Intermolecular interactions and thermodynamic properties of LLM-105

Fan Jun-Yu; Gao Nan; Wang Peng-Ju; Su Yan

doi:10.7498/aps.73.20231696

Abstract

2,6-diamino-3,5-dinitropyrazine-1-oxide (LLM-105) is a typical high-energy and low-sensitivity energetic material (EM), which has excellent detonation performance and thermal stability. In the quasi-harmonic approximation, the dispersion corrected density functional theory is used to study the intermolecular interactions and thermodynamic properties of energetic LLM-105 crystal. By introducing the zero-point energy and temperature effect corrections, PBE-D3 dispersion correction scheme can significantly improve the calculation accuracy of structural parameters at an experimental temperature (294 K). The temperature dependent lattice parameters and thermal expansion coefficients exhibit strong anisotropy, especially the thermal expansivity in b-axis orientation (intermolecular layers) is much higher than that in the ac plane (intramolecular layers). Through Hirshfeld surface and fingerprint analysis, it is found that the intermolecular interactions of LLM-105 are mainly O···H hydrogen bonding interactions. The change of intermolecular interactions will result in the rotation of nitro group, which can contribute to forming new hydrogen-bonding interaction pattern. Mulliken population analysis shows that the bond order of C—NO₂ bond is more sensitive to the change of temperature, so this bond may be a trigger bond for the high-temperature decomposition reaction of LLM-105.The fundamental thermodynamic properties of EMs can not only provide key parameters for mesoscopic or macroscopic thermodynamic simulations, but also gain theoretical insights into the temperature effects of EMs. Specific heat capacity reflects the amount of heat to be supplied to heating the matter and it is important to make the risk assessment of EMs during storage or when exposed to external thermal stimuli. Herein, the basic thermodynamic parameters, such as heat capacity, entropy, bulk modulus and elastic constants under different conditions are predicted. Among them, the calculated heat capacity and entropy describe the nonlinear behaviors within a temperature range of 0 to 500 K, and the calculated isobaric heat capacity C_p(T) is in good agreement with the available experimental measurements. The elasticity of material describes the macroscopic response of crystal to external force, and the bulk modulus B₀ of molecular crystal can be determined through the equation of state, which is an important parameter for evaluating material stiffness. The bulk modulus under adiabatic condition is in reasonable agreement with experimental value, and the evolution of bulk modulus with temperature reflects the softening behavior of LLM-105 at temperature. Furthermore, the complete set of second-order elastic constants (SOECs) of LLM-105 is calculated and 13 independent SOECs (C₁₁, C₁₂, C₁₃, C₁₅, C₂₂, C₂₃, C₂₅, C₃₃, C₃₅, C₄₄, C₄₆, C₅₅, C₆₆) are predicted. With the increasing temperature, all elastic constants gradually decrease due to the weakening of intermolecular interactions of LLM-105. Overall, these results will fundamentally provide a deep understanding of temperature effects and serve as a reference for the experimental measurement of the thermodynamic parameters of EMs.

Keywords:

Authors and contacts

1.
Department of Physics, Taiyuan Normal University, Jinzhong 030619, China
2.
School of Materials Science and Engineering, Taizhou University, Taizhou 318000, China
3.
Zhejiang Laboratory, Hangzhou 311100, China
4.
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Ministry of Education, Dalian University of Technology, Dalian 116024, China

Corresponding author: Su Yan, su.yan@dlut.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 91961204, 12004272), the Applied Basic Research Programs of Natural Science Foundation of Shanxi Province, China (Grant No. 202203021212194), and the Fundamental Research Funds for the Central Universities of China (Grant No. DUT20ZD207).

FullText HTML : translate this paragraph

References

[1]	王泽山 2006 含能材料概论 (哈尔滨: 哈尔滨工业大学出版社) 第4—8页 Wang Z S 2006 Introduction to Energetic Material (Harbin: Harbin Institute of Technology Press) pp4–8
[2]	Zhang C Y, Wang X C, Huang H 2008 J. Am. Chem. Soc. 130 8359 Google Scholar
[3]	Jiao F B, Xiong Y, Li H Z, Zhang C Y 2018 Cryst. Eng. Comm 20 1757 Google Scholar
[4]	Gilardi R D, Butcher R J 2001 Acta Crystallogr. E 57 o657 Google Scholar
[5]	Tarver C M, Urtiew P A, Tran T D 2005 J. Energ. Mater. 23 183 Google Scholar
[6]	Averkiev B B, Antipin M Y, Yudin I L, Sheremetev A B 2002 J. Mol. Struc. 606 139 Google Scholar
[7]	Li G, Zhang C Y 2020 J. Hazard. Mater. 398 122910 Google Scholar
[8]	Miao M S, Sun Y H, Zurek E, Lin H Q 2020 Nat. Rev. Chem. 4 508 Google Scholar
[9]	Gump J C, Stoltz C A, Mason B P, Freedman B G, Ball J R, Peiris S M 2011 J. Appl. Phys. 110 073523 Google Scholar
[10]	Stavrou E, Riad Manaa M, Zaug J M, et al. 2015 J. Chem. Phys. 143 144506 Google Scholar
[11]	Xu Z L, Su H, Zhou X Q, et al. 2019 J. Phys. Chem. C 123 1110 Google Scholar
[12]	Xu Z L, Chen Q, Li X D, et al. 2020 J. Phys. Chem. C 124 2399 Google Scholar
[13]	Manaa M R, Kuo I F, Fried L E 2014 J. Chem. Phys. 141 064702 Google Scholar
[14]	Wang J K, Xiong Y, Li H Z, Zhang C Y 2018 J. Phys. Chem. C 122 1109 Google Scholar
[15]	Wang X, Zeng Q, Li J S, Yang M L 2019 ACS Omega 4 21054 Google Scholar
[16]	Wu Q, Yang C H, Pan Y, Xiang F, Liu Z C, Zhu W H, Xiao H M 2013 J. Mol. Model. 19 5159 Google Scholar
[17]	Zong H H, Zhang L, Zhang W B, Jiang S L, Yu Y, Chen J 2017 J. Mol. Model. 23 275 Google Scholar
[18]	Yuan W S, Hong D, Luo Y X, Li X H, Liu F S, Liu Z T, Liu Q J 2023 Spectrochim. Acta A 303 123170 Google Scholar
[19]	Yu Q, Zhao C D, Li J S 2022 J. Therm. Anal. Calorim. 147 12965 Google Scholar
[20]	Li J Y, Zhang H B, Wen M P, Xu J J, Liu X F, Sun J 2016 J. Energ. Mater. 34 170 Google Scholar
[21]	Ma R, Sun W C, Picu C R 2021 Int. J. Solids Struct. 232 111170 Google Scholar
[22]	Menikoff R, Sewell T D 2002 Combust. Theor. Model. 6 103 Google Scholar
[23]	Chen J, Qiao Z Q, Wang L L, Nie F D, Yang G C, Huang H 2011 Mater. Lett. 65 1018 Google Scholar
[24]	Yu Q, Zhao C D, Liao L Y, Li H Z, Sui H L, Yin Y, Li J S 2020 Phys. Chem. Chem. Phys. 22 13729 Google Scholar
[25]	Jiang J, Liu J Y, Chen Y H, Wu Q H, Ju Z Y, Zhang S H 2021 Mol. Simulat. 47 678 Google Scholar
[26]	Xiao Q, Sui H L, Hao X F, Chen J, Yin Y, Yu Q, Yang X L, Ju X 2020 Spectrochim. Acta A 240 118577 Google Scholar
[27]	Dovesi R, Orlando R, Erba A, et al. 2014 Int. J. Quantum Chem. 114 1287 Google Scholar
[28]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[29]	Grimme S 2011 Wires. Comput. Mol. Sci. 1 211 Google Scholar
[30]	Monkhorst H J, Pack J D 1976 Phys. Rev. B 13 5188 Google Scholar
[31]	Fischer T H, Almlof J 1992 J. Phys. Chem. 96 9768 Google Scholar
[32]	Fan J Y, Su Y, Zheng Z Y, Zhao J J 2021 J. Phys. Condens. Mat. 33 275702 Google Scholar
[33]	Parwani A V 2009 CrystEngComm 11 19 Google Scholar
[34]	Spackman M A, McKinnon J J 2002 CrystEngComm 4 378 Google Scholar
[35]	Weese R K, Burnham A K, Turner H C, Tran T D 2007 J. Therm. Anal. Calorim. 89 465 Google Scholar
[36]	Ma H X, Song J R, Zhao F Q, Gao H X, Hu R Z 2008 Chin. J. Chem. 26 1997 Google Scholar
[37]	Drebushchak V A 2020 J. Therm. Anal. Calorim. 142 1097 Google Scholar
[38]	Hooks D E, Ramos K J, Bolme C A, Cawkwell M J 2015 Propell. Explos. Pyrot. 40 333 Google Scholar
[39]	Fan J Y, Su Y, Zhang Q Y, Zhao J J 2019 Comp. Mater. Sci. 161 379 Google Scholar
[40]	位付景, 张伟斌, 董闯, 陈华 2023 物理学报 72 096201 Google Scholar Wei F J, Zhang W B, Dong C, Chen H 2023 Acta Phys. Sin. 72 096201 Google Scholar
[41]	Stevens L L, Velisavljevic N, Hooks D E, Dattelbaum D M 2008 Propell. Explos. Pyrot. 33 286 Google Scholar
[42]	Nye J F 1985 Physical Properties of Crystals: Their Representation by Tensors and Matrices (New York: Oxford University Press) pp131–148

Cited By

图 1 (a) 实验温度下, 不同泛函结合不同色散修正方案计算的F-V曲线; (b) PBE泛函结合D3色散修正方案、零点能效应和温度效应计算的LLM-105的晶胞体积以及与实验的误差

Figure 1. (a) F-V curves calculated at experimental temperature using different functional and dispersion correction schemes; (b) PBE functional combined with D3 dispersion correction scheme, zero point energy effect and temperature effect calculation of LLM-105 cell volume and experimental error.

DownLoad: Full-Size Img PowerPoint

图 2 计算和实验测量^[11]的LLM-105晶胞体积(a)和晶格参数(b)随温度的演化; (c) LLM-105的热膨胀系数随温度的演化; (d) LLM-105的分子结构和层间堆垛示意图, 其中粉色代表氢原子, 棕色代表碳原子, 蓝色代表氮原子, 红色代表氧原子

Figure 2. Calculated cell volume (a) and lattice parameters (b) at elevated temperature, compared with experimental values^[11]; (c) variation of thermal expansion coefficient of LLM-105 with temperature increasing; (d) diagram of molecular structure and interlayer stacking of LLM-105, where the pink, brown, blue and red balls represent hydrogen, carbon, nitrogen and oxygen atoms, respectively.

DownLoad: Full-Size Img PowerPoint

图 3 LLM-105晶体的 Hirshfeld表面、指纹图及其分子间相互作用对表面的相对贡献, 灰色区域表示LLM-105 晶体的 Hirshfeld 表面及其映射的二维指纹图, 其中蓝色和红色区域分别表示原子周围的正、负静电势

Figure 3. Hirshfeld, fingerprint, and intermolecular interactions of LLM-105 crystal, the gray area represents the Hirshfeld surface of LLM-105 crystal and its mapped two-dimensional fingerprint, where the blue and red areas represent the positive and negative electrostatic potential of the atoms, respectively.

DownLoad: Full-Size Img PowerPoint

图 4 (a) 0—500 K范围内, LLM-105晶体分子间相互作用的变化量; (b) 层间分子的质心距离随温度的变化; (c) LLM-105分子中化学键布居数随温度的变化; (d) 计算的硝基(—NO₂)与C—N环平面所成二面角随温度的变化

Figure 4. (a) Changes of intermolecular interactions of LLM-105 crystals at 0–500 K; (b) evolution of centroid distance of interlayer molecules under temperature; (c) variation of the bond population in LLM-105 molecules with temperature increasing; (d) calculated dihedral angle between nitro group (—NO₂) and C—N ring plane with temperature increasing.

DownLoad: Full-Size Img PowerPoint

图 5 等容(黑色曲线)和等压(红色曲线)条件下LLM-105晶体的热容(a)和熵(b)随温度的变化

Figure 5. Heat capacity (a) and entropy (b) of LLM-105 crystal as a function of temperature under constant-volume (black line) and constant-pressure (red line) conditions, respectively.

DownLoad: Full-Size Img PowerPoint

图 6 (a) 等温(B_t)和绝热(B_s)条件下LLM-105晶体的体模量随温度的变化; (b) LLM-105晶体的二阶弹性常数完全集及其随温度的变化

Figure 6. (a) Variation of bulk modulus of LLM-105 crystal under isothermal (B_t) and adiabatic (B_s) conditions with temperature increasing; (b) the complete set of second-order elastic constants of LLM-105 crystal and its temperature dependence.

DownLoad: Full-Size Img PowerPoint

表 1 实验温度下色散修正方案(PBE-D2/D3)结合零点能和温度效应计算的LLM-105结构参数及其与实验值之间的偏差

Table 1. Structural parameters of LLM-105 calculated by dispersion correction scheme (PBE-D2/D3) combined with zero-point energy and temperature effects at experimental temperature. The deviation is given under each calculated value.

	V/Å³	a/Å	b/Å	c/Å	β/(^°)
Expt.^[4]	748.16	5.716	15.850	8.414	101.041
PBE-D2	745.83	5.62	16.30	8.25	99.46
	(–0.31%)	(–1.60%)	(2.82%)	(–1.96%)	(–1.57%)
PBE-D3	748.26	5.74	15.81	8.39	100.39
	(0.01%)	(0.36%)	(–0.25%)	(–0.32%)	(–0.65%)

DownLoad: CSV

[1]	王泽山 2006 含能材料概论 (哈尔滨: 哈尔滨工业大学出版社) 第4—8页 Wang Z S 2006 Introduction to Energetic Material (Harbin: Harbin Institute of Technology Press) pp4–8
[2]	Zhang C Y, Wang X C, Huang H 2008 J. Am. Chem. Soc. 130 8359 Google Scholar
[3]	Jiao F B, Xiong Y, Li H Z, Zhang C Y 2018 Cryst. Eng. Comm 20 1757 Google Scholar
[4]	Gilardi R D, Butcher R J 2001 Acta Crystallogr. E 57 o657 Google Scholar
[5]	Tarver C M, Urtiew P A, Tran T D 2005 J. Energ. Mater. 23 183 Google Scholar
[6]	Averkiev B B, Antipin M Y, Yudin I L, Sheremetev A B 2002 J. Mol. Struc. 606 139 Google Scholar
[7]	Li G, Zhang C Y 2020 J. Hazard. Mater. 398 122910 Google Scholar
[8]	Miao M S, Sun Y H, Zurek E, Lin H Q 2020 Nat. Rev. Chem. 4 508 Google Scholar
[9]	Gump J C, Stoltz C A, Mason B P, Freedman B G, Ball J R, Peiris S M 2011 J. Appl. Phys. 110 073523 Google Scholar
[10]	Stavrou E, Riad Manaa M, Zaug J M, et al. 2015 J. Chem. Phys. 143 144506 Google Scholar
[11]	Xu Z L, Su H, Zhou X Q, et al. 2019 J. Phys. Chem. C 123 1110 Google Scholar
[12]	Xu Z L, Chen Q, Li X D, et al. 2020 J. Phys. Chem. C 124 2399 Google Scholar
[13]	Manaa M R, Kuo I F, Fried L E 2014 J. Chem. Phys. 141 064702 Google Scholar
[14]	Wang J K, Xiong Y, Li H Z, Zhang C Y 2018 J. Phys. Chem. C 122 1109 Google Scholar
[15]	Wang X, Zeng Q, Li J S, Yang M L 2019 ACS Omega 4 21054 Google Scholar
[16]	Wu Q, Yang C H, Pan Y, Xiang F, Liu Z C, Zhu W H, Xiao H M 2013 J. Mol. Model. 19 5159 Google Scholar
[17]	Zong H H, Zhang L, Zhang W B, Jiang S L, Yu Y, Chen J 2017 J. Mol. Model. 23 275 Google Scholar
[18]	Yuan W S, Hong D, Luo Y X, Li X H, Liu F S, Liu Z T, Liu Q J 2023 Spectrochim. Acta A 303 123170 Google Scholar
[19]	Yu Q, Zhao C D, Li J S 2022 J. Therm. Anal. Calorim. 147 12965 Google Scholar
[20]	Li J Y, Zhang H B, Wen M P, Xu J J, Liu X F, Sun J 2016 J. Energ. Mater. 34 170 Google Scholar
[21]	Ma R, Sun W C, Picu C R 2021 Int. J. Solids Struct. 232 111170 Google Scholar
[22]	Menikoff R, Sewell T D 2002 Combust. Theor. Model. 6 103 Google Scholar
[23]	Chen J, Qiao Z Q, Wang L L, Nie F D, Yang G C, Huang H 2011 Mater. Lett. 65 1018 Google Scholar
[24]	Yu Q, Zhao C D, Liao L Y, Li H Z, Sui H L, Yin Y, Li J S 2020 Phys. Chem. Chem. Phys. 22 13729 Google Scholar
[25]	Jiang J, Liu J Y, Chen Y H, Wu Q H, Ju Z Y, Zhang S H 2021 Mol. Simulat. 47 678 Google Scholar
[26]	Xiao Q, Sui H L, Hao X F, Chen J, Yin Y, Yu Q, Yang X L, Ju X 2020 Spectrochim. Acta A 240 118577 Google Scholar
[27]	Dovesi R, Orlando R, Erba A, et al. 2014 Int. J. Quantum Chem. 114 1287 Google Scholar
[28]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[29]	Grimme S 2011 Wires. Comput. Mol. Sci. 1 211 Google Scholar
[30]	Monkhorst H J, Pack J D 1976 Phys. Rev. B 13 5188 Google Scholar
[31]	Fischer T H, Almlof J 1992 J. Phys. Chem. 96 9768 Google Scholar
[32]	Fan J Y, Su Y, Zheng Z Y, Zhao J J 2021 J. Phys. Condens. Mat. 33 275702 Google Scholar
[33]	Parwani A V 2009 CrystEngComm 11 19 Google Scholar
[34]	Spackman M A, McKinnon J J 2002 CrystEngComm 4 378 Google Scholar
[35]	Weese R K, Burnham A K, Turner H C, Tran T D 2007 J. Therm. Anal. Calorim. 89 465 Google Scholar
[36]	Ma H X, Song J R, Zhao F Q, Gao H X, Hu R Z 2008 Chin. J. Chem. 26 1997 Google Scholar
[37]	Drebushchak V A 2020 J. Therm. Anal. Calorim. 142 1097 Google Scholar
[38]	Hooks D E, Ramos K J, Bolme C A, Cawkwell M J 2015 Propell. Explos. Pyrot. 40 333 Google Scholar
[39]	Fan J Y, Su Y, Zhang Q Y, Zhao J J 2019 Comp. Mater. Sci. 161 379 Google Scholar
[40]	位付景, 张伟斌, 董闯, 陈华 2023 物理学报 72 096201 Google Scholar Wei F J, Zhang W B, Dong C, Chen H 2023 Acta Phys. Sin. 72 096201 Google Scholar
[41]	Stevens L L, Velisavljevic N, Hooks D E, Dattelbaum D M 2008 Propell. Explos. Pyrot. 33 286 Google Scholar
[42]	Nye J F 1985 Physical Properties of Crystals: Their Representation by Tensors and Matrices (New York: Oxford University Press) pp131–148

[1]	Hu Min-Li, Fang Fan, Fan Qun-Chao, Fan Zhi-Xiang, Li Hui-Dong, Fu Jia, Xie Feng. Theoretical study on macroscopic thermodynamic properties of NO⁺ ion system. Acta Physica Sinica, 2023, 72(16): 165101. doi: 10.7498/aps.72.20230541
[2]	Zhu Cheng, Chen Xian-Hui, Wang Cheng, Song Ming, Xia Wei-Dong. Calculation of thermodynamic properties and transport coefficients of Ar-C-Si plasma. Acta Physica Sinica, 2023, 72(12): 125202. doi: 10.7498/aps.72.20222390
[3]	Jian Jun, Lei Jiao, Fan Qun-Chao, Fan Zhi-Xiang, Ma Jie, Fu Jia, Li Hui-Dong, Xu Yong-Gen. Theoretical study on thermodynamic properties of NO gas. Acta Physica Sinica, 2020, 69(5): 053301. doi: 10.7498/aps.69.20191723
[4]	Zhao Yu-Na, Cong Hong-Lu, Cheng Shuang, Yu Na, Gao Tao, Ma Jun-Gang. First-principles study of lattice dynamical and thermodynamic properties of Li₂NH. Acta Physica Sinica, 2019, 68(13): 137102. doi: 10.7498/aps.68.20190139
[5]	Deng Shi-Jie, Zhao Yu-Hong, Hou Hua, Wen Zhi-Qin, Han Pei-De. Structural, mechanical and thermodynamic properties of Ti2AlX (X= C, N) at high pressure. Acta Physica Sinica, 2017, 66(14): 146101. doi: 10.7498/aps.66.146101
[6]	Wu Ruo-Xi, Liu Dai-Jun, Yu Yang, Yang Tao. First-principles investigations on structure and thermodynamic properties of CaS under high pressures. Acta Physica Sinica, 2016, 65(2): 027101. doi: 10.7498/aps.65.027101
[7]	Li He-Ling, Wang Juan-Juan, Yang Bin, Shen Hong-Jun. Investigation of thermodynamic properties of weakly interacting Fermi gas in weakly magnetic field by using the N-E-V distribution and pseudopotential method. Acta Physica Sinica, 2015, 64(4): 040501. doi: 10.7498/aps.64.040501
[8]	Chen Xin-Long, Men Fu-Dian, Tian Qing-Song. Effect of anomalous magnetic moment on thermodynamic properties of weakly interacting Fermi gas in weak magnetic field. Acta Physica Sinica, 2015, 64(8): 080501. doi: 10.7498/aps.64.080501
[9]	Li Xiao-Feng, Liu Zhong-Li, Peng Wei-Min, Zhao A-Ke. Elastic and thermodynamic properties of CaPo under pressure via first-principles calculations. Acta Physica Sinica, 2011, 60(7): 076501. doi: 10.7498/aps.60.076501
[10]	Men Fu-Dian, Wang Bing-Fu, He Xiao-Gang, Wei Qun-Mei. Thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic field. Acta Physica Sinica, 2011, 60(8): 080501. doi: 10.7498/aps.60.080501
[11]	Li Shi-Na, Liu Yong. First-principles calculation of elastic and thermodynamic properties of copper nitride. Acta Physica Sinica, 2010, 59(10): 6882-6888. doi: 10.7498/aps.59.6882
[12]	Xu Bu-Yi, Chen Jun-Rong, Cai Jing, Li Quan, Zhao Ke-Qing. Theoretical study on the structure，spectra and thermodynamic property of 2-（toluene-4-sulfonylamino）-benzoic. Acta Physica Sinica, 2009, 58(3): 1531-1536. doi: 10.7498/aps.58.1531
[13]	Chen Yi, Shen Jiang. Structural and thermodynamic properties of Fe based compounds with NaZn13-type. Acta Physica Sinica, 2009, 58(13): 141-S145. doi: 10.7498/aps.58.141
[14]	Liu Na-Na, Song Ren-Bo, Sun Han-Ying, Du Da-Wei. The electronic structure and thermodynamic properties of Mg2Sn from first-principles calculations. Acta Physica Sinica, 2008, 57(11): 7145-7150. doi: 10.7498/aps.57.7145
[15]	Li Quan, Zhu Zheng-He. The potential energy function and thermodynamic properties of AuZn and AuAl for the ground states and low-lying excited states. Acta Physica Sinica, 2008, 57(6): 3419-3424. doi: 10.7498/aps.57.3419
[16]	Song Hai-Feng, Liu Hai-Feng. Theoretical study of thermodynamic properties of metal Be. Acta Physica Sinica, 2007, 56(5): 2833-2837. doi: 10.7498/aps.56.2833
[17]	Yuan Du-Qi. The influence of weak interaction on thermodynamic properties and the stability of imperfect Bose gas. Acta Physica Sinica, 2006, 55(4): 1634-1638. doi: 10.7498/aps.55.1634
[18]	Men Fu-Dian. Thermodynamic properties of a weakly interacting Fermi gas in weak magnetic field. Acta Physica Sinica, 2006, 55(4): 1622-1627. doi: 10.7498/aps.55.1622
[19]	Su Guo-Zhen, Chen Li-Xuan. Thermodynamic properties of a weakly interacting Fermi gas. Acta Physica Sinica, 2004, 53(4): 984-990. doi: 10.7498/aps.53.984
[20]	Zhang Ya-Nan, Yan Shi-Lei. Thermodynamic properties of random transverse mixed Ising spin system with cryst al field. Acta Physica Sinica, 2003, 52(11): 2890-2895. doi: 10.7498/aps.52.2890

Catalog

Vol. 73, Issue 4 - 2024-02-20

Metrics

Abstract views: 2985
PDF Downloads: 97
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板