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Atoms in crystals will generate thermal diffuse scattering during thermal vibration. Thermal diffuse scattering analysis has great potential applications in condensed matter physics and material science research. Scandium oxide (Sc2O3) has unique physical and chemical properties, which make it have high research and application value. In this work, X-ray diffraction experiment is performed on Sc2O3 at room temperature of 26 ℃. The thermal diffuse scattering intensity exhibits a clear vibrational shape. The full diffraction back-based intensity equation of Sc2O3 is expanded, and the theoretical value of the thermal diffuse scattering intensity is calculated until the full diffraction back-based intensity spectrum of the 14th nearest atom (r = 0.3816 nm) is calculated. By fitting the theoretical value to the experimental value, we can see the inter-atomic thermal vibration correlation effect μ values corresponding to the nearest neighbor atom to the 7th nearest neighbor atom, the values of distance r from the nearest neighbor atom to the 7th nearest neighbor atom are 0.2067, 0.2148, 0.2161, 0.2671, 0.2945, 0.3229 and 0.3265nm, respectively, corresponding to their inter-atomic thermal vibration correlation effect μ values of 0.64, 0.63, 0.62, 0.61, 0.60, 0.58 and 0.57. Research result shows that the intensity of thermal diffuse scattering in Sc2O3 is closely related to the atomic thermal vibration, the most significant influence on the vibration shape of thermal diffuse scattering intensity is the thermal vibration correlation effect between the 7th nearest atom Sc1-Sc2. Inter-atomic thermal vibration correlation effect μ values will provide important parameters for studying the mechanical and thermal properties of materials, laying the foundation for the next-step calculating specific heat and interatomic force constant, and thus playing a crucial role in the use and development of materials.
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Keywords:
- Sc2O3 /
- thermal diffuse scattering /
- inter-atomic thermal vibration correlation effect values
[1] Kulikov B P, Baranov V N, Bezrukikh A I, Deev V B, Motkov M M 2018 Metallurgist 61 1115
Google Scholar
[2] 唐冲冲, 常化强, 包晓刚, 刘贵清, 刘奎仁 2012 中国稀土学报 30 680
Tang C C, Chang H Q, Bao X G, Liu G Q, Liu K R 2012 CJCR 30 680
[3] Masanori H, Kouji Y, Hiromasa Y, Toru H O 2008 J. Alloys Compd. 474 124
Google Scholar
[4] Zhigachev A O, Rodaev V V, Zhigacheva D V, Lyskov N V, Shchukina M A 2021 Ceram. Int. 47 32490
Google Scholar
[5] Pu Y C, Li S R, Yan S, Huang X, Wang D, Ye Y Y, Liu Y Q 2019 Fuel 241 607
Google Scholar
[6] Nakazono K, Takahashi R, Yamada Y, Sato S 2021 Mol. Catal. 516 111996
Google Scholar
[7] Sha H Y, He Z Z, Li C, Wang X Y, Jiang Q, Zeng F M, Su Z M 2019 Opt. Mater. 93 39
Google Scholar
[8] Xue J M, Li F, Liu Y Q, Yang F, Hou Z X 2023 Appl. Surf. Sci. 613 155984
Google Scholar
[9] Zhang C, Zhou Z X, Tang Z M, Ballo D, Wang C, Jian G 2022 J. Alloys Compd. 889 161622
Google Scholar
[10] Poirot N, Bregiroux D, Boy P, Autret-Lambert C, Belleville P, Bianchi L 2015 Ceram. Int. 41 3879
Google Scholar
[11] Toshiyuki M T I J L 2005 J. Am. Ceram. Soc. 88 817
Google Scholar
[12] Jiang B X, Hu C, Li J, Kou H, Shi Y, Liu W B, Pan Y B 2011 J. Rare Earths 29 951
Google Scholar
[13] Lu S Z, Yang Q H 2012 Chin. Phys. B 21 047801
Google Scholar
[14] Lu X, Jiang B X, Li J, Liu W B, Wang L, Ba X B, Hu C, Liu B L, Pan Y B 2013 Ceram. Int. 39 4695
Google Scholar
[15] Ma M Z, Dong L L, Jing W, Xu T, Kang B, Hou F 2019 Proceedings of the 11th International Conference on High-Performance Ceramics Kunming, China, May 25–29, 2019 p012080
[16] Dai Z F, Liu Q, Hrenia D, Dai J W, Wang W, Li J 2018 Opt. Mater. 75 673
Google Scholar
[17] Wang Y, Sun X D, Qiu G M 2007 J. Rare Earths 25 68
Google Scholar
[18] Jacobsohn L G, Serivalsatit K, Quarles C A, Ballato J 2015 J. Mater. Sci. 50 3183
Google Scholar
[19] 杨丛松, 陈博, 王芳, 郑剑平, 赵建翔 2017 稀有金属 41 163
Google Scholar
Yang C S, Chen B, Wang F, Zheng J P, Zhao J X 2017 Chin. Rare Metals 41 163
Google Scholar
[20] Kong P F, Pu Y T, Ma P, Zhu J L 2020 Thin Solid Films 714 138357
Google Scholar
[21] Sakuma T, Shimoyama T, Basar K, Xianglian, Takahashi H, Arai M, Ishii Y 2005 Solid State Ion 176 2689
Google Scholar
[22] Arai M, Sakuma T 2001 J. Phys. Soc. Jpn. 70 144
Google Scholar
[23] Beni G, Platzman P M 1976 Phys. Rev. B 14 1514
[24] Basar K, Xianglian, Sakuma T, Takahashi H, Igawa N 2009 ITB J. Sci. (Bdg.) 41 50
Google Scholar
[25] Sakuma T, Makhsun, Sakai R, Xianglian, Takahashi H, Basar K, Igawa N, Sergey A D 2015 AIP Conf. Proc. 1656 020002
Google Scholar
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Google Scholar
[27] Sakuma T, Mohapatra S R, Uehara H, Sakai R, Xianglian, Takahashi H, Igawa N, Basar K 2011 Atom Indonesia 36 121
Google Scholar
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Google Scholar
[29] Sakuma T, Xianglian, Siagian S, Basar K, Takahashi H, Igawa N, Kamishima O 2010 J. Therm. Anal. Calorim. 99 173
Google Scholar
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Google Scholar
[31] 香莲, 赵敏兰, 佐久间隆, 井川直樹 2015 原子与分子物理学报 32 499
Google Scholar
Xianglian, Zhao M L, Sakuma T, Igawa N 2015 J. at. Mol. Sci. 32 499
Google Scholar
[32] Xianglian, Sakuma T, Mohapatra S R, Uehara H, Takahashi H, Kamishima O, Igawa N 2012 Mol. Simul. 38 448
Google Scholar
[33] Xianglian, Basar K, Honda H, Siagian S, Ohara K, Sakuma T, Takahashi H, Igawa N, Ishii Y 2007 Solid State Ion 179 776
Google Scholar
[34] 郭田田, 香莲, 包文秀, 包桂芝 2018 光散射学报 30 182
Google Scholar
Guo T T, Xianglian, Bao W X, Bao G Z 2018 J. Light Scatter. 30 182
Google Scholar
[35] Sakuma T 1992 J. Phys. Soc. Jpn. 61 4041
Google Scholar
[36] 刘泽朋, 王瑞刚, 香莲, 包桂芝 2023 内蒙古民族大学学报(自然科学版) 38 199
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 (a) Sc2O3的X射线衍射实验图谱; (b) Sc2O3的热漫散射强度图谱
Figure 1. (a) X-ray diffraction experimental pattern of Sc2O3; (b) thermal diffuse scattering intensity spectrum of Sc2O3.
图 2 (a)原子独立振动产生的热漫散射强度; (b) Sc2O3的Compton散射强度; (c) Sc2O3全衍射背底强度
Figure 2. (a) Thermal diffuse scattering intensity generated by independent vibration each atom; (b) Compton scattering intensity of Sc2O3; (c) Sc2O3 full diffraction back-base intensity.
图 3 Sc2O3的全衍射背底强度的计算结果和实验结果的对比 (a)最近邻原子; (b)第1—3近邻原子; (c)第1—5近邻原子; (d) 第1—7近邻原子; (e)第1—9近邻原子; (f)第1—11近邻原子
Figure 3. Comparison between calculated and experimental results of the total diffraction back-base intensity of Sc2O3: (a) The nearest neighbor atom; (b) the 1–3 nearest neighbor atomic; (c) the 1–5 nearest neighbor atomic; (d) the 1–7 nearest neighbor atomic; (e) the 1–9 nearest neighbor atom; (f) the 1–11 nearest neighbor atom.
图 4 Sc2O3晶体结构模型和原子间距离
Figure 4. Sc2O3 crystal structure model and interatomic distance.
图 5 Sc2O3粉末晶体的各原子间相关效应产生的热漫散射强度
Figure 5. Thermal diffuse scattering intensity generated by the inter atomic correlation effect of Sc2O3 powder crystal.
图 6 Sc2O3在室温26 ℃下原子间热振动相关效应值μ与原子间距离r的图谱, 以及与其他几种材料的的对比
Figure 6. Spectrum of the inter-atomic thermal vibration correlation effect values μ and the inter-atomic distance r of Sc2O3 at room temperature of 26 ℃. The corresponding spectra of other materials are also given.
参数 物理意义 参数 物理意义 $k$ 仪器参数 $ {N_0} $ 晶体内晶胞数 ${u_i}$ 单位晶胞内i原子数 ${Z_{{r_{s\left( i \right)s'\left( j \right)}}}}$ 配位数 ${f_i}$ i原子散射因子 ${r_{s\left( i \right)s'\left( j \right)}}$ 晶胞内i原子与j原子间距 ${B_i}$ i原子各向同性温度因子 ${\sigma _{{\text{incoh}}}}$ 非干涉性的原子散射截面 Ie 单个电子散射强度 θ 散射角 λ 入射光线波长 
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表 2 Sc2O3的晶体结构参数
Table 2. Crystal structure parameters of Sc2O3.
x y z B/nm2 Sc1 0.2500 0.2500 0.2500 0.003329 Sc2 0.4649 0 0.2500 0.011085 O 0.3928 0.1528 0.3802 0.009657
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表 3 Sc2O3的原子间热振动相关效应值μ
Table 3. Interatomic thermal vibration related effect values of Sc2O3.
原子间距离r/nm 配位数Z 原子间相关效应值$ \mu $ O-Sc2 0.2067 2 0.64 O-Sc2 0.2148 1 0.63 O-Sc1 0.2161 1 0.62 O-O 0.2671 4 0.61 O-O 0.2945 1 0.60 O-O 0.3229 2 0.58 Sc1-Sc2 0.3265 6 0.57
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[1] Kulikov B P, Baranov V N, Bezrukikh A I, Deev V B, Motkov M M 2018 Metallurgist 61 1115
Google Scholar
[2] 唐冲冲, 常化强, 包晓刚, 刘贵清, 刘奎仁 2012 中国稀土学报 30 680
Tang C C, Chang H Q, Bao X G, Liu G Q, Liu K R 2012 CJCR 30 680
[3] Masanori H, Kouji Y, Hiromasa Y, Toru H O 2008 J. Alloys Compd. 474 124
Google Scholar
[4] Zhigachev A O, Rodaev V V, Zhigacheva D V, Lyskov N V, Shchukina M A 2021 Ceram. Int. 47 32490
Google Scholar
[5] Pu Y C, Li S R, Yan S, Huang X, Wang D, Ye Y Y, Liu Y Q 2019 Fuel 241 607
Google Scholar
[6] Nakazono K, Takahashi R, Yamada Y, Sato S 2021 Mol. Catal. 516 111996
Google Scholar
[7] Sha H Y, He Z Z, Li C, Wang X Y, Jiang Q, Zeng F M, Su Z M 2019 Opt. Mater. 93 39
Google Scholar
[8] Xue J M, Li F, Liu Y Q, Yang F, Hou Z X 2023 Appl. Surf. Sci. 613 155984
Google Scholar
[9] Zhang C, Zhou Z X, Tang Z M, Ballo D, Wang C, Jian G 2022 J. Alloys Compd. 889 161622
Google Scholar
[10] Poirot N, Bregiroux D, Boy P, Autret-Lambert C, Belleville P, Bianchi L 2015 Ceram. Int. 41 3879
Google Scholar
[11] Toshiyuki M T I J L 2005 J. Am. Ceram. Soc. 88 817
Google Scholar
[12] Jiang B X, Hu C, Li J, Kou H, Shi Y, Liu W B, Pan Y B 2011 J. Rare Earths 29 951
Google Scholar
[13] Lu S Z, Yang Q H 2012 Chin. Phys. B 21 047801
Google Scholar
[14] Lu X, Jiang B X, Li J, Liu W B, Wang L, Ba X B, Hu C, Liu B L, Pan Y B 2013 Ceram. Int. 39 4695
Google Scholar
[15] Ma M Z, Dong L L, Jing W, Xu T, Kang B, Hou F 2019 Proceedings of the 11th International Conference on High-Performance Ceramics Kunming, China, May 25–29, 2019 p012080
[16] Dai Z F, Liu Q, Hrenia D, Dai J W, Wang W, Li J 2018 Opt. Mater. 75 673
Google Scholar
[17] Wang Y, Sun X D, Qiu G M 2007 J. Rare Earths 25 68
Google Scholar
[18] Jacobsohn L G, Serivalsatit K, Quarles C A, Ballato J 2015 J. Mater. Sci. 50 3183
Google Scholar
[19] 杨丛松, 陈博, 王芳, 郑剑平, 赵建翔 2017 稀有金属 41 163
Google Scholar
Yang C S, Chen B, Wang F, Zheng J P, Zhao J X 2017 Chin. Rare Metals 41 163
Google Scholar
[20] Kong P F, Pu Y T, Ma P, Zhu J L 2020 Thin Solid Films 714 138357
Google Scholar
[21] Sakuma T, Shimoyama T, Basar K, Xianglian, Takahashi H, Arai M, Ishii Y 2005 Solid State Ion 176 2689
Google Scholar
[22] Arai M, Sakuma T 2001 J. Phys. Soc. Jpn. 70 144
Google Scholar
[23] Beni G, Platzman P M 1976 Phys. Rev. B 14 1514
[24] Basar K, Xianglian, Sakuma T, Takahashi H, Igawa N 2009 ITB J. Sci. (Bdg.) 41 50
Google Scholar
[25] Sakuma T, Makhsun, Sakai R, Xianglian, Takahashi H, Basar K, Igawa N, Sergey A D 2015 AIP Conf. Proc. 1656 020002
Google Scholar
[26] Wada T, Sakuma T, Sakai R, Uehara H, Xianglian, Takahashi H, Kamishima O, Igawa N, Sergey A D 2012 Solid State Ion 225 18
Google Scholar
[27] Sakuma T, Mohapatra S R, Uehara H, Sakai R, Xianglian, Takahashi H, Igawa N, Basar K 2011 Atom Indonesia 36 121
Google Scholar
[28] Sakuma T, Xianglian, Shimizu N, Mohapatra S R, Isozaki N, Uehara H, Takahashi H, Basar K, Igawa N, Kamishima O 2010 Solid State Ion 192 54
Google Scholar
[29] Sakuma T, Xianglian, Siagian S, Basar K, Takahashi H, Igawa N, Kamishima O 2010 J. Therm. Anal. Calorim. 99 173
Google Scholar
[30] Basar K, Siagian S, Xianglian, Sakuma T, Takahashi H, Igawa N 2008 Nucl. Instrum. Methods Phys. Res. A 600 237
Google Scholar
[31] 香莲, 赵敏兰, 佐久间隆, 井川直樹 2015 原子与分子物理学报 32 499
Google Scholar
Xianglian, Zhao M L, Sakuma T, Igawa N 2015 J. at. Mol. Sci. 32 499
Google Scholar
[32] Xianglian, Sakuma T, Mohapatra S R, Uehara H, Takahashi H, Kamishima O, Igawa N 2012 Mol. Simul. 38 448
Google Scholar
[33] Xianglian, Basar K, Honda H, Siagian S, Ohara K, Sakuma T, Takahashi H, Igawa N, Ishii Y 2007 Solid State Ion 179 776
Google Scholar
[34] 郭田田, 香莲, 包文秀, 包桂芝 2018 光散射学报 30 182
Google Scholar
Guo T T, Xianglian, Bao W X, Bao G Z 2018 J. Light Scatter. 30 182
Google Scholar
[35] Sakuma T 1992 J. Phys. Soc. Jpn. 61 4041
Google Scholar
[36] 刘泽朋, 王瑞刚, 香莲, 包桂芝 2023 内蒙古民族大学学报(自然科学版) 38 199
Google Scholar
Liu Z P, Wang R G, Xianglian, Bao G Z 2023 J. Inner Mongolia Minzu Univ. (Nat. Sci.) 38 199
Google Scholar
[37] Rietveld H M 1967 Acta Cryst. 22 151
Google Scholar
[38] Izumi F, Ikeda T 2000 Mater. Sci. Forum 399 198
Google Scholar
[39] Lonsdale K 1962 International Tables for X-Ray Crystallography (Vol. III) (United Kingdo: Published by International Union of Crystallography) pp72–103
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