Study on spectroscopic parameters and molecular constants of HCl(X~1Σ~+) molecule by using multireference configuration interaction approach

Equilibrium internuclear separations, harmonic frequencies and potential energy curves (PECs) of HCl($X^{1}\Sigma ^{ + })$ molecule are investigated by using the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in combination with a series of correlation-consistent basis sets in the valence range. The PECs are all fitted to the Murrell--Sorbie function, and they are used to accurately derive the spectroscopic parameters ($D_{\rm e}$, $D_{0}$, $\omega_{\rm e}\chi_{\rm e}$, $\alpha_{\rm e}$ and $B_{\rm e})$. Compared with the available measurements, the PEC obtained at the basis set, aug-cc-pV5Z, is selected to investigate the vibrational manifolds. The constants $D_{0}$, $D_{\rm e}$, $R_{\rm e}$, $\omega_{\rm e}$, $\omega_{\rm e}\chi_{\rm e}$, $\alpha_{\rm e}$ and $B_{\rm e}$ at this basis set are 4.4006~eV, 4.5845~eV, 0.12757~nm, 2993.33~cm$^{ - 1}$, 52.6273~cm$^{ - 1}$, 0.2981~cm$^{ - 1}$ and 10.5841~cm$^{ - 1}$, respectively, which almost perfectly conform to the available experimental results. With the potential determined at the MRCI/aug-cc-pV5Z level of theory, by numerically solving the radial Schr\"{o}dinger equation of nuclear motion in the adiabatic approximation, a total of 21 vibrational levels are predicted. Complete vibrational levels, classical turning points, inertial rotation and centrifugal distortion constants are reproduced, which are in excellent agreement with the available Rydberg--Klein--Rees data. Most of these theoretical vibrational manifolds are reported for the first time to the best of our knowledge.     

Relation between quantum NOT gate speed and asymmetry of the potential of RF-SQUID

This paper investigates the relationship between the speed of a quantum not gate and the asymmetry of the potential in an interactive system formed by a two-level RF-SQUID qubit and a classical microwave pulse. The RFSQUID is characterized by an asymmetric double well potential which gives rise to diagonal matrix elements that describe the interaction of the SQUID with the microwave pulse. And the diagonal matrix elements account for the interaction of the microwave pulse with the SQUID. The results indicate that, when the angular frequency of the microwave field is chosen as near resonate with the transition （0） ←→ （1）, i.e. ω1 -ω0 ≈ ωm, （1） the gate speed is decided by three factors, the Rabi frequency, the difference of the diagonal matrix elements between the two levels, and the angular frequency of the applied microwave pulse ωm; （2） the gate speed descends when the asymmetry of the potential is considered.     

Radiation-induced resistance oscillation and instability in GaAS/AlGaAS heterostructure under transverse magnetic fields

This paper studies dynamics of a modulation-doped GaAs/AlGaAs heterostructure under transverse magnetic fields and microwave radiations. It finds that negative differential conductivity, due to the real-space electron transfer and delayed dielectric relaxation of the interface potential barrier, can lead to complex behaviours when a relatively small magnetic field is applied. Quasiperiodicity, frequency-locking and the routes from period-doubling to chaos are found. Under a large magnetic field, however, two time-independent homogeneous steady states exist; and the longitudinal resistance of the system shows an interesting oscillation with period tuned by the ratio of microwave radiation frequency w to the cyclotron frequency Wc and local minima at ω/ωc = integer ＋ 1/4.     

A note on localized transition in the spin-boson model by variational calculation

We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as J（w） = π∑k Ck^2 = π/2αw^8w^1-s, there exists a universal sc for both kinds of variational schemes, the localized transition happens only for 2 s ≤ sc, moreover, the localized transition is discontinuous for s 〈 sc while a continuous transition always occurs when s = sc. At T = 0, we have sc = 1, while for T ≠ 0, sc = 2 which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.     

Gravitational frequency-shift and deflection of light by quintessence

In this paper we investigate the gravitational frequency-shift and deflection of light in the Schwarzschild black hole space-time surrounded by quintessence. With the analytical and numerical methods, we find that the gravitational frequency-shift of light in the Schwarzschild black hole space-time surrounded by the quintessence increases as the values of the normalization factor $c$ increases, but the gravitational frequency-shift of light decreases with the quintessential state parameter $ \omega_{\rm q}$ increasing. We also calculate the deflection of light by quintessence and find that the deflection rate decreases as the values of the quintessential parameters $c$ and $ \omega_{\rm q}$ increase.     

Spin polarization effect for Cr2 molecule

Density functional theory （DFT） （B3P86） of Gaussian 03 has been used to optimize the structure of the Cr2 molecule, a transition metal element molecule. The result shows that the ground state for the Cr2 molecule is a 13- multiple state, indicating that there exists a spin polarization effect in the Cr2 molecule. Meanwhile, we have not found any spin pollution because the wave function of the ground state does not mingle with wave functions of higher-energy states. So the ground state for Cr2 molecule being a 13-multiple state is indicative of spin polarization effect of the Cr2 molecule among transition metal elements, that is, there are 12 parallel spin electrons in the Cr2 molecule. The number of non-conjugated electrons is greatest. These electrons occupy different spatial orbitals so that the energy of the Cr2 molecule is minimized. It can be concluded that the effect of parallel spin in the Cr2 molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell Sorbie potential functions with the parameters for the ground state and other states of the Cr2 molecule are derived. The dissociation energy De for the ground state of the Cr2 molecule is 0.1034eV, equilibrium bond length Re is 0.3396 nm, and vibration frequency we is 73.81cm^-1. Its force constants f2, f3 and f4 are 0.0835, -0.2831 and 0.3535 aJ. nm^-4 respectively. The other spectroscopic data for the ground state of the Cr2 molecule ωeχe, Be and αe are 1.2105, 0.0562 and 7.2938 x 10^-4cm^-1 respectively.     

Ru thickness-dependent interlayer coupling and ultrahigh FMR frequency in FeCoB/Ru/FeCoB sandwich trilayers

The antiferromagnetic (AFM) interlayer coupling effective field in a ferromagnetic/non-magnetic/ferromagnetic (FM/NM/FM) sandwich structure, as a driving force, can dramatically enhance the ferromagnetic resonance (FMR) frequency. Changing the non-magnetic spacer thickness is an effective way to control the interlayer coupling type and intensity, as well as the FMR frequency. In this study, FeCoB/Ru/FeCoB sandwich trilayers with Ru thickness ($t_{\rm Ru}$) ranging from 1 Å to 16 Å are prepared by a compositional gradient sputtering (CGS) method. It is revealed that a stress-induced anisotropy is present in the FeCoB films due to the B composition gradient in the samples. A $t_{\mathrm{Ru}}$-dependent oscillation of interlayer coupling from FM to AFM with two periods is observed. An AFM coupling occurs in a range of $2 {\rm Å} \le t_{\rm Ru} \le 8 {\rm Å}$ and over 16 $\mathrm{Å}$, while an FM coupling is present in a range of $t_{\rm Ru}< 2$ Å and $9 {\rm Å} \le t_{\rm Ru} \le 14.5 Å$. It is interesting that an ultrahigh optical mode (OM) FMR frequency in excess of 20 GHz is obtained in the sample with ${t}_{\mathrm{Ru}}= 2.5 \mathrm{Å}$ under an AFM coupling. The dynamic coupling mechanism in trilayers is simulated, and the corresponding coupling types at different values of $t_{\mathrm{Ru}}$ are verified by Layadi's rigid model. This study provides a controllable way to prepare and investigate the ultrahigh FMR films.     

Spin polarization effect for Co2 molecule

The density functional theory (DFT)(b3p86) of Gaussian 03 has been used to optimize the structure of the Co$_{2}$ molecule, a transition metal element molecule. The result shows that the ground state for the Co$_{2}$ molecule is a 7-multiple state, indicating a spin polarization effect in the Co$_{2}$ molecule. Meanwhile, we have not found any spin pollution because the wavefunction of the ground state is not mingled with wavefunctions of higher-energy states. So for the ground state of Co$_{2}$ molecule to be a 7-multiple state is the indicative of spin polarization effect of the Co$_{2}$ molecule, that is, there exist 6 parallel spin electrons in a Co$_{2}$ molecule. The number of non-conjugated electrons is the greatest. These electrons occupy different spacial orbitals so that the energy of the Co$_{2}$ molecule is minimized. It can be concluded that the effect of parallel spin in the Co$_{2}$ molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell--Sorbie potential functions with the parameters for the ground state and the other states of the Co$_{2}$ molecule are derived. The dissociation energy $De$ for the ground state of Co$_{2}$ molecule is 4.0489eV, equilibrium bond length $R_{\rm e}$ is 0.2061~nm, and vibration frequency $\omega _\e $ is 378.13~cm$^{ - 1}$. Its diatomic molecule force constants $f_2$, $f_3$, and $f_4$ are 2.4824~aJ$\cdot$nm$^{ - 2}$, -7.3451~aJ$\cdot$nm$^{ - 3}$, and 11.2222~aJ$\cdot$nm$^{ - 4 }$respectively(1~aJ=$10^{-18}$~J). The other spectroscopic data for the ground state of Co$_{2}$ molecule $\omega_{\e}\chi _{\e}$, $B_{\e}$, and $\alpha_{\e}$ are 0.7202~cm$^{-1}$, 0.1347~cm$^{-1 }$, and 2.9120$\times $ 10$^{-1}$~cm$^{-1}$ respectively. And $\omega_{\e}\chi _{\e}$ is the non-syntonic part of frequency, $B_{\e}$ is the rotational constant, $\alpha_{\e}$ is revised constant of rotational constant for non-rigid part of Co$_2$ molecule.     

用H-W-ECR CVD系统实现氢化非晶硅薄膜在氢气环境下同时进行化学热退火和光诱导退火的研究

为了研究氢化非晶硅薄膜的稳定性，我们设计了一个在原子氢气氛中热退火的同时进行光诱导退火的实验(TLAH)。实验装置是由传统的微波电子回旋共振化学气相沉积系统改造而成为热丝辅助微波电子回旋共振化学气相沉积系统。为了对这一退火方法进行比较，对样品还进行了热退火、热退火同时进行光诱导退火。同时，为了定量地分析光电导衰退，我们假设光电导衰退遵循扩展指数规律：1/σph=1/σs-(1/σs-1/σ0)exp[-(t/τ)β]，这里扩展指数参数β 和时间常数 τ 可从与 lnt 的线性关系中截距和斜率得到, 式中光电导饱和值σs可以通过在对数坐标系中表示的光电导和光照时间关系进行高斯拟合得到。实验结果显示：TLAH 方法可以提高氢化非晶硅薄膜的稳定性、改善其微结构和光电特性，同时还发现，光学带隙明显减小、荧光光谱显著地朝着低能方向移动。     

Electron tunnelling phase time and dwell time through an associated delta potential barrier

The electron tunnelling phase time τP and dwell time τD through an associated delta potential barrier U(x) = ξδ(x) are calculated and both are in the order of 10^-17～10^-16s. The results show that the dependence of the phase time on the delta barrier parameter ξ can be described by the characteristic length lc = h^2/meξ and the characteristic energy Ec=meξ^2/h^2 of the delta barrier, where me is the electron mass, lc and Ec are assumed to be the effective width and height of the delta barrier with lcEc=ξ, respectively. It is found that TD reaches its maximum and τD = τp as the energy of the tunnelling electron is equal to Ec/2, i.e. as lc =λDB, λDB is de Broglie wave length of the electron.     

The influence of collision energy on magnetically tuned 6Li-6Li Feshbach resonance

The effect of collision energy on the magnetically tuned $^{6}$Li-$^{6}$Li Feshbach resonance (FR) is investigated theoretically by using the coupled-channel (CC) method for the collision energy ranging from 1 μ$ {\rm K} \cdot {k}_{\rm B}$ to 100 μ$ {\rm K} \cdot {k}_{\rm B}$. At the collision energy of 1 μ$ {\rm K} \cdot {k}_{\rm B}$, the resonance positions calculated are 543.152 Gs (s wave, the unit $1 {\rm Gs}=10^{-4} {\rm T}$), 185.109 Gs (p wave $|m_{l}| = 0$), and 185.113 Gs (p wave $|m_{l}| = 1$), respectively. The p-wave FR near 185 Gs exibits a doublet structure of 4 mGs, associated with dipole-dipole interaction. With the increase of the collision energy, it is found that the splitting width remains the same (4 mGs), and that the resonance positions of s and p waves are shifted to higher magnetic fields with the increase of collision energy. The variations of the other quantities including the resonance width and the amplitude of the total scattering section are also discussed in detail. The thermally averaged elastic rate coefficients at $T=10$, 15, 20, 25 K are calculated and compared.     

Critical radius and dipole polarizability for a confined system

The analytical transfer matrix method (ATMM) is applied to calculating the critical radius $r_{\rm c}$ and the dipole polarizability $\alpha_{\rm d}$ in two confined systems: the hydrogen atom and the Hulth\'{e}n potential. We find that there exists a linear relation between $r_{\rm c}^{1/2}$ and the quantum number $n_{r}$ for a fixed angular quantum number $l$, moreover, the three bounds of $\alpha_{\rm d}$ ($\alpha_{\rm d}^{K}$, $\alpha_{\rm d}^{B}$, $\alpha_{\rm d}^{U}$) satisfy an inequality: $\alpha_{\rm d}^{K}\leq\alpha_{\rm d}^{B}\leq\alpha_{\rm d}^{U}$. A comparison between the ATMM, the exact numerical analysis, and the variational wavefunctions shows that our method works very well in the systems.     

Observation of four-wave mixing in caesium atoms using a noncycling transition

In this paper the generation of four-wave mixing (FWM) signal using a noncycling transition of caesium atoms is investigated when the pumping laser is locked to the transition $6{\rm S}_{1/2}F=4\to6{\rm P}_{3/2}F'=4$, and meanwhile the probe frequency is scanned across the $6{\rm S}_{1/2}F=4 \to6{\rm P}_{3/2}$ transition. The efficiency of the four-wave mixing signal as a function of the intensity of the pumping beams and the detuning of the pumping beams is also studied. In order to increase the detection efficiency, a repumping laser which is resonant with $6{\rm S}_{1/2} F=3\to 6{\rm P}_{3/2}F'=4$ transition is used. A theoretical model is also introduced, and the theoretical results are in qualitative agreement with experimental ones.     

Two-electron and one-photon transitions in highly charged nickel-like ions

This paper calculates the transition wavelengths and probabilities of the two-electron and one-photon (TEOP) transition from the $(3{\rm s}^{-1}_{1/2}4{\rm d}_{j})_{J=1,2}$ to $(3{\rm p}^{-1}_{3/2}4{\rm s}_{1/2})_{J=1}$ and the $(3{\rm p}^{-1}_{1/2}4{\rm s}_{1/2})_{J=1}$ to $(3{\rm d}^{-1}_{j}4{\rm d}_{j'})_{J=1,2}$ for highly charged Ni-like ions with atomic number $Z$ in the range $47\leq Z\leq92$. In the calculations, the multi-configuration Dirac--Fock method and corresponding program packages GRASP92 and REOS99 were used, and the relativistic effects, correlation effects and relaxation effects were considered systematically. It is found that the TEOP transitions are very sensitive to the correlation of electrons, and the probabilities will be enhanced sharply in some special $Z$ regions along the isoelectronic sequence. The present TEOP transition wavelengths are compared with the available data from some previous publications, good agreement is obtained.     

The role of hydrogen in hydrogenated microcrystalline silicon film and in deposition process with VHF-PECVD technique

The role of hydrogen in hydrogenated microcrystalline silicon ($\mu $c-Si:H) thin films in deposition processes with very high frequency plasma-enhanced chemical vapour deposition (VHF-PECVD) technique have been investigated in this paper. With \textit{in situ} optical emission spectroscopy (OES) diagnosis during the fabrication of $\mu $c-Si:H thin films under different plasma excitation frequency $\nu _{\rm e }$ (60MHz--90MHz), the characteristic peak intensities ($I_{{\rm SiH}^*}$, $I_{{\rm H}\alpha^*}$ and $I_{{\rm H}\beta ^*}$) in SiHVHF-PECVD技术 氢化微晶硅 光发射光谱 薄膜学VHF-PECVD technique, hydrogenated microcrystalline silicon, role of hydrogen, optical emission spectroscopyProject supported by the Natural Science Foundation of Guangdong Province, China (Grant No 05300378), the State Key Development Program for Basic Research of China (Grant Nos G2000028202 and G2000028203) and the Program on Natural Science of Jinan University, Guangzhou, China (Grant No 51204056).2005-11-252005-11-252006-01-05The role of hydrogen in hydrogenated microcrystalline silicon （μc-Si：H） thin films in deposition processes with very high frequency plasma-enhanced chemical vapour deposition （VHF-PECVD） technique have been investigated in this paper. With in situ optical emission spectroscopy （OES） diagnosis during the fabrication of μc-Si：H thin films under different plasma excitation frequency Ve （60MHz-90MHz）, the characteristic peak intensities （IsiH＊, IHα＊ and IHβ＊ ） in SiH4＋H2 plasma and the ratio of （IHα＊ ＋ IHβ＊ ） to IsiH＊ were measured; all the characteristic peak intensities and the ratio （IHα＊ ＋ IHβ＊ ）/IsiH＊ are increased with plasma excitation frequency. It is identified that high plasma excitation frequency is favourable to promote the decomposition of SiH4＋H2 to produce atomic hydrogen and SiHx radicals. The influences of atomic hydrogen on structural properties and that of SiHx radicals on deposition rate of μc-Si：H thin films have been studied through Raman spectra and thickness measurements, respectively. It can be concluded that both the crystalline volume fraction and deposition rate are enhanced with the increase of plasma excitation frequency, which is in good accord with the OES results. By means of FTIR measurements, hydrogen contents of μc-Si：H thin films deposited at different plasma excitation frequency have been evaluated from the integrated intensity of wagging mode near 640 cm^-1. The hydrogen contents vary from 4% to 5%, which are much lower than those of μc-Si：H films deposited with RF-PECVD technique. This implies that μc-Si：H thin films deposited with VHF-PECVD technique usually have good stability under light-soaking.     

Improved calculation of relic gravitational waves

In this paper, we have improved the calculation of the relic gravitational waves (RGW) in two aspects. First, we investigate the transfer function by taking into consideration the redshift-suppression effect, the accelerating expansion effect, the damping effect of free-streaming relativistic particles, and the damping effect of cosmic phase transition, and give a simple approximate analytic expression, which clearly illustrates the dependence on the cosmological parameters. Second, we develop a numerical method to calculate the primordial power spectrum of RGW in a very wide frequency range, where the observed constraints on $n_{\rm s}$ (the scalar spectral index) and $P_{\rm S}(k_0)$ (the amplitude of primordial scalar spectrum) and the Hamilton--Jacobi equation are used. This method is applied to two kinds of inflationary models, which satisfy the current constraints on $n_{\rm s}$, $\alpha$ (the running of $n_{\rm s}$) and $r$ (the tensor--scalar ratio). We plot them in the $r-{\it\Omega}_{\rm g}$ diagram, where ${\it\Omega}_{\rm g}$ is the strength of RGW, and study their measurements from the cosmic microwave background (CMB) experiments and laser interferometers.     

Spin polarization effect for Os2 molecule

Density functional Theory （DFT） （B3p86） of Gaussian03 has been used to optimize the structure of Os2 molecule. The result shows that the ground state for Os2 molecule is 9-multiple state and its electronic configuration is ^9∑^＋g, which shows spin polarization effect of Os2 molecule of transition metal elements for the first time. Meanwhile, we have not found any spin pollution because the wavefunction of the ground state does not mingle with wavefunctions with higher energy states. So, the fact that the ground state for Os2 molecule is a 9-multiple state is indicative of spin polarization effect of Os2 molecule of transition metal elements. That is, there exist 8 parallel spin electrons. The non-conjugated electron is greatest in number. These electrons occupy different spacious tracks, so that the energy of Os2 molecule is minimized. It can be concluded that the effect of parallel spin of Os2 molecule is larger than the effect of the conjugated molecule, which is obviously related to the effect of electron d delocalization. In addition, the Murrell-Sorbie potential functions with the parameters for the ground state ^9∑^＋g and other states of Os2 molecule are derived. Dissociation energy De for the ground state of Os2 molecule is 3.3971eV, equilibrium bond length Re is 0.2403nm, vibration frequency ωe is 235.32cm^-1. Its force constants f2, f3, and f4 are 3.1032×10^2aJ·nm^-2, -14.3425×10^3aJ·nm^-3 and 50.5792×10^4aJ·nm^-4 respectively. The other spectroscopic data for the ground state of Os2 molecule ωexe, Be and ae are 0.4277cm^- 1, 0.0307cm^- 1 and 0.6491 × 10^-4cm^-1 respectively.     

Effect of substrate temperature on the growth and properties of boron-doped microcrystalline silicon films

Highly conductive boron-doped hydrogenated microcrystalline silicon (\mu c-Si:H) films are prepared by very high frequency plasma enhanced chemical vapour deposition (VHF PECVD) at the substrate temperatures $T_{\rm S})$ ranging from 90$^\circ$C to 270$^\circ$C. The effects of $T_{\rm S}$ on the growth and properties of the films are investigated. Results indicate that the growth rate, the electrical (dark conductivity, carrier concentration and Hall mobility) and structural (crystallinity and grain size) properties are all strongly dependent on $T_{\rm S}$. As $T_{\rm S}$ increases, it is observed that 1) the growth rate initially increases and then arrives at a maximum value of 13.3 nm/min at $T_{\rm S}$=210$^\circ$C, 2) the crystalline volume fraction ($X_{\rm c})$ and the grain size increase initially, then reach their maximum values at $T_{\rm S}$=140$^\circ$C, and finally decrease, 3) the dark conductivity ($\sigma _{\rm d})$, carrier concentration and Hall mobility have a similar dependence on $T_{\rm S}$ and arrive at their maximum values at $T_{\rm S}$=190$^\circ$C. In addition, it is also observed that at a lower substrate temperature $T_{\rm S}$, a higher dopant concentration is required in order to obtain a maximum $\sigma _{\rm d}$.     

Geometric quantities of lower doubly excited bound states of helium

Expectation values of single electron and interelectronic geometric quantities such as $\langle r\rangle$, $\langle r_{12}\rangle$, $\langle r_<\rangle$, $\langle r_>\rangle$, $\langle \cos\theta_{12}\rangle$ and $\langle \theta_{12}\rangle$ are calculated for doubly excited $2{\rm p}n{\rm p}\,{}^1P^{\,\rm e}\,(3\leq n\leq5),\, 2{\rm p}n{\rm p}\,{}^3\!P^{\,\rm e}\,(2\leq n\leq5)$ and $2{\rm p}n{\rm d}\,{}^{1,3}D^{\,\rm o}\,(3\leq n\leq5)$ states of helium using Hylleraas-$B$-spline basis set. The energy levels converge to at least 10 significant digits in our calculations. The extrapolated values of geometric quantities except for $\langle \theta_{12}\rangle$ reach 10 significant digits as well; $\langle \theta_{12}\rangle$ reaches at least 7 significant digits using a multipole expansion approach. Our results provide a precise reference for future research.     

Effect of pressure evolution on the formation enhancement in dual interacting vortex rings

In the biological locomotion, the ambit pressure is of particular importance to use as a means of propulsion. The multiple vortex rings have been proved to generate additional thrust by interaction, but the mechanism of this thrust enhancement is still unknown. This study examines the effect of ambit pressure on formation enhancement in interacting dual vortex rings. The vortex rings, which have the same formation time, are successively generated in a piston-cylinder apparatus. The finite-time Lyapunov exponent (FTLE) visualizes the flow fields as an indication of Lagrangian coherent structures (LCSs), and the pressure field is calculated based on the digital particle image velocity (DPIV). We extract the back pressure of the rear vortex in dual vortices and the back pressure circulation $\varGamma_{\rm b}$, which is defined as a form of overpressure circulation $\varGamma_{\rm p}$. The $\varGamma_{\rm b}$ has a positive linear relationship with $\varGamma_{\rm p}$. A critical interval distance $d_{\rm cr}^*$ in a range of 0.32-0.42 is found where $\varGamma_{\rm b}$ and $\varGamma_{\rm p}$ reach the maximum synchronously, leading to a full-interaction mode. Moreover, an over-interaction mode and an under-interaction mode are proposed when the dimensionless interval distance $d^*$ is smaller or larger than $d_{\rm cr}^*$. To conclude, the high back pressure caused by vortex interaction can enhance the formation of vortex rings and lead to high thrust.