光滑粒子流体动力学模型发射π介子源的视像分析
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摘要
强度干涉学是研究高能重离子碰撞系统时空结构的一种重要工具,最近一种较新的强度干涉学方法――视像法(IMAGINGTechnology)已被越来越多的研究者所采用。这种方法不同于以往的干涉学方法,它不依赖于发射源的模型假设,并可以直接由实验数据给出发射源的非常直观的几何信息。
为了用视像法来研究一个更真实的发射源,本文使用了NeXSPheRIO模型,它是一种高能重离子碰撞的光滑粒子流体动力学模型,这个模型非常适合高能核核碰撞中较大初始能量涨落、大形变的系统演化的模拟。
本文先介绍了光滑粒子流体动力学的基本原理,给出了相对论重离子碰撞的光滑粒子流体动力学的运动方程,并对NeXSPheRIO模型做了简要介绍。之后介绍了视像法的基本原理,以及在实际应用时该方法的处理。本文用NeXSPheRIO模型对√sNN =200GeVAu+Au碰撞系统的演化进行了模拟。研究了不同碰撞参数,不同碰撞能量,不同冻结温度等情况下粒子发射源的时空演化,并对由该模型得到的2π关联数据进行了一维和三维视像法的分析。研究发现相同碰撞参数下系统半径并不一定相同,并且系统的半径随碰撞参数的增加而减小;碰撞能量对于碰撞系统的半径影响并不明显;冻结温度对源的半径影响很大,较高的冻结温度的碰撞系统其半径较小等等。
Intensity Interferometry is an important tool for studying time-space structure ofhighly energy heavy-ion collisions. Recently, IMAGING Technology, a new method ofIntensity Interferometry, is adopted by more and more researchers. Different from usualIntensity Interferometry methods, IMAGING Technology does not depend on the modelof surce, then we can be directly calculated with the experimental data by this approach,and the consequence of which could give a intuitionistic geometric information of source.
Here, for researching a real source, we use NeXSPheRIO model, which is a modelof Smoothed Particle Hydrodynamics for highly energy heavy-ion collisions. And thismodel is adapt for simulation of high-energy nuclear collisions system which exist largedeformation and large ?uctuation of initial energy density distribution.
In this paper, we introduced the basic principles of Smooth Particle Hydrodynam-ics.Gives motion equation of the smooth particle hydrodynamics for relativistic heavy-ion collision, and made a brief introduction to NeXSPheRIO model (A smooth particlehydrodynamics model for relativistic heavy ion collisions ). Then we introduced the ba-sic principles of IMAGING Technology, and how to deal with actual problems by thismethod. Finally, we simulated the evolution of√sNN = 200 GeV Au+Au collisionssystem by NeXSPheRIO model, and researched time-space evolution of particle sourcewith different collision parameters, collision energy and freeze out temperature, then wedid imaging, including 1D and 3D, to the 2πcorrelation data. Researching found thatthe system radius is different with the same collision parameter, and the system radiusincrease with the collision parameter decreases; The in?uence of collision energy is notobvious to the collision system radius; Changing of freezing out temperature is a largerin?uence on the system radius, higher the freezing temperature of the system and smallerits radius.
为了用视像法来研究一个更真实的发射源,本文使用了NeXSPheRIO模型,它是一种高能重离子碰撞的光滑粒子流体动力学模型,这个模型非常适合高能核核碰撞中较大初始能量涨落、大形变的系统演化的模拟。
本文先介绍了光滑粒子流体动力学的基本原理,给出了相对论重离子碰撞的光滑粒子流体动力学的运动方程,并对NeXSPheRIO模型做了简要介绍。之后介绍了视像法的基本原理,以及在实际应用时该方法的处理。本文用NeXSPheRIO模型对√sNN =200GeVAu+Au碰撞系统的演化进行了模拟。研究了不同碰撞参数,不同碰撞能量,不同冻结温度等情况下粒子发射源的时空演化,并对由该模型得到的2π关联数据进行了一维和三维视像法的分析。研究发现相同碰撞参数下系统半径并不一定相同,并且系统的半径随碰撞参数的增加而减小;碰撞能量对于碰撞系统的半径影响并不明显;冻结温度对源的半径影响很大,较高的冻结温度的碰撞系统其半径较小等等。
Intensity Interferometry is an important tool for studying time-space structure ofhighly energy heavy-ion collisions. Recently, IMAGING Technology, a new method ofIntensity Interferometry, is adopted by more and more researchers. Different from usualIntensity Interferometry methods, IMAGING Technology does not depend on the modelof surce, then we can be directly calculated with the experimental data by this approach,and the consequence of which could give a intuitionistic geometric information of source.
Here, for researching a real source, we use NeXSPheRIO model, which is a modelof Smoothed Particle Hydrodynamics for highly energy heavy-ion collisions. And thismodel is adapt for simulation of high-energy nuclear collisions system which exist largedeformation and large ?uctuation of initial energy density distribution.
In this paper, we introduced the basic principles of Smooth Particle Hydrodynam-ics.Gives motion equation of the smooth particle hydrodynamics for relativistic heavy-ion collision, and made a brief introduction to NeXSPheRIO model (A smooth particlehydrodynamics model for relativistic heavy ion collisions ). Then we introduced the ba-sic principles of IMAGING Technology, and how to deal with actual problems by thismethod. Finally, we simulated the evolution of√sNN = 200 GeV Au+Au collisionssystem by NeXSPheRIO model, and researched time-space evolution of particle sourcewith different collision parameters, collision energy and freeze out temperature, then wedid imaging, including 1D and 3D, to the 2πcorrelation data. Researching found thatthe system radius is different with the same collision parameter, and the system radiusincrease with the collision parameter decreases; The in?uence of collision energy is notobvious to the collision system radius; Changing of freezing out temperature is a largerin?uence on the system radius, higher the freezing temperature of the system and smallerits radius.
引文
1 D. J. Blaizot. High Energy Nuclear Collisions[C]//EPS-HEP’99, International Euro-physics Conference on High Energy Physics. Tampere Finland, 1999:22–25.
2 L. Landau. On the Multiple Production of Particles in High Energy Collisions[J].Izv. Akad. Nauk SSSR Ser. Fiz., 1953, 17:51–64.
3 L. Landau, S. Belenkij. The Hydrodynamic Theory of Multiple Formation of Parti-cles[J]. Usp. Fiz. Nauk, 1955, 56:309.
4 S. Belenki, L. Landau. Hydrodynamic Theory of Multiple Production of Particles[J].Nuovo Cimento Suppl, 1956, 3:15–31.
5 D. T. Haar. Collected Papers of L. D. Landau[M]. New York: Gordon and Breach,1965.
6 R. Clare, D. Strottman. Relativistic Hydrodynamics and Heavy Ion Reactions[J].Phys. Rep., 1986, 141(4):177–280.
7 L. Csernai. Introduction to Relativistic Heavy Ion Collisions[M]. New York: JohnWiley & Sons, 1994.
8 H. Sto¨cker, W. Greiner. High Energy Heavy Ion Collisions―probing the Equationof State of Highly Excited Hardronic Matter[J]. Phys. Rep., 1986, 137:277–392.
9 C. Adler, Z. Ahammed, C. Allgower, et al. Pion Interferometry of√sNN = 130 GeVAu+Au Collisions at RHIC[J]. Phys. Rev. Lett., 2001, 87:082301.
10 K. Adcox, S. S. Adler, N. N. Ajitanand, et al. Transverse-mass Dependence of Two-pion Correlations in Au+Au Collisions at√sNN = 130 GeV[J]. Phys. Rev. Lett.,2002, 88:192302.
11 S. S. Adler, S. Afanasiev, C. Aidala, et al. Bose-einstein Correlations of ChargedPion Pairs in Au+Au Collisions at√sNN = 200 GeV[J]. Phys. Rev. Lett., 2004,93:152302.
12 J. Adams, M. M. Aggarwal, Z. Ahammed, et al. Pion Interferometry in Au+AuCollisions at√sNN = 200 GeV[J]. Phys. Rev., 2005, C71:044906.
13 U. A. Wiedemann, U. Heinz. Particle Interferometry for Relativistic Heavy-ion Col-lisions[J]. Phys. Rept., 1999, 319:145–230.
14 S. Solf, S. A. Bass, A. Dumitru. Pion Interferometry at Rhic: Probing a ThermalizedQuark-Gluon Plasma? [J]. Phys. Rev. Lett., 2001, 86:3981–3984.
15 R. M. Weiner. Boson Interferometry in High Energy Physics[J]. Phys. Rept., 2000,327:249–346.
16 S. Pratt. Correlations and Fluctuations, a Summary of Quark Matter 2002[J]. Nucl.Phys., 2003, A715:389c–398c.
17 W. N. Zhang, M. J. Efaaf, C. Y. Wong. Pion Interferometry for a Granular Source ofQuark-Gluon Plasma Droplets[J]. Phys. Rev., 2004, C70:024903.
18 E. Witten. Cosmic Separation of Phases[J]. Phys. Rev., 1984, D30:272–285.
19 L. P. Csernai, J. I. Kapusta. Nucleation of Relativistic First-order Phase Transi-tions[J]. Phys. Rev., 1992, D46:1379–1390.
20 S. Alamoudi, D. G. Barci, D. Boyanovsky, et al. Dynamical Viscosity of NucleatingBubbles[J]. Phys. Rev., 1999, D60:125003.
21 J. Randrup. Spinodal Decomposition During the Hadronization Stage at RHIC ? [J].Phys. Rev. Lett., 2004, 92:122301.
22 W. N. Zhang, Y. Y. Ren, C. Y. Wong. Analysis of Pion Elliptic Flow and Hanbury-Brown-Twiss Interferometry in a Granular Quark-gluon Plasma Droplet Model[J].Phys. Rev., 2006, C74:024908.
23 J. D. Bjorken. Highly Relativistic Nucleus-nucleus Collisions: The Central RapidityRegion[J]. Phys. Rev., 1983, D27:140–151.
24 D. A. Brown, P. Danielewicz. Imaging of Sources in Heavy-ion Reactions[J]. Phys.Lett., 1997, B398:252–258.
25 D. A. Brown, A. Enokizono, M. Heffner, et al. Imaging Three DimensionalTwo-particle Correlations for Heavy-ion Reaction Studies[J]. Phys. Rev., 2005,C72:054902.
26 G. R. Liu, M. B. Liu.光滑粒子流体动力学――一种无网格粒子法[M].湖南大学出版社,2005.
27 J. Monaghan. Smoothed Particle Hydrodynamics[J]. Annu. Rev. Astron. Astrophys.,1992, 30:543–574.
28 H. T. Elze, Y. Hama, T. Kodama, et al. Variational Principle for Relativistic FluidDynamics[J]. J. Phys., 1999, G25:1935–1957.
29 C. Aguiar, T. Kodama, T. Osada, et al. Smoothed Particle Hydrodynamics for Rela-tivistic Heavy Ion Collisions[J]. J. Phys., 2001, G27:75–94.
30 F. Grassi, Y. Hama, T. Kodama, et al. Results on Transverse Mass Spectra Obtainedwith NeXSPheRIO[J]. J. Phys. G: Nucl. Part. Phys, 2005, 31:S1041–S1044.
31 R. Andrade, F. Grassi, Y. Hama, et al. On the Necessity to Include Event-by-EventFluctuations in Experimental Evaluation of Elliptical Flow[J]. Phys. Rev. Lett., 2006,97:202302.
32 C. E. Aguiar, Y. Hama, T. Kodama, et al. Event-by-Event Fluctuations in Hydrody-namical Description of Heavy-ion Collisions[J]. Nucl. Phys., 2002, A698:639–642.
33 R. Andrade, F. Grassi, Y. Hama, et al. NeXSPheRIO Results on Elliptic Flow atRHIC and Connection with Thermalization[J]. Eur. Phys. J., 2006, A29:23–26.
34 J. O. Socolowski, F. Grassi, Y. Hama, et al. Fluctuations of the Initial Conditions andthe Continuous Emission in the Hydrodynamical Description of Two-Pion Interfer-ometry[J]. Phys. Rev. Lett., 2004, 93:182301.
35 H. J. Drescher, F. M. Liu, S. Ostapchenko, et al. Initial Condition for Quark-gluonPlasma Evolution[J]. Phys. Rev., 2002, C65:054902.
36 K. Werner, H. Drescher, S.Ostapchenko, et al. The Nexus Model[C]//Invited talkgiven at the International Workshop on the Physics of the Quark Gluon Plasma.Palaiseau, France: [arXiv.org:hep-ph/0209198], 2001.
37 Y. Hama, T. Kodama, O. S. Jr. Topics on Hydrodynamic Model of Nucleus-nucleusCollisions[J]. Braz. J. Phys., 2005, 35:24–51.
38 D. H. Boal, C.-K. Gelbke, B. K. Jennings. Intensity Interferometry in SubatomicPhysics[J]. Rev. Mod. Phys., 1990, 62:553–602.
39 H. Beker, H. B?ggild, J. Boissevain, et al. Kaon Interferometry in Heavy-ion Colli-sions at the CERN Sps[J]. Z. Phys. C, 1994, 64:209–217.
40 H. B?ggild, J. Boissevain, M. Cherney, et al. Identified Pion Interferometry in Heavy-ion Collisions at CERN[J]. Phys. Lett., 1993, B302:510–516.
41 H. B?ggild, J. Boissevain, M. Cherney, et al. Directional Dependence of the PionSource in High-energy Heavy-ion Collisions[J]. Phys. Lett., 1995, B349:386–392.
42 D. A. Brown, F. Wang, P. Danielewicz. Implications of the Unusual Structure in thepp Correlation from Pb+Pb Collisions at 158 AGeV[J]. Phys. Lett., 1999, B470:33–38.
43 D. A. Brown, P. Danielewicz. Optimized Discretization of Sources Imaged in Heavy-ion Reactions[J]. Phys. Rev., 1998, C57:2474–2483.
44 S. E. Koonin. Proton Pictures of High-energy Nuclear Collisions[J]. Phys. Lett.,1997, B70:43–47.
45 S. Pratt, T. Csorgo, J. Zimanyi. Detailed Predictions for Two-pion Correlations inUltrarelativistic Heavy-ion Collisions[J]. Phys. Rev., 1990, C42:2646–2652.
46 G. F. Bertsch. Meson Phase Space Density in Heavy Ion Collisions from Interfer-ometry[J]. Phys. Rev. Lett., 1994, 72:2349–2350.
47 D. A. Brown, S. Y. Panitkin, G. F. Bertsch. Extracting Particle Freeze-out Phase-space Densities and Entropies from Sources Imaged in Heavy-ion Reactions[J].Phys.Rev., 2000, C62:014904.
48 D. A. Brown, P. Danielewicz. Observing Non-Gaussian Sources in Heavy-ion Reac-tions[J]. Phys. Rev., 2001, C64:014902.
49 P. Danielewicz, S. Pratt. Analysis of Low-momentum Correlations with CartesianHarmonics[J]. Phys. Lett., 2005, B618:60–67.
50 P. Danielewicz, S. Pratt. Analyzing Correlation Functions with Tesseral and Carte-sian Spherical Harmonics[J]. Phys. Rev., 2007, C75:034907.
51 J. Applequist. Traceless Cartesian Tensor Forms for Spherical Harmonic Functions:New Theorems and Applications to Electrostatics of Dielectric Media[J]. J. Phys.,1989, A22:4303–4330.
52 J. Applequist. Maxwell-Cartesian Spherical Harmonics in Multipole Potenials andAtomic Orbitals[J]. Theor. Chem. Acc., 2002, 107:103–115.
53 Z. Chajecki, M. Lisa. Global Conservation Laws and Femtoscopy of Small Sys-tems[J]. Phys.Rev., 2008, C78:064903.
54 A. Kisiel, D. A. Brown. Efficient and Robust Calculation of Femtoscopic CorrelationFunctions in Spherical Harmonics Directly from the Raw Pairs Measured in Heavy-ion Collisions[J]. submitted to Phys. Rev. C [arXiv:0901.3527v1], 2009.
2 L. Landau. On the Multiple Production of Particles in High Energy Collisions[J].Izv. Akad. Nauk SSSR Ser. Fiz., 1953, 17:51–64.
3 L. Landau, S. Belenkij. The Hydrodynamic Theory of Multiple Formation of Parti-cles[J]. Usp. Fiz. Nauk, 1955, 56:309.
4 S. Belenki, L. Landau. Hydrodynamic Theory of Multiple Production of Particles[J].Nuovo Cimento Suppl, 1956, 3:15–31.
5 D. T. Haar. Collected Papers of L. D. Landau[M]. New York: Gordon and Breach,1965.
6 R. Clare, D. Strottman. Relativistic Hydrodynamics and Heavy Ion Reactions[J].Phys. Rep., 1986, 141(4):177–280.
7 L. Csernai. Introduction to Relativistic Heavy Ion Collisions[M]. New York: JohnWiley & Sons, 1994.
8 H. Sto¨cker, W. Greiner. High Energy Heavy Ion Collisions―probing the Equationof State of Highly Excited Hardronic Matter[J]. Phys. Rep., 1986, 137:277–392.
9 C. Adler, Z. Ahammed, C. Allgower, et al. Pion Interferometry of√sNN = 130 GeVAu+Au Collisions at RHIC[J]. Phys. Rev. Lett., 2001, 87:082301.
10 K. Adcox, S. S. Adler, N. N. Ajitanand, et al. Transverse-mass Dependence of Two-pion Correlations in Au+Au Collisions at√sNN = 130 GeV[J]. Phys. Rev. Lett.,2002, 88:192302.
11 S. S. Adler, S. Afanasiev, C. Aidala, et al. Bose-einstein Correlations of ChargedPion Pairs in Au+Au Collisions at√sNN = 200 GeV[J]. Phys. Rev. Lett., 2004,93:152302.
12 J. Adams, M. M. Aggarwal, Z. Ahammed, et al. Pion Interferometry in Au+AuCollisions at√sNN = 200 GeV[J]. Phys. Rev., 2005, C71:044906.
13 U. A. Wiedemann, U. Heinz. Particle Interferometry for Relativistic Heavy-ion Col-lisions[J]. Phys. Rept., 1999, 319:145–230.
14 S. Solf, S. A. Bass, A. Dumitru. Pion Interferometry at Rhic: Probing a ThermalizedQuark-Gluon Plasma? [J]. Phys. Rev. Lett., 2001, 86:3981–3984.
15 R. M. Weiner. Boson Interferometry in High Energy Physics[J]. Phys. Rept., 2000,327:249–346.
16 S. Pratt. Correlations and Fluctuations, a Summary of Quark Matter 2002[J]. Nucl.Phys., 2003, A715:389c–398c.
17 W. N. Zhang, M. J. Efaaf, C. Y. Wong. Pion Interferometry for a Granular Source ofQuark-Gluon Plasma Droplets[J]. Phys. Rev., 2004, C70:024903.
18 E. Witten. Cosmic Separation of Phases[J]. Phys. Rev., 1984, D30:272–285.
19 L. P. Csernai, J. I. Kapusta. Nucleation of Relativistic First-order Phase Transi-tions[J]. Phys. Rev., 1992, D46:1379–1390.
20 S. Alamoudi, D. G. Barci, D. Boyanovsky, et al. Dynamical Viscosity of NucleatingBubbles[J]. Phys. Rev., 1999, D60:125003.
21 J. Randrup. Spinodal Decomposition During the Hadronization Stage at RHIC ? [J].Phys. Rev. Lett., 2004, 92:122301.
22 W. N. Zhang, Y. Y. Ren, C. Y. Wong. Analysis of Pion Elliptic Flow and Hanbury-Brown-Twiss Interferometry in a Granular Quark-gluon Plasma Droplet Model[J].Phys. Rev., 2006, C74:024908.
23 J. D. Bjorken. Highly Relativistic Nucleus-nucleus Collisions: The Central RapidityRegion[J]. Phys. Rev., 1983, D27:140–151.
24 D. A. Brown, P. Danielewicz. Imaging of Sources in Heavy-ion Reactions[J]. Phys.Lett., 1997, B398:252–258.
25 D. A. Brown, A. Enokizono, M. Heffner, et al. Imaging Three DimensionalTwo-particle Correlations for Heavy-ion Reaction Studies[J]. Phys. Rev., 2005,C72:054902.
26 G. R. Liu, M. B. Liu.光滑粒子流体动力学――一种无网格粒子法[M].湖南大学出版社,2005.
27 J. Monaghan. Smoothed Particle Hydrodynamics[J]. Annu. Rev. Astron. Astrophys.,1992, 30:543–574.
28 H. T. Elze, Y. Hama, T. Kodama, et al. Variational Principle for Relativistic FluidDynamics[J]. J. Phys., 1999, G25:1935–1957.
29 C. Aguiar, T. Kodama, T. Osada, et al. Smoothed Particle Hydrodynamics for Rela-tivistic Heavy Ion Collisions[J]. J. Phys., 2001, G27:75–94.
30 F. Grassi, Y. Hama, T. Kodama, et al. Results on Transverse Mass Spectra Obtainedwith NeXSPheRIO[J]. J. Phys. G: Nucl. Part. Phys, 2005, 31:S1041–S1044.
31 R. Andrade, F. Grassi, Y. Hama, et al. On the Necessity to Include Event-by-EventFluctuations in Experimental Evaluation of Elliptical Flow[J]. Phys. Rev. Lett., 2006,97:202302.
32 C. E. Aguiar, Y. Hama, T. Kodama, et al. Event-by-Event Fluctuations in Hydrody-namical Description of Heavy-ion Collisions[J]. Nucl. Phys., 2002, A698:639–642.
33 R. Andrade, F. Grassi, Y. Hama, et al. NeXSPheRIO Results on Elliptic Flow atRHIC and Connection with Thermalization[J]. Eur. Phys. J., 2006, A29:23–26.
34 J. O. Socolowski, F. Grassi, Y. Hama, et al. Fluctuations of the Initial Conditions andthe Continuous Emission in the Hydrodynamical Description of Two-Pion Interfer-ometry[J]. Phys. Rev. Lett., 2004, 93:182301.
35 H. J. Drescher, F. M. Liu, S. Ostapchenko, et al. Initial Condition for Quark-gluonPlasma Evolution[J]. Phys. Rev., 2002, C65:054902.
36 K. Werner, H. Drescher, S.Ostapchenko, et al. The Nexus Model[C]//Invited talkgiven at the International Workshop on the Physics of the Quark Gluon Plasma.Palaiseau, France: [arXiv.org:hep-ph/0209198], 2001.
37 Y. Hama, T. Kodama, O. S. Jr. Topics on Hydrodynamic Model of Nucleus-nucleusCollisions[J]. Braz. J. Phys., 2005, 35:24–51.
38 D. H. Boal, C.-K. Gelbke, B. K. Jennings. Intensity Interferometry in SubatomicPhysics[J]. Rev. Mod. Phys., 1990, 62:553–602.
39 H. Beker, H. B?ggild, J. Boissevain, et al. Kaon Interferometry in Heavy-ion Colli-sions at the CERN Sps[J]. Z. Phys. C, 1994, 64:209–217.
40 H. B?ggild, J. Boissevain, M. Cherney, et al. Identified Pion Interferometry in Heavy-ion Collisions at CERN[J]. Phys. Lett., 1993, B302:510–516.
41 H. B?ggild, J. Boissevain, M. Cherney, et al. Directional Dependence of the PionSource in High-energy Heavy-ion Collisions[J]. Phys. Lett., 1995, B349:386–392.
42 D. A. Brown, F. Wang, P. Danielewicz. Implications of the Unusual Structure in thepp Correlation from Pb+Pb Collisions at 158 AGeV[J]. Phys. Lett., 1999, B470:33–38.
43 D. A. Brown, P. Danielewicz. Optimized Discretization of Sources Imaged in Heavy-ion Reactions[J]. Phys. Rev., 1998, C57:2474–2483.
44 S. E. Koonin. Proton Pictures of High-energy Nuclear Collisions[J]. Phys. Lett.,1997, B70:43–47.
45 S. Pratt, T. Csorgo, J. Zimanyi. Detailed Predictions for Two-pion Correlations inUltrarelativistic Heavy-ion Collisions[J]. Phys. Rev., 1990, C42:2646–2652.
46 G. F. Bertsch. Meson Phase Space Density in Heavy Ion Collisions from Interfer-ometry[J]. Phys. Rev. Lett., 1994, 72:2349–2350.
47 D. A. Brown, S. Y. Panitkin, G. F. Bertsch. Extracting Particle Freeze-out Phase-space Densities and Entropies from Sources Imaged in Heavy-ion Reactions[J].Phys.Rev., 2000, C62:014904.
48 D. A. Brown, P. Danielewicz. Observing Non-Gaussian Sources in Heavy-ion Reac-tions[J]. Phys. Rev., 2001, C64:014902.
49 P. Danielewicz, S. Pratt. Analysis of Low-momentum Correlations with CartesianHarmonics[J]. Phys. Lett., 2005, B618:60–67.
50 P. Danielewicz, S. Pratt. Analyzing Correlation Functions with Tesseral and Carte-sian Spherical Harmonics[J]. Phys. Rev., 2007, C75:034907.
51 J. Applequist. Traceless Cartesian Tensor Forms for Spherical Harmonic Functions:New Theorems and Applications to Electrostatics of Dielectric Media[J]. J. Phys.,1989, A22:4303–4330.
52 J. Applequist. Maxwell-Cartesian Spherical Harmonics in Multipole Potenials andAtomic Orbitals[J]. Theor. Chem. Acc., 2002, 107:103–115.
53 Z. Chajecki, M. Lisa. Global Conservation Laws and Femtoscopy of Small Sys-tems[J]. Phys.Rev., 2008, C78:064903.
54 A. Kisiel, D. A. Brown. Efficient and Robust Calculation of Femtoscopic CorrelationFunctions in Spherical Harmonics Directly from the Raw Pairs Measured in Heavy-ion Collisions[J]. submitted to Phys. Rev. C [arXiv:0901.3527v1], 2009.