Structures, crystal,

For all possible crystals there are seven basic or primitive unit cells, which are shown in Fig. 1.2. We will represent the lengths of the sides as a, b and c and the angles as 

Rgure 1.2 The seven possible primitive unit cells with atoms or molecules only at each corner of the unit cell. Drug molecules will typically form triclinic, monoclinic and orthorhombic unit cells. 

The structures in Fig. 1.2 have atoms or molecules only at each comer of the unit cell. It is possible to find unit cells with atoms or molecules also at the centre of the top or bottom faces (end-centred), at the centre of every face (face-centred) or with a single atom in the centre of the cell (body-centred), as in Fig. 1.3. 

Note that these variations do not occur with every type of unit cell we find 

Face-centred cubic and orthorhombic Body-centred, cubic tetragonal and orthorhombic 

FIGURE 21.12 The bcc structure. An atom is located at the center of each cubic cell (orange) as well as at each corner of the cube (blue). The atoms are reduced slightly in size to make positions clear. 

The crystal lattice is an abstract construction whose points of intersection describe the underlying symmetry of a crystal. To flesh out the description of a particular solid state structure, we must identify some structural elements that are pinned to the lattice points. These structural elements can be atoms, ions, or even groups of atoms as we see in this and the next chapter. We begin with some illustrative simple cases. Some of the chemical elements crystallize in particularly simple solid structures, in which a single atom is situated at each point of the lattice. 

Polonium is the only element known to crystallize in the simple cubic lattice, with its atoms at the intersections of three sets of equally spaced planes that meet at right angles. Each unit cell contains one Po atom, separated from each of its six nearest neighbors by 3.35 A. 

The alkali metals crystallize in the body-centered cubic (bcc) structure at atmospheric pressure (Fig. 21.12). A unit cell of this structure contains two lattice points, one at the center of the cube and the other at any one of the eight corners. A single alkali-metal atom is associated with each lattice point. An alternative way to visualize this is to realize that each of the eight atoms that lie at the corners of a bcc unit cell is shared by the eight unit cells that meet at those corners. The contribution of the atoms to one unit cell is therefore 8 X = 1 atom, to which is added the atom that lies wholly within that cell at its center. 

When the angles are all 90° (so that their cosines are 0), this formula reduces to the simple result V = abc for the volume of a rectangular box. If the mass of the unit cell contents is known, the theoretical cell density can be computed. This density must come close to the measured density of the crystal, a quantity that can be 

The structure of a crystal lattice is similar to packing billiard balls in a box. By forming a lattice the particles will be arranged in such a way as to leave as little as possible space empty, or, in other words, the available space is filled as effectively as possible. The structure in one layer is shown in figure 4.1. 

As you can see, a very regular pattern arises when the centres of the spheres are linked. In figure 4.1 six centres are linked and together they form a regular hexagon. It is however also possible to link three centres to form an equilateral triangle or four which will result in a parallelogram. 

So how is the second layer stacked onto the first one It will be evident that a sphere in the second layer will be placed in the cavity between three spheres in the first layer. The centres of the four spheres thus form the angles of a trilateral pyramid (figure 4.2). 

The third layer of spheres can now be placed on top of the second in two ways.These three layers are called A, B and C, the spheres of a layer with the same letter are placed vertically above each other. By placing the third layer vertically above the first and the fourth above the second, a packing ABABA. arises, a so-called hexagonal close-packed structure, abbreviated hep. A second possibility is a packing of the type ABCABC. a so-called cubic close-packed structure, abbreviated ccp. Both types of packing are shown in figure 4.3. 

The crystal structures of many compounds can be described in a simplified way as an hep or a ccp with part of the cavities filled with other particles. 

The only thermodynamically stable crystallographic modification of alumina is a-Al203, or comndum. Corundum has a hexagonal crystal lattice with the cell 

Except for the thermodynamically stable a modification, there exist also numerous metastable modifications, denoted y, %, T], i, e, 5, 6, and k. These modifications are often used as supports for catalysts. All metastable modifications have a partially deformed closely packed hexagonal oxygen sublattice with various configurations of interstitial aluminum atoms. On approaching the equilibrium, the crystal lattice becomes more ordered until the stable a modification is formed. The type of metastable polymorph influences the morphology of the formed a-Al203 partides. 

In this chapter, crystal structure of ZnO encompassing lattice parameters, electronic band structure, mechanical properties, including elastic constants and piezoelectric constants, lattice dynamics, and vibrational processes, thermal properties, electrical properties, and low-field and high-field carrier transport is treated. 

1) The term zinc blende originated from compounds such as ZnS, which could be in cubic or hexagonal phase. But the term has been used ubiquitously for compound semiconductors with cubic symmetry. The correct term that should be used for the cubic phase of ZnO GaN is actually sphalerite. To be consistent with the diction throughout the literature even at the expense of bordering inaccuracy, the term zinc blende is used throughout this book. 

2) Also called Seignette salt - named after Pier Seignette from La Rochelle, France, who first 

Strukturiericht, the original crystallographic reports. From 1919to 1939 (Vols 1-8), they were published in Germany. Since then, they have been published in the United States under the name Structure Reports, Acta Crystallographica Section E, by the International Union of Crystallography. 

The nearest-neighbor bond lengths along the c-direction (expressed as h) and off c-axis (expressed as foi) can be calculated as 

Almost all the iron oxides, hydroxides and oxide hydroxides are crystalline. The degree of structural order and the crystal size are, however, variable and depend on the conditions under which the crystals were formed. All Fe oxides display a range of crystallinities except for ferrihydrite and schwertmannite which are poorly crystalline. 

Certain iron oxides are isostructural with other metal oxides. Goethite, for example, is isostructural with diaspore (a-AlOOH) and hence, is sometimes referred to as having the diaspore structure. Iron oxides and their isostructural metal oxides are listed in Table 2.1. 

A crystalline solid can be described by three vectors a, b and c, so that the crystal structure remains invariant under translation through any vector that is the sum of integral multiples of these vectors. Accordingly, the direct lattice sites can be defined by the set 

It is common also to define a set of reciprocal lattice vectors , b, c, such as 

According to the definitions given by Eqs. (1.1) to (1.3), the product G R = 2jrx integer. Therefore each vector of the reciprocal lattice is normal to a set of planes in the direct lattice, and the volume of a unit cell of the reciprocal lattice is related to the volume of the direct lattice I4 by 

Some physical properties of semiconductor electrodes depend on the orientation of the crystal, and surface properties vary from one crystal plane to the other. It is therefore very important in studies of surface and interface effects that the proper surface is selected. A semiconductor crystal can be cut by sawing or by cleavage. Cleavage in 

Semiconductor Electrochemistry, Zweite Auflage. Rttdiger Memming. 

Simple cubic Body-centrered cubic Face-centered cubic 

In the tetragonal system the spacing equation naturally involves both a and c since these are not generally equal  

Interplanar spacing equations for all systems are given in Appendix 3. 

So far we have discussed topics from the field of mathematical (geometrical) crystallography and have said practically nothing about actual crystals and the atoms of which they are composed. In fact, all of the above was well known long before the discovery of x-ray diffraction, i.e., long before there was any certain knowledge of the interior arrangements of atoms in crystals. 

It is now time to describe the structure of some actual crystals and to relate this structure to the point lattices, crystal systems, and symmetry elements discussed above. The cardinal principle of crystal structure is that the atoms of a crystal are set in space either on the points of a Bravais lattice or in some fixed relation to those points. It follows from this that the atoms of a crystal will be arranged periodically in three dimensions and that this arrangement of atoms will exhibit many of the properties of a Bravais lattice, in particular many of its symmetry elements. 

The simplest crystals one can imagine are those formed by placing atoms of the same kind on the points of a Bravais lattice. Not all such crystals exist but, fortunately for metallurgists, many metals crystallize in this simple fashion, and Fig. 2-14 shows two common structures based on the body-centered cubic (BCC) and face-centered cubic (FCC) lattices. The former has two atoms per unit cell and the latter four, as we can find by rewriting Eq. (2-1) in terms of the number of atoms, rather than lattice points, per cell and applying it to the unit cells shown. 

In fee lattiees, the tetrahedral and oetahedral sites are surrounded by regular polyhedra. For hep lattiees, the polyhedra are distorted if the ratio of lattiee parameters efa deviates from the ideal value of 1.633. In a bee lattiee, the polyhedra are greatly distorted. For the O sites, two metal atoms are mueh eloser to the interstitial site than the other four metal atoms. Therefore, the O sites are subdivided into O, Oj, and sites aeeording to the direction of the fourfold symmetry axis. In the same way, 

Intermetallics are not treated in this review, therefore, thermodynamic data regarding borides are not included in this chapter. 

Depending on the temperature, nickel borates can form crystals and glassy phases with different degrees of polymerisation, which are very often difficult to identify from the crystallographic point of view [66GME]. The numerous water-free and water- 

So far no direct determination of thermodynamic data for water-free solid nickel borates has been reported in the literature. One of the reasons for this is probably the difficulty in obtaining well-defined crystalline phases. The only available data, published in a compilation of thermodynamic data for borate systems [74SLO/JON], are estimated values of the standard enthalpy of formation and the entropy for Ni0-2B203 at 

The value of (Ni0-2B203, s, 298.15 K) = - (62.8 20.0) kJ-moT reported by [74SLO/JON] was obtained by a linear relationship between the enthalpy of reaction of the crystalline oxides and the ratio of cationic charge z to radius r. In view of the uncertainties with respect to the ionic radii selected, for (i) the z r versus A,//° correlation and for (ii) the Ni ion, this value has not been accepted by this review. 

No evidence of Ni(B02)2 4H20 formation was foimd in a solubility study on the system NiCl2-ZnCl2-H3B03-H20 [89BAL]. 

Neutton diffiaction was performed on the D4c diffractometer [10] at the ILL at a wavelength of 0.4972 A. Glass samples of the same composition, for example all BTO samples, were loaded into a 5 nun inner diameter vanadium can. The total mass of samples at each composition is shown in Table 4.1. The average glass sample radius was L7 mm for the pure BTO and L4 mm for the doped samples. An anpty vanadium can, boron powder in a vanadium can, the empty belljar, and a nickel standard were also measured for calibralion and correction purposes. All meas-uronaits were made at room temperature. A sample packing fraction of 0.51 0.05 was determined from liquid volume measurements. All samples were corrected for 

4 Rare Earth Doped Barium Titanate Glass 

In order to correct for the low r attractive behaviour of the Buckingham potential, a 

BHM type potential was fitted to it. This differed from the Buckingham potential by less than 5 % at 2 A r 6.7 A. All potentials were taken from Bush and Catlow [5], except the Yb +-potential which was taken from Lewis and Catlow [13]. In the absence of an Er +-0 potential the potential was used, as Y and Er are isomorphs 

Molecular dynamics simulations were performed using the DL POLY 2 [28] molecular dynamics package and the Buckingham potentials shown in Table 4.4. Due to the C term in the 0-0 potential, the short range behaviour becomes attractive below 1.5 A. This was corrected by fitting the 0-0 potential with a Bom-Huggins-Mayer type potential [11]  

1 In a dose-packed layer each sphere (atom) is in contact with six others such that the neighboring six spheres just touch one another. Why are six spheres, not five, or seven or some other number, required for dose packing around each sphere  

2 List the contents (number and type of atoms, ions, and/or molecules) of the conventional unit cell in each of the following crystalline solids  

3 Crystalline CaS (density2.S8gcm ) has been shown by the powder method to have the NaCl type of structure. 

4 The cubic unit cell of an alloy is illustrated in the figure on the right. 

6 The mineral perovskite has a cubic unit cell of edge 3.84 A with Ca, Ti, and O atoms located at its corners, body-center, and face-centers, respectively. 

Cellulose II can be obtained by sweUing cellulose I samples with alkali (known as mercerization typically, 21.5% NaOH aq. solution at 20 °C for 24 h) or by regeneration from cellulose solutions into precipitates, which is the typical process for the technical spinning of man-made cellulose fibers. 

Cellulose III] and Hfj] are converted from the corresponding cellulose I and II by immersing them in liquid ammonia (—78°C). The unit cells for both crystalline structures resemble each other, but the meridional reflections especially differ in X-ray investigations. However, the molecules in these two stractures pack in quite different manners parallel arrangements in III and antiparallel ones in IIIii this is concluded from the fact that III and IIIn can easily be returned to 

1 At what temperature would diethyl ether have a vapor pressure of 250 muiHg Use the vapor pressure at 18°C and A//vap given in Sample Problem 12.2. 

3 Using the graph, estimate the vapor pressure of the liquid at 100°C. 

4 Using the result from question 12.2.3 and another point from the graph, estimate A/Cap for the liquid. 

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In certain crystals, e.g. in quartz, there is chirality in the crystal structure. Molecular chirality is possible in compounds which have no chiral carbon atoms and yet possess non-superimposable mirror image structures. Restricted rotation about the C=C = C bonds in an allene abC = C = Cba causes chirality and the existence of two optically active forms (i)... 

The crystal structure determines not only the arrangement of atoms in the lattice but also the external form of the crystal. 

Laue pattern The symmetrical array of spots obtained on a photographic plate exposed to a non-homogeneous beam of X-rays after its passage through a crystal. The patterns constitute the earliest, although one of the most difficult, methods of investigating crystal structure by means of X-rays. 

Figure 1. shows the measured phase differenee derived using equation (6). A close match between the three sets of data points can be seen. Small jumps in the phase delay at 5tt, 3tt and most noticeably at tt are the result of the mathematical analysis used. As the cell is rotated such that tlie optical axis of the crystal structure runs parallel to the angle of polarisation, the cell acts as a phase-only modulator, and the voltage induced refractive index change no longer provides rotation of polarisation. This is desirable as ultimately the device is to be introduced to an interferometer, and any differing polarisations induced in the beams of such a device results in lower intensity modulation. 

Calculate the surface energy at 0 K of (100) planes of radon, given that its energy of vaporization is 35 x 10 erg/atom and that the crystal radius of the radon atom is 2.5 A. The crystal structure may be taken to be the same as for other rare gases. You may draw on the results of calculations for other rare gases. 

A number of substances such as graphite, talc, and molybdenum disulfide have sheetlike crystal structures, and it might be supposed that the shear strength along such layers would be small and hence the coefficient of friction. It is true... 

Knowing the lattice is usually not sufficient to reconstruct the crystal structure. A knowledge of the vectors (a, b, c) does not specify the positions of the atoms within the unit cell. The positions of the atoms withm the unit cell is given by a set of vectors i., = 1, 2, 3... u where n is the number of atoms in the unit cell. The set of vectors, x., is called the basis. For simple elemental structures, the unit cell may contain only one atom. The lattice sites in this case can be chosen to correspond to the atomic sites, and no basis exists. 

This structure is called close packed because the number of atoms per unit volume is quite large compared with other simple crystal structures. 

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

The induction energy is inlierently non-additive. In fact, the non-additivity is displayed elegantly in a distributed polarizability approach [28]. Non-additive induction energies have been found to stabilize what appear to be highly improbable crystal structures of the alkalme earth halides [57]. 

Dulmage W J and Lipscomb W N 1951 The crystal structures of hydrogen cyanide, HON Acta Crystallogr. 4 330... 

Another example of epitaxy is tin growdi on the (100) surfaces of InSb or CdTe a = 6.49 A) [14]. At room temperature, elemental tin is metallic and adopts a bet crystal structure ( white tin ) with a lattice constant of 5.83 A. However, upon deposition on either of the two above-mentioned surfaces, tin is transfonned into the diamond structure ( grey tin ) with a = 6.49 A and essentially no misfit at the interface. Furtliennore, since grey tin is a semiconductor, then a novel heterojunction material can be fabricated. It is evident that epitaxial growth can be exploited to synthesize materials with novel physical and chemical properties. 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

Figure Bl.8.4. Two of the crystal structures first solved by W L Bragg. On the left is the stnicture of zincblende, ZnS. Each sulphur atom (large grey spheres) is surrounded by four zinc atoms (small black spheres) at the vertices of a regular tetrahedron, and each zinc atom is surrounded by four sulphur atoms. On the right is tire stnicture of sodium chloride. Each chlorine atom (grey spheres) is sunounded by six sodium atoms (black spheres) at the vertices of a regular octahedron, and each sodium atom is sunounded by six chlorine atoms.

The first crystal structure to be detennined that had an adjustable position parameter was that of pyrite, FeS2 In this structure the iron atoms are at the comers and the face centres, but the sulphur atoms are further away than in zincblende along a different tln-eefold synnnetry axis for each of the four iron atoms, which makes the unit cell primitive. 

As the number of atoms in the asyimnetric unit increases, the solution of a structure by any of these phase-independent methods becomes more difficult, and by 1950 a PhD thesis could be based on a single crystal structure. At about that time, however, several groups observed that the fact that the electron density must be non-negative everywhere could be exploited to place restrictions on possible phases. The first use of this fact was by D Marker and J S Kasper [24], but their relations were special cases of more general relations introduced by J Karle and H Hauptman [25]. Denoting by A. the set of indices h., k., /., the Karle-Hauptman condition states that all matrices of the fonu... 

Flarker D 1936 The application of the three-dimensional Patterson method and the crystal structures of proustite, Ag,AsS, and pyrargyrite, Ag,SnS, J. Chem. Phys. 4 381-90... 

Satellite transition MAS NMR provides an alternative method for detennining the interactions. The intensity envelope of the spimiing sidebands are dominated by site A2 (using the crystal structure nomenclature) which has the smallest Cq, resulting in the intensity for the transitions of this site being spread over the smallest... 

Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

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