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Residual Entropy of Square Ice

  • Elliott H. Lleb
Chapter

Abstract

At low temperatures, ice has a residual entropy, presumably caused by an indeterminacy in the positions of the hydrogen atoms. While the oxygen atoms are in a regular lattice, each O-H-O bond permits two possible positions for the hydrogen atom, subject to certain constraints called the “ice condition.” The statement of the problem in two dimensions is to find the number of ways of drawing arrows on the bonds of a square planar net so that precisely two arrows point into each vertex. If N is the number of molecules and (for large N) W N is the number of arrangements, then S = Nk lnW. Our exact result is W = (4/3)3/2.

Keywords

Real Axis Transfer Matrix Large Eigenvalue Diagonal Term Horizontal Arrow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935); L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, New York, 1960), 3rd ed.; L. K. Runnels, Sci. Am. 215, 118 (1966).Google Scholar
  2. W. F. Giauque and M. F. Ashley, Phys. Rev. 43, 81 (1933).ADSCrossRefGoogle Scholar
  3. W. F. Giauque and E. L. Stout, J. Am. Chem. Soc. 58, 1144 (1936).ADSCrossRefGoogle Scholar
  4. E. L. MacDougall and W. F. Giauque, J. Am. Chem. Soc. 58, 1032 (1936).CrossRefGoogle Scholar
  5. J. F. Nagle, J. Math. Phys. 7, 1484 (1966).MathSciNetADSCrossRefGoogle Scholar
  6. J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515 (1933).Google Scholar
  7. Earlier estimates were made by E. A. DiMarzio and F. H. Stillinger, J. Chem. Phys. 40, 1577 (1964); H. Takahasi, Proc. Phys. Math. Soc. (Japan) 23, 1069 (1941).Google Scholar
  8. F. Rys, Hely. Phys. Acta 36, 537 (1963); J. C. Slater, J. Chem. Phys. 9, 16 (1941); see also J. F. Nagle [J. Math. Phys. 7, 1492 (1966)] for useful pictures and a bibliography of earlier work. The solution to the F model is reported in E. H. Lieb, Phys. Rev. Letters 18, 1046 (1967) and the RDP model 19, 108 (1967).Google Scholar
  9. A summary of this work was given in E. H. Lieb, Pbys. Rev. Letters 18, 692 (1967).Google Scholar
  10. C. N. Yang and C. P. Yang, Phys. Rev. 150, 321 (1966); 150, 327 (1966). References to equations in these two papers will be denoted (YYI.21) and (YYII.21), respectively. The reader is also referred to these papers for a bibliography of previous work. See also J. des Cloizeaux and M. Gaudin, J. Math. Phys. 7, 1384 (1966).Google Scholar
  11. See, for example, M. E. Fisher, Arch. Rat. Mech. Analysis 17, 377 (1964). It is to this paper that we are most indebted.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Elliott H. Lleb
    • 1
  1. 1.Department of PhysicsNortheastern UniversityBostonUSA

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