Macromolecular Crystallography

Section 3.1, Light Scattering

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LIGHT

Light and matter: a driving wave and a simple oscillator...

This model for light and matter allows a consistent explanation of a very large part of the subjects which can be included under the general term “Spectroscopy”.

Several different representations and concepts:

wavelet.png

Ray direction of wave propagation, or direction of a beam of photons.

Wave of wavelength λ, each wave broad and long enough to perform as expected.

Photons specific quantum energy 1/λ, a photon = 1 quantum, a chunk of a wave broad and long enough to perform as expected.
BEWARE: The particle description of a photon is very misleading for scattering/diffraction effects: deflection of balls by other balls or objects is NOT the way photons seem to interact with atoms and produce scattered photons.

Light has both electric and magnetic properties, i.e. electromagnetic wave. One can almost always ignore the magnetic component and still adequately explain experimental phenomena.

Light interacts with matter: The driving wave interacts with an oscillator. Considering the sinusoidally varying electric field of the wave, the oscillator must involve an oscillating dipole, this could be an induced dipole in a polarizable object. So we are getting a measure of polarizability of a medium, e.g., molecules.

Energy can be transfered from (or to) this oscillator, for this introduction we will often ignore most processes except re-radiation of light. (i.e. we can go through an absorption band, but ignore the part that is actual absorption). Absorption is just the loss of energy from the oscillator before it re-radiates. Absorption thus can be thought of as energy lost in the process of getting the oscillator started and keeping it going.

Oscillator (oscillating dipole) can radiate energy (light) So we must consider certain points:

1) Must investigate properties of original interaction

2) Must know character of re-emitted light: is it different from original wave? If so, how?

3) Must investigate how light waves interact with each other since we have opened the possibility of various light waves: e.g. original wave and emitted waves from various oscillators.

Avoid quantum mechanics: except note that any system, including our oscillators, can exist in only certain energy levels; so energy is handled only in discrete chunks which will be important occasionally to consider. But, the picture of a driving wave and a simple oscillator does, in fact, explain much of the observed light scattering phenomena!



EMspectrum.jpg


COMBINATION OF [X-RAY : LIGHT] WAVES

ABCDwaves.png

Waves A and B are exactly in phase with one another, and they combine constructively to give a wave of twice their amplitude.

Waves B and C are exactly out of phase, and they cancel completely, combining to give a wave of zero amplitude.

Waves C and D are somewhat out of phase, and they combine to give a wave with an intermediate phase and less than twice their original amplitude.

adding-waves.2.kin shown in KiNG



VECTOR NOTATION FOR WAVES

The length of the vector represents the amplitude of the wave, and the direction of the vector represents the phase of the wave.

ABCDcombinedWavesCircles.png

The construction of adding the two vectors head to tail gives the correct amplitude and phase for the combined wave, as above. These are just vectors in the complex plane and can be written as:

vectorEquation.jpg

φ factor (eiφ); eiφ = cos(φ) + isin(φ)



REPRESENTATION OF LIGHT WAVES

Plane polarized light: the simplest case

planePolarizedWaveY.png planePolarizedPhaseY.png
planePolarizedWaveX.png planePolarizedPhaseX.png

UNPOLARIZED LIGHT

Electric field vectors in all x,y directions unpolarizedEvectors.png


each component vector30deg.png or vector60deg.png can be treated as composed of x and y
components in phase with each other: vector30degxy.png vector60degxy.png


The real components add directly xyRealComponents.png
only if φ angle same (i.e. could define φy = φx = 0)


Thus can treat unpolarized light as the resultant of two plane polarized rays in phase. Any arbitrary photon in an unpolarized light ray can be broken into two components, and a great number of photons will then average out to give equal intensities in the two directions of polarized light chosen.

xyPolarizedWave.png     xyPolarizedVectors.png


complex planes xyComplexPlaneVecs.png phase vectors φy = φx = 0)



UNPOLARIZED LIGHT SCATTERING FROM A SINGLE PARTICLE

Point Particle (single oscillator)

Straight through:

scatteringStraight.png Perhaps a phase shift with respect to the original wave but otherwise no effect.


As seen from an angle, i.e. those photons scattered in a particular direction.
For example: direction of scattering in the plane of the paper:

scatteringAtAngle.png

F•scattered = F•original
F↕scattered = F↕original cos(θ)
F↕scattered is a measure of the projection of the dipole motion ⊥ to the direction of scatter.


Ioriginal = F2original
Iscattered = I•scattered + I↕scattered (add components)
Iscattered = F•2scattered + F↕2scattered
The amount of polarization and total Intensity, I = F2, will vary as a function of θ
In the simple case at θ=90°, the ray is completely polarized.
Later we will find that we can describe diffraction as a "reflection" from a plane (a plane of oscillators) and see that this simple 2D diagram works as a projection to explain the general case for crystals.

"X-RAY DIFFRACTION", B. E. Warren, MIT, 1969
Chapter 1 "X-RAY SCATTERING BY ATOMS"
"1.1 CLASSICAL SCATTERING BY A FREE ELECTRON
When an x-ray beam falls on an atom, two processes may occur: (1) the beam may be absorbed with an ejection of electrons from the atom, or (2) the beam may be scattered. We shall first consider the scattering process in terms of classical theory. The primary beam is an electromagnetic wave with electric vector varying sinusoidally with time and directed perpendicular to the direction of propagation of the beam. This electric field exerts forces on the electrons of the atom producing accelerations of the electrons. Following classical electromagnetic theory, an accelerated charge radiates. This radiation, which spreads out in all direction from the atom, has the same frequency as the primary beam, and it is called scattered radiation."



UNPOLARIZED LIGHT SCATTERING FROM A SINGLE OBJECT (tumbling in space)
(DIGRESSION FORM CRYSTALLOGRAPHY)

GENERAL CASE: single object tumbling in solution

For scattering from discrete objects tumbling in solution we would need to consider the general case of scattering in all directions.
scatteringTumbling.png

Scattering of Io unpolarized = scattering components 0.5 Io(x) + 0.5 Io(y),
equivalent but for different effective polarization
so one has intensity = 0.5 Iosin2θ1; and the other has Intensity = 0.5 Iosin2θ2.
Thus: Itotal scattered = Ix + Iy = 0.5 Io(sin2θ1 + sin2θ2) = 0.5 Io(1 + cos2θ)


This describes Intensity in any one direction (i.e. for a particular solid angle)

Field strength = Amplitude per unit area (on a sphere at distance r) decreases as 1/r

Intensity per unit area decreases as 1/r2

This is important since measuring scatter from isolated objects the 1/r term is needed to match observed scatter at the measuring device. (Note that for scattering from a crystal the allowed direction of scatter is limited and the measuring device can capture all the light scattered in a particular direction, so there is not a fall-off of intensity just due to radial spread.)

Now we have some of the geometrical terms in the equation for light scattering from particles whose diameter is << λ ; i.e., where we can treat the particle as a single oscillator. In practice that diameter is : d < λ/20.

Small Angle X-ray Scattering (SAXS)

Solution scattering gets especially interesting when the wavelength of the light is of the same length as features within the tumbling molecule, i.e. just the same relations that make crystal scattering really useful for detailed structure determination. In addition to the Intensity equations above, must be added intensity scattered from features within the molecules averaged by tumbling. Rather than building a model into the image obtained from crystallography, the best that can be done is to propose shapes for the molecule and see if the data is consistent with what would be observed from such shapes tumbling in solution. Of course, if one already knows a great deal about the possible structures that a particular molecule might have, like opening/closing between subunits, or donuts around DNA, then the models can be quite detailed. Later, we will derive the general scattering equation - though the crystallographer does not have to describe the tumble averaging.



LIGHT AS A WAVE

Interaction of light and matter

Analogy: Damped Driven Simple Harmonic Motion
Example: bound "electron" and driving "wave"
(styrofoam ball on yarn driven by hand wave...)

Phase lag of an oscillator, scattered wave always in phase with oscillator:

phaseLag.jpg

(Quantum mechanics has its own way of getting broad absorption bands.)
Identify the “oscillators” and we have the field of Absorption Spectroscopy.

Points to note:

1. Oscillator acts as a radiation source: radiated wave is in phase with the oscillator.

2. The energy that can be pumped into the system, and thus the amount reradiated increases as ω → ω0

3. For visible light ω < ω0, so oscillator and emitted wave in phase (0°) with the driving wave.

For x-rays ω > ω0 for most electrons in atoms, so oscillator and resulting scattered wave are 180° out of phase with the driving wave. Since one is usually concerned with the scattered wave, this is often defined so that the scattered wave is at 0° and the driving wave at 180°.

4. Usual to treat phase of actual scattered light as having a usual component at 0° and an anomalous component with 90° phase difference (lag). For x-rays with scattered wave redefined to be 0° this is 90°phase advance, see later for a way to draw this).



PHASE SHIFT

Phase lag of an oscillator, scattered wave always in phase with oscillator:

(refractive index η is a measure of interaction)

phaseShift.png

η is a measure of the interaction of light with matter: interaction affects relative phase of the oscillator with respect to the driving wave and a probability of feeding energy into the oscillator. The more energy in the oscillator, the more can be lost to other processes like absorption.

In any real situation there may be many types of oscillators in the medium interacting with the light but perhaps only one type near enough to resonance to have a significant anomalous scattering component. Also, in common situations almost all of the effective oscillators will have a resonance frequency either greater than the light frequency, as is the case for visible light; or much smaller, as is the case for x-rays. Any odd oscillator which happens to be far on the other side of resonance will make its unique and minor contribution 180° out of phase with the bulk of the scattered radition and thus very slightly diminish the scattered intensity.


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