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Unformatted text preview: Molecular Dynamics simulations Lecture 07: Interaction models for metals Dr. Olli Pakarinen Original lecture notes by Dr. Jani Kotakoski, 2010 University of Helsinki Fall 2011 M ETALS I By definition, a metal is a material which is kept together by metallic bonding I Metallic bonding is a result of nearlyfree electrons of a group of atoms which bind with the positively charged atomic cores I Of all elements, 80% are metals [image from A. Kuronen] C RYSTAL STRUCTURES FOR METALS I The most typical crystal structure for a metal is the hexagonal close packed (HCP) I The next most common is face centered cubic (FCC), whereas the third one is the body centered cubic (BCC) I With potentials describing these three lattices, already ∼ 60 % of all stable singleelement structures are covered I Both HCP and FCC have a packing density of 0.74 (filling of space with hard spheres) I No structure, regular or not, has a denser packing HCP I Hexagonal close packed lattice vectors are: a = ( a , , ) α = β = 90 ◦ b = ( a / 2 , √ 3 2 a , ) γ = 120 ◦ c = ( , , c ) I The two atoms in the primitive cell are located at: ( , , ) & ( 2 3 , 1 3 , 1 2 ) I Atom in the HCP structure has 12 nearest neighbors I If the binding energy would depend only on the nearestneighbor distance (i.e., a pair potential), there would be no difference in the energy of HCP and FCC I Examples of materials with HCP lattice: He, Be, Mg, Ti, Zn, Cd, Co, Y, Zr, Gd and Lu FCC I FCC’s lattice vectors are, of course, cubic: a = ( a , , ) b = ( , a , ) c = ( , , a ) I The atoms are located at: ( , , ) , ( , 1 2 , ) ( 1 2 , 1 2 , ) , ( , 1 2 , 1 2 ) I Again, there are 12 nearest neighbors and a 0.74 packing I FCC metals tend to be soft and ductile over a wide temperature range I Examples include: Au, Ag, Cu, Al and γFe BCC I Also BCC has, by definition, a cubic lattice: a = ( 1 , , ) b = ( , 1 , ) c = ( , , 1 ) I The atoms are located at: ( , , ) ( 1 2 , 1 2 , 1 2 ) I Each atom has 8 nearest neighbors and 6 nextnearest neighbors I Packing density is 0.68 I BCC metals are typically strong and reasonably ductile I Examples include: W, Cr, αFe and K Effective Medium Theory I In the effective medium theory (EMT) the interactions between the particles are assumed to depend on an effective environment I In this case, the total energy is written as U tot = X i F [ n i ( R i )] + 1 2 X i , j Φ ( R i R j ) (1) I Here, F is a functional of the local electron density, and Φ is (typically) a purely repulsive pair potential for the atom cores I In principle, EMT obtained from ab initio calculations can be directly used as a potential I However, more typically, EMT is used as the starting point for developing a more flexible potential I Within EMT, the real material is modeled as jellium which consists of 1) a homogeneous electron gas, formed from the free electrons 2) a constant positive background density (ions) I Hence, both electrons and ions are smeared to a respective effective medium I Placing (i.e., “Placing (i....
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 Winter '12
 Kotakoski
 The Land

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