1. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Ch2. Wave Diffraction and Reciprocal Lattice 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. 2-1 Diffraction of Waves by Crystals  Crystal structure studied by the diffraction of photons, neutrons, and electrons (keV) (0.01eV) (100eV)  Diffraction depends on the crystal structure and on the wavelength – in crystal, the λ of radiation ≤ lattice constant (a) “diffracted beam” (different from the incident, refracted direction)  Consider.. ① reflection ② θ 회절 반사 – 경로차 beam ①, beam ② : 2dsinθ – for constructive interference : 2dsinθ=nλ (n : 정수 )→ sinθ ≤ 1 → λ ≤ 2d – Bragg refraction can 2d≥λ : can not use visible light ( 가시광선보다 파장이 짧아야 한다 ) – Bragg law : consequence of the periodicity of the lattice a d

3. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 이중슬릿 확장 : 발 ( 격자 ) 라고 불리우는 수 mm 당 수천개의 실틈 ( 슬릿 ) 을 갖고있다. 광원분석 등 에돌이 선 너비 극대 : (Reference) 에돌이 발 ( 회절격자 )

4. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr (Reference) X-ray 생성원리  대부분의 원자의 화학적, 전기적, 자기적, 광학적 성질은 최외각 전자에 의해 결정  For Pb (lead, 82) 최외각 전자 ε=7.4eV ( 최외각 전자 하나를 제거하기 위해 필요한 energy ) e - ε=88eV (K-shell 전자 하나를 제거하기 위해 필요한 energy ) Incident photon ejected K-shell el. if L β x-ray 방출 K α x-ray 방출 N M L K e.g.) for Cu, (z=29) K-shell I.E. : 8.99 keV K α x-ray : 8.06 keV (transition from L to K shell)

5. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr (Reference) Experimental Techniques Used in Collecting XRD data (1)  The scattering angle 2θbetween the diffracted and incident beams. By substituting sinθ into Bragg’s law, one determines the interplanar spacing(d) as well as the orientation of the plane responsible for the diffraction.  The intensity I of the diffraction beam. This quantity determines the cell-structure factor, F hkl, and hence gives information concerning the arrangement of atoms in the unit cell. 1.The rotating-crystal method ① For analysis of the structure of a single crystal ② A monochromatic incident beam of wavelength λ ③ The specimen is then rotated until a diffraction condition obtains (λand θ satisfy Bragg’s law)

6. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr (Reference) Experimental Techniques Used in Collecting XRD data (2) ① For a rapid determination of the symmetry and orientation of a single crystal ② A white x-ray beam (spectrum of continuous wavelength) is made to fall on the crystal, which has a fixed orientation relative to the incident beam ③ Cannot determine the actual values of the interplanar spacings – only their ratios ④ Can determine the shape but not the absolute size of the unit cell 2. The Laue method 3. The powder method ① For the specimen which is not a single crystal (polycrystalline) ② A monochromatic beam impinges on the specimen, and the diffracted beams are recorded on a cylindrical film surrounding it

7. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2-2 Scattered Wave Amplitude  Bragg law : condition for the constructive interference ( 보강간섭 ) of diffracted beam  Diffracted beam 의 scattering intensity 의 결정 or 해석 crystal 은 어떠한 translation ( ) 에 대해서도 invariant ( 즉, crystal 내부의 국소적 (local) 물리 성질 - 예로써 전하밀도, 자기 moment 밀도 등 - 은 에 대해 invariant 하다 )  Fourier Analysis : 모든 형태의 주기적 파형은 sine 과 cosine 의 파형 혹은 e ikx 의 함수로 표현 가능 idea : crystal 의 주기성

8. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2-3 Fourier Analysis (1)  Consider electron number density, n : invariant under T 즉, n (r + T)=n (r ) : 격자나 원자, 전자 등의 분포의 주기성 때문에  주기성 문제를 단순하게 취급하기 위해서 1 차원 결정을 고려 Fourier 전개를 위한 이상적 상황을 제공 n(x) : 1 차원 el. # density can be expanded in a Fourier series of sines and cosines a a a x n(x) The factor 2π/a : ensures that n(x) has the period “a” p: positive integers C p, S p : 실수 상수 (Fourier 계수 )

9. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2-3 Fourier Analysis (2)  Compact and generalized form of n(x)  3 차원의 경우로 확장시키면,  Inversion of Fourier Series 3 차원 역격자 (reciprocal lattice) vector ↔ 2πp / a (1 차원 ) Invariant under T n G : determines the x-ray scattering intensity

10. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2-4 Reciprocal Lattice Vectors (1)  앞 절의 3-D Fourier series 를 계속 진행하려면 we must know G  Construct b 1, b 2, b 3 (the axis vectors of the reciprocal lattice) – If are primitive vectors, then are primitive vectors in reciprocal lattice – and so forth i.e.,  Crystal structure ① crystal lattice ( 실제 공간, real space) ② reciprocal lattice ( 역공간, reciprocal space)  Diffraction pattern of crystal : a map of the reciprocal lattice ( 추후 증명 ) (where : integers) 역격자 (reciprocal lattice) 공간에서 필요한 단위벡터

11. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2-4 Reciprocal Lattice Vectors (2)  Dimension of direct lattice (or crystal lattice) : [ length ] Dimension of reciprocal lattice : [ 1/ length ]  Reciprocal lattice : a lattice in the Fourier space associated with the crystal  파동 방정식, k (wave vector) → dimension : (2π/λ) : 1/ length “drown in Fourier space” ∴ Fourier space 상의 모든 위치는 wave 를 표현한다. Integer! This result proves that the Fourier representation of a function n(r) periodic in the crystal lattice can contain components only at the reciprocal lattice vector

12. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.5 Diffraction Condition (1)  Theorem : the set of reciprocal lattice vectors G determines the possible x-ray reflection  Consider.. 1.The phase difference between incident beam and scattered beam from volume elements (dV), r apart : 2.The amplitude of the scattered wave 3.Total amplitude of scattered EM wave 4. 경로차 : rsinΦ → phase difference : 2π (rsinΦ/λ ) Incident beam : Scattered beam : dV 0

13. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.5 Diffraction Condition (2)  Define scattering amplitude F :  In elastic scattering of photon, hω is conserved. ; Constructive interference condition (Bragg law)  The diffracted condition for a constructive interference is  If G is a reciprocal lattice vector, so is –G : another statement of the Bragg condition..eq ①..eq ② [H.W] problem 1 in Ch2 → 결과 : d (h k l) = 2π/ │ G │ (for elastic scattering)

14. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.5 Diffraction Condition (3)  Using eq ① ;  Generalize, 2dsinθ = nλ  Various statements of the Bragg condition : ① 2dsinθ = nλ ② ③ Laue Equations ; ; geometrical representation – Consider.. : Δk lies on a certain cone 조건도 동일 Δk : satisfy eqs. ①, ②, and ③ ①②③①②③ (h k l) planes

15. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.6 Brillouin Zone  Significant for “diffraction condition”  정의 : ‘Wigner-Seitz’ primitive cell in the reciprocal lattice  Gives a vivid geometrical interpretation of the diffraction condition : – 예 ) Any k1 vector reached to plane1, satisfy the diffraction condition :  Construction of B.Z. ( 예 : 2D square lattice) – 1 st, 2 nd,3 rd … B.Z : same area – Can be translated into 1 st B.Z. by G  [ 참고 ] 역사적으로, B.Z, is not the language of x-ray, but is an essential part of energy band structure (p.40) Plane 1 Plane 2 ↔ 2dsinθ = nλ ( Fig. 9(a) 설명 )

16. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.7 Reciprocal Lattice to SC Lattice

17. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.8 Reciprocal Lattice to BCC Lattice Primitive vectors of a fcc lattice in reciprocal lattice Primitive translational vector of bcc lattice

18. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.9 Reciprocal Lattice to FCC Lattice Primitive vectors of a bcc lattice in reciprocal lattice 4π/a a

19. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.10 Fourier Analysis of the Basis (1) ▪ 즉, basis 원자들에 의한 r 위치의 원자의 전자 분포 0

20. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.10 Fourier Analysis of the Basis (2)  The structure factor  Define atomic form factor  And

21. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.11 Structure Factor of the BCC Lattice E.g.) Metallic sodium : bcc – no diffraction pattern : (1 0 0), (3 0 0), (1 1 1) – exist diffraction pattern : (2 0 0), (1 1 0), (2 2 2) For bcc, identical atoms at (0,0,0) and ( ½, ½, ½ ) and f is atomic form factor

22. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.12 Structure Factor of the FCC Lattice  Assumed that the basis of the fcc structure referred to the cubic cell has identical atoms at 0 0 0 ; 0 ½ ½ ; ½ 0 ½ ; ½ ½ 0 → “ f ” is the same

23. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.13 Atomic Form Factor(1) : f  A measure of the scattering power of the jth atom in the unit cell  Depends on the number and distribution of atomic electrons, and the λ and θ of the radiation  Consider that the integral extended over the el. concentration associated with a single atom and the el. distribution is spherically symmetric about the origin

24. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr 2.13 Atomic Form Factor(2)  If the same total el. density were concentrated at r=0 ∴ f : the ratio of the radiation amplitude scattered by the actual el. distribution in an atom to that scattered by one el. localized at a point → overall el. distribution in a solid as seen in x-ray diffraction is fairly closed to that of the appropriate free atoms ; 꼭 설명 1.Thermal motion does not broaden a diffraction line, but reduces the intensity 2.Broadening of diffraction line due to the lose of crystallization

25. Electronic Materials Research Lab in Physics, http://smartpolymer.korea.ac.kr Reference ① vector ㅗ (h k l) plane in real space ② interplanar distance, (h k l) plane H.W. (#1) 연습문제 1 번 Can show the relations (#2) 연습문제 5 번 ㅗ ㅗ