• Reference system: an atom in a FCC crystal. The lattice constant of the reference crystal should match the density of the actual system. Use a measure for the local electron density as the map.
K.W. Jacobsen, M. Puska and J. K. Nørskov, Surf. Sci. 366, 394 (1996); Phys. Rev. B 35, 7423 (1987).
Effective medium theory (EMT)
• •
Blue lines: Electron density from blue atoms. Green line: Sum of contributions from blue atoms. This is the embedding density of red atom.
The Sutton-Chen potential provides a reasonable description of various bulk properties, with an approximate many-body representation of the delocalized metallic bonding. However, it does not include any directional terms, which are likely to be important for transition metals with partially occupied d shells.
Electron density ρ
ion ion
EAM框架下内聚能的一般形式
镶嵌原子势的内聚能表述
电子密度和镶嵌函数
原子的电子密度函数
0.02 (a)
0.02 (b)
ρ(R ) (10 e/nm )
ρ(R) (10 e/nm )
3
3
0.01
←
0.01
←
Pt
3
ACu → 0.00 2.0 3.0
EAM和SC势比较适用于没有成键取向结构的密堆金属； TB和FS势更适用于具有体心立方结构的过渡金属； EMT势的势函数计算往往过于复杂。
Effective medium theory (EMT)
• Energy differences are easier to calculate than energies. Choose a reference system with a known energy, and concentrate on the energy difference.
Glue potential for Au
Phonon dispersion of Au. Lines: glue potential; dots: experiments
Force matching method
http://www.ud.infn.it/~ercolessi/forcematching.html
• • • • • •
Effective medium theory (EMT): Phys. Rev. B26, 2875 (1982). EAM potential: Phys. Rev. Lett. 50, 1285 (1983). Finnis-Sinclair (FS) potential: Philos. Mag. A 50, 45 (1984). Glue potential: Phys. Rev. Lett. 57, 719 (1986). Sutton-Chen (SC) potential: Philos. Mag. Lett. 61, 139 (1990). Tight-binding (TB) potential: Phys. Rev. B 48, 22 (1993).
镶嵌原子方法(embedded atom method, EAM)
EAM的物理图像
镶嵌原子方法的物理思想起源于 Friedel提出的原子嵌入能概念， 即：原子的凝聚能主要取决于该原 子所占据位置的局域电子密度。后 来也和密度范函理论建立了联系。 ion ion ion ion ion ion ion ion
Original ideal: J. K. Nørskov, Phys. Rev. B 26, 2875 (1982).
Effective medium theory (EMT)
Eref: the energy of the reference system; EAS is the correction. • The energy of an atom (and thus its force) is a function of the position of the atoms in a local neighbourhood. • Lessons from quantum mechanics are built into the functional form.
Sutton-Chen potential
• • • •
C: a dimensionless parameter ε: a parameter with dimensions of energy, α: lattice constant, m, n: positive integers with n>m
Effective medium theory (EMT)
Effective medium theory (EMT)
Phonon dispersion relation for Al. Lines: EMT, dots: experiment
Finnis-Sinclair potential
The FS potential incorporates the band character of metallic cohesion and allows to obtain correct values for the vacancy formation and cohesive energy. It reproduces the lattice constants, bulk modulus and surface energy accurately.
Philos. Mag. Lett. 61, 139 (1990).
Tight-binding potential
• Cohesive energy due to d electron
Friedel model of d band
EF=W(Z-5)/10
•
Bandwidth W computed from 2rd moment of density of state for d band
Pd Ni 2.0
→
0.00
4.0 5.0 6.0
3.0
4.0
5.0
6.0
R (0.1 nm)
R (0.1 nm)
“Training” of parameters
拟合EAM参数采用的典型实验数据
EAM在金属中的成功应用
基于镶嵌原子思想的各种经验势
基于镶嵌原子的思想, 先后发展出了描述原子间相互作用势的有效介 质理论(EMT)、镶嵌原子方法(EAM) 、紧束缚势(TB) 、二级矩近似 (SMA) 和胶体模型(Glue model) 等。它们的区别在于镶嵌能的非线性 表述不同或是镶嵌密度的测度不一样。
Glue potential for Au
F. Ercolessi, M. Parrinello and E. Tosatti, Philos. Mag. A 58, 213 (1988)
Glue potential for Au
Glue potential for Au
Lattice parameter of Au as function of temperature. Line: MD simulation using glue potential; Dots: experiment
计算材料学第三章 原子间相互作用势
赵纪军
三束实验室，物理学院&高科技研究院 Email: zhaojj@，电话：84709748
计算材料学第三章
3-2. 金属键合的经验势描述
描述金属键合的基本要求 • The main physical point to model is that bonds become weaker when the local environment becomes more crowded (consequence of the Pauli principle). • The potentials must be able to reproduce the energy difference between FCC, HCP and BCC.
Philos. Mag. A 50, 45 (1984).
Glue potential
In 1983-1986, Ercolessi et al. developed the glue model, a formulation containing a density-dependent many-body term in addition to usual two-body interactions. This term allows to mimic the "gluing" character of the cohesion due to conduction electrons in metals: ions have a low energy as long as they are immersed in the "electron sea", while the exact position of neighbouring ions is relatively unimportant. This led us to write a Hamiltonian where a short-ranged "density function" is attached to atoms, so that for each atom in the system we can compute an effective coordination defined as the sum of all the density contributions coming from neighboring atoms. The energy of this atom will then depend non-linearly upon this effective coordination. The non-linearity of the energy dependence upon coordination essentially models this physical fact, ultimately a consequence of Pauli's principle: the strength of individual bonds decreases as the local environment becomes more crowded. In contrast, with two-body potentials the strength of individual bonds does not depend on the environment. This feature is crucial to capture the physics of bonding in metals. http://www.ud.infn.it/~ercolessi/potentials/