The normalized reduced form and cell mathematical tools for lattice analysis--symmetry and similarity.To intelligently and effectively use crystallographic crys·tal·log·ra·phy n. The science of crystal structure and phenomena. crys  tal·log databases,
mathematical and computer tools are required that can elucidate e·lu·ci·date v. e·lu·ci·dat·ed, e·lu·ci·dat·ing, e·lu·ci·dates v.tr. To make clear or plain, especially by explanation; clarify. v.intr. To give an explanation that serves to clarify. diverse types of intra- and interlattice relationships. Two such tools are the normalized reduced form In social science and statistics, particularlly econometrics, a reduced form equation is a method of dealing with endogeneity. A reduced form equation is defined by James Stock & Mark Watson (2007) in the following way: and normalized reduced cell. Practical experience has revealed that the first tool--the normalized reduced form--is very helpful in establishing lattice (theory) lattice - A partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. This definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not metric symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. as it enables one to readily deduce de·duce tr.v. de·duced, de·duc·ing, de·duc·es 1. To reach (a conclusion) by reasoning. 2. To infer from a general principle; reason deductively: significant relationships between the elements of the reduced form. Likewise research with crystallographic databases has demonstrated that the second tool--the normalized reduced cell--plays a vital role in determining metrically met·ri·cal adj. 1. Of, relating to, or composed in poetic meter: metrical verse; five metrical units in a line. 2. Of or relating to measurement. similar lattices. Knowledge of similar lattices has practical value in solving structures, in assignment of structure types, in materials design, and in nano-technology. In addition to using the reduced cell, it is recommended that lattice-matching strategies based on the normalized reduced cell be routinely carried out in database searching, in data evaluation, and in experimental work. Key words: identification; lattice-matching strategies; lattice relationships; lattice similarity; metric lattice; normalized reduced cell and form; symmetry. ********** 1. Introduction The various crystallographic databases [1] now available constitute a large, comprehensive, and rapidly growing scientific resource, serving as an invaluable source of data for the intelligent design of materials, for crystal engineering, and for nanotechnology. To evaluate data entering these databases and to intelligently and effectively use this resource, diverse mathematical tools are required that can establish intralattice relationships or elucidate various types of interlattice relationships. Two such tools are the normalized reduced form and the normalized reduced cell--tools that are ideal for elucidating certain types of intra- and interlattice relationships. For example, with the normalized reduced form, one can determine lattice-metric symmetry and deduce other types of intralattice relationships. With the normalized reduced cell, one can determine metrically similar lattices (1) via lattice matching techniques against the lattices in the crystallographic databases. Practical experience has revealed that these tools are very useful for routine and complex lattice analyses. Before proceeding with applications of these tools, it is necessary to define the normalized reduced cell and form. 1.1 Definitions The reduced cell is a unique primitive cell In geometry, solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell, is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2D, 3D, or other dimensions. of the lattice, which is based on the three shortest lattice translations. For the precise mathematical definition of the reduced cell and form and for procedures to calculate this cell, see [2] and NBS (National Bureau of Standards) See NIST. NBS - National Bureau of Standards: part of the US Department of Commerce, now NIST. Technical Note 1290 [3]. The normalized reduced cell of a lattice is determined simply by dividing the cell edges of the reduced cell by the a-cell edge. The normalized reduced form is calculated from the normalized reduced cell and is defined by the vector dot products of the normalized reduced cell edge vectors: a * a b * b c * c b * c a * c a * b As an example, consider the reduced cell for a typical triclinic crystal structure reported in the recent literature [4]: [a.sub.t] = 9.6907[Angstrom angstrom (ăng`strəm), abbr. Å, unit of length equal to 10−10 meter (0.0000000001 meter); it is used to measure the wavelengths of visible light and of other forms of electromagnetic radiation, such as ultraviolet ] [b.sub.t] = 10.3119[Angstrom] [c.sub.t] = 11.2549[Angstrom] [[alpha].sub.t] = 63.954[degrees] [[beta].sub.t] = 70.282[degrees] [[gamma].sub.t] = 87.414[degrees] The corresponding normalized reduced cell and form are: Cell: a = 1.0000 b = 1.0641 c = 1.1614 [alpha] = 63.954[degrees] [beta] = 70.282[degrees] [gamma] = 87.414[degrees]
Form: a * a b * b c * c 1.000 1.132 1.349
=
b * c a * c a * b 0.543 0.392 0.048
The fact that there is no specialization A career option pursued by some attorneys that entails the acquisition of detailed knowledge of, and proficiency in, a particular area of law. As the law in the United States becomes increasingly complex and covers a greater number of subjects, more and more attorneys are (2) in the normalized reduced form shows that the metric lattice is triclinic. 2. Discussion and Applications The reduced form and cell have long been used in lattice metric symmetry determination and identification, respectively. Although the reduced form can be used in the symmetry checks discussed below, the normalized reduced form has the advantage in that it makes the interrelationships--and specialization--of the elements of the reduced form more transparent. Recognition of matrix-element specialization is a basis of symmetry determination as well as for investigations of many other lattice-related phenomena. Likewise, although Crystal Data Determinative Ratios [5] may be used to locate similar lattices within a given crystal system, the normalized reduced cell provides the logical basis for a far more powerful and comprehensive lattice-matching technique which is crystal system independent and conceptually parallel to techniques based on matching reduced cells. Details of the application of the normalized reduced form and cell for symmetry determination and for the determination lattice-metric similarity are outlined below. 2.1 Symmetry Determination via the Normalized Reduced Form (NRF NRF National Retail Federation NRF NATO Response Force NRF National Research Foundation (South Africa) NRF Neighbourhood Renewal Fund (urban renewal funding package in the UK) NRF Nouvelle Revue Française ) The normalized reduced form (NRF) is a practical tool, which can be used--in conjunction with the matrix method [6]--for metric symmetry determination. With the NRF one can readily determine the metric symmetry of the lattice by matching it against a table of the 44 reduced forms [7]. To illustrate, Table 1 herein presents the 13 reduced forms corresponding to the centered monoclinic mon·o·clin·ic adj. Of or relating to three unequal crystal axes, two of which intersect obliquely and are perpendicular to the third. monoclinic Adjective Crystallog lattices. Typical examples of the NRFs are given that have been derived from cell constants published in recent issues of Acta Crystallographica Section E. Once normalized, the pattern of the relationships of dot products in the NRF is easy to ascertain. From the examples, one can see that it is especially easy to determine the reduced form number (first column) by matching a given NRF against the characteristic reduced form matrices presented in the second through fourth columns in Table 1. Once the reduced form number is known, one can consult the reference table of the 44 reduced forms [7] to obtain the appropriate transformation matrix to determine the conventional cell. In the last column, the frequency of occurrence (for the first 2.4 years that Acta Crystallographica Section E has been in existence) of each reduced form is given. The frequencies reveal that reduced form numbers 39, 27, 10, 37, and 14 are the most common for the centered monoclinic lattices. The crystal symmetry can never exceed the metric symmetry, but it can be less. However, by analyzing the crystallographic databases, it has been observed that the metric and crystal symmetry are almost always the same [20, 21]. This coincidence of crystal and metric symmetry continues to hold true in recently published structures. For example, a detailed analysis of the NRFs, for 205 centered monoclinic cells published in Acta Crystallographica Section E, revealed that in every case the crystal and metric symmetry are identical. This fact provides a basis for a reliable method for evaluation of the symmetry of crystalline Like a crystal. It implies a uniform structure of molecules in all dimensions. For example, phase change technology, widely used for rewritable optical discs, uses crystalline spots (bits) to reflect the laser beam. Amorphous, non-crystalline bits do not reflect light. compounds [20, 21]. For example, cases in which metric symmetry exceeds the crystal symmetry represent either misidentified symmetry [22] or something unusual in the crystal structure [23]. Furthermore, from inspection of the NRF, one may ascertain extra relationships (not required by one of the 44 reduced forms) among the dot products. The experimentalist (or user of cell data in the crystallographic databases) should be aware that any extra specialization in the NRF may signify sig·ni·fy v. sig·ni·fied, sig·ni·fy·ing, sig·ni·fies v.tr. 1. To denote; mean. 2. To make known, as with a sign or word: signify one's intent. an important fact: for example, that one has inadvertently determined a derivative cell of a lattice of higher symmetry. Finally, as an integral part of routine practice, it is suggested that the normalized reduced form be determined and checked against a table of reduced forms [7] to ascertain the highest possible metric symmetry, to check for extra specialization, and to determine the transformation matrix to a conventional cell. 2.2 Lattice Similarity Determination via Normalized Reduced Cells The reduced cell has long played a practical role in lattice-matching strategies [24, 25, and 26]. Likewise, the normalized reduced cell can play a useful role in lattice-matching techniques. Lattice-matching methods based on the reduced cell are used to locate lattices or derivative lattices that have the same metric parameters. Lattice-matching techniques based on the normalized reduced cell are designed to find lattices that have similar metric parameters. The two strategies are conceptually analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development. a·nal·o·gous adj. . To understand how the normalized reduced-cell strategy works, first we summarize sum·ma·rize intr. & tr.v. sum·ma·rized, sum·ma·riz·ing, sum·ma·riz·es To make a summary or make a summary of. sum the reduced cell strategy, which is in common use. 2.2.1 Lattice-Matching Procedure Based on Reduced Cells The basic identification strategy is to check the lattice of the unknown (or an existing lattice) against all lattices in a database for a match and then to exclude unwanted matches on the basis of chemical information. In this scheme, for example, an unknown crystal is selected and mounted on a single-crystal diffractometer A Diffractometer (Main Entry: dif·frac·tom·e·ter Pronunciation: di-"frak-'tä-m&-t&r Function: noun) is a measuring instrument for analyzing the structure of a usually crystalline substance from the scattering pattern produced when a beam of radiation or particles (as X rays or and a unit cell is determined and reduced. The reduced cell is then checked against the file of known materials. If desired, one calculates derivative lattices, which are also reduced and checked against the file of known lattices. Experience has shown that identification based on matching reduced cells is very straightforward and reliable. In fact research with the crystallographic databases has shown that the reduced cell coupled with the element types uniquely defines a material. Currently this identification strategy is used in association with several crystallographic databases that are distributed to the scientific community. It has also been integrated into automated single-crystal x-ray diffractometers [27]. Similarly, a registration-identification procedure based on reduced cells is used in the addition of new compounds to the Cambridge Crystallographic Database [28]. Further details on lattice matching, on a computer program for lattice matching, and on the calculation of derivative lattices have been published as an NBS Technical Note [25] and in Acta Crystallographica [26]. 2.2.2 Lattice-Matching Technique Based on Normalized Reduced Cells In a manner strictly parallel to the above, the normalized reduced cell can be used instead of the reduced cell in lattice-matching techniques. This is illustrated in Fig. 1 in which the normalized reduced cell has replaced the reduced cell. Here the basic search strategy is the same as above except that the normalized reduced cell is checked against the file (database) of normalized reduced cells for known materials. If desired, one calculates derivative lattices, which are also reduced, normalized and checked against the file of known lattices represented by their respective normalized reduced cells. The set of matches can be further restricted using chemical or other types of data. As most materials crystallize crys·tal·lize also crys·tal·ize v. crys·tal·lized also crys·tal·ized, crys·tal·liz·ing also crys·tal·iz·ing, crys·tal·liz·es also crys·tal·iz·es v.tr. 1. in the low symmetry crystal systems (e.g., over 90% of organic and organometallic organometallic /or·ga·no·me·tal·lic/ (-me-tal´ik) consisting of a metal combined with an organic radical, used particularly for a compound in which the metal is linked directly to a carbon atom. compounds crystallize in the triclinic, monoclinic, and orthorhombic or·tho·rhom·bic adj. Of or relating to a crystalline structure of three mutually perpendicular axes of different length. orthorhombic systems), this type of lattice matching generally produces a limited and meaningful set of matches. With this technique, the experimentalist can find metrically similar lattices. Knowledge of similar lattices has practical value in solving structures, in relating structures, in assignment of structure types, in materials design, and in nanotechnology. For example, information gained from a similarity search is valuable in the development of materials having a desired physical property. If a given compound has the desired property, one can find all compounds with similar lattices, some of which may exhibit the specified property to a greater extent. 3. Conclusion The normalized reduced form and cell represent practical mathematical tools for the analysis of intra- and interlattice relationships. With respect to intralattice relationships, experience with thousands of lattices has revealed that the normalized reduced form is a very useful tool for the evaluation of lattice-metric symmetry as well as for the determination of other significant relationships between the elements of the reduced form. Likewise, with respect to interlattice relationships, experience has shown that the normalized reduced cell is an excellent tool to determine metrically similar lattices. [FIGURE 1 OMITTED] In deducing significant interlattice relationships, one can systematically run the normalized reduced cell and reduced cell search in parallel with each other. It is suggested that such dual searching be routinely carried out--in data evaluation, in searching crystallographic databases, and in determining crystal structures--to ascertain the manner in which an extant ex·tant adj. 1. Still in existence; not destroyed, lost, or extinct: extant manuscripts. 2. Archaic Standing out; projecting. or new lattice is related to the field of existing lattices. First, to find lattices that are metrically the same, the reduced cell can be matched against a file of reduced cells of known materials, and second, to find lattices that are metrically similar, the normalized reduced cell can be checked against a file of normalized reduced cells.
Table 1. Metric classification of the 13 reduced forms that correspond
to the centered monoclinic lattices. For each of the 13 generic reduced
form matrices, a typical example of a normalized reduced form is given
in columns 5-7. The cell data used in the calculations as well as the
frequency data given in column 11 are based on data published in recent
issues of Acta Crystallographica, Sect. E
Reduced Reduced form matrix
form No.
a = b
10 a*a a*a c*c
b*c b*c a*b
14 a*a a*a c*c
-|b*c| -|b*c| -|a*b|
17 a*a a*a c*c
-|b*c| -|a*c| -(a*a-|b*c|-|a*c|)
b = c
20 a*a b*b b*b
b*c a*c a*c
25 a*a b*b b*b
-|b*c| -|a*c| -|a*c|
a [less than or equal to] b [less than or equal to] c
27 a*a b*b c*c
b*c a*a/2 a*a/2
28 a*a b*b c*c
a*b/2 a*a/2 a*b
29 a*a b*b c*c
a*c/2 a*c a*a/2
30 a*a b*b c*c
b*b/2 a*b/2 a*b
37 a*a b*b c*c
-|b*c| -a*a/2 0
39 a*a b*b c*c
-|b*c| 0 -a*a/2
41 a*a b*b c*c
-b*b/2 -|a*c| 0
43 a*a b*b c*c
-[b*b-|a*b|]/2 -[a*a-|a*b|]/2 -|a*b|
Reduced Normalized Type Bravais Ref. Freq.
form No. reduced form matrix lattice
a = b
10 1.00 1.00 1.40 + MC (a) [8] 28
0.24 0.24 0.46
14 1.00 1.00 1.10 - MC [9] 21
-0.22 -0.22 -0.18
17 1.00 1.00 5.65 - MC [10] 14
-0.29 -0.41 -0.30
b = c
20 1.00 1.27 1.27 + MC [11] 13
0.26 0.19 0.19
25 1.00 1.59 1.59 - MI [12] 12
-0.30 -0.43 -0.43
a [less than or equal to] b [less than or equal to] c
27 1.00 1.32 3.61 + MC [13] 32
0.47 0.50 0.50
28 1.00 1.31 1.95 + MC [14] 4
0.16 0.50 0.32
29 1.00 3.16 4.39 + MC [15] 5
0.20 0.40 0.50
30 1.00 2.20 2.75 (b) + MC N/A 0
1.10 0.15 0.30
37 1.00 1.10 1.55 - MI [16] 22
-0.52 -0.50 0.00
39 1.00 1.26 1.71 - MC [17] 47
-0.30 0.00 -0.50
41 1.00 1.24 9.36 - MI [18] 5
-0.62 -0.36 0.00
43 1.00 2.48 5.47 - MI [19] 2
-1.14 -0.40 -0.20
(a) For each example, the first symbol "M" stands for monoclinic, and
the second symbol "C or I" represents the centering of the conventional
cell of the lattice.
(b) Created for illustrative purposes; an actual example was not found
in Acta Crystallogr., Sect. E.
Acknowledgment acknowledgment, in law, formal declaration or admission by a person who executed an instrument (e.g., a will or a deed) that the instrument is his. The acknowledgment is made before a court, a notary public, or any other authorized person. The author gratefully acknowledges a long-term and productive collaboration with Vicky Lynn Karen on the mathematical properties of lattices and their practical applications. Accepted: February 4, 2004 Available online: http://www.nist.gov/jres (1) Lattices that are defined by reduced cells that have corresponding angles Corresponding angles are formed when a given transversal line crosses two parallel lines. The corresponding angles are congruent and these congruent angles can be used to determine the degrees of the other angles of the parallel lines. equal and corresponding edges proportional. (2) A reduced form is said to be specialized spe·cial·ize v. spe·cial·ized, spe·cial·iz·ing, spe·cial·iz·es v.intr. 1. To pursue a special activity, occupation, or field of study. 2. if there is a simple mathematical relationship between two or more of the matrix elements (e.g. a*a = b*b; b*c = 1/2 b*b; b*c = 1/2 a*c). Table 1 gives the types of specialization found in the reduced forms corresponding to the centered monoclinic lattices. 4. References [1] Special Issue on Crystallographic Databases, F. Allen and J. Glusker, eds., Acta Cryst. B58, 317-422 (2002). [2] A. Santoro and A. D. Mighell, Determination of Reduced Cells, Acta Cryst. A26, 124-127 (1970). [3] V. L. Karen and A. D. Mighell, NIST* Lattice: A Program to Analyze Lattice Relationships, NIST Tech. Note 1290 (1991). [See also Technical Note 1214 (1985).] [4] A. J. Blake, M. Felloni, P. Hubberstey, M. Schroder and C. Wilson, Hexakis(dimethyl sulfoxide dimethyl sulfoxide (DMSO) Colourless, nearly odourless liquid organic compound. It mixes in all proportions with water, ethanol, and most organic solvents and dissolves a wide variety of compounds (but not aliphatic hydrocarbons). )nickel nickel, metallic chemical element; symbol Ni; at. no. 28; at. wt. 58.69; m.p. about 1,453°C;; b.p. about 2,732°C;; sp. gr. 8.902 at 25°C;; valence 0, +1, +2, +3, or +4. (II) dinitrate dimethyl sulfoxide disolvate, Acta Cryst. E57, m556-m557 (2001). [5] Crystal Data Determinative Tables (Nat. Hist.) tables presenting the specific character of minerals, plants, etc., to assist in determining the species to which a specimen belongs. See also: Determinative , 3rd Ed., Vol.1 (1972), Vol. 2 (1973), Vols. 3-4 (1978), Vols. 5-6 (1983). U.S. Department of Commerce, National Bureau of Standards National Bureau of Standards: see National Institute of Standards and Technology. National Bureau of Standards - National Institute of Standards and Technology , and the JCPDS--International Centre for Diffraction Data, Swarthmore, PA. [6] V. L. Karen (Himes) and A. D. Mighell, A Matrix Approach to Symmetry, Acta Cryst. A43, 375-384 (1987). [7] Mighell, A. D. (2001). J. Res. Natl. Inst. Stand. Technol. 106, 983-995. This reference includes an updated version of the reference table of 44 reduced forms which was originally published by A. D. Mighell, A. Santoro, and J. D. H. Donnay, in International Tables for X-ray Crystallography X-ray crystallography, the study of crystal structures through X-ray diffraction techniques. When an X-ray beam bombards a crystalline lattice in a given orientation, the beam is scattered in a definite manner characterized by the atomic structure of the lattice. , Vol. I., 3rd Ed., Birmingham, Kynoch Press (1969) pp. 530-535. [8] A. W. Xu, Y. P. Cai, L. Z. Zhang, C. Y. Su, and B. S. Kang, [8,8'-(Propane-1,3-diyldioxy)diquinoline-[[kappa Kappa Used in regression analysis, Kappa represents the ratio of the dollar price change in the price of an option to a 1% change in the expected price volatility. Notes: Remember, the price of the option increases simultaneously with the volatility. ].sup.4] N,O,O',N'] silver(I) trifluoromethane-sulfonate, Acta Cryst. E58, m770-m771 (2002). [9] M, Kato, A. Toshikawa, and S. Kishi, [mu]-Guanidinidobis[(terpyridine)platinum(II)] tris(hexaflurophosphate) acetonitrile acetonitrile /ac·e·to·ni·trile/ (as?e-to-ni´tril) a colorless liquid with an etherlike odor used as an extractant, solvent, and intermediate; ingestion or inhalation yields cyanide as a metabolic product. solvate Noun 1. solvate - a compound formed by solvation (the combination of solvent molecules with molecules or ions of the solute) chemical compound, compound - (chemistry) a substance formed by chemical union of two or more elements or ingredients in definite proportion , Acta Cryst. E58, m248-m250 (2002). [10] G. Margraf, H.-W. Lerner, and M. Bolte, Tetramethylphosphonium chloride chloride (klōr`īd, klôr`–), chemical compound containing chlorine. Most chlorides are salts that are formed either by direct union of chlorine with a metal or by reaction of hydrochloric acid (a water solution of hydrogen chloride) hydrate hydrate (hī`drāt), chemical compound that contains water. A common hydrate is the familiar blue vitriol, a crystalline form of cupric sulfate. Chemically, it is cupric sulfate pentahydrate, CuSO4·5H2O. , Acta Cryst. E58, o546-o547 (2002). [11] D. S. Bohle and D. Stasko, [N-(2-Oxidophenyl)salicylaldiminato-[[kappa].sup.3] N,O,O']-(2.2':6',2"-terpyridine-[[kappa].sup.3]N)copper(II), Acta Cryst. E58, m340-m341 (2002). [12] A. L. Hector and T. A. Mayer, Poly[[hexaacetatodiethanoltrimagnesium(II)] diethanol solvate], a polymeric polymeric /poly·mer·ic/ (pol?i-mer´ik) exhibiting the characteristics of a polymer. pol·y·mer·ic adj. 1. Having the properties of a polymer. 2. acetatebridged magnesium magnesium (măgnē`zēəm, –zhəm), metallic chemical element; symbol Mg; at. no. 12; at. wt. 24.305; m.p. about 648.8°C;; b.p. about 1,090°C;; sp. gr. 1.738 at 20°C;; valence +2. chain structure, Acta Cryst. E58, m628-m630 (2002). [13] S. L. Raishbrook, P. Turner, and R. Codd, A potential synthon A synthon is a concept in retrosynthetic analysis. It is defined as a structural unit within a molecule which is related to a possible synthetic operation. The term was coined by E.J. Corey. for models of vanadium vanadium (vənā`dēəm), metallic chemical element; symbol V; at. no. 23; at. wt. 50.9415; m.p. about 1,890°C;; b.p. 3,380°C;; sp. gr. about 6 at 20°C;; valence +2, +3, +4, or +5. Vanadium is a soft, ductile, silver-grey metal. haloperoxidase: (3,5-dimethylpyrazole)-bis[2-hydroxy-2-methylbutanoato(1-)]-oxovanadium(IV), Acta Cryst. E58, m737-m739 (2002). [14] R. J. Butcher, G. M. Mockler, and O. McKern, (Piperidine pi·per·i·dine n. A strongly basic, colorless liquid from which certain phenothiazine antipsychotics are derived. [kappa]N)[N-(salicylidene)phenyl-alaninato-[[kappa].sup.3] O,N,O']copper(II), Acta Cryst. E59, m61-m63 (2003). [15] L. Esser, Z. Otwinowski, and H. Kim, 2,5-Dibromo-6-isopropyl-3-methyl-p-benzoquinone, Acta Cryst. E58, o170-o171 (2002). [16] R. A. Howie and S. M. S. V. Wardell, Bis(2-carboethoxyethyl)diiodotin at 120 K, Acta Cryst. E58, m257-m259 (2002). [17] S. F. Soh, C. S. Lai, and E. R. T. Tiekink, (4,7-Dimethyl-1,10-phenanthroline)-bis(O-isopropropyldithiocarbonato)zinc(II), Acta Cryst. E58, m641-m643 (2002). [18] J. Jovanovic, M. Schurmann, H. Preut, and M. Spiteller, Second modification of (E)-2,3,2', 3'-tetrahydro-[1,1']-biindenylidene, Acta Cryst. E57, o1100-o1101 (2001). [19] P. Harris and P. Kofod, A methyl-coordinated [Rh.sup.III] ion in methylpenta-amminerhodium(III)-chloropentaammine-rhodium(III)-dithionate (0.73/2.27/3), Acta Cryst. E58, m460-m462 (2002). [20] A. D. Mighell and J. R. Rodgers, Lattice Symmetry Determination, Acta Cryst. A36, 321-326 (1980). [21] V. L. Karen (Himes) and A. D. Mighell, A Matrix Method for Lattice Symmetry Determination, Acta Cryst. A38, 748-749 (1982). [22] R. E. Marsh, M. Kapon, S. Hu, and F. H. Herbstein, Some 60 new space-group corrections, Acta Cryst. B58, 62-77, (2002). [23] H. Mansikkamaki, M. Nissinen, and K. Rissanen, Encapsulation (1) In object technology, the creation of self-contained modules that contain both the data and the processing. See object-oriented programming. (2) The transmission of one network protocol within another. of diquats by resorcinarenes: a novel staggered anion-solvent mediated me·di·ate v. me·di·at·ed, me·di·at·ing, me·di·ates v.tr. 1. To resolve or settle (differences) by working with all the conflicting parties: hydrogen bonded hydrogen bond n. A chemical bond in which a hydrogen atom of one molecule is attracted to an electronegative atom, especially a nitrogen, oxygen, or fluorine atom, usually of another molecule. capsule capsule In botany, a dry fruit that opens when ripe. It splits from top to bottom into separate segments known as valves, as in the iris, or forms pores at the top (e.g., poppy), or splits around the circumference, with the top falling off (e.g., pigweed and plantain). , Chem. Commun. 1902-1903 (2002). [24] A. D. Mighell, The Reduced Cell: Its Use in the Identification of Crystalline Materials, J. Appl. Cryst. 9, 491-498 (1976). [25] V. L. Karen (Himes) and A. D. Mighell, NBS*LATTICE: A Program to Analyze Lattice Relationships, Natl. Bur. Stand. (U.S.) Tech. Note 1214 (1985). [26] A. D. Mighell and V. L. Karen, Compound Identification and Characterization A rather long and fancy word for analyzing a system or process and measuring its "characteristics." For example, a Web characterization would yield the number of current sites on the Web, types of sites, annual growth, etc. Using Lattice-Formula Matching Techniques, Acta Cryst. A42, 101-105 (1986). [27] S. K. Byram, C. F. Campana, J. F. Fait, and R. A. Sparks. Using NIST Crystal Data within Siemens' Software for Four-Circle and Smart CCD CCD in full charge-coupled device Semiconductor device in which the individual semiconductor components are connected so that the electrical charge at the output of one device provides the input to the next device. Diffractometers, J. Res. Natl. Inst. Stand. Technol. 101, 295 (1996). [28] J. R. Rodgers, and A. D. Mighell, Registration-Identification of Crystalline Materials Based on Lattice and Empirical Formula empirical formula: see formula. . J. Chem. Information and Computer Sciences 21(1), 42-47 (1981). Alan D. Mighell National Institute of Standards and Technology National Institute of Standards and Technology, governmental agency within the U.S. Dept. of Commerce with the mission of "working with industry to develop and apply technology, measurements, and standards" in the national interest. , Gaithersburg, MD 20899-8520 alan.mighell@nist.gov About the author: Alan D. Mighell has been a research scientist at NIST since 1964. His research interests include structural crystallography and the design and development of mathematical procedures for materials identification, for establishing lattice relationships, and for the evaluation of crystallographic data. He is in the Ceramics Division of the NIST Materials Science and Engineering Materials science and engineering A multidisciplinary field concerned with the generation and application of knowledge relating to the composition, structure, and processing of materials to their properties and uses. Laboratory. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce. |
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