1. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Ch3. Crystal Bonding Prof. J. Joo (jjoo@korea.ac.kr) Department of Physics, Korea University http://smartpolymer.korea.ac.kr Solid State Physics Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr

2. 3.1 Introduction  결정 격자를 유지시키는 힘 : 핵의 positive charge 와 전자의 negative charge 와 의 “ 잡아당기는 Coulomb force” Magnetic force → (very) weak Gravitational force → negligible  새로운 용어 : “Van der Waals force” “exchange energy” “covalent bonds”  “Cohesive energy” : that must be added to the crystal to separate its components into neutral free atoms at rest, at infinite separation  “Lattice energy” (for ionic crystals) : that must be added to the crystal to separate its component ions into free ions at rest, at infinite separation  Tables provide an important information e.g.) Table 1 : cohesive energy Table 2 : melting points Table 3 : bulk modulus ( 부피 탄성률 ) : [N/m 2 ] & compressibilities ( 압축률 ) : [m 2 /N] closely related to cohesive energy

3. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Reference.. (Fig.1) The Principle Types of Crystalline Binding

4. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Reference.. Holliday 일반물리학 (p762)

5. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.2 Crystals of Inert Gases  He, Ne, Ar, Kr, Xe (Properties : Table 4)  Important properties 1.Weakly bound → low melting temp. (< 170K) 2.Very high ionization energy (≈10~25eV) ( ∵ outmost el. Shells are filled) 3. Fcc structure except He 3 and He 4  He, Ne 등 결정 고체를 이루기 위해서 주변 atom 의 el. 의 분포에 의한 미시적 영향력 Van der Waals-London Interaction or “ Fluctuating dipole forces”  Consider two identical inert gas atoms “Van der Waals force” e- r 1 p r 2 p R The atoms induce dipole moments each other → 결과적으로 두 개의 atom 사이에 attractive interaction 유발 → 뒤의 유도과정 … ( R > r )

6. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.3 Van der Waals – London Interaction (1) atom 1 atom 2 R X1X1 (K 1 ) X2X2 (K 2 ) → 엄밀하게 양자역학적으로 에너지를 계 산하여야 하지만, 현상론적으로 (phenomenologically), 고려되는 계는 “coupled oscillation” → Due to Coulomb interaction

7. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.3 Van der Waals – London Interaction (2)  “Coupled Oscillation” → due to Coulomb interaction 즉, 원래 진동수의 변화를 유발시킴 ( 가정 ) 두 개의 동일 atom → k 1 = k 2 = k, x 1 ≈ x 2 ≡ x ∴ The energy of the coupled system is ≡k' 새로운 진동수와 관련 : Taylor 전 개

8. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.3 Van der Waals – London Interaction (3) 상수 : positive 중요한 의미 (due to induced dipole) 1.Quantum effect 2.Dipole coupling 에 의해서 계 (system) 의 energy 가 낮아짐 3. 두 개의 원자의 charge density 의 “overlap” 에 의해 이루어진 것이 아님

9. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.4 Repulsive Interaction (1)  Consider 2 H-atoms and push together until the protons are almost in contact → similar to atomic He el. configuration a) Total spin 0 Total el. Energy -78.98 eV → the electrons are ↑↓ (antiparallel) the Pauli principle has no effect b) Total spin 1 Total el. Energy -59.38 eV Pauli exclusion principle Increases the repulsion : +19.6 eV repulsive potential = : 실험적 사실 H H He 1S ↑ 1S ↑ 1S ↓ + → H H He 1S ↑ 1S ↑ 2S ↑ + → 1S 2S “Lennard–Jones Potential” for the crystal of inert gas

10. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.4 Repulsive Interaction (2) repulsive potential van der Waals potential “ 일반적 표현 ” : “ 두 개의 atom 이 가까울 수록 repulsive potential 이 exponential 하게 증가 ”

11. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.5 Equivalent Lattice Constants  If neglecting the kinetic energy, the cohesive energy of an inert gas is given by the sum of all pairs of atoms # of atomsdistance to nearest neighbor where : distance between reference atom i and any other atom j : depending on the structure ( 예 ) fcc structure  안정된 위치, 즉 atomic vibration 에 대해서 평행 상태의 위치 “R 0 /σ=1.09” → 고유한 값 (fcc inert gas) (ref.)  Cohesive energy :

12. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.6 Ionic Crystals  Positive ion 과 negative ion 사이의 정전기적 상호작용 → “ ionic bonding”  ( 예 ) NaCl, Li F 1S 2 2S 2 2P 5 → F - : 1S 2 2S 2 2P 6 1S 2 2S 1 → Li + : 1S 2  Some distortion near the region of contact with neighboring atoms – Fig.7  Ionic bonding 의 세기 should know ① ionization energy ( 이온화 에너지, Table 5) ② el. affinity ( 전자친화력, Table 6)  ( 예 ) NaCl (quick estimation) the energy per molecule unit of crystal sodium chloride is 7.9+3.6-5.1 = 6.4eV Lattice energy More exact calculation : next slide

13. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.7 Electronic or Madelung Energy (1)  Van der Waals 상호작용 → small contribution (1-2% in cohesive energy)  Main contribution to the binding energy : electrostatic and called by Madelung energy  만약 U ij : 이온 i 와 이온 j 사이의 상호작용이라면 “central field repulsive interaction” “Coulomb potential” ① λ,ρ : 실험적 상수 (lattice constant, 압축률에 의해 결정 ) ② R -12 (inert gas) 대신 exponential 감소가 더욱 general representation

14. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.7 Electronic or Madelung Energy (2)  Suppose N-molecules or 2N-ions U tot = N·U i : each pair of interactions  Introduce P ij such that r ij ≡ P ij R n – n separation 구조와 관련  가정 : repulsive 상호작용은 단지 n - n 만 있다면, # of n.n.

15. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.7 Electronic or Madelung Energy (3)  At the equilibrium separation ; i.e., ∴ Total energy at R=R 0 where : Madelung energy (attractive energy & repulsive Coulomb inter. → ionic bonding 을 표현 ) from repulsive term

16. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.8 Evaluation of the Madelung Constant The first calculation of the Coulomb energy constant ‘α’ was made by Madelung → Madelung constant (α) n.n. distance the distance of jth ion from the ref. i ion 3 Dimensional calculation : very difficult (hardly converge) (E.g.) 3D Ionic crystal (based upon until charge and referred to the n.n. distance) NaCl : 1.748 CsCl : 1.763 ZnS : 1.683 !! Table.7 : properties of Alkali Halide crystal with the NaCl structure For KCl

17. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.9 Covalent Crystals (1)  Covalent bond : one or more pairs of electrons are shared by the two atoms → strong bond such as in the ionic crystals → strong directional properties : 1.C, Si, Ge : directional properties such as tetrahedral angles 2.CH 4 (tetrahedral methane) → the electrons forming the bond tend to be partly localized in the region between two atoms joined by the bond → the spins of the two electrons are antiparallel H H H H C

18. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.9 Covalent Crystals (2)  One of the simple e.g. for covalent bond : “H 2 molecule” → the strongest binding → two electrons are “antiparallel” → spin-dependent Coulomb energy “exchange interaction”  Pauli pr. ① strong repulsive interaction between atoms with filled shells ② if the shells are not filled, el. overlap can be accommodated and the bond will be shorter (e.g.) Cl 2 → ~2Å solid Ar → ~3.76Å  C, Si, Ge (hybridization) → for C : 1s 2 2s 2 2p 2 → to form “tetrahedral system” of covalent bonds → promote to the electronic configuration 1s 2 2s2p 3 (sp 3 hybridization) 전자친화도 : 진공 중에서 중성원자가 전자와 결합할 때 방출하는 에너지 = 음이온에 서 전자하나를 떼어내는데 필요한 에너지 “Negative energy corresponds to binding”

19. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.10 Metals  Metals 1.high electrical conductivity (≥10 4 Ω -1 cm -1 ) 2.large number of el. (high electron density) → free to move 3.el. available to move → “conduction electron” “valence electrons” of the atom became the “conduction el.” of the metal  The attractive interaction between the electron gas and the positive ions → “metallic bonding” : ~1eV “not stronger than covalent bond, ionic bond”

20. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 3.11 Hydrogen Bonds  Under certain condition, a hydrogen atom is attracted by strong force to two atoms → forming hydrogen bond : ~0.1eV → for electronegative atoms : F. O. N Note> 극단적인 case : bare proton ( H + ) forms the hydrogen bond → hydrogen connects two atoms ( HF 2 - )

21. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Reference..