Diffraction T Ishikawa Part 2 Dynamical Diffraction 2252021

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Diffraction T. Ishikawa Part 2, Dynamical Diffraction 2/25/2021 JASS 02 1

Diffraction T. Ishikawa Part 2, Dynamical Diffraction 2/25/2021 JASS 02 1

Introduction n n 2/25/2021 In the 1 st part, we dealt with “Kinematical Theory”

Introduction n n 2/25/2021 In the 1 st part, we dealt with “Kinematical Theory” where the scattered x-rays suffer no additional scattering. The 2 nd part is designed to give basic ideas of “Dynamical Diffraction” observed with perfect crystals as a result of multiple scattering. JASS 02 2

Basic Idea Kinematical Diffraction Dynamical Diffraction 2/25/2021 JASS 02 3

Basic Idea Kinematical Diffraction Dynamical Diffraction 2/25/2021 JASS 02 3

Maxwell Equation (1/2) E: electric field D: electric displacement H: magnetic field B: magnetic

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 e 0: permittivity of vacuum m 0: permeability of vacuum c: electric susceptibility c: speed of light in vacuum 2/25/2021 JASS 02 4

Maxwell Equation (2/2) For periodically oscillating electromagnetic field; jtrue = 0, rtrue = 0.

Maxwell Equation (2/2) For periodically oscillating electromagnetic field; jtrue = 0, rtrue = 0. For non-magnetic materials, M=0 so that B = m 0 H. 2/25/2021 JASS 02 5

Polarization P = electric dipole moment in unit volume c(r) have the periodicity of

Polarization P = electric dipole moment in unit volume c(r) have the periodicity of crystal lattice 2/25/2021 JASS 02 6

Electromagnetic Wave in Periodic Medium Bloch Theorem Incident Plane Wave in Vacuum Waves inside

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 2/25/2021 JASS 02 7

Some Mathematic. . 2/25/2021 JASS 02 8

Some Mathematic. . 2/25/2021 JASS 02 8

Mathematics (cont’d) (*) 2/25/2021 JASS 02 9

Mathematics (cont’d) (*) 2/25/2021 JASS 02 9

Basic Equations for Dynamical Theory Condition for the equation (*) should be valid for

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 2/25/2021 JASS 02 10

Boundary Conditions (1/3) z vacuum Fields in vacuum: (Ea, Da) Fields in crystal: (E,

Boundary Conditions (1/3) z vacuum Fields in vacuum: (Ea, Da) Fields in crystal: (E, D) 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 2/25/2021 JASS 02 11

Boundary Conditions (2/3) Wavefield : Superposition of plane waves Wave Vector in Crystal: Kg

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 2/25/2021 vacuum wave JASS 02 12

Boundary Conditions (3/3) Boundary Condition at z=H (Crystal Surface) 2/25/2021 JASS 02 13

Boundary Conditions (3/3) Boundary Condition at z=H (Crystal Surface) 2/25/2021 JASS 02 13

Two-Wave Approximation (1/3) Under usual experimental conditions, only two waves with K 0 (incident

Two-Wave Approximation (1/3) Under usual experimental conditions, only two waves with K 0 (incident direction) and Kg (diffracted direction) are strong inside the crystal. Wavefield in crystal: Basic Equation: Averaged refractive index of crystal: 2/25/2021 Polarization Factor JASS 02 14

Two-Wave Approximation (2/3) Condition for the basic equation, dispersion surface dispersion sphere to have

Two-Wave Approximation (2/3) Condition for the basic equation, dispersion surface dispersion sphere to have non-trivial solutions is Ko O Kg g G By introducing new parameters: 2/25/2021 JASS 02 15

Two Wave Approximation (3/3) y For non-absorbing crystals, Tg Lo When we introduce a

Two Wave Approximation (3/3) y For non-absorbing crystals, Tg Lo When we introduce a new parameter L as , To x Near the point Lo, Dispersion surfaces form Hyperbolla 2/25/2021 JASS 02 16

Amplitude Ratio j = 1, 2 2/25/2021 JASS 02 17

Amplitude Ratio j = 1, 2 2/25/2021 JASS 02 17

Diffraction Geometry Symmetric Laue Case 2/25/2021 Symmetric Bragg Case JASS 02 18

Diffraction Geometry Symmetric Laue Case 2/25/2021 Symmetric Bragg Case JASS 02 18

Symmetric Laue Case Dispersion sphere of vacuum wave (radius K) Starting point of wave

Symmetric Laue Case Dispersion sphere of vacuum wave (radius K) Starting point of wave vector Ko: P Laue point: L Deviation from Bragg Condition 2/25/2021 JASS 02 19

Symmetric Laue Case: Deviation Parameter W Solving above equations, we can get Usually W=Ws

Symmetric Laue Case: Deviation Parameter W Solving above equations, we can get Usually W=Ws 2/25/2021 upper sign: j=1, lower sign: j=2 JASS 02 20

Symmetric Laue Case: Amplitude Ratio Here, For non-absorbing crystals, 2/25/2021 JASS 02 21

Symmetric Laue Case: Amplitude Ratio Here, For non-absorbing crystals, 2/25/2021 JASS 02 21

Symmetric Bragg Case (1/2) Between L 1 and L 2, z has no intersections

Symmetric Bragg Case (1/2) Between L 1 and L 2, z has no intersections with dispersion surfaces. Total Reflection Region Deviation from Bragg Condition 2/25/2021 JASS 02 22

Symmetric Bragg Case (2/2) Dqo : Deviation from geometrical Bragg angle by refraction upper

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 2/25/2021 JASS 02 23

Rocking Curves Use monochromatic plane wave as an incident beam; Rocking the sample crystal

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. 2/25/2021 JASS 02 24

Rocking Curve: Symmetric Laue Case (1/3) Incident Wave Ka, Eoa Crystal Wave z=0 o-wave

Rocking Curve: Symmetric Laue Case (1/3) Incident Wave Ka, Eoa Crystal Wave z=0 o-wave Kg 2, Eg 2 Ko 2, E 02 z=HK , E g 1 Kga, Ega Ko 1, Eo 1 Ka, Eda g-wave Outgoing Wave o-wave g-wave 2/25/2021 JASS 02 25

Rocking Curve: Symmetric Laue Case (2/3) Boundary condition at z = 0 (incident surface)

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), 2/25/2021 JASS 02 26

Rocking Curve: Symmetric Laue Case (3/3) 2/25/2021 JASS 02 27

Rocking Curve: Symmetric Laue Case (3/3) 2/25/2021 JASS 02 27

Rocking Curve: Symmetric Bragg Case (1/3) Boundary condition at z = 0 Ka, Eoa

Rocking Curve: Symmetric Bragg Case (1/3) Boundary condition at z = 0 Ka, Eoa Kga, Ega z=0 Ko 1, Eo 1: W<-1 Ko 2, Eo 2: W>1 2/25/2021 Kg 1, Eg 1: W<-1 upper sign: W<-1, lower sign: W>1 Kg 2, Eg 2: W>1 JASS 02 28

Rocking Curve: Symmetric Bragg Case (2/3) Another solution will give a divergent solution 2/25/2021

Rocking Curve: Symmetric Bragg Case (2/3) Another solution will give a divergent solution 2/25/2021 JASS 02 29

Rocking Curve: Symmetric Bragg Case (3/3) Rocking curve (Darwin Curve) |W|<1: All incident energies

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 q. B by Range of total reflection (-1<W<1) Darwin Width, ~microradian order 2/25/2021 JASS 02 30

Absorbing Crystal Absorption: Anomalous dispersion term into atomic scattering factor Centrosymmetric Crsyatls A new

Absorbing Crystal Absorption: Anomalous dispersion term into atomic scattering factor Centrosymmetric Crsyatls A new parameter k is defined as 2/25/2021 JASS 02 31

Symmetric Laue Case: Absorbing Crystal (1/2) Oscillating Term, Hardly to be observed experimentally without

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) 2/25/2021 JASS 02 32

Symmetric Laue Case: Absorbing Crystal (2/2) Forward Diffraction Thin Crystal 2/25/2021 Thick Crystal JASS

Symmetric Laue Case: Absorbing Crystal (2/2) Forward Diffraction Thin Crystal 2/25/2021 Thick Crystal JASS 02 33

Symmetric Bragg Case: Absorbing Crystal Rocking curve for a symmetric Bragg case diffraction from

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 2/25/2021 JASS 02 34

Summary n n n Very quick scan of x-ray diffraction theory was attempted. You

Summary n n n 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 u Handbook on Synchrotron Radiation Vol. 3, North-Holland, 1991. u 2/25/2021 JASS 02 35

Thank you for your attention. Acknowledgement Some materials presented here are originally prepared by

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. 2/25/2021 JASS 02 36

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