Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Crystallography using X-Ray based methods Jakob Odersky Simon L¨ owe EPFL December 19, 2012 Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Table of Contents Basic properties of X-Rays 1 Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet Identifying a substance using a polycrystalline method 2 Identifying a substance using a monocrystalline method 3 Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Why X-Rays? Why X-Rays? Penetration of “soft matter” Stopped by “hard matter“ Wavelength same order as atom dimension ( ≃ ˚ A) → Cristal diffraction Figure: http://www.2mcctv.com/blog/wp-content/uploads/2012/10/electromagnetic-spectrum.jpg , Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Production of X-Rays Production of X-Rays Emission of electrons by a heated cathode (e.g. Tungstene) Acceleration due to an external potential V 0 Induction of electron transitions in anode (Cu, Mo, ...) Emission of X-Rays Figure: http://en.wikipedia.org/wiki/File:WaterCooledXrayTube.svg , 15.12.12 Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Study of the spectrum Two types of spectra Continuous spectrum → Strong deceleration of the electrons in the anode results in emission of Bremsstrahlung → Duane-Hunt law yields λ min (˚ A) = 12 . 398 V 0 (kV) Discrete spectrum → Incoming electrons induce transition from lower energy state E 1 to higher energy state E 2 → Deexcitation process emits photon of wavelength E 2 − E 1 = hc hc λ = ∆ E Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Study of the spectrum Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Elastic scattering from a single electron Figure: Taken from the Petr Leiman’s lectures. Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Elastic scattering from a single electron Electromagnetic wave scattering Given an incident wave for electron at r e : E ( r e , t ) = E 0 exp(2 π i ( k 0 r e − ν t )) Scattered spherical wave at R for R ≫ r e : E ( R , t ) = E or exp( − 2 π i Sr e ) where S = k − k 0 the scattering vector Generalization to an extended object: F ( S ) = F [ ρ ]( S ) where ρ the electron density → the recorded diffraction pattern is an image of the reciprocal space Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Elastic scattering from a single electron Figure: Taken from the Petr Leiman’s lectures. Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Diffraction Diffraction Diffraction is scattering dominated by interference effects Destructive interference in most directions Constructive interference in well defined directions → Problem: what are these directions? Appears when there is internal order in the sample → crystals Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet What is a crystal? Crystal A crystal is characterized by : homogeneity invariance of the sides microscopic periodicity anisotropy of physical properties Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Crystal Structure Figure: http://4.bp.blogspot.com/-NB-CjJtE2Zg/Tja397z4-II/AAAAAAAAAKY/qxOrzU2cQuY/s1600/ space-lattice-unit-cell-represenatation.jpeg , 17.12.12 Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Diffraction conditions Laue Conditions and Bragg’s Law One can show that the diffraction condition are: Sa = h Sb = k Sc = l where h,k,l are the Miller indexes Furthermore, it is easy to show that the reciprocal lattice vectors correspond to the scattering vectors which can give rise to diffraction S = H ⇒ H = 2 sin θ and λ H = n ( h a ∗ + k b ∗ + l c ∗ ) ⇒ H = nr hkl = n d hkl → This yields Bragg’s Law: 2 d hkl sin θ = n λ Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Ewald Sphere Figure: Taken from the Petr Leiman’s lectures. Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet A first application Determining the thickness of a nickel sheet Absorption of X-Rays in matter: I ( x ) = I 0 exp( − µ x ) where µ depends on the matter and the wavelength � I 0 � ln I Knowing µ one can determine the thickness: d Ni = µ Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet A first application Determining the thickness of a nickel sheet Absorption of X-Rays in matter: I ( x ) = I 0 exp( − µ x ) where µ depends on the matter and the wavelength � I 0 � ln I Knowing µ one can determine the thickness: d Ni = µ For the K β 1 transition of copper, the mass attenuation coefficient is µ m ≃ 2 . 79 · 10 2 [ cm ] 2 [ g ] − 1 ⇒ µ = ρ Cu µ m ≃ 2483 . 1[ cm ] − 1 This yields: d thick = 25[ µ m ] d thin = 6 . 39[ µ m ] Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
Production of X-Rays Basic properties of X-Rays Study of the spectrum Identifying a substance using a polycrystalline method (X-Ray) Diffraction Identifying a substance using a monocrystalline method Crystals Determining the thickness of a nickel sheet Determining the thickness of a nickel sheet Jakob Odersky, Simon L¨ owe Crystallography using X-Ray based methods
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