Main

The scientific potential of the 45Sc resonance, together with the possibility of its resonant excitation by photons from modern accelerator-based sources of hard X-rays (no radioactive parent isotope is available for 45Sc), was identified more than 30 years ago11. However, earlier attempts to induce resonant excitation at third-generation synchrotron radiation sources were not successful, mainly because of the lack of sufficient spectral flux. This flux constraint was overcome only recently with the advent of narrow-band XFELs working at a high repetition rate, such as the European XFEL (EuXFEL)12,13.

Long-lived nuclear isomeric states with resonance quality factors comparable to or exceeding those used in modern atomic optical clocks1 have the potential for revolutionizing quantum metrology, clock technology, chronometric geodesy and gravimetry and enable fundamental physics tests that rely on the measurement of time or frequency with utmost precision, such as the search for time–space variation of the fundamental constants, dark matter, violation of the Lorentz invariance and Einstein’s equivalence principle7,8,9,14,15,16,17. These isomeric states could also lead to the improvement of resolution of the coherent nuclear forward scattering spectroscopy by orders of magnitude11 and to compact long-lived nuclear quantum memory18.

The main advantage of nuclear transitions over atomic transitions in these applications is their lower sensitivity to perturbations caused by electric and magnetic fields because of the tiny size of nuclei and small magnitudes of nuclear electromagnetic moments. This could lead to higher accuracy of the nuclear clocks and could make it possible to use macroscopic ensembles of nuclear oscillators in bulk solids. Using large ensembles, along with a higher transition frequency, would yield stability that is orders of magnitude higher. Moreover, ultra-low temperatures would not be required because of the Mössbauer effect that suppresses thermal line broadening. For these reasons, the concept of a nuclear clock, especially of a Mössbauer nuclear clock, has recently drawn considerable attention2,3,4,6.

Most long-lived nuclear isomeric states have resonance energies above 30 keV (fig. 1 of ref. 2) and hence cannot be resonantly driven by the existing sources of coherent radiation. Among them is a 88-keV transition in 109Ag with a 57-s lifetime that was investigated with radioactive sources for almost 60 years15,19 without yielding conclusive evidence yet for resonance detection. Other approaches have used bremsstrahlung to populate indirectly long-lived isomeric states such as the 39-keV level of 103Rh using higher-energy broadband excited states20. Fortunately, some lower-energy isomers exist that qualify for coherent resonant electromagnetic excitation. The most prominent is 229Th, which has an extraordinarily low transition energy of about 8 eV (refs. 2,4,5,6,7,8,9). This transition is especially attractive because it is within the reach of ultraviolet light frequency combs. Notable progress was achieved recently in a more precise determination of its energy4,5,6. Still, exciting this resonance with photons remains an open challenge2,3,4,5,6,7,8,9. An alternative, although indirect, route to populate the isomeric state could proceed using X-ray excitation of the 29.2-keV broadband second excited state of 229Th (ref. 3).

Fig. 1: Excitation and decay scheme of the 12.4-keV first excited state of the 45Sc nucleus.
figure 1

The nonradiative internal conversion decay channel dominates, resulting in the emission of Sc characteristic X-ray fluorescent photons and Auger electrons. Kα (4.09 keV) and Kβ (4.46 keV) fluorescent photons were detected in this study.

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Now, with the availability of modern X-ray free-electron lasers (XFELs)12,13,21,22,23,24,25, several lower-energy and long-lived nuclear isomeric resonances are within reach. One of the most promising is 45Sc, which has a transition energy of E0 = 12.4 keV, an isomer lifetime of τ0 = 0.47 s and a natural linewidth of Γ0 = ħ/τ0 = 1.4 feV, resulting in an extremely high quality factor of Q = E0/Γ0 ≃ 1019 (see Table 1 for the nuclear parameters of 45Sc and Fig. 1 for a schematic of the nuclear transitions). This quality factor is six orders of magnitude larger than that of the 14.4-keV resonance of 57Fe—a workhorse of Mössbauer spectroscopy26,27,28—and surpasses other measurable Mössbauer resonances by orders of magnitude29,30. A high-quality resonance of this kind would enable the measurement of gravitational red shift or gravitational time dilation at displacements much shorter than the 1-mm record established on the basis of a Sr atomic clock31. Furthermore, 45Sc is a stable isotope with 100% natural abundance and is readily available either as ultra-pure Sc metal or as Sc2O3, in which its 12.4-keV transition has a high Lamb–Mössbauer factor fLM ≃ 0.75 at room temperature. All of these facts make 45Sc superior to any other candidate for a Mössbauer nuclear clock.

Table 1 Parameters of 45Sc, taken from or based on ref. 32 unless otherwise stated, with updated values measured or used in this work highlighted in bold
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Here we report on the resonant excitation of the long-lived ultra-narrow 12.4-keV nuclear state of 45Sc using XFEL pulses. In this experiment, the transition energy, previously known to an uncertainty of only about ±50 eV (ref. 32), was determined with more than 100 times higher accuracy as \({\mathrm{12,389.59}}_{+0.12{\rm{(syst)}}}^{\pm 0.15{\rm{(stat)}}}\,{\rm{eV}}\). Furthermore, we obtained estimates for the resonance cross-section and for the coefficients of internal conversion of this transition. The high average spectral flux of the incident X-rays and the very low background noise in the detection of the nuclear decay products were crucial for the success of the experiment.

Approach

X-rays generated by EuXFEL were used to excite 45Sc nuclei in a solid-state target from the ground into the isomeric state. The pulsed source provided 12.4-keV X-rays with the highest spectral density and a pulse duration much less than the isomer lifetime τ0 = 0.47 s. Nuclear decay products were detected with a characteristic delay of about τ0 after the excitation. To achieve an utmost reduction of detection background after irradiation, the detectors were offset from the beam path, and the target was mechanically moved out of the X-ray beam into this offset position after each pulse train to record the delayed fluorescence signal. Two low-noise Si solid-state drift detectors were used to detect nuclear decay (Fig. 2 and section ‘Experimental challenges’).

Fig. 2: Schematic of the experiment designed to resonantly excite 45Sc nuclei from the ground state to the long-lived ultra-narrow 12.4-keV excited state using XFEL pulses and to detect the 45Sc resonance.
figure 2

Upstream and downstream X-ray counters Du and Dd detect time-delayed nuclear decay products—both inelastic Kα,β fluorescence and coherent elastic NFS. To minimize the detection background, the decay product detectors are offset from the beam path and the sample is moved to the detectors after irradiation with each pulse train. The resonance energy is measured with X-ray single-shot and Bond spectrometers.

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As 45Sc exhibits a rather high coefficient of internal conversion α, in the range of about 400−650 (that is, the probability of decay via internal conversion greatly exceeds the probability of a radiative decay; Table 1), we used this incoherent nuclear decay channel for resonance detection. Nuclear decay via internal conversion results in the ejection of an electron from an inner shell (predominantly 1s, that is, K shell). The resulting core hole is filled by the emission of characteristic Kα (4.09 keV) and Kβ (4.46 keV) X-rays, corresponding to the 2p−1s and 3p−1s transitions, respectively, and/or emission of Auger electrons. Although Auger electrons dominate (the K-shell fluorescence yield is only ωK = 0.19 for Sc)33, we chose to detect X-ray fluorescence, as X-ray detectors (unlike electron detectors) do not require a special environment. Moreover, along with the 4-keV photons, they can simultaneously detect 12.4-keV forward-scattered photons with a high spectral resolution. It is expected that this set-up will produce elastically and coherently scattered delayed 12.4-keV photons in the forward direction, a phenomenon known as nuclear-resonant forward scattering (NFS)34,35,36. Observation of such photons would not only enable detection of the 45Sc resonant excitation but also—if the time dependence of the NFS could be measured—provide information on the resonance linewidth11. However, the probability of detecting 12.4-keV photons was expected to be significantly lower than that of 4-keV photons, because of both the high internal conversion coefficient and the anticipated large yet unknown broadening of the resonance line. Therefore, we focused on the detection of 4-keV photons.

Because the energy of the 12.4-keV state was known before the experiment to an accuracy of only ±50 eV (ref. 32), the energy of the incident X-rays had to be scanned over a comparatively large range of 100 eV and accurately measured in parallel with detecting the nuclear decay products. The relative photon energy was continuously measured with a single-shot spectrometer (SSS) and then related to the absolute energy scale using Bond spectrometer measurements (Fig. 2 and section ‘Absolute resonance energy’).

Although the procedure seems simple, many challenges had to be overcome to realize the experiment, as detailed in the section ‘Experimental challenges’.

Initially, two energy scans were performed over a range of ±50 eV around 12.40 keV. After the candidate 45Sc resonance energy E0 was located, the subsequent scans were constrained to a smaller range of ±5 eV around this value. The pure data acquisition time was around 30 h, during which about 1020 photons from the EuXFEL were directed to the 45Sc targets. In total, about 93 nuclear decay events were detected. Although the number of the nuclear decay events detected was relatively small (about 93), the number of 45Sc nuclei excited in the experiment was estimated to be much larger around 7 × 103 (Methods). Because of the extremely low detector background, this number was sufficient to reveal the 45Sc resonance and determine the resonance energy with high accuracy.

Resonant excitation of 45Sc

Figure 3 shows the two-dimensional plots of counts from the upstream and downstream decay detectors, Du and Dd, respectively, with a plot of all detected photons with energies Ef ≲ 26 keV at all incoming photon energies Ei (Fig. 3a), and close-ups of regions of interest (ROIs) around the energies of the fluorescence and NFS photons (Fig. 3b,c). In Fig. 3a, the vertical red lines indicate the constrained scan range of ±5 eV around the 45Sc resonance energy E0. Because of this scan-range reduction, the density of the detected (background) photons seems to be larger between the vertical red lines.

Fig. 3: Counts from the Du and Dd detectors plotted as the energy Ef of the detected X-ray photons versus the incident X-ray photon energy (expressed as the difference of the incident energy Ei and the resonance energy E0).
figure 3

The photons were recorded in a time window of about 20−80 ms after every pulse-train excitation. Each photon is shown by a blue circle for Du and a yellow circle for Dd. a, All detector counts with Ef ≲ 26 keV for incident energies in a range of −40 eV ≲ Ei − E0 ≲ 40 eV around the 45Sc resonance energy E0. Vertical red lines indicate a 10-eV ROI around Ei = E0. Horizontal dark yellow lines indicate a 1-keV ROI around Ef = 12.4 keV, whereas the horizontal magenta lines indicate a 1-keV ROI around Ef = 4.3 keV, the approximate energy of the fluorescence photons. b, Close-up of the 4.3-keV ROI, showing two clusters of counts centred at the energies of Sc Kα and Kβ fluorescence. c, Close-up of the 12.4-keV ROI.

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Figure 3b shows a close-up of the 4.3-keV ROI in the reduced incident energy Ei scan range. It shows two clusters of about 93 counts that are centred at detector energy values Ef that correspond to the energies of Sc Kα (4.09 keV) and Kβ (4.46 keV) fluorescence. Because these events were recorded with at least a 20-ms delay after the arrival of the XFEL pulses on the 45Sc target, this observation presents clear evidence of the detection of decay products of the long-lived 12.4-keV excited state of 45Sc nuclei resonantly excited by XFEL pulses.

Figure 3c shows a close-up of the 12.4-keV ROI in the same Ei range. There are just a couple of counts at these energies. Thus, as anticipated, an NFS signal could not be detected in this experiment (see section ‘Nuclear forward scattering’).

Resonance energy

Figure 4 shows the data of Fig. 3b but integrated over Ef in the detection ROI. The graph shows all delayed Sc K-shell fluorescence counts as a function of the energy of the incoming X-ray photons. A clear resonance peak with a high signal-to-noise ratio of around 65 is observed, centred at E0. The measurements performed with the Bond-type X-ray spectrometer37 (Fig. 2 and section ‘Absolute resonance energy’) enabled us to relate the relative to the absolute energy scale with high accuracy and determine the energy of 45Sc resonance as \({E}_{0}={\mathrm{12,389.59}}_{+0.12{\rm{(syst)}}}^{\pm 0.15{\rm{(stat)}}}\,{\rm{eV}}\). We note that the peak width is 1.32(12) eV, reflecting the energy width of the XFEL pulses.

Fig. 4: Time-delayed Sc K-shell fluorescence recorded by detectors Du and Dd as a function of incoming X-ray photon energy (expressed as the difference of the incident energy Ei and the resonance energy E0).
figure 4

The solid black line shows the resonance line obtained from the data evaluation (see section ‘Data analysis’). The coloured dots show exemplary binned data, with different incident energy bin sizes and shifts of the bin grid. The spectral width of 1.32(12) eV reflects the spectral width of the incoming XFEL radiation.

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Count rates, internal conversion coefficients and resonance cross-section

The scale on the left of Fig. 4 presents normalized counts: that is, the integral over the spectrum equals the total number of the delayed K-shell fluorescence photons (approximately 93) recorded in the experiment.

The scale on the right presents the nuclear-resonance-assisted K-shell fluorescence yield that would be measured for one incident resonant photon (in the natural linewidth Γ0) on the target. The peak yield derived from the experimental data is \({\Sigma }_{{\rm{K}}}^{\exp }=0.44(10)\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\) (for details, see section ‘Incoherent scattering count rates’). However, the theory outlined in that section predicts a peak yield of \({\Sigma }_{{\rm{K}}}=0.23\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\)—smaller by a factor of 1.9(4)—when the internal conversion coefficients α and αK adapted from ref. 32 (Table 1) are used in the calculations.

A better agreement between the theoretical and experimental peak yields is achieved if smaller values of α and αK are used. In particular, using the internal conversion coefficients predicted by the state-of-the-art internal conversion theory (https://bricc.anu.edu.au/)38, the nuclear-resonance-assisted K-shell fluorescence yield becomes \({\Sigma }_{{\rm{K}}}=0.39\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\), which is in good agreement with the experiment. The updated α and αK values and a nuclear resonance cross-section σ0 calculated using these internal conversion coefficients are shown in bold in Table 1.

Discussion

Taken together, the resonant excitation of the 45Sc resonance and the accurate measurement of the resonance energy reported here open new prospects in ultra-high-precision spectroscopy, nuclear clock technology and extreme metrology in the regime of hard X-rays. This work further demonstrates that the state-of-the-art high-repetition-rate narrow-band XFELs provide a promising experimental platform for the study of ultra-long-lived nuclear resonances at energies of hard X-rays. The improved accuracy of the resonance-energy determination sets the stage for the next step: an observation of the time dependence of the coherent NFS. This observation would not only provide the actual spectral width of the 45Sc resonance11 but also immediately enable important applications in extreme metrology and ultra-high-precision coherent NFS spectroscopy, because the narrowness of the 45Sc resonance is expected to surpass all currently accessible Mössbauer resonances by orders of magnitude. An example could be an interferometric measurement of gravitational red shift in the time domain, which could potentially reduce the required height difference to the submillimetre range (compared with 22.6 m in the historic Pound–Rebka experiment using 57Fe39). Coherent NFS from 45Sc could also be used for measuring extremely weak couplings in solids that result in energy shifts or splitting much below the currently accessible 1-MHz range.

The observation of 45Sc NFS will require extra steps such as cooling the nuclear target to exclude the thermal line broadening, minimizing or suppressing inhomogeneous broadening and maximizing the NFS signal by increasing the target thickness (see section ‘Nuclear forward scattering’).

We foresee the development of a 45Sc-based nuclear clock in the future. Achieving this goal will require a further increase in the resonant spectral flux using improved narrow-band 12.4-keV X-ray sources and frequency combs stretching up to this energy. There are two possible routes to this end. The first route is based on using an X-ray free-electron-laser oscillator (XFELO)40 and a hard X-ray comb generated by a nuclear-resonance-stabilized XFELO41. The realization of such devices is presently in progress42,43. Specifications for the X-ray source and for the nuclear clock procedure conceptually outlined in ref. 41 will become more explicit after the observation of 45Sc NFS and optimization of the 45Sc target for maximum NFS signal strength and the smallest linewidth of the actual 45Sc resonance. The second route is to use intracavity high harmonic generation (HHG) as the frequency comb44. Recently, the cut-off photon energy in HHG has been pushed into the kiloelectron volt range, that is, to 5.2 keV (ref. 45), and the intracavity HHG was spread into the extreme ultraviolet light range46, with an expected spectral width of each harmonic of the order of 0.1 Hz (refs. 47,48), which would perfectly match the spectral width of the 45Sc resonance.

Methods

Experimental challenges

Although the experimental approach seems simple, many challenges had to be overcome to use our measurement and detection scheme at an XFEL facility, as explained below.

Spectral flux

An X-ray photon spectral flux of at least about 0.1 ph s−1 \({\Gamma }_{0}^{-1}\) ≃ 0.7 × 1014 ph s−1 eV−1 is required to detect the 45Sc resonance in a reasonable time. These spectral flux values are becoming possible with the advent of narrow-band self-seeded hard X-ray FELs22,24,25,54,55 operating at a high repetition rate, resulting in up to three orders of magnitude higher average spectral flux compared with values at storage-ring-based sources of hard X-rays56. The EuXFEL recently became the first to our knowledge operational facility of this kind12,13.

The experiment was performed at the Materials Imaging and Dynamics (MID) instrument57 at the EuXFEL. Figure 2 shows the schematic of the experiment. The EuXFEL operated as a self-seeding XFEL—that is, X-rays from the first half of the XFEL undulator were used to seed the second half through a diamond-crystal-based (wake) monochromator13,22,25,55,58,59.

The EuXFEL generated 12.4-keV X-ray pulses with an average rate of 4 kHz, an average pulse energy of about 350 μJ, a spectral width of around 1.3 eV (full width at half maximum), an average X-ray power of about 1.4 W, a photon flux of about 7.0 × 1014 ph s−1 and a spectral flux of about 5 × 1014 ph s−1 eV−1 or about 0.7 ph s−1 \({\Gamma }_{0}^{-1}\). The actual spectral flux transported to 45Sc targets was \({F}_{{\Gamma }_{0}}\simeq 0.31(6)\,{{\rm{ph}}}^{-1}\,{{\rm{s}}}^{-1}\,{\Gamma }_{0}^{-1}\) because of attenuation on the way from the source to the target by a factor of A = 0.44(9).

Background suppression

The low signal strength estimated for this experiment calls for a detector system with a noise floor not exceeding 10−3 ph s−1 in the presence of the huge incoming photon flux. This noise floor requirement has been met by using temporal, energy, spatial and polarization degrees of freedom to discriminate detector signals against background counts.

The EuXFEL provides a unique time sequence of X-ray pulses, which is perfectly suited for the detection of the 45Sc resonance with a very good signal-to-noise ratio. X-rays arrive in submillisecond pulse trains having a periodicity of 100 ms (Fig. 2). In this experiment, each pulse train consisted of 400 X-ray pulses spaced by 440 ns, with each pulse having a duration of a few tens of femtoseconds. The resulting pulse-train duration of 176 μs is much shorter than the natural lifetime τ0 of 45Sc. This pulse-train sequence is ideal for efficient excitation and detection of the  45Sc resonance, because it enables temporal discrimination of detector signals against a huge incident flux during the excitation process: during the dark time between the pulse trains, the millisecond-delayed 45Sc resonance decay products can be detected in near-complete absence of background signals.

Additional background reduction is achieved by using energy-resolving Si solid-state drift detectors (Amptek X123, labelled Du and Dd in Fig. 2) with an energy resolution of 150 eV in the X-ray regime below 15 keV. This detector choice facilitates sharp energy discrimination of the 4-keV Kα,β characteristic fluorescent photons against all other X-ray energies.

Equally essential for background suppression is the relation between the irradiation position and the detection position. The detectors are placed at around 12-mm offset from the incident beam, and the 45Sc target is shifted swiftly back and forth from the beam path to a position between the detectors every 100 ms by a dedicated linear motion system (Fig. 2). The sample dwells at the irradiation position for around 10 ms and at the detector position for about 60 ms; the transit time between positions is approximately 15 ms. The periodic motion is synchronized with the arrival of the 12.4-keV X-ray pulse trains. The detector offset is in the horizontal plane to suppress X-ray Thomson scattering, because the electric vector of the incident photon polarization is also in the horizontal plane.

As a result of these measures, the total detector background count rate in the experiment was as low as about 2 × 10−4 ph s−1.

The 45Sc target

The choice of the 45Sc target was dictated by the necessity of achieving high nuclear resonance optical thickness, low absorption of the nuclear decay products, resilience to radiation damage, simplicity in fabrication and integration into the motion and detection system. Commercially available Sc-metal foil with a thickness L = 25 μm was chosen as a target; the foil has a number density N = 3.98 × 1022 cm−3 of 45Sc nuclei and an area number density ρ = NL = 1 × 1020 cm−2. With this choice, the nuclear resonance optical thickness had a substantial value of σ0 ρ = 1.87 (see Table 1 for σ0). The target thickness of 25 μm was chosen to maximize the collection efficiency of the 4-keV photons as it corresponds to one attenuation length (LK ≃ 23 μm) of 4-keV photons in Sc metal.

Radiation damage

Every X-ray pulse train with an energy of about 65 mJ heats the 25-μm thick Sc-metal target by an estimated temperature of about 100 K. The heat conductivity of Sc metal is low—15.8 W m−1 K−1, comparable to that of stainless steel. Radiation damage to the target is thus inevitable. Special measures were taken to avoid damage, including water and air cooling, defocusing of the beam (the beam footprint was about 2 mm2) and periodic replacement of the target. Two targets had to be replaced in the course of the experiment because of partial radiation damage.

Uncertainty of the resonance energy

Because of the ±50 eV uncertainty in the 45Sc resonance energy before this experiment, the EuXFEL photon energy had to be scanned over a rather large energy range around the anticipated 12.40-keV value while simultaneously detecting the nuclear decay products (EuXFEL currently delivers photons in a self-seeding mode in the energy range of 6–14.4 keV). Such scans require simultaneous and accurate measurement of the energy of the incident photons. An SSS60,61,62,63 installed in the endstation of MID (Fig. 2) measures the spectral distribution and its peak photon energy in relative units (SSS scale). The peak value on the SSS energy scale is related to the absolute incident photon energy scale Ei with an accuracy of ≲0.3 eV by measuring selected photon energies using a Bond-type spectrometer37 (Fig. 2 and section ‘Absolute resonance energy’).

Incoherent scattering count rates

The attenuation of monochromatic X-rays of a spectral flux F0 propagating through a target of thickness L consisting of resonant nuclei with a number density N can be calculated as

$$F(E,L)={F}_{0}\exp \left[-\sigma (E)NL-L/{L}_{0}\right],$$
(1)

where the energy dependence of the total resonance cross-section σ(E) of both resonant elastic (radiative) scattering and inelastic (internal conversion) scattering is given by the Breit–Wigner formula:

$$\sigma (E)={\sigma }_{0}\frac{{\Gamma \Gamma }_{0}/4}{{(E-{E}_{0})}^{2}+{(\Gamma /2)}^{2}},\quad \Gamma ={\Gamma }_{0}+\Delta \Gamma ,$$
(2)
$${\sigma }_{0}=\frac{4{\rm{\pi }}}{{k}^{2}}\,\frac{2{I}_{{\rm{e}}}+1}{2(2{I}_{{\rm{g}}}+1)}\,\frac{1}{1+\alpha }.$$
(3)

Here σ0 is the total nuclear resonance cross-section; E0 is the resonance energy; Γ0 = Γ1 + Γ2 = Γ1 (1 + α) is the total resonance width, which comprises the elastic and inelastic widths Γ1 and Γ2, respectively; α is the internal conversion coefficient; k = E0/ħc; ħ is Planck’s constant; c is the speed of light in vacuum; and L0 is the photoabsorption length of the incident X-rays in the target. The standard expression for the Breit–Wigner cross-section64 is modified here to take into account an additional broadening ΔΓ to the natural linewidth of the resonance. For other notations, see Table 1.

Using equations (1)–(3), a total incoherent nuclear-resonant yield Σ0 can be calculated. This yield is a relative number reflecting the contributions of all decay products incoherently scattered into 4π by the resonant nuclei in the target and normalized to resonant photon flux F0Γ0 (incident photon flux within Γ0). It can be calculated as

$${\Sigma }_{0}=\frac{1}{{F}_{0}{\Gamma }_{0}}\int {\rm{d}}E\,\sigma (E)N{\int }_{0}^{L}{\rm{d}}x\,F(E,x).$$
(4)

If we assume that the resonant absorption length \({L}_{{\rm{R}}}={[{\sigma }_{0}N({\Gamma }_{0}/\Gamma )]}^{-1}\) becomes large compared with both L and L0 because of a large degree of line broadening (that is, Γ ≫ Γ0), equation (4) can be approximated as

$${\Sigma }_{0}=\frac{{\rm{\pi }}}{2}{\sigma }_{0}\,N{L}_{0}[1-\exp (-L/{L}_{0})].$$
(5)

This equation, however, ignores the possible losses of the nuclear decay products in the target, which will be addressed later.

A partial nuclear-resonance-assisted K-shell X-ray fluorescence yield measured in this experiment can be calculated as

$${\Sigma }_{{\rm{K}}}={\Sigma }_{0}\frac{{\alpha }_{{\rm{K}}}{\omega }_{{\rm{K}}}}{1+\alpha }.$$
(6)

Here αK is a partial K-shell internal conversion coefficient (α = αK + αL + …) and ωK is the K-shell X-ray fluorescence yield. In the case of Sc atoms, ωK = 0.19(1); that is, the fluorescence yield is much smaller than the Auger yield33,65.

In the particular case of the Sc-metal target of thickness L = 25 μm used in the experiment with number density N = 3.98 × 1022 cm−3 of 45Sc nuclei in the Sc metal, L0 = 60 μm and with values of σ0 and αK/(1 + α) ≃ 0.75 (Table 1), we obtain \({\Sigma }_{{\rm{K}}}=0.23\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\) for the anticipated partial nuclear-resonance-assisted K-shell fluorescence yield per incident photon within the natural linewidth. The target thickness L was chosen in the experiment to be about one photoabsorption length of K-shell fluorescent photons (LK ≃ 23 μm) to maximize the nuclear-resonance-assisted K-shell fluorescence yield.

However, from the experimental data, we obtain \({\Sigma }_{{\rm{K}}}^{\exp }=0.44(10)\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\), which is a factor of 1.9(4) larger than the theoretically predicted value. The \({\Sigma }_{{\rm{K}}}^{\exp }\) value was obtained from a raw experimental count rate \({N}_{{\rm{K}}}=0.0162(17)\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\) by correcting it for (1) X-ray beam attenuation of A = 0.44(9) along the beam path at the MID beamline and (2) reduced detection efficiency Deff = 0.084(7) in our experiment. The detection efficiency was lower because of a limited solid angle of detection, limited data acquisition time and absorption of K-shell fluorescent photons in the target and in the air.

The large discrepancy indicates possible overvaluation of the magnitudes of the internal conversion coefficients α and αK in equations (5) and (6). One of the reasons for the overvaluation is that the value of ωK = 0.144(4) applied in refs. 51,53 to determine αK was substantially smaller than the presently accepted value of ωK = 0.19(1) (refs. 33,65).

The internal conversion theory (https://bricc.anu.edu.au/)38 provides smaller values for the internal conversion coefficients for the 12.4-keV to ground-state transition in 45Sc: α = 424, αK = 363 and αK/(1 + α) = 0.854. Using these values results in larger values of the resonant cross-section, σ0 = 1.9 × 10−20 cm2, and of the partial nuclear-resonance-assisted K-shell fluorescence yield, \({\Sigma }_{{\rm{K}}}=0.39\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\). The latter value is in good agreement with the experimental value of \({\Sigma }_{{\rm{K}}}^{\exp }=0.44(10)\,{{\rm{ph}}}_{{\rm{K}}}/{{\rm{ph}}}_{{\Gamma }_{0}}\).

Although the number of the detected nuclear decay events was relatively small (Nd ≃ 93), the number of 45Sc nuclei excited in the experiment was estimated to be much larger amounting to Nd(1 + α)/(αKωKDeff) ≃ 7 × 103.

Nuclear forward scattering

The time dependence of NFS F(t) in the limit of an optically thin nuclear-resonant target irradiated with a broadband pulsed X-ray source at t = 0 is given by11,66

$$F(t)=F(0)\,\exp \left(-{\Gamma }_{\xi }t/\hbar \right),$$
(7)
$$F(0)=4{F}_{{\Gamma }_{0}}\,{\xi }^{2}\,\frac{{\Gamma }_{0}}{{\Gamma }_{\xi }}\exp \left(-L/{L}_{0}\right),$$
(8)

where

$$\xi ={\sigma }_{0}\,NL{f}_{{\rm{LM}}}/4,\quad {\Gamma }_{\xi }=\Gamma +\xi {\Gamma }_{0},\quad \Gamma ={\Gamma }_{0}+\Delta \Gamma ,$$
(9)

and fLM is the Lamb–Mössbauer factor—the probability of recoil-free elastic nuclear-resonant absorption and emission (Mössbauer effect). In Sc metal, it is rather high, with fLM ≃ 0.73 at room temperature T ≃ 300 K, and even at T ≃ 1,000 K it is substantial: fLM ≃ 0.35. These numbers are calculated using the Debye model assuming a Debye temperature of ΘD = 355 K for Sc metal67.

To first order, the Mössbauer effect leads to an unbroadened nuclear resonance line despite atomic vibrations in crystals. However, the second-order Doppler effect can lead both to a temperature-dependent red shift68,69 and to broadening70 of the nuclear resonance line. Typically such thermal broadening is negligible for standard Mössbauer resonances such as that of 57Fe. However, it is not negligible for the 45Sc resonance. According to the theory70, the thermal broadening of the 45Sc line should be ΔΓth ≃ 63Γ0 at T ≃ 2ΘD; ΔΓth ≃ 27Γ0 at T ≃ ΘD/3; and ΔΓth ≃ 0.7Γ0 at T ≃ ΘD/5. Only at T ≃ ΘD/10 does the broadening become ΔΓth ≃ 0.005 Γ0, that is, much less than the natural linewidth. In our experiment, the incident beam heats the target to a temperature much higher than room temperature, most probably resulting in ΔΓth > 50Γ0. Many different mechanisms could lead to resonance broadening; however, thermal broadening is unavoidable even in perfectly homogeneous targets, unless they are cooled to T ≃ ΘD/10, which corresponds to about 35 K for a Sc-metal target.

To determine whether NFS photons can be detected in resonance excitation with an XFEL, we estimate the NFS peak count rate F(0) in our experiment using equations (7)–(9). We assume that in equation (1) the incident spectral flux density on the target is \({F}_{{\Gamma }_{0}}\simeq 0.3\,{\rm{ph}}\,{{\rm{s}}}^{-1}{\Gamma }_{0}^{-1}\); (2) the target sample temperature is high, with T ≳ 2ΘD, resulting in fLM ≃ 0.35 and a thermal resonance broadening ΔΓth > 50Γ0; (3) the nuclear target thickness L = 25 μm and therefore the thickness parameter ξ = 0.16 (with σ0 = 1.87 × 10−20 cm2; see Table 1); and (4) an attenuation factor in the nuclear target \(\exp (-L/{L}_{0})=0.66\). As a result, we obtain for the peak NFS count rate a very small value of F(0) = 0.0004 ph s−1. We also note that the NFS signal decays rapidly, with a decay time constant of less than τ0/50.

If we repeat this calculation for a lower temperature T ≃ ΘD/10, at which ΔΓth ≃ 0 and fLM = 0.9, and choose a target thickness L = 2L0 = 120 μm optimized for NFS11 yielding ξ = 2.0 and \(\exp (-L/{L}_{0})=0.14\), with these values we would obtain a peak count rate of F(0) = 0.23 ph s−1 that is 600 times higher than for T ≳ 2ΘD (assuming there is no other broadening mechanism than the thermal broadening).

This analysis explains why NFS was not detected in our experiment, which we had optimized for the observation of fluorescent photons following internal conversion. It also shows that in the case of negligible inhomogeneous broadening, a marked improvement in the NFS signal is expected if the nuclear target is cooled to T ≃ ΘD/10 and the target thickness is optimized for NFS.

A relatively small inhomogeneous broadening ΔΓ ≲ 500Γ0 is crucial for the observation of NFS. The inhomogeneous broadening can be suppressed by using solid-state magnetic nuclear resonance techniques71,72,73 generalized for the nuclear gamma-resonance case74,75,76 or by time reversal triggered by magnetic field inversion77.

Absolute resonance energy

Absolute values of the wavelengths λ and energies E = hc/λ of X-ray photons are usually measured in terms of the accurately known Si crystal lattice parameter aSi (refs. 78,79) in X-ray Bragg diffraction experiments on high-purity flawless Si single crystals. In this study, we use the crystal lattice parameter aSi = 543.101990(1) pm of Si at 22.5 °C and the linear thermal expansion coefficient γ = 2.581(2) × 10−6 K−1 (ref. 79). The λ-to-aSi relationship is established using Bragg’s law as

$$2{d}_{H}\sin \theta =\lambda (1+{w}_{H}),\quad {w}_{H}={w}_{H}^{(s)}\frac{b-1}{b},$$
(10)

where θ is the glancing angle of incidence between the wavevector K0 of the incoming X-rays and the diffracting atomic planes with diffraction vector H = (h k l) and interplanar distance \({d}_{H}={a}_{{\rm{Si}}}/\sqrt{{h}^{2}+{k}^{2}+{l}^{2}}\) at the Bragg reflection peak. Equation (10) is a generalized Bragg’s law80. It takes into account a small but potentially important refraction correction wH in the crystal in a general case of a scattering geometry with a nonzero asymmetry angle η between the crystal normal \(\widehat{{\bf{z}}}\) and H that is characterized by the asymmetry factor \(b=-\,\sin (\theta +\eta )/\sin (\theta -\eta )\) (Extended Data Fig. 1). The value of wH is determined through its value \({w}_{H}^{(s)}\) in the symmetric scattering geometry with η = 0 (b = −1). To relate λ to aSi the angle θ has to be accurately measured, which is often not straightforward.

In the present study, we apply Bond’s method37, which suggests a measurement of the small angle β = β1 + β2 between two Bragg-reflecting positions of the crystal instead of θ, where βi = θi + ηi − π/2 (Extended Data Fig. 1). The smaller the β, the higher the accuracy of the method.

To this end, we choose two Bragg reflections Hi = (8 0 0) and H2 = (0 8 0) from a Si crystal slab cut parallel to the (1 1 0) planes with an accuracy of Δ ≲ 0.35 mrad. In this case, η = η1 + η2 = π/2, and ηi ≃ π/4 to an accuracy of Δ/2. For the nominal photon energy E = 12.40 keV, the angle θ ≃ 47.5°, which results in a favourably small β ≃ 5°.

Because Δ is small, the following parameters can be considered equal for both reflections with high accuracy: bH = −23.0(2), (bH − 1)/bH = 1.043 and wH = 6.12 × 10−6, assuming \({w}_{H}^{(s)}=5.87\times 1{0}^{-6}\). Under these assumptions, Bragg’s equation in our case reduces to

$$\lambda =\frac{hc}{E}=\frac{{a}_{{\rm{Si}}}}{4(1+{w}_{H})}\sin (\pi /4+\beta /2).$$
(11)

By measuring the angle β between the two Bragg-reflecting peak positions of the Si crystal and using equation (11), we determine the absolute peak values of the photon energy distributions delivered by the XFEL and relate them to the corresponding relative energies measured by the SSS. In this way, the relationship between the relative and the absolute energy scales is established.

The spectral resolution \(\Delta {E}_{{\rm{B}}}=\sqrt{2}\Delta {\Theta }_{i}{D}_{H}\simeq 38\,{\rm{meV}}\) of the Bond spectrometer is defined mainly by the angular spread ΔΘi ≃ 2.4 μrad of the XFEL beam and the Bragg reflection DuMond tangent (dispersion) DH = 11.4 meV μrad−1. Contributions related to the intrinsic energy width ΔEH = 7.7 meV and the angular width ΔθH = 0.7 μrad of each Bragg reflection are smaller.

Following this procedure, the energy of the 45Sc nuclear transition is determined as \({E}_{0}={\mathrm{12,389.59}}_{+0.12{\rm{(syst)}}}^{\pm 0.15{\rm{(stat)}}}\,{\rm{eV}}\). The uncertainties are because of the following main contributions that are larger than the spectrometer resolution ΔEB: (1) A crystal temperature uncertainty of 23.0 ± 0.5 °C, a relative energy error and fit errors add together to a statistical error of ±150 meV. (2) Angular misalignment of the crystal in the off-diffraction planes results in measuring smaller energies. Crystal alignment errors of ±0.12° yaw and ±0.25° roll—rotation about \(\widehat{{\bf{z}}}\) and \(\widehat{{\bf{y}}}\) axes, respectively—result in an approximately +120 meV systematic error derived from numerical calculations using SHADOW81.

Data analysis

To determine resonance energy and width, the experimental raw counts in the incident and detector energy ROIs (Fig. 3b) were normalized using an X-ray gas intensity monitor of MID, binned in incident energy and subsequently fitted using a Gaussian line profile, taking into account the Poisson uncertainty in the experimental count rate. We found that the fit result depends on the bin size and the shift of the binning grid relative to the candidate resonance energy. Therefore, we repeated this analysis for bin sizes between 0.45 eV and 1.11 eV and bin shifts within ±(bin size)/2 around the resonance candidate. This analysis results in a distribution of resonance energies, which we fitted using a Gaussian function. The centre energy of this Gaussian fit is taken as the 45Sc resonance energy reported in the text, and the width (1 standard deviation) as its uncertainty. Similarly, the initial fits with different bin sizes and bin shifts result in a distribution of 45Sc resonance widths. An analogous procedure was applied to this distribution to obtain the 45Sc resonance width and its uncertainty reported in the text. We note that the apparent side bands at Ei − E0 ≃ −3.5 eV and Ei − E0 ≃ 3 eV are artefacts due to noise counts and binning procedure.