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13 November, A.D. 2026 Exponential Growth Or Decay of Such a Population with a Time Constant of L/Ao

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13 November, A.D. 2026 Exponential Growth Or Decay of Such a Population with a Time Constant of L/Ao and structure of semiconduc- in the "normal" semiconductors considered I have tried here to put the emphasis position here. on those basic properties that are com- tors on the one hand and parameters 11. P. Fielding, G. Fischer, E. Mooser, J. Phys. mon to all semiconductors and that such as energy gap and charge-carrier Chem. Solids 8, 434 (1959). 12. H. Winston, Phys. Rev. 94, 328 (1954). distinguish them from other solids. It mobility on the other. 13. S. Geller and J. H. Wernick, Acta Cryst. 12, is interesting as well as surprising to 46 (1959). 14. U. Winkler, Helv. Phys. Acta 28, 633 (1955). see how the many and various semi- References and Notes 15. P. Junod. E. Mooser, H. Schade, ibid. 29, conducting compounds are all governed 193 (1956). 1. A. H. Wilson, Proc. Roy. Soc. (London) 16. E. Mooser and W. B. Pearson, Phys. Rev. by the same simple chemical and struc- A133, 458 (1931). 101, 492 (1956). 2. , ibid. A134, 277 (1931). tural rules. These rules present a chal- 7a, 744 (1952). 17. , J. Chem. Phys. 26, 893 (1957). 3. H. Welker, Z. Naturforsch. G. Busch, F. Hulliger, U. Winkler, Helv. lenge to the theoretician, who has yet 4. The zinc blende structure can be considered 18. a superlattice of the diamond structure. Phys. Acta 27. 195 (1954). to interpret them in a rigorous way. 5. E. Mooser and W. B. Pearson, Progress in 19. J. Appel, Z. Naturforsch. 9a, 265 (1954). They present a challenge also to the Semiconductors, vol. 5 (Heywood, London, 20. E. Mooser and W. B. Pearson, J. Phys. Chem. in press). Solids 7, 65 (1958). experimentalist because they introduce 6. C. H. L. Goodman and R. W. Douglas, 21. G. Busch, E. Mooser, W. B. Pearson, Helv. and un- Physica 20, 1107 (1954). Phys. Acta 29, 192 (1956). him to large families of new 7. 0. G. Folberth and H. Pfister, Semiconduc- 22. E. Mooser and W. B. Pearson, Acta Cryst. explored semiconducting materials. And tors and Phosphors (Interscience, New York, 12, 1015 (1959). 1958), p. 474; I. G. Austin, C. H. L. Good- 23. F. Laves, Theory of Alloy Phases (American the challenge is all the greater since it man, A. E. S. Pengelly, J. Electrochem. Soc. Society for Metals, Cleveland, Ohio, 1956), is to be expected that, as our knowledge 103, 609 (1956); C. H. L. Goodman, Nature p. 124. 179, 828 (1957). 24. U. Dehlinger, Theoretische Metallkunde of semiconductors and their properties 8. E. Mooser and W. B. Pearson, J. Electronics (Springer, Berlin, 1955). increases, the chemical and structural 1, 629 (1956). 25. Since the principal quantum number of the 9. C. H. L. Goodman, J. Phys. Chem. Solids 6, valence shell of an atom coincides with the rules will be reflected in at present 305 (1958). number of the period in which the atom 10. While the mechanism of electronic conduction stands, one readily finds from Fig. 3 that, for largely unknown but much-sought-for in molecular compounds is not fully under- example, n = 4 for AlSb and n - 4.75 for relationships between the chemical com- stood at present, it certainly differs from that CuInTe-2. rate of change of N, the number of elements in the population, is given by dN==YoN-Oo N = ao N dt (1) where ao = yo - Oo may be called the Doomsday: Friday, productivity of the individual element. Depending upon whether ao : 0, inte- gration of Eq. 1 gives the well-known 13 November, A.D. 2026 exponential growth or decay of such a population with a time constant of l/ao. At this date human population will approach infinity In reality, alas, the situation is not if it grows as it has grown in the last two millenia. that simple, inasmuch as the two param- eters describing fertility and mortality may vary from element to element and, Heinz von Foerster, Patricia M. Mora, Lawrence W. Amiot moreover, fertility may have different values, depending on the age of a par- ticular element. To derive these distribution func- Among the many different aspects tion and has to be excluded from the tions from observations of the behavior which may be of interest in the study population count ("death"). of a population as a whole involves the of biological populations (1) is the one Under conditions which come close use of statistical machinery of consider- in which attempts are made to estimate to being paradise-that is, no environ- able sophistication (3, 4). the past and the future of such a popu- mental hazards, unlimited food supply, However, so long as the elements lation in terms of the number of its and no detrimental interaction be- live in our hypothetical paradise, it is elements, if the behavior of this popula- tween elements-the fate of a biologi- in principle possible, by straightforward tion is observable over a reasonable cal population as a whole is completely mathematical methods, to extract the period of time. determined at all times by reference to desired distribution functions, and the All such attempts make use of two the two fundamental properties of an fate of the population as a whole, with fundamental facts concerning an in- individual element: its fertility and its all its ups and downs, is again de- dividual element of a closed biological mortality. Assume, for simplicity, a fic- termined by properties exclusively at- population-namely, (i) that each ele- titious population in which all elements tributable to individual elements. If ment comes into existence by a sexual behave identically (equivivant popula- one foregoes the opportunity to de- or asexual process performed by an- tion, 2) displaying a fertility of /0 off- scribe the behavior of a population in other element of this population spring per element per unit time and all its temporal details and is satisfied ("birth"), and (ii) that after a finite having a mortality Oo = l/tm, derived The authors are members of the staff of the de- time each element will cease to be a from the life span for an individual ele- partment of electrical engineering, University of distinguishable member of this popula- ment of tm units of time. Clearly, the Illinois, Urbana. 4 NOVEMBER 1960 1291 with a general account of its develop- choice of a is a simple linear depend- the living conditions beyond those ment over long stretches of time, the ence of the form found in a "natural setting." problem reduces to solving Eq. 1, ex- The human population may serve as cept that N, yo, and 9o have to be re- a = ao-al N (2) a typical example, as evidenced by its placed by appropriate mean values which, inserted in Eq. 1, results after steady social build-up during historical (y, 9) taken over several generations, integration in what demographers pre- time, its vigorous urbanization in recent over all ages, and over all elements. fer to call the "logistic growth curve," centuries, and its extensive development The difficulties encountered in es- displaying a "sigmoid" shape, if N is of the means of mass communication tablishing the distribution functions for plotted linearly against linear time in recent decades. y and 9 from observations of the be- (6; 7, p. 67). The choice of this par- Since a, the productivity, reflects in havior of the population as a whole ticular function is usually justified by a sense the living standard of the popu- should not be confounded with the our general observation that populations lation, one is tempted to hypothesize predicament which arises if one drops do not grow beyond all measures but that the productivity of populations the fictitious assumption that the ele- settle down to a stationary value N., comprised of elements capable of mu- ments are all thriving in a hypothetical which is given at once for a = 0 from tual communication is a monotonic paradise. While the former difficulties Eq. 2 as N. = ao/ a,. Furthermore, rea- increasing function of the number of can be overcome by "merely" develop- sonable fits of the resulting function elements. Tentatively, let a be a weak ing the appropriate mathematical ap- have been observed with actual bio- function of N: paratus to cope with this intricate logical populations-for example, fruit problem, the difficulties in the latter flies in milk bottles (8), bacterial col- a = ao Nl" (3) case are of a different kind, since now onies in petri dishes (7, p. 71), and so where ao and k 1 are later to be de- the fate of the population is not any on. termined from experiment. Inserting longer solely dependent upon the two Regardless of whether or not the Eq. 3 into Eq. 1, and integrating, yields, intrinsic properties of the elements- simple expression given in Eq. 2 is still with the integration constant deter- their fertility and their mortality. Haz- valid if the mechanisms of the inter- mined (t = ti . N = N1) at once the ards in the environment, competition action between environment and pop- desired dependence of N (t): between elements for limited food sup- ulation are analyzed more carefully, ply, the abundance of predators or there seems to be strong evidence that, N=N1 to -t ) (4) prey-to name just a few factors- for instance, in sexually reproducing may all act on either mortality or species the advantages of having mates where the characteristic date to replaces fertility or on both, and in the absence more readily available in larger popu- a collection of constants: of further insight into these mechan- lations is more than counterbalanced isms, Eq.
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