Class crystal

Pyroelectrics. Pyroelectric ceramics are materials that possess a uoique polar axis and are spontaneously polarized ia the abseace of an electric field. Pyroelectrics are also a subset of piezoelectric materials. Ten of the 20 crystal classes of materials that display the piezoelectric effect also possess a unique polar axis, and thus exhibit pyroelectricity. In addition to the iaduced charge resultiag from the direct pyroelectric effect, a change ia temperature also iaduces a surface charge (polarizatioa) from the piezoelectric aature of the material, and the strain resultiag from thermal expansioa. 

Table 3. Classification of Mixed-Metal Oxide Inorganic Pigments According to Crystal Class ...

Table 3 is arranged by crystal class (14). The crystal class of a given pigment is determined almost solely by the ratio of the ionic sizes of the cation and the anion and thek respective valences. Hence, for any given stoichiometry and ionic size ratio, only one or two stmctures will be possible. In some classes (spkiel, zkcon), a wide range of colors is possible within the confines of that class. Pigments within a given class usually have excellent chemical and... 

Hie US Armed Forces requirements for oxalic acid are covered by Federal Spec 0-0-690a, Oxalic Acid, Dihydrate, Technical , (July 1, 1968). It details three classes class 1—large crystals, class 2—small crystals, and class 3— powder. It requires a min assay of 99.0% by wt as H2C204.2H20, a max ash content of 0.20% by wt, and the following particle size characteristics using US Standard sieves ... 

Table 3.1 The 32 crystal classes and the corresponding crystal systems...

State the crystal classes and crystal systems to which the following space groups belong ... 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

Crystals can only be piezoelectric when they are non-centrosymmetric. In addition, they may not belong to the crystal class 4 32. The effect is thus restricted to 20 out of the 32 crystal classes. 

No ferroelectricity is possible when the dipoles in the crystal compensate each other due to the crystal symmetry. All centrosymmetric, all cubic and a few other crystal classes are... 

Table 19.1 Crystal classes permitting ferroelectric crystals...

In crystalline solids, the Raman effect deals with phonons instead of molecular vibration, and it depends upon the crystal symmetry whether a phonon is Raman active or not. For each class of crystal symmetry it is possible to calculate which phonons are Raman active for a given direction of the incident and scattered light with respect to the crystallographic axes of the specimen. A table has been derived (Loudon, 1964, 1965) which presents the form of the scattering tensor for each of the 32 crystal classes, which is particularly useful in the interpretation of the Raman spectra of crystalline samples. 

Table 15 The crystallographic point groups (crystal classes).

TrR = 2 cos y 1 = n, a positive or negative integer, or zero. Then, cos y can only be integer or half-integer, and only axes of orders 1,2,3,4 and 6 are possible (see problem 18). With this limitation it is found that only 32 groups can.be formed from foe operations that describe the symmetry of a unit cell. These point groups constitute the 32 crystal classes shown in Table 15. 

The unit cell is defined by the lengths (a, b, and c) of the crystal axes, and by the angles (a, f>, and y) between these. The usual convention is that a defines the angle between the b- and c-axes, p the angle between the a- and c-axes, and y the angle between the a- and 6-axes. There are seven fundamental types of primitive unit cell (whose characteristics are provided in Table 7.1), and these unit cell characteristics define the seven crystal classes. If the size of the unit cell is known (i.e., a, (i, y, a, b, and c have been determined), then the unit cell volume (V) may be used... 

The seven crystal classes, defined from their fundamental unit cells... 

Crystallographic nomenclature (Bravais lattices, crystal classes, space groups) The following information is generally included in a usual crystallographic description ... 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

Lattice equivalent ( translationengleich , abbreviation t). M contains all the translations of G, the crystal class of M is of lower symmetry than that of G. 

Globular proteins were much more difficult to prepare in an ordered form. In 1934, Bernal and Crowfoot (Hodgkin) found, that crystals were better preserved if they were kept in contact with their mother liquor sealed in thin-walled glass capillaries. By the early 1940s crystal classes and unit cell dimensions had been determined for insulin, horse haemoglobin, RNAase, pepsin, and chymotrypsin. Complete resolution of the structures required identification of the crystal axes and some knowledge of the amino acid sequence of the protein—requirements which could not be met until the 1950s. 

Thus the reciprocal lattice axes are perpendicular to the (100), (010) and (001) planes in the real-space lattice. In cubic, tetragonal and orthorhombic crystals it is also trae that they are parallel to the [100], [010] and [001] directions, but this is not tme in other crystal classes. The general formulae for the reciprocal space... 

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