Alfred E. Attard Henry C. Lee
Department of Forensic Science and Center for Applied Research University of New Haven West Haven, CT 06516
X-Ray Crystallographic Computations Using a Programmable Calculator
Hand-held programmable calculators provide convenience and so~histicationin analvsis a t relatively low cost (1-6). T o illustrate the range of usefulness of p r o g k m a b l e calculators in providing computational assistance in chemical analysis a few~crystallo&aphicprograms have been developed. These programs are hased on the Hewlett-Packard HP-25 Programmable Calculator which uses the Reverse Polish Notation (RPN).
LP = 1 + cos220 sin 20
Temperature and Scale Factors
Program 6 computes the temperature factor, B, and the scale factor, K, from the experimental data. If the structure is known, the structure factor, F,, can be calculated by program 4. B and K are then evaluated from the following relation (9)
Atomic Scattering Factors
Program 1computes the atomic scattering factor using the Gaussian form of Moore (7):
+ B c b x a+ C
f(r)= Ae-"2
Program 2 computes the atomic scattering factor hased on the polynominal series of Ferguson and Kinvan (8): f(z) = oo + o l x + azx2+ . . . + a 7 x 7
In both programs, x = sin OIX. Appropriate atomic parameters for the self-consistent or variational model and for the Thomas-Fermi-Dirac statistical model are obtained from these references (7,8). Interplanar Spacing Program 3 computes the interplanar spacing, d, and sin
BIX
for the following crystal structures: cubic, tetragonal, hexaeonal. rhombohedral. orthorhomhic. monoclinic. and tricliiic fo; any given set of'~illerindices h; k, 1. This pr&ram reauires the initial storaee of suitable parameters for the appropriate crystal structure.
Structure Factor Program 4 computes the real and imaginary components of the structure factor as well as the square of the structure
where IFoI2is the experimentally observed intensity corrected for absorption, Lorentz-polarization, and multiplicity (if the specimen is a powder). Where the structure is not known and B must be known in advance of the structure determination, the following relation is used (9)
where fj is the atomic scattering factor of the j-th atom. The sum is over all the atoms in the unit cell. Either program 1or 2 mav he used for calculation of the atomic scatterine factor. program inputs are therefore sin28/X2and either l F o l 2 / 1 ~ , 1 2 or I F 0 1 ~ l Z ffor j ~ each observed diffraction maximum. Program output consists of K.B. and the coefficient of determination, r2.This program is an adaptation of a linear regression analysis. The coefficient of determination, r2, is a statistical measure of the "aoodness of fit." and in an analvsis of the relation y = ar + b , r 2 is given by (10):
factor. The structure factor is F(hk1) = A(hk1) + i B(hk1) where Afhkl) = Z f cos 2n (hx + ky + l z ) B(hk1) = Z f sin 2 r (hx + ky + 12) Program inputs are the unit cell coordinates and atomic scattering factor for each atom in the unit cell. The program output consists of A(hkl), B(hkl), a i d IF(hkl)I2.This program does not employ symmetry-reduced calculations but uses the general noncentrosymmetric calculation.
Literature Cited
Lorentz-PolarizationFactor
Program 5 computes the LP factor for the powder method and for the equatorial single crystal method. The program input is the diffraction angle, 8. Program output is either the LP factor for the Debye-Scherrer powder method (9) LP=
The square root of the coefficient of determination is the well-known correlation coefficient (11). The value of r2 will he between 0 and 1and indicates how closely the equation fits the experimental data. The closer r 2 is to 1,the better is the fit of the data. We have described here a few crystallographic programs which are suitahle fur the analvsis ut'x-rav diffraction data in the laboratory by students under optimalieamins conditions. These programs are capable of operation on any compatible RPN machine. The program listings and documentation are available, free of charge, from A. E. Attard.
+
1 cos22.9 . smZOcas 0
or the L P factor for the zero-level single crystal method (9)
650 1 Journal of Chemical Education
(1) Attard.A. E., md Lec.H. C.,J. CHEM. EDUC.. 55,428 (19781. (2) Attard, A. E., and h e , H. C., J. Chromafogr. Sei., 16,514 (19781. (3) Schmidt, S. A.. Am. J. Physics, 45.79 (1977). (4) Munich. J.,J. CHEM.EDUC.,54.421 (1977). (5) Simans,S.,Am. JPhysics, 45,1W7 (1977). (61 Smith, J.W.."SeientifieAnalysis on tho Paeket Caleulafor" John Wiley&Sona, Inc.. New York. 1975. (7) Moore. F.~..,AcfoCryst.. 16.1169 (1963). (8) Fenuson, L.F.,and K i m , J. E., CompuferPhysio.Camm., 5,328 (1973). (9) StouGG.H.,md Jemm, L. H.,“X-Ray StrumveDeteminstioflTheMaemilknCo., Landon, 1968. (10) "HP-25 Applicstiona Pmgrams."Hewlott-Psekard.Cupertino, California, 1975. (11) E d w a r d s , A . L . ; ' A o l n t r d " ~ o " t o L m m ~ i i "mdccIstiii111W.H.FFF~ & Co., San Franeiseo. 1976.
