平衡晶格常數(shù)及體積模量

1、平衡晶格常數(shù)及體彈模量的模擬計算（0K）一、實驗原理1.1 平衡晶格常數(shù)通過分子動力學(xué)模擬，在給定條件下，計算晶體結(jié)構(gòu)最穩(wěn)定，也就是體系能量最小時所對應(yīng)的晶格間距，即為平衡晶格常數(shù)。1.2 體彈模量在彈性變形范圍內(nèi),物體的體應(yīng)力與相應(yīng)體應(yīng)變之比的絕對值稱為體彈模量。表達式為式中，P 為體應(yīng)力或物體受到的各向均勻的壓強，dV/V為體積的相對變化。對于立方晶胞，總能量可以表示為=ME，E 為單個原子的結(jié)合能，M 為單位晶胞內(nèi)的原子數(shù)。晶胞體積可以表示為V=a3，那么壓強P為故體彈模量可以表示為根據(jù)實驗第一部分算出的平衡晶格常數(shù)，以及能量與晶格間距的函數(shù)關(guān)系，可以求得對應(yīng)晶格類型的體積模量。二、擬合

2、方法2.1 多項式擬合使用下式對計算數(shù)據(jù)進行擬合，計算系數(shù)a、b、c。平衡晶格常數(shù)即為-b/2a，二階導(dǎo)數(shù)即為2a。2.2 Birch-Murnaghan方程擬合Birch-Murnaghan方程如下通過這種方法可以直接擬合得出平衡晶格常數(shù)及體彈模量。三、操作步驟3.1 步驟及解釋 $ cp -r share/1_lattice &復(fù)制文件夾$ cd 1_lattice &依次進入包含某一元素運行文件的文件夾中$ cd Cu (or Al, Si, Fe, Mg)$ gedit in.lattice &編輯運行文件$ lmp < in.lattice &使

3、用lammps運行文件$ A.i686 a0.cfg &使用ayomeye觀察晶體結(jié)構(gòu)$ gnuplot plot.2nd.gnu (plot.bm.gnu) &擬合數(shù)據(jù)并作圖3.2 實際步驟（以Cu為例）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_lattic

4、e$ cd Cu/user022cluster Cu$ lsuser022cluster Cu$ lmp < in.latticeuser022cluster Cu$ A.i686 a0.cfguser022cluster Cu$ gnuplot plot.bm.gnu四、模擬數(shù)據(jù)4.1 Mg 多項式擬合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

5、(0.09833%) Birch-Murnaghan方程擬合HCP Lattice:E_0 = -1.52868631023185 eVV_0 = 22.76161517799 Angstrom*3B_0 = 36.0301849559129 GPa4.2 Al4.2.1 多項式擬合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 B

6、irch-Murnaghan方程擬合FCC Lattice:a_0 = 4.04527130437683 AngstromV_0 = 16.5494273213047 Angstrom*3B_0 = 77.7803912766984 GPa4.3 Si 4.3.1 多項式擬合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

7、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方程擬合4.4 Fe 4.4.1 多項式擬合Final set of parameters Asymptotic Standard Error= =a = 7.12859 +/- 0.001914 (0.02686%)b = -40.7092 +/- 0

8、.01093 (0.02686%)c = 53.9969 +/- 0.01561 (0.02891%)4.4.2 Birch-Murnaghan方程擬合BCC Lattice:a_0 = 2.85532720281661 AngstromV_0 = 11.6395892035167 Angstrom*3B_0 = 177.84840115414 GPa4.5 Cu 4.5.1 多項式擬合Final set of parameters Asymptotic Standard Error= =a = 3.48337 +/- 0.005239 (0.1504%)b = -25.1497 +/- 0.

9、03783 (0.1504%)c = 41.9049 +/- 0.06829 (0.163%)4.5.2 Birch-Murnaghan方程擬合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 多項式擬合結(jié)果及實驗數(shù)值2ndStructureMExpta0(Å)ExptB(GPa)MD a0(Å)Relative

10、error（a0）%MDB(GPa)Relative error（B）%MgHCP,4/(3𝑐/𝑎)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擬合數(shù)值及實驗數(shù)值bmStructureMExpta0(

11、97;)ExptB(GPa)MDa0(Å)Relative error（a0）%MDB (GPa)Relative error（B）%MgHCP,4/(3𝑐/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 晶格常數(shù)模擬值與實驗值的相對誤差Relative error（a0）%MgAlSiFeCu2nd0.7710.08700.3720.112bm0.7690.09200.3720.114表 4 體彈模量模擬值與實驗值的相對誤差Relative error（B）%MgAlS