Diffraction I: Directions of Diffracted Beams

Wenjea J. TsengDepartment of Materials Science and Engineering

National Chung Hsing UniversityURL: http://audi.nchu.edu.tw/~wenjea/

Email: wenjea@dragon.nchu.edu.tw

Chapter 3

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This chapter aims to combine the physics of x-rays (Chapter 1) and the geometry of crystals (Chapter 2) together and discuss the phenomenon of x-ray diffraction.

d is the opening b/t the slit.

Von Laue was a Germanphysicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.

Von Laue

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Use of CuS crystals in his first diffraction experiment (1912).

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Braggs Law

X Rayleigh Scattering

William L. Bragg William H. Bragg

Both won the Nobel Prize in Physics in 1915.

(1912)

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DiffractionDiffraction is due essentially to the existence of certain phase relationships between two or more waves. For the sake of simplicity, consider now a beam of x-rays , such as beam 1, proceeding from left to right. The beam is also assumed to be plane-polarized so that the electric field vector E always in one plane.

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Two conclusions may be drawn from the illustration on last page:

1. Differences in the length of the path traveled lead to differences in phase.

2. The introduction of phase differences produces a change in amplitude.

Remarks

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Differences in the path difference of various rays may be illustrated when we consider how a crystal diffracts x-rays, i.e.,

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0cos PKPKPRQK

In the directions 11 and 1a1a, the path difference is

In the directions 11 and 22, the path difference is

sinsin '' ddLNQML

This is also the path difference for the overlapping rays scattered by S and P. Scattered rays 1 and 2 will be completely in phase if

sin2 'dn

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sin2sin2'

dnd

Bragg Law

This relation was first formulated by W. L. Bragg and is known as the Bragg Law. Note that n is called the order of reflection and it may take on any integral value consistent. The n is equal to the number of wavelengths in the path difference between rays scattered by adjacent planes. Therefore, the rays scattered by all the atoms in all the planes are therefore completely in phase and reinforce one another to form a diffracted beam in particular directions. Note that in all other directions, the scattered rays are out of phase and annul one another to form destructive interference completely.

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Equivalence of (a) a second-order 100 reflection and (b) a first-order 200 reflection.

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Remarks

1. By atoms arranged periodically in crystals, in a very few directions, those satisfying the Bragg law, the scattering is strong and is called diffraction. Amplitudes add.

2. In most directions, those not satisfying the Bragg law, there is no scattering because the scattered rays cancel one another.

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Diffraction DirectionsWhat determines the possible angles 2 in which a given crystal can diffract a beam of monochromatic x-rays?

sin2d

222 lkhad

For cubic crystals

)(4

sin 22222

2 lkha

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Diffraction Directions(continue)What determines the possible angles 2 in which a given crystal can diffract a beam of monochromatic x-rays?

sin2d

2

2

2

22

2

1cl

akh

d

For tetragonal crystals

2

2

2

2222

4sin

cl

akh

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Diffraction Methods(1) Diffraction occurs whenever Braggs law is satisfied. (2) The Braggs law puts a very stringent condition on and for

any given crystal.

In general, with monochromatic radiation (i.e., is fixed) on an arbitrary setting of a single crystal will not produce any diffraction beams unless the and values satisfy the Braggs law exactly. Therefore, some way of satisfying the Braggs law are devised as follows:

Method Laue Variable Fixed

Rotating crystal Fixed Variable (in part)Powder Fixed Variable

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Laue Method

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Laue Method (continue)

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Rotating-Crystal Method

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Powder MethodIn the powder method, the crystal to be examined is reduced to a very fine powder or already is in the form of loose or consolidated microscopic grains. The sample in a suitable holder is placed in a beam of monochromatic x-rays. Each particle is a tiny crystal or assemblage of smaller crystals, oriented randomly with respect to the incident beam of a fixed .

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Debye-Scherrer Powder Method

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Debye-Scherrer Powder Method (continue)

(111) (200)(220)

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Remarks on the Diffraction MethodsHistorically, the above diffraction methods all began with the development of simple film camera where the sample to be examined in mostly millimeter size was placed at the center or at an appropriate position so that the incident x-ray can radiate the sample. The methods have been largely replaced by powder diffractometer to be explained in later classes; nonetheless, some factors still exist to select a camera technique rather than the powder diffractometer. For example, lack of adequate sample quantity and the lower cost involved in the camera methods.

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Other Applications (1)The Braggs law can be applied in two ways. 1. Using x-rays of known and measure the spacing d of

various planes in a crystal are then determined.2. A crystal with planes of known spacing d can be used to

measure .

sin2d

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X-ray SpectroscopyX-ray spectroscopy is a gathering name for several spectroscopictechniques for characterization of materials by using x-ray excitation.

When an electron from the inner shell of an atom is lost due to some sort of excitation, it is replaced with an electron from the outer shell; the difference in energy is emitted as an X-ray photon which has a wavelength that is characteristic for the element. Analysis of the X-ray emission spectrum produces qualitative results about elemental composition of the specimen.

c.f. Appendix 7 of the text.

hchvE

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weak

(in angstroms)

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X-rays can be excited by a high-energy beam of charged particles such as electrons (as in electron microscope) or protons (see PIXE), or a beam of X-rays (see X-ray fluorescence, or XRF). These methods enable elements from the entire periodic table to be analyzed, with the exception of H, He and Li. In electron microscopy electron beam excites X-rays; there are two main techniques for analysis of spectrum of characteristic X-ray radiation: Energy-dispersive X-ray spectroscopy and Wavelength dispersive X-ray spectroscopy.

X-ray Spectroscopy (continue)

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Energy Dispersive X-ray SpectroscopyElectron beam excitation is used in electron microscopes, scanning electron microscopes (SEM) and scanning transmission electron microscopes (STEM). X-ray beam excitation is used in X-ray fluorescence (XRF) spectrometers. A detector is used to convert X-ray energy into voltage signals; this information is sent to a pulse processor, which measures the signals and passes them onto an analyzer for data display and analysis.

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Energy Dispersive X-ray Spectroscopy (continue)

Example

Limitations: (1) There are energy peak overlaps among different elements. (2) EDS cannot detect the lightest elements, typically below the atomic number of Na for detectors equipped with a Be window. Polymer-based thin windows allow for detection of light elements, depending on the instrument and operating conditions.

Only elemental info.

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Wavelength Dispersive X-ray SpectroscopyWavelength-dispersive X-ray spectroscopy (WDXRF or WDS) is a method used to count the number of X-rays of a specific wavelength diffracted by a crystal. The wavelength of the impinging X-ray and the crystal's lattice spacings are related by Bragg's law and produce constructive interference if they fit the criteria of Bragg's law.

The analytical crystal and the detector can be moved around an arc known as the Rowland Circle. This grants the operator the ability to change the angle between the sample, the crystal, and the detector, thereby changing the X-ray wavelength that would satisfy Braggs law.

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Wavelength Dispersive X-ray Spectroscopy (continue)

Limitations: (1) WDS cannot determine elements below atomic number 5 (Boron). (2) Despite the improved spectral resolution of elemental peaks, some peaks exhibit significant overlaps that result in analytical challenges (e.g., VK and TiK). (3) WDS analyses are not able to distinguish among the valence states of elements (e.g. Fe2+ vs. Fe3+).

Comparison of WDS and EDS

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Other Applications (2)X-ray diffraction allows measurement of the lattice parameters of a crystals unit cell, and therefore its volume, together with the number of atoms in the cell. Theoretical density or x-ray density of pure elements and compounds can then be determined based on the weight of atoms in the cell dividing by the volume of the unit cell.

cellunitofvolumecellunitinatomsofweight

densityrayx

3cmgofunitin

NVA

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Additional Topic: The Reciprocal LatticeWhile Laues experiment allowed a visual understanding of the diffraction phenomena from crystals after x-ray irradiation, Ewald developed a useful reciprocal lattice concept for describing the observed diffraction phenomena.

Paul Peter Ewald

(January 23, 1888 August 22, 1985)

A German physicist who also inspired Max von Laue for his diffraction experiment on crystals.

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The Reciprocal Lattice (continue)Ewald perceive the Bragg planes in a space lattice by removing a dimension from the 3D arrangement by representing each 2D plane as a vector: dhkl is defined as the perpendicular distance from the origin of a unit cell to the first plane in the family (hkl), as illustrated for the (110) plane below.

d110 is represented as a vector with magnitude equal to the value of d in angstrom and in the direction from the origin of the unit cell meeting the plane at a right angle (90o).

The Reciprocal Lattice (continue)

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A diagram may be made by representing each dhkl vector as a point, at its tip, in the appropriate crystallographic coordinate system.

ExampleThe vectors are approaching the origin according to the reciprocals of the dhkl values.

Ewald first proposed that instead of plotting the dhklvectors, the reciprocal of these vectors should be plotted.

The Reciprocal Lattice (continue)

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hklhkl d

d 1 a, b, and c axes become a*, b*, and c*, , and angles become , , and

The reciprocal lattice

*** lckbhadhkl

The Reciprocal Lattice (continue)

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The Ewald sphere of reflection with a crystal in the center and its associated reciprocal lattice tangent to the sphere, at the point where the x-ray beam exits.

Real-space lattice Reciprocal lattice

The Reciprocal Lattice (continue)

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The Ewald sphere of reflection with the crystal rotated so that the (230) reciprocal lattice point touches it, permitting it to diffract.

With rotation in real-space lattice

The Reciprocal Lattice (continue)

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Reciprocal-lattice treatment of the rotating-crystal method.

The Reciprocal Lattice (continue)

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Reciprocal-lattice treatment of the powder diffraction method.

The Reciprocal Lattice (continue)

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Reciprocal-lattice treatment of the Laue diffraction method.

The Reciprocal Lattice (continue)

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Diffraction pattern of ammonium oxylate () crystal.