平衡晶格常數(shù)及體積模量

1、平衡品格常數(shù)及體彈模量的模擬計算（0K）一、實驗原理平衡晶格常數(shù)通過分子動力學模擬，在給定條件下，計算晶體結構最穩(wěn)定，也就是體系能量最小時所對應的晶格間距，即為平衡晶格常數(shù)。體彈模量在彈性變形范圍內,物體的體應力與相應體應變之比的絕對值稱為體彈模量。表達式為dP一研式中，p為體應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為單個原子的結合能，M為單位晶胞內的原子數(shù)。晶胞體積可以表示為丫=2八3,那么壓強P為deMdEdV3ada故體彈模量可以表示為_MdzEB9出da2工口根據(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)關系，可以

2、求得對應晶格類型的體積模量。二、擬合方法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,二階導數(shù)即為2a。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$cp-rshare/1_lattice&復制文件夾$cd1_lattice&依次進入包含某一元素運行文件的文件夾中$cdCu(orAl,Si,Fe,Mg)$geditin.lattice&編輯運行文件$Impin.lattice&使用lammps運行文件$A.i686aO.cfg&使用ayomey

3、e觀察晶體結構$gnuplotplot.2nd.gnu(plot.bm.gnu)&擬合數(shù)據(jù)并作圖3.2實際步驟(以Cu為例)user022cluster$cp-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、模擬數(shù)據(jù)4.1Mg4.1Mg4.1.1多項式擬合-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況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方程擬合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多項式擬合-3.4L06-3+4106-3+41(：-74+0254j：34.煙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方程擬合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多項式擬合yaluNe/atonyaluNe/atonLatticespacingandenergy-4+33E4Latticespacing(flrgatroni)5+4L55+4255+4355+445-4+33G7I5+41FinalsetofparametersAsymptoticStandardErrora=1.93485b=-21.0163c=52.73314.3.2Birch-Murnaghan方程擬合+/-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多項式擬合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方程擬合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多項式擬合結果及實驗數(shù)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擬合數(shù)值及實驗數(shù)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晶格常數(shù)模擬值與實驗值的相對誤差Relativeerror(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