Tunable x-ray radiation sources are of wide importance for imaging and spectroscopy in fundamental science, medicine, and industry. The growing demand for highly tunable, high-brightness lab-scale x-ray sources motivates research of new mechanisms of x-ray generation. Parametric x-ray radiation (PXR) is a mechanism for tunable x-ray radiation from free electrons traversing crystalline materials. Although PXR has been investigated over decades, it remained limited in usages due to the low flux and strict dependence on fixed crystal properties. Here, we find new effects hiding in the PXR mechanisms, which provide control over the radiation polarization and spatial and temporal distribution. The radiation can form ultrashort pulses and delta-pulse trains, which makes the new effects fundamentally different from all conventional mechanisms of x-ray generation. We show how these new effects can be created from free-electron interactions with van der Waals materials. Furthermore, we consider free electrons traversing near material edges, which provides an additional degree of tunability in angular distribution and polarization of PXR. Our findings enable us to utilize recent breakthroughs in the atomic-scale design of 2D material heterostructures to provide platforms for creating tunable x-ray pulses.

## I. INTRODUCTION

Energy tunability of x-ray sources is a key factor in numerous applications, such as core-level spectroscopy^{1} and various x-ray imaging techniques.^{2} The energy tunability is usually achieved in undulator facilities, such as synchrotrons and free-electron lasers.^{3} However, the operation of such facilities necessitates immense resources in terms of space, energy, and safety measures, which limit their accessibility. Most laboratory-scale sources take the form of an x-ray tube in which free electrons are used to induce bremsstrahlung or core transitions that produce x rays. These sources cannot provide energy tunability or radiation directionality, which are of importance in a wide variety of usages.

These limitations motivate research into new physical mechanisms for x-ray generation with the potential to create laboratory-scale x-ray sources that are tunable and directional. In particular, research of x-ray-generation mechanisms has led to the development of laser-driven Compton-based x-ray sources^{4} that use the micrometer-scale periodicity of light, which is much smaller than the millimeter/centimeter scales of conventional undulator facilities. Undulators with micrometer-scale periodicity enable significantly smaller accelerations than the conventional facilities while still resulting in tunable x-ray generation. Recent proposals rely on shrinking the undulating periodicity even further by leveraging the high electromagnetic confinement in graphene surface plasmons,^{5} metasurfaces,^{6} metamaterials,^{7} and nanophotonic vacuum fluctuations.^{8} A very recent experiment showed the feasibility and promise of shrinking the undulating periodicity to its absolute limit: using the atomic crystal periodicity of van der Waals (vdW) materials to generate tunable and monochromatic x-ray radiation.^{9}

Parametric x-ray radiation (PXR) is the mechanism of radiation generation from the fundamental interaction of charged particles and crystals. PXR is a type of polarized radiation, similar to Cherenkov and transition radiation (TR). The energy of the radiation is detracted from the particle as part of its inelastic scattering in the crystal,^{9} which is generally described by the Bethe formula^{10} for the mean energy loss and by the Landau–Vavilov formula^{11} for the energy-loss probability distribution. Specifically, PXR is related to the energy transfer from the incident charged particle to the bound electrons of the material, creating polarization currents from which x-ray radiation is emitted. This mechanism of x-ray emission differs fundamentally from other energy-loss processes. For example, bremsstrahlung radiation is produced by the decelerating charged particle when deflected by the Coulomb potential of atomic nuclei and bound electrons.^{12} There, the kinetic energy of the particle can be directly converted into radiation, while in PXR, the kinetic energy transfers first to the structure’s polarization current and from there to radiation.

PXR has the potential to be an effective compact, tunable, monochromatic x-ray source, complementary in some of its aspects to more complex sources, such as free-electron lasers.^{13–16} Nevertheless, it has not been used in practical applications so far due to its limited flux and strict dependence on the fixed crystal properties.^{17} Moreover, PXR was considered so far only as a monochromatic x-ray source, where the emission frequency depends on the emission angle. However, as our work now reveals, the PXR interaction contains surprising new aspects that were not explored before: generating ultrafast x-ray pulses with complex temporal and spatial features. Such novel features have prospects for attoscience and ultrafast optics in the x-ray spectrum.

Here, we show how PXR can be specially designed to generate ultrashort pulses and delta-pulse trains with a controllable period, angular distribution, and polarizability. We specifically show that vdW materials and 2D heterostructures are excellent platforms for our proposal, having unique symmetry properties that provide additional control over PXR emission. Importantly, we find that electron interaction with crystal edges can enhance the emission, change its spatial distribution, and modify the polarization for specific polar angles. Furthermore, we find the regimes for which the emission reaches the ultimate flux theoretically possible for x-ray radiation from any inelastic energy-loss process, as described by the Bethe formula.^{11} Our work provides quantitative results that consider density effects and the impact of incident electron beam divergence on the PXR emission quality. Finally, we compare PXR in vdW materials with competing mechanisms of x-ray generation, bremsstrahlung, and TR and find the conditions for which PXR dominates the x-ray emission.

The generation of x-ray pulses has been shown in a broad range of facilities, e.g., synchrotron radiation light sources that are pulsed at the sub-ns scale (and below), free-electron lasers,^{18,19} high harmonic generation (HHG) in gas, which can generate attosecond pulses in the soft-x-ray region,^{20} and inverse-Compton sources.^{21} All these sources apply strong electromagnetic fields to extract pulsed x-ray emission from the electrons. In contrast, the PXR approach does not require any external field—it is a passive source that relies on the electron–crystal interaction. The ultrafast pulsed nature of x-ray emission that we predict (reaching few attosecond durations and below) arises from the wide bandwidth (BW) of the radiation spectrum, created under specific conditions in PXR. These findings complement previous studies of PXR, which have not considered so far its ultrafast nature.

## II. RESULTS

### A. The resonant conditions of parametric x-ray radiation (PXR)

We find that the intensity of any PXR source can be explained by a combination of two fundamentally different constructive interference conditions (Fig. 1): (1) along the electron’s trajectory, similar to the Smith–Purcell (SP) effect;^{22} and (2) transverse to the electron’s trajectory, which is unique to PXR and exploits the interference between the transverse plane (TP) atoms, e.g., the atoms in the atomic layer of the vdW material. If these conditions are simultaneously satisfied, we get a considerable enhancement of the x-ray emission. Provided the right choice of crystal geometry and particle velocity, such an enhancement can make PXR into the dominant mechanism of x-ray generation and even the dominant mechanism of energy loss of charged particles in the crystal.

### B. General theory

We present the main aspects of the PXR effect by deriving its electric field and Poynting vector and studying them at certain cross sections to identify the resonant angles. Our model assumptions are like previous works: we assume that the incident electron elastic scattering caused by the interaction with the atomic nuclei and atomic electrons is small compared to the material’s depth and interaction length. Therefore, the electron trajectory can be treated as a straight line (paraxial assumption).^{23} Moreover, we neglect other physical phenomena, such as Cherenkov radiation and scattering events that alter the crystal nuclei, since their cross sections are much smaller than the considered mechanism.^{24} Finally, we assume that the electron traverses the material with a constant velocity, as its mean energy loss due to inelastic collisions with atoms in the material is much smaller than its kinetic energy, as given by the Bethe formula.^{10} In the case of ultra-relativistic incident electrons (*γ* ≫ 1) with a normal incidence angle (*θ*_{e} = 0), PXR emission can be derived fully analytically, as we show in Sec. I of the supplementary material.

Traditionally, PXR was treated based on consideration of electromagnetic wavefields inside the crystal and their dispersion relation.^{25} The main difference between this conventional PXR formalism and our new approach is that the former is described in reciprocal space, whereas the latter is described in real space. The derivation results are equivalent. Nevertheless, the derivation in real space holds several advantages: (I) In the case of non-symmetric materials and finite crystals (in the transverse directions), the formalism in real space is easier to implement. For instance, an electron passing near the edge of the target material is more suitable to consider in the real space rather than in reciprocal space. (II) Our approach is of particular value here for isolating the effect of the transverse planes, which is of special importance for vdW materials. (III) In the case of non-ultrarelativistic electrons, the transverse plane electric field of the electrons decay exponentially in space; therefore, only the dipoles near the electron’s trajectory contribute to the PXR emission, which requires a much smaller number of terms to be calculated if we use real space. Thus, in this case, the real space method is less complex than the reciprocal space method. (IV) Our work assumes that the target crystal is thin (<1 *µ*m); therefore, the emitted x-ray field does not get significantly re-scattered or reabsorbed, and the kinematical approximation holds.^{25} In contrast, the reciprocal space approach has an advantage in thicker crystals, since it provides a complete description of the emitted x ray, i.e., the PXR dynamical theory (specifically, the exact description of the extinction and reabsorption of the emitted x ray^{25} can only be given by the PXR dynamical theory).

Here, we focus on the specific case of a 2D hexagonal lattice, which is the most relevant for vdW materials; however, all the results can be generalized to the arbitrary 3D lattice (Sec. 12 of the supplementary material). We begin our analysis with an electron of velocity *β* = *v*_{e}/*c* and wavevector $ke=k0\beta z\u0302=\omega \beta cz\u0302$ traversing a 3D lattice structure spanned by vectors *r*_{g} = *g*_{1}*u*_{1} + *g*_{2}*u*_{2} + *g*_{3}*u*_{3}, where $g=g1,g2,g3\u2208Z3$. To arrive at conditions of efficient x-ray generation, we assume that one of the lattice vectors (*u*_{3} without loss of generality) consolidate with the electron trajectory in the $z\u0302$ axis (the zone axis). We notice that the electron nearfield in free space can be written in the following way^{12} [$\gamma =1\u2212\beta 2\u22121/2$]:

where *ω* is the angular frequency of the emitted radiation, *q* is the electron’s charge, $k\u0302\Vert =z\u0302$ is the momentum direction of the incident electron, $k\u0302\u22a5$ is the perpendicular plane to the electron trajectory, $b=k\u0302\Vert \xd7r$ is the distance between point ** r** and the electron trajectory, and

*K*

_{i}are the modified Bessel functions of the second kind. This electric field is induced on the atomic electrons in which effective dipoles are created given by the atomic polarizability $\alpha ij\omega Ej\omega =Pi\omega $.

^{26}

The electric field induced by the 3D atomic dipoles^{27} equals (at the far-field)

where $fl\omega \u225c\alpha l\omega \omega 2\mu 04\pi r0$ is the scattering factor of the dipoles in the *l*th layer,^{28} $r0=14\pi \u03f50q2mec2$ is the classical radius of the electron, $kph=k0sin\theta ph\u2061cos\phi phx\u0302+sin\theta ph\u2061sin\phi phy\u0302+cos\theta phz\u0302$ is the photon wavevector, and $I\u2194$ is the unit dyad. To have a constructive interference from the dipoles in Eq. (2), all the terms in the scalar product $kph\u2212ke\u22c5rg$ should be integer multiplication of 2*π*. In particular, for a hexagonal lattice, for which the lattice vectors are given by $u3=dzz\u0302$, $u1=dxy32x\u0302+12y\u0302$, and $u2=dxyy\u0302$, the interference conditions have the following form:

Conditions (3a)–(3c) are identical to Bragg conditions for diffraction as given in terms of reciprocal lattice vectors, where the indices $m1,m2,n$ are the Miller indices.^{25} Equation (3c) is the known Smith–Purcell condition for constructive interference from the dipoles parallel to the incident electron trajectory.^{22} Equations (3a) and (3b) correspond to constructive interference of the dipoles in the perpendicular plane to the incident electron trajectory. These conditions can be met simultaneously and are shown in Fig. 1(c) for the hexagonal lattice with azimuthal angle *φ*_{ph} = 0 and normal incidence electron *θ*_{e} = 0.

The electric field in Eq. (1) decays exponentially for $b\rho max>1$, where $\rho max\u225c\gamma v\omega $.^{12} Therefore, the number of dipoles contributing to constructive interference is proportional to $\u221d\pi \rho max2Acell$, where *A*_{cell} is the primitive cell area of the 2D lattice in the transverse plane. To achieve the most intense radiation, we should increase the number of contributing dipoles in the transverse plane, i.e., $\rho max2\u226bAcell$. Under these conditions, the sum term in Eq. (2) can be transformed into an integral, and the PXR electric field is then given by (see Sec. 1 of the supplementary material)

The energy per unit solid angle per angular frequency is

where

Here, Θ_{1}, Θ_{2}, and Θ_{3} represent the phase in Eq. (3) modulus 2*π*, taken in the range of $\u2212\pi ,\pi $. Θ is a dimensionless parameter (phase) that indicates the distance from a resonant point and relates to the intensity of the radiation. *ξ* is an additional dimensionless parameter that indicates the phase between the resonant conditions and relates to the emission polarization. *N* is the number of layers of the vdW material.

Equation (5) is similar to the classical result of PXR as can be found in Ref. 25, excluding the Debye–Waller factor, which describes the decrease in diffracted intensity due to thermal lattice fluctuation. However, as a recent research suggests,^{29,30} the decrease in diffracted intensity for the WSe_{2} material due to this effect is relatively small (up to ∼30%); thus, the Debye–Waller term in our case can be excluded. Equation (5) has several important properties [Figs. 1(d) and 1(e)]. First, the intensity grows as $\u221d\rho max2/Acell\u221d\gamma 2$. This is a significant improvement relative to the SP effect, which increases only logarithmically with *ρ*_{max} and *γ* (see Sec. 12 of the supplementary material). Second, the scattering factor term $f\omega 2sinN2\Theta 3/sin12\Theta 32$ accounts for the dipoles’ interference in the $z\u0302$ direction with *N* layers of interaction. For Θ_{3} ≪ 1, it increases quadratically with the number of layers (∝*N*^{2}) and the emission frequency width decreases by ∝*N*^{−1}. Third, the angular dependence goes as $\u221d\rho max\Theta dxy/\rho max\Theta dxy2+12$ and its width is proportional to ∝*γ*^{−1}. In other words, compared with the SP effect, the PXR peak is much more intense and has a smaller divergence.

### C. Generation of delta-pulse trains and sub-attosecond pulses

General integers $n,m1,m2$ that satisfy Eq. (3) are denoted by PXR of order $n,m1,m2$. The special case of *n* = *m*_{1} = *m*_{2} = 0 is called forward emission PXR, as it is satisfied for *θ*_{ph} = 0. The second special case of *m*_{1} = *m*_{2} = 0 and *n* > 0 is called backward emission PXR, as it is satisfied for *θ*_{ph} = *π*. Figure 2 summarizes the different emission regimes for different solid angles.

Forward emission PXR is the case of a small phase difference between the dipoles in the transverse plane and in the $z\u0302$ axis. For this case, the peak angle is at $sin\theta ph\u22451/\gamma $ with a peak width of ∼2/*γ*, which is a directional radiation that consolidates with the incident electron trajectory. In this case, the frequency spectrum is continuous [Fig. 2(b)]. When looking at the time domain profile of the forward emission PXR, we find an ultrashort pulse with a duration of $\Delta t=Ndz\gamma 2\u2061c\u221dN/\gamma 2$ [as]; therefore, a sub-attosecond pulses can be created by designing an environment with *N* < *γ*^{2}.

Backward emission PXR is the case of a small phase difference between the dipoles in the transverse plane, yet a phase difference between dipoles in the $z\u0302$ axis that forms an integer multiplication of 2*π*. In this case, the spectrum consists of delta-pulse trains with period $d\omega =\pi n\beta cdz$. The time domain profile of the backward emission PXR is a delta-pulse train with an overall duration of $dzN2c\u221dN[as]$, a period of $2dz\beta c\u22483[as]$, and an individual pulse duration in the train of $Ndz\gamma 2c\u221dN/\gamma 2[as]$. Each pulse duration is the same as in the forward emission PXR, yet multiple such pulses appear separated on the attosecond scale. Therefore, by designing the material thickness *N* and the inter-lattice distance *d*_{z}, one can control the delta-pulse duration and period.

### D. Controlling the spatial shape of the radiation and edge PXR

PXR of other orders has a different emission mechanism [Figs. 2(c) and 2(d)]. It has narrow, monochromatic energy with two lobes separated by a distance of 2/*γ*. By an appropriate design of either the emission angle, the incident electron energy, the crystal thickness, or the inter-lattice distance, one can control the spatial shape and polarization of the emission [Fig. 3(c)]. The lobes have a frequency width of $\Delta \omega \u22452cdzN1\beta \u2212cos\theta ph$ and their centers are separated by $\delta \omega =\omega Rsin\theta ph1/\beta \u2212cos\theta ph1\gamma $. The emission shape and polarization are influenced by the ratio between *δω* and Δ*ω*. For the overlapping case (*δω* < Δ*ω*), the emission shape is a donut with radial polarization. For the non-overlapping case (*δω* > Δ*ω*), the emission shape is a two-lobe peak with linear polarization. Notice that *δω* depends on the incident electron energy *γ*, the emission frequency *ω*_{R}, and the emission angle *θ*_{ph}, whereas the lobe width Δ*ω* is proportional to the material thickness *d*_{z}*N* and the emission angle *θ*_{ph}. The most dominant parameter, which influences the spatial shape and polarization, is the emission angle. For polar angles *θ*_{ph} near the poles, the emission is radially polarized with a donut shape. As the polar angle *θ*_{ph} gets closer to the equator, the emission becomes linearly polarized with two lobes. The additional parameters *γ*, *N*, *ω*_{R} also impact the spatial shape and polarization but not as significantly as the polar emission angle [Fig. 3(c)].

The two-lobe angular shape occurs due to the symmetry properties of the material’s dipoles. For an electron that penetrates the target material, the angle between the electron and the material dipoles is distributed uniformly in the range[−π, π); therefore, opposite dipoles cancel each other, and a zero is created in the resonant point [Fig. 3(a)]. When changing the angle away from the resonant condition (even slightly), the cancellation of the opposite dipoles no longer occurs, and the peak intensity is achieved.

Another approach to avoid the cancellation of opposite dipoles is to break the symmetry by an edge effect [Fig. 3(a)]. When the incident electron passes near the edge of the target material, it creates asymmetry in one of the transverse plane axes. The peak intensity remains roughly the same as in case of a bulk interaction, yet with substantial differences in the polarization and emission shape: First, the emitted x-ray spectrum has a single peak, in contrast to the two-lobe peak in the regular bulk case [Fig. 3(b)]. The peak location is found exactly in the resonant point that satisfies Eq. (3). In particular, for backward emission PXR, the peak intensity is now created at *θ*_{ph} = *π* (instead of *θ* = *π* − 1/*γ*). For forward emission PXR, the peak intensity is created at *θ*_{ph} = 0 (instead of *θ*_{ph} = 1/*γ*). Moreover, for emission angles near the equator, the intensity of edge PXR is higher than the regular bulk PXR and is much narrower, which gives additional improvement to the emitted brilliance [Fig. 3(b)].

An additional unique aspect of edge PXR is its spatial distribution—having a dipole shape with azimuthal polarization instead of a donut shape with radial polarization in the regular bulk case [Fig. 3(c)]. The emission angular shape becomes narrower when closer to the equator by a factor of cos *θ*_{ph} relative to the regular bulk PXR. The edge PXR has an azimuthal polarization without dependence on the polar angle in contrast to the regular bulk PXR, for which the polarization alters from radial to linear. Edge PXR has two additional potential advantages: The x-ray reabsorption in the crystal can be reduced,^{31,32} and the incident electron has a potentially longer travel distance without being scattered by collisions inside the crystal.^{33} Despite the donut shape of the regular bulk PXR and dipole shape of the edge PXR, the orbital angular momentum (OAM) of the two emission types is 0 (see Sec. 5 of the supplementary material).

## III. DISCUSSION

Several aspects should be considered regarding the effect of the material geometry on PXR-type radiation. (I) Each crystalline structure has different conditions that need to be met, as can be seen from Eq. (3). For non-crystalline structures, the general conditions are not met, except for the forward emission PXR, which is independent of the material’s geometry. (II) It can be shown that the resonant conditions have an azimuthal rotation symmetry, which depends on the 3D lattice structure (supplementary material, Sec. 1). For a hexagonal lattice, the symmetry holds under azimuthal rotations of *π*/3. (III) A relative advantage of vdW materials or general superlattice structures over monocrystals is in the relatively large and more tunable spacing *d*_{z}.^{9} This degree of freedom allows us to control the delta-pulse train periodicity and pulse durations. Moreover, a larger *d*_{z} makes the Smith–Purcell condition [Eq. (3) and Fig. 1(d)] easier to satisfy, which implies that in the same photon frequency range, more harmonies would appear. Similarly, the number of intersection points between SP and TP will increase, giving extra tunability for the emission angle and emission frequency.

The derivation of PXR emission in Eq. (5) is based on a single electron excitation. However, in certain experimental realizations, the electron excitation contains multiple electrons, i.e., electron bunches with bunch duration *τ*_{bunch}, which is typically much longer than the attosecond scale of the PXR emission. The electron bunch duration dictates the temporal shape of the PXR emission and suppresses the high photon energies as $\u221dNesinc\omega \tau bunch$, where *N*_{e} is the number of electrons in a single bunch (see Sec. 13 of the supplementary material). The electron bunch duration also adds uncertainty to the arrival time of the attosecond x-ray pulse; this is relevant even when each bunch contains only a single electron. We can think about this uncertainty as jitter in the electron pulse, which can generally arise from the mechanisms creating the electron (e.g., the nature of photoemission) or from the jitter in the laser pulse creating the electron. One way to avoid the effective broadening and the uncertainty in the arrival of the x-ray pulse is to use a short enough electron bunch. Recent works showed how attosecond electron pulses can be created using electron–laser interactions that perform electron shaping.^{34–38} This approach also gives control over the arrival time of the electron, which unlocks the possibility to synchronize between the PXR emission and the laser driving the attosecond electron pulses.^{39}

We can estimate the typical broadening of the x-ray pulse from having multiple electrons. Consider a moderate electron current of ∼1 *µ*A, i.e., electrons flux of ∼10^{13} (electrons/s). For the case of a uniform temporal distribution in the electron beam, the average distance between electrons is ∼100 fs, which is much longer than the characteristic temporal shape of x-ray emission (attosecond scale). Thus, for moderate current continuous-wave electron pulses, we do not expect the number of electrons to broaden each individual attosecond PXR x-ray pulse (more in Sec. 13 of the supplementary material). However, for PXR created from electron pulses, the average distance between electrons is smaller, and then x-ray pulse broadening is more likely. Then, methods for creating shorter electron bunches become more important.

Apart from creating short electron bunches, we consider another approach that utilizes the attosecond pulse duration of the PXR x-ray pulses without pre-bunching the electrons. Using a coincidence measurement, for example, by measuring the electron arrival time to a detector (after generating the PXR), we can correlate the electron with the emission of the x-ray pulse. In this option, the temporal resolution is dictated by the electron detector temporal resolution (e.g., often limited to ps timescales due to the electronic synchronization). For better time resolution, we can design pump–probe schemes between the attosecond x-ray pulse and the electron pulse (or another pulse that is synchronized with the electron or created by it). In such schemes, the time resolution is only limited by the ability to delay one pulse relative to the other because any jitter in the pulse creating the process cancels out (since it affects both the electron and the x-ray pulses in the same way). Note that this coincidence measurement approach is also applicable with a continuous source of electrons (e.g., thermionic or field emission as in electron microscopes).

An important conclusion from this discussion is that due to the sensitivity of PXR to the electron beam quality, the preferable electron sources for producing PXR radiation are ones not limited by space charge, as in continuous sources with moderate current or accelerators with high repetition rates. As an example, an acceleration energy of ∼100 MeV and an average electron current of 1 *μ*A, and with target material vdW (WSe_{2}) with 100 nm thickness can achieve a photon flux of ∼10^{7} (photons/s/0.1%BW), which is of the same order as the soft x-ray source generated by high-harmonic generation (HHG).^{20} The photon flux can be increased even further by increasing the electron source current, yet sample cooling should be used to evaporate the heat (the high heat conductivity of certain vdW materials is an advantage here^{9}). An additional advantage of PXR over HHG is that PXR emits in both the soft and hard x-ray regions, whereas HHG emission is only in the soft x-ray region.

Density effects should be considered as they cause corrections to PXR from ultra-relativistic particles. In dense media, many atoms are situated between the incident electron and an atom with impact parameter *b* from the electron. These atoms, influenced by the fast particle’s fields, will produce perturbing fields at the chosen atom’s position, modifying its response to the fields of the fast electron. In other words, each atom is affected by its neighbors in a way that alters its polarizability—shifted relative to its free-space value.^{12} Analytically, the density effect is represented by substituting $\gamma 2\u219211\u2212\beta 2\u03f5\omega $ in Eqs. (4) and (5), where $\u03f5\omega $ is the electric permittivity of the material.^{12} Under the approximation $\u03f5\omega \u22451\u2212\omega p2\omega 2$, where *ω*_{p} is the plasma oscillation frequency, an analytical expression for PXR with the density effect can be obtained (see Sec. 6 of the supplementary material). The density effect impacts both the peak intensity and the angular divergence, setting an upper limit to these quantities, which does not change even when a further increase in electron acceleration energies is performed [Fig. 4(a)]. This effect is considerable for emission frequencies below *ω* < *γω*_{p} and causes a decrease in the intensity by a factor of $1+\gamma \omega p\omega 2$ and an increase in the beam divergence by a factor of $1+\gamma \omega p\omega 2$.

The density effect has a considerable impact on emission brilliance (an important measure of the quality of an x-ray source^{40}). Brilliance considers the number of photons produced per second, their angular divergence and angular spreads, the cross-sectional area of the beam, and the photons falling within a bandwidth (BW) of 0.1% of the central frequency. When examining PXR, both the peak intensity increases as $\u221d\rho max2Acell$ and the angular divergence decreases as ∝*ρ*_{max} [Eq. (5) and Fig. 1(e)]; therefore, the overall brilliance increases as $\rho max3$ (see Sec. 10 of the supplementary material). However, when considering the density effect, it causes saturation both in the peak intensity and in the angle divergence, which limits the brilliance [Figs. 4(b) and 4(c)]. However, despite this effect, PXR brilliance is still up to five orders of magnitude higher than the SP effect.

Another important aspect to consider is the incident electron beam divergence. So far, we have assumed that the incident electron’s momentum is $ke=\omega \beta cz\u0302$, which consolidates with the *u*_{3} lattice vector. However, the electron beam has a finite divergence angle $0\u2264\theta e\u2264\theta emax$, which results in fluctuations of the initial wavevector by $ke=\omega \beta ccos\theta ez\u0302+sin\theta ex\u0302$. $\theta emax$ is the maximal incident angle of the impinging electron. To take the divergence angle into account, we average the PXR intensity over all the possible electron angles. Under the paraxial assumption, $lz\u2061sin\theta e\u226a\rho max=\gamma ve\omega $, where *l*_{z} indicates the sample length, an analytical solution can be derived (see Sec. 7 of the supplementary material). The effect of the divergence angle on PXR is severe for $\gamma >1/\theta emax$ and causes a spread of the emitted x-ray beam [Fig. 4(d)]. This result implies that the x-ray beam divergence cannot be improved more than the incident electron beam divergence. However, for typical values of *γ* and $\theta emax$ found in experimental setups,^{21} this effect is much less severe than the density effect.

To quantify the absolute upper limit on the possible enhancements of PXR, we consider the total energy that can be transferred from the charged particle to the crystal. We compare the total PXR emission energy and the energy loss of the incident electron due to inelastic collisions. The latter is given by the Bethe formula^{10} and gives an upper limit on the total energy of PXR radiation. It can be shown that over a wide range of parameters, most of the energy lost by the incident electron goes to PXR radiation (e.g., $\Delta EPXR\Delta EBethe\u22450.7$; see Sec. 8 of the supplementary material), and most of its flux goes to the forward emission. This is a significant result since it implies that forward emission PXR energy reaches practically the upper limit achievable due to inelastic energy-loss processes. However, forward emission PXR is broadband; therefore, if a monochromatic source is desired, only a small portion of the total energy goes to a monochromatic x-ray generation.

To observe the phenomena we predicted, both the conditions given by Eq. (3) and of $\rho max2\u226bAcell$ should be fulfilled. There are several interesting regimes of parameters in which these conditions can be met. The first regime is of ultra-relativistic electron beams that are brought to interact with the crystalline structure. This can be achieved using wakefield accelerators^{41} that reach GeV energies in a relatively compact lab.^{42} Such systems recently showed accelerated electron beams of high enough quality to enable electron diffraction, which implies low-enough divergences that could also enable the same electron beam to be used for our PXR-type experiments. The second regime can be achieved by using lower photon frequencies and moderate electron energies with DC\RF guns in acceleration energies on the MeV scale.^{43} In this case, we can find significant enhancement of PXR radiation in the hard UV and soft x-ray regions. To enhance these characteristics for the second case, a large inter-layer spacing *d*_{z} should be used as the vdW material.

Additional mechanisms compete with PXR, possibly concealing it, and thus we also compare the PXR emission to the two most important competitors: bremsstrahlung and TR. Bremsstrahlung is produced by the decelerating charged particle when deflected by the Coulomb potential, whereas TR is emitted when a charged particle passes through an interface between two different media. A comparison between the different emission mechanisms is shown in Fig. 5 and an analytical comparison is discussed in Sec. 10 of the supplementary material. For the sake of simplicity, the working assumption in our work is that the target crystal is thin (∼100 nm); therefore, attenuation effects are insignificant in the hard x-ray spectrum for such material thicknesses.^{25} Both bremsstrahlung and TR emission peaks are along the electron trajectory; therefore, they compete only with forward direction PXR and not with other emission angles. As the backward emission is unique to PXR, the delta-pulse train temporal shape is generated only by PXR. In the forward direction, both bremsstrahlung and PXR have broadband emission, which imply that both have ultrashort pulses in the forward direction. However, the TR spectrum decays as ∝*ω*^{−4} compared with PXR decay of ∝*ω*^{−2}; thus, the TR pulse duration is significantly longer than PXR and bremsstrahlung (the pulse duration is inverse of the spectral width, and thus, the difference in the spectral width alters the pulse duration). TR is concentrated around *θ*_{ph} = 1/*γ* and is strong for *ω* < *γω*_{p} and is suppressed significantly for *ω* > *γω*_{p}. TR intensity also increases quadratically with Lorentz factor ∝*γ*^{2}, similar to PXR emission. For incident electron energies of up to 150 MeV, PXR is more dominant than TR, whereas TR is more dominant above this incident electron energy and for low energy photon emission. In order to compare PXR with bremsstrahlung, we use the Bethe–Heitler formula for describing bremsstrahlung emission.^{44} This radiation is concentrated around $\theta ph\u22641\gamma $. Moreover, due to density effects, its spectrum is suppressed up to *ω* = *γω*_{p}. Its peak intensity increases ∝*γ*^{2}; however, it increases only linearly with the material thickness and not quadratically as in PXR. If we compare it with PXR, then as can be seen in Fig. 5 for emission angle *θ*_{ph} = 1/*γ* and up to *ω* < *γω*_{p}, PXR is more dominant than bremsstrahlung, whereas above *ω* > *γω*_{p}, bremsstrahlung is more dominant. Altogether, we find a wide range of parameters for which PXR is dominant.

Finally, to examine the importance of having two transverse dimensions (rather than only one) for enhancing the PXR, we shall also compare PXR with the 2D Smith–Purcell,^{45} where there is a one transverse dimension of dipoles. The main difference is that the intensity of the resonant 2D Smith–Purcell increases only as $\u221dln\rho maxdx2$, which is higher than the 1D SP effect but much smaller than the 3D PXR (see Sec. 10 of the supplementary material). The use of two transverse dimensions for constructive interference is crucial for the quadratic dependence on *ρ*_{max} and *γ*.

## IV. OUTLOOK

To summarize, we propose a novel mechanism for x-ray generation with strong resonant features: Given the right conditions, the suggested mechanism creates attosecond-scale x-ray pulses and delta-pulse trains of sub-attosecond peaks. We also characterized the practical limits (e.g., density effect) and provided quantitative predictions. Altogether, we find three different types of emission, depending on the polar angle *θ*_{ph}: (1) In the forward direction (*θ*_{ph} = 0), we find broadband emission that corresponds to ultrashort donut-shaped pulses with radial polarization. (2) In the backward direction (*θ*_{ph} = *π*), we find a delta-pulse train. (3) In all other emission angles, we find a two-lobe-shaped emission with a narrow spectrum. Moreover, the x-ray emission resonances are symmetric to rotations that depend on the structure’s symmetry (e.g., for hexagonal lattices, it is *π*/3). We show how the spatial shape and polarization of the emitted x-ray beam can be manipulated by different variables—incident electron energy, material thickness, emission angle, and emission frequency. Using an incident electron that traverses along the target material’s edge reveals additional properties, such as much higher intensity for polar angles near the equator, azimuthal polarization, and radiation asymmetries.

Additional aspects should be considered for future work, for example, combining electron micro-bunching. Micro-bunching techniques can generate nano-modulated electrons via emittance exchange,^{46} laser–plasma interactions,^{47} or electromagnetic intensity gratings.^{48} Interestingly, the conditions in Eq. (3) can be met not only by ultra-relativistic electrons but also by using micro-bunched electrons with moderate relativistic energy. The same resonant effects will also occur in this case.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the full mathematical derivation of all key equations in the main text and further discussion, such as PXR theory for bunched electron sources, PXR polarization, and edge PXR.

## ACKNOWLEDGMENTS

We thank Yaniv Kurman and Xihang Shi for stimulating discussions.

This project was supported by the Israel Science Foundation (Grant No. 830/19) and an ISF-NRF grant.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Collected Papers of L. D. Landau*(Pergamon, 1965), pp. 417–424.

_{2}and simulations based on electrostatic potentials that include bonding effects

_{2}