Lecture 4: Diffraction

PHYS 430/603 material

Laszlo Takacs

UMBC Department of Physics

How to determine the atomic structure of materials?

• The distance between atoms in a solid is about 0.25 nm while the wavelength of visible light is 500 nm. For light, any condensed matter seems continuous.

• To “see” atoms, we need radiation with wavelength shorter than the atomic distance, < 0.25 nm

– X-rays: E = hf = hc/ ≈ 5 keV, easy to obtain from characteristic radiation or from synchrotron light source. E.g. Cu K ≈ 0.154 nm.

– Neutrons: E = p2/2m = h2/2m2 ≈ 0.0131 eV. The corresponding temperature of random motion T = 2E/kB ≈ 304 K ≈ Room temperature.

– Electrons: E = h2/2m2 ≈ 24.1 eV. We usually use much higher energies in an electron microscope, much smaller.

• We can “see” atomic structure in several ways:– Using diffraction– Atomic resolution electron microscopy– Field ion microscopy– Scanning microscopies

Constructive interference is observed, if the path difference between the two interfering rays is an integer multiple of the wavelength.

Young’s experiment Bragg’s lawd sin() = nd sin() = n, usually n = 1

If this condition is not satisfied, partial cancellation is possible for two (or a finite number of) scattering slits or lattice planes. The cancellation is complete for an infinite (very large) number of slits - diffraction grating; or lattice planes - ideal crystal.

Debye - Scherrer methodConsider a set of lattice planes with interplanar distance d. Braggs law specifies the corresponding angle . Constructive interference occurs only if the angle between the incoming X-ray beam and the lattice planes is and it exits so that the diffraction angle is 2 and the incoming and diffracted beams are in the same plane perpendicular to the lattice planes (like for a mirror.)

In a powder sample, grains with any orientation exist with about the same probability, thus rather than nothing (wrong ) or just a spot, a cone of diffracted rays is obtained. A separate cone will exist for every possible set of lattice planes, giving a series of arcs on the film.

The Debye-Sherrer method used to be the workhorse of materials science. A major disadvantage is the use of film, that requires chemical processing and reading the film. It also limits accuracy.

Debye-Scherrer cameras

Example: Diffraction pattern of polycrystalline Al using Cu K radiation

• Al is fcc, a = 0.4050 nm; l = 1.• D = a / sqrt (h2 + k2 + l2), for fcc h, k, l are either all even or all odd

due to interference form atoms within a single unit cell. = 0.1542 nm, the average from Cu K1 and K2.

Plane h2+k2+l2 d sin(theta) 2*theta

{1 1 1} 3 0.2338 0.3297 38.50(2 0 0} 4 0.2025 0.3807 44.76{2 2 0} 8 0.1432 0.5384 65.16{3 1 1} 11 0.1221 0.6314 78.31{2 2 2} 12 0.1169 0.6595 82.52{4 0 0} 16 0.1013 0.7615 99.19{3 3 1} 19 0.0929 0.8298 112.16{4 2 0} 20 0.0906 0.8514 116.72{4 2 2} 24 0.0827 0.9326 137.69{3 3 3} 27 0.0779 0.9892 163.14{5 1 1}{4 4 0} 32 0.0716 1.0769 #NUM!

Braggs law is sufficient to predict where (at what angle) diffraction can occur. It cannot predict the intensities; for that the diffraction process has to be analyzed in full detail:

• X-rays are scattered by electrons as point charges. Even the intensity of scattering on a single electron is angle dependent.

• Atoms contain several electrons with some probability density distribution. The inter-atomic diffraction is not fully constructive, it is described by the atomic form factor.

• Lattice planes are not ideal mirrors, the reflection from them is a diffraction phenomenon form atoms within the plane. This is described by another form factor. If the unit cell contains more than one formula unit, there are “missing” peaks, angles at which the interference is destructive from a unit cell.

• There are numerous additional factors, like absorption, finite grain size, lattice distortions, and imperfections of the X-ray optics.

Evaluation is carried out by:• Comparing to a standard pattern - JCPDF card, basically fingerprinting.• Determining the structure of an unknown material by solving the

reverse problem. Very difficult from powder data.

The spectrum of an X-ray tube with Mo anode and the effect of filtering. K remains a doublet, but can be separated numerically, or a monochromator can be used.Other frequently used anode materials are Cu and Cr.

E = hc(= 1.2398, if measured in keV and nm)

X-ray optics of a powder diffractometer

The principle is identical to the Debye-Sherrer method, but detection is electronically, rather than on a film.F: anode bombarded with electrons; SS: Soller slits to limit axial divergence (out of plane)Div: divergence slitP: sampleR: receiving slitQ: anti-scatter slitFrom here the X-ray beam enters the detector.

The diffraction angle is defined by F, P, and R.

Focusing geometry

The diffraction angle is about the same from a large area of the sample, if the sample is tangent to the circle formed by the anode, sample, and receiving slit.

Laue method: Determination of the orientation and symmetry of a single crystal

“White” X-rays are used; Braggs law is always satisfied for one wavelength or the other. Only the orientation of the lattice planes matters.

Reflections from planes belonging to the same zone show up as hyperbolas.

Rotational symmetry result in symmetrical pattern.

Determination of orientation/ texture

• The wavelength is fixed. The angle from source to sample to detector is set to the Bragg angle corresponding to the distance between the desired set of lattice planes.

• If the sample is a single crystal, reflection is detected only if the orientation of the lattice planes satisfies the basic mirror rules - its normal in the plane of the beam, angles on both sides equal.

• For a polycrystal, the intensity is proportional to the fraction of properly oriented crystallites. Intensity as a function of orientation is measured.