Lowest-order relativistic interaction between lattice vibrations and internal degrees of freedom of a nucleus

A moving nucleus experiences a Lorentz contraction and spin rearrangement due to relativity. A nucleus that oscillates in a molecule or solid due to vibrations will undergo minor relativistic modifications which are a result of the vibrations, indicative of a relativistic phonon-nuclear interaction. The derivation of the lowest-order interaction from the many-particle Dirac model is reviewed. The Dirac model with a realistic potential model is not covariant, which is a source of concern. The lowest-order phonon-nuclear interaction obtained from a covariant two-body Bethe–Salpeter model is found to be similar to the interaction obtained from the Dirac model, supporting the notion that the interaction is not an artifact. Matrix elements of the lowest-order interaction are expressed in terms of one-body operators, which facilitates evaluation and allows for quantitative estimates of the magnitude.


Introduction
There has long been interest in physical effects that result from interactions between nuclei and their atomic, molecular and solid state surroundings. An early example is the Szillard-Chalmers effect (used for isotope separation), in which neutron capture on an isotope which subsequently decays can be separated from the molecule (or solid) by the recoil resulting from the nuclear decay [1]. Another early example is the Migdal effect, in which an atomic electron is ionized following the sudden perturbation of the nucleus as occurs in alpha and beta decay [2,3] (of current interest in dark matter searches in which nuclear recoil from scattering with a weakly interacting particle would result in electron ionization [4]). Recoiless gamma emission and absorption in solids [5] has proven to Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. useful in many areas [6]. The gamma line from a nucleus in a molecule can have satellite structure due to phonon exchange [7]. Nuclear excitation induced by an electronic transition has been observed [8]. There have been many works in which relativistic and cooperative effects involving nuclei in atoms and molecules have been studied [9,10]. Our interest in this paper concerns a relativistic effect which appears to have overlooked in the past, in which a nucleus that oscillates in a molecule or solid experiences an interaction between the center of mass motion and internal nuclear degrees of freedom.
A clean separation of the center of mass and relative degrees of freedom can be done for nonrelativistic many-particle nuclear models [11][12][13], but this clean separation does not extend in general to the relativistic case. For most applications the lack of a clean separation is unimportant. Relativistic structure calculations are done in the rest frame, and in a covariant model a Lorentz boost can be used in principle for the construction of a wave function in a moving frame [14]. To an observer in that moving frame the nucleus looks the same as it does in the rest frame.
This motivates the question whether there are observable consequences resulting from this lack of a clean separation in the relativistic problem.
Suppose that the center of mass of the nucleus oscillates as a consequence of being part of a molecule or solid that vibrates. Even though the nuclear velocity is small compared to the speed of light, when the nucleus moves there will be a finite Lorentz boost of its wave function relative to the ground state appropriate for whatever velocity and direction it has at the time. It would be possible to describe the Lorentz boosted wave function in each case in terms of a superposition of its rest frame basis states. When the velocity and direction changes, the Lorentz boost is changed, which means that the corresponding superposition of rest frame basis states will also change. These dynamics are consistent with a coupling of the center of mass velocity (and momentum) and the internal nuclear degrees of freedom. Because of this, a relativistic interaction is expected that couples vibrational degrees of freedom with internal nuclear degrees of freedom.
A derivation of low-order contributions to relativistic phonon-nuclear interaction is possible starting from a manyparticle Dirac model, as reviewed below. One issue associated with this approach is that even though a Dirac phenomenology has been widely used [15][16][17][18][19][20][21], nucleons are not Dirac particles [22]. Worrisome is that the many-particle Dirac model is not covariant when realistic nuclear potentials are used [23], so there is room for concern that the interaction derived might be an artifact. This provides motivation for developing a version of the low-order interaction from a covariant two-body Bethe-Salpeter model, which is also discussed below. The resulting lowest-order interaction is close to what is obtained from the Dirac model, which supports the notion that the interaction is not an artifact.
For the interaction to have observable consequences, it must be in experiments where phonon exchange is important. The lowest-order relativistic phonon-nuclear interaction has E1/M2 multipolarity, which means that it does not contribute to transitions between degenerate nuclear states. At secondorder this relativistic phonon-nuclear coupling can contribute (weakly) with higher-order M1/E2 multipolarity. But this interaction involves two-phonon exchange, which could contribute in the presence of strong THz vibrations, conditions under which there is competition with other stronger multiphonon processes.
The lowest transition energies between ground states and excited nuclear states of stable nuclei are in the keV range; hence, it is not possible to see an excited state decay through the emission of a few phonons since the highest (lattice) phonon energies are less than 1 eV. Because of this, the lowestorder observable processes involving the relativistic phononnuclear coupling involve excitation transfer [24,25]. In resonant excitation transfer, an excited nucleus at one site makes a transition to the ground state, exchanging a phonon in the process; and a ground state nucleus at another site makes a transition to an excited state at the same energy, exchanging a phonon with the same phonon mode.
We have carried out experiments in which radioactive 57 Co beta decays to produce excited 57 Fe, which is stimulated mechanically resulting in excitation transfer of the nuclear excitation. Resonant excitation transfer produces phase coherence in nearby nuclei, which in the presence of an ordered lattice leads to angular anisotropy in the emitted gamma radiation. We have evidence that this effect occurs on the 122 keV and 136 keV lines in our samples [26][27][28]. Magnetic dipole and electric quadrupole interactions are responsible for excitation transfer in these experiments, and not the relativistic phononnuclear interaction under discussion. Since the strength of the interaction of the relativistic phonon-nuclear interaction can be comparable to the magnetic interaction in these experiments, it should be possible to observe the effect in new excitation transfer experiments involving M2 transitions (which are longer lived than E1 transitions, and which do not respond to local electric fields).
In what follows, the lowest-order relativistic phononnuclear interaction is derived from a many-particle Dirac model in section 2, and in section 3 a similar interaction is derived from a two-body Bethe-Salpeter model. The interaction Hamiltonian is somewhat complicated since two-body operators involving the nuclear potential appear. Matrix elements of the interaction can be expressed in terms of much simpler one-body operators by taking advantage of an identity, which is discussed in section 4. Conclusions appear in section 5. A series of Appendices deal with technical issues, including: difficulties with the Dirac model (appendix A); relative and center of mass operators (appendix B); the absence of the interaction in a nonrelativistic model (appendix C); roles of acceleration and phonon exchange (appendix D); a unitary transformation that eliminates the lowest-order interaction for some potential models (appendix E); and estimates for the magnitude of the interaction (appendix F).

Relativistic phonon-nuclear coupling from a Dirac model
Center of mass and relative degrees of freedom are mixed together at the outset in a many-particle Dirac model due to the presence of α and β matrices. If such a model were covariant, then an acceptable description could in principle be obtained making use of a suitable unitary transformation which separates the different degrees of freedom, and in the process isolating terms that couple them. In what follows, we pursue such an approach. However, since the many-particle Dirac model is not covariant, there is a concern that the coupling term that results might be artifactual.
An important issue concerns what kind of description to use for the center of mass degrees of freedom. Nuclei in a lattice or molecule move slowly, so that a nonrelativistic description is appropriate. A clean separation of center of mass and relative degrees of freedom is problematic in a Dirac formalism in general due to the intermixing, but with a nonrelativistic center of mass wave function the separation is straightforward. The unitary transformation used below leaves the relative problem as a many-particle Dirac model in the rest frame, with nuclear motion accounted for by coupling between the different degrees of freedom. Because the Dirac model is not covariant, there is no motivation to develop a systematic framework in which higher-order interactions can be developed systematically. The focus here is only the lowest-order term that couples the different degrees of freedom, largely since this lowest-order interaction has not received much attention in the literature (other than to require that it must be able to be eliminated with a rotation for uniform motion in free space). To test whether this interaction is observable, experiments would be expected to focus on this lowest-order (and hence strongest) interaction.

Many-particle Dirac model
To be specific, a many-nucleon Dirac model is considered for a nucleus is that interacts with other atoms in a molecule or solid according tô where j and k refer to the N nucleons in the nucleus under study, and whereĤ other describes other particles the nucleus interacts with. For simplicity the discussion is restricted to two-body interactions, and there is no difficulty extending the rotation that follows to three-body interactions. Technical issues specific to the many-particle Dirac model are discussed in appendix A.

Center of mass position and momentum
The center of mass position, momentum and mass are taken to be in anticipation that the center of mass problem for a slowmoving nucleus will be nonrelativistic. The center of mass position and momentum will be oscillatory in a molecule or solid, so here they are written as operators which are considered to operate on the relevant vibrational degrees of freedom.
For simplicity the relative positions and momenta are used in terms of normal coordinates as discussed in appendix B, allowing the model Hamiltonian to be written aŝ where the ξ j andπ j are the relative positions and momenta.

Partial Foldy-Wouthuysen transformation
A separation of the nonrelativistic center of mass degrees of freedom and relativistic internal degrees of freedom can be done making use of a modification of the Foldy-Wouthuysen transformation [29][30][31]

Transformed Hamiltonian
The transformed model that results can be written in the formĤ where the scalar center of mass Hamiltonian iŝ The relative Hamiltonian for the internal degrees of freedom isĤ which can be thought of as a generalized Fermi-Kemmer-Yang Hamiltonian [33,34]. This kind of model has been considered for nuclear structure calculations [35][36][37].
Coupling between the center of mass and relative degrees of freedom in the transformed version of the problem is [32] Since the center of mass momentum operatorP in this context is a phonon operator involving quantized vibrational modes of the molecule or solid, the part of the interaction involving the potential is recognized as the lowest-order contribution to the relativistic phonon-nuclear interaction. TheP · β jπj terms sum together to give a very small contribution since and should be regarded as a higher-order contribution; however, in what follows there is a simplification if the term is kept. Other higher-order terms resulting from the transformation are denoted as · · · in equation (6).

An interpretation
Intuition as to what the partial Foldy-Wouthuysen transformation does in this context is useful. Suppose that Φ[0] is the ground state of the nucleus in the rest frame, then to lowest order the ground state in the moving frame in the Dirac model is approximately It is seen that moving wave function is modified, which is how the Dirac model treats changes due to the Lorentz boost in terms of rest frame basis states. A comparable expression for the transformed wave function is which is the same at lowest order (since in a nonrelativistic reduction β differs from the identity by terms of order |π| 2 /(2mc) 2 ). This suggests an interpretation in which the partial Foldy-Wouthuysen transformation modifies the ground state to include excited states consistent (at low order) with how (Lorentz-boosted) moving states are described in the manyparticle Dirac model.

Relativistic coupling from a Bethe-Salpeter model
A concern is that the lowest-order phonon-nuclear interaction derived above might be an artifact, since the many-particle Dirac model is not covariant. This provides motivation to consider the separation of center of mass and relative degrees of freedom in a covariant Bethe-Salpeter model. Once again the different degrees of freedom are mixed, and making use of a scalar description for the center of mass degrees of freedom is helpful. The same partial Foldy-Wouthuysen transformation as used above achieves a separation of the center of mass and relative degrees of freedom at low order. The resulting coupling is very similar to what was derived from the Dirac model above, which argues against the interaction being an artifact.

Model
It is helpful in this context to start with a Bethe-Salpeter model using a notation similar to that used for the Dirac model above [38] ( As was the case above, interest is focused on problems in which a nucleus oscillates as a result of vibrations of the molecule or solid. An external field could be added as a step in this direction, but from the discussion above it is clear that what matters is not an explicit inclusion of other bodies and their interactions in the model, but instead what interactions appear when the center of mass and relative degrees of freedom are separated. The identification of the center of mass momentum as an operator of the relevant vibrational degrees of freedom here is sufficient.

Center of mass position and momentum
The center of mass position, momentum and mass are taken to be where, as above, the center of mass dynamics are considered to be nonrelativistic. The center of mass positionR and momentumP operators act on vibrational degrees of freedom.
The two-body Bethe-Salpeter equation can be written as To reduce the complexity of the notation, the dependence of Ψ on the vibrational degrees of freedom is suppressed.

Partial Foldy-Wouthuysen transformation
A similar partial Foldy-Wouthuysen transformation as was used on the many-particle Dirac model above, can be applied here This transformation provides for a separation of the center of mass and relative degrees of freedom for the Bethe-Salpeter model here much the same as above.
The rotated version of the Bethe-Salpeter model is where the transformation of the mass and kinetic energy terms is exact, and the transformation of the potential operator is given explicitly only to lowest order.

Discussion
The partial Foldy-Wouthuysen transformation in this case leads to a model in which the center of mass degrees of freedom are nonrelativistic, which can be seen by rewriting as The energies E 1 and E 2 at low order each pick up scalar contributions corresponding to half of the center of mass energy (so the total energy now includes the nonrelativistic center of mass energy). The center of mass part of the problem is scalar, as was the case in the many-particle Dirac model. TheP ·π j terms from above that appear in the phononnuclear interaction of equation (9) above have counterparts here The potential terms that appear in the phonon-nuclear interaction above also have a counterpart here The conclusion from this discussion is that the low-order phonon-nuclear interaction found using a Dirac model have close analogs in the Bethe-Salpeter model, which supports the notion that this interaction is not an artifact.

Matrix element in terms of one-body operators
The lowest-order phonon-nuclear matrix element from the Dirac model between two rest frame nuclear states is made up of a contribution involving one-body operators, and a contribution involving two-body operators, according to The evaluation of the two-body matrix elements involves a significant computational effort, which motivates the question as to whether it is possible to replace it with simpler one-body matrix elements.

Identity
A useful identity that accomplishes this is Upon setting the operatorQ in this identity tô and takingĤ to be the rest frame Hamiltonian appropriate for the rest frame basis states of the interaction matrix element, the operator identity becomes

Elimination of two-body operators
This identity can be used to develop an expression for the interaction matrix element in terms of one-body operators according to The lowest-order contribution from a nonrelativistic reduction can be written as where Φ +++··· (j) is the positive mass energy sector component of Φ j . This greatly reduces the computational effort required for the evaluation of the lowest-order phonon-nuclear interaction.

Discussion
Changes in the internal degrees of freedom are produced when the nucleus moves as a consequence of relativistic invariance, resulting in a coupling between relative and center of mass degrees of freedom for a nucleus that oscillates in a vibrating molecule or solid. The derivation of the lowest-order interaction from a many-particle Dirac model is reviewed, leading to the interaction of equation (9). A similar result is obtained from a two-body Bethe-Salpeter model in equation (18), which suggests that the coupling from the Dirac model is not an artifact due to the lack of covariance. The evaluation of matrix elements of the interaction Hamiltonian is simplified by the elimination of two-body terms, as given in equation (26). Johnson took time to review an earlier version of the paper (which we very much appreciate), and raised a number of issues [39], which are discussed in appendices C-F. One issue is whether the interaction is a result of acceleration alone, in which case there would be expected a nonrelativistic version of the interaction. In appendix C, a comparable nonrelativistic model is considered in which the same acceleration is present, but no coupling between the center of mass and relative degrees of freedom is found. However, the inclusion of the lowest-order relativistic correction to the kinetic energy results in interaction which is of higher order than the one considered in the text. A unitary transformation us used in appendix E (equation (E.8)) that results in a version of the interaction written in terms of the acceleration.
It was suggested that acceleration is a requirement for internal nuclear transitions to occur, but that the model under consideration does not describe this. In appendix D, a model is discussed in which the center of mass momentum operator is written in terms of creation and annihilation operators, so that a transition is mediated by phonon exchange (which is more important in applications we are interested in than acceleration). Since phonon exchange comes with a change in vibrational energy, the interaction is consistent with a picture in which acceleration occurs.
The possibility of a low-order interaction between the center of mass and relative degrees of freedom was disputed, motivating the discussion in appendix E in which the associated issues are considered. For potentials that commute with the Dirac β operators the lowest order interaction for a nucleus moving at constant velocity can be rotated out. A similar unitary transformation under conditions where the nucleus is oscillating leads to a rotated version of the interaction which is proportional to the center of mass acceleration. This provides additional clarification that the effect under discussion is not a consequence of acceleration alone, but depends in addition on commutations involving the potential.
Whether the interaction is observable or not depends on the magnitude of the interaction. In appendix F, a rough estimate for the magnitude of the interaction is developed for E1 transitions, and numerical values tabulated for low-energy E1 transitions involving the ground state. The interaction strength for these transitions is comparable to the interaction strength for M1 coupling for the low-lying excited states of 57 Fe currently under study in our lab in excitation transfer experiments. A comparable interaction strength would be expected for the interaction in the case of M2 transitions, which make better candidates for excitation transfer experiments intended to demonstration the relativistic phonon-nuclear interaction since interference from competing electric dipole or magnetic dipole interactions are avoided.
This work has focused on the relativistic phonon-nuclear interaction at lowest order, and issues connected to this interaction. It remains for research in the future to make use of this interaction for specific examples (different nuclei, different crystals and crystal structures, different optical or acoustic vibrational models, and different applications such as possible line broadening at elevated temperature and excited state nuclear excitation transfer).
Note that the coupling between the center of mass degrees of freedom and relative degrees in this work is a consequence of use of a Dirac type of formalism. In a molecule or solid in which the concept of moving composite is generalized from the nucleus to include electrons which move with it, then it should be possible for there to be a phonon-electron version of the interaction.

Data availability statement
No new data were created or analysed in this study.

Appendix A. Issues connected with the Dirac model
The many-particle Dirac model suffers from known problems, with additional issues when used for nucleons. In this appendix a brief review of some of the most important issues is given.

A.1. Brown-Ravenhall disease
It is recognized that many-particle models suffer from Brown-Ravenhall disease [40], in which positive mass energy states are coupled to negative mass energy states. Consequently, all positive energy states are unstable in this model. To address this projection operators are used as comes about naturally in a derivation from field theory [41,42]. Alternatively, trial wave functions can be built up from combinations of individual positive energy particle orbitals. In practice, many-particle Dirac models have been and continue to be widely used, and are found to work well for many problems [43].

A.2. Lack of Poincaré invariance
Historically the many-particle Dirac model has not been used to describe a composite in motion, in part because the model is not Poincaré invariant (for physically realistic potentials) [44][45][46], and in part due to the development of much more powerful covariant quantum field theories (such as QED, QCD). A conclusion of this paper is that the lack of Poicaré invariance does not prevent the Dirac model from being used to obtain the lowest-order phonon-nuclear interaction.

A.3. Nucleons as Dirac particles
Multi-particle Dirac models for nucleons have been used to model nuclei [19,47], even though neither neutrons or protons are Dirac particles [22]. Recognizing this, the approach in this work is to focus on the basic Dirac model taking advantage of its simplicity, leaving for discussions elsewhere augmentations [48][49][50].

Appendix B. Relative and center of mass operators
For a separation of the center of mass and relative degrees of freedom, a specification of the center of mass positions and momenta are required.

B.1. Linear transformation
A linear transformation between particle coordinates determines the transformation between momenta [51], since the relative and center of mass momentum operators satisfŷ The specification of the relative momenta and coordinates is not unique. One example iŝ where M is the total mass The corresponding center of mass and relative coordinates are This is similar to the center of mass and relative momenta and coordinates used in [52].

B.2. Auxiliary position and momentum variables
The composite is assumed to contain N particles, so that there are N − 1 relative coordinates and one center of mass coordinate, and similarly N − 1 relative momenta and one center of mass momentum. It is nevertheless convenient to define a relative coordinate and momentum for particle N which satisfies

B.3. Hamiltonian in terms of relative and center of mass variables
Consider a many-particle Dirac model for a relativistic composite of the form Written in terms of the center of mass coordinates and momenta this becomeŝ (B.8) The use of consistent position and momentum operators was not made clear in previous work [32].

Appendix C. Absence of coupling in a nonrelativistic model
No nuclear dynamics are expected in connection for a nucleus at constant velocity, which suggests that acceleration is important. Because of this it has been argued that transitions should be expected in response to acceleration, in which case transitions should occur in a nonrelativistic model [39].
To investigate, a nonrelativistic many-particle model is considered given bŷ Making use of center of mass and relative coordinates leads tô where the cross terms involvingP ·π j cancel. No transitions occur in the nonrelativistic model in response to acceleration due to the clean separation between the center of mass and relative degrees of freedom. In a scalar model that includes the lowest-order relativistic correction to the kinetic energŷ the lowest-order phonon-nuclear interaction iŝ This interaction is of higher order than what is considered in the text.

Appendix D. Phonon exchange and acceleration
It was argued that acceleration is needed for transitions to occur, but this is not part of the model under consideration in the text [39].
To pursue this issue, the center of mass momentum operator is considered in the representative case of a nucleus that is part of a monatomic lattice. For a nucleus at site l the momentum operator can be written aŝ l . Theâ † k,s andâ k,s operators are creation and annihilation operators that act on the vibrational degrees of freedom of the lattice wave function.
In a classical version of the model, the nucleus at site l would either be at rest, or oscillating. At rest there is no acceleration, but also no velocity or momentum. If the nucleus is not at rest, then it accelerates. In the quantum version of the model the situation is somewhat more complicated, and the notion of phonon exchange is more important that the notion of acceleration. This can be underscored by writing the interaction Hamiltonian in the form where the interaction strengths are and where a and b sum over the different rest frame (nonrelativistic) basis states of the nucleus. For a (lowest-order) transition to occur between rest frame nuclear state Φ a and Φ b , there needs to be the creation or annihilation of a phonon in one of the modes. A situation can be imagined in which the velocity of the nucleus is constant temporarily. In this special case the lattice kinetic energy will be changed if a phonon is created or annihilated, which could be considered to occur within the context of a generalized acceleration.

Appendix E. Coupling at lowest order
The existence of an interaction at lowest order has been disputed [39]: 'It is relevant in this regard that a coupling appears at lowest order, but it must get canceled out at higher order; otherwise there would be a coupling between the rest frame and the internal dynamics.' To pursue this, a unitary transformation is considered which rotates out the interaction in the case of constant velocity in free space.

E.1. Constant velocity model
A model for this can be specified based on with the center of mass momentum P now taken to be a constant. An isolated nucleus with (constant) velocity much less than c is described by the rotated model of equation (6) adapted to this situation with the lowest-order contribution to the interaction

E.2. Unitary transformation
The low-order interaction in this case is removed with a second unitary transformation for potentials that satisfy Although not general, this condition is satisfied in the case of the lowest-order relativistic one-pion exchange potential based on a pseudovector pion-nucleon interaction [53]. The elimination of the lowest-order interaction through a unitary transformation is consistent with the interaction introducing no transitions at lowest-order in a moving frame with constant velocity.

E.3. Acceleration in a molecule or solid
A generalization of the unitary transformation above is possible in the case where the nucleus is oscillating in a molecule or solid, where it similarly the lowest-order interaction. Since the unitary transformation involves the center of mass momentum, it does not commute with the local (environmental) potential, which is a function of the center of mass position. This results in a low-order interaction in the twicerotated frame of the form where U(R) is the (environmental) potential seen by the nucleus (which is also a function of other nuclear coordinates, where this dependence is suppressed to simplify the notation). In the case of a scalar potential model for U(R) this results in This twice rotated interaction is proportional to the acceleration − 1 M (∇ R U); however, there are other terms involved as well. The conclusion is that acceleration is a prerequisite for nuclear transitions to occur, but that acceleration alone is not sufficient (as discussed in appendix C for a nonrelativistic model).

E.4. Interaction based on momentum or acceleration?
It might be asked whether it would be better to work with the lowest-order interaction in the simpler form proportional toP, or to make use of this somewhat more complicated version in which the acceleration appears explicitly. A related problem concerns a charged particle interacting with the radiation field. In this case there is no radiation when the charged particle propagates in free space at constant velocity. The interaction in the Coulomb gauge is which is proportional to the momentum, and does not involve (at least explicitly) the acceleration. It is possible to rotate out the lowest order interaction in free space. Similarly, it is also possible to rotate it out in the presence of a potential leaving a dressed interaction in the rotated problem in terms of acceleration.
In both cases, the calculations are simpler making use of an interaction proportional to the momentum.

Appendix F. Magnitude of the interaction
Rough estimates for the magnitude of the lowest-order phonon-nuclear interaction are useful, in part to understand how it compares with other interactions, and in part to address the issue of observability. Conceptually straightforward might be the evaluation of the associated matrix element based on simple harmonic oscillator wave functions, or wave function derived from a deformed potential model. This approach has not been particularly successful for the (closely related) evaluation of radiative decay rates for low-energy E1 decays to the ground state, which are potential candidate transitions for studying phonon-nuclear coupling. Better in connection with the discussion here would be the development of estimates connected to the (observed) radiative decay rate, which is pursued below.
We recall that E1 transitions in nuclei are electric dipole transitions, analogous to the 2p − 1s decay in atomic hydrogen. One unit of angular momentum is exchanged, which necessitates a change of parity between the initial and final nuclear states. Similarly, M1 transitions in nuclei are magnetic dipole transitions (with no parity change), analogous to a simple electronic spin flip transition in an atom induced by a magnetic field. An E2 nuclear transition is a quadrupole transition, which would be like a 3d − 1s decay in atomic hydrogen. Excitation of an M2 transition in a nucleus could in principle occur as a result of the magnetic field under conditions where the wavelength is short so that the spatial dependence of the field leads to angular momentum exchange along with a spin flip (in a nonrelativistic picture where spin-orbit interactions are ignored). In nuclei electric dipole transitions tend to be fastest, and magnetic dipole transitions much slower. E2 and M2 transitions are slower still, with electric transitions being faster than magnetic transition. Higher-order M3 and E3 transitions are known which are even slower. Unlike electronic transitions in atoms, nuclear transitions are often observed to have mixed mulipolarity (such as E1+M2).
In equation (27) was given a reduction of the phononnuclear interaction matrix element in terms of nonrelativistic nuclear wave functions according to The σ × π interaction has E1 and M2 multipolarity, including terms which do not alter the nucleon spin. This suggests that a rough estimate for E1 transitions could be obtained making use of q * π, where q * is the effective charge. This results in the estimate The radiative decay rate for an E1 transition in the dipole approximation is This can be used to obtain In an excitation transfer experiment, vibrations would be stimulated to maximize phonon exchange. Since the coupling is weak, vibrations could be driven at large amplitude subject to the constraint of not melting or damaging the sample. To to produce the estimates in table 1. Data for decay rates and internal conversion coefficients needed to estimate the radiative decay rates were from the BNL NUDAT2 website. For a particular transition, the interaction matrix element is of the form where a 12 is a dimensionless vector. Estimates for the magnitude |a| are included in table 1.
The magnitude of the phonon-nuclear matrix element estimated this way is on the order of a few hundred neV, which seems small. However, this is comparable to the magnitude of the phonon-nuclear coupling matrix elements associated with M1 interactions in models for excitation transfer experiments with the 14.4 keV transition in 57 Fe. This is encouraging, suggesting that excitation transfer mediated by this new interaction may be observable.
The focus here has been on E1 transitions for which the rough estimate is likely to be best. But an unambiguous demonstration of the interaction for E1 transitions is complicated due to interactions with the local oscillating electric field. Consequently, low-energy M2 transitions would be preferred in an experiment, where the magnitude of the phonon-nuclear interaction would be expected to reach a few hundred neV.