952 Analyst, October, 1979, VoE. 104, p p . 952-960 Quantitative Determination of Lead Dioxide Polymorphs by X-ray Powder Diffractometry P. R. Skidmore and R. R. Schwarz" Berec Group Limited, Group Technical Centre, St. Ann's Road, London, N15 3TJ The X-ray diffractometric methods used by previous workers to analyse positive plate material from lead-acid batteries €or a-, 8- and amorphous lead dioxides are critically appraised. In addition two alternative approaches, not previously applied to this problem, are assessed. *4n internal standard method is considered to be the best. Keywords : X-ray diflraction ; quantitative phase analysis ; lead dioxide ; lead sulphate ; lanthanum hydroxide. The object of this work was to develop a reliable method for the quantitative phase analysis of lead dioxide present in the positive plates of lead-acid batteries. Because the electro- chemical properties of the two crystalline forms of lead dioxide found in positive plate material, cc-lead dioxide (orthorhombic) and 13-lead dioxide (tetragonal), differ considerably, plate composition markedly affects battery performance and durabi1ity.l-4 I t has also been found that in some samples a certain amount of the lead dioxide content determined by chemical analysis cannot be detected as crystalline material by X-ray diffraction.132 Further evidence for this amorphous form has come from microscopic examination and thermal analysis of positive plates5 In addition to the lead dioxide the plates may contain lead sulphate and an organic binder.Several previous studies attempted to correlate X-ray diffraction line intensities with the amount of each phase pre~ent.ll~s~-~ Reappraisal of this work revealed its limitations and two different approaches were considered, the internal standard methodlOJ1 and the double dilution method.12 Experimental The X-ray diffraction patterns were obtained with a Philips X-ray diffractometer.Apparatus and Reagents A broad focus tube giving copper Kcc radiation was used at 60 mA and 45 kV and a nickel /?-filter was placed in the diffracted beam. Slits used were : divergence, +' ; receiving, 0.2 mm; scatter, 8'. The samples were milled in a McCrone micronising mill using alumina grinding elements. The material was packed into an aluminium holder, backed with Selotape, and pressed from the back against a glass microscope slide to give a plane surface.The linear regression lines were calculated using a least-squares program on a Texas Instruments programmable 57 calculator. Preparation of the lead dioxide standard samples The work by Bagshaw et al.13 on the preparation of the polymorphs of lead dioxide revealed that different preparation methods do not give samples with uniform X-ray diffraction properties. They differ in their degree of orientation, crystallinity, crystallite size and purity. The practical importance of this was shown by Dodson3 who found that different preparations of lead dioxide gave different X-ray diffraction calibration graphs. Moreover, even the purest lead dioxide samples are slightly non-stoicheiometric and probably contain a certain amount of amorphous lead dioxide.Wiesener and co-workers1p2 obtained values for the total crystalline lead dioxide in excess of lOOyo for many of their samples, probably because of the high amorphous content of their standard materials. Thus, it must be concluded that, despite taking every care with the preparation of the lead dioxide poly- morphs, it is possible that two sources of systematic error will remain: different degrees of preferred orientation of the standards and of the samples to be analysed and the presence of amorphous lead dioxide in the supposedly completely crystalline standards for pure a- and * Retired.SKIDMORE AND SCHWARZ 953 pure P-lead dioxide. To minimise these sources of error, the preparations chosen as standards for our work had the intensities of their diffraction lines as nearly as possible in the same ratios as the positive plate material and had narrow diffraction lines, indicating that they were highly crystalline.The two crystalline polymorphs of lead dioxide were prepared by several of the methods given by Bagshaw et aZ.13 For cc-lead dioxide these were: (i) (ii) (iii) oxidation of lead acetate by ammonium per~ulphatel~; electroformation of a hand-pasted plate in 0.7 M sodium sulphate solution with two steel counter electrodes at 0.15 A for 2 d ; oxidation of orthorhombic lead monoxide by fusion with a sodium nitrate - sodium chlorate mixture. Method (i) gave material with broadened diffraction lines, method (ii) gave an impure sample and so the product of method (iii) was chosen as the standard for the calibration lines.After difficulty was experienced in preparing reproducible samples by method (iii), a short investi- gation was undertaken. Yellow lead monoxide (orthorhombic) was reacted with fused sodium nitrate and sodium chlorate at 330 "C for 10 min and the product was washed and dried. I t contained a mixture of Pb,O, and PbO,,,, identified by their Xaray diffraction patterns.15 After repeating the fusion, which introduced no new compounds, the product was washed with 3 M nitric acid overnight. This procedure did not remove all the PbO1.55 from the fusion product, although it did appear to remove the Pb,O,. The source of the inconsistencies previously noted in a-lead dioxide preparations seemed to be this contamination with PbO,.,,.However, boiling the fusion product with 500ml of 5 M nitric acid for 15 min gave an a-lead dioxide virtually free from PbO,.,,. No P-lead dioxide was detected in this material, but prolonged boiling (for more than 1.5 h) did lead to a small amount of @-lead dioxide contamination. Oxidation of tetragonal lead monoxide by the sodium nitrate - sodium chlorate fusion gave only PbO,.,,. This sample was partly converted into a-lead dioxide by boiling with 5~ nitric acid but its diffraction pattern was less crystalline than that from yellow lead monoxide. The suspension was then heated to 60 "C and filtered. For /3-lead dioxide the preparation methods used were : (iv) reaction of red lead (Pb,O,) with nitric acid; (v) hydrolysis of lead tetraacetate; (vi) electroformation of a hand-pasted plate in 3.8 M sulphuric acid with a platinum counter electrode at 1 A for 24 h. Method (v) gave a very finely divided material that was difficult to handle and method Method (iv) was used to prepare the standard P-lead dioxide.(vi) gave an impure sample. Lanthanum hydroxide internal standard The choice of an internal standard material was difficult because many otherwise desirable compounds exhibit diffraction lines that overlap those of a- and P-lead dioxide. In particular, corundum (A1203), which was suggested as suitable by Chungll and which has been used by the Joint Committee on Powder Diffraction Standards (J.C.P.D.S.) in their recent semi-quantitative work, cannot be used. Material marketed as lanthanum oxide (La,O,) was chosen because it was found to have a clear line at 15.8 "20 (Cu Ka radiation).However, when the pattern obtained was com- pared with the J.C.P.D.S. card file it was found that this material was in fact lanthanum hydroxide.16 A survey of several batches of so-called lanthanum oxide from different manufacturers (Hopkin and Williams, BDH Chemicals and Koch-Light) revealed that even when the bottles were previously unopened the sample would consist of a mixture of La(OH), and La203.17 Lanthanum oxide rapidly absorbs water on exposure to air giving lanthanum hydroxide, causing a marked increase in volume of the material. A previous referencels to the use of lanthanum oxide in quantitative X-ray diffraction analysis (as a "heavy absorber" to overwhelm the effects of the rest of the matrix) noted that different batches of "lanthanum oxide" showed a variable stability to hydration.I t is probable that here, too, the author was using lanthanum oxide - hydroxide mixtures. Lanthanum hydroxide precipitated from solution is gelatinous and gives a poorly resolved954 Analyst, vol. 104 X-ray diffraction pattern with broad peaks. The internal standard material was therefore prepared by stirring the lanthanum oxide - hydroxide mixture obtained commercially with water for 30 min and drying the product over pliosphorus pentoxide or in an oven at 110 "C. We have found the diffraction patterns of various batches of lanthanum hydroxide prepared in this way to be indistinguishable, but re-calibration would normally be carried out with each new batch as it is brought into use.SKIDMORE AND SCHWARZ : QUANTITATIVE DETERMINATION OF Preparation of the X-yay s9ecirnen To achieve reproducibility in any X-ray diffraction analytical method, the preparation of the specimen must aim to miniinise preferred orientation, to give maximum intensities by reducing primary extinction and microabsorption and to mix any added material (for example, an internal standard) uniformly into the sample. In this study a small ball mill was used to grind the sample, moistened with propan-2-01, and at the same time to mix in the added material. Klug and AlexanderlO point out that it is desirable to reduce the size of the crystallites to about 5 p m but that caution must be exercised concerning prolonged grinding. Apart from the general danger of introducing stress into the lattice and producing very small particles and thus causing the broadening of the diffraction lines,1° it has been suggested that over-grinding may cause a transformation of P-lead dioxide into a-lead dio~ide,19~Jg*~~ the coating of particles of cc-lead dioxide by P-lead dioxide,' or the thermal decomposition of cc-lead dioxide.6 An investigation of the optimum milling time was there- fore carried out. The effect on diffraction line intensities of milling a 1 + 1 + 1 mixture of lanthanum hydroxide, cc-lead dioxide and P-lead dioxide is shown in Fig.1. No dramatic peak-height fluctuations are seen, indicating that the mixing-in of the lanthanum hydroxide is rapidly achieved, The initial increase in intensity of one of the P-lead dioxide lines paralleled by a fall in intensity of the other indicates that the i3-lead dioxide sample shows a small amount of preferred orientation that can be removed by reducing the particle size.X La (OH), (100) Y a(110) v La (OH), (1 10) 0 1 2 4 7 10 Mi I I i ng ti me/m in Fig. 1. Effect of milling time on the X-ray diffraction line intensities on a 1 + 1 + 1 mixture of a-lead dioxide, /3-lead dioxide and lanthanum hydroxide. Measurement of difracted line intensity Peak area is a better measure of the diffracted intensity than peak height, particularly for materials such as the lead dioxides, which can show considerable peak broadening because of the small size of the crystallites. The areas were calculated from step scan data, which gave the counts accumulated in 100 s taken at every 0.05 "20.The background was estimated using the regions just before and just beyond the peak and was subtracted from the total number of counts under the peak to give the peak area.October, 1979 LEAD DIOXIDE POLYMORPHS BY X-RAY POWDER DIFFRACTOMETRY Procedures 955 Direct intensity method Wash the mixture out of the mill with propan-2-01 as required and remove the solvent. intervals of 0.05" and a step time of 100 s. the total counts and subtracting the background. calibration line. Weigh 1 g of the sample into the McCrone mill, add 3 ml of propan-2-01 and mill for 4 min. Obtain the X-ray diffraction pattern from 21 to 27.5 "26 by a step scan procedure using Calculate the area under the peaks by adding Obtain the phase composition from the Internal standard method Proceed as in the direct intensity method, obtaining the diffraction pattern between 14 and 27.5 "26.Weigh 1 g of the sample and 0.5 g of lanthanum hydroxide into the McCrone mill. Double dilution method and 27.5 "26 only. mix thoroughly in an agate pestle and mortar. 21.5 and 27.5 "26. Carry out the internal standard method, but obtain the diffraction pattern between 21.5 Add a further 1 g of lanthanum hydroxide to 1 g of the internal standard mixture and Obtain the diffraction pattern again between Calibration lines 1 g of lead dioxide and treat as the sample in the methods above. Weigh out the a- and P-lead dioxide standards in appropriate amounts to give a total of Lead sulphate cowection factor Weigh out lead sulphate (PbSO,, Hopkin and Williams, G.P.R.grade) and the ,&lead dioxide standard in appropriate amounts to give a total of 1 g and treat as the sample in the internal standard method above, obtaining the diffraction pattern between 14 and 24.5 "20. Results and Discussion Most quantitative calibration methods are derived from the well known equation given by Klug and AlexanderlO : where Iij is the intensity of the ith line of component J , x, is the weight fraction of J , ,ii* is the total matrix mass absorption coefficient and k , is a constant containing the instru- mental factors and the density of J . Rearranging equation (1) to give an expression for xJ yields xJ = KiJP*IiJ . . . . . . . . - * (2) where Ki, is the reciprocal of k,. Direct Intensity - Weight Fraction Calibrations component (p;) of the sample multiplied by their respective weight fractions : The total matrix mass absorption coefficient, p*, is the sum of the coefficients of each N - (3) The mass absorption coefficient of a compound at a particular wavelength depends solely on the atoms it contains and so there is only one mass absorption coefficient for any lead dioxide.956 Analyst, Vol, 104 Hence, as the positive plate material always consists mainly of lead dioxide, the matrix mass absorption coefficient can be assumed to be constant and the approximation can be made that SKIDMORE AND SCHWARZ : QUANTITATIVE DETERMINATION OF .. . . - (4) ~j cc Iij . . .. This approach was used by Wiesenei- et aZ.l and gave straight-line calibration graphs.We repeated the calibration with four mixtures of our standard lead dioxide (Table I) and also obtained straight-line graphs of intensity against weight fraction with correlation coefficients better than 0.97. Although the method works well for mixtures of standard lead dioxides, it is considered necessary by Wiesener et aZ.l to wash the plate material to remove lead(I1) compounds before X-ray analysis to ensure that the X-ray sample contains only lead dioxide in line with the initial assumption in relation (4). Evidence suggests, however, that some of the lead dioxide will dissolve during this procedure,21 possibly leading to a change in the a : ,8 ratio. More- over, although an external standard was used to indicate instrumental effects such as variations in incident beam intensity, there are other effects such as variations in the sample packing or its position in the goniometer for which an external standard does not compensate.The total crystalline lead dioxide is then given by Sekido and YokohQ use equation (4) with proportionality constants ka and kb. T , = kaIia -t kbIjp . . . . . . . . .. For a large set of Ii, and I j p data, these authors used a least-squares numerical method to calculate k a and kb by minimising the function where T , is the total lead dioxide in each sample determined chemically. However, when the material contains unknown and variable amounts of amorphous lead dioxide, T , > T , by an unknown and variable amount for each sample and equation (6) cannot be used. TABLE I DATA FROM a- AND P-LEAD DIOXIDE STANDARD MIXTURES of E-PbO, (xR) (1,) of P-PbO, (XP) (Id XPl% I p l I u Intensity of ( 1 10) line Intensity of (1 10) line Weight fraction of a-PbO,, counts Weight fraction of P-PbO,, counts 0.201 2.45 x 105 0.799 3.77 x 106 3.98 15.35 0.334 3.59 x 105 0.666 2.60 x lo6 1.99 7.25 0.500 7.06 x 105 0.500 2.23 x lo6 1.00 3.16 0.801 9.74 x 105 0.199 1.10 x 106 0.248 1.13 Intensity Ratio - Weight Fraction Ratio Calibration equation (2) Taking the ratio of the intensities of two components of a mixture, J and K , gives from Dodson3 and, following his paper, Kordes,6 Fedorova et aZ.' and Dugdales plotted a semi- logarithmic graph of x, against log(Ijp/Ii,).By comparing this relation with equation (7) it can be seen that the graph of x, against Ijp/Ii, would not be expected to give a straight line. Only a brief, unsatisfactory derivation of this relationship has been given,6 and it is possible that the logarithmic relationship was introduced to give the appearance of direct proportionality to the graph.We constructed a calibration graph of q/x. against I j p / I i a from the same four mixtures used for the direct intensity - weight fraction calibrations (Table I) and obtained a straight line (correlation coefficient 0.998). Because the cc-lead dioxide is in effect being used as a standard for the ,&lead dioxide, variations in matrix effects or in instrumental performance are eliminated, but no estimate of the total crystallinity of the material can be obtained.October, 1979 LEAD DIOXIDE POLYMORPHS BY X-RAY POWDER DIFFRACTOMETRY 957 Because of the limitations of these direct methods of relating intensity and concentration, two further methods were applied to this analysis, the internal standard method and the double dilution method.Internal Standard Method standard S added in a known constant amount. Internal standard analysis is based on equation (7) where component K is the internal . . . . . . (8) That is: X J -7, - = c x - . . X S 4 s where C is the proportionality constant. TABLE I1 CALIBRATION DATA FOR INTERNAL STANDARD METHOD Weight of ct-PbO, Weight of La(OH), Intensity of x-PbO, (1 10) line Intensity of La(OH), (111) line Weight of P-PbO, Intensity of P-PbO, (1 10) line Weight of La(OH), Intensity of La(OH), (111) line 0.061 3 0.2509 0.495 8 0.751 1 0.904 4 1.001 0.997 6 1.253 1.498 1.749 1.952 0.075 38 0.211 6 0.367 7 0.6142 0.726 9 0.7785 0.779 3 0.993 6 1.273 1.510 1.619 1.945 1.758 1.493 1.251 6 1.100 0.966 1 1.005 0.742 3 0.493 2 0.229 2 0.065 1 4.417 3.948 3.228 2.745 2.157 2.017 2.304 1.675 1.074 0.503 9 0.1372 Eleven standard mixtures (Table 11) were made up and an X-ray diffraction pattern of a Least-squares typical mixture is given in Fig.2, showing the lines used in the analysis. regression analysis on the data in Table I1 gave the following lines: for u- + 0.0320 weight a-PbO, - intensity a-PbO, weight La(OH), - ''''O weight P-Pbo, = 0.444 x intensity P-PbO, + o.0279 weight La(0H) intensity La(0H) , intensity La(OH), for p- Angle, "28 Fig. 2. X-ray diffraction pattern of a 1 + 1 + 1 mixture of or-lead dioxide, /$lead dioxide and lanthanum hydroxide.958 SKIDMORE AND SCHWARZ : QUANTITATIVE DETERMINATION OF Analyst, Vol.104 both with correlation coefficients of 0.997. The two small positive intercepts imply that the background was slightly overestimated, probably because it is rising under both the peaks. The situation is shown, exaggerated, in Fig. 3. The background, estimated as a straight line between the averages of the two regions b, and b,, lies above the actual curved back- ground. Thus, the internal standard method gives satisfactory calibration lines and has several advantages over the methods used by previous workers. It provides a compensation factor for both matrix and instrumental variability. Because the crystalline forms are determined separately the results can be used to estimate the non-crystalline lead dioxide by subtraction from the lead dioxide content determined chemically. Contents of cc- or ,&lead dioxide below about 2% will therefore not be detected. I Angle," 28 Fig.3. Over-estimation of the background below the peaks. Internal Standard Method in the Presence of Lead Sulphate The presence of lead sulphate in some positive plate samples causes a problem because its (111) line is superimposed on the or-lead dioxide (011) line. Rather than attempt to remove lead sulphate from the matrix by washing, it should be possible to make a correction for its presence, using the (011) line, which is free from overlap, to estimate the lead sulphate con tent . The total intensity of the compound line I i , is given by x,/x, can be obtained from another line l of component K : Thus, substituting from equation (10) into equation (9) and rearranging: The ratio of proportionality constants e/d is given by .... .. .. . . .. . . .. A set of mixtures was prepared containing lead sulphate, P-lead dioxide and lanthanum hydroxide to give a calibration line from equation (12) for the estimation of the ratio e/d. A point from lead sulyhate alone was also included. Performing a least-squares regression on the data given in Table I11 gives a straight line with a gradient of 0.432 and an intercept of 11 797 counts (correlation coefficient 0.997). The intercept arises from the overestimationOctober, 1979 LEAD DIOXIDE POLYMORPHS BY X-RAY POWDER DIFFRACTOMETRY TABLE I11 CALIBRATION DATA FOR THE LEAD SULPHATE CORRECTION 959 Intensity of PbSO, Intensity of PbSO, Weight of PbSO, Intensity of PbSO, (011) line (011) line, counts (111) line, counts Weight of La(OH), Intensity of La(OH), (111) line _ _ _ _ _ _ _ _ 17 175 45426 97 586 146 909 158213 179 050 202 166 252 183 337 301 1 055 491 23061 38 094 53 305 81 107 90 669 86 628 97 393 116553 132 323 474583 0.090 6 0.204 1 0.397 9 0.598 2 0.51 1 7 0.667 7 0.798 2 0.9950 1.208 - 0.046 1 0.1242 0.255 2 0.406 4 0.329 1 0.446 6 0.5157 0.647 5 0.876 8 __ of the background and should be neglected to avoid the systematic error of including it twice in the determination of the a-lead dioxide content.Least- squares regression on the data in Table I11 gives The data can also be used to determine the lead sulphate content of the sample.intensity PbSO, (01 1) - 0.0288 weight PbSO4 = 0.724 weight La(0H) intensity La(0H) with a correlation coefficient of 0.996. is negative, implying that the background is underestimated. In this instance the background correction constant Double Dilution Method The internal standard method cannot compensate for the effects of microabsorption that arise because diffraction from the. sample and from the standard take place within different particles. The double dilution method, which should eliminate both matrix and micro- absorption effects, has been reviewed by Clark and Preston.12 One portion of “diluent,” a compound absent from the sample matrix, is added, the intensity of a line of each com- ponent is measured, a further portion of diluent is added and the intensities of the same lines are measured again. Then, where Iijf and Iij” are the intensities when one and two portions, respectively, of diluent are added.Several of the mixtures used for the internal standard calibration were diluted with a further 1 g of lanthanum hydroxide and the diffraction patterns obtained again. Although Clark and Preston recommend that the diluent should be amorphous and should have a low mass absorption coefficient, this standardisation was successfully carried out with a crystalline high mass absorption material. The results are given in Table IV. TABLE IV CALIBRATION DATA FOR THE DOUBLE DILUTION METHOD F ( I ) = I&J’I&J’’/(IU’ - IiJ”). Weight fraction, cc F(1,) Weight fraction, /3 F(1p) 0.125 1.19 0.875 34.24 0.249 2.20 0.751 28.18 0.498 4.85 0.502 20.34 0.628 5.99 0.372 16.25 0.752 6.05 0.248 8.90 0.884 7.14 0.116 2.29960 SKIDMORE AN11 SCHWARZ The equations of the lines are calculated to be == 0.122 F(I,) - 0.0342 for cx with a correlation coefficient of 0.984 and xp = 0.0245 F(1p) + 0.0278 for ,f3 with a correlation coefficient of 0.993, where F ( I ) is the right-hand side of relation (13).The background corrections shown by the intercepts are significantly large and limit the precision of the method and its application at low levels. The two mixing-in stages that are required for this method result in greater random errors than for the internal standard method and make the analysis more time consuming. Large random errors also arise from the counting and particle statistics in the more dilute sample when the content of one phase is low.Because of these sources of error, the double dilution method is considered to be inferior to the internal standard method. Conclusion This investigation has demonstrated that internal standard X-ray diffraction provides a more reliable method for the phase analysis of lead dioxide in positive plate material from lead-acid batteries than those used by previous workers, Analysis of positive plates of lead-acid batteries for a- and P-lead dioxide are now carried out routinely by the internal standard method in this laboratory and an estimate of the amorphous lead dioxide content of the plates is obtained from the difference between the crystallographic and chemical determination.The authors thank the Directors of Rerec Group Limited for permission to publish this work and D. M. Holton for her help with the experimental work. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. References Wiesener, K., Hoffmann, W., and Rademacher, (I., Electrochim. Acta, 1973, 18, 913. Wiesener, K., and Reinhardt, P., 2. Phys. Chem., 1975, 256, 285. Dodson, V. H., J . Electrochem. Soc., 1961, 108, 406. Ness, P., Electrochim. Acla, 1967, 10, 161. Caulder, S. M., and Simon, A. C., J . Electrochem. SOC., 1974, 121, 1546. Kordes, D., Chemie-Ingr-Tech., 1966, 38, 638. l?edorova, N. N., Aguf, 1. A., Levinson, L. M., and Dasoyan, M. A., 2av. Lab., 1964, 30, 727. Dugdale, I., in Collins, D. H., Edztor, in the discussion of the paper by Acton, R. G., “Power Sources,” Sekido, S., and Yokoh, T., J . Electrochem. SOC. Japan, 1963, 31, 15. Klug, H. P., and Alexander, L. E., “X-ray Diffraction Procedures,” Second Edition, Wiley- Chung, F. H., A h . X-ray Analysis, 1974, 17, 106. Clark, N. H., and Preston, R. J . , X-ray Spectrom., 1974, 3, 21. J3agshaw, N. E., Clarke, R. L., and Halliwell, B., J . Appl. Chern., 1966, 16, 180. Angstadt, K. T., Venuto, C. J . , and Ruetschi, P., J . Electrochem. SOC., 1962, 109, 177. Joint Committee on Powder Diffraction Standards, Powder Diffraction File card numbers 23-331 Joint Committee on Powder Diffraction Standards, Powder Diffraction File card number 6-0585. Joint Committee on l’owder Difiraction Standards, Powder Diffraction File card number 5-602. Flinter, B. H., Neues Jahrb. Maneral Monatsh., 1973, 5, 216. Jenkins, R., A d v . X-ray Analysis, 1974, 17, 32. Weigelt, D., and Schrader, R., 2. Anorg. Allg. Chem., 1970, 372; 228. Reutschi, P., Sklarchuk, J., and Angstadt, 13. T., zn Collins, D. H., Editor, “Batteries,” Volume 1 , Pergamon Press, Oxford, 1966, p. 142. Interscience, New York, 1974. and 27-1200. Pergamon Press, Oxford, 1962, p. 89. Received December 22nd, 1978 Accepted April 20th, 1979