1. Generation of and Properties of X-rays
Before we can talk about X-ray diffraction, we must briefly discuss the generation and properties of X-rays

2. A New Kind of Rays Wilhelm Conrad Röntgen
German physicist who produced and detected Röntgen rays, or X-rays, in 1895.
He determined that these rays were invisible, traveled in a straight line, and affected photographic film like visible light, but they were much more penetrating.
X-rays were first discovered by Röntgen, a German physicist in 1895, , an achievement that earned him the first Nobel Prize in Physics in 1901.
The slide title is taken from the name of Rontgen’s original paper, “On A New Kind of Rays”
The first radiograph he took was of his wife’s hand. The picture here is of his colleague, Albert von Kolliker.
These rays were invisible, traveled in a straight line, and affected photographic film like visible light. Much more penetrating.
They were called X-rays because of their unknown nature.

3. Properties of X-Rays Electromagnetic radiation (l = 0.01 nm – 10 nm)
Wavelengths typical for XRD applications:
0.05 nm to 0.25 nm or
0.5 to 2.5 Å
1 nm = 10-9 meters = 10 Å
E = ħc / l
X-ray region from 10 to .01 nm
X-rays are a form of electromagnetic radiation.
Electromagnetic radiation has characteristics of both waves (according to classical theory) and particles (according to quantum theory).
Particles of electromagnetic radiation are called photons and have an energy associated it of hc/lambda (h is Planck's constant and c is the speed of light)
This is the region of X-rays used in XRD. It is in between the so-called Hard (shorter wavelength) and Soft (higher wavelength) X-ray region. 3

4. Generation of Bremsstrahlung Radiation
X-ray
Fast incident
electron
nucleus
Atom of the anode material
electrons
Electron
(slowed down and changed direction)
Bremsstrahlung radiation means “braking” radiation.
Electron deceleration releases radiation across a spectrum of wavelengths.
The braking radiation represents a continuum (white radiation).
“Braking” radiation.
Electron deceleration releases radiation across a spectrum of wavelengths.

5. Generation of Characteristic RadiationK L K K L M
Emission
Photoelectron
Electron
Incoming electron knocks out an electron from the inner shell of an atom.
Designation K,L,M correspond to shells with a different principal quantum number.
Here the electrons suddenly decelerate upon colliding with the metal target. If enough energy is contained within the electron, it is able to knock out an electron from the metal atom’s inner shell.
This is an unstable state, and this vacancy is quickly filled by an electron from a higher shell. This process releases a “quantum” or photon of radiation that has a wavelength (or energy) characteristic of the energy difference between shells.
The designations K, L, and M correspond to the quantum number q = 1, 2, 3
α and β indicates the shell that the “filling” electron is from relative the vacancy shell.
Ka indicates the photon that is released from an electron transition from the L shell to the K shell

6. Generation of Characteristic Radiation
Bohr`s model
Not every electron in each of these shells has the same energy. The shells must be further divided.
K-shell vacancy can be filled by electrons from 2 orbitals in the L shell, for example.
The electron transmission and the characteristic radiation emitted is given a further numerical subscript.
In reality, the picture a little more complicated. All electrons in each of these shells do not have exactly the same energy. The shells themselves must be further divided to indicate this.
So, in this example a K-shell vacancy can be filled by 2 different L-shell electrons with slightly different energies. These shells are designated 1 and 2. So when we talk about Kα radiation here we are actually talking about both the Kα1 and Kα2 wavelengths (and sometimes more).

7. KKK000
Generation of Characteristic Radiation
Energy levels (schematic) of the electrons M
Intensity ratios
KKK000 L
The L level is actually made up of 3 different energy levels .
K
K
K
K K

8. Emission Spectrum of an X-Ray Tube
Braking = continuous spectra
Characteristic = line spectra
At 8.5 kV, no characteristic line is produced.
As accelerating voltage is increased, characteristic lines appear and grow in intensity, but so does the braking radiation. Although the characteristic radiation is much stronger, all of these wavelengths can participate in XRD and at the very least cause undesirable background.

9. Emission Spectrum of an X-Ray Tube: Close-up of Ka
As we will see later, we must have a source of monochromatic X-rays for X-ray diffraction. We must try to eliminate white radiation and Kβ radiation to use only the Kα.
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley

10. Sealed X-ray Tube Cross Section
The tube itself is evacuated and sealed to insulate the anode and cathode, prevent contamination, and generally extend the tube’s lifetime.
The cathode is a tungsten filament, and the anode is a metal target (Cu, Co, Cr, Fe, or Mo). The filament current of ~ 3 amps produces electrons that are accelerated to the target by the voltage across the cathode and anode. Tube current is a measure of the flow of electrons from the filament to the target.
X-rays are emitted in all directions, but they only exit the tube at 2 or more Be windows. The windows maintain the evacuated environment, but are very transparent to X-rays.
This process produces considerable heat, which necessitates water cooling.
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley
Sealed tube
Cathode / Anode
Beryllium windows
Water cooled 10

11. Characteristic Radiation for Common X-ray Tube Anodes
Ka1 (100%)
Ka2 (50%)
Kb (20%) Cu Å Å Å Mo Å Å Å
Note how anodes made of a different material have Ka lines with different wavelengths.
To get a different wavelength, simply change the composition of the anode!!

12. Modern Sealed X-ray Tube
Tube made from ceramic
Beryllium window is visible.
Anode type and focus type are labeled.
Tube now made of ceramic. Beryllium windows

13. Sealed X-ray Tube Focus Types: Line and Point
Target
Filament
Spot
Take-off angle
The X-ray beam’s cross section at a small take-off angle can be a line shape or a spot, depending on the tube’s orientation.
The take-off angle is the target-to-beam angle, and the best choice in terms of shape and intensity is usually ~6°.
A focal spot size of 0.4 × 12 mm:
0.04 × 12 mm (line)
0.4 × 1.2 mm (spot)
Target
Line
The filament is shaped to produce a line focus on the target material.
The beam’s cross section viewed at a small take-off angle can be line- or spot-shaped, depending on the tube’s orientation with respect to the focal spot.
The take-off angle is the target-to-beam angle. Best angle is typically 6 degrees in terms of shape and X-ray beam intensity.
Focal spot size = 0.4 × 12 mm results in beam cross-section of 0.04 × 12 mm for line and 0.4 × 1.2 mm for point.

14. Interaction of X-rays with Matter
The previous discussion was concerned with the generation of X-rays for use as sources; now we must discuss the various ways in which X-rays interact with matter (samples).

15. Interactions with Matterd
incoherent scattering
Co (Compton-Scattering)
coherent scattering
Pr (Bragg-scattering)
wavelength Pr
absorption
Beer´s law I = I0*e-µd
intensity Io
fluorescence
> Pr
Of all the effects produced by the passage of X-rays through matter, we are primarily interested in the coherent scattering of X-rays by atoms.
These are the X-rays that we measure in diffraction experiments, as the scattered X-rays carry information about the electron distribution in materials.
photoelectrons

16. Coherent Scattering e-
Incoming X-rays are electromagnetic waves that exert a force on atomic electrons.
The electrons will begin to oscillate at the same frequency and emit radiation in all directions.
X-rays are electromagnetic waves that will exert a force on charged particles (in this case, the electrons of atoms). The electrons will oscillate with the same frequency as the incoming radiation and emit radiation in all directions with this same frequency.

17. Constructive and Deconstructive Interference=
The addition (superposition) of 2 or more waves that result in a new pattern.
Superposition of 2 waves and how they add together. The resulting wave has an amplitude that is dependent on the relative phase of the original waves. =

18. Coherent Scattering by an Atom
Coherent scattering by an atom is the sum of this scattering by all of the electrons.
Electrons are at different positions in space, so coherent scattering from each generally has different phase relationships.
At higher scattering angles, the sum of the coherent scattering is less. 2q
Electrons are at different positions in space, so only in the forward direction does the scattering from each electron add constructively.
At higher scattering angles, the destructive interference between scattering at different electron positions becomes larger.
The amount of scattering for each type of element (atom) as a function of scattering angle is referred to as the scattering factor.

19. Atomic Scattering Factor
Scattering factor is used as an indication of the strength of scattering of an atom in particular direction.
Scattering is a maximum in the forward scattering direction and decreases with scattering angle.
Atomic scattering factors have been determined for each wavelength of radiation.
f= Amplitude of wave scattered by atom
Amplitude of wave scattered by one electron

20. X-ray Diffraction by Crystals
We will now discuss the interaction of monochromatic X-rays with crystals.

21. Diffraction of X-rays by Crystals
The science of X-ray crystallography originated in 1912 with the discovery by Max von Laue that crystals diffract X-rays.
Von Laue was a German physicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.
The science of X-ray crystallography originated in 1912 with
the discovery by von Laue that crystals diffract X-rays. Since that time, single-crystal X-ray diffraction has developed into the most powerful method known for obtaining the atomic arrangement in the solid state.
X-ray crystallographic structure determination can be applied to a wide range of structure sizes, from very small molecules and simple salts, to complex minerals, synthetically prepared inorganic and organometallic complexes, natural products and to biological macromolecules, such as proteins and even viruses.
Max Theodor Felix von Laue
(1879 – 1960)

22. X-ray Diffraction Pattern from a Single-crystal Sample
Rotation Photograph
This image happens to be a 360 rotation in phi, collected on a CCD detector with all goniometer angles set to zero.
Note that the X-ray diffraction pattern from a single crystal is a set of spots.

23. Diffraction of X-rays by Crystals
After Von Laue's pioneering research, the field developed rapidly, most notably by physicists William Lawrence Bragg and his father William Henry Bragg. In , the younger Bragg developed Bragg's law, which connects the observed scattering with reflections from evenly-spaced planes within the crystal.
William Henry Bragg
After Von Laue's pioneering research, the field developed rapidly, most notably by physicists William Henry Bragg and his son William Lawrence Bragg.
In , the younger Bragg developed Bragg's law, which connects the observed scattering with reflections from evenly spaced planes within the crystal.
William Lawrence Bragg

24. Bragg’s Law
Look at X-rays that scatter coherently from 2 of these parallel planes (or lines) of lattice points separated by a distance d.
X-rays scattering coherently from 2 of the parallel planes separated by a distance d.
Incident angle and reflected (diffracted angle) are given by q. 24

25. Bragg’s Law
The condition for constructive interference is that the path difference leads to an integer number of wavelengths.
Bragg condition  concerted constructive interference from periodically-arranged scatterers.
This occurs ONLY for a very specific geometric condition.
The condition for constructive interference from all lattice points is that the path difference is equal to an integer number of wavelengths, AND that the incident angle and exit angle are equal.
This is written as the Bragg equation.
The diffraction peak only occurs for a very specific geometric condition.
Concerted constructive interference 25

26. n = 2d sin() d Bragg’s Law  
We can think of diffraction as reflection at sets of planes running through the crystal.
Only at certain angles 2 are the waves diffracted from different planes a whole number of wavelengths apart (i.e., in phase).
At other angles the waves reflected from different planes are out of phase and cancel one another out.
We can think of diffraction as reflection at sets of planes running through the crystal. Only at certain angles 2θ are the waves diffracted from different planes a whole number of wavelengths apart (i.e., in phase). At other angles, the waves reflected from different planes are out of phase and cancel one another out.

27. Reflection Indices z y x
These planes must intersect the cell edges rationally, otherwise the diffraction from the different unit cells would interfere destructively.
We can index them by the number of times h, k and l that they cut each edge.
The same h, k and l values are used to index the X-ray reflections from the planes. x
These planes must intersect the cell edges rationally, otherwise the diffraction from the different unit cells would interfere destructively.
We can index them by the number of times h, k and l that they cut each edge.
The same h, k and l values are used to index the X-ray reflections from the planes.
Planes (or )

28. Examples of Diffracting Planes and their Miller Indicesb c
Method for identifying diffracting planes in a crystal system.
A plane is identified by indices (hkl) called Miller indices, that are the reciprocals of the fractional intercepts that the plane makes with the crystallographic axes (abc).
Miller indices: The reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
Miller indices (hkl) The plane makes fractional intercepts 1/h, 1/k, 1/l with the axes.

29. Diffraction Patterns
Two successive CCD detector images with a crystal rotation of one degree per image:
When Bragg’s Law is satisfied through the appropriate combinations of crystal orientation angles with respect to the incident X-ray beam, reflections appear on the detector.
This example shows two successive CCD detector images with a crystal rotation of one degree per image.
A complete dataset consists of the measurement of all independent intensities in the diffraction pattern out to a specified resolution limit.
For each X-ray reflection (black dot), indices h,k,l can be assigned and an intensity I = F 2 measured

30. Reciprocal Space
The immediate result of the X-ray diffraction experiment is a list of X-ray reflections hkl and their intensities I.
We can arrange the reflections on a 3D grid based on their h, k and l values. The smallest repeat unit of this reciprocal lattice is known as the reciprocal unit cell; the lengths of the edges of this cell are inversely related to the dimensions of the real-space unit cell.
This concept is known as reciprocal space; it emphasizes the inverse relationship between the diffracted intensities and real space.
The immediate result of the X-ray diffraction experiment is a list of X-ray reflections hkl and their intensities I.
We can arrange the reflections on a 3D grid based on their h, k and l values.
The smallest repeat unit of this reciprocal lattice is known as the reciprocal unit cell; the lengths of the edges of this cell are inversely related to the dimensions of the real-space unit cell.
This concept is known as reciprocal space; it emphasizes the inverse relationship between the diffracted intensities and real space.

31. The Crystallographic Phase Problem
An Argand diagram showing the vector nature of the structure factor F(hkl). F(hkl) = A(hkl) + iB(hkl).
A(hkl) is the real component, and B(hkl) is the imaginary component.
α(hkl) is the phase angle

32. The Crystallographic Phase Problem
In order to calculate an electron density map, we require both the intensities I = F 2 and the phases a of the reflections hkl.
The information content of the phases is appreciably greater than that of the intensities.
Unfortunately, it is almost impossible to measure the phases experimentally!
In order to calculate an electron density map, we require both the intensities I = F 2 and the phases a of the reflections hkl.
The information content of the phases is appreciably greater than that of the intensities.
Unfortunately, it is almost impossible to measure the phases experimentally!
This is known as the crystallographic phase problem and would appear to be unsolvable!

33. Crystal Structure Solution by Direct Methods
Early crystal structures were limited to small, centro-symmetric structures with ‘heavy’ atoms. These were solved by a vector (Patterson) method.
The development of ‘direct methods’ of phase determination made it possible to solve non-centro-symmetric structures on ‘light atom’ compounds

34. Rapid Growth in Number of Structures in Cambridge Structural Database

35. The Structure Factor F and Electron Density ∫
Fhkl = V xyz exp[+2i(hx+ky+lz)]dV
xyz = (1/V) hkl Fhkl exp[-2i(hx+ky+lz)]
F and  are inversely related by these Fourier transformations.
Note that  is real and positive, but F is a complex number: in order to calculate the electron density from the diffracted intensities I = F2, we need the PHASE () of F.
Unfortunately, it is almost impossible to measure  directly!
Fourier transforms are a general class of calculations that may be applied to many practical problems.
All Fourier transforms involve three-dimensional summations in two directions.
Most of the modern spectroscopic tools commonly used by chemists utilize Fourier transforms in some way.
For FT-IR, we measure interferograms and transform them to IR spectra.
For FT-NMR, we use pulse sequences, relaxation times in the measurement of experimental data, and then transform the data to an NMR spectrum.
These spectroscopic methods use Fourier transforms in a time domain.
In contrast, X-ray diffraction utilizes Fourier transforms in a space domain.
For X-ray diffraction, we use Fourier transform mathematics to go back and forth from reciprocal space (diffraction pattern) to real space (electron density).

36. Real Space and Reciprocal Space
Unit Cell (a, b, c, , , )
Electron Density, (x, y, z)
Atomic Coordinates – x, y, z
Thermal Parameters – Bij
Bond Lengths (A)
Bond Angles (°)
Crystal Faces
Reciprocal Space
Diffraction Pattern
Reflections
Integrated Intensities – I(h,k,l)
Structure Factors – F(h,k,l)
Phase – (h,k,l)

37. Diffraction pattern on a
X-Ray Diffraction
X-ray beam
  1Å
(0.1 nm)
This diagram is a conceptual view of the experimental portion of crystal structure determination.
The single-crystal specimen to be analyzed is in the center of the picture. This sample is typically about from to mm in size.
When this crystal is irradiated with monchromatic X-rays with a wavelength of approximately 1 Angstrom (left), it produces a diffraction pattern which may be recorded on a two-dimensional detector such as a CCD (right).
As the crystal is rotated, spots will appear on the detector whenver Bragg‘s Law is satisfied.
The diffraction pattern is the Fourier transform of the crystal structure.
~ (0.2mm)3 crystal
~1013 unit cells, each ~ (100Å)3
Diffraction pattern on a
CCD detector