Unified treatment for accurate and fast evaluation of the Fermi－Dirac functions

doi:10.1088/1674-1056/19/5/050501

Abstract A new analytical approach to the computation of the Fermi-Dirac (FD) functions is presented, which was suggested by previous experience with various algorithms. Using the binomial expansion theorem, these functions are expressed through the binomial coefficients and familiar incomplete Gamma functions. This simplification and the use of the memory of the computer for calculation of binomial coefficients may extend the limits to large arguments for users and result in speedier calculation, should such limits be required in practice. Some numerical results are presented for significant mapping examples and they are briefly discussed.

Keywords: statistical semiconductors plasma physics nuclear physics

Received: 02 October 2009
Accepted manuscript online:

PACS:	05.30.Fk	(Fermion systems and electron gas)
	02.30.Lt	(Sequences, series, and summability)
	02.30.Mv	(Approximations and expansions)
	02.30.Sa	(Functional analysis)

Cite this article:

I. I. Guseinov and B. A. Mamedov Unified treatment for accurate and fast evaluation of the Fermi－Dirac functions 2010 Chin. Phys. B 19 050501

[1]	Blakemore J S 1962 Semiconductor Statistics (New York: Pergamon) p234
[2]	Blakemore J S 1982 Solid-State Electron. 25 1067
[3]	Dingle R B 1956 J. Appl. Res. B 6 225
[4]	Marshak A H, Shibib M A, Fossum J G and Lindholm F A 1981 IEEE Trans. Electron Devices ED-28 293
[5]	Stoner E C 1939 Phil. Mag. 28 257
[6]	Constantinescu D H and Moruzzi C 1978 Phys. Rev. D 18 1820
[7]	Gong Z, Dappen W and Zejda L 2001 Astrophys. J. 546 1178
[8]	Chandrasekhar S 1939 An Introduction to the Study of Stellar Structure (New York: Dover) p178
[9]	Clayton D D 1968 Principles of Steller Evolution and Nucleosyn-thesis (New York: McGraw-Hill) p247
[10]	Rhodes P 1950 Prog. R. Soc. London Ser. A 204 396
[11]	Wu Z Q, Li S C and Han G X 1996 JQSRT 56 623
[12]	Wilson B G and Chen M H 1999 JQSRT 61 813
[13]	Lampe M 1968 Phys. Rev. 170 306[13a] Lampe M 1968 Phys. Rev. 174 276
[14]	Hubbard W B and Lampe M 1969 Astrophys. J. Suppl. Ser. 18 297
[15]	Rinker G A 1985 Phys. Rev. B 31 4220
[16]	Fullerton L W and Rinker G A 1986 Comput. Phys. Commun. 39 181
[17]	Hite D, Boykin T B, Singh N and Shen D 2005 Am. J. Phys. 73 856
[18]	Li G Q, Shi J Q and Gao Q 1990 Nucl. Phys. A 515 273
[19]	Loss D and Schoeller H 1989 J. Stat. Phys. 54 765
[20]	Adawi I 1975 J. Stat. Phys. 12 263
[21]	Elton L R B 1961 Nuclear Sizes (New York: Oxford University Press) p65
[22]	Grypeos M E, Lalazissis S A, Mossen S E and Panos C P 1991 J. Phys. G: Nuc. Part. Phys. 17 1093
[23]	Bohr A and Mottelson B 1969 Nuclear Structure (New York: Benjamin)
[24]	Lukyanov V K 1995 J. Phys. G: Nuc. Part. Phys. 21 145
[25]	Sommerfeld A 1928 Z. Phys. 47 1
[26]	McDougall J and Stoner E C 1938 Philos. Trans. R. Soc. London Ser. A 237 67
[27]	Johnson V A and Shipley F M 1953 Phys. Rev. 90 523
[28]	Glasser M L 1964 J. Math. Phys. 5 1150
[29]	Battocletti F E 1965 Proc. IEEE 53 2162
[30]	Jones E L 1966 Proc. IEEE 54 708
[31]	Hill R N 1970 Am. J. Phys. 38 1440
[32]	Joyce W B and Dixon R W 1977 Appl. Phys. Lett. 31 354
[33]	Joyce W B 1978 Appl. Phys. Lett. 32 680
[34]	Gautschi W 1979 ACM Trans. Math. Software 5 482
[35]	Selvakumar C R 1982 Proc. IEEE 70 516
[36]	Abidi S T and Noor Mohammad S 1984 J. Appl. Phys. 56 3341
[37]	Kiess E 1987 Am. J. Phys. 55 1006
[38]	Ell C, Blank R, Benner S and Haung H 1989 J. Opt. Soc. Am. B 6 2006
[39]	Fernandez Velica F J 1984 Phys. Rev. A 30 1194
[40]	Didonato A R and Morris Jr A H 1987 ACM Trans. Math. Software 13 318
[41]	Gong H V 1991 Solid-State Electron. 34 489
[42]	Sagar R P 1991 Comput. Phys. Commun. 66 271
[43]	Cong H V and Doan-Khanh B 1992 Solid-State Electron. 35 949
[44]	Goano M 1993 Solid-State Electron. 36 217
[45]	Smith A W and Rohatgi A 1993 J. Appl. Phys. 73 7030
[46]	Antia H M 1993 Astrophys. J. Suppl. Ser. 84 101
[47]	Mohankumar N and Natarajan A 1996 Astrophys. J. 458 233
[48]	Uehling E A and Uhlenbeck G 1933 Phys. Rev. Lett. 43 552
[49]	Aparicio J M 1998 Astrophys. J. Suppl. Ser. 117 627
[50]	Lukyanov V K 1995 J. Phys. G 21 145
[51]	Goano M 1995 ACM T. Math. Software. 21 221
[52]	Ohsugi I J, Kojima T and Nishida I 1988 J. Appl. Phys. 63 5179
[53]	Chang T Y and Izabelle A 1989 J. Appl. Phys. 65 2162
[54]	Reser B I 1996 J. Phys.: Condens. Matter 8 3151
[55]	Gong Z, Dappen W and Zejda L 2001 Astrophys. J. 546 1178
[56]	Gong Z, Zejda L, Dappen W and Aparicio J M 2001 arXiv: astro-ph/0102329
[57]	Grypeos M, Koutroulos C, Lukyanov V and Shebeko A 1998 J. Phys. G: Nuc. Part. Phys. 24 1913
[58]	Mohankumar N, Kannan T and Kanmani S 2005 Comput. Phys. Commun. 168 71
[59]	Bhagat V, Bhattacharya R and Roy D 2003 Comput. Phys. Commun. 155 7
[60]	Garoni T M, Frankel N E and Glasser M L 2001 J. Math. Phys. 42 1860
[61]	Lether F G 2001 J. Sci. Comput. 16 69
[62]	Rzadkowski G and Lepkowski S 2008 J. Sci. Comput. 35 63
[63]	Gradshteyn I S and Ryzhik I M 1980 Tables of Integrals, Sums, Series and Products 4th ed. (New York: Academic Press)
[64]	Guseinov I I and Mamedov B A 2006 JQSRT 102 251
[65]	Mamedov B A 2005 JQSRT 94 507
[66]	Guseinov I I and Mamedov B A 2004 J. Math. Chem. 36 341
[67]	Guseinov I I and Mamedov B A 2005 J. Math. Chem. 38 311
[68]	Mamedov B A 2008 Comput. Phys. Commun. 178 673
[69]	Banuelos A, Depine R A and Mancini R C 1981 J. Math. Phys. 22 452

[1]	Crystal and electronic structure of a quasi-two-dimensional semiconductor Mg₃Si₂Te₆ Chaoxin Huang(黄潮欣), Benyuan Cheng(程本源), Yunwei Zhang(张云蔚), Long Jiang(姜隆), Lisi Li(李历斯), Mengwu Huo(霍梦五), Hui Liu(刘晖), Xing Huang(黄星), Feixiang Liang(梁飞翔), Lan Chen(陈岚), Hualei Sun(孙华蕾), and Meng Wang(王猛). Chin. Phys. B, 2023, 32(3): 037802.
[2]	Variational quantum simulation of thermal statistical states on a superconducting quantum processer Xue-Yi Guo(郭学仪), Shang-Shu Li(李尚书), Xiao Xiao(效骁), Zhong-Cheng Xiang(相忠诚), Zi-Yong Ge(葛自勇), He-Kang Li(李贺康), Peng-Tao Song(宋鹏涛), Yi Peng(彭益), Zhan Wang(王战), Kai Xu(许凯), Pan Zhang(张潘), Lei Wang(王磊), Dong-Ning Zheng(郑东宁), and Heng Fan(范桁). Chin. Phys. B, 2023, 32(1): 010307.
[3]	Effect of observation time on source identification of diffusion in complex networks Chaoyi Shi(史朝义), Qi Zhang(张琦), and Tianguang Chu(楚天广). Chin. Phys. B, 2022, 31(7): 070203.
[4]	Voter model on adaptive networks Jinming Du(杜金铭). Chin. Phys. B, 2022, 31(5): 058902.
[5]	Origin of anomalous enhancement of the absorption coefficient in a PN junction Xiansheng Tang(唐先胜), Baoan Sun(孙保安), Chen Yue(岳琛), Xinxin Li(李欣欣), Junyang Zhang(张珺玚), Zhen Deng(邓震), Chunhua Du(杜春花), Wenxin Wang(王文新), Haiqiang Jia(贾海强), Yang Jiang(江洋), Weihua Wang(汪卫华), and Hong Chen(陈弘). Chin. Phys. B, 2021, 30(9): 097804.
[6]	Atomic and electronic structures of p-type dopants in 4H-SiC Lingyan Lu(卢玲燕), Han Zhang(张涵), Xiaowei Wu(吴晓维), Jing Shi(石晶), and Yi-Yang Sun(孙宜阳). Chin. Phys. B, 2021, 30(9): 096806.
[7]	Water and nutrient recovery from urine: A lead up trail using nano-structured In₂S₃ photo electrodes R Jayakrishnan, T R Sreerev, and Adith Varma. Chin. Phys. B, 2021, 30(5): 056103.
[8]	Synaptic plasticity and classical conditioning mimicked in single indium-tungsten-oxide based neuromorphic transistor Rui Liu(刘锐), Yongli He(何勇礼), Shanshan Jiang(姜珊珊), Li Zhu(朱力), Chunsheng Chen(陈春生), Ying Zhu(祝影), and Qing Wan(万青). Chin. Phys. B, 2021, 30(5): 058102.
[9]	First-principles investigation of the valley and electrical properties of carbon-doped α-graphyne-like BN sheet Bo Chen(陈波), Xiang-Qian Li(李向前), Lin Xue(薛林), Yan Han(韩燕), Zhi Yang(杨致), and Long-Long Zhang(张龙龙). Chin. Phys. B, 2021, 30(5): 057101.
[10]	Restricted Boltzmann machine: Recent advances and mean-field theory Aurélien Decelle, Cyril Furtlehner. Chin. Phys. B, 2021, 30(4): 040202.
[11]	Constructing refined null models for statistical analysis of signed networks Ai-Wen Li(李艾纹), Jing Xiao(肖婧, and Xiao-Ke Xu(许小可). Chin. Phys. B, 2021, 30(3): 038901.
[12]	First-principles study of the co-effect of carbon doping and oxygen vacancies in ZnO photocatalyst Jia Shi(史佳), Lei Wang(王蕾), and Qiang Gu(顾强). Chin. Phys. B, 2021, 30(2): 026301.
[13]	Statistical potentials for 3D structure evaluation: From proteins to RNAs Ya-Lan Tan(谭雅岚), Chen-Jie Feng(封晨洁), Xunxun Wang(王勋勋), Wenbing Zhang(张文炳), and Zhi-Jie Tan(谭志杰). Chin. Phys. B, 2021, 30(2): 028705.
[14]	Selected topics of quantum computing for nuclear physics Dan-Bo Zhang(张旦波), Hongxi Xing(邢宏喜), Hui Yan(颜辉), Enke Wang(王恩科), and Shi-Liang Zhu(朱诗亮). Chin. Phys. B, 2021, 30(2): 020306.
[15]	Simulation of the gravitational wave frequency distribution of neutron star-black hole mergers Jianwei Zhang(张见微), Chengmin Zhang(张承民), Di Li(李菂), Xianghan Cui(崔翔翰), Wuming Yang(杨伍明), Dehua Wang(王德华), Yiyan Yang(杨佚沿), Shaolan Bi(毕少兰), and Xianfei Zhang(张先飞). Chin. Phys. B, 2021, 30(12): 120401.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Unified treatment for accurate and fast evaluation of the Fermi－Dirac functions

Cite this article:

Online attention