Evidence for MeV Radiation from a Large-Amplitude Crystalline Undulator

The generation of intense, tunable photon beams at sub-Å wavelengths remains a major goal in modern accelerator and radiation physics. High-brilliance radiation in the hard-X-ray and gamma-ray domains is required for applications ranging from nuclear photonics and nuclear-structure studies to particle physics and advanced imaging. Conventional magnetic undulators, whose periods are typically on the centimeter scale, efficiently generate radiation up to the hard-X-ray domain. Extending this concept toward much higher photon energies, however, requires increasingly energetic electron beams and large accelerator and undulator systems. Laser-assisted schemes, such as inverse Compton scattering [1, 2] and laser–ion concepts including the Gamma Factory proposal [3, 4], offer alternative routes to gamma-ray production, but generally rely on high-performance beams combined with intense, high-repetition-rate laser systems.

Crystalline undulators (CU) offer a distinct approach. In an aligned crystal, the interplanar potential can confine charged particles [5], giving rise to microscopic channeling oscillations and their associated radiation [6] (see Figure 1a). In the original CU concept, crystallographic planes are additionally bent with a controlled spatial periodicity (see Figure 1b). A channeled particle then follows the periodically curved planes, so that the bending motion is superimposed on the intrinsic channeling oscillations and generates an additional radiation component [7, 8]. Whereas the characteristic scale of channeling radiation is determined primarily by the microscopic interplanar potential, CU radiation introduces additional geometrical parameters set by the engineered deformation, namely the bending period $\lambda_u$ and amplitude $A_u$. Since crystalline bending periods can be orders of magnitude shorter than those of magnetic undulators, this additional radiation component can reach the MeV range at multi-GeV beam energies.

This concept was developed into the large-amplitude, long-period (LALP) regime, characterized by $A_u \gg d$ and $\lambda_u \gg\lambda_{\mathrm{ch}}$, where $d$ is the interplanar spacing and $\lambda_{\mathrm{ch}}$ is the characteristic channeling-oscillation period [9, 10, 11, 12]. In this regime, the particle follows the periodically curved planes while undergoing much smaller intrinsic channeling oscillations. The engineered bending therefore introduces a distinct transverse motion and a corresponding radiation component on a spectral scale separated from that of channeling radiation. Under suitable conditions, the CU contribution can become comparable to or exceed the intrinsic channeling-radiation component.

Realizing the large-amplitude regime experimentally is demanding because a sufficiently large and uniform periodic bulk deformation must be produced while preserving stable channeling throughout the active volume. Different technologies have been explored, including acoustically bent crystals [13], graded Si$_{1-x}$Ge$_x$ structures [14, 15], surface grooving [16, 17, 18], and periodically patterned surface stresses [19, 20, 21]. An early measurement with 10 GeV positrons at U70-IHEP in a grooved Si crystal revealed a spectral enhancement qualitatively consistent with an undulator contribution [22]. However, the limited signal-to-background ratio and the possible overlap with channeling radiation prevented an unambiguous identification of the undulator component.

Complementary approaches have explored small-amplitude periodic perturbations, $A_u \ll d$, first through resonant coupling between the induced motion and channeling radiation [23, 24]. This line of research later led to the small-amplitude, short-period (SASP) regime proposed in Ref. [25, 26] and investigated experimentally with strained $\mathrm{Si}_{1-x}\mathrm{Ge}_x$ crystals using electron beams at MAMI B [27]. More recently at MAMI, a four-period boron-doped diamond superlattice produced a narrow radiation feature around 1.3 MeV with 855 MeV electrons [28]. This device featured a sub-interplanar modulation amplitude with a 5.0 $\mu$m period, intermediate between the SASP and LALP regimes. Thus, these experiments have explored electron-based configurations with small-amplitude modulations, rather than the conventional LALP regime. Positrons offer a particularly favorable route to CU. Indeed, under channeling, positrons remain confined between atomic planes, i.e. they experience weaker nuclear scattering than for electrons and therefore exhibit much longer dechanneling lengths. This enables the use of thicker crystals with many more undulator periods, resulting in higher intensity and narrower spectral lines, which both contribute to the generation of high-brilliance photon beams. Although high-energy positron beams are currently available at fewer facilities than for electron beams, upcoming sources such as the FCC-ee injector [29] and the proposed CEBAF positron upgrade at Jefferson Lab [30] will provide intense, high-quality positron beams, opening a clear path toward positron-driven CUs as compact, high-brilliance MeV photon sources.

Here we report evidence for CU radiation in the LALP regime using a ten-period, stressor-engineered Si(110) CU and a 35 GeV/$c$ positron beam. The periodic bulk deformation is characterized independently of the radiation measurement by combining high-resolution X-ray diffraction with finite-element modeling. The resulting geometry fixes the expected radiation energy before comparison with the photon data, enabling a direct quantitative test of the connection between crystal geometry, relativistic particle dynamics, and MeV photon emission.

For a periodically bent trajectory, the energy of the $n$th harmonic observed at a polar angle $\vartheta$ is

with, for an approximately harmonic planar potential,

The first term represents the transverse motion imposed

by the periodically bent planes, whereas the second accounts for the intrinsic channeling oscillations [9, 10].

The CU was fabricated by patterning a 400-nm tensile film of $\mathrm{Si}_3\mathrm{N}_4$ in strips on opposite faces of a 160 $\mu$m-thick Si(110) wafer as originally proposed in Ref. [20] and recently optimized in Ref. [21]. The residual stress arises from differential thermal contraction after deposition. The realistic bending profile developed in [21] was included in the simulations [31] of the radiation spectra based on the channeling module [32] of the MBN Explorer software package [33], showing the possibility to observe an undulator peak.

The device employed in this study was fabricated using a refined version of the technique described in Refs. [20, 21]. The detrimental effects of higher spatial harmonics in the bending profile were analysed in Ref. [34], where a sinusoidal surface deformation was proposed as a means of suppressing them. Here, we provide a practical fabrication solution based on a two-dimensional stressor pattern optimized to produce a nearly sinusoidal bending profile throughout the crystal thickness. This stressor-engineering method is the subject of a patent application [35]. The strip geometry was optimized by finite-element modeling (FEM) to produce an approximately sinusoidal deformation across the crystal thickness while suppressing higher spatial harmonics (see Figure 2(a,Figure 2b)).

High-resolution X-ray diffraction at 8 keV independently probed the periodic displacement. Figure 2(c) compares the measured near-surface profile with the FEM prediction using the COMSOL code at the corresponding sampling depth. Their agreement supports the use of FEM to infer the deformation throughout the volume crossed by the positrons. At the crystal midplane, the calculated profile is nearly sinusoidal, with $A_u \simeq1.85$ nm and $\lambda_u = 334$ $\mu$m. A weak long-wavelength curvature, attributed to a small imbalance between the stressor layers on the two surfaces, is retained in the deformation used for the radiation simulations. For the measured deformation and a 35 GeV/$c$ positron beam, the bending contribution corresponds to an undulator parameter $K_u = 2\pi\gamma A_u/\lambda_u \simeq2.38$, yielding $K_{\mathrm{eff}} \simeq2.77$ when the channeling contribution is included. Eqs. (1) and (2) predict an on-axis fundamental energy of

These parameters place the device well within the large-amplitude, long-period regime. The bending amplitude is nearly one order of magnitude larger than the Si(110) interplanar spacing, $A_u/d_{110} \simeq9.6$. The long-period condition can be quantified from the Lindhard critical angle for planar channeling, $\theta_c = \sqrt{2U_0/(pv)} \simeq\sqrt{2U_0/E}$, which for 35 GeV/$c$ positrons in Si(110) and $U_0 \simeq22$ eV gives $\theta_c \simeq35$ $\mu$rad. In the harmonic approximation, the corresponding channeling-oscillation period is $\lambda_{\mathrm{ch}} \simeq\pi d_{110}/\theta_c \simeq17$ $\mu$m, so that $\lambda_u/\lambda_{\mathrm{ch}} \simeq20$.

The experiment was performed at the CERN SPS H2 beam line [Fig. 2(d)]. Two upstream double-sided silicon microstrip detectors [36] reconstructed each incident positron trajectory, and the crystal was mounted on a precision goniometer. Downstream, a dipole magnet swept away charged particles, and the forward radiation passed through two coaxial 40-cm-long collimators (3 mm and 10 mm apertures) before reaching a 23-cm-long trapezoidal BGO calorimeter (2.1 and 2.9 cm$^2$ front and back faces). The first collimator, 28.26 m from the crystal, defined an angular half-acceptance of 53 $\mu$rad $\simeq3.6/\gamma$. The planar channeling orientation was found with a 120 GeV electron beam, whose smaller critical angle allows a more precise alignment, by rotating the crystal while recording the full radiation spectrum.

A differential measurement was performed using two orientations of the same crystal. In the planar-channeling configuration, the periodically bent Si(110) planes were aligned with the incoming positron beam. In the reference configuration, the crystal was rotated by 1.6 mrad, approximately $45\theta_c$. Identical incident phase-space selections were applied to the two data sets: reconstructed tracks were required to cross a common fiducial region and to enter within $\pm\theta_c$ of the mean beam direction.

Figure 3 (top) shows the energy recorded by the forward BGO calorimeter for each positron, on an event-by-event basis. Since the detector integrates the energy carried by all photons entering its angular acceptance, the distributions represent total forward radiative energy loss rather than an inclusive single-photon spectrum. Owing to the restricted forward acceptance, however, spectral structures in the emitted radiation remain visible. Both configurations exhibit a smoothly falling continuum. In planar alignment, a pronounced enhancement emerges between about 6 and 8 MeV, and the yield remains slightly higher up to the end of the displayed range.

To test the CU interpretation, we performed detector-level Monte Carlo simulations with Geant4 [37], using the G4ChannelingFastSimModel [38, 39] for particle transport and radiation in the oriented crystal and that recently was extended to the case of CU [29, 39]. Radiation emission is evaluated within the Baier–Katkov quasiclassical formalism [40, 41]. The FEM-derived deformation profile of the fabricated crystal was implemented in the simulation. The incident positron phase space was sampled from the tracker data, including its resolution, while the experimental collimation and calibrated BGO response were included in the detector-level simulation. The planar-channeling simulation reproduces the position and overall shape of the observed peak excess, whereas no localized feature remains in the misaligned configuration [Fig. 3, center]. Below approximately 5 MeV, the measured yield exceeds the simulation in both configurations, possibly because of residual beam-line background not included in the model. The simulation also predicts a broader second-harmonic contribution at higher energies, for which only a weak indication is present in the data due to limited statistics.

Figure 3 (bottom) shows the simulated single-photon spectra for both the oriented CU and the misaligned configuration. In the CU case the photon spectrum is strongly enhanced, whereas the enhancement is less pronounced in the energy-deposition distribution. This happens because the intense channeling radiation in the harder part of the spectrum shifts events towards higher total energy losses, which depletes the soft part of the energy-loss distribution. We then repeated the simulation for a straight Si crystal of the same thickness in the same planar-channeling configuration, keeping the beam, channeling, collimation and detector conditions identical. In this case the localized structure near 7 MeV disappears, and the spectrum is close to that of the misaligned crystal. This indicates that the feature cannot be attributed to conventional channeling radiation alone, but requires a periodically bent crystal. Indeed, for the present parameters, planar channeling radiation features a broad spectrum peaking in the hundred-MeV range and contributes negligibly in the 4–17 MeV region.

To quantify the excess, the deposited-energy spectrum measured in the misaligned configuration (Figure 3, top) was fitted with a smooth continuum, $f_{\mathrm{mis}}(E)=a/E+b$. The aligned spectrum was fitted with the same continuum plus a Gaussian of mean $\mu$ and standard deviation $\sigma_{\mathrm{G}}$, providing a phenomenological parametrization of the enhancement. The fit yields $\mu= 6.8 \pm0.3$ MeV and $\sigma_{\mathrm{G}} = 0.7 \pm0.3$ MeV. Under the null hypothesis that, in the absence of an undulator contribution, the aligned continuum coincides with the misaligned continuum, the aligned yield was compared bin by bin with $f_{\mathrm{mis}}$ within the window $[4.5,9]$ MeV. This interval was determined a priori, based on Geant4 prediction. The comparison used a $\chi^2$ test with six degrees of freedom, accounting for both Poisson counting uncertainties and uncertainties in the fitted background. The resulting p-value corresponds to a significance of $4.05\sigma$.

The measured peak position agrees with both the Geant4 simulation and the 7.2 MeV fundamental predicted from the XRD-validated deformation model, which fixes $A_u$ and $\lambda_u$ independently of the radiation data. The slight downward shift and broadening of the peak arise from the finite resolution, the wide collimation angle, and the calorimetric measurement of the total forward energy, all included in the simulation.

The 53 µrad acceptance, approximately $3.6/\gamma$, integrates photons emitted at different angles and hence, according to Eq. (1), at different energies, which broadens the observed fundamental. To illustrate the spectrum for a collection geometry typical of undulator sources, we simulated the same ten-period crystal for the 20 GeV positron beam foreseen at the FCC-ee injector [29], with an angular half-acceptance of $1/\gamma\simeq25$ µrad. As shown in Figure 4, the reduced acceptance yields a pronounced fundamental above 4 MeV together with a weaker higher-harmonic structure. Using the nominal beam intensity and transverse phase-space parameters of Ref. [29], the simulated peak spectral brilliance would be approximately $1.8 \times10^{21}$ photons s$^{-1}$ mm$^{-2}$ mrad$^{-2}$ (0.1% BW)$^{-1}$, comparable to the design peak brilliance of advanced inverse-Compton gamma-ray sources in the MeV domain [1, 2, 42, 43] and exceeding that of operating facilities such as HIγS [44].

The 20 GeV projection illustrates how the experimentally tested stressor-engineering concept can be extended beyond the parameters of the present device. The $\lambda_u = 334$ µm period realized here should be regarded as an experimentally tested operating point rather than a technological limit. The lithographic stressor approach provides a straightforward route to extending the present ten-period pattern to several tens of periods, potentially yielding a narrower and more intense harmonic structure, with the useful number of periods primarily constrained by dechanneling. Shorter bending periods provide a complementary route: when $A_u$ is co-optimized to control $K_{\mathrm{eff}}$, they can shift the fundamental toward higher photon energies or allow a given photon energy to be reached at lower beam energy. They can also increase the number of oscillations within the available channeling length, provided that the increased curvature does not substantially shorten the dechanneling length. For positrons this constraint is much less severe than for electrons, which makes the positron-based LALP scheme a natural route toward thicker crystals with more undulator periods and higher brilliance. Achieving the large-amplitude regime at shorter periods requires stronger bending stresses, for which coactive thin films provide a promising route [45]. Because the active bulk itself is periodically deformed, no separate straight backing section, such as that used in the recent diamond-superlattice realization [28], is required, avoiding a distinct channeling-radiation background.

In summary, our measurement provides evidence for CU radiation from 35 GeV/$c$ positrons channeled through a ten-period, stressor-engineered Si(110) crystal, quantitatively consistent with the engineered crystal geometry. The planar-channeling distribution exhibits a localized excess at $6.8 \pm0.3$ MeV, in agreement with Monte Carlo simulations and consistent with the 7.2 MeV fundamental predicted analytically, with a statistical significance of $4.05\sigma$. No corresponding feature is observed after crystal misalignment, and in simulations it disappears when the periodic bending is removed while planar channeling is retained. Owing to the longer dechanneling length of positrons, this approach opens a favorable route toward thicker CUs with more periods and, as projected for the FCC-ee injector, a peak brilliance comparable to inverse-Compton sources. These results establish a quantitative link between engineered crystal geometry and tunable MeV radiation.

Acknowledgments

This work was supported by the European Union through the EIC Pathfinder Open project TECHNO-CLS (Grant Agreement No. 101046458) and by the INFN CSN5 projects CORAL and GEANT4INFN. We acknowledge ISCRA for awarding this project access to the LEONARDO supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CINECA (Italy). We gratefully acknowledge the CERN SPS and PS accelerator and beam-line staff and the CERN Geodetic Metrology (BE-GM) team for their excellent support during the experiment. The stressor-engineering technology used in this work is the subject of Italian patent application IT 102024000022335, jointly owned by INFN, the University of Padua, and the University of Ferrara [35].

References