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.

## I. INTRODUCTION

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.

^{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 by

^{7,8}

*ℏω*

_{B}is the emitted Bragg energy;

*d*

_{hkl}is the d-spacing of the Bragg plane, which corresponds to Miller indices (hkl);

*ɛ*

_{0}is the constant part of the medium permittivity (

*ɛ*

_{0}≈ 1 for x-rays);

*θ*

_{B}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 = 2

*θ*

_{B}. This relation allows PXR energy tunability in experiments by rotating the PXR crystal, i.e., altering the Ω and

*θ*

_{B}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 ∼$105\u2212106photonss\xb7mm2$, 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.

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.

## II. RESULTS

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.

*χ*

_{g}), the thermal load on the crystal (captured by the Debye–Waller factor), and the geometrical factor (

*f*

_{geo}). In the framework of PXR kinematical theory, the photon distribution emitted from a single electron is given by

^{15}

*α*is the fine-structure constant;

*ω*

_{B}is the emitted PXR photon energy;

*c*is the speed of light;

*θ*

_{B}is the Bragg angle;

*e*

^{−2W}is the Debye–Waller factor, which captures thermal effects;

*χ*

_{g}is the Fourier expansion of the electric susceptibility, describing the diffraction efficiency; $N\theta x,\theta y$ is the PXR angular dependence; and

*f*

_{geo}is the geometrical factor that describes the PXR photon self-absorption during the emission process. The PXR photon energy (

*ω*

_{B}) and the Bragg angle are related by the condition for constructive interference between the material’s dipoles [Eq. (1)].

*θ*

_{x}is the angle in the diffraction plane,

*θ*

_{y}is the angle perpendicular to

*θ*

_{x}in the diffraction plane, and $\theta ph2=\gamma e\u22122+\omega p\omega 2$, where

*ω*

_{p}is the plasma frequency of the material. Equation (3) is valid for relativistic electron beams (

*γ*

_{e}≫ 1, where

*γ*

_{e}is the Lorentz factor of the electron) and for small scattering angles relative to the Bragg angle (i.e.,

*θ*

_{x},

*θ*

_{y}≪ 1). In this work, we focus on the regime where the density effect is negligible, i.e., electron energies that satisfy 1 ≪

*γ*

_{e}≪

*ω*/

*ω*

_{p}, leading to $\theta ph\u2248\gamma e\u22121$. 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 GeV

^{16,17}to electron beam energies of a few MeV

^{18,19}and below.

^{20–24}Notably, lower electron energies lead to a larger PXR beam divergence due to the emission dependence of $\theta ph\u2248\gamma e\u22121$. Therefore, the optimal electron energy should be determined considering the experimental geometry and the required field of view.

### A. Heat dissipation

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 (*τ*_{pulse}) much shorter than the thermal conductivity timescale.

#### 1. Heat transfer from electrons to matter

^{25}

*Z*is the material atomic number,

*N*is the material density,

*v*

_{e}=

*βc*is the electron velocity,

*γ*

_{e}is the Lorentz factor,

*m*is the electron rest mass,

*ℏ*⟨

*ω*⟩ is the mean excitation potential,

*T*

_{max}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 $\u223c2(MeVcm2/g)$ (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.

*T*) in this volume can be expressed by (supplementary material, Sec. I B)

*dE*

_{e}/

*dx*⟩ is the average electron energy loss per unit length given by the Bethe–Bloch formula,

*ρ*is the PXR crystal mass density,

*C*

_{p}is the PXR crystal specific heat capacity,

*Q*

_{pulse}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 (

*Q*

_{pulse}) 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 *Q*_{pulse} = 700 nC and beam dimensions of *A* = 1 mm^{2}. 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).

#### 2. Regimes of heat diffusion

*T*(

**,**

*r**t*) can be derived from the heat equation,

^{26}

*P*

_{source}is the power per unit volume deposited in the crystalline by the electron beam, and

*P*

_{sink}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, $\sigma =5.67\xd710\u22128Wm2K4$ is the Stefan–Boltzmann constant,

*L*is the material thickness, and

*T*

_{env}is the environment temperature. For the time between the electron pulses,

*P*

_{source}= 0.

*L*≪

*L*

_{HD}, the black-body radiation is the dominant heat dissipation mechanism, whereas for

*L*≫

*L*

_{HD}, 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 cm^{2}. When the material thickness is *L* = 100 *µ*m, thermal conduction governs, leading to a relatively slow dissipation process. In contrast, when the material thickness is *L* = 1 *µ*m, 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 (*L* ≫ *L*_{HD}). 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 ($A=\pi Rbeam2$) 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/*ρC*_{p}. The typical thermal diffusion coefficients of the examined materials in this paper are 0.5–1 (cm^{2}/s) (Table I). We define the characteristic diffusion time by $\tau D\u225cRbeam2/4D$ as the time elapsed from the end of the electron pulse until the temperature in the center of the beam dropped to $Tmax1\u2212e\u22121$. The timescale of the heat diffusion process depends on the beam area $\tau D\u221dRbeam2$, 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.

Material and . | Diffusion . | Optimal repetition . | Maximal pulse . | Maximal average . |
---|---|---|---|---|

Bragg plane . | coefficient (cm^{2} 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 and . | Diffusion . | Optimal repetition . | Maximal pulse . | Maximal average . |
---|---|---|---|---|

Bragg plane . | coefficient (cm^{2} 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, *T*_{D}, the material temperature, *T*, and the d-spacing of the diffraction plane of interest, *d*_{hkl}.

*u*

^{2}(

*T*). This quantity is given by

^{13}

*M*is the material mass and

*k*

_{B}is the Boltzmann constant. The Debye–Waller term

*W*is calculated from

*u*

^{2}(

*T*) and the reciprocal lattice vector

*τ*= 2

*π*/

*d*

_{hkl}using the relationship $e\u22122W=exp\u2212\tau 2u2(T)$.

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.

*τ*

_{pulse}and a pulse charge of

*Q*

_{pulse}traverses the target crystal with a repetition rate of

*f*

_{R}= 1/

*τ*

_{R}. We assume the pulse duration is significantly shorter than the thermal diffusion characteristic time, i.e.,

*τ*

_{pulse}≪

*τ*

_{D}. During the electron pulse, the crystal temperature at the impact area increases by Δ

*T*=

*T*

_{max}−

*T*

_{min}[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

*R*

_{beam}, the repetition rate

*f*

_{R}, and the pulse charge

*Q*

_{pulse}as a function of the target material type and the dimensions. The optimized values are given by (supplementary material, Sec. I F)

This result leads to intriguing outcomes. First, the optimal maximal temperature is lower than the melting temperature and depends on the inter-lattice distance $\u221ddhkl2$. 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 (*I*_{opt} = Q_{pulse}*f*_{R}) does not depend on the beam area since the optimal repetition rate is $fR\u221d1/Rbeam2$, and the optimal pulse charge is $Qpulse\u221dRbeam2$. 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 *µ*A. The optimal electron beam charge per pulse for this beam dimension is between 2 and 10 *µ*C, 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}

### B. Overcoming PXR self-absorption

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 *f*_{geo} 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.

#### 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 (*f*_{geo}) and the Fourier expansion of the electric susceptibility (*χ*_{g}) compete [Eq. (2)]. The geometrical factor scales with *f*_{geo} ∝ *L*_{abs} ∝ *ω*^{3}/*Z*^{4}, while the scattering factor scales with $\chi g2\u221dZ2/\omega 4$, leading to a dependence of the PXR yield on $NPXR\u221dfgeo\chi g2\u221d1/Z2$. Therefore, in the conventional PXR setup, lighter materials are preferable to produce more PXR photons.

^{30}In particular, the scattering angle are modeled with Gaussian probability with zero mean scattering and a standard deviation given by

^{31}

*E*

_{e}is the electron energy,

*L*is the material thickness, and

*X*

_{0}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 *L*_{opt} ≈ 0.1*X*_{0}, 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

*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

*d*

_{xy}is the transverse plane length of the crystal.

^{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

*R*

_{beam}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}

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 ($\theta ph\u223c3\gamma e\u22121$). 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.

## III. DISCUSSION

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.

### A. High average current electron sources

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 *µ*m. Therefore, employing an electron source with a higher repetition rate is advantageous for enhancing the brightness of the PXR scheme.

### B. Heat dissipation through crystal movement

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}

### C. Challenges with the enhanced PXR schemes

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.

### D. Optimizing the PXR source signal-to-noise ratio

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}

### E. Applications of PXR and comparison with other compact x-ray sources

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.

## IV. SUMMARY AND OUTLOOK

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 *µ*A. 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 $>106photonss\xb7mm2$ for an x-ray beam spot size of 100 mm^{2} 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.

## SUPPLEMENTARY MATERIAL

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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

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*Proceedings of the European Particle Accelerator Conference 2006*

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