2015 Vol. 39, No. 8
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2015, 39(8): 082001. doi: 10.1088/1674-1137/39/8/082001
Abstract:
We present physics opportunities and topics with the ss states (strangeonia) that can be studied with the BESⅢ detector operating at the BEPCⅡ collider. Though the φ and η/η' states have long been established experimentally, only a handful of strangeonia are well known, in contrast to the rich cc charmoium system. An overview of the ss states and their experimental status is presented in this paper. The BESⅢ experiment has collected the world's largest samples of J/ψ, ψ(2S), ψ(3770), and direct e+e- annihilations at energies below the J/ψ and above 3.8 GeV, and will continue to accumulate high quality, large integrated luminosity in the τ-charm energy region. These data, combined with the excellent performance of the BESⅢ detector, will offer unprecedented opportunities to explore the ss system. In this paper we describe the experimental techniques to explore strangeonia with the BESⅢ detector.
We present physics opportunities and topics with the ss states (strangeonia) that can be studied with the BESⅢ detector operating at the BEPCⅡ collider. Though the φ and η/η' states have long been established experimentally, only a handful of strangeonia are well known, in contrast to the rich cc charmoium system. An overview of the ss states and their experimental status is presented in this paper. The BESⅢ experiment has collected the world's largest samples of J/ψ, ψ(2S), ψ(3770), and direct e+e- annihilations at energies below the J/ψ and above 3.8 GeV, and will continue to accumulate high quality, large integrated luminosity in the τ-charm energy region. These data, combined with the excellent performance of the BESⅢ detector, will offer unprecedented opportunities to explore the ss system. In this paper we describe the experimental techniques to explore strangeonia with the BESⅢ detector.
2015, 39(8): 083101. doi: 10.1088/1674-1137/39/8/083101
Abstract:
Using a conventional constituent-quark model, I=1/2 scalar κ, vector K*(892), and axial vector K1 mesons are studied in the asqtad-improved staggered fermion with wall-source and point-sink interpolators. The mass ratio of mκ/mK*(892) is numerically confirmed to apparently vary with quark mass, and the experimental ordering mK*(892) > mκ holds elegantly when the light u/d quark masses are sufficiently small, while the valence strange quark mass is fixed to its physical value. We also get reasonable signals for the K1 meson suggested by the SCALAR Collaboration from lattice QCD. The computations are conducted with the MILC Nf=3 flavor gauge configurations at three lattice spacings: a≈ 0.15, 0.12, and 0.09 fm.
Using a conventional constituent-quark model, I=1/2 scalar κ, vector K*(892), and axial vector K1 mesons are studied in the asqtad-improved staggered fermion with wall-source and point-sink interpolators. The mass ratio of mκ/mK*(892) is numerically confirmed to apparently vary with quark mass, and the experimental ordering mK*(892) > mκ holds elegantly when the light u/d quark masses are sufficiently small, while the valence strange quark mass is fixed to its physical value. We also get reasonable signals for the K1 meson suggested by the SCALAR Collaboration from lattice QCD. The computations are conducted with the MILC Nf=3 flavor gauge configurations at three lattice spacings: a≈ 0.15, 0.12, and 0.09 fm.
2015, 39(8): 083102. doi: 10.1088/1674-1137/39/8/083102
Abstract:
Supersymmetry (SUSY) may be one of the most favored extensions of the Standard Model (SM), but so far at the LHC no evidence of SUSY particles has been observed. An obvious question is whether they have already emerged but escaped our detection, or whether they do not exist at all. We propose that the future ILC may provide sufficient energy and luminosity to produce SUSY particles as long as they are not too heavy. Superflavor symmetry associates production rates of SUSY mesinos with those of regular mesons, because both contain a heavy constituent and a light one. In this work, we estimate the production rate of SUSY mesinos near their production threshold and compare it with BB production. Our analysis indicates that if SUSY mesinos with masses below s/2 (s is the ILC energy) exist, they could be observed at the future ILC or even the proposed CEPC in China.
Supersymmetry (SUSY) may be one of the most favored extensions of the Standard Model (SM), but so far at the LHC no evidence of SUSY particles has been observed. An obvious question is whether they have already emerged but escaped our detection, or whether they do not exist at all. We propose that the future ILC may provide sufficient energy and luminosity to produce SUSY particles as long as they are not too heavy. Superflavor symmetry associates production rates of SUSY mesinos with those of regular mesons, because both contain a heavy constituent and a light one. In this work, we estimate the production rate of SUSY mesinos near their production threshold and compare it with BB production. Our analysis indicates that if SUSY mesinos with masses below s/2 (s is the ILC energy) exist, they could be observed at the future ILC or even the proposed CEPC in China.
2015, 39(8): 083103. doi: 10.1088/1674-1137/39/8/083103
Abstract:
In this work, we use the Born-Oppenheimer approximation, where the potential between atoms can be approximated as a function of distance between the two nuclei, to study the four-quark bound states. By this approximation, Heitler and London calculated the spectrum of the hydrogen molecule, which includes two protons (heavy) and two electrons (light). Generally, the observed exotic mesons Zb(10610), Zb(10650), Zc(3900) and Zc(4020) (Zc(4025)) may be molecular states made of two physical mesons and/or diquark-anti-diquark structures. Analogous to the Heitler-London method for calculating the mass of the hydrogen molecule, we investigate whether there exist energy minima for these two structures. Contrary to the hydrogen molecule case where only the spin-triplet possesses an energy minimum, there exist minima for both of these states. This implies that both molecule and tetraquark states can be stable objects. Since they have the same quantum numbers, however, the two states may mix to result in the physical states. A consequence would be that partner exotic states co-existing with Zb(10610), Zb(10650), Zc(3900) and Zc(4020) (Zc(4025)) are predicted and should be experimentally observed.
In this work, we use the Born-Oppenheimer approximation, where the potential between atoms can be approximated as a function of distance between the two nuclei, to study the four-quark bound states. By this approximation, Heitler and London calculated the spectrum of the hydrogen molecule, which includes two protons (heavy) and two electrons (light). Generally, the observed exotic mesons Zb(10610), Zb(10650), Zc(3900) and Zc(4020) (Zc(4025)) may be molecular states made of two physical mesons and/or diquark-anti-diquark structures. Analogous to the Heitler-London method for calculating the mass of the hydrogen molecule, we investigate whether there exist energy minima for these two structures. Contrary to the hydrogen molecule case where only the spin-triplet possesses an energy minimum, there exist minima for both of these states. This implies that both molecule and tetraquark states can be stable objects. Since they have the same quantum numbers, however, the two states may mix to result in the physical states. A consequence would be that partner exotic states co-existing with Zb(10610), Zb(10650), Zc(3900) and Zc(4020) (Zc(4025)) are predicted and should be experimentally observed.
2015, 39(8): 084001. doi: 10.1088/1674-1137/39/8/084001
Abstract:
In order to investigate the temperature effect on the stopping power of liquid lithium material for keV D+, the excitation functions of the α-particle yields for the 6Li(d,α)4He reaction in liquid lithium (495--600 K) have been measured for the bombarding energies from 50 to 70 keV by 2.5 keV steps. The observations show that the thick-target α-yield increases statistically as lithium temperature increases. These phenomena revealed that the only possible reason is a temperature effect on the stopping power, i.e., increasing temperature resulting in a lower stopping power. As the lithium temperature increased from 495 to 600 K, the energy loss of deuterons decreased about 6.7% in the energy region of E<70 keV.
In order to investigate the temperature effect on the stopping power of liquid lithium material for keV D+, the excitation functions of the α-particle yields for the 6Li(d,α)4He reaction in liquid lithium (495--600 K) have been measured for the bombarding energies from 50 to 70 keV by 2.5 keV steps. The observations show that the thick-target α-yield increases statistically as lithium temperature increases. These phenomena revealed that the only possible reason is a temperature effect on the stopping power, i.e., increasing temperature resulting in a lower stopping power. As the lithium temperature increased from 495 to 600 K, the energy loss of deuterons decreased about 6.7% in the energy region of E<70 keV.
2015, 39(8): 084101. doi: 10.1088/1674-1137/39/8/084101
Abstract:
Energy levels and the reduced probability of E2-- transitions for ytterbium isotopes with proton number Z=70 and neutron numbers between 100 and 106 have been calculated through phenomenological (PhM) and interacting boson (IBM-1) models. The predicted low-lying levels (energies, spins and parities) and the reduced probability for E2-- transitions results are reasonably consistent with the available experimental data. The predicted low-lying levels (gr--, β1-- and γ1-- band) produced in the PhM are in good agreement with the experimental data compared with those by IBM-1 for all nuclei of interest. In addition, the phenomenological model was successful in predicting the β2--, β3--, β4--, γ2-- and 1+-- band while it was a failure with IBM-1. Also, the 3+-- band is predicted by the IBM-1 model for 172Yb and 174Yb nuclei. All calculations are compared with the available experimental data.
Energy levels and the reduced probability of E2-- transitions for ytterbium isotopes with proton number Z=70 and neutron numbers between 100 and 106 have been calculated through phenomenological (PhM) and interacting boson (IBM-1) models. The predicted low-lying levels (energies, spins and parities) and the reduced probability for E2-- transitions results are reasonably consistent with the available experimental data. The predicted low-lying levels (gr--, β1-- and γ1-- band) produced in the PhM are in good agreement with the experimental data compared with those by IBM-1 for all nuclei of interest. In addition, the phenomenological model was successful in predicting the β2--, β3--, β4--, γ2-- and 1+-- band while it was a failure with IBM-1. Also, the 3+-- band is predicted by the IBM-1 model for 172Yb and 174Yb nuclei. All calculations are compared with the available experimental data.
2015, 39(8): 084102. doi: 10.1088/1674-1137/39/8/084102
Abstract:
A set of optimal proton optical potential parameters for p+184W reactions are obtained at incident proton energy up to 250 MeV. Based on these parameters, the reaction cross-sections, elastic scattering angular distributions, energy spectra and double differential cross sections of proton-induced reactions on 184W are calculated and analyzed by using theoretical models which integrate the optical model, distorted Born wave approximation theory, intra-nuclear cascade model, exciton model, Hauser-Feshbach theory and evaporation model. The calculated results are compared with existing experimental data and good agreement is achieved.
A set of optimal proton optical potential parameters for p+184W reactions are obtained at incident proton energy up to 250 MeV. Based on these parameters, the reaction cross-sections, elastic scattering angular distributions, energy spectra and double differential cross sections of proton-induced reactions on 184W are calculated and analyzed by using theoretical models which integrate the optical model, distorted Born wave approximation theory, intra-nuclear cascade model, exciton model, Hauser-Feshbach theory and evaporation model. The calculated results are compared with existing experimental data and good agreement is achieved.
2015, 39(8): 084103. doi: 10.1088/1674-1137/39/8/084103
Abstract:
Fusion reactions with a weakly bound projectile are studied using the double-folding model along with a repulsive interaction modifying term. Using this modified potential, including nuclear matter incompressibility effects, the fusion reaction cross sections and suppression parameters are calculated for 9Be+209Bi, 208Pb, 29Si and 27Al reactions. The results show that applying these effects at energies near the Coulomb barrier improves the agreement between the calculated and experimental cross sections, and modifies the mean values of the suppression parameter.
Fusion reactions with a weakly bound projectile are studied using the double-folding model along with a repulsive interaction modifying term. Using this modified potential, including nuclear matter incompressibility effects, the fusion reaction cross sections and suppression parameters are calculated for 9Be+209Bi, 208Pb, 29Si and 27Al reactions. The results show that applying these effects at energies near the Coulomb barrier improves the agreement between the calculated and experimental cross sections, and modifies the mean values of the suppression parameter.
2015, 39(8): 084104. doi: 10.1088/1674-1137/39/8/084104
Abstract:
A modified explanation of the cold nuclear matter (CNM) effects on J/ψ production in p+A collisions is presented in this paper. The advantage of the modified explanation is that all the CNM effects implemented in this model have clear physical origins and are mostly centered on the idea of multiple parton scattering. With the CNM effects presented in this paper, we calculated the nuclear modification factor RpA in J/ψ production under different collision energies. The results are compared with the corresponding experiment data and the factors calculated with classic nuclear effects. The factors calculated with CNM effects presented in this paper can accurately reproduce almost all existing J/ψ measurements in p-A collisions, which is much better than results obtained with the factors calculated with classic nuclear effects. The new model is therefore a more suitable approach to explain CNM effects in the hardproduction of quarkonium.
A modified explanation of the cold nuclear matter (CNM) effects on J/ψ production in p+A collisions is presented in this paper. The advantage of the modified explanation is that all the CNM effects implemented in this model have clear physical origins and are mostly centered on the idea of multiple parton scattering. With the CNM effects presented in this paper, we calculated the nuclear modification factor RpA in J/ψ production under different collision energies. The results are compared with the corresponding experiment data and the factors calculated with classic nuclear effects. The factors calculated with CNM effects presented in this paper can accurately reproduce almost all existing J/ψ measurements in p-A collisions, which is much better than results obtained with the factors calculated with classic nuclear effects. The new model is therefore a more suitable approach to explain CNM effects in the hardproduction of quarkonium.
2015, 39(8): 086001. doi: 10.1088/1674-1137/39/8/086001
Abstract:
In order to realize real-time fusion neutron spectrum diagnosis for the HL_2A Tokamak, a Bonner Sphere Spectrometer (BSS) array has been developed, consisting of eight polyethylene spheres (PS) with embedded 3He proportional counters. To validate its spectrometric capability, spectrum measurement of an 241Am-Be neutron source was carried out and is described. The Monte Carlo code Geant4 was used to calculate the response functions, taking this interference into consideration. Finally, the neutron spectrum was unfolded in the energy range from 10-9 MeV to 20 MeV. The unfolded spectrum has remarkable consistency with the ISO 8529-1 standard 241Am-Be neutron spectrum which is a preliminary demonstration that this BSS is reliable and practical.
In order to realize real-time fusion neutron spectrum diagnosis for the HL_2A Tokamak, a Bonner Sphere Spectrometer (BSS) array has been developed, consisting of eight polyethylene spheres (PS) with embedded 3He proportional counters. To validate its spectrometric capability, spectrum measurement of an 241Am-Be neutron source was carried out and is described. The Monte Carlo code Geant4 was used to calculate the response functions, taking this interference into consideration. Finally, the neutron spectrum was unfolded in the energy range from 10-9 MeV to 20 MeV. The unfolded spectrum has remarkable consistency with the ISO 8529-1 standard 241Am-Be neutron spectrum which is a preliminary demonstration that this BSS is reliable and practical.
2015, 39(8): 086002. doi: 10.1088/1674-1137/39/8/086002
Abstract:
High-purity germanium (HPGe) detectors are well suited to analyse the radioactivity of samples. In order to reduce the environmental background for an ultra-low background HPGe spectrometer, low-activity lead and oxygen free copper are installed outside the probe to shield from gamma radiation, with an outer plastic scintillator to veto cosmic rays, and an anti-Compton detector to improve the peak-to-Compton ratio. Using Geant4 tools and taking into account a detailed description of the detector, we optimize the sizes of these detectors to reach the design requirements. A set of experimental data from an existing HPGe spectrometer was used to compare with the simulation. For the future low-background HPGe detector simulation, considering different thicknesses of BGO crystals and anti-coincidence efficiency, the simulation results show that the optimal BGO thickness is 5.5 cm, and the peak-to-Compton ratio of 40K is raised to 1000 when the anti-coincidence efficiency is 0.85. In the background simulation, 15 cm oxygen-free copper plus 10 cm lead can reduce the environmental gamma rays to 0.0024 cps/100 cm3 Ge (50 keV--2.8 MeV), which is about 10-5 of the environmental background.
High-purity germanium (HPGe) detectors are well suited to analyse the radioactivity of samples. In order to reduce the environmental background for an ultra-low background HPGe spectrometer, low-activity lead and oxygen free copper are installed outside the probe to shield from gamma radiation, with an outer plastic scintillator to veto cosmic rays, and an anti-Compton detector to improve the peak-to-Compton ratio. Using Geant4 tools and taking into account a detailed description of the detector, we optimize the sizes of these detectors to reach the design requirements. A set of experimental data from an existing HPGe spectrometer was used to compare with the simulation. For the future low-background HPGe detector simulation, considering different thicknesses of BGO crystals and anti-coincidence efficiency, the simulation results show that the optimal BGO thickness is 5.5 cm, and the peak-to-Compton ratio of 40K is raised to 1000 when the anti-coincidence efficiency is 0.85. In the background simulation, 15 cm oxygen-free copper plus 10 cm lead can reduce the environmental gamma rays to 0.0024 cps/100 cm3 Ge (50 keV--2.8 MeV), which is about 10-5 of the environmental background.
2015, 39(8): 086003. doi: 10.1088/1674-1137/39/8/086003
Abstract:
A neutron-TPC (nTPC) is being developed for use as a fast neutron spectrometer in the fields of nuclear physics, nuclear reactor operation monitoring, and thermo-nuclear fusion plasma diagnostics. An nTPC prototype based on a GEM-TPC (Time Projection Chamber with Gas Electron Multiplier amplification) has been assembled and tested using argon-hydrocarbon mixture as the working gas. By measuring the energy deposition of the recoil proton in the sensitive volume and the angle of the proton track, the incident neutron energy can be deduced. A Monte Carlo simulation was carried out to analyze the parameters affecting the energy resolution of the nTPC, and gave an optimized resolution under ideal conditions. An alpha particle experiment was performed to verify its feasibility, and to characterize its performance, including energy resolution and spatial resolution. Based on the experimental measurement and analysis, the energy resolution (FWHM) of the nTPC prototype is predicted to be better than 3.2% for 5 MeV incident neutrons, meeting the performance requirement (FWHM<5%) for the nTPC prototype.
A neutron-TPC (nTPC) is being developed for use as a fast neutron spectrometer in the fields of nuclear physics, nuclear reactor operation monitoring, and thermo-nuclear fusion plasma diagnostics. An nTPC prototype based on a GEM-TPC (Time Projection Chamber with Gas Electron Multiplier amplification) has been assembled and tested using argon-hydrocarbon mixture as the working gas. By measuring the energy deposition of the recoil proton in the sensitive volume and the angle of the proton track, the incident neutron energy can be deduced. A Monte Carlo simulation was carried out to analyze the parameters affecting the energy resolution of the nTPC, and gave an optimized resolution under ideal conditions. An alpha particle experiment was performed to verify its feasibility, and to characterize its performance, including energy resolution and spatial resolution. Based on the experimental measurement and analysis, the energy resolution (FWHM) of the nTPC prototype is predicted to be better than 3.2% for 5 MeV incident neutrons, meeting the performance requirement (FWHM<5%) for the nTPC prototype.
2015, 39(8): 086004. doi: 10.1088/1674-1137/39/8/086004
Abstract:
Aiming at the observation of cosmic-ray chemical composition in the "knee" energy region, we have been developing a new type of air-shower core detector (YAC, Yangbajing Air shower Core detector array) to be set up at Yangbajing (90.522° E, 30.102° N, 4300 m above sea level, atmospheric depth: 606 g/m2) in Tibet, China. YAC works together with the Tibet air-shower array (Tibet-Ⅲ) and an underground water Cherenkov muon detector array (MD) as a hybrid experiment. Each YAC detector unit consists of lead plates of 3.5 cm thickness and a scintillation counter which detects the burst size induced by high energy particles in the air-shower cores. The burst size can be measured from 1 MIP (Minimum Ionization Particle) to 106 MIPs. The first phase of this experiment, named "YAC-Ⅰ", consists of 16 YAC detectors each with a size of 40 cm×50 cm and distributed in a grid with an effective area of 10 m2. YAC-Ⅰ is used to check hadronic interaction models. The second phase of the experiment, called "YAC-Ⅱ", consists of 124 YAC detectors with coverage of about 500 m2. The inner 100 detectors of 80 cm× 50 cm each are deployed in a 10×10 matrix with a 1.9 separation; the outer 24 detectors of 100 cm×50 cm each are distributed around these to reject non-core events whose shower cores are far from the YAC-Ⅱ array. YAC-Ⅱ is used to study the primary cosmic-ray composition, in particular, to obtain the energy spectra of protons, helium and iron nuclei between 5×1013 eV and 1016 eV, covering the "knee" and also connected with direct observations at energies around 100 TeV. We present the design and performance of YAC-Ⅱ in this paper.
Aiming at the observation of cosmic-ray chemical composition in the "knee" energy region, we have been developing a new type of air-shower core detector (YAC, Yangbajing Air shower Core detector array) to be set up at Yangbajing (90.522° E, 30.102° N, 4300 m above sea level, atmospheric depth: 606 g/m2) in Tibet, China. YAC works together with the Tibet air-shower array (Tibet-Ⅲ) and an underground water Cherenkov muon detector array (MD) as a hybrid experiment. Each YAC detector unit consists of lead plates of 3.5 cm thickness and a scintillation counter which detects the burst size induced by high energy particles in the air-shower cores. The burst size can be measured from 1 MIP (Minimum Ionization Particle) to 106 MIPs. The first phase of this experiment, named "YAC-Ⅰ", consists of 16 YAC detectors each with a size of 40 cm×50 cm and distributed in a grid with an effective area of 10 m2. YAC-Ⅰ is used to check hadronic interaction models. The second phase of the experiment, called "YAC-Ⅱ", consists of 124 YAC detectors with coverage of about 500 m2. The inner 100 detectors of 80 cm× 50 cm each are deployed in a 10×10 matrix with a 1.9 separation; the outer 24 detectors of 100 cm×50 cm each are distributed around these to reject non-core events whose shower cores are far from the YAC-Ⅱ array. YAC-Ⅱ is used to study the primary cosmic-ray composition, in particular, to obtain the energy spectra of protons, helium and iron nuclei between 5×1013 eV and 1016 eV, covering the "knee" and also connected with direct observations at energies around 100 TeV. We present the design and performance of YAC-Ⅱ in this paper.
2015, 39(8): 087001. doi: 10.1088/1674-1137/39/8/087001
Abstract:
A stochastic cooling system is under design and construction at HIRFL-CSRe (Heavy Ion Research Facility in Lanzhou - experimental Cooling Storage Ring), with the aim of cooling secondary particles produced at HIRFL-RIBLL2 (2nd Radioactive Ion Beam Line in Lanzhou). The optical layout of CSRe has been optimized to meet the requirements of a stochastic cooling system. In this paper, a particle tracking method is used to investigate both transverse and longitudinal cooling on the basis of the modified optical layout, demonstrating how it can be used to optimize stochastic cooling parameters. Simulation results indicate that the particle tracking method is an innovative and reasonable method to study stochastic cooling. It also has the advantage of discovering the influence of Twiss parameters at the pickups and kickers, which will be explored in further studies.
A stochastic cooling system is under design and construction at HIRFL-CSRe (Heavy Ion Research Facility in Lanzhou - experimental Cooling Storage Ring), with the aim of cooling secondary particles produced at HIRFL-RIBLL2 (2nd Radioactive Ion Beam Line in Lanzhou). The optical layout of CSRe has been optimized to meet the requirements of a stochastic cooling system. In this paper, a particle tracking method is used to investigate both transverse and longitudinal cooling on the basis of the modified optical layout, demonstrating how it can be used to optimize stochastic cooling parameters. Simulation results indicate that the particle tracking method is an innovative and reasonable method to study stochastic cooling. It also has the advantage of discovering the influence of Twiss parameters at the pickups and kickers, which will be explored in further studies.
2015, 39(8): 087002. doi: 10.1088/1674-1137/39/8/087002
Abstract:
The alignment tolerance of multipoles on a girder is better than ± 30 μm in the storage ring of the High Energy Photon Source (HEPS) which will be the next project at IHEP (Institute of High Energy Physics). It is difficult to meet the precision when only using the traditional optical survey method. In order to achieve this goal, a vibrating wire alignment technique with high precision and sensitivity is considered to be used in this project. This paper presents some preliminary research works about theory, scheme design and achievements.
The alignment tolerance of multipoles on a girder is better than ± 30 μm in the storage ring of the High Energy Photon Source (HEPS) which will be the next project at IHEP (Institute of High Energy Physics). It is difficult to meet the precision when only using the traditional optical survey method. In order to achieve this goal, a vibrating wire alignment technique with high precision and sensitivity is considered to be used in this project. This paper presents some preliminary research works about theory, scheme design and achievements.
2015, 39(8): 088101. doi: 10.1088/1674-1137/39/8/088101
Abstract:
There is an empirical formula describing the relationship between the peak magnetic field and the undulator structure parameters for a uniform-parameter hybrid undulator. In this paper, we investigate the relationship for a linearly tapered undulator through numerical calculation by using the code RADIA, and check it with the empirical formula. The results imply that this empirical formula is also effective for linearly tapered undulators at a big enough scope for the requirements of normal FEL experiments. Therefore, for a linearly tapered undulator, we can use the empirical formula to design the variation of the undulator gap. For the tapering rate demanded by normal FEL experiments, the gap of a linearly tapered undulator increases almost linearly, and the tapering rate will keep constant while adjusting the undulator gap with the same variation for each undulator period.
There is an empirical formula describing the relationship between the peak magnetic field and the undulator structure parameters for a uniform-parameter hybrid undulator. In this paper, we investigate the relationship for a linearly tapered undulator through numerical calculation by using the code RADIA, and check it with the empirical formula. The results imply that this empirical formula is also effective for linearly tapered undulators at a big enough scope for the requirements of normal FEL experiments. Therefore, for a linearly tapered undulator, we can use the empirical formula to design the variation of the undulator gap. For the tapering rate demanded by normal FEL experiments, the gap of a linearly tapered undulator increases almost linearly, and the tapering rate will keep constant while adjusting the undulator gap with the same variation for each undulator period.
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