
Citation: | LING Xuyu, LIU Fusheng, WANG Yigao. Influence of Initial Porosity on Shock Chemical Reaction of Nibium-Silicon Powder Mixture[J]. Chinese Journal of High Pressure Physics, 2020, 34(3): 034101. doi: 10.11858/gywlxb.20190851 |
稀土六硼化物LaB6具有较低的逸出功和低蒸发率等优异的电子发射性能,同时还具有熔点高、化学稳定性强、硬度高等特点,因此自Lafferty发现LaB6的热发射性能以来一直是电子材料及器件领域的研究热点[1-3]。人们对LaB6晶体材料的制备和表征及其多元硼化物开展了较多的研究工作,例如:张宁等[4]通过悬浮区熔法制备了高质量LaB6单晶体;包黎红等[5]制备了La0.6Ce0.4B6材料,硬度达到2.31 GPa,1873 K下的最高发射电流密度达40.7 A/cm2;刘洪亮等[6]通过测试发现,LaB6晶体材料(100)晶面具有最佳的发射性能,最大发射电流密度在1773 K时可达40.4 A/cm2。然而,高致密度的LaB6晶体材料的烧结制备较为困难(致密度较低,80%~92%),所得材料的力学性能较差[7]。当前,关于LaB6晶体材料力学性能的理论研究报道较少,为此本研究基于密度泛函理论,结合Birch-Murnaghan物态方程,系统研究LaB6晶体材料的弹性性质及其他力学性能,以期为LaB6的深入研究和应用提供参考。
计算中,将芯电子及核视为原子核,用Vanderbilt超软势近似其对外层电子的作用,外层电子设为La(5s25p65d16s2)、B(2s2 2p1),电子波函数采用平面波基矢组[8]。电子交换关联项采用广义梯度近似法(General Gradient Approximation,GGA)中的PBE(Perdew Burke Ernzerhof)泛函近似。首先对晶格结构进行充分弛豫,在此过程中固定晶体的对称性,允许原子在3个方向上弛豫。考虑到La d电子的在位库仑相互作用,计算时将其作用项设置为2.5 eV。计算弹性常数时,每个原子的能量收敛精度设置为2×10–6 eV,最大受力收敛精度设置为0.06 eV/nm;应力-应变计算中,原子最大位移的收敛精度设置为2×10–5 nm;电子结构计算中,电子平面波矢组基矢截止能量设为240 eV,布里渊区k点的自动生成采用Monkhorst-Pack法,k点网格密度为4×4×4。
根据胡克定律,固体材料在弹性形变范围内所受的应力与应变之间符合
S=Cε(1) |
(1) |
式中:S为应力,
[S1S2S3S4S5S6]=[C11C12C13C14C15C16C21C22C23C24C25C26C31C32C33C34C35C36C41C42C43C44C45C46C51C52C53C54C55C56C61C62C63C64C65C66][ε1ε2ε3ε4ε5ε6](2) |
(2) |
由于晶体具有结构对称性(见图1),因此弹性常数张量矩阵可化为一个具有21个独立分量的矩阵[8],考虑到LaB6为立方晶系,空间群为
计算分析得到的LaB6晶体材料的晶格参数列于表1。从表1中可以看出,初始结构经过充分弛豫之后,计算得到的晶格参数与实验值之间的相对误差均小于3%,说明计算分析过程所用参数较为合理。
Method | a/nm | b/nm | c/nm | α/(°) | β/(°) | γ/(°) |
Experiment | 0.415 49 | 0.415 49 | 0.415 49 | 90 | 90 | 90 |
Calculation | 0.420 205 | 0.420 205 | 0.420 205 | 90 | 90 | 90 |
通过计算得到LaB6晶体材料的弹性常数参数C11、C12和C44分别为436.92、22.37和47.64 GPa,
C11>0,C44>0,C11>|C12|,C11+2C12>0 |
(3) |
表明本研究所用晶格结构为力学稳定的晶体结构。
分别采用Voigt法、Reuss法和Hill法,根据以下公式,计算LaB6晶体材料的体弹性模量和剪切弹性模量
BV=(C11+2C12)/3 |
(4) |
GV=(C11−C12+3C44)/5 |
(5) |
BR=(C11+2C12)/3 |
(6) |
GR=5(C11−C12)C44/[4C44+3(C11−C12)] |
(7) |
BH=(BR+BV)/2 |
(8) |
GH=(GR+GV)/2 |
(9) |
式中:B为体弹性模量,G为剪切弹性模量,下标V、R、H分别表示Voigt法、Reuss法和Hill法。计算结果列于表2,可见LaB6晶体具有较大的体弹性模量。由于LaB6晶体在a、b、c方向上具有各向同性,因此采用Voigt方法和Reuss方法计算所得体弹性模量具有相同的数值。
BV/GPa | BR/GPa | BH/GPa | GV/GPa | GR/GPa | GH/GPa |
160.55 | 160.55 | 160.55 | 111.49 | 68.85 | 90.17 |
杨氏模量E可用于衡量固体材料的刚度,其值越大,刚性越强。泊松比
采用以下公式计算LaB6晶体材料的杨氏模量E、泊松比
E=9BG/(3B+G) |
(10) |
γ=(3B−2G)/(6B+2G) |
(11) |
AB=(BV−BR)/(BV+BR) |
(12) |
AG=(GV−GR)/(GV+GR) |
(13) |
H=0.92λ−1.137G0.708 |
(14) |
λ=B/G |
(15) |
表3列出了计算分析结果,可以看出:LaB6晶体材料的杨氏模量为227.85 GPa,远大于一些金属的杨氏模量,与钢材的杨氏模量接近,表明LaB6不易发生弹性形变,刚性较强;其泊松比为0.26,表明LaB6具有一定的脆性;体剪弹性模量上限值的比值λ为1.44,小于1.75,与泊松比计算分析结果吻合;体弹性模量各向异性因子AB=0,剪切弹性模量各向异性因子AG=0.24,表明LaB6的体弹性具有各向同性,而剪切弹性具有一定的各向异性;理论硬度H达到11.56 GPa,表明LaB6抵抗剪切形变的能力较强。
E/GPa | γ | λ | AB | AG | H/GPa | vl/(km·s–1) | vt/(km·s–1) | vm/(km·s–1) |
227.85 | 0.26 | 1.44 | 0 | 0.24 | 11.56 | 7.72 | 4.38 | 4.87 |
采用以下公式[13-14]计算LaB6晶体材料的纵波弹性波速vl、剪切弹性波速vt和平均弹性波速vm
vl=√(B+43G)1ρ,vt=√Gρ,vm=[13(2v3t+1v3l)]−13 |
(16) |
计算结果如表3所示。LaB6晶体材料的3支弹性波中,有1支纵波和2支横波。从表3可以看出,LaB6晶体的纵波弹性波速(7.72 km/s)较大,剪切弹性波速(4.38 km/s)相对较小,平均波速达到4.87 km/s。在LaB6晶体材料的长波极限,声学波的纵波支速度较大,横波支速度相对较小,表明原子相对运动的振动波速较大。
计算得到的LaB6晶体材料的能带结构和分态密度如图2所示。从图2中可以看出,LaB6具有较窄的带隙,带隙宽度为0.20 eV,费米能级穿过价带,表明LaB6呈金属性。从分态密度图可以看出,曲线具有多个峰值,表明LaB6内部电子具有较强的局域性,这也是稀土La化合物特有的性质。结合能带结构和分态密度可以看出,LaB6材料价带顶的能带由p、d和s电子形成,导带底的能带由p和d电子形成,其中p态电子对价带顶和导带底的形成起最重要的作用。计算分析得到的LaB6晶体材料各轨道的电荷转移分布情况如表4所示。可见,La的s轨道和p轨道的电子转移至d轨道和B原子上,B得到La的电子,其s轨道的电子转移至p轨道,因此La呈2.53价而B呈–0.42价,表明La和B之间具有较强的共价键成分。La和B的这种结合也表明LaB6具有较高的体弹性模量、杨氏模量和硬度,同时其离子键成分使其具有一定的脆性。
Atom | Charge distribution/e | |||
s orbital | p orbital | d orbital | Total charge | |
B | 0.88 | 2.54 | 0.00 | –0.42 |
La | 1.50 | 5.46 | 1.51 | 2.53 |
基于密度泛函理论和Birch-Murnaghan物态方程,系统分析了LaB6晶体材料的弹性常数参数、体弹性模量、剪切弹性模量和力学性能。结果表明:LaB6晶体材料具有较大的
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Method | a/nm | b/nm | c/nm | α/(°) | β/(°) | γ/(°) |
Experiment | 0.415 49 | 0.415 49 | 0.415 49 | 90 | 90 | 90 |
Calculation | 0.420 205 | 0.420 205 | 0.420 205 | 90 | 90 | 90 |
BV/GPa | BR/GPa | BH/GPa | GV/GPa | GR/GPa | GH/GPa |
160.55 | 160.55 | 160.55 | 111.49 | 68.85 | 90.17 |
E/GPa | γ | λ | AB | AG | H/GPa | vl/(km·s–1) | vt/(km·s–1) | vm/(km·s–1) |
227.85 | 0.26 | 1.44 | 0 | 0.24 | 11.56 | 7.72 | 4.38 | 4.87 |
Atom | Charge distribution/e | |||
s orbital | p orbital | d orbital | Total charge | |
B | 0.88 | 2.54 | 0.00 | –0.42 |
La | 1.50 | 5.46 | 1.51 | 2.53 |
Method | a/nm | b/nm | c/nm | α/(°) | β/(°) | γ/(°) |
Experiment | 0.415 49 | 0.415 49 | 0.415 49 | 90 | 90 | 90 |
Calculation | 0.420 205 | 0.420 205 | 0.420 205 | 90 | 90 | 90 |
BV/GPa | BR/GPa | BH/GPa | GV/GPa | GR/GPa | GH/GPa |
160.55 | 160.55 | 160.55 | 111.49 | 68.85 | 90.17 |
E/GPa | γ | λ | AB | AG | H/GPa | vl/(km·s–1) | vt/(km·s–1) | vm/(km·s–1) |
227.85 | 0.26 | 1.44 | 0 | 0.24 | 11.56 | 7.72 | 4.38 | 4.87 |
Atom | Charge distribution/e | |||
s orbital | p orbital | d orbital | Total charge | |
B | 0.88 | 2.54 | 0.00 | –0.42 |
La | 1.50 | 5.46 | 1.51 | 2.53 |