This section will briefly introduce the history of nuclear resonance vibrational spectroscopy (NRVS) with respect to Mössbauer spectroscopy. Then the mathematical framework for the technique is established  followed by practical aspects with an emphasis on data analysis specifically at the SPring8 synchrotron.
Mössbauer Spectroscopy relies on disconnecting the vibrational motions of the Mössbauer nucleus: recoilfree absorption and emission.^{13} The logical evolution of this idea was to realize and differentiate phonons that are coupled to the nuclear excitation.^{4,5} However, the resonances in Mössbauer spectroscopy are typically on the order of neV and vibrational quanta are on the order of meV. This implies a Doppler shifted source would need to move with a velocity 6 orders of magnitude faster  which is widely considered impractical.
To observe vibrational modes, a source with high flux, high monochromation, and high tuneability is necessary. Fortunately, third generation synchrotron xray light sources meet these requirements.^{6} In the mid 1990's the first synchrotronbased vibrational Mössbauer spectra were collected concomitantly at the advanced photon source (APS) and SPring8^{7,8} and, the aptly named, nuclear resonance vibrational spectroscopy (NRVS) was born.
NRVS is, in many ways, the antithesis of Mössbauer spectroscopy. If we start with a freely recoiling atom the energy of the observed feature would be offset from the resonance by the recoil energy:
For ^{57}Fe at a 14.4125 keV incident photon, the recoil energy is 1.956 meV. Lipkin has shown that the first moment of the vibrational excitation spectrum is the recoil energy.^{9}
Where is the excitation function and is the recoil energy. This relationship gives a useful method for normalizing actual spectra. It follows that the individual vibrational features can be modeled within a harmonic approximation^{10} and the following section of mathematical treatment comes from Sturhahn^{11} as a modification of the previously established Green function formalism in a Debye framework.^{12}
It is critical to establish the nuclear absorption cross section for the elastic process (Mössbauer) and the inelastic process (NRVS) as both contribute to the excitation spectrum. First we start with Fermi’s golden rule  the transition rate is as follows:
Where and represent the intermediate state and initial states respectively and both are metastates such that =. The Diracdelta function ensures that , the monochromatic xray energy, is equal to , the difference in energy between states. The interaction Hamiltonian, , relies on the nuclear current operator and transverse vector potential of the xray field .
Treating the monochromatic xray field as a plane wave with wavevector the transverse vector potential becomes:
Then it follows that the equation 5.4 can be rewritten as:
Where is the Fourier image of the nuclear current operator in the centerofmass frame. The expectation value for the interaction Hamiltonian is then:
Finally, we obtain the energydependent absorption cross section, by inserting 5.7 into 5.3.
With the first term being the normalized excitation probability which is related to Van Hove’s atomic selfcorrelation function by:^{13}
Then integration of 5.8 in all space yields:
where we have introduced the familiar notation of , the nuclear level energy width, and the nuclear resonant cross section.^{14} is again the excitation probability, and for the elastic peak is estimated as the LambMössbauer factor . Here we assume a Debye model, and the inelastic portion extends through the Debye energy, . Then for the inelastic energydependent cross section we find.
Sturhahn showed this model, when values for metallic ^{57}Fe are substituted in, yields a nuclear inelastic cross section six orders of magnitude smaller than the elastic nuclear resonance and three orders of magnitude smaller than the photoelectric cross section.^{11} Thus arises the problem of sorting out the much smaller nuclear inelastic signal (NRVS) from the unrelated electronic scattering events.
Fortunately, the natural lifetime of the nuclear excited state (^{57}Fe: 141 ns) is a much longer time scale than the femtosecond time scale electronic scattering events. Therefore, we can discriminate between NRVS related scattering and electronic scattering by the delay between the incident excitation pulse and measured scattering in the time domain.
The ^{57}Fe NRVS related scattering has three primary components: the nuclear relaxation (14.4 keV), the electron ejected during internal conversion, and the k_{α} emission from corehole filling following internal conversion (6.4 keV). The conversion ratio for ^{57}Fe is 8.56 and therefore, in theory, only 10.5% of the released photons after excitation are at 14.4 keV. In practice, conversion electrons themselves are typically not measured due to low detection efficiencies and high vacuum requirements.
Another implication of equations 5.10 and 5.11 is that there will be differences in penetration depth for the elastic peak compared to the inelastic region by the incident beam. This complicates determination of the spectral normalization factor () as now there is a penetration suppression factor such that the measured intensity is:
Here, we take advantage of Lipkin’s sum rule^{9} and recognize that the first moment of the spectrum depends only on the recoil energy of the nuclear isotope. Reintroducing equation 5.2 with the normalization term we obtain:
Using equation 5.12 and 5.13 together we can calculate the penetration suppression factor:
Next, we can further interpret the spectrum by approximating the interatomic potential V as quadratic with respect to interatomic displacements (the harmonic approximation). The total excitation probability becomes:
Where is the excitation probability of a phonon of order. The single phonon contribution to the probability is:
Where is the partial vibrational density of states (PVDOS) and is the thermodynamic beta. Here it is critical to note that from equation 5.16 the excitation probability for a single phonon event is limited by , this implies high energy single phonon features will be difficult to observe. The general excitation probability is then described with the recursive equation for :
Integration of equation 5.17 for all energy space affords the total excitation probability as a function of phonon order:
Here we can see dominance in the first order contribution to the probability, and also that for very low values of LambMössbauer factor, , we see a rise in the higher order phonon contribution. We can simplify the convolution in 5.17 to multiplication through Fouriertransformation.
Then the Fourierimage of the total energydependent excitation probability follows as:
Now we can extract the single phonon excitation simply by solving for .
Where is the inverse Fouriertransform operator. This form of the function is more amenable as it is dependent on the excitation spectrum and a physical characteristic, . Now we can relate equation 5.21 to 5.16 to solve for the PVDOS using accessible properties/observables , , and .
For a single degree of probe atom translational freedom, distributed through the lattice vibrational modes, is normalized to unity. From equation 5.9 we see that the excitation probability depends on the dot product of the wavevector and the atomic motion vector . However, if the sample is assumed isotropic in a form that allows sampling of all three degrees of freedom then the total isotropic PVDOS normalizes to three. Also from equation 5.9 we see the selection rule for NRVS  the dot product implies the probe atom motion must overlap with the incident photon wavevector. The term accounts for the vibrational features on the positive side of the resonance (phonon creation) and the negative side of the elastic resonance (phonon annihilation).
Figure 5.1 A schematic diagram for ^{57}Fe NRVS transitions  which can involve either the formation or annihilation of phonons. The temperature dependent ratio of creation to annihilation events is determined by the nuclear ground state populations (detailed balance).
The two values are related by the Boltzmann relationship (referred to as the detailed balance). Using equation 5.23 we can fit the sample temperature based on the ratio of and .
For a strict harmonic approximation, we can also describe the density of states by converting the approximately independent vibrational modes from normal mode coordinates to Cartesian coordinates.
Where are the transformation coefficients called normal mode composition factors, and are the Cartesian coordinate vectors atomic and atomic mass; for the probe atom and normal mode . In this framework we can estimate the PVDOS intensity using the relationship:
Where is the line shape function wrapping the energy terms from 5.22 and is the projection of the normal mode composition factors onto the wavevector . Again, if we assume the system is isotropic and all three degrees of freedom can be probed by the incident photon, then we can use the expectation value:
The transformation into normal mode composition factors allows for a straightforward prescription to estimate the PVDOS intensity through normal mode analysis using density functional theory (DFT); the normal mode composition factors can be extracted from diagonalization of the Hessian force matrix used in normal mode analysis.^{15}
This section will focus on the practice of NRVS with ^{57}Fe at beamline BL9XU at the SPring8 synchrotron  other variations can be abstracted from this description. The synchrotron storage ring must operate with a bunch pattern that is amenable to the lifetime of the isotope of interest (e.g. the storage ring must have a bunch spacing close to the nuclear lifetime: which for ^{57}Fe is 141ns). The photon beam generated from fermion undulation is then monochromated by a high heat load monochromator (HHLM) to roughly 1eV. This is followed by a tuneable high resolution monochromator (HRM) to about 1 meV then an ionization chamber (for normalization) before encountering the sample (Figure 5.2).


Figure 5.2 Top: Schematic setup for NRVS at BL9 at SPring8  from^{6}. Bottom: The time structure scheme for NRVS detection at SPring8 operating in the ‘C’ bunch mode, with 70ps pulses spaced 153ns apart, with the detector measuring from 20ns to 153ns.
Scattering is detected by a fourelement avalanche photodiode (APD) array. Each of the elements sums into a timetoamplitude converter (TAC). Discrimination between the unrelated electronic scattering from the NRVS scattering by delaying the detection acceptance window to a time starting at roughly 20ns after the incident pulse and ending before the next pulse. Further, the TAC signaltonoise is improved through electronic anticoincidence.
Although absolute NRVS intensity is proportional to Fe atom motion, from equation 5.16 we are reminded that better signal for single phonon creation events (and inhibition of higher order phonons) can be achieved through lower temperatures. As such, NRVS is typically performed within a cryostat at very low nominal temperatures (<10K) with actual temperatures ranging from 40100K as determined by the spectral detailed balance.
The collected spectrum is roughly equivalent to equation 5.12 which relies on equation 5.16. It is generally favorable to analyze the spectrum in terms of PVDOS to remove the 1/E intensity dependence for single phonon events (equation 5.16). Sturhahn created a program called PHOENIX that is the de facto standard to convert raw NRVS data to PVDOS (among many other features not discussed here)^{16} using the Fourierlog method and will be described below.^{17}
For actual experimental data the measured intensity transforms from equation 5.12 to:
Here we have distributed the normalization constant from 5.12 into and and have introduced the experimental monochromator resolution function . Next, the resolution function must be precisely fit to the spectrum. The high statistics of the elastic peak give an obvious target to fit the resolution function, however, the contribution to the area under the elastic peak also comes from the single phonon contribution  here estimated as from 5.16 as a Debye solid  so the following function is fitted:
All c terms are variable fitting parameters. The term allows for adjusting the position of the resonance, scales the height of the function to the elastic peak, and the term adjusts the single phonon contribution. The function provides the resolution function as:
Where is a generic shape function:
All c values are fit using a least squares minimization. Next the first moment of the spectrum is normalized to the recoil energy according to equation 5.13 determining . From the eventual deconvolution indicated in equation 5.21, we see that the LambMössbauer factor must be determined by the normalized difference in fitted elastic contribution and inelastic contribution.
Finally, PHOENIX performs the Fourierlog deconvolution^{17} a modification of equation 5.21:
Here is the Fourier image of the resolution function and is the Fourier image of the intensity spectrum. Likewise, PHOENIX calculates higher order phonons. The term is a mollifier function that convolutes with the subtracted spectrum to limit Fourier artifacts.
The final PVDOS is calculated identically to equation 5.22.
From a practical standpoint the energy axes of multiple scans are aligned based on their position of the elastic resonance, then a calibration factor is applied. Typically, this factor can be determined by the position of three prominent peaks for [Et_{4}N]FeCl_{4} (Figure 5.3).
Figure 5.3 The ^{57}Fe
NRVS PVDOS for [Et_{4}N]FeCl_{4}. The vibrational modes are
labeled. Modes that involve no Fe motion, such as E and A, are not observed in
the spectrum.
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