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。1.2 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=a3，那么压强P为故体弹模量可以表示为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。平衡晶格常数即为-b/2a，二阶导数即为2a。2.2 Birch-Murnaghan方程拟合Birch-Murnaghan方程如下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/1_lattice &复制文件夹$ cd 1_lattice &依次进入包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$ lmp in.lattice &使用lammps运行文件$ A.i686 a0.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Cu为例）user022cluster $ cp -r share/1_lattice/ user022cluster $ ls1_lattice 2_point bin Desktop shareuser022cluster $ cd 1_lattice/user022cluster 1_lattice$ lsAl Cu Fe Mg Siuser022cluster 1_lattice$ cd Cu/user022cluster Cu$ lsin.lattice jin_copper_lammps.setfl plot.2nd.gnu plot.bm.gnuuser022cluster Cu$ lmp in.latticeuser022cluster Cu$ A.i686 a0.cfguser022cluster Cu$ gnuplot plot.2nd.gnuuser022cluster Cu$ gnuplot plot.bm.gnu四、模拟数据4.1 Mg 4.1.1 多项式拟合Final set of parameters Asymptotic Standard Error= =a = 2.27117 +/- 0.002085 (0.09179%)b = -14.464 +/- 0.01328 (0.0918%)c = 21.4997 +/- 0.02114 (0.09833%)4.1.2 Birch-Murnaghan方程拟合HCP Lattice:E_0 = -1.52868631023185 eVa_0 = 3.18431542679217 AngstromV_0 = 22.76161517799 Angstrom*3B_0 = 36.0301849559129 GPaB_0= -0.7614642075025434.2 Al4.2.1 多项式拟合Final set of parameters Asymptotic Standard Error= =a = 2.20808 +/- 0.005565 (0.252%)b = -17.8654 +/- 0.04503 (0.252%)c = 32.7261 +/- 0.09108 (0.2783%)4.2.2 Birch-Murnaghan方程拟合FCC Lattice:E_0 = -3.41065714040381 eVa_0 = 4.04527130437683 AngstromV_0 = 16.5494273213047 Angstrom*3B_0 = 77.7803912766984 GPaB_0= 7.110798815569124.3 Si 4.3.1 多项式拟合Final set of parameters Asymptotic Standard Error= =a = 1.93485 +/- 0.001141 (0.05897%)b = -21.0163 +/- 0.01239 (0.05896%)diamond Lattice:E_0 = -4.33660000718975 eVa_0 = 5.43095170306466 AngstromV_0 = 20.0234005455525 Angstrom*3B_0 = 101.425444944596 GPaB_0= 2.85073370817905c = 52.7331 +/- 0.03365 (0.06381%)4.3.2 Birch-Murnaghan方程拟合4.4 Fe 4.4.1 多项式拟合Final set of parameters Asymptotic Standard Error= =a = 7.12859 +/- 0.001914 (0.02686%)b = -40.7092 +/- 0.01093 (0.02686%)c = 53.9969 +/- 0.01561 (0.02891%)4.4.2 Birch-Murnaghan方程拟合BCC Lattice:E_0 = -4.1224351934112 eVa_0 = 2.85532720281661 AngstromV_0 = 11.6395892035167 Angstrom*3B_0 = 177.84840115414 GPaB_0= 1.418948577322464.5 Cu 4.5.1 多项式拟合Final set of parameters Asymptotic Standard Error= =a = 3.48337 +/- 0.005239 (0.1504%)b = -25.1497 +/- 0.03783 (0.1504%)c = 41.9049 +/- 0.06829 (0.163%)4.5.2 Birch-Murnaghan方程拟合FCC Lattice:E_0 = -3.49001356518869 eVa_0 = 3.6098665239098 AngstromV_0 = 11.7601656929196 Angstrom*3B_0 = 137.631357881733 GPaB_0= 4.22407949095044表 1 多项式拟合结果及实验数值2ndStructureMExpta0()ExptB(GPa)MD a0()Relative error（a0）%MDB(GPa)Relative error（B）%MgHCP,4/(3/)3.209, c/a=1.623353.184,c/a=1.6280.77136.0242.925AlFCC, 44.049724.0450.08777.7347.964SiDiamond,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bmStructureMExpta0()ExptB(GPa)MDa0()Relative error（a0）%MDB (GPa)Relative error（B）%MgHCP,4/(3/)3.209, c/a=1.623353.184 ,c/a=1.6280.76936.0302.943AlFCC, 44.049724.0450.09277.7808.028SiDiamond,85.431985.4310101.4253.495FeBCC, 22.8661682.8550.372177.8485.862CuFCC, 43.6141423.6100.114137.6313.077表 3 晶格常数模拟值与实验值的相对误差Relative error（a0）%MgAlSiFeCu2nd0.7710.08700.3720.112bm0.7690.09200.3720.114表 4 体弹模量模拟值与实验值的相对误差Relative error（