Gamma rays of energy 14.4 keV from excited 57Fe nuclei show a very narrow energy width of 4.67 neV by the Mössbauer effect. Mössbauer gamma rays are utilised as probe beams in unique quasi-elastic scattering spectroscopy with neV-energy resolution. The technique enables measurements of atomic/molecular dynamics on timescales between nanoseconds and microseconds for various condensed matter systems, such as supercooled liquids, glasses and soft materials. The microscopic dynamics is measured in time domain or energy domain based on synchrotron radiation using a time-domain interferometer or a nuclear Bragg monochromator, respectively. We introduce state-of-the-art spectroscopic techniques, application results and future perspectives of quasi-elastic Mössbauer gamma ray scattering based on synchrotron radiation.
- Mössbauer gamma ray
- synchrotron radiation
- quasi-elastic scattering
- glass transition
- slow dynamics
The recoilless nuclear excitation of a gamma ray and its reversal process of recoilless gamma ray emission were first reported by Mössbauer . These phenomena occur in solids when the recoil momentum of gamma rays in absorption and emission processes is taken up by the whole crystal. Consistently, this physical phenomenon is referred to as the Mössbauer effect . For 57Fe nuclei, the excitation energy to the first excited state is 14.4 keV, whereas the uncertainty width of the excited state is relatively very narrow. Therefore, the gamma rays emitted from the excited 57Fe nuclei by the Mössbauer effect show an energy ∼14.4 keV and a natural energy width ∼4.67 neV. The photon emitted by the nuclei is called the gamma ray because it originates at the nucleus. However, Mössbauer gamma rays have lower energy than gamma rays involved in astronomy physics and are, instead, closer to the energy range of hard X-rays. In this chapter, we refer to such gamma rays as Mössbauer gamma rays. In these cases, the ratio of the gamma rays’ energy to the natural energy width reaches , indicating that the Mössbauer gamma rays exhibit very high monochromaticity. The surrounding electrons affect nuclear excitation energies through hyperfine interactions. Therefore, electronic states around the specific Mössbauer nuclei can be selectively studied from the measured nuclear excitation energies via the Mössbauer effect. This spectroscopic technique, known as Mössbauer spectroscopy, has been widely used for more than 40 elements and 70 nuclear species (referred to as the Mössbauer nuclear species) to resolve various challenges in the fields of chemistry, physics, geology and biology .
Microscopic dynamics in condensed matter, which do not contain Mössbauer nuclear species, have been studied since the 1960s with Mössbauer gamma rays . In these experiments, the Mössbauer effect is utilised to generate the monochromatic gamma rays from a radioactive isotope (RI) source, and a quasi-elastic scattering experiment is performed for some samples . In this chapter, we refer to the methods as quasi-elastic gamma ray scattering (QEGS) spectroscopy based on conventional nomenclature, such as inelastic/quasi-elastic neutron/X-ray scattering though this method has often been referred to as the Rayleigh-scattering of Mössbauer radiation method. The neV-energy resolution of the gamma rays from 57Fe nuclei allows the dynamics to be measured on timescales of about 100 ns. However, the measurements require much longer times because gamma rays from RI sources do not have parallel beams with enough brilliance for the QEGS experiment.
Recently, synchrotron radiation (SR)-based QEGS spectroscopic techniques using a 57Fe-nuclear Bragg monochromator (NBM) [4, 5] and a time-domain interferometer (TDI) of 57Fe gamma rays  have been developed. These methods have enabled much faster measurements of the atomic/molecular dynamics than RI-based QEGS spectroscopy, owing to the high brilliance and directionality of the SR source. To date, alloys, supercooled molecular liquids, polymers, ionic liquid, liquid crystals and polymer nanocomposite systems have been studied by SR-based QEGS spectroscopy.
In this chapter, we consider Mössbauer gamma rays from 57Fe nuclei because the gamma ray is most frequently used for QEGS spectroscopy. The length scales of the density correlation function currently observable by SR-based QEGS spectroscopy using TDI range from 0.1 to 6 nm, and the fluctuation timescales vary from few nanoseconds to sub-microseconds, as shown in Figure 1. The figure demonstrates how QEGS spectroscopy enables us to study density fluctuations, which are quite difficult to study by conventional spectroscopies in the microscopic range. Many unsolved issues are related to these time and length scales, including microscopic activation processes, which are related to the nature of the glass transition, start to occur in glass-forming materials in the time and length scales with cooling.
This chapter is organised as follows: In section 2, basic concepts of quasi-elastic scattering are introduced, and QEGS spectroscopic techniques are explained. In section 3, experimental results of application studies on several supercooled glass formers are described. In section 4, we conclude this chapter by describing future perspectives of QEGS.
2. Quasi-elastic scattering spectroscopy using Mössbauer gamma rays
In this section, we introduce the quasi-elastic scattering technique using Mössbauer gamma rays. In section 2.1, basic concepts of the quasi-elastic scattering technique are described. In section 2.2, we introduce energy-domain spectroscopic techniques of QEGS using Mössbauer gamma rays from conventional RI and SR sources. In section 2.3, time-domain measurement techniques of QEGS spectroscopy using single-line and multi-line TDI are described.
2.1 Introduction to quasi-elastic scattering
In this scattering process, gamma rays with wavevector
We introduce the spatial correlation function of the electron density as , where denotes the equilibrium average over and position , and
We introduce the time and space correlation function describing the microscopic structural dynamics. Its
2.2 Energy-domain spectroscopy of QEGS
In this section, we consider QEGS-based energy-domain spectroscopic techniques using Mössbauer gamma rays from conventional RI and SR sources. Figure 3a shows the common experimental design of the technique [8, 9]. In the setup, monochromatic Mössbauer gamma rays impinge on the sample. The quasi-elastic broadening of the scattered gamma ray’s energy is analysed by the 57Fe-Mössbauer absorber, as explained below. As Figure 3b shows, is observed as a transmittance-type spectrum , which is conceptually written as where is the resolution function.
2.2.1 RI-based QEGS spectroscopy: Rayleigh-scattering Mössbauer radiation
Rayleigh-scattering Mössbauer radiation (RSMR) spectroscopy is a conventional QEGS spectroscopic technique that uses RI as the source of the gamma ray probe. RSMR spectroscopy has been used to study microscopic dynamics in glass formers, proteins and liquid crystals as summarised in a review by Champeney . In this method, monochromatic Mössbauer gamma rays (e.g., from a radioactive 57Co source with an energy of 14.4 keV and an energy width of 4.67 neV) are sent to the sample. A broadening of the energy width of the quasi-elastically scattered gamma rays from a sample is detected by an absorption spectroscopy method commonly used in Mössbauer spectroscopy (Figure 3a). A transmittance-type energy spectrum is obtained by scanning the velocity of a movable 57Fe gamma ray absorber with a single-line excitation profile. The absorber acts as the energy analyser, since its velocity determines the relative energy shift c via the Doppler effect, where c is the speed of light. RSMR measurements require ample measuring time (at least several weeks) to obtain a spectrum with enough statistics for analysis because the RI source emits gamma rays in all directions, and limited flux is introduced to the sample.
2.2.2 SR-based QEGS spectroscopy using 57Fe-nuclear Bragg monochromator
The QEGS-based energy-domain spectroscopic technique using an SR source was developed with the 57Fe-NBM [4, 5]. NBM is used for a specific condition, in which conventional X-ray diffraction by electrons is forbidden, while nuclear resonant diffraction with nuclear excitation and deexcitation processes is allowed. In such cases, we can detect almost pure Mössbauer gamma rays on a 10 neV-energy width scale due to the specific Bragg angle selectively from a very intense incident SR. Therefore, the SR-NBM system is often called as synchrotron Mössbauer source . The SR-based QEGS experiment has higher efficiency than conventional RSMR using RI because the monochromatic gamma rays from the NBM exhibited high directivity . Moreover, the energy width of the Mössbauer gamma ray probe could be controlled to be much larger than the natural-line width (i.e., up to μeV) . This unique characteristic of SR-based QEGS spectroscopy using NBM allows us to measure microscopic dynamics up to sub-nanosecond timescales.
2.3 Time-domain measurement of QEGS
The time-domain spectroscopy of QEGS is achieved using TDI. In this section, we introduce time-domain spectroscopic techniques.
2.3.1 SR-based QEGS using single-line TDI
The measurement principles of QEGS using the simplest TDI (usually referred to as single-line TDI) are described here. We discuss TDI using Mössbauer gamma rays from 57Fe because it exhibits the highest utility among nuclear species potentially available for TDI. Figure 4a shows the schematic experimental setup [6, 12, 13].
First, we consider the nuclear forward scattering (NFS) case, which often provides a calibration for the QEGS measurement because it is not affected by the dynamics of the sample. In the upper panel of Figure 4a, we show the experimental design for the NFS experiment using TDI. The incident SR crosses two identical materials with a single-line 57Fe nuclear excitation profile corresponding to the nuclear time response function ultimately detected by the detector. Most of the SR beam crosses the 57Fe materials without any interaction. A small portion (typically ∼10−6) of SR excites the 57Fe nuclei in the materials, causing the gamma rays to emit when the excited 57Fe nuclei decay. The gamma rays travel undeflected towards the forward detector because of the high directivity inherited from the incident SR. The gamma rays can be distinguished from the much more intense SR because they are delayed from the SR pulse by a typical delay time coincident with the lifetime of the excited 57Fe nuclei (∼100 ns). The upstream material is moved with a constant velocity to change the relative nuclear excitation energy through the Doppler effect and consequently the energy spectrum of the gamma rays at the detector position shows two peaks due to the difference in the gamma ray energy between the two materials (see the upper panel of Figure 4b). The time resolution of the detector is typically 1 ns, which is much shorter than the lifetime of excited 57Fe, which enables to measure the time spectrum of the delayed gamma rays with high precision. The upper panel of Figure 4c shows the corresponding time spectrum. We can see the decay of the gamma rays’ intensity on the timescale of excited 57Fe. On the time spectrum, there is a beating pattern caused by the interference of the gamma rays with two peaks in the energy spectrum.
Next, we consider the QEGS case, corresponding to the scattering of the sample at a finite angle. In the lower panel of Figure 4a, we show the QEGS experimental design. The incident SR is scattered by a sample and detected by the detector. Two identical materials with a single-line 57Fe nuclear excitation profile are placed on the beam path in front of and behind the sample. This system is called the single-line TDI because each material that emits gamma rays (here, referred to as single-line emitter) shows a single-line nuclear excitation profile. A typical energy spectrum of gamma rays at the detector position is shown in the lower panel of Figure 4b. The gamma rays from the upstream emitter (denoted as ‘up’ in Figure 4b) are quasi-elastically scattered by the sample and the energy width is broadened as . However, the energy width of the gamma rays from the downstream emitter (denoted as ‘down’ in Figure 4b) is not broadened because it is emitted by the sample after the scattering process.
Next, we considered the time spectrum of the gamma rays obtained by the detector for the QEGS case. When the energy shift is sufficiently large (), the radiative coupling effect can be neglected [6, 12, 13]. Additionally, we can assume that the incident SR showed a temporal pulse structure with negligible width. In such cases, the electric field at detector position at an angle corresponding to can be written as
where is the angular frequency of the beating pattern. We ignored the coefficient of the transmittance because it does not affect the final spectrum shape. The first, second and third terms of Eq. (1) represent the electric field amplitudes of the prompt SR, gamma rays emitted from the upstream and downstream emitters, respectively. The delayed gamma rays’ measurement for part of the obtained time spectrum is written as
In Eq. (2), is the intermediate scattering function normalised by , that is, and the static structure factor is [6, 12, 13]. We assume that the scattering from a sample with a macroscopic number of atoms was measured with an acquisition time long enough to provide a reliable determination of the relevant average ensembles. For an NFS experiment under the same emitter conditions, the NFS time spectrum is expressed by Eq. (2) with and . Examples of time spectra for NFS and QEGS cases with a relaxation time of 100 ns are shown in the upper and lower panels of Figure 4c, respectively. For the actual fitting of the spectra, the time resolution of the detector and constant background noise would need to be considered.
Next, we considered the meaning of the time spectrum. The broadening of the gamma rays by an energy width reflects the dynamics in a sample. The broadening induces the distribution of the beat frequency in a time domain and this effect is seen as the relaxation of the beating pattern with the relaxation time in the simplest case. Further consideration revealed that the relaxation time of the beating pattern coincides with the relaxation time of the density correlation in the sample (namely, the intermediate scattering function) [6, 12, 13]. This analysis is a basic interpretation of how the time spectrum reflects the dynamics in a sample. We note that an intrinsic relaxation of caused by an external vibration, for example, should also be considered for the actual dynamics study.
2.3.2 SR-based QEGS spectroscopy using multi-line TDI
Here, we consider QEGS spectroscopy using multi-line TDI . In this case, emitters with several nuclear excitation energies are used for TDI. We assume again that the two emitters show different excitation energies from each other. Generally, the nuclear time response functions in emitters are different from each other in multi-line cases. Therefore, we introduce the time response functions for the upstream and downstream emitters as and , respectively. In such cases, we obtain the expression
As an example of multi-line TDI, we considered α-iron foils as emitters, where the nuclear excitations are allowed for six different energies without an external magnetic field. Figure 5a shows an experimental setup using α-iron emitters. When the magnetic field is applied to the α-iron foils, as shown in Figure 5a, the transitions allowed in the two emitters are selected to be different from each other. Consequently, the gamma rays’ energy emitted from these two emitters is different, as shown in Figure 5b, where the gamma rays from the upstream and downstream emitters are denoted as ‘up’ and ‘down’, respectively. Examples of the energy spectra of gamma rays for cases without atomic motion and motion with a relaxation time of 100 ns are shown. Figure 5c depicts the corresponding time spectra. The beating pattern changes following the decay of. By introducing the multi-line condition, the interference beating pattern of the gamma rays on the time spectrum becomes more complex than the single-line case. However, the incident SR can be more effectively utilised for experiments and the gamma rays’ count rate increases. Additionally, it can be shown that the time spectrum changes more drastically, reflecting the dynamics . These properties of the multi-line TDI greatly improve the measurement efficiency in comparison to the single-line method.
2.3.3 SR-based QEGS using TDI considering energy resolution of incident SR
Here, we consider the effect of the energy width of the incident SR on the gamma rays’ time spectrum obtained by the QEGS experiment. After the first induced heat load from the Si(111) monochromator, the SR showed a relatively broad energy profile; an energy width of the eV order could be considered white for the QEGS system. However, the incident SR is usually further monochromatised by using a high-resolution monochromator (HRM). This device generates typical energy widths in the meV range to suppress radiation damage to the system [6, 12, 13, 14]. The meV-energy interval is equivalent to or smaller than the energy scale of phonons in samples. Therefore, a portion of the incident SR transfers a larger amount of energy to the sample by interacting with the phonons. We found that the inelastic scattering process affects the intensity ratio of the gamma rays from the upstream and downstream emitters. Considering this effect, we modify Eq. (3) as
where is the factor reflecting the sample dynamics on a meV-energy scale . It was confirmed that the QEGS time spectrum obtained using TDI with multi-line gamma rays could be nicely analysed using Eq. (4) . Additionally, we showed that QEGS spectroscopy using HRM originally has two resolution functions on neV- and meV-energy scales. By using multi-line TDI in the condition , dynamical information, such as the elastic scattering intensity, can be obtained simultaneously on nanosecond and sub picosecond timescales .
3. Application results of SR-based QEGS using TDI
To date, SR-based QEGS spectroscopy has been used to study glass-forming molecular liquids [15, 16, 17, 18, 19], polymers , polymer nanocomposites , ionic liquids , alloys  and liquid crystals .
3.1 Microscopic dynamics in glass formers
The general mechanism of the liquid-glass transition phenomenon, which has not been revealed, has attracted much interest. It is widely accepted that a relaxation process, known as the α process, is closely related to glass transitions [25, 26, 27]. Therefore, atomic and molecular dynamics of supercooled glass formers have been energetically investigated to understand glass transitions. The temperature (
The other challenging task in these systems is understanding the origin of the dynamical change of the α process, which starts to occur at a temperature of ∼1.2 upon cooling, where is the glass transition temperature. The changing temperature is recognised as the dynamical crossover temperature . In addition to the α process, various processes have been observed in relaxation maps, which summarise the temperature dependence of processes in glass formers. Among the various relaxation processes subjected to a thorough scrutiny, it is worth mentioning the Johari-Goldstein (JG)-β process, which emanates from the α process in relaxation maps and, instead, follows Arrhenius behaviour even below the glass transition temperature, where is the activation energy and R is the gas constant . Recently, the JG-β process was believed to commonly exist in supercooled glass formers and relate to the nature of the glass transition mechanism . The branching temperature of the JG-β process from the α process is frequently seen near the dynamical crossover temperature . This synchronism is believed to be an intrinsic feature of supercooled glass formers. However, the dynamical crossover and branching phenomena are far from being understood fully. Conventional methods, such as dielectric relaxation spectroscopy, do not provide spatial-scale information on the dynamics, and the α and JG-β processes are not clearly discerned around and . Therefore, has been estimated as a crossing point of the α-relaxation time and an extension of the JG-β relaxation time by assuming the Arrhenius law .
Understanding the microscopic dynamics around and is indispensable to elucidating the glass transition mechanism. SR-based QEGS spectroscopy is a method ideally suited to understand the microscopic dynamics in deeply supercooled glass formers around and and its evolution towards the glass transition. This technique enables to measure the atomic/molecular dynamics with specification of its spatial scale on a nanosecond/microsecond timescale, where the JG-β process commonly occurs . We performed SR-based QEGS experiments using single-line and multi-line TDI on various glass formers. We introduce the results on
3.2 Results on
We studied OTP using single-line 57Fe gamma rays TDI for the QEGS measurements [16, 20]. Detectors were placed at angles corresponding to
Figure 6 depicts the temperature dependence of . At
In the larger
3.3 Results on polybutadiene
As mentioned in section 3.2, the nature of glass transition is still not fully understood despite thorough investigative efforts . In the last three decades, extensive studies on glass transitions have been performed theoretically, experimentally and by computer simulations. One of the most important experimental results constructed relaxation time maps of several glass-forming materials  by predicting the decoupling of the JG-β process from the α process. Extensive experimental studies have been performed to reveal the decoupling mechanism using various techniques such as NMR , dielectric relaxation (DR)  and neutron spin echo (NSE) [42, 43, 44]. We performed QEGS measurements using single-line TDI on polybutadiene (PB), which is a typical glass-forming polymer, to decouple the JG-β process from the α process .
The sample used in this experiment was 1,4-cis-trans-polybutadiene (PB), which is never crystallised because of the microstructure of cis:trans:vinyl = 47:46:7. The
The average relaxation time obtained from the fitting curve is shown in Figure 7 as a function of the inverse of absolute temperature 1/
An extended mode coupling theory (eMCT) has been proposed to account for hopping processes . This theory predicts a dynamical transition from the α process to a local, hopping-dominated, relaxation process at . In other words, this transition corresponds to the switch of the temperature dependence from the VFT law to the Arrhenius law. In the eMCT framework, the transition from the α process to the JG-β process corresponds to the transition from the hydrodynamic continuous motion to the hopping motion. The fact that the transition above the first peak occurs near supports this interpretation. In the present experiment, however, we observed that the α process persisted even below
The question still remains as to why the α process lasts even below
3.4 Results on polybutadiene with nano-silica
Tyre rubber has been continuously developed to improve various aspects of its performance, such as its grip, fuel consumption and wear resistance, by adding fillers such as silica nanoparticles and cross-linking agents [46, 47]. However, the microscopic mechanisms behind these improvements are still not fully elucidated and a better understanding is needed to further improve tyre products. Many studies have shown that confined polymer layers around nanoparticles affect the rubber’s macroscopic properties [48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. Molecular-scale dynamics studies have also revealed that the presence of nanoparticles slows down the microscopic segmental α-relaxation motion and increases its heterogeneity [52, 53]. However, we still do not have a complete picture of the microscopic dynamics for these systems. Additionally, the effect of the particle size on the microscopic dynamics has not been elucidated.
To elucidate the effect of nanoparticles on the microscopic α-relaxation dynamics of polymers, we studied the microscopic dynamics of a polybutadiene (PB) and silica nanoparticle mixture by SR-based QEGS using multi-line TDI. Two types of samples were used for this experiment: pure 1,4-PB and 1,4-PB nanocomposites with silica nanoparticles. Two PB nanocomposites, PB-silica20 and PB-silica100, were prepared with 20 vol% of silica nanoparticles with average diameters of 20 and 100 nm, respectively. The glass transition temperature
Figure 8 shows the obtained wide-angle X-ray scattering (WAXS) profile of the two nanoparticle samples. From these WAXS results, we confirmed that the position of the main peak, mainly reflecting the intermolecular correlation of the PB, had changed very little and was covered by the
Next, for the PB nanocomposites with silica nanoparticles, the polymer dynamics was studied through the analysis of the relaxation time extracted from the intermediate scattering function, while also considering its non-relaxing component originating from the stable nanoparticles. For the polymer nanocomposite systems, it is known that the contribution of the α-relaxation of polymers to the intermediate scattering function can be treated as a KWW function [48, 49]. Therefore, we used the function to fit the normalised intermediate scattering function for the time spectra of PB-silica100 and PB-silica20, where is the contribution of the non-relaxing component. By fitting the time spectra obtained for PB-silica100 at 250 K, we determined that the contribution of the non-relaxing component was = 0.22 ± 0.07 at
Figure 9 shows the temperature dependence obtained for . The α-relaxation times of pure PB obtained by dielectric relaxation spectroscopy (depicted as a line in Figure 9) demonstrate that our results are consistent with the dielectric relaxation spectroscopy results . The temperature dependencies of obtained for PB-silica20 and PB-silica100 also show divergent behaviour, although the VFT parameters appear to be different compared to pure PB. At 250 K, the α-relaxation times obtained at
4. Conclusions and perspectives
Quasi-elastic scattering techniques using Mössbauer gamma rays are promising approaches for revealing nanosecond and microsecond dynamics directly from the microscopic viewpoint. Currently, quasi-elastic scattering systems using the gamma rays TDI have been developed and utilised for application studies. Additionally, by using a band-width variable 57Fe-NBMs, we expect that the timescale of measurable dynamics will be expanded (e.g., up to sub 100 pico-second). Developing techniques that expand the timescales of measurements (i.e., between sub 100 pico-seconds and sub-microseconds), such as energy-domain quasi-elastic scattering systems combined with time-domain quasi-elastic scattering systems, is highly desirable.
Moreover, various new X-ray-based techniques are proposed for studying microscopic dynamics, based on focusing monochromators , or X-ray echo spectroscopy  or free electron lasers (e.g., four-wave mixing experiments) . The combination of these new X-rays (and gamma rays)-based techniques expands the timescales of the measurements significantly (e.g., from femtoseconds to microseconds). Future studies will open new methodologies for depicting the microscopic structural dynamics of condensed matter by X-rays.