1. 1. Detector 2. Crystal diffraction conditions
2-7-05

2. Detector in X-ray crystallography: Image plates (IP)
The imaging plate (IP) is a reusable 2-D X-ray detector.
Imaging plates are sensitive not only to X-rays, but other types of radiation as well: gamma radiation, alpha and beta particles, neutrons and electron beams, and serve as the primary radiation detector in many fields
Advantages: high sensitivity, low noise, wide dynamic range, good linearity, good spatial resolution, modest cost.
It consists of a photostimulable phosphor powder in an organic binder with a thickness between 25 and 150 mm deposited on a flexible polymer support film of about 250 mm thickness.
The family of compounds comprising BaFX:Eu2+ (X = Cl, Br) have been known to have high luminescence efficiency for X-ray excitation.
Phosphor
Supporting film
A mar detector
Dia. = 345mm

3. Detector in X-ray crystallography: Image plates (IP)
The IP is exposed to the x-ray source and stores the impinging X-rays as a latent image in the phosphor.
This image is recovered by scanning a laser beam across the IP, causing photostimulated luminescence (PSL).
This PSL is recorded by a photomultiplier that is scanned across the IP at the same time as the laser.
The resultant signal is digitized and stored in a file for data processing.

4. Image plates are commonly used in academic laboratories
Disadvantage: decay, high background, slow readout.
Advantages: high sensitivity, low noise, wide dynamic range, good linearity, good-to-high spatial resolution, modest cost.

5. Detector in X-ray crystallography: CCD
Charge Coupled Device (CCD) was invented in the late 1960s by researchers at Bell Labs, The CCD's  superb ability to detect light has turned it into the industry-standard image sensor technology.
CCD Basics - CCD imaging is performed in a three-step process:        1. Exposure, which converts light into an electronic charge at discrete sites called pixels        2. Charge transfer, which moves the packets of charge within the silicon substrate        3. Charge-to-voltage conversion and output amplification.

6. Detector in X-ray crystallography: CCD
The three main components of the Quantum CCD detector are phosphor screen (to convert X-rays to light), fiber-optic taper (to demagnify the light image down to the size of the CCD chip), and CCD chip to detect the light image as an electric charge image.  The electric charge image is read  out of the CCD chip and digitized (converted to binary numbers) then fed into a computer.
An ADSC CCD detector used in many major synchrotron stations

7. Detector in X-ray crystallography: CCD
The principal advantages of CCDs are their sensitivity, dynamic range and linearity.
The sensitivity, or quantum efficiency, is simply the fraction of photons incident on the chip which are detected. It is common for CCDs to achieve a quantum efficiency of about 80%. Compare this figure with only a few percent for photographic plates.
A typical dynamic range (that is, the ratio of the brightest accurately detectable signal to the faintest) is about 105. The corresponding figures for a photographic plate are a range of less than about 1000.
Within this dynamic range the response is essentially linear: the size of the signal is simply proportional to the number of photons detected, which makes calibration straightforward.
Fast readout of digitized images
Disadvantages: High costs and small areas

8. A typical diffraction pattern

9. Diffraction geometry: an outline
Diffraction basics
Laue equations
Braggs law
Reciprocal space and diffraction
Diffraction methods
– Laue photographs
– Oscillation photographs

10. X-ray is an electromagnetic wave
The electric component: E=Acos(wt + a)
Three component: amplitude, frequency, initial phase angle

11. Interference between waves
Waves with the same frequency can interfere:
(a) in phase
(b) out of phase
(c) partially out of phase

12. Double slit experiment

13. Diffraction at a single slit

14. The envelop function

15. Diffraction and sampling

16. Diffraction from crystals
A crystal is a three dimensional diffraction grating
The lattice periodicity of the crystal determines the sampling regions of the diffraction pattern Where the peaks appear
The unit cell contents (the distribution of electron densities) give you the envelope Function
The intensity of the peaks

17. A wave can be represented as a vector in a Argand diagram
The electric component of an electromagnetic wave is:
A cos(wt+a) = A cosa coswt – A sina sinwt
= A cosa coswt + A sina cos(wt+90°)
Acosa
Asina A
Real
Imaginary
The above vector corresponds to a wave Acosa + iAsina = Aexp[ia]

18. Scattering for a system with two electrons
where
What is S???
S is perpendicular to the imaginary reflecting plane

19. Scattering for a system with two electrons: origin shift does not change wave intensity
The amplitude and intensity of wave T does not change

20. Scattering by an atom Atomic scattering factor is:
Assuming centrosymmetric electron cloud distribution
The electron cloud of an atom
f is real if centrosymmetry is assumed
f for a carbon atom

21. Scattering by a unit cell
For each atom, the scattering is:
A unit cell with three atoms
For the whole unit cells, we will have:
Where F(S) is called structure factor
Structure factor F is the sum of scattering by all atoms

22. Scattering by a crystals
A crystal contains a large number of unit cells
For a general unit cell in a crystal, the structure factor is:
The total wave scattered by the crystal is:

23. Scattering by a crystals
A crystal does not diffract X-ray unless
Not constructive interference unless all waves have the same phase
This is the famous Laue condition

24. Laue conditions specify scattering conditions
What is S???
S is perpendicular to the imaginary reflecting plane
What does Laue condition mean???
a, b, c are unit vectors of crystal unit cell axes
For a particular crystal unit, X-ray will only be scattered along discrete directions

25. Diffraction geometry: Bragg’s law
A Crystal contain many lattice planes
Lattice plane reflect incident X-ray
Scattered X-rays interfere with each other
Constructive interference results only when:
2dsinq=nl
This is the Bragg’s law

26. Diffraction geometry: Bragg’s law
From
Diffraction angle for a particular lattice plane can be calculated from a particular lattice plane, or vice versa.

27. Real space and reciprocal space

28. Real space and reciprocal space
Imaginary reciprocal lattices are created to help us understand diffraction geometry
bcsina
acsinb
absing
a* =
b* =
c* = V V V

29. Real space vs. reciprocal space
In addition:
a*a=1
V=1/V*

30. The construction of a reciprocal lattice
The reciprocal unit cell dimension is inversely related to the direct unit cell dimensions.
Each reflection falls onto to a reciprocal lattice point
The length of a reciprocal vector is inversely related to the distance between crystal lattice planes that give rise to that particular reflections.

31. Physical meaning of reciprocal lattice
If Laue conditions are met, the end point of the vector S(h,k,l) are located in the reciprocal lattice points.
Proof:
S can always be written as S = X  a* + Y  b* + Z  c*
Therefore,
a S = X ( a*  a) + Y  (b*  b) + Z  (c*  c)
= X
Because Laue condition states: a S = h, therefore,
X=h

32. The construction of Ewald sphere
Ewald sphere is a geometric construction that allows one to visualize the diffraction directions and the properties of Bragg's law

33. The construction of Ewald sphere
Ewald sphere is a geometric construction that allows one to visualize the diffraction directions and the properties of Bragg's law
Diffraction occurs when a reciprocal lattice point intersects the Ewald sphere.

34. Conclusions for reciprocal lattice
The reciprocal lattice rotates exactly as the crystal does.
The Ewald sphere allows us to visualize a diffraction experiment
The diffracted beam will "travel" from the center of the Ewald sphere through the point where its reciprocal lattice point intersects the sphere.
The radius of the Ewald sphere is 1/l Diffraction only occurs when a reciprocal lattice point intersects the Ewald sphere