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Structural Biology Methods
Fall 2022
Lecture #2

Electromagnetic wave
Distance
E – electric field strength
t – time
z – position
A – amplitude
λ – wavelength
2π – to convert relative distance to angles

Distance
E (t=0; z) = A cos (2p z/l)
Electric field strength
M (t=0; z) = A sin (2p z/l)
E (t=0; z) = A cos (2p z/l)
F
F = A cos(2p z/l) + i A sin(2p z/l)

•F = A cosa + i A sina
•F = A exp(ia)
a
A
Real axis
A - wave amplitude
a - wave phase angle
Argand_diagram.jpg
F
A
Wave as a vector

=



=



scattering_from_one_atom.jpg
X-rays scatter from electrons in all directions
Primary beam
Secondary beams
coherent_beam.jpg

scattering_from_multiple_atoms.jpg
X-ray scattering from several electrons
Primary beam
wave_interactions.jpg coherent_beam.jpg
When do electrons scatter “in phase” so that waves add constructively?

Molecule is composed of many electrons
Each electron will scatter secondary radiation uppon exposure to x-rays
The scattered secondary beams will interact and cause interference
The scattering from a molecule is dependent on number of and  distances between electrons
In other words, scattering from molecule is dependent on its structure
If we would know the amplitudes and phases of scattered molecule, we could calculate the structure
of molecule...

There is no path and PHASE DIFFERENCE when rays reflect from a plane
coherent_beam.jpg coherent_beam.jpg


nl = 2d sinq
Bragg’s law:
There is NO PHASE DIFFERENCE if the path differences are equal to whole number multiplies of
wavelength
w
wave_interactions.jpg coherent_beam.jpg coherent_beam.jpg
sinq = w/d
2w = nl

Picture 7 Picture 8



nl = 2d sinq
Bragg’s law:
sinq = w/d
2w = nl
There is NO PHASE DIFFERENCE if the path differences are equal to prime number multiplies of
wavelength (l)
w
wave_interactions.jpg

scattering_from_one_atom.jpg
X-rays scatter from electrons in all directions
Primary beam
Secondary beams
coherent_beam.jpg

Dot product



System of two electrons






-s0















Scattering by an atom






Scattering by an atom depends of the length of |S| (resolution)



Scattering by a unit cell






Scattering by a crystal












Diffraction Conditions



Bragg planes are identical to lattice planes



=>






Reciprocal lattice and Ewald construction






Expected end of lecture #2






























Wave as a vector
•F=Acosa+iAsina
•F=Aexp(ia)
a
A
Real axis
A- wave amplitude
a- wave phase
Argand_diagram.jpg
F
A

Phase problem
α
phase_problem.jpg
α
α




•F = A cosa + i A sina
•F = A exp(ia)
a
A
Real axis
A - wave amplitude
a - wave phase angle
Argand_diagram.jpg
F
A
Wave as a vector







Wave description
E(t=0; z) = A cos (2πz/λ)
E – electric field strength
t – time
z – position
A – amplitude
λ – wavelength
2π – conversion to angles
ν – frequency
c – speed of light
ω – angular velocity
ν = c/λ         ω=2πν