Parametric x-ray radiation (PXR) is a prospective mechanism for producing directional, tunable, and quasi-coherent x-rays in laboratory-scale dimensions, yet it is limited by heat dissipation and self-absorption. Resolving these limits, we show the PXR source flux is suitable for medical imaging and x-ray spectroscopy. We discuss the experimental feasibility of these findings for a compact commercial PXR source.

X-ray sources have been detrimental for many applications since their discovery by Röntgen in 1895,1 from medical diagnosis and treatment to electronic inspection, food security, pharmaceutical quality control, and border security. Despite their widespread use, the x-ray generation mechanism in laboratory-scale facilities has remained relatively unchanged since the first x-ray tubes, i.e., electrons accelerate from a cathode and impact a target anode placed in a vacuum tube. The two main mechanisms in such x-ray tubes are bremsstrahlung and characteristic x-ray radiation. Recent advances have increased x-ray tube brightness by micro-focused sources and liquid-jet anodes,2 enabling new applications such as phase-contrast imaging and high-resolution diffraction.3,4 Notwithstanding these advances, the fundamental limitations of x-ray tubes remained the same—their low-brightness, broadband and isotropic emission.

Parametric x-ray radiation (PXR) is a promising mechanism for producing compact and tunable x-rays due to its high spectral yield and large field-of-view.5 PXR is produced from the interaction between relativistic electrons and a periodic crystal structure.6 When a collimated electron source beam impacts a crystal, it induces polarization currents on the target material atoms. Each induced material atom acts as a radiating dipole. When the Bragg condition of constructive interference between the dipole array holds, an intense, directional, and quasi-monochromatic x-ray beam is emitted at a large angle relative to the electron trajectory. The expression for the PXR emission energy is given by7,8
EPXR=ωPXR=2πcdhklsinθB1ϵ0βcosΩ,
(1)
where ℏωB is the emitted Bragg energy; dhkl is the d-spacing of the Bragg plane, which corresponds to Miller indices (hkl); ɛ0 is the constant part of the medium permittivity (ɛ0 ≈ 1 for x-rays); θB is the angle between the incident electron and the Bragg plane; Ω is the emission angle of the PXR photon relative to the electron beam; and β = v/c is the normalized velocity of the electron beam. Bragg’s law is satisfied for the condition Ω = 2θB. This relation allows PXR energy tunability in experiments by rotating the PXR crystal, i.e., altering the Ω and θB angles.9 The PXR photon energy is effectively independent of the incident electron energy for relativistic electrons with energy above 5 MeV. The photon energy is determined solely by the spacing between the crystal planes and the experimental geometry that depicts the angles.

PXR has been extensively investigated over decades,10 both theoretically and experimentally, and has been demonstrated in practical applications such as phase-contrast imaging using differential-enhanced imaging, x-ray absorption fine structure, x-ray fluorescence, and computed tomography.11 Despite significant research progress, the main limitation preventing the widespread use of PXR is its limited flux. For example, practical mammography imaging requires an x-ray beam rate of ∼105106photonss · mm2, yet the maximal PXR flux achieved in recent experiments is more than two orders of magnitudes lower.12 

Here, we identify the underlying mechanisms limiting the PXR flux and propose methods to break these limits (Fig. 1). In particular, we derive upper bounds for the PXR sources’ performance, showing how they can reach adequate levels for in vivo imaging applications. Two parameters determine the PXR source flux—the electron source current (i.e., the number of electrons that pass through the crystalline per time unit) and the yield (i.e., the average number of photons produced per electron). The usable electron beam current is limited since it heats the PXR crystal.13 The yield of PXR is high relative to other electron-driven sources,5 and yet, it is substantially reduced by the self-absorption of the emitted x-ray photons within the PXR crystal.14 Below, we show how novel designs of PXR structures, as well as optimization of the electron source parameters, can break these limits.

FIG. 1.

Parametric x-ray radiation (PXR) source: limitations and proposed solutions. (a) A typical parametric x-ray source setup includes (right to left): an electron source and acceleration structure, a PXR crystal that produces a direction and quasi-monochromatic x-ray beam in a large emission angle relative to the electron trajectory, a crystal monochromator for filtering the PXR beam from the bremsstrahlung noise floor (supplementary material, Sec. III), and a collimator with an exit window. (b) At non-zero temperatures, the PXR crystal heats up from the electrons. This phenomenon amplifies the thermal vibration within the crystal lattice, which in turn contributes to a phase mismatch between the lattice dipoles, leading to the PXR yield reduction. (c) We propose to optimize the heat dissipation within the PXR crystal using a pulsed electron source. The electron source parameters that are used as part of the optimization are the pulse charge (Qpulse), the repetition rate (fR = 1/τR), and the electron beam spot size (A=πRbeam2). (d) The PXR photons self-absorption within the crystal. Close to the crystal surface, the amount of PXR photons produced is linear with the material thickness. However, PXR photons that emit in deeper regions must traverse through the entire crystal, contributing significantly less than PXR photons produced at the surface of the crystal. Hence, the material absorption length limits the PXR yield. (e) We propose two geometrical schemes to mitigate the PXR photons’ self-absorption: the stacked multiple crystal PXR and the edge PXR. The first scheme is composed of multiple crystals, each thinner than the absorption length. The distance between the crystals is long enough such that the trajectory of the emitted photon would not go through the adjacent crystal. In the edge PXR crystal scheme, the electron beam passes near the edge of the crystal, such that the emitted photon travels a distance lower than the absorption length of the material.

FIG. 1.

Parametric x-ray radiation (PXR) source: limitations and proposed solutions. (a) A typical parametric x-ray source setup includes (right to left): an electron source and acceleration structure, a PXR crystal that produces a direction and quasi-monochromatic x-ray beam in a large emission angle relative to the electron trajectory, a crystal monochromator for filtering the PXR beam from the bremsstrahlung noise floor (supplementary material, Sec. III), and a collimator with an exit window. (b) At non-zero temperatures, the PXR crystal heats up from the electrons. This phenomenon amplifies the thermal vibration within the crystal lattice, which in turn contributes to a phase mismatch between the lattice dipoles, leading to the PXR yield reduction. (c) We propose to optimize the heat dissipation within the PXR crystal using a pulsed electron source. The electron source parameters that are used as part of the optimization are the pulse charge (Qpulse), the repetition rate (fR = 1/τR), and the electron beam spot size (A=πRbeam2). (d) The PXR photons self-absorption within the crystal. Close to the crystal surface, the amount of PXR photons produced is linear with the material thickness. However, PXR photons that emit in deeper regions must traverse through the entire crystal, contributing significantly less than PXR photons produced at the surface of the crystal. Hence, the material absorption length limits the PXR yield. (e) We propose two geometrical schemes to mitigate the PXR photons’ self-absorption: the stacked multiple crystal PXR and the edge PXR. The first scheme is composed of multiple crystals, each thinner than the absorption length. The distance between the crystals is long enough such that the trajectory of the emitted photon would not go through the adjacent crystal. In the edge PXR crystal scheme, the electron beam passes near the edge of the crystal, such that the emitted photon travels a distance lower than the absorption length of the material.

Close modal

We specifically propose a path to optimizing the electron source current through an in-depth analysis of the thermal load and the heat dynamics in the PXR crystal. Within this context, we derive an upper bound for the average electron beam current that can effectively traverse the PXR crystalline material. Furthermore, we explore the practical feasibility of utilizing state-of-the-art electron sources to attain the required average current levels.

To enhance the PXR yield, we introduce and evaluate advanced PXR geometrical configurations intended to mitigate the issue of PXR self-absorption within the crystal. This approach results in a substantial enlargement of PXR yield, particularly for the lower PXR energy ranges. We discuss the experimental possibilities and challenges associated with implementing these geometric schemes.

The two results sub-sections focus on the two main challenges limiting PXR sources and suggest possible solutions. The first is optimizing the heat dissipation, and the second is designing different geometries to overcome the PXR self-absorption within the crystal. Our proposals and findings constitute a promising route for a practical PXR source construction for wide range of applications.

The PXR yield depends on several factors, including the target material, the crystal geometry, the diffraction efficiency (χg), the thermal load on the crystal (captured by the Debye–Waller factor), and the geometrical factor (fgeo). In the framework of PXR kinematical theory, the photon distribution emitted from a single electron is given by15 
dNPXRdθxdθy=α4πωBcsin2θBfgeoχg2e2WNθx,θy,
(2)
where α is the fine-structure constant; ωB is the emitted PXR photon energy; c is the speed of light; θB is the Bragg angle; e−2W is the Debye–Waller factor, which captures thermal effects; χg is the Fourier expansion of the electric susceptibility, describing the diffraction efficiency; Nθx,θy is the PXR angular dependence; and fgeo is the geometrical factor that describes the PXR photon self-absorption during the emission process. The PXR photon energy (ωB) and the Bragg angle are related by the condition for constructive interference between the material’s dipoles [Eq. (1)].
The PXR angular dependence, Nθx,θy, is given by
Nθx,θy=θx2cos22θB+θy2θx2+θy2+θph22,
(3)
where θx is the angle in the diffraction plane, θy is the angle perpendicular to θx in the diffraction plane, and θph2=γe2+ωpω2, where ωp is the plasma frequency of the material. Equation (3) is valid for relativistic electron beams (γe ≫ 1, where γe is the Lorentz factor of the electron) and for small scattering angles relative to the Bragg angle (i.e., θx, θy ≪ 1). In this work, we focus on the regime where the density effect is negligible, i.e., electron energies that satisfy 1 ≪ γeω/ωp, leading to θphγe1. We identify this regime as the most suitable for the high-quality x-rays required for applications. Generally, PXR has been demonstrated with electron beam energies ranging from ultra-relativistic beams of several GeV16,17 to electron beam energies of a few MeV18,19 and below.20–24 Notably, lower electron energies lead to a larger PXR beam divergence due to the emission dependence of θphγe1. Therefore, the optimal electron energy should be determined considering the experimental geometry and the required field of view.

Naturally, increasing the PXR source brightness involves efficiently transmitting a large number of electrons through the PXR crystal while minimizing the beam spot size. However, this approach has a drawback: the electron flux deposits energy within the crystal, resulting in considerable heating. This heating, in turn, induces thermal vibrations that affect the PXR yield. Thus, to attain the highest achievable PXR flux, we optimize the heat load imposed on the PXR crystal. Our optimization process involves estimating the PXR crystal’s temperature as a function of various factors, including the PXR crystal itself, the electron source’s current, repetition rate, and spot size. Subsequently, we derive an optimal upper limit for the current density traversing the target. We assume the electron source is pulsed, with a pulse duration (τpulse) much shorter than the thermal conductivity timescale.

1. Heat transfer from electrons to matter

Relativistic electrons lose a small fraction of their kinetic energy when passing through the PXR crystal. The energy loss goes partially into radiation emission (i.e., bremsstrahlung) and partially into heat. To estimate the energy loss of the electron that transfers into heat, we calculate the inelastic collision-stopping power, describing the average energy loss per unit length due to Coulomb collisions. The Bethe–Bloch formula describes the mean electron energy loss due to this process,25 
dEedx=4πNZz2e4mve2ln2γe2mve2Tmaxω2β2δβγ2,
(4)
where Z is the material atomic number, N is the material density, ve = βc is the electron velocity, γe is the Lorentz factor, m is the electron rest mass, ω⟩ is the mean excitation potential, Tmax is the maximum energy transfer in a single collision, and δ is the Fermi’s density correction. The typical values of the mean energy loss are 2(MeVcm2/g) (supplementary material, Sec. I). According to Eq. (4), consequently, the electron energy loss increases linearly with the atomic number Z, i.e., heavier materials carry a higher heat load.
The heat from a single accelerator pulse is deposited in a volume determined by the size of the electron beam spot and the thickness of the PXR crystal. Assuming the cooling is negligible during the pulse, the temperature load (ΔT) in this volume can be expressed by (supplementary material, Sec. I B)
ΔT=dEe/dxCpρQpulseA,
(5)
where ⟨dEe/dx⟩ is the average electron energy loss per unit length given by the Bethe–Bloch formula, ρ is the PXR crystal mass density, Cp is the PXR crystal specific heat capacity, Qpulse is the total charge per second, and A is the electron beam spot area. The temperature load depends not only on the number of electrons that impact the crystalline (Qpulse) but also on the active beam area (A). As the electron beam is more concentrated, the heat load increases.

In Fig. 2(a), we illustrate the temperature load characteristics for various materials (W, Mo, Cu, Si, and HOPG) as a function of the incident electron energy, assuming an electron beam with a pulse charge of Qpulse = 700 nC and beam dimensions of A = 1 mm2. Analyzing the temperature dependency on the incident electron energy reveals that at lower energies, electrons experience more substantial energy losses. For incident electron energies larger than ∼1 MeV, the energy loss shows a logarithmic increase with electron energy. Considering the PXR target materials, tungsten displays the highest temperature load due to its substantial electron energy loss, yet its melting temperature is higher relative to the other materials examined. Graphite is the optimal choice for heat dissipation, characterized by both a low-temperature load and a high melting temperature (supplementary material Sec. I).

FIG. 2.

Heat dissipation impact on parametric x-ray radiation for different crystals. (a) An incident electron beam with an energy of Ee and a pulse charge of Qpulse impacts the target crystal, assuming the pulse duration is much shorter than the dissipation process characteristic time. During this process, energy is transferred to the crystal and converted to heat [Eq. (5)]. The heat is dissipated in two ways—through thermal conduction and black-body radiation. The graph displays the temperature rise of an incident electron with a pulse charge of Qpulse = 700 nC and a beam area of A = 1 mm2 as a function of the incident electron energy and different materials. (b) Two extreme cases for heat dissipation: thermal conduction dominance and black-body radiation dominance, determined by crystal thickness [Eq. (7)]. Thinner materials show faster heat dissipation, allowing the potential absorption of higher electron beam currents. (c) Similar to the rotating anode in x-ray tubes, the target crystalline can be translated to increase the incident electron beam current. (d) The spatial heat dissipation process after the electron pulse arrival is presented. The characteristic time for the heat dissipation is Rbeam2/4D.

FIG. 2.

Heat dissipation impact on parametric x-ray radiation for different crystals. (a) An incident electron beam with an energy of Ee and a pulse charge of Qpulse impacts the target crystal, assuming the pulse duration is much shorter than the dissipation process characteristic time. During this process, energy is transferred to the crystal and converted to heat [Eq. (5)]. The heat is dissipated in two ways—through thermal conduction and black-body radiation. The graph displays the temperature rise of an incident electron with a pulse charge of Qpulse = 700 nC and a beam area of A = 1 mm2 as a function of the incident electron energy and different materials. (b) Two extreme cases for heat dissipation: thermal conduction dominance and black-body radiation dominance, determined by crystal thickness [Eq. (7)]. Thinner materials show faster heat dissipation, allowing the potential absorption of higher electron beam currents. (c) Similar to the rotating anode in x-ray tubes, the target crystalline can be translated to increase the incident electron beam current. (d) The spatial heat dissipation process after the electron pulse arrival is presented. The characteristic time for the heat dissipation is Rbeam2/4D.

Close modal

2. Regimes of heat diffusion

During the time between the accelerator pulses, the heat is both thermally conducted in the direction of the edge of the crystalline and partially dissipated by the black-body radiation through the crystalline surfaces [Fig. 2(b)]. The temperature profile in the crystalline T(r, t) can be derived from the heat equation,26 
ρCpTtκ2T=PsourcePsinkεσLT4Tenv4,
(6)
where κ is the heat conductivity, Psource is the power per unit volume deposited in the crystalline by the electron beam, and Psink is the power per unit volume that is cooled at the edge of the crystalline. The last term in Eq. (6) represents the black-body radiation, where ε is the material emissivity, σ=5.67×108Wm2K4 is the Stefan–Boltzmann constant, L is the material thickness, and Tenv is the environment temperature. For the time between the electron pulses, Psource = 0.
The heat diffusion equation shows two heat dissipation regimes of extreme behaviors—the first is characterized by thermal conduction as the dominant heat dissipation process, while the second is characterized by the dominance of black-body radiation. The material thickness L and the electron beam active area A affect these behaviors. The black-body radiation becomes the dominant heat dissipation mechanism for thin materials and sufficiently large electron beam dimensions. In such cases, the active area emitting black-body radiation exceeds the thermal conductivity within the heat load volume. We will define a characteristic length scale that governs which behavior dominates
LHD=εσATmax34κ,
(7)
when LLHD, the black-body radiation is the dominant heat dissipation mechanism, whereas for LLHD, the thermal conduction is the dominant one.

Figure 2(b) shows the heat dissipation process for the different regimes. To illustrate, we use a tungsten PXR crystal and examine the temperature profile as a function of the crystal thickness under the following assumptions: the initial temperature is T = 2500 K, and the electron beam active area is A = 1 cm2. When the material thickness is L = 100 µm, thermal conduction governs, leading to a relatively slow dissipation process. In contrast, when the material thickness is L = 1 µm, the black-body radiation dominates, significantly accelerating the heat dissipation.

For the rest of this work, we assume thermal conductivity is the dominant regime, as it sets a more stringent bound on the electron beam current impacting the target crystal. This assumption holds in most of the experimental cases (LLHD). Nevertheless, when working in the black-body radiation regime (i.e., thin materials), the potential electron beam current can be an order of magnitude higher than in the case of only thermal conductance. This factor is especially advantageous for materials whose absorption length is in the order of ∼ μm with a high melting temperature, such as tungsten, which allows for higher electron beam currents.

Figure 2(d) shows the spatial temperature profile after the electron pulse transition through the PXR crystal. Immediately after the end of the electron pulse, the temperature in the active beam area (A=πRbeam2) reaches its maximum and is determined by the temperature load [as defined by Eq. (5)]. At this stage, the thermal diffusion process initiates. The heat diffusion rate is characterized by the thermal diffusion coefficient D ≜ κ/ρCp. The typical thermal diffusion coefficients of the examined materials in this paper are 0.5–1 (cm2/s) (Table I). We define the characteristic diffusion time by τDRbeam2/4D as the time elapsed from the end of the electron pulse until the temperature in the center of the beam dropped to Tmax1e1. The timescale of the heat diffusion process depends on the beam area τDRbeam2, i.e., a larger beam area leads to a longer dissipation time. This indicates that enlarging the pulse charge by increasing the beam area (while maintaining the pulse charge density) will proportionally extend the dissipation process.

TABLE I.

Optimal electron source current for an electron beam radius of 1 mm.

Material andDiffusionOptimal repetitionMaximal pulseMaximal average
Bragg planecoefficient (cm2 s−1)rate (Hz)charge (μC)current (mA)
HOPG [002] 0.93 372 10 3.77 
Si [111] 0.92 365 4.4 1.61 
Cu [111] 1.163 465 2.12 0.99 
Mo [110] 0.537 214 3.1 0.66 
W [110] 0.695 278 2.2 0.62 
Material andDiffusionOptimal repetitionMaximal pulseMaximal average
Bragg planecoefficient (cm2 s−1)rate (Hz)charge (μC)current (mA)
HOPG [002] 0.93 372 10 3.77 
Si [111] 0.92 365 4.4 1.61 
Cu [111] 1.163 465 2.12 0.99 
Mo [110] 0.537 214 3.1 0.66 
W [110] 0.695 278 2.2 0.62 

3. The effect of heat load on the PXR yield

So far, we have addressed the broader concept of heat dissipation within a crystal without specifically examining its impact on the PXR yield. To address this aspect, the vibrations of the crystal atoms influenced by the Debye–Waller factor must be considered.27 These vibrations arise from two distinct phenomena. The first is purely quantum mechanical and arises from the uncertainty principle. These vibrations are independent of temperature and occur even at absolute zero temperatures. For this reason, they are known as zero-point fluctuations. At finite temperatures, elastic waves (or phonons) are thermally excited in the crystal, thereby increasing the amplitude of the vibrations. Those thermal vibrations cause PXR phase loss between the lattice dipoles, leading to a decrease in the PXR yield. This effect depends on the material-specific Debye temperature, TD, the material temperature, T, and the d-spacing of the diffraction plane of interest, dhkl.

The first quantity of interest is the mean square amplitude of the thermal vibration of the crystal, u2(T). This quantity is given by13 
u2T=324MkBTD1+4TTD20TD/Tyey1dy,
(8)
where M is the material mass and kB is the Boltzmann constant. The Debye–Waller term W is calculated from u2(T) and the reciprocal lattice vector τ = 2π/dhkl using the relationship e2W=expτ2u2(T).

Figure 3(b) shows the influence of the Debye–Waller factor on the PXR yield. A higher temperature load leads to an exponential reduction in the PXR yield, emphasizing the trade-off between the electron beam current and the Debye–Waller factor. In other words, a higher electron beam leads to a higher crystal temperature, which results, in turn, in a drop in the PXR yield. Due to this phenomenon, target temperature optimization is essential for achieving a maximal PXR flux.

FIG. 3.

Optimizing the PXR source by designing the electron beam source to reduce heat dissipation. (a) A pulsed electron source with parameters including the pulse duration (τpulse), pulse current (Ipulse), and source repetition rate (fR = 1/τR) interacts with the PXR crystal. During each pulse, the crystal temperature rises from Tmin to Tmax and then exponentially returns to Tmin during the relaxation time. (b) The Debye–Waller factor (DWF) impacts the PXR yield. As the temperature increases, the yield decreases exponentially, with a strong dependence on the d-spacing (dhkl) of the Bragg planes. (c) Determining the optimized PXR temperature and electron beam current. Graphite exhibits excellent heat dissipation due to its high Debye–Waller factor (DWF). (d) Finding the optimal repetition rate. The optimized repletion rate depends on the incident electron beam radius and the diffusion coefficient of the material.

FIG. 3.

Optimizing the PXR source by designing the electron beam source to reduce heat dissipation. (a) A pulsed electron source with parameters including the pulse duration (τpulse), pulse current (Ipulse), and source repetition rate (fR = 1/τR) interacts with the PXR crystal. During each pulse, the crystal temperature rises from Tmin to Tmax and then exponentially returns to Tmin during the relaxation time. (b) The Debye–Waller factor (DWF) impacts the PXR yield. As the temperature increases, the yield decreases exponentially, with a strong dependence on the d-spacing (dhkl) of the Bragg planes. (c) Determining the optimized PXR temperature and electron beam current. Graphite exhibits excellent heat dissipation due to its high Debye–Waller factor (DWF). (d) Finding the optimal repetition rate. The optimized repletion rate depends on the incident electron beam radius and the diffusion coefficient of the material.

Close modal
In Fig. 3(a), we outline the scheme, where an electron pulse with a duration of τpulse and a pulse charge of Qpulse traverses the target crystal with a repetition rate of fR = 1/τR. We assume the pulse duration is significantly shorter than the thermal diffusion characteristic time, i.e., τpulseτD. During the electron pulse, the crystal temperature at the impact area increases by ΔT = TmaxTmin [Eq. (5)]. Following the end of the electron pulse, the crystal temperature drops exponentially, with the assumption that only thermal conductivity takes place in the heat dissipation process at this stage. The objective is to optimize the values of the electron beam dimensions Rbeam, the repetition rate fR, and the pulse charge Qpulse as a function of the target material type and the dimensions. The optimized values are given by (supplementary material, Sec. I F)
Topt=minMkBTD2dhkl212π22,Tmelt,fR=4D/Rbeam2,
(9)
and the optimal electron source current is given by
Iopt=4πexp1κToptdEe/dx.
(10)

This result leads to intriguing outcomes. First, the optimal maximal temperature is lower than the melting temperature and depends on the inter-lattice distance dhkl2. As the inter-lattice distance decreases, the optimal temperature drops. This drop can be intuitively understood as the thermal vibrations are more severe for lower inter-lattice distances, where the relative phase shift is inversely proportional to the inter-lattice distance. Second, the optimal current (Iopt = QpulsefR) does not depend on the beam area since the optimal repetition rate is fR1/Rbeam2, and the optimal pulse charge is QpulseRbeam2. In other words, as the beam spot size increases (indicating a lower heat load density), the heat dissipation timescale increases by the same factor, reducing the possible electron source repetition rate. Figures 3(c) and 3(d) show the PXR flux dependence on the pulse charge and the repetition rate, respectively.

4. Optimal PXR conditions given heat limitations

Table I summarizes the optimal repetition rate, charge pulse, and average current for HOPG, Si, Cu, Mo, and W, considering an electron beam source with a radius of 1 mm. Overall, the optimized electron source current is in the range of ∼500–3000 µA. The optimal electron beam charge per pulse for this beam dimension is between 2 and 10 µC, depending on the material and the Bragg plane. Notably, the pulse charge density aligns with values employed in previous experiments. However, the repetition rate values are between 200 and 400 Hz, two orders of magnitude higher than those used in previous experiments.11,28

Thicker PXR crystals, while having more crystal layers contributing to stronger PXR emission, also exhibit higher x-ray absorption due to their thickness. Thus, the emitted PXR photons are self-absorbed within the crystal, limiting the contribution of all crystal layers to the PXR intensity [Fig. 4(a)]. The geometrical term fgeo in Eq. (2) captures this effect and sets an upper bound on the PXR yield. This limitation is especially significant for high-Z materials with shorter absorption lengths. In this section, we propose two different PXR schemes to overcome this limitation by reducing the distance the emitted PXR photons traverse within the crystal, resulting in a considerable PXR yield enlargement.

FIG. 4.

Optimizing PXR yield by resolving self-absorption. (a) The x-ray attenuation length. The radiating dipoles produce PXR photons through all the crystal layers. However, photons produced in the depth of the crystal must traverse the entire crystal length, leading to their attenuation and a weaker contribution to the beam intensity relative to photons produced near the crystal surface. The absorption length depends on the x-ray energy and the crystal material; heavier materials and lower PXR energies have shorter absorption lengths. (b) The electron multiple scattering effect. The electrons slightly deviate from their initial trajectory due to the electrostatic forces applied by the material atoms. The normalized scattering length is defined by the distance the electron goes through the material until the angular divergence is 10 mrad. Increasing the electron energy leads to longer scattering lengths. (c) The stacked multiple crystal PXR scheme. The crystals are stacked upon each other with two fabrication conditions: (1) Each crystal should be thinner than the absorption length, and (2) the distance between the crystals should be larger than the escape path of the emitted photon. The assumption in the PXR flux derivation is a conservative estimation of incoherent summation between the crystals. (d) The edge PXR scheme. The electron beam passes within the crystal in parallel to the crystal edge. In this scheme, the beam spot size should be smaller than the attenuation length of the material to overcome the self-absorption of the emitted PXR photons.

FIG. 4.

Optimizing PXR yield by resolving self-absorption. (a) The x-ray attenuation length. The radiating dipoles produce PXR photons through all the crystal layers. However, photons produced in the depth of the crystal must traverse the entire crystal length, leading to their attenuation and a weaker contribution to the beam intensity relative to photons produced near the crystal surface. The absorption length depends on the x-ray energy and the crystal material; heavier materials and lower PXR energies have shorter absorption lengths. (b) The electron multiple scattering effect. The electrons slightly deviate from their initial trajectory due to the electrostatic forces applied by the material atoms. The normalized scattering length is defined by the distance the electron goes through the material until the angular divergence is 10 mrad. Increasing the electron energy leads to longer scattering lengths. (c) The stacked multiple crystal PXR scheme. The crystals are stacked upon each other with two fabrication conditions: (1) Each crystal should be thinner than the absorption length, and (2) the distance between the crystals should be larger than the escape path of the emitted photon. The assumption in the PXR flux derivation is a conservative estimation of incoherent summation between the crystals. (d) The edge PXR scheme. The electron beam passes within the crystal in parallel to the crystal edge. In this scheme, the beam spot size should be smaller than the attenuation length of the material to overcome the self-absorption of the emitted PXR photons.

Close modal

1. Analysis of x-ray scattering and self-absorption in crystals

Any x-ray beam undergoes attenuation when interacting with a thick target material due to photoelectric absorption, Compton scattering, and elastic scattering.29 The same phenomenon holds for the emitted PXR photons within the crystal. For materials examined in this work and PXR energies below 70 keV, photoelectric absorption is the most significant attenuation factor.29 Initially, the production of PXR photons per unit length remains constant as the electron traverses the crystal. However, PXR photons that must traverse through the entire crystal will contribute significantly less than PXR photons produced at the surface of the crystal. Consequently, the material absorption length limits the PXR yield.

Figure 4(a) presents the absorption length for various materials and x-ray energies. Typically, heavier materials or lower x-ray energies result in shorter absorption lengths, limiting the standard PXR scheme. High-Z materials exhibit increased diffraction efficiency but have shorter attenuation lengths, i.e., the geometrical factor (fgeo) and the Fourier expansion of the electric susceptibility (χg) compete [Eq. (2)]. The geometrical factor scales with fgeoLabsω3/Z4, while the scattering factor scales with χg2Z2/ω4, leading to a dependence of the PXR yield on NPXRfgeoχg21/Z2. Therefore, in the conventional PXR setup, lighter materials are preferable to produce more PXR photons.

However, even if the self-absorption limitation is overcome, the PXR intensity cannot increase linearly with the material thickness without further restrictions. In this case, the main limiting factor becomes the electron beam scattering [Fig. 4(b)]. When an electron goes through the PXR crystal, it slightly deviates from its initial trajectory due to the electrostatic forces applied by the material atoms. This scattering process has a random walk profile, for which the likelihood and the degree of electron scattering is a probability function of the crystal thickness and the mean free path.30 In particular, the scattering angle are modeled with Gaussian probability with zero mean scattering and a standard deviation given by31 
σθms=13.6MeVEeLX01+0.038lnLX0,
(11)
where Ee is the electron energy, L is the material thickness, and X0 is the radiation length.

Figure 4(b) shows the scattering angle standard deviation of various materials and electron energies. The electron beam multiple scattering broadens the angular distribution of PXR, leading to a reduction in PXR brightness. To assess this broadening’s impact, we employ the Potylitsyn method.32 It involves convolving the Gaussian distribution representing electron scattering with the angular shape of PXR [Eq. (3)]. We derive the optimal material thickness to be approximately Lopt ≈ 0.1X0, assuming the absence of self-absorption (see Sec. II E of the supplementary material). Beyond this crystal thickness, the increase in PXR flux becomes negligible, reducing the source's brightness. This typical thickness significantly exceeds the absorption length, particularly for lower PXR energies. Consequently, PXR geometry schemes capable of mitigating self-absorption limitations will yield substantial benefits in these spectral ranges.

2. Proposed schemes for enhanced PXR

To cope with the PXR self-absorption limitation, we propose two schemes: the first scheme is a stacked multiple crystal structure [Fig. 4(c)], and the second is an edge PXR structure [Fig. 4(d)]. To overcome the PXR self-absorption in the first scheme, two conditions should be fulfilled: (1) The thickness of each crystal should be thinner than the absorption length. (2) The distance between the crystals should be large enough such that the escape path of the emitted photon does not go through the adjacent crystal. These conditions are summarized as follows:
L/cosφLabs,d/tanφdxy,
(12)
where L is the crystal thickness, φ = Ω is the emission angle of the photon relative to the incident electron, d is the distance between the crystals, and dxy is the transverse plane length of the crystal.
The second scheme of an edge PXR structure is also called grazing PXR or extremely asymmetric diffraction (EAD) PXR.33–35 This scheme is based on the transmission of the electron beam within the crystal, yet parallel to and near the crystal edge. In this scheme, the electron spot size should be smaller than the absorption of the PXR crystal yet large enough to enable parallel motion. The condition that this structure should satisfy is
2Rbeam/sinφLabs,
(13)
where Rbeam is the beam spot radius. This structure has been examined experimentally for a silicon crystal, where a PXR yield gain of a factor of 5 was reported, fitting well with the theoretical expected gain (Fig. 5).36 
FIG. 5.

Optimal PXR performance using our proposed optimization mechanisms to bypass the current limitations of PXR sources. Photon rate comparison between a current state-of-the-art PXR scheme (a) and enhanced PXR schemes with optimal electron source current and optimal geometry (b) for different materials. The spectrum is split into regions for different applications. The dashed line marks the photon rate needed for practical applications. For the lower PXR energy regions, the special PXR structures have a more significant gain since the absorption length is shorter in these regions. The material thickness taken for the conventional PXR scheme graph is the absorption length of the material, and the thickness taken for the special PXR scheme graph is the minimum between 0.1X0 and 10 mm, where X0 is the material radiation length. The incident electron energy is 60 MeV. The detector’s angular aperture used for flux derivation is the PXR beam divergence (θph3γe1). The electron source currents used for the derivation are based on Table I.

FIG. 5.

Optimal PXR performance using our proposed optimization mechanisms to bypass the current limitations of PXR sources. Photon rate comparison between a current state-of-the-art PXR scheme (a) and enhanced PXR schemes with optimal electron source current and optimal geometry (b) for different materials. The spectrum is split into regions for different applications. The dashed line marks the photon rate needed for practical applications. For the lower PXR energy regions, the special PXR structures have a more significant gain since the absorption length is shorter in these regions. The material thickness taken for the conventional PXR scheme graph is the absorption length of the material, and the thickness taken for the special PXR scheme graph is the minimum between 0.1X0 and 10 mm, where X0 is the material radiation length. The incident electron energy is 60 MeV. The detector’s angular aperture used for flux derivation is the PXR beam divergence (θph3γe1). The electron source currents used for the derivation are based on Table I.

Close modal

Except for the yield gain, this geometry produces a different PXR spatial shape. An electron penetrating the target material in a conventional manner excites the material dipoles symmetrically, causing the dipole fields to cancel each other at the resonant point defined by the Bragg condition.37 Therefore, the conventional PXR geometry produces either a double lobe or a donut shape with a hole at the center. In contrast, the edge PXR scheme breaks this symmetry since the angle between the incident electron and the dipoles is distributed only in half of the plane. Thus, the edge PXR geometry produces a beam with a peak intensity exactly at the resonant point.37 

Figure 5 shows the PXR photon rate, comparing a standard PXR scheme and enhanced PXR schemes for different PXR materials. The electron source currents used for the derivation are based on Table I. The x-ray spectrum is divided into the target applications, i.e., x-ray crystallography (<15 keV), mammography (10–25 keV), chest and head radiography (40–50 keV), and abdomen and pelvis radiography (50–70 keV).38 The dashed line represents the photon rate necessary for in-vivo imaging. The target's angular aperture used for flux derivation is the PXR beam divergence (θph3γe1). The enhanced PXR schemes gain up to two orders of magnitude in flux relative to a conventional PXR structure. The gain is considerable for lower x-ray energies due to the higher self-attenuation in this region. For higher x-ray energies, the flux decreases due to lower diffraction efficiency. Overall, the PXR flux levels are adequate for the practical applications attributed to this optimization.

In this section, we discuss the potential experimental opportunities and challenges, focusing on the two key aspects of enhancing the average electron source current and exploiting advanced PXR geometries. To begin, we explore the availability of electron sources that align with the criteria for achieving the desired average current. In addition, we consider a scheme involving the movement of the PXR crystal, analogous to the principles of a rotating-anode x-ray tube, aiming to further increase the average electron source current. These discussions shed light on the practical considerations and challenges associated with realizing these advancements.

As the peak current of the electron source increases, electron beam instabilities become a concern. This phenomenon is commonly referred to as beam blow-up (BBU) or beam break instability,39 arising from the interaction between the electron beam and the cavity modes of the accelerating cells.40 Under these conditions, the electron beam experiences density and velocity perturbations, increasing its emittance and energy spread. Therefore, to mitigate the electron BBU instabilities, a higher repetition rate with a lower peak current in each pulse is preferable.41 Notably, the next-generation x-ray FEL electron sources are designed to operate at a high repetition rate, often reaching 1 MHz.42 It is important to emphasize that even if there is a moderate decline in electron beam quality, it can still meet the requirements for the PXR source. In contrast to the stringent demands placed on x-ray FEL electron sources,43 the requirements for the PXR scheme are comparatively more relaxed.37 

State-of-the-art and next-generation electron sources meet these requirements.44–46 According to the relationship between the optimal repetition rate and the electron beam spot size [Eq. (9)], the electron beam spot size should be lower for higher repetition rates. For example, the optimal beam spot size for an electron source with a repetition rate of 1 MHz and a pulse charge of ∼1 nC is 40 µm. Therefore, employing an electron source with a higher repetition rate is advantageous for enhancing the brightness of the PXR scheme.

X-ray tubes experience similar heating challenges as in PXR. A solution used in certain x-ray tubes is based on a rotating anode.27,47,48 This solution increases the effective heat dissipation area since the electron beam interacts with different positions of the target material. The PXR heat dissipation solution can adopt a similar principle, as shown in Fig. 2(d). However, a fundamental distinction lies in the target material movement. In the case of the PXR source, the modification should involve translation rather than rotation, as altering the orientation of the PXR crystal would affect the direction of x-ray emission.

An additional considerable difference between the x-ray tube and the PXR source is the alignment precision. While precise alignment is unnecessary for x-ray tubes, it is a critical factor for a PXR source. The alignment process can be similar to the double crystal monochromator scheme used in synchrotron facilities,49 where large, perfect crystals are available. These wafers can be translated much like a rotating anode so that the electron beam is concentrated near the outer edge of the wafer. However, further study should explore possible artifacts (such as blurring) of a moving crystal target, as this approach has not been previously employed in PXR production.

Finally, it is essential to consider the heat conduction occurring outside the PXR crystal. In the analysis of temperature dynamics, we assumed the surface of the PXR crystal was held at the environment temperature. When the thermal wave arrives at the surface of the PXR crystal, it either radiates by black-body radiation or is thermally conducted to an assembled material. The second option has better heat dissipation from the PXR crystal. Thus, a high-conductance material could be attached to the PXR crystal edges to act as a heat sink [Fig. 2(c)]. For example, a rotating x-ray tube anode uses molybdenum for this purpose.48 

Implementing the proposed PXR schemes faces a few challenges. PXR emission from a single crystal is quasi-coherent, with spatial and temporal coherence originating from the Bragg constructive interference condition. The stacked multiple-crystal PXR scheme creates multiple incoherent PXR beams unless the different crystals are exactly aligned and spaced in the same orientation such that the Bragg condition is still fulfilled. Due to the difficulty of achieving such an exact alignment, we showed the advantage of PXR under the conservative estimate without this alignment. Then, there is the potential for a blurring artifact in the final image, similar to the mosaicity effect of a single crystal.27 The crystal mosaicity represents the imperfection in the lattice translation throughout the crystal. For the generation of a high-quality beam, it is essential that the orientation of the different mosaic blocks be distributed within an angular range of 0.01°–0.1°.27 Consequently, the alignment of the crystals must be managed to ensure that the angular range of misalignment remains below the typical mosaicity threshold. This poses a limitation on the precision of the structure’s fabrication. However, even for larger blurring, image-processing techniques can mitigate this artifact.38 

In the other approach that we analyzed, the edge PXR scheme, the quasi-coherence of the PXR beam is preserved since the emission is from a single crystal. The demand for precise alignment of the electron beam to the crystal edge (“grazing” interaction) may present its own challenges. Nevertheless, earlier experiments demonstrated the feasibility of such an interaction.36 Even under these limitations, the PXR source flux will still grow substantially using the enhanced geometrical structures (relative to conventional PXR). This advantage is especially significant for heavy materials and lower x-ray energies.

In addition to enhancing the yield of the PXR source, optimizing its signal-to-noise ratio (SNR) is crucial for the x-ray image quality. Although PXR is quasi-monochromatic, it competes with broadband mechanisms, i.e., bremsstrahlung and transition radiation created simultaneously from the same crystal. If this background radiation is intense, it will produce a noisy image.50 

Two methods exist for coping with this challenge. The first method (discussed further in Sec. III of the supplementary material) optimizes the target angular aperture. While bremsstrahlung and transition radiation mostly emit in the forward direction, parallel to the incident electron trajectory, PXR emits at a large angle to the trajectory. Therefore, by increasing the emission angle while preserving the target within the angular aperture of the PXR emission, the noise is minimized. The second method is based on filtering the PXR beam using a crystal monochromator with the same parameters as the PXR crystal (i.e., the same crystal, Bragg plane, and Bragg angle). This scheme is possible due to the unique spatial dispersion of the PXR beam, which overlaps with the transfer function of the crystal monochromator, allowing the use of a crystal monochromator with only low attenuation.51 

PXR was demonstrated in various applications, including K-edge imaging, computed tomography (CT), and phase-contrast imaging using differential-enhanced imaging (DEI).11 It can potentially serve biomedical imaging with a quasi-monochromatic and directional beam, reducing radiation dose while improving contrast.

PXR has several advantages compared to other compact x-ray sources. PXR produces beams that are significantly more quasi-monochromatic and directional than those generated by x-ray tubes. This distinction goes beyond existing compact sources and offers substantial reductions in radiation doses. In addition, PXR’s energy tunability through crystal rotation adds flexibility in selecting desired x-ray energies. The emission of a quasi-coherent beam with spatial dispersion enables filtration methods that retain most of the beam flux.51 Compared to inverse Compton scattering, PXR requires lower acceleration energies, and its larger field of view supports a shorter distance between the source and target, leading to a more compact imaging setup. Operationally, PXR simplifies procedures by eliminating the need for temporal synchronization, requiring only geometrical calibration. Altogether, the improvements suggested in this work may make PXR the most promising laboratory-scale source for applications requiring coherence and directionality in a compact source.

PXR is a promising source of quasi-coherent hard x-rays, obtainable using electrons of a few tens of MeV, much below the typical acceleration used in synchrotrons and free-electron lasers. Our study analyzed the heat dynamics of the PXR source, deriving the optimal electron source current that can traverse through the PXR crystal. Through optimization of the electron charge pulse and repetition rate relative to the Debye–Waller factor, we identify the optimal average electron current values ranging from 500 to 3000 µA. A possible way to further increase the average electron source current is to use a moving PXR crystal, leading to a larger volume in which the heat is deposited. Overall, the thermal optimization allows using average electron source currents that exceed previous experiment benchmarks by at least two orders of magnitude.

Furthermore, we analyzed PXR geometries designed to overcome the inherent limitation of PXR photons’ self-absorption within the crystal. In these schemes, the emitted PXR photons traverse a shorter distance than the absorption length before exiting the crystal. The resulting yield gain exceeds the standard PXR geometry by up to an additional order of magnitude, especially for denser materials and lower emission energies.

Overall, the optimization process of the heat dissipation and the enhanced PXR structures enhance the PXR beam flux by more than three orders of magnitude compared with previous experiments. The resulting flux is >106photonss·mm2 for an x-ray beam spot size of 100 mm2 on target, allowing for in vivo imaging applications, e.g., mammography. Applications that require good coherence over a large field of view would benefit from the proposed PXR source, which is harder to achieve with other approaches, such as inverse Compton scattering.

Future research efforts should aim to employ a PXR machine equipped with higher electron beam current levels, ranging from hundreds of microamperes to a few milliamperes. Such a PXR machine could have an x-ray beam of high enough brightness to enable in vivo imaging. Since this scheme will demonstrate PXR at higher electron beam currents than ever before, it is imperative to carefully investigate potential artifacts that may arise, such as blurring. In addition, it is essential to undertake further experimental assessments of the PXR yield enhancement via advanced PXR geometries across a broad spectrum of PXR energies and various crystal materials. Such experiments hold promise for the practical utility of the PXR mechanism for commercial applications.

The supplementary material includes a full mathematical derivation of all key equations in the main text and further discussion.

The authors have no conflicts to disclose.

Amnon Balanov: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Alexey Gorlach: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Ido Kaminer: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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

1.
W. C.
Röntgen
, “
On a new kind of rays
,”
Nature
53
,
274
(
1896
).
2.
O.
Hemberg
,
M.
Otendal
, and
H. M.
Hertz
, “
Liquid-metal-jet anode electron-impact x-ray source
,”
Appl. Phys. Lett.
83
,
1483
(
2003
).
3.
M.
Endrizzi
, “
X-ray phase-contrast imaging
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
878
,
88
(
2018
).
4.
A.
Momose
, “
Recent advances in X-ray phase imaging
,”
Jpn. J. Appl. Phys.
44
,
6355
(
2005
).
5.
V. G.
Baryshevsky
and
I. D.
Feranchuk
, “
A comparative analysis of various mechanisms for the generation of X-rays by relativistic particles
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
228
,
490
(
1985
).
6.
V. G.
Baryshevsky
,
I. D.
Feranchuk
, and
A. P.
Ulyanenkov
, “
Electromagnetic radiation from a charged particle in crystals: Qualitative consideration
,” in
Parametric X-Ray Radiation in Crystals: Theory, Experiment and Applications
, edited by
V. G.
Baryshevsky
,
I. D.
Feranchuk
and
A. P.
Ulyanenkov
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
,
2005
), pp.
1
17
.
7.
T.
Akimoto
,
M.
Tamura
,
J.
Ikeda
,
Y.
Aoki
,
F.
Fujita
,
K.
Sato
,
A.
Honma
,
T.
Sawamura
,
M.
Narita
, and
K.
Imai
, “
Generation and use of parametric X-rays with an electron linear accelerator
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
459
,
78
(
2001
).
8.
M. L.
Ter-Mikaelian
,
High-Energy Electromagnetic Processes in Condensed Media
(
John Wiley & Sons
,
1972
).
9.
B.
Sones
,
Y.
Danon
, and
R. C.
Block
, “
X-ray imaging with parametric X-rays (PXR) from a lithium fluoride (LiF) crystal
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
560
,
589
(
2006
).
10.
V. G.
Baryshevsky
and
I. D.
Feranchuk
, “
The X-ray radiation of ultrarelativistic electrons in a crystal
,”
Phys. Lett. A
57
,
183
(
1976
).
11.
Y.
Hayakawa
et al, “
Element-sensitive computed tomography by fine tuning of PXR-based X-ray source
,”
Nucl. Instrum. Methods Phys. Res., Sect. B
355
,
251
(
2015
).
12.
Y.
Hayakawa
,
Y.
Takahashi
,
T.
Kuwada
,
T.
Sakae
,
T.
Tanaka
,
K.
Nakao
,
K.
Nogami
,
M.
Inagaki
,
K.
Hayakawa
, and
I.
Sato
, “
X-ray imaging using a tunable coherent X-ray source based on parametric X-ray radiation
,”
J. Instrum.
8
,
C08001
(
2013
).
13.
K. Y.
Amosov
,
B. N.
Kalinin
,
A. P.
Potylitsin
,
V. P.
Sarychev
,
S. R.
Uglov
,
V. A.
Verzilov
,
S. A.
Vorobiev
,
I.
Endo
, and
T.
Kobayashi
, “
Influence of temperature on parametric X-ray intensity
,”
Phys. Rev. E
47
,
2207
(
1993
).
14.
V. G.
Baryshevsky
,
I. D.
Feranchuk
,
A. O.
Grubich
, and
A.
Ivashin
, “
Theoretical interpretation of parametric X-ray spectra
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
249
,
306
(
1986
).
15.
K.-H.
Brenzinger
et al, “
Investigation of the production mechanism of parametric X-ray radiation
,”
Z. Phys. A: Hadrons Nucl.
358
,
107
(
1997
).
16.
A. N.
Didenko
,
B. N.
Kalinin
,
S.
Pak
,
A. P.
Potylitsin
,
S. A.
Vorobiev
,
V. G.
Baryshevsky
,
V. A.
Danilov
, and
I. D.
Feranchuk
, “
Observation of monochromatic X-ray radiation from 900 MeV electrons transmitting through a diamond crystal
,”
Phys. Lett. A
110
,
177
(
1985
).
17.
V. G.
Baryshevsky
,
V. A.
Danilov
,
O. L.
Ermakovich
,
I. D.
Feranchuk
,
A.
Ivashin
,
V. I.
Kozus
, and
S. G.
Vinogradov
, “
Angular distribution of parametric X-rays
,”
Phys. Lett. A
110
,
477
(
1985
).
18.
Y.
Adishev
,
V.
Zabaev
,
V.
Kaplin
,
S.
Razin
,
S. R.
Uglov
,
S.
Kuznetsov
, and
Y. P.
Kunashenko
, “
Parametric X-ray radiation generated by 5.7-MeV electrons in a pyrolytic-graphite crystal
,”
Phys. At. Nucl.
66
,
420
(
2003
).
19.
V.
Alekseev
,
A.
Eliseyev
,
E.
Irribarra
,
I.
Kishin
,
A.
Kubankin
, and
R.
Nazhmudinov
, “
Parametric X-ray radiation in polycrystals
,”
Probl. At. Sci. Technol.
122
,
187
(
2019
).
20.
S.
Huang
,
R.
Duan
,
N.
Pramanik
,
M.
Go
,
C.
Boothroyd
,
Z.
Liu
, and
L. J.
Wong
, “
Multicolor X-rays from free electron–driven van Der Waals heterostructures
,”
Sci. Adv.
9
,
eadj8584
(
2023
).
21.
M.
Shentcis
et al, “
Tunable free-electron X-ray radiation from van Der Waals materials
,”
Nat. Photonics
14
,
686
(
2020
).
22.
X.
Shi
,
M.
Shentcis
,
Y.
Kurman
,
L. J.
Wong
,
F. J.
García de Abajo
, and
I.
Kaminer
, “
Free-electron-driven X-ray caustics from strained van Der Waals materials
,”
Optica
10
,
292
(
2023
).
23.
S.
Huang
,
R.
Duan
,
N.
Pramanik
,
C.
Boothroyd
,
Z.
Liu
, and
L. J.
Wong
, “
Enhanced versatility of table-top X-rays from van der Waals structures
,”
Adv. Sci.
9
,
2105401
(
2022
).
24.
X.
Shi
,
Y.
Kurman
,
M.
Shentcis
,
L. J.
Wong
,
F. J.
García de Abajo
, and
I.
Kaminer
, “
Free-electron interactions with van Der Waals heterostructures: A source of focused X-ray radiation
,”
Light Sci. Appl.
12
,
148
(
2023
).
25.
E.
Segrè
,
H.
Staub
,
H. A.
Bethe
, and
J.
Ashkin
,
Experimental Nuclear Physics
(
John Wiley & Sons
,
New York
,
1953
), Vol.
1
.
26.
J. R.
Cannon
,
The One-Dimensional Heat Equation
(
Cambridge University Press
,
Cambridge
,
1984
).
27.
J.
Als-Nielsen
and
D.
McMorrow
,
Elements of Modern X-Ray Physics
, 2nd ed. (
John Wiley & Sons
,
2011
).
28.
Y.
Hayakawa
et al, “
Computed tomography for light materials using a monochromatic X-ray beam produced by parametric X-ray radiation
,”
Nucl. Instrum. Methods Phys. Res., Sect. B
309
,
230
(
2013
).
29.
J. H.
Hubbell
and
S. M.
Seltzer
, “
Tables of X-ray mass attenuation coefficients and mass energy-absorption coefficients 1 keV to 20 MeV for elements Z = 1 to 92 and 48 additional substances of dosimetric interest
,” (National Institute of Standards and Technology, Gaithersburg, MD, 1995).
30.
H. A.
Bethe
, “
Molière’s theory of multiple scattering
,”
Phys. Rev.
89
,
1256
(
1953
).
31.
J.
Beringer
,
J. F.
Arguin
,
R. M.
Barnett
,
K.
Copic
,
O.
Dahl
,
D. E.
Groom
,
C. J.
Lin
,
J.
Lys
,
H.
Murayama
,
C. G.
Wohl
et al, “
Review of particle physics
,”
Phys. Rev. D
86
,
010001
(
2012
).
32.
A. P.
Potylitsin
, “
Influence of beam divergence and crystal mosaic structure upon parametric X-ray radiation characteristics
,” arXiv:cond-mat/9802279 (
1994
).
33.
I. D.
Feranchuk
and
S. I.
Feranchuk
, “
Grazing incidence parametric X-ray radiation from the relativistic electron beam moving in parallel to the superlattice surface
,”
Eur. Phys. J. Appl. Phys.
38
,
135
(
2007
).
34.
A. N.
Eliseev
,
A. S.
Kubankin
,
R. M.
Nazhmudinov
,
N. N.
Nasonov
,
V. I.
Sergienko
,
A. V.
Subbotin
,
G. G.
Subbotin
, and
V. A.
Khablo
, “
Observation of the enhancement of parametric radiation under conditions of the grazing incidence of relativistic electrons on the crystal surface
,”
JETP Lett.
90
,
438
(
2009
).
35.
O. D.
Skoromnik
,
V. G.
Baryshevsky
,
A. P.
Ulyanenkov
, and
I. D.
Feranchuk
, “
Radical increase of the parametric X-ray intensity under condition of extremely asymmetric diffraction
,”
Nucl. Instrum. Methods Phys. Res., Sect. B
412
,
86
(
2017
).
36.
Y.
Hayakawa
,
K.
Hayakawa
,
M.
Inagaki
,
T.
Kuwada
,
K.
Nakao
,
K.
Nogami
,
T.
Sakai
,
I.
Sato
,
Y.
Takahashi
, and
T.
Tanaka
, “
Geometrical effect of target crystal on PXR generation as a coherent X-ray source
,”
Int. J. Mod. Phys. A
25
,
174
188
(
2010
).
37.
A.
Balanov
,
A.
Gorlach
, and
I.
Kaminer
, “
Temporal and spatial design of X-ray pulses based on free-electron–crystal interaction
,”
APL Photonics
6
,
70803
(
2021
).
38.
W. R.
Hendee
,
E. R.
Ritenour
, and
K. R.
Hoffmann
,
Medical Imaging Physics
, 4th ed. (
John Wiley & Sons
,
2003
).
39.
V. K.
Neil
,
R. K.
Cooper
, and
L. S.
Hall
, “
Further theoretical studies of the beam breakup instability
,”
Part. Accel.
9
,
213
(
1979
).
40.
Y.
Tang
,
T. P.
Hughes
,
C. A.
Ekdahl
, and
K. C. D.
Chan
, “
BBU calculations for beam stability experiments on DARHT-2
,” in
Proceedings of the European Particle Accelerator Conference 2006
(JACoW, 2006), pp.
1
23
.
41.
J. E.
Coleman
,
D. C.
Moir
,
C. A.
Ekdahl
,
B. T.
McCuistian
,
J. B.
Johnson
,
G. A.
Sullivan
, and
M. T.
Crawford
, “
Limitations of increasing the intensity of a relativistic electron beam
,” in
Proceedings of the Particle Accelerator Conference (JACoW,
Pasadena, CA
,
2013
), p.
484
.
42.
P.
Musumeci
,
J.
Giner Navarro
,
J. B.
Rosenzweig
,
L.
Cultrera
,
I.
Bazarov
,
J.
Maxson
,
S.
Karkare
, and
H.
Padmore
, “
Advances in bright electron sources
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
907
,
209
(
2018
).
43.
C.
Pellegrini
,
A.
Marinelli
, and
S.
Reiche
, “
The physics of X-ray free-electron lasers
,”
Rev. Mod. Phys.
88
,
015006
(
2016
).
44.
A.
Opanasenko
,
V.
Mytrochenko
,
V.
Zhaunerchyk
, and
V.
Goryashko
, “
Design study of a low-emittance high-repetition rate thermionic rf gun
,”
Phys. Rev. Accel. Beams
20
,
053401
(
2017
).
45.
V. V
Mitrochenko
, “
Thermionic RF gun with high duty factor
,” in
Proceedings of the 1997 Particle Accelerator Conference (Cat. No. 97CH36167)
(
IEEE
,
1997
), Vol.
3
, pp.
2817
2819
.
46.
H. P.
Bluem
,
D.
Dowell
,
A. M. M.
Todd
, and
L. M.
Young
, “
High brightness thermionic electron gun performance
,” in
Proceedings, 50th Advanced ICFA Beam Dynamics Workshop on Energy Recovery Linacs (ERL’11)
,
Tsukuba, Japan
,
2011
.
47.
R. W.
Wood
, “
Note on ‘focus tubes’ for producing x-rays
,”
London, Edinburgh Dublin Philos. Mag. J. Sci.
41
,
382
(
1896
).
48.
E.
Krestel
, “
The X-ray tube
,” in
Imaging Systems for Medical Diagnostics
(
Siemens
,
Berlin and Munich
,
1990
), pp.
222
246
.
49.
J.
Hrdý
, “
Double crystal monochromator for synchrotron radiation with decreased radiation power density
,”
Rev. Sci. Instrum.
63
,
459
(
1992
).
50.
M.
Johnson
and
D.
Mcnabb
, “
Comparison of techniques to reduce bremsstrahlung background radiation from monoenergetic photon beams
,” Report No. UCRL-TR-222530 (Lawrence Livermore National Laboratory,
2006
).
51.
Y.
Hayakawa
,
I.
Sato
,
K.
Hayakawa
,
T.
Tanaka
,
A.
Mori
,
T.
Kuwada
,
T.
Sakai
,
K.
Nogami
,
K.
Nakao
, and
T.
Sakae
,
Nucl. Instrum. Methods Phys. Res., Sect. B
252
,
102
(
2006
).