 10–3 eV2. In the case of the normal massspectrum (the small mass splitting belongs to the twolightest neutrinos) the phase of the atmospheric�neu�trino oscillation advances, whereas it is retarded forthe inverted mass spectrum (the small mass splittingbelongs to the two heaviest neutrinos), by a measur�able amount for every solar�neutrino oscillation.

(4) Oscillating Mössbauer could be used to

search for the conversion → νsterile [27] involvingadditional mass eigenstates. Sterile neutrinos (νsterile)would not show the weak interaction of the standardmodel of elementary particle interactions. Thereforesuch a conversion would manifest itself by the disap�pearance of The results of the LSND (LiquidScintillator Neutrino Detector) experiment [28] indi�cate a mass splitting of Δm2 ≈ 1 eV2 [6]. In this context,several experiments have been performed by the Mini�BooNE collaboration. However, the results are notconclusive [29]. For a value of Δm2 ≈ 1 eV2, the oscil�lation length for Mössbauer of 2.6 keV would beonly ~1 cm!

CONCLUSIONS

Several basic aspects of the 3H–3He system havebeen considered for a possible observation of Möss�bauer It will not be possible to reach the naturallinewidth because of homogeneous broadening due tostochastic relaxation processes and inhomogenousbroadening mainly due to the variations of bindingenergies and their direct influence on the energy of the

In addition, the probability for phononless emis�sion and detection will drastically be reduced due tolattice expansion and contraction after the transfor�mation of the nucleus. The observation of Mössbauer

of the system 3H–3He imbedded in Nb metal, willmost probably not be possible. The rare�earth system163Ho–163Dy offers several advantages, in particular alarge probability of phononless emission and detec�tion. However, magnetic relaxation processes andtechnological requirements still have to be investigatedin detail to decide if the 163Ho–163Dy system is a suc�

νe

Δm122 m2

2 m12

Δm232 m3

2

m22

νe

νe

νe.

νe

νe.

νe.

νe

cessful alternative. In conventional Mössbauer exper�iments with the 25.65 keV γ transition in 161Dy anexperimental linewidth of Γexp ≈ 5 × 10–8 eV wasachieved. If such a value could also be reached withMössbauer of the 163Ho–163Dy system, highlyinteresting experiments concerning basic questions inneutrino physics could be performed.

ACKNOWLEDGMENTS

It is a great pleasure to thank S. Bilenky, Joint Insti�tute for Nuclear Research, Dubna, Russia, for numer�ous fruitful discussions. This work was supported byfunds of the Deutsche Forschungsgemeinschaft DFG(Transregio 27: Neutrinos and Beyond), the MunichCluster of Excellence (Origin and Structure of theUniverse), and the Maier�Leibnitz�Laboratorium(Garching).

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