Diffraction method

Neutron diffraction studies have the advantage of being able to determine guest and host (both O and H/D) positions. With the difficulty of preparing single crystals of gas hydrates, most diffraction studies are performed on powder samples. Powder x-ray and neutron diffraction can be used with Rietveld analysis of the data for detailed structure determination (Rawn et al., 2003 Hester et al., 2006a). 

Small angle neutron scattering instruments are specifically designed to examine disordered materials, such as to determine hydration structures during hydrate formation (Koh et al., 2000 Buchanan et al., 2005 Thompson et al., 2006), or to study kinetic inhibitor adsorption onto a hydrate surface (Hutter et al., 2000 King et al., 2000). 

Neutron spectroscopy (also referred to as inelastic neutron scattering) has been used to measure the extent of guest-host interactions in a hydrate lattice, which help to explain the anomalous thermal behavior of hydrates (e.g., low thermal 

These embrace X-ray diffraction, neutron diffraction and electron diffraction. The first two of these are almost entirely used in the study of crystalline solids, while electron diffraction is of most value (to inorganic chemists at least) for structure determinations of gaseous substances. X-ray diffraction has been used to obtain structural information for species in solution, and electron diffraction has applications in the 

The reader may wonder why - given the completeness with which an X-ray analysis should reveal the structure of a crystalline solid and the extent to which the method has been automated - inorganic chemists should bother with any other method of characterisation for solid compounds. It is true that many inorganic substances are nowadays characterised by X-ray crystallography alone. There are, however, cases 

If the substance under scrutiny contains hydrogen atoms whose location is important, X-ray analysis will usually have to be supplemented by other methods. Very light atoms contribute very little to the scattering of X-rays and - especially if much heavier atoms are present - may be effectively invisible to the crystallographer. There are a number of other circumstances where, even if good crystals are available, X-ray crystallography may fail to yield the desired structural information. 

It might be thought that the electron density maps which arise from an X-ray analysis should provide detailed information about electron distribution and hence bonding. The electron densities obtained from routine X-ray studies are not sufficiently accurate for such purposes. However, the finer details of electron density distributions in crystals can be obtained in favourable cases by very careful and accurate X-ray diffraction studies. Neutron diffraction (see below) can also provide such information. 

Accurate maps depicting the unpaired electron density can be obtained, and the results compared with theoretical calculations. 

Scattered waves from neighbouring atoms interfere in exactly the same way and unless the atoms are ordered as in a crystal, the total diffraction pattern is a function of the radial distribution of scattering density (atoms) only. This is the mechanism whereby diffraction patterns arise during gas-phase electron diffraction, scattering by amorphous materials, and diffraction 

X-ray Crystallography This diffraction technique is the most widely used and respected for crystal/ molecular structure determination. 

The phases can be calculated if all atomic positions in the unit cell are known from the alternative expression for the structure factor  

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

A further complication is that molecules in different blocks, or even neighbouring unit cells can have different orientations. If this happens in an orderly fashion the size of the unit cell is increased by an integral factor and a superstructure appears. Not if the alternative orientations appear at random. In that case the size of the unit cell remains the same and the different molecular images are crystallographically superimposed in the same space, with coincident atomic positions only occurring by accident. 

Laue method Rotating-crystal method Powder method 

The Laue method was the first diffraction method ever used, and it reproduces von Laue s original experiment. A beam of white radiation, the continuous spectrum from an x-ray tube, is allowed to fall on a fixed single crystal. The Bragg angle 6 is therefore fixed for every set of planes in the crystal, and each set picks out and diffracts that particular wavelength which satisfies the Bragg law for the particular values of d and 9 involved. Each diffracted beam thus has a different wavelength. 

There are two variations of the Laue method, depending on the relative positions of source, crystal, and film (Fig. 3-5). In each, the film is flat and placed perpendicular to the incident beam. The film in the transmission Laue method (the original Laue method) is placed behind the crystal so as to record the beams diffracted in the forward direction. This method is so called because the diffracted beams are partially transmitted through the crystal. In the back-reflection Laue method the film is placed between the crystal and the x-ray source, the incident beam passing through a hole in the film, and the beams diffracted in a backward direction are recorded. 

In either method, the diffracted beams form an array of spots on the film as shown in Fig. 3-6. This array of spots is commonly called a pattern, but the term is not used in any strict sense and does not imply any periodic arrangement of the 

The spots lying on any one curve are reflections from planes belonging to one zone. This is due to the fact that the Laue reflections from planes of a zone all lie on the surface of an imaginary cone whose axis is the zone axis. As shown in Fig. 3-7(a), one side of the cone is tangent to the transmitted beam, and the angle of inclination of the zone axis (Z.A.) to the transmitted beam is equal to the semi-apex angle of the cone. A film placed as shown intersects the cone in an 

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The detection of residual austenite in fact requires average frequency, however for comparison reasons (reference) with a different recognized method, it is recommended to use high frequency, as with high frequency of eddy currents the penetration depth is comparable in the diffraction method and eddy current method. 

Segregation of bearings, with regard to residual austenite was performed with the aid of WIROTEST 202 and WIROTEST 12 finish. Selected rings with defined indications were subject to metalographic tests, in order to state whether residual austenite occurs, and then using the diffraction method, the percentage content of residual austenite. 

STM found one of its earliest applications as a tool for probing the atomic-level structure of semiconductors. In 1983, the 7x7 reconstructed surface of Si(l 11) was observed for the first time [17] in real space all previous observations had been carried out using diffraction methods, the 7x7 structure having, in fact, only been hypothesized. By capitalizing on the spectroscopic capabilities of the technique it was also proven [18] that STM could be used to probe the electronic structure of this surface (figure B1.19.3). 

Surface reconstructions have been observed by STM in many systems, and the teclmique has, indeed, been used to confmn the missing row structure in the 1 x 2 reconstruction of Au(l 10) [28]. As the temperature was increased within 10 K of the transition to the disordered 1 1 phase (700 K), a drastic reduction in domain size to -20-40 A (i.e. less than the coherence width of LEED) was observed. In this way, the STM has been used to help explain and extend many observations previously made by diffraction methods. 

Flowever, it is necessary to first discuss the meaning of diffraction , because this concept can be interpreted in several ways. After these fiindamental aspects are dealt with, we will take a statistical and historical view of the field. It will be seen that many different diffraction methods are available for surface structural detemiination. 

From the above descriptions, it becomes apparent that one can include a wide variety of teclmiques under the label diffraction methods . Table Bl.21.1 lists many techniques used for surface stmctural detemiination, and specifies which can be considered diffraction methods due to their use of wave interference (table Bl.21.1 also explains many teclmique acronyms commonly used in surface science). The diffraction methods range from the classic case of XRD and the analogous case of FEED to much more subtle cases like XAFS (listed as both SEXAFS (surface extended XAFS) and NEXAFS (near-edge XAFS) in the table). 

Table 81.21.1. Surface stmctural detemiination methods. The second colunni indicates whether a technique can be considered a diffraction method, in the sense of relying on wave interference. Also shown are statistics of surface stmctural detemiinations, extracted from the Surface Stmcture Database [14], up to 1997. Counted here are only detailed and complete stmctural determinations, in which typically the experiment is simulated computationally and atomic positions are fitted to experiment. (Some stmctural detemiinations are perfomied by combining two or more methods those are counted more than once in this table, so that the colunnis add up to more than the actual 1113 stmctural detemiinations included in the database.)...

There are several approaches to gain the required surface sensitivity with diffraction methods. We review several of these here, emphasizing the case of solid/vacuum interfaces some of these also apply to other interfaces. 

As the table shows, a host of other teclmiques have contributed a dozen or fewer results each. It is seen that diffraction teclmiques have been very prominent in the field the major diffraction methods have been LEED, PD, SEXAFS, XSW, XRD, while others have contributed less, such as NEXAFS, RHEED, low-energy position diffraction (LEPD), high-resolution electron energy loss spectroscopy (HREELS), medium-energy electron diffraction (MEED), Auger electron diffraction (AED), SEELFS, TED and atom diffraction (AD). 

The major non-diffraction method is IS, which is described in chapter BT23. 

The diffraction pattern observed in LEED is one of the most connnonly used fingerprints of a surface structure. Witii XRD or other non-electron diffraction methods, there is no convenient detector tliat images in real time the corresponding diffraction pattern. Point-source methods, like PD, do not produce a convenient spot pattern, but a diffrise diffraction pattern that does not simply reflect the long-range ordermg. 

As in the case of ions we can assign values to covalent bond lengths and covalent bond radii. Interatomic distances can be measured by, for example. X-ray and electron diffraction methods. By halving the interatomic distances obtained for diatomic elements, covalent bond radii can be obtained. Other covalent bond radii can be determined by measurements of bond lengths in other covalently bonded compounds. By this method, tables of multiple as well as single covalent bond radii can be determined. A number of single covalent bond radii in nm are at the top of the next page. 

The many possible oxidation states of the actinides up to americium make the chemistry of their compounds rather extensive and complicated. Taking plutonium as an example, it exhibits oxidation states of -E 3, -E 4, +5 and -E 6, four being the most stable oxidation state. These states are all known in solution, for example Pu" as Pu ", and Pu as PuOj. PuOl" is analogous to UO , which is the stable uranium ion in solution. Each oxidation state is characterised by a different colour, for example PuOj is pink, but change of oxidation state and disproportionation can occur very readily between the various states. The chemistry in solution is also complicated by the ease of complex formation. However, plutonium can also form compounds such as oxides, carbides, nitrides and anhydrous halides which do not involve reactions in solution. Hence for example, it forms a violet fluoride, PuFj. and a brown fluoride. Pup4 a monoxide, PuO (probably an interstitial compound), and a stable dioxide, PUO2. The dioxide was the first compound of an artificial element to be separated in a weighable amount and the first to be identified by X-ray diffraction methods. 

The formation of such materials may be monitored by several techniques. One of the most useful methods is and C-nmr spectroscopy where stable complexes in solution may give rise to characteristic shifts of signals relative to the uncomplexed species (43). Solution nmr spectroscopy has also been used to detect the presence of soHd inclusion compound (after dissolution) and to determine composition (host guest ratio) of the material. Infrared spectroscopy (126) and combustion analysis are further methods to study inclusion formation. For general screening purposes of soHd inclusion stmctures, the x-ray powder diffraction method is suitable (123). However, if detailed stmctures are requited, the single crystal x-ray diffraction method (127) has to be used. 

The mineralogical, structural, physical, and thermodynamic properties of the various crystalline alumiaa hydrates are Hsted ia Tables 1, 2, and 3, respectively. X-ray diffraction methods are commonly used to differentiate between materials. Density, refractive iadex, tga, and dta measurements may also be used. 

For x-ray investigations, the diffractometer method is generally used. The lattice constants indicate purity or composition of soHd solutions the rapid counting-tube goniometric method can be used at the manufacturing plant for quaUty control. The rotating-crystal and neutron diffraction methods are sometimes used for stmcture elucidation. 

Wyckoff, H.W., Hirs, C.H.W., Timasheff, S.N. Diffraction methods for biological macromolecules. Methods Enzymol. 114 330-386, 1985. 

X-ray studies indicate that the vinyl chloride polymer as normally prepared in commercial processes is substantially amorphous although some small amount of crystallinity (about 5% as measured by X-ray diffraction methods) is present. It has been reported by Fuller d in 1940 and Natta and Carradini in 1956 that examination of the crystalline zones indicates a repeat distance of 5.1 A which is consistent with a syndiotactic (i.e. alternating) structure. Later studies using NMR techniques indicate that conventional PVC is about 55% syndiotactic and the rest largely atactic in structure. 

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