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Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA 1. X-Ray Diffraction 2. Electron Diffraction 3. Neutron Diffraction Friday, July 1, 2016 Prof. Dr. Abdul Majid 1 Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Crystal Diffraction Electron Diffraction Friday, July 1, 2016 Prof. Dr. Abdul Majid 2 Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Crystal Diffraction Bragg’s Law ππ = ππ πΊππ π½ Friday, July 1, 2016 Prof. Dr. Abdul Majid 3 Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Crystal Diffraction Bragg’s Law ππ = ππ πΊππ π½ Friday, July 1, 2016 Prof. Dr. Abdul Majid 4 Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Electron Diffraction Properties of Electron According to de Broglie duality relation where p is the momentum, λ is the wavelength, and h is Planck’s constant h = 6.626 x 10-34 J.s = 4.136x10-15eV.s To observe wave-like behavior, we require some kind of grating where the “distance between slits” is of order the wavelength. At typical laboratory energies, the electron’s de Broglie wavelength is of order one Angstrom (10–8 cm), about the same size as the interatomic spacing in common crystals. The regular atomic arrays in crystals are thus perfectly scaled gratings for creating a “matter wave” diffraction pattern, measuring their wavelength, and verifying de-Broglie relation. As an added bonus, with the principle verified, the diffraction patterns then become powerful tools for the study of crystal structure. Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Electron Diffraction The de Broglie Wavelength vs. Voltage Electron is accelerated through high voltage V. Its energy and momentum are then given by Solving for the momentum: Thus, a 150 V electron has a de Broglie wavelength of 1 Angstrom, and the wavelength should vary in inverse proportional to the accelerating voltage. Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Electron Diffraction Single Crystal Diffraction pattern of Structure Powder Diffraction The Debye-Scherrer technique. Diffraction pattern of Au Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Electron Diffraction Diffraction Pattern of Structure Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Electron Diffraction There are three particularly important features of diffraction using high energy electrons: (1) Since l is very small, Bragg angles are also small, so the Bragg Law can be simplified to: l = 2dqB (2) The diameter of the Ewald sphere is very large compared to the size of the unit cell in the reciprocal lattice. (3) Lenses are able to focus the diffraction pattern and to change the camera length, which is equivalent to moving the film in an x-ray experiment. Electron energy / keV Wavelength / pm Diffracting planes qB 50 5.355 Cu111 0.75 100 3.701 Al200 0.54 300 1.969 Si200 0.32 Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Neutron Crystal Diffraction The diffraction maximum traces a circle in the projection plane. The circle is the base of a cone whose half-angle is given by where R is the radius of the circle, and L is the distance from the target to the screen. Combining this with the Bragg condition, and assuming R<<L gives and thus, if L and λ are known, measurement of the radius yields d, the distance between Bragg planes. Electron diffraction becomes a tool for measuring inter-atomic distances in crystals Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Neutron Crystal Diffraction According to Bragg law nl = 2dSinq nl = d(2q) π = tan(2q) πΏ As OR π = 2q πΏ ππ nl = πΏ for R<<L According to de Broglie duality relation β π = π= nl= π βππ β 2πππ πβ 2πππ = ππ πΏ πβπΏ = π 2πππ Geometry of electron diffraction in a TEM. L is the effective camera length, T is the transmitted spot, D is the diffracted spot, R is the distance between T and D, and 2q is the Bragg angle. The angle 2q is very small due to the small wavelength of the electrons, so Bragg’s Law reduces to 2dq ≈ π. Additionally, it can be seen that tan(2q) = R/L, which reduces to 2q ≈ R/L. Combining these two equations yields Rd ≈Lπ. Majmaah University Department of Physics College of Science, Al-Zulfi KINGDOM OF SAUDI ARABIA Neutron Crystal Diffraction nl= πβ 2πππ π βππ = = ππ πΏ πβπΏ π 2πππ In the case of a cubic crystal,