平衡晶格常数及体积模量

1、平衡品格常数及体弹模量的模拟计算（0K）一、实验原理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。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丫=2八3,那么压强P为deMdEdV3ada故体弹模量可以表示为_MdzEB9出da2工口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

2、求得对应晶格类型的体积模量。二、拟合方法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。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$cp-rshare/1_lattice&复制文件夹$cd1_lattice&依次进入包含某一元素运行文件的文件夹中$cdCu(orAl,Si,Fe,Mg)$geditin.lattice&编辑运行文件$Impin.lattice&使用lammps运行文件$A.i686aO.cfg&使用ayomey

3、e观察晶体结构$gnuplotplot.2nd.gnu(plot.bm.gnu)&拟合数据并作图3.2实际步骤(以Cu为例)user022cluster$cp-rshare/1_lattice/user022cluster$ls1_lattice2_pointbinDesktopshareuser022cluster$cd1_lattice/user022cluster1_lattice$lsAlCuFeMgSiuser022cluster1_lattice$cdCu/user022clusterCu$lsin.latticejin_copper_lammps.setflplot.2nd.gn

4、uplot.bm.gnuuser022clusterCu$lmpin.latticeuser022clusterCu$A.i686a0.cfguser022clusterCu$gnuplotplot.2nd.gnuuser022clusterCu$gnuplotplot.bm.gnu四、模拟数据4.1Mg4.1Mg4.1.1多项式拟合-l+5277-l+5278-1.5273-l+528Latticesjztcingandenergy-1+52SL-1,5282-1.5263FinalsetofparametersAsymptoticStandardErrorT晟T晟TT-1x5270-l+5

5、279-1x526-1+5281-1.5282-l+5283-lr52S5-1.520G-lr52S722+S22+422.522+G22.722+822.923况123Ja=2.27117+/-0.002085(0.09179%)b=-14.464+/-0.01328(0.0918%)c=21.4997+/-0.02114(0.09833%)4.1.2Birch-Murnaghan方程拟合EquationofStateofHCPfcPaBirdiFitHCPLattice:E_0=-1.52868631023185eVa_0=3.18431542679217AngstromV_0=22.76

6、161517799Angstrom*3口B_0=36.0301849559129GPacB_0=-0.7614642075025434.2Al4.2.14.2.1多项式拟合-3.4L06-3+4106-3+41(：-74+0254j：34.烟4.M41454+C6丸。55464+0054.07Latticespacing(Angstrom)-3+40K-3+40Q9-3+41-3+4101-3+41Q2-3+41Q3-3t4LML-att-iccspacing己ndenergy口JJFFi6LI_LIJFinalsetofparametersAsymptoticStandardError2.2

7、0808-17.86542.20808-17.865432.7261(0.252%)(0.252%)(0.2783%)+/-0.005565+/-0.04503+/-0.091084.2.2Birch-Murnaghan方程拟合Lattice:-3.41065714040381eV-3,41.02-3.4郎4.0452713043768316.549427321304777.78039127669847.11079881556912Equationo-FStateafFCC三口荻fg3ulu-3.4L01T.4LMFCCE_0a_0V_0B_0B_0AngstromAngstrom*3GPa4

8、.3Si4.3Si4.3.1多项式拟合yaluNe/atonyaluNe/atonLatticespacingandenergy-4+33E4Latticespacing(flrgatroni)5+4L55+4255+4355+445-4+33G7I5+41FinalsetofparametersAsymptoticStandardErrora=1.93485b=-21.0163c=52.73314.3.2Birch-Murnaghan方程拟合+/-0.001141(0.05897%)+/-0.01239(0.05896%)+/-0.03365(0.06381%)EquationofGtite

9、oFdi。3nd-4.3358-4.3353-4t336-4.3361-4,3362-4H3363-4H3364-4.3365-4.3366diamondLattice:E_0=-4.33660000718975eVa_0=5.43095170306466AngstromIBlanondBirchfit-V_0=20.0234005455525Angstrom*3B_0=101.425444944596GPaB_0=2.8507337081790519+815,8515tg19t95202Ot0S20tl20+152Ot220+34.4Fe4.4Fe4.4.1多项式拟合Latt-icaspci

10、ngandenergyLatticesp.aclna(flngstrcfii)FinalsetofparametersAsymptoticStandardError5-3McqerREJSLL+/-0.001914+/-0.01093+/-0.001914+/-0.01093+/-0.01561(0.02686%)(0.02686%)(0.02891%)a=7.12859b=-40.7092c=53.99694.4.2Birch-Murnaghan方程拟合bquation+ttate口卜BLL-4+1195ECCeiBirchFit-4t-4t12E0=-4.1224351934112eV-4

11、.1205a_0=2.85532720281661Angstrom-4.1205V_0=11.6395892035167Angstrom*3B0=177.84840115414GPa-1-11.5511.611.6511.71k711.811.3511.911.9512表1多项式拟合结果及实验数值2ndStructureMExptao肉ExptB(GPa)MDao(A)Relativeerror(旬)%MDB(GPa)Relativeerror(B)%MgHCP,4/(V3*c/a)3.209,c/a=1.623353.184,c/a=1.6280.77136.0242.925AlFCC,44

12、.049724.0450.08777.7347.964SiDiamond,85.431985.4310101.4763.547FeBCC,22.8661682.8550.372177.7785.820CuFCC,43.6141423.6100.112137.4233.223表2bm拟合数值及实验数值bmStructureMExptao(A)ExptB(GPa)MDao(A)Relativeerror(a0)%MDB(GPa)Relativeerror(B)%MgHCR4/(V3*c/a)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Relativeerror(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