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Compton scattering is defined as an inelastic scattering process in which the interaction between strong laser fields and electrons in matter leads to photon emission. In recent years, with the rapid development of X-ray free-electron lasers, the intensity of X-ray lasers has steadily increased, and the photon energy in Compton scattering process has risen correspondingly. Previous studies focus on single-photon Compton scattering of free electrons. However, the mechanism of non-relativistic X-ray photon scattering by bound electrons remains to be elucidated. Therefore, we develop a frequency-domain theory based on non-perturbative quantum electrodynamics to investigate single-photon Compton scattering of bound electrons in strong X-ray laser fields. Our results show that the double-differential probability of Compton backscattering decreases with the increase of incident photon energy. This work establishes a relationship between Compton scattering and atomic ionization in high-frequency intense laser fields, thereby providing a platform for studying atomic structure dynamics under high-intensity laser conditions.
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Keywords:
- Compton scattering /
- frequency-domain theory /
- bound electron
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图 1 不同核电荷数类氢离子的波长与康普顿散射DDP谱图. 入射光子能量${\omega _1} = 17.4\;{\mathrm{keV}}$, 散射角$\theta = 133.75 ^\circ $, 其中实心曲线为QED理论计算的结果, 空心曲线是Eisenberger利用IA理论得到的结果
Fig. 1. Wavelength and double-differential probability spectra of Compton scattering for hydrogen-like ions with different nuclear charge numbers Z. Incident photon energy ${\omega _1} = 17.4\;{\mathrm{keV}}$, scattering angles $\theta = 133.75 ^\circ $. The solid curves represent results calculated using QED theory, while the hollow curves are those obtained by Eisenberger using the impulse approximation.
图 2 (a) 入射、散射光的波矢和偏振的几何示意图, xoz平面由入射激光的偏振$ ({\epsilon}_{1}) $和波矢量(k1)定义, 散射光子方向由球坐标$\left( {\theta , \phi } \right)$表征; (b)—(d) 康普顿散射的DDP关于散射光子能量$ {\mathrm{\omega }}_{2} $和散射角$\theta $的分布, 白色点表示自由电子模型预测值; (e)—(g) 实心和空心曲线分别是利用频域理论和Klein-Nishina公式计算得到的康普顿散射微分概率关于散射角的分布; 入射光子能量分别为500 eV (b), (e), 1 keV (c), (f), 10 keV (d), (g), 激光强度$I = 4 \times 1{0^{20}}\;\mathrm{W/cm}^2$
Fig. 2. (a) Geometric diagram of wave vectors and polarization states of incident and scattered light. The xoz plane is defined by the polarization $ \left({\epsilon}_{1}\right) $and wave vector (k1) of the incident laser, while the direction of the scattered photon is characterized by spherical coordinates $\left( {\theta , \phi } \right)$. (b)–(d) The double-differential cross-section of Compton scattering as a function of scattered photon energy and scattering angle, with white dots indicating predictions from the free-electron model. (e)–(g) Solid and hollow curves represent differential probability distributions of Compton scattering versus scattering angle, calculated using frequency-domain theory and the Klein-Nishina formula, respectively. The incident photon energies are 500 eV (b), (e) , 1 keV (c), (f) and 10 keV (d), (g) with laser intensity $I = 4 \times 1{0^{20}}\;\mathrm{W/cm}^2$.
图 3 动量空间中基态电子的密度分布随散射光能量和散射角的变化 (a) ${\omega _1} = 1\;{\mathrm{keV}}$; (b) ${\omega _1} = 10\;{\mathrm{keV}}$; 激光强度$I = $$ 4 \times 1{0^{20}}\;\mathrm{W/cm}^2$, 图中虚线表示(19)式自由电子模型预测散射光子能量值随散射角的变化
Fig. 3. Variation of the electron density distribution in momentum space for ground-state electrons as a function of scattered photon energy and scattering angle with laser intensity $I = 4 \times 1{0^{20}}\;\mathrm{W/cm}^2$: (a) ${\omega _1} = 1\;{\mathrm{keV}}$; (b) ${\omega _1} = 10\;{\mathrm{keV}}$. Dashed lines indicate the scattered photon energy versus scattering angle predicted by the free-electron model in Eq. (19).
图 4 实心曲线是利用频域理论计算的康普顿散射DDP与散射角的关系, 其中$ {\omega _1} = 1\;{\mathrm{keV}}, 5\;{\mathrm{keV}}, 10\;{\mathrm{keV}} $, ${\omega _2} \approx{\omega _1} - $$ {I_{\text{P}}}$, 空心曲线是利用汤姆孙公式计算的微分概率与散射角的关系
Fig. 4. Dependence of the double-differential probability of Compton scattering on scattering angle, calculated using frequency-domain theory (solid curve). $ {\omega _1} =1\;{\mathrm{keV}},5\;{\mathrm{keV}}, $$ 10\;{\mathrm{keV}} $, ${\omega _2} \approx {\omega _1} - {I_{\text{P}}}$. The hollow curve shows the differential cross-section derived from the Thomson scattering formula as a function of the scattering angle.
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