phasing_handout1 - Intensities of the Reflections With the...

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Intensities of the Reflections With the help of Braggs law and the Ewald construction, we can calculate the place of a reflection on the detector, provided we know the unit cell dimensions. Indeed, the position of a spot is determined alone by the metric symmetry of the unit cell. The intensity of a spot, however, depends on the contents of the unit cell (and, of course, on exposure time, crystal size, etc. ). Reflections are weakened by the radius of the atoms (atomic form factor) and the thermal motion of the atoms (temperature factor U). Both these effects are stronger at high resolution. d ½ ½ δ Removed due to copyright re strictions . Please see: Massa, Werner. Crystal Structure Determination. 2nd ed. Translated into English by R. O. Gould. New York, NY : Springer, 2004. ISBN: 3540206442.
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Structure Factors Every atom in the unit cell contributes to every reflection according to its chemical nature and its relative position. Owing to this shift in position relative to the other atoms, the photons contributed by each atom in the unit cell have a phase shift relative to those from other atoms. a b y 2 x 2 π 2 0 ∆Φ ( ) i i i c i b i a i i lz ky hx + + = ∆Φ + ∆Φ + ∆Φ = Φ 2 ) ( ) ( ) (
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Structure Factors ( ) i i i c i b i a i i lz ky hx + + = ∆Φ + ∆Φ + ∆Φ = Φ π 2 ) ( ) ( ) ( This makes the structure factor a complex number: [] ( ) i i i i i i i f i f F Φ + Φ = Φ = sin cos exp Every atom i in the unit cell contributes to every structure factor F ( hkl ) (that is reflection) according to its position in the cell and its chemical nature (different values for f i !): ( ) ( ) + + + + + = i i i i i i i i lz ky hx i lz ky hx f hkl F 2 sin 2 cos ) (
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Complex Numbers and the Argand Plane In general: Complex numbers have both real and imaginary components: They have the general form x = a + ib , where . 1 = i i b a Euler’s equation gives a different perspective for complex numbers: ( )( ) α sin cos i e i = = functions: and in a Taylor’s series about α = 0 and matching the expressions term for term. i + = Another check is that both expressions obey the same differential equation: y ’ = iy . R i α 1 The expression is apparently a complex number represented by a vector of unit length and angle α in the complex plane. The expression is a complex number of magnitude (length) r and angle α . i e i re R This equation can be verified by expanding the α= ) sin( ) cos( ) ( i y e y ) (
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Complex Numbers and the Argand Plane A given complex number may be specified by either its real and I maginary components ( a and b ) or its magnitude and phase ( r and α ), with the following relationships between them: R R i b a ( ) α cos r a = ( ) sin r b = i α r () 2 / 1 2 2 b a r + = ( ) a b 1 tan = Structure factors are complex quantities. If the magnitude of a particular reflection is measured but the phase has not been determined yet, the possible values of that structure factor can be represented by a circle of radius in the complex plane.
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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phasing_handout1 - Intensities of the Reflections With the...

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