平衡晶格常数及体积模量

1、平衡晶格常数及体积模量平衡晶格常数及体弹模量的模拟计算(0K)一、实验原理1.1平衡晶格常数通过分子动力学模拟，在给定条件下，计算晶体结构最稳定，也就是体系 能量最小时所对应的晶格间距，即为平衡晶格常数。2体弹模量.在弹性变形范围内,物体的体应力与相应体应变之比的绝对值称为体弹模 量。表达式为dP人一研式中，P为体应力或物体受到的各向均匀的压强，dV/V为体积的相对变化。 对于立方晶胞，总能量可以表示为£ =ME, E为单个原子的结合能，M为单位晶 胞内的原子数。晶胞体积可以表示为V=a'3,那么压强P为de M dE p dV 3a2 da故体弹模量可以表示为_ M d2E

2、“ 9屋 da2 0根据实验第一部分算出的平衡晶格常数，以及能量与晶格间距的函数关系, 可以求得对应晶格类型的体积模量。二、拟合方法2.1 多项式拟合使用下式对计算数据进行拟合，计算系数a、b、c。fO) = ax2 +2为 +(;平衡晶格常数即为-b/2a,二阶导数即为2ao2. 2 Birch-Murnaghan 方程拟合B i rch-Murnaghan 方程如下-1)曰("_ i)通过这种方法可以直接拟合得出平衡晶格常数及体弹模量。三、操作步骤3. 1步骤及解释$ cp -r share/l_lattice &复制文件夹$ cd l_lattice&依次进入包

3、含某一元素运行文件的文件夹中$ cd Cu (or Al, Si, Fe, Mg)$ gedit in. lattice&编辑运行文件$ Imp < in. lattice&使用 lammps 运行文件$ A. 1686 aO. cfg&使用ayomeye观察晶体结构$ gnuplot plot. 2nd. gnu (plot> bm. gnu) &拟合数据并作图3.2实际步骤(以史为例)user022$cluster $ cp -r share/l_lattice/ user0226cluster Is l_lattice 2_point bin

4、Desktop shareuser022©cluster cd l_lattice/userO220cluster l_lattice$ IsAl Cu Fe Mg SiuserO220clustercd Cu/user0226cluster Cu$ Isin. lattice jin_copper_lammps. setfl plot. 2nd. gnu plot. bm. gnuuser022©cluster Cu$ Imp < in. latticeuser022cluster Cu$ A. i686 aO. cfguser022©cluster Cu

5、!$ gnuplot plot, 2nd. gnu user0226cluster Cu$ gnuplot plot. bm. gnu四、模拟数据4.1 Mg4.1.1多项式拟合-1.6277-1.5278-L5279-14528-1.5281.-1.52824-5283T.5284-1.5285-1.5286-1.5287Lattice speclng axi energy5,侬 5,173J753.183,1853r193J953,23.201Lattice gc，ng (仔脚竹。冷Final set of parametersAsymptotic Standard Error+/- 0.

6、 002085+/- 0. 01328+/- 0. 02114(0. 09179%)(0. 0918%)(0. 09833%)a=2. 27117b=-14. 464c=21, 49974.1. 2 Birch-Mumaghan 方程拟合Equation of Stoic of HCPT.KJTT-1.S278-1.5279-L528-1,5281-1.S292-1.883T.融4-1.5285-1.5227心 C Birch FitHCP Lattice:E_0-1.52868631023185 eV1: /XX22.322/2522.622-?22.222.92323.123JVolune

7、/aton4.2 Al4. 2.1多项式拟合Lattice spacing end energy4.34.0454.C54.0554.064.0第4j)7Lattice spacing (Angstron)4.0254.034妖-3.4C63-3.4039-331-3.4101T.4IQ2-5,4105T.41M-3.4105-3.4106-3.4107Final set of parametersAsymptotic Standard Error+/- 0.005565+/- 0. 04503+/- 0. 09108(0. 252%)(0. 252%)(0. 2783%)a=2. 20808

8、b=-17. 8654c=32. 72614. 2. 2 Birch-Murnaghan 方程拟合Equation of Stte of FCC16.316.2516,416 J5 16.516电LB.G 16.65 LGJ 16.75LG.JVolurre/aton (R"5)=3三SP'I&UUJFCC IE_0 = a_0 = v_o =B_0 = B 0'=Lattice spring and energyFinal set of parametersAsymptotic Standard Error=1. 93485=-21. 0163=52. 73

9、31+/- 0.001141+/- 0.01239+/- 0.03365(0.05897%)(0. 05896%)(0. 06381%)4. 3. 2 Birch-Mumaghan 方程拟合Equation oF State oF diamnd4.3 Si4. 3.1多项式拟合-4.3S3-4.3S3-4.335YAI-4.3ZE2-4.33E34造4Y.礴T.zes-4 4S3G7 11*1115.415J155J25.4255.435.4355X45.4455,机-4 .0 5g-4.3359-4.336Bianond ©Birch Fit-4.3361-4.3362-4.356

10、3-4.3364-4.5365-4 .召 66Gw) 8-<6，gl5diamond Lattice:E_0=,-4.33660000718975 eV19. G519.919.952020.0520.1Volune/aton (白人3)20.1520.22D.2Lattice spacing (firQstron)4.4 Fe4. 4.1多项式拟合suIXSQfstwls-4.1195-4.12-4.1205-4,121-4.1215-4,122-4.1225Lattice spHcinm and Bnerg”£352,842.8452 历2.8552.862.8652,87

11、2.87Lattice spacing (firestrcm)Final set of parametersAsymptotic Standard Errora=7. 12859b=-40. 7092c=53. 99694. 4. 2 Birch-Murnaghan 方程拟合+/- 0.001914+/- 0. 01093+/- 0.01561(0. 02686%)(0. 02686%)(0.02891%)-4.1195-4.1205tquation of State oF BCLA3 ws堂 6J2UJY.121-4.1215B11iIIlliECC dBirch fit fBCC Latt

12、ice:§E_0 = -4.1224351934112 eV/a_0 = 2. 85532720281661-4.122-4.122511.411.4511.5 ILES ll.G 11.65 LI .711.7511.8U.93 USVolune/aiom4.5 Cu4. 5.1多项式拟合3.595.5S53.65-6O53.613.6153.623.6253.63Lattice spacing (firQstron)Lattice spacing and snor9y-3.0893) W0JJW6G-WUUJ-3,4902+/- 0. 005239+/- 0. 03783+/-

13、0. 06829(0. 1504%)(0.1504%)(0. 163%)Final set of parametersAsymptotic Standard Errora=3. 48337b=-25.1497c=41. 90494. 5. 2 Birch-Mumaghan 方程拟合-3.498G>>)三3B*6,<nLUJEquation of State of FiCFCC Lattice:-3.4888-3J8AE 0-3.4892-3.4894-3.489S-3.4898-3,49-3.4902-3. 49001356518869F声Birch fitLI.8Volum

14、c/otom (AA3)表1多项式拟合结果及实验数值2ndStruc tureMExpt a。(A )Expt B(GPa )MD a0(A )Relati ve error (a0) %MD B(G Pa)Relati ve error (B) %MgHCP, 4/ (V»c/q3. 20Q353. 18 A0. 77136.2. 925bmSti-uc tureMExpt 噌 1.)62 3Expt B(GPa )能 窃 1.)62 8Relati ve errorB (GP a)Relati ve errorA 1A f AA f Ax vvQ / 7vrjryD 7 70Al

15、FCC, 44. 049724. 0450. 08777.7347. 964SiDiamo nd, 85. 431985. 4310101.4763. 547FeBCC, 22. 8661682. 8550. 372177.7785. 820CuFCC, 43.6141423.6100. 112137,4233. 223表2 bm拟合数值及实验数值MgHCP, 4/ (V3*c/a)3. 20 9, c/a=1.623353. 18 4 , c/a=1.62 80. 76936.0302. 943AlFCC, 44. 049724. 0450. 09277.7808. 028SiDiamo nd, 85. 431985. 4310101.4253. 495FeBCC, 22. 8661682. 8550. 372177.8485. 862CuFCC, 43. 6141423.61 00. 114137.6313. 077表3晶格常数模拟值与实验值的相对误差Relative error (a0) %MgAlSiFeCu2nd0. 7710. 08700. 3720. 112bm0. 7690. 0920