February 1973 AS. Nucleation Ramachandra Behaviour in the Murty and Bh. of Iodide Presence V. Ramana and of soluble Murty Iodate 61 Systems salts By A.S. Ramachandra Murty and Bh. V. Ramana Murty Indian Instituteof TropicalMeteorology,Poona-5,India. Abstract Using a simple drop-freezingtechnique as in earlier studies, the ice nucleationactivitiesof certain iodide and iodate systems,with and without solublesalts, have been investigated. It has been found that, of the systemstested, the iodide systemscommonlyused in the conventional cloud seeding,which are AgI.NaI and AgI.KI, are the least active. Also, the iodate system used in the pyrotechnicseeding,which is AgIO3+KIO3,is not the most active. Betternucleating systemshave beennamedfor possibleuse in both the types of seeding. The comparativeactivities noticed of the systemstested have been explainedon certain qualitativeconsiderations. 1. Introduction Identification of particles which will nucleate supercooled drops to ice crystals, at warm temperatures, is of primary importance in studies of weather modification. Acetone solution of AgI using NaI or KI or NH4I as solubilizer and pyrotechnic mixture using AgI03 or AgI03 + KIO3 are the commonly reported silver iodide and iodate systems respectively for production of such particles. From the experiments, using wind tunnel and cloud chamber facility for measurement, the silver iodide system using NH4I was found to be more active for ice nucleation than the one using NaI (St. Amand et al., 1971). The present authors, using a different technique of measurement, which is drop-freezing, and unaware of the results referred to above, reported on the nucleation activities of silver iodide systems (Ramachandra Murty and Ramana Murty, 1971) which also pointed out that the one using NH4I is most active followed by those using NaI and KI. This feature is now confirmed. Using that same technique, the authors, in an earlier study, examined the nucleation activities of silver iodide suspensions containing certain soluble salts; and the study suggested that whereas the presence of sulphates improves the nucleating activity of the silver iodide suspension, that of the chlorides decelerates it (Ramachandra Murty and Ramana Murty, 1972a). This aspect is now thoroughly examined. Also, the nucleation activity of iodate systems has been studied. These investigations have enabled the sorting out, according to activity, of the various iodide and iodate systems which have been tested. The results obtained are presented below. While no explanation had been offered while reporting the results referred to earlier (Ramachandra Murty and Ramana Murty, 1971 and 1972 a), attempt has been made now to explain qualitatively the results obtained so far. 2. Measurements The equipment and the technique of measurement are as described earlier (Ramachandra Murty and Ramana Murty, 1971 and 1972b). The salient features are as follows : Drops of diameter 2.2 mm from the systems (aqueous suspensions) to be tested are placed in equal numbers of 30 each in two identical metal dishes. The dishes are sealed with glass covers and then cooled by transferring both of them simultaneously on to a cold stage which has been kept cooled, initially, to any of the pre-determined temperatures -3°C, - 5°C and -10°C . The number of drops freezing at successive one-minute time intervals is noted with naked eye until all the drops taken in both the dishes have frozen. The fractions of drops frozen at successive time intervals and the total time taken for all the 30 drops to freeze, for each of the systems tested, formed the basic data for the study. For the initial temperatures of the cold stage which are referred to above, i.e. -3°C, -5°C and -10°C, the dishes attained, 62 Journal of the Meteorological Society of Japan Vol. 51, No. 1 in the quasi-equilibrium conditions, temperatures which are respectively -2.7°C, -4.7°C and -9 .8°C. The temperatures are measured by thermocouples, using SWG-36 copper-constantan wires, at successive one-minute time intervals. cloud chamber (Mason and Van Den Heuvel, 1959). The results on AgI and PbI2 have been obtained earlier and reported (Ramachandra Murty and Ramana Murty, 1972b). These along with the results now obtained on CuS are presented in figure 1 and summarised in Table 1. These 3. Reliability of measurements experiments have been conducted with the cold The reliability of the present measurements for stage initially at -10°C. the purpose of comparing the ice nucleation The temperatures found of the first freezing activities of substances has been ensured. This is event for the suspension drops containing AgI, done by testing on substances about the threshold PbI2 and CuS are -4.2°C, -6.5°C and -6.8°C nucleation temperatures of which there is general respectively, and these values agree closely with agreement. The substances considered are AgI, the accepted threshold temperatures referred to PbI2 and CuS for which the threshold temperatures above for these substances. Also, the total time quoted are -4°C, -6°C and -6°C (vide Mason, taken for all the drops to freeze was found to 1971). These substances have been tested using be minimum in the case of AgI and maximum aqueous suspensions of 1 % concentration, for the in the case of CuS. The findings confirm that threshold nucleation temperatures found at 1 the temperature of the first freezing event as well concentration for different substances in bulk as the total time taken for all the drops to freeze, water (millimeter drops) agreed closely with those as from the present study, serve as indicators found for the respective substances in conventional of the comparative nucleation activities of given substances. 4. Fig. 1. Fraction of drops frozen with the cold stage initially at -10°C. Temperatures intervals. Table 1. Freezing characteristics of suspension drops of various composition when the * Generally are given at 5 minute cold stage initially accepted threshold was at -10°C. temperature. Nucleation activities of AgI in the presence of sulphate and chloride The effects of sulphates and chlorides on the nucleation activity of the aqueous suspension of AgI have been examined. The sulphates tested are of ammonium, lithium, sodium, potassium, caesium and magnesium and the chlorides tested are of sodium, potassium and magnesium. The results are presented in figure 2 and summarized in table 2 and these refer to the cold stage temperature initially at -5°C. It is seen that, whereas the sulphates improve the nucleation activity of AgI, chlorides do the opposite. (NH4)2SO4 has been found to improve the activity of AgI most, followed by Cs2SO4. Li2SO4 has improved its activity least. S. Nucleation activity of AgI in the presence of other iodides (solubilizers) The effects of various other iodides on the nucleation activity of the aqueous suspension of AgI have been examined. The iodides considered are NH4I, LiI, NaI, KI and CsI. The results are presented in table 3, and these refer to the cold stage temperature initially at -10°C. Except for ammonium iodide, the other iodides decelerated the nucleation activity of AgI. Lithium iodide decelerated it least and caesium iodide most. February 1973 A.S. Fig. 2. Table 2. Freezing drops Ramachandra Fraction characteristics of various of drops Murty and Bh. frozen with cold stage of suspension composition V. Table Ramana initially 3. is 10 gm per litre salt 0.1 M. have now experiments stage * Plus symbol '+' alone iodates at -5°C. The figures AgI03 than containing have not are found +KI03. and results 3b the AgI. with either shown These the cold obtained and to be better to be The combinations + KI03 are found The are summarised triple alone All ice Also, ice to be better AgI03 than the KI three nucleants than NaTO3 and nucleants than +NaIO3 and ice nucleants AgI03 + NaI03 combination combination or the both better combination ice nucleant The NaI freezing. iodides. found AgI03. a better 3a with in 4. are AgI. along conducted corresponding KI03 tested been in Systems the been have initially presented in table It has been pointed out that the smoke obtained by burning the mixture AgI03 +KI03 is efficient for ice-formation, being better than the smoke obtained by burning AgI03 alone (Odencrantz, 1969). In this operation, the contribution due to KIO3 separately is, however, not known. The substances AgIO3, KI03 and NaIO3 and combinations thereof, as well as the substances NaI and of suspension The concentration of AgI suspension is 2 gm per litre. and of Iodates versus Iodides characteristics drops of various composition when the cold stage initially was at -10°C. KI, 6. 63 at -5°C. Freezing when the cold stage initially was at -5°C. The concentration of AgI in the suspension the soluble Murty AgI03 AgIO3 + NaIO3 indicates that , in the given system, compounds are present, by weight, only in the stated ratio. For instance, in the system AgI+(NH4)2SO4, the compounds AgI and (NH4)2SO4 are present in the ratio 1 : 1. ** Dot symbol ' . ' indicates that, in the given system, compounds are present, by weight, in the stated molar ratio, For instance, in the system 2AgI.NH4I, the compounds AgI and NH4I are pre present in the ratio of 2 moles of AgI to one mole of NH4I. is 64 Journal of the Meteorological Society of Japan Table 4. Freezing Vol. 51, No. 1 characteristics drops of the cold of suspension various composition when stage temperature initially was at -5°C. Fig. 3(a). Fraction of drops frozen with cold stage initially at -5°C. No freezing event was observed in the case of drops containing NaI and KI. + KI03 is found double combination 7. to be even better AgI03 + NaIO3. Nucleation activity different sulphates of iodide than systems the with In an attempt to know the extent to which the nucleation activity of a given iodide system could be improved by the addition of a sulphate to it, measurements have been made with various combinations thereof. The sulphates and the iodide systems considered are Na2SO4, K2S04, Fig. 3(b). Fraction of drops frozen with cold stage initially at -5°C. Fig. 4(a). (NH4)2SO4 and AgI.NaI, AgI.KI, AgI.NH4I respectively. The experiments have been conducted with the cold stage initially at - 3°C on systems containing NH4I (these systems show freezing even at about -1°C), and at -10°C on systems containing NaI and KI (these systems show Fraction of drops frozen with cold stage initially at - 3°C . No freezing event was observed in the case of drops containing AgI alone. February 1973 A.S. Ramachandra Murty and Bh. V. Ramana Murty 65 freezing only below -5°C). The results obtained are presented in figures 4a to 4c and summarized in table 5. The nucleation activity of the three silver iodide systems (NH4I.AgI, NaI.AgI and KI.AgI) improved with the addition of a sulphate to them. Of the three sulphates considered, that of ammonium rendered the iodide systems most efficient. The most efficient triple combination is the one which contained AgI, NH4I and (NH4)2SO4. Fig. 4(b). Fig. 4(c). Table 5. Fraction of drops frozen with the cold stage initially at -10°C. Fraction of drops frozen with the cold stage initially at -10°C. Freezing characteristics of suspension drops of various composition. The concentration of AgI in the suspension is 0.5 gm per litre. 8. Discussion 8. 1. Materials for producing efficient ice nuclei Among the substances tested in the present study, the following is the choice for producing efficient ice nuclei. If a single substance alone is to be preferred, it is AgIO3. If it is a combination of two, its ice nucleation activity can be more than that of the single substance chosen; and that combination is AgIO3 + NaIO3. If it is a combination of three, its ice nucleating activity can be more than that of even the double combination chosen; and that combination is AgIO3 + NaIO3 +KIO3. The iodates are used in cloud seeding by pyrotechnic methods. The combination of the iodates which is reportedly used by such methods is the double combination AgIO3 + KIO3 (vide Simpson et al., 1970) and this is now seen to be inferior to the double combination AgIO3 +NaIO3 which is itself inferior to the triple combination AgIO3 + NaIO3 + KIO3. The latter two combinations could, therefore, advantageously replace the former in pyrotechnic methods. But, there is one general drawback in the use of iodates. Iodates may be reduced to iodides at temperatures at which pyrotechnic smokes are produced (Simpson et al., 1970). There is no guarantee, therefore, that the pyrotechnic smokes of the iodates produced in the free air would behave as actively as the aqueous suspensions of the corresponding iodates. The alternatives which are suggested by the present authors, therefore, are AgI + NH4I for the double combination and AgI + NH4I + (NH4)2SO4 for the triple combination; and these will be free from such drawback. No doubt, it has been presumed that in pyrotechnic seeding it is the pure silver iodide smoke produced which activates ice-formation at as warm a temperature as -2°C to - 3°C (Elliott et al., 1971). But, it has been shown that the 66 Journal of the Meteorological Society of Japan Vol. 51, No. 1 nucleating activity of silver iodide particles made below to the available values of the ionic decreases with increase of sample purity (Corrin refractions (cc per gram ion) and the molar et al., 1967). In view of the fact that the iodate refractions (cc per gram mole) as the case may be. suspensions have now been found to be better 8. 3. Effects of adding sulphate and chloride to ice nucleants than the corresponding iodide drops o f AgI suspension suspensions, it is considered that the iodates can be used definitely beneficially in the place of Water molecule is polar with a formal positive iodides, only in 'Spray-Seeding'. charge (0.33e) on hydrogen and negative charge (0.66e) on oxygen. AgI is a covalent solid and, 8. 2. Ionic polarizability and ice nucleation therefore, when AgI particle alone is present in Effects noticed of sulphates, chlorides and other the water drop, water molecules can aggregate iodides (solubilizers) on the nucleation activity of on the particle both from their O- and H+ ends. silver iodide and the order in which the iodide Consider the situation when the particle is present systems, with and without sulphate, and the iodate in the drop along with NaCl in one case and systems exhibited nucleation activity, require to be along with Na2SO4 in another case. The ions explained. While it is difficult to visualise a in the ionic pairs formed in the case of NaCI, rigorous explanation for the features noticed, the Na+(H2O) Cl-, will be closer to each other than authors have the following qualitative remarks to those formed in the case of Na2SO4, 2 Na+ (10 H2O) SO42-. In the case of Na2SO4, they will offer. Ice-formation, on the nucleating substrate, may be far separated by the large number of be facilitated by the possible aggregation of water intervening water molecules as against by one molecules around the adsorbed solute ions present suchh molecule in the case of NaCl. Because of on the substrate. The extent to which aggregation the closeness of the ions in the ionic pairs in the of water molecules takes place around the adsorbed case of NaCl, the pairs exert nearly equal and ion may depend, among other factors, upon the opposite forces on the water molecules, making polarizability of that ion, its electronic configura- less possible for them to aggregate on the AgI tion and the polarizing power of the oppositely particle either from their O- end or H end. charged ion in the pair. It may be noted that, These features would cause the nucleation activity according to Weyl (1951), a liquid surface consists of AgI + NaCl to be less than that of either AgI of the most polarizable ions because polarizability +Na2SO4 or AgI alone. The SO42- has two negatively charged oxygen permits an adjustment of the force-fields of the surface ions and, as a result, a lowering of the atoms and two S = O groups with lone pair of surface free energy. The implication of this pos- electrons. Water molecule has a lone pair of tulate, as also discussedby Hosler (1951), is that a electrons on its oxygen atom. Consequently, small supercooled drop can be made to crystallize water molecules will be bound strongly to SO42by the addition of foreign molecules or ions that group. The addition of a sulphate to AgI in reduce the surface free energy of the drop. Since suspension could, therefore, improve the nucleathe amount of surface free energy is a function tion activity of AgI in accordance with observaof the polarizability of the ions, the presence of tion. more polarizable ions or molecules on the substrate 8. 4. Comparative effects of different sulphates will reduce the amount of supercooling necessary to form ice crystals. Based on these consideraOf the sulphates studied, (NH4)2 SO4 Improved tions it is found possible to account for the the activity of AgI most. The reason for the observed features in the present study. observed feature is considered as follows. All For a series of ions carrying the same charge the cations have structure breaking effect except it is known that, in general, the polarizability NH4+ which has it the least because of its increases and the polarizing power decreases with charge being directed tetrahedrally along the N-H the size of the ion. The value of molar refraction bonds. of a given substance gives its polarizability. As The remaining sulphates tested improved the the values of neither the ionic radii nor molar nucleation activity of AgI in the following order : This is polarizabilities for all the substances tested are Cs2SO4 > K2SO4 > Na2SO4 > Li2SO4. readily available to the authors, references are also the order in which the values of molar February 1973 A.S. Ramachandra Murty and Bh. V. Ramana Murty 67 polarizabilitythis is the factor which is con- refraction for the iodate and iodide ions reported sidered responsible for the improvement of the respectively are 17.86 and 17.53 (Heydweiller, nucleation activity -of these sulphates lie. The 1925). The iodates should, therefore, lead to values of the molar refraction, as calculated from more nucleation than the corresponding iodides. the data of the physical constants, are 27.7, 19.1, Though the difference between reported values of 14.8 and 13.7 respectively for these sulphates. refraction of the iodate and iodide ions is small, The values of molar refraction which were it may be noted that it is in the right direction reported for Na2SO4 and Li2SO4 are respectively being consistent with what is anticipated on the 14.7 and 13.8 (vide Heydweiller, 1925). basis of the present findings. It is important to note that the substances 8. 5. Effects of other iodides on AgI systems NaIO3 and KIO3, which are highly soluble, are When any of the iodides LiI, NaI, KI or CsI found to be better ice nucleants than even AgI. is added to the AgI system, the resulting complex If it is considered that the observed ice nucleaion formed by adsorption (I-Ag-I)-1 could take tion activity of AgI is due to the combined effect water molecules from only the H+ end. But, in of its polarizability and its crystal structure, the the system containing AgI alone, which is neutral, above finding suggests that the iodate ion, on it has been pointed out that AgI takes water account of its larger polarizability itself, has been molecules from both the H+ end and the 0- end. surpassing AgI inspite of the combined effect of The nucleation activity of the AgI system should, the polarizability and crystal structure of AgI. therefore, be less with any one of the above The role of AgI in ice nucleation due to its iodides than without it. However, the nucleation polarizability was pointed out long ago (Weyl, activity (of the AgI system) will be affected dif- 1951). ferently by the different iodides depending upon 8. 7. Effects of other iodates on AgIO3 systems the polarizing power of the corresponding cation. For the same charge, the smaller size cation will When KIO3 or NaIO3 is present in the system have larger polarizing power (because of the containing AgIO3, the polarizability of the comlarger charge density), and so reduces the plex compound formed, by adsorption, will be polarizability of the complex anion, (I-Ag-I)-1, by more than that of the system containing AgIO3 a lesser extent. As the polarizing powers of the alone because polarizability is an additive cations Li+, Na+, K+ and Cs+ decrease in sequence property. However, between the combinations the values of the polarizing power respectively with NaIO3 and with KIO3 there is a difference, are 1.6, 1.0, 0.57 and 0.37 (vide Ladd and Lee, which is similar to what has been pointed out, 1969) - the net polarizability of the complex as in the case of AgI combinations with NaI anion (I-Ag-I)-1 formed in these cases also and with KI. Both Na+ and K+ reduce the decreases in the same sequence. The observed polarizability of the corresponding anion IO3- in sequence of the nucleation activities of the AgI the corresponding complex compounds formed, systems, which progressively decreased from Li+ Na+ reducing it less than K+ on account of the to Cs, supports this conjecture. former's larger polarizing power. Also, the net However, when NH4I is added to the AgI polarizability of the system containing the triple system, the AgI forms a coordination complex combination, AgIO3 + NaIO3 + KI3, will be more (NH3-Ag-I)0 which can take water molecules from than that of the system containing the double both the H+ end and the O- end. Also, this combination, AgIO3 + NaIO3. These consideracoordination complex, on account of its larger tions lend support to the comparative nucleation size than AgI, will have larger polarizability than activities as observed in the case of the iodate AgI. These features could lead to more nuclea- systems. tion activity in the case of AgI +NH4I than in the case of AgI alone, in agreement with observa- 9. Conclusion tions. 8. 6. Iodates versus Iodides The polarizability of the iodate ion is more than that of the iodide ion. The values of ionic The study pointed out that iodates are better ice-forming nucleants than the corresponding iodides. They improve in their ice nucleating activity by the addition of certain soluble compounds to them in suspension. Ten of the systems 68 Journal of the Meteorological Society of Japan selected from the many tested in the present study, when arranged in the order of their nucleation activity, can be listed as follows : 1. (a) Ag103 + NaI03 + KI03 (b) AgI +NH4I + (NH4)2SO4 2. (a) AgIO3 + NaIO3 (b) AgI + (NH4)2SO4 3. AgIO3 + KIO3 4. AgI + NH4I 5. AgIO3 6. NaIO3 7. KIO3 8. AgI 9. AgI +NaI 10. AgI + KI It is seen from the above that the systems most commonly used for conventional cloud seeding, which are AgI + NaI and AgI + KI, occupy only the bottom most levels in the list. The use of AgI + NH4I, which is now gaining ground in the conventional cloud seeding, should now be adhered to in the place of AgI + NaI and AgI + KI. The use of the triple iodate combination AgIO3 + NaIO3 + KIO3 should be considered for the pyrotechnic seeding instead of the double iodate combination AgIO3 + KIO3. Further, the testing of the triple combination NH4I + AgI + (NH4)2SO4, for use in the conventional and pyrotechnic seedings, should receive consideration. That the presence of some soluble salts raises the freezing point of water drops is well known (Mason, 1954; Hosler and Hosler, 1955; Pruppacher, 1963; DePena et al., 1962; Sano and Uzu, 1965), though there has been no satisfactory explanation propounded about it. What is more important in the context of weather modification is to exploit this effect of the soluble salts for developing suitable mixes which can be used for producing efficient ice-forming nuclei. This aspect should now receive attention. Vol. 51. No. 1 DePena, R. G., J. V. Ireborne, and E. M. DeAchavel, 1962: The freezing of supercooled droplets of electrolytic solutions. J. Atmos. Sci., 19, 302-308. Elliot, R. D., P. St. Amand, and J. R. Thomson, 1971: Santa Barbara pyrotechnic cloud seeding test results 1967-70. J. Appl. Meteor., 10, 785795. Heydweiller, V. A. 1925: Optische Untersuchungen an wasserigen Elektrolytlosungen. Physikalische Zeitschrift, 26, 526--556. Hosler, C. L., 1951: On the crystallization of supercooled clouds. J. Meteor., 8, 326-331. and C. B. Hosler, 1955: An investigation of freezing of water in capillaries. Trans. Amer. Geophy. Union, 36, 126-132. Ladd, M. F. C. and W. H. Lee, 1969: Modern Physical Chemistry : an Introduction. Penguin Books Ltd. England, p. 113. Mason, B. J., 1954: Progress in cloud physics research. A progress report on recent investigations at Imperial College, London. Archiv fur Meteorologie, Geophysik and Bioklimatologie, Series A, Band 6, 1-52. -,1971: The Physics of Clouds, Second Edition. Clarendon Press, Oxford, p. 216. and A. P. Van Den Heuvel, 1959 -, The properties and behaviour of some artificial ice nuclei. Proc. Phys. Soc., 74, 744-755. Odencrantz, F. K., 1969: Freezing of water droplets: Nucleation efficiency at temperatures above - 5°C. J. Appl. Meteor., 8, 322-325. Pruppacher, H. R., 1963: Some relations between the supercooling and the structure of aqueous solutions. J. Chem. Phy., 39, 1589-1594. Ramachandra Murty, A. S., and Bh. V. Ramana Murty, 1971: Ice nucleation at warmer temperatures by silver iodide and silver iodide systems. Submitted to Journal de Researches Atmospheriques. 1972a: Freezing characteristics of rain water drops with different solutes and their implication on anomalous ice crystal concentrations in clouds. Tellus, 24, 150-160. 1972b: Ice nucleation by ordinary Portland cement. Tellus, 24, 581-585. Sano, I. and Y. Uzu, 1965: An experimental investigaAcknowledgements tion on the freezing of droplets of water and The authors express their deep sense of grataqueous solutions, with particular reference to the itude to Professor S.K.K. Jatkar, formerly of the influence of the solute. Proceedings of the InterPoona University, for the valuable suggestions national Conference on Cloud Physics, May 24received from him in course of the work. June 1, 1965, Tokyo and Sapporo, 181-184. Simpson, J., W. L. Woodley, H. A. Friedman, T. W. References Slusher, R. S. Scheffee, and R. L. Steel, 1970: An airborne pyrotechnic cloud seeding system and Corrin, M. L., J. A. Nelson, B. Cooley, and B. its use. J. App!. Meteor., 9, 109-122. Rosenthal, 1967: The preparation of 'pure' AgI for nucleation studies. J. Atmos. Sci., 24, 594-595. St. Amand, P., W. G. Finnegan, and L. Burkhardt, February 1973 A.S. Ramachandra Murty 1971: Understanding of use of the simple and complex ice nuclei generated from pyrotechnics and acetone burners. J. Weather Modification, 3, 31-48. and Bh. V. Ramana Weyl, W. A. some J. Murty 1951: Surface of its physical Colloid, Sci., 69 structure and chemical of water and manifestations. 6, 389-405. 可 溶 性 塩 を 加 え た 場 合 の 沃 素 化 合 物 系 及 び 沃 素 酸 化 合 物 系 の氷 晶 化 能 力 A.S.ラ マ チ ャ ソ ドラ ムル テ ィ ・Bh. V. R.ラ マナ ムルテ ィ イ ソ ド熱帯気象研究所,プ ーナ す でに示 した簡 易水滴 凍結法 に より,沃 素化合物系 と沃 素酸化 合物系 の氷晶化能力を調べた.実 験 は,可 溶性塩 を 加えた場合 と加えない場合について行な った.調 べた系の中で,雲 への種 まきに よ く用い られ るAgI・NaI系 とAgI・ KI系 の氷晶化能力が最 も悪 く,ま たAgIO3+KIO3系 のそれ も最 も良 くはなか った.本 論文 では,実 際の種 まきに 使えそ うな氷晶核物質の系を示 した他,調 べ た系 の氷晶化能力の相対的差異について定性的考察 を試 みた.
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