2011 Vol. 35, No. 3
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2011, 35(3): 209-213. doi: 10.1088/1674-1137/35/3/001
Abstract:
DJpsiFDC is an event generator package for the process gg→J/ψJ/ψ. It generates events for primary leading-order 2→2 processes. The package could generate a Les Houches Event (LHE) document and this could easily be embedded into detector simulation software frameworks. The package is produced in Fortran code.
DJpsiFDC is an event generator package for the process gg→J/ψJ/ψ. It generates events for primary leading-order 2→2 processes. The package could generate a Les Houches Event (LHE) document and this could easily be embedded into detector simulation software frameworks. The package is produced in Fortran code.
2011, 35(3): 214-222. doi: 10.1088/1674-1137/35/3/002
Abstract:
In the supersymmetric left-right model, the light neutrino masses are given by the Type-Ⅱ seesaw mechanism. A duality property of this mechanism indicates that there exist eight possible Higgs triplet Yukawa couplings which result in the same neutrino mass matrix. In this paper, we work out the one-loop renormalization group equations for the effective neutrino mass matrix in the supersymmetric left-right model. The stability of the Type-Ⅱ seesaw scenario is briefly discussed. We also study the lepton-flavor-violating processes (τ→ μγ and τ→eγ) by using the reconstructed Higgs triplet Yukawa couplings.
In the supersymmetric left-right model, the light neutrino masses are given by the Type-Ⅱ seesaw mechanism. A duality property of this mechanism indicates that there exist eight possible Higgs triplet Yukawa couplings which result in the same neutrino mass matrix. In this paper, we work out the one-loop renormalization group equations for the effective neutrino mass matrix in the supersymmetric left-right model. The stability of the Type-Ⅱ seesaw scenario is briefly discussed. We also study the lepton-flavor-violating processes (τ→ μγ and τ→eγ) by using the reconstructed Higgs triplet Yukawa couplings.
2011, 35(3): 223-227. doi: 10.1088/1674-1137/35/3/003
Abstract:
Quantum field theory is a fundamental tool in particle and nuclear physics. Elemental particles are assumed to be point particles and, as a result, the loop integrals are divergent in many cases. Regularization and renormalization are introduced in order to get the physical finite results from the infinite, divergent loop integrations. We propose new quantization conditions for the fields whose base is very natural, i.e., any particle is not a point particle but a solid one with three dimensions. With this solid quantization, divergence could disappear.
Quantum field theory is a fundamental tool in particle and nuclear physics. Elemental particles are assumed to be point particles and, as a result, the loop integrals are divergent in many cases. Regularization and renormalization are introduced in order to get the physical finite results from the infinite, divergent loop integrations. We propose new quantization conditions for the fields whose base is very natural, i.e., any particle is not a point particle but a solid one with three dimensions. With this solid quantization, divergence could disappear.
2011, 35(3): 228-232. doi: 10.1088/1674-1137/35/3/004
Abstract:
I show the formulation of de Sitter Special Relativity (dS-SR) based on Dirac-Lu-Zou-Guo's discussions. dS-SR quantum mechanics is formulated, and the dS-SR Dirac equation for hydrogen is suggested. The equation in the earth-QSO framework reference is solved by means of the adiabatic approach. It's found that the fine-structure "constant" α in dS-SR varies with time. By means of the t-z relation of the ΛCDM model, α's time-dependency becomes redshift z-dependent. The dS-SR's predictions of Δα/α agree with data of spectra of 143 quasar absorption systems, the dS-space-time symmetry is SO(3,2) (i.e., anti-dS group) and the universal parameter R (de Sitter ratio) in dS-SR is estimated to be R≈2.73×1012 ly. The effects of dS-SR become visible at the cosmic space-time scale (i.e., the distance ≥ 109 ly). At that scale, dS-SR is more reliable than Einstein SR. The α-variation with time is evidence of SR with de Sitter symmetry.
I show the formulation of de Sitter Special Relativity (dS-SR) based on Dirac-Lu-Zou-Guo's discussions. dS-SR quantum mechanics is formulated, and the dS-SR Dirac equation for hydrogen is suggested. The equation in the earth-QSO framework reference is solved by means of the adiabatic approach. It's found that the fine-structure "constant" α in dS-SR varies with time. By means of the t-z relation of the ΛCDM model, α's time-dependency becomes redshift z-dependent. The dS-SR's predictions of Δα/α agree with data of spectra of 143 quasar absorption systems, the dS-space-time symmetry is SO(3,2) (i.e., anti-dS group) and the universal parameter R (de Sitter ratio) in dS-SR is estimated to be R≈2.73×1012 ly. The effects of dS-SR become visible at the cosmic space-time scale (i.e., the distance ≥ 109 ly). At that scale, dS-SR is more reliable than Einstein SR. The α-variation with time is evidence of SR with de Sitter symmetry.
2011, 35(3): 233-242. doi: 10.1088/1674-1137/35/3/005
Abstract:
The time evolution of vacuum energy density is investigated in the coherent states of inflationary universe using a linear invariant approach. The linear invariants we derived are represented in terms of annihilation operators. On account of the fact that the coherent state is an eigenstate of an annihilation operator, the wave function in the coherent state is easily evaluated by solving the eigenvalue equation of the linear invariants. The expectation value of the vacuum energy density is derived using this wave function. Fluctuations of the scalar field and its conjugate momentum are also investigated. Our theory based on the linear invariant shows that the vacuum energy density of the universe in a coherent state is decreased continuously with time due to nonconservative force acting on the coherent oscillations of the scalar field, which is provided by the expansion of the universe. In effect, our analysis reveals that the vacuum energy density decreases in proportion to tβ where β is 3/2 for radiation-dominated era and 2 for matter-dominated era. In the case where the duration term of radiation-dominated era is short enough to be negligible, the estimation of the relic vacuum energy density agrees well with the current observational data.
The time evolution of vacuum energy density is investigated in the coherent states of inflationary universe using a linear invariant approach. The linear invariants we derived are represented in terms of annihilation operators. On account of the fact that the coherent state is an eigenstate of an annihilation operator, the wave function in the coherent state is easily evaluated by solving the eigenvalue equation of the linear invariants. The expectation value of the vacuum energy density is derived using this wave function. Fluctuations of the scalar field and its conjugate momentum are also investigated. Our theory based on the linear invariant shows that the vacuum energy density of the universe in a coherent state is decreased continuously with time due to nonconservative force acting on the coherent oscillations of the scalar field, which is provided by the expansion of the universe. In effect, our analysis reveals that the vacuum energy density decreases in proportion to tβ where β is 3/2 for radiation-dominated era and 2 for matter-dominated era. In the case where the duration term of radiation-dominated era is short enough to be negligible, the estimation of the relic vacuum energy density agrees well with the current observational data.
2011, 35(3): 243-247. doi: 10.1088/1674-1137/35/3/006
Abstract:
Gaussian modifications of the neutrino energy loss (NEL) by electron capture on the strongly screening nuclides 55Co and 56Ni are investigated. The results show that in strong electron screening (SES), the NEL rates decrease without modifying the Gamow-Teller (G-T) resonance transition. For instance, the NEL rates of 55Co and 56Ni decrease more than two and three orders of magnitude for ρ7=5.86, T9≤5, Ye=0.47, Δ=6.3, respectively. In contrast, due to Gaussian modification, the NEL rates increase about two orders of magnitude in SES. Due to SES, the maximum values of the C-factor (in %) on NEL of 55Co, 56Ni are of the order of 99.80%, 99.56% at ρ7=5.86 Ye=0.47 and 99.60%, 99.65% at ρ7=106 Ye=0.43, respectively.
Gaussian modifications of the neutrino energy loss (NEL) by electron capture on the strongly screening nuclides 55Co and 56Ni are investigated. The results show that in strong electron screening (SES), the NEL rates decrease without modifying the Gamow-Teller (G-T) resonance transition. For instance, the NEL rates of 55Co and 56Ni decrease more than two and three orders of magnitude for ρ7=5.86, T9≤5, Ye=0.47, Δ=6.3, respectively. In contrast, due to Gaussian modification, the NEL rates increase about two orders of magnitude in SES. Due to SES, the maximum values of the C-factor (in %) on NEL of 55Co, 56Ni are of the order of 99.80%, 99.56% at ρ7=5.86 Ye=0.47 and 99.60%, 99.65% at ρ7=106 Ye=0.43, respectively.
2011, 35(3): 248-254. doi: 10.1088/1674-1137/35/3/007
Abstract:
This presented study is to make comparison of cross sections to produce 117Sb and 90Nb via different reactions with particle incident energy up to 70 MeV as a part of systematic studies on particle-induced activations on enriched Sn, Y2O3 and ZrO2 targets, theoretical calculation of production yield, calculation of required thickness of target and suggestion of optimum reaction to produce Antimony-117 and Niobium-90.
This presented study is to make comparison of cross sections to produce 117Sb and 90Nb via different reactions with particle incident energy up to 70 MeV as a part of systematic studies on particle-induced activations on enriched Sn, Y2O3 and ZrO2 targets, theoretical calculation of production yield, calculation of required thickness of target and suggestion of optimum reaction to produce Antimony-117 and Niobium-90.
2011, 35(3): 255-258. doi: 10.1088/1674-1137/35/3/008
Abstract:
By means of the AKK08 fragmentation function, the net-proton transverse momentum (pT) spectra in A+A collisions are studied with two phenomenological models based on the Color Glass Condensate formalism. After a χ2 analysis of the experimental data from BRAHMS, the normalization constant C is extracted at RHIC energies of √sNN=62.4 and 200 GeV, and the theoretical results of the net-proton pT spectra at selected rapidities are also given. It is shown that the theoretical results are in good agreement with the experimental data. Finally, assuming the constant C should have an exponent dependence of √sNN, we also predict the theoretical results of net-proton pT spectra at LHC energies of √sNN=2.76, 3.94, and 5.52 TeV.
By means of the AKK08 fragmentation function, the net-proton transverse momentum (pT) spectra in A+A collisions are studied with two phenomenological models based on the Color Glass Condensate formalism. After a χ2 analysis of the experimental data from BRAHMS, the normalization constant C is extracted at RHIC energies of √sNN=62.4 and 200 GeV, and the theoretical results of the net-proton pT spectra at selected rapidities are also given. It is shown that the theoretical results are in good agreement with the experimental data. Finally, assuming the constant C should have an exponent dependence of √sNN, we also predict the theoretical results of net-proton pT spectra at LHC energies of √sNN=2.76, 3.94, and 5.52 TeV.
2011, 35(3): 259-263. doi: 10.1088/1674-1137/35/3/009
Abstract:
The centrality and energy dependence of rapidity correlation patterns are studied in Au+Au collisions by using AMPT with string melting, RQMD and UrQMD models. The behaviors of the short-range correlation (SRC) and the long-range correlation (LRC) are presented clearly by two spatial-position dependent correlation patterns. For centrality dependence, UrQMD and RQMD give similar results as those in AMPT, i.e., in most central collisions, the correlation structure is flatter and the correlation range is larger, which indicates a long range rapidity correlation. A long range rapidity correlation showing up in RQMD and UrQMD implies that parton interaction is not the only source of long range rapidity correlations. For energy dependence, AMPT with string melting and RQMD show quite different results. The correlation patterns in RQMD at low collision energies and those in AMPT at high collision energies have similar structures, i.e. a convex curve, while the correlation patterns in RQMD at high collision energies and those in AMPT at low collision energies show flat structures, having no position dependence. Long range rapidity correlation presents itself at high energy and disappears at low energy in RQMD, which also indicates that long range rapidity correlations may come from some trivial effects, rather than the parton interactions.
The centrality and energy dependence of rapidity correlation patterns are studied in Au+Au collisions by using AMPT with string melting, RQMD and UrQMD models. The behaviors of the short-range correlation (SRC) and the long-range correlation (LRC) are presented clearly by two spatial-position dependent correlation patterns. For centrality dependence, UrQMD and RQMD give similar results as those in AMPT, i.e., in most central collisions, the correlation structure is flatter and the correlation range is larger, which indicates a long range rapidity correlation. A long range rapidity correlation showing up in RQMD and UrQMD implies that parton interaction is not the only source of long range rapidity correlations. For energy dependence, AMPT with string melting and RQMD show quite different results. The correlation patterns in RQMD at low collision energies and those in AMPT at high collision energies have similar structures, i.e. a convex curve, while the correlation patterns in RQMD at high collision energies and those in AMPT at low collision energies show flat structures, having no position dependence. Long range rapidity correlation presents itself at high energy and disappears at low energy in RQMD, which also indicates that long range rapidity correlations may come from some trivial effects, rather than the parton interactions.
2011, 35(3): 264-268. doi: 10.1088/1674-1137/35/3/010
Abstract:
System size is more than a geometrical quantity in relativistic heavy ion collisions; it is closely related to evolution process, i.e. a different system size corresponds to a different evolution process, and whether QGP is produced depends on the system size. We propose that the system size should be under the same level when comparing the measurements from different colliding nuclei. The equivalence of the peripheral collisions of Au-Au and the central collisions of smaller nuclei is studied using the Monte Carlo method. Comparing the transverse overlapping area of the colliding nuclei, the number of participant nucleons and the number of nucleon-nucleon binary collisions in various colliding nuclei, we give an estimate of the correspondence in system size. This is helpful in the experimental comparison of the measurements from different colliding nuclei.
System size is more than a geometrical quantity in relativistic heavy ion collisions; it is closely related to evolution process, i.e. a different system size corresponds to a different evolution process, and whether QGP is produced depends on the system size. We propose that the system size should be under the same level when comparing the measurements from different colliding nuclei. The equivalence of the peripheral collisions of Au-Au and the central collisions of smaller nuclei is studied using the Monte Carlo method. Comparing the transverse overlapping area of the colliding nuclei, the number of participant nucleons and the number of nucleon-nucleon binary collisions in various colliding nuclei, we give an estimate of the correspondence in system size. This is helpful in the experimental comparison of the measurements from different colliding nuclei.
Application of the parametric formulae for electromagnetic showers in unconverted γπ0 discrimination
2011, 35(3): 269-277. doi: 10.1088/1674-1137/35/3/011
Abstract:
In the LHC experiment, the neutral pions produced during jet fragmentation are the background sources for all physics channels with high-energy photons in their final state. In this paper, the application of the three-dimensional parametric formula for electromagnetic (EM) showers, which we developed in the Alpha Magnetic Spectrometer II experiment, is presented to distinguish the unconverted photons from the neutral pions. With the constructed electromagnetic calorimeter (ECAL) in a GEANT4 simulation, the parametric formulae were validated and the unconverted γπ0 discrimination was performed with the Toolkit for Multivariate Data Analysis (TMVA) package in ROOT for different transverse energies ranging from 15 GeV to 75 GeV, which is the most sensitive region for light Higgs (with mass ～120 GeV) searches with the channel H→γγ. With this discrimination method and the selected transverse energy region, we can reject π0 with the efficiency from ～40% (65--75 GeV) to ～90% (15--25 GeV) when keeping 90% γ efficiency.
In the LHC experiment, the neutral pions produced during jet fragmentation are the background sources for all physics channels with high-energy photons in their final state. In this paper, the application of the three-dimensional parametric formula for electromagnetic (EM) showers, which we developed in the Alpha Magnetic Spectrometer II experiment, is presented to distinguish the unconverted photons from the neutral pions. With the constructed electromagnetic calorimeter (ECAL) in a GEANT4 simulation, the parametric formulae were validated and the unconverted γπ0 discrimination was performed with the Toolkit for Multivariate Data Analysis (TMVA) package in ROOT for different transverse energies ranging from 15 GeV to 75 GeV, which is the most sensitive region for light Higgs (with mass ～120 GeV) searches with the channel H→γγ. With this discrimination method and the selected transverse energy region, we can reject π0 with the efficiency from ～40% (65--75 GeV) to ～90% (15--25 GeV) when keeping 90% γ efficiency.
2011, 35(3): 278-281. doi: 10.1088/1674-1137/35/3/012
Abstract:
A few traditional pulse-forming circuits are implemented in a commercial 0.13 μm digital complementary-metal-oxide-semiconductor (CMOS) technology. These circuits, based on a coplanar waveguide, are analyzed and compared through CadenceTM Spectre simulations. The results show that these traditional pulse-forming-line (PFL) based circuits can be implemented in standard CMOS technology for short pulse generations. Further work is needed to explore the potential of the circuit techniques and to minimize parasitic effects.
A few traditional pulse-forming circuits are implemented in a commercial 0.13 μm digital complementary-metal-oxide-semiconductor (CMOS) technology. These circuits, based on a coplanar waveguide, are analyzed and compared through CadenceTM Spectre simulations. The results show that these traditional pulse-forming-line (PFL) based circuits can be implemented in standard CMOS technology for short pulse generations. Further work is needed to explore the potential of the circuit techniques and to minimize parasitic effects.
2011, 35(3): 282-287. doi: 10.1088/1674-1137/35/3/013
Abstract:
The CDEX (China Dark matter EXperiment) Collaboration will carry out a direct search for WIMPs (Weakly Interacting Massive Particles) using an Ultra-Low Energy Threshold High Purity Germanium (ULE-HPGe) detector at the CJPL (China JinPing deep underground Laboratory). A complex shielding system was designed to reduce backgrounds and a detailed GEANT4 Monte Carlo simulation was performed to study the achievable reduction of γ rays induced by radionuclides and neutron backgrounds by D(γ,n)p reaction. Furthermore, the upper level of allowed radiopurity of shielding materials was estimated under the constraint of the expected goal. Compared with the radiopurity reported by other low-background rare-event experiments, it indicates that the shielding used in the CDEX can be made out of materials with obtainable radiopurity.
The CDEX (China Dark matter EXperiment) Collaboration will carry out a direct search for WIMPs (Weakly Interacting Massive Particles) using an Ultra-Low Energy Threshold High Purity Germanium (ULE-HPGe) detector at the CJPL (China JinPing deep underground Laboratory). A complex shielding system was designed to reduce backgrounds and a detailed GEANT4 Monte Carlo simulation was performed to study the achievable reduction of γ rays induced by radionuclides and neutron backgrounds by D(γ,n)p reaction. Furthermore, the upper level of allowed radiopurity of shielding materials was estimated under the constraint of the expected goal. Compared with the radiopurity reported by other low-background rare-event experiments, it indicates that the shielding used in the CDEX can be made out of materials with obtainable radiopurity.
2011, 35(3): 288-292. doi: 10.1088/1674-1137/35/3/014
Abstract:
In this paper, 2-D Particle-In-Cell simulations are made for Laser Wakefield Accelerations (LWFA). As in a real experiment, we perform plasma density scanning for typical 100 TW laser facilities. Several basic laws for self-injected acceleration in a bubble regime are presented. According to these laws, we choose a proper plasma density and then obtain a high quality quasi-monoenergetic electron bunch with a rms energy of more than 650 MeV and a bunch length of less than 1.5 \upmu m.
In this paper, 2-D Particle-In-Cell simulations are made for Laser Wakefield Accelerations (LWFA). As in a real experiment, we perform plasma density scanning for typical 100 TW laser facilities. Several basic laws for self-injected acceleration in a bubble regime are presented. According to these laws, we choose a proper plasma density and then obtain a high quality quasi-monoenergetic electron bunch with a rms energy of more than 650 MeV and a bunch length of less than 1.5 \upmu m.
2011, 35(3): 293-295. doi: 10.1088/1674-1137/35/3/015
Abstract:
In order to calculate the effect on the beam caused by an irregular accelerator element, we have expanded and improved the Linear and Electrostatic Accelerator Dynamics Simulation (LEADS) code. To achieve better calculation precision, the element was divided into lots of equal intervals. In order to simplify the calculation process, a one-dimensional field is simulated and the Lorenz equation is used directly. A one-dimensional field can be imported into the LEADS code. The heteromorphic quadrupole is invented and its field is simulated and optimized using the POISSON code. As examples, the effect on the beam caused by the heteromorphic quadrupole and octupole is simulated.
In order to calculate the effect on the beam caused by an irregular accelerator element, we have expanded and improved the Linear and Electrostatic Accelerator Dynamics Simulation (LEADS) code. To achieve better calculation precision, the element was divided into lots of equal intervals. In order to simplify the calculation process, a one-dimensional field is simulated and the Lorenz equation is used directly. A one-dimensional field can be imported into the LEADS code. The heteromorphic quadrupole is invented and its field is simulated and optimized using the POISSON code. As examples, the effect on the beam caused by the heteromorphic quadrupole and octupole is simulated.
2011, 35(3): 296-300. doi: 10.1088/1674-1137/35/3/016
Abstract:
In the ILC (International Linear Collider) main linac, low emittance preservation is the most important issue for beam dynamics study. As the main sources of emittance dilution, the dispersive and wakefield effects were studied in this paper. The theoretical calculations and numerical simulations of these two effects on single-bunch emittance dilution, without any misalignment errors, are presented in detail.
In the ILC (International Linear Collider) main linac, low emittance preservation is the most important issue for beam dynamics study. As the main sources of emittance dilution, the dispersive and wakefield effects were studied in this paper. The theoretical calculations and numerical simulations of these two effects on single-bunch emittance dilution, without any misalignment errors, are presented in detail.
2011, 35(3): 301-307. doi: 10.1088/1674-1137/35/3/017
Abstract:
A resonant buildup of beam-induced fields in a superconducting radio frequency (RF) cavity may make a beam unstable or a superconducting RF cavity quench. Higher-order mode (HOM) couplers are used for damping higher-order modes to avoid such a resonant buildup. A coaxial HOM coupler based on the TTF (TESLA Test Facility) HOM coupler has been designed for the superconducting RF cavities at the Proton Engineering Frontier Project (PEFP) in order to overcome notch frequency shift and feed-through tip melting issues. In order to confirm the HOM coupler design and finalize its structural dimensions, two prototype HOM couplers have been fabricated and tested. Low-power testing and measurement of the HOM couplers has shown that the HOM coupler has good filter properties and can fully meet the damping requirements of the PEFP low-beta superconducting RF linac.
A resonant buildup of beam-induced fields in a superconducting radio frequency (RF) cavity may make a beam unstable or a superconducting RF cavity quench. Higher-order mode (HOM) couplers are used for damping higher-order modes to avoid such a resonant buildup. A coaxial HOM coupler based on the TTF (TESLA Test Facility) HOM coupler has been designed for the superconducting RF cavities at the Proton Engineering Frontier Project (PEFP) in order to overcome notch frequency shift and feed-through tip melting issues. In order to confirm the HOM coupler design and finalize its structural dimensions, two prototype HOM couplers have been fabricated and tested. Low-power testing and measurement of the HOM couplers has shown that the HOM coupler has good filter properties and can fully meet the damping requirements of the PEFP low-beta superconducting RF linac.
2011, 35(3): 308-312. doi: 10.1088/1674-1137/35/3/018
Abstract:
Laser-beam interaction in an undulator is commonly suggested in the development of free electron laser (FEL) schemes. In this paper, a three-dimensional algorithm is developed to assist in laser-beam interaction simulation in an undulator, which is built on the basis of the fundamentals of electrodynamics, i.e. the electron's behavior is determined by the magnetic field and the laser electric field in the time domain. On the basis of the algorithm, the detuning effect in a laser heater, the carrier envelope phase effect of a few-cycle laser in attosecond X-ray FEL schemes and output wavelength tuning in a high gain harmonic generation FEL are numerically discussed.
Laser-beam interaction in an undulator is commonly suggested in the development of free electron laser (FEL) schemes. In this paper, a three-dimensional algorithm is developed to assist in laser-beam interaction simulation in an undulator, which is built on the basis of the fundamentals of electrodynamics, i.e. the electron's behavior is determined by the magnetic field and the laser electric field in the time domain. On the basis of the algorithm, the detuning effect in a laser heater, the carrier envelope phase effect of a few-cycle laser in attosecond X-ray FEL schemes and output wavelength tuning in a high gain harmonic generation FEL are numerically discussed.
2011, 35(3): 313-317. doi: 10.1088/1674-1137/35/3/019
Abstract:
The multi-objective optimization of inverse planning based on the Pareto solution set, according to the multi-objective character of inverse planning in accurate radiotherapy, was studied in this paper. Firstly, the clinical requirements of a treatment plan were transformed into a multi-objective optimization problem with multiple constraints. Then, the fast and elitist multi-objective Non-dominated Sorting Genetic Algorithm (NSGA-Ⅱ) was introduced to optimize the problem. A clinical example was tested using this method. The results show that an obtained set of non-dominated solutions were uniformly distributed and the corresponding dose distribution of each solution not only approached the expected dose distribution, but also met the dose-volume constraints. It was indicated that the clinical requirements were better satisfied using the method and the planner could select the optimal treatment plan from the non-dominated solution set.
The multi-objective optimization of inverse planning based on the Pareto solution set, according to the multi-objective character of inverse planning in accurate radiotherapy, was studied in this paper. Firstly, the clinical requirements of a treatment plan were transformed into a multi-objective optimization problem with multiple constraints. Then, the fast and elitist multi-objective Non-dominated Sorting Genetic Algorithm (NSGA-Ⅱ) was introduced to optimize the problem. A clinical example was tested using this method. The results show that an obtained set of non-dominated solutions were uniformly distributed and the corresponding dose distribution of each solution not only approached the expected dose distribution, but also met the dose-volume constraints. It was indicated that the clinical requirements were better satisfied using the method and the planner could select the optimal treatment plan from the non-dominated solution set.
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