金属晶格稳定性与热学性质第一原理研究
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摘要
本文的方向是应用基于密度泛函理论及密度泛函微扰理论的第一原理方法研究金属在压力下的相稳定性及热学性质。材料需要在不同的环境中使用,而压力和温度是描述环境的两个重要参量,因而研究材料在不同压力下的相稳定性与热学性质具有重要的理论意义与实用价值。
应用基于密度泛函理论的平面波赝势方法计算了部分金属(W、Pt、Be、Mg、Sc、Y、Ti、Zr)在零温下不同结构的总能随压力变化关系,根据焓值得出相转变压力,得到了状态方程。计算了电子结构如态密度曲线及电子布居数,对可能的相变途径进行了讨论。利用密度泛函微扰理论计算了这些金属部分结构的声子谱、弹性常数,据此讨论了不同结构在压力下的力学稳定性。
利用密度泛函微扰理论在准谐近似下计算了部分金属的自由能与温度关系,由此得出它们的热学性质,如热膨胀系数、定容及定压摩尔热容,讨论了电子对热膨胀及热容的贡献,根据Gruneisen关系讨论了各向异性热膨胀的起因。
本文的研究结果表明:
1.在压力下Be将发生hcp→9R→bcc相变,根据焓值判断转变压力分别为4570KBar及5480 KBar。9R相在压力下的出现是其它理论工作未曾考虑的。与其它理论计算不同,本文的计算表明零点能对转变压力的影响可以忽略。Mg在压力下直接发生hcp→bcc相变,转变压力为520 KBar,理论计算与实验符合。根据弹性常数及声子谱计算,Be与Mg的fcc及hcp相在所计算的压力范围内都是力学稳定的。低压下Be的bcc相弹性常数虽然满足稳定性要求,但短波长的横模声子T_([110])[ξξ0]频率为虚数,不满足稳定性要求,因而早期理论工作仅根据弹性常数判断稳定性是不充分的。
2.在压力下Sc将发生hcp→9R→dhcp→distorted-fcc→Sc-V相变,根据焓值判断转变压力分别为170 KBar、320 KBar、730 KBar及2440 KBar。总能计算表明Y的基态为9R结构,Y在压力下将发生9R→dhcp→distorted-fcc→Sc-V相变,转变压力为70KBar、520 KBar及2200 KBar。计算的零压下Y平衡体积9R相总能仅比hcp相低1meV,而本文的计算收敛精度也是1 meV,因而需要更精确的计算方能做出判断。可以肯定的是零温零压下这两相的总能或焓相差极小,室温下的hcp稳定相应该是热稳定的效应。本文对Sc-V相的计算是其他理论工作没有的。对Sc计算得出的Sc-V相将稳定于2440 KBar以上,与实验得出的2400 KBar相吻合。对Y在高压将出现Sc-V相的结论则有待实验验证。
3.总能与焓的计算得出W随着压力增加,将发生bcc→fcc相变,转变压力为10840KBar。对Pt的计算表明在14950 KBar将发生fcc→bcc相变。声子谱计算还表明Pt的fcc及hcp相在所计算的压力范围(0～16000 KBar)是力学稳定的。
Ti随着压力增加,将发生ω→γ→δ→bcc相变,而实验结论为hcp→ω→γ→δ相变。理论计算给出零温下稳定态为ω结构,与实验测得的有限温度下hcp→ω相变数据外推至零温结果一致。实验未能观察到δ→bcc相变,很可能是实验未能保持静水压力,切应力使δ相保持稳定。对Zr的理论计算结果则与实验符合得很好,随着压力增加,将发生hcp→ω→bcc相变。
4.在密度泛函微扰理论及准谐近似基础上计算的材料热学性质,包括各向异性热学性质,在较宽的温度范围内与实验符合。计算表明Be有一定的各向异性线膨胀,而Mg的性质接近各向同性。对非谐振动各向异性及弹性各向异性的分析表明Be在α方向有更大的热膨胀,主要是因为α方向的柔顺系数更大。Mg的非谐振动与弹性在α与c方向相差很小,因而在相当大的温区有着几乎各向同性的热膨胀。Y的弹性各向异性很小,其各向异性热膨胀源于非谐振动的各向异性。根据弹性的各向异性,Ti和Zr似乎都应在α方向有更大的热膨胀,但它们在c方向的非谐振动更强,Ti的弹性各向异性很小,因而Ti在c方向的热膨胀更大。对Zr,高温时非谐振动的各向异性起了主要作用,使其在c方向的热膨胀更大,而在低温时,弹性各向异性起主要作用,使其α方向有更大的热膨胀。
5.对热容的计算表明在电子常数较大时电子对热容的贡献不能忽略。在这些金属中只有Be由于电子常数很小,电子的贡献几乎可以完全忽略。Mg与W的电子常数不大,温度不太高时电子的贡献较小,但由于对W的计算温度范围为0～2000 K,因而必须考虑电子对热容的贡献。至于Pt、Y、Ti、Zr,由于电子常数较大,电子对热容的贡献很大,甚至在室温就有显著的贡献。
The dissertation is devoted to the study of phase stabilities under pressure and thermal properties of some metals from first principles within density functional theory (DFT)and density functional perturbation theory(DFPT).Materials are always used under certain circumstance which is often described by pressure and temperature,so the study of phase stabilities under pressure and thermal properties is of great theoretical and practical significance.
The pressure dependencies of total energy and enthalpy of different structures for some metals(W,Pt,Be,Mg,Sc,Y,Ti,Zr)are calculated with plane wave pseudopotential methods based on density functional theory,and the phase transition pressures are determined.The equations of state(EOS)are obtained.The electronic structures such as density of state(DOS)and electronic population are calculated,and different possible transition paths are discussed.The phonon spectra and elastic constants of some structures are calculated with the density functional perturbation theory(DFPT),and the mechanical stabilities of some structures under pressure are discussed according to these calculations.
The thermal properties of some metals are investigated by performing DFT and DFPT calculations within the quasiharmonic approximation.The temperature dependence of various quantities such as the anisotropic thermal expansion and the heat capacity are calculated.The electronic contribution to the thermal expansion and heat capacity are discussed.
The results of this paper show that:
1.Be undergoes a transition from hcp to bcc via the intermediate phase 9R at 4570 KBar and 5480 KBar respectively.It has never been considered in earlier work that 9R phase might occur under pressure.Unlike earlier work,our calculation shows zeropoint vibration can be neglected when determining transition pressure.Mg undergoes a transition from hcp to bcc at 520 KBar.The calculated and experimental results agree satisfactorily.According to the calculated elastic constants and phonon spectra,both fcc and hcp phases of Be and Mg are mechanical stable under high pressure.The bcc phase of Be is not mechanical stable under ambient pressure because of the softening of its[ξξ0]transverse phonon modes,although its elastic constants satisfies the conditions of mechanical stability.It shows that earlier work which judges phase stability only by elastic constants is not sufficient.
2.The structures of Sc follow a systematic sequence,from hcp→9R→dhcp→distorted-fcc→Sc-V.The transition pressures are 170 KBar,320 KBar,730 KBar,and 2440 KBar,respectively.Sc-V phase has never been considered in earlier work.The total energy and enthalpy show that 9R phase is the ground state of Y,and discover successive transitions to dhcp,distorted-fcc,Sc-V at 70 KBar,520 KBar,and 2200 KBar, respectively.The observed hcp phase under ambient temperature might be due to the thermal effect.
3.W undergoes a transition from bcc to fcc at 10840 KBar and Pt from fcc to bcc at 14950 KBar.The calculations of phonon spectra and bcc→fcc and bcc→hcp phase transform path are agreement with the results of total energy and enthalpy.The fcc and hcp phases of Pt are mechanical stable under pressure(0～16000 KBar)according to the phonon spectra.
Ti undergoesω→γ→δ→bcc transition under pressure according to calculation and the experiment results are hcp→ω→γ→δ.Theωphase is found to be the ground state of Ti and it shows successive transitions toγ,δ,bcc.However,the theoreticalδ→bcc occurs at 1310 KBar,which is not in agreement with the experiment result thatδphase is stable up to 2200 KBar.We suggest that the stability ofδmay be due to nonhydrostatic conditions.
Zr undergoes a transition from hcp toωand bcc.The theoretical and experimental results agree satisfactorily.
4.The calculated thermal properties are in good agreement with available experimental data in a wide range of temperature.The different anisotropy of thermal expansion is due to their different anisotropy in elasticity and anharmonicity of vibrations.Be shows larger thermal expansion in a-axis because of its smaller elastic rigidness in the same direction,while the larger thermal expansion in c-axis for Y can be attributed to the larger anharmonicity of vibrations in the same direction.The anisotropic thermal expansion of Ti and Zr are due to their anisotropic Gruneisen parameters,while Mg has almost isotropic thermal expansion because of its isotropic Gruneisen parameters and elasticity.
5.The calculated specific heat at constant pressure shows that the electronic contribution is negligible for simple metals Be and Mg but not for transition metals,because the electronic constants are very small in considering range of temperature for Be and Mg but large for transition metals.We can expect that the electronic contribution to specific heat is important when the electronic constant is large or the temperature is very high.
应用基于密度泛函理论的平面波赝势方法计算了部分金属(W、Pt、Be、Mg、Sc、Y、Ti、Zr)在零温下不同结构的总能随压力变化关系,根据焓值得出相转变压力,得到了状态方程。计算了电子结构如态密度曲线及电子布居数,对可能的相变途径进行了讨论。利用密度泛函微扰理论计算了这些金属部分结构的声子谱、弹性常数,据此讨论了不同结构在压力下的力学稳定性。
利用密度泛函微扰理论在准谐近似下计算了部分金属的自由能与温度关系,由此得出它们的热学性质,如热膨胀系数、定容及定压摩尔热容,讨论了电子对热膨胀及热容的贡献,根据Gruneisen关系讨论了各向异性热膨胀的起因。
本文的研究结果表明:
1.在压力下Be将发生hcp→9R→bcc相变,根据焓值判断转变压力分别为4570KBar及5480 KBar。9R相在压力下的出现是其它理论工作未曾考虑的。与其它理论计算不同,本文的计算表明零点能对转变压力的影响可以忽略。Mg在压力下直接发生hcp→bcc相变,转变压力为520 KBar,理论计算与实验符合。根据弹性常数及声子谱计算,Be与Mg的fcc及hcp相在所计算的压力范围内都是力学稳定的。低压下Be的bcc相弹性常数虽然满足稳定性要求,但短波长的横模声子T_([110])[ξξ0]频率为虚数,不满足稳定性要求,因而早期理论工作仅根据弹性常数判断稳定性是不充分的。
2.在压力下Sc将发生hcp→9R→dhcp→distorted-fcc→Sc-V相变,根据焓值判断转变压力分别为170 KBar、320 KBar、730 KBar及2440 KBar。总能计算表明Y的基态为9R结构,Y在压力下将发生9R→dhcp→distorted-fcc→Sc-V相变,转变压力为70KBar、520 KBar及2200 KBar。计算的零压下Y平衡体积9R相总能仅比hcp相低1meV,而本文的计算收敛精度也是1 meV,因而需要更精确的计算方能做出判断。可以肯定的是零温零压下这两相的总能或焓相差极小,室温下的hcp稳定相应该是热稳定的效应。本文对Sc-V相的计算是其他理论工作没有的。对Sc计算得出的Sc-V相将稳定于2440 KBar以上,与实验得出的2400 KBar相吻合。对Y在高压将出现Sc-V相的结论则有待实验验证。
3.总能与焓的计算得出W随着压力增加,将发生bcc→fcc相变,转变压力为10840KBar。对Pt的计算表明在14950 KBar将发生fcc→bcc相变。声子谱计算还表明Pt的fcc及hcp相在所计算的压力范围(0～16000 KBar)是力学稳定的。
Ti随着压力增加,将发生ω→γ→δ→bcc相变,而实验结论为hcp→ω→γ→δ相变。理论计算给出零温下稳定态为ω结构,与实验测得的有限温度下hcp→ω相变数据外推至零温结果一致。实验未能观察到δ→bcc相变,很可能是实验未能保持静水压力,切应力使δ相保持稳定。对Zr的理论计算结果则与实验符合得很好,随着压力增加,将发生hcp→ω→bcc相变。
4.在密度泛函微扰理论及准谐近似基础上计算的材料热学性质,包括各向异性热学性质,在较宽的温度范围内与实验符合。计算表明Be有一定的各向异性线膨胀,而Mg的性质接近各向同性。对非谐振动各向异性及弹性各向异性的分析表明Be在α方向有更大的热膨胀,主要是因为α方向的柔顺系数更大。Mg的非谐振动与弹性在α与c方向相差很小,因而在相当大的温区有着几乎各向同性的热膨胀。Y的弹性各向异性很小,其各向异性热膨胀源于非谐振动的各向异性。根据弹性的各向异性,Ti和Zr似乎都应在α方向有更大的热膨胀,但它们在c方向的非谐振动更强,Ti的弹性各向异性很小,因而Ti在c方向的热膨胀更大。对Zr,高温时非谐振动的各向异性起了主要作用,使其在c方向的热膨胀更大,而在低温时,弹性各向异性起主要作用,使其α方向有更大的热膨胀。
5.对热容的计算表明在电子常数较大时电子对热容的贡献不能忽略。在这些金属中只有Be由于电子常数很小,电子的贡献几乎可以完全忽略。Mg与W的电子常数不大,温度不太高时电子的贡献较小,但由于对W的计算温度范围为0～2000 K,因而必须考虑电子对热容的贡献。至于Pt、Y、Ti、Zr,由于电子常数较大,电子对热容的贡献很大,甚至在室温就有显著的贡献。
The dissertation is devoted to the study of phase stabilities under pressure and thermal properties of some metals from first principles within density functional theory (DFT)and density functional perturbation theory(DFPT).Materials are always used under certain circumstance which is often described by pressure and temperature,so the study of phase stabilities under pressure and thermal properties is of great theoretical and practical significance.
The pressure dependencies of total energy and enthalpy of different structures for some metals(W,Pt,Be,Mg,Sc,Y,Ti,Zr)are calculated with plane wave pseudopotential methods based on density functional theory,and the phase transition pressures are determined.The equations of state(EOS)are obtained.The electronic structures such as density of state(DOS)and electronic population are calculated,and different possible transition paths are discussed.The phonon spectra and elastic constants of some structures are calculated with the density functional perturbation theory(DFPT),and the mechanical stabilities of some structures under pressure are discussed according to these calculations.
The thermal properties of some metals are investigated by performing DFT and DFPT calculations within the quasiharmonic approximation.The temperature dependence of various quantities such as the anisotropic thermal expansion and the heat capacity are calculated.The electronic contribution to the thermal expansion and heat capacity are discussed.
The results of this paper show that:
1.Be undergoes a transition from hcp to bcc via the intermediate phase 9R at 4570 KBar and 5480 KBar respectively.It has never been considered in earlier work that 9R phase might occur under pressure.Unlike earlier work,our calculation shows zeropoint vibration can be neglected when determining transition pressure.Mg undergoes a transition from hcp to bcc at 520 KBar.The calculated and experimental results agree satisfactorily.According to the calculated elastic constants and phonon spectra,both fcc and hcp phases of Be and Mg are mechanical stable under high pressure.The bcc phase of Be is not mechanical stable under ambient pressure because of the softening of its[ξξ0]transverse phonon modes,although its elastic constants satisfies the conditions of mechanical stability.It shows that earlier work which judges phase stability only by elastic constants is not sufficient.
2.The structures of Sc follow a systematic sequence,from hcp→9R→dhcp→distorted-fcc→Sc-V.The transition pressures are 170 KBar,320 KBar,730 KBar,and 2440 KBar,respectively.Sc-V phase has never been considered in earlier work.The total energy and enthalpy show that 9R phase is the ground state of Y,and discover successive transitions to dhcp,distorted-fcc,Sc-V at 70 KBar,520 KBar,and 2200 KBar, respectively.The observed hcp phase under ambient temperature might be due to the thermal effect.
3.W undergoes a transition from bcc to fcc at 10840 KBar and Pt from fcc to bcc at 14950 KBar.The calculations of phonon spectra and bcc→fcc and bcc→hcp phase transform path are agreement with the results of total energy and enthalpy.The fcc and hcp phases of Pt are mechanical stable under pressure(0～16000 KBar)according to the phonon spectra.
Ti undergoesω→γ→δ→bcc transition under pressure according to calculation and the experiment results are hcp→ω→γ→δ.Theωphase is found to be the ground state of Ti and it shows successive transitions toγ,δ,bcc.However,the theoreticalδ→bcc occurs at 1310 KBar,which is not in agreement with the experiment result thatδphase is stable up to 2200 KBar.We suggest that the stability ofδmay be due to nonhydrostatic conditions.
Zr undergoes a transition from hcp toωand bcc.The theoretical and experimental results agree satisfactorily.
4.The calculated thermal properties are in good agreement with available experimental data in a wide range of temperature.The different anisotropy of thermal expansion is due to their different anisotropy in elasticity and anharmonicity of vibrations.Be shows larger thermal expansion in a-axis because of its smaller elastic rigidness in the same direction,while the larger thermal expansion in c-axis for Y can be attributed to the larger anharmonicity of vibrations in the same direction.The anisotropic thermal expansion of Ti and Zr are due to their anisotropic Gruneisen parameters,while Mg has almost isotropic thermal expansion because of its isotropic Gruneisen parameters and elasticity.
5.The calculated specific heat at constant pressure shows that the electronic contribution is negligible for simple metals Be and Mg but not for transition metals,because the electronic constants are very small in considering range of temperature for Be and Mg but large for transition metals.We can expect that the electronic contribution to specific heat is important when the electronic constant is large or the temperature is very high.
引文
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