2015 Vol. 39, No. 4
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2015, 39(4): 041001. doi: 10.1088/1674-1137/39/4/041001
Abstract:
Using data samples collected at center-of-mass energies of √s=4.009, 4.230, 4.260, and 4.360 GeV with the BES0 detector operating at the BEPC/ collider, we perform a search for the process e+e-→γχc,j (J =0, 1, 2) and find evidence for e+e-→γχc1 and e+e-→γχc2 with statistical significances of 3.0σ and 3.4σ, respectively. The Born cross sections σB(e+e-→γχc,j), as well as their upper limits at the 90% con dence level (C.L.) are determined at each center-of-mass energy.
Using data samples collected at center-of-mass energies of √s=4.009, 4.230, 4.260, and 4.360 GeV with the BES0 detector operating at the BEPC/ collider, we perform a search for the process e+e-→γχc,j (J =0, 1, 2) and find evidence for e+e-→γχc1 and e+e-→γχc2 with statistical significances of 3.0σ and 3.4σ, respectively. The Born cross sections σB(e+e-→γχc,j), as well as their upper limits at the 90% con dence level (C.L.) are determined at each center-of-mass energy.
2015, 39(4): 041002. doi: 10.1088/1674-1137/39/4/041002
Abstract:
We present several possible hadronic states found in coupled-channel models within the on-shell approximation. The interaction potential is constructed as a sum of the tree-level Feynman diagrams calculated with the effective Lagrangians. Based on the recent empirical data, we illustrate the possible existence of several baryonic and mesonic states with definite quantum numbers in the model. We give their properties for the purpose of further study and discuss the potential of finding them in future experiments.
We present several possible hadronic states found in coupled-channel models within the on-shell approximation. The interaction potential is constructed as a sum of the tree-level Feynman diagrams calculated with the effective Lagrangians. Based on the recent empirical data, we illustrate the possible existence of several baryonic and mesonic states with definite quantum numbers in the model. We give their properties for the purpose of further study and discuss the potential of finding them in future experiments.
2015, 39(4): 043101. doi: 10.1088/1674-1137/39/4/043101
Abstract:
It is shown that a novel anomaly associated with transverse Ward-Takahashi identity exists for a pseudo-tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiyah-Singer index theorem.
It is shown that a novel anomaly associated with transverse Ward-Takahashi identity exists for a pseudo-tensor current in QED, and the anomaly gives rise to a topological index of a Dirac operator in terms of an Atiyah-Singer index theorem.
2015, 39(4): 043102. doi: 10.1088/1674-1137/39/4/043102
Abstract:
We calculate the D-brane superpotentials for two compact Calabi-Yau manifolds X14(1,1,2,3,7) and X8(1,1,1,2,3) which are of non-Fermat type in the type II string theory. By constructing the open mirror symmetry, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.
We calculate the D-brane superpotentials for two compact Calabi-Yau manifolds X14(1,1,2,3,7) and X8(1,1,1,2,3) which are of non-Fermat type in the type II string theory. By constructing the open mirror symmetry, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.
2015, 39(4): 043103. doi: 10.1088/1674-1137/39/4/043103
Abstract:
A scheme is proposed for quantum information splitting of a two-qubit Bell state by using a four-qubit entangled state as a quantum channel. In the scenario, it is supposed that there are three legitimate parties, say Alice, Bob and Charlie. Alice is the sender of quantum information. Bob and Charlie are two agents. Alice first performs GHZ state measurement and tells Bob and Charlie the measurement results via a classical channel. It is impossible for Bob to reconstruct the original state with local operations unless help is obtained from Charlie. If Charlie allows Bob to reconstruct the original state information, he needs to perform a single-qubit measurement and tell Bob the measurement result. Using the measurement results from Alice and Charlie, Bob can reconstruct the original state. We also consider the problem of security attacks. This protocol is considered to be secure.
A scheme is proposed for quantum information splitting of a two-qubit Bell state by using a four-qubit entangled state as a quantum channel. In the scenario, it is supposed that there are three legitimate parties, say Alice, Bob and Charlie. Alice is the sender of quantum information. Bob and Charlie are two agents. Alice first performs GHZ state measurement and tells Bob and Charlie the measurement results via a classical channel. It is impossible for Bob to reconstruct the original state with local operations unless help is obtained from Charlie. If Charlie allows Bob to reconstruct the original state information, he needs to perform a single-qubit measurement and tell Bob the measurement result. Using the measurement results from Alice and Charlie, Bob can reconstruct the original state. We also consider the problem of security attacks. This protocol is considered to be secure.
2015, 39(4): 044001. doi: 10.1088/1674-1137/39/4/044001
Abstract:
Spallation neutrons were produced by the irradiation of Pb with 250 MeV protons. The Pb target was surrounded by water which was used to slow down the emitted neutrons. The moderated neutrons in the water bath were measured by using the resonance detectors of Au, Mn and In with a cadmium (Cd) cover. According to the measured activities of the foils, the neutron flux at different resonance energies were deduced and the epithermal neutron spectra were proposed. Corresponding results calculated with the Monte Carlo code MCNPX were compared with the experimental data to check the validity of the code. The comparison showed that the simulation could give a good prediction for the neutron spectra above 50 eV, while the finite thickness of the foils greatly effected the experimental data in low energy. It was also found that the resonance detectors themselves had great impact on the simulated energy spectra.
Spallation neutrons were produced by the irradiation of Pb with 250 MeV protons. The Pb target was surrounded by water which was used to slow down the emitted neutrons. The moderated neutrons in the water bath were measured by using the resonance detectors of Au, Mn and In with a cadmium (Cd) cover. According to the measured activities of the foils, the neutron flux at different resonance energies were deduced and the epithermal neutron spectra were proposed. Corresponding results calculated with the Monte Carlo code MCNPX were compared with the experimental data to check the validity of the code. The comparison showed that the simulation could give a good prediction for the neutron spectra above 50 eV, while the finite thickness of the foils greatly effected the experimental data in low energy. It was also found that the resonance detectors themselves had great impact on the simulated energy spectra.
2015, 39(4): 044101. doi: 10.1088/1674-1137/39/4/044101
Abstract:
The 62Sm, 64Gd, 66Dy, 70Yb, 72Hf and 74W nuclei are classified as deformed nuclei. Low-lying bands are one of the most fundamental excitation modes in the energy spectra of deformed nuclei. In this paper a theoretical analysis of the experimental data within the phenomenological model is presented. The energy spectra of ground states are calculated. It is found that the low-lying spectra of ground band states are in good agreement with the experimental data.
The 62Sm, 64Gd, 66Dy, 70Yb, 72Hf and 74W nuclei are classified as deformed nuclei. Low-lying bands are one of the most fundamental excitation modes in the energy spectra of deformed nuclei. In this paper a theoretical analysis of the experimental data within the phenomenological model is presented. The energy spectra of ground states are calculated. It is found that the low-lying spectra of ground band states are in good agreement with the experimental data.
2015, 39(4): 044102. doi: 10.1088/1674-1137/39/4/044102
Abstract:
In the context of the combined model of evolution-dominated hydrodynamics + leading particles, we discuss the pseudorapidity distributions of charged particles produced in p-p collisions. A comparison is made between the theoretical predictions and experimental measurements. The combined model works well in p-p collisions in the whole available energy region from √s=23.6 to 900 GeV.
In the context of the combined model of evolution-dominated hydrodynamics + leading particles, we discuss the pseudorapidity distributions of charged particles produced in p-p collisions. A comparison is made between the theoretical predictions and experimental measurements. The combined model works well in p-p collisions in the whole available energy region from √s=23.6 to 900 GeV.
2015, 39(4): 044103. doi: 10.1088/1674-1137/39/4/044103
Abstract:
Based on the isospin-dependent quantum molecular dynamics model along with the GEMINI model, heavy-ion collisions at intermediate energies are studied. We calculate the production cross sections of different fragments for reactions of 112Sn+112Sn and 124Sn+124Sn at different beam energies. The species and production cross sections of neutron-rich isotopes are generally dependent on the isospin of the system and the incident energies. The isotopes 48Ca and 54Ca are more productive for the neutron-rich system at 30 to 150 MeV/nucleon.
Based on the isospin-dependent quantum molecular dynamics model along with the GEMINI model, heavy-ion collisions at intermediate energies are studied. We calculate the production cross sections of different fragments for reactions of 112Sn+112Sn and 124Sn+124Sn at different beam energies. The species and production cross sections of neutron-rich isotopes are generally dependent on the isospin of the system and the incident energies. The isotopes 48Ca and 54Ca are more productive for the neutron-rich system at 30 to 150 MeV/nucleon.
2015, 39(4): 046001. doi: 10.1088/1674-1137/39/4/046001
Abstract:
A Gas Electron Multiplier (GEM) detector with an effective area of 300 mm×300 mm has been constructed using a novel self-stretching technique, which allows a highly flexible and efficient GEM detector assembly free of glue or spacers. This makes the re-opening and repair of the GEM detectors possible and significantly reduces the scrap rate in the mass production of large-area GEM detectors. With the technique, the assembly time can be limited to a few hours, a factor of ten improvement compared to that using gluing techniques. The details of design and assembly procedure of the 300 mm×300 mm GEM detector are described in this paper. This detector was tested with 8 keV X-rays for the effective gain, energy resolution and performance uniformity. The results show that the typical energy resolution is 20% at an effective gain of about 10^4, with fairly good uniformity.
A Gas Electron Multiplier (GEM) detector with an effective area of 300 mm×300 mm has been constructed using a novel self-stretching technique, which allows a highly flexible and efficient GEM detector assembly free of glue or spacers. This makes the re-opening and repair of the GEM detectors possible and significantly reduces the scrap rate in the mass production of large-area GEM detectors. With the technique, the assembly time can be limited to a few hours, a factor of ten improvement compared to that using gluing techniques. The details of design and assembly procedure of the 300 mm×300 mm GEM detector are described in this paper. This detector was tested with 8 keV X-rays for the effective gain, energy resolution and performance uniformity. The results show that the typical energy resolution is 20% at an effective gain of about 10^4, with fairly good uniformity.
2015, 39(4): 046002. doi: 10.1088/1674-1137/39/4/046002
Abstract:
The inorganic CsI(Tl) crystal scintillator is a candidate anti-compton detector for the China Dark matter Experiment. Studying the intrinsic radiopurity of the CsI(Tl) crystal is an issue of major importance. The timing, energy and spatial correlations, as well as the capability of pulse shape discrimination provide powerful methods for the measurement of intrinsic radiopurities. The experimental design, detector performance and event-selection algorithms are described. A total of 359×3 kg-days data from three prototypes of CsI(Tl) crystals were taken at China Jinping Underground Laboratory (CJPL), which offers a good shielding environment. The contamination levels of internal isotopes from 137Cs, 232Th and 238U series, as well as the upper bounds of 235U series are reported. Identification of the whole α peaks from U/Th decay chains and derivation of those corresponding quenching factors are achieved.
The inorganic CsI(Tl) crystal scintillator is a candidate anti-compton detector for the China Dark matter Experiment. Studying the intrinsic radiopurity of the CsI(Tl) crystal is an issue of major importance. The timing, energy and spatial correlations, as well as the capability of pulse shape discrimination provide powerful methods for the measurement of intrinsic radiopurities. The experimental design, detector performance and event-selection algorithms are described. A total of 359×3 kg-days data from three prototypes of CsI(Tl) crystals were taken at China Jinping Underground Laboratory (CJPL), which offers a good shielding environment. The contamination levels of internal isotopes from 137Cs, 232Th and 238U series, as well as the upper bounds of 235U series are reported. Identification of the whole α peaks from U/Th decay chains and derivation of those corresponding quenching factors are achieved.
2015, 39(4): 046201. doi: 10.1088/1674-1137/39/4/046201
Abstract:
A novel simple method based on pulse area analysis (PAA) is presented for acquisition of nuclear spectra by a digitizer. The PAA method can be used as a substitute for the traditional method of pulse height analysis (PHA). In the PAA method a commercial digitizer was employed to sample and sum in the pulse, and the area of the pulse is proportional to the energy of the detected radiation. The results of simulation and experiment indicate the great advantages of the PAA method, especially as the count rate is high and the shaping time constant is small. When the shaping time constant is 0.5 μs, the energy resolution of PAA is about 66% better than that of PHA.
A novel simple method based on pulse area analysis (PAA) is presented for acquisition of nuclear spectra by a digitizer. The PAA method can be used as a substitute for the traditional method of pulse height analysis (PHA). In the PAA method a commercial digitizer was employed to sample and sum in the pulse, and the area of the pulse is proportional to the energy of the detected radiation. The results of simulation and experiment indicate the great advantages of the PAA method, especially as the count rate is high and the shaping time constant is small. When the shaping time constant is 0.5 μs, the energy resolution of PAA is about 66% better than that of PHA.
2015, 39(4): 047001. doi: 10.1088/1674-1137/39/4/047001
Abstract:
A low loss- (LL) type 500 MHz 5-cell superconducting niobium prototype cavity with a large beam aperture has been developed successfully including the optimization, the deep drawing and electron beam welding, the surface treatment and the vertical testing. The performance of the fundamental mode was optimized and the higher order modes were damped by adopting an enlarged beam pipe for propagation. Surface preparation or treatment including mechanical polishing, buffered chemical polishing and high pressure rinsing with ultra-pure water and so on was carried out carefully to ensure a perfect inner surface condition. The vertical testing results show that the accelerating voltage higher than 7.5 MV was obtained while the quality factor was better than 1× 109 at 4.2 K. No obvious multipacting or field emission was found during the test. However, a quench happened while increasing the field a little higher than 7.5 MV that at present limited the cavity performance.
A low loss- (LL) type 500 MHz 5-cell superconducting niobium prototype cavity with a large beam aperture has been developed successfully including the optimization, the deep drawing and electron beam welding, the surface treatment and the vertical testing. The performance of the fundamental mode was optimized and the higher order modes were damped by adopting an enlarged beam pipe for propagation. Surface preparation or treatment including mechanical polishing, buffered chemical polishing and high pressure rinsing with ultra-pure water and so on was carried out carefully to ensure a perfect inner surface condition. The vertical testing results show that the accelerating voltage higher than 7.5 MV was obtained while the quality factor was better than 1× 109 at 4.2 K. No obvious multipacting or field emission was found during the test. However, a quench happened while increasing the field a little higher than 7.5 MV that at present limited the cavity performance.
2015, 39(4): 047002. doi: 10.1088/1674-1137/39/4/047002
Abstract:
SLED (SLAC Energy Doubler) is a crucial component for the C-band microwave acceleration unit of a soft X-ray Free Electron Laser (SXFEL). To study the behavior of SLED, a mathematical model is commonly built and analyzed. In this paper, a new method is proposed to build the model of SLED at the Shanghai Institute of Applied Physics. With this method, the parameters of the two cavities can be analyzed separately. Also it is suitable to study parameter optimization of SLED and analyze the effect from the parameters variations. Simulation results of our method are also presented.
SLED (SLAC Energy Doubler) is a crucial component for the C-band microwave acceleration unit of a soft X-ray Free Electron Laser (SXFEL). To study the behavior of SLED, a mathematical model is commonly built and analyzed. In this paper, a new method is proposed to build the model of SLED at the Shanghai Institute of Applied Physics. With this method, the parameters of the two cavities can be analyzed separately. Also it is suitable to study parameter optimization of SLED and analyze the effect from the parameters variations. Simulation results of our method are also presented.
2015, 39(4): 047003. doi: 10.1088/1674-1137/39/4/047003
Abstract:
The China Spallation Neutron Source/Rapid Cycling Synchrotron (CSNS/RCS) accelerates a high-intensity proton beam from 80 MeV to 1.6 GeV. Since the beam current and beam power is high, the beam loading is a severe problem for the stability of the circulating beam in the RCS. To study the beam loading effect in the CSNS/RCS theoretically, the RLC circuit model of the rf cavity, the method of Fast Fourier Transform and the method of Laplace transform have been employed to obtain the impedance of the rf system, the beam spectrum and the beam-induced voltage, respectively. Based on these physical models, the beam dynamics equations have been revised and a beam loading model has been constructed in the simulation code ORIENT. By using the code, the beam loading effect on the rf system of the CSNS/RCS has been investigated. Some simulation results have been obtained and conclusions have been drawn.
The China Spallation Neutron Source/Rapid Cycling Synchrotron (CSNS/RCS) accelerates a high-intensity proton beam from 80 MeV to 1.6 GeV. Since the beam current and beam power is high, the beam loading is a severe problem for the stability of the circulating beam in the RCS. To study the beam loading effect in the CSNS/RCS theoretically, the RLC circuit model of the rf cavity, the method of Fast Fourier Transform and the method of Laplace transform have been employed to obtain the impedance of the rf system, the beam spectrum and the beam-induced voltage, respectively. Based on these physical models, the beam dynamics equations have been revised and a beam loading model has been constructed in the simulation code ORIENT. By using the code, the beam loading effect on the rf system of the CSNS/RCS has been investigated. Some simulation results have been obtained and conclusions have been drawn.
2015, 39(4): 047004. doi: 10.1088/1674-1137/39/4/047004
Abstract:
A new special coupler with a kind of bowl-shaped ceramic window for a proton linear accelerator named the Chinese Accelerator Driven System (C-ADS) at the Institute of Modern Physics (IMP) has been simulated and constructed and a continuous wave (CW) beam commissioning through a four-meter long radio frequency quadruple (RFQ) was completed by the end of July 2014. In the experiments of conditioning and beam, some problems were promoted gradually such as sparking and thermal issues. Finally, two new couplers were passed with almost 110 kW CW power and 120 kW pulsed mode, respectively. The 10 mA intensity beam experiments have now been completed, and the couplers during the operation had no thermal or electro-magnetic problems. The detailed design and results are presented in the paper.
A new special coupler with a kind of bowl-shaped ceramic window for a proton linear accelerator named the Chinese Accelerator Driven System (C-ADS) at the Institute of Modern Physics (IMP) has been simulated and constructed and a continuous wave (CW) beam commissioning through a four-meter long radio frequency quadruple (RFQ) was completed by the end of July 2014. In the experiments of conditioning and beam, some problems were promoted gradually such as sparking and thermal issues. Finally, two new couplers were passed with almost 110 kW CW power and 120 kW pulsed mode, respectively. The 10 mA intensity beam experiments have now been completed, and the couplers during the operation had no thermal or electro-magnetic problems. The detailed design and results are presented in the paper.
2015, 39(4): 047005. doi: 10.1088/1674-1137/39/4/047005
Abstract:
Significant beam loss caused by the charge exchange processes and ion impact-induced outgassing may restrict the maximum number of accelerated heavy ions during the high intensity operation of an accelerator. In order to control beam loss due to charge exchange processes and confine the generated desorption gas, tracking of the beam loss distribution and installation of absorber blocks with low-desorption rate material at appropriate locations in the main Cooler Storage Ring (CSRm) at the Institute of Modern Physics, Lanzhou, will be performed. The loss simulation of uranium ions with electron-loss is presented in this report and the conclusion is that most charge changed particles are lost in the second dipole of the super-period structure. The calculation of the collimation efficiency of the CSRm ring will be continued in the future.
Significant beam loss caused by the charge exchange processes and ion impact-induced outgassing may restrict the maximum number of accelerated heavy ions during the high intensity operation of an accelerator. In order to control beam loss due to charge exchange processes and confine the generated desorption gas, tracking of the beam loss distribution and installation of absorber blocks with low-desorption rate material at appropriate locations in the main Cooler Storage Ring (CSRm) at the Institute of Modern Physics, Lanzhou, will be performed. The loss simulation of uranium ions with electron-loss is presented in this report and the conclusion is that most charge changed particles are lost in the second dipole of the super-period structure. The calculation of the collimation efficiency of the CSRm ring will be continued in the future.
2015, 39(4): 048001. doi: 10.1088/1674-1137/39/4/048001
Abstract:
A VUV beamline at the Shanghai Synchrotron Radiation Facility (SSRF) for angle-resolved photoelectron spectroscopy (ARPES) measurements is designed. To increase the resolution and bulk sensitivity, a photon energy as low as 7 eV is desired. Because the reflectivity for p-polarized photons strongly decreases when the photon energy is below 30 eV, the design of a high flux beamline for low energy VUV photons is a challenge. This work shows a variable including angle Varied Line-space Plane-Grating Monochromator (VLPGM) with varied grating depth (VGD) which can achieve both high resolution and high flux with broad energy coverage.
A VUV beamline at the Shanghai Synchrotron Radiation Facility (SSRF) for angle-resolved photoelectron spectroscopy (ARPES) measurements is designed. To increase the resolution and bulk sensitivity, a photon energy as low as 7 eV is desired. Because the reflectivity for p-polarized photons strongly decreases when the photon energy is below 30 eV, the design of a high flux beamline for low energy VUV photons is a challenge. This work shows a variable including angle Varied Line-space Plane-Grating Monochromator (VLPGM) with varied grating depth (VGD) which can achieve both high resolution and high flux with broad energy coverage.
2015, 39(4): 048002. doi: 10.1088/1674-1137/39/4/048002
Abstract:
A feasible and convenient method is proposed to suppress higher-harmonics for a varied-line-spacing plane grating monochromator in the soft X-ray region. Related calculations and experiments demonstrate that decreasing the included angle slightly by changing the parameter of the exit arm length can significantly improve light purity. This method is suitable and has been used for experiments of detector calibration in beamline 4B7B at the Beijing Synchrotron Radiation Facility (BSRF).
A feasible and convenient method is proposed to suppress higher-harmonics for a varied-line-spacing plane grating monochromator in the soft X-ray region. Related calculations and experiments demonstrate that decreasing the included angle slightly by changing the parameter of the exit arm length can significantly improve light purity. This method is suitable and has been used for experiments of detector calibration in beamline 4B7B at the Beijing Synchrotron Radiation Facility (BSRF).
2015, 39(4): 048003. doi: 10.1088/1674-1137/39/4/048003
Abstract:
X-ray magnetic circular dichroism (XMCD) has become an important and powerful tool because it allows the study of material properties in combination with elemental specificity, chemical state specificity, and magnetic specificity. A new soft X-ray magnetic circular dichroism apparatus has been developed at the Beijing Synchrotron Radiation Facility (BSRF). The apparatus combines three experimental conditions: an ultra-high-vacuum environment, moderate magnetic fields and in-situ sample preparation to measure the absorption signal. We designed a C-type dipole electromagnet that provides magnetic fields up to 0.5 T in parallel (or anti-parallel) direction relative to the incoming X-ray beam. The performances of the electromagnet are measured and the results show good agreement with the simulation ones. Following film grown in situ by evaporation methods, XMCD measurements are performed. Combined polarization corrections, the magnetic moments of the Fe and Co films determined by sum rules are consistent with other theoretical predictions and experimental measurements.
X-ray magnetic circular dichroism (XMCD) has become an important and powerful tool because it allows the study of material properties in combination with elemental specificity, chemical state specificity, and magnetic specificity. A new soft X-ray magnetic circular dichroism apparatus has been developed at the Beijing Synchrotron Radiation Facility (BSRF). The apparatus combines three experimental conditions: an ultra-high-vacuum environment, moderate magnetic fields and in-situ sample preparation to measure the absorption signal. We designed a C-type dipole electromagnet that provides magnetic fields up to 0.5 T in parallel (or anti-parallel) direction relative to the incoming X-ray beam. The performances of the electromagnet are measured and the results show good agreement with the simulation ones. Following film grown in situ by evaporation methods, XMCD measurements are performed. Combined polarization corrections, the magnetic moments of the Fe and Co films determined by sum rules are consistent with other theoretical predictions and experimental measurements.
2015, 39(4): 048101. doi: 10.1088/1674-1137/39/4/048101
Abstract:
The free electron laser (FEL) gain formulas for a non-resonant case are studied, and some new rigorous analytical formulas are given explicitly. For the mono-energetic and non-resonant electron beam, the exact expression of the solution of the FEL characteristic cubic equation is obtained with a form much more simple than that in the literatures, and the gain length as the function of the detuning parameter is explicitly given. Then the gain for different detuning parameters and from low to high can be easily calculated. A simplified approximation formula is also given for the exponential gain calculation in the non-resonant case. For the case of the electron beam with an energy spread, the solution of the characteristic cubic equation is given explicitly for rectangular energy distribution and Lorentz distribution, respectively. Moreover the explicit expression also can be used for the solution of the characteristic cubic equation including the impact of the space charge. The transition from the low gain to the high gain is analyzed. The variations of the gain bandwidth and of the detuning parameter for the maximum gain are demonstrated. The applicable ranges of the small signal gain formula and the exponential gain formula are analyzed.
The free electron laser (FEL) gain formulas for a non-resonant case are studied, and some new rigorous analytical formulas are given explicitly. For the mono-energetic and non-resonant electron beam, the exact expression of the solution of the FEL characteristic cubic equation is obtained with a form much more simple than that in the literatures, and the gain length as the function of the detuning parameter is explicitly given. Then the gain for different detuning parameters and from low to high can be easily calculated. A simplified approximation formula is also given for the exponential gain calculation in the non-resonant case. For the case of the electron beam with an energy spread, the solution of the characteristic cubic equation is given explicitly for rectangular energy distribution and Lorentz distribution, respectively. Moreover the explicit expression also can be used for the solution of the characteristic cubic equation including the impact of the space charge. The transition from the low gain to the high gain is analyzed. The variations of the gain bandwidth and of the detuning parameter for the maximum gain are demonstrated. The applicable ranges of the small signal gain formula and the exponential gain formula are analyzed.
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