Question

Describe the powder X-ray diffraction experiment. List the basic components in your diagram and discuss the features/roles of each component. Include Bragg’s Law as well as coherent and incoherent scattering. What types of information can be obtained using this technique? Pros and cons with respect to single crystal XRD?

Answer 1

Diffraction

Diffraction is a event that occurs when light contact with an obstacle. The waves of light can either bend around the obstacle, or in the case of a slit, can travel through the slits. The resulting diffraction pattern will show areas of constructive interference, where two waves interact in phase, and destructive interference, where two waves interact out of phase. Calculation of the phase difference can be explained by examining by below fig 1

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In the figure below, two parallel waves, BD and AH are striking a gradient at an angle θo. The incident wave BD travels farther than AH by a distance of CD before reaching the gradient. The scattered wave (depicted below the gradient) HF, travels father than the scattered wave DE by a distance of HG. So the total path difference between path AHGF and BCDE is CD - HG. To observe a wave of high intensity (one created through constructive interference), the difference CD - HG must equal to an integer number of wavelengths to be observed at the angle psi, CD−HG=nλCD−HG=nλ, where λλ is the wavelength of the light. Applying some basic trigonometric properties, the following two equations can be shown about the lines:

CD=xcos(θo)CD=xcos⁡(θo) and HG=xcos(θ)HG=xcos⁡(θ)

xx = the distance between the points where the diffraction repeats. Combining the two equations,

x(cosθo−cosθ)=nλ

Bragg's Law

Diffraction of an x-ray beam, occurs when the light interacts with the electron cloud surrounding the atoms of the crystalline solid. Due to the periodic crystalline structure of a solid, it is possible to describe it as a series of planes with an equal interplaner distance. As an x-ray's beam hits the surface of the crystal at an angle , some of the light will be diffracted at that same angle away from the solid (see figure 2 below). The remainder of the light will travel into the crystal and some of that light will interact with the second plane of atoms. Some of the light will be diffracted at an angle thetatheta, and the remainder will travel deeper into the solid. This process will repeat for the many planes in the crystal. The x-ray beams travel different pathlengths before hitting the various planes of the crystal, so after diffraction, the beams will interact constructively only if the path length difference is equal to an integer number of wavelengths (just like in the normal diffraction case above). In the figure below, the difference in path lengths of the beam striking the first plane and the beam striking the second plane is equal to BG + GF. So, the two diffracted beams will constructively interfere (be in phase) only if BG+GF=nλBG+GF=nλ. Basic trigonometry will tell us that the two segments are equal to one another with the interplaner distance times the sine of the angle θθ. So we get:

BG=BC=dsinθ(1)(1)BG=BC=dsin⁡θ

Thus,

2dsinθ=nλ(2)(2)2dsin⁡θ=nλ

This equation is known as Bragg's Law, named after W. H. Bragg and his son, W. L. Bragg; who discovered this geometric relationship in 1912. {C}{C}Bragg's Law relates the distance between two planes in a crystal and the angle of reflection to the x-ray wavelength. The x-rays that are diffracted off the crystal have to be in-phase in order to signal. Only certain angles that satisfy the following condition will register:

sinθ=nλ2d(3)(3)sin⁡θ=nλ2d

For historical reasons, the resulting diffraction spectrum is represented as intensity vs. 2θ.

Instrumentation:

the components of x-ray diffraction; a source, sample holder, and signal converter/readout are required.

The Source

x-ray tubes provides a means for generating x-ray radiation in most analytical instruments. An evacuated tube houses a tungsten filament which acts as a cathode opposite to a much larger, water cooled anode made of copper with a metal plate on it. The metal plate can be made of any of the following metals: chromium, tungsten, copper, rhodium, silver, cobalt, and iron. A high voltage is passed through the filament and high energy electrons are produced. The machine needs some way of controlling the intensity and wavelength of the resulting light. The intensity of the light can be controlled by adjusting the amount of current passing through the filament; essentially acting as a temperature control. The wavelength of the light is controlled by setting the proper accelerating voltage of the electrons. The voltage placed across the system will determine the energy of the electrons traveling towards the anode. X-rays are produced when the electrons hit the target metal. Because the energy of light is inversely proportional to wavelength (E=hc=h(1/λE=hc=h(1/λ), controlling the energy, controls the wavelength of the x-ray beam.

X-ray Filter

Monochromators and filters are used to produce monochromatic x-ray light. This narrow wavelength range is essential for diffraction calculations. For instance, a zirconium filter can be used to cut out unwanted wavelengths from a molybdenum metal target (see figure 4). The molybdenum target will produce x-rays with two wavelengths. A zirconium filter can be used to absorb the unwanted emission with wavelength Kβ, while allowing the desired wavelength, Kα to pass through.

Sample Holder

The sample holder for an x-ray diffraction unit is simply a needle that holds the crystal in place while the x-ray diffractometer takes readings.

Needle Sample Holder

The sample holder for an x-ray diffraction unit is simply a needle that holds the crystal in place while the x-ray diffractometer takes readings.

Pros and cons with respect to single crystal XRD

pros: 1. nondestructive 2. no separate standards need 3. provides very detaied structural information

cons: 1. must have single polished crystals 2. sample set up time consuming 3. data collection takes 4hr.

coherent radiation means that the phases of two ( or more ) waves representing the radiation differ by a known constant.

Incoherence means that the phase differences are unknown/random.