lunes, 15 de febrero de 2010

Line Broadening Analysis


Abel A. Colmenares E.

Diffraction Line-Profile Shape by Synchrotron and Laboratory X-ray Sources


Precise knowledge of diffraction line-profile shape is of utmost importance in x-ray powder diffraction, especially in line-broadening analysis, Rietveld refinement, and other whole-powderpattern-fitting programs. In this regard, laboratory x-ray sources were researched extensively in the past, but synchrotron radiation remains inadequately characterized, despite its increasingly frequent recent use. Most of the line-profile models rely on a milestone study of Caglioti, Paoletti, and Ricci' that was developed for neutron diffraction and later adapted in the synchrotron case.2 Basic studies of synchrotron powder diffraction were undertaken by Cox et aZ.3w,h o also gave a comprehensive review4 of the field. Synchrotron radiation is inherently advantageous to laboratory sources for line-broadeningstudies for many reasons: naturally high beam collimation provides a superior resolution, the wavelength of a monochromatic beam can be easily tuned, and line shape is generally simpler and controlled to our preference. Most important, however, is the high resolution, that is, the narrow instrumental line profile implies a high sensitivity to the small physical broadening.

For the laboratory measurements, we used a horizontal goniometer in divergent Bragg- Brentano flat-plate geometry with both incident and diffracted Soller slits to minimize axial beam divergence, 2 mm divergent and 0.2 mm receiving slits. Cu Kq, radiation was scanned with a cooled germanium solid-state detector. Synchrotron-radiation measurements were performed on the X3B 1 beamline at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory. The triple-axis parallel geometry included Si channel- 111 -cut monochromator, flat specimen, Ge 111 -cut analyzer crystal, and proportional detector (Figure 1). Typical NIST SRM LaB, diffraction lineprofiles are presented in Figure 2. At this diEaction angle, synchrotron radiation gives four times smaller line width and 2.5 times larger peak-to-background ratio, despite twice as large a background count. Both line profiles are closely approximated with the Voigt function or its pseudo-Voigt and Pearson VII approximations.4 However, it is still a matter of debate5 why the line profiles tend to be almost pure Lorentz functions at high angles, the same effect that is observed for laboratory sources. Therefore, it is desirable to study the overall effect of geometrical aberrations on the difiaction-line shape.

Figure 1: Schematic view of X3Bl NSLS beamline in the (vertical) equatorial plane. M: monochromator crystal; ES: equatorial slit; S: specimen; A: analyzer crystal; D: detector.

Figure 2 Diffraction-line profiles of MST SRM LaE16 obtained at laboratory and synchrotron (NSLS) x-ray sources. P/B denotes the peak-to-background ratio.


The main equatorial instrumental factors affecting the diffraction-line profile and/or position are the following:
(i) Source height (vertical angular distribution of the polychromatic beam) is approximated with the Gauss function at the bending magnet. It depends on the storage-ring electron (positron) relativistic factor y, the photon energy c, and the critical photon energy ec (5.04 keV at NSLS):

where the vertical (equatorial, for it is in the scattering plane) divergence

Here, v, E, and m, are the electron (positron) speed, energy, and rest mass, respectively, and c is the speed of light.
(ii) Equatorial slit width

(iii) Normalized Darwin Bragg-reflection shape6 of the monochromator and analyzer (perfect) crystals (rocking curve):

Here, s defines the region for a perfect reflection (without absorption) from a crystal.
(iv) Specimen effects that cause important aberrations in laboratory divergent geometry, such as transparency, flat surface, and its missetting, are negligible in synchrotron parallel geometry with the analyzer crystal.
The most important axial aberration is a divergence, which sometimes causes severe asymmetry at low angles. The effect on powder lime shapes was considered by van Laar and Yelon7 and recently applied to high-resolution synchrotron diffractometers by Finger, Cox, and Jephcoat.
The total diiaction-line profile results from a convolution of all the contributions, which has to be accomplished numerically. However, for most purposes, a simple estimation of line widths as a fimction of diffraction angle may suffice. Wavelength dispersion follows from the Bragg law:

Here, the shape of perfect Bragg reflection is approximated with the Gauss function. They depend on the structure factor, polarization, absorption, and temperature.

To recognize the relative importance of various contributions, we estimate the angular resolution at the X3B 1 NSLS beamline with 8 keV photon energy, that is, the approximate Cu Ka wavelength:

Figure3: PwHMrofsplit-P earson VII fits to the line profiles of LaB, and different broadening models presented with lines.


Abel A. Colmenares E.

Application of Neutron Diffraction in everyday life

The Power of Polarised Neutrons

Both neutrons and electrons have a spin of 1/2, so have a magnetic moment and can interact magnetically with matter. The strength of the interaction depends not only on the size of the electronic magnetic moments, but also on their relative orientations. Beams of neutrons with all their spins parallel – a polarised beam – can therefore be used to study materials in which the electronic magnetic moments are ordered in a structurally interesting way. The intensities of the neutrons scattered by the sample (usually a single crystal), are measured in a detector.

However, the best way to extract precise information about the sample's magnetic properties is to compare scattering patterns obtained with neutron beams polarised in opposite directions. A device called a spin flipper reverses the polarisation of the incident neutron beam (figure 1), so that the ratio between the scattered intensities for the two orientations can be measured – the flipping ratio. Precise measurements of this ratio can be used to map the distribution of magnetic moments within a crystal – as was shown by Clifford Shull and his students at MIT in pioneering experiments in the 1960s investigating the classic ferromagnetic metals iron, cobalt and nickel. They demonstrated that although the electrons responsible for the magnetism are also involved in conducting electricity they are well-localised in space.

Since then, this method has been used to look at magnetisation in many materials. One of the more exotic compounds studied is a large molecule containing 12 manganese atoms linked to acetate groups, commonly known as Mn12-Ac, and which has unusual low-temperature magnetic properties. The molecule contains an inner core of four manganese ions, each with three unpaired electrons, surrounded by an outer ring of eight manganese ions with a total of 32 unpaired electrons. The overall spin of the complex is obtained by summing all the magnetic moments in the molecule. Magnetisation measurements give a net spin of 10, which suggests that the spins on the outer manganese atoms (total spin,16) are oriented antiparallel to those in the inner ones (total spin, -6). Polarised neutron scattering confirmed this picture, and the magnetisation values at the various manganese sites agree well with those predicted theoretically.

Polarised neutron diffraction has also been used to pinpoint the unpaired electrons responsible for magnetism in an organic compound containing no metals. It consists of a fluorocarbon ring attached to a ring of one carbon, two sulfur and two nitrogen atoms. The unpaired electrons are found on the latter, fivemembered ring (figure 2).

Polarisation analysis

Adding a second flipper and a polarisation analyser between the sample and detector allows us to carry out measurements that specifically distinguish magnetic scattering from the normal nuclear variety. Data from four combinations of measurements made by switching each flipper can be obtained: the intensities of scattered polarised neutrons both parallel and antiparallel to the incident polarised neutrons, and the same with the incident polarisation reversed. This technique can be used to study antiferromagnetism in which the direction of magnetic moments in a crystal alternate.

More sophisticated measurements of the magnitude and direction of the scattered polarisation for various orientations of the incident polarisation will give the absolute magnetic configuration of materials such as magneto-electric crystals (in which an electric field induces magnetisation or vice-versa). For example, such polarimetric experiments on he magneto-electric crystal chromium oxide (Cr2O3) determine the absolute orientation, relative to the surrounding oxygen atoms, of the oppositely directed moments on pairs of chromium ions. Figure 3 shows the configuration stabilised by cooling in different combinations of electric and magnetic fields.

D3 diffractometer used for
polarised neutron diffraction

Figure 1: A simple polarised neutron diffractometer. The neutrons from a reactor source are reflected by a magnetised crystal which selects neutrons with a particular wavelength and polarised parallel to the magnetisation direction. These neutrons are then scattered by a magnetised single crystal sample and the intensities of the reflected neutrons are recorded.

Figure 2: The magnetisation distribution in the CNSSN ring of the organic
magnet, p-O2NC6F4CNSSN

Figure 3: The crystal structure of chromium oxide showing the
moment orientations stabilised by cooling in (a) parallel
and (b) antiparallel magnetic (H) and electric (E) fields

Abel A. Colmenares E.

Application of Neutron Diffraction in everyday life

Fuel from the Ocean Floor

Industrialised countries are constantly searching for new sources of fossil fuels. One potential candidate are gas hydrates found on the ocean floor and in arctic regions.These crystalline compounds consist of networks of water molecules in which are caged small gas molecules – methane, for example.

People have known of gas hydrates, also called clathrate hydrates, for almost 200 years, but it was only a few decades ago that they were discovered in Nature – and in great abundance! In fact, marine sediments probably contain 10,000 billion tonnes of methane hydrate – considerably exceeding sources of coal, oil and gas (methane, of course, is natural gas).Methane hydrates are therefore likely to be of major economic importance, once we know how to extract them safely. Gas hydrates are also interesting for other reasons. They can cause blockages in gas pipelines.Hydrates containing carbon dioxide could be used to trap the gas at the bottom of the ocean thus reducing levels in the atmosphere; in fact, it is likely that water on Mars is largely stored as carbon dioxide hydrate. Air hydrates found in the deeper parts of polar ice sheets reveal how the composition of air has changed over the past million years.

Not surprisingly, there is intense research interest in gas hydrates. One important aspect is to understand how their stability changes with pressure and temperature. Gas hydrates exist only at high pressure and/or low temperatures – as are found at depths of several hundred metres in the sea or in permafrost regions. Experiments on gas hydrates, therefore, have to be done under similar conditions.This has proved extremely challenging and has meant that a number of properties of gas hydrates have not been well established.

How stable are gas hydrates?

For this reason, our research group at the University of Göttingen decided, a few years ago, to investigate the stability of gas hydrates.We first prepared the hydrates as polycrystalline materials containing large and small 'cages' (below left). Since the constituent atoms are lightweight, neutron powder diffraction is the ideal technique to study them.

One of the major unknowns is how the filling up of the large and small cages with gas molecules depends on pressure and temperature.We had assumed that it obeyed a well-established thermodynamic theory but it had never been rigorously proven.Using the D2B instrument at ILL,we were able to test the theory by following the changes in composition of several gas hydrates with increasing pressure.

We found that although the theory predictions were broadly followed, there were deviations in all cases, and in some cases the theory failed completely.We were very surprised to find that the large cages in the nitrogen hydrate contained two nitrogen molecules, violating one of the basic assumptions of the theory, although it can be modified to allow for this double occupancy. In other cases such as the carbon dioxide hydrate, the behaviour was quite different, and the theoretical description has to be changed considerably. These findings are extremely important to chemical engineers developing techniques to handle gas hydrates.We also looked at the compressibility of gas hydrates, which is important in detecting them using seismic profiling of the ocean floor.

Neutron techniques will continue to probe these and other questions such as unravelling how their fascinating structures form and decompose.

"Icy sponges filled with natural gas may be the next source of the world's energy"

The detector bank of the high resolution powder difractometer D2B

An electron micrograph of methane hydrate
showing its sponge-likestructure

The structure of methane hydrate. Methane molecules occupying
small and large cages are shown in green and yellow respectively


Abel A. Colmenares E.

Complimentary Techniques


Abel A. Colmenares E.