http://www.tms.org/pubs/journals/JOM/9908/Ress/Ress9908.html

TABLE OF CONTENTS 

The literature is full of reports of lattice parameters and bond angles that are variable or imprecise for chemicals. This imprecision is a result of the analysis techniques used and the imprecision in the samples and measurement process. Research has shown that fuzzy logic is a unique tool for representing imprecision; this article extends fuzzy research into the area of molecular modeling.
a  b 
Figure 1. (a) The crisp set 3; (b) the fuzzy set . 
In normal set theory, an object is either a member of a set or not (i.e., there are only two states), and the set is often referred to as a crisp set. Fuzzy sets, on the other hand, have degrees of membership to that set. Thus, it is possible for an object to have partial membership in a set. This forms the basis of fuzzyset theory. Figure 1 shows a comparison of a crisp set and a fuzzy set. Notice that the number two is not a member of the crisp set 3, but it is a member of the fuzzy set . The vertical axis is called the degree of membership (a) and is normalized such that a [0, 1]. For example, the number 2.5 has a degree of membership of 0.5 in the fuzzy set in Figure 1.
Figure 2. An example of triangular fuzzy numbers, where = <2,3,5>. 
The left and right spreads of a TFN are a common measure of the TFN's variability or imprecision. In Figure 2, the left spread is 3 – 2 = 1; the right spread is 5 – 3 = 2.
One reason why TFNs are well suited to modeling and design is because their arithmetic operators and functions are developed, which allow fast operation on equations. These arithmetic operators include the following examples, where the following TFNs are given:
= <a_{1}, a_{2}, a_{3}> and = <b_{1}, b_{2}, b_{3}> 
and the following values are used for illustration:
= <2, 4, 5> and = <1, 2, 3> 
Addition
=
+
is defined as
= <c_{1}, c_{2}, c_{3}> = <a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}>  (1) 
e.g., = <2 + 1, 4 + 2, 5 + 3> = <3, 6, 8> 
Subtraction
=
–
is defined as
= <c_{1}, c_{2}, c_{3}> = <a_{1} – b_{3}, a_{2} – b_{2}, a_{3} – b_{1}>  (2) 
e.g., = <2 – 3, 4 – 2, 5 – 1> = <–1, 2, 4> 
Multiplication
=
x
is defined as
= <c_{1}, c_{2}, c_{3}> = <a_{1} x b_{1}, a_{2} x b_{2}, a_{3} x b_{3}>  (3) 
e.g., = <2 x 1, 4 x 2, 5 x 3> = <2, 8, 15> 
Division
= ÷ is defined as
= <c_{1}, c_{2}, c_{3}> = <a_{1} ÷ b_{3}, a_{2} ÷ b_{2}, a_{3} ÷ b_{1}>  (4) 
e.g., = <2 ÷ 3, 4 ÷ 2, 5 ÷ 1> = <0.667, 2, 5> 
Exponentiation
=
^{k} (where k is the crisp scalar with, for example, a value of 0.5) is defined as
= <c_{1}, c_{2}, c_{3}> = <a_{1}^{k}, a_{2}^{k}, a_{3}^{k}>  (5) 
e.g., = <2^{0.5}, 4^{0.5}, 5^{0.5}> = <1.414, 2, 2.236> 
In addition to these five arithmetic operators, recent research into fuzzy trigonometric functions has produced fuzzy versions of the cosine, sine, tangent, cotangent, secant, and cosecant functions (plus their respective inverses) that work with TFNs and the above arithmetic operators.^{4} Together, the arithmetic operators and trigonometric functions enable the modeling of chemical compounds and the calculation of fuzzy bond lengths and fuzzy bond angles.
Figure 3. A example of fuzzy lines where (top) point A is fixed, point B is fuzzy, and (bottom) point A is fuzzy, and point B is fuzzy. Also shown are values for the fuzzy endpoints. 
One solution is to take the average value, but that, in effect, creates a biased value with no retention of the variability of the data. Another approach is to create fuzzy lattice parameters and apply them to the Wyckoff coordinates. The fuzzy lattice parameters are created by searching the literature for experiments performed under the same conditions and then extracting the minimum, average, and maximum values for each lattice parameter—thus creating TFNs. The end result is a fuzzy unit cell, with fuzzy bond lengths and fuzzy bond angles, that incorporates the variability found in literature. However, two interesting phenomena arise in creating fuzzy unit cells—the concept of fuzzy lines and fuzzy vertices.
a 
b 
Figure 4. The (a) 2D and (b) 3D fuzzy vertices. 
Using values extracted from literature,^{6–12} the fuzzy lattice parameters are
= <5.773, 5.781, 5.785> Å
= <11.550, 11.602, 11.642> Å
and the anion displacement parameter is
The "Fuzzy" Applets page also contains a Java applet for calculating the fuzzy density of a chalcopyrite compound. While most of the properties of chemical compounds have indirect relationships with the lattice parameters, density has a direct relationship. For example, if considering CuInSe_{2}, the fuzzy lattice parameters can be used to calculate a fuzzy representation of the volume of the unit cell. From a standard periodic table of elements, the mass of copper, indium, and selenium is used to find the total mass of the fuzzy unit cell, which is 2.234 x 10^{–21} grams. As density is the ratio of mass over volume, the fuzzy density for CuInSe_{2} is <5.733, 5.761, 5.803> g/cm^{3}. Thus, the density may be as low as 5.733 g/cm^{3} or as high as 5.803 g/cm^{3}, but we would expect it to be nominally 5.761 g/cm^{3}.
= <3.870, 3.882, 3.890> Å
= <11.666, 11.687, 11.708> Å
Using the fuzzy lattice parameters and the Wyckoff coordinates for the Pmmm space group,^{20} a fuzzy unit cell for the YBa_{2}Cu_{3}O_{7} compound was constructed (Figure 6, an animation).
Table I. Selected Fuzzy Bond Angles and Lengths for CuInSe_{2}^{13} and YBa_{2}Cu_{3}O_{7}^{21}  

Compound  Bond Angles (°)  Crisp (°)  Bond Lengths (Å)  Crisp (Å)  
CuInSe_{2}  
Cu_{1}SeCu_{2}  <109.593, 113.999, 115.877>  114.870      
Cu_{1}SeIn_{1}  <106.269, 109.141, 111.512>  108.902      
Cu_{1}SeIn_{2}  <106.374, 109.421, 112.127>  109.462      
Cu_{2}SeIn_{1}  <106.405, 109.421, 112.175>  109.462      
Cu_{2}SeIn_{2}  <107.499, 109.141, 110.624>  108.902      
In_{1}SeIn_{2}  <103.127, 105.329, 109.073>  104.760      
SeCu_{1}      <2.390, 2.441, 2.527>  2.4337  
SeCu_{2}      <2.400, 2.441, 2.512>  2.4337  
SeIn_{1}      <2.488, 2.575, 2.632>  2.5893  
SeIn_{2}      <2.499, 2.575, 2.619>  2.5893  
YBa_{2}Cu_{3}O_{7}  
Cu(m)O(m)Cu(m)  <160.54, 164.29, 168.71>  164.27      
Cu(m)O(m')Cu(m)  <161.70, 163.89, 166.16>  163.92      
Cu(t)O(t')      <1.919, 1.941, 1.959>  1.942  
Cu(t)O(b)      <1.819, 1.833, 1.846>  1.846  
Cu(m)O(b)      <2.291, 2.322, 2.354>  2.298  
Cu(m)O(m)      <1.912, 1.933, 1.954>  1.929  
Cu(m)O(m')      <1.947, 1.960, 1.974>  1.961  
BaO(t')      <2.870, 2.892, 2.914>  2.876  
BaO(m)      <2.958, 2.973, 2.989>  2.982  
BaO(m')      <2.944, 2.964, 2.985>  2.959  
BaO(b)      <2.727, 2.747, 2.767>  2.741  
YO(m)      <2.389, 2.408, 2.427>  2.409  
YO(m')      <2.389, 2.408, 2.427>  2.385 
It is believed that this research is a necessary first step to representing the imprecision found in unit cells and will, hopefully, lead to a better understanding of chemical properties.
References
1. L.A. Zadeh, "Fuzzy Sets," Information and Control, 8 (3) (1965), pp. 338–353.
2. HJ. Zimmermann, Fuzzy Set Theory and Its Application, 2nd ed. (Boston, MA: Kluwer Academic Publications, 1991).
3. T.J. Ross, Fuzzy Logic with Engineering Applications (New York: McGrawHill, 1995).
4. D.A. Ress, "Development of Fuzzy Trigonometric Functions to Support Design and Manufacturing" (Dissertation thesis, Department of Industrial Engineering, North Carolina State University, 1999).
5. W.G. Wyckoff, Crystal Structure (New York: WileyInterscience Publishers, 1963).
6. J. E. Jaffe and A. Zunger, "Electronic Structure of the Ternary Chalcopyrite Semiconductors CuAlS_{2}, CuGaS_{2}, CuInS_{2}, CuAlSe_{2}, CuGaSe_{2}, and CuInSe_{2}," Physical Review B, 28 (10) (1983), 58225847.
7. P. Kistaiah and K.S. Satyanarayanamurthy, "Temperature Behaviour of the Tetragonal Distortion and Thermal Expansion of GuIIVI2 Chalcopyrite Semiconductors," J. Less Comm. Met., 105 (1985), pp. 37–54.
8. J.E. Jaffe and A. Zunger, "Theory of Band Gap Anomaly in ABC_{2} Chalcopyrite Semiconductors," Physical Review B, 29 (4) (1984), pp. 1882–1906.
9. H.Y. Ueng and H.L. Wang, "Defect Structure of the Nonstoichiometric CuIIIIVI2 Chalcopyrite Semiconductors,"NonStoichiometry in Semiconductors, ed. K.J. Bachmann, H.L. Hwang, and C. Schwab (Netherlands: Elsevier Science Publishers B.V., 1992), pp. 69–79.
10. B.R. Pamplin, T. Kiyosawa, and K. Masumoto, "Ternary Chalcopyrite Compounds," Prog. Crystal. Growth Charact., 1 (1979), pp. 331–387.
11. J.L. Shay and J.H. Wernick, Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications (New York: Pergamon Press, 1975).
12. L.I. Berger and V.D. Prochukan, Ternary DiamondLike Semiconductors (New York: Consultants Bureau, 1969).
13. K.S. Knight, "The Crystal Structure of CuInSe_{2} and CuInTe_{2}," Materials Research Bulletin, 27 (1992), pp. 161–167.
14. M.A. Beno et al., 1987. "Structure of the SinglePhase HighTemperature Superconductor YBa_{2}Cu_{3}O_{7}," Applied Physics Letters, 51 (1987), pp. 57–59.
15. J.J. Capponi et al., "Structure of the 100K Superconductor Ba_{2}YCu_{3}O_{7} between (5÷300)K by Neutron Powder Diffraction," Europhys. Lett., 3 (12) (1987), pp. 1301–1307.
16. F. Beech et al., "Neutron Study of the Crystal Structure of the Superconductor Ba_{2}YCu_{3}O_{9}," Physical Review B, 35 (16) (1987), pp. 8778–8781.
17. J.E. Greedan, A.H. O'Reilly, and C.V. Stager, "Oxygen Ordering in the Crystal Structure of the 93K Superconductor YBa_{2}Cu_{3}O_{7} Using Powder Neutron Diffraction at 298 and 79.5K," Physical Review B, 35 (16) (1987), pp. 8770–8773.
18. Y. LePage et al., "RoomTemperature Structure of the 90K Bulk Superconductor YBa_{2}Cu_{3}O_{8}," Physical Review B, 35 (13) (1987), pp. 7245–7248.
19. T. Siegrist et al., "Crystal Structure of the HighT_{c} Superconductor Ba_{2}YCu_{3}O_{9}," B, 35 (13) (1987), pp. 7137–7139.
20. C.P. Poole, Jr, T. Datta, and H.A. Farach, Copper Oxide Superconductors (New York: John Wiley & Sons, 1988).
21. P. Bordet et al., "Crystal Structure of Y_{0.9}Ba_{2.1}Cu_{3}O_{9}, a Compound Related to the HighT_{c} Superconductor YBa_{2}Cu_{3}O_{7}," Nature, 327 (1987), pp. 687–689.
ABOUT THE AUTHOR
David A. Ress is with the Materials Process Design Branch of the Air Force Research Laboratory at WrightPatterson Air Force Base.
For more information, contact D.A. Ress, Air Force Research Laboratory, MLMR, 2977 P Street, Suite 13, WPAFB, Ohio 454337746; email 75112.444@compuserve.com.
Direct questions about this or any other JOM page to jom@tms.org.
Search  TMS Document Center  Subscriptions  Other Hypertext Articles  JOM  TMS OnLine 
