Diffraction T. Ishikawa Part 2, Dynamical Diffraction 1/16/2019 JASS02

Introduction In the 1st part, we dealt with “Kinematical Theory” where the scattered x-rays suffer no additional scattering. The 2nd part is designed to give basic ideas of “Dynamical Diffraction” observed with perfect crystals as a result of multiple scattering. 1/16/2019 JASS02

Basic Idea Kinematical Diffraction Dynamical Diffraction 1/16/2019 JASS02

Maxwell Equation (1/2) E: electric field D: electric displacement H: magnetic field B: magnetic induction r: charge density j: current density P: polarization M: magnetization e0: permittivity of vacuum m0: permeability of vacuum c: electric susceptibility c: speed of light in vacuum 1/16/2019 JASS02

Maxwell Equation (2/2) For periodically oscillating electromagnetic field; jtrue = 0, rtrue = 0. For non-magnetic materials, M=0 so that B = m0H. 1/16/2019 JASS02

Polarization Polarization P = electric dipole moment in unit volume c(r) have the periodicity of crystal lattice 1/16/2019 JASS02

Electromagnetic Wave in Periodic Medium Bloch Theorem Incident Plane Wave in Vacuum Waves inside Periodic Medium u(r) has periodicity of crystal lattice u(r) can be expanded in a Fourier Series with reciprocal lattice vector, g. Bloch Wave 1/16/2019 JASS02

Some Mathematic.... 1/16/2019 JASS02

Mathematics (cont’d) (*) 1/16/2019 JASS02

Basic Equations for Dynamical Theory Condition for the equation (*) should be valid for arbitrary r gives the basic equation for dynamical diffraction theory: Since the basic equation is well approximated by 1/16/2019 JASS02

Boundary Conditions (1/3) z Fields in vacuum: (Ea, Da) Fields in crystal: (E, D) vacuum z=H crystal Boundary conditions from Maxwell Equations: Continuity of tangential components of Electric Fields Et =Eat Continuity of normal components of Electric Displacements Dz =Daz t: tangential component, z: z(=normal) component 1/16/2019 JASS02

Boundary Conditions (2/3) Wavefield : Superposition of plane waves Wave Vector in Crystal: Kg Wave Vector in Vacuum: Km Kgt = Kmt Km crystal wave Kg vacuum wave 1/16/2019 JASS02

Boundary Conditions (3/3) Boundary Condition at z=H (Crystal Surface) 1/16/2019 JASS02

Two-Wave Approximation (1/3) Under usual experimental conditions, only two waves with K0 (incident direction) and Kg (diffracted direction) are strong inside the crystal. Wavefield in crystal: Basic Equation: Averaged refractive index of crystal: Polarization Factor 1/16/2019 JASS02

Two-Wave Approximation (2/3) Condition for the basic equation, dispersion surface O G dispersion sphere to have non-trivial solutions is Ko Kg g By introducing new parameters: 1/16/2019 JASS02

Two Wave Approximation (3/3) y For non-absorbing crystals, Tg Lo x When we introduce a new parameter L as To , Near the point Lo, Dispersion surfaces form Hyperbolla 1/16/2019 JASS02

Amplitude Ratio Amplitude Ratio j = 1, 2 1/16/2019 JASS02

Diffraction Geometry Symmetric Laue Case Symmetric Bragg Case 1/16/2019 JASS02

Symmetric Laue Case Dispersion sphere of vacuum wave (radius K) Starting point of wave vector Ko: P Laue point: L Deviation from Bragg Condition 1/16/2019 JASS02

Symmetric Laue Case: Deviation Parameter Deviation Parameter W Solving above equations, we can get upper sign: j=1, lower sign: j=2 Usually W=Ws 1/16/2019 JASS02

Symmetric Laue Case: Amplitude Ratio Here, For non-absorbing crystals, 1/16/2019 JASS02

Symmetric Bragg Case (1/2) Between L1 and L2, z has no intersections with dispersion surfaces. Total Reflection Region Deviation from Bragg Condition 1/16/2019 JASS02

Symmetric Bragg Case (2/2) Dqo : Deviation from geometrical Bragg angle by refraction upper sign: j=1, lower sign: j=2 Deviation parameter, W Amplitude Ratio upper sign: j=1, lower sign: j=2 1/16/2019 JASS02

Rocking Curves Use monochromatic plane wave as an incident beam; Rocking the sample crystal around the Bragg angle; We can observe so-called rocking curve. 1/16/2019 JASS02

Rocking Curve: Symmetric Laue Case (1/3) Incident Wave Ka,Eoa Crystal Wave z=0 o-wave Kg2, Eg2 Ko2, E02 g-wave z=H Kg1, Eg1 Ko1, Eo1 Outgoing Wave Ka,Eda Kga,Ega o-wave g-wave 1/16/2019 JASS02

Rocking Curve: Symmetric Laue Case (2/3) Boundary condition at z = 0 (incident surface) Boundary condition at z = H (exit surface) upper sign: j=1, lower sign: j=2 At W=0 (exact Bragg condition), 1/16/2019 JASS02

Rocking Curve: Symmetric Laue Case (3/3) 1/16/2019 JASS02

Rocking Curve: Symmetric Bragg Case (1/3) Boundary condition at z = 0 Ka,Eoa Kga,Ega z = 0 Kg1, Eg1: W<-1 Ko1, Eo1: W<-1 upper sign: W<-1, lower sign: W>1 Kg2, Eg2: W>1 Ko2, Eo2: W>1 1/16/2019 JASS02

Rocking Curve: Symmetric Bragg Case (2/3) Another solution will give a divergent solution 1/16/2019 JASS02

Rocking Curve: Symmetric Bragg Case (3/3) Rocking curve (Darwin Curve) |W|<1: All incident energies are reflected back. Total Reflection Center of total reflectiuon, W=0, is deviated from geometrical Bragg angle qB by Range of total reflection (-1<W<1) Darwin Width, ~microradian order 1/16/2019 JASS02

Absorbing Crystal Absorption: Anomalous dispersion term into atomic scattering factor Centrosymmetric Crsyatls A new parameter k is defined as 1/16/2019 JASS02

Symmetric Laue Case: Absorbing Crystal (1/2) Oscillating Term, Hardly to be observed experimentally without very good plane wave averaging Bloch Wave a small absorption Bloch Wave b large absorption Anomalous Transmission (Borrman Effect) 1/16/2019 JASS02

Symmetric Laue Case: Absorbing Crystal (2/2) Forward Diffraction Thin Crystal Thick Crystal 1/16/2019 JASS02

Symmetric Bragg Case: Absorbing Crystal Rocking curve for a symmetric Bragg case diffraction from a semi-infinite absorbing crystal (with centrosymmetry) k = 0 k = 0.1 1/16/2019 JASS02

Summary Very quick scan of x-ray diffraction theory was attempted. You may need reference text books. References Dynamical Theory of X-Ray Diffraction, A. Authie, Oxford University Press, 2001 Handbook on Synchrotron Radiation Vol. 3, North-Holland, 1991. 1/16/2019 JASS02

Thank you for your attention. Acknowledgement Some materials presented here are originally prepared by Prof. Seishi Kikuta for his textbook written in Japanese. Some ppt materials have been prepared by Dr. Shunji Goto. Discussion in preparing the lecture with Drs. Shunji Goto, Kenji Tamasaku and Makina Yabashi is appreciated. 1/16/2019 JASS02